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English Pages [999] Year 2016
Practical Electrical Engineering
Sergey N. Makarov • Reinhold Ludwig Stephen J. Bitar
Practical Electrical Engineering
Sergey N. Makarov ECE Department Worcester Polytechnic Institute Worcester, Washington, USA
Reinhold Ludwig ECE Department Worcester Polytechnic Institute Worcester, Massachusetts, USA
Stephen J. Bitar Worcester Polytechnic Institute Worcester, Massachusetts, USA
ISBN 978 3 319 21172 5 ISBN 978 3 319 21173 2 (eBook) DOI 10.1007/978 3 319 21173 2 Library of Congress Control Number: 2015950619 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acidfree paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
To Antonina, Margot, and Juliette
Contents
1
From Physics to Electric Circuits . . . . . . . . . . . . . . . . . . . . . . 1.1 Electrostatics of Conductors . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Charges, Coulomb Force, and Electric Field . . . . 1.1.2 Electric Potential and Electric Voltage . . . . . . . . . 1.1.3 Electric Voltage Versus Ground . . . . . . . . . . . . . 1.1.4 Equipotential Conductors . . . . . . . . . . . . . . . . . . 1.1.5 Use of Coulomb’s Law to Solve Electrostatic Problems . . . . . . . . . . . . . . . . . . . . 1.2 SteadyState Current Flow and Magnetostatics . . . . . . . . . 1.2.1 Electric Current . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Difference Between Current Flow Model and Electrostatics . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Physical Model of an Electric Circuit . . . . . . . . . 1.2.4 Magnetostatics and Ampere’s Law . . . . . . . . . . . 1.2.5 Origin of Electric Power Transfer . . . . . . . . . . . . 1.3 Hydraulic and Fluid Mechanics Analogies . . . . . . . . . . . . 1.3.1 Hydraulic Analogies in the DC Steady State . . . . 1.3.2 Analogies for AlternatingCurrent (AC) Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Analogies for Semiconductor Circuit Components . . . . . . . . . . . . . . . . . . . . .
1 3 3 4 5 7 9 11 11 11 13 14 16 18 18 19 20
Part I DC Circuits: General Circuit Theory—Operational Ampliﬁer 2
Major Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Resistance: Linear Passive Circuit Element . . . . . . . . . . 2.1.1 Circuit Elements Versus Circuit Components . . 2.1.2 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 υi Characteristic of the Resistance: Open and Short Circuits . . . . . . . . . . . . . . . . . 2.1.4 Power Delivered to the Resistance . . . . . . . . . . 2.1.5 Finding Resistance of Ohmic Conductors . . . . .
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29 31 31 31
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Contents 2.1.6
3
Application Example: Power Loss in Transmission Wires and Cables . . . . . . . . . . . . . 2.1.7 Physical Component: Resistor . . . . . . . . . . . . . . 2.1.8 Application Example: Resistive Sensors . . . . . . . 2.2 Nonlinear Passive Circuit Elements . . . . . . . . . . . . . . . . . 2.2.1 Resistance as a Model for the Load . . . . . . . . . . 2.2.2 Nonlinear Passive Circuit Elements . . . . . . . . . . 2.2.3 Static Resistance of a Nonlinear Element . . . . . . 2.2.4 Dynamic (SmallSignal) Resistance of a Nonlinear Element . . . . . . . . . . . . . . . . . . . 2.2.5 Electronic Switch . . . . . . . . . . . . . . . . . . . . . . . 2.3 Independent Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Independent Ideal Voltage Source . . . . . . . . . . . . 2.3.2 Circuit Model of a Practical Voltage Source . . . . 2.3.3 Independent Ideal Current Source . . . . . . . . . . . . 2.3.4 Circuit Model of a Practical Current Source . . . . 2.3.5 Operation of the Voltage Source . . . . . . . . . . . . . 2.3.6 Application Example: DC Voltage Generator with Permanent Magnets . . . . . . . . . . 2.3.7 Application Example: Chemical Battery . . . . . . . 2.4 Dependent Sources and TimeVarying Sources . . . . . . . . . 2.4.1 Dependent Versus Independent Sources . . . . . . . 2.4.2 Deﬁnition of Dependent Sources . . . . . . . . . . . . 2.4.3 Transfer Characteristics . . . . . . . . . . . . . . . . . . . 2.4.4 TimeVarying Sources . . . . . . . . . . . . . . . . . . . . 2.5 Ideal Voltmeter and Ammeter: Circuit Ground . . . . . . . . . 2.5.1 Ideal Voltmeter and Ammeter . . . . . . . . . . . . . . . 2.5.2 Circuit Ground: Fluid Mechanics Analogy . . . . . 2.5.3 Types of Electric Ground . . . . . . . . . . . . . . . . . . 2.5.4 Ground and Return Current . . . . . . . . . . . . . . . . 2.5.5 Absolute Voltage and Voltage Drop Across a Circuit Element . . . . . . . . . . . . . . . . . . Circuit Laws and Networking Theorems . . . . . . . . . . . . . . . . 3.1 Circuit Laws: Networking Theorems . . . . . . . . . . . . . . . . 3.1.1 Electric Network and Its Topology . . . . . . . . . . . 3.1.2 Kirchhoff’s Current Law . . . . . . . . . . . . . . . . . . 3.1.3 Kirchhoff’s Voltage Law . . . . . . . . . . . . . . . . . . 3.1.4 PowerRelated Networking Theorems . . . . . . . . . 3.1.5 Port of a Network: Network Equivalence . . . . . . 3.2 Series and Parallel Network/Circuit Blocks . . . . . . . . . . . 3.2.1 Sources in Series and in Parallel . . . . . . . . . . . . . 3.2.2 Resistances in Series and in Parallel . . . . . . . . . . 3.2.3 Reduction of Resistive Networks . . . . . . . . . . . . 3.2.4 Voltage Divider Circuit . . . . . . . . . . . . . . . . . . . 3.2.5 Application Example: Voltage Divider as a Sensor Circuit . . . . . . . . . . . . . . . . . . . . . .
39 41 42 46 46 47 48 49 50 52 52 54 55 57 58 59 61 64 64 64 66 67 69 69 70 71 71 72 89 91 91 93 95 98 99 100 100 102 104 105 107
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Contents 3.2.6
4
Application Example: Voltage Divider as an Actuator Circuit . . . . . . . . . . . . . . . . . . . . 3.2.7 Current Limiter . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Current Divider Circuit . . . . . . . . . . . . . . . . . . . 3.2.9 Wheatstone Bridge . . . . . . . . . . . . . . . . . . . . . . 3.3 Superposition Theorem and Its Use . . . . . . . . . . . . . . . . . 3.3.1 Linear and Nonlinear Circuits . . . . . . . . . . . . . . 3.3.2 Superposition Theorem or Superposition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Y (Wye) and Δ (Delta) Networks: Use of Superposition . . . . . . . . . . . . . . . . . . . . . 3.3.4 T and Π Networks: TwoPort Networks . . . . . . . 3.3.5 General Character of Superposition Theorem . . . Circuit Analysis and Power Transfer . . . . . . . . . . . . . . . . . . . 4.1 Nodal/Mesh Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Importance of Circuit Simulators . . . . . . . . . . . . 4.1.2 Nodal Analysis for Linear Circuits . . . . . . . . . . . 4.1.3 Supernode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Mesh Analysis for Linear Circuits . . . . . . . . . . . 4.1.5 Supermesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Generator Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Equivalence of Active OnePort Networks: Method of Short/Open Circuit . . . . . . . . . . . . . . 4.2.2 Application Example: Reading and Using Data for Solar Panels . . . . . . . . . . . . . . . . . . . . . 4.2.3 Source Transformation Theorem . . . . . . . . . . . . 4.2.4 Thévenin’s and Norton’s Theorems: Proof Without Dependent Sources . . . . . . . . . . . 4.2.5 Application Example: Generating Negative Equivalent Resistance . . . . . . . . . . . . . . . . . . . . 4.2.6 Summary of Circuit Analysis Methods . . . . . . . . 4.3 Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Maximum Power Transfer . . . . . . . . . . . . . . . . . 4.3.2 Maximum Power Efﬁciency . . . . . . . . . . . . . . . . 4.3.3 Application Example: Power Radiated by a Transmitting Antenna . . . . . . . . . . . . . . . . . 4.3.4 Application Example: Maximum Power Extraction from Solar Panel . . . . . . . . . . . . . . . . 4.4 Analysis of Nonlinear Circuits: Generic Solar Cell . . . . . . 4.4.1 Analysis of Nonlinear Circuits: Load Line Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Iterative Method for Nonlinear Circuits . . . . . . . . 4.4.3 Application Example: Solving the Circuit for a Generic Solar Cell . . . . . . . . . . . . . . . . . . .
109 111 112 113 116 116 117 119 121 122 139 141 141 141 145 146 147 149 149 150 151 153 158 159 160 160 162 163 164 167 167 169 170
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Contents 5
Operational Ampliﬁer and Ampliﬁer Models . . . . . . . . . . . . . 5.1 Ampliﬁer Operation and Circuit Models . . . . . . . . . . . . . 5.1.1 Ampliﬁer Operation . . . . . . . . . . . . . . . . . . . . . 5.1.2 Application Example: Operational Ampliﬁer Comparator . . . . . . . . . . . . . . . . . . . . 5.1.3 Ampliﬁer Circuit Model . . . . . . . . . . . . . . . . . . 5.1.4 IdealAmpliﬁer Model and First SummingPoint Constraint . . . . . . . . . . . . . . . . . 5.2 Negative Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Idea of the Negative Feedback . . . . . . . . . . . . . . 5.2.2 Ampliﬁer Feedback Loop: Second SummingPoint Constraint . . . . . . . . . . . . . . . . . 5.2.3 Ampliﬁer Circuit Analysis Using Two SummingPoint Constraints . . . . . . . . . . . . . . . . 5.2.4 Mathematics Behind the Second SummingPoint Constraint . . . . . . . . . . . . . . . . . 5.2.5 Current Flow in the Ampliﬁer Circuit . . . . . . . . . 5.2.6 MultipleInput Ampliﬁer Circuit: Summing Ampliﬁer . . . . . . . . . . . . . . . . . . . . . . 5.3 Ampliﬁer Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Choosing Proper Resistance Values . . . . . . . . . . 5.3.2 Model of a Whole Voltage Ampliﬁer Circuit . . . . 5.3.3 Voltage Ampliﬁer Versus Matched Ampliﬁer . . . 5.3.4 Cascading Ampliﬁer Stages . . . . . . . . . . . . . . . . 5.3.5 Ampliﬁer DC Imperfections and Their Cancellation . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 DCCoupled SingleSupply Ampliﬁer: VirtualGround Circuit . . . . . . . . . . . . . . . . . . . . 5.4 Difference and Instrumentation Ampliﬁers . . . . . . . . . . . . 5.4.1 Differential Input Signal to an Ampliﬁer . . . . . . . 5.4.2 Difference Ampliﬁer: Differential Gain and CommonMode Gain . . . . . . . . . . . . . . . . . 5.4.3 Application Example: Instrumentation Ampliﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Instrumentation Ampliﬁer in Laboratory . . . . . . . 5.5 General Feedback Systems . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 SignalFlow Diagram of a Feedback System . . . . 5.5.2 ClosedLoop Gain and Error Signal . . . . . . . . . . 5.5.3 Application of General Theory to Voltage Ampliﬁers with Negative Feedback . . . . . . . . . . 5.5.4 Voltage, Current, Transresistance, and Transconductance Ampliﬁers with the Negative Feedback . . . . . . . . . . . . . . . . . . . . . .
189 191 191 194 195 197 199 199 199 201 205 206 207 209 209 211 212 215 217 220 222 222 223 225 228 230 230 230 232
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Contents Part II 6
7
Transient Circuits
Dynamic Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Static Capacitance and Inductance . . . . . . . . . . . . . . . . . . 6.1.1 Capacitance, SelfCapacitance, and Capacitance to Ground . . . . . . . . . . . . . . . . . . . 6.1.2 Application Example: ESD . . . . . . . . . . . . . . . . 6.1.3 ParallelPlate Capacitor . . . . . . . . . . . . . . . . . . . 6.1.4 Circuit Symbol: Capacitances in Parallel and in Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.5 Application Example: How to Design a 1μF Capacitor? . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 Application Example: Capacitive Touchscreens . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.7 SelfInductance (Inductance) and Mutual Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.8 Inductance of a Solenoid With and Without Magnetic Core . . . . . . . . . . . . . . . . . . . 6.1.9 Circuit Symbol: Inductances in Series and in Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.10 Application Example: How to Design a 1mH Inductor? . . . . . . . . . . . . . . . . . . . . . . . 6.2 Dynamic Behavior of Capacitance and Inductance . . . . . . 6.2.1 Set of Passive Linear Circuit Elements . . . . . . . . 6.2.2 Dynamic Behavior of Capacitance . . . . . . . . . . . 6.2.3 Dynamic Behavior of Inductance . . . . . . . . . . . . 6.2.4 Instantaneous Energy and Power of Dynamic Circuit Elements . . . . . . . . . . . . . . . 6.2.5 DC Steady State . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Behavior at Very High Frequencies . . . . . . . . . . 6.3 Application Circuits Highlighting Dynamic Behavior . . . . 6.3.1 Bypass Capacitor . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Blocking Capacitor . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Decoupling Inductor . . . . . . . . . . . . . . . . . . . . . 6.3.4 Ampliﬁer Circuits With Dynamic Elements: Miller Integrator . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Compensated Miller Integrator . . . . . . . . . . . . . . 6.3.6 Differentiator and Other Circuits . . . . . . . . . . . . Transient Circuit Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 7.1 RC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 EnergyRelease Capacitor Circuit . . . . . . . . . . . . 7.1.2 Time Constant of the RC Circuit and Its Meaning . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Continuity of the Capacitor Voltage . . . . . . . . . . 7.1.4 Application Example: Electromagnetic Railgun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents 7.1.5
7.2
7.3
7.4
7.5
7.6
Application Example: Electromagnetic Material Processing . . . . . . . . . . . . . . . . . . . . . 7.1.6 Application Example: Digital Memory Cell . . . 7.1.7 EnergyAccumulating Capacitor Circuit . . . . . . RL Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 EnergyRelease Inductor Circuit . . . . . . . . . . . . 7.2.2 Continuity of the Inductor Current . . . . . . . . . . 7.2.3 EnergyAccumulating Inductor Circuit . . . . . . . 7.2.4 EnergyRelease RL Circuit with the Voltage Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Application Example: Laboratory Ignition Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Switching RC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 About Electronic Oscillators . . . . . . . . . . . . . . 7.3.2 Bistable Ampliﬁer Circuit with the Positive Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Triggering . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Switching RC Oscillator . . . . . . . . . . . . . . . . . 7.3.5 Oscillation Frequency . . . . . . . . . . . . . . . . . . . 7.3.6 Circuit Implementation: 555 Timer . . . . . . . . . . SingleTimeConstant (STC) Transient Circuits . . . . . . . 7.4.1 Circuits with Resistances and Capacitances . . . . 7.4.2 Circuits with Resistances and Inductances . . . . . 7.4.3 Example of a NonSTC Transient Circuit . . . . . 7.4.4 Example of an STC Transient Circuit . . . . . . . . 7.4.5 Method of Thévenin Equivalent and Application Example: Circuit with a Bypass Capacitor . . . . Description of the SecondOrder Transient Circuits . . . . 7.5.1 Types of SecondOrder Transient Circuits . . . . . 7.5.2 SeriesConnected SecondOrder RLC Circuit . . 7.5.3 Initial Conditions in Terms of Circuit Current and Capacitor Voltage . . . . . . . . . . . . . 7.5.4 Step Response and Choice of the Independent Function . . . . . . . . . . . . . . . . . . . 7.5.5 Parallel Connected SecondOrder RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . Step Response of the Series RLC Circuit . . . . . . . . . . . . 7.6.1 General Solution of the Secondorder ODE . . . . 7.6.2 Derivation of the Complementary Solution: Method of Characteristic Equation . . . . . . . . . . 7.6.3 Finding Integration Constants . . . . . . . . . . . . . 7.6.4 Solution Behavior for Different Damping Ratios . . . . . . . . . . . . . . . . . . . . . . . 7.6.5 Overshoot and Rise Time . . . . . . . . . . . . . . . . . 7.6.6 Application Example: Nonideal Digital Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents Part III 8
9
AC Circuits
SteadyState AC Circuit Fundamentals . . . . . . . . . . . . . . . . . 8.1 Harmonic Voltage and Current: Phasor . . . . . . . . . . . . . . 8.1.1 Harmonic Voltages and Currents . . . . . . . . . . . . 8.1.2 Phase: Leading and Lagging . . . . . . . . . . . . . . . 8.1.3 Application Example: Measurements of Amplitude, Frequency, and Phase . . . . . . . . . . 8.1.4 Deﬁnition of a Phasor . . . . . . . . . . . . . . . . . . . . 8.1.5 From Real Signals to Phasors . . . . . . . . . . . . . . . 8.1.6 From Phasors to Real Signals . . . . . . . . . . . . . . . 8.1.7 Polar and Rectangular Forms: Phasor Magnitude . . . . . . . . . . . . . . . . . . . . . . . 8.1.8 Operations with Phasors and Phasor Diagram . . . 8.1.9 Shorthand Notation for the Complex Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Concept of Impedance . . . . . . . . . . . . . . . . 8.2.2 Physical Meaning of Impedance . . . . . . . . . . . . . 8.2.3 Magnitude and Phase of Complex Impedance . . . 8.2.4 Application Example: Impedance of a Human Body . . . . . . . . . . . . . . . . . . . . . . . 8.3 Principles of AC Circuit Analysis . . . . . . . . . . . . . . . . . . 8.3.1 AC Circuit Analysis: KVL, KCL, and Equivalent Impedances . . . . . . . . . . . . . . . . 8.3.2 Complete Solution for an AC Circuit: KVL and KCL on Phasor Diagram . . . . . . . . . . . 8.3.3 Source Transformation . . . . . . . . . . . . . . . . . . . 8.3.4 Thévenin and Norton Equivalent Circuits . . . . . . 8.3.5 Summary of AC Circuit Analysis at a Single Frequency . . . . . . . . . . . . . . . . . . . . 8.3.6 Multifrequency AC Circuit Analysis: Superposition Theorem . . . . . . . . . . . . . . . . . . . Filter Circuits: Frequency Response, Bode Plots, and Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 FirstOrder Filter Circuits and Their Combinations . . . . . . 9.1.1 RC Voltage Divider as an Analog Filter . . . . . . . 9.1.2 HalfPower Frequency and Amplitude Transfer Function . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Bode Plot, Decibel, and RollOff . . . . . . . . . . . . 9.1.4 Phase Transfer Function and Its Bode Plot . . . . . 9.1.5 Complex Transfer Function: Cascading Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.6 RL Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . 9.2 Bandwidth of an Operational Ampliﬁer . . . . . . . . . . . . . . 9.2.1 Bode Plot of the OpenLoop Ampliﬁer Gain . . . .
389 391 391 393 396 396 398 399 399 401 404 405 405 407 408 410 411 411 412 413 415 417 417 433 435 435 440 441 444 445 448 451 451
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Contents 9.2.2
10
UnityGain Bandwidth Versus GainBandwidth Product . . . . . . . . . . . . . . . . . 9.2.3 Model of the OpenLoop AC Gain . . . . . . . . . . 9.2.4 Model of the ClosedLoop AC Gain . . . . . . . . . 9.2.5 Application Example: Finding Bandwidth of an Ampliﬁer Circuit . . . . . . . . . . . . . . . . . . 9.2.6 Application Example: Selection of an Ampliﬁer IC for Proper Frequency Bandwidth . . . . . . . . . 9.3 Introduction to Continuous and Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Meaning and Deﬁnition of Fourier Transform . . 9.3.2 Mathematical Properties of Fourier Transform . . 9.3.3 Discrete Fourier Transform and Its Implementation . . . . . . . . . . . . . . . . . . . . . 9.3.4 Sampling Theorem . . . . . . . . . . . . . . . . . . . . . 9.3.5 Applications of Discrete Fourier Transform . . . . 9.3.6 Application Example: Numerical Differentiation via the FFT . . . . . . . . . . . . . . . 9.3.7 Application Example: Filter Operation for an Input Pulse Signal . . . . . . . . . . . . . . . . . 9.3.8 Application Example: Converting Computational Electromagnetic Solution from Frequency Domain to Time Domain . . . . . SecondOrder RLC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Theory of SecondOrder Resonant RLC Circuits . . . . . . 10.1.1 SelfOscillating Ideal LC Circuit . . . . . . . . . . . 10.1.2 Series Resonant Ideal LC Circuit . . . . . . . . . . . 10.1.3 Series Resonant RLC Circuit: Resonance Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Quality Factor Q of the Series Resonant RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5 Bandwidth of the Series Resonant RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.6 Parallel Resonant RLC Circuit: Duality . . . . . . 10.2 Construction of SecondOrder RLC Filters . . . . . . . . . . 10.2.1 SecondOrder BandPass RLC Filter . . . . . . . . 10.2.2 SecondOrder LowPass RLC Filter . . . . . . . . . 10.2.3 SecondOrder HighPass RLC Filter . . . . . . . . . 10.2.4 SecondOrder BandReject RLC Filter . . . . . . . 10.2.5 SecondOrder RLC Filters Derived from the Parallel RLC Circuit . . . . . . . . . . . . . 10.3 RLC Circuits for NearField Communications and Proximity Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 NearField Wireless Link . . . . . . . . . . . . . . . . . 10.3.2 Transmitter Circuit . . . . . . . . . . . . . . . . . . . . . 10.3.3 Receiver Circuit . . . . . . . . . . . . . . . . . . . . . . .
. 452 . 453 . 454 . 455 . 456 . 458 . 458 . 460 . 461 . 463 . 464 . 464 . 466
. . . . .
467 481 483 483 485
. 486 . 488 . . . . . . .
490 493 496 496 499 500 502
. 504 . . . .
506 506 507 508
xiv
Contents 10.3.4
11
Application Example: NearField Wireless Link in Laboratory . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Application Example: Proximity Sensors . . . . . . AC Power and Power Distribution . . . . . . . . . . . . . . . . . . . . . 11.1 AC Power Types and Their Meaning . . . . . . . . . . . . . . . . 11.1.1 Instantaneous AC Power . . . . . . . . . . . . . . . . . . 11.1.2 Timeaveraged AC Power . . . . . . . . . . . . . . . . . 11.1.3 Application Example: rms Voltages and AC Frequencies Around the World . . . . . . . . 11.1.4 rms Voltages for Arbitrary Periodic AC Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.5 Average AC Power in Terms of Phasors: Power Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.6 Average Power for Resistor, Capacitor, and Inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.7 Average Power, Reactive Power, and Apparent Power . . . . . . . . . . . . . . . . . . . . . 11.1.8 Power Triangle . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.9 Application Example: Wattmeter . . . . . . . . . . . . 11.2 Power Factor Correction: Maximum Power Efﬁciency and Maximum Power Transfer . . . . . . . . . . . . . . . . . . . . 11.2.1 Power Factor Correction . . . . . . . . . . . . . . . . . . 11.2.2 Application Example: Automatic Power Factor Correction System . . . . . . . . . . . . . . . . . . 11.2.3 Principle of Maximum Power Efﬁciency for AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Principle of Maximum Power Transfer for AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 AC Power Distribution: Balanced ThreePhase Power Distribution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 AC Power Distribution Systems . . . . . . . . . . . . . 11.3.2 Phase Voltages: Phase Sequence . . . . . . . . . . . . . 11.3.3 Wye (Y) Source and Load Conﬁgurations for ThreePhase Circuits . . . . . . . . . . . . . . . . . . . 11.3.4 Application: Examples of ThreePhase Source and the Load . . . . . . . . . . . . . . . . . . . . . 11.3.5 Solution for the Balanced ThreePhase WyeWye Circuit . . . . . . . . . . . . . . . . . . . . . . . . 11.3.6 Removing the Neutral Wire in LongDistance Power Transmission . . . . . . . . . . . . . . . . . . . . . . 11.4 Power in Balanced ThreePhase Systems: Deltaconnected ThreePhase Circuits . . . . . . . . . . . . . . . 11.4.1 Instantaneous Power . . . . . . . . . . . . . . . . . . . . . 11.4.2 Average Power, Reactive Power, and Apparent Power . . . . . . . . . . . . . . . . . . . . .
510 511 523 525 525 526 528 529 531 532 533 535 538 539 539 542 542 544 546 546 547 549 550 551 553 556 556 558
xv
Contents 11.4.3
12
Application Example: Material Consumption in ThreePhase Systems . . . . . . . . . . . . . . . . . . . 11.4.4 Balanced DeltaConnected Load . . . . . . . . . . . . . 11.4.5 Balanced DeltaConnected Source . . . . . . . . . . . Electric Transformer and Coupled Inductors . . . . . . . . . . . . . 12.1 Ideal Transformer as a Linear Passive Circuit Element . . . 12.1.1 Electric Transformer . . . . . . . . . . . . . . . . . . . . . 12.1.2 Ideal OpenCircuited Transformer: Faraday’s Law of Induction . . . . . . . . . . . . . . . . 12.1.3 Appearance of Transformer Currents . . . . . . . . . 12.1.4 Ampere’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.5 Ideal Loaded Transformer . . . . . . . . . . . . . . . . . 12.1.6 Ideal Transformer Versus Real Transformer: Transformer Terminology . . . . . . . . . . . . . . . . . 12.1.7 Mechanical Analogies of a Transformer . . . . . . . 12.2 Analysis of Ideal Transformer Circuits . . . . . . . . . . . . . . 12.2.1 Circuit with a Transformer in the Phasor Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Referred (Or Reﬂected) Source Network in the Secondary Side . . . . . . . . . . . . . . . . . . . . 12.2.3 Referred (Or Reﬂected) Load Impedance to the Primary Side . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Transformer as a Matching Circuit . . . . . . . . . . . 12.2.5 Application Example: Electric Power Transfer via Transformers . . . . . . . . . . . . . . . . . 12.3 Some Useful Transformers . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Autotransformer . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Multiwinding Transformer . . . . . . . . . . . . . . . . . 12.3.3 CenterTapped Transformer: SingleEnded to Differential Transformation . . . . . . . . . . . . . . 12.3.4 Current Transformer . . . . . . . . . . . . . . . . . . . . . 12.4 RealTransformer Model . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Model of a Nonideal LowFrequency Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Model Parameters and Their Extraction . . . . . . . 12.4.3 Analysis of Nonideal Transformer Model . . . . . . 12.4.4 Voltage Regulation and Transformer Efﬁciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.5 About HighFrequency Transformer Model . . . . . 12.5 Model of Coupled Inductors . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Model of Two Coupled Inductors . . . . . . . . . . . . 12.5.2 Analysis of Circuits with Coupled Inductors . . . . 12.5.3 Coupling Coefﬁcient . . . . . . . . . . . . . . . . . . . . . 12.5.4 Application Example: Wireless Inductive Power Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.5 Application Example: Coupling of Nearby Magnetic Radiators . . . . . . . . . . . . . .
558 560 560 577 579 579 579 583 583 584 586 589 590 590 590 592 593 595 598 598 599 600 602 604 604 604 606 609 610 612 612 613 616 618 622
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Contents Part IV 13
14
Digital Circuits
Switching Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Switch Concept . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Switch Position in a Circuit . . . . . . . . . . . . . . . . 13.1.3 MOSFET Switches and Threshold Voltage . . . . . 13.1.4 Sketch of Transistor Physics . . . . . . . . . . . . . . . 13.2 Power Switching Circuits . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Switching Quadrants . . . . . . . . . . . . . . . . . . . . . 13.2.2 Switching a Resistive Load . . . . . . . . . . . . . . . . 13.2.3 Switching a DC Motor . . . . . . . . . . . . . . . . . . . 13.2.4 OneQuadrant Switch for a DC Motor . . . . . . . . 13.2.5 Half HBridge for a DC Motor . . . . . . . . . . . . . . 13.2.6 Full HBridge for a DC Motor . . . . . . . . . . . . . . 13.2.7 Application Example: PulseWidth Modulation (PWM) Motor Controller PWM Voltage Form . . . . . . . . . . . . . . . . . . . . . 13.3 Digital Switching Circuits . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 NOT Gate or Logic Inverter . . . . . . . . . . . . . . . . 13.3.2 NOR Gate and OR Gate . . . . . . . . . . . . . . . . . . 13.3.3 NAND Gate and AND Gate . . . . . . . . . . . . . . . . 13.3.4 Simple Combinational Logic Circuits: Switching Algebra . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Universal Property of NAND Gates: De Morgan’s Laws . . . . . . . . . . . . . . . . . . . . . . 13.3.6 Logic Circuit Analysis and Application Example: Logic Gate Motor Controller . . . . . . . . 13.3.7 Logic Circuit Synthesis . . . . . . . . . . . . . . . . . . . 13.3.8 The Latch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AnalogtoDigital Conversion . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Digital Voltage and Binary Numbers . . . . . . . . . . . . . . . . 14.1.1 Introduction: ADC and DAC Circuits . . . . . . . . . 14.1.2 Analog Voltage Versus Digital Voltage . . . . . . . . 14.1.3 Bit Rate, Clock Frequency: Timing Diagram . . . . 14.1.4 Binary Numbers . . . . . . . . . . . . . . . . . . . . . . . . 14.1.5 Hexadecimal Numbers . . . . . . . . . . . . . . . . . . . . 14.1.6 ASCII Codes and Binary Words . . . . . . . . . . . . . 14.1.7 Tristate Digital Voltage . . . . . . . . . . . . . . . . . . . 14.2 DigitaltoAnalog Converter . . . . . . . . . . . . . . . . . . . . . . 14.2.1 DigitaltoAnalog Converter . . . . . . . . . . . . . . . . 14.2.2 Circuit (A BinaryWeightedInput DAC) . . . . . . . 14.2.3 Underlying Math and Resolution Voltage . . . . . . 14.2.4 DAC FullScale Output Voltage Range, Resolution, and Accuracy . . . . . . . . . . . . . . . . . 14.2.5 Other DAC Circuits . . . . . . . . . . . . . . . . . . . . .
641 643 643 644 645 647 650 650 651 651 652 653 655
657 660 660 661 664 668 669 670 672 673 689 691 691 692 695 698 700 702 704 707 707 707 708 711 714
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Contents 14.3
15
SampleandHold Circuit: Nyquist Rate . . . . . . . . . . . . . . 14.3.1 AnalogtoDigital Converter . . . . . . . . . . . . . . . . 14.3.2 A Quick Look at an Analog Sinusoidal Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 SampleandHold Voltage . . . . . . . . . . . . . . . . . 14.3.4 SampleandHold Circuit (SH Circuit) . . . . . . . . 14.3.5 Nyquist Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 AnalogtoDigital Converter . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Flash ADC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 ADC Resolution in Bits, FullScale Input Voltage Range, and Voltage Resolution . . . 14.4.3 ADC Equation and Quantization Error . . . . . . . . 14.4.4 SuccessiveApproximation ADC . . . . . . . . . . . . Embedded Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Architecture of Microcontrollers . . . . . . . . . . . . . . . . . . . 15.1.1 A Generic Microcontroller . . . . . . . . . . . . . . . . . 15.1.2 Central Processing Unit . . . . . . . . . . . . . . . . . . . 15.1.3 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.4 Input and Output Devices . . . . . . . . . . . . . . . . . 15.1.5 Timers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.6 Buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.7 Universal Synchronous/Asynchronous Receiver/Transmitter (USART) . . . . . . . . . . . . . 15.2 Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Organization of Memory . . . . . . . . . . . . . . . . . . 15.2.2 Types of Memory . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Flash Memory in Embedded Devices . . . . . . . . . 15.3 Arduino Uno: An Embedded Microcontroller . . . . . . . . . 15.3.1 What Is Arduino? . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Arduino IDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Getting Started with Arduino IDE . . . . . . . . . . . 15.3.4 Arduino Language, Program Storage, and Basic Program Setup . . . . . . . . . . . . . . . . . . 15.3.5 Compiling and Uploading Code to Arduino Uno . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Basic Arduino Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Assignment Statements and Their Features . . . . . 15.4.3 Arithmetic Operations . . . . . . . . . . . . . . . . . . . . 15.4.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.5 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.6 Objects. Application Example: A Servomotor . . . 15.4.7 Interfacing with IO Pins . . . . . . . . . . . . . . . . . . . 15.5 More Advanced Arduino Programming . . . . . . . . . . . . . . 15.5.1 Conditional Statements . . . . . . . . . . . . . . . . . . . 15.5.2 Switch Statements . . . . . . . . . . . . . . . . . . . . . . .
717 717 717 719 720 721 724 724 726 726 728 745 747 747 748 749 750 750 750 751 753 753 755 756 757 757 757 758 759 761 762 762 763 764 765 767 768 769 771 771 773
xviii
Contents 15.5.3 15.5.4 15.5.5 15.5.6 15.5.7
Part V 16
Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arrays and Strings . . . . . . . . . . . . . . . . . . . . . . Serial Communication . . . . . . . . . . . . . . . . . . . Interrupts. Application Example: Emergency Motor Stop . . . . . . . . . . . . . . . . . . Square Wave and PWM Generation with Arduino . . . . . . . . . . . . . . . . . . . . . . . . .
. 774 . 777 . 779 . 780 . 782
Diode and Transistor Circuits
Electronic Diode and Diode Circuits . . . . . . . . . . . . . . . . . . 16.1 Diode Operation and Classiﬁcation . . . . . . . . . . . . . . . . 16.1.1 Circuit Symbol and Terminals . . . . . . . . . . . . . 16.1.2 Three Regions of Operation . . . . . . . . . . . . . . . 16.1.3 Mechanical Analogy of Diode Operation . . . . . 16.1.4 ForwardBias Region: Switching Diode . . . . . . 16.1.5 ReverseBias Region: Varactor Diode . . . . . . . . 16.1.6 Breakdown Region: Zener Diode . . . . . . . . . . . 16.1.7 Other Common Diode Types . . . . . . . . . . . . . . 16.2 Diode Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 IdealDiode Model: Method of Assumed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 ConstantVoltageDrop Model . . . . . . . . . . . . . 16.2.3 Exponential Model in the ForwardBias Region and Its Use . . . . . . . . . . . . . . . . . . . . . 16.2.4 LoadLine Analysis . . . . . . . . . . . . . . . . . . . . . 16.2.5 Iterative Solution . . . . . . . . . . . . . . . . . . . . . . . 16.2.6 Linearization About a Bias Point: SmallSignal Diode Model . . . . . . . . . . . . . . . . 16.2.7 Superposition Principle for SmallSignal Diode Model . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Diode Voltage Regulators and Rectiﬁers . . . . . . . . . . . . 16.3.1 Voltage Reference and Voltage Regulator . . . . . 16.3.2 Voltage Regulator with Zener Diode . . . . . . . . . 16.3.3 HalfWave Rectiﬁer . . . . . . . . . . . . . . . . . . . . . 16.3.4 FullWave Rectiﬁer with a Dual Supply . . . . . . 16.3.5 Diode Bridge Rectiﬁer . . . . . . . . . . . . . . . . . . . 16.3.6 Application Example: Automotive BatteryCharging System . . . . . . . . . . . . . . . . . 16.3.7 Application Example: Envelope (or Peak) Detector Circuit . . . . . . . . . . . . . . . . . . . . . . . 16.4 Diode WaveShaping Circuits . . . . . . . . . . . . . . . . . . . . 16.4.1 Diode Clamper Circuit (DC Restorer) . . . . . . . . 16.4.2 Diode Voltage Doubler and Multiplier . . . . . . . 16.4.3 Positive, Negative, and Double Clipper . . . . . . . 16.4.4 Transfer Characteristic of a Diode Circuit . . . . .
. . . . . . . . . .
795 797 797 797 798 798 800 801 801 804
. 804 . 808 . 809 . 810 . 811 . 811 . . . . . . .
813 814 814 815 817 819 820
. 821 . . . . . .
823 827 827 828 830 832
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Contents 17
18
Bipolar Junction Transistor and BJT Circuits . . . . . . . . . . . . 17.1 Physical Principles and Operation Laws . . . . . . . . . . . . . 17.1.1 Physical Structure: Terminal Voltages and Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . 17.1.3 Operating Regions . . . . . . . . . . . . . . . . . . . . . . 17.1.4 Active Region . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.5 Saturation Region and Cutoff Region . . . . . . . . . 17.1.6 Transistor v–i Dependencies . . . . . . . . . . . . . . . . 17.1.7 Early Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.8 The pnp Transistor . . . . . . . . . . . . . . . . . . . . . . 17.2 LargeSignal Circuit Models of a BJT . . . . . . . . . . . . . . . 17.2.1 LargeSignal Circuit Model of a BJT . . . . . . . . . 17.2.2 LargeSignal DC Circuit Model of a BJT . . . . . . 17.2.3 Method of Assumed States . . . . . . . . . . . . . . . . 17.2.4 Transistor Circuit Analysis Using the Method of Assumed States . . . . . . . . . . . . . . 17.2.5 DC Transistor Bias Circuits . . . . . . . . . . . . . . . . 17.2.6 βIndependent Biasing and Negative Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.7 Common DiscreteCircuit Bias Arrangement . . . . 17.2.8 Other Bias Circuits . . . . . . . . . . . . . . . . . . . . . . 17.3 Practical BJT Circuits at DC . . . . . . . . . . . . . . . . . . . . . . 17.3.1 ConstantCurrent Sources: Active Region of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Voltage Follower (Voltage Buffer): Active Region of Operation . . . . . . . . . . . . . . . . 17.3.3 BJT Switches: Saturation Region . . . . . . . . . . . . 17.3.4 Application Example: Automotive BJT Dome Light Switch . . . . . . . . . . . . . . . . . . 17.3.5 Application Example: Door Lock BJT Switch and Darlington Pair . . . . . . . . . . . . . . . . 17.4 SmallSignal Transistor Ampliﬁer . . . . . . . . . . . . . . . . . . 17.4.1 Generic VoltageGain Ampliﬁer . . . . . . . . . . . . . 17.4.2 Simpliﬁed Model of the BJT CommonEmitter Ampliﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 SmallSignal BJT Analysis and Superposition . . . 17.4.4 Analysis of SmallSignal CommonEmitter Ampliﬁers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.5 Application Example: Transistor Ampliﬁer Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . MOS FieldEffect Transistor (MOSFET) . . . . . . . . . . . . . . . . 18.1 Principle of Operation and Threshold Voltage . . . . . . . . . 18.1.1 Physical Structure: Terminal Voltages and Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1.2 Simpliﬁed Principle of Operation . . . . . . . . . . . . 18.1.3 NMOS Capacitor . . . . . . . . . . . . . . . . . . . . . . .
851 853 853 854 856 857 859 861 863 863 866 866 868 870 871 873 874 876 878 880 880 882 884 886 887 889 889 890 892 894 898 919 921 921 923 924
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Contents 18.1.4
18.2
18.3
18.4
Voltage Across the Oxide Layer Before and at the Onset of Strong Inversion . . . . . . . . . . 18.1.5 Voltage Across the Semiconductor Body . . . . . . . 18.1.6 Threshold Voltage . . . . . . . . . . . . . . . . . . . . . . . 18.1.7 PMOS Transistor . . . . . . . . . . . . . . . . . . . . . . . 18.1.8 Oxide Thicknesses and Capacitances in CMOS Processes . . . . . . . . . . . . . . . . . . . . . . 18.1.9 Family Tree of FETs . . . . . . . . . . . . . . . . . . . . . Theoretical Model of a MOSFET . . . . . . . . . . . . . . . . . . 18.2.1 Test Circuit and Operating Regions . . . . . . . . . . 18.2.2 Linear Subregion of Triode Region at Strong Inversion . . . . . . . . . . . . . . . . . . . . . . 18.2.3 Nonlinear Subregion of Triode Region at Strong Inversion . . . . . . . . . . . . . . . . . . . . . . 18.2.4 Saturation Region . . . . . . . . . . . . . . . . . . . . . . . 18.2.5 The vi Dependencies . . . . . . . . . . . . . . . . . . . . 18.2.6 PMOS Transistor . . . . . . . . . . . . . . . . . . . . . . . 18.2.7 LargeSignal MOSFET Model in Saturation . . . . 18.2.8 Device Parameters in CMOS Processes . . . . . . . MOSFET Switching and Bias Circuits . . . . . . . . . . . . . . . 18.3.1 Triode Region for Switching Circuits: Device Parameter Extraction . . . . . . . . . . . . . . . 18.3.2 ResistorSwitch Model in Triode Region . . . . . . . 18.3.3 Application Example: Output Resistance of Digital Logic Gates . . . . . . . . . . . . . . . . . . . . 18.3.4 MOSFET Circuit Analysis at DC . . . . . . . . . . . . 18.3.5 Application Example: Basic MOSFET Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MOSFET Ampliﬁer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.1 MOSFET CommonSource Ampliﬁer . . . . . . . . . 18.4.2 Voltage Transfer Characteristic . . . . . . . . . . . . . . 18.4.3 Principle of Operation and QPoint . . . . . . . . . . . 18.4.4 MOSFET Biasing for Ampliﬁer Operation . . . . . 18.4.5 SmallSignal MOSFET Model and Superposition . . . . . . . . . . . . . . . . . . . . . . . 18.4.6 MOSFET Transconductance . . . . . . . . . . . . . . . 18.4.7 Analysis of CommonSource MOSFET Ampliﬁer . . . . . . . . . . . . . . . . . . . . . .
Erratum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
925 926 928 929 930 931 932 932 933 934 935 937 939 939 940 942 942 944 945 947 951 953 953 953 955 956 956 957 958 E1
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973
xxi
Chapter 1
Chapter 1: From Physics to Electric Circuits
Overview Prerequisites:  Knowledge of university physics: electricity and magnetism Objectives of Section 1.1:  Show that the electric voltage and the electric potential may be treated as two equivalent quantities  Define the electric voltage work per unit charge in the form of a line integral and show its independence on the integration path for conservative fields  Relate voltage to the potential energy of the electric field  Introduce threedimensional potential distributions and realize the guiding function of metal wires  Formulate and understand major conditions of electrostatics of conductors  Visualize surface charge distributions in the electrostatic case
Objectives of Section 1.2:  Introduce electric current density as a function of the applied electric field  Visualize steadystate current flow in a single conductor along with the associated electric potential/voltage distribution  Visualize electric and magneticfield distributions for a twowire DC transmission line  Obtain initial exposure to the Poynting vector  Realize that electric power is transferred via Poynting vector even in DC circuits  Indicate a path toward circuit problems where the field effects become important Objectives of Section 1.3:  Review basic hydraulic (fluid mechanics) analogies for DC circuit elements  Present major hydraulic analogies for dynamic circuit elements in AC circuits  Briefly discuss hydraulic analogies for semiconductor components Application Examples:  Human body subject to applied voltage  Human body in an external electric field
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_1
I1
Chapter 1
From Physics to Electric Circuits
Keywords: Electricity, Electric ﬁeld intensity, Electric ﬁeld, Electric ﬁeld magnitude, Lines of force, Electric potential, Electric voltage, Line integral, Contour integral, Conservative ﬁeld, Potential energy of the electric ﬁeld, Voltage drop, Voltage difference, Ground reference, Neutral conductor, Common conductor, Voltage versus ground, Equipotential lines, Volumetric charge density, Surface charge density, Gauss’ theorem, Equipotential surface, Selfcapacitance, Electrostatic discharge, Effect of electrostatic discharge on integrated circuits, Boundary element method, Electric current density, Material conductivity, Transmission line, Direct current (DC), Electric load, Ideal wire, Kirchhoff’s voltage law (KVL), Magnetic ﬁeld, Magneticﬁeld intensity, Ampere’s law, Cross (vector) product, Poynting vector, Poynting theorem, Wireless communications, Wireless power transfer, Fluid mechanics analogy of an electric circuit, Hydraulic analogy of an electric circuit, Voltage source (hydraulic analogy), Resistance (hydraulic analogy), Current source (hydraulic analogy), Capacitance (hydraulic analogy), Inductance (hydraulic analogy), Electric transformer (hydraulic analogy), NMOS transistor (hydraulic analogy), Bipolar junction transistor (hydraulic analogy)
I2
Chapter 1
Section 1.1: Electrostatics of Conductors
Section 1.1 Electrostatics of Conductors This introductory chapter is optional in the sense that the reader does not need its content as a prerequisite for the subsequent chapters. The aim of this chapter is to illustrate why electric circuits trace their origin to electromagnetic ﬁelds. The chapter highlights several ﬁeld concepts which form the theoretical foundation of electric circuits. At the same time it makes clear why, for the majority of electric circuits, the electric and magnetic ﬁelds are often ignored without affecting the ﬁnal results. When this is the case, the electric circuits and components follow useful and simple hydraulic analogies discussed below.
1.1.1 Charges, Coulomb Force, and Electric Field Electric Charges The smallest electric charge is known as the elemental charge of an electron, q 1:60218 10 19 C (coulombs). In electrical engineering, we deal with much larger charges, which, for this reason, are assumed to be inﬁnitely divisible. There are no positive movable charges in metal conductors. Therefore, when we talk about positive charges, it is implied that we have a lack of electrons at a certain location, e.g., at the surface. Oppositely, the negative charge is the excess of electrons at a certain location. Deﬁnition of the Electric Field Electrostatics plays a key role in explaining the operations of electric capacitors and all semiconductor devices. The word electricity is derived from the Greek word for amber. Probably Thales of Miletus was the ﬁrst who discovered, about 600 B.C., that amber, when rubbed, attracts light objects. An electrostatic force acting on a charge q is known as the Coulomb force. This Coulomb force is a vector; it is measured in newtons (or N) ~ F
q~ E ½N
ð1:1Þ
Equation (1.1) is the deﬁnition of the electric ﬁeld intensity vector, ~ E, often called the electric ﬁeld. This electric ﬁeld is created by other (remote or nearby) charges. In the general case, the electric ﬁeld exists both in free space and within material objects, whether conductors or dielectrics. The electric ﬁeld ~ E is measured in volts (V) per q meter (V/m). The ﬁeld magnitude, E E2 þ E2 þ E2 ~ E, has the same units. From x
y
z
Eq. (1.1), 1V
1 N 1m 1 C
1 J 1 C
ð1:2Þ
I3
Chapter 1
Section 1.1: Electrostatics of Conductors
1.1.5 Use of Coulomb’s Law to Solve Electrostatic Problems The theory of electrostatics solely relies upon the distribution of surface charges, since no other charges exist. An important case is a conductor subject to voltage V applied from one terminal of the voltage source; the connection position does not matter. The other terminal is usually located very far away, ideally at inﬁnity; it is customarily assigned a value of 0 V. As a result, the conductor acquires an extra positive charge Q if V is positive. The quantities of interest are the value of Q itself and the resulting surface charge distribution. The ratio Q/V is the selfcapacitance of the conductor. Consider a complicated conductor a human body. The idea of the solution is simple and elegant. The entire body surface is divided into many small elements with a constant charge density each. We assign an unknown charge qi to every such element with number i (i 1, . . . , N ) and center position ~ ri . Charge qi is very similar to the point charge. It follows from Coulomb’s law that it generates the electric potential in space given by V ð~ rÞ
qi 4πε0 j~ r
ð1:6Þ
~ ri j
Here, ε0 8:85419 10 12 F=m is the electric permittivity of air. The net voltage of the jth element is the sum of all such voltage contributions, i.e., V ~ rj
N X i 1
q i 4πε0 ~ rj
~ ri
1 V,
j
1, ::, N
ð1:7Þ
Equation (1.7) forms a system of N equations for N unknown charges. The diagonal terms need a special treatment. By solving this algebraic system of equations using linear algebra, we obtain the unknown charges; their sum is the net charge Q. Figure 1.4 shows the surface charge distribution found this way for two human subjects. Emphasize that the charge (and the strongest electric ﬁeld) concentrates at the sharpest parts of the body: elbows, hands, feet, and the head. The total excess body charge Q in both cases is approximately 50 10 12 C. This is a very small charge; the same charge is stored in a 50 pF capacitor at 1 V. The method of this example is widely used in electrostatic simulations including modeling electrostatic discharge and its effect on integrated circuits.
I9
Chapter 1
From Physics to Electric Circuits a)
surface charge density 11
b)
2
x10 C/m 6
5
4
3
2
1
Fig. 1.4. Surface charge distribution over the human body based on an applied voltage of 1 V. Both subjects are ECE graduate students.
A human beneath the power line is subject to an electric ﬁeld. A similar solution applies given that the body surface is still the equipotential surface. Figure 1.5 shows the surface charge distribution for two human subjects in a vertical electric ﬁeld of 1 V/m. The negative charges concentrate close to the head, whereas the positive charges concentrate in the lower body. Figure 1.5 also shows the electric potential distribution around the body. Dense equipotential lines mean a high local electric ﬁeld. The local ﬁeld may exceed the external ﬁeld by a factor of 10 or more. All results are linearly scaled with the applied electric ﬁeld.
a)
surface charge density 1.6 V
11
2
x10 C/m
1.5 V 1.4 V 1.3 V
8 6
1.2 V 1.1 V
4
b) 1.6 V 1.5 V 1.4 V 1.3 V 1.2 V 1.1 V 1.0 V 0.9 V
1.0 V
2
0.8 V
0.9 V
0
0.7 V
0.8 V 0.7 V
2 4
0.6 V
0.6 V 0.5 V 0.4 V
6 0.5 V 0.4 V 0.3 V 0.2 V
0.3 V 8
0.2 V
10
0.1 V
Fig. 1.5. Human body subject to an applied electric ﬁeld: surface charge and potential distribution.
I10
Chapter 1
From Physics to Electric Circuits
1. Everywhere on the surface Sc, the current density component perpendicular to the surface is zero, i.e., ~j ~ n 0, where ~ n is the unit normal vector to the surface, and dot denotes the scalar product of two vectors. In other words, no current can ﬂow from the conductor into air. 2. On the surface of electrodes Se, the voltage is given: for example, +0.5 V on the left electrode and 0.5 V on the right electrode. Alternatively, the inﬂowing current, ~j ~ n, may be given.
How Does the Conductor “Guide” the Electric Field? We consider the current ﬂow in a conducting cylinder with two circular electrodes shown in Fig. 1.6b. The electrostatic counterpart of the problem is given by the same coaxial electrode pair in air, see Fig. 1.6a. The electrodes have the radius a; they are separated by 25a. The electrode voltages are 0.5 V. The exact value of the cylinder conductivity does not matter; the same results will be obtained. The electrostatic problem and the steadystate current problem are both solved as described in the previous section. Results of both solutions are given in Fig. 1.6 where the equipotential lines and the electric ﬁeld vectors are plotted. Some general observations from this ﬁgure are worthy of note: 1. The currentcarrying conductor “guides” the electric ﬁeld as shown in Fig. 1.6a which, otherwise, would be spread out in space; see Fig. 1.6b. 2. In the long conductor, the electric ﬁeld and the electric current are both directed along the conductor axis; they are uniform across any conductor cross section, which is simultaneously an equipotential surface. In other words, current ﬂow in the long conductor is one dimensional, like water ﬂow in a pipe. This is also true if the conductor is bent or has a noncircular cross section 3. The voltage decreases linearly along the conductor from the most positive to the most negative value. The voltage drop per unit length is constant; it is only a function of the applied voltage. It is seen from Fig. 1.6b that within in a currentcarrying conductor of length l: E
V l
ð1:9Þ
where E is the magnitude of the electric ﬁeld (its direction is along the conductor axis), and V is the voltage across the conductor (1 V in the present case). Equation (1.9) is a simpliﬁed version of Eq. (1.4) for uniform ﬁelds. In many textbooks, it is used to derive Ohm’s law. Note that the electric ﬁeld in Fig. 1.6b is not continuous across the conductorair interface. A component of the electric ﬁeld perpendicular to the conductor boundary suddenly appears. This component is due to the surface charges on the conductor air interface (not shown in the ﬁgure).
I12
Part I DC Circuits: General Circuit Theory—Operational Amplifier
Chapter 2
Chapter 2: Major Circuit Elements
Overview Prerequisites:  Knowledge of university physics: electricity and magnetism (optional)  Knowledge of vector calculus (optional) Objectives of Section 2.1:  Realize the difference between circuit elements and circuit components  Review (derive) Ohm’s law  Become familiar with the i characteristic of the resistance including limiting cases  Realize the importance of ohmic losses in long cables  Become familiar with discrete fixed resistors and with resistive sensing elements Objectives of Section 2.2:  Realize the meaning of a passive nonlinear circuit element and its i characteristic  Define two resistance types (static and dynamic) for a nonlinear passive circuit element  Present two examples of nonlinear elements: ideal diode and a threshold switch Objectives of Section 2.3:  Introduce the concept of independent voltage and current sources and become familiar with their i characteristics  Introduce the concept of practical voltage/current sources including their i characteristics  Obtain initial exposure to the operation principles of voltage sources including specific examples Objectives of Section 2.4:  Become familiar with the concept of a dependent source  Become familiar with four major types of dependent sources  Obtain initial exposure to transfer characteristics of dependent sources  Become familiar with ideal timevarying and AC sources
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_2
II29
Chapter 2
Major Circuit Elements
Objectives of Section 2.5:  Formalize the meaning of voltmeter and ammeter from the viewpoint of open and short circuits  Obtain a clear understanding of the circuit ground and its role in the circuit  Review different ground types Application Examples:  Power loss in transmission lines and cables  Resistive sensing elements  DC voltage generator with permanent magnets  Chemical battery Keywords: Circuit elements, Circuit components, υi characteristic, Resistance, Polarity, Voltage difference, Voltage drop, Voltage polarity, Passive reference conﬁguration, Ohm’s law, Linear passive circuit element, Conductance, Siemens, mho, Short circuit, Open circuit, Ohmic conductor, Mobility of charge carriers, Material conductivity, Material resistivity, Electric circuit, American Wire Gauge (AWG), Resistor, Fixed resistors, Surface Mound Devices (SMD) (resistance value, geometry size), Variable resistor, Potentiometer, Resistive sensors, Photoresistor, Photocell, Negative temperature coefﬁcient (NTC), Thermistor equation, Thermistor constant, Thermocouple, PeltierSeebeck effect, Strain gauge, Strain sensitivity, Gauge factor, Strain gauge equation, Potentiometric position sensor, Nonlinear passive circuit elements, Nonohmic circuit elements, Radiation resistance, Ideal diode, Shockley equation, Static resistance, Dynamic resistance, Smallsignal resistance, Differential resistance, Incremental resistance, (DC) Operating point, Quiescent point, Electronic switch, Solidstate switch, Switch threshold voltage, Twoterminal switch, Threeterminal switch, Unidirectional switch, Bidirectional switch, Independent ideal voltage source, Active reference conﬁguration, Nonlinear passive circuit elements, Nonohmic circuit elements, Radiation resistance, Static resistance, Ideal diode, Shockley equation, Dynamic resistance, Smallsignal resistance, Differential resistance, Incremental resistance, (DC) Operating point, Quiescent point, Electronic switch, Solidstate switch, Switch threshold voltage, Twoterminal switch, Threeterminal switch, Unidirectional switch, Bidirectional switch, Independent ideal voltage source, Active reference conﬁguration, Practical voltage source, Maximum available source current, Maximum available source power, Opencircuit source voltage, Shortcircuit circuit current, Internal source resistance, Independent ideal current source, Practical current source, Charge separation principle, Faraday’s law of induction, Lorentz force, Instantaneous generator voltage, Average generator voltage, Battery voltage, Battery capacity, Battery energy storage, Dependent sources, Voltagecontrolled voltage source, Currentcontrolled voltage source, Voltagecontrolled current source, Currentcontrolled current source, Opencircuit voltage gain, Transresistance, Transconductance, Shortcircuit current gain, Voltage ampliﬁer, Current ampliﬁer, Transresistance ampliﬁer, Transconductance ampliﬁer, Transfer characteristic, AC voltage source, Ideal voltmeter, Ideal ammeter, Earth ground, Chassis ground, Common (neutral) terminal (ground), Forward current, Return current, Absolute voltages in a circuit
II30
Chapter 2
Section 2.1: Resistance: Linear Passive Circuit Element
Section 2.1 Resistance: Linear Passive Circuit Element 2.1.1 Circuit Elements Versus Circuit Components Circuit Elements Similar to mechanical mass, spring, and damper used in analytical dynamics, circuit elements are simple hypothetic ideal models. Every circuit element is characterized by its unique voltage/current dependence called the υi characteristic. Most of the υi characteristics reﬂect general physical laws. A list of the circuit elements includes: 1. 2. 3. 4. 5. 6. 7. 8. 9.
Resistance Capacitance Inductance Ideal electric transformer Voltage source (independent and dependent) Current source (independent and dependent) Ideal switch Ideal (Shockley) diode Logic gates (NOT, AND, OR).
Circuit elements may be linear (resistance) or nonlinear (ideal diode), passive (resistance) or active (voltage source), static (resistance) or dynamic (capacitance/inductance), or both. Although all circuit elements studied here are static ones, the extension to the case of timevarying voltage and current is often trivial.
Circuit Components Circuit components are numerous hardware counterparts of the circuit elements. Examples of the circuit components include resistor, capacitor, inductor, battery, etc. The circuit components may be modeled as combinations of the ideal circuit elements with one dominant desired element (e.g., resistance) and several parasitic ones (e.g., parasitic inductance and capacitance of a physical resistor). Another example is a battery, which is modeled as an ideal voltage source in series with a (small) resistance. In practice, we attempt to model any existing or newly discovered circuit component as a combination of the wellknown circuit elements. The same is valid for more complicated structures targeted by electrical, mechanical, and biomedical engineers. An example is a human body, the response of which is modeled as a combination of resistance and capacitance. 2.1.2 Resistance Symbols and Terminals Figure 2.1 shows the circuit symbol for resistance with current direction and voltage polarity: positive voltage applied to the left terminal and a negative voltage applied to the right terminal cause a current to ﬂow from left to right, as depicted in Fig. 2.1b. As a circuit element, the resistance is fully symmetric: terminals 1 and 2 in Fig. 2.1 may be interchanged without affecting its operation. Thus, the resistance does not have a polarity. II31
Chapter 2
Major Circuit Elements
2.1.3 υi Characteristic of the Resistance: Open and Short Circuits Figure 2.2 plots the linear dependence given by Eq. (2.3) for two distinct resistances. The corresponding plot is known as the υi characteristic (or the υi dependence). We use small letters υi to maintain consistency with the following study of timevarying circuits. The υi characteristic is the “business card” of the circuit element every circuit element has its own υi characteristic. Once the υi characteristic is known, the circuit element is characterized completely. The following is true with reference to Fig. 2.2a: 1. 2. 3. 4.
The slope of the υi dependence for the resistance is equal to 1/R or G. Smaller resistance leads to a stepper υi dependence (large currents). Larger resistance leads to a ﬂatter υi dependence (small currents). The negative part of the υi dependence simply means simultaneous switching voltage polarity and current direction, respectively, in Fig. 2.1. a) I
b)
smaller resistance
I
short circuit
larger resistance 0
0 V
open circuit V
Fig. 2.2. υi Characteristics for resistances and for the short and open circuits, respectively.
Open and Short Circuits Two limiting cases of the resistance υi characteristics are the short circuit and the open circuit, as seen in Fig. 2.3. R 0
R
short circuit open circuit
Fig. 2.3. Transformation of resistance to a short and an open circuit, respectively.
When R ! 0, the resistance becomes a short circuit, or an ideal wire. There is no voltage drop V across the wire, but any current I can ﬂow through it. Therefore, the υi characteristic of the short circuit is the straight vertical line in Fig. 2.2b. When R ! 1, the resistance becomes an open circuit, or an ideal vacuum gap. There is no current I through the gap at any value of the applied voltage V. Therefore, the υi characteristic of the open circuit is the straight horizontal line in Fig. 2.2b. II34
Chapter 2 I
Aqn υ
Section 2.1: Resistance: Linear Passive Circuit Element ð2:9Þ
Here, nq is the volumetric charge density of free charges q with concentration n in coulombs per cubic meter, C/m3, and υ is the magnitude of the average charge velocity in m/s. In the onedimensional model of the current ﬂow, the average velocity vector is directed along the xaxis seen in Fig. 2.5. Since the electrons have been historically assigned a negative charge, the electric current direction is opposite to the direction of electron motion in a conductor. The electron carries an elemental charge of q 1.60218 10 19 C. Because A and nq are constants for a given conductor, the electric current is simply associated with the charge’s mean velocity υ.
Finding Average Carrier Velocity In order to ﬁnd υ, we use the following method. The total voltage drop V applied to a sufﬁciently long, conducting cylinder is uniformly distributed along its length following the equally spaced equipotential surfaces; this is schematically shown in Fig. 2.5. This fact has been proved in Chapter 1. Such a voltage distribution corresponds to a constant uniform electric ﬁeld within the cylinder, which is also directed along the cylinder axis. The magnitude of the ﬁeld, E, with the units of V/m, is given by E
V =l
ð2:10Þ
The electric ﬁeld creates a Coulomb force acting on an individual positive charge q. The Coulomb force is directed along the ﬁeld; its magnitude F is given by F
qE
ð2:11Þ
The key is a linear relation between the charge velocity υ and force F or, which is the same, a linear relation between the charge velocity υ and the applied electric ﬁeld E, i.e., υ
μE
ð2:12Þ
where μ is the socalled mobility of charge carriers, with the units of m2/(Vs). Carrier mobility plays an important role in semiconductor physics. With the help of Eqs. (2.10) and (2.12), the expression for the total current Eq. (2.9) is transformed to l l l V I I RI, σ qnμ, R ð2:13Þ Aqnμ Aσ Aσ This is the expression for the resistance of a cylindrical conductor. Material conductivity σ is measured in S/m. Its reciprocal is the material resistivity ρ 1=σ measured in m. Ω
II37
Chapter 2
Section 2.1: Resistance: Linear Passive Circuit Element
The word thermistor is a contraction of the words “thermal” and “resistor.” As a second example of a resistance subject to ambient conditions, we will consider a photoresistor (photocell) shown in Fig. 2.9b. The ﬁnal example is a strain gauge shown in Fig. 2.9c.
Fig. 2.9. Sensing elements which change their resistances when ambient conditions change.
Thermistor The thermistor changes its resistance as temperature increases or decreases. Generalpurpose thermistors are made out of metal oxides or other semiconductors. Successful semiconductor thermistors were developed almost simultaneously with the ﬁrst transistors (1950s). For a metaloxide thermistor, its resistance decreases with increasing temperature. Increasing the temperature increases the number of free carriers (electrons) and thus increases the sample conductivity (decreases its resistance). Shown in Fig. 2.9a is a very inexpensive NTC negative temperature coefﬁcient leaded thermistor. According to the manufacturer’s datasheet, it reduces its resistance from approximately 50 kΩ at room temperature (about 25 C) by 4.7 % for every degree Celsius (or Kelvin) and reaches about 30 kΩ at body temperature according to the thermistor equation: 1 1 ð2:17Þ R1 R2 exp B T1 T2 where T1, T2 are two absolute temperatures always given in degrees K. Temperature T2 corresponds to a room temperature of 25 C so that R2 R25 C, temperature T1 is the observation temperature, and B is the thermistor constant, which is equal to 4200 K in the present case. Equation (2.17) is a nontrivial result of the solidstate physics theory. We emphasize that Eq. (2.17) is more accurate than the temperature coefﬁcient of the thermistor the above referenced value of 4.7 %. Typical applications include temperature measurement, control, compensation, power supply fan control, and printed circuit board (PCB) temperature monitoring. Inexpensive thermistors operate from 30 C to approximately +130 C. At higher temperatures, thermocouples should be used.
Thermocouple Figure 2.9 does not show one more important temperature sensor the thermocouple which is used to measure large temperatures and large temperature differences. It operates
II43
Chapter 2
Major Circuit Elements
based on a completely different principle. The thermocouple does not signiﬁcantly change its resistance when temperature changes. Instead, it generates an electric current and the associated voltage, when the junction of the two metals is heated or cooled, known as the PeltierSeebeck effect; this voltage can be correlated to temperature. Therefore, the thermocouple, strictly speaking, is not a resistive sensor.
Photoresistor (Photocell) An idea similar to the thermistor design applies. Quanta of light incident upon the photocell body create new free charge carriers new electronhole pairs in a semiconductor. If the concentration of free charges increases, the resistance of the sample decreases according to Eq. (2.13). The resistance is inversely proportional to the concentration. The photocell in Fig. 2.9b is characterized by very large nonlinear variations of the resistance in response to ambient light. Strain Gauge The strain gauge measures mechanical strain. The operation is based on Eq. (2.13), which deﬁnes the resistance through material conductivity σ, the length of the resistor l, and its cross section A. When the resistor, which may be a trace on the base of a metal alloy, is stretched, its length l increases and its cross section A decreases. Hence, the resistance R increases due to both of these effects simultaneously; changes in the resistance may be made visible for small strains. Shown in Fig. 2.9c is an inexpensive uniaxial strain gauge with a nominal resistance of 350 Ω; typical resistances are 120, 350, 600, 700, and 1000 Ω. The gauge changes its resistance R in proportion to the strain sensitivity SG of the wire’s resistance, also called the gauge factor (GF). For a strain gauge, the relative resistance variation, ΔR/R, is estimated based on known values of the strain sensitivity, SG, and strain, ε. The strain gauge equation has the form ΔR=R
SGε
ð2:18Þ
The dimensionless strain sensitivity SG varies around 2. The strain (a relative elongation) is a dimensionless quantity. It is measured in microstrains, με, where one με is 10 6. Typical strain values under study are on the order of 1000 με. Using Eq. (2.18) this yields a relative resistance variation as small as 0.2 %. Because of this, the circuits for the strain measurements should be designed and built with great care. Temperature compensation efforts are also required. Since the relative resistance changes are very small, the strain gauge is a linear device: the strain is directly proportional to resistance variations.
Potentiometric Position Sensor Another general resistive sensor is a potentiometric (or potentiometer) position sensor. Its operation becomes apparent when we rotate the potentiometer dial in Fig. 2.8a. A change in the resistance is directly proportional to the rotation angle. The resistance variation can
II44
Chapter 2
Section 2.1: Resistance: Linear Passive Circuit Element
be converted to voltage variation and then measured. Similar potentiometer sensors for measuring linear motion also exist.
Sensitivity of Resistive Sensors One major difference between different resistive sensing elements is a very different degree of the relative resistance variations. For the photocell, the resistance variation is up to 100 times. For the thermistor, the resistance variation can be as much as 50 %. For the strain gauge, the resistance variations do not exceed 0.5 %. Circuit Symbols There are several similar but not identical standards for circuit symbols related to resistance: International standard IEC 60617, American ANSI standard Y32 (IEEE Std 315), etc. Figure 2.10 shows popular circuit symbols for variable resistances. a) generic variable resistor
b) potentiometer
c)
thermistor
d)
photoresistor
e)
strain gauge
Fig. 2.10. Circuit symbols for variable resistances: (a) generic variable resistance, (b) potentiometer, (c) thermistor, (d) photoresistor, and (e) strain gauge.
II45
Chapter 2
Major Circuit Elements
G 0:5 A=mV and G 500 S are equivalent. Emphasize that the transconductance has nothing in common with the conductance (inverse resistance) of a passive resistor. Equation (2.43) is also valid irrespective of the circuits connected to the dependent source to the right and to the left in Fig. 2.31b. In this sense, the voltagecontrolled current source is again the ideal circuit element. Such a source is a transconductance ampliﬁer.
CurrentControlled Voltage Source This dependent source is shown in Fig. 2.31c. The source voltage or the output voltage follows the input current through the resistance in Fig. 2.31c according to a linear law: υout
Riin
ð2:44Þ
where the constant R with units of V=A Ω is called the transresistance of the dependent source, similar to the name resistance. For example, the expressions R 5V=mA and R 5000 Ω are equivalent. Emphasize that the transresistance has nothing in common with the resistance of a passive resistor. Equation (2.44) is again valid irrespective of the circuits connected to the dependent source to the right and to the left in Fig. 2.31c. In this sense, the currentcontrolled voltage source is also an ideal circuit element. Such a source is a transresistance ampliﬁer.
CurrentControlled Current Source The last dependent source is shown in Fig. 2.31d. The source current or the output current follows the input current according to a linear law: iout
A iin
ð2:45Þ
where the dimensionless constant A is called the shortcircuit current gain of the dependent source. However, units of A/A or A/mA are often used. For example, the expressions A 0:5 A=mA and A 500 are equivalent. We repeat that Eq. (2.45) is valid irrespective of the circuits connected to the dependent source to the right and to the left in Fig. 2.31d the voltagecontrolled voltage source is the ideal circuit element. Such a source is a current ampliﬁer.
2.4.3 Transfer Characteristics The dependent sources do not possess the υi characteristic. Instead, a transfer characteristic of the source is used, which relates the output voltage or current to the input voltage or current. For example, the transfer characteristic of the voltagecontrolled voltage source follows Eq. (2.42). It is a straight line in the υin, υout plane (the xyplane), with the slope equal to A. Other linear transfer characteristics are obtained similarly.
II66
Chapter 2
Section 2.5: Ideal Voltmeter and Ammeter: Circuit Ground
Section 2.5 Ideal Voltmeter and Ammeter: Circuit Ground 2.5.1 Ideal Voltmeter and Ammeter The ubiquitous voltmeter and ammeter are devices designed to measure voltages and currents. Both devices are usually assembled in one unit known as a digital multimeter (DMM). From the circuit point of view, the ideal voltmeter is an open circuit which conducts zero current as shown in Fig. 2.34. An ideal ammeter is a short circuit which conducts any current with zero resistance see the same ﬁgure. In reality, the voltmeter will conduct a small leakage current, and the ammeter will exhibit a small resistance. a a
+V 
+A 
b
a open circuit
b
b
a short circuit
b
Fig. 2.34. Circuit equivalencies for ideal voltmeter and ammeter.
These features guarantee that the connection of the measurement device will not change the circuit operation. Figure 2.35 shows the proper connection of the voltmeter and ammeter to measure current through circuit element A and voltage across this element. I
I +A 
X
+
V

+V 
Fig. 2.35. Correct connection of voltmeter and ammeter for voltage and current measurements.
The ammeter is always connected in series with element X. In other words, to connect the ammeter we must break the circuit either before or after element X. Since the ammeter has no resistance, it acts just like an ideal wire and thus does not perturb the electric circuit. On the other hand, the voltmeter is always connected in parallel with element X. The circuit current I in Fig. 2.35 cannot ﬂow through the voltmeter, which acts as an open circuit. As required, it will ﬂow through element X. We conclude that an ideal voltmeter does not perturb the circuit either. Generally, voltage measurements are simpler to perform than current measurements.
Wrong Connections of Ammeter and Voltmeter The ammeter connected in parallel will short out the element A: the current will ﬂow through the ammeter. If the element A were a load, there would no longer be a load resistance in the circuit. And with no attached load, the power supply will deliver the II69
Chapter 2
Section 2.5: Ideal Voltmeter and Ammeter: Circuit Ground
2.5.3 Types of Electric Ground Figure 2.37 shows three different types of electric ground connections. The ﬁrst one is the earth ground. A true earth ground, as deﬁned by the National Electrical Code (USA), physically consists of a conductive pipe or rod driven into the earth to a minimum depth of 8 feet. Obviously, it is not always possible to physically connect the circuit directly to the earth. Some examples include a cell phone, an automobile, or an airplane. Earth ground
0V
Chassis ground
0V
Common (neutral) ground
0V
Fig. 2.37. Different ground types: earth ground, chassis ground, and common (neutral) ground.
The second ground type is the chassis ground. It is the physical metal frame or structure of an automobile, an airplane, a desktop computer, a cell phone, or other electrical devices; the term case is very similar in meaning. The chassis ground primarily involves a connection to the metal case. It is implied that the case should eventually discharge due to contact with other objects or with earth. The term ground plane for planar printed circuits, which is usually the copper bottom of a printed circuit board, is equivalent to chassis ground. The third ground type in Fig. 2.37 is the common terminal or common ground. The word common is typical for many circuits including the ampliﬁer circuits considered next, when a dualpolarity power supply is used. Here two identical batteries are connected in series, plus to minus. The common terminal of the dual power supply so designed serves as the reference ground; even a metal case is not necessarily required. The AC analog of the common ground is the neutral terminal of your wall plug. Frequently, different ground types may be interconnected. For example, the neutral terminal of the wall plug should be connected to earth ground at a certain location. The chassis ground of a large truck may be connected to the physical ground by a little ﬂexible strip nearly touching the asphalt.
2.5.4 Ground and Return Current We have already seen that electric current in a circuit always ﬂows in closed loops. This is a simple and yet a very critical property of an electric circuit. The steadystate current that ﬂows to a load is sometimes called forward current, whereas the current that returns to the power supply is the return current. Can the chassis ground itself be used as a part of this loop for the return current? The answer is yes, and Fig. 2.38a depicts this situation as an example. Here a 9 V battery is powering an incandescent light bulb. For the chassis ground in Fig. 2.38a, the circuit is correctly drawn, but putting two wires into the soil, as shown in Fig. 2.38b, will fail owing to the high resistance of the earth. The use of the ground to establish a path for the return current is quite common for the chassis ground (automotive electronics) and also for the common ground. However, it should not be attempted for the earth ground connection. Emphasize that, in many II71
Chapter 2
Major Circuit Elements
circuit diagrams, the difference between the chassis ground, the common ground, and the true earth ground is often ignored. Namely, the symbol of the earth ground used in the circuit often implies either the chassis ground or the common ground, i.e., the (physically grounded or not) current return path. a) I
 +
 +
=
9V
9V
metal chassis
short circuit
b) I
 + 9V
0V
I
 +
=
9V
0V open circuit
physical ground (soil)
Fig. 2.38. (a) The return current path for the chassis ground is metal; it can be replaced by a wire. (b) There is no current return path, since soil (dry or wet) is a very poor conductor. The circuit is therefore open and not functioning.
2.5.5 Absolute Voltage and Voltage Drop Across a Circuit Element The electric ground serves as a voltage reference point in a circuit. It allows us to use two types of voltages in the circuit: 1. The absolute voltage at a certain circuit node 2. The voltage drop or simply the voltage across a circuit element Figure 2.39 shows the concept. Voltages Va,b,c,d are absolute voltages measured versus ground at nodes a, b, c, d in Fig. 2.39. Voltages VA,B,C give the voltage drop across the circuit elements A, B, and C. Indeed, the ideal wires remain the equipotential surfaces (have the same absolute voltage). Taking into account the polarity of the voltages VA,B,C shown in Fig. 2.39, one has for the node voltages Va
0 V, V b
Va þVA
10 V, V c
Vb
VB
5 V, V d
Vc
VC
0V
ð2:48Þ
Note that both voltage types absolute voltage and voltage across a circuit element are often denoted by the same letter V (in the DC case) and may be easily misplaced. Both of them are widely used in electric circuit analyses. The hint is that the voltage across a circuit element always has the polarity labeled with sign, whereas the absolute voltage often has not. II72
Chapter 2
Problems
Problem 2.6. The power absorbed by a resistor from the ECE laboratory kit is 0.2 W. Plot the υi characteristics of the corresponding resistance to scale given that the DC voltage across the resistor was 10 V. I, mA 8 4 0 V, volts 4 8 4
2
0
2
4
Problem 2.7. The number of free electrons in copper per unit volume is n ¼ 8:46 1028 m13 . The charge of the electron is 1.60218 10 19 C. A copper wire of cross section 0.25 mm2 is used to conduct 1A of electric current. A. Sketch the wire, the current direction, and the direction of electron motion. B. How many coulombs per one second is transported through the conductor? C. How fast do the electrons really move? In other words, what is the average electron velocity?
B. Solve task A when the wire cross section is increased to 2.5 mm2. Problem 2.11. Determine the total resistance of the following conductors: A. A cylindrical silver rod of radius 0.1 mm, length 100 mm, and conductivity 6.1107 S/m. B. A square graphite bar with the side of 1 mm, length 100 mm, and conductivity 3.0104 S/m. C. A semiconductor doped Si wafer with the thickness of 525 μm. Carrier mobility is μ ¼ 0:15 m2/(Vs). Carrier concentration is n ¼ 1023 m 3 . Carrier charge is 1.6 10 19 C. The resistance is measured between two circular electrodes with the radius of 1 mm each, which are attached on the opposite sides of the wafer. Assume uniform current ﬂow between the electrodes. a)
b)
c)
Problem 2.8. Repeat the above problem when the conductor’s cross section is increased to 5 mm2. Problem 2.9. A copper wire having a length of 1000 ft and a diameter of 2.58826 mm is used to conduct an electric current of 5 A. A. What is wire’s total resistance? Compare your answer to the corresponding result of Table 2.2. B. What is the power loss in the wire? Problem 2.10 A. A copper wire having a length of 100 m and a cross section of 0.5 mm2 is used to conduct an electric current of 5 A. What is the power loss in the wire? Into what is this power loss transformed?
Problem 2.12. A setup prepared for a basic wireless powertransfer experiment utilizes a square multiturn loop schematically shown in the ﬁgure, but with 40 full turns. A #22 gauge copper wire with the diameter of 0.645 mm is used. Total loop resistance, R, is needed. Please assist in ﬁnding the loop resistance (show units).
Problem 2.13. Estimate resistance, Rn (show units), of the nside of a Si pnjunction diode in
II77
Chapter 2
Major Circuit Elements Justify your answer.
+
+
I V
V

Problem 2.23. Find the dynamic (smallsignal) resistance r of a nonlinear passive circuit element—the ideal diode—when the operating DC point V0, I0 is given by the solutions to the previous problem. Consider all four cases.
Problem 2.24. A nonlinear passive circuit element is characterized by the υi characteristic in the form I ¼ I S p V =V S 2 with I S ¼ 1 A and
I
1þðV =V S Þ
V S ¼ 1 V. Plot the υi characteristic to scale. Next, ﬁnd the static element resistance R0, the element current I0, and the corresponding dynamic element resistance r when (A) V0 ¼ 0.1 V, (B) V0 ¼ 1.0 V, and (C) V0 ¼ 5.0 V.
V
2.3 Independent Sources
I, A
1
0
2.3.1 Independent Ideal Voltage Source 2.3.2 Circuit Model of a Practical Voltage Source
0 V, volts
1 5
1 mA
A)
+
VA=20 V 1 mA
B)
A

VA=20 V
+
Problem 2.26. A DC circuit shown in the following ﬁgure includes two interconnected passive elements: an ideal diode and a resistance. One possible circuit solution is given by an intersection of two υi characteristics marked by a circle in the same ﬁgure. This solution predicts a nonzero circuit current and a positive voltage across both circuit elements. A. Is this solution an artifact (a mistake has been made somewhere)? B. Is this solution true (the circuit so constructed might function)?
A
1 mA
C)
A
+
Problem 2.25. Repeat the previous problem when I S ¼ 0:5 A and V S ¼ 0:5 V. All other parameters remain the same. Consider the following DC operating points: (A) V0 ¼ 0.05 V, (B) V0 ¼ 0.50 V, and (C) V0 ¼ 2.50 V.
VA= 20 V

0

5
Problem 2.27. In the following ﬁgure, determine if the element is a resistance or a voltage source. Find the power delivered to element A or taken from element A in every case.
Problem 2.28. Based on voltage and current measurements, determine if the circuit element is a resistance or a voltage source. Readings of the ammeter and voltmeter are shown in the following ﬁgure.
II80
Chapter 2
Major Circuit Elements
Find current I in every case. (a) The negative terminal is left disconnected; (b) the negative terminal is connected to the positive terminal through the resistor; (c) both terminals are connected to chassis ground. a)
7.5 kW
 +
2.5.5 Absolute Voltage and Voltage Drop Across a Circuit Element Problem 2.55. Determine if the circuit element shown in the following ﬁgure is a resistance, a voltage source, or a wire (short circuit). Absolute voltages at points a and b are measured versus ground. I
I
9V
Va
Vb
a
b
0V 0V
b) 7.5 kW
 +
1. V a 3 V, V b 3V, I 1 A 2. V a 3 V, V b 1V, I 1A 3. V a ¼ 2 V, V b ¼ 5 V, I ¼ 2 A:
I
9V
c)
Problem 2.56. Determine if the circuit element shown in the following ﬁgure is a resistance, a voltage source, or a wire (short circuit). Absolute voltages at points a and b are measured versus ground.
7.5 kW
 +
I
9V
I
Problem 2.54. What is the voltmeters’ (ammeter’s) reading in the ﬁgure below?
5V
+ 
+V 
#1
Vb
a
b 0V
10 W
a)
Va
V
+ 
#2
+ 
#2
1. V a 6 V, V b 3 V, I 1 A 2. V a 1 V, V b 1 V, I 1A 3. V a ¼ 7V, V b ¼ 5V, I ¼ 2 A:
0V 10 W
b) 5V
+ 
+V 
A
#1
0V
II86
Chapter 3
Chapter 3: Circuit Laws and Networking Theorems Overview Prerequisites:  Knowledge of circuit elements, their υi characteristics, and Ohm’s law (Chapter 2) Objectives of Section 3.1:  Understand the meaning of an electric network and its topology (nodes, branches, loops, meshes)  Review the Kirchhoff’s current law, its use and value  Review the Kirchhoff’s voltage law, its use and value  Become familiar with the Tellegen’s theorem and Maxwell’s minimum heat theorem Objectives of Section 3.2:  Be able to combine sources and resistances in series and parallel  Practice in the reduction of resistive networks using series/parallel equivalents  Realize the function and applications of the voltage divider circuit  Realize the function of the current divider circuit  Understand the function and applications of the Wheatstone bridge Objectives of Section 3.3:  Understand the role and place of linear circuit analysis  Learn the superposition theorem  Understand the decisive value of superposition theorem for linear circuit analysis  Learn about immediate applications of the superposition theorem  Obtain the initial exposure to Y and Δ networks and to T and Π networks Application examples:  Voltage divider as a sensor circuit  Voltage divider as an actuator circuit  Superposition theorem for a cell phone
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_3
III89
Chapter 3
Circuit Laws and Networking Theorems
Keywords: Electric network, Branches of electric network, Nodes of electric network, Loops of electric network, Meshes of electric network, Essential mesh, Branch currents, Branch voltages, Series connection, Parallel connection, Shunt connection, Kirchhoff’s current law, Kirchhoff’s voltage law, Maxwell’s minimum heat theorem, Tellegen’s theorem, Power conservation law for electric networks, Oneport network, Equivalent electric networks, Equivalent electric circuits, Series battery bank, Battery pack, Dualpolarity power supply, Common ground of the dualpolarity power supply, Virtual ground of the dualpolarity power supply, Parallel battery bank, Series and parallel combinations (of resistances of conductances), Equivalent resistance, Equivalent circuit element, Reduction of resistive networks, Voltage divider circuit, Voltage division rule, Sensor circuit sensitivity, Maximum sensitivity of the voltage divider circuit, Current limiter, Currentlimiting resistor, Current divider circuit, Current division rule, Wheatstone bridge (deﬁnition of difference signal difference voltage balanced), Linear circuit (deﬁnition of homogeneity additivity superposition), Nonlinear circuit (deﬁnition of linearization dynamic or smallsignal resistance), Superposition theorem, Superposition principle, Y network, Δ network, Twoterminal networks, Threeterminal networks, Conversion between Y and Δ networks, Replacing a node by a loop, Δ to Y transformation, Y to Δ transformation, Balanced Y network, Balanced Δ network, Star to delta transformation, T network, T pad, Π network, Π pad, Twoterminal network (deﬁnition of input port output port)
III90
Chapter 3
Section 3.1: Circuit Laws: Networking Theorems
Section 3.1 Circuit Laws: Networking Theorems Electric components are interconnected to design functional electric circuits that can perform speciﬁc tasks like driving a motor or monitoring a power plant. Interconnected circuit components form an electric network. In turn, any electric network is solved using two simple yet very general laws: Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL). Series, parallel, and other combinations of any circuit elements, whether linear or not, can be explained and solved using these laws.1 They were established by Gustav Kirchhoff (1824 1887), a German physicist and mathematician, in 1845, while Kirchhoff was a 21yearold student at the University of Koenigsberg in East Prussia. The circuit analysis of all circuits is based on KCL and KVL.
3.1.1 Electric Network and Its Topology An electric network can be studied from a general mathematical point of view. If the speciﬁc electrical properties are abstracted, there remains a geometrical circuit, characterized by sets of nodes, branches, and loops. These three items form the topology of an electric network, and the interconnection of its elements can be represented as a graph. The study of electric network topology makes it possible to: A. Identify identical circuit blocks in electric circuits which might be drawn in a variety of ways. Examples include series/parallel connections, as well as wye (Y or T) and delta (Δ or Π) blocks as considered in this chapter. B. Analyze general properties of very large electric circuits such as electric power grids. For example, there are important relationships between the power grid topology and risk identiﬁcation and mitigation. C. Relate electric circuits to other disciplines. For example, there is a remarkable ability of electric networks to model the dynamical behavior of complicated biological systems.
Nodes, Branches, Loops, and Meshes Consider an electric network with four arbitrary circuit elements A to D shown in Fig. 3.1a. A branch is a twoterminal circuit element. All four twoterminal elements in Fig. 3.1a are therefore branches.
The KCL and KVL concepts are so powerful they even ﬁnd applications in equivalent form in magnet systems such as transformers and motors. For example, the magnetic ﬂux in a yoke with air gaps can be modeled according to KCL and KVL.
1
III91
Chapter 3
Circuit Laws and Networking Theorems
Section 3.2 Series and Parallel Network/Circuit Blocks 3.2.1 Sources in Series and in Parallel SeriesConnected Battery Bank The simultaneous use of KCL and KVL allows us to analyze the behavior of combinations of active circuit elements and establish their equivalence. The physical counterparts are various battery banks, which are interconnections of the identical batteries. Figure 3.8 shows a series battery bank, also called a battery pack, with two or more batteries connected in series. The battery symbol implies an ideal voltage source. a) IB a
+ 9V

V
IB
+
+
a
18 V
+

9V


b)
b
IB
a
IB
IB
IB
b
+
a
+
1.5 V 1.5 V
V
6V
1.5 V
IB

1.5 V

b b
IB
Fig. 3.8. Series combinations of battery cells and their equivalent representations.
We intend to ﬁnd the resulting voltage and current of this combination. To determine the equivalent voltage, we close the circuit loop shown in Fig. 3.8a by introducing a virtual circuit element with terminals a and b, with an unknown voltage V between these terminals. This element simulates the rest of the circuit, which closes the current path. KVL for the loop shown in Fig. 3.8a results in V þ9 Vþ9 V
0)V
18 V
ð3:9Þ
Using KCL, we obtain the same current ﬂows throughout the lefthanded circuit of Fig. 3.8a; this current is exactly equal to IB.3 Therefore, the series combination of two batteries in Fig. 3.8a is equivalent to one battery bank that provides double the voltage, or 18 V, compared to the unit cell. However, it delivers a current of a single cell. The same 3
If the realistic battery cells are capable of delivering different currents when short circuited, then the lowest cell current will ﬂow under short circuit condition.
III100
Chapter 3
Section 3.2: Series and Parallel Network/Circuit Blocks
method can now be applied to multiple battery cells connected in series; one such battery bank is shown in Fig. 3.8b.
DualPolarity Voltage Power Supply Two batteries, or other voltage sources, connected in series can be used as a dualpolarity power supply as shown in Fig. 3.9. The middle terminal gives us the virtual ground for the circuit or the common ground. Both negative and positive voltages with respect to the common port can now be created in the circuit. Such a source is of particularly importance for operational ampliﬁer circuits and for transistor circuits. Every multichannel laboratory power supply may operate as a dual power supply. The common terminal may (but does not have to) additionally be connected to the earth ground.
+
+
IB
9V

common
+
=
9V

IB
Fig. 3.9. A dualpolarity power supply constructed with two battery cells.
ParallelConnected Battery Bank As an alternative to the seriesconnected battery bank, we also investigate the parallel battery bank shown in Fig. 3.10. To determine the equivalent voltage of the combination, we again close the circuit loop by introducing a virtual circuit element with terminals a and b and with an unknown voltage V between these terminals. The use of KVL gives V
ð3:10Þ
9 V
Thus, the voltage of the battery bank in Fig. 3.10 is still equal to the unit cell voltage. However, applying KCL to both nodes shown in black in Fig. 3.10 indicates that the current doubles. Therefore, the parallel combination of two batteries is equivalent to one battery bank that provides the same voltage of 9 V as one unit cell but at twice the current strength. 2IB
a
2IB
+
+ 9V

b
9V
IB

IB
9V

V
+
+
a
2IB
b
Fig. 3.10. A parallel combination of battery cells and its equivalent single battery representation.
III101
Chapter 3
Circuit Laws and Networking Theorems
Series Versus Parallel Connection What is the difference between series and parallel combinations of two 9V batteries? First, let us ﬁnd the power delivered to the circuit. We assume I B 1 A in both cases, even though the speciﬁc value of the current is not really important. For the series combination in Fig. 3.8a, the delivered power is 18 V 1 A 18 W. For the parallel combination in Fig. 3.10, the delivered power is again 9 V 2 A 18 W. Thus, as far as the power rating is concerned, there is no difference. You should, however, remember that we always deliver power to a load. For the series combination, the implied load resistance is 18 V/1 A 18 Ω. For the parallel combination, the anticipated load resistance should be 9 V/2 A 4.5 Ω. Thus, it is the load resistance that determines which combination should be used. This question is of great practical importance. Combinations of Current Sources Combinations of current sources are studied similarly. They are important for photovoltaic and thermoelectric semiconductor devices. 3.2.2 Resistances in Series and in Parallel Series Connection After the sources have been analyzed, we turn our attention to series and parallel combinations of resistances (or conductances). Their physical counterparts are various circuit loads, for example, individual households connected to the same power grid or individual motors driven by the same source. Figure 3.11 depicts the series combination of two resistances R1 and R2. I
+
a
R1
V1
I
+
+

+ R2
V2

b
Req=R1+R2
V

V
I
I
Fig. 3.11. Two resistances in series and their equivalent single resistance representation.
Note that the current direction for both resistances corresponds to a passive reference conﬁguration: current ﬂows “down the voltage hill.” Again, we close the circuit loop by introducing a virtual circuit element with terminals a and b and with an unknown voltage V between those terminals. This virtual circuit element, which simulates the rest of the circuit, allows us to close the current path. KVL for the loop shown in Fig. 3.11 results in V
V1 þ V2
IR1 þ IR2
I ðR1 þ R2 Þ
IReq ) Req
R1 þ R2
ð3:11aÞ
III102
Chapter 3
Section 3.2: Series and Parallel Network/Circuit Blocks
Thus, two resistances can be replaced by one equivalent resistance, which is the sum of the individual resistances. It follows from Eq. (3.11a) that the equivalent resistance gives us the same circuit current (and the same power into the load) as the original resistance combination does, for any applied voltage. This is the formal description of the equivalent resistance to be generalized later. Equation (3.11a) can easily be extended to any arbitrary number of resistances connected in series. Equation (3.11a) may also be formulated in terms of conductances, the reciprocals of resistances, 1 Geq
1 1 þ G1 G2
ð3:11bÞ
Parallel Connection Next, we consider the parallel combination of resistances shown in Fig. 3.12. KVL for the loop shown in the ﬁgure and for another loop between two resistances indicates that the voltages across every resistance are equal to V. KCL applied to either node shown in black results in I
I
I2
+
R1 V
R2 V


V
I
Req=R1R2/(R1+R2)
V

+
+
+
I1
I
Fig. 3.12. Two resistances connected in parallel and the equivalent resistance.
I
I1 þ I2
V V þ R1 R 2
V 1 ) Req Req
1 1 þ ) Geq R1 R2
G1 þ G2
ð3:12Þ
Therefore, the parallel combination of two resistances is equivalent to one resistance, which has a value equal to the reciprocal of the sum of the reciprocal values of both. The equivalent resistance again gives us the same circuit current as the original resistance combination does, for any applied voltage. Emphasize that the equivalent resistance is always smaller in value than each of the resistances to be combined in parallel. Note that the conductances simply add up for the parallel combinations. Equation (3.12) can again easily be extended to any arbitrary number of resistances connected in parallel.
Meaning of Equivalent Circuit Element In summary, the study of series/parallel active and passive circuit elements leads us to the following simple deﬁnition of an equivalent circuit element, either passive or active. The equivalent circuit element possesses the same υi characteristic as the υi III103
Chapter 3
Section 3.2: Series and Parallel Network/Circuit Blocks
The voltage divider with multiple resistances R1, R2, . . ., RN is solved in the form VS
V1 þ V2 þ . . . þ VN ) I
and V i
VS R1 þ R2 þ . . . þ RN
ð3:15Þ
Ri VS R1 þ R2 þ . . . þ RN
3.2.5 Application Example: Voltage Divider as a Sensor Circuit Consider a resistive sensing element (thermistor, strain gauge, photoresistor, etc.) denoted by R2(x) in Fig. 3.17. The element changes its resistance R2(x) when an external parameter x changes. Parameter x could be temperature, pressure, humidity, solar radiation, or any other physical parameter that undergoes process changes. A simple sensor conﬁguration is a direct connection to a voltage source and to the DMM for voltage measurements; see Fig. 3.17a. No matter how the sensor resistance changes, the sensor will always output the source voltage. A solution to the third problem is a voltage divider circuit shown in Fig. 3.17b. The extra resistance R1 is ﬁxed. According to Eq. (3.14), voltage V2 in Fig. 3.17b varies depending on the inﬂuence of R2(x); V 2 ðxÞ
R 2 ðx Þ VS R1 þ R2 ðxÞ
ð3:16Þ
a)
b)
+
V2
R1
+

R2
V2=VS
+ 
V
VS
+ 
+
+ 

VS
V1
R2
V2
+ 
V

Fig. 3.17. (a) Incorrect sensor circuit. (b) A sensor circuit on the basis of a resistive voltage divider where R2(x) changes its resistance depending on the process parameter x.
The variable voltage V2(x) is measured by the voltmeter. The dependence of V2 on R2 is clearly nonlinear in Eq. (3.16). Although Eq. (3.16) can be linearized by choosing a sufﬁciently large R1 to make the denominator nearly constant, we will show later that such an operation greatly decreases device sensitivity. Let us assume that the external parameter x in Eq. (3.16) changes from a lower limit x1 to an upper limit x2, i.e., x1 x x2 . As a result, the sensing resistance changes monotonically, but not necessarily linearly, from R0 R2 ðx1 Þ to R00 R2 ðx2 Þ. We also assume that if x1 x x2 then R0 > R00 . The sensor circuit’s sensitivity, S, is given by
III107
Chapter 3
S
V 2 ðx1 Þ x2
Circuit Laws and Networking Theorems V 2 ðx 2 Þ x1
V units of x
ð3:17Þ
The sensitivity is expressed in terms of voltage variation per one unit of x. A higher sensitivity implies a larger voltage variation and thus provides a better sensor resolution and improved robustness against noise.
Design of the Sensor Circuit for Maximum Sensitivity Let us pose the following question: what value should the ﬁxed resistance R1 assume in order to achieve the highest sensitivity of the voltage divider sensor? It is clear that R1 cannot be very small (otherwise the voltage reading will always be VS and the sensitivity will be zero) and that R1 cannot be very large (otherwise the voltage reading will be always 0 V and the sensitivity will be zero). The sensitivity is thus a positive function that is zero at R1 0 and at R1 1. According to the extreme value theorem, a global maximum should exist between these two values. We denote the unknown resistance R1 with variable t, substitute V2 from Eq. (3.14), and rewrite Eq. (3.17) in the form 0 VS R R00 ð3:18aÞ S x2 x1 t þ R0 t þ R00 It is convenient to transform this result into a simpler expression S V S S 0 f ðt Þ, where a constant S0 is called the intrinsic sensitivity of the resistive sensing element, and f(t) is the sole function of the ﬁrst resistance, i.e., 0 R R00 t S0 , f ðt Þ ð3:18bÞ 0 x2 x1 ðR þ t ÞðR00 þ t Þ This function f(t) is to be maximized. At the function’s maximum, the derivative of f(t) versus t should be zero. Using the quotient rule for the differentiation of a fraction, it follows from Eq. (3.18b) that f 0 ðt Þ
R0 R00
t2
ðR0 þ t Þ2 ðR00 þ t Þ
ð3:18cÞ
2
The ﬁnal result following from the condition f 0 ðt Þ p R0 R00 t R1
0 is surprisingly simple ð3:18dÞ
In other words, the ﬁxed resistance of the voltage divider circuit should be equal to the geometric mean of two extreme resistances of the sensing element itself.
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Chapter 3
Circuit Laws and Networking Theorems
Section 3.3 Superposition Theorem and Its Use 3.3.1 Linear and Nonlinear Circuits Linear Circuit The superposition theorem studied in this section is only valid for linear circuits. A linear circuit is a circuit that includes only linear circuit elements (elements with a linear or straight υi characteristic): 1. 2. 3. 4. 5.
Resistance Capacitance (υi relationship is timedependent but still linear) Inductance (υi relationship is timedependent but still linear) Voltage source (independent and linear dependent) Current source (independent and linear dependent)
Every linear circuit satisﬁes both the homogeneity and additivity properties. To explain those properties, we consider a linear circuit with an input parameter x (input voltage or current) and an output parameter f(x) (output voltage or current). A function f is the characteristic of the circuit itself; it must be a linear function. Namely, when an input parameter is a linear superposition ax1 þ bx2 of two individual stimuli x1, x2, the output is also a linear superposition of two individual responses, i.e., f ðax1 þ bx2 Þ
af ðx1 Þ þ bf ðx2 Þ
ð3:28Þ
For example, if we double all source strengths in a linear circuit, voltages across every passive circuit element and currents through every circuit element will also double.
Nonlinear Circuit and Circuit Linearization A nonlinear circuit will include nonlinear circuit elements, e.g. elements with a nonlinear υi characteristic. Any circuit with semiconductor components (such as diodes, transistors, solar cells) is a nonlinear circuit. Since the vast majority of electronic circuits include semiconductor components, a legitimate question to ask is what value do the linear circuits have in this case? One answer is given by a linearization procedure, which makes it possible to reduce the nonlinear circuit to a linear one, in a certain domain of operating parameters. Mathematically, circuit linearization means that a nonlinear relationship V(I) is expanded into a Taylor series, dV V ðI Þ V 0 þ ðI I 0 Þ þ . . . ð3:29Þ dI V V 0 , I I 0 about a certain operating point V0, I0, where only the constant and the linear terms are retained. The derivative in Eq. (3.29), the socalled dynamic or smallsignal resistance r, is now used in place of the familiar resistance R for the linear ohmic circuit elements.
III116
Chapter 3
Circuit Laws and Networking Theorems
Conversion Between Y and Δ Networks A problem of signiﬁcant practical importance is the conversion between the Y and Δ networks in Fig. 3.27a, b. This conversion is equivalent to replacing a node by a loop (strictly speaking, by a mesh) and vice versa in a more complicated circuit or network. Such a replacement may signiﬁcantly simplify the overall circuit analysis. The conversion is established based on the superposition theorem. Two arbitrary networks are equivalent if their υi characteristics are the same. In other words, by connecting three arbitrary sources to terminals 1, 2, and 3 of the Y network, we must obtain terminal voltages and currents identical to those of the Δ network with the same sources. We select three current sources I1, I2, I3 in Fig. 3.27c, d. The solution with three sources is obtained as a superposition of three partial solutions, with two sources opencircuited at a time. Let’s keep the source I1 and replace I2, I3 by open circuits ﬁrst. Voltages V12 for both networks will be the same when the equivalent resistances R12 between terminals 1 and 2 will be the same. A similar treatment holds for terminals 1 and 3 (source I3 is kept) and terminals 2 and 3 (source I2 is kept), respectively. Therefore, with reference to Fig. 3.27c, d, we have R12
R1 þ R3
Rb ðRa þ Rc Þ
R13
R1 þ R2
Rc ðRa þ Rb Þ
R23
R2 þ R3
Ra ðRb þ Rc Þ
Rb ðRa þ Rc Þ Ra þ Rb þ Rc Rc ðRa þ Rb Þ Ra þ Rb þ Rc Ra ðRb þ Rc Þ Ra þ Rb þ Rc
ð3:33aÞ ð3:33bÞ ð3:33cÞ
Next, we add Eq. (3.33a) and Eq. (3.33b) and subtract from this result Eq. (3.33c). This gives us the expression for R1. To obtain R2, we add Eq. (3.33b) and Eq. (3.33c) and subtract Eq. (3.33a). R3 is obtained by adding Eq. (3.33a) and Eq. (3.33c) and subtracting Eq. (3.33b). The result has the form of Δ to Y transformation: R1
Rb Rc , Ra þ Rb þ Rc
R2
Ra Rc , Ra þ Rb þ Rc
R3
Ra Rb Ra þ Rb þ Rc
ð3:34Þ
The inverse transformation, Y to Δ transformation, follows Ra Rc
R1 R2 þ R1 R3 þ R2 R3 , Rb R1 R1 R2 þ R1 R3 þ R2 R3 R3
R1 R2 þ R1 R3 þ R2 R3 , R2
ð3:35Þ
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Chapter 3
Circuit Laws and Networking Theorems
1. Should the batteries be connected in series or in parallel? 2. What is a typical capacity of the AAA battery? 3. How much energy in watt hours is initially stored in each battery? Problem 3.22. A load has the resistance of 1.5 Ω and requires the applied voltage of 3 V. A number of battery cells are given, each of which is rated for 1.5 V. Each cell may deliver no more than 1 A of current: 1. Construct and draw a battery bank that could be used to drive the load. 2. Is the solution to the problem unique? Problem 3.23. Repeat the previous problem when the required load voltage is changed to 6 V.
R15 R14 R13
... R2 R1
+V 
12 V
Problem 3.26. Find the equivalent resistance between terminals a and b. 100 W
a
3.2.2 Resistances in Series and in Parallel 3.2.3 Reduction of Resistive Networks Problem 3.24. Determine the equivalent resistance between terminals a and b. a) a
200 W
200 W
b
Problem 3.27. Find the equivalent resistance between terminals a and b. 100 W
a
1 kW
3 kW
3 kW
3 kW 100 W
150 W
b b) a
100 W
b
1 kW
1 kW
1 kW
b
Problem 3.25. The equivalent electric circuit for a car rear window defroster is shown in the ﬁgure. All resistances are equal: R1 ¼ . . . ¼ R15 ¼ 10 Ω. Determine the heat power (power delivered to the defroster).
Problem 3.28. Find the equivalent resistance between terminals a and b. 500 W
1kW
a 2 kW
1 kW
b 1kW
500 W
Problem 3.29. Determine the equivalent resistance between terminals a and b.
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Chapter 3
Circuit Laws and Networking Theorems
Combined Voltage and Current Dividers Problem 3.61. For the circuit shown in the following ﬁgure: A. Find currents i, i1, i2 (show units). B. Find power P delivered by the voltage source to the circuit. C. Find voltages V1, V2.
i2
1 kW
0.25 kW
0.3 kW
4 kW
V2

Problem 3.62. Find voltage V across the 20kΩ resistance and current i for the circuit shown in the following ﬁgure:
P=?
Problem 3.65. Determine current i (show units) in the circuit that follows: 20 V
5 kW
+
4 kW
i2
+
+ 
+ 

+
i1 15 V
i1 10 V
1 kW

V1 8 kW
i
A. Current i2 through the 0.25kΩ resistance B. Power P absorbed by the 0.3kΩ resistance
2 kW
20 kW
20 kW
i 25 V
+
5 kW
5 kW

+
6 kW 20 kW
V
i
Problem 3.66. Determine current i and voltage υ for the circuit shown in the ﬁgure (show units).

5 kW
24 V
5k
600 W 15 V
+ 
i1 1 kW
i2 0.25 kW
700 W
Problem 3.64. For the circuit shown in the ﬁgure below, determine:
i
30 k
10 k
3k

P=?
+

Problem 3.63. For the circuit that follows, determine: A. Current i2 through the 0.25 kΩ resistance B. Power P absorbed by the 600 Ω resistance
+
30 kW
5k
1k
3.3 Superposition Theorem and Its Use 3.3.2 Superposition Theorem or Superposition Principle Problem 3.67. For the circuit shown in the ﬁgure below, determine the absolute voltage (versus chassis ground) and the electric current at the circuit point a.
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Chapter 3
Circuit Laws and Networking Theorems
A. Draw the corresponding Y network. B. Label its terminals. C. Determine and label the corresponding resistance values.
A. Draw the corresponding T network. B. Label its terminals and ports. C. Determine and label the corresponding resistance values.
10 kW
20 kW 1
3
1
3 10 kW
25 kW
port #1 20 kW
port #2
10 kW
2
4 ground connection
2
Problem 3.74. Convert the twoport network shown in the following ﬁgure from T to Π: 1. Draw the corresponding Π network. 2. Label its terminals and ports. 3. Determine and label the corresponding resistance values.
Problem 3.76. For the bridge network shown below, determine the equivalent resistance between terminals a and b. 3.75 kW a 2.5 kW
25 kW 1
port #1
5 kW
20 kW 3
10 kW
10 kW
port #2
20 kW 2
10 kW
4 ground connection
b
Problem 3.75. Convert the twoport network shown in the following ﬁgure from Π to T:
III138
Chapter 4
Circuit Analysis and Power Transfer
Keywords: Nodal analysis, Mesh analysis (meshcurrent analysis), Supernode, Supermesh, Method of short/ open circuit (deﬁnition of, opencircuit network voltage, shortcircuit network current), Source transformation theorem, Circuit equivalent (see equivalent circuit), Thévenin’s theorem (formulation, proof, special cases), Thévenin equivalent, Norton’s theorem, Norton equivalent, R2R ladder network, Negative equivalent (Thévenin) resistance, Maximum power theorem (principle of maximum power transfer), Power efﬁciency, Analysis of nonlinear circuits, Load line (deﬁnition, method of), Iterative method for nonlinear circuits (deﬁnition of, explicit iterative scheme, implicit iterative scheme), Solar cell (cSi, opencircuit voltage, shortcircuit photocurrent density, ﬁll factor, characteristic equation of), Solar panel (series cell connection, opencircuit voltage, shortcircuit photocurrent, ﬁll factor, maximum power load voltage, maximum power load current)
IV140
Chapter 4
Section 4.1: Nodal/Mesh Analysis
Section 4.1 Nodal/Mesh Analysis 4.1.1 Importance of Circuit Simulators The series and parallel equivalents along with Y and Δ transformations provide a practical tool for solving simple circuits involving typically only a few elements. However, for more elaborate circuits, circuit simulators such as SPICE (Simulation Program with Integrated Circuit Emphasis) and its various modiﬁcations become indispensable tools for the professional engineer. SPICE was developed by the Electronics Research Laboratory at the University of California, Berkeley, and ﬁrst presented in 1973. These circuit simulators are quite general and allow us to model circuits with passive and active elements including semiconductor components such as diodes, transistors, and even solar cells. Since those elements typically exhibit nonlinear current voltage behaviors, elaborate solution strategies are needed. The circuit simulators use quite interesting algorithms: they often operate in the time domain, even for DC circuits. For example, a solution for a DC circuit is obtained as the steadystate limit of a transient solution, for voltage and/or current sources turned on at a certain time instance. The key of the timedomain approach is its inherent ability to solve nonlinear problems, with passive and active circuit elements. In this section, we are unable to discuss in detail the principles of the numerical circuit simulation. However, we will provide the foundation of the nodal analysis (or node analysis) and the mesh analysis (or the meshcurrent analysis), which are two important features of a professional circuit simulator. The nodal and mesh analyses in its pure form do not involve timedomain methods. They are primarily applicable only to linear circuits, also referred to as linear networks. 4.1.2 Nodal Analysis for Linear Circuits The nodal analysis is a general method of solving linear networks of arbitrary complexity, which is based on KCL and Ohm’s law. Let us consider a circuit shown in Fig. 4.1a, which is a resistive bridge circuit with a bridging resistance. This circuit may be solved using Δ to Y conversion; see, for instance, example 3.15 of Chapter 3. Here, we prefer to use the nodal analysis directly. The nodal analysis operates with the absolute values of the node voltages in the circuit with respect to ground reference. It may be divided into a number of distinct steps: 1. A ground reference needs to be assigned ﬁrst: a node where the voltage is set to 0 V. To this end, we ground the negative terminal of the voltage power supply. 2. Next, we select nontrivial (also called nonreference) nodes for which we do not know the voltages. These are nodes 1 and 2 in Fig. 4.1b. The two additional nodes are eliminated from the analysis since the voltages there are already known. 3. We label absolute node voltages versus ground reference as V1, V2 see Fig. 4.1c.
IV141
Chapter 4
Section 4.2: Generator Theorems
Section 4.2 Generator Theorems 4.2.1 Equivalence of Active OnePort Networks: Method of Short/Open Circuit In Chapter 3, we considered passive linear networks with only resistances, and we have transformed them into equivalent circuits. Active linear networks, which include sources and resistances simultaneously, can undergo similar transformations. We know that two electric singleport networks are equivalent when their terminal υi characteristics are identical. For passive resistive networks studied in Chapter 3, we connected arbitrary source(s) across the network terminals and checked the resulting υi characteristics. For active networks with sources and resistances, we can use the same method. Alternatively, we could connect arbitrary resistance(s) across the network terminals and check either the resulting voltage or current. A test resistance to be connected will be denoted here by R. If for two networks the voltages across the resistance R (or currents through it) coincide for all values of R, the networks are equivalents. Method of Short/Open Circuit In general, testing all possible values of resistance R connected to terminals a and b of a network in Fig. 4.7 is not necessary. Note that an active linear network may ultimately have only two elements: a source and a resistance. To uniquely determine the two elements (their values), only two equations are necessary. It is therefore customary to check only two (limiting) values of the test resistance: R ! 1 and R
ð4:13Þ
0 test of short circuit current a arbitrary electric network
VOC

b
a
+
test of open circuit voltage
arbitrary electric network
ISC b
Fig. 4.7. Method of short/open circuit for an active, oneport network.
Conveniently, this corresponds to open and shortcircuit conditions. In the ﬁrst case, the voltage between terminals a and b is the opencircuit network voltage VOC. In the second case, the current ﬂowing from terminal a to terminal b is the shortcircuit network current ISC. The pair VOC, ISC is key for the method of short/open circuit. This method states that two active linear circuits are equivalent when their VOC and ISC coincide. Network equivalency relates not only to the linear active networks with two components, but, as will be shown soon, it is valid for all active linear networks.
IV149
Chapter 4
Section 4.2: Generator Theorems
Since the entire circuit is still linear, the υi characteristic of the current source in Fig. 4.12a must have the form of a linear function, V
AI þ B;
ð4:17Þ
where V is the voltage across the current source. A and B are some “constant” coefﬁcients, which do not depend on I, but do depend on the network parameters. Our goal is to ﬁnd A and B, respectively. First, we check the value I 0 when the external current source is turned off, i.e. replaced by an open circuit. From Eq. (4.17), voltage V equals B. On the other hand, it equals of VOC the original network. Therefore, B
V OC
ð4:18Þ
Now, let us turn off all the internal sources. The network becomes an equivalent resistance Req. The constant B (its opencircuit voltage) is zero. Equation (4.17) therefore yields V AI, for any value of I. On the other hand, for the current source I connected to the resistance Req, it must be V Req I. Comparing the two expressions, we obtain A
Req
ð4:19Þ
The simpler network in Fig. 4.12b is also described by the υi characteristic in the form of Eq. (4.17). In this case, B V OC V T , A Req RT . We ﬁnally compare two υi characteristics, V V
Req I þ V OC f or linear active network and RT I þ V T f or Thevenin equivalent;
ð4:20Þ
and establish the Thévenin’s theorem. A test voltage source could be used in place of the current source in Fig. 4.12, with the same result obtained. The physical background of the Thévenin’s theorem is thus the fact that the terminal response of any linear network is a linear υi characteristic a linear function with only two independent coefﬁcients, A and B. A simpler network with exactly two independent parameters VT and RT is just right to model this response.
Equivalence of Arbitrary Linear Networks with Identical VOC, ISC On one hand, linear active networks with only two elements (a source and a resistance) are equivalent when their VOC, ISC coincide. On the other hand, any active linear network is equivalent to a linear network with only two elements. Therefore, we conclude that two arbitrary linear networks are equivalent when their VOC and ISC coincide.
IV155
Chapter 4
Circuit Analysis and Power Transfer
electromagnetic radiation from the antenna. Thus, the antenna, if properly matched to the power source, will radiate 50 % of the total power as electromagnetic waves into space. Now a young electrical engineer decides to “modify” the handset by cutting the monopole antenna and leaving only one third of its length. In this case, the antenna’s radiation resistance is reduced to one ninth of its original value. How does this affect the radiated signal? To answer this question, we ﬁnd the instantaneous load power, which also follows Eq. (4.24), i.e., P L ðt Þ
RL V 2T ðt Þ
ð4:32Þ
ðRT þ RL Þ2
The ratio of the power levels for the two antenna conﬁgurations does not depend on time: PL short PL original
50=9 ð50 þ 50=9Þ
2
=
50 ð50 þ 50Þ
2
0:0018 0:0050
0:36
ð4:33Þ
Thus, for the shorter antenna, we will only achieve about 36 % of the radiated power compared to the original handset. In practice, this estimate becomes even lower due to the appearance of a very signiﬁcant antenna capacitance.
4.3.4 Application Example: Maximum Power Extraction from Solar Panel Every solar panel has the measured data for VOC and ISC listed on its backside. For linear circuits, V T V OC , RT V OC =I SC . If the solar panel were a linear circuit, the maximum extracted power would be exactly equal to 0.25VOCICS according to Eq. (4.26). Fortunately, this is not the case. The maximum extracted power is signiﬁcantly greater than this value. However, it is still less than the “best” possible value of VOCICS. To quantify the maximum power output, every solar panel has another set of measured data, VMP and IMP, also listed on its backside. VMP stands for maximum power load voltage and IMP stands for the maximum power load current. The maximum extracted power is the product VMPIMP, which is always less than VOCICS. The ratio of these two powers, F
V MP I MP < 1; V OC I SC
ð4:34Þ
is known as the ﬁll factor of the solar panel (or solar module). We will derive the theoretical value of the ﬁll factor in the next section. Table 4.2 lists some experimental data for crystalline (cSi) solar panels. The experimental ﬁll factor not only accounts for the nonlinear physics of the cell, but it also includes some resistive losses in an individual cell and in the entire solar module. Equation (4.34) approximates the ﬁll factor of a cell too.
IV164
Chapter 4
Problems
B. How is the maximum load power different, compared to the previous problem? Problem 4.53. The heating element of an electric cooktop has two resistive elements, R1 ¼ 50 Ω and R2 ¼ 100 Ω, that can be operated separately, in series, or in parallel from a certain voltage source that has a Thévenin (rms) voltage of 120 V and internal (Thévenin) resistance of 30 Ω. For the highest power output, how should the elements be operated? Select and explain one of the following: 50 Ω only, 100 Ω only, series, or parallel.
Problem 4.56. Using two Web links http://powerupco.com/site/ http://www.affordablesolar.com/, identify the solar panel that has the greatest ﬁll factor to date. Problem 4.57. A 10W BSP1012 PV cSi module shown in the ﬁgure has 36 unit cells connected in series, the shortcircuit current of 0.66 A, and the opencircuit voltage of 21.3 V. The maximum power parameters are V MP ¼ 17:3 V and I MP ¼ 0:58 A.
Problem 4.54. You are given two speakers (rated at 4 Ω and 16 Ω, respectively) and an audio ampliﬁer with the output resistance (impedance) equal to 8 Ω.
8W
4W
16 W
A. Sketch the circuit diagram that gives the maximum acoustic output with the available components. Explain your choice. B. Sketch the circuit diagram for the maximum power efﬁciency. Explain your choice.
4.3.4 Application Example: Maximum Power Extraction from Solar Panel Problem 4.55. A. Describe in your own words the meaning of the ﬁll factor of a solar cell (and solar module). B. A 200W GE Energy GEPVp200 cSi panel has the following reading on its back: V OC ¼ 32:9 V, I SC ¼ 8:1 A, V MP ¼ 26:3 V, and I MP ¼ 7:6 A. What is the module ﬁll factor? What is approximately the ﬁll factor of the individual cell?
A. Estimate the area of the single solar cell using the common photocurrent density value for cSi solar cells (show units). B. Estimate the opencircuit voltage for the single cell. C. Estimate the ﬁll factor of the module and of the cell. D. Estimate the value of the equivalent load resistance R required for the maximum power transfer from the module to the load. Problem 4.58. A 20W BSP2012 PV cSi module shown in the ﬁgure has 36 unit cells connected in series, the shortcircuit current of 1.30 A, the opencircuit voltage of 21.7 V, the maximum power voltage VMP of 17.3 V, and
IV185
Chapter 4
Circuit Analysis and Power Transfer
the maximum power current IMP of 1.20 A. Repeat the four tasks of the previous problem.
Outline parameters of a solar module (number of cells, cell area, and overall area) which is capable of driving the motor at the above conditions and estimate the overall panel size.
Problem 4.59. A REC SCM220 220watt 20V cSi solar panel shown in the ﬁgure has the following readings on the back: the shortcircuit current ISC of ~8.20 A, the opencircuit voltage VOC of ~36.0 V, the maximum power voltage VMP of ~28.7 V, and the maximum power current IMP of ~7.70 A. Repeat the four tasks of problem 4.57.
Problem 4.60. A 14.4W load (a DC motor) rated at 12 V is to be driven by a solar panel. A cSi photovoltaic sheet material is given, which has the opencircuit voltage of 0.6 V and the photocurrent density of J P ¼ 0:03 A=cm2 .
Problem 4.61. A custom 100W load (a DC motor) rated at 24 V is to be driven by a solar panel. A cSi photovoltaic sheet material is given, which has the opencircuit voltage of 0.6 V and the photocurrent density of J P ¼ 0:03 A=cm2 . Outline parameters of a solar module (number of cells, cell area, and overall area) which is capable of driving the motor at the above conditions and estimate the overall panel size. Problem 4.62. You are given: the generic ﬁll factor F ¼ 0:72 for a cSi solar panels, the generic opencircuit voltage V OC ¼ 0:6V of a cSi cell, and the generic photocurrent density J P ¼ 0:03 W=cm2 . A. Derive an analytical formula that expresses the total area Amodule in cm2 of a solar panel, which is needed to power a load, in terms of the required load power PL. B. Test your result by applying it to the previous problem. Problem 4.63. You are given a lowcost lowpower ﬂexible (with the thickness of 0.2 mm) aSi laminate from PowerFilm, Inc., with the following parameters: a ﬁll factor of F ¼ 0:61, a singlecell opencircuit voltage of V OC ¼ 0:82 V, and a photocurrent density of J P ¼ 0:0081 A=cm2 . A. Derive an analytical formula that expresses the total module area Amodule in cm2,
IV186
Chapter 5
Chapter 5: Operational Ampliﬁer and Ampliﬁer Models Overview Prerequisites:  Knowledge of major circuit elements (dependent sources) and their i characteristics (Chapter 2)  Knowledge of basic circuit laws (Chapter 3) and Thévenin equivalent (Chapter 4) Objectives of Section 5.1:  Learn and apply the model of an operational amplifier including principle of operation, opencircuit gain, power rails, and input and output resistances  Correlate the physical operational amplifier with the amplifier circuit model  Establish the idealamplifier model  Learn the first practical amplifier circuit the comparator Objectives of Section 5.2:  Understand and apply the concept of negative feedback to an operational amplifier circuit  Construct three canonic amplifier circuit configurations with negative feedback: the noninverting amplifier, the inverting amplifier, and the voltage follower  Understand the current flow in the amplifier circuit including the power transfer from the power supply to the load Objectives of Section 5.3:  Choose the proper resistance values for the feedback loop and learn how to cascade multiple amplifier stages  Learn about input/output resistances of the amplifier circuit and establish load bridging and load matching conditions important in practice  Find ways to eliminate the DC imperfections of the amplifier that become very apparent at high amplifier gains  Use an amplifier IC with a single voltage supply (a battery) Objectives of Section 5.4:  Obtain the initial exposure to differential signals and difference amplifiers  Build an instrumentation amplifier  Connect an instrumentation amplifier to a resistive sensor
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_5
V189
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models
Objectives of Section 5.5:  Learn a general feedback system including closedloop gain and error signal  Apply the general feedback theory to voltage amplifier circuits  Construct current, transresistance, and transconductance amplifiers with the negative feedback Application Examples: Operational amplifier comparator Instrumentation amplifier in laboratory
Keywords: Operational ampliﬁer: (abbreviation opamp, integrated circuit, dual inline package, noninverting input, inverting input, output terminal, power terminals, offsetnull terminals, differential input voltage, opencircuit voltage gain, openloop voltage gain, openloop conﬁguration, closedloop conﬁguration, power rails, voltage transfer characteristic, railtorail, comparator, digital repeater, zerolevel detector, circuit model, input resistance, output resistance, ideal ampliﬁer, idealampliﬁer model, marking, summing point, commonmode input signal, differential input signal, summingpoint constraints, ﬁrst summingpoint constraint, second summingpoint constraint, sourcing current, sinking current, DC imperfections, input offset voltage, input bias current, input offset currents), Negative feedback, Feedback loop, Feedback as a dynamic process, Noninverting ampliﬁer, Inverting ampliﬁer, Voltage follower (buffer) ampliﬁer, Summing ampliﬁer, Digitaltoanalog converter, Binary counter, DCcoupled ampliﬁer, ACcoupled ampliﬁer, Capacitive coupling of an ampliﬁer, Gain tolerance of an ampliﬁer, Circuit model of a voltage ampliﬁer, Input resistance of ampliﬁer circuit, Output resistance of ampliﬁer circuit, Load bridging (impedance bridging), Load matching (impedance matching), Cascading ampliﬁer stages, Virtualground (integrated) circuit, Differential voltage of a sensor, Commonmode voltage of a sensor, Differential sensor, Singleended sensor, Difference ampliﬁer, Differential ampliﬁer circuit gain, Commonmode ampliﬁer circuit gain, Commonmode rejection ratio (CMRR), Unity commonmode gain stage, Instrumentation ampliﬁer, Load cell, Current ampliﬁer using opamp, Transconductance ampliﬁer using opamp, Transresistance ampliﬁer using opamp, Howland current source (Howland current pump) Linear feedback system: (forward gain? openloop gain, feedback gain, feedback factor, summing node, difference node, closedloop gain, error signal)
V190
Chapter 5
Section 5.1: Ampliﬁer Operation and Circuit Models
Section 5.1 Ampliﬁer Operation and Circuit Models The lowpower ampliﬁer integrated circuit (IC) is arguably the most widely employed discrete circuit component encountered in common electronic audio, control, and communication systems. Among ampliﬁers, the differential input, highgain ampliﬁer called the operational ampliﬁer (or simply opamp) has become a popular choice in many circuit applications. At this point, it is impossible for us to understand the internal operation of the ampliﬁer IC without basic knowledge of semiconductor electronics, especially the junction transistor studied in the following chapters. Fortunately, the circuit model of an operational ampliﬁer does not require knowledge of the IC fabrication steps, nor does it require an understanding of the internal transistor architecture. Conceptually, operational ampliﬁers can be introduced early in the book, which enables us to immediately proceed toward our goal of designing and building practical circuits.
5.1.1 Ampliﬁer Operation Symbol and Terminals After the ampliﬁer chip is fabricated as an integrated circuit and the bond wires are attached, it is permanently sealed in a plastic package. Often the encasing is done in a dual inline (DIPN) package with N denoting the number of IC pins. Figure 5.1 on the right shows an example of a DIP package. One IC chip may contain several independent individual ampliﬁers. We start analyzing the ampliﬁer model by ﬁrst labeling the terminals and introducing the ampliﬁer circuit symbol (a triangle) as shown in Fig. 5.1 on the left. The ampliﬁer is typically powered by a dualpolarity voltage power supply with three terminals: V CC and common (ground) port of 0 V, see Section 3.2. The index C refers to the collector voltage of the internal transistors. +VCC v+ v
+
+

=
vout
VCC common
v v+
VCC
offset null
common
NC +VCC vout
Fig. 5.1. Terminals of the operation ampliﬁer (left); they also denote pins of the ampliﬁer IC package (see a common LM 741 chip on the right). All voltages are referenced with respect to a common port of the dualpolarity voltage supply.
The ampliﬁer has a total of ﬁve terminals, notably: 1. A noninverting input with the input voltage υþ with respect to common 2. An inverting input with the input voltage υ with respect to common 3. An output terminal with the output voltage υout with respect to common V191
Chapter 5
Section 5.2: Negative Feedback
Section 5.2 Negative Feedback In most practical circuits, the ampliﬁer IC is not used in openloop conﬁguration. Engineers have modiﬁed the openloop condition into a negative feedback loop in order to set the gain to a desired value and ensure the ampliﬁer’s stability. This section provides you with all the essential knowledge needed to design an ampliﬁer with negative feedback. The mathematical model introduced in this section is based on two conditions imposed at the ampliﬁer’s input; we term them the summingpoint constraints: a) No electric current ﬂows into or out of the ampliﬁer inputs. b) The differential voltage at the ampliﬁer’s input is zero. The ﬁrst summingpoint constraint has already been introduced in the previous section. The second summingpoint constraint has yet to be derived. We will show that the two summingpoint constraints, along with KCL and KVL, will enable us to solve any ampliﬁer circuit that involves a negative feedback, no matter what the speciﬁc nature of the feedback loop is and regardless of whether it is DC, AC, or a transient circuit.
5.2.1 Idea of the Negative Feedback The idea of the negative feedback goes way back we may say almost to the Stone Age. Take a wooden rod of 1 2 feet in length. Hold the rod in the vertical position at the tip of your ﬁnger. You will probably succeed. Now, close your eyes and try to do the same. You will most likely fail. The reason for the failure is a breakdown of the feedback loop. This loop is created by visual control of the rod’s position; you automatically apply a compensating acceleration to the bottom tip of the rod when it begins to fall. Another good example is driving a car and trying to stay in the center of the lane. The negative feedback for electronic ampliﬁers was ﬁrst invented and realized by Harold S. Black (1898 1983), a 29yearold American electrical engineer at Bell Labs. To many electrical engineers, this invention is considered perhaps the most important breakthrough of the twentieth century in the ﬁeld of electronics because of its wide applicability. We will construct simple ampliﬁer circuits of a given gain, using a resistive feedback loop. Being able to perform this task is already critical from the practical point of view. 5.2.2 Ampliﬁer Feedback Loop: Second SummingPoint Constraint We construct the feedback loop, as shown in Fig. 5.7, by connecting the output to the inverting input terminal. This was exactly the idea of Harold Black. The shadowed box in the feedback loop may represent one or more circuit elements. The feedback loop may be a simple wire, a resistance, a network of circuit elements (resistances, inductances, capacitances), etc. The negative feedback simply means that the output voltage, or rather a portion of it, is returned back to the inverting input.
V199
Chapter 5
Section 5.2: Negative Feedback vx 10V
5V 2.5V 0V 0
1
2
3
time, µs
Fig. 5.8. Dynamics of the differential input voltage as a function of time.
Both Table 5.1 and Fig. 5.8 make clear that the differential voltage υx decays to zero very rapidly, once the feedback loop is introduced. Hence we arrive at the second summingpoint constraint, which is valid only for the ampliﬁers with the negative feedback loop: the differential input voltage to the ampliﬁer is exactly equal to zero. The second summingpoint constraint is a close approximation to reality. Its accuracy depends on the value of the openloop gain of the ampliﬁer. If the openloop gain were inﬁnite, the second summingpoint constraint would be exact.
5.2.3 Ampliﬁer Circuit Analysis Using Two SummingPoint Constraints Next, we will solve an ampliﬁer circuit with negative feedback using the two summingpoint constraints (SPC): (i) no current into or out of the input ampliﬁer terminals and (ii) the differential input voltage is zero. The method of two summingpoint constraints is an accurate solution method for a wide variety of ampliﬁer circuits with the negative feedback. For ampliﬁer circuits with a single input, we will denote the input voltage to the ampliﬁer circuit by υin. Voltage υin may be equal to υþ or to υ , depending on ampliﬁer type to be used. Noninverting Ampliﬁer The ﬁrst ampliﬁer conﬁguration is the socalled noninverting ampliﬁer shown in Fig. 5.9. The feedback loop contains one resistance R2. Another resistance R1 shunts the inverting input to ground. The input voltage to the ampliﬁer circuit is the voltage υin with respect to ground, or common in this case, which implies the use of the dualpolarity voltage power supply. The output voltage with respect to common is υout. We apply the ﬁrst summingpoint constraint and KCL to the node “*” in Fig. 5.9 and obtain i1
ð5:11Þ
i2
Equation (5.11) is further transformed using Ohm’s law in the form υ* 0 R1
υout υ* R2
ð5:12Þ
V201
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models vin
+ vx
v* common
+
+

vout
i2

R2
i1 R1
common
common
Fig. 5.9. Circuit diagram of the noninverting ampliﬁer. A dual power supply is not shown.
The second summingpoint constraint yields υ*
since υx υin R1
ð5:13Þ
υin 0. Equation (5.12) thus reads υout υin υout ) R2 R2
υin υin þ R1 R2
As a result, we ﬁnd that the voltage inputtooutput relation becomes R2 υin 1þ υout R1
ð5:14Þ
ð5:15Þ
The ampliﬁer circuit is solved: we have expressed the output voltage in terms of the input voltage and a resistor ratio. Equation (5.15) is the basic result in ampliﬁer theory. It shows that the feedback loop allows us to precisely control the gain with two arbitrary resistances. One chooses the proper resistance combination to achieve any ﬁnite gain between one (setting R2 0) and the openloop (inﬁnite) gain (setting R1 0). In the last case, the negative input terminal becomes grounded; the feedback loop is irrelevant and can be replaced by an open circuit so that the ampliﬁer again becomes the comparator. The gain expression R2 ACL 1 ð5:16Þ 1þ R1 is called the closedloop gain of the ampliﬁer; it clearly relates the output voltage to the input voltage. Equation (5.16) is a dramatic illustration of the negative feedback. We started with an ampliﬁer having a very large yet loosely predictable openloop gain. Through applying the negative feedback, we arrived at a gain that is much smaller than the openloop gain; however, it is controllable and stable. Equation (5.16) can be derived more simply using the voltage divider concept. Namely, resistors R1, R2 form a voltage divider between 0 V and the output voltage. Hence, the voltage at node (*) may be found. Equating this voltage to the input voltage gives us Eq. (5.16). V202
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models
negative ampliﬁer terminal is directly connected to the output. According to the ampliﬁer equation, Eq. (5.1) of the previous section, we have υout
Aðυþ
υ Þ
Aðυin
υout Þ
ð5:22Þ
Solving Eq. (5.22) for the output voltage yields υout
A υþ in
υin
A υin 1þA
ð5:23Þ
Using a Maclaurin series expansion, we obtain with A >> 1 the result A 1þA
1 1 1 þ 1=A
1=A 1
ð5:24Þ
which is consistent with Eq. (5.21) and is very accurate since typically A > 105. A similar derivation holds for the noninverting (or the inverting) ampliﬁer conﬁguration. With this in mind, the second SPC is clearly optional. Instead, the ampliﬁer deﬁnition Eq. (5.1) may be used, along with the condition of the high openloop gain. However, it is rather tedious to repeat the asymptotic analysis every time; so we prefer to use the accurate and simple summingpoint constraint. The ﬁnite value of the opencircuit gain A becomes important for highspeed ampliﬁers with the feedback loop; see Chapter 10.
5.2.5 Current Flow in the Ampliﬁer Circuit The current ﬂow in the complete ampliﬁer circuit is illustrated in Fig. 5.14. The output current through the load resistance RL of the ampliﬁer circuit in Fig. 5.14 is provided by the dualpolarity power supply. In this sense, the ampliﬁer is also a “valve” (similar to its building block, the transistor), which “opens” the power supply in response to the lowpower (or virtually nopower) input voltage signal. In Fig. 5.14, you should note that standard resistor values (5 % or 1 % tolerance) may be slightly different from the values used in this ﬁgure for convenience. We consider the positive input voltage of 100 mV in Fig. 5.14a ﬁrst. The noninverting ampliﬁer has a closedloop gain ACL equal to 50. The output voltage is thus +5 V, which is the push mode. The load current of 10 mA is found from Ohm’s law. The feedback current of 0.1 mA is found using the second SPC and Ohm’s law. The power supply current is the sum of both. The feedback current controls the gain of the ampliﬁer, and the load current drives the load. The overall ampliﬁer circuit efﬁciency (neglecting the loss in the IC itself) depends on the ratio of these two currents. Therefore, we should keep the feedback current small. The power current path is shown in Fig. 5.14a by a thick trace. The current at node A can only enter the upper power supply. Thus, it is the upper power supply being used. The ampliﬁer is operating in the “push” mode, i.e., the ampliﬁer sources the current. When the input voltage is negative as in Fig. 5.14b, the lower power supply is delivering power. Now, the ampliﬁer sinks the current; it is operating in the “pull” mode. V206
Chapter 5
Section 5.3: Ampliﬁer Circuit Design
Section 5.3 Ampliﬁer Circuit Design Now that the theory of the negative feedback loop has been established, we can turn our attention to the laboratory. Our hope is to be immediately successful with our designs. However, a number of questions will arise almost instantly. They raise issues such as how to choose the resistor values, how to connect the sensor as part of the input load, and how to use an ampliﬁer chip with a single power supply (a battery).
5.3.1 Choosing Proper Resistance Values There are several rules regarding how to choose resistances R1, R2 controlling the feedback loop in both noninverting and inverting conﬁgurations. They are: 1. Resistances R1, R2 cannot be too small. Imagine that in Fig. 5.14 of Section 5.2, the resistor values are changed to R1 1 Ω, R2 49 Ω. The same noninverting gain will be achieved and the same output voltage will be obtained. However, the feedback loop current now becomes 100 mA instead of 0.1 mA. The generalpurpose opamp chips are not capable of delivering such large currents. Furthermore, the ohmic losses in the feedback loop become high. Therefore, one should generally use R1 , R2 50
100 Ω
ð5:29aÞ
2. Resistances R1, R2 cannot be too large. Let us assume that resistance R2 equals 100 MΩ. This means that this physical resistor and the feedback loop represent almost an “open circuit.” Unwanted electromagnetic signals may couple into such a circuit through the related electric ﬁeld difference across its terminals. This effect is known as capacitive coupling. Furthermore, the very large resistances increase the parasitic effect of the input offset current. Plus, very large resistances are unstable their values depend on moisture, temperature, etc. Therefore, one should generally use R1 , R2 1 MΩ
ð5:29bÞ
3. When a precision design is not warranted, inexpensive 5 % tolerance resistors may be used. Otherwise, 1 % or even 0.1 % tolerance resistors are employed. Moreover, in lieu of ﬁxed resistors, we may use one or two potentiometers to make the gain adjustable. 4. The load resistance should be sufﬁciently large in order not to overdrive the ampliﬁer. A good choice is RL 100 Ω
ð5:29cÞ
This requires an output current of 20 mA at υout 2 V when RL is exactly 100 Ω. If RL < 100 Ω, the ampliﬁer output voltage may decrease compared to the expected value due to ampliﬁer’s inability to source/sink sufﬁcient current.
V209
Chapter 5
Section 5.3: Ampliﬁer Circuit Design b)
a)
+
vin
LM741
+

+VCC
vout
R2
+ 1

R1
5
VCC 10 k
+VCC
VCC R
RP
R
Fig. 5.25. (a) Output DC offset voltage for the LM741 is reduced to zero by adjusting the potentiometer placed between its offsetnull pins and (b) a similar operation performed with the virtual ground of the feedback loop.
Input Bias and Offset Currents In reality, for the opamp to operate, there will be very small currents into the input terminals (into transistor bases), typically on the order of 100 nA. When those currents ﬂow through the feedback resistances, they create corresponding voltages which appear as an output offset voltage as well. Let a current of 100 nA ﬂow into the negative terminal of the ampliﬁer in Fig. 5.24. If the input to the ampliﬁer is grounded, this current must ﬂow through resistance R2. Therefore, it will create the extra output DC voltage of υout
þ 100 nA R2
ð5:45Þ
when the input voltage to the ampliﬁer υin is exactly zero. To appreciate its value, we can use a resistance R2 1 MΩ as an example. This yields υout
ð5:46Þ
0:1 V
at the output. Fortunately, the currents ﬂowing into the ampliﬁer are nearly the same for either terminal. Therefore, their average (the input bias current) considerably exceeds their difference (the input offset current). There is a way to eliminate the larger effect of the input bias current. It consists of modifying the circuits for the noninverting and inverting ampliﬁer by adding one extra resistance R R1 R2 as shown in Fig. 5.26. b) R1
+ vin

+
vout

vin
R2 R1
R2

R=R1R2
+
a)
+ vout

R=R1R2
Fig. 5.26. Cancellation of the effect of input bias currents.
V219
Chapter 5 υ*
υb þ
Operational Ampliﬁer and Ampliﬁer Models
R1 ðυout R1 þ R2
υb Þ
ð5:52aÞ
and, according to the second voltage divider, υ*
υa þ
R3 ð0 V R3 þ R4
υa Þ
ð5:52bÞ
Both expressions must be equal to each other due to the second SPC (the differential input voltage in a negative feedback ampliﬁer is zero). Therefore, R1 ðυout υb Þ R1 þ R2 R1 R1 υout R1 þ R2 R1 þ R2
υb þ
R3 υa υa ) R3 þ R4 R3 1 υb R3 þ R4
To create a voltage difference, i.e., υa side of Eq. (5.52c), we select R2 R1
R4 R3
1 υa
ð5:52cÞ
υb , between the input voltages on the righthand
ð5:52dÞ
as the necessary condition. Then, both factors in parentheses on the righthand side of Eq. (5.52c) become equal. This yields the basic equation of the difference ampliﬁer, υout
R2 ðυ a R1
υb Þ
R2 υD R1
ð5:53Þ
Equation (5.53) is a simple, yet highly useful result for ampliﬁer circuit design. Namely, once the ampliﬁer in Fig. 5.29 is connected to the sensor in Fig. 5.28, the differential voltage υD υa υb is ampliﬁed with the gain of R2/R1 (the differential ampliﬁer circuit gain). At the same time, the undesired commonmode voltage υCM 0:5ðυa þ υb Þ is completely rejected, i.e., ampliﬁed with a gain of 0, no matter what speciﬁc values the input voltages have versus ground. In other words, the commonmode ampliﬁer circuit gain is zero. Note that the ratio of two gains (differential gain versus commonmode gain) is an important characteristic of the differenceampliﬁer circuit. It is called the commonmode rejection ratio (CMRR). In our case, this ratio is clearly inﬁnity. Unfortunately, in reality, this value is ﬁnite though quite large. One obvious reason is a possible mismatch in resistance ratios in Eq. (5.52d), which will not allow us to obtain Eq. (5.53) exactly. A certain portion of υCM 0:5ðυa þ υb Þ will be present at the output.
V224
Chapter 5
Section 5.4: Difference and Instrumentation Ampliﬁers
ampliﬁer in the second stage. The difference ampliﬁer becomes mainly responsible for rejecting the commonmode signal and the ampliﬁcation of the differential signal. +
vb

R2 R4
R3
R1
vb*

+
va*
R3
R4
0V
+
vout
R1
R2

+
va
0V
Fig. 5.32. An important step toward the instrumentation ampliﬁer: we add noninverting ampliﬁcation stages at the input.
However, another problem arises there. The two noninverting ampliﬁers amplify voltages υa, υb (close to 3.75 V in the present case). Therefore, at any appreciable gain (say, ACL 1 þ R4 =R3 10), they simply saturate and will not function! To avoid this issue, we use a simple yet critical change shown in Fig. 5.33 where we remove the commonport connection from the noninverting stage. The circuit in Fig. 5.33 behaves completely differently compared to the original circuit in Fig. 5.32. We no longer have the output voltages υ a , υ b given by υa *
ACL υa ,
υa *
ACL υa ,
ACL
1þ
R4 R3
ð5:55Þ
Instead, those voltages now become υa *
υa þ
R4 ðυ a 2R3
υb Þ,
υb *
υb
R4 ðυ a 2R3
υb Þ
ð5:56Þ
The details of the derivation are seen in Fig. 5.33. The currents and voltages labeled in this ﬁgure are obtained using two summingpoint constraints. The key observation is that the absolute voltages υa, υb are no longer ampliﬁed but are simply passed through. Only the differential voltage υD υa υb is ampliﬁed. The circuit in Fig. 5.33 is also a “difference ampliﬁer,” and it may be called the unity commonmode gain stage. The ﬁnal step in the construction of the instrumentation ampliﬁer is to connect both stages together. Figure 5.34 gives the ﬁnal circuit that can be employed in conjunction with the Wheatstone bridge for the strain gauge shown in Fig. 5.31. Here, a quad opamp chip (LM148 series) is used; it has four individual ampliﬁers inside the chip. For our circuit, we need three of them. The circuit is powered by a 7:5 V dual supply. According to Eqs. (5.53) and (5.56), the overall (differential) gain in Fig. 5.34 becomes V227
Chapter 5
Section 5.4: Difference and Instrumentation Ampliﬁers
The states in Fig. 5.35 have been achieved by a proper tuning of the potentiometer in the Wheatstone bridge. Thus, we have built a simple, yet useful, uncalibrated, uniaxial, stressmonitoring system. Frequently, the output of an instrumentation ampliﬁer is connected to an analogtodigital converter (ADC) and then to a computer system.
Fig. 5.35. Operation of the instrumentation ampliﬁer with the strain gauge attached to a metal slab. (a) A “positive” bending moment is applied and (b) a “negative” bending moment is applied. The oscilloscope resolution is 1 V per division in every case.
Load Cell and Other Uses The circuit in Fig. 5.34 gives us the idea of a commercial load cell. Strain gauges are commercially available in prefabricated modules such as load cells that measure force, tension, compression, and torque. All four resistors of the Wheatstone bridge may be strain gauges. Load cells typically use a fullbridge conﬁguration and contain four leads for bridge excitation and measurement. The manufacturers provide calibration and accuracy information. However, the load cells do not normally include the instrumentation ampliﬁer itself. Another mechanical engineering example where the differential ampliﬁer is quite useful is a thermocouple. When measuring a thermocouple in a noisy environment, the noise from the environment appears as an offset on both input leads, making it a commonmode voltage signal. Many other examples indeed exist, particularly in biomedical engineering. The instrumentation ampliﬁer is used to amplify an output signal from virtually any analog differential sensor instrument. Also note that instrumentation ampliﬁers with precision resistors are available as separate integrated circuits. Those ICs have a much better performance than an instrumentation ampliﬁer wired on the protoboard. Other instrumentation ampliﬁer types exist, which are different from the topology of the instrumentation ampliﬁer circuit in Fig. 5.34. In principle, it is possible to design an instrumentation ampliﬁer circuit with only two ampliﬁer gain stages. The summary to this chapter provides an example used in practice. V229
Chapter 5
Problems 5.1 Amplifier operation and circuit models 5.1.1 Ampliﬁer Operation
Problem 5.1. An operational ampliﬁer has ﬁve terminals. A. Sketch the ampliﬁer symbol. B. Name each of the opamp terminals and describe its function in one sentence per terminal. C. Can the ampliﬁer IC have more than ﬁve terminals? Explain. Problem 5.2. You may wonder about the meaning of the two letters preceding ampliﬁer marking, e.g., LM741. Each of the semiconductor companies has its own abbreviation, e.g., LM for an ampliﬁer designed and manufactured by the National Semiconductor Corporation (acquired by Texas Instruments in 2011), AD for an ampliﬁer manufactured by Analog Devices, MC for STMicroelectronics, TL for Texas Instruments, etc. The same chip, e.g., LM741, may be manufactured by several semiconductor chip makers. The part number is given by a numerical code that is imprinted on the top of the package. An MC1458 ampliﬁer IC chip is shown in the ﬁgure below. This IC is a dual operational ampliﬁer. In other words, one such IC package contains two separate operational ampliﬁers.
Problems Problem 5.3. What is the minimum number of pins required for: A. The dual operational ampliﬁer (the corresponding IC package contains two separate operational ampliﬁers)? B. The quad operational ampliﬁer (the corresponding IC package contains four separate operational ampliﬁers)? Problem 5.4. An operational ampliﬁer has an opencircuit gain of A ¼ 2 105 and is powered by a dual source of 10 V. It is operated in the opencircuit conﬁguration. What is the ampliﬁer’s opencircuit output voltage υout if A. υþ ¼ 0V, υ ¼ 0V B. υþ ¼ þ1V, υ ¼ þ1V C. υþ ¼ þ1V, υ ¼ 0V D. υþ ¼ 0V, υ ¼ 1V E. υþ ¼ þ1mV, υ ¼ 0V F. υþ ¼ 1mV, υ ¼ 0V G. υþ ¼ 10μV, υ ¼ 0V H. υþ ¼ 0V, υ ¼ 10μV Problem 5.5. Based on the solution to Problem 5.4, why do you think the operational ampliﬁer is seldom used in the openloop conﬁguration, at least in analog electronics? Problem 5.6. Using the website of the National Semiconductor Corporation, determine the maximum and minimum supply voltages (operating with the dualpolarity power supply) for the following ampliﬁer’s ICs: A. LM358 B. LM1458 C. LM741 Which ampliﬁer IC from the list may be powered by two AAA batteries?
A. Download the ampliﬁer’s datasheet from http://www.datasheetcatalog.com B. Redraw the ﬁgure to this problem in your notes and label the pins for the noninverting input, the inverting input, and the output of the operational ampliﬁer #1. C. Label pins for þV CC and V CC .
Problem 5.7. Plot to scale the output voltage of the operation ampliﬁer with an opencircuit gain A ¼ 5 104 when the noninverting input voltage υþ changes from 2 mV to +2 mV and the inverting input voltageυ is equal to 1 mV. The ampliﬁer is powered by a 12V dual voltage supply. Label the axes.
V239
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models
Problem 5.8. Repeat the previous problem for A ¼ 5 105 .
5.1.2 Operational Ampliﬁer Comparator
Problem 5.9. In a circuit shown in the ﬁgure below, an operational ampliﬁer is driven by a 10V dual power supply (not shown). The opencircuit DC gain of the ampliﬁer is A ¼ 1, 000, 000. Sketch to scale the output voltage to the ampliﬁer when a) V threshold ¼ 30 mV b) V threshold ¼ þ30 mV Assume that the ampliﬁer hits the power rails in saturation. A=1,000,000
v+
Problem 5.10. Based on the solution to the previous problem, why do you think the operational ampliﬁer in the openloop conﬁguration may be useful for digital circuits? Problem 5.11. Solve Problem 5.9 when the input voltage is applied to the inverting input and the threshold voltage is applied to the noninverting input. Problem 5.12. The circuit shown in the ﬁgure is a zerolevel detector. An operational ampliﬁer in the openloop conﬁguration is driven by a 10V dual power supply (not shown). The opencircuit ampliﬁer gain is 100,000. Sketch the output voltage to scale. Assume that the ampliﬁer hits the power rails in saturation.
vout A =100,000
+

Vthreshold Input voltage v+, mV
v out

0V
80
Input and output voltages, V
60
15
40
10
20
5
0
0
20
5
40
+
v in
v in
10
0
10
20 30 time, ms
a) Output voltage vout, V
40
50
10 5 0 5 10 15 10
20 30 time, ms
b)
0
10
20 30 time, ms
40
50
Vthreshold= 30mV
15
0
15
40
50
Output voltage vout, V
Vthreshold=+30mV
10
40
Problem 5.13. In a circuit shown in the ﬁgure below, an operational ampliﬁer is driven by a 15V dual power supply (not shown). The opencircuit gain of the ampliﬁer is A ¼ 100, 000. Sketch to scale the output voltage to the ampliﬁer when a) V threshold ¼ 0 mV b) V threshold ¼ þ4 mV
15 10 5 0 5 10 15 0
20 30 time, ms
50
V240
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models
Determine the output voltage, Vout. You are not allowed to use any of the materials of the next section! Hint: Denote the unknown voltage at node * by υ*, express υ* in terms of υout, and then solve for υout. R2=500 R1=100
i2
v out

+
1V i1
common
Problem 5.21. An ECE laboratory project uses the LM358 ampliﬁer IC. A. What semiconductor company has developed this chip? B. Is the chip from the lab project necessarily manufactured by this company? (See http://www.datasheetcatalog.com/ for manufacturers’ datasheets related to this product.) C. Use the DigiKey distributor’s website and estimate average cost for this ampliﬁer chip (DIP8 package) in today’s market. Problem 5.22. An ECE laboratory project uses the TL082 ampliﬁer IC. A. What semiconductor company has developed this chip? B. Is the chip from the lab project necessarily manufactured by this company? (See http://www.datasheetcatalog.com/ for manufacturers' datasheets related to this product.) C. Use the DigiKey distributor’s website to estimate the average cost for this ampliﬁer chip (DIP8 package) in today’s market.
5.2 Negative Feedback 5.2.2 Ampliﬁer Feedback Loop. Second SummingPoint Constraint 5.2.3 Ampliﬁer Circuit Analysis Using Two Summingpoint Constraints Problem 5.23 A. Name the two summingpoint constraints used to solve an ampliﬁer circuit. B. Which summingpoint constraint remains valid without the negative feedback?
Noninverting Ampliﬁer Problem 5.24 A. Draw the circuit diagram of the basic noninverting ampliﬁer conﬁguration. B. Accurately derive the expression for the ampliﬁer gain in terms of the resistances, assuming an ideal operational ampliﬁer. Problem 5.25. Using the two summingpoint constraints, solve the idealampliﬁer circuit shown in the ﬁgure if the input voltage has the value of 2 mV. A. Label and determine the currents in the feedback loop. B. Determine the output voltage of the ampliﬁer versus the common port. vin
+ 2 mV

pin 5
+

pin 7
+
vx
pin layout LM1458(#2)

vout
+
pin 6

R2=51 kW common
R1=100 W
common
V242
Chapter 5
Problems R2=33 kW pin layout LM1458 (#1)
R1=1 kW
pin 2

vin
+
pin 1
vout
+
Problem 5.26. Determine the output voltage of the ideal operational ampliﬁer shown in the ﬁgure. The ampliﬁer is driven by a 10V dual power supply.
0.5V

+
pin 3

pin layout LM1458 (#1)
vin
+ 0.5 V

pin 3
+
pin 1
+
vx


vout
common common
+
pin 2

Voltage Follower
R2=100 kW common
R1=1 kW
common
Inverting Ampliﬁer Problem 5.27 A. Draw the circuit diagram of the basic inverting ampliﬁer conﬁguration. B. Give the expression for the ampliﬁer gain in terms of the resistances, assuming an ideal operational ampliﬁer.
Problem 5.30 A. Using only the ﬁrst summingpoint constraint (SPC), solve the circuit shown in the ﬁgure, i.e., determine the output voltage of the ampliﬁer versus the common port. B. What function does this ampliﬁer have? Why is it important? pin layout LM1458 (#1)
vin
+
+
Problem 5.28. Using the two summingpoint constraints, solve the idealampliﬁer circuit shown in the ﬁgure that follows if the input voltage is 1 mV. A. Label and determine the currents in the feedback loop. B. Determine the output voltage of the ampliﬁer versus the common port. R2=10 kW R1=100 W
+
+
1 mV

pin 5
pin 7
1 mV

vout pin 1

+
pin 2

common
Exercises on the Use of the Negative Feedback Problem 5.31. (A review problem) For three basic idealampliﬁer circuits:
pin layout LM1458 (#2)
pin 6

vin
pin 3
vout
+

common common
Problem 5.29. Determine the output voltage of the ideal operational ampliﬁer shown in the ﬁgure. The ampliﬁer is driven by a 10V dual power supply.
Inverting ampliﬁer Noninverting ampliﬁer Voltage follower
(each includes negative feedback) present 1. A circuit diagram 2. Expression for the ampliﬁer gain Problem 5.32. Determine the output voltage of ampliﬁer conﬁgurations shown in the ﬁgure that follows. The ampliﬁer is powered by a 9V dualpolarity voltage power supply. Assume an ideal operational ampliﬁer.
V243
Chapter 5
Problems R2=5.1 kW
a) R1=100 W
vout

vin
+ b) R1=100 W
vout

vin
R2=5.1 kW
A. Find the value of the output current iout if the input current is 1 mA, R1 ¼ 9 kΩ, R2 ¼ 1 kΩ. B. Why do you think this ampliﬁer type is known as the current ampliﬁer? To answer this question quantitatively, analytically express the output current iout (current through the load) in terms of the unknown input current iout and two arbitrary resistor values, R1,2.
+ iin
RL=51 W

+
c)
R2=5.1 kW
RL
iout
R1
R1=100 W
vout

vin
R2
+ R1=100 W
RL=51 W
R2=5.1 kW
d) R1=100 W
vout

vin
5.2.4 Mathematics Behind the Second SummingPoint Constraint
+ R1=100 W
RL=51 W
Problem 5.36. An inverting ampliﬁer that achieves highgain magnitude with a smaller range of resistance values is shown in the ﬁgure below. Find its output voltage υout vs. ground (or common port) and the resulting ampliﬁer gain.
Problem 5.38 A. Derive an expression for the closedloop gain of the noninverting ampliﬁer based only on the þdeﬁnition of the output voltage υout ¼ A υin υin , without using the second summingpoint constraint. B. Determine the exact gain value when A ¼ 2 105 R1 ¼ 1 kΩ,R2 ¼ 9 kΩ +
vin
1 kW vin=1 mV
v* vout
+
vout
*
R2
R1

+
1 kW

10 kW
10 kW
Problem 5.37. The ampliﬁer circuit shown in the ﬁgure employs negative feedback.
Problem 5.39 A. Derive an expression for the closedloop gain of the inverting ampliﬁer based only on the deﬁnition of the output voltage υout
V245
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models
¼ A υþ in υin , without using the second summingpoint constraint. B. Determine the exact gain value when
vin=1 V
vout
+

RL=100 W
A ¼ 2 105
R2=4 kW
R1 ¼ 1 kΩ,R2 ¼ 10 kΩ
+
R1=1 kW
common 0 V
5.2.5 Current Flow in the Ampliﬁer Circuit
Hint: Change the polarity of the input voltage and the voltage sign if you have trouble operating with negative values.

Problem 5.40. The ampliﬁer circuit shown in the ﬁgure below is powered by a 9V dualpolarity voltage power supply. A. Redraw the ampliﬁer schematic in your notes. B. Show the current direction in every wire of the circuit by an arrow and write the corresponding current value close to each arrow.
+
Problem 5.42. The ampliﬁer shown in the ﬁgure below is powered by a 9V dualpolarity voltage power supply. A. Redraw the ampliﬁer schematic in your notes. B. Show the current direction in every wire of the circuit by an arrow and write the corresponding current value close to each arrow. R2=10 kW
vout
+
vin R1=1 kW
vout

100 mV RL=500 W
R2=49 kW
+100 mV
+
+

+ RL=100 W
common 0 V
+
+ 
R1=1 kW
+


Problem 5.41. Repeat the previous problem for the circuit shown in the ﬁgure below.
Problem 5.43. Repeat the previous problem when the input voltage to the ampliﬁer is 100 mV.
V246
Chapter 5
Problems
5.2.6 MultipleInput Ampliﬁer Circuit: Problem 5.46. By solving the ampliﬁer circuit shown in the following ﬁgure, Summing Ampliﬁer Problem 5.44. By solving the ampliﬁer circuit shown in the ﬁgure, ﬁll out the table that follows. Assume that R0 ¼ RF , R1 ¼ RF =2, and R2 ¼ RF =4.
D2
R2
iF
0V
2R
vout
υout, V
D 0, V 0 5 0 5 0 5 0 5
R
0V
0V
Problem 5.47. State the limitations on the feedback resistances and the output load resistance of an ampliﬁer circuit. Problem 5.48. The noninverting ampliﬁer shown in the ﬁgure below has been wired in laboratory. A. Do you have any concerns with regard to this circuit? B. If you do, draw the corrected circuit diagram. +

v in=100mV
2R
R
c
+ 10 10

2R
c
R
5.3.1 Choosing Proper Resistance Values
+
2R
b
b
D2=5 V
0V a
R
5.3 Amplifier Circuit Design
Problem 5.45. The ampliﬁer circuit shown in the ﬁgure below employs negative feedback. This conﬁguration is known as a threebit digitaltoanalog converter (DAC) on the base of an R/2R ladder. By solving the ampliﬁer circuit, determine its output voltage in terms of resistances R, RF, given the input voltages D0 ¼ 0 V, D1 ¼ 0 V, D2 ¼ 5 V.
2R
a
vout
common
D1, V 0 0 5 5 0 0 5 5
D1=0 V
RF
2R 0V
common
D0=0 V
2R
+ 2R
D2, V 0 0 0 0 5 5 5 5
D2

R1
D1
+
D1
D0
RF

R0
+
D0
determine its output voltage in terms of resistances R, RF, given the input voltages D0 ¼ 0 V, D1 ¼ 5 V, D2 ¼ 0 V.
iF
1
RF
common
vout
v out

5.3.2 Model of a Whole Ampliﬁer Circuit 5.3.3 Input Load Bridging or Matching Problem 5.49. For three basic ampliﬁer circuits
0V
0V
Inverting ampliﬁer Noninverting ampliﬁer Voltage follower
V247
Chapter 5
Operational Ampliﬁer and Ampliﬁer Models Problem 5.79. Find the output voltage to the differenceampliﬁer circuits shown in the ﬁgures below. Assume the idealampliﬁer model and exact resistance values. a)
Problem 5.74. The model of an input signal from a threeterminal sensing device is shown in the ﬁgure below. What are the differential and commonmode voltages at terminals a and b?
1V
a
+ 
0.1 V
+ 
0.1 V
5 kW 1 kW
0.5 V
50 kW

0V
b)
5 kW 1 kW 1V

+
1V
+
10 kW
1V
+ 
v out

5.4.1 Differential Input Signal to an Ampliﬁer
+
5.4 Difference and Instrumentation Amplifiers
10 kW 25 kW
v out
+

b 0V 0V
Problem 5.75. The Wheatstone bridge in Fig. 5.27 is connected to V CC rails instead of þV CC and ground. Furthermore, R2 ¼ 1:1R1 , RðxÞ ¼ 1:1R3 , and V CC ¼ 6 V. What are the differential and commonmode voltages? Problem 5.76. The Wheatstone bridge in Fig. 5.27 is connected to ground and V CC rails instead of þV CC and ground. Given that R2 ¼ 1:05R1 , RðxÞ ¼ 1:05R3 , and V CC ¼ 6 V, determine the differential and commonmode voltages.
5.4.2 Difference Ampliﬁer Problem 5.77. Design a difference ampliﬁer with a differential gain of 20. Present the circuit diagram and specify one possible set of resistor values. In the circuit diagram, label the input voltages as υa, υb and express the output voltage in terms of υa, υb. Problem 5.78. Repeat the previous problem for a differential gain of 100.
Problem 5.80. Your technician needs to control a process using two sensors with output voltages υ1 and υ2, respectively. The weighted difference in sensor reading, υ ¼ 1υ1 0:5υ2 , is critical for the product quality. The technician reads voltage υ1 and then voltage υ2 and then uses a calculator to ﬁnd υ. Help the technician, i.e., sketch for him a differenceampliﬁer circuit that will directly output υ to the DMM. The negative terminal of the DMM is always grounded. Specify one possible set of resistor values. Problem 5.81. Repeat the previous problem when the weighted difference in sensor reading, υ ¼ 10υ1 5υ2 , needs to be processed. Problem 5.82. For the circuit shown in the ﬁgure, ﬁnd the output voltage if the input voltages are υb ¼ 1 V and υa ¼ 1:01 V, respectively. Assume the idealampliﬁer model and exact resistance values.
V252
Chapter 5
Problems 5.4.3 Instrumentation Ampliﬁer
10R
R vb
v out

+
+

va
10R
R
0V
Problem 5.83 A. For the circuits shown in the ﬁgures below, ﬁnd the output voltage if the input voltages are υa ¼ 1 V and υb ¼ 1 V, respectively Assume the ideal ampliﬁer and exact resistance values. B. What is the value of the commonmode gain in every case? a)
2R
R
vb v out

+
+

va
2R
1.05R
0V
b) 2R
1.05R
Problem 5.85 A. Why is the original difference ampliﬁer not used as an instrumentation ampliﬁer? B. Why is the circuit in Fig. 5.31 not used as the instrumentation ampliﬁer? Problem 5.86 A. Find the differential gain and the commonmode gain for the ampliﬁer circuit shown in Fig. 5.32. The differential output voltage is υa * υb *, and the commonmode output voltage is 0:5ðυa * þ υb *Þ. B. Find the differential gain and the commonmode gain for the ampliﬁer circuit shown in Fig. 5.31. Problem 5.87. Design an instrumentation ampliﬁer with a differential gain of 210. Present the corresponding circuit diagram and specify one possible set of resistance values. In the circuit diagram, label the input voltages as υa, υb and express the output voltage in terms of υa, υb.
vb v out

+
+

va
2R
R
0V
Problem 5.84. For the differenceampliﬁer circuit shown in the ﬁgure below, ﬁnd the differentialmode resistance (impedance) to the ampliﬁer. The differentialmode resistance is deﬁned as the ratio of a voltage of a power supply placed between terminals a and b to the current that ﬂows through this power supply.
Problem 5.88. Design an instrumentation ampliﬁer with a differential gain of 1010. Present the corresponding circuit diagram and specify one possible set of resistor values. In the circuit diagram, label the input voltages as υa, υb and express the output voltage in terms of υa, υb. Problem 5.89. The following voltages are measured: υa ¼ 3:750 V and υb ¼ 3:748 V. Find voltages versus circuit ground (common port of the dual supply) for every labeled node in the circuit shown in the ﬁgure below. The ampliﬁer circuit is powered by a 10 V dual supply. Assume exact resistance values and the idealampliﬁer model.
XR
R
v out

+
+
R
XR
0V
V253
Chapter 5
vb
2

100 kW
100 kW
6
1
+ 1 kW
10 kW
1.05 kW
5
3
100 kW
100 kW
va
3
Problem 5.90. Repeat the previous problem when node 1 is grounded. Problem 5.91. The following voltages are measured: υa ¼ 5:000 V and υb ¼ 5:001 V. Find voltages versus circuit ground (common port of the dual supply) for every labeled node in the circuit shown in the ﬁgure below. The ampliﬁer circuit is powered by a 10 V dual supply. Assume exact resistance values and the idealampliﬁer model.
Problem 5.94 A. Find the output voltage for the ampliﬁer circuit shown in the ﬁgure below. B. Denote the input voltage of 0.1 V by υa, the input voltage of 0.08 V by υb, the 10kΩ resistor by R1, the 40kΩ resistor by R2, and the 10kΩ resistor by RG. Express the output voltage in the general form, in terms of two input voltages and the resistances. +


v out
40 kW
100 kW
25 kW 2
4
1 kW
1 kW

1
6
4 kW
+ 1 kW 3
va
25 kW
+
0.1 V
+
6
5
100kW
+

1 kW


100kW
4
1 kW
0.95 kW
4

10 kW
25 kW 2
1
vb

100 kW
1 kW
va
+

+
+
vb
Operational Ampliﬁer and Ampliﬁer Models
1 kW
5
0.08 V
10 kW
+

25 kW
+ 40 kW
10 kW
Problem 5.92. Repeat the previous problem when node 1 is grounded. Problem 5.93. The following voltages are measured: υa ¼ 5:000 V and υb ¼ 5:001 V. Find voltages versus circuit ground (common port of the dual supply) for every labeled node in the circuit shown in the ﬁgure below. The ampliﬁer circuit is powered by a 10 V dual supply. Assume exact resistance values and the idealampliﬁer model.
5.5 General Feedback Systems 5.5.1 Signalﬂow Diagram of a Feedback System 5.5.2 ClosedLoop Gain and Error Signal Problem 5.95. The block diagram of Fig. 5.35 is applied to a voltage ampliﬁer.
V254
Chapter 5 A. Given the input signal xin ¼ 10 mV, the error signal xe ¼ 1 μV, and the output signal xout ¼ 1 V, determine the openloop gain and the feedback factor. B. Given the ratio of input to error signal xin =xe ¼ 100 and the feedback factor of 0.1, determine the openloop gain. Problem 5.96. The openloop gain A in Fig. 5.35 varies between two extreme values of A ¼ 10, 000 2, 000 (20 % gain variation) depending on the system parameters. The forward gain block is used in the closedloop conﬁguration with the feedback factor β of 0.1. Determine the two extreme values of the closedloop gain, ACL. Problem 5.97. The openloop gain A in Fig. 5.35 is 100,000. The forward gain block is used in the closedloop conﬁguration with the feedback factor β of 1. Determine the error signal, xe, if the input voltage signal is 1 mV. Problem 5.98. The closedloop gain of a noninverting ampliﬁer circuit with R1 ¼ 1 kΩ, R2 ¼ 100 kΩ is 99. Determine the opencircuit gain A of the ampliﬁer chip.
Problems Problem 5.100. The circuit shown in the ﬁgure that follows is a feedback transconductance ampliﬁer. Express iout in terms of υin. transconductance amplifier
+

v in
RL
R2
+ R1
iout
RF
Problem 5.101. The ampliﬁer circuit shown in the ﬁgure that follows is the Howland current source widely used in biomedical instrumentation; its output is the current through the load resistance. A. Classify the ampliﬁer circuit in terms of four basic ampliﬁer topologies and mention the most important circuit features. B. Derive its gain equation iout ¼ ðυa υb Þ =R2 given that R1 =R3 ¼ R2 =R4 . R3 R1 vb
+

va
R2
5.5.3 Application of General Theory to Voltage Ampliﬁers with Negative Feedback 5.5.4 Voltage, Current, Transresistance, and Transconductance Ampliﬁers with the Negative Feedback
iout =iL
R L R4
Problem 5.99. Derive the gain Eq. (5.69) for the ampliﬁer circuits shown in Fig. 5.37.
V255
Part II Transient Circuits
Chapter 6
Chapter 6: Dynamic Circuit Elements
Overview Prerequisites:  Knowledge of basic circuit theory (Chapters 2 and 3)  Knowledge of operational amplifiers with negative feedback (Chapter 5) Objectives of Section 6.1:  Define types of capacitance encountered in electric circuits  Define selfinductance and mutual inductance from the first principles  Define field energy stored in a capacitor/inductor  Be able to combine capacitances/inductances in series and in parallel  Understand construction of practical capacitors/inductors  Understand fringing effect and its use in sensor circuits Objectives of Section 6.2:  Derive dynamic equations for capacitance/inductance from the first principles  Establish how the capacitance may create large transient currents  Establish how the inductance may create large transient voltages  Define instantaneous energy and power of dynamic circuit elements  Establish the behavior of dynamic circuit elements in the DC steady state and at a very high frequency Objectives of Section 6.3:  Obtain initial exposure to bypass/blocking capacitor and decoupling inductor  Obtain initial exposure to amplifier circuits with dynamic circuit elements Application Examples: Electrostatic discharge and its effect on integrated circuits How to design a 1F capacitor? How to design a 1mH inductor? Capacitive touchscreens Bypassing a DC motor
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_6
VI259
Chapter 6
Dynamic Circuit Elements
Keywords: Capacitance, Capacitance of two conductors, Selfcapacitance, Capacitance to ground, Capacitance of two equal conductors separated by large distances, Energy stored in a capacitance, Electrostatic discharge (ESD), ESD effect on integrated circuits, Device under test (DUT), Parallelplate capacitor (base formulas, fringing effect, fringing ﬁelds), Capacitor (absolute dielectric permittivity, relative dielectric permittivity, dielectric strength, normalized breakdown voltage, electrolytic, tantalum, ceramic, marking, set of base values), Capacitive touch screens (selfcapacitance method, mutualcapacitance method), Magnetic ﬂux density, Magnetic ﬁeld, Absolute magnetic permeability, Relative magnetic permeability, Magnetic induction, Magnetic ﬂux, Selfinductance, Inductance, Mutual inductance, Energy stored in an inductance, Solenoid (air core, toroidal magnetic core, straight magnetic core, short, fringing ﬁelds), Inductor (marking, set of base values, also see solenoid), Dynamic equation for capacitance (deﬁnition, derivation, ﬂuid mechanics analogy), Capacitance (instantaneous energy, instantaneous power, behavior in the DC steady state, behavior at very high frequencies), Dynamic equation for inductance (deﬁnition, derivation, ﬂuid mechanics analogy), Inductance (instantaneous energy, instantaneous power, behavior in the DC steady state, behavior at very high frequencies), Bypass capacitor, Decoupling capacitor, Shunt capacitor, Snubber RC circuit, Decoupling inductor, Inductor choke, Transient circuit, Ampliﬁer circuits with dynamic circuit elements, Active ﬁlters, Miller integrator (circuit, DC gain, compensation, time constant), Analog pulse counter, Analog computer, Differentiator ampliﬁer (circuit, gain at very high frequencies), Active differentiator
VI260
Chapter 6
C
Section 6.1: Static Capacitance and Inductance
ε0 A d
ð6:6Þ
where ε0 is the dielectric permittivity of vacuum if the capacitor is situated in vacuum. For Eq. (6.6) to hold, the plates do not have to be square. If the capacitor does not have a highε dielectric inside, Eq. (6.6) is a good approximation only if d is very small compared to the dimensions of the plates. Otherwise, the fringing effect must be taken into account. The fringing effect is illustrated in Fig. 6.4a, b. Fringing means that the electric ﬁeld extends outside the physical capacitor. The electric ﬁeld outside the capacitor possesses certain extra energy. Therefore, according to Eq. (6.3) where voltage V is ﬁxed, the capacitance must increase compared to the nonfringing case. Figure 6.4c in the summary of this chapter present numerically found capacitance values Cexact for the parallelplate capacitor with fringing. These values have been accurately computed using a rigorous numerical adaptive procedure. Figure 6.4c predicts a nearly linear increase of the ratio Cexact/C as a function of the separation distance. Therefore, the wrong result, C ! 0 when d ! 1, which is predicted by Eq. (6.6), is corrected. Instead, one will have C ! 0:5 C self when d ! 1. a)
b)
Potential (voltage) distribution, V 0.3
+1 V
0.4
0.5 0.6 0.7 0.8 0.9
lines of force (Efield)
1 V
c) 3.5
equipotential lines
Cexact/C
3
09 08 07
2.5
06
2
05 04 03
1.5 1 0
0.2
0.4
0.6
0.8 d/a 1
Fig. 6.4. (a) Equipotential lines and lines of force for a capacitor with d=a ¼ 0:2 in the central crosssectional plane (the plates are at 1 V). (b) Fringing electric ﬁeld for the same capacitor observed in the central crosssectional plane (the plates are at 1 V). (c) Ratio of the accurate capacitance values (found numerically) to the values predicted by Eq. (6.6).
The fringing ﬁeld of capacitors is utilized in capacitive touch screens. In this case, the signiﬁcant fringing ﬁeld is a desired effect. Therefore, conﬁgurations other than the parallelplate capacitor are used. These conﬁgurations will be studied later in this section.
VI265
Chapter 6
Dynamic Circuit Elements
Well, such a capacitor will certainly occupy a signiﬁcant fraction of a lecture hall and is hardly practical. How, then, do manufactures design a capacitor of 1 μF? The ﬁrst step is to use a dielectric material sandwiched within the capacitor. A dielectric medium increases charges stored on the two metal capacitor plates depending on εr 1, the relative dielectric permittivity of the dielectric medium. Table 6.1 gives us a list of permittivities for a number of dielectric materials. For each material, a dielectric strength, or normalized breakdown voltage, is also given. This is actually the maximum electric ﬁeld (notice the unit of V/m) that the capacitor can handle. It is for this reason that capacitors carry a voltage rating that you should not exceed in your circuit. From a practical point of view, the higher the capacitance, the lower the voltage rating. The wellknown dilemma with the capacitor is that a decrease in the separation distance increases the capacitance and the stored energy. However, as already mentioned, it simultaneously decreases the maximum applied voltage due to the dielectric breakdown effect. For our capacitor, we will again use the mica dielectric material listed in Table 6.1. Equation (6.6) now transforms to A
dC ε0 εr
10 3 10 6 8:854 10 12 7
ð6:11bÞ
16 m2
Table 6.1. Relative dielectric permittivity and dielectric strength of some common materials. Material Air Aluminum oxide Fused silica (glass) Gallium arsenide (GaAs) Germanium (Ge) crystal Mica Nylon Plexiglas Polyester Quartz Rutile (titanium dioxide) Silicon (Si) crystal Styrofoam Teﬂon Water (distilled, deionized)
Relative permittivity 1.0 8.5 3.8 13 16 7.0 3.8 3.4 3.4 4.3 100–200 12 1.03–1.05 2.2 ~80
Dielectric strength in V/m 0.4–3.0 106 Up to 1000 106 470–670 106 (or lower) ~10 106 Up to 400 106 ~20 106 ~30 106 8 106 (fused quartz) 10–25 106 ~30 106 87–173 106 65–70 106
Even though the result looks a bit better, it is still far from practical. However, what if we try to make the dielectric layer very thin? An oxide is a dielectric, so could we just oxidize one top aluminum plate with a very thin (i.e., d 10 μm) oxide layer and pressﬁt it to
VI268
Chapter 6
Section 6.1: Static Capacitance and Inductance
the second plate? The result becomes (the relative dielectric constant of 8.5 is now that for aluminum oxide from Table 6.1) A
hC ε0 εr
10 5 10 6 8:854 10 12 8:5
13 cm2
ð6:11cÞ
Electrolytic Capacitors Once such a thin ﬁlm is rolled into a cylinder, it will clearly become a compact design, similar in size to a 1μF electrolytic capacitor routinely used in the laboratory. Unfortunately, one problem still remains: the permanent oxide layer is fragile and rough in shape. A better idea is to chemically grow such a layer using a socalled anodization process. This process occurs when the aluminum foil is in contact with an electrolyte as a second conductor and an appropriate voltage is applied between them. This is the smart idea behind an electrolytic capacitor. And this is also the reason why an electrolytic capacitor is polarized. The term electrolytic capacitor is applied to any capacitor in which the dielectric material is formed by an electrolytic method; the capacitor itself does not necessarily contain an electrolyte. Along with aluminum capacitors, tantalum capacitors (both wet and dry) are also electrolytic capacitors. Ceramic Capacitors A competitor to the electrolytic capacitor is a nonpolarized ceramic capacitor. Ceramic capacitors consist of a sandwich of conductor sheets alternated with ceramic material. In these capacitors the dielectric material is a ceramic agglomerate whose relative static dielectric permittivity, εr, can be changed over a very wide range from 10 to 10,000 by dedicated compositions. The ceramic capacitors with lower εr values have a stable capacitance and very low losses, so they are preferred in highprecision circuits and in highfrequency and RF electronic circuits. Typically, these “fast” ceramic capacitors have very small capacitances, on the order of pF and nF, and they can hold a high voltage. At the same time, the “slow” ceramic capacitors may have values as high as 1 μF. Therefore, the task of the above example can be solved with the ceramic capacitor as well. Capacitor Marking Figure 6.7 shows two examples of ceramic capacitors, with 100pF and 1.0μF capacitance from two different companies. To read the capacitance in the ﬁgure, we use the following rule: 101 10 101 pF 100 pF, and 105 10 105 pF 1 μF. Indeed, 473 47 103 pF 47 nF, and so forth. The tolerance letters may be present: F 1 %, G 2 %, J 5%, K 10 %, and M 20 %. Also, the voltage rating should be given.
VI269
Chapter 6
Dynamic Circuit Elements
Fig. 6.7. Left, ceramic capacitors of 100 pF. Right, radial leaded ceramic capacitors of 1.0 μF.
A standardized set of capacitance base values is deﬁned in the industry. The capacitance of any (electrolytic or not) capacitor can then be derived by multiplying one of the base numbers 1.0, 1.5, 2.2, 3.3, 4.7, or 6.8 by powers of ten. Therefore, it is common to ﬁnd capacitors with capacitances of 10, 15, 22, 33, 47, 68, 100, 220 μF, and so on. Using this method, values ranging from 0.01 to 4700 μF are customary in most applications. The value of the capacitance and the allowed maximum voltage are prominently written on the case of the electrolytic capacitor so reading those does not constitute any difﬁculties.
6.1.6 Application Example: Capacitive Touchscreens Capacitive touchscreens use the fringing ﬁeld of a capacitor studied previously. Many small capacitors with a signiﬁcant fringing ﬁeld are involved. If a conducting ﬁnger (an extra conductor) is placed in the fringing ﬁeld, the corresponding capacitance changes. There are two possible solutions called the selfcapacitance method and the mutualcapacitance method, respectively. The difference is in the measurement nodes for the capacitance. In the ﬁrst case, the capacitance is measured between the touch pad electrode and a ground. In the second case, the capacitance is measured between two pad electrodes, neither of which is grounded. Both methods may be combined. SelfCapacitance Method Consider a human ﬁnger in the proximity of a touchscreen as shown in Fig. 6.8a. The touchscreen itself may be a lattice of circular touch pads surrounded by a ground plane and separated from it by an airgap ring see Fig. 6.8b. When the ﬁnger is not present, each pad has capacitance CP to ground, which is called a parasitic capacitance. When the (grounded) ﬁnger appears in the vicinity of the touchpad, there appears another capacitance, CF, which is called the ﬁnger capacitance. Figure 6.8a indicates that both capacitances are in parallel so that the resulting ground capacitance increases as CP ! CP þ CF > CP
ð6:12Þ
VI270
Chapter 6
Dynamic Circuit Elements
For the series conﬁguration in Fig. 6.12a, the same current I is applied to every inductance. Given that the equivalent inductance is also subject to current I and must possess the same magneticﬁeld energy deﬁned by Eq. (6.18), one has E
1 Leq I 2 2
1 1 1 L1 I 2 þ L2 I 2 þ L3 I 2 ) Leq 2 2 2
L1 þ L2 þ L3
ð6:25Þ
The parallel conﬁguration in Fig. 6.12b may be analyzed given the condition of equal magnetic ﬂux through each inductance. Since this condition is related to Faraday’s law of induction, we postpone the corresponding discussion until the next section.
6.1.10 Application Example: How to Design a 1mH Inductor? Again, after this theoretical excursion, let us design a practical inductor. Our goal is to construct a 1mH inductance. According to Eq. (6.21) the required number of turns is s Ll N ð6:26Þ μ0 A We will select a coil length of l Equation (6.26) then yields s 10 3 5 10 2 N 356 4π 2 10 7 10 4
5 cm and an (average) coil radius of r
1 cm.
ð6:27Þ
Such a coil can be wound on a former in the laboratory with a sufﬁciently thin wire, say AWG 28. A different approach to reducing the number of turns while maintaining, or even increasing, the inductance is to use a magnetic material within the coil, as shown in Fig. 6.11. Table 6.2 lists the magnetic permeability for a number of magnetic materials. The simplest magnetic core is an iron core. However, it is lossy since an alternating magnetic ﬁeld creates socalled eddy currents in the conducting core, which are dissipated into heat. One solution to this problem is to use thin insulated sheets of iron, or laminations. Various ferrites (oxides of iron, or other metals) are an alternative to iron, which are ceramics and known as good electric insulators. Other types of losses may occur there, explanations of which go beyond the scope of our text. Once a soft ferrite with μr 100 is used in the design of the 1mH inductor, the number of turns necessary p to achieve the same inductance decreases by μr 10. In our example, it becomes equal to 36 turns instead of over 300 turns. However, the magnetic core cannot be a short rod like that shown in Fig. 6.11c; it must form a closed loop shown in Fig. 6.11b.
VI276
Chapter 6
Section 6.1: Static Capacitance and Inductance
Table 6.2. Relative permeability of some common materials. Material Air Magnetic iron Iron powder Nickel Permalloy (78.5 % nickel + 21.5 % iron) Soft ferrites with low losses at frequencies up to 100 MHz Hard ferrites with low losses up to 1 MHz
Relative static permeability 1.0 200 2–75 100 8000 20–800 1000–15,000
Inductor Marking Leaded inductors have color codes, similar to resistors. A standardized set of inductance base values is deﬁned in the industry. The inductance of any inductor can then be derived by multiplying one of the base numbers 1.0, 1.1, 1.3, 1.5, 1.9, 2.2, 2.7, 3.3, 3.9, 4.7, 5.8, 6.8, or 8.2 by powers of ten. Therefore, it is common to ﬁnd inductances with values of 1.0, 2.7, 6.8 μH, and so on. Using this method, values ranging from 0.01 to 100 μH are customary in most applications.
VI277
Chapter 6
Dynamic Circuit Elements
Section 6.2 Dynamic Behavior of Capacitance and Inductance 6.2.1 Set of Passive Linear Circuit Elements The three elements, resistance, capacitance, and inductance, constitute the fundamental set of passive circuit elements for any linear electric circuit. This is very similar to a mechanical system consisting of dashpot, spring, and mass, which form the basic set of any linear kinematic system. Having discussed the underlying DC concepts of capacitance and inductance, we now turn our attention to their dynamics. A simple example of dynamic behavior is given by a vacuum cleaner. If one manually unplugs the working vacuum cleaner from the wall outlet (please avoid doing so), a profound spark may appear. On the other hand, turning off the vacuum cleaner normally produces no spark. The reason for the spark is that breaking the current through a dynamic circuit element an inductance which models the coil of the motor, creates very large transient voltages. One reason for studying transients is the wish to avoid such sparks and to properly design the electric switch. The use of two dynamic circuit elements the capacitance and inductance is enormous, especially in power systems. Every electric motor is basically an inductance; most power motors need a power correction circuit that in turn requires a shunt capacitance. Some motors need starting capacitors or surge capacitors for large motors. On the other hand, the capacitance of logic gates is responsible for the socalled propagation delay. This delay determines a very important measure of the performance of a digital system, such as a computer, which is the maximum speed of operation. Thus, the capacitances and inductances are just everywhere, like mass and spring systems present everywhere in mechanical engineering. However, they are becoming most apparent when we consider a transient behavior, an alternating current, or highfrequency digital and communication circuits. 6.2.2 Dynamic Behavior of Capacitance Both capacitance and inductance are passive circuit elements, which means that, like resistance, they do not deliver a net power increase to the circuit. Indeed, after charging, the capacitor is able to power a circuit, usually for a short period of time. At the end of the discharge cycle, it needs to be recharged. Therefore, we use the passive reference conﬁguration for the capacitance in Fig. 6.13a. Lowercase letters denote timevarying voltage and current. The dynamic behavior of the capacitance is described by the wellknown voltagetocurrent relation (dynamic equation), which plays the role of “Ohm’s law” for the capacitance iC
C
dυC dt
ð6:28Þ
VI278
Chapter 6
Section 6.2: Dynamic Behavior of Capacitance and Inductance
So, if the capacitance is associated with a spring, the inductance is associated with the mass, and the resistance is associated with a dash pot (damping element), then the entire electric circuit containing dynamic elements is nothing else but a mechanical system. Is this correct? Clearly the same analysis methods are applicable to both systems, electrical and mechanical! The model of an entire building in terms of lumped mechanical elements is in theory the same as the model of a complicated electric circuit. Both models can be analyzed by using the theory of linear systems, and both models follow the same control theory. A more difﬁcult issue is related to nonlinear circuit elements.
6.2.4 Instantaneous Energy and Power of Dynamic Circuit Elements An elegant derivation of the energy stored in a capacitance can also be obtained by integrating the power delivered (or taken) by the capacitance. The instantaneous electric power pC(t) can be written in the form pC ðt Þ
υC iC
υC C
dυC dt
1 dυ2C C 2 dt
ð6:36Þ
The stored energy is then the time integration of the power, i.e., ðt E C ðt Þ
pC ðt 0 Þdt 0
0
ðt
1 dυ2C 0 dt C 2 dt 0
1 2 C υC ðt Þ 2
υ2C ð0Þ
ð6:37Þ
0
where the lower limit, υC(0), is the initial state of the capacitance. Suppose that υC ð0Þ i.e., the capacitance is initially uncharged and has zero stored energy. Then, E C ðt Þ
1 2 Cυ ðt Þ 2 C
0,
ð6:38Þ
Equation (6.38) is the formal proof of the corresponding static result, Eq. (6.3), postulated in the previous section. We can derive the energy stored in the inductance using the same method by integrating the power. The instantaneous power supplied to or obtained from the inductance has the form pL ðt Þ
υL iL
iL L
diL dt
1 di2L L 2 dt
ð6:39Þ
The energy stored in the inductance is the integral of Eq. (6.39), i.e., ðt E L ðt Þ
0
pL ðt Þdt
0
0
Suppose that iL ð0Þ energy. Then,
ðt
1 di2L 0 dt L 2 dt 0
1 2 L i ðt Þ 2 L
i2L ð0Þ
ð6:40Þ
0
0, i.e., the inductance is initially “uncharged” or has no stored
VI283
Chapter 6
Section 6.3: Application Circuits Highlighting Dynamic Behavior
Inductances might be used instead of capacitances; the ampliﬁer circuits so constructed will be either a differentiator or an integrator. We again pass the corresponding analysis to the homework exercises. However, the physical inductors tend to have a signiﬁcant series resistance and are more bulky. Last but not least, we may ask ourselves a question: as long as the ampliﬁer circuits can perform multiplication, addition (or subtraction), and integration (or differentiation), can we now build an analog computer, which operates with analog voltages and replaces its digital counterpart at least for simple computational tasks? The answer is yes, we can. In fact, this was done a long time ago, in the mid1960s.
VI293
Chapter 6
Problems
Problem 6.42. Determine the equivalent resistance between terminals a and b for the circuit shown in the following ﬁgure in the DC steady state. a 10 F 50 50 10 H 50
6.3.4 Ampliﬁer Circuits with Dynamic Elements: Miller Integrator 6.3.5 Compensated Miller Integrator 6.3.6 Differentiator and Other Circuits Problem 6.45. The input voltage to the Miller integrator circuit with the ideal ampliﬁer shown in the ﬁgure is a series of rectangular voltage pulses. Each is 50 mV tall and 8 ms wide. Given that the initial value of the output voltage is zero, how many voltage pulses are necessary to reach the negative output voltage threshold of 8 V?
b
10nF

vout(t)

vin(t) 100k
+
+
Problem 6.43. Determine the equivalent resistance between terminals a and b for the circuit shown in the following ﬁgure in the DC steady state.
+

a
10 H 100 10 H
100 b
6.3 Application Circuits Highlighting Dynamic Behavior 6.3.1 Bypass Capacitor 6.3.2 Blocking Capacitor 6.3.3 Decoupling Inductor Problem 6.44. Describe the purpose of a A. Bypass capacitor B. Blocking capacitor C. Decoupling inductor in your own words. Specify the placement of each component: in series or in parallel with the source.
10nF vin(t) 200k
+


100
Problem 6.46. The input voltage to the Miller integrator circuit with the ideal ampliﬁer shown in the ﬁgure is a series of rectangular voltage pulses. Each is 50 mV tall and 16 ms wide. Given that the initial value of the output voltage is zero, how many voltage pulses are necessary to reach the negative output voltage threshold of 9.2 V?
+
10 F
vout(t)
+

Problem 6.47. How would you modify the circuit to the previous problem when the positive threshold voltage of +9.2 V should be reached at the output? Problem 6.48. For the circuit shown in the following ﬁgure, express the output voltage, υout(t), as a function of time in terms of the input voltage, υin(t), and circuit parameters R, C. Assume the ideal ampliﬁer.
VI303
Chapter 6
Problems
a)
0, DC Rf R
C

vin
vout
Problem 6.55. For two circuits shown in the ﬁgure that follows, obtain an analytical expression for the output voltage as a function of time and circuit parameters when the input voltage has the form υin ðt Þ ¼ 1expðαt Þ ½mV .
+
a)
b)
Rf
0, DC
R
C
L
vout

vin
R
+
b)
R
vin
C
vout

Rin
R Rin
L

vin
+
c)
vout
vout
+
Rin

vin
+
d) Rf L

R
vout
+
vin
VI305
Chapter 7
Chapter 7: Transient Circuit Fundamentals
Overview Prerequisites:  Knowledge  Knowledge  Knowledge  Knowledge
of of of of
firstorder ordinary differential equations (calculus) Thévenin/Norton equivalent circuits (Chapter 4) constitutive relations for dynamic circuit elements (Chapter 6) basic amplifier theory (Chapter 5)
Objectives of Section 7.1:  Demonstrate the universal character of the KVL/KCL as applied to any electric circuit including transient circuits  Establish the general character of the time constant τ RC for RC circuits  Establish the continuity of the capacitor voltage and its role in circuit ODEs  Solve any firstorder transient RC circuit configuration and understand the practical meaning of the RC circuit using different application examples Objectives of Section 7.2:  Demonstrate the universal character of the KVL/KCL as applicable to any electric circuit including transient circuits  Establish the general character of the time constant τ L=R for RL circuits  Establish the continuity of the inductor current and its role in circuit ODEs  Solve any firstorder transient RL circuit configuration and understand the practical meaning of the RL circuit using an application example Objectives of Section 7.3:  Obtain initial exposure to a bistable amplifier circuit with positive feedback  Understand the principle of operation of a relaxation oscillator RC timer on the base of the bistable amplifier circuit  Establish oscillation frequency and voltage amplitudes from the relaxation oscillator; demonstrate the corresponding laboratory setup  Briefly discuss the 555 timer IC
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_7
VII307
Chapter 7
Transient Circuit Fundamentals
Objectives of Section 7.4:  Define the singletimeconstant (STC) transient circuit  Be able to classify any transient circuit with dynamic elements of the same type as either a singletimeconstant circuit or a more complicated circuit  Solve an example of a nonSTC circuit  Convert an arbitrary transient circuit with one capacitance or one inductance to the basic RC/RL firstorder circuit  Solve a firstorder transient circuit with a harmonic forcing function Objectives of Section 7.5:  Understand topology and classification for the secondorder transient circuits  Convert a transient circuit with a series/parallel LC block to the standard secondorder RLC series/parallel transient circuits  Introduce two major RLC circuit parameters: damping coefficient and undamped resonant frequency  Introduce the step response of a secondorder transient circuit as a solution with a DC source and a switch. Understand the general value of the step response  Properly select the independent function (capacitor voltage or inductor current) for the standard form of the step response with zero initial conditions Objectives of Section 7.6:  Use the method of characteristic equation for secondorder transient circuits  Understand the meaning of overdamped, critically damped, and underdamped circuits  Use the value of damping ratio ς to distinguish between three different cases of circuit behavior  Obtain the complete analytical solution for the step response of the RLC circuit  Apply this solution for modeling a nonideal (realistic) digital waveform Application Examples:  Electromagnetic railgun  Electromagnetic material processing  Digital memory cell  Laboratory ignition system  RC timer or clock circuit in laboratory  Transient circuit with a bypass capacitor  Modeling and origin of the nonideal digital waveform
VII308
Chapter 7
Transient Circuit Fundamentals
Keywords: Transient RC circuit, Transient RL circuit, Energyrelease RC/RL circuit, Energyaccumulating RC/RL circuit, Time constant of RC circuit, Time constant of RL circuit, Relaxation time, Voltage continuity across the capacitor, Fluid mechanics analogy of transient RC circuit, Lorentz force, Selfinduced Lorentz force, Railgun, Electromagnetic material processing, Electromagnetic forming, Current continuity through the inductor, Fluid mechanics analogy of transient RL circuit, Forced response, Electronic ignition system, Piezoelectric effect, Clock frequency, Clock signal, Positive feedback, Linear oscillators, Switching oscillators, Switching RC oscillator, Astable multivibrator, Relaxation oscillator, Bistable ampliﬁer circuit (operation, threshold voltage, mechanical analogy, triggering, trigger signal), Digital memory element, Inverting Schmitt trigger, Noninverting Schmitt trigger, 555 timer IC, Singletimeconstant circuits (deﬁnition, classiﬁcation of, examples of, with general sources), STC circuits, NonSTC circuits (deﬁnition, examples of), Series RLC circuit (generic representation, qualitative description, mechanical analogy, step response, duality), Parallel RLC circuit (generic representation, qualitative description, mechanical analogy, step response, duality), Secondorder ODE (homogeneous, nonhomogeneous, initial conditions, in terms of current, in terms of voltage, forcing function, general solution, forced response, particular solution, complementary solution natural response, step response), Damping coefﬁcient, Neper, Time constant of the decay envelope, Undamped resonant frequency, Step response, Impulse response, Damping ratio, Natural frequency, Overdamped circuit, Critically damped circuit, Underdamped circuit, Overshoot, Undershoot, Rise time, Fall time, Ringing, Nonideal digital waveform
VII309
Chapter 7
Transient Circuit Fundamentals
Section 7.1 RC Circuits The ﬁrstorder RC circuits explored in this section involve the process of discharging or charging a capacitor. This is a timedependent, or transient, circuit behavior, and to understand it, we are required to solve dynamic circuit equations. Mathematically, this implies the solution of ﬁrstorder ordinary differential equations (ODEs) with time as one independent variable. Fortunately, KVL and KCL remain valid for any static or dynamic circuit. These laws can be employed to derive the circuit equations. After that, it is either solved analytically for simple circuits or numerically for realistic RC circuits.
7.1.1 EnergyRelease Capacitor Circuit The circuit in Fig. 7.1 depicts a capacitor, C, that has been charged to a certain voltage V0
υC ðt 0Þ
ð7:1Þ
prior to use. Through a switch, the capacitor is connected to a load, represented by a resistor R 10 Ω. The switch shown in Fig. 7.1 may be a transistor switch. We assume that the switch closes and thereby connects the load to the capacitor at t 0. Our goal is to ﬁnd all circuit parameters, plus the power delivered to the load as functions of time. t=0
+
+
C=10 F vC

iC
iR
vR

R=10
Fig. 7.1. Discharging a capacitor through a load resistor.
The solution to this dynamic circuit is based on applying KVL and KCL, which are valid for all electric circuits. Using KCL gives the result: iC
ð7:2Þ
iR
at any instance of time, t. Since both circuit elements in Fig. 7.1 are passive, we can apply the constitutive relations between currents and voltages without changing the sign: iC
zﬄﬄ}ﬄﬄ{ dυC C dt
iR
z}{ υR R
υC R
ð7:3Þ
This is true because KVL states for any positive time, t > 0,
VII310
Chapter 7 υR
Section 7.1: RC Circuits
υC
ð7:4Þ
Equation (7.3) therefore yields C
dυC υC þ dt R
0)
dυC υC þ dt τ
0,
τ
RC
ð7:5Þ
This is the famous ﬁrstorder transient circuit equation. Here, τ carries units of seconds since R is recorded in Ω and C is given in F A s/Vand is called the time constant or the relaxation constant of the circuit. It is the only constant that is present in the ﬁrstorder differential equation. The solution of an ODE of this type has the generic form t υC ðt Þ Kexp ð7:6Þ τ This fact is proven by direct substitution. The constant K is determined from the initial condition, Eq. (7.1), which yields K
ð7:7Þ
V0
Thus, the circuit voltages have the same form t υC ðt Þ υR ðt Þ V0 exp ; t0 τ
ð7:8aÞ
for nonnegative values of t. However, although the capacitor voltage is equal to V0 at t < 0, the resistor voltage is exactly zero at t < 0, since the switch was open. The current through the load resistor is i R ðt Þ
υ R ðt Þ R
t V0 exp ; R τ
t0
ð7:8bÞ
and is zero for negative t. We recall that the capacitor current is the negative of the load current. The instantaneous power delivered to the load resistance is expressed in the form pR ð t Þ
υR ðt ÞiR ðt Þ
t V0 2 exp 2 ; R τ
t0
ð7:8cÞ
Equations (7.8a c) provide the complete solution of the circuit shown in Fig. 7.1. What is the most remarkable and perhaps most important property of the solution? The answer to this question is linked to the amount of power that can be discharged in a ﬁnite amount of time. Let us examine Eq. (7.8c) more closely. When the load resistance, R, becomes small, the delivered power can reach an arbitrarily high value at small positive t. Expressed in another way, when discharged through a small resistance, the (ideal) capacitor delivers an extremely high power pulse during a short period of time!
VII311
Chapter 7
Section 7.1: RC Circuits
Fig. 7.5. (a) Electromagnetic forming of metal joints. The current in the windings generates a magnetic ﬂux that induces eddy currents in the metal. Their product is the Lorentz force. (b) Electromagnetically reformed door panel compared with the production geometry.
SelfInduced Lorentz Force Electromagnetic forming processes shown in Fig. 7.5 do not use permanent magnets. Instead, the socalled selfinduced Lorentz force is employed. The idea is to generate the magnetic ﬂux B with the same current iC. Figure 7.5a shows the related concept used in noncontact electromagnetic forming of metal joints. The high discharge current, iC, creates a strong timevarying magnetic ﬁeld, both inside and outside of the coils in Fig. 7.5a. In turn, the timevarying magnetic ﬁeld induces socalled eddy currents in the metal sample. The product of these eddy currents and the magnetic ﬁeld gives rise to a Lorentz force according to Eq. (7.11). This force is strong enough to deform the joints. 7.1.6 Application Example: Digital Memory Cell This completely different example investigates a digital circuit that stores binary information. Figure 7.6 shows a schematic of a dynamic randomaccess memory (DRAM) memory cell. The cell stores its bit of information as charge deposited on the cell capacitor C. When the cell is storing a logic 1, the capacitor is charged to a positive voltage V0; when a logic zero is stored, the capacitor is discharged to a zero voltage. Because of leakage effects, there is always a nonzero resistance R to ground (not shown in the ﬁgure). Thus, the cell circuit becomes that of Fig. 7.1. The capacitor will discharge and must be refreshed periodically. During refresh, the capacitor voltage is restored to V0 if necessary. The refresh operation is in fact performed every 5 10 ms!
VII317
Chapter 7
Section 7.1: RC Circuits
at any positive time instance t. Therefore, Eq. (7.13) yields C
dυC υC þ dt R
VS dυC υC ) þ R dt τ
VS , τ
τ
RC
ð7:15Þ
Equation (7.15) has now an excitation term (the power supply voltage) on its righthand side. Consequently, the solution to this equation is called a forced response. Equation (7.15) is known as an inhomogeneous ﬁrstorder differential equation. This is in contrast to the homogeneous differential equation (7.5). Nonetheless, Eq. (7.15) still remains a ﬁrstorder transient equation with the same time constant τ. The solution to any ﬁrstorder ordinary differential equation of that type has the generic form t υC ðt Þ K 1 exp ð7:16aÞ þ K2 τ This fact can be checked by direct substitution. The two terms containing the exponential factor will cancel out after differentiation. The remaining terms in Eq. (7.15) yield K2 τ
VS ) K2 τ
ð7:16bÞ
VS
The constant parameter K1 can be determined from the initial condition, υC ðt Since exp(0) 1 in Eq. (7.16a), we conclude
0Þ
K1 þ K2
ð7:16cÞ
0 ) K1
VS
Thus, the circuit voltages have the form h t i t υC ðt Þ V S 1 exp , υR ðt Þ V S exp τ τ
0.
ð7:16dÞ
The resistor voltage has the same form as the resistor voltage in Eq. (7.8a) for the discharging capacitor. However, the capacitor voltage has not. The capacitor current is iC ðt Þ
C
dυC ðt Þ dt
t VS exp R τ
ð7:16eÞ
It is equivalent to Eq. (7.8b), the discharge current for the RC circuit. What is the most remarkable property of the solution given? According to Eq. (7.16d), we always need a certain amount of time to charge the capacitor. It is clear that this time will be on the order of the time constant τ. Moreover, from a formal point of view, the capacitor voltage will never exactly reach the source voltage (the exact equality only occurs at t ! 1), see Fig. 7.8.
VII319
Chapter 7
Transient Circuit Fundamentals
Section 7.3 Switching RC Oscillator The time constant τ of an RC circuit provides a natural time scale. It is widely employed for timing purposes. Some of you may have already used microcontroller starter kits. The microcontroller manual discussing the various settings will likely feature a topic entitled “RC oscillator.” Here you will discover that the microcontroller clock frequency can be controlled by an external resistor R and capacitor C. How is this possible? We have just seen that the RC circuit discharges the capacitor through the resistor, but how can it be used to create a periodic clock signal at a given frequency? The present section aims to augment an RC circuit with an ampliﬁer circuit and establish a clock circuit.
7.3.1 About Electronic Oscillators An electronic oscillator is a circuit that has an output, but no input in the common sense. It generates a certain periodic waveform at the output. The period, amplitude, and shape of this waveform are determined by the circuit topology. The “heart” of any oscillator circuit is an ampliﬁer block with some sort of a positive feedback. The positive feedback is the opposite of the negative feedback. A part of the output ampliﬁer’s voltage is fed back into the input with the sign plus. All oscillator circuits may be divided into linear oscillators, which create sinusoidal waveforms, and switching oscillators, which create square and other periodic nonharmonic waveforms. The subject of this section is a switching oscillator circuit, which is called an astable multivibrator or a relaxation oscillator. This circuit uses the comparator ampliﬁer but with a positive feedback loop and a transient RC block. It is perhaps the simplest and yet efﬁcient oscillator circuit. 7.3.2 Bistable Ampliﬁer Circuit with the Positive Feedback Saturation Mechanism Consider the circuit shown in Fig. 7.17. At ﬁrst sight, it is similar to the inverting ampliﬁer conﬁguration. However, the ampliﬁer polarity is interchanged, which means that the feedback is now positive. The circuit has no input: both potential inputs are grounded. Since there is no current into the ampliﬁer itself (the ﬁrst summingpoint constraint still applies), two resistors of the feedback loop form a voltage divider between the output voltage υout and ground. Therefore, the voltage at node (+) becomes υþ
R1 υout R1 þ R2
ð7:30Þ
VII330
Chapter 7
Section 7.3: Switching RC Oscillator
R1 a
R2
+
+

0V v+=
R1 v R1+R2 out b
+ vout
0V
Fig. 7.17. A bistable ampliﬁer circuit.
To analyze the circuit in Fig. 7.17, we again consider the feedback as a fast dynamic process with a very short delay in the feedback loop. We assume that R1 R2 for simplicity. It means that 50 % of υout is returned back to the noninverting input in, say, 1 μs. The openloop ampliﬁer’s gain will be A 106 ; the ampliﬁer hits the power rails υout V CC in saturation. The initial value of υþ will be 0 V, and the initial value of υout will be 1 μV (at the noise level). Table 7.1 shows the dynamics of the feedback process where the ampliﬁer operates as υout Aυþ , but it takes 1 μs to return 50 % of υout. It follows from Table 7.1 that the ampliﬁer will be very quickly saturated; its output will be the positive rail voltage υout þV CC . All other positive initial values of υout will lead to the same result. Simultaneously, all negative initial values of υout will lead to the saturation at the negative rail υout V CC . Table 7.1. Dynamics of the output voltage for the bistable ampliﬁer circuit. Time, μs 0 1 2
υþ 0V 0:5 10 6 V 0.25 V
υout 10 6 V 0.5 V þV CC
Two Stable States The key point is that once the saturation state υout
þV CC , υþ
þ
R1 V CC R1 þ R2
ð7:31Þ
has been reached, the ampliﬁer circuit will exist in this state indeﬁnitely, despite all the subsequent electric noise. To prove this fact, we may introduce a small perturbation in υout and/or in υþ ; the circuit will quickly return to the solution given by Eq. (7.31). Quite similarly, once the opposite saturation state υout
V CC υþ
R1 V CC R1 þ R2
ð7:32Þ
VII331
Chapter 7
Transient Circuit Fundamentals
555 Timer Integrated Circuit Rectangular pulse forms at lower frequencies are routinely created by the wellknown 555 timer IC (integrated circuit). The 555 timer operates conceptually similarly to the relaxation oscillator circuit described above; it is more versatile though. The 555 timer creates a waveform of relatively sharp and clean rectangular voltage pulses whose frequency is controlled by an external capacitor and resistor. The duty cycle (ratio of the positive phase duration to the signal period) can also be controlled. The 555 timer is perhaps one of the most popular integrated circuits ever built.
VII336
Chapter 7
Section 7.4: SingleTimeConstant (STC) Transient Circuits
Section 7.4 SingleTimeConstant (STC) Transient Circuits 7.4.1 Circuits with Resistances and Capacitances Consider a transient circuit with an arbitrary number of capacitances and resistances. The circuit has an independent voltage source (or sources) and a switch. Instead of the voltage source, some capacitors may be charged prior to closing the switch. A singletimeconstant transient circuit (STC circuit) is that which solution has the form t υðt Þ K 1 exp ð7:36Þ þ K2 τ for any branch voltage in the circuit. In other words, only one exponential function is involved, similar to the basic RC circuits studied previously. Here, τ is the only time constant of the circuit. The STC transient circuits are frequently encountered in practice, in particular, in the study of transistor ampliﬁers. The STC transient circuits include: 1. Transient circuits with only one capacitance C. According to Thévenin’s theorem, the network of resistances and source(s) seen by the capacitor is reduced to its Thévenin equivalent. As a result, we obtain the circuit shown in Fig. 7.7. Its time constant is given by τ
RT C
ð7:37Þ
where RT is Thévenin resistance the equivalent resistance of the network with the independent voltage source(s) shorted out. 2. Transient circuits with only one resistance R. Thévenin’s theorem may be applied again, this time to the network of capacitances and source(s) seen by the resistance. As a result, we again obtain the circuit from Fig. 7.7. Its time constant is given by τ
RC T
ð7:38Þ
where CT is the equivalent capacitance of the network with the independent voltage source(s) shorted out. 3. Transient circuits with an arbitrary number of capacitances and resistances given that the solutions for different capacitor voltages obtained by the simultaneous use of KVL and KCL are all linear functions of each other. Consider a circuit with multiple capacitances. Assume that N is the ﬁnal number of capacitances after all possible series/parallel combinations. For the STC condition to hold, there should be N 1 independent closed loops that include only capacitances (and possibly independent voltage source(s)) but do not include resistances. This useful result has been conﬁrmed by the authors based on an extensive circuit analysis.
VII337
Chapter 7
Transient Circuit Fundamentals a)
VS
b) 1.0VS
t=0
+ 
R1
C1
i
+ v1 
+

c)
v1
i2 C 2 v2
R2
capacitor voltages v1(t) and v2(t)
0.0VS
i1
circuit current i(t) and current I0
2I0
v2 I0
1.0VS 1.5VS
5 0 time in terms of
0
0
10
0
0
0
0
5
10
0
time in terms of
0
0
Fig. 7.25. A nonSTC circuit with two resistances and two capacitances and its solution behavior.
υ 1 ðt Þ
V S þ K 1 exp
α1 t α2 t þ K 2 exp τ0 τ0
ð7:48Þ
where K1, K2 are two constants determined by the initial conditions The solution for voltage υ2(t) is found using the second of equations (7.46): α1 t α2 t þ ð1 þ α2 ÞK 2 exp ð7:49Þ υ2 ðt Þ ð1 þ α1 ÞK 1 exp τ0 τ0 The initial conditions imply that both capacitors are uncharged prior to closing the switch. This gives K 1 ð1 þ α2 Þ=ðα1 α2 ÞV S and K 2 ð1 þ α1 Þ=ðα2 α1 ÞV S . With this in mind, the solution is complete. Figure 7.25b, c shows the behavior of the two capacitor voltages. A truly remarkable point is that instantaneous voltage across the second capacitor in Fig. 7.25b exceeds the (absolute) source voltage. Moreover, the instantaneous circuit current in Fig. 7.25c exceeds the initial circuit current I 0 V S =R1 by 2.5 times. Those distinct features are observed for other secondorder transient circuits studied further.
7.4.4 Example of an STC Transient Circuit In Fig. 7.26, we present an example of a rather complicated circuit, which still follows the STC circuit model. The proof is based on the observation that node (*) in Fig. 7.26 connects three branches: two of which are exactly the inductances and the remaining one is the current source. According to KCL, this circuit again implies that both inductances cannot have zero current simultaneously prior to closing the switch. VII342
Chapter 7
Section 7.5: Description of the SecondOrder Transient Circuits
7.5.3 Initial Conditions in Terms of Circuit Current and Capacitor Voltage In contrast to the ﬁrstorder transient circuits, there are two equally possible choices of the independent function for the RLC circuit in Fig. 7.30:  Circuit (or inductor) current i(t)  Capacitor voltage, υC(t) The circuit current remains continuous over time (inductor “inertia”) and so does the capacitor voltage (capacitor “inertia”). We have chosen the circuit current and obtained the secondorder ODE Eq. (7.53). You may wonder if this is really the best choice. The answer is nontrivial and is hidden in the initial conditions. Any ﬁrstorder ODE needs one initial condition. Any secondorder ODE needs two initial conditions. Let us establish these initial conditions for the circuit current ﬁrst. Following the current continuity through the inductor, the circuit current must be zero at t 0, that is, i ðt
0Þ
ð7:65aÞ
0
Hence, the ﬁrst initial condition is established. The second one is that of the initial capacitor voltage equal to zero. According to Eq. (7.60) and Eq. (7.65a), di di VS υL ðt ¼ 0Þ þ υR ðt ¼ 0Þ þ υC ðt ¼ 0Þ ¼ V S ) L ðt ¼ 0Þ ¼ V S ) ðt ¼ 0Þ ¼ ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} dt dt L Riðt 0Þ 0
0
ð7:65bÞ Thus, the initial circuit current is zero, whereas its ﬁrst derivative is not. This is a drawback of the electric current formulation given by Eq. (7.63) for the series RLC circuit. On the other hand, the capacitor voltage and its derivative (which is proportional to the capacitor/inductor/circuit current) are both zero at t 0, which leads to a simpler “universal” homogeneous formulation of the initial conditions, i.e., υ C ðt
0Þ
0,
dυC ðt 0Þ dt
ð7:66Þ
0
At the same time, the secondorder ODE for the capacitor voltage becomes inhomogeneous, i.e., at t 0 d 2 υ C ðt Þ dυC ðt Þ þ 2α þ ω20 υC ðt Þ dt 2 dt
ω20 V S
ð7:67Þ
It is worth noting that this equation has exactly the same form as Eq. (7.63), but with the nonzero righthand side. The derivation of Eq. (7.67) is similar to the derivation of Eq. (7.63); it is suggested as a homework problem.
VII349
Chapter 7
Transient Circuit Fundamentals
7.5.4 Step Response and Choice of the Independent Function The selection between current and voltage formulations reﬂects our desire to convert the circuit differential equation into a standard form, which will allow us to use powerful tools of signals and systems theory. What should be preferred: the homogeneous secondorder ODE Eq. (7.63) for the circuit current augmented with inhomogeneous initial conditions Eqs. (7.65a, b), or the inhomogeneous secondorder ODE Eq. (7.67) for the capacitor voltage augmented with the homogeneous initial conditions Eq. (7.66)? The answer is as follows. If the initial conditions are all zero, the only remaining excitation is the forcing function: the righthand side of Eq. (7.67). We consider a unit step function, u(t), deﬁned by (see Fig. 7.32a)
0 t 0. Differentiation over time and division by C yield the expected homogeneous secondorder ODE: d 2 υ ðt Þ 1 dυðt Þ 1 þ þ υ ðt Þ 2 dt RC dt LC
0
ð7:71Þ
Eq. (7.71) is written in the form of Eq. (7.63): d 2 υ ðt Þ dυðt Þ þ 2α þ ω20 υðt Þ 2 dt dt
ð7:72Þ
0
if we deﬁne the damping coefﬁcient α and the undamped resonant frequency ω0 as α
1 , 2RC
ω0
1 p LC
ð7:73Þ
Initial Conditions and Choice of Independent Function The initial conditions for Eq. (7.72) are that the voltage across the capacitor and the inductor current must be continuous. Therefore, they must have the form υ ðt
0Þ
0,
dυ ðt dt
0Þ
IS C
ð7:74Þ
The voltage derivative is not zero at the initial time moment. Eq. (7.74) is similar to Eqs. (7.65a, b). It has been stated that the step response of the secondorder system is generally calculated with the homogeneous (zero) initial conditions. The secondorder circuit ODE written in terms of the inductor current d 2 i L ðt Þ diL ðt Þ þ 2α þ ω20 υL ðt Þ 2 dt dt
ω20 I S
ð7:75Þ
possesses zero initial conditions, i.e., i L ðt
0Þ
0,
diL ðt 0Þ dt
0
ð7:76Þ
Those are the preferred conditions for the step response calculations. The derivation of Eq. (7.75) is similar to the derivation of Eq. (7.72); it is left as a homework problem.
VII352
Chapter 7
Transient Circuit Fundamentals
Section 7.6 Step Response of the Series RLC Circuit 7.6.1 General Solution of the Secondorder ODE Solution for Step Response The starting point is Eq. (7.69) of the previous section for the capacitor voltage of the series RLC circuit augmented with the homogeneous initial conditions, i.e., d 2 υ C ðt Þ dυC ðt Þ þ 2α þ ω20 υC ðt Þ Auðt Þ, 2 dt dt dυC ðt 0Þ 0 υC ðt 0Þ 0, dt
A
ω20 V S
ð7:78aÞ ð7:78bÞ
Similar to the ﬁrstorder transient circuits with arbitrary sources, the general solution is also given by the sum of two parts: a particular solution of the inhomogeneous equation (7.78a), let us call it xp(t), and a complementary solution, let us call it xc(t), of the homogeneous equation (7.78a). Homogeneous implies that the righthand side of the ODE equals zero. The particular solution is known as the forced response and the complementary solution is known as the natural response. As a result, the total solution is υC ðt Þ
xp ðt Þ þ xc ðt Þ
ð7:79Þ
For the circuit with the DC voltage source and the switch acting as a step excitation, the particular solution is trivial. It is proved by direct substitution: xp ðt Þ
V S ) υ C ðt Þ
V S þ x c ðt Þ
ð7:80Þ
The complementary solution carries information about the entire circuit and requires care.
Solution in Arbitrary Case What if the righthand side of Eq. (7.78a) is an arbitrary function of time? How is the solution obtained? We have already established that any such solution can be obtained on the basis of the step response. In general, the solution is expressed in terms of a convolution integral, which involves an arbitrary righthand side of the secondorder ODE and the time derivative of the step response. This interesting and fundamental question is studied further in signals and systems theory. 7.6.2 Derivation of the Complementary Solution: Method of Characteristic Equation Similar to the ﬁrstorder transient circuits, we seek a complementary solution (natural response) of the homogeneous Eq. (7.78a) in the most general exponential form
VII354
Chapter 7 xp ðt Þ
Section 7.6: Step Response of the Series RLC Circuit
Kexpðst Þ
where K and s are two arbitrary constants. The substitution yields 2 s þ 2α s þ ω20 Kexpðst Þ 0
ð7:81Þ
ð7:82Þ
For a nontrivial solution, the characteristic equation s2 þ 2αs þ ω20 0 must be satisﬁed, that is, 8 s ! > 2 > ς 1 > > α 1 p >
< 2 2 ς2 α þ pα ω0 2 2 s or s ! ð7:83Þ s þ 2αs þ ω0 ¼ 0 ) s1, 2 ¼ ¼ 1, 2 > 2 α α2 ω20 > ς 1 > > > : α 1þ ς2 where the new constant ζ α=ω0 is the damping ratio of the RLC circuit. Formally, this constant has units of 1/rad; it is often considered dimensionless. We must distinguish between three separate cases depending on the value of the damping ratio: Case A This situation (overdamping) corresponds to ς > 1. In this case, s1,2 are both real and negative. Since the original ODE is linear, the general solution is simply the combination of two independent decaying exponential functions:
xc ðt Þ
K 1 expðs1 t Þ þ K 2 expðs2 t Þ
ð7:84aÞ
Case B This case (critical damping) corresponds to ς ¼ 1. Both roots s1,2 become identical. Therefore, a solution in the form of Eq. (7.84a) with two independent constants can no longer be formed. Only one independent constant may be available. Fortunately, another solution in the form t exp(s1t) exists in this special case. This fact is proved by direct substitution. Thus, the general solution becomes
xc ðt Þ
K 1 expðs1 t Þ þ K 2 texpðs1 t Þ
ð7:84bÞ
Case C This case (underdamping) corresponds to ς < 1. Both roots s1,2 become complex. This means that our initial simple guess Eq. (7.81) is no longer correct. One can prove by direct substitution that the general solution now has the oscillating form
xc ðt Þ
K 1 expð α t Þ cos ωn t þ K 2 expð α t Þ sin ωn t
ð7:84cÞ
p where ωn ¼ ω20 α2 is the (radian) natural frequency of the circuit. 1/α is also called the time constant or the time constant of the decay envelope. The complementary solution in the form of Eqs. (7.84) always contains two independent integration constants. They should be used to satisfy the initial conditions, which complete the solution.
VII355
Chapter 7
Transient Circuit Fundamentals
7.6.3 Finding Integration Constants According to Eqs. (7.79) and (7.80), the capacitor voltage is υC ðt Þ xc ðt Þ þ V S where xc(t) is given by Eqs. (7.84). The integration constants may be found using Eq. (7.78b), which dictates that both the capacitor voltage and its derivative must vanish at the initial time t 0. We then have from Eqs. (7.84) Case A: K 1 þ K 2 þ V S 0, s1 K 1 þ s2 K 2 0 Case B: K 1 þ V S 0, s1 K 1 þ K 2 0 Case C: K 1 þ V S 0, αK 1 þ ωn K 2 0
ð7:85aÞ ð7:85bÞ ð7:85cÞ
The solution of Eqs. (7.85) has the form Case A:
K1
Case B:
K1
Case C:
K1
s2 V S s1 V S , K2 s 1 s2 s 2 s1 V S , K 2 s1 V S α V S, K 2 VS ωn
ð7:86aÞ ð7:86bÞ ð7:86cÞ
Equations (7.83) through (7.86) complete the step response solution for the series RLC circuit. The circuit may behave quite differently depending on the value of the damping ratio ς.
7.6.4 Solution Behavior for Different Damping Ratios We consider the series RLC circuit shown in Fig. 7.35. We will choose round numbers L 1 mH, C 1 nF. These values approximately correspond to an RLC transient circuit operating in the 100 kHz 1 MHz frequency band.
+ 10 V=VS
R vR

t=0
+ 
i(t) 1 mH
1 nF

+ vL 
vC +
Fig. 7.35. RLC series circuit; the resistance value R may be varied.
VII356
Chapter 7
Problems
Problems
t=0 R=100
C=10 F

7.1 RC Circuits
+ vC
iC
vR
iR
7.1.1 EnergyRelease Capacitor Circuit 7.1.2 Time Constant of an RC Circuit and Its Meaning 7.1.3 Continuity of the Capacitor Problem 7.3. Prove that Eq. (7.6) is the solution to Eq. (7.5) (both from Section 7.1) using Voltage direct substitution and the differentiation that
Dynamic circuit element
+
vC

iC
Equivalent circuit at DC (short or open) Relation between voltage and current (passive reference conﬁguration) Expression for the time constant of a transient circuit that includes the dynamic element (C) and a resistor R.
+
Problem 7.1. For the capacitor as a dynamic circuit element, develop: 1. Equivalent circuit at DC 2. Relation between voltage and current 3. Expression for the time constant of a transient circuit that includes the dynamic element and a resistor R
follows. Problem 7.4. A. Show that the time constant, τ, of an RC circuit has the units of seconds. B. To obtain the slow discharge rate of lesser instantaneous power into the load, should the load resistance be small or large? Problem 7.5. A 100μF capacitor discharges into a load as shown in the following ﬁgure. The load resistance may have values of 100 Ω, 10 Ω, and 1 Ω. The capacitor is charged to 20 V prior to t ¼ 0. t=0 C=100 μF
+
Problem 7.2. Using KCL and KVL, derive the differential equation for the circuit shown in the following ﬁgure, keeping the same labeling for the voltages and the currents. A. Is the ﬁnal result different from Eq. (7.5) of Section 7.1? B. Could you give an example of a certain voltage and/or current labeling (by arbitrarily changing polarities and directions in the ﬁgure) that causes the differential equations to change?
RL
vC
τ¼

A. Find time constant τ and the maximum instantaneous power delivered to the load resistor in the very ﬁrst moment for every resistor value—ﬁll out the Table that follows. Instantaneous load power right after the switch closes RL 100 Ω 10 Ω 1Ω
τ, s
pL ðt ¼ þ0Þ, W
B. Do the instantaneous power values from the Table depend on the capacitance value?
VII367
Chapter 7
Transient Circuit Fundamentals
Problem 7.6. A 100μF capacitor, shown in the following ﬁgure, discharges into a 10Ω load resistor. The capacitor is charged to 15 V prior to t ¼ 0. A. Find the time constant of the circuit (show units). B. Express the voltage across the capacitor as a function of time and sketch it to scale versus time over time interval from 2 ms to 5 ms.
vC(t), V 20
15
10
5
10
05
0
05
10
15
20
25
t, ms
05
10
15
20
25
t, ms
pR(t), W 80
t=0
60
R=10
C=100 F
+
+
vC
vR
40


20
10
05
0
vC(t), V 20
15
10
5
2
1
0
1
2
3
4
t, ms
Problem 7.7. A 100μF capacitor, shown in the following ﬁgure, discharges into a 5Ω load resistor. The capacitor is charged to 20 V prior to t ¼ 0. A. Find an expression for the voltage across the capacitor as a function of time and sketch it to scale versus time over the interval from –2τ to 5τ. B. Repeat the exercise for instantaneous power delivered to the resistor. t=0
+ vC

C=100 F
R=5
+
Problem 7.8. In the circuit shown in the following ﬁgure, the capacitor is charged to 20 V prior to t ¼ 0. A. Find an expression for the voltage across the capacitor as a function of time and sketch it to scale versus time over the interval from –2τ to 5τ. B. Repeat for instantaneous power delivered to the rightmost resistor. t=0
+
vC

C=1 F R=50
R=50
Problem 7.9. Present the text of a MATLAB script (or of any software of your choice) in order to generate Fig. 7.2d of Section 7.1. Attach the ﬁgure so generated to the homework report.
vR

VII368
Chapter 7
Problems
Problem 7.10. Prove that the integral of the load power in Fig. 7.2d given by Eq.(7.8c) is exactly equal to the energy stored in the charged capacitor, E C ¼ 12 CV 20 prior to t ¼ 0.
B ¼ 0:3 T. The accelerating object has a length of 2 cm. Plot to scale the Lorentz force as a function of discharge current over the interval 0 < iC < 1000 A.
Problem 7.11. A. Create the generic capacitor voltage discharge curve similar to Fig. 7.2a but for an arbitrary capacitor powering an arbitrary load resistor over the time interval from – 2τ to 5τ. The capacitor is charged to V0 prior to t ¼ 0. To do so, ﬁnd the capacitor voltage as a fraction of V0 for every unit of τ and ﬁll out the Table that follows.
Problem 7.14. An electromagnetic capacitor accelerator needs to create an average force of 5 N over 2 ms on a moving object with length of 1 cm. The load (armature plus object) resistance is 1 Ω, and the external magnetic ﬁeld is B ¼ 0:25 T. Determine: A. The required capacitor voltage prior to discharge B. The required capacitance of the capacitor (bank of capacitors) Hint: Assume that the average force acts over the time interval τ. Its value is approximately equal to 60 % of the initial force value.
Capacitor voltage in terms of V0 2τ
t υC(t)
τ
τ
0
2τ
3τ
4τ
5τ
B. Repeat the same task for Fig. 7.2d related to load power. Find the load power in terms of the maximum power just after closing the switch.
25
5
20
4
15
3
10
2
5
1
0 2
1
0
1
2 time, ms
3
4
Capacitor current, A
Capacitor voltage, V
Problem 7.12. For an unknown energyrelease RC circuit, capacitor voltage and capacitor current were measured in laboratory before and after closing the switch at t ¼ 0 as shown in the ﬁgure that follows. Approximate R and C.
Problem 7.15. Solve the previous problem when: A. The average force increases to 50 N B. The average force increases to 500 N Problem 7.16. The world's largest capacitor bank is located in Dresden, Germany. The pulsed, capacitive power supply system was designed and installed for studying high magnetic ﬁelds by experts from Rheinmetall Waffe Munition. The bank delivers 200 kA of discharge current in the initial time moment (just after the switch closes). The time constant is 100 ms. Estimate the bank capacitance if the charging voltage is 200 kV.
0 5
7.1.4 Application Example: Electromagnetic Railgun 7.1.5 Application Example: Electromagnetic Material Processing Problem 7.13. An electromagnetic capacitor accelerator with permanent magnets has
VII369
Chapter 7
Transient Circuit Fundamentals
7.1.7 EnergyAccumulating Capacitor Problem 7.19. For the circuit shown in the following ﬁgure: Circuit Problem 7.17. A 1μF capacitor shown in the following ﬁgure is charged through the 1kΩ load resistor. The initial capacitor voltage is zero. A. Find the time constant of the circuit (show units). B. Express the voltage across the capacitor as a function of time and sketch it to scale versus time over the interval from 2 ms to 5 ms. +
iR
+ 
R=0.5
+
+ 
R=1 k vR
15 V=VS
t=0
+
vC
C=1 F iC

vC(t), V
iC
50 V=VS
C=33 F

t=0
A. Find an expression for the capacitor voltage, υC, and the capacitor current, iC, including the value of time constant. B. Sketch the capacitor voltage, υC, and the capacitor current, iC, to scale versus time over the time interval from –2τ to 5τ.
vC
iC

C=33 F
Problem 7.20. Sketch your own ﬂuidﬂow counterpart of the charging circuit shown in the ﬁgure and establish as many analogies between electrical (R, C, VS) and mechanical parameters of your drawing as possible.
20
t=0
R
15
10
VS 5
2
1
0
1
2
3
4
R=1
25 V=VS
+ 
+ C=47 F iC
C
t, ms
Problem 7.18. For the circuit shown in the following ﬁgure: A. Find an expression for the capacitor voltage, υC, and the capacitor current, iC, including the value of time constant. B. Sketch the capacitor voltage, υC, and the capacitor current, iC, to scale versus time over the interval from –2τ to 5τ. t=0
+ 
vC

Problem 7.21. For the circuit shown in the following ﬁgure: A. How much time does it take to charge the capacitor to 10 V? B. To 25 V? t=0
+ 
R=50
+ 25 V=VS
C=47 F iC
vC

Problem 7.22. For the circuit shown in the ﬁgure, how much time does it take to charge
VII370
Chapter 7
Transient Circuit Fundamentals
Problem 7.28. In the circuit that follows, the capacitor, C1, is initially uncharged. The switch is closed at t ¼ 0.
Dynamic circuit element
+
vL

iL t=0
R1
+ 
R2
VS C1
Give answers to the following questions based on known circuit parameters C1, R1, R2, VS: A. What is the current through resistor R2 as a function of time? B. What is the maximum current through resistor R1? C. What is the current through resistor R1 at a time long after the switch closes? D. What is the charge, Qþ ðt Þ, of the capacitor, C1, as a function of time? The switch is then opened a very long time after it has been closed – reset the time to t ¼ 0. A. What is the charge Qþ ðt Þ of the capacitor, C1, as a function of time? B. What is the current through resistor R2 as a function of time? Specify the current direction in the ﬁgure.
7.2 RL Circuits
Equivalent circuit at DC (short or open) Relation between voltage and current (passive reference conﬁguration) Expression for the time constant of a transient circuit that includes the dynamic element (L) and a resistor R
τ¼
Problem 7.30. A. Using KCL and KVL, derive the differential equation for the inductor current in the circuit shown in the ﬁgure that follows, keeping the same labeling for the voltages and the currents. B. Is the ﬁnal result different from Eq. (7.19) of Section 7.2? R=1 kΩ
L=1 mH
+ vL

+ iL
iR
vR

Problem 7.31. Prove that Eq. (7.20) is the solution to Eq. (7.19) using direct substitution and the corresponding differentiation.
7.2.1 EnergyRelease Inductor Circuit Problem 7.32. 7.2.2 Continuity of the Inductor Current A. Show that the time constant τ has units Problem 7.29. For the inductor as a dynamic circuit element, present: 1. Equivalent circuit at DC 2. Relation between voltage and current 3. Expression for the time constant of a transient circuit that includes the dynamic element and a resistor, R
of seconds for the RL circuit. B. To ensure a slower energy release rate of the inductor, should the load resistance be small or large? C. To ensure a faster energy release rate of the inductor, should the load resistance be small or large?
VII372
Chapter 7
Transient Circuit Fundamentals
7.2.4 EnergyRelease RL Circuit with 7.3 Switching RC Oscillator the Voltage Supply 7.3.2 Bistable Ampliﬁer Circuit with the 7.2.5 Application Example: Laboratory Positive Feedback Ignition Circuit Problem 7.43. A 270μH inductor shown in the 7.3.3 Triggering
following ﬁgure releases its energy into the 510Ω load resistor. The power supply voltage is 20 V. The switch opens at t ¼ 0. A. Present an expression for the inductor current as a function of time and sketch it to scale versus time over the interval from 1 μs to 2.5 μs. B. Repeat the same task for the resistor voltage. R0=51
t=0 L=270 H
+ 
Problem 7.45. The bistable ampliﬁer circuit shown in the following ﬁgure (inverting Schmitt trigger) exists in the positive stable state. Ampliﬁer’s power supply rails are 12 V. Determine output voltage when the applied trigger signal is 1. υin ¼ 6 V 2. υin ¼ 2 V 3. υin ¼ 4 V Assume that the ampliﬁer hits the power rails in saturation. 3 kW
R=510 1 kW
20 V=VS
+
+

vout
0V
Problem 7.44. The circuit for the previous problem is converted to the energyaccumulating RL circuit by inversing the switch operation. Assume that the switch was open prior to t ¼ 0. The switch closes at t ¼ 0. A. Derive an expression for the inductor current as a function of time. B. Repeat the same task for the voltage across resistor R. C. Could this circuit generate large voltage spikes, similar to the circuit from the previous problem? R0=51
t=0 L=270 H R=510
+ 
20 V=VS
vin
Problem 7.46. Repeat the previous problem when the initial stable state of the ampliﬁer circuit is negative. Problem 7.47. The bistable ampliﬁer circuit shown in the following ﬁgure (noninverting Schmitt trigger) exists in the positive stable state. Ampliﬁer’s power supply rails are 15 V. Determine output voltage when the applied trigger signal is A. υin ¼ 1 V B. υin ¼ 2 V C. υin ¼ 4 V Assume that the ampliﬁer hits the power rails in saturation. 3 kW vin
1 kW
+

+ vout
0V
VII376
Chapter 7
Problems
Problem 7.48. Repeat the previous problem when the initial stable state of the ampliﬁer circuit is negative.
7.4 SingleTimeConstant (STC) Transient Circuits
7.4.1 Circuits with Resistances and Capacitances Problem 7.49. A clock circuit (relaxation oscil 7.4.2 Circuits with Resistances and lator circuit) shown in the following ﬁgure is Inductances 7.3.4 Switching RC Oscillator 7.3.5 Oscillation Frequency
powered by a 10V power supply. 300 kW 100 kW
+

+ vC

C
+ vout

R
Problem 7.52. Determine whether or not the transient circuits shown in the following ﬁgure are the STC circuits. If this is the case, express the corresponding time constant in terms of the circuit parameters. A)
R1 t=0
R3
0V
Vout 10V
+ 
7 5V
VS
R2
R4
C
5 0V 2 5V 0V
t
t=0
B)
2 5V
C1
5 0V 7 5V 10V
Sketch to scale the capacitor voltage, υC, as a function of time. Assume that υC ðt ¼ 0Þ ¼ 0. Assume the ideal ampliﬁer model. The speciﬁc values of R and C do not matter; they are already included in the time scale. Problem 7.50. An RC clock circuit is needed with the oscillation frequency of 1 kHz and amplitude of the capacitor voltage of 4 V. Determine one possible set of circuit parameters R1, R2, R given that the capacitance of 100 nF is used. The power supply voltage of the ampliﬁer is 12 V. Assume that the ampliﬁer hits the power rails in saturation. Problem 7.51. A relaxation oscillator circuit may generate nearly triangular waveforms at the capacitor. Which values should the feed1 back factor β ¼ R1RþR attain to make it possible? 2
+ 
C)
VS C2 t=0
R1
R1 C1
+ 
D)
+ 
VS C2
R2
C1
R1
C2
R2
t=0
VS
Problem 7.53. Repeat the previous problem for the circuits shown in the following ﬁgure.
VII377
Chapter 7
Problems
B. Find the value of the undamped resonant frequency, ω0 (show units). C. Find the value of the damping ratio, ζ (show units). D. Find the particular solution (forced response). E. Outline the form of the complementary solution (natural response). F. Which value should the circuit resistance have for a critically damped circuit?
Given ﬁxed L and C, which values of resistance (large or small) lead to the overdamped circuit? Problem 7.93. For the series RLC circuit shown in the ﬁgure below, ﬁll out the table of circuit parameters. t=0 100
10V=VS t=0
5V=VS
+ 
1 mH
Problem 7.91. Repeat the previous problem for the series RLC circuit shown in the ﬁgure below.
5V=VS
1 mH C
Table of circuit parameters
1 nF
t=0
+ 
300
C, μF 0.01 0.1 0.4 1.0
ζ
Circuit type (overdamped, critically damped, underdamped)
300
+ 
10 mH 1 F
Problem 7.92. For the series RLC circuit shown in the following ﬁgure, ﬁll out the table of circuit parameters.
Given ﬁxed L and R, which values of capacitance (large or small) lead to the overdamped circuit? Problem 7.94. For the series RLC circuit shown in the following ﬁgure, ﬁll out the table of circuit parameters. t=0
t=0
R
10V=VS 10V=VS
100
+ 
+ 
L 1 nF
1 mH 1 F
Table of circuit parameters R, Ω 25 50 75 100
ζ
Circuit type (overdamped, critically damped, underdamped)
Table of circuit parameters L, μH 0.1 1 2.5 10
ζ
Circuit type (overdamped, critically damped, underdamped)
VII383
Chapter 7
Transient Circuit Fundamentals
Given ﬁxed R and C, which values of inductance (large or small) lead to the overdamped circuit?
Problem 7.98. Repeat the previous problem for the circuit shown in the following ﬁgure. t=0
Problem 7.95. Show that underdamped solution and critically damped solutions coincide with each other when ς ! 1.
10V=VS
1 mH
+ 
6 100 F
7.6.3 Finding Integration Constants 7.6.4 Solution Behavior for Different Damping Ratios Problem 7.99. In the circuit shown in the ﬁgure Problem 7.96. For the circuit shown in the ﬁgure below: t=0
below, the capacitor was charged to 10 V prior to closing the switch.
60 t=0
5V=VS
+ 
6
1 H 1 mH
1 nF 100 F
A. Determine damping coefﬁcient α, undamped resonant frequency ω0, and damping ratio ζ. B. Determine constants K1, K2. C. Write solution for the capacitor voltage with all constants deﬁned. D. Calculate and plot to scale capacitor voltage at 0, 0.05, 0.1, 0.2, and 0.3 μs. Problem 7.97. For the circuit shown in the following ﬁgure: A. Determine damping coefﬁcient α, undamped resonant frequency ω0, and damping ratio ζ. B. Determine constants K1, K2. C. Write the solution for the capacitor voltage with all constants deﬁned. D. Calculate capacitor voltage at 0, 1, 2, 3, 4, and 5 ms and plot it to scale versus time. t=0
1V=VS
100 F
+ 
10

vC +
A. How are the circuit equation and initial conditions different from Eqs. (7.78a, b)? B. Determine damping coefﬁcient α, undamped resonant frequency ω0, and damping ratio ζ. C. Determine constants K1, K2. D. Write the solution for capacitor voltage with all constants deﬁned. E. Calculate the capacitor voltage at 0, 1, 2, 3, 4, and 5 ms and plot it to scale versus time.
7.6.5 Overshoot and Rise Time 7.6.6 Application: Nonideal Digital Waveform
Problem 7.100. The following ﬁgure shows the underdamped step response for a series RLC circuit. The DC source has the voltage of 10 V. Using the ﬁgure: A. Estimate the overshoot percentage. B. Estimate the rise time. C. Do these estimates (approximately) agree with Eqs. (7.88a, b)?
1 mH
VII384
Chapter 7
Problems Problem 7.103. For the circuit shown in the following ﬁgure:
Capacitor voltage, V 16 14 12
a)
10
R
8
vS(t)
6
+ 
L C
4 2 0 0
0.5
1
1.5
2
2.5
3
3.5
4
vS(t)
4.5 t, ms
b)
Problem 7.101. Capacitor voltage is measured in a series RLC circuit as shown in the ﬁgure to the previous problem. Given R ¼ 2 Ω, estimate circuit inductance L and circuit capacitance C. Problem 7.102. The ﬁgure that follows shows the distorted rectangular waveform (capacitor voltage) for the circuit shown in Fig. 7.33. The DC source has the voltage of 10 V. A. Using the ﬁgure, estimate the overshoot and undershoot percentages. B. Using the ﬁgure, estimate the rise time and the fall time. C. Do these estimates (approximately) agree with Eqs. (7.88a, 7.88b)? 15
Capacitor voltage, V

vC +
VS
t 0
T
A. Determine the step response υC(t) for the circuit shown in ﬁgure (a) given that L ¼ 1 μH, C ¼ 1 nF; V S ¼ 10 V, and R ¼ 75 Ω. B. Express the solution υpulse C (t) for the voltage pulse shown in ﬁgure (b) in terms of the step response. C. Given T ¼ 0:5 μs, calculate the solution for the voltage pulse over the time interval from 0 to 0.7 μs in steps of 0.1 μs and plot it to scale. Problem 7.104. Repeat the previous problem assuming T ¼ 0:2 μs. Calculate the solution for the voltage pulse over the time interval from 0 to 0.5 μs in steps of 0.05 μs and plot it to scale versus time.
10 5 0 5 10 0
1
2
3
4
5
6
7
8
9
t, s
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Part III AC Circuits
Chapter 8
Chapter 8: SteadyState AC Circuit Fundamentals Overview Prerequisites:  Knowledge of DC circuit analysis (Chapters 2, 3, and 4)  Knowledge of dynamic circuit elements (Chapter 6, optionally Chapter 7)  Knowledge of complex arithmetic and calculus Objectives of Section 8.1:  Apply and work with the major parameters of the steadystate AC signals: amplitude, frequency, and phase  Establish the concept of phase leading or lagging for AC voltages and currents  Become familiar with the major function of the oscilloscope measure periodic (AC) voltages in a circuit  Understand the meaning of the phasor as a representation of the real signal  Convert real signals to phasors and vice versa  Perform basic operations with phasors  Be able to construct the phasor diagram for real signals and restore the real signals from the phasor diagram  Become familiar with the phasor (angle) notation Objectives of Section 8.2:  Provide the complete mathematical derivation of complex impedances  Apply the impedance concept to resistor, capacitor, and inductor  Understand the meaning of magnitude and phase of the complex impedance Objectives of Section 8.3:  Understand and apply the AC circuit analysis with phasors and impedances  Appreciate the value of the phasor diagram as a tool for AC circuit analysis  Transfer major circuit theorems to steadystate AC circuits  Be able to solve multifrequency AC circuits using superposition principle Application examples:  Measurements of amplitude, frequency, and phase  Impedance of a human body
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_8
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Chapter 8
SteadyState AC Circuit Fundamentals
Keywords: Steadystate AC circuit, Signals, Harmonic, Amplitude, PkPk value, Angular frequency, Phase, Period, Steadystate AC voltage, Steadystate alternating current, Leading signal, Lagging signal, Oscilloscope, Phasor, Phasor voltage, Phasor current, Phasor diagram, Phasor notation, Angle notation, Complex impedance, Impedance of the resistor, Impedance of the capacitor, Impedance of the inductor, Reactance, KVL in phasor form, KCL in phasor form, Source transformation in the frequency domain, Thévenin’s theorem for steadystate AC circuits, Norton theorem for steadystate AC circuits, Superposition principle for multifrequency AC circuits
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Chapter 8
Section 8.1: Harmonic Voltage and Current: Phasor
Section 8.1 Harmonic Voltage and Current: Phasor Whether you use a kitchen appliance or operate a piece of industrial machinery, they are powered by alternating current (AC) power supplies. It may seem odd at ﬁrst to create power sources with alternating polarities that result in ﬂuctuating voltage and current directions through resistors, inductors, and capacitors. Consider the following situation: current ﬂows through a coil from left to right with increasing and then decreasing magnitude, then changing direction, and again increasing and decreasing in magnitude before the process repeats itself. A beneﬁt of using periodic, sinusoidal waveforms is related to the relative ease of stepping up or down AC voltages and currents with little, ideally no, power losses via a transformer. The economic importance of AC circuits can hardly be overstated; the bulk of the residential and industrial power demand is produced, transformed, and distributed via AC circuits. To understand the basics of AC circuit analysis, we start with the key characteristics of sinusoidal waveforms such as frequency, phase, and amplitude. Special attention is paid to the phase. We deﬁne the meaning of leading and lagging phase for two steadystate AC voltages or currents. We note that in electrical engineering, AC voltages are often called signals.
8.1.1 Harmonic Voltages and Currents In steadystate AC circuits, all voltages and currents measured across or through the elements are periodic and in the ideal case harmonic (i.e., sine or cosine) functions of time. These voltages and currents have the same frequency but different phases and amplitudes. Interestingly, the word harmonic originates from the reference to music sounds of pure, single tones, pitches, or frequencies. The word steadystate means that the circuit frequency, phases of all voltages and currents, and amplitudes of all voltages and currents do not change over time. Transient effects are entirely excluded from our consideration. Similarly, the AC voltage and currents are called steadystate AC voltage and steadystate alternating current. It is common to use terms AC voltage and AC power, but the term AC current does not make much sense, even though it might be a part of the electrical engineering jargon. Figure 8.1 shows an AC harmonic signal. All voltages and currents in an AC circuit will have exactly this form, regardless of whether they are measured over a resistor, capacitor, or inductor. For this ﬁgure we consider the voltage υ(t). The current i(t) could be treated in an identical manner. As a harmonic function, the steadystate AC voltage can be written in the form υ ðt Þ
V m cos ðω t þ φÞ
ð8:1aÞ
where Vm is the voltage amplitude (maximum absolute voltage), with the unit of volts. ω is the angular frequency, with the unit of rad/s. φ is the phase, with the unit of radians. VIII391
Chapter 8
SteadyState AC Circuit Fundamentals
8.1.3 Application Example: Measurements of Amplitude, Frequency, and Phase Different parameters of the steadystate AC voltage or of a general periodic voltage studied in the section are readily measured with the oscilloscope. The oscilloscope always measures the timeamplitude response of the voltage, not the current. Parameters include frequency, amplitude (or peaktopeak value), the rms voltage, the mean voltage (voltage averaged over a period), etc. Figure 8.4 provides a practical measurement example. Several voltage signals can be measured simultaneously with several oscilloscope channels. The phase is not measured for a single signal, only the phase difference between two and more signals can be measured. The oscilloscope has two commonly used settings. In the ﬁrst setting called DC coupled, any voltage supplied to the channel will cause a deﬂection of the trace. As a result, the actual value of the input voltage with respect to oscilloscope ground can be measured. In the second setting called AC Coupled, any DC offset to the periodic signal is eliminated via an internal coupling capacitor. The result, after a short transient period, is that the trace will settle at 0 V regardless of the magnitude of an applied DC voltage. This conﬁguration is used whenever it is desired to ignore the constant DC component of an applied voltage waveform and only observe the AC component of the waveform with zero mean.
Fig. 8.4. Front panel of an inexpensive dualchannel digitalstorage oscilloscope. Note the measured signal parameters.
8.1.4 Deﬁnition of a Phasor This topic is critical for the steadystate AC circuit analysis. We are about to introduce the method of solving AC circuits based on the use of socalled phasors or complex numbers. Working with phasors allows us to “cancel” out the frequency dependence and the time dependence. This is possible because in a linear system the harmonic behavior is the same for all circuit components, we only need to keep the amplitude and phase information for every sinusoid. The application of phasors will eventually allow us to reduce the AC circuit to an equivalent “DC” circuit that is solved using standard tools. However, there is no “free lunch”; the voltages and currents in the resulting “DC” circuit appear to be VIII396
Chapter 8
Section 8.1: Harmonic Voltage and Current: Phasor
complex numbers. They carry information of two parameters: the phase and the amplitude of a sinusoid. The use of a complex number relies on two independent parameters, real and imaginary parts (or magnitude and phase); it is ideally suited to represent AC signals while entirely eliminating timedomain harmonics. The starting point of the phasor concept is rooted in Euler’s formula in the form e jα
cos α þ j sin α
ð8:3aÞ
The identity expresses the complex exponent in terms of the realvalued cosine and sine p functions. Here, j 1 is the imaginary unit and α is an arbitrary, real number. Note p that α can be equal to ωt or to ω t þ φ. In electrical engineering, the symbol j 1 is preferred over the mathematical symbol i which may be confused with the electric current. In terms of real and imaginary parts, we obtain ð8:3bÞ cos α Re e jα , sin α Im e jα Why do we need the complex exponent instead of cosine and sine? To answer this question, let us study the following identity: e jðωtþφÞ
e jωt e jφ
ð8:3cÞ
We will write the current or the voltage in the form of Eq. (8.3c) and use the multiplicative property of the exponent. Then, the factor ejωt can be exactly canceled out in every term in the underlying differential equation. This is a major simpliﬁcation, because the ODE becomes an algebraic equation. The key is the function Re() or the real part of a complex number: υðt Þ V m cos ðωt þ φÞ Re V m e jðωtþφÞ ð8:4Þ iðt Þ I m cos ðωt þ ψ Þ Re I m e jðωtþψ Þ The remaining complex number available after cancellation of the time factor ejωt is called a phasor: the phasor voltage V and the phasor current I: V
V m e jφ ,
I
I m e jψ
ð8:5Þ
Equation (8.5) as a deﬁnition tells us that the phasor is a complex number comprised of two parameters: amplitude and phase. For related operations with complex numbers, you can see the chapter summary. You should notice that frequency is no longer present since it remains the same for all circuit elements, and it is equal to the known frequency of the voltage power supply. The phasor has the same units as the original quantity: the phasor voltage has units of volts, and the phasor current has units of amperes. The theory and
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Chapter 8
Section 8.2: Impedance
Section 8.2 Impedance 8.2.1 The Concept of Impedance We can avoid solving ODEs for the AC circuits entirely if we establish a relation between the phasor voltage and the phasor current for the inductor and capacitor, similar to Ohm’s law for the resistor. We consider three basic AC circuit elements shown in Fig. 8.9 in a passive reference conﬁguration and determine their phasor voltages and phasor currents. resistance
+
v(t)
inductance

+
i(t)
v(t)

capacitance
+
v(t)
i(t)
i(t)
Fig. 8.9. Three AC circuit elements.
Resistance For the resistance, voltage will be given by a cosine function shown in Fig. 8.10a. One has υ
Ri,
υ ðt Þ
V m e jφ
V
V m cos ðωt þ φÞ ) iðt Þ V m jφ e I R
Vm cos ðω t þ φÞ R
ð8:15Þ
Capacitance For the capacitance, voltage will again be given by a cosine function shown in Fig. 8.10b. One has dυ , υðt Þ V m cos ðωt þ φÞ dt ) i ðt Þ ω CV m sin ðωt þ φÞ ωCV m cos ðωt þ φ þ π=2Þ V V m e jφ I ωCV m e jφþjπ=2 i
C
ð8:16Þ
Inductance For the inductance, current will be given by a cosine function shown in Fig. 8.10c. One has di , dt ) υ ðt Þ
υ I
L
I m e jφ
iðt Þ
I m cos ðωt þ φÞ
ωLI m sin ðωt þ φÞ V
ωLI m cos ðωt þ φ þ π=2Þ
ð8:17Þ
ωLI m ejφþjπ=2
Impedance Z of an arbitrary linear circuit element or of an arbitrary oneport network comprised of such elements is deﬁned by Z
V I
ð8:18Þ
VIII405
Chapter 8
SteadyState AC Circuit Fundamentals
Thus, the phasors for voltage and current are linked via a constant, which can be real or complex. In order to emphasize that this constant is not exactly a resistance, the constant is called the impedance. Substituting Eqs. (8.15), (8.16), and (8.17) into Eq. (8.18), we obtain: V I V Capacitance : ZC I V Inductance : ZL I Resistance :
ZR
R ½Ω e jπ=2 ω C
1
1 ∠ ωC
e jπ=2 ωL
ω L∠90
90
1 jωC
½Ω
ð8:19Þ
jωL ½Ω
Voltage(V)  dashed curve and current(A)  solid curve 2
a) resistance
1
0
ZR
R 0
voltage and current are in phase
1
2 2
b) capacitance
1
0
1 C
ZC
90
1
voltage lags current by 90 deg 2 2
c)
1
inductance 0
ZL
L 90
voltage leads current by 90 deg 1 2 0
2
3
4
t
Fig. 8.10. Voltage and current sinusoids for resistor, inductor, and capacitor (at phase zero).
VIII406
Chapter 8
SteadyState AC Circuit Fundamentals
Note that the magnitudes of complex impedances in Fig. 8.12 are all equal to 2 Ω: jZR j
jZR j
jZR j
2 Ω
ð8:23Þ
8.2.4 Application Example: Impedance of a Human Body The impedance is not only the characteristic of an electric circuit but also of an arbitrary conducting object, which may be modeled as a combination of ideal resistance, inductance, and capacitance. They can be connected in series, in parallel, or a combination of series and parallel. As an example, we consider here the impedance of a human body in the frequency range from 10 kHz to 3 MHz. The impedance magnitude is shown in Fig. 8.13. It was measured for about 400 human subjects of ages between 18 and 70 years and then averaged. The subject stood on a large aluminum sheet as a ground plane. The electrode was a cylindrical brass rod for making a grasping contact with the hand. In its simplest form, the concept of impedance measurement implies the simultaneous measurements of harmonic voltage and current and then the extraction of the amplitude ratio (magnitude of the impedance) and the phase difference (impedance phase). Impedance magnitude, 660
Tenyearold child
600 540 480 420
Fema e
360
Male 300 1
10
2
3
10 10 frequency, kHz
4
10
Fig. 8.13. Magnitude of the average impedance of the human body. From: I. Chatterjee et al., “Human Body Impedance and Threshold Currents for Perception and Pain for Contact Hazard Analysis in the VLFMF Band,” IEEE Trans Biomedical Eng., May 1986.
Figure 8.13 indicates that the impedance magnitude decreases with frequency. Therefore, the human body impedance, at least at relatively low frequencies, behaves similarly to the impedance of a capacitor, where the magnitude also decreases with frequency. A purely resistive component is also present. The impedance measurements have been used for the extraction of various biomedical data such as assessment of a fatfree mass.
VIII410
Chapter 8
SteadyState AC Circuit Fundamentals
the impedances may be combined as if they were simple resistors. The series/parallel equivalents for the impedances are equally applicable. After plugging in numbers for ω, R, C, VS, the phasor current and phasor voltages can be found and converted back to time domain. This is the method of solving in AC circuits.
8.3.2 Complete Solution for an AC Circuit: KVL and KCL on Phasor Diagram Let us assume that the power supply voltage in Fig. 8.14a has the form υS ðt Þ V m cos ωt, and its amplitude and frequency are given by V m 5 V, ω 1000 rad=s. Further we know that C 1 μF, R 1 kΩ in Fig. 8.14a. The solution to this AC circuit includes several steps as discussed above. First, we convert the circuit to the phasor/impedance form as shown in Fig. 8.14b. All currents/voltages are replaced by their phasors, and all circuit elements are replaced by their impedances. Next, we solve the phasor circuit in Fig. 8.14b as if it were a “DC” circuit. The element impedances are ZR
1000 ½Ω,
ZC
1 jω C
1 j1000 1 10
j1000 ½Ω
6
ð8:27Þ
Note that the impedances here are written in rectangular instead of polar form. We must now ﬁnd the equivalent impedance. From the series impedance combination: p Zeq ZR þ ZC 1000 j1000 10002 þ 10002 ∠ arctanð1Þ 1414∠ 45 ½Ω ð8:28Þ The phasor current (circuit current) has the form I
VS Zeq
5 1414∠
45
3:54∠45 ½mA
ð8:29Þ
The phasor voltages across the resistor and the capacitor are found according to Ohm’s law, that is: VR
ZR I
VC
ZC I
1000 0:00354∠45 j1000 0:00354∠45
3:54∠45 3:54∠
½V
90 ∠45
ð8:30Þ 3:54∠
45 ½V
ð8:31Þ
The phasor voltages are plotted in the phasor diagram as depicted in Fig. 8.15.
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Chapter 8
Summary
Table 1. Some basic operations with complex numbers. e j0 ¼ 1, e jπ=2 ¼ j, e jπ=2 ¼ j, e jπ ¼ 1, e jπ ¼ 1, je jα j ¼ 1 j ¼ ∠90 , 1j ¼ ∠ 90 , j2 ¼ 1, 1j ¼ j, jjj ¼ 1 V m e jφ ¼ V m ∠φ ¼ V m ð cos φ þ j sin φÞ, V m e jφ ¼ V m ∠ ϕ ¼ V m ð cos φ j sin φÞ jV m e jφ j ¼ jV m e jφ j ¼ V m p x þ jy ¼ V m e jφ , V m ¼ x2 þ y2 , φ ¼ arctan yx , x 0 y x ¼ x2 þy 2 jx2 þy2 p V m ¼ x2 þ y2 , φ ¼ arctan yx , 1 xþjy
1 xþjy
¼ V1m e
jφ
,
ðx þ jyÞ* ¼ x jy,
*
ðV m e jφ Þ ¼ V m e
jφ
,
x0
ðV m ∠φÞ* ¼ V m ∠ φ
V m ∠φ I m ∠ψ ¼ ðV m e jφ ÞðI m e jψ Þ ¼ V m I m e jðφþψ Þ ¼ V m I m ∠ðφ þ ψ Þ V m ∠φ I m ∠ψ
¼ VI mmee jψ ¼ VI mm e jðφ jφ
ψÞ
¼ VI mm ∠ðφ ψ Þ
Table 2. Selected trigonometric identities. sin α ¼ cos ðα π=2Þ, sin α ¼ cos ðα þ π=2Þ, cos 2α ¼ 2 cos 2 α 1, sin 2α ¼ 2 sin α cos α cos ðα þ βÞ ¼ cos α cos β sin α sin β cos α cos β ¼ 0:5ð cos ðα þ βÞ þ cos ðα βÞÞ sin ðα þ βÞ ¼ sin α cos β þ cos α sin β sin α sin β ¼ 0:5ð cos ðα βÞ cos ðα þ βÞÞ sin α cos β ¼ 0:5ð sin ðα þ βÞ þ sin ðα βÞÞ αþβ α β sin α þ sin β ¼ 2 sin αþβ cos α 2 β cos α þ cos β ¼ 2 cos 2 cos 2 2 arctanφ ¼ π2 arctanφ1 , φ > 0 arctanφ ¼ π2 arctanφ1 , φ < 0 q C 1 cos ωt þ C 2 sin ωt ¼ C 21 þ C 22 cos ðωt þ φÞ, φ ¼ arctan CC 21 C1, C2 > 0 C 1 cos ðω t þ φÞ C 2 cos ðωt þ ψ Þ ¼ 0:5C 1 C 2 ð cos ðφ ψ Þ þ cos ð2ωt þ φ þ ψ ÞÞ C 1 cos qðωt þ φÞ þ C 2 cos ðω t þ ψ Þ ¼ C cos ðωt þ φÞ C¼
C 21 þ C 22 þ 2C 1 C 2 cos ðφ ψ Þ
C 2 sin ðφ ψ Þ 0 C 1 þ C 2 cos ðφ ψ Þ > 0 ϕ ¼ φ arctan þ π C 1 þ C 2 cos ðφ ψ Þ < 0 C 1 þ C 2 cos ðφ ψ Þ
VIII421
Chapter 8
SteadyState AC Circuit Fundamentals
0.8
8.1 Harmonic Voltage and Current: Phasor
0.4
8.1.1 Harmonic Voltages and Currents 8.1.2 Phase: Leading and Lagging Problem 8.1 A. Write a general expression for the AC harmonic voltage signal (steadystate AC voltage) using the cosine function. B. Identify amplitude, angular frequency, and phase. C. Write relations between the angular frequency, frequency, and the period. Problem 8.2 A. Determine frequency in Hz, angular frequency in rad/s, and amplitude of the harmonic voltage signal shown in the ﬁgure below (show units for every quantity). B. Write the AC voltage in the form of a cosine function with the corresponding amplitude, frequency, and phase. voltage, V
voltage V
Problems
0
0.4
0.8 0
1
0
1
2 30
45 time, s
1.5 time, ms
2
2.5
3
Problem 8.4. Repeat problem 8.2 for a harmonic voltage signal with a DC offset shown in the ﬁgure below. voltage, V 10.0
5.0
0
5.0
0
15
1
10.0
2
0
0.5
60
75
90
5
10
15 time, s
20
25
30
Problem 8.5 A. Determine frequency in Hz, angular frequency in rad/s, amplitude, and phase (versus the base cos ωt signal) of the harmonic voltage shown in the ﬁgure below (show units for every quantity). B. Write the AC voltage in the form of a cosine function with the corresponding amplitude, frequency, and phase.
Problem 8.3. Repeat problem 8.2 for the voltage signal shown in the ﬁgure below.
VIII422
Chapter 8
Problems voltage, V
voltage, V
3 2
2 1
1
0
0
1
1 2
2
3 0
15
30
45 time, s
60
75
90
Problem 8.6. Repeat problem 8.5 for the voltage signal shown in the ﬁgure below. voltage, V 2
0
5
10
15
30
35
40
Problem 8.9. An AC voltage in a circuit is given by υðt Þ ¼ 10 cos ð2π50tÞ ½V. Using software of your choice, plot the voltage to scale over the time interval of two periods, i.e., 0 t 2T . Label the axes. Problem 8.10. An AC voltage in a circuit is given by the voltage expression υðt Þ ¼ 10 cos ð1000t π=3Þ ½V. Using software of your choice, plot the voltage to scale over the time interval of four periods, i.e., 0 t 4T . Label the axes.
1
0
1
2 0
15
30
45 time, s
60
75
90
Problem 8.7. Repeat problem 8.5 for the voltage signal shown in the ﬁgure below.
Problem 8.11. The reference voltage is shown by a solid curve in the ﬁgure; the AC voltage under study is shown by a dashed curve. Determine if the AC voltage under study leads or lags the reference voltage, and, if so, by how many degrees. voltage, V
voltage, V 3
3
2
2
1
1
0
0
1
1
2
2
3
20 25 time, ms
0
5
10
15
20 25 time, ms
30
35
40
3
0
5
10
15
20 25 time, ms
30
35
40
Problem 8.8. Repeat problem 8.5 for the voltage signal shown in the ﬁgure below.
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Chapter 8
SteadyState AC Circuit Fundamentals
Problem 8.12. Repeat problem 8.11 for the voltage signal shown in the ﬁgure below. voltage, V 3 2 1 0 1 2 3
0
5
10
15 20 25 time, ms
30
35
40
Problem 8.13. Repeat problem 8.11 for the voltage signal shown in the ﬁgure below. voltage, V
in V, and phase in radians (versus the base cosine signal) of the voltage signal in the form υðt Þ ¼ 5 sin ð100 t þ 225 Þ ½V. Hint: Convert the signal to the base cosine form ﬁrst. Problem 8.17. The AC voltage is given by a combination of two sinusoids: A. υðt Þ ¼ 1sin ðω t þ π=2Þ 2sin ðωt π=2Þ B. υðt Þ¼ 1 sin ðωt þ π=2Þ 2 sin ðωt π=3Þ Convert this voltage to the basic cosine form υðt Þ ¼ V m cos ðωt þ φÞ and determine the amplitude and the phase (versus the base cosine signal). Hint: Trigonometric identities may be found in the summary to this chapter.
8.1.4 Deﬁnition of a Phasor 8.1.5 From Real Signals to Phasors 8.1.6 From Phasors to Real Signals Problem 8.18. Determine the phasors for the realvalued AC voltages and currents. Show units. Express all phase angles in radians.
3 2 1
υ ðt Þ υ ðt Þ i ðt Þ i ðt Þ
0 1
10 cos ðω t þ π=3Þ ½V 3 cos ðωt 30 Þ ½V 12 cos ðω t þ π=6Þ ½A 1 cos ðω t π=2Þ ½A
2 3
0
5
10
15
20 25 time, s
30
35
40
Problem 8.14. Determine the frequency in Hz, period in s, amplitude in V, and phase in degrees (versus the base cosine signal) of the voltage signal in the form υðt Þ ¼ 15 sin ð100 t þ 45 Þ ½V. Hint: Convert the signal to the base cosine form ﬁrst. Problem 8.15. Determine the frequency in Hz, period in s, amplitude in V, and phase in radians (versus the base cosine signal) of the voltage signal in the form υ ðt Þ ¼ 15 sin ð1000 t 35 Þ ½V. Hint: Convert the signal to the base cosine form ﬁrst.
Problem 8.19. Determine the phasors for the realvalued AC voltages and currents. Use the shorthand notation ∠ for the complex exponent. Show units. Express all phase angles in degrees.
υðt Þ υðt Þ iðt Þ iðt Þ
10 sin ðω t þ π=3Þ ½V 3 cos ð100t 30 Þ ½V 12 sin ðωt þ π=6Þ ½A cos ðω t Þ þ sin ðωt Þ ½A
Problem 8.20. The phasors of the AC voltage and current are given by
V V I I
5∠π=3 ½V 3∠π ½V 2:1∠45 ½A 1∠ 180 ½A
Problem 8.16. Determine the frequency in Hz, period in s, amplitude in V, peaktopeak value
VIII424
Chapter 8
Problems
The AC source has the angular frequency ω. Restore the corresponding realvalued voltages and currents. Show units; express all phase angles in radians. Problem 8.21. The phasors of the AC voltage and current are given by
V V I I
10∠π=2 ½V 15∠ π=3 ½V 20∠ 16 ½A j∠45 ½A
The AC source has the angular frequency ω. Restore the corresponding realvalued voltages and currents as functions of time. Show units. Express all phase angles in radians. Problem 8.22. The phasors of the AC voltage and current are given by
V V I I
1 ½V ∠π þ ∠ π ½V 2∠45 ½A ∠45 þ ∠ 45 ½A
6
Im
4 2 6
4
2
0
2
4
Re 6
2 4 6
Problem 8.24. Repeat problem 8.23 for V ¼ 4expðjπ=6Þ, but convert this number to the rectangular form. Problem 8.25. Repeat problem 8.23 for V ¼ 25=ð3 þ j4Þ. Problem 8.26. Phasors of three AC voltage signals are shown in Fig. 8.8. Every division in the ﬁgure corresponds to 1 V. The AC source has the angular frequency ω. Restore the corresponding realvalued voltages in time domain. Show units. Express all phase angles in radians.
Problem 8.27. The phasors of the AC voltage The AC source has the angular frequency ω. and current are given in the rectangular form: Restore the corresponding realvalued voltages and currents. Show units; express all phase V 3 þ j2 ½V angles in degrees. I 2 þ j3 ½A The AC source has the angular frequency ω.
8.1.7 Polar and Rectangular Forms: Restore the corresponding realvalued voltages Phasor Magnitude and currents. Show units, express all phase angles 8.1.8 Operations with Phasors and Phasor in radians. Diagram Problem 8.23. A complex number V is given by V ¼ 4 þ j 2. A. Convert it into polar form; express the phase angle in degrees. B. Plot the number on the phasor diagram. C. If this number is a phasor of a voltage signal with the units of volts, what is the voltage signal in time domain? Assume the angular frequency ω.
Problem 8.28*. Solve the previous problem using MATLAB. Present the corresponding MATLAB script. Problem 8.29. The phasors of the AC voltage and current are given in the rectangular form:
V V V
ð3 þ j2Þ2 ½V ð 2 þ j3Þð7 þ jÞ ½V 1 ½V 0:2 þ j0:1 VIII425
Chapter 8
SteadyState AC Circuit Fundamentals
three unknown circuit elements are shown in the ﬁgure below. a)
Problem 8.45. Voltages (dashed curves) and currents (solid curves) for three unknown circuit elements are shown in the ﬁgure below.
=1000 rad/sec
2
a)
0.5 0 0.5 1 1.5
0
2
3
4
t
2
0.5 1
0
2
3
4
t
4
t
4
t
=100 rad/sec
2 1.5
1
voltage(V) or current(A)
voltage(V) or current(A)
0
2
b)
0.5 0 0.5 1 1.5
1 0.5 0 0.5 1 1.5
2
0
2
3
4
t
2
=10000 rad/sec
2
c)
1.5
0
2
3
=10000 rad/sec 2 1.5
1
voltage(V) or current(A)
voltage(V) or current(A)
0.5
=100 rad/sec
1.5
c)
1
1.5
2
b)
=1000 rad/sec
2 1.5
1
voltage(V) or current(A)
voltage(V) or current(A)
1.5
0.5 0 0.5 1 1.5
1 0.5 0 0.5 1 1.5
2
0
2
3
4
t
Determine: A. The type of the element (resistor, capacitor, or inductor) B. The value of the corresponding resistance, inductance, or capacitance Note that the angular frequency is different in every case.
2
0
2
3
Determine: A. The type of the element (resistor, capacitor, or inductor) B. The value of the corresponding resistance, inductance, or capacitance
VIII428
Chapter 8
Problems
Problem 8.46. Phasor voltages and currents for three unknown circuit elements are shown in the ﬁgure below. Determine the type of the element (R, L, or C) and the value of R, L, or C. a)
a)
1000 rad/sec 3 V 2
1 Re (V or A) I
1000 rad/sec 3
V
Im (V or A)
0
1
2
Im (V or A)
2 1
I
0
1
Re (V or A)
b)
2
=3000 rad/sec 3
I
Im (V or A)
2 1 Re (V or A)
b)
0 =3000 rad/sec 3
Im (V or A)
V
1
2
V
2 1
I Re (V or A)
0
1
2
c)
=20000 rad/sec 3
Im (V or A)
2 1 Re (V or A)
c)
I =20000 rad/sec 3
0
1
2 V
Im (V or A)
2 1 Re (V or A) 0
1
2 I
V
Problem 8.47. Phasor voltages and phasor currents for three unknown circuit elements are shown in the ﬁgure below. Determine the type of the element (R, L, or C) and the value of R, L, or C when appropriate.
Problem 8.48*. The following MATLAB script plots the realvalued signals in time domain corresponding to the phasor voltage V ¼ 5∠30 ½V and to the phasor current I ¼ 2∠ 60 ½A for an inductor. clear all f ¼ 2e6; % frequency, Hz T ¼ 1/f; % period, sec dt ¼ T/100; % sampling int. t ¼ [0:dt:2.5*T]; % time vector vL ¼ 5*cos(2*pi*f*t+pi/6); % voltage iL ¼ 2*cos(2*pi*f*tpi/3); % current t ¼ t/T; % time in periods
VIII429
Chapter 8
SteadyState AC Circuit Fundamentals Problem 8.51. Find Zeq in the polar form for the circuit element combination shown in the ﬁgure below when ω ¼ 100, 000 rad=s, C ¼ 100 nF, L ¼ 1 mH, R ¼ 100 Ω.
plot(t, vL, ’b’); grid on; hold on; plot(t, iL, ’r’); xlabel(’t/T’); ylabel(’voltage/current’)
Modify the script and plot the realvalued voltages and currents corresponding to the phasors shown in the phasor diagrams for Problem 8.46. Problem 8.49. Repeat the previous problem for the phasors shown in the phasor diagrams for Problem 8.47.
8.3 Principles of AC Circuit Analysis 8.3.1 AC Circuit Analysis: KVL, KCL, and Equivalent Impedances 8.3.2 Complete Solution for an AC Circuit: KVL and KCL on Phasor Diagram Problem 8.50. For the AC circuit element combinations shown in the ﬁgure that follows, a)
b)
R
Z eq
C
c)
L
R
Z eq
L
C
A. Find the equivalent impedance Zeq in polar form given that ω ¼ 10000 rad=s, C ¼ 0:1 μF, L ¼ 100 mH, R ¼ 1 kΩ. B. Plot the result for the partial impedances and for Zeq on the corresponding phasor diagram.
R
C
L
Problem 8.52. For three circuit element combinations shown in the ﬁgure below, ﬁnd Zeq given that ω ¼ 2000 rad=s, C ¼ 5 μF, L ¼ 50 mH, R ¼ 1 kΩ. a)
Z eq
R
C
L
C
L
b)
R
Z eq
C
c)
Z eq
R
Z eq
Z eq
R
L
Problem 8.53. A complex impedance of any circuit may be written in the form Z ¼ R þ jX where R is called the resistance and X is called the electrical reactance or simply the reactance. An engineer measures a reactance of 2 Ω over an inductor at 60 Hz. What is the inductance? Problem 8.54. The same engineer measures a reactance of 1 kΩ over a capacitor at 60 Hz. What is the capacitance? Problem 8.55. For the circuit shown in the ﬁgure below, υS ðtÞ ¼ 10 cos ωt ½V, ω ¼ 10, 000 rad=s, C ¼ 1 μF, R ¼ 100 Ω: A. Find phasor current I and phasor voltages, VR, VC, and construct the voltage phasor diagram for phasors VR, VC, VS.
VIII430
Chapter 9
Chapter 9: Filter Circuits: Frequency Response, Bode Plots, and Fourier Transform Overview Prerequisites:  Knowledge  Knowledge  Knowledge  Knowledge  Knowledge
of of of of of
complex arithmetic superposition principle for linear circuits (Chapter 3) harmonic voltage and current behavior (Chapter 8) phasor/impedance method for AC circuit analysis (Chapter 8) an operational amplifier with negative feedback (Chapter 5)
Objectives of Section 9.1:  Establish the concept of a firstorder analog filter as a twoport network  Understand the difference between highpass and lowpass filters  Understand the effect of filter termination  Become familiar with the fundamental filter characteristics including transfer function, break frequency, rolloff, and high/lowfrequency asymptotes  Understand the construction of the Bode plot including decibels; become familiar with some of the jargon used by electrical engineers  Establish the close agreement between firstorder RC and RL filters; become familiar with the concept of cascaded filter networks Objectives of Section 9.2:  Establish the model for the openloop gain of an operational amplifier as a function of frequency  Understand the meaning of datasheet parameters such as unitygain bandwidth and gainbandwidth product  Establish the model for the closedloop gain of an operational amplifier as a function of frequency from first principles  Find the frequency bandwidth for any practical operational amplifier circuit using the datasheet
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_9
IX433
Chapter 9
Filter Circuits: Frequency Response, Bode Plots. . .
Objectives of Section 9.3:  Obtain an introductory exposure to the continuous Fourier transform and be able to compute the transform for simple examples including the meaning of a sinc function  Be able to relate continuous and discrete Fourier transform via the Riemann sum approximation  Be able to define sampling points of the DFT in both time and frequency domain  Understand the structure and ordering of the DFT frequency spectrum including its relation to negative frequencies  Apply the DFT to a filter with a given transfer function and generate the discrete frequency spectrum of the output signal  Apply the DFT to filter operation with input pulse or nonperiodic signals  Apply the DFT (FFT and IFFT) in MATLAB Application examples:  Effect of a load connected to the filter  Effect of nextstage filter load  Finding bandwidth of an amplifier circuit using the datasheet  Selection of an amplifier IC for proper frequency bandwidth  Numerical differentiation via the FFT  Filter operation for an input pulse signal  Converting computational electromagnetic solution from frequency domain to time domain
Keywords: Analog ﬁlter, RC ﬁlter, RL ﬁlter, Port, Twoport network, Firstorder highpass ﬁlter, Firstorder lowpass ﬁlter, Filter termination, Amplitude transfer function, Phase transfer function, Power transfer function, Complex transfer function, Frequency response, Break frequency, Halfpower frequency, 3dB frequency, Corner frequency, Bode plot, Decibel, Rolloff, Highfrequency asymptote, Lowfrequency asymptote, Frequency band, Passband, Stopband, Decade, Octave, Power gain, Openloop ampliﬁer gain, Unitygain bandwidth, Gainbandwidth product, Internal compensation, Openloop AC gain, Closedloop AC gain, Ampliﬁer circuit bandwidth, Fourier transform continuous (direct inverse Fourier spectrum, direct inverse Fourier spectrum, bandlimited spectrum, reversal property, sinc function, mathematical properties, amplitudemodulated signal, Parseval’s theorem, energy spectral density), Fourier transform discrete (Fast digital signal processing (DSP), sampling points, sampling interval, sampling frequency, sampling theorem, Riemann sum approximation, rectangle rule, fundamental frequency, direct (DFT), inverse (IDFT), standard form, reversal property, structure of discrete spectrum, numerical differentiation, ﬁlter operation for pulse signals)
IX434
Filter Circuits: Frequency Response, Bode Plots. . .
Chapter 9
General Solution Let us ﬁrst convert the circuit in Fig. 9.1a to a phasor form as shown in Fig. 9.1b. We assume that υS ðt Þ V m cos ω t; therefore VS V m . Next, we solve the resulting “DC circuit” in the complex domain. The voltage division yields 1 jωC
VC
ZC Vm ZR þ ZC
Rþ
Vm
1 Vm 1 þ jωRC
1 V m ½V 1 þ jω τ
ð9:1aÞ
VR
ZR Vm ZR þ ZC
R Vm 1 R þ jωC
jωRC Vm 1 þ jωRC
jωτ V m ½V 1 þ jω τ
ð9:1bÞ
1 jωC
where τ RC is exactly the same time constant that appears for transient circuits in Chapter 7. Converting Eq. (9.1a) and (9.1b) into polar form gives VC VR
1 q V m ∠φC , 2 1 þ ðωτÞ ωτ q V m ∠φR , 1 þ ðωτÞ2
φC φR
tan π 2
1
tan
ðωτÞ 1
ðω τÞ
ð9:1cÞ ð9:1dÞ
After the polar form has been obtained, the realvalued voltages are found in the form υC ðt Þ
V mC cos ðωt þ φC Þ ½V,
V mC
υR ðt Þ
V mR cos ðωt þ φR Þ ½V,
V mR
1 q V m ½V 2 1 þ ðωτÞ ωτ q V m ½V 1 þ ðωτÞ2
ð9:1eÞ
The general solution of the RC circuit in Fig. 9.1a is now complete. The key observations are that the amplitudes of the resistor voltage and the capacitor voltage now become functions of frequency.
Qualitative Analysis The circuit in Fig. 9.1 is a voltage divider. The supply voltage (or the input voltage to the ﬁlter) is divided between the capacitor and the resistor. Which voltage dominates at low frequencies and which at high frequencies? To answer those questions, we consider Eq. (9.1e). When ω ! 0,
IX436
Chapter 9
V mC V mR
Section 9.1: FirstOrder Filter Circuits and Their Combinations
1 q Vm ! Vm 1 þ ðωτÞ2 ωτ q Vm ! 0 1 þ ðωτÞ2
ð9:2aÞ
Therefore, at low frequencies, the capacitor voltage dominates; it is approximately equal to the supply voltage. This fact is quite clear because the capacitor acts like an open circuit for DC, implying that the capacitor voltage “sees” nearly all the supply voltage. On the other hand, when ω ! 1, V mC V mR
1 q Vm ! 0 2 1 þ ðωτÞ ωτ q Vm ! Vm 2 1 þ ðωτÞ
ð9:2bÞ
Therefore, at high frequencies, the resistor voltage dominates; it approximately equals the supply voltage. This fact is also easy to understand because the capacitor acts like a short circuit for a highfrequency AC, jZC j 1=ðω C Þ ! 0, so that the capacitor voltage is nearly zero and all the supply voltage is “seen” by the resistor.
Filter Concept: TwoPort Network Now, we will explore the concept of an analog lowpass RC ﬁlter. We consider the power supply AC voltage as the input voltage υin(t) into the ﬁlter. We consider the capacitor voltage as the output voltage υout(t) of the ﬁlter. According to Eq. (9.2a, 9.2b), υout ðt Þ υin ðt Þ υout ðt Þ 0
at low frequencies at high frequencies
The circuit so constructed passes voltage signals with lower frequencies (like the human voice) but cuts out voltage signals with higher frequencies (like noise). Figure 9.2 on the left depicts the corresponding circuit transformation. This transformation implies that the input voltage is acquired from another circuit block and the output voltage is passed to another circuit block. The qualitative ﬁlter description is complete. You should note that both circuits on the right of Fig. 9.2 are called twoport networks. A port is nothing else but a pair of voltage terminals, either related to the input voltage or to the output voltage, respectively. Can we construct an RC ﬁlter that passes high frequencies but cuts out low frequencies? In other words, can we create a socalled highpass ﬁlter? The solution is simple and elegant; the output voltage is now the resistor voltage, not the capacitor voltage. Figure 9.2b shows the corresponding circuit transformation.
IX437
Chapter 9
Filter Circuits: Frequency Response, Bode Plots. . .
should be much greater than the ﬁlter’s resistance R. Put in approximate mathematical terms: R=RL 1. The lowresistance load (e.g., a loudspeaker) would simply short out the capacitor! To avoid this effect, a buffer ampliﬁer may have to be inserted between the load and the ﬁlter.
9.1.2 HalfPower Frequency and Amplitude Transfer Function LowPass Filter We are going to show how to construct a lowpass RC ﬁlter for a particular application. The design engineer needs to know at approximately which frequency the signal should be cut out. It is a common agreement to choose this frequency so that the amplitude of the p output voltage is exactly 1= 2 0:707 of the input voltage amplitude Vm. In other words, the output ﬁlter power, which is proportional to the square of the output voltage, becomes exactly half of the input power. The corresponding frequency is called the break frequency or halfpower frequency of the lowpass ﬁlter. According to Eq. (9.1e), the break frequency ωb f b is found using the amplitude of the output (capacitor) voltage in the following way: 1 1 1 ωb 1 1 q ¼ p ) ωb τ ¼ 1 ) ωb ¼ ) f b ¼ ¼ ¼ 2π τ 2πτ 2πRC 2 1 þ ðω b τ Þ2
½Hz
ð9:5Þ
Expressed in terms of the break frequency, the amplitude of the output voltage to the q voltage across the capacitor in Eq. (9.1e), has the form V m = 1 þ ð f =f b Þ2 since ωτ f =f b . With the input voltage amplitude to the ﬁlter being Vm, the ratio of the two amplitudes is the amplitude transfer function of the lowpass ﬁlter Hm. This transfer function is given by H m ðf Þ
1 q 1 2 1 þ ð f =f b Þ
ð9:6aÞ
We note that the transfer function is dimensionless (or has the units of V/V). For a given input voltage, the amplitude transfer function allows us to determine the output voltage amplitude. The behavior of Eq. (9.6a) is such that the amplitude transfer function is always less than one: the output voltage cannot exceed the input voltage.
HighPass Filter The break frequency, ωb or fb, of the highpass ﬁlter has the meaning of reducing the p voltage amplitude by a factor of 1= 2 and reducing the signal power by the factor of ½. According to Eq. (9.1e) for the resistor voltage, it is found using the equality q p ωb τ= 1 þ ðωb τÞ2 1= 2, which gives us exactly the same value as the break IX440
Filter Circuits: Frequency Response, Bode Plots. . .
Chapter 9
5
Hm, dB
Amplitude Bode plot  combined filter
0 5 10 15 20 25 30 35 40 45 100
101
102
103 104 frequency, Hz
105
106
Fig. 9.8. Solid curve: Eq. (9.10) for the cascaded ﬁlters. Dashed curve: the exact solution with the opencircuited capacitor C2.
In Fig. 9.8, the exact transfer function may exceed 0 dB. In other words, the voltage gain of the combined (still passive) ﬁlter may be greater than one. How is it possible? The answer is that, in contrast to the circuits in Fig. 9.2, the circuit in Fig. 9.7 is in fact already a secondorder circuit. Secondorder circuits may experience a resonance behavior where the circuit voltages across individual elements may (very considerably) exceed the original supply voltage. This effect, called voltage multiplication, is of great practical importance and will be considered in detail in Chapter 10 devoted to secondorder AC circuits. Note that that the true power gain of a passive ﬁlter of any order and any topology is always less than one (less than 0 dB). Only electronic ampliﬁers may have a positive, and often high, power gain; this is discussed in the next section.
9.1.6 RL Filter Circuits The RL circuits are used for the same ﬁltering purposes as the RC circuits. Figure 9.9 depicts the concept. It may be demonstrated that the corresponding circuit theory and Eqs. (9.6a, 9.6b) for the transfer functions become equivalent to ﬁrstorder RC ﬁlter circuits under the following conditions: 1. The time constant τ RC is replaced by the time constant τ L=R, similar to the corresponding operation for the ﬁrstorder transient circuits. The break frequency f b 1=ð2πτÞ remains the same. 2. The role of the capacitor and inductor are interchanged. For example, the RL circuit in Fig. 9.9a is a ﬁrstorder highpass ﬁlter because the inductor voltage, which is the output ﬁlter voltage, is exactly zero for a DC signal. However, it becomes a ﬁrstorder lowpass ﬁlter if the inductor is replaced by a capacitor, as shown in Fig. 9.9a.
IX448
Chapter 9
Section 9.2: Bandwidth of an Operational Ampliﬁer
Section 9.2 Bandwidth of an Operational Ampliﬁer The operational ampliﬁer circuits introduced earlier are implicitly assumed to operate equally well for any frequency of the input signal. In reality this is not true. An operational ampliﬁer may operate only over a certain frequency band, and the associated frequency bandwidth is perhaps the most critical device parameter. Frequently we do not realize how severe this limitation can be and how difﬁcult it is to build a highfrequency or radiofrequency ampliﬁer. As an example, we should point out that none of the common ampliﬁer ICs studied in introductory ECE classes can be used as a frontend ampliﬁer for an AM radio receiver (520 1610 kHz), even if the noise levels were low. Indeed, highfrequency ampliﬁers with larger frequency bandwidths exist. A case in point is the accessible LM7171 chip. Key to understanding the ampliﬁer frequency behavior is the theory of the ﬁrstorder RC ﬁlters developed in the previous section.
9.2.1 Bode Plot of the OpenLoop Ampliﬁer Gain OpenLoop Ampliﬁer Gain and Its Relation to the Previous Results The (amplitude) frequency response of an operational ampliﬁer is simply a plot of its gain magnitude versus frequency of the input AC voltage signal. This response is usually a Bode plot. The problem is that the gain of the ampliﬁer (both open loop and closed loop) generally decreases with increasing frequency. We consider the openloop gain (gain without the feedback loop) ﬁrst. The openloop gain magnitude will be denoted here by AOL AOL ð f Þ. Note that in Chapter 5 we have already introduced the opencircuit gain, A, of an ampliﬁer at DC without the feedback loop. What is the relation between AOL and A introduced previously? The answer is given by the equality A AOL ð f 0Þ as long as the ampliﬁer is open circuited. OpenLoop Gain Behavior The openloop gain decreases with increasing frequency of the input signal. Figure 9.11 shows the frequency response of an openloop ampliﬁer on a loglog scale. You may recall that the loglog scale used in this ﬁgure is simply the Bode plot introduced in the previous section. This ﬁgure is typical for the LM741 ampliﬁer IC and similar generalpurpose devices. Comparing the Bode plot in Fig. 9.11 with the Bode plot of the RC ﬁlter in Fig. 9.4 of the previous section, we discover that the ampliﬁer’s gain as a function of frequency is virtually identical to the transfer function of the RC ﬁlter for the same break frequency of 10 Hz, as seen in Fig. 9.11! In both cases, we have a rolloff of 20 dB per decade. Obviously, the scale is different. Why is this so? This occurs because the ampliﬁer ICs are usually internally compensated, which means incorporating a simple RC ﬁlter network (in practice, it may be a single capacitor C) into the IC chip itself. This process is called internal compensation of the ampliﬁer. The goal of such a modiﬁcation is to ensure that the ampliﬁer circuit will be stable. Stability refers to the ampliﬁer’s immunity to spontaneous oscillations. These undesired oscillations occur when the input IX451
Filter Circuits: Frequency Response, Bode Plots. . .
Chapter 9
frequency excites internal resonances, similar to a mechanical massspring system, that continue ad inﬁnitum. 106
Openloop gain AOL
105
100 20 dB per decade roll off
104
80
103
60
102
40
101
20
BW
fb
1 1
10
1
Openloop gain [dB] 20log10 (AOL)
120 3dB or 0.707A OL (0)
10
2
103
104
105
106
0 107
Frequency of input voltage, Hz
Fig. 9.11. Bode plot of the openloop gain magnitude for the LM741type ampliﬁer IC. Note the logarithmic scale on the left and the corresponding scale in dB on the right. The frequency bandwidth given by the break frequency fb is only 10 Hz.
9.2.2 UnityGain Bandwidth Versus GainBandwidth Product The ampliﬁer gain in Fig. 9.11 decreases by a factor of 0.1 (or 20 dB) gain rolloff per frequency decade. The decay already starts at a relatively low break frequency of 10 Hz p where the DC openloop gain drops by the factor of 0.707 or 1= 2. The corresponding p value in dB is 20log10 1= 2 3 dB. The gain continues to decrease further and reaches unity at the frequency of 1 MHz. This frequency is equal to the unitygain bandwidth (BW) of the ampliﬁer, i.e., for the ampliﬁer IC depicted in Fig. 9.11: BW
ð9:12Þ
1 MHz
A remarkable observation from Fig. 9.11 is that the gainbandwidth product (sometimes denoted by GBW or GB in datasheets) remains constant over the band for every particular gain value. The gainbandwidth product is equal to the length of every single arrow (in Hz) in Fig. 9.11 times the corresponding gain value (dimensionless), that is, f f f
102 Hz ) GBW 103 Hz ) GBW 104 Hz ) GBW
102 104 103 103 104 102
106 Hz 106 Hz 106 Hz
BW, BW, BW;
ð9:13Þ
etc. Thus, the gainbandwidth product is exactly equal to the unitygain bandwidth BW; it is frequently speciﬁed in the manufacturer datasheet. In what follows, we will use the unitygain bandwidth as the major parameter of interest. Note that instead of, or along
IX452
Chapter 9
Section 9.2: Bandwidth of an Operational Ampliﬁer
with, the unitygain bandwidth, the rise time of an ampliﬁer may be speciﬁed in the datasheet. Approximately, we can state that BW 0:35=rise time ½Hz.
9.2.3 Model of the OpenLoop AC Gain The openloop gain dependence on the frequency has the form of a lowpass ﬁlter. We could therefore describe the openloop gain in a complex form that is identical to the complex transfer function of the lowpass ﬁlter given, for example, by Eqs. (9.9a, b) of the previous section. The openloop AC gain in complex phasor form states AOL ð f Þ
AOL ð0Þ , 1 þ jð f =f b Þ
AOL ð0Þ is the openloop DC gain
ð9:14Þ
For example, AOL ð0Þ 105 in Fig. 9.11. According to Eq. (9.14), the openloop AC gain is a complexvalued frequencydependent transfer function. This circumstance is reﬂected in a phase difference between the output and input voltages. To be consistent with Fig. 9.11 and with the previous DC ampliﬁer analysis, the magnitude of the complex gain function in Eq. (9.14) is denoted by the same symbol, AOL, i.e., jAOL j
AOL ð f Þ
AOL ð0Þ q 1 þ ð f =f b Þ2
ð9:15Þ
The Bode plot applied to Eq. (9.15) will give us exactly the dependence shown in Fig. 9.11. According to Eq. (9.15), the unitygain bandwidth satisﬁes the equality 1
q
AOL ð0Þ 1 þ ðBW=f b Þ
2
ð9:16Þ
q Since BW =f b 1, one has 1 þ ðBW =f b Þ2 BW=f b with a high degree of accuracy. Therefore, according to Eq. (9.16), BW
AOL ð0Þf b
ð9:17Þ
Looking at Fig. 9.11, we observe a very signiﬁcant decrease of the openloop gain, even in the audio frequency range. For example, the openloop gain decreases by a factor of 1000 in the audio range from 10 Hz to 10 kHz. Does it mean that the LM741 or any generalpurpose ampliﬁer cannot be used in this range? The general answer is that the operational ampliﬁer is mostly used with a negative feedback loop. When the closedloop DC gain is not very high (say 10), the corresponding closedloop AC gain appears to be nearly constant over a much wider bandwidth (say up to 100 kHz). This critical result will be proved mathematically shortly.
IX453
Chapter 9 V*
Vin R1 Vin R1
Section 9.2: Bandwidth of an Operational Ampliﬁer V*
Vin þ Vout =AOL R 1 1 1 Vout AOL R2 R2
Vout R2
1 AOL R1
)
Vout =AOL R2
Vout
)
ð9:19Þ
It follows from Eq. (9.19) that the output phasor voltage to the ampliﬁer and the closedloop ampliﬁer phasor gain ACL become Vout
R2 R1 1 þ
Vin V ) ACL out R 1 Vin 2 AOL 1 þ R1
R2 R1 1 þ
1 AOL
1
1þ
R2 R1
ð9:20Þ
Next, we substitute Eq. (9.15) into Eq. (9.20) and rearrange terms to obtain the form ACL ð f Þ
R2 h R1 1 þ
1 AOL ð0Þ
1 þ RR21
1 i
þ AOL1ð0Þ 1 þ RR21 jff
ð9:21Þ
b
The ﬁrst term in the denominator on the righthand side of Eq. (9.20) is one with a high degree of accuracy since AOL ð0Þ 105 108 . This approximation is valid for any realistic resistor values. Therefore, we again arrive at the ﬁrstorder lowpass ﬁlter response: ACL ð f Þ ACL ð0Þ
1þj
R2 , R1
ACL ð0Þ loop f =f closed b
f
closed loop b
, AOL ð0Þf b 1 þ R2 =R1
ð9:22Þ BW 1 þ R2 =R1
but with a very different break frequency fbclosed loop. A similar treatment holds for the noninverting ampliﬁer conﬁguration. The result is identical to Eq. (9.22); however, the closedloop DC gain ACL(0) is now given by ACL ð0Þ
1þ
R2 R1
ð9:23Þ
9.2.5 Application Example: Finding Bandwidth of an Ampliﬁer Circuit The relation reported in Eq. (9.22) is perhaps the most important single result with regard to the AC behavior of operational ampliﬁers. It reveals that the closedloop AC gain has conceptually the same RC ﬁlter response as the openloop gain; see Eq. (9.15). However,
IX455
Section 9.3: Introduction to Continuous and Discrete. . .
Chapter 9
We emphasize that the properties listed in Table 9.2 also apply to the discrete Fourier transform studied below, but the corresponding indexing of discrete frequencies has to be carefully arranged.
9.3.3 Discrete Fourier Transform and Its Implementation Direct Discrete Fourier Transform Present and future demands are such that we must process continuous signals by discrete methods. Perhaps the most important method is the discrete Fourier transform (DFT) and its fast versions: fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT). Let f(t) be a continuous pulse signal which is the source of the data. We assume that f(t) is zero outside of the interval 0 t < T . Let f ðt n Þ, n 0, . . . , N 1 be its values at N uniformly distributed sampling points tn ΔT n, n 0, . . . , N 1 within the interval of interest. Here T N
ΔT
ð9:29Þ
is the sampling interval. Then, the integral of the direct Fourier transform in Eq. (9.25a) may be found using the rectangle rule (or the Riemann sum approximation) F ðωÞ
ΔT
N 1 X
e
jωnΔT
f ðt n Þ
ð9:30Þ
n 0
We could in principle evaluate this expression at any value of ω. However, with only N data points to start with, only N ﬁnal outputs will be signiﬁcant. We choose those N uniformly distributed frequency sampling points as ωm ω0 m, m 0, . . . , N 1, where ω0
2π T
ð9:31Þ
is the fundamental frequency (with one period over the interval T). Let F ðωm Þ, m 0, . . . , N 1 be the values of F(ω) at the frequency sampling points. Then, Eq. (9.30) gives F ð ωm Þ
ΔT
N 1 X
e
2π j N mn f ðt n Þ,
m
0, . . . , N
1
ð9:32Þ
n 0
Inverse Discrete Fourier Transform (IDFT) A very similar operation is applied to the integral of the inverse Fourier transform given by Eq. (9.25b). We ﬁrst assume that F(ω) is zero outside of the interval 0 ω < N ω0; in other words, it is bandlimited. Then, the corresponding integral in Eq. (9.25b) is again approximated using the rectangle rule so that the ﬁnal result has the form IX461
Chapter 9 a)
Filter Circuits: Frequency Response, Bode Plots. . . b) 1
Coil current, kA
0.5
0 0
0.1
0.2
0.3
0.4
3.5
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
time, ms
c) Electric field (Ez), mV/m
3 2.5 2 1.5 1 0.5 0 0.5 0
0.1
0.2
0.3
0.4
0.5
time, ms
Fig. 9.18. Timedomain computational solution for the induced electric ﬁeld within a human body obtained from the frequencydomain data via the FFT.
IX468
Chapter 9
Sampling theorem
Filter Circuits: Frequency Response, Bode Plots. . . 1. Any signal bandlimited to ωmax can be reproduced exactly using the N discrete Fourier transform if ωmax ω0 2 2. Alternatively, the sampling interval must satisfy inequality 1 1 ωmax ΔT , f max ¼ 2π 2 f max F * ½N m ¼ F ½m
Structure of discrete Fourier spectrum
Equivalent frequency samples for negative frequencies Transfer function multiplication
+
N N N F ½0, F ½1, . . . , F 1 ,F ,F þ 1 , . . . , F ½ N 1 ¼ 2 2 2
N N N 1 ,F , F* 1 , . . . , F * ½1 F ½0, F ½1, . . . , F 2 2 2 N N N ω0 , 1 ω0 , 2 , . . . , ω0 0, ωm , . . . , 2 2 2
N N N N 1 F 1 , H F , 2 2 2 2
N * N H 1 F þ 1 , . . . , H* ½1F ½N 1 2 2
HF ! H½0F ½0, H½1F ½1, . . . , H
IX472
Chapter 9
Problems
C
vin(t)
9.1.1 RC Voltage Divider as an Analog Filter
Problem 9.5. The input voltage to the RC ﬁlter circuit shown in the ﬁgure is V in ðtÞ ¼ 5 cos ω t ½V. The ﬁlter has the following parameters: C ¼ 1 μF and R ¼ 100 Ω. The ﬁlter operates in the frequency band from 100 Hz to 50 kHz. The ﬁlter is connected to a load with the load resistance of 1 MΩ. By solving the corresponding AC circuit, determine the output voltage amplitude across the load (and its percentage versus the input voltage amplitude) with and without the load at f ¼ 100 Hz, f ¼ 1592 Hz, and f ¼ 50 kHz.
+
R
+
Problem 9.2 A. Draw a schematic of the lowpass analog RC ﬁlter. Show the input and output ports. B. Repeat the same task for the highpass analog RC ﬁlter.
Load
Problem 9.6. Repeat the previous problem when the load resistance changes from 1 MΩ to 100 Ω (decreases). Problem 9.7. Repeat Problem 9.5 for the ﬁlter circuit shown in the following ﬁgure. Assume the load resistance of 100 Ω. C
+ vout(t)
vin(t)=5cos(wt) [V]
+
+
+
vout(t)
R
vout(t)
Load

C

vin(t)
C

R
vin(t)=5cos(wt) [V]

Problem 9.3. The input voltage to the ﬁlter circuit shown in the following ﬁgure is a combination of two harmonics, υin ðtÞ ¼ 1 cos ω1 t þ 1 cos ω2 t, with the amplitude of 1 V each. The ﬁlter has the following parameters: R ¼ 100 kΩ and C ¼ 1:59 nF. Determine the output voltage υout(t) to the ﬁlter given that f 1 ¼ 100 Hz and f 2 ¼ 100 kHz. Express all phase angles in degrees.
vout(t)
R

Problem 9.1 A. Explain the function of an analog RC ﬁlter. B. Write the capacitor and resistor voltages υR(t) and υC(t) of the series RC circuits in the general form, as functions of the AC angular frequency. C. Which circuit element (or which voltage) dominates at low frequencies? At high frequencies?
+
+
9.1 FirstOrder Filter Circuits and Their Combinations
following ﬁgure. All other parameters remain the same.

Problems


Problem 9.4. Repeat the previous problem for the ﬁlter circuit shown in the
IX473
Filter Circuits: Frequency Response, Bode Plots. . .
Problem 9.9. Given R ¼ 100 kΩ and C ¼ 1:59 nF, determine the break frequency of the lowpass RC ﬁlter and of the highpass RC ﬁlter, respectively. Problem 9.10. List all possible alternative names for the break frequency. Problem 9.11. Write the amplitude transfer function for the lowpass RC ﬁlter. Repeat for the highpass RC ﬁlter. Indicate units (if any). Problem 9.12. The input signal to a highpass RC ﬁlter includes a 60Hz component. Its amplitude is to be reduced by a factor of 10. What break frequency should the ﬁlter have?
Problem 9.18. What do engineers mean by one decade? One octave? Problem 9.19. For the ﬁlter circuit shown in the following ﬁgure, given that R ¼ 100 kΩ and C ¼ 159 pF: A. Create the amplitude Bode plot by ﬁnding transfer function values for (at least) every decade. B. Label the break frequency. C. Label the ﬁlter passband. C
Problem 9.13. The input signal to a lowpass RC ﬁlter includes a 10kHz component. Its amplitude is to be reduced by a factor of 5. What break frequency should the ﬁlter have?
vin(t)
Problem 9.15. It is known that H m ð f ÞdB ¼ 0, 6, 20 ½dB. Find the corresponding values of Hm( f ).
R
0
Problem 9.14. Describe the meaning of the Bode plot in your own words.
+
Problem 9.8. A. Describe the physical meaning of the (halfpower) break frequency in your own words. B. Give the expression for the break frequency in terms of circuit parameters of an RC ﬁlter. Is it different for lowpass and highpass ﬁlters?
between the two corresponding decibel measures in dB? B. When the ratio p of the amplitudes of two signals is 1/ 2, what is the difference between the two corresponding decibel measures in dB? C. When the ratio p of the amplitudes of two signals is 20, what is the difference between the two corresponding decibel measures in dB? D. When the ratio of the powers of two signals is 1000, what is the difference between the two corresponding decibel measures in dB?
+
9.1.2 HalfPower Frequency and Amplitude Transfer Function 9.1.3 Bode Plot, Decibel, and Rolloff
vout(t)

Chapter 9
Hm, dB
5 10 15 20 25 30
Problem 9.16. The following values are given H m ð f Þ ¼ 1, 0:707, 0:1, and 100. Find the corresponding values of Hm( f )dB. Problem 9.17 A. When the ratio p of the amplitudes of two signals is 2, what is the difference
35 40 45 2 10
10
3
4
10 frequency, Hz
10
5
10
6
Problem 9.20. Repeat the previous problem with R ¼ 100 kΩ and C ¼ 53 pF.
IX474
Chapter 9
Problems
Problem 9.21. For the ﬁlter circuit shown in the following ﬁgure, assume the values R ¼ 10 kΩ and C ¼ 1:59 nF. A. Create the amplitude Bode plot by ﬁnding the transfer function values for (at least) every decade. B. Label the break frequency. C. Label the ﬁlter passband.
Problem 9.24. An amplitude Bode plot for a certain RC ﬁlter is shown in the ﬁgure below. A. Approximately determine the ﬁlter’s capacitance, C, for a given R ¼ 100 kΩ. Describe each step of your approach. B. Suggest a way to verify your solution.
+
+
R
vin(t)
C

0
vout(t)
Hm, dB
5 10 15 2
20
10
3
10
4
10 frequency, Hz
5
10
25 30 35 40 45 2 10
10
3
4
10 frequency, Hz
10
5
10
6
Problem 9.22. Repeat the previous problem with R ¼ 100 kΩ and C ¼ 53 pF. Problem 9.23. An amplitude Bode plot for a certain RC ﬁlter is shown in the ﬁgure below. A. Approximately determine the ﬁlter’s resistance R if it is known that C ¼ 265 pF. Describe each step of your approach. B. Suggest a way to verify your solution. Hm, dB
Problem 9.25. Prove analytically that the amplitude transfer functions of the lowpass ﬁlter and the highpass ﬁlter are the mirror reﬂections of each other about the break frequency in the Bode plot.
9.1.4 Phase Transfer Function and Its Bode Plot Problem 9.26. Write the phase transfer function for the lowpass RC ﬁlter. Repeat for the highpass RC ﬁlter. Show units. Problem 9.27. The input voltage to a lowpass RC ﬁlter has a zero phase. At what frequency in terms of the break frequency fb is the phase shift at the output equal to 1 , 45 , and 89 ? Problem 9.28. The input voltage to a highpass RC ﬁlter has a zero phase. At what frequency in terms of the break frequency fb is the phase shift at the output equal to 5 , 45 , and 85 ?
2
10
3
10
4
10 frequency, Hz
5
10
Problem 9.29. A lowpass RC ﬁlter has the break frequency of 10 kHz. Create the phase Bode plot by ﬁnding the transfer function values for (at least) every decade.
IX475
Chapter 9
Problems
Problem 9.34. For the ﬁlter circuit shown in the ﬁgure below, assume the values R1 ¼ 628 Ω and L1 ¼ 10 mH. A. Create the amplitude Bode plot by ﬁnding transfer function values for (at least) every decade. B. Determine the rolloff per decade in dB. Assume that the loading effect of the ﬁlter stages is negligibly small (e.g., a buffer ampliﬁer stage is used).
+
+
L1
+
L1
R1
vout1(t)
R1
vout(t)

vin(t)

Hm, dB
0
Problem 9.38. Frequency response of an ampliﬁer is characterized by the openloop DC gain AOL ð0Þ ¼ 1:41 106 and the break frequency of f b ¼ 20 Hz. Numerically calculate the gainbandwidth product for the ampliﬁer at: A. 20 Hz, B. 2 kHz, C. 2 MHz.
9.2.3 Model of the OpenLoop AC Gain Problem 9.39. Frequency response of an ampliﬁer is characterized by the openloop DC gain AOL ð0Þ ¼ 106 and the break frequency of f b ¼ 20 Hz. Plot the openloop gain magnitude in dB over the range of frequencies (the frequency band) from 1 Hz to 10 MHz on the loglog scale (the Bode plot) and label the axes.
20
40
60
80
100
Problem 9.37. Using a manufacturing company’s website (usually it is a more accurate frequently updated source) or the corresponding datasheet, ﬁnd the unitygain bandwidth for the following ampliﬁer ICs: A. TL082 B. LM741 C. LM7171
2
10
3
10
4
10
5
10
6
10
Problem 9.40. In the previous problem, ﬁnd the unitygain bandwidth BW of the ampliﬁer.
frequency, Hz
Problem 9.35. The transfer function of a ﬁlter 1þjð f =1000Þ . Create its circuit is given by H ð f Þ ¼ 1þ ð f =1000Þ2 amplitude and phase Bode plots in the frequency band from 10 Hz to 100 kHz by ﬁnding transfer function values for (at least) every decade.
9.2 Bandwidth of an Operational Amplifier
Problem 9.41. Internally compensated LM358series ampliﬁers have the unitygain bandwidth (BW) of 1 MHz. The typical largesignal DC voltage gain at room temperature is 100 V/mV. A. Find the openloop DC gain in dB and the openloop break frequency fb. B. Find the openloop gain at 100 Hz, 1 kHz, and 10 kHz.
9.2.1 Bode Plot of the Openloop Ampli Problem 9.42. The openloop gain magnitude of an internally compensated highfrequency ﬁer Gain ampliﬁer has been given as 9.2.2 Unitygain Bandwidth Versus GainBandwidth Product AOL ð100 HzÞ ¼ 0:9 106 ,
Problem 9.36. An ampliﬁer has the unitygain bandwidth BW of 5 MHz. What exactly does this mean? Explain and provide equations.
AOL ð1 MHzÞ ¼ 1:0 102
IX477
Filter Circuits: Frequency Response, Bode Plots. . .
Chapter 9
at room temperature. Determine: A. 3dB break frequency, B. DC openloop gain, C. Unitygain bandwidth BWof the ampliﬁer. Problem 9.43. Repeat the previous problem for AOL ð100 HzÞ ¼ 0:5 106 ,
A. An inverting ampliﬁer with the gain of 1, B. An inverting ampliﬁer with the gain of 10, C. A noninverting ampliﬁer with the gain of 100, D. A voltage follower (buffer ampliﬁer). constructed using the same IC.
AOL ð1 MHzÞ ¼ 1:0 102 :
9.2.6 Application Example: Selection of 9.2.4 Model of the Closedloop AC Gain an Ampliﬁer IC for Proper Frequency 9.2.5 Application Example: Finding Bandwidth Problem 9.46. An inverting ampliﬁer with a Bandwidth of an Ampliﬁer Circuit Problem 9.44. An ampliﬁer with the openloop gain described by the ﬁrstorder RC circuit response with AOL ð0Þ ¼ 105 and f b ¼ 20 Hz is used in a closedloop inverting conﬁguration with R2 =R1 ¼ 9 and R2 =R1 ¼ 99, respectively. A. Using the template that follows, create the Bode plots for the corresponding frequency response (closedloop gain), G( f ), in the band from 10 Hz to 10 MHz on the same graph. Plot the gain values for (at least) every decade. B. Also on the same graph, plot the openloop gain as a function of frequency. C. Determine the bandwidth of the closedloop ampliﬁer so constructed in every case. Bode plot
100
60 gain dB
Problem 9.47. A noninverting ampliﬁer with a gain of 31 and a bandwidth of at least 90 kHz is needed. Which ampliﬁer chip is appropriate for this circuit (and which is not)? A. TL082 B. LM7171 C. LM8272
9.3 Introduction to Continuous and Discrete Fourier Transform
80
9.3.1 Meaning and Deﬁnition of Fourier Transform
40 20 0 20 1 10
gain of 20 and a bandwidth of at least 200 kHz is needed. Which ampliﬁer chip is appropriate for this circuit (and which is not)? A. LM358 B. TL082 C. LM741 D. LM7171 E. LM8272
2
10
3
10
4
5
10 10 frequency, Hz
6
10
7
10
Problem 9.45. The unitygain bandwidth of an ampliﬁer IC is 1 MHz. Determine the bandwidth of the following ampliﬁer circuits:
Problem 9.48. Establish all values of the angular frequency ω in Fig. 9.14b at which the Fourier spectrum F(ω) of a rectangular pulse crosses the frequency axis (becomes zero). Express your result in terms of pulse duration T. Problem 9.49 A. Establish the value of the Fourier transform F(ω) for the pulse shown in the following ﬁgure at ω ¼ 0.
IX478
Chapter 9
Problems
B. Establish the complete pulse spectrum F(ω) at all values of angular frequency ω. f(t)
T/2
Vm
0
T/2
t
Problem 9.50 Establish the Fourier transform F(ω) for the following voltage signals π in time domain: f ðtÞ ¼ A sin t , 2 t < 2 A. 2 f ðt Þ ¼ 0, otherwise π f ðt Þ ¼ A cos t , 2 t < 2 B. 2 f ðt Þ ¼ 0, otherwise
Problem 9.53. The Fourier transform of f(t) is F(ω). What is the Fourier transform of f ðt Þ? Problem 9.54. The function f(t)cos ω0t is an amplitudemodulated signal: a highfrequency carrier cos ω0t, which is transmitted wirelessly, has a lowfrequency envelope f(t), which carries information and is being demodulated at the receiver. If the Fourier transform of f(t) is F(ω), what is the Fourier transform of f(t)cos ω0t? Problem 9.55. If f(t) represents the voltage across a 1Ω load, then f2(t) is the power deliv1 ð f 2 ðtÞdt is the total ered to the load and 1
energy delivered to the load. Prove Parseval’s theorem, 1 ð
f ðt Þdt 2
1
Problem 9.51 Show that for an arbitrary real voltage signal f(t): A. The real part of F(ω) is an even function of angular frequency ω. B. The imaginary part of F(ω) is an odd function of angular frequency ω. C. The magnitude of F(ω) is an even function of angular frequency ω. D. Replacing ω by ω generates the complex conjugate of F(ω); in other words, F ðωÞ ¼ F * ðωÞ.
1 2π
1 ð
jF ðωÞj2 dω; 1
which relates the total energy to an integral of the energy spectral density, jF ðωÞj2 ¼ F ðωÞF * ðωÞ, of the signal. Hint: Use the reversal property of the Fourier transform given by Eq. (9.26). Problem 9.56. Based on Parseval’s theorem established in the previous problem, ﬁnd the 1 ð value of the integral sinc2 ðt Þdt. 1
9.3.3 Discrete Fourier Transform and Its 9.3.2 Mathematical Properties of Fou Implementation rier Transform Problem 9.57. You are using the discrete FouProblem 9.52. The Fourier transform of f(t) is F(ω). What is the Fourier transform of ðt 2 2 d f ðt Þ=dt 2 f ðτÞdτ? 1
rier transform of length 8 (N ¼ 8) for a signal f ðt Þ ¼ sin t over a time interval from 0 to 2π s. A. Compute all sampling points in the time domain. B. Compute all sampling points in the frequency domain.
IX479
Chapter 9
Filter Circuits: Frequency Response, Bode Plots. . .
C. Compute equivalent frequency samples using negative frequencies. D. Compute all discrete samples f [n]. E. Compute all discrete samples F[m] using the deﬁnition of the discrete Fourier transform. Explain the physical meaning of their values. F. Repeat the previous step using function fft of MATLAB. Compare both sets of F[m]. G. Restore all discrete samples f [n] using the deﬁnition of the inverse discrete Fourier transform. Compare them with the exact function values. H. Repeat the previous step using function ifft of MATLAB. Compare both sets of f [n]. Problem 9.58. Repeat the previous problem for the signal f ðt Þ ¼ cos t. All other parameters remain the same. Problem 9.59. For Problem 9.57, establish and prove a discrete version of Parseval’s theorem formulated in Problem 9.55. Problem 9.60. An input signal to a ﬁlter has a discrete frequency spectrum F ½m, m ¼ 0, . . . , N 1 computed via the FFT. You are given ﬁlter transfer function H computed at N2 þ 1 frequency points of the FFT, H½m, m ¼ 0, . . . , N =2. Compute the discrete spectrum of the ﬁlter’s output to be fed into the IFFT.
9.3.6 Application Example: Numerical Differentiation via the FFT 9.3.7 Application Example: Filter Operation for an Input Pulse Signal
Problem 9.61*. Present the text of a MATLAB script that numerically differentiates the input signal f ðt Þ ¼ sin t over the time interval from 0 to 4π s using the FFT with 4096 sampling points and plot the resulting signal derivative. Problem 9.62. Repeatthe previousproblem for
the signal f ðt Þ ¼ exp ðt 2π Þ2 . All other parameters remain the same. Problem 9.63. A monopolar pulse f ðt Þ ¼ exp 2ðt 5Þ2 , 0 t < 10 s is an input to a series combination of two identical ﬁrstorder highpass ﬁlters. Find the output of the ﬁlter combination when the (angular) break frequency is given by: A. ω0 ¼ 0:5 rad=s B. ω0 ¼ 10 rad=s Use the FFT and IFFT with N ¼ 64. Plot the ﬁlter output and explain the output signal behavior in every case. 9.64. A bipolar pulse 2 f ðt Þ ¼ ð5 t Þexp 2ðt 5Þ , 0 t < 10 s is Problem
an input to a ﬁrstorder lowpass ﬁlter. Find the ﬁlter output when its (angular) break frequency is given by: A. ω0 ¼ 0:5 rad=s B. ω0 ¼ 5 rad=s Use the FFT and IFFT with N ¼ 64. Plot the ﬁlter output along with the input signal on the same graph and explain the output signal behavior in both cases.
IX480
Chapter 10
Chapter 10: SecondOrder RLC Circuits
Overview Prerequisites:  Knowledge of complex arithmetic  Knowledge of phasor/impedance method for AC circuit analysis (Chapter 8) Objectives of Section 10.1:  Learn the concept of a resonant circuit and its relation to other engineering disciplines  Understand the internal dynamics of the series/parallel RLC resonator including voltage and current behavior near the resonant frequency  Establish the meaning and be able to calculate resonant frequency, quality factor, and bandwidth of the secondorder resonant circuits  Establish and quantify the duality between series and parallel RLC resonators Objectives of Section 10.2:  Construct four major types of the secondorder RLC filters  Relate all filter concepts to the corresponding circuit diagrams  Specify two filter design parameters: the undamped resonant frequency and the quality factor  Realize the advantages of the secondorder filters versus the firstorder filters Objectives of Section 10.3:  Become familiar with the concept of the nearfield wireless link  Apply the theory of the series resonant RLC circuit to the basic design of the nearfield wireless transmitter and receiver  Understand the operation of proximity sensors based on resonant RLC circuits Application examples:  Nearfield wireless link in undergraduate laboratory  Proximity sensors
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_10
X481
Chapter 10
SecondOrder RLC Circuits
Keywords: Selfoscillating LC circuit, Series resonant RLC circuit, Parallel resonant RLC circuit, Series RLC tank circuit, Parallel RLC tank circuit, Undamped resonant frequency, Resonant frequency, Quality factor of the series resonant RLC circuit, Quality factor of the parallel resonant RLC circuit, Quality factor (general deﬁnition, interpretation, mechanical analogy, tradeoff between Qfactor and inductance value), Bandwidth of the series resonant RLC circuit, Bandwidth of the parallel resonant RLC circuit, Halfpower bandwidth, Upper halfpower frequency, Lower halfpower frequency, Duality of series/parallel RLC circuits, Ideal ﬁlter, Cutoff frequency, Secondorder bandpass RLC ﬁlter, Secondorder lowpass RLC ﬁlter, Secondorder bandreject (or bandstop or notch) RLC ﬁlter, Secondorder highpass RLC ﬁlter, Quality factor of the ﬁlter circuit, Center frequency of the bandpass ﬁlter, Lower and upper halfpower frequencies, Butterworth response, Quality factor of the nonideal inductor, Voltage multiplier circuit, Voltage multiplication, Nearﬁeld wireless link, Horseshoe coil
X482
Chapter 10
Section 10.1: Theory of SecondOrder Resonant RLC Circuits
Section 10.1 Theory of SecondOrder Resonant RLC Circuits In this section, we study the last group of standard AC circuits the resonators. They are secondorder AC circuits in LC or LCR conﬁguration. The term second order means that the circuits will be described by secondorder differential equations if we work in the time domain. The value of a resonator circuit in electronics cannot be overstated. In order to proceed with any type of wireless communication, we ﬁrst need to create a highfrequency AC signal as part of a resonator circuit. Beyond highfrequency circuits, resonators are often used in power electronics and as sensors. In this section, we apply the phasor/impedance method to analyze resonator circuits. We will discover that the most important characteristic is the resonant frequency. Another important parameter is the quality factor, which also determines the resonator bandwidth.
10.1.1 SelfOscillating Ideal LC Circuit The circuit shown in Fig. 10.1a includes an inductor and a capacitor and there is no power source connected to the circuit. The circuit is also ideal, which means that there is no resistance. In other words, the parasitic resistance of the inductor, parasitic resistance of the capacitor, and the wire resistance are all neglected. We assume that the power supply (voltage or current) was disconnected at t 0, after the resonator was excited. The steadystate alternating current and the AC voltages across the circuit elements are sought once the oscillation process has been stabilized, i.e., at t ! 1.
Fig. 10.1. Selfoscillating ideal LC circuit and its phasor representation.
When we apply the phasor/impedance method to the circuit in Fig. 10.1a, we obtain the circuit shown in Fig. 10.1b. KVL in phasor form yields (note the passive reference conﬁguration) VL þ VC
0 ) ZL I þ ZC I
0 ) ðZL þ ZC ÞI
0
ð10:1Þ
Generally, Eq. (10.1) requires the phasor current I to be zero. Obviously, if the phasor current is zero, then the real current is also zero and so are the voltages across the inductor and the capacitor. The circuit is not functioning. However, you should note that, if X483
Chapter 10 ZL þ ZC
SecondOrder RLC Circuits ð10:2Þ
0
in Eq. (10.1), the phasor current does not have to be zero and may have any value depending on the initial excitation. Equation (10.2) is satisﬁed at only one single frequency f0: 1 0 ) ðmultiply by jÞ ) jω0 C 1 1 p p ) f0 LC 2π LC
jω0 L þ ω0
ω0 L þ
1 ω0 C
0) ð10:3Þ
which is the undamped resonant frequency of the LC circuit. Equation (10.3) is perhaps the most important result of resonator theory. Once Eq. (10.3) is satisﬁed, the solution for the circuit current is obtained in the form: I
I0 ) i ð t Þ
I m cos ω0 t
ð10:4aÞ
The current amplitude Im may be arbitrary; it is determined by the initial excitation. The voltages are found accordingly: VL VC
ZL I ) υL ðt Þ Z C I ) υ C ðt Þ
ω0 LI m cos ðω0 t þ 90 Þ 1=ðω0 C ÞI m cos ðω0 t 90 Þ
ð10:4bÞ
The ideal selfoscillating LC circuit in Fig. 10.1 can oscillate indeﬁnitely long. What is the physical basis of selfoscillations in an LC circuit? To answer this question, let us take a closer look at Eqs. (10.4). When the circuit current is at its maximum, the magnetic ﬁeld energy stored in the inductor also has reached its maximum. Since the voltages are shifted by π/2 versus the current, they are exactly zero at that time instance. The zero capacitor voltage means that no energy of the electric ﬁeld is stored in the capacitor. All of the energy stored in the circuit is concentrated in the inductor. When the circuit current reaches zero, the situation becomes the opposite: the capacitor stores the entire circuit energy, and the inductor does not have any stored energy. As time progresses, the process continues so that the current ﬂows back and forth in the circuit charging and discharging the capacitor (and in certain sense the inductor) periodically. Figure 10.2 shows the ideal mechanical counterpart of the circuit in Fig. 10.1. A massive wheel with a rotational inertia represents the inductor and the ﬂexible membrane, the capacitor. The ﬂuid ﬂows back and forth either rotating the wheel (increasing its rotational energy) or bending the membrane (increasing its release energy).
X484
Chapter 10
Section 10.1: Theory of SecondOrder Resonant RLC Circuits
provides one physical interpretation of the quality factor: it determines the maximum amplitude of the resonant oscillations. A higher Qfactor results in larger amplitudes. Yet another, perhaps even more important, interpretation is related to the “sharpness” of the resonance at frequencies close to ω0. What is the physical meaning of Eq. (10.9)? Why does the Qfactor increase with increasing the inductance but not the capacitance? To answer these questions, consider the ﬂuid mechanics analogy in Fig. 10.2. The high Q implies a massive wheel (note: high inductance is equivalent to high wheel mass). Simultaneously, it implies a large membrane stiffness (the small capacitance, which is inversely proportional to the stiffness). The mechanical resonator so constructed will be less susceptible to losses at resonance but will not resonate at all if the driving force has a frequency far away from the resonance. A general deﬁnition of the quality factor also applicable to mechanical engineering and physics is as follows. The quality factor is 2π times the ratio per cycle of the energy stored in the resonator to the energy supplied by a source, while keeping the signal amplitudes constant at the resonant frequency. According to Eq. (10.6c), the instantaneous energies stored in the inductor and capacitor are given by 2 1 2 Vm L E L ðt Þ Li ðt Þ cos 2 ðω0 t Þ, R 2 2 ð10:11Þ 2 1 2 Vm 1 sin 2 ðω0 t Þ Cυ ðt Þ E C ðt Þ R 2ω20 C 2 C L, the coefﬁcients in front of the cosine squared and sine squared terms Since 1= ω20 C are equal. It means that even though both energies vary over time, their sum is a constant: 2 Vm L E L ðt Þ þ E C ðt Þ ð10:12aÞ R 2 The energy dissipated in the resistance is the integral of the instantaneous absorbed power over the period; this integral is equal to 2π=ω ð 0
Ediss 0
V 2m cos 2 ðω0 t Þdt R
V 2m 2R
2π=ω ð 0
ð1 þ cos ð2ω0 t ÞÞdt 0
πV 2m ω0 R
ð10:12bÞ
The ratio of the two energies times 2π precisely equals Eq. (10.9).
X489
Chapter 10 ω0
SecondOrder RLC Circuits
1 p , f0 LC
1 p 2π LC
ð10:19Þ
which coincides with the resonant frequency of the series RLC tank circuit and with the undamped resonant frequency of the LC circuit. Thus, at resonance, Zeq R and one has V RI m for the phasor voltage in Fig. 10.6b. Knowing the phasor voltage, we can establish the phasor currents. The corresponding realvalued voltage and currents at resonance take on the forms υ ðt Þ iC ðt Þ
RI m cos ðω0 t ω0 L RI m ðω0 C Þ cos ðω0 t þ 90 Þ
RI m cos ðω0 t Þ,
i L ðt Þ
90 Þ,
ð10:20Þ
The amplitude of the circuit voltage, along with the amplitudes of inductor and capacitor currents in Eq. (10.20), reaches a maximum at resonance. Next, we wish to introduce the Qfactor of the circuit, similar to Eq. (10.8) for the series resonator, that is, υðt Þ i C ðt Þ
RI m cos ðω0 t Þ,
iL ðt Þ
I m Q cos ðω0 t
90 Þ,
I m Q cos ðω0 t þ 90 Þ
Comparing Eq. (10.20) with Eq. (10.21), we obtain a different expression: p RC R ω0 RC Q p ω0 L L=R
ð10:21Þ
ð10:22Þ
which is exactly the reciprocal of the Qfactor of the series RLC circuit. This means that a highQ parallel resonant circuit will require higher capacitances than inductances. Fortunately, all the results related to the series resonant RLC circuit can directly be converted to the parallel RLC resonant circuit using the substitutions: υðt Þ ! Riðt Þ,
iL ðt Þ ! υC ðt Þ=R,
iC ! υL ðt Þ=R
ð10:23aÞ
Here, the lefthand side corresponds to the parallel RLC circuit, whereas the righthand side corresponds to the series RLC circuit. Furthermore, we need to replace Vm by RIm and interchange the role of two partial time constants: L $ RC R
ð10:23bÞ
in the original solution for the series RLC circuit; see Eq. (10.22). The solution so constructed will match exactly the solution of the parallel RLC circuit depicted in
X494
Chapter 10
SecondOrder RLC Circuits
0
Bode plot
H(f), dB
1 2 3 dB
3 4 5 6 7
half power bandwidth (stopband)
8 9
Phase transfer funct on, degrees
10 160 140 120 100 80 60 40 20 0 5
6
10
f, Hz
10
7
10
Fig. 10.14. Amplitude and phase Bode plots for the bandreject series RLC ﬁlter.
10.2.5 SecondOrder RLC Filters Derived from the Parallel RLC Circuit All secondorder ﬁlters considered so far are derived from the series RLC circuit, with the same quality factor given by Q ω0 L=R. The natural structure after shorting out the input voltage source is shown in Fig. 10.15a. A complementary group of these ﬁlter circuits exists; after shorting out the input voltage source, its natural structure is that of the parallel RLC circuit seen in Fig. 10.15b. These circuits operate quite similarly, but all of them have the quality factor of the parallel RLC resonator, that is, Q ω0 RC. a)
b)
L
R
L
C
R C
Fig. 10.15. (a) Series RLC circuit with no excitation and (b) parallel RLC circuit with no excitation.
X504
Chapter 10
Section 10.2: Construction of SecondOrder RLC Filters
For the ﬁlter circuits derived from the parallel RLC circuit, the resonant frequency still has to satisfy the condition that a real circuit impedance is “seen” by the voltage source. The resonant frequency found this way either does not equal the undamped p resonant frequency f 0 1= 2π LC or does not exist at all. However, the structure of the ﬁlter equations is not affected by this result. Only the undamped resonant frequency f0 appears to be important for the voltage transfer function, which indeed remains the same for any ﬁlter circuit containing one inductance and one capacitance.
X505
Chapter 10
SecondOrder RLC Circuits
frequency of the AC source. When the resonant frequency is close to the AC frequency, the circuit voltage is large. However, when the resonant frequency deviates from the source frequency, the circuit voltage becomes smaller. The change in the circuit voltage is detected. A second method is to make the tank circuit selfresonant, by using an ampliﬁer with a positive feedback. A resonant circuit so built does not need an AC power supply. It oscillates exactly at f0 when there is no object to be detected. When the object is present the oscillation frequency changes. The change in the AC frequency is encoded by another electronic circuit. Using selfresonant tank circuits is perhaps the most common method in practice. A third method is based on the effect of the resistance in the tank circuit. When a metal object is placed close to the coil, the coil’s series resistance signiﬁcantly increases, due to the socalled eddy current losses (for all metals) and, possibly, hysteresis losses (for magnetic metals such as iron, nickel, steel alloys, etc.). The increase in the resistance leads to smaller voltage oscillations in the selfresonant circuit. The circuit may be tuned in such a way as to stop oscillating at a given value of the extra resistance. Great sensitivity may be achieved with this method. Figure 10.22 shows the inductor assembly in a resonant sensor for an automatic trafﬁc light. The inductor now is a singleturn (or multiturn) pavement loop. When a vehicle is located above the loop, its (self) inductance L decreases. This leads to an increase in the resonance frequency. The change in frequency, not the change in the amplitude, is typically detected and encoded. The latter is used to indicate the presence of a vehicle and to adjust the trafﬁc light control. Most vehicle detectors based on loop inductors operate with frequencies from 10 to 100 kHz. A (simpliﬁed) equivalent tank circuit for the trafﬁc light control is shown in Fig. 10.23b. We note the series resistance R, which is the parasitic resistance of the loop. The parasitic resistance includes both the effect of the passing vehicle and of the ground.
Fig. 10.22. Multiple vehicle detection loops after installation at an intersection. Courtesy of the US Trafﬁc Corporation, Loop Application Note of 3/10/03.
X512
Chapter 10
SecondOrder RLC Circuits
Summary TERM Series and parallel RLC resonators
Resonant frequency Quality factor of the resonant circuit Bandwidth of the resonant circuit Halfpower lower and upper frequencies Other RLC resonators
L+RC
Series RLC circuit
Parallel RLC circuit
p p ω0 ¼ 1= LC , f 0 ¼ 1= 2π LC Coincides with the undamped resonant frequency p L=R 1 ¼ ω0 ðL=RÞ ¼ Q¼ p ω0 RC RC dimensionless
p p ω0 ¼ 1= LC , f 0 ¼ 1= 2π LC Coincides with the undamped resonant frequency p RC 1 Q¼p ¼ ω0 RC ¼ ω0 ðL=RÞ L=R dimensionless
B fU fL ¼ s f L, U ¼ f 0
f0 1 ¼ ½Hz Q 2π ð L=RÞ 1
1 1þ 2 2Q ð2QÞ
Circuit diagram
B fU fL ¼
!
s ½Hz
f L, U ¼ f 0
f0 1 ¼ ½Hz Q 2π RC
! 1 1þ ½Hz ð2QÞ2 2Q 1
Resonant frequency s 1 RC 1 ω0 ¼ ðRC Þ L=R Different from the undamped resonant frequency 1 1 r ðL=RÞ RC 1 L=R Different from the undamped resonant frequency r 1 L=R ω0 ¼ 1 ðL=RÞ RC Different from the undamped resonant frequency ω0 ¼
C+LR
(R+L)C
(continued)
X514
Chapter 10
Summary
p RLC ﬁlter circuits derived from the series RLC circuit: f 0 ¼ 1= 2π LC , Q ¼ 1=ðω0 RC Þ Bandpass 1 ﬁlter H0 ð f Þ ¼ f f 0 1 þ jQ f0 f Lowpass ﬁlter
Highpass ﬁlter
Bandreject ﬁlter
Hð f Þ ¼ Q
f0 H0 ð f Þ f
Hð f Þ ¼ Q
f H0 ð f Þ f0
f f0 H0 ð f Þ Hð f Þ ¼ jQ f0 f
p RLC ﬁlter circuits derived from the parallel RLC circuit: f 0 ¼ 1= 2π LC , Q ¼ ω0 RC Bandpass 1 ﬁlter based H0 ð f Þ ¼ f f on parallel 0 1 þ jQ f0 f RLC circuit Lowpass ﬁlter based on parallel RLC circuit
Hð f Þ ¼ Q
f0 H0 ð f Þ f
Highpass ﬁlter based on parallel RLC circuit
Hð f Þ ¼ Q
f H0 ð f Þ f0
Bandreject ﬁlter based on parallel RLC circuit
f f0 H0 ð f Þ Hð f Þ ¼ jQ f0 f
(continued)
X515
Chapter 10
SecondOrder RLC Circuits
Nearﬁeld wireless transmitter/receiver
Resonant circuit at the transmitter (TX) The series capacitor forms the series RLC circuit and increases the amplitude of the p magnetic ﬂux density anywhere in space by the factor 1 þ Q2, Q ¼ ω0 ðL=RÞ
Resonant circuit at the receiver (RX) The shunt capacitor again forms the series RLC circuit and increases the amplitude of the output voltage by the factor Q, Q ¼ ω0 ðL=RÞ
X516
Chapter 10
Problems
Problem 10.9. A series resonant LC circuit is driven by a laboratory AC voltage source with an amplitude V m ¼ 12 V and an internal resistance of 50 Ω (a function generator). Which value should the ratio L/C have to obtain the amplitude of the capacitor voltage equal to 200 V at the resonance? Problem 10.10. A series resonant RLC circuit is needed with the resonant frequency of 1 MHz and a Qfactor of 100. The circuit resistance is 10 Ω. Determine the necessary values of L and C. Problem 10.11. Describe the physical meaning of the resonance bandwidth of the series resonant RLC circuit in your own words. Problem 10.12. A series resonant RLC circuit has the resonant frequency of 1 MHz and the quality factor of 10. Create the Bode plot for the amplitude of the circuit current normalized by its maximum value at the resonance over frequency band from 0.5 to 2 MHz. H(f), dB
5 10 15 20 25 1MHz
a
L
R
C
b
Problem 10.16. Repeat the previous problem for the circuit shown in the ﬁgure that follows. a
C
R
L
b
10.1.6 Parallel Resonant RLC Circuit: Duality
Bode plot
0
0 5MHz
Hint: The resonance is deﬁned by the condition of the purely real equivalent impedance between terminals a and b. In other words, Im Zeq ¼ 0 at the resonance.
2MHz
Problem 10.13. Determine the bandwidth, B, of the series resonant RLC circuit with the resonant frequency of 1 MHz and a Qfactor of 100. Problem 10.14. A series resonant RLC circuit is needed with the resonant frequency of 1 MHz and the bandwidth of 10 kHz. Given the circuit resistance of 10 Ω, determine L and C. Problem 10.15. For the RLC circuit block shown in the ﬁgure, establish the resonant frequency in terms of component values.
Problem 10.17. For a generic parallel RLC resonant circuit with the supply current iS ðt Þ ¼ I m cos ωt, resistance R, inductance L, and capacitance C, give the expressions (show units) for: A. Equivalent circuit impedance at the resonance B. Resonant frequency C. Quality factor of the resonant circuit Problem 10.18. In the parallel resonant RLC circuit shown in the ﬁgure that follows, given I m ¼ 0:5 A, L ¼ 30 μH,C ¼ 0:43 μF, R ¼ 50 Ω: A. Determine the resonant frequency and the Qfactor. B. Determine resonant phasor currents IR, IL, and IC; construct the phasor diagram. C. Determine the realvalued resistor current iR(t) and the inductor/capacitor currents iL(t), iC(t) at the resonance.
X519
Chapter 10 Problem 10.25*. Generate Fig. 10.4 of this section, the Bode plots for the lowpass ﬁlter using MATLAB.
Problems
0
Bode plot
H(f), dB
1 2 3
3 dB
4 5
8 9 10 104
105
f, Hz
10.2.6. SecondOrder RLC Filters Derived from the Parallel RLC Circuit
Problem 10.30. Secondorder RLC ﬁlters may be constructed either on the basis of the series RLC circuit or on the basis of the parallel RLC circuit. Thep undamped resonant frequency, f 0 ¼ 1= 2π LC , which is present in the ﬁlter equations, remains the same in either case. However, the quality factor does not. Three unknown secondorder RLC ﬁlter circuits are shown in the ﬁgure that follows. A. Determine the ﬁlter function (bandpass, lowpass, highpass, or bandreject). B. By analyzing ﬁlter’s natural structure (after shorting out the input voltage source), determine the expression for the ﬁlter quality factor. a)
+
C
+
L
vin(t)
R vout(t)
b)
vin(t)
+
+
L
R vout(t)
C

vin(t)
L
c)
+
+
C

Problem 10.29. A bandreject RLC ﬁlter is required with the center (resonant) frequency of 100 kHz and the halfpower bandwidth, B, of 20 kHz. A. Create its amplitude Bode plot in the frequency band from 10 kHz to 1 MHz. B. Label the ﬁlter passband. C. Determine the necessary values of L and C given R ¼ 20 Ω.
106
R
vout(t)

Problem 10.28 A. Draw the circuit diagram of the secondorder RLC bandreject ﬁlter. Label R, L, and C. B. Show the input and output ports (input and output voltages) C. Deﬁne the resonant frequency and the Qfactor of the ﬁlter circuit.
7

Problem 10.27. A highpass RLC ﬁlter is required with the passband from 0 to 1 MHz. Create amplitude Bode plots for the ﬁlter in the frequency band from 100 kHz to 10 MHz given the resonant frequency of the ﬁlter circuit of 1 MHz and A. Q ¼ 10p B. Q ¼ 1= 2 C. Q ¼ 0:1
6

Problem 10.26 A. Draw the circuit diagram of the secondorder RLC highpass ﬁlter. Label R, L, and C. B. Show the input and output ports (input and output voltages) C. Deﬁne the resonant frequency and the Qfactor of the ﬁlter circuit. D. Which Qfactor is required for the maximally ﬂat response? E. What is the ﬁlter’s halfpower frequency for the maximally ﬂat response?
X521
Chapter 11
Chapter 11: AC Power and Power Distribution Overview Prerequisites:  Knowledge of complex arithmetic  Knowledge of basic circuit analysis (Chapters 3 and 4)  Knowledge of phasor/impedance method for AC circuit analysis (Chapter 8) Objectives of Section 11.1:  Find average AC power for a resistive load and understand the rms values  Express average power for any AC load in terms of power angle and power factor  Express average power in terms of phasors/impedances  Define major AC power types: average power, reactive power, complex power, and apparent power  Be able to construct the power triangle and classify the load power factor Objectives of Section 11.2:  Be able to perform power factor correction of an inductive load (AC motor)  Learn about maximum power efficiency technique in general  Derive and test a simple condition for maximum power transfer to a load from an arbitrary AC source Objectives of Section 11.3:  Learn the structure of power distribution systems  Establish the concept of the threephase power transmission system  Understand the meaning and realization of threephase source and threephase load  Solve for phase and line voltages and line currents in the threephase balanced wyewye system  Establish the meaning and the role of the neutral conductor in the wyewye power distribution system Objectives of Section 11.4:  Establish that the instantaneous power in balanced threephase systems is constant  Extend the concepts of reactive power, complex power, and apparent power to the threephase systems  Compare conductor material consumption in singlephase and threephase systems  Become familiar with deltaconnected threephase sources and loads  Establish equivalency between delta and wye topologies with no ground © Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_11
XI523
Chapter 11
AC Power and Power Distribution
Application Examples:  rms voltages and AC frequencies around the world  Wattmeter  Automatic power factor correction system  Examples of threephase source and the load  Conductor material consumption in threephase systems
Keywords: Time averaging, Average power, rms voltage, rms current, AC fuse, Root mean square, Sawtooth wave, Triangular wave, Noise signals, Power angle, Power factor, Reactance, Capacitive reactance, Inductive reactance, Active power, True power, Reactive power, Complex power, Apparent power, VAR (voltamperes reactive), VA (voltamperes), Power triangle, Lagging power factor, Leading power factor, Wattmeter, Wattmeter current coil, Wattmeter potential coil, AC power conservation laws, Power factor correction, Power factor correction capacitor, PFC capacitor, Principle of maximum power efﬁciency for AC circuits, Principle of maximum power transfer for AC circuits, Impedance matching, Singlephase twowire power distribution system, Singlephase threewire power distribution system, Neutral conductor, Neutral wire, Splitphase distribution system, Polyphase distribution systems, Threephase fourwire power distribution system, Phase voltages, Linetoneutral voltages, abc phase sequence, Positive phase sequence, acb phase sequence, Negative phase sequence, Balanced phase voltages, Wye (or Y) conﬁguration, Balanced threephase source, Wyeconnected source, Wyeconnected load, Wyewye distribution system, Phase impedances, Load impedances per phase, Balanced threephase load, Synchronous threephase AC generator, Alternator, Rotor, Stator, Synchronous AC motor, Rotating magnetic ﬁeld, Linetoline voltages, Line voltages, Line currents, Superposition principle for threephase circuits, Perphase solution, Total instantaneous load power of the threephase system, Average load power of the balanced threephase system, Reactive load power of the balanced threephase system, Complex load power of the balanced threephase system, Balanced deltaconnected load, Balanced deltaconnected source, Deltadelta distribution system
XI524
Chapter 11
Section 11.1: AC Power Types and Their Meaning
Section 11.1 AC Power Types and Their Meaning The present section studies the basics of AC power. We begin with the rootmeansquare (rms) representation of AC voltages and currents. The rms concept enables us to develop a DC equivalent representation, which compares AC to DC conditions in terms of power delivered to the load. It is important to understand that the rms concept is a general power concept; it applies not only to periodic AC circuits but virtually to any circuits, even with nonperiodic power sources, like noise power sources. Further results presented in this section are primarily intended for power electronic circuits; they have an equal applicability to radiofrequency communication circuits.
11.1.1 Instantaneous AC Power We consider an arbitrary load with resistance R, load current i(t), and load voltage υ(t), in the passive reference conﬁguration. The instantaneous power delivered into the load is given by pðt Þ
υðt Þiðt Þ
υ 2 ðt Þ R
ð11:1Þ
according to Ohm’s law. If we use the load voltage in the form υðt Þ then pð t Þ
υðt Þiðt Þ
V 2m cos 2 ω t R
V m cos ωt ½V,
V 2m ð1 þ cos 2ωt Þ 2R
ð11:2Þ
where we applied the trigonometric identity cos 2 ω t 0:5ð1 þ cos 2ωt Þ. Interestingly, the load power is not constant; it varies in time, and the behavior is shown in Fig. 11.1 for a load voltage amplitude V m 3V, frequency f 50 Hz, and load resistance R 5 Ω. 6
6 T
4
4
voltage V
2
2
0
0
2
2 AC voltage
4
4
6 0
5
10
15
20 25 time, ms
30
load power W
instan aneous AC power
35
6 40
Fig. 11.1. Power (solid line) for a load voltage υðt Þ ¼ 3 cos ð2π50t Þ ½V (dotted line).
XI525
Chapter 11
AC Power and Power Distribution
11.1.2 Timeaveraged AC Power An interesting question arises when we have to determine the bill for the variable AC power in Fig. 11.1. As far as the utility power is concerned, a consumer would prefer to pay for the minimum amount of power. The power minima in Fig. 11.1 occur at t 0:25T , 0:75T , 1:25T , etc:; here T is the period of the AC voltage signal. Since the power is exactly zero at its minima, we would pay nothing. On the other hand, a utility would prefer to charge for the maxima of the power, which occur at t 0, 0:5T , 1:0T , 1:5T , etc: A fair solution is clearly somewhere in the middle. It is based on time averaging the load power and then charging the consumer for the average (or mean) power as indicated in Fig. 11.1 by the shaded rectangle. Thus, we are interested in the averaged instantaneous power of Eq. (11.2). The time averaging is always done over a full period T of the AC voltage signal. The notation for the timeaverage value is often denoted by an overbar. Thus, the deﬁnition reads:
P
ðT 1 pðt Þ pðt Þdt T
ð11:3Þ
0
where P is now the average power delivered to the load. We note that the average power times the period T gives us the energy E (in J or more often in Wh, 1 Wh 3600 J) delivered to the load per period, i.e., E T P:
rms Voltage and rms Current Using Eq. (11.2) we obtain from Eq. (11.3) ðT
P
V 2m
V 2m
1 T 2R
1 ð1 þ cos 2ω t Þ dt T 2R
0T 1 ð ðT @ 1 dt þ cos 2ω tdt A 0
0
ð11:4Þ
0
The ﬁrst integral yields a nonzero contribution, whereas the second integral is exactly equal to zero, due to fact that the average of the sine or cosine functions over a period, or multiple periods, is zero. Thus, ðT
ðT 1 dt
0
T,
cos 2ωtdt
1 sin 2ωtj0T 2ω
1 sin ð4π t=T Þj0T 2ω
0
ð11:5Þ
0
Inserting these values into Eq. (11.4) results in P
V 2m 2R
V 2rms , R
V rms
Vm p 2
ð11:6Þ
XI526
Chapter 11
Section 11.1: AC Power Types and Their Meaning
11.1.5 Average AC Power in Terms of Phasors: Power Angle For arbitrary dynamic circuit elements, the power analysis is carried out in terms of phasors. Consider element A in Fig. 11.4 which has realvalued voltage and current given by υðt Þ iðt Þ
V m cos ðωt þ φÞ I m cos ðωt þ ψ Þ P? v(t)

a
+
ð11:11Þ
b
A i(t)
Fig. 11.4. Arbitrary circuit element in the passive reference conﬁguration.
When element A is a resistor, the phases in Eq. (11.11) are the same, and ﬁnding the average power is straightforward. However, when element A is an inductor, capacitor, or a combination of resistor and inductor/capacitor, the situation becomes different. In this case, the phases of voltage and current in Eqs. (11.11) do not necessarily coincide. By deﬁnition:
P
pðt Þ
ðT 1 υðt Þiðt Þdt T
ðT 1 V m I m cos ðωt þ φÞ cos ðω t þ ψ Þdt T
0
ð11:12Þ
0
To manipulate the cosine expression in Eq. (11.12), we can use the trigonometric identity cos ðωt þ φÞ cos ðω t þ ψ Þ 0:5 cos ðφ ψ Þ þ 0:5 cos ð2ωt þ φ þ ψ Þ. The integral of the second term in Eq. (11.12) will be equal to zero since it is the integral of the plain cosine function over two periods. The result then has the form:
P
pðt Þ
ðT 1 υðt Þiðt Þdt T
V mI m cos ðφ 2
ψÞ
V mI m cos θ 2
V rms I rms cos θ
0
ð11:13Þ Equation (11.13) is of great importance for power electronics since it introduces the socalled power angle θ θ
φ
ψ,
90o θ þ90o
ð11:14Þ
and the power factor PF
cos ðφ
ψÞ
cos θ
ð11:15Þ XI531
Chapter 11
Q
Section 11.1: AC Power Types and Their Meaning
Im V I* 2
X jIj2 2
jZjjIj2 sin θ 2
jVjjIj sin θ 2
V rms I rms sin θ
½VAR ð11:21bÞ
The physical units of the reactive power are also watts. However, to underscore the fact that this power is not an active useful power, the units of VAR (voltamperes reactive) are used. The reactive power ﬂows back and forth from the source to the load, through an electric line but does not do real work. The last power type is the complex power S that is simply S
V I* 2
P þ jQ
½VA
ð11:21cÞ
The complex power is measured in voltamperes (VA). The magnitude of the complex power S jSj is called the apparent power. We can see that the apparent power is given by S
jSj
jZjjIj2 2
jVjjIj 2
V rms I rms
½VAR
ð11:21dÞ
The apparent power is the “best possible” load power that can be obtained if one measures current and voltage and ignores the phase shift between them. Equations (11.21) and (11.22) raise the obvious question: why do we need so many AC power types? The answer is that a purely resistive load (the power angle θ equals zero) is merely a dream and not realistic. Any AC load generally has a signiﬁcant reactive impedance part. So does an electric motor, a small antenna in your cellphone, and even a household electric heater whose heating spiral is a series combination of a resistance and a small, but often visible, inductance. Therefore, we always deal with active and reactive power; the sum of their squares is the square magnitude of the apparent power. The reactive power increases the electric current ﬂowing in the circuit and thus increases the unrecoverable losses in (sometimes very long) power lines, which have a ﬁnite resistance. Therefore, our goal is to decrease the percentage of the reactive power and thus decrease the net power loss. The three power deﬁnitions show us how to accomplish this task. In the next section, we will need to decrease the power angle θ by modifying the load through adding other circuit components; in other words, we are attempting to load match the circuit.
11.1.8 Power Triangle Since cos 2 θ þ sin 2 θ 1, the three powers (average, reactive, and apparent) are interconnected by the relation S2
P2avg þ Q2
½W
ð11:22Þ
XI535
Chapter 11
AC Power and Power Distribution
11.2.4 Principle of Maximum Power Transfer for AC Circuits The principle of maximum power transfer is perhaps less important for residential power distribution systems where efﬁciency counts. However, it is critical for radiofrequency and communication circuits, which are conceptually the same AC circuits but operating at much higher frequencies. With reference to Fig. 11.12a, the following question should now be asked: at which value of the load impedance ZL RL þ jX L is the average (true) power delivered to the load maximized? The phasor current in Fig. 11.12a is given by I
VT ½A ZL þ ZT
ð11:34Þ
so that the average power delivered to the load becomes P
R L j Ij 2 2
RL jVT j2
0:5RL jVT j2
2jZL þ ZT j2
ðRL þ RT Þ2 þ ðX L þ X T Þ2
½W
ð11:35Þ
Let us take a closer look at Eq. (11.35); in order to reach the maximum true power, the load reactance XL should be equal to the generator reactance XT taken with the opposite sign so that X L þ X T 0. This yields for the average load power P
0:5RL jVT j2 ðRL þ RT Þ2
½W
ð11:36Þ
Consequently, the problem reduces to the maximum power transfer of a DC circuit as studied in Chapter 2. The corresponding condition for the maximum load power is RL
ð11:37Þ
RT
This condition, augmented by the equality for the reactances XL
ð11:38Þ
XT
leads to a simple, yet very useful result for the maximum power transfer to the load: ZL
Z*T ) Pmax
1 jVT j2 8 RT
½W
ð11:39Þ
We note that the load impedance should be the complex conjugate of the generator impedance. Along with the maximum power transfer, Eq. (11.39) assures that there is no reﬂection of radiofrequency waves propagating along the circuit transmission lines from XI544
Chapter 11
AC Power and Power Distribution
Section 11.3 AC Power Distribution: Balanced ThreePhase Power Distribution System 11.3.1 AC Power Distribution Systems Representative AC power distribution systems are shown in Fig. 11.13. A singlephase twowire power distribution system is depicted in Fig. 11.13a. It consists of a generator p with a voltage amplitude of Vm, an rms value of V rms V m = 2, and a phase φ connected through two conductors to a load with impedance Z. The previous analysis of AC power was solely restricted to this conﬁguration. An extension is the singlephase threewire power distribution system shown in Fig. 11.13b. Such a system contains two identical AC sources of the same amplitude and phase connected to two (Z1, Z2) loads or to one (Z) load through two outer conductors and the neutral conductor (or neutral wire). This system is the common household distribution system. It allows us to connect both 120V and 240V appliances as shown in Fig. 11.13b; we sometimes called it the splitphase distribution system. The neutral wire is usually physically grounded. In contrast to those two cases, the power distribution systems shown in Fig. 11.13c, d are the polyphase distribution systems in the sense that they use AC sources with different phases. For example, Fig. 11.13c illustrates a twophase threewire distribution system with two voltage sources; the second one lags the former by 90 . Finally, Fig. 11.13d shows the most important and practical threephase fourwire power distribution system with three sources and three load impedances Z1, Z2, and Z3. Generally, the threephase system also uses a (grounded) neutral wire. We will show that this wire may be omitted for balanced power distribution circuits, with the earth itself acting as the neutral conductor. This is important for longdistance, highpower transmissions. Power systems designed in this way are grounded at critical points to ensure safety. Today, a vast majority of electric power is generated and distributed via the threephase power systems. Why is this so? You will soon learn that in contrast to the singlephase systems, the instantaneous power in balanced threephase systems is constant or independent of time rather than pulsating. This circumstance results in more uniform power transmission and less vibration of electric machines. Furthermore, threephase AC motors have a nonzero starting torque in contrast to the singlephase motors. Last but not least, it will be shown that the threephase system surprisingly requires a lesser amount of wire compared to the singlephase system.
XI546
Section 11.3: AC Power Distribution: Balanced ThreePhase. . .
Chapter 11
single phase two wire system
a)
single phase three wire system
b)
+ V  m
A
a
+ V  m
Z
Z1 n
neutral
N
+ V  m
Z2 B
b
c)
Z
three phase four wire system
d)
A
a
+
two phase three wire system a
+

Vm
N
90 b
C
cb

Z2
Z2
120
+
Vm
+
+
B
b

Z1
 Vm 0 n
Z1
Vm 0
A
Vm
B
Z3
120 n
neutral
N
Fig. 11.13. Various AC power distribution systems. N or n indicates the neutral line.
11.3.2 Phase Voltages: Phase Sequence The voltage sources in the threephase system in Fig. 11.13d are set between lines a, b, c and the neutral line n. Those voltages are called phase voltages or linetoneutral voltages. The phase voltages are 120 out of phase. One possible scenario for the realvalued phase voltages is υan ðt Þ υcn ðt Þ Van
V m cos ðωt Þ, υbn ðt Þ V m cos ðω t þ 120 Þ V m,
Vbn
V m∠
V m cos ðωt
120 ,
Vcn
120 Þ,
V m ∠ þ 120
ð11:41aÞ ð11:41bÞ
Phase voltage υan leads phase voltage υbn, which in turn leads υcn. This set of voltages is shown in Fig. 11.14. It has a positive or abc phase sequence since the voltages reach their peak values in the order abc as seen in Fig. 11.14. Simultaneously, the phasor voltages are obtained from each other by clockwise rotation in the phasor diagram. This is shown in Fig. 11.15a.
XI547
Chapter 11
Section 11.3: AC Power Distribution: Balanced ThreePhase. . .
in the motor mode. Changing the phase sequence from abc to acb will reverse the direction of the magnetic ﬁeld rotation and thus reverse the direction of the motor rotation! This method is used in practice since it requires interchanging only two connections.
Residential Household Another example of the load impedance per phase is related to a typical residential household in the USA. A single phase of a threephase residential distribution system is normally used to power them up; see Fig. 11.18. This single phase still has a high rms voltage (4800 V or 7200 V). A stepdown centertap transformer is used to decrease this voltage level to the desired level of 120 240 V and provide the neutral contact necessary for the threewire singlephase residential system shown in Fig. 11.13b. This transformer case is also seen in Fig. 11.18. In the USA, a polemounted transformer in a suburban setting may supply one to three houses.
Fig. 11.18. Threephase to threewire residential system connected via a stepdown transformer. From the pole transformer, the residential power system serving two houses is run down the pole underground. Cape Cod, MA.
11.3.5 Solution for the Balanced ThreePhase WyeWye Circuit Phase Voltages and Line Voltages A threephase balanced circuit (wyewye conﬁguration) which includes the source and the load is shown in Fig. 11.19. We place the nodes n and N at the originally anticipated center positions. The positive phase sequence of 0, 120 , þ120 is assumed. The sum of phase voltages is to be found ﬁrst. In the phasor form, Van þ Vbn þ Vcn
V m ð1 þ 1∠ 120 þ 1∠þ120 Þ V m ð1 1Þ 0
V m ð1 þ 2 cos 120 Þ
ð11:43Þ
XI551
Chapter 11
Section 11.3: AC Power Distribution: Balanced ThreePhase. . .
Line Currents: PerPhase Solution The currents Ia,b,c in Fig. 11.19 are called line currents. To ﬁnd the line currents, the circuit may be solved separately for every phase using the superposition principle. The superposition principle implies shorting out two of the three voltage sources at a time. This method applies to both balanced and unbalanced circuits. Shorting out voltage sources Vbn and Vcn leads to a singlephase equivalent circuit, shown in Fig. 11.20, since the two remaining source impedances will be shorted out by the neutral wire. As long as the system is balanced, the same equivalent circuit will be derived for every other phase. a
Van
Ia
+
ZL

n
Fig. 11.20. A singlephase equivalent circuit by shorting out Vbn and Vcn.
Applying this method to every phase, we obtain Ia Ic
Van =Z Vcn =Z
where I m I a þ Ib þ Ic
I m ∠ θ, Ib I m ∠120 θ
I m ∠ 120
Vbn =Z
θ,
ð11:45Þ
V m =Z. The sum of the line currents is given by ðVan þ Vbn þ Vcn Þ=Z
0
ð11:46Þ
according to Eq. (11.43). Thus, the sum of the balanced line currents is also exactly zero, either in phasor form or in the time domain. Equations (11.43), (11.44), (11.45), and (11.46) hold for any phase sequence, with or without the common phase shift.
11.3.6 Removing the Neutral Wire in LongDistance Power Transmission Equation (11.46) for the line currents has an important implication. By taking into account Eq. (11.46), KCL for node n in Fig. 11.19 yields In
ð I a þ Ib þ Ic Þ
0
ð11:47Þ
Equation (11.47) states that the neutral conductor in the balanced circuit carries no current. Such a wire could in principle be removed from the balanced circuit without affecting the rest of it. Removing the neutral conductor is economically beneﬁcial in longdistance highvoltage power transmission, which utilizes the balanced circuits. In highvoltage power lines, the conductors in multiples of three are used; see Fig. 11.21.
XI553
Chapter 11
AC Power and Power Distribution
the threephase system consumes 75 % less conductor material compared to the singlephase system. The key is the absence of the neutral wire (or, possibly, using a much thinner neutral wire). Other examples for particular loads might result in even more dramatic savings.
11.4.4 Balanced DeltaConnected Load Along with the wyeconnected load, an important example of the threephase load is the deltaconnected load, which is shown in Fig. 11.27a in the balanced conﬁguration. The balanced deltaconnected load is common, along with a balanced wyeconnected load. The deltaconnected load inherently does not have a neutral port. This load may be converted to the wyeconnected load shown in Fig. 11.27b by using the YΔ transformation algorithm established in Chapter 3. This algorithm equally applies to the impedance circuits. The algorithm considerably simpliﬁes when the loads are balanced (load resistances or impedances are equal). With reference to Fig. 11.27, one has ΖY
1 ΖΔ $ ΖΔ 3
ð11:58Þ
3ΖY
for phase impedance transformation. Here, indexes Y and Δ refer to the wyeconnected and deltaconnected loads, respectively.
a)
b)
Delta connected load
Wye connected load
a
a
Z
b
Z
b
1 Z 3
Z
c
c
1 Z 3
1 Z 3
Fig. 11.27. Deltaconnected load versus ad.
11.4.5 Balanced DeltaConnected Source The balanced deltaconnected source is shown in Fig. 11.28b. In its original conﬁguration, it is not using the ground terminal or a neutral conductor. The deltaconnected source so wound is generally less common and less safe than the wyeconnected source. It may be created by the threephase generator shown in Fig. 11.17 of the previous section, assuming the three individual coil windings aa0 , bb0 , and cc0 are interconnected in a closed loop.
XI560
Section 11.4: Power in Balanced ThreePhase Systems. . . b) Delta connected source
Wye connected source
+
b
Vab
a b


+
Vca

+

Vbc

+
Vcn
a
+
Vbn
Van
+
a)

Chapter 11
c
c
Fig. 11.28. Wyeconnected source versus deltaconnected source.
The balanced wyeconnected source without a neutral or ground conductor can be easily converted to the balanced delta source and vice versa. The concept is shown in Fig. 11.28. The line voltages Vab, Vbc, Vca between nodes a b, b c, and c a of the wye source become the phase voltages of the delta source. The relation between two voltage types is given by Eq. (11.44) of the previous section, that is (positive phase sequence), Vab
p
3Van ∠30 ,
Vbc
p 3Vbn ∠30 ,
Vca
p
3Vcn ∠30
ð11:59Þ
Thus, according to Eq. (11.59) and Fig. 11.28, the phase voltages of the equivalent p deltaconnected source Vab, Vbc, Vca are greater in amplitude by a factor of 3 1:73 as compared to the phase voltages Van, Vbn, Vcn of the equivalent wyeconnected source in Fig. 11.28. The line voltages of the deltaconnected source coincide with its phase voltages given lossless conductors and coincide with the line voltages of the wyewye source; all of them are simply Vab, Vbc, Vca. Indeed, the sum of the phase voltages for the deltaconnected source is still equal to zero according to Eq. (11.43) of the previous section. Hence, there is no current circulation in the (ideal) delta loop in Fig. 11.28b. Transformations given by Eqs. (11.58) and (11.59) allow us to consider four distinct sourceload conﬁgurations in the threephase systems: wyewye, wyedelta, deltawye, and deltadelta. All of them may be reduced to the wyewye circuit or solved independently. Figure 11.29 shows one such conﬁguration: a balanced deltadelta distribution system. In the deltadelta system, the line voltages coincide with the phase voltages, whereas the line currents Ia, Ib, and Ic are different from the load (or phase) currents IAB, IBC, and ICA. This is in contrast to the wyewye system where the line and phase voltages are different, but the line and load currents remain the same.
XI561
Chapter 11
AC Power and Power Distribution
Summary rms Voltages and currents in terms of sine/cosine amplitudes and in the general case v u ðT u Vm Im u1 2 υ ðt Þ dt For sinusoidal signals: V rms ¼ p , I rms ¼ p General periodic case: V rms ¼ t T 2 2 0
1 Average power for resistive load: P ¼ V m I m ¼ V rms I rms , V rms ¼ RI rms 2 Power angle θ and power factor PF υðt Þ ¼ V m cos ðωt þ φÞ iðt Þ ¼ I m cos ðω t þ ψ Þ
) θ ¼ φ ψ,
90 θ þ90 PF ¼ cos ðφ ψ Þ ¼ cos θ
V mI m cos θ ¼ V rms I rms cos θ (zero for L and C) 2 Re V I* , Z ¼ jZj∠θ Average power and power angle in terms of phasors: P ¼ 2 Average power P, reactive power Q, complex power S, and apparent power S Re V I* RjIj2 jZjjIj2 jVjjIj ¼ cos θ ¼ cos θ ¼ V rms I rms cos θ ½W ¼ P¼ 2 2 2 2 Im V I* X jIj2 jZjjIj2 jVjjIj ¼ sin θ ¼ Q¼ sin θ ¼ V rms I rms sin θ ½VAR ¼ 2 2 2 2 Average power for arbitrary load: P ¼
S¼
V I* ¼ P þ jQ ½VA 2
S ¼ j Sj ¼
jZjjIj2 jVjjIj ¼ ¼ V rms I rms ½VAR 2 2 Power triangle (lagging/leading power factor)
AC power conservation laws For any network of N loads connected in series, parallel, or in general: S ¼ S1 þ S2 þ . . . SN , P ¼ P1 þ P2 þ . . . PN , Q ¼ Q1 þ Q2 þ . . . QN (continued)
XI564
Chapter 11
Summary Power factor correction
C¼
L ðω LÞ2 ) ) Z ¼ R þ R ω2 L2 þ R2
L¼
without capacitor
1 þ ω2 R2 C 2 1 )Z¼Rþ ) 2 ω C RðωC Þ2 without inductor
zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{ 2 jωCV m I ¼ RðωC Þ V m2 þ 1 þ ðωR C Þ 1 þ ðωR C Þ2 ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}
zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{ jωLV m I ¼ 2 RV m 2 2 R þ ðω LÞ R þ ðωLÞ2 ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} with capacitor
with inductor
P remains exactly the same, Q becomes zero, PF becomes 100 % Maximum power transfer P¼
0:5RL jVT j2 ðRL þ RT Þ2 þ ðX L þ X T Þ2
Pmax ¼ at
½ W
1 jVT j2 ½ W 8 RT
ZL ¼ Z*T
Some equivalent drawings of the same balanced three phase four wire wye–wye power distribution system
(continued)
XI565
Chapter 11
AC Power and Power Distribution
Major parameters of the balanced threephase fourwire wye–wye power distribution system Positive phase sequence Van ¼ V m , Vbn ¼ V m ∠ 120 , Vcn ¼ V m ∠ þ120 Negative phase sequence Van ¼ V m , Vbn ¼ V m ∠ þ120 , Vcn ¼ V m ∠ 120 Current in the neutral wire: In ¼ 0 Per phase solution: Ia ¼ I m ∠ θ, Ib ¼ I m ∠ 120 θ, Ic ¼ I m ∠ 120 θ I m ¼ V m =Z, Z ¼ Z∠θ Vab ¼ Van Vbn ¼ Line voltages (positive phase sequence): Vbc ¼ Vbn Vcn ¼ Vca ¼ Vcn Van ¼
p p p
3Van ∠30 ,
3Vbn ∠30 , 3Vcn ∠30
3 Instantaneous/average load power: pðt Þ ¼ P ¼ V m I m cos θ ¼ 3V rms I rms cos θ ¼ const 2 3 Apparent load power: S ¼ V m I m ¼ 3V rms I rms 2 Some common wye distribution systems 3Phase, 4Wire 208Y=120 V ðUSÞ Line : V rms ¼ 208 V, V m ¼ 294 Phase : V rms ¼ 120 V, V m ¼ 170 3Phase, 4Wire 400Y=230 V ðEU, OthersÞ Line : V rms ¼ 400 V, V m ¼ 566 Phase : V rms ¼ 230 V, V m ¼ 325 Wye load to delta load conversion
V V V V
Wye source to delta source conversion
Vab ¼
p
3Van ∠30 ,
Vbc ¼
p
3Vbn ∠30 ,
Vca ¼
p
3Vcn ∠30 (positive phase sequence)
XI566
Chapter 11
Problems 11.1 AC Power Types and Their Meaning 11.1.1 Instantaneous AC Power 11.1.2 TimeAveraged AC Power Problem 11.1. An AC voltage signal across a resistive load with R ¼ 10 Ω is given by: A. υðt Þ ¼ V m cos 1000t ½V B. υðt Þ ¼ V m sin 60t ½V C. υðt Þ ¼ V m cos ð60t þ 45 Þ ½V where V m ¼ 10 V. Determine the average AC power into the load in every case. Problem 11.2. An alternating current through a resistive load with R ¼ 100 Ω is given by: A. iðtÞ ¼ I m cos 106 t ½V B. iðt Þ ¼ I m cos 37t ½V C. iðt Þ ¼ I m sin ð2011t þ 45 Þ ½V where I m ¼ 1 A. Determine the average AC power into the load in every case. Problem 11.3. An rms voltage across a resistive load with R ¼ 100 Ω is given by: A. V rms ¼ 5 V B. V rms ¼ 100 V C. V rms ¼ 0 V Determine the average power into the load in every case. Problem 11.4. An rms current through a resistive load with R ¼ 1 kΩ is given by: A. I rms ¼ 1 A B. I rms ¼ 100 μA C. I rms ¼ 0 A Determine the average power into the load in every case. Problem 11.5. An AC voltage signal is given by: A. υðt Þ ¼ V m cos ðω t þ φÞ ½V B. υðt Þ ¼ 1V þ V m cos ðωt þ φÞ ½V C. υðt Þ ¼ 1 V V m sin ðωt þ φÞ ½V where V m ¼ 1 V, ω ¼ 100 rad=s, and φ ¼ π=2 rad. Find the timeaverage voltage υðt Þ in every case.
Problems Problem 11.6. Present a mathematical proof of the fact that the expression for the average V2 power, P ¼ 2Rm , holds for an AC voltage signal given by υðt Þ ¼ V m cos ðωt þ φÞ ½V where φ is an arbitrary phase.
11.1.3 Application Example: rms Voltages and AC Frequencies Around the World
Problem 11.7. A 100 Ω resistive load is connected to an AC wall plug in: A. Peoples Republic of China B. India C. USA D. Germany Determine the average power delivered to the load in every case. Also determine the rms load current in every case. Problem 11.8. What do you think is a major A. Advantage B. Disadvantage of having a higher AC voltage?
11.1.4 rms Voltages for Arbitrary Periodic AC Signals Problem 11.9. Determine the average power delivered to a 100 Ω resistive load when the applied periodic voltage signal has the form υðt Þ ¼ ð5t þ 0:01Þ=T ½V over one period T ¼ 0.01 s. This signal is known as the sawtooth or the triangular wave: A. Use the analytical calculation of the rms voltage. B. Use the rms voltage found numerically, based on a MATLAB script or any software of your choice. Problem 11.10. Determine the average power delivered to a 100 Ω resistive load when the p applied periodic voltage has the form υðt Þ ¼ t =T ½V over one period T ¼ 0.01 s: A. Use the analytical calculation of the rms voltage. B. Use the rms voltage found numerically, based on a MATLAB script or any software of your choice.
XI567
Chapter 11
AC Power and Power Distribution
Problem 11.11. Of the two periodic voltage signals shown in the ﬁgures below, a) 6 voltage, V 4 2
The periodic voltage on the top graph is the cosine function. The periodic voltage on the bottom graph is given by υðt Þ ¼ 3:2
105 ðt 0:005Þ2 4 ½V over the time interval from 0 to T. a)
6 voltage, V
0
4
2
2
4
0
6
2
0
5
10
15
20 25 time, ms
30
35
40 4
b) 6 voltage, V
6
4
b)
2
0
5
10
15
20 25 time, ms
30
35
40
30
35
40
6 voltage, V
0
4
2
2
4
0
6
2
0
5
10
15
20 25 time, ms
30
35
40
which signal delivers more average power into a resistive load? The periodic voltage on the top graph is the cosine function. Explain your answer and provide an analytical proof (ﬁnd the rms voltages and the average power in every case). Problem 11.12. Of the two periodic signals shown in the ﬁgures that follow, which signal delivers more average power into a resistive load? Explain your answer and provide: A. An analytical proof—ﬁnd the rms voltage and the average power in every case B. A numerical proof (use MATLAB or any software of your choice).
4 6 0
5
10
15
20 25 time, ms
Problem 11.13. Of the two periodic signals shown in the ﬁgure that follows, which signal delivers more average power into a resistive load? The periodic voltage on the top graph is the cosine function. Explain your answer and provide: A. An analytical proof—ﬁnd the rms voltage and the average power in every case B. A numerical proof (use MATLAB or any software of your choice).
XI568
Chapter 11
Problems Problem 11.39. Repeat the previous problem for the circuit shown in the ﬁgure below:
+
Vm

n
Z
Z
+
Vm
120
N
+

Vm 0

Problem 11.36. Given Vbn ¼ 120 ∠45 ½V, ﬁnd Van and Vcn assuming: A. The positive abc phase sequence B. The negative acb phase sequence Express your result in phasor form. Make sure that the phase ranges from 180 to +180 . To check the solution, you may want to use the corresponding phasor diagram shown in the ﬁgure for the previous problem.
120
Z
+


+
11.3.3 Wye (Y) Source and Load Problem 11.40. Repeat Problem 11.38 for the Conﬁgurations for Threephase Circuits circuits shown in the ﬁgure that follows: 11.3.4 Application: Examples of Threephase Source and the Load Z2 11.3.5 Solution for the Balanced ThreePhase WyeWye Circuit Vm 0 + Z1 Z1 11.3.6 Removing the Neutral Wire in Z2 LongDistance Balanced Highpower Vm 120 Transmission Z Problem 11.37 A. Draw the circuit diagram for a generic threephase fourwire balanced wyewye power distribution system. B. Labelphasevoltages and phaseimpedances (load impedances per phase). C. Label line currents. Problem 11.38. A threephase circuit is shown in the ﬁgure that follows: A. Is it a balanced wyewye circuit? B. If not, show your corrections on the ﬁgure.
+
Vm 0

+

Vm
120

Vm
120
2Z1
Z1
Z1
Vm
120
1
Z2
Problem 11.41. Prove that Eq. (11.28) of this chapter for line voltages also holds for the negative phase sequence to within the substitution 30 ! 30 . Problem 11.42. The local electric service in the European Union is provided by a threephase fourwire abcn wye system with the line voltages equal to 400 V rms each (socalled Niederspannungsnetz): A. Determine the rms phase voltages. B. By connecting terminals abcn in any sequence of your choice, could you in principle obtain the rms voltages higher than 400 V? Problem 11.43. Determine line currents in the balanced threephase wyewye circuit shown in the ﬁgure that follows. You are given the acb sequence of phase voltages Van ¼ 170∠0 ½V,
XI573
+
Chapter 11
AC Power and Power Distribution
Vbn ¼ 170∠120 ½V, Vbn ¼ 170∠120 ½V, and load impedance per phase, Z ¼ 8 þ j30 Ω. Plot phasor currents on the phasor diagram that follows. Ia
b
n

Vcn
Z
Z
Ib


+
+
a
Vbn
Van
N
Ic
11.4 Power in Balanced ThreePhase Systems: Deltaconnected ThreePhase Circuits 11.4.1 Intantaneous Power 11.4.2 Average Power, Power, and Apparent Power
Problem 11.45. In a threephase balanced wyewye system, the rms phase voltages are 120 V, and the rms line currents are 10 A. The impedance has the power angle of θ ¼ 75 . Find: 1. The instantaneous load power 2. The average load power
Z
+ c
Im
Problem 11.46. In a threephase balanced wyewye system, the rms line voltages are 400 V, and the rms line currents are 10 A. The impedance has the power angle of θ ¼ 60 . Find: 1. The instantaneous load power 2. The average load power
Re
Problem 11.44. In the balanced threephase wyewye circuit shown in the ﬁgure that follows, the power line resistance and inductance are additionally included into consideration. The threephase source operates at 60 Hz; R ¼ 2 Ω, L ¼ 9:6 mH. You are given the abc sequence of phase voltages Van ¼ 170∠0 ½V, Vbn ¼ 170∠ 120 ½V, Vbn ¼ 170∠120 ½V, and load impedance per phase, Z ¼ 7 þ j30 Ω. A. Determine line currents. B. Plot phasor line currents on the phasor diagram to the previous problem.
Problem 11.47. In the threephase system shown in the ﬁgure that follows, Z ¼ 40∠60 . The sources have the relative phases 0, 120 , þ 120 . The rms line voltages are 208 V. Determine: A. The type of the threephase system B. Instantaneous power delivered to the threephase load C. Average power delivered to the threephase load a
L

Z
+


Ib
Z
Z
c
Z

Vcn
b
Z

R
Z
+
+
+
L
+
+
a
Vbn
Van
Ia

R
Reactive
R
c
Ic
L
Problem 11.48. A balanced wyewye threephase system in the ﬁgure that follows uses lossless transmission lines and operates at 60 Hz.
XI574
Chapter 11
Problems a)
120 0
120
120
+
a
120
30 60
120
c


Z
Z
b)
n
N
+
Z
a

+ c

Vcn

b
N
+
+
a
Vbn
+
n
Van
30 60


30 60 b
+
The linetoneutral voltages have the amplitudes of 170 V,V m ¼ 170 V. Every phase impedance is a 92mH inductance in series with a 20 Ω resistance. Find the instantaneous load power.


C
c
Problem 11.53. A threephase balanced deltadelta system is shown in the ﬁgure that follows. Its wyewye equivalent is sought, which is shown in the same ﬁgure. For the wyewye system, write the corresponding voltage and impedance values in the phasor form close to every circuit element. a)
400 30
+

Problem 11.50. A threephase induction motor is modeled by a balanced wye load. The motor (active) power is 2.5 kW; the line current is 10 A rms, and the phase voltage of a threephase wye source is 120 V rms. Determine the power factor of the motor.
A
B
+
+
Problem 11.49. In the previous problem: A. Determine the load average power, reactive power, and the apparent power. B. Do these powers coincide with the corresponding source measures?
b
b
A
90 60
+
400 150
400
90
90 60
90 60 C
c
b)
B

+
a
b


+
n
N

+
Problem 11.52. A threephase balanced wyewye system is shown in the ﬁgure that follows. Its deltadelta equivalent is sought, which is shown in the same ﬁgure. For the deltadelta system, write the corresponding voltage and impedance values in the phasor form close to every circuit element.

11.4.4 Balanced DeltaConnected Load 11.4.5 Balanced DeltaConnected Source
a
+
Problem 11.51. In the previous problem, the motor (active) power is 9 kW; the line current is 15 A rms, and the line voltage of a threephase wye source is 400 V rms. Determine the power factor of the motor.
c
XI575
Chapter 12
Chapter 12: Electric Transformer and Coupled Inductors Overview Prerequisites:  Knowledge of complex arithmetic  Knowledge of basic circuit analysis (Chapters 3 and 4)  Knowledge of self and mutual inductances (Chapter 6)  Knowledge of phasor/impedance method for AC circuit analysis (Chapter 8) and of basic AC power analysis (Chapter 11) Objectives of Section 12.1:  Derive the ideal transformer model from the first principles  Understand the role of the ideal magnetic core  Understand the role of Faraday’s law and Ampere’s law  Prepare the background for introducing magnetic circuits  Understand and apply the dot convention  Relate ideal transformer model to a model with dependent sources Objectives of Section 12.2:  Be able to analyze electric circuits with ideal transformer  Learn about load and source reflections  Learn about impedance matching via transformers  Learn about electric power transfer via transformers Objectives of Section 12.3:  Derive equations for useful transformer types autotransformer, multiwinding transformer, and centertapped transformer from the first principles  Understand the role of the centertapped transformer for singleended to differential transformation and for power division Objectives of Section 12.4:  Understand the physical background of the Steinmetz model and relate the model parameters to real transformers  Be able to analyze the nonideal transformer model  Define transformer voltage regulation and power efficiency  Briefly discuss the highfrequency transformer model
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_12
XII577
Chapter 12
Electric Transformer and Coupled Inductors
Objectives of Section 12.5:  Introduce the model of two coupled inductors from the first principles  Learn how to analyze electric circuits with coupled inductors  Learn about the useful conversion to the Tnetwork of uncoupled inductances  Obtain the basic exposure to wireless inductive power transfer including its major features and challenges Application examples:  Electric power transfer via transformers  Wireless inductive power transfer  Coupling of nearby magnetic radiators
Keywords: Electric transformer (primary winding, secondary winding, circuit symbol, isolation transformer, instrumentation transformer, current transformer, clamp on ammeter, potential transformer, exciting current, magnetizing current, magnetizing inductance, magnetizing reactance, power conservation, stored energy, turns ratio, stepup transformer, stepdown transformer, highvoltage side, lowvoltage side, transformer rating, dot convention, dotted terminals, voltage polarity, current reference directions, summary of reference directions, mechanical analogies), Ideal transformer model (ideal magnetic core, ideal opencircuited transformer, ideal transformer equations, ideal transformer equations in phasor form, power conservation, stored energy, model in terms of dependent sources), Ampere’s law (linked current, for ideal magnetic core, for multiwinding transformer), Referred (reﬂected) source network in the secondary, Referred (reﬂected) load impedance in the primary, Load reﬂection, Source reﬂection, Reﬂected resistance, Reﬂected inductance, Reﬂected capacitance, Transformer as a matching circuit, Matching realvalued impedances, Matching arbitrary complex impedances, Partial matching condition, Power transfer via transformers (for ﬁxed load voltage, sendingend voltages, for ﬁxed source voltage), Autotransformer (step down, step up, circuit symbol, ideal transformer equations), Multiwinding transformer (ideal circuit equations, Ampere’s law, telephone hybrid circuit), Centertapped transformer (ideal transformer equations, singleended to differential transformation, 180 power divider, 180 power splitter), Real transformer (nonideal lowfrequency model, Steinmetz model, Steinmetz parameters (magnetizing reactance, core loss resistance, primary leakage reactance, secondary leakage reactance, primary ohmic resistance, secondary ohmic resistance), nonideal transformer model terminology/analysis, voltage regulation, transformer power efﬁciency, nonideal highfrequency model), Model of two coupled inductors (equations, circuit symbol, circuit analysis, solution for N coupled inductors, stored energy, conversion to Tnetwork), Mutual inductance (deﬁnition, of ideal transformer, of two coaxial coils), Coupling coefﬁcient (deﬁnition, largest possible value, trend), Wireless inductive power transfer (application, basic model, examples of), Mutual coupling for nearby magnetic radiators (arrays of magnetic radiators), Lenz’s law
XII578
Chapter 12
Section 12.1: Ideal Transformer as a Linear Passive. . .
Section 12.1 Ideal Transformer as a Linear Passive Circuit Element 12.1.1 Electric Transformer General An electric transformer is a simple and versatile device, which targets AC power transfer from one electric circuit to another. These circuits are coupled via a timevarying magnetic ﬂux linking two or more coils. There is no direct electric connection between the coils. The transformer cannot transfer power between the DC circuits. Analysis of transformers involves many principles that are basic to the understanding of electric machines. Transformers are primarily used to: 1. 2. 3. 4.
Change the voltage level in power electronics AC circuits. Insulate one AC (or RF) circuit from another (isolation transformers). Match the impedance of the source and the load in electronic circuits. Measure AC voltages and currents (instrumentation transformers).
As another everyday application example, we mention various DC power supplies (AC to DC converters or adapters), both switching and linear. These DC supplies power PCs, printers, modems, cordless phones, video game consoles, etc. at your home. Lowfrequency (bulky) or highfrequency (smaller) transformers are very important parts of these supplies, irrespective of their particular construction.
Function Although the transformer typically consists of two coupled inductors see Fig. 12.1 its function is principally different from that of the familiar inductance. While the inductance is an energystorage (and energyrelease) circuit element, the ideal transformer, as a new circuit element, never stores any instantaneous energy. It does not possess any inductance (or impedance in general) either. Approach The model a twowinding electric transformer introduced in this section does yet not use the concept of a magnetic circuit. Instead, we accurately formulate and employ Faraday’s law and Ampere’s law directly. The same transformer model in the framework of magnetic circuits is revisited in the next chapter. Through the text, we use opposite reference directions for the transformer currents; the equal directions are also discussed in the text. 12.1.2 Ideal OpenCircuited Transformer: Faraday’s Law of Induction We will perform transformer analysis in several steps illustrated in Fig. 12.1. XII579
Section 12.1: Ideal Transformer as a Linear Passive. . .
Chapter 12
law of induction see Eq. (6.19) and (6.33) this ﬂux is uniquely determined by the winding voltage (which is equal to the source voltage) in the form υ 1 ðt Þ
N1
dΦðt Þ dt
ð12:2Þ
Exciting Current and Inductance for the Ideal Magnetic Core An inductor current iΦ(t) which would be present in Fig. 12.1a is called the exciting current (or magnetizing current). Accordingly, inductance Lm is called the magnetizing inductance. The magnetizing inductance is found using Eq. (6.23). The exciting current is then found using Eq. (6.19) of the same chapter, which is valid in both static and dynamic cases. This gives Lm
μ0 μr AN 21 , l
i Φ ðt Þ
N1
Φ ðt Þ Lm
ð12:3Þ
where A is the core cross section shown in Fig. 12.1. When the relative magnetic permeability μr of the core is very high, the coil inductance Lm is very large. Therefore, the corresponding inductor current iΦ(t) is quite small. An ideal magnetic core assumes that μr ! 1. Therefore, according to Eq. (12.3), Lm ! 1,
i Φ ðt Þ
0,
μr ! 1
ð12:4Þ
Equation (12.4) corresponds to the ideal transformer model. The primary winding becomes an open circuit of inﬁnite inductance as shown in Fig. 12.1b. However, the ﬁnite magnetic ﬂux Φ(t) is still established in the core. There is no contradiction here since a negligible exciting current iΦ(t) is necessary to establish the ﬁnite ﬂux Φ(t) in a core with the inﬁnitely high permeability (inﬁnitely high inductance). The situation is somewhat similar to an operational ampliﬁer with the negative feedback where the negligible input different voltage controls the largesignal ampliﬁer operation.
Using Faraday’s Law a Second Time: Relation Between Transformer Voltages As a next step, another coil with N2 turns the secondary winding of the transformer can be added as shown in Fig. 12.1c. The core ﬂux Φ(t) also links the secondary winding and creates an opencircuit voltage at its terminals. No ﬂux leakage in air is permitted in the ideal model. Faraday’s law is applied a second time, which yields υ 2 ðt Þ
N2
dΦðt Þ dt
ð12:5Þ
The plus sign implies the dot convention to be discussed shortly. From Eqs. (12.2) and (12.5), the voltage ratio becomes
XII581
Chapter 12
Section 12.1: Ideal Transformer as a Linear Passive. . .
3. No other loss in the core called the iron loss. The iron losses would include hysteresis loss and eddy current loss. Real transformers studied in Section 12.4 deviate from this ideal circuit model often very signiﬁcantly. The ratio of output power to input power is called the efﬁciency of the transformer. For large power transformers, the efﬁciency can be in excess of 98 %. For RF (radiofrequency) transformers, the efﬁciency is typically much lower. Two methods of analysis can be used to study realistic transformers: 1. An extended equivalent circuit model that includes the present ideal transformer model plus extra inductances and resistances, see Section 12.4. 2. A different mathematical model of magnetically coupled circuits with self and mutual inductances, see Section 12.5.
Terminology Engineers have adopted a special terminology when dealing with transformers: A. The ratio N1 : N2 is the turns ratio of the transformer. A transformer with a primary winding of 100 turns and a secondary winding of 200 turns has a turns ratio of 1:2. A transformer with a primary winding of 200 turns and a secondary winding of 150 turns has a turns ratio of 4:3. B. When N 2 > N 1 , the transformer increases the input AC voltage; it is called a stepup transformer. C. When N 2 < N 1 , the transformer decreases the input AC voltage; it is called a stepdown transformer. D. The winding with a higher number of turns is the highvoltage (HV) side of the transformer. E. The winding with a smaller number of turns is the lowvoltage (HV) side of the transformer. In Figs. 12.1, 12.2, and 12.3, we have used the opposite current reference directions for the two dotted terminals. Quite often, the same reference directions are employed. This is to underscore the fact that either winding may serve as the input of the transformer. Sign minus should then be inserted into Eq. (12.10) which relates i1 and i2.
Transformer Rating Power transformers seldom drive purely resistive loads. Therefore, their power rating is given in VA (voltamperes) or kVA instead of watts, identical to the complex power deﬁned in Section 11.1. More precisely, this is the apparent load power deﬁned by Eq. (11.21d). Consider a popular example of a transformer that carries the following information on a nameplate or in a reference manual: 10 kVA, 1100:110 V. The voltage rating means the one transformer winding (highvoltage side) is rated for 1100 V, whereas another (lowvoltage side) for 110 V. The turns ratio is the voltage ratio, N 1 : N 2 10. The corresponding current ratings are 9.09 A rms and 90.9A rms, respectively. XII587
Chapter 12
Electric Transformer and Coupled Inductors
Section 12.2 Analysis of Ideal Transformer Circuits 12.2.1 Circuit with a Transformer in the Phasor Form Consider a generic transformer circuit shown in Fig. 12.6a in the frequency domain. The circuit is given in the phasor form assuming a harmonic signal source. A source circuit with the phasor voltage VS and impedance ZS is connected to a load with impedance ZL via an ideal transformer studied in the previous Section. Since any linear AC source network can be represented in the form of its Thévenin equivalent, and any linear passive source can be replaced by the equivalent impedance, Fig. 12.6a is a rather general interpretation of the transformer setup with linear networks in Fig. 12.4. When written in the phasor form, the ideal transformer model given by Eqs. (12.6) and (12.10) does not need a special treatment. We simply replace the realvalued voltages and currents by phasors: υ2 i2
N2 N2 υ1 ) V2 V1 N1 N1 N1 N1 i1 ) I2 I1 N2 N2
ð12:14aÞ ð12:14bÞ
In power electronics, phasor voltage and phasor current in Fig. 12.6a are often expressed in terms of rms values times the phasor (the complex exponent). This is in contrast to the previous analysis where we have used the amplitude of a sinusoidal function times the phasor. The circuit analysis remains the same, but the factor of 2 in the expressions for the power disappears. We will mention this convention every time when required.
12.2.2 Referred (Or Reﬂected) Source Network in the Secondary Side What voltage and impedance does the load see in Fig. 12.6a? In other words, what is the Thévenin equivalent circuit of the source and the transformer combined? To answer this question, we ﬁnd Thévenin equivalent voltage VT of the oneport network with terminals a and b in Fig. 12.6a as its opencircuit voltage. Using Eqs. (12.14a, b) and setting ZL 1 in Fig. 12.6a yields I2
0 ) I1
0 ) VT V2
N2 V1 N1
N2 VS N1
ð12:15Þ
The Thévenin equivalent impedance ZT is found by dividing the opencircuit voltage by the shortcircuit current ISC I2 . Setting ZL 0 in Fig. 12.6a, one ﬁnds the shortcircuit current ZL
0 ) V2
V1
0 ) ISC
I2
N1 I1 N2
N 1 VS N 2 ZS
ð12:16Þ
XII590
Chapter 12
Section 12.2: Analysis of Ideal Transformer Circuits
Matching Arbitrary Complex Impedances Can a transformer match two arbitrary complex impedances? Unfortunately, it cannot. The transformer operates as an impedance multiplier; it multiplies (or divides) by a real number. On the other hand, the complex impedance match requires two complex conjugate impedances. A transformer could often provide a “better” match (see the summary to this chapter) but cannot perform impedance matching in full. Other circuit elements (capacitance or inductance) may be necessary to complete the task. 12.2.5 Application Example: Electric Power Transfer via Transformers Circuit with a Fixed Load Voltage Figure 12.8a shows a circuit for transmitting electric power over a long transmission line with the total resistance R and the total inductance L. The circuit in Fig. 12.8a is converted to a phasor form ﬁrst. We consider the phasors in terms of rms values. Two competing schemes are studied: transmission without transformers (see Fig. 12.8b) and a transmission scheme with two 1:20 and 20:1 ideal transformers see Fig. 12.8c. In order to compare the performance of two circuits (with and without transformers), it is assumed that the load phasor voltage VL (V rms) and the load phasor current IL (V rms) have the same values in both cases. This guarantees us the same average power delivered to the load. The power loss (ohmic loss) in the line resistance in Fig. 12.8b is Ploss
RjIL j2
ð12:24Þ
The power loss in the line resistance in Fig. 12.8c is decreased by a factor of 400: Ploss
1 RjIL j2 400
ð12:25Þ
since the line current is exactly 20 times less than in the ﬁrst case. This result is independent of the particular values of R and L. Simultaneously, the line voltages increase by a factor of 20, but the load voltage still remains the same due to the stepdown transformer. Thus, using a pair of transformers allows us to choose an economically optimum voltage for transmitting a given amount of power. The line sees a high voltage of the secondary of the ﬁrst transformer while the load essentially sees the source voltage. Not only does the use of transformers greatly decrease the line loss, but it also potentially allows us to use smaller source voltages (sendingend voltages). The required source power also decreases.
XII595
Chapter 12
i1
N 2 i2
Section 12.3: Some Useful Transformers 0 ) i2
1 i1 N2
ð12:36Þ
which is a particular case of the ideal transformer equation with N 1 1. Thus, by measuring current i2 or the associated voltage, the unknown current i1 can be established. Note that Fig. 12.15 and Eq. (12.36) only demonstrate the very basic concept; the practical current transformer design is signiﬁcantly more elaborated. Another type of instrumentation transformers potential transformers is used for accurate AC voltage measurements.
XII603
Chapter 12
Electric Transformer and Coupled Inductors
Section 12.4 RealTransformer Model 12.4.1 Model of a Nonideal LowFrequency Transformer Figure 12.16 shows a linear circuit model of a practical lowfrequency (60 or 50 Hz) twowinding transformer. This circuit is also known as the Steinmetz model and its parameters as Steinmetz parameters. The complete model will be explained in several steps: 1. Consider the opencircuited transformer ﬁrst. The primary winding is characterized by a large but ﬁnite magnetizing inductance, Lm, which was deﬁned in Eq. (12.3). This is the standard inductance expression for a long solenoid with the magnetic core. It is shown in Fig. 12.16a. 2. A small amount of magnetic ﬂux is still situated outside the core so that Eq. (12.3) needs to be reﬁned. The total inductance as in Fig. 12.16 of the primary winding is therefore somewhat larger: L1
Lm þ Ll1
ð12:36Þ
where a small addition Ll1 is called the leakage inductance, see Fig. 12.16a. 3. A practical primary winding has a certain ohmic resistance, R1, which is placed in series with L1. Simultaneously, the core loss (hysteresis and eddy current loss) in the magnetic material consumes some extra current. It is modeled by an equivalent resistance Rc, which is placed in parallel with Lm. Rc is often called the core loss resistance. The resulting equivalent circuit in Fig. 12.16b is also an equivalent circuit of a nonideal inductor with the magnetic core. 4. Finally, the secondary winding is added and a load is connected to the transformer. This results in the complete equivalent circuit model of Fig. 12.16c. Two new circuit parameters Ll2, and R2 are the leakage resistance and the ohmic resistance, respectively, of the secondary winding. In general, all model parameters, and especially Lm and Rc, are frequency dependent.
12.4.2 Model Parameters and Their Extraction Table 12.2 lists typical equivalent circuit values for three distinct power transformers of different power ratings and compares them with the ideal transformer model. Not the inductances themselves but the rather reactance values are given at angular frequency ω 2πf where f 50 or 60 Hz.
XII604
Chapter 12
Section 12.5: Model of Coupled Inductors
di1 di2 þM dt dt di1 di2 þ L2 þM dt dt
υ1
þL1
υ2
ð12:50Þ
of two coupled inductors. Here the coefﬁcient M > 0 is the mutual inductance of two coils, which is also measured in henries. You can see from the second expression in Eqs. (12.50) that the mutual inductance determines the voltage induced in inductor #2 due to changes of the electric current in inductor #1. Alternatively, from the ﬁrst expression in Eqs. (12.50), the same mutual inductance determines the voltage induced in inductor #1 due to changes of the electric current in inductor #2. The signs in Eq. (12.50) are important. They follow a few rules: 1. Eqs. (12.50) corresponds to the dot convention and voltage polarities/current directions shown in Fig. 12.19. 2. If one of the current reference directions, say the direction of i2, is selected oppositely, we will have to use the minus sign in the respective terms in Eqs. (12.50). 3. If one of the voltage polarities changes, we have will have to use the minus sign where required in Eqs. (12.50). Figure 12.19 also shows the circuit symbol for two coupled inductors. This symbol does not include the magnetic core; event if it is present in reality. Also, the mutual inductance is shown by an arrow.
12.5.2 Analysis of Circuits with Coupled Inductors Solving circuits with the coupled inductors requires care. Consider, for example, a simple circuit shown in Fig. 12.20 in frequency domain. We cannot apply the impedance relations following from Eq. (12.49). For two coupled inductors, V1 6 þjωL1 I1 and V2 6 jω L2 I2 . Instead, we should convert Eqs. (12.50) to the phasor form ﬁrst and obtain V1 V2
jωL1 I1 þ jω M I2 jωM I1 þ jω L2 I2
ð12:51Þ
These are two equations for four unknowns V1, I1, V2, I2. The two remaining equations are KVL for the left part of the circuit and KVL for the right part of the circuit, i.e., V1
þVS and V2
RL I2
ð12:52Þ
XII613
Chapter 12
Section 12.5: Model of Coupled Inductors
Solution for N Coupled Inductors The solution method for two coupled inductors may be straightforwardly extended to the case of N coupled inductors. In this case, Eqs. (12.51) will involve a matrix of self and mutual inductances (the inductance matrix L) on the size N N. KVL applied to every individual inductor circuit gives N remaining algebraic equations. The coupled system of 2N algebraic equations is then solved for unknowns I1, . . ., IN and V1, . . ., VN. The systems of coupled inductors are used in many applications, mostly for (multiple output) switchedmode power conversion. They could also be used for sensor and other purposes in bioelectromagnetics and other disciplines see Fig. 12.21.
Fig. 12.21. An array of coupled inductors (small coils) located on top of a humanhead phantom. This hypothetic setup was tested for applications related to brain stimulation.
Finding Mutual Inductance(s) Despite the dynamic nature of Eqs. (12.50), the mutual inductance deﬁned previously by Eq. (6.16) is inherently a static quantity. It may be computed from the corresponding 3D magnetostatic analysis (often quite complicated). We will brieﬂy review this question at the end of this section. Right now, however, we will establish the highest possible value of M, which is achieved for the ideal transformer. Conversion to TNetwork It is may be convenient to replace a circuit with two coupled inductors by a circuit without magnetic coupling. This can be done using either a Tnetwork or a Πnetwork of three inductances see Section 3.3 of Chapter 3. Figure 12.22 illustrates a conversion to the Tnetwork for the circuit from Fig. 12.20. For leftmost inductance La, rightmost inductance Lb, and shunt inductance Lc of the Tnetwork, one has La
L1
M,
Lb
L2
M,
Lc
M
ð12:57Þ
The proof is based on establishing υ i relationships for both twoport networks in Fig. 12.22 and demonstrating their identity.
XII615
Chapter 12
Electric Transformer and Coupled Inductors
10 10 10 10 10
received voltage, mV
2
1
0
1
2
0
0.4
0.8
1.6
1.2
2
distance d, m
Fig. 12.26. Voltage amplitude in coil #2 when the current amplitude in coil #1 is 100 mA.
12.5.5 Application Example: Coupling of Nearby Magnetic Radiators The mutual coupling between the two inductors (a transmitter and a receiver) is key for the wireless inductive power transfer. At the same time, the mutual coupling may have a negative effect when the transmitter includes two or more independently driven coils assembled in a coil array. The mutual coupling between the individual transmit coils may reduce the individual coil current, i.e., reduce the resulting total magnetic ﬁeld. Circuit with Two Identical Radiators The circuit shown in Fig. 12.27 formalizes the problem. It models the coupling effect between two nearby identical magnetic radiators. This circuit is important for nearﬁeld wireless power transfer with coil arrays including medical applications. Our goal is to express the source phasor current IS through the circuit parameters for two distinct cases: A. Mutual coupling is absent. B. Mutual coupling is present; the mutual reactance is X M > 0. jXM RS
RS
IS VS
IS
+ 
jX1
jX1
+ 
VS
Fig. 12.27. Modeling mutual coupling of two transmit coils.
To solve the circuit, we convert the two coupled inductors to a Tnetwork. The resulting circuit is shown in Fig. 12.28.
XII622
Chapter 12
Section 12.5: Model of Coupled Inductors
RS
j(X1Xm)
j(X1Xm)
RS
IS VS
IS
+ 
jXM
+ 
VS
Fig. 12.28. Equivalent circuit with uncoupled inductors.
When the mutual coupling is absent, the central inductor becomes a wire; both currents are given by IS
VS RS þ jX 1
ð12:70Þ
When the mutual coupling is present, we can use KVL for the either circuit loop, which gives VS þ RS IS þ jðX 1
X M ÞIS þ 2jX M IS
0
ð12:71Þ
Therefore, IS
VS RS þ jðX 1 þ X M Þ
ð12:72Þ
Note that the sign in front of XM may vary depending on coil orientation. For example, it is positive for two coaxial coils (or loops) with inphase currents (or inphase ﬂuxes) and is negative otherwise.
Tuned Radiators The individual radiator circuit should be tuned to the operating frequency to maximize the circuit current/magnetic ﬁeld. The tuning is typically achieved by a series capacitor with reactance X 1 , which exactly cancels the inductor’s reactance þX 1 . If this is the case, Eq. (12.73) is transformed to IS
VS ) j IS j RS þ jX M
jVS j q R2S þ ðX M Þ2
ð12:73Þ
Thus, for the tuned radiators, the mutual coupling reduces the circuit current and the magnetic ﬁeld, irrespective of the coil orientation. When XM is small compared to the source resistance, this effect is of little value. Altogether, it can be eliminated by adjusting the value of the tuning capacitors.
XII623
Chapter 12
Problems transformer shown in the following ﬁgure has the form υS ðtÞ ¼ 5 cos ωt ½V. The source current into the dotted terminal is iS ðt Þ ¼ 10 cos ωt ½A. You are given N 1 ¼ 200, N 2 ¼ 50. Determine voltage υ2(t) and current i2(t) in the secondary. i2
+
i1
Reprinted from Micro Hydropower Systems Canada 2004, ISBN 0662358805
N1
+
v2(t)
N2
v1(t)
The system uses a single phase induction generator with the rms voltage of 240 V. The system serves four small houses, each connected to the generator via a separate transmission line with the same length of 3000 m. Each line uses AWG#10 solid aluminum wire with a diameter of 2.59 mm. The house load in every house is an electric range with the resistance of 20 Ω. Determine total active power delivered by the generator, PS, total power loss in the transmission lines, Ploss, and total active useful power, PL: 1. When no transformers are used; 2. When a 1:5 stepup transformer is used in powerhouse and a 5:1 stepdown transformer is used at home. Problem 12.32. Solve the previous problem when the distributed line inductance is additionally taken into account. The inductance per unit length of a twowire line is given by μ0 d 1 ln þ where a is the wire radius and d is a 4 π the separation distance. Assume the separation distance of 1 m. The operation frequency is 50 Hz.
12.3 Some useful transformers 12.3.1 Autotransformer Problem 12.33. A voltage source connected to the primary winding of an ideal stepup auto


Problem 12.34. Solve the previous problem for the circuit shown in the following ﬁgure. i1
+
i2
N1 v1(t)
+
N2
v2(t)


Problem 12.35. For the circuit shown in the following ﬁgure, VS ¼ 100∠45 ½V, ZS ¼ 5 þ j5 Ω, ZL ¼ 5 þ j5 Ω, and N 1 : N 2 ¼ 4 : 1. Find phasor current of the source, IS. Express your result in polar form. Assume the ideal autotransformer. ZS IS N1
VS
+ 
N2
ZL
Problem 12.36. Find the equivalent input impedance, Zin, for two autotransformer circuits shown in the following ﬁgure.
XII635
Chapter 12
Electric Transformer and Coupled Inductors
a) N1
Zin
N2
ZL
for every individual load are shown in the ﬁgure. Determine: A. Turns ratio, N1 : N2 of the transformer B. rms value of input current i1 i1
N1:N2
b)
+ N1
ZL
N2
Zin
12.3.2 Multiwinding Transformer 12.3.3 CenterTapped Transformer: SingleEnded to Differential Transformation Problem 12.37. In the circuit shown in the following ﬁgure, N 1 ¼ 2N 2 ¼ N 3 . How is the instantaneous power partitioned between the two secondary windings if both of them are terminated into the same load resistances? To answer this question, express both p2(t) and p3(t) in terms of p1(t). i2 i1
+ v1(t)

i3
N1 N3
+ v2(t)
N2

+ v3(t)

Problem 12.38. In the circuit of the previous problem, N 1 ¼ 2N 2 ¼ 2N 3 . How is the instantaneous power partitioned between the two secondary windings if winding #2 is terminated into resistance R and winding #3 is terminated into resistance 3R? Problem 12.39. Determine the turns ratio for the centertapped transformer in Fig. 12.13. Problem 12.40. A household is using an ideal centertapped distribution transformer shown in the following ﬁgure. All voltage values are the rms values. The resistive loads include a TV, a microwave, and a kitchen range. The powers
120 V
200 W TV
120 V
1500 W MW
2400 V

10kW Range
Problem 12.41. Determine phasor currents IR1, IR2, and IR0 for the centertapped balanced transformer circuit shown in the following ﬁgure. You are given the source phasor voltage VS ¼ 10∠0 ½V, resistance values R ¼ 50 Ω, and turns ratio N 1 : N 2 ¼ 1. N1:N2
IR1 R
VS
+ 
IR0
R
IR2
Problem 12.42. Solve the previous problem in a general form, i.e., express phasor currents IR1, IR2, and IR0 in the (generally unbalanced) circuit shown in the following ﬁgure in terms of given circuit parameters VS, Z1, Z2, and turns ratio a ¼ N 1 : N 2 . N1:N2
IR1 Z1
VS
+ 
IR0
Z2
IR2
12.3.4 Current Transformer Problem 12.43. Determine current i2(t) in an ideal current transformer shown in the ﬁgure given that i1 ðt Þ ¼ 10 cos ωt ½A. Count the number of turns.
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Chapter 12
Electric Transformer and Coupled Inductors
frequency domain. Given the rated load with a power factor of 0.7 lagging, determine: A. Percentage regulation B. Percentage efﬁciency for transformer #1 model from Table 12.2 I2
1
a I2
IS VS
+ 
RC jXm
+
+
aE2
E2


Problem 12.58. In the circuit shown in the following ﬁgure, determine source phasor current IS given that RL ¼ 1 Ω, RS ¼ 2 Ω, and VS ¼ 15∠0 ½V. j3
a=N1/N2
R1+jXl1
Problem 12.57. Solve Problem 12.55 when the mutual reactance is equal to 3 Ω.
RS
R2+jXl2
+ VL
ZL

IS VS
+ 
j5
j4
RL
Problem 12.53. Repeat the previous problem for transformer #2 model from Table 12.2.
12.5.4 Application Example: Wireless Problem 12.54. Repeat Problem 12.52 for Inductive Power Transfer 12.5.5 Application Example: Coupling transformer #3 model from Table 12.2. of Nearby Magnetic Radiators
12.5 Model of Coupled Inductors 12.5.1 Model of Two Coupled Inductors 12.5.2 Analysis of Circuits with Coupled Inductors 12.5.3 Coupling Coefﬁcient Problem 12.55. Solve a circuit with two coupled inductors in the ﬁgure below in frequency domain: A. Determine the load phasor current and the source phasor current. B. Determine phasor voltages V1 and V2 given that VS ¼ 10∠0 ½V and RL ¼ 10 Ω. I1
Problem 12.59. Two small coaxial ceramiccore coils with r1 ¼ r2 ¼ 2:0 cm and with N 1 ¼ N 2 ¼ 100 are separated by 1 m. What is the voltage signal induced in the second coil (RX) if the current in the ﬁrst coil (TX) is given by i1 ¼ 100 mA sin ðωtÞ ? The operation frequency is 1 MHz. Problem 12.60. Repeat the previous problem when the operation frequency changes to 10 MHz. Problem 12.61. In the circuit shown in the following ﬁgure, ﬁnd source phasor current IS in time domain iS(t) given that υS ðt Þ ¼ 10 cos ωt ½V.
I2
j4
4
+
+ 
VS
V1 j4

+ j4
V2
j1
j4
j4
4
IS RL

VS
IS
+ 
j3
j3
+ 
VS
Problem 12.56. Solve the previous problem when the mutual inductance is exactly zero.
XII638
Part IV Digital Circuits
Chapter 13
Chapter 13: Switching Circuits
Overview Prerequisites:  Knowledge of basic circuit analysis Objectives of Section 13.1:  Understand the functionality of a semiconductor transistor switch  Characterize the operation of a transistor switch by differentiating between the groundside pulldown switch (NMOS transistor) and the powerside pullup switch (PMOS transistor)  Appreciate the value of MOSFET threshold voltage  Solve simple switching circuits Objectives of Section 13.2:  Become familiar with simple switching motor controllers and load controller switches  Track the operation of the Hbridge and the half Hbridge motor controllers  Obtain initial exposure to pulsewidth modulation (PWM) and motor speed control Objectives of Section 13.3:  Establish the relation between symbols for logic gates and underlying electric circuits on transistor level  Review basic logic gates  Obtain initial exposure to Boolean algebra and logic circuit analysis and synthesis  Understand the functionality of a semiconductor memory cell Application Examples:  PWM motor controller  Logic gate motor controller
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_13
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Chapter 13
Switching Circuits
Keywords: Electronic switch, Switch control voltage, Groundside switch, Powerside switch, Series switch, Pulldown switch, Pullup switch, Metaloxidesemiconductor (MOS) transistors, NMOS transistors, PMOS transistors, Complementary transistors, CMOS circuits, Switching transistors, Switching diagram, Transistor threshold voltage, Matched switching transistors, Control switching circuits, Switching quadrants, Half Hbridge, Full Hbridge, Motor speed controller, Motor control states (forward mode, reverse mode, free run to a stop, motor brake), Onequadrant switch, Forbidden states, Pulsewidth modulation (PWM), Duty cycle of PWM, Average supply voltage of PWM, Logic inverter, NOT gate, Truth table, NOR gate, OR gate, NAND gate, AND gate, Switching algebra, Boolean algebra, Boolean expressions, Laws of Boolean algebra (commutative law of addition, commutative law of multiplication, associative law of addition, associative law of multiplication, distributive law), De Morgan’s laws, Exclusive OR (XOR) gate, Exclusive NOR (XNOR) gate, Logic circuit analysis, Logic circuit synthesis, Hardware description language (HDL), Sumofproducts approach, Productofsums approach, Karnaugh maps, Static random access memory (SRAM), Latch, Static RAM cell, Access transistors
XIII642
Chapter 13
Switching Circuits
3. The electronic switch may use two distinct transistor types: the socalled metaloxidesemiconductor ﬁeldeffect transistors (MOSFETs) studied in Chapter 18 and the bipolar junction transistors (BJTs) studied in Chapter 17, respectively. 4. In this chapter, we will always implement MOSFET transistors since they are speciﬁcally used in digital circuits including microprocessors and computers. 5. The most important feature of the switch is that it consumes virtually no input power. Namely, the input current Iin into the control terminal in Fig. 13.1b is zero or close to zero, in contrast to the control voltage Vin. An electronic switch is an important part of many analog circuits including power conversion circuits (DC to DC, AC to DC, etc.), DC and AC motor drives, etc. The switch is capable of turning on and off large line currents between terminals a and b. For example, a properly designed electronic switch may in principle allow us to turn on a 1MW power plant with a single 9V battery. On the other hand, an electronic switch is also the heart of any digital circuit. We could in principle build lowpower switches using operational ampliﬁers studied in Chapter 5. However, a powerful, simple, versatile, and by far the fastest switch is a singletransistor switch.
13.1.2 Switch Position in a Circuit Depending on the switch position in a circuit, we distinguish between: 1. A groundside switch 2. A powerside switch 3. A series switch All three switching conﬁgurations are quite intuitive; they are shown in Fig. 13.2. Resistor RL designates a load. The switch position dictates the type of transistor to be used and the acceptable values of control voltages. The details are given in the following text. For example, the groundside switch is implemented with an ntype transistor (MOSFET or BJT). Such a transistor conducts by negative carriers electrons. The switch is normally open (which means that it is open at zero control voltage), but is closed at higher control voltage values. In contrast to that, the powerside switch is implemented with a ptype transistor (MOSFET or BJT). Such a transistor conducts by positive carriers holes. The switch is normally closed (closed at zero control voltage), but is open at such control voltage values that are close to the supply voltage. The series switch (the switch between two power blocks of a larger circuit) may use either transistor type. However, the control voltage must be higher than the voltage to be switched. From the viewpoint of a simple resistive load RL in Fig. 13.2, it really does not matter where the control switch is exactly located: on the ground side or on the power side. Hence, either type of the switch may be chosen. However, more sophisticated loads such as motors or solenoids are controlled by several switches that are located both on the ground side and on the power side and, thus, have quite distinct features.
XIII644
Chapter 13
Section 13.2: Power Switching Circuits
13.2.7 Application Example: PulseWidth Modulation (PWM) Motor Controller PWM Voltage Form If someone needs speed control, an obvious way may be to vary load voltage using a voltage divider with a variable resistor. For example, a 12V Mabuchi DC motor RS380PH3270 may operate at supply voltages from 4.5 V to 15 V, not necessarily at exactly the nominal voltage of 12 V. However, this method results in higher losses in the divider circuit. This method might also result not only in the decrease of the motor speed but also the motor current Ia and the instantaneous motor torque. Another way of controlling the speed is the pulsewidth modulation (PWM). In that case, the supply voltage to the motor is varied as a rectangular periodic waveform shown in Fig. 13.16.
14
VS, V
12
toff
ton
10
VDC
8 6
T
4 2 0 0
0.02
0.04
0.06
0.08 0.1 time, sec
Fig. 13.16. Pulsewidth modulation of the supply voltage to the motor.
The motor operates at its nominal voltage (12 V in Fig. 13.16) during the ON phase (with the duration ton). The power supply is disconnected from the motor during the OFF phase with the duration toff. The period of the periodic waveform in Fig. 13.16 is given by T
t on þ t off
The frequency (measured in hertz, 1 Hz
ð13:3Þ 1=s) of the waveform is given by
1 ð13:4Þ ½Hz T The duty cycle d (fraction) or D (percentage) of the periodic wave form is given by f
t on t on ð13:5Þ , D 100 % T T When both ON and OFF phases are equal, the duty cycle is said to be exactly 50 %. The average supply voltage VDC of PWM in Fig. 13.16 is given by d
XIII657
Chapter 13
Switching Circuits
Section 13.3 Digital Switching Circuits A digital circuit is the same electric circuit except for the fact that it performs a different function. For example, previously we have used transistors to control the motor. Now, we will use the same transistors and even in a similar conﬁguration, in order to perform logic operations and arithmetic operations. An immediate question to ask is how an electric circuit may be used to operate with numbers because we used to think that the circuit operates with voltages and currents only. The simple answer here is that virtually any digital circuit, including the computer that you are using right now, employs electric voltages as a carrier of information. A digital circuit consists of logic gates. A logic gate is a switching circuit that we shall study in this section. Any logic gate is an extension or a generalization of a logic inverter, which is considered ﬁrst.
13.3.1 NOT Gate or Logic Inverter Consider the circuit shown in Fig. 13.19. It includes the PMOS transistor (normally ON) as the powerside switch and the NMOS transistor (normally OFF) as a groundside switch. The circuit is powered by a 5V power The two transistors are matched supply. and have equal (absolute) threshold voltages V Tp V Tn 1 V. The circuit is identical to the half Hbridge considered in the previous section, but it serves a different purpose. 5 V=VS G
PMOS
Vin
Vout NMOS G
0V
Fig. 13.19. CMOS logic inverter or NOT gate.
The input voltage to the circuit follows the ﬁrst column of Table 13.3. Further, we shall use the chart from Fig. 13.4 of Section 13.1 in order to study the circuit behavior. When the input voltage is low (0 V), the NMOS switch is OFF and the PMOS switch is ON. The output voltage is 5 V. When the input voltage is high (5 V), the situation changes to the opposite: the NMOS switch is ON and the PMOS switch is OFF. The output voltage is 0 V. Hence the output voltage follows the second column of Table 13.3. Table 13.3. Output voltage of the logic inverter versus the input voltage. Vin 0V 5V
Vout 5V 0V
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Chapter 13
Section 13.3: Digital Switching Circuits
If we denote the 0V voltage by value 0 (low or false) and the 5V voltage by value 1 (high or true), Table 13.3 is converted to the socalled truth table Table 13.4. Table 13.4. Truth table for the logic circuit in Fig. 13.19—the logic inverter. We substitute 0 instead of 0 V and 1 instead of 5 V. Vin 0 1
Vout 1 0
The circuit thus performs logic inversion (substitutes zero instead of one or false instead of true and vice versa). The symbol for this is the logic NOT gate shown in Fig. 13.20. 5 V=VS G
PMOS
Vin
NOT gate Vout
=
Vin
Vout
NMOS G
0V
Fig. 13.20. Symbol for the NOT gate along with the corresponding circuit diagram.
Note that the input voltage may slightly vary around zero volts or around ﬁve volts. As long as these variations do exceed 1 V (do not exceed threshold voltages of the transistors), the circuit shall still output exactly the results shown in Tables 13.3 and 13.4. The corresponding task is suggested as a homework problem. This property of the NOT gate is critical it means that the present logic operation is immune to electric noise.
13.3.2 NOR Gate and OR Gate NOR Gate Consider the circuit with four transistors shown in Fig. 13.21. This circuit may be considered as an extension of the logic inverter shown in Fig. 13.19. Now, one has two input voltages V1 and V2 instead of one input voltage Vin. The circuit is powered by a 5V power supply. All four transistors (two PMOS and two NMOS transistors) are matched and have equal threshold voltages V Tp V Tn 1 V.
XIII661
Chapter 13
Switching Circuits 5V=VS
V1
V1
PMOS
V2
PMOS
NMOS
V2
0V
Vout
NMOS
0V
Fig. 13.21. CMOS NOR logic gate.
The input voltages to the circuit follow the ﬁrst columns of Table 13.5. Further, we shall use the chart from Fig. 13.4 of Section 13.1 in order to study the circuit behavior. When the input voltages V1 and V2 are both low (0 V), the NMOS switches are OFF and the PMOS switches are ON. The output voltage is therefore 5 V. When the input voltage V1 is high (5 V), irrespectively of the value of the input voltage V2, the leftmost NMOS transistor is ON, so that the output voltage is always 0 V. A similar situation occurs when V2 is high (5 V). The output voltage is always 0 V. Hence the output voltage follows the last column of Table 13.5. Table 13.5. Output voltage of the NOR gate versus the input voltages. V1 0V 0V 5V 5V
V2 0V 5V 0V 5V
Vout 5V 0V 0V 0V
If we again denote the 0V voltage by value 0 (low or false) and the 5V voltage by value 1 (high or true), Table 13.5 is converted to the truth table Table 13.6. Table 13.6. Truth table for the logic circuit in Fig. 13.21—the NOR gate. V1 0 0 1 1
V2 0 1 0 1
Vout 1 0 0 0
It follows from Table 13.6 that this circuit does not perform the logic OR operation (outputs true when at least one input is true). Rather, it does exactly the opposite. XIII662
Chapter 13
Switching Circuits
means that they contradict the operation of the two logic gates. Thus, any arbitrary initial voltage distribution will be very quickly transformed to one of the stable states. As long as power is present, the latch can remain in any of the stable states indeﬁnitely long. In other words, the latch circuit memorizes the initial state, due to the effect of the positive feedback. a)
VS
PMOS
G1
PMOS Vout1
Vin2 Vout2
b)
VS
G2
NMOS
G1
Vin1
Vout1
Vin2
Vout2
=
Vin1 NMOS
G2
Fig. 13.32. Basic latch consisting of two inverters. Table 13.15. Stable states of the latch circuit. Note the inversion of all voltages for two different states. State #1 #2
Vin1 1 0
Vin2 0 1
Vout1 0 1
Vout2 1 0
With the two stable states, the latch circuit is capable of storing one bit of data. One state is then designated as 0 (LOW) and another as 1 (HIGH). It now remains to design a mechanism by which the state can be written and read. This is accomplished in a static RAM memory cell, which uses two additional access (or pass) NMOS transistors.
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Chapter 13
Summary
Summary Transistor switches Groundside NMOS transistor switch (normally OFF)—pull down switch A. Closes when V in > V tn ; B. For efﬁcient operation, the control voltage Vin must be as high as possible in the ON position; C. Complement to PMOS switch Powerside PMOS transistor switch (normally ON)—pull up switch A. Opens when V in > V S V tp ; B. For efﬁcient operation, the control voltage Vin must be as low as possible in the ON position; C. Complement to NMOS switch Transistor switching diagram A. Vin—control voltage; B. Assume V tn ¼ V Th ; C. Assume V tp ¼ V Th ; D. Typical values of VTh are in the range from 0.4 to 4 V E. VTh depends on transistor geometry and composition
Transistor motor controllers Hbridge
Controls a DC motor load enabling: – Forward mode; – Reverse mode; – Free run to a stop; – Brake states Available commercially as an Hbridge IC. Simpler modiﬁcation—half Hbridge (continued)
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Chapter 13
Switching Circuits
Basic pulse width modulation (PWM) waveform Period (sec) T ¼ t on þ t off Frequency (Hz) f ¼ 1=T t on t on 100 % Duty cycle d ¼ , D ¼ T T Average supply voltage VDC ¼ dVS
Basic logic gates and switching (Boolean) algebra Logic inverter (NOT gate)
For singletransistor design, resistance R should be very large NAND gate
AND gate
(continued)
XIII676
Chapter 13
Summary
NOR gate
OR gate
Exclusive OR (XOR) gate and exclusive NOR (XNOR) gates
Switching (Boolean) algebra AþB¼BþA AB¼BA A þ ðB þ C Þ ¼ ðA þ B Þ þ C A ðB C Þ ¼ ðA BÞ C A ðB þ C Þ ¼ A B þ A C A þ 0 ¼ A, A þ 1 ¼ 1, A 0 ¼ 0, A 1 ¼ 1, A þ A ¼ A, A þ A ¼ 1, A A ¼ A, A A ¼ 0, A ¼ A De Morgan’s laws: A þ B ¼ A B, A B ¼ A þ B (continued)
XIII677
Chapter 13
Switching Circuits
Latch (Transistor memory)
XIII678
Chapter 13
Problems
Problems 13.1 Principle of Operation 13.1.1 Switch Concept 13.1.2 Switch Position in a Circuit 13.1.3 MOSFET Switches 13.1.4 Sketch of Transistor Physics Problem 13.1. Describe the function and major properties of an electronic switch in your own words.
Find the output voltage Vout to each circuit if: i. V in ¼ 0:0 V ii. V in ¼ 0:2 V iii. V in ¼ 1:6 V iv. V in ¼ 1:8 V Problem 13.4. Two digital circuits shown in the ﬁgure below operate with a singlesupply voltage V S ¼ 5:0 V. The transistor threshold voltages are V Tn ¼ V Tp ¼ 1:0 V. a) rest of a circuit
Problem 13.2. A. Describe the meaning of the groundside switch, the powerside switch, and the series switch. Why do we distinguish between those switch types? B. Draw the circuit symbol for a switching NMOS transistor used in the groundside switch. Draw the circuit symbol for a PMOS transistor used in the powerside switch. Designate the line current direction in both cases.
Vout Vin
NMOS
b)
VS
Vin
PMOS
0V
Vout rest of a circuit
Problem 13.3. Two digital circuits shown in the following ﬁgure operate with a singlesupply voltage V S ¼ 1:8 V. The transistor threshold voltages are V Tn ¼ V Tp ¼ 0:5 V, which correspond to a 0.18μm CMOS process. Note: Every “CMOS process” is a manufacturing process for tiny MOSFETs used in digital circuits. A 0.18μm CMOS means that the channel length (gate width) of the MOSFET in Fig. 13.5 is greater than or equal to 0.18 μm. a) rest of a circuit Vout Vin
NMOS
Find the output voltage Vout to each circuit if: i. V in ¼ 0:0 V ii. V in ¼ 0:7 V iii. V in ¼ 5 V iv. V in ¼ 4:6 V Problem 13.5. For the circuit shown in the ﬁgure below, V S ¼ 2:5 V and V Tn ¼ V Tp ¼ 0:5 V (0.25μm CMOS process). Determine the output voltage Vout when: A. V in ¼ 0:0 V B. V in ¼ 0:2 V C. V in ¼ 2:3 V D. V in ¼ 2:5 V E. V in ¼ 1:0 V VS
0V G
b)
VS
Vin
PMOS
Vin
Vout rest of a circuit
PMOS Vout NMOS
G
0V
XIII679
Chapter 13
Switching Circuits
Problem 13.6. For the circuit shown in the ﬁgure below, V S ¼ 2:5 V and V Tn ¼ V Tp ¼ 0:5 V (0.25μm CMOS process). Determine the output voltage Vout when: A. V 1 ¼ 0:0 V, V 2 ¼ 0:0 V B. V 1 ¼ 0:7 V, V 2 ¼ 0:2 V C. V 1 ¼ 2:3 V, V 2 ¼ 2:3 V D. V 1 ¼ 2:3 V, V 2 ¼ 0:0 V E. V 1 ¼ 1:0 V, V 2 ¼ 1:0 V VS V1
rest of a circuit
V
0V
Problem 13.10. For the circuit shown in the ﬁgure below, determine all possible values of the voltage V. The transistor’s threshold voltage is V Tp ¼ 0:5V; the supply voltage is 2.5 V.
PMOS
VS
Vout V2
NMOS V rest of a circuit
0V
Problem 13.7. For the circuit shown in the ﬁgure below, V S ¼ 2:5 V and V Tn ¼ 0:5 V. Determine the output voltage Vout when: A. V in ¼ 0:0 V B. V in ¼ 0:2 V C. V in ¼ 2:3 V D. V in ¼ 2:5 V
Problem 13.11. A circuit shown in the ﬁgure below is used to measure the threshold voltage of the NMOS transistor. Could you explain why? Hint: Determine all possible values of the voltage V. VS R(1 M
The output terminal is disconnected (current cannot ﬂow into this terminal).
or larger) V~VTn
VS
R 0V Vout Vin 0V
Problem 13.8. Draw two alternative circuit symbols for an NMOS switch. Repeat for the PMOS switch. Problem 13.9. For the circuit shown in the following ﬁgure, determine all possible values of the voltage V. The transistor's threshold voltage is 0.5 V.
Problem 13.12. A circuit shown in the ﬁgure below is used to measure the threshold voltage of the PMOS transistor. Could you explain why? Hint: Determine all possible values of the voltage V. VS
V~VS VTp
R(1 M
or larger)
0V
XIII680
Chapter 13
Switching Circuits
V1
+
PMOS
E

+ 
B. Motor is in reverse mode (current direction through the motor is from right to left). C. Motor suddenly stops (motor terminals are shorted out, which creates the braking effect). D. Motor runs freely to a stop (motor is disconnected from the power supply).
VS=12 V V2
NMOS
0V
Problem 13.20. In the half Hbridge circuit shown in the ﬁgure to the previous problem, the control voltages V1, V2 are either 0 V or 12 V. If we denote the 0V control voltage by 0 (low) and the 12V control voltage by 1 (high), all possible combinations of the control voltages are covered by the table that follows (the table in fact lists all binary numbers from 0 to 3). Fill out the table using four states: A. Forward mode (current direction is from left to right) B. Brake C. Free run to a stop D. Forbidden (short circuit) V1 0 0 1 1
V2 0 1 0 1
State
Problem 13.21. Suggest and sketch a schematic of the half Hbridge where three switching states (ON, brake, free run to a stop) are realized with only NMOS transistors. Problem 13.22. For the Hbridge shown in the following V S ¼ 12 V and ﬁgure, V Tn ¼ V Tp ¼ 3:5 V. The control voltages may switch from 0 V to 12 V. Establish at least one set of particular values for four control voltages V1, V2, V3, V4 that ensures the following motor operations: A. Motor is in forward mode (current direction through the motor is from left to right).
V1
+ 
V3 PMOS
PMOS
NMOS
NMOS
VS=12 V V2
V4
0V
Problem 13.23. In the Hbridge circuit shown in the ﬁgure to the previous problem, the control voltages V1, V2, V3, V4 are either 0 V or 12 V. If we denote the 0V control voltage by digit 0 (low) and the 12V control voltage by digit 1 (high), all possible combinations of the control voltages are covered by the table that follows (the table in fact lists all binary numbers from 0 to 15). Fill out the table using ﬁve states: A. Forward mode (current direction is from left to right) B. Reverse mode (current direction is from right to left) C. Brake D. Free run to a stop E. Forbidden (short circuit)
13.2.7 Application Example: PulseWidth Modulation (PWM) Motor Controller Problem 13.24. For a PWM form shown in the following ﬁgure, determine: A. Period, T (show units) B. Frequency, f (show units) C. Duty cycle, d (also give its percentage D) D. Average supply voltage, VDC
XIII682
Chapter 13
14
Problems
VS, V
Problem 13.26* A. Compile a MATLAB script to generate the ﬁgure to Problem 13.24 above. Attach the script and the ﬁgure to the homework. B. Compile two MATLAB scripts to generate the two ﬁgures to Problem 13.25 above. Attach the scripts and the ﬁgures to the homework.
12 10 8 6 4 2 0 0
0.02
0.04 0.06 time, sec
0.08
0.1
Problem 13.25. For a PWM form shown in the following ﬁgure, determine A. Period, T (show units) B. Frequency, f (show units) C. Duty cycle, d (also give its percentage D) D. Average supply voltage, VDC a) 14
VS , V
12 10 8
4 2 0 0
10
20
30
40
50
60
70
80
90
100
time, msec
14
13.3.1 NOT Gate or Logic Inverter 13.3.2 NOR Gate and OR Gate 13.3.3 NAND Gate and AND Gate Problem 13.27 A. Draw the symbol for the logic inverter (NOT gate). B. Draw the corresponding circuit diagram. C. Given the input voltage to the NOT gate, ﬁll out the table that follows. The transistors used in the circuit are matched and have equal threshold voltages V Tp ¼ V Tn ¼ 1 V. Vin 0.5 V 4.7 V
6
b)
13.3 Digital Switching Circuits
VS , V
Problem 13.28. For the circuit shown in the ﬁgure, ﬁll out the table that follows. The transistors used in the circuit are matched and have equal threshold voltages V Tp ¼ V Tn ¼ 1 V. Vin
12
Vout
Vout
10 8
Vin 0V 5V
6 4
Vout
2 0 0
10
20
30
40
50
60
time, msec
70
80
90
100
Problem 13.29. For the circuit shown in the ﬁgure, ﬁll out the table that follows. The transistors used in the circuit are matched
XIII683
Chapter 13
Switching Circuits
and equal have V Tp ¼ V Tn ¼ 1V.
threshold
Vin
voltages
Vout
Vin 0.3 V 4.2 V
Vout
Problem 13.32. Draw the symbol for the logic OR gate and present the corresponding truth table. Problem 13.33. Draw the symbol for the logic NOR gate and present the corresponding truth table. Problem 13.34. Draw the circuit diagram: A. For the NOR gate B. For the OR gate
Problem 13.30. The circuit shown in the following ﬁgure is a logic gate. However, it utilizes only one transistor. The current cannot ﬂow into the output terminal (it is disconnected in the ﬁgure). Construct: A. Table of Vout versus V in ¼ 0, 0:5, 4:5, 5 V B. The truth table given that the source voltage is 5 V and the threshold voltage is 1 V. What logic gate is it? What value of the resistor R would you choose to minimize circuit loss and the load impact? 5 V=VS
R Vout Vin 0V
Problem 13.31. Repeat the previous problem for the circuit shown in the following ﬁgure. 5 V=VS
Vin Vout
Problem 13.35. For the NOR gate with input voltages V1 and V2, ﬁll out the table that follows. The transistors used in the circuit are and have equal threshold voltages matched V Tp ¼ V Tn ¼ 1 V. V1 0.1 V 0.5 V 4.5 V 4.1 V
V2 0.7 V 4.9 V 0.1 V 4.4 V
Vout
Problem 13.36. For the OR gate with input voltages V1 and V2, ﬁll out the table that follows. The transistors used in the circuit are and have equal threshold voltages matched V Tp ¼ V Tn ¼ 1 V. V1 0.3 V 0.2 V 4.1 V 5.3 V
V2 0.1 V 4.1 V 0.1 V 4.4 V
Vout
Problem 13.37. The following ﬁgure is an internal electric circuit of a logic gate. It has three inputs and one output. 1. Fill out the truth table. 2. Draw the symbol of the corresponding logic gate.
R
0V
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Chapter 13
Problems 5 V=VS
V1 5 V=VS
V2 V3
V1
Vout
V2
V3
Problem 13.38. A freshman ECE student attends class if at least one of the following conditions is satisﬁed: 1. He/she feels that this lecture might be useful. 2. The lecture is not early in the morning. 3. His/her friends might be present there too. Every morning he/she “votes” by simultaneously pushing any appropriate combination of three 5V buttons (V1, V2, and V3) placed in parallel. A logic circuit is needed that lights a green LED (outputs 5 V) when there is time to go to the lecture. A. Draw the corresponding logic circuit in the symbolic form (in the form of logic gates). B. Draw the MOSFET representation of that logic circuit. C. Present the corresponding truth table. Problem 13.39. Draw the symbol for the logic AND gate and present the corresponding truth table. Problem 13.40. Draw the symbol for the logic NAND gate and present the corresponding truth table. Problem 13.41. Draw representation A. For the NAND gate B. For the AND gate
the
MOSFET
Problem 13.42. For the NAND gate with input voltages V1 and V2, ﬁll out the table that follows. All transistors used in the circuit are and have equal threshold voltages matched V Tp ¼ V Tn ¼ 1 V. V1 0.9 V 0.1 V 4.1 V 4.5 V
V2 0.9 V 5.1 V 0.1 V 4.7 V
Vout
Problem 13.43. For the AND gate with input voltages V1 and V2, ﬁll out the table that follows. The transistors used in the circuit are and have equal threshold voltages matched V Tp ¼ V Tn ¼ 1 V. V1 0.1 V 0.2 V 5.1 V 5.0 V
V2 0.1 V 4.2 V 0.1 V 4.6 V
Vout
Problem 13.44. A senior ECE student attends class if all of the following conditions are satisﬁed: 1. He/she feels that this lecture might be useful. 2. The lecture is not early in the morning. 3. His/her friends might be present there too. Every morning he/she “votes” by simultaneously pushing any appropriate combination of three 5V buttons (V1, V2, and V3) placed in parallel. A logic circuit is needed that lights a green LED (outputs 5 V) when there is time to go to the lecture. A. Draw the corresponding logic circuit in the symbolic form (in the form of logic gates). B. Draw the MOSFET representation of that logic circuit. C. Present the corresponding truth table.
How many transistors are we using in every case?
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Chapter 13
Problems
Problem 13.54. If AND gates are substituted in place of NAND gates in the logic circuit in Fig. 13.30c, will the motor controller still function properly? Explain why yes or why no. Problem 13.55. If NOR gates are substituted in place of NAND gates in the logic circuit in Fig. 13.30c, will the motor controller still function properly? Explain why yes or why no. Problem 13.56. Given the logic circuit and the input waveforms in the following ﬁgure, draw the output waveform on the same ﬁgure. Hint: Construct the truth table of the logic circuit ﬁrst.
V1 V2
Vout
V1 V2 Vout
V1 V2
Problem 13.58. Given the logic circuit and the input waveforms in the following ﬁgure, draw the output waveform on the same ﬁgure. Hint: Construct the truth table of the logic circuit ﬁrst.
Vout
Problem 13.59. Given the logic circuit and the input waveforms in the following ﬁgure, draw the output waveform on the same ﬁgure. Hint: Construct the truth table of the logic circuit ﬁrst.
V1 V2
V1
Vout
V2
Problem 13.57. Given the logic circuit and the input waveforms in the following ﬁgure, draw the output waveform on the same ﬁgure. Hint: Construct the truth table of the logic circuit ﬁrst.
Vout
V1 V2
V1
Vout V2
V1 V2 Vout
Vout
Problem 13.60. A small county board is composed of three commissioners. Each commissioner votes on measures presented to the board by pressing a 5V button indicating whether the commissioner votes for or against a measure. If two or more commissioners vote for a measure, it passes. You are asked to help with a logic circuit that takes the three votes as inputs and
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Chapter 13 lights a green LED (outputs 5 V) to indicate that a measure passed. You can use AND, NAND, and NOT logic gates, as many of them as you need. A. Present the corresponding logic circuit in the symbolic form (in the form of logic gates). B. Present the corresponding truth table. C. How many transistors does your circuit include? Problem 13.61. A small county board is composed of three commissioners. Each commissioner votes on measures presented to the board by pressing a 5V button indicating whether the commissioner votes for or against a measure. If two or more commissioners vote for a measure, it passes. You are asked to help with a logic circuit that takes the three votes as inputs and lights a green LED (outputs 5 V) to indicate that a measure passed. You can use OR, NOR, and NOT logic gates, as many of them as you need. A. Present the corresponding logic circuit in the form of logic gates. B. Present the corresponding truth table. C. How many transistors does your circuit include?
13.3.8 The Latch
Switching Circuits among all other memories. The ﬁgure that follows shows a SRAM memory cell including: 1. The latch with four transistors. 2. A word line WL connected through two access NMOS transistors M5, M6. They are always turned on (become the short circuit) when the selected cell’s word line is raised high (to VS or another high voltage). 3. A bit line BL and its counterpart, another bit line BL. Word Line (WL)
VS
M2
PMOS
NMOS
PMOS
NMOS
G1
G2 M5 NMOS
M6 NMOS
M3
M4
BL
BL
The following ﬁgure shows another attempt to design the SRAM memory cell with only four NMOS transistors. Will this design function? Why yes or why no?
Problem 13.62. Draw the circuit diagram of a basic latch and explain its operation in your own words. Problem 13.63. Most of the transistors are used in semiconductor memories. There are several types of semiconductor memories. One of them is the static RAM (SRAM) or the static random access memory. RAM means that every data bit is accessible any time unlike hard disk memory. SRAM cells provide the fastest operation
VS M1
Word Line (WL)
VS
VS 1G
G2
1G
G1 M6
M5 M3 BL
M4 BL
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Chapter 14: AnalogtoDigital Conversion
Overview Prerequisites:  Knowledge of basic circuit analysis  Knowledge of transistor switches (Chapter 13) Objectives of Section 14.1:  Relate hardware meaning and mathematical meaning of digital voltage  Convert between binary, decimal, and hexadecimal numbers  Understand parallel and series representation of digital voltage  Become familiar with clock frequency and timing diagram of digital circuits  Learn binary representation of ASCII characters  Obtain initial exposure to tristate digital voltage Objectives of Section 14.2:  Appreciate the necessity of the digitaltoanalog converter  Design simple hardware realization(s) of digitaltoanalog converters  Relate circuit structures to the corresponding mathematical operations with binary numbers  Become familiar with resolution, accuracy, and voltage range of a DAC  Learn two basic DAC constructions: binaryweighted input and R/2R ladder Objectives of Section 14.3:  Understand the necessity for sampling analog voltages  Design simple hardware realization(s) of the sampleandhold circuit  Understand the value of the Nyquist rate Objectives of Section 14.4:  Design simple hardware realization(s) of the analogtodigital converters: flash ADC and successiveapproximation ADC  Become familiar with key parameters of ADCs: resolution in bits, fullscale voltage range, and voltage resolution  Obtain initial exposure to ADC speed, throughput rate, and conversion time
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_14
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Keywords: Analogtodigital converter (ADC), Digitaltoanalog converter (DAC), Analog voltage, Digital voltage, Analog computer, Binary number system, Binary number, Least signiﬁcant bit (LSB), Most signiﬁcant bit (MSB), Parallel representation of a binary number, Serial representation of a binary number, Binary line codes, Unipolar NRZ line code, Polar NRZ line code, Unipolar RZ line code, Manchester line code, Bit rate, RS232 interface, Clock frequency, Timing diagram of a digital circuit, Asynchronous transmission, Synchronous transmission, Hexadecimal numbers, ASCII codes, Digital word, Bit, Byte, Nibble, Tristate buffer, Tristate digital voltage, Data bus, Output enable, Chip select, Summing ampliﬁer, Binaryweightedinput DAC, R/2R ladder DAC, DAC scaling voltage factor, Scaled digitaltoanalog conversion, DAC resolution voltage, Binary counter, DAC reconstruction ﬁlter, DAC postﬁlter, DAC equation, DAC fullscale output voltage range, DAC quantization levels, DAC external voltage reference, DAC resolution, DAC relative accuracy, Sampleandhold voltage, Sampling interval, Sampling rate, Sampling frequency, Acquisition (sample) time, Sampleandhold circuit, Trackandhold circuit, Track/store circuit, Nyquist rate, Nyquist frequency, Digital signal processing, NyquistShannon sampling theorem, Flash ADC, Successiveapproximation ADC, Successiveapproximation register, ADC fullscale measurement voltage range, ADC encoder block, ADC resolution in bits, ADC voltage resolution, ADC resolution accuracy, ADC equation, Midrise coding scheme, Midtread coding scheme, ADC quantization error, ADC quantization noise, ADC conversion time, ADC speed
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Chapter 14
Section 14.1: Digital Voltage and Binary Numbers
Section 14.1 Digital Voltage and Binary Numbers 14.1.1 Introduction: ADC and DAC Circuits Consider a typical block diagram of a digital signal processor (DSP) see Fig. 14.1. The diagram includes an analogtodigital converter (ADC) interfacing with a digital processor. The ADC converts an analog input voltage into data in the form of binary codes which are processed mathematically. The digital processor performs mathematical operations with binary numbers. The output to the processor is converted back to the realworld voltage using the digitaltoanalog converter (DAC). Every time you use your cell phone, you are using a DSP, and this is only one example of its application. A generalpurpose microprocessor possesses a similar structure.
Analog input voltage vin(t)
ADC circuit
D0 D1 D2 D3 D4 D5 D6 D7
Digital processor
D0 D1 D2 D3 D4 D5 D6 D7
DAC circuit
Analog output voltage vDAC(t)
Interface (control) circuit
Fig. 14.1. Structure of a DSP including different circuit blocks.
In this chapter, we will study the ADC circuit(s), the DAC circuit(s), and some useful interface/control circuits. Those basic digital circuits utilize and extend the generic ampliﬁer concept studied previously. In this sense, the present chapter offers further solidiﬁcation of the ampliﬁer theory and practice. Section 14.1 introduces the meaning of a digital voltage and its relation to binary number system. It shows how to read a binary word and how to understand its circuit representation. Along with this, we brieﬂy review hexadecimal numbers and demonstrate how to convert between different number systems using MATLAB. Section 14.2 studies two basic digitaltoanalog converter conﬁgurations. Resolution voltage, accuracy, and fullscale output voltage range are the most important features of a digitaltoanalog converter (DAC) chip. It may be amazing to discover how the corresponding electronic circuits follow the mathematical formulas for number conversion. In this section, we also introduce the useful concept of a binary counter and give a number of examples. Section 14.3 analyzes the ﬁrst block of the analogtodigital converter a sampleandhold circuit. The meaning of the sampling rate and the fundamental concept of the Nyquist rate are naturally introduced in this context. Section 14.4 is devoted to two basic analogtodigital converters. Resolution voltage, fullscale input voltage range, quantization error, and a slower speed of the A to D conversion are explained. Indeed, the present chapter does not exactly follow modern XIV691
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digital hardware, neither does it provide the reader with comprehensive digital fundamentals including the control logic. Only the basic circuit concepts will be illustrated here, based on simpler yet functionally similar circuits.
14.1.2 Analog Voltage Versus Digital Voltage Analog voltage is any instantaneous (usually continuous) voltage in a circuit. The term digital voltage is not quite precise. Strictly speaking, digital voltage is the same as analog voltage, which, however, may have only two states high and low at any time instant and/or at any particular node in the circuit. The digital voltage concept is closely related to the switching concept: the MOSFET switch studied in Chapter 13 may have only two states: on or off. The output voltage of the switch (the switching voltage) is thus exactly a digital voltage; it may be either high (switch is on) or low (switch is off). Using a number of such switches together allows us to process and store information in the circuit in the form of digital voltages. Thus, the analog circuit becomes a digital machine. Analog Voltage Consider the (decimal) number 10. How could we represent it in a computer? One way is shown in Fig. 14.2. We simply form a voltage divider circuit or an ampliﬁer circuit or another analog circuit, with exactly one output. We’d strive to have 10 V at that output, as precisely as possible. This is the simplest and most intuitive way of assigning the voltage to a number. V
V 10 V
rest of circuit 0V
t
Fig. 14.2. Analog output voltage V of 10 V corresponding to a number 10.
However, the accuracy of this representation heavily depends on resistor tolerance, ampliﬁer gain tolerance, temperature variations, etc. Such an idea basically corresponds to an analog computer, which is brieﬂy considered below.
Digital Voltage and Binary Number System The second less intuitive but much more versatile way to represent a certain number is shown in Fig. 14.3. We still try to represent (decimal) number 10. But instead of one output voltage V, we now introduce four output voltages denoted by D3,2,1,0. However, each output can no longer have arbitrary voltage values. The output voltages may be either low or high, say, 0 V or 5 V, respectively. XIV692
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Section 14.1: Digital Voltage and Binary Numbers
14.1.3 Bit Rate, Clock Frequency: Timing Diagram Bit Rate While the parallel digital output is only characterized by high and low states at any time instant, the serial bit stream may have a huge variety of forms or codes by which the 0 s and 1 s are represented. Some of them (the socalled line codes used for wired or wireless data transmission) are shown in Fig. 14.6. The ﬁrst one is the unipolar (0 5 V) NRZ (nonreturn to zero) code that has also been drawn in Figs. 14.4 and 14.5. The second code is similar in form, but it utilizes both positive (+5 Vhigh) and negative ( 5 Vlow) voltages versus ground. The third one has the duty cycle of 50 % (returns to zero in the middle of bit width). The last code represents binary one by a 5 V twophase pulse and binary 0 by a 5 V reversed twophase pulse. LSB
0
1
0
0
1
1
1
0
MSB
5V
Unipolar NRZ code 0V 5V
Polar NRZ code
0V 5V 5V
Unipolar RZ code 0V 5V
Manchester code
0V 5V
T
T
T
T
T
T
T
T
t
Fig. 14.6. Some serial line codes. All codes have the same bit width, T, and bit rate, fb.
Despite all those differences, the codes shown in Fig. 14.6 convey the same information binary number 01110010 in the same amount of time and thus have the same bit width, T. It is clear that the capacity of a serial digital data stream depends on the speed with which the bits (regardless of format) are transferred through a path. This capacity is determined by a bit rate. The bit rate is the number of bits conveyed or processed per second. As the name implies, the bit rate, fb, for any serial bit stream in Fig. 14.6 is fb
1 T
ð14:4Þ
where T is the bit width. Although the bit rate, fb, in Eq. (14.4) has the units of frequency or hertz, it is rather measured in bits/s or bps. The reason for this is in that the serial bit stream conveying nontrivial information is never a periodic waveform see Fig. 14.6. Therefore, we cannot use the meaning of frequency.
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Section 14.1: Digital Voltage and Binary Numbers
corresponding to a bit stream in Fig. 14.6 or in Fig. 14.7 is shown in Fig. 14.8. Therefore, one determines clock frequency, f, in Fig. 14.8, f
1 T
ð14:5Þ
in hertz. If the bit width and the clock period coincide, the bit rate is equal to the clock frequency. 5V
Clock
0V
T
T
T
T
T
T
T
t
T
Fig. 14.8. Clock signal for a bit stream in Fig. 14.7. The duty cycle of the periodic waveform is 50 %.
We note that the clock frequency is critical not only for digital IO but also for the functioning of the computer itself. A computer or a microprocessor processes streams of digital data. The logic gates introduced in Chapter 13 make it possible to perform logical and arithmetical operations on binary numbers. As the logic gates are comprised of individual transistors (MOSFETs), their parasitic capacitances delay and distort clock/ bit pulses. As a result, there is always a limiting clock frequency that dictates both the maximum serial IO speed and the maximum computer speed. Eventually, those speeds are determined by the response of switching transistors introduced in Chapter 13. This question is of great practical importance; it requires detailed knowledge of MOS transistors introduced in Chapter 18 and the detailed knowledge of transient RC circuits studied in Chapter 7.
Timing Diagram Figure 14.9 below illustrates a timing diagram: the clock signal and the actual synchronized serial bit stream (the unipolar NRZ code). The timing diagram is a necessary attribute of many digital device datasheets. A case in point is an analogtodigital converter studied in Section 14.3 of this chapter. LSB
MSB
5V 0V 5V 0V
1
3
5
7
9
11
13
t, s
Fig. 14.9. Timing diagram for a binary bit steam.
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Section 14.1: Digital Voltage and Binary Numbers
Fig. 14.10. An analog computer from Electronic Associates, Inc. West Long Branch, New Jersey, with 156 individual ampliﬁers (Analog Computer Museum by Doug Coward).
The culprits turned out noise and even the accuracy of the analog circuit components. Imagine what tolerance the ﬁnal result could have if you would perform 100 multiplications using 1 %accurate resistors? The digital approach is principally different: the bit values are deﬁned by logic threshold voltage levels, which allow us for a wide variation of circuit voltages within those margins. For example, a bit value 1 for a 0to5V logic may correspond to any voltage between 2.6 V and 5.0 V and a bit value zero to any voltage between 0 V and 0.4 V. Another, perhaps even more important point is the existence of fast digital memory (ROM and RAM), which is more advanced than the analog memory carriers. Still, analog computers and especially hybrid computers are used in certain military and commercial applications even today.
Binary Numbers Values that are in the ones and zeros format are said to be binary. Bi means “two,” so a binary number can only have one of two values per digit, as opposed to decimal numbers, which can have one of ten values per digit. The conversion of integer or fractional binary numbers to decimal numbers is rather straightforward; a few examples have been given at the beginning of this section. When a binary fraction is present, we perform the conversion following Eq. (14.6), that is, 1 1 27 26 25 24 23 22 21 20 2 1 2 2 2 3 2 4 ¼ 153:5625decimal ¼ 128 þ 16 þ 8 þ 1 þ þ 1 0 0 1 1 0 0 1 : 1 0 0 1 2 16
ð14:6Þ On the other hand, the conversion of decimal numbers to binary numbers is a bit more involved. To do so, we repeatedly divide the decimal number by two until the quotient is zero. Equation (14.7) gives an example for decimal number 153. The resulting binary number is given by the remainder column; we read it as shown in Eq. (14.7).
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Section 14.1: Digital Voltage and Binary Numbers
capacitor) circuit components as shown in Chapter 8. When a digital device generates high and low voltages only, a tristate buffer is added in order to achieve an extra third state: the HighZ state. This buffer may either be internal or external to a chip. The concept is explained in Fig. 14.12. First, in Fig. 14.12a we consider the standard buffer ampliﬁer that uses a singlepolarity power supply. When connected to an output of a digital device, the ampliﬁer simply transfers high and low digital voltages, without adding new states. Therefore, its output D still has only two states high and low. The next step is to add a switching circuitry shown in Fig. 14.12b to the ampliﬁer. When control voltage enable is high, both the NMOS and PMOS switches will be on see Chapter 15 and note an inverter connected to the PMOS. Nothing really changes compared to the previous case. However, when control voltage enable is low, both switches will be off. This means that the ampliﬁer will be disconnected from the power supply completely. In other words, its output D becomes an entirely open circuit because current can ﬂow nowhere (the current cannot ﬂow in/out the ampliﬁer input(s) and into the power rails). a)
buffer amplifier
tri state buffer
b)
power rail power rail
VS
VS
HIGH or LOW
+

PMOS HIGH or LOW
HIGH or LOW
+ D Vin

HIGH or LOW or HIGH Z D
NMOS
power rail ENABLE
power rail
Fig. 14.12. Concept of a tristate buffer.
Thus, the output to the ampliﬁer, D, in Fig. 14.12b achieves the HighZ state. The corresponding truth table is Table 14.3. Table 14.3. Truth table for the tristate buffer in Fig. 14.12b. Input Input to the ampliﬁer 0 1 0 1
Enable 0 0 1 1
Output D HighZ HighZ 0 1
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Chapter 14
Section 14.2: DigitaltoAnalog Converter
Section 14.2 DigitaltoAnalog Converter 14.2.1 DigitaltoAnalog Converter Conversion between analog and digital voltage signals is done by means of a digitaltoanalog converter (DAC) and an analogtodigital converter (ADC). Both DAC and ADC are electric circuits that perform the corresponding operations. In this subsection we consider the idea of the DAC circuit. Its goal is to convert a sequence of binary numbers (digital voltages) generated by a processor or by a computer into a realworld voltage signal, which could be used as an input to a control system, to an audio ampliﬁer, etc. The place of the DAC in the generic block diagram of a DSP is shown in Fig. 14.14.
Analog input voltage vin(t)
ADC circuit
D0 D1 D2 D3 D4 D5 D6 D7
Digital processor
D0 D1 D2 D3 D4 D5 D6 D7
DAC circuit
Analog output voltage vDAC(t)
Interface (control) circuit
Fig. 14.14. Place of the DAC in the DSP block diagram.
14.2.2 Circuit (A BinaryWeightedInput DAC) Digitaltoanalog conversion is conceptually simple. A 4bit binaryweightedinput DAC at the base of an ampliﬁer is shown in Fig. 14.15. The circuit implies four digital input voltages (data lines) D3, D2, D1, D0 and one analog output voltage υDAC. The parallel path of four binary signals in Fig. 14.15 is called a data bus, D0 always represents the least signiﬁcant bit (LSB). The converter in Fig. 14.15 has the form of a summing ampliﬁer. The summing ampliﬁer is further connected to an (optional) inverting ampliﬁer stage having the gain of minus one. The summing ampliﬁer is also an inverting ampliﬁer with the negative feedback but with multiple inputs. Its operation may be explained as follows. According to the KCL and to the ﬁrst summingpoint constraint (no current into the ampliﬁer), one has for the feedback current with reference to Fig. 14.15
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AnalogtoDigital Conversion
LSB
iF 16RW
0 V D0
RF
+
2RW
5 V D3
i3
0V
MSB
R

0V
4RW
0 V D2

5 V D1
R
i1
v1
vDAC
+
8RW
0V
Fig. 14.15. Binaryweightedinput digitaltoanalog converter. Index w indicates weighted resistances.
iF
i3 þ i2 þ i1 þ i0
ð14:12aÞ
On the other hand, the second summingpoint constraint (the differential voltage to the ampliﬁer is zero and the inverting input is the virtual ground) yields i3
D3 , i2 2RW
D2 , i1 4RW
D1 , i0 8RW
D0 16RW
ð14:12bÞ
in terms of input voltages D3, D2, D1, D0. Therefore, voltage υ1 in Fig. 14.15 is found from Eq. (14.12a) that is now written in the form: 0 υ1 D3 D2 D1 D0 iF i3 þ i2 þ i1 þ i0 þ þ þ RF 2RW 4RW 8RW 16RW ð14:12cÞ 1 3 2 1 0 D3 2 þ D2 2 þ D1 2 þ D0 2 16RW Consequently, the output voltage to the entire converter becomes υDAC
υ1
RF D3 23 þ D2 22 þ D1 21 þ D0 20 ; 16RW
ð14:12dÞ
which is the ﬁnal result for the binaryweightedinput DAC shown in Fig. 14.15.
14.2.3 Underlying Math and Resolution Voltage It should be emphasized that Eq. (14.12d), which describes the circuit operation, is precisely the hardware realization of the mathematical formula for binarytodecimal conversion given, for example, by Eq. (14.1) at the beginning of the previous section. In order to prove this fact, Eq. (14.12d) may be rewritten in the form ð14:13aÞ υDAC Q d 3 23 þ d 2 22 þ d 1 21 þ d 0 20 where the scaling voltage factor Q is given by XIV708
Chapter 14
Section 14.2: DigitaltoAnalog Converter
14.2.4 DAC FullScale Output Voltage Range, Resolution, and Accuracy Using the (mostly illustrative) binaryweightedinput DAC circuit as a starting point, we now intend to obtain general facts about DAC resolution and accuracy, which are applicable to any DAC chip, operating over an arbitrary voltage range. DAC Equation When all binary numbers are counted in ascending order starting with 0000 toward the maximum 4bit number of 1111, the circuit in Fig. 14.15 produces an analog output voltage increasing in steps see Fig. 14.16b above. In order to obtain the range of variation of the output voltage in a general case, we may want to rewrite Eq. (14.13) as d3 d2 d1 d0 υDAC E þ þ þ ð14:15aÞ 2 4 8 16 which is the commonly used DAC equation. Here, the constant E with the units of volts, E
2N Q,
N
ð14:15bÞ
4
is the fullscale output voltage range of the DAC, and Q is its resolution voltage. For a 4bit DAC, the maximum binary number is 1111, and Eq. (14.15a) yields υDAC jmax
2N 1 E 2N
15 EE 16
ð14:15cÞ
Clearly, the maximum output voltage approaches E more and more precisely as the number of bits, N, increases. Equation (14.15) no longer relies upon the speciﬁc DAC circuitry. They are valid for any DAC chip. Equation (14.15b) indicates that the resolution voltage and the fullscale output voltage range are simply related to each other by a factor of 2N. This factor is exactly the number of distinct binary words or quantization levels for a DAC with N inputs (input bits). Note that Eq. (14.15a) is perhaps the most useful DAC formula, often present in the datasheets.
Setting Output Voltage Range In a realistic DAC chip, the combinations of voltage sources and resistors shown in Fig. 14.15 (resistor “current sources”) are replaced with the transistorbased “current sources.” Therefore, it is not necessary to change resistor values in order to obtain different voltage ranges and resolution voltages, as might appear at ﬁrst sight from Eqs. (14.13b) and (14.15b). Instead, the output voltage range, E, and simultaneously the resolution voltage Q are simply controlled by setting an external voltage reference, which precisely coincides with the desired value of E. A case in point is a DAC0808 chip (an 8bit DAC) from National Semiconductor Corp. (Texas Instruments). XIV711
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AnalogtoDigital Conversion Output
4.0 V
Output
3.0 V
4 bit DAC vDAC
Vout=2.5e5*t
2.0 V
6 bit DAC vDAC
1.0 V
0.0 V 0
2
4
6
8
10
12
14 16 0 time, s Output
2
4
6
8
10
12
14 16 time, s
4.0 V
3.0 V
8 bit DAC vDAC
2.0 V
1.0 V
0.0 V 0
2
4
6
8
10
12
14 16 time, s
Fig. 14.17. Comparative resolution of the 4bit DAC, the 6bit DAC, and the 8bit DAC. In all three cases, the fullscale output voltage range (4 V) is the same. The linear time dependence, υout ðt Þ ¼ 2:5 105 t ðVÞ, over the time interval from 0 to 16 μs is approximated.
14.2.5 Other DAC Circuits R/2R Ladder DAC An alternative to the circuit in Fig. 14.15 is an R/2R ladder DAC, which is shown (without the inverter) in a 4bit conﬁguration in Fig. 14.18. The resistive ladder is a circuit, which has a similar performance when more sections are added. This circuit requires a more careful analysis based on Thévenin equivalents, but the ﬁnal result exactly coincides with Eq. (14.15a), that is (after adding the inverter), d3 d2 d1 d0 υDAC E þ þ þ ð14:17aÞ 2 4 8 16 where the fullscale voltage range is given by E
2N Q
RF D R
ð14:17bÞ
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of Fig. 14.18 is redrawn in the form of the equivalent voltage sources D3, D2, D1, D0 referenced to ground in Fig. 14.19, top. b R
c R
iF
d RF
R
0V

a
Vout
+
2R 2R D0 +
2R D1 +
2R D2 +
2R D3 + 0V
0V R D0/2 + R D1/2+D0/4 + R D2/2+D1/4+D0/8 + R D3/2+D2/4+D1/8+D0/16 +
Fig. 14.19. Solving R/2R ladder DAC using the method of Thévenin equivalent while adding one ladder section at a time.
The resulting circuit block on the left of the ampliﬁer (the ladder) contains only voltage sources and resistors. The method of Thévenin equivalent is applied in order to convert it to a form of a single voltage source VT in series with a resistance RT. The bruteforce calculation of Thévenin resistance for the entire ladder block is rather simple. However, a calculation of Thévenin voltage VT is not. The resistance calculation (with shorted out voltage sources) gives RT R. But what about VT? The idea (which is also applicable to other ladder networks) is to apply Thévenin equivalent in steps, adding one single section a,b,c,d of the ladder block at a time, starting with the leftmost section a in Fig. 14.19. Every such step is analyzed straightforwardly; it gives Thévenin voltages and resistances shown in Fig. 14.19. The last step in Fig. 14.19 followed by solving the inverting ampliﬁer circuit leads us exactly to Eq. (14.17) if an extra inverter is added.
PWM DAC One should mention a digital PWM (pulsewidth modulation) code, which is digitally stored and then converted to an analog signal by means of an analog RC ﬁlter studied in Chapter 9. Such a technique is becoming increasingly popular today. XIV716
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Section 14.3: SampleandHold Circuit: Nyquist Rate
Section 14.3 SampleandHold Circuit: Nyquist Rate 14.3.1 AnalogtoDigital Converter The analogtodigital converter (ADC) studied in this and next sections should translate a continuously varying analog voltage (voice, electromagnetic signal, readout of a sensor) into a continuous stream of binary numbers (and equivalent digital voltages) passed to a processor. The digitaltoanalog converter (DAC) studied in the previous section performs an inverse operation. The place of the ADC in the generic block diagram of a DSP (digital signal processor) is shown in Fig. 14.20. The ADC circuit analysis is in general much more involved than the DAC circuit analysis. ADC design is an exciting and growing area of the electrical engineering with many hundreds of engineers employed.
Analog input voltage vin(t)
ADC circuit
D0 D1 D2 D3 D4 D5 D6 D7
Digital processor
D0 D1 D2 D3 D4 D5 D6 D7
DAC circuit
Analog output voltage vDAC(t)
Interface (control) circuit
Fig. 14.20. Place of the ADC in the DSP block diagram.
14.3.2 A Quick Look at an Analog Sinusoidal Voltage First, an input analog voltage to the circuit in the ﬁgure above should be analyzed. The simplest and simultaneously the most important case of the analog voltage is a pure sine or cosine function shown in Fig. 14.21 and also called the harmonic. The sinusoidal voltages are critical in AC circuit analysis studied previously. Why are we interested in a sinusoidal voltage input also in this chapter? The reason is that, according to the method of Fourier analysis studied in Chapter 9, all existing continuous voltages υin(t), including voltage signals corresponding to the human voice, may be expanded into a sum of such multiple sinusoidal functions. Every sinusoidal function will have its own frequency, phase, and amplitude. The sinusoidal voltage can be written in the form
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Section 14.3: SampleandHold Circuit: Nyquist Rate
there is C hold 0:02 μF. Those circuits may be used to hold a given voltage value from any particular sensor in a robot or elsewhere. a)
input buffer
sample
output buffer
vin(t)
+
+

Chold
reset
0V
b) input buffer
0V
sample G
output buffer
vin(t)
+

vSH(t)

D
+
vSH(t)

S D
reset
G
Chold S 0V
0V
Fig. 14.23. Sampleandhold circuit. (a) Circuit with “sample” and “reset” switches. (b) Switches replaced by MOSFET transistors.
However, usually, the sampleandhold circuit is an internal part of the ADC chip as explained further in the next section. Another popular modiﬁcation is the trackandhold circuit suggested as one of the homework problems. Could an ADC function without the sampleandhold circuit? Yes, it could. This is exactly the way how ﬁrst ADCs have been made. However, it means that the voltage value to be converted will not be held; it will change during the conversion process, which may lead to wrong bits.
14.3.5 Nyquist Rate How fast should we sample? The answer seems to be trivial: as fast as possible in order to acquire the most precise replica of the input signal. However, a very fast sampling rate may be either impossible in practice for ultrafast signals, or it may lead to huge and unrealistic memory consumptions. On the other hand, reducing the sampling rate, while still keeping the major information about the analog voltage behavior, allows us to proceed with a realistic circuit design and realistic memory requirements. Therefore, a minimum acceptable sampling rate, fS min, for a given analog voltage signal is of great practical importance. In order to establish this minimum value, we will again consider the sinusoidal voltage of 1 MHz shown in Fig. 14.24. We introduce the Nyquist rate, fN, of this voltage signal, which is two times its frequency, i.e., XIV721
Chapter 14 fN
AnalogtoDigital Conversion ð14:21Þ
2f
We further analyze three distinct cases in Fig. 14.24: 1. When the sampling frequency is higher than the Nyquist rate see Fig. 14.24a then the sampleand hold voltage generally follows the signal shape. After proper ﬁltering (smoothing the stairs in Fig. 14.24a), it will very well replicate the original analog sinusoid. 2. When the sampling frequency is exactly equal to the Nyquist rate see Fig. 14.24b then the sampleand hold voltage becomes exactly the pulse train. And yet, after proper ﬁltering, this pulse train may be converted to the sinusoidal function of the same frequency the reconstruction may be successful. 3. However, when the sampling frequency is less than the Nyquist rate see Fig. 14.24c then the sampleandhold voltage may become just a straight line. No matter what do we do further, the signal information is entirely lost! sample
a)
hold
1 0.5
fS=2fN
0 0.5 1
b)
hold
sample 1 0.5
vin(t), V vSH(t), V 0
fS=fN
0.5 1
c)
hold
sample 1 0.5 0
fS=0.5fN
0.5 1 0
1
t, s
2
Fig. 14.24. Effect of sampling rate on the sampleand hold voltage.
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Chapter 14
AnalogtoDigital Conversion
Section 14.4 AnalogtoDigital Converter The sampleandhold (SH) circuit studied previously is only the “frontend” of the analogtodigital converter. Even if this circuit is available, the A to D converter itself still needs to be designed. Therefore, the ADC block in Fig. 14.25 still needs to be studied. This section completes the analogtodigital converter model. We will consider only two basic ADC circuit concepts (ﬂash and successive approximation) leaving many others.
Analog input voltage vin(t)
ADC circuit
D0 D1 D2 D3 D4 D5 D6 D7
Digital processor
D0 D1 D2 D3 D4 D5 D6 D7
DAC circuit
Analog output voltage vDAC(t)
Interface (control) circuit
Fig. 14.25. DSP block diagram. The ADC converter still needs to be completed.
14.4.1 Flash ADC Circuit The next step in ADC design is to convert the sampleandhold voltage υSH(t) to the digital voltage itself. Figure 14.26 shows the concept of a ﬂash A to D converter. This is the fastest A to D conversion method. All data are essentially processed in parallel. The sampleandhold circuit is not shown in Fig. 14.26, but it is implied. The circuit in Fig. 14.26 is intended to encode the sampleandhold voltage into 3bit binary numbers or 3bit digital voltages D2, D1, D0. It includes (from left to right): 1. A voltage divider with eight equal resistors, R. The voltage divider subdivides the reference voltage, E, into eight levels corresponding to eight possible 3bit words: 000, 001, 010, 011, 100, 101, 110, and 111. The reference voltage E to the ADC chip simultaneously determines the fullscale measurement voltage range of the ADC studied below. 2. Seven openloop comparator ampliﬁers (the comparator for a 000 condition is not needed) compare the input voltage with the seven nontrivial voltage levels of the voltage divider. The comparators generate high voltage when υSH(t) exceed the corresponding voltage divider level and low voltage otherwise. 3. The output of the comparator block the second column of Table 14.4. 4. Finally, an ADC encoder block. This circuit converts binary words from the comparator block to the binary numbers. It may be constructed with logic gates. XIV724
Chapter 14
Section 14.4: AnalogtoDigital Converter Reference voltage Input from sample and hold circuit vSH(t)
E=10V R
+

8.75V
R
+

7.5V
R
Priority encoder
+

6.25V
7 6 5 4 3 2 1 0
R
+

5.0V
R
+

3.75V
R
Three bit parallel output D0 D1 D2
+

2.5V
R
0V
+

1.25V
R
Enable pulses 0V
Fig. 14.26. A 3bit ﬂash ADC.
Operation Table 14.4 lists circuit parameters for the voltage divider analysis in Fig. 14.26. Table 14.4. Output of comparators (second column) and output of the entire ADC chip (third column).
Voltage range of sampleandhold voltage υSH, V 8.75–10 7.5–8.75 6.25–7.5 5–6.25 3.75–5 2.5–3.75 1.25–2.5 0–1.25
Output of the comparator block (7–0) 11111110 01111110 00111110 00011110 00001110 00000110 00000010 00000000
Output of the priority encoder (binary number d 2d 1d 0) 111 110 101 100 011 010 001 000
Voltage decoded back from d2d1d0 and resolution Q, Qð4d 2 þ 2d 1 þ 1d 0 Þ V 8.75 7.5 6.25 5 3.75 2.5 1.25 0
Quantization error, max 1.25 V or Q 1.25 V or Q 1.25 V or Q 1.25 V or Q 1.25 V or Q 1.25 V or Q 1.25 V or Q 1.25 V or Q
XIV725
Chapter 14
Problems D. Decode the corresponding binary number given the LSB/MSB positions (present its decimal equivalent). MSB
LSB 5V 0V 5V 0V
Problem 14.7. A Tektronix oscilloscope is measuring a voltage clock signal. The oscilloscope window is shown in the following ﬁgure. A. Determine the clock period. B. Determine the clock frequency (show units). C. Determine the duty cycle of the clock waveform. D. Determine PkPk (peaktopeak) voltage of the clock signal.
2
4
6
8
10
12
14
t, s
Problem 14.9. The following ﬁgure shows a timing diagram: the clock signal and the actual synchronized serial bit stream (the polar NRZ code). A. Determine the clock period. B. Determine the clock frequency (show units). C. Determine the bit rate of the bit stream (show units). D. Decode the corresponding binary number given the LSB/MSB positions (present its decimal equivalent). LSB
MSB
5V 0V 5V 0V
Problem 14.8. The following ﬁgure shows a timing diagram: the clock signal and the actual synchronized serial bit stream (the unipolar NRZ code). A. Determine the clock period. B. Determine the clock frequency (show units). C. Determine the bit rate of the bit stream (show units).
5 V
0.1
0.3
0.5
0.7
0.9
1.1
1.3
t, ns
14.1.4 Binary Numbers 14.1.5 Hexadecimal Numbers 14.1.6 ASCII Codes and Binary Words 14.1.7 Tristate Digital Voltage Problem 14.10. Name two major reasons why the analog computer was surpassed by the digital computer.
XIV735
Chapter 14
AnalogtoDigital Conversion
Problem 14.11. Describe in your own words the meaning of: 1. A digital word 2. A nibble 3. A byte
Problem 14.17. Write down the year of your birth. Without a calculator or MATLAB, convert this decimal number to: A. Binary number B. Hexadecimal number
Problem 14.12. Without a calculator or MATLAB, convert the following binary numbers to decimal numbers: A. 1010 B. 101010 C. 11.1 D. 10.001
Problem 14.18. Using the ASCII conversion table, write the string USA in terms of: A. Three decimal numbers B. Three hexadecimal numbers C. Three binary numbers
Problem 14.13. Using either a calculator or MATLAB, convert the following binary numbers to decimal numbers: A. 1000001.111111 B. 0001111.000010 Problem 14.14. Without a calculator or MATLAB, convert the following decimal numbers to binary numbers: A. 19 B. 10 C. 1960 D. 14.25 Problem 14.15. Using either a calculator or MATLAB, convert decimal numbers that follow to binary numbers. The desired degree of precision is six bits after the binary point: A. 133.33 B. 999.125 C. 256.256 Problem 14.16 A. How many bits are necessary to represent decimal number 300,000,000 in binary form? B. How many bytes are necessary?
Problem 14.19. Without a calculator or MATLAB, convert hexadecimal numbers that follow to decimal numbers and binary numbers, respectively: 1. 1 2. 12 3. 1A 4. AAA 5. ECE 6. BE 7. CE Problem 14.20. Using either a calculator or MATLAB, convert hexadecimal numbers that follow to decimal numbers: 1. 3F7 2. EE.25 3. 555h 4. 0x555 Problem 14.21. Convert binary numbers that follow to hexadecimal numbers: 1. 01 2. 11110101 3. 1000000110000000 4. 1111111110000001 5. 1111111111111111.1111
XIV736
Chapter 14
Problems
Problem 14.22. The DAC circuit from Fig. 14.15 is operating using the virtualground condition of the inverting ampliﬁer in order to add up weightedinput currents. With this in mind, a beginning ECE student removes the ampliﬁer from the circuit and puts a common ground reference at the summing node instead. Will the circuit work?
D1, V 0 0 5 5 0 0 5 5
Problem 14.25. Repeat the previous problem for the DAC shown in the following ﬁgure. iF
Problem 14.24. A 3bit binaryweighted digitaltoanalog converter (DAC) is shown in the ﬁgure. The circuit does not include the inverter. Fill out the table that follows: iF 10 k
80 k
D0

20 k
D2
0V
+
40 k
D1
Vout
5k
80 k
D0 40 k
Problem 14.23 A. For an 8bit binaryweightedinput DAC, express its output voltage through the resolution voltage (LSB voltage), Q, and binary digits d7, d6, . . ., d0, which corresponds to the input voltages. B. Find the output voltage when the LSB voltage is 20 mV and d 7 ¼ d 6 . . . ¼ d 2 ¼ 1, d 1 ¼ 0, d 0 ¼ 1.
υDAC, V
D0, V 0 5 0 5 0 5 0 5
D1 20 k
0V

14.2.1 Digital to Analog Converter (DAC) 14.2.2 Circuit (A BinaryWeightedInput DAC) 14.2.3 Underlying Math and Resolution Voltage
D2, V 0 0 0 0 5 5 5 5
vDAC
+
14.2 Digital to Analog Converter
D2
Problem 14.26. For a 4bit binaryweightedinput DAC, the input voltages D3, D2, D1, D0 are either 0 V or 5 V. The resolution voltage Q of 10 mV is required: A. Present the circuit diagram of a DAC; label the input voltages. B. Specify one set of possible resistor values. Problem 14.27. For a 5bit binaryweightedinput DAC, the input voltages D4, D3, D2, D1, D0 are either 0 V or 2.5 V. The resolution voltage Q of 1 mV is required: A. Present the circuit diagram of a DAC; label the input voltages. B. Specify one set of possible resistor values. Problem 14.28. Design a 4bit binaryweightedinput DAC circuit which attempts to output the analog voltage in the form of a linear time dependence, υout ðt Þ ¼ 5 105 t ðVÞ over time interval from 0 to 4 μs. The input to the DAC is a binarycounter sequence of all 4bit
XIV737
Chapter 14
AnalogtoDigital Conversion
binary numbers. The bit width is 0.25 μs; high and low voltages are 5 V and 0 V, respectively: A. Present the corresponding circuit diagram; specify required DAC resolution voltage (LSB voltage) and necessary resistor values. B. Plot the input digital voltages to scale versus time. C. Plot the output voltage to the DAC to scale versus time. a)
RW=
D1 D2 D3
RW=
b)
RF=
Input
5V 0V
D0
5V
5V 0V
0V
D1
5V 0V 5V
D2
0V 0
c)
Circuit diagram
Q=
RF=
Input
b) D0
a)
Circuit diagram
Q=
binary numbers. The bit width is 1 μs; high and low voltages are 5 V and 0 V, respectively: A. Present the corresponding circuit diagram; specify the required DAC resolution voltage (LSB voltage) and necessary resistor values. B. Plot the input digital voltages to scale versus time. C. Plot the output voltage to the DAC to scale versus time.
1
vDAC
2
3
4
5V 0V 5V 0V
time, s
Output
2.0 V
0
c)
1
2
3
4
5
6
7
8
time, s
vDAC
Output
4.0 V
1.5 V
1.0 V 2.0 V 0.5 V
0.0 V
0.0 V 0
1
2
3
4
time, s
Problem 14.29. Design a 3bit binaryweightedinput DAC circuit which attempts to output the analog voltage in the form of a linear time dependence, υout ðt Þ ¼ 5 105 t ðVÞ over time interval from 0 to 8 μs. The input to the DAC is a binarycounter sequence of all 3bit
0
1
2
3
4
5
6
7
8
time, s
Problem 14.30. A 4bit binaryweightedinput DAC has the resolution voltage, Q, of 0.125 V and the input voltages shown in the following ﬁgure. Pin D3 is accidentally connected to ground. Plot the output voltage of a DAC to scale versus time.
XIV738
Chapter 14
Problems
a) D0 D1 D2 D3
14.2.4 DAC Fullscale Output Voltage Range, Resolution, and Accuracy
Input 5V 0V 5V 0V 5V 0V 5V 0V 0
b)
1
vDAC
2.0 V
2
3
4
time, s
Output
Problem 14.32 A. For an 8bit DAC chip, express its output voltage through the fullscale output voltage range, E, and binary digits d7, d6, . . ., d0, which corresponds to the input voltages. B. Find the output voltage when the fullscale output voltage range, E, is 6 V and d 7 ¼ d 6 . . . ¼ d 2 ¼ 1, d 1 ¼ 0, d 0 ¼ 1.
1.5 V
1.0 V
0.5 V
0.0 V 0
1
2
3
4
time, s
Problem 14.31. Repeat the previous problem for the input voltages shown in the following ﬁgure. a)
Input D0 D1 D2 D3
5V 0V 5V 0V 5V 0V 5V 0V 0
1
vDAC
2.0 V
2
3
4
time, s
b)
Output
1.5 V
1.0 V
0.5 V
0.0 V 0
1
2
3
4
time, s
Problem 14.33 A. For a 10bit DAC chip, express its output voltage through the fullscale output voltage range, E, and binary digits d9, d8, . . ., d0, which corresponds to the input voltages. B. Find the output voltage when the fullscale output voltage range, E, is 10 V and d 9 ¼ d 8 . . . ¼ d 3 ¼ 1, d 2 ¼ d 1 ¼ 0, d0 ¼ 1 C. Find the resolution voltage of the DAC, Q. Problem 14.34. A 6bit DAC and a 8bit DAC use a 6 and 8bit binarycounter input sequences in order to produce the analog voltage in the form of a linear time dependence, υout ðt Þ ¼ 5 105 t ðVÞ, over the same time interval from 0 to 10 μs. Determine: 1. DAC resolution in bits (quantization levels) 2. Fullscale output voltage range, E 3. DAC voltage resolution, Q 4. Necessary bit rate, fb Problem 14.35. An 8bit DAC and a 10bit DAC use an 8 and 10bit binarycounter input sequences in order to produce the analog voltage in the form of a linear time dependence, υout ðt Þ ¼ 5 106 t ðVÞ, over the same time interval from 0 to 1 μs. Determine: 1. DAC resolution in bits (quantization levels) 2. Fullscale output voltage range, E 3. DAC voltage resolution, Q 4. Necessary bit rate, fb
XIV739
Chapter 14
AnalogtoDigital Conversion
Problem 14.36. For a 3bit R/2R ladder DAC, the input voltages D2, D1, D0 are either 0 Vor 5 V. The resolution voltage Q of 100 mV is required: A. Present the circuit diagram of a DAC; label the input voltages. B. Specify one set of possible resistor values.
14.3 SampleandHold Circuit. Nyquist Rate
14.3.1 Analog to Digital Converter (ADC) 14.2.5 Other DAC Circuits 14.3.2 A Quick Look at an Analog SinuProblem 14.37. By solving the ampliﬁer circuit, soidal Voltage
determine the output voltage of the 3bit R/2R ladder DAC shown in the following ﬁgure given that D2 ¼ 0 V, D1 ¼ 5 V, D0 ¼ 0 V. D0=0 V
D1=5 V
D2=0 V iF
2R
2R
RF
2R
Vout
0V 2R
b
R
+
a
c
R
Problem 14.40 A. Determine frequency in Hz, angular frequency in rad/sec, phase, and amplitude of the harmonic voltage signal shown in the following ﬁgure. B. Write the voltage in the form of a cosine function with the corresponding amplitude, frequency, and phase. voltage, V 2
0V
0V 1
Problem 14.38. By solving the ampliﬁer circuit, determine the output voltage of the 4bit R/2R ladder DAC in Fig. 14.18 given that D3 ¼ 0 V, D2 ¼ 5 V, D1 ¼ 0 V, D0 ¼ 0 V.
0
1
2
D0=0 V
D1=0 V
D2=5 V
D3=0 V
0
15
30
iF
2R
2R
2R
RF
2R
vDAC
0V a
R
b
R
c
+
2R
d
R
0V
45 time, s
60
75
90
Problem 14.41. Determine frequency in Hz, angular frequency in rad/s, and amplitude of the harmonic voltage signal shown in the following ﬁgure. voltage, V
0V 2
Problem 14.39. An important step in the analysis of the R/2R ladder network is ﬁnding Thévenin equivalent for the circuit shown in the ﬁgure. Find the Thévenin equivalent and draw the corresponding circuit.
1
0
1
2
R
a
R Dx
0
15
30
45 time, s
60
75
90
2R
+ 
Dy
+ 
b
Problem 14.42 A. Determine frequency in Hz, angular frequency in rad/s, phase, and amplitude of
XIV740
Chapter 14
Problems
the harmonic voltage signal shown in the ﬁgure. B. Write the AC voltage in the form of a cosine function, with the corresponding amplitude, frequency, and phase.
Problem 14.45. For the voltage signal shown in the following ﬁgure, determine: A. Frequency, f, and amplitude, Vm, of the analog voltage B. Sampling interval, TS, and sampling rate, fS, for the sampleandhold voltage
voltage, V
vin(t), V vSH(t), V
2
1
2 0
1 1
0 2
1 0
15
30
45 time, s
60
75
90
2
0
14.3.3 SampleandHold Voltage Problem 14.43. For the voltage signal shown in the following ﬁgure, determine: A. Frequency, f, and amplitude, Vm, of the analog voltage B. Sampling interval, TS, and sampling rate, fS, for the sampleandhold voltage
t, s 1
Problem 14.46*. Plot the ﬁgure to the previous problem using MATLAB, introduce a title, and label the axes. Present the text of the corresponding MATLAB script.
14.3.4 SampleandHold (SH Circuit)
Circuit
Problem 14.47. Draw the schematic of the sampleandhold circuit. Explain its operation in steps.
vin(t), V vSH(t), V 1 0.5 0 0.5 1 0
0.5
1
2 t, s
Problem 14.44*. Plot the ﬁgure to the previous problem using MATLAB, introduce a title, and label the axes. Present the text of the corresponding MATLAB script.
Problem 14.48. The circuit shown in the ﬁgure is another modiﬁcation of the sampleandhold circuit. Assuming that the capacitor responds instantaneously: A. Explain the circuit operation. B. Sketch its output voltage to scale versus time in the ﬁgure the follows. The switching control voltage is shown on the top of the ﬁgure. The switch is closed at high control voltage and is open at low control voltage.
XIV741
Chapter 14
AnalogtoDigital Conversion
input buffer
vin(t), V vSH(t), V
output buffer hold (open)
vin(t)
fS=fN
vSH(t)
+
+


Chold
0V 0
vin(t), V vSH(t), V?
switch closed switch open 1 0.5
1
2 t, s
Problem 14.52. An audio signal (containing all analog sinusoids with frequencies between 20 Hz and 20 kHz) is recorded using a simpliﬁed sampleandhold circuit (the reset switch is omitted) shown in the following ﬁgure.
0
input buffer 0.5
output buffer sample
vin(t)
vSH(t)
+
1
+

0
1
2
3
4

5 t, s
Chold
14.3.5 Nyquist Rate Problem 14.49. An analog voltage is a combination of three sinusoidal harmonics with frequencies 1 MHz, 0.5 MHz, and 0.2 MHz. The voltage amplitudes of the individual sinusoids are 1 V, 1 V, and 5 V. What is the limit on minimum acceptable sampling rate of the sampleandhold circuit? Problem 14.50. An analog voltage is a combination of four sinusoidal harmonics with frequencies 0.5 MHz, 0.2 MHz, 1 MHz, and 1.2 MHz. The voltage amplitudes of the individual sinusoids are 5 V, 1 V, 1 V, and 0 V. What is the limit on minimum acceptable sampling rate of the sampleandhold circuit? Problem 14.51. Using the ﬁgure below, could you demonstrate when the sampling at exactly the Nyquist rate may not be successful?
0V
The sample switch closes every: A. 227 μs B. 22.7 μs C. 2.27 μs then opens momentarily. Which case should be preferred for the minimum memory requirement (for an audio CD)?
14.4 Analog to Digital Converter 14.4.1 Flash ADC Problem 14.53. How many comparators would we need for an 8bit ﬂash ADC? For a 12bit ﬂash ADC?
XIV742
Chapter 14
Problems
Problem 14.54 A. Draw a complete circuit diagram of a 2bit ﬂash ADC with the fullscale measurement voltage range of 4 V. B. Fill out the following table: Range of sampleandhold voltage υSH, V
Output of comparator block (3–0)
Output of the priority encoder (binary number d 1d 0)
3–4 2–3 1–2 0–1
Problem 14.55 A. Draw a complete circuit diagram of a 3bit ﬂash ADC with the fullscale measurement voltage range of 8 V. B. Fill out the following table: Range of sampleandhold voltage υSH, V 7–8 6–7 5–6 4–5 3–4 2–3 1–2 0–1
Output of comparator block (7–0)
Output of the priority encoder (binary number d 3d 2d 1d 0)
14.4.2 ADC Resolution in Bits, Fullscale Input Voltage Range, and Voltage Resolution Problem 14.56. For a 6bit ADC determine: A. Resolution in bits (quantization levels) B. Voltage resolution, Q C. Relative accuracy percentage assuming a 1 LSB error when the fullscale measurement voltage range is 8 V. Problem 14.57. For an 8bit ADC determine: A. Resolution in bits (quantization levels) B. Voltage resolution, Q C. Relative accuracy percentage assuming a 1 LSB error when the fullscale measurement voltage range is 10 V.
14.4.3 ADC Equation and Quantization Error
Problem 14.58. A 5bit ﬂash ADC follows a midrise coding scheme. The reference voltage is 12 V: 1. Present the ADC equation. 2. Determine ADC quantization error. 3. Find ADC output code when the sampleandhold voltage, υSH, is 3.1 V. Problem 14.59. An 8bit ﬂash ADC follows a midtread coding scheme. The reference voltage is 5 V: 1. Present the ADC equation. 2. Determine ADC quantization error. 3. Find ADC output code when the sampleandhold voltage, υSH, is 0.2 V.
XIV743
Chapter 14
AnalogtoDigital Conversion
Problem 14.60. A 3bit ﬂash ADC is constructed as shown in the following ﬁgure: A. Fill out the table below, which describes the ADC operation. Express all absolute voltage values in terms of the resolution voltage, Q. B. Based on this table, estimate the quantization error of the ADC in terms of Q. C. Is this ADC design is better than the midrise coding scheme? Reference voltage E=8V 1.5R
vSH(t)
+

Voltage range of υSH, in terms of Q
Output of comp. block (7–0)
Output of priority encoder (binary number d 2d 1d 0)
Voltage decoded back, in terms of Q
6.5Q–8Q 5.5Q–6.5Q 4.5Q–5.5Q 3.5Q–4.5Q 2.5Q–3.5Q 1.5Q–2.5Q 0.5Q–1.5Q 0–0.5Q
R
14.4.4 SuccessiveApproximation ADC
+

R
+

R
+

R
+

R
7 6 5 4 3 2 1 0
+

R
+

0.5R
Problem 14.61. Draw the circuit diagram for a 3bit successiveapproximation ADC.
Priority encoder
0V
D0 D1 D2
Problem 14.62. A 4bit successiveapproximation ADC has the input voltage of υSH ¼ 1:05 V; the DAC resolution voltage is 0.1 V. Determine the sequence of binary states and the ﬁnal ADC output. Problem 14.63. A 5bit successiveapproximation ADC has the input voltage of υSH ¼ 3:1 V; the DAC resolution voltage is 0.2 V. Determine the sequence of binary states and the ﬁnal ADC output.
0V
XIV744
Chapter 15
Chapter 15: Embedded Computing
Overview Prerequisites:  Knowledge of binary and hexadecimal numerical representations and conversions  Knowledge of basic digital logic circuits  Knowledge of basic circuitry Objectives of Section 15.1:  Understand highlevel architecture of a generic computer  Define elemental parts of an embedded computers, i.e., CPU, memory, I/O peripherals, buses  Understand architecture and function of the CPU, memory, and I/O Objectives of Section 15.2:  Understand the organization of memory  Describe how data is stored in little and bigendian microprocessors  Describe and understand various categories and types of memory Objectives of Section 15.3:  Understand capabilities of Arduino Uno  Install opensource Arduino IDE and driver software  Learn how to write Arduino sketches and upload code to Arduino Uno Objectives of Section 15.4:  Understand basic data types available on Arduino Uno  Perform basic operations on data types using arithmetic operations on Arduino  Create functions to simplify code and reduce repeated instructions  Understand libraries and their functions on Arduino  Control a servomotor using Arduino Uno board
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_15
XV745
Chapter 15
Embedded Computing
Objectives of Section 15.5:  Understand and create code with conditional statements  Understand and implement code with switch statements  Be able to control loops of a sketch and incite repetition  Use strings and arrays in a program where appropriate  Print out messages to the serial monitor for debugging  Understand interrupts and their advantages and disadvantages  Be able to generate a square wave with a controllable duty cycle on a pin Application examples:  Servomotor control  Emergency motor stop
Keywords: CPUs: Realtime, Von Neumann/Princeton architecture, Hierarchy, ALU (arithmetic logic unit), CPU functions (fetch, decode, execute, writeback), Opcode, Operands, Instruction set, RISC (Reduced Instruction Set Computing), Byte, Address, Kilobyte, Megabyte, Gigabyte, Kilobinary, Megabinaries, Gigabinaries, Bus, Bus width, Address bus, Data bus, Rule of shared buses; Memory: Address, Big endian, Little endian, Volatile, Nonvolatile, RAM (random access memory), ROM (readonly memory), EEPROM (electronically erasable programmable readonly memory), Flash memory, PROM (programmable readonly memory), EPROM (erasable programmable readonly memory), Address space; Arduino: Arduino, Arduino Uno, IDE (integrated development environment), File menu, Edit menu, Sketch menu, Tools menu, Functions, Return value, Void, Setup(), Loop(), Comment, Primitive data types, Int (integer), Float (ﬂoating point), Double (double precision), Boolean (true or false), Char (character), Variables, Type speciﬁer, Declaration, Assignment statement, Typecasting, Function header, Body of the function, Argument, Arrays, Strings, Library, Encapsulation, Access functions, Header ﬁle, Function prototypes, Script ﬁle, Servo library, Servomotor, Servo object, Conditional statements, State programming, Switch statement, For loop, Pseudocode, Inﬁnite loop, Increment operator, Decrement operator, While loop, Element, Index, Serial communication, Baud rate, Polling, Interrupts, ISR (interrupt service routine), Interrupt queue, Debounced
XV746
Chapter 15
Section 15.1: Architecture of Microcontrollers
Section 15.1 Architecture of Microcontrollers Embedded computers are literally everywhere in modern life. Embedded computing is the incorporation of computing devices like microcontrollers into the design of a product or larger system that is not itself a computer. On any given day, we interact with and depend on dozens of small computers to make coffee, run cell phones, take pictures, run the dishwasher, control elevators, stop the car, and so on. For every PCstyle computer made each year, there are approximately 50 100 embedded computer devices produced. Consider the contents of an average student’s backpack. It is likely to contain a notebookstyle personal computer (PC). However, it is just as likely to contain a digital music player, a cell phone, a handheld graphing scientiﬁc calculator, a keyless entry car key, and maybe an ebook reader! Each of these devices relies on the computing capability embedded within it. A key feature of an embedded computer is that it typically only performs a single function or small set of very tightly coupled functions. It is not a generalpurpose device like a PC. Another feature of an embedded system is that it can enable operation of a complex function by a nonexpert. For example, one can live a full and complete life without knowing the details of the Moving Pictures Expert Group Audio Layer3 encoding standard (i.e., MPEG3). To use a digital music player, we simply press the play button. Embedded computers are often resource constrained and must exhibit very high reliability. They must perform their assigned task in real time or very close to real time and must work ﬂawlessly for years powered only by a small battery. To achieve these goals, embedded computers are usually a mix of tightly integrated hardware and software. An engineer must understand and appreciate both the hardware and the software aspects of embedded computers to use them effectively. This chapter presents a highlevel overview of microcontrollers and how they can be interfaced with control hardware components in an electronic circuit.
15.1.1 A Generic Microcontroller Figure 15.1 is a block diagram view of a generic computer. Any computer system, whether large or small, will contain three functional subblocks: the central processing unit (CPU), memory, and input/output (or I/O) devices. The general architecture in Fig. 15.1 is called the Von Neumann or Princeton architecture, and it dates from 1946. In earlier mainframe and PCstyle computers, the CPU, memory, and I/O devices were each implemented as separate circuits often on separate circuit boards and were connected through wire buses. Modern microcontrollers are complete computer systems implemented within a single integrated circuit (IC).
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Embedded Computing Control Lines Address Bus Data Bus
Memory
CPU
I/O Devices
Fig. 15.1. Architecture of a generic computer.
The hierarchy of hardware and software for generalpurpose computing (e.g., on a PC) is shown in Table 15.1. Table 15.1. Hierarchy of hardware and software for generalpurpose computing. Applications (MultiSim, MATLAB, Google Earth, etc.) Operating system (Windows 10, Linux, or Mac OS X) Hardware Abstraction Layer (device drivers, basic I/O system—BIOS) Hardware (CPU, memory, I/O devices)
This hierarchy is simpliﬁed in an embedded system where a single custom application interfaces directly with the microcontroller’s hardware see Table 15.2. Table 15.2. Hierarchy of an embedded system. Application (single, custom application) Hardware (CPU, memory, I/O devices)
15.1.2 Central Processing Unit The CPU is the “brain” of a computer. The CPU contains the control unit, the arithmetic logic unit (ALU), and bus interface circuitry as seen in Fig. 15.2. It controls the operation of memory and the I/O devices and executes program instructions. Conceptually, a CPU performs four functions: fetch, decode, execute, and writeback. First the CPU fetches an instruction from the part of memory where the program is stored. An instruction is simply a multibyte binary code. The instruction is then decoded within the control unit to extract its opcode (operational code which speciﬁes operations to be performed) and the operands (quantities on which operations are performed). The sample format of this process can be seen in Fig. 15.3. A CPU can only understand and execute a ﬁxed number of operations which is called its instruction set.
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Section 15.1: Architecture of Microcontrollers
initiate operations by the memory or peripherals. There will also be an address bus which used by the CPU to specify where data should be read from or written to and a data bus which conveys binary data between the CPU and memory or between the CPU and the I/O devices. Notice in Fig. 15.1 that a single bidirectional data bus goes between memory, the CPU, and the I/O peripherals. It is the control lines from the CPU that establish who has control of the bus. Only one device is allowed to place data on the data bus at any given time. The fundamental rule of shared buses is that there can be only one bus driver.
d7
a)
b)
8 Memory
CPU
CPU
Memory
D70 d0
Fig. 15.4. (a) Explicitly showing all eight connections required to transfer 1 byte makes schematics cluttered; (b) multibit buses are drawn as a single, thicker line with bus width indicated above.
15.1.7 Universal Synchronous/Asynchronous Receiver/Transmitter (USART) Parallel Data Transmission Within a microcontroller, data is passed along parallel interfaces (buses) between the CPU and memory or between the CPU and I/O peripherals. However, parallel data transmission is not practical for the “long” haul (i.e., off chip) for several reasons. Chief among them are that the length of parallel links is usually limited and that there simply aren’t enough pins available on a microcontroller’s package to support parallel interfaces. Serial Data Transmission In serial communications, the bits that comprise a binary data word are sent sequentially along a single data line. Serial data transfer is typically slower than parallel data transfer but a minimal serial link requires only two data lines, transmit (TX) and receive (RX), regardless of word size. Also, there are serial devices available to communicate over a great range of distances. Serial data links can be made to communicate among integrated circuits on the same printed circuit board or made to communicate over many kilometers through an RF modem or satellite phone.
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Chapter 15
Section 15.2: Memory
Section 15.2 Memory 15.2.1 Organization of Memory Memory is a group of sequential locations where binary data is stored. Typically, in a microcontroller, each memory location holds 8 bits or 1 byte of data. Each location in memory has a unique address which the CPU uses to read to and write from that location see Fig. 15.5 as an example. You may think of each memory location as mailbox. The address number on the outside of the mailbox has nothing to do with what is actually in the mailbox; it just identiﬁes where the mail is to be placed. Memory addresses serve the same purpose. The address identiﬁes the location at which some data is stored but does not convey any information about what that data may be. Frequently, a single word of data is longer than 1 byte. For example, a 16bit binary number would require 2 bytes of memory to store. Similarly, the output of a 12bit ADC would require 2 bytes of storage even though the most signiﬁcant 4 bits of the second byte are not used. Multibyte data is stored in successive memory locations and the address associated with the whole word is the address where the ﬁrst byte from the word is stored.
xxxAh
... 15
14
..Bits.. .
9
8
xxx9h
7
6
..Bits..
1
0
xxx8h
Byte
xxx7h
Byte
xxx6h
Word (High Byte)
xxx5h
Word (High Byte)
xxx4h
...
xxx3h
Fig. 15.5. Byte storage in memory.
The question is which byte is stored ﬁrst in memory, the byte containing the most signiﬁcant 8 bits of the word or the byte containing the least signiﬁcant bits? For that matter, within a byte which bit is the most signiﬁcant and which is the least? By convention, the bits within a byte are labeled left to right as bits 7 through 0 with bit 7 representing the most signiﬁcant bit and bit 0 being the least signiﬁcant. XV753
Chapter 15
Section 15.3: Arduino Uno: An Embedded Microcontroller
Section 15.3 Arduino Uno: An Embedded Microcontroller 15.3.1 What Is Arduino? The Arduino is an opensource platform designed to be a lowcost, ﬂexible, and easytoprogram embedded microprocessor. Technically, the word Arduino refers to the name of the hardware, while the actual embedded microprocessors are named differently. The Arduino Uno, for example, is one of the lowestcost and easily accessible variations of the Arduino boards. For the purpose of this textbook chapter, the examples and programming will be done on the Arduino Uno. The Arduino Uno’s brain (the R3 Uno board) is derived from an ATmega328 microcontroller. While the board is powered by a supply voltage of 7 12 V, the actual board itself uses 5V logic levels and features a 10bit analogtodigital convertor (ADC). The ATmega328 chip embedded into the board gives the Uno 14 programmable digital IO pins and 6 analog pins. The board also features 32 KB of ﬂash storage (with a dedicated 0.5 KB for the bootloader to load the Arduino code), 1 KB of EEPROM, and 2 KB of SRAM. The Arduino Uno board also runs at a clock speed of 16 MHz. The Uno is surprisingly low cost for the actual board and the onboard microcontroller. Besides the Uno, there are many other variations of Arduino boards which are tailored to certain applications, such as the Arduino LilyPad for being sewn on clothing, the Arduino Robot board for robotics, and the Arduino Esplora for facilitating development of videogame controllers. One feature about all of the Arduino boards is that the Arduino software to program and compile code for the boards is completely free online. This allows for easy program creation. Also available online is a reference for all the syntax for programs and countless examples provided by the makers of the Arduino and the community as a whole; fostering a sense of creativity and creating a knowledge base for projects, tutorials, and code examples. 15.3.2 Arduino IDE Now with a better understanding of what the Arduino is, the most logical question is how does one interface with the Arduino? This is accomplished through the use of the USB connection on the board. When the Uno is connected to the computer, the computer attempts to install the driver but will most likely fail in the process. This is normal as the driver required to run the Arduino must be installed with the integrated development environment (IDE) supplied by the makers of the Arduino board. An IDE is a program designed to aid in the programming of a certain language by providing a visual representation of a centralized collection of coding resources and ﬁles. The Arduino IDE provides a program editor, compiler, and uploading tool for the Arduino. The Arduino IDE is essentially an IDE for the C language, with a few prewritten libraries.
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Chapter 15
Section 15.4: Basic Arduino Syntax
15.4.2 Assignment Statements and Their Features The code in Fig. 15.12 sets the integer constant MOTOR PIN to have a value of 9 using the assignment statement. This value can be referenced later in the code by simply typing MOTOR PIN instead of having to remember the value 9. This also aids in reusability of the code as the programmer can simply change the value of MOTOR PIN at the declaration and this change will percolate through the code where the name was used (instead of having to change all the places where 9 was used in code). The constant identiﬁer in front of the type identiﬁer tells the compiler that the value of MOTOR PIN should not change. The compiler will then generate a warning if future code segments try to alter the value of MOTOR PIN. This allows for basic error avoidance. Line 2 creates a constant ﬂoating point value of 2.2 that is called KG TO LBS. This name could be used in the code by simply writing KG TO LBS. Line 3 declares a Boolean variable called is stop button pressed that can only take on the values true or false. This type of variable is only stored in 1 byte of data (8 bits). Line 4 declares a new integer variable called button presses and sets its value to 0. This is an example of a single declaration. The declaration of an integer sets aside 2 bytes of memory on the Arduino Uno. The next line shows a double declaration of integer variables. Line 5 declares two new integer variables current mode and num switches. A note about these two variables is that while num switches contains the value 2, current mode has not been initialized. The value will most likely be 0, but this is not guaranteed. Thus, the safest course of action is to initialize the variable to a known value as otherwise the variable uses whatever was last in the memory location where the variable was stored. For example, as this code stands now, num switches will only take on the value of 2 when the code is ﬁrst uploaded to the Arduino or when power is applied to the Arduino (either externally or with the USB). Line 6 declares and initializes a ﬂoating point variable called foo to have the value of 17.77. This value could be changed by typing foo ¼ 9.83; or by a more detailed assignment statement. Floating points on the Arduino are stored in 4 bytes as are doubles. Thus on the Arduino, ﬂoats and doubles can be considered the same. In particular, line 7 is equivalent to float conversion factor ¼ 4617.88;. An interesting note about ﬂoating point values and integer values is that an integer value may be converted to a ﬂoating point automatically if for instance the following line of code is written: float needed value
¼ 9; ==stores the value 9:0 in needed value
ð15:3Þ
If the opposite is attempted: int not a float
7:77; ==try to store 7:77 in an int
ð15:4Þ
then a problem exists because a ﬂoating point value (with numbers to the right of the decimal point) is trying to be stored an integer with a lesser precision. How does the Arduino handle this conundrum? One could say rounding would solve this, but the
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Embedded Computing
A speciﬁc function that returns a ﬂoating point representing the number of pounds from an input of kilograms is shown in Fig. 15.15. return_type functionName(parameter 1, parameter 2 … parameter n) { // Body of function Return data type //if needed }
Fig. 15.14. The generic structure of a function in Arduino.
The function deﬁned in Fig. 15.15 has one ﬂoat output parameter and functions in Arduino only may have 1 output parameter. The two lines before the deﬁnition of the //float convertKgToLbs(float kgs) //takes in a float that is the mass in kg and returns the value in lbs float convertKgToLbs(float kgs) { const float KG_TO_LBS = 2.2; //conversion factor return (kgs * KG_TO_LBS); //return the value after conversion }
Fig. 15.15. A function for converting kilograms to pounds.
function are called the function header and exist to document the functions’ inputs and outputs and what the function does. Any parameters to the function must be separated by a comma. The return data type, speciﬁed as ﬂoat here, could be any data type that has been deﬁned previously. Only the data type identiﬁer is included though. The actual body of the function in Fig. 15.15 only consists of the declaration of a constant integer called KG TO LBS which is the conversion factor in this case (2.2 lbs in 1 kg). The actual work of the function is done in the line return (kgs * KG TO LBS);. This line converts the parameter kgs supplied to the function to pounds by multiplying by KG TO LBS and then returns the value. The function can be either placed at the top of the ﬁle or in a separate ﬁle that is included with the Arduino sketch that uses the function. A function cannot be declared within another function. In order to call the function convertKgToLbs (in Arduino), the following line is used: float totalPounds
¼
convertKgToLbsð4:7Þ; ==convert 4:7 kg to lbs
ð15:6Þ
The above line converts 4.7 kg to pounds and then stores the result in the ﬂoat variable called totalPounds. The value 4.7 is the argument to the function. If the function did not have a return value, the function call would simply be: convertKgToLbs(4.7);. In this case, the function deﬁnition must use the reserved word “void” instead of any data type before the function name.
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Embedded Computing
The value of the pin then becomes either high (5 V) or low (0 V). On analog pins, the writing command creates a PWM signal with a controllable duty cycle from about 5 % to around 95 %. This is achieved by the command analogWriteðvalueÞ; ==where value is from 0 to 255 ðinclusiveÞ
ð15:13Þ
A value of 0 corresponds to the lowest duty cycle and a value of 255 corresponds to the highest duty cycle. With this information, more complex Arduino programs can be written that interface external circuitry, LEDs, and other sensors. There are numerous examples within the Arduino IDE for all types of analog writing to pins and reading of sensors, which can be found under the File ! Examples menu. Reading through the code under this menu will help with understanding the Arduino syntax.
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Chapter 15
Section 15.5: More Advanced Arduino Programming
Section 15.5 More Advanced Arduino Programming 15.5.1 Conditional Statements The previous section is enough to get simple programs working on the Arduino. However, these programs are limited in functionality and in complexity as they lack the real control statements and functional blocks that afford greater detail in programs. Some of these control statements are called conditional statements. These statements create some logical expressions with the output of true or false. The basic structure of conditional statements in Arduino is shown in Fig. 15.17. Line Line Line Line Line Line Line Line Line Line Line Line Line
1. if(logical expression 1) { 2. //Some code to run if logical expression 1 is True 3. } 4. else if (logical expression 2) { 5. //Different code to run if logical expression 2 is True 6. } 7. //more else if statements 8. else if (logical expression n) { 9. //Even more code to run if logical expression n is True 10. } 11. else { 12. //Code to run if all above are false 13. }
Fig. 15.17. The basic structure of conditional statements.
In Fig. 15.17, there can be an arbitrary number of conditional statements. However, there must be at least one if statement before any else if or else statements. Also there can only be one else statement for any block of conditional statements. This makes intuitive sense after thinking that one does not say “or else” without ﬁrst stating the “if” clause of a demand. The Arduino starts at line 1 and check logical expression 1 ﬁrst. If logical expression 1 is true, then the Arduino runs the code between the curly braces (brackets) on lines 1 and 3. After executing that code, the Arduino skips execution of the code until after line 13. Using this technique, it is possible to explicitly lay out the control of the program through conditional statements. This can lead to state programming using variables as ﬂags, but state machine programming is outside the scope of this text. If logical expression 1 is false, then the Arduino jumps to line 4 and checks logical expression 2 and processes the code in curly brackets on lines 4 and 6 if logical expression 2 were true, etc. If the Arduino processes all of the conditional statements in Fig. 15.17 and none of them are true, then control would jump to the curly brackets following the else statement on line 13. The curly brackets are only needed if the conditional code is more than a single line.
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Chapter 15
Section 15.5: More Advanced Arduino Programming
called with an argument of 10 after the pin was switched HIGH and then after the pin was switched LOW again. The code to create the 50kHz square wave is given in Fig. 15.24. Line Line Line Line Line Line Line Line Line Line
1. const int SQUARE_WAVE_PIN = 4; 2. void setup() { 3. pinMode(SQUARE_WAVE_PIN, OUTPUT); 4. } 5. void loop() { 6. digitalWrite(SQUARE_WAVE_PIN, HIGH); 7. delayMicroseconds(10); //wait 10 microseconds 8. digitalWrite(SQUARE_WAVE_PIN, LOW); 9. delayMicroseconds(10); 10.}
Fig. 15.24. Program that generates a 50kHz square wave on pin 4.
The program in Fig. 15.24 produces a square wave with a frequency of around 33 kHz. Why is there a disparity between theory and application? The reason is in hardware switching times. The transistors that regulate on and off of digital pins require some ﬁnite time to change state and process the instructions from the code. What happens when there is not a delay? In theory, the pin should be able to change state inﬁnitely fast. This obviously cannot happen as things cannot change instantaneously due to the hardware constraints and software latency. In the Arduino, the fastest frequency achievable is around 100 kHz. This shows that the hardware switching time is about 5 μs as the 100kHz square wave has a period of around 10 μs. An astute observation would be that this switching time is on the same order of magnitude as the delay times of the 50kHz square wave. Thus, in order to compensate for hardware switching times and produce the 50kHz signal, the delays could be modiﬁed as shown in Fig. 15.25. The numbers there are the closest integers since delayMicroseconds does not accept ﬂoating point values. Line Line Line Line Line Line Line Line Line Line
1. const int SQUARE_WAVE_PIN = 4; 2. void setup() { 3. pinMode(SQUARE_WAVE_PIN, OUTPUT); 4. } 5. void loop() { 6. digitalWrite(SQUARE_WAVE_PIN, HIGH); 7. delayMicroseconds(4); //wait 4 microseconds 8. digitalWrite(SQUARE_WAVE_PIN, LOW); 9. delayMicroseconds(6); 10.}
Fig. 15.25. Program that generates ~50kHz square wave.
In order to change the duty cycle of a square signal in the servo library, the analogWrite function can be employed. The analogWrite function can be called by supplying the pin number and the value to write. This value is in the range of 0 255 which represent duty cycles from about 5 % to about 100 %. If the analogWrite function is used, the servomotor must be connected to an analog pin.
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Chapter 15
Embedded Computing
Problems 15.1 Architecture of Microcontrollers Problem 15.1. A. List some common IO devices. B. Draw the schematic of a CPU. C. Draw the schematic of a basic computer. Problem 15.2. Convert the following numbers to bits: A. 14 GB B. 27 MB C. 42 KB D. 0.97 TB Problem 15.3. A. Draw a diagram showing a CPU communicating with memory over a 16bit bus and label each of the data connections. B. Repeat the previous task but use the abbreviated bus notation.
15.2 Memory Problem 15.4. Store the value 57984 (decimal) in memory starting at address 0100h using littleendian notation. Repeat with bigendian notation. Present the corresponding tables.
Address Byte value 0202h 22h 0201h FFh 0200h A0h 01FFh ....
Problem 15.7. What is the decimal value stored in memory in the following table starting from address 0200h using littleendian notation? Repeat with bigendian notation. Address Byte value 0204h ... 0203h 77h 0202h 04h 0201h BBh 0200h 08h 01FFh ....
Problem 15.8. Describe each of the following types of memory and their application: A. RAM B. ROM C. EEPROM D. PROM E. EPROM F. Flash Problem 15.9.
Problem 15.5. Store the value 372110 in memory starting at address 0400h using littleendian notation. Repeat with bigendian notation. Present the corresponding tables.
A. Describe the difference between volatile and nonvolatile memory. B. Describe the advantages and disadvantages of ﬂash memory.
Problem 15.6. What is the decimal value stored in memory in the following table starting from address 0200h using littleendian notation? Repeat with bigendian notation.
Problem 15.10. How wide must a data bus be to access: A. 1 GB of memory? B. 1.44 MB of memory? C. 1 TB of memory? D. 22 KB of memory?
Address Byte value 0204h ... 0203h 01h (continued)
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Chapter 15
Problems
15.3 Arduino Uno: An Embedded Microcontroller Problem 15.11. A. Write the barebones minimum that is required to create an Arduino sketch. B. Write the corresponding C code. Problem 15.12. What type of memory and how much of it is available for programs on the Arduino Uno? Problem 15.13. Describe the process of writing and uploading a program for the Arduino.
15.4 Basic Arduino Syntax Problem 15.14. Describe the rationale behind a function header and write an example for a function that accepts several integers and returns the maximum. Do not worry about implementation of the function. Problem 15.15. Design a function that will return the integer value of a character that is passed to the function (Hint: use typecasting). Problem 15.16. Write a function for the Arduino that takes in the resistance and current in a circuit branch and ﬁnds and returns the voltage drop across the resistance. Problem 15.17. Write a function for the Arduino that takes in the value of two resistors and ﬁnds the equivalent parallel combination of the two (returned as a double). Problem 15.18. Write a function for the Arduino that returns the value of the transfer function for a given frequency for the following lowpass ﬁlter with R ¼ 100 kΩ and C ¼ 1.59 nF. Present the corresponding code. R + Vin(t) –
+ C
Vout(t) –
Problem 15.19. Write a function that will conﬁgure the pins in the following conﬁguration. Also if needed enable the pull up resistors on any input pins: Pin 2 ¼ INPUT Pin 3 ¼ OUTPUT Pin 4 ¼ OUTPUT Pin 5 ¼ INPUT Pin 6 ¼ INPUT Pin 7 ¼ OUTPUT Pin A0 ¼ INPUT Pin A2 ¼ OUTPUT Problem 15.20. Design and implement a code that reads in an analog voltage value across a variable resistor (potentiometer) and converts the resulting value into the duty cycle value of a square wave on pin 3. The potentiometer should be on pin A4. Discuss the corresponding normalization procedure. Problem 15.21. Write a program that reads in an analog voltage value from a light sensor (phototransistor) on analog pin A0 and converts this value to the corresponding binary number to be displayed on 8 LEDs. Discuss the corresponding normalization procedure. Problem 15.22. If the output of analogWrite is a duty cycle ranging from 5 to 95 % corresponding to 0–255, convert the following values to their complement: A. 67 % B. 23 C. 44 % D. 233 Problem 15.23. The ADC on the Arduino takes in an input signal in the range of 0–5 V and converts it to a decimal value in the range of 0–1023. Convert the following values to their complements: A. 3.7 V B. 898 C. 1.2 V D. 469 Problem 15.24. Write a program that counts in binary from 0 to 255 on eight external LEDs on
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Chapter 15
Problems
Problem 15.35. Translate the following while loops into for loops: int counter = 7; while(counter 7) { index = 9; }
a)
b)
Problem 15.36. Translate the following for loops into while loops: a) for(int row = 0; row < 6; row++) { delay(9); }
b) for(int count=278;count!=0;count=2) { delay(17); }
Problem 15.37. Find and ﬁx the errors in the following loops: a) for(int count=278;count!=0;count=3) { delay(2); }
b) int number_of_times = 11; while(number_of_times > 0) { delay(11); }
Problem 15.38. Write a function that will cycle through the string “Elizabeth” and will copy the string into a blank string that has ten entries (remember the terminating null character at the end of the string).
Problem 15.39. Write a sketch that will print “An Arduino says Hello world!” to the serial monitor the following number of times: A. 7 B. 12 C. 190 Problem 15.40. Write a sketch that can print out the binary representation of the numbers 0–64 to the serial monitor. Hint: Use a function to convert the number to binary then print it out. Problem 15.41. Write a function that will cycle through an array and print the values with the corresponding indices to the serial monitor. The array should be {7, 29, 444, 42, 69, 8, 10020}. Problem 15.42. Design a program that uses a potentiometer to cycle through letters in the following character array (string) and print the result to the serial monitor. char name[8] ¼ “Stephan”; Problem 15.43. Compare and contrast the use of interrupts versus polling of inputs. Problem 15.44. Discuss the issue of latency in code: what it is, what causes latency, and how to reduce it? Problem 15.45. Create a program that turns the Arduino into a voltmeter (0–5 V only!) using two analog pins A0 and A1. The Arduino should print out the voltage at the two pins and the difference between the two pins in both polarities. The voltage should be expressed in volts, not in the decimal numbers from the ADC. Hint: Write a function that performs the function of a DAC in software to print out values. Problem 15.46. Implement a code that uses an external 2bit ﬂash ADC to read values on four digital pins and convert the result to a 2bit binary.
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Uno
3
Problem 15.60. Create a program that will generate a square wave on pin 10 of the Arduino with the following frequencies:
A. B. C. D.
25 kHz 7 kHz 300 Hz 2 Hz
Problem 15.61. Write Arduino code that generates a 20Hz square wave on pin 7 without using delay or delayMicroseconds function.
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Part V Diode and Transistor Circuits
Chapter 16
Chapter 16: Electronic Diode and Diode Circuits Overview Prerequisites:  Knowledge of basic circuit analysis Objectives of Section 16.1:  Understand basic principles of diode operation and its mechanical analogy  Learn diode symbols  Learn three operation regions of a diode  Become familiar with common diode types and their functions Objectives of Section 16.2:  Solve diode circuits using the idealdiode model  Solve diode circuits using the constantvoltagedrop model  Solve diode circuits using the exponential model and loadline method  Learn about smallsignal diode model and incremental resistance  Establish applications of the superposition principle to diode circuits Objectives of Section 16.3:  Establish the concept of voltage reference/voltage regulator  Analyze the voltage regulator with the Zener diode  Become familiar with three major rectifier types: halfwave rectifier, fullwave rectifier, and bridge rectifier  Study selected applications of diode rectifier circuits Objectives of Section 16.4:  Become familiar with clamper and voltage multiplier diode circuits  Learn the function and topology of diode clipper circuits including hard and soft clippers Application Examples:  Automotive batterycharging system  Envelope detector circuit for AM radio
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_16
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Chapter 16
Electronic Diode and Diode Circuits
Keywords: Smallsignal diode, Forwardbias diode operation region, Reversebias diode operation region, Breakdown diode operation region, Zener breakdown voltage, Shockley’s equation, Shockley’s idealdiode equation, Thermal voltage, Saturation current, Ideality factor of the diode, Builtin voltage of pnjunction, Switching diode, Maximum operating frequency, Varactor diode, Tuning diode, Zener diode, Schottky barrier diode, PIN diode, Photodiode, LED, Ideal diode, Idealdiode model, Method of assumed states, Constantvoltagedrop model, Exponential diode model, Load line, Loadline method, DC operating point, Quiescent point, Bias point, Iterative method, Linearization procedure, Smallsignal diode resistance, Incremental resistance, Smallsignal diode model, Superposition principle for smallsignal diode model, Method of Thévenin equivalent for diode circuits, Diode voltage reference, Diode voltage regulator, Forwardbias voltage regulator, Zener voltage regulator, Piecewiselinear diode model, Dynamic Zener resistance, Incremental Zener resistance, Line regulation, Halfwave diode rectiﬁer, Conduction angle, Fullwave diode rectiﬁer, Bridge diode rectiﬁer, Envelope detector circuit, Peak detector circuit, Amplitude envelope, Modulating signal, Modulation depth, Demodulation, Linear region of operation of the envelope detector, Squarelaw region of operation of the envelope detector, Radiofrequency power meter, Diode clamper circuit, DC restorer circuit, Diode voltage multiplier, Diode voltage doubler, Diode voltage tripler, Diode voltage quadrupler, Positive clipper, Negative clipper, Double clipper, ESD protection clipper, Zener diode clipper, Hard limiter, Soft limiter
XVI796
Chapter 16
Section 16.1: Diode Operation and Classiﬁcation
Section 16.1 Diode Operation and Classiﬁcation 16.1.1 Circuit Symbol and Terminals An electronic diode is the most basic twoterminal semiconductor device. The common diode is simply a sealed semiconductor silicon pnjunction shown in Fig. 16.1a. The diode symbol shown in Fig. 16.1b has a prominent arrow that indicates the proper direction of the current. The positive side (where the current enters and the voltage is more positive) is called anode and the negative side (where the current leaves) is called cathode. This terminology has been adopted from vacuum tubes, which were used as diodes in the past (circa 1910 1960). Thus, the diode, in contrast to the resistor, capacitor, and inductor, is a polarized device. The current iD and the voltage across the diode υD follow the passive reference conﬁguration, seen in Fig. 16.1b, since the diode is a passive device. The voltage υD is also called the terminal voltage. The generalpurpose silicon and germanium diodes usually have one or two prominent black rings printed on their package terminations indicating the diode’s cathode as depicted in Fig. 16.1c b)
c)
+
anode (+)
p
n
cathode()
vD
=

a)
cathode
= anode
iD
Fig. 16.1. (a) Internal composition of a Si diode, (b) circuit symbol, and (c) physical counterpart.
16.1.2 Three Regions of Operation Figure 16.2 demonstrates an experimentally measured volt ampere (or v i) characteristic of a common smallsignal (i.e., lowpower) 1N4148 silicon switching diode. Such diodes are manufactured by many semiconductor companies. In Fig. 16.2, we plot the diode current iD versus the voltage across the diode, υD. A closer look at Fig. 16.2 indicates that the diode has three distinct operating regions: 1. The forwardbias region characterized by the inequality υD > 0 2. The reversebias region characterized by the inequality V Z0 < υD < 0 3. The breakdown region characterized by the inequality υ < V Z0. The two vertical asymptotes shown in Fig. 16.2 correspond to Zener breakdown voltage VZ0 and to a certain threshold voltage VD0, which is close to the builtin voltage of the pnjunction, VBI. The three regions of operation and the associated constants will be described in detail in the following sections.
XVI797
Chapter 16
Section 16.1: Diode Operation and Classiﬁcation
negative (and even quite small positive) terminal voltages. The gap between the charged “capacitor plates” is determined by the terminal voltage, which controls the resulting capacitance. As a result, we obtain a voltagecontrolled capacitor that has numerous applications in electronic communication circuits. The varactor diode is optimized to increase capacitance variations in response to the applied voltage. The circuit symbol for the varactor diode is seen in Fig. 16.5, where the schematic shows the builtin capacitor. anode (+)
cathode( )
Fig. 16.5. Circuit symbol for a varactor (variable capacitor) diode.
+
iD

vD
b)

a)
vD
+
16.1.6 Breakdown Region: Zener Diode The steepness of the v i diode curve in Fig. 16.2 in the breakdown region is utilized in the design of voltage regulators. The corresponding Zener breakdown voltage VZ0 is on the order of 75 100 V for switching diodes, which makes their use in the reversebias region impractical. A diode with a much smaller VZ0, on the order of 5 20 V and specially designed to operate in the reversebias region, is called a Zener diode. The siliconmade Zener diode features a heavily doped pnjunction. The breakdown voltage is adjusted by a proper doping composition. Figure 16.6 compares circuit symbols for the switching diode and Zener diode, respectively. Under normal operating conditions in the reversebias region, the cathode of the Zener diode is more positive and the diode current ﬂows from cathode to anode. Therefore, both the diode voltage and the diode current in Fig. 16.6b have positive values. The Shockley’s equation is not used in the breakdown region. Instead, the behavior of the Zener diode is described by a piecewiselinear diode model. The diode breakdown is not destructive; the diode successfully functions in the breakdown region.
iD
Fig. 16.6. Switching (a) versus Zener (b) diode. Note reversed voltage polarity/current direction.
16.1.7 Other Common Diode Types Schottky Diode The Schottky (barrier) diode does not use a pnjunction. Instead, a junction of a metal (anode) and an ntype semiconductor (cathode) are formed. Schottky diodes exhibit lower forwardbias voltages (0.15 to 0.5 V) and ultrafast switching speeds, since they are majoritycarrier (conduction) devices, in contrast to the “slow” diffusioncurrentbased pnjunctions. Schottky diodes may employ different semiconductors including Si,
XVI801
Chapter 16
Electronic Diode and Diode Circuits
GaAs, etc. in contact with metal (molybdenum, platinum, chromium). The corresponding circuit symbol is shown in Fig. 16.7. anode (+)
cathode( )
Fig. 16.7. Circuit symbol for a Schottky diode.
PIN Diode A Si PIN diode is a pin semiconductor junction. In contrast to the standard junction, there exists a region of intrinsic (or lightly doped) silicon (ilayer) of low conductivity between doped p and n layers. When the diode voltage is high and positive, this region becomes ﬁlled with charge carriers. Consequently, the PIN diode becomes a variable, voltagecontrolled resistor. This makes it useful as a switch or attenuator for radiofrequency signals. Another application is related to photodetection. In this case, the diode voltage is typically negative. When there is no ambient light, the region of intrinsic silicon has no charge carriers; hence, the diode is not conducting. When the ambient light is present, a photon collides with a single electron in the lattice. When the photon energy is sufﬁciently high, the electron leaves the crystal lattice and becomes a free carrier. Hence, the diode becomes conducting and operates as a photodetector. Photodiode The idea of the photodiode can be explained on the basis of the PIN diode. The lightinduced carriers support a reverse current through the diode, the socalled photocurrent. Its intensity can be measured; it is proportional to the intensity of the incident light. LightEmitting Diode A LightEmitting Diode (LED) performs the opposite function of the photodiode. When a free conduction electron ﬁnds its place in the crystal lattice (becomes a valence electron), it loses energy, which is irradiated as a quantum of visible light. The LED junction is not a silicon junction, but is made of gallium arsenide (GaAs), another semiconductor material. Several different compounds may be involved and the junction becomes the heterojunction. The corresponding circuit symbol is shown in Fig. 16.8 (the symbol for the photodiode has the oppositely directed small arrows). Any generalpurpose LED may operate as a solar cell generate a nonzero voltage across its terminals when illuminated by light. A simple experiment is to connect an LED to the DMM and measure the voltage across it with and without the ambient light (cover it with your hand). anode (+)
cathode( )
Fig. 16.8. Circuit symbol for an LED.
XVI802
Chapter 16
Electronic Diode and Diode Circuits iD,mA 5 Q  DC operating point (bias point)
tangent at Q
id,mA
2.5
1/rd 0 0.55
0.75
vD,V vd,mV
Fig. 16.16. Linearization procedure for a switching diode with n ¼ 1:7, I S ¼ 1:1 nA at 25 C.
Consequently, the DC operating point of the diode is called the bias point (quiescent point). Further, a very weak AC signal υd(t), id(t) is added. Though weak, this signal contains information to be processed. The diode voltage and diode current become υD ðt Þ i D ðt Þ
V D þ υ d ðt Þ I D þ i d ðt Þ
j υ d ðt Þj < V T
ð16:7aÞ
The inequality in Eq. (16.7a) means that the AC signal amplitude should be much less than 25 mV. The linearization concept states that this weak AC signal will satisfy not the complicated nonlinear diode expression, but the familiar linear Ohm’s law in the form: υ d ðt Þ
r d i d ðt Þ
ð16:7bÞ
where rd is the smallsignal diode resistance or incremental resistance determined by the slope of the v i dependence in Fig. 16.16. Thus, the diode becomes a resistor for small signals, which greatly simpliﬁes the AC analysis. Our goal is to ﬁnd this smallsignal resistance as a function of VD and ID. To do so, we use the asymptotic expansion (Taylor series) with regard to the small parameter υd/VT: !! υD VD υd VD υd υd 2 exp exp exp 1þ þO exp nV T nV T nV T nV T nV T nV T ð16:7cÞ and neglect all terms on the order of (υd/nVT)2 or less denoted by the symbol O. Substitution into Shockley’s equation yields
XVI812
Chapter 16 a
Section 16.3: Diode Voltage Regulators and Rectiﬁers
a
R=1 k
ID,mA 10
VD
RLVL

rZ
=
VZT VZ0

4
2
VD,V
+
ID
+
+ 
+
D1
VZ0

I VS
b
IL
1/rZ 0V
IZT
c
a
R=1 k
+
ID VS
+ 
17 VL
+ 
4.76 V

0V
Fig. 16.19. (a) Shunt voltage regulator on the base of a Zener diode. (b). Piecewiselinear diode model for the Zener diode in the breakdown region. (c) The circuit from (a) with the Zener diode replaced by its equivalent circuit and with the opencircuited load.
16.3.3 HalfWave Rectiﬁer A diode rectiﬁer is perhaps the most important application of the diode. Mostly large AC currents are rectiﬁed, i.e., converted to DC currents. The diode rectiﬁer forms an input stage for electronic DC power supplies including switchingmode power supplies. Furthermore, diode rectiﬁers form a basis for batterycharging circuits including automotive applications. The diode rectiﬁer is also an important part of wireless energyharvesting and communication devices. In particular, any passive RFID tag uses a diode rectiﬁer to convert the (very weak) AC power of a received electromagnetic signal to DC electric power. The concept of a simple (halfwave) diode rectiﬁer is shown in Fig. 16.20. An AC power supply with sinusoidal voltage (either positive or negative with respect to ground) is connected to a load via a diode in series. We assume that υS ðt Þ V m sin ωt. Let us use the idealdiode model ﬁrst. When the voltage versus ground is positive, the diode is in the ON state and can be replaced by a wire. All electric power is transferred to the load and the current through the load ﬂows from top to bottom. When the voltage is negative, the diode is OFF, and the load acquires no current. Hence, the current through the load always ﬂows in one direction (or does not ﬂow at all).
XVI817
Chapter 16
Electronic Diode and Diode Circuits
Diode Circuit Transformation Figure 16.26a shows part of a threephase diode rectifying circuit that rectiﬁes a voltage from one of the three stator coils of an alternator (threephase voltage generator studied in Chapter 11) and charges the battery. Every stator coil is an independent voltage source that is offset 120 with regard to the others. The complete rectifying circuit includes six diodes; every voltage power supply uses four of them. Figure 16.26a shows an equivalent circuit for one such power supply (one phase); the charging battery is replaced by a resistor load. Two other circuits are identical. The circuit uses chassis ground; this is typical in automotive applications. A nodeby node analysis (the nodes are marked as * and ** in Fig. 16.26a) shows that the circuit in Fig. 16.26a may be converted to the circuit shown in Fig. 16.26b. This circuit is “almost” the familiar fullwave rectiﬁer from Fig. 16.24, except that the load is now connected to the chassis ground instead of the opposite end of the bridge. The opposite bridge end is also grounded. We remember, however, that the chassis ground is just a metal case, and it conducts the electric current exactly as an ordinary metal wire does. Thus, we can restore the missing connection and put the load in the middle of the bridge to obtain exactly the rectiﬁer circuit of Fig. 16.24. ThreePhase Diode Rectiﬁer with DeltaConnected Alternator Figure 16.26c shows the complete rectifying system for a deltaconnected automotive alternator. Every independent power supply υab(t), υbc(t), and υca(t) is connected to its own bridge rectiﬁer circuit shown in Fig. 16.26a, b. Those circuits share common diodes so that the total number of diodes is six. We reduce each circuit to the standard bridge rectiﬁer model in Fig. 16.25. We can apply the method of assumed states to each individual circuit based on the idealdiode model. The individual power supply voltages are given by υab ðt Þ V m cos ðωt Þ, υbc ðt Þ V m cos ðωt 120 Þ, and υca ðt Þ V m cos ðωt þ 120 Þ. The rectiﬁed load voltages have the form shown in Fig. 16.23b to within a phase shift. Although every individual diode circuit is nonlinear, the currents add up at the load so that their linear superposition applies for the net load voltage and current. Figure 16.26d shows the rectiﬁed load voltage found as the sum of the three voltage contributions; the hardware prototype of the rectiﬁer is shown below. We emphasize that modeling and simulation of automotive electrical power systems facilitate efﬁcient design of the next generation of vehicles.
XVI822
Chapter 16
Problems 16.1 Diode Operation and Classification 16.1.1 Circuit Symbol and Terminals 16.1.2 Three Regions of Operation 16.1.3 Mechanical Analogy of Diode Operation 16.1.4 ForwardBias Region: Switching Diode 16.1.5 ReverseBias Region: Varactor Diode 16.1.6 Breakdown Region: Zener Diode 16.1.7 Other Diode Types Problem 16.1. Draw the circuit symbol for the diode, labeling the anode and the cathode. In what direction does the electric current ﬂow? Problem 16.2. A package for a smallsignal 1N4148 Si switching diode (yellow package) is shown in the ﬁgure. Where is its anode, on the left or on the right?
Problem 16.3. A. Sketch the typical v–i diode curve B. Indicate three regions of diode operation and write the name of each region on the ﬁgure. Problem 16.4. Determine thermal voltage, which is present in Shockley equation at: A. 0 C B. 10 C C. 20 C D. 40 C Problem 16.5. Plot the v–i characteristic of a diode with: A. n ¼ 1:0, I S ¼ 1:1 nA B. n ¼ 2:0, I S ¼ 1:1 nA at room temperature of 25 C on the same ﬁgure. Use the ﬁgure that follows as a template. Label
Problems each curve. Use the value of 1.6021810 19 C for the electron charge and the value of 1.3806610 23 J/K for the Boltzmann constant. iD, mA 10
5
0 0.5
0
0.5
vD, V
1
Problem 16.6. Plot the v–i characteristic of a diode with: A. n ¼ 1:0, I S ¼ 1 nA B. n ¼ 1:0, I S ¼ 0:01 nA at room temperature of 25 C on the same ﬁgure. Use the ﬁgure to the previous problem as a template. Label each curve. Use the value of 1.6021810 19 C for the electron charge and the value of 1.3806610 23 J/K for the Boltzmann constant. Problem 16.7. For a diode with n ¼ 2:0, the following measurement is taken: υD ¼ 0:65 V and iD ¼ 1 mA. Given thermal voltage of 26 mV, determine diode’s saturation current IS. Problem 16.8. For a diode with I S ¼ 1 pA, the following measurement is taken: υD ¼ 0:62 V and iD ¼ 1 mA. Given thermal voltage of 26 mV, determine diode’s ideality constant, n. Problem 16.9. At which forwardbias voltage in terms of VT does the diode conduct a current of 104IS given the ideality factor of two? Problem 16.10. A diode with n ¼ 2:0 is to be used as a temperature sensor in the forwardbias region (it is a common diode application). Determine: A. The corresponding change in the diode voltage (initial value, ﬁnal value, and the difference) when the temperature rises from 20 to 60 C
XVI837
Chapter 16
Problems
The diode current must be ﬁxed at 1 mA. The power supply voltage is ﬁxed at 9 V. A. Present the corresponding circuit diagram and specify the component (s) values. B. Label the sensor output voltage.
3 kΩ
1 kΩ
5V
+ 2 kΩ
Problem 16.19. A diode circuit is shown in the ﬁgure that follows. Find the values of the diode current and the voltage across the diode, assuming that the diode is ideal.
I
1 kΩ
0V
Problem 16.22. Assuming the idealdiode model, ﬁnd current I for the circuit shown in the ﬁgure below.
10 V
R=1 kΩ
+
ID
VD
R=1 kΩ

0V
0V
+ 3 kΩ
Problem 16.20. For the diode circuit shown in the following ﬁgure, determine the values of the diode current and the voltage across the diode, assuming that the diode is ideal.
I
5 kΩ
0V
Problem 16.23. For the circuit shown in the ﬁgure below, determine circuit current I, assuming that the diode is ideal. The ground path is simultaneously the current return path.
10V R=1 kΩ
+
10 V I
VD
R=2 kΩ
5 kΩ
2 kΩ 5V

ID 0V
3 kΩ
1 kΩ
2 kΩ
1 kΩ
0V
Problem 16.21. Assuming the idealdiode model, ﬁnd current I for the circuit shown in the ﬁgure below.
0V
Problem 16.24. For the circuit below ﬁnd circuit current I, assuming that both diodes are ideal. The ground path is simultaneously the current return path.
XVI839
Chapter 16
Electronic Diode and Diode Circuits +15 V
I
+
I
10 kΩ
1 kΩ
V

5 kΩ
I +3 mA
2 kΩ
10 kΩ
V 3V
0V
3 mA
Problem 16.25. For the circuit shown in the ﬁgure below, ﬁnd circuit current I, assuming that the diodes are ideal. The ground path is simultaneously the current return path. I
Problem 16.28. Sketch I versus V to scale for the circuit shown in the ﬁgure. Assume the idealdiode model and allow V to range from 5 V to 5 V.
+15 V
+
10 kΩ
+3 V
I
1kΩ
V
5 kΩ
3V
+ 
I 2 kΩ
10 kΩ +3 mA
V
0V
Problem 16.26. The circuit shown in the ﬁgure below can be used as a signaling system using one wire plus a common ground return. At any moment, the input has one of three voltage values shown in the ﬁgure. What is the status of the lamps for each input value?
5V
+5 V 3 mA
Problem 16.29. For the circuit shown in the ﬁgure below, ﬁll out Table 16.5. 5V
V
5V 0V 5V
D1
D2
1 kΩ
red 0V
green 0V
D1
+
V1 V2
D2
VOUT

0V
Problem 16.27. Sketch I versus V to scale for the circuit shown in the following ﬁgure. Assume the idealdiode model and allow V to range from 3 V to 3 V.
XVI840
Chapter 16
Problems
VOUT
Problem 16.30. A freshman ECE student attends class if all of the following conditions are satisﬁed: A. He/she feels that this lecture might be useful. B. There are no other more important things to do. C. The way to the Department is cleaned up from snow. Every morning he/she “votes” by simultaneously pushing any appropriate combination of three 5V buttons (A, B, C) placed in parallel. A simple diode circuit is needed that lights a green LED when there is time to go to the lecture. Problem 16.31. A small county board is composed of three commissioners. Each commissioner votes on measures presented to the board by pressing a 5V button indicating whether the commissioner votes for or against a measure. If two or more commissioners vote for a measure, it passes. You are asked to help with a an idealdiode circuit that takes the three votes as inputs and lights a green LED to indicate that a measure passed. You can use as many diodes/resistors as you need. 1. Explain your reasoning for building the diode circuit. 2. Present the appropriate circuit diagram.
16.2.2. ConstantVoltageDrop Model
+
V2 (V) 0 5 0 5
I 1kΩ
V
I +3 mA
3V
+3 V
V
3 mA
Problem 16.34. Sketch I versus V to scale for the circuit shown in the ﬁgure using: A. Idealdiode model B. Constantvoltagedropdiode model with the turnon voltage of 1 V Allow V to range from 5 V to 5 V. +
V1 (V) 0 0 5 5
I 1kΩ
V

Table 16.5. Output voltage of the diode circuit as a function of two input voltages.
Problem 16.33. Sketch I versus V to scale for the circuit shown in the following ﬁgure using: A. Idealdiode model B. Constantvoltagedropdiode model with the turnon voltage of 1 V Allow V to range from 3 V to 3 V.

What type of logic gate is it?
I +3 mA 5V
+5 V
V
Problem 16.35. Sketch I versus V to scale for the circuit shown in the ﬁgure using: A. Idealdiode model B. Constantvoltagedropdiode model with the turnon voltage of 1 V Allow V to range from 5 V to 5 V.
Problem 16.32. What is the constantvoltagedropdiode model? Draw the corresponding v–i diagram.
XVI841
Chapter 16
Electronic Diode and Diode Circuits Determine the ideality factor and the saturation current of Shockley equation at this temperature.
+
1 kΩ
V 3V
+ 
I +3 mA
V 5V
+5 V 3 mA
Problem 16.40. In the circuit shown in the ﬁgure below, the diode is described in terms of the exponential forwardbias model with the Shockley equation plotted in the ﬁgure. A. Graphically determine the solution—the DC operating point VD, ID using the loadline method. B. Compare the obtained diode current with that found in the constantvoltagedrop model.
Problem 16.36. Present equivalent circuit for the twodiode conﬁguration shown in the ﬁgure, assuming the constantvoltagedropdiode model with the turnon voltage of 1 V.
Problem 16.37. Using the constantvoltagedropdiode model, you need to design a circuit for the diode temperature sensor described in Problem 16.10. The diode current must be ﬁxed at 1 mA. The power supply voltage is ﬁxed at 9 V. A. Present the corresponding circuit diagram and specify the component (s) values. B. Label the sensor output voltage.
500 Ω
ID
+
a)
2V
+ 
VD

I
0V iD, mA
b) 5.0
Shockley equation
2.5
0 0
0.5
1
1.5
2
2.5
3
vD, V
16.2.3 Exponential Model in the Forward Problem 16.41. Repeat the previous problem for the circuit shown in the ﬁgure that follows. Bias Region and Its Use 16.2.4 LoadLine Analysis 666 Ω I 16.2.5 Iterative Solution 3V
+ 
VD

Problem 16.39. A 1N4148 diode manufactured by Hitachi has a current of 0.15 mA at 0.6 V and a current of 1.5 mA at 0.7 V, all at minus 25 .
+
D
Problem 16.38. A 1N4148 diode manufactured by Fairchild has a current of 0.7 mA at 0.6 V and a current of 8 mA at 0.725 V, all at 25 . Determine the ideality factor and the saturation current of Shockley equation at this temperature.
0V
Problem 16.42. In the circuit shown in the ﬁgure that follows, the diode is described in terms of the exponential forwardbias model where the ideality factor and the saturation
XVI842
Chapter 16
Electronic Diode and Diode Circuits
Problem 16.49. Repeat the previous problem when Thévenin resistance of the source changes to 1 kΩ.
the corresponding circuit using a 1N4733A Zener diode to provide a stable 5V reference. R1
R=500 Ω
+
VS
+ 
ID
VL

0V
Problem 16.52. In a typical 12V automotive application, battery voltage may vary between 10.5 and 14.1 V. The ECM (engine control module) determines fuel delivery and spark advance to control emissions based on several sensors connected to the engine. Many of these sensors require a stable 5V reference that can be achieved through the use of a Zener diode. The ﬁgure that follows shows
ID
VB
+
Problem 16.51. A 1N5231B Zener diode with the test point V ZT ¼ 5:1 V, I ZT ¼ 20 mA and with the dynamic resistance rZ ¼ 17 Ω is used in the voltage regulator circuit shown in the ﬁgure below. A. Determine load voltage for the regulator circuit given that V S ¼ 9 V 1 V and that the load has a very high (inﬁnite) resistance. B. Determine line regulation
D1
+
16.3.2 Voltage regulator with Zener diode
IL
I
+
Problem 16.50. You are given a (variable) voltage source VS represented by its Thévenin equivalent with resistance RT and a load represented by its equivalent resistance of RL. A forwardbias diode voltage regulator is used to keep the load voltage constant. Using the constantvoltagedropdiode model, answer two questions: A. What is the maximum possible regulated load voltage if RL ¼ RT ? B. What is the maximum possible regulated load voltage if RL ¼ 10RT ?
VD
RL VL



0V
The Zener diode has a reference (test) voltage of 5.1 V at a reference (test) current of 49 mA. A. Choose a value for resistor R1 to limit the current through the Zener diode to approximately 50 mA with the sensor disconnected (switch OPEN). B. Given that the battery voltage may vary between 10.5 and 14.1 V, determine how much the Zener voltage (voltage across the Zener diode) ﬂuctuates with the sensor disconnected (switch OPEN). Note: This Zener diode has a dynamic resistance of 7 Ω at the test current of 49 mA. C. What minimum load resistance can be connected to the circuit without the voltage drop more than 0.5 V from 5 V? D. Plot load voltage as function of the load resistance in the range 0–1000 Ω for two extreme battery voltages. E. If the switch is closed and the load resistance is 100 Ω, what is the power efﬁciency of this circuit for two extreme values of the battery voltage? Note: Efﬁciency percentage ¼ PLOAD/ PBAT 100 %. Problem 16.55. Using software of your choice (MATLAB is recommended), plot the output of a halfwave diode rectiﬁer to scale over a time period from 0 to 8 s when the input voltage is given by
υS ðt Þ
V m sin ωt þ 0:5V m sin 2ωt
with Signal frequency—0.5 Hz Signal amplitude—V m ¼ 9 V. Assume the ideal diode.
XVI844
Chapter 16
Electronic Diode and Diode Circuits
16.3.7 Application example: Envelope (or peak) detector circuit Problem 16.61. A. Explain the function of the envelope detector in your own words. B. What is the difference between linear and squarelaw regions of operation for the envelope detector? C. When does the envelope detector operate in the linear region? In the squarelaw region? Problem 16.62. Design an envelope detector given the carrier frequency of f ¼ 1:7 MHz. The modulation is a human voice, with the maximum passing modulation frequency of 20 kHz. A. Draw the circuit diagram of the envelope detector. B. Specify one possible set of values for RL, C. Problem 16.63. Design an envelope detector (specify one possible set of values for RL, C) given the carrier frequency of f ¼ 10 MHz. The modulation is a digital signal, with the bit rate of 100 kbps. Hint: The equivalent frequency of the digital signal is the bit rate in Hz. Problem 16.64. A MATLAB script that follows models an envelope detector circuit in Fig. 16.27 by solving the exact circuit ODE given by Eq. (16.19b)
dυout dt 1 ¼ ðRL I S exp½ðυin þ V bias τ υout Þ
υout Þ=V T
% Input signal % carrier freq., Hz f = 1e6; % modulation freq., Hz fm = 2e4; Am = 0.5; % time array t = linspace(0, 4/fm, 1e6); % envelope E = 1 + Am*cos(2*pi*fm*t); % input signal ampl., V Vm = 0.10; % input signal, V vin = Vm*E.*cos(2*pi*f*t); % Envelope detector % capacitance, F C = 10e9; % resistance, Ohm R = 5e4; % time constant, sec tau = R*C; % Boltzmann constant [J/K] k = 1.38066e23; % electron charge [C] q = 1.60218e19; % temperature [K] T = 298; % thermal voltage [V] VT = k*T/q; % saturation current, A Is = 1e9; % bias voltage, V Vbias = 0.0; % Numerical solution % (firstorder Euler) vout = zeros(size(t)); dt = t(2) t(1); iD = zeros(size(t)); % starting voltage vout(1) = Vbias + vin(1) 0.60; for n = 1:length(t)1 iD(n) = Is*(exp((vin(n)... +Vbiasvout(n))/VT )1); vout(n+1) = vout(n) +... dt/tau*(R*iD(n)  vout(n)); end % Graphics t = t(end/2:end); vout = vout(end/2:end); vin = vin(end/2:end); subplot(1,2,1); plot(t, vin+Vbias); grid on; axis square subplot(1,2,2); plot(t, vout); grid on; axis square
with a particular set of design parameters of your choice. Which bias voltage (0, 1, 4, or 8 V) is most beneﬁcial for the performance of your circuit? Justify your answer.
XVI846
Chapter 16
Problems
A. Idealdiode model B. Constantvoltage drop model Label the endpoint voltages.
Problem 16.77. Repeat Problem 16.76 for the diode circuit shown in the ﬁgure below. Assume R1 ¼ 1 kΩ, R2 ¼ 1 kΩ, and R3 ¼ 1 kΩ. Label the endpoint voltages.
a)
+
R1
+
D1
vin(t)
+
vout(t) R2

R1
D1
D2
vin(t)


+ vout(t)
R2
R3

b) vout,V 3 2 1 0 1 2 3 3
2
1
0
1
2
vin,V
3
XVI849
Chapter 17

Bipolar Junction Transistor and BJT Circuits
Analyze most typical commonemitter smallsignal BJT amplifier circuits Obtain the initial exposure to smallsignal transistor amplifier bandwidth
Application Examples:  Automotive BJT dome light switch  Door lock BJT switch and Darlington pair  Transistor amplifier bandwidth
Keywords: Bipolar junction transistor (BJT), Transfer resistor, npnjunction transistor, pnp junction transistor, Emitter of a BJT, Base of a BJT, Collector of a BJT, Emitterbase junction (EBJ), Collectorbase junction (CBJ), Baseemitter voltage of a BJT, Basecollector voltage of a BJT, Collectoremitter voltage of a BJT, Commonemitter conﬁguration, Commonbase conﬁguration, Commoncollector conﬁguration, Active operating region of a BJT, Saturation operating region of a BJT, Cutoff operation region of a BJT, Saturation (scale) current of a transistor, Transistor test circuits, (forward) Commonemitter current gain, (forward) Commonbase current gain, Reverse commonemitter current gain, EbersMoll model, Forced beta, Forced current gain, BJT v–i dependencies, Early effect, Early voltage, (ﬁrstorder) Largesignal BJT circuit model, (ﬁrstorder) πtype largesignal BJT circuit model, Largesignal DC circuit model of a BJT, Method of assumed states, DC transistor bias circuits, Basebias BJT bias circuit, Fixedbase BJT bias circuit, BJT bias circuit with emitter resistance, BJT fourresistor bias circuit, Discretecircuit transistor ampliﬁers, Integratedcircuit transistor ampliﬁers, BJT bias circuit with dualpolarity power supply, Constantcurrent BJT source, Constantcurrent LED driver, Constantvoltage BJT source, Emitter follower BJT voltage source conﬁguration, BJT DC voltage buffer (voltage follower), Voltagecontrolled BJT switch, Currentcontrolled BJT switch, Groundside switch, Powerside switch, Control side of the BJT switch, Load (power) side of the BJT switch, Body control module, Driver BJT module, Darlington BJT pair, Superbeta transistor, Sziklai BJT pair, Opencircuit voltage gain of the generic voltage ampliﬁer, Input resistance of the generic voltage ampliﬁer, Output resistance of the generic voltage ampliﬁer, Voltage transfer characteristic of commonemitter ampliﬁer, DC operating point of the transistor ampliﬁer, Quiescent (Q) point of the transistor ampliﬁer, Quiescentpoint parameters, Separation of DC and AC quantities in transistor ampliﬁer, Smallsignal voltage gain of BJT voltage ampliﬁer, Smallsignal input/ output resistance of BJT voltage ampliﬁer, Smallsignal ground of the BJT ampliﬁer, Smallsignal baseemitter resistance of the BJT, Smallsignal transconductance of the BJT, Smallsignal approximation, Smallsignal transistor circuit model, Hybridπ BJT model, Basebias conﬁguration of BJT commonemitter ampliﬁer, Emitter resistance conﬁguration of BJT commonemitter ampliﬁer, Fourresistor bias conﬁguration of BJT commonemitter ampliﬁer, Capacitively coupled load, Capacitively coupled input signal, Transistor ampliﬁer bandwidth, Miller effect, Amplitude frequency response of transistor ampliﬁer, Midband of BJT ampliﬁer frequency response, Low end of BJT ampliﬁer frequency response, High end of BJT ampliﬁer frequency response
XVII852
Chapter 17
Bipolar Junction Transistor and BJT Circuits
Although the concentration decreases when the distance along the base increases, it does not imply that the minority carriers disappear. They simply move faster when approaching the collector. The transistor collector current is the diffusion current of excess minority carriers in the base. The current density per unit area is found as J C qDn dΔn=dx where Dn is the base diffusion constant of the minority carriers. The total collector current is the current density times junction area A. Using Eq. (17.5) yields υBE qn2 Dn 1 , IS A i iC I S exp ð17:6Þ VT N A0 W Equation (17.6) is the idealdiode Shockley equation. The saturation current IS in Eq. (17.6) has another name, the scale current, which underscores the fact that it scales linearly with the emitterbase junction area A. Typical values are in the range I S 10 12 10 15 A.
Base Current When the emitterbase junction is forward biased, some holes are injected from the ptype base into the emitter. These holes constitute the base current iB shown in Fig. 17.5. The base current is an inevitable side effect of the junction transistor. Exactly the same method of the diffusion equation applies. The ﬁnal result has the form υBE 1 ð17:7Þ iB I SB exp VT The saturation current ISB has the form of second Eq. (17.6) related to the emitter region.
Relation Between Transistor Currents: CommonEmitter Current Gain According to Eqs. (17.6) and (17.7), transistor currents are directly proportional to each iC
βiB , iE
ðβ þ 1Þ iB
ð17:8Þ
where the dimensionless constant β is the (forward) commonemitter current gain of the transistor. The current gain cannot be controlled precisely due to uncertainties of the manufacturing process. Typical values are β 20 200, but much higher values may be obtained. The current gain is the most important DC parameter of the junction transistor. Consider a generalpurpose smallsignal (which means lowpower) npn 2N3904 Si transistor. The current gain from the device’s datasheet ranges from a minimum of 30 100 (for different collector currents) to a maximum of 300 (at room temperature). Along with β, another parameter α is of interest, called the commonbase current gain: iC
αiE , α
β βþ1
ð17:9Þ XVII858
Chapter 17 a
Section 17.1: Physical Principles and Operation Laws
April 4, 1950
2,502,488
W.SHOCKLEY SEMICONDUCTOR AMPLIFIER
Filed Sept. 24, 1948
21
24
22
+ –
10
13
c
23 11
15
N
14
P
19
12
25
+
–
17
RL
18
b Sept. 25, 1951
2,569,347
W.SHOCKLEY
CIRCUIT ELEMENT UTILIZING SEMICONDUCTIVE MATERIAL 3 Sheets–Sheet 1
Filed June 26, 1948
Fig 3 54
52
51
53
55
C
56 E
N
P
57 –
+
N B
58
RL –
+
Fig. 17.10. (a) An early concept of a semiconductor ampliﬁer on the basis of a single pnjunction; (b) evolution of the pnjunction concept resulted in the invention of the bipolar junction transistor. Both ﬁgures are from the original patents. (c) William Bradford Shockley (seating), John Bardeen and Walter Brattain Bell Labs, June 1948.
XVII865
Chapter 17
Bipolar Junction Transistor and BJT Circuits
17.2.2 LargeSignal DC Circuit Model of a BJT In the DC steady state, the circuit parameters are ﬁxed. The steep exponential dependencies of the BJT transistor may be replaced by turnon voltages, similar to the constantvoltagedrop diode model studied in Chapter 16. Figure 17.13 outlines the concept. This results in a simpliﬁed largesignal DC BJT circuit model, which allows us to determine the operation region (active, saturation, or cutoff) and estimate major circuit parameters. This model works reasonably well when it is necessary to establish the DC operating point of a transistor ampliﬁer circuit and for the qualitative analysis of transistor power circuits. Short circuit (a “wire”): any current
IB
and constant voltage drop VBE
open circuit (no current)
0
VBE
vBE
Fig. 17.13. Largesignal DC circuit model of the EBJ.
Figure 17.14 shows the DC circuit model for three different transistor regions. 1. In the active region, V BE 0:7 V (turnon voltage of the EBJ), and the CBJ is reverse biased or has a small positive bias below its turnon voltage of V BC 0:5 V (turnon voltage of the CBJ is smaller due to more shallow collector doping). Therefore, V CE V BE V BC > 0:2 V. This data falls within the wider active region of Table 17.1. Indeed, I C βI B . 2. In the saturation region, V BE 0:7 V (turnon voltage of the EBJ) and V BC 0:5 V (turnon voltage of the CBJ). Therefore, V CE V BE V BC 0:2 V. This data again falls within the wider saturation region of Table 17.1. In saturation, I C < βI B . The particular value of the collector current is determined by the rest of the circuit. The transistor is most efﬁcient as a switch (has the lowest relative power loss) in the saturation region. 3. In the cutoff region, both junctions are reverse biased or have a small positive bias so that V BE 0:5 V, V BC 0:4 V according to Table 17.1. We round the last value to 0.5 V to better memorize it. All terminal currents are zeros; the transistor is an open circuit.
XVII868
Chapter 17
Bipolar Junction Transistor and BJT Circuits
transistor by a marionette shown in Fig. 17.18 and assign some positions (voltages) to its arms and legs, then the bias circuit plays the role of the “puppet master.” Our goal now is to design a simple activeregion bias circuit that is typical for transistor ampliﬁers. The following conditions should be met: 1. Provide a constant DC collector current IC that remains nearly the same at large β variations (and temperature variations). 2. Make sure that the transistor has enough output voltage swing (a sufﬁciently wide dynamic range) in the active region.
17.2.6 βIndependent Biasing and Negative Feedback The ﬁrst such circuit is the basebias or ﬁxedbase transistor circuit studied previously and shown again in Fig. 17.19a. The name comes from the fact that the base current in the active region, I B ðV BB 0:7 VÞ=RB , is ﬁxed. The ampliﬁer’s output will be the collector voltage VC versus ground, which coincides with VCE. The general solution is obtained by applying the KVL to the baseemitter loop and has the form: V BB 0:7 V RC IC β ) V C V CE V CC β ðV BB 0:7 VÞ ð17:19Þ RB RB For the particular component values shown in Fig. 17.19a and β 100, the output voltage VC has been determined in Examples 17.4 and 17.5, respectively. It approximately satisﬁes the equality V C V CC =2
ð17:20Þ
which is desired for the maximum output voltage swing. As a competitor, we suggest using an alternative bias circuit with emitter resistance shown in Fig. 17.19b, which employs the emitter resistance RE instead of the base resistance RB. In fact, the emitter resistance is one way of introducing the negative feedback in a transistor circuit. The feedback loop is as follows: a larger base current increases the voltage drop across the emitter resistance. By KVL, the higher emitter voltage drop decreases the VBE and thus attempts to decrease the base current, which is the negative feedback. The general solution for the circuit in Fig. 17.19b in the active region has the form IE IC
V BB
0:7 V
) RE β V BB 0:7 V ) VC βþ1 RE
V CC
β RC ðV BB β þ 1RE
ð17:21Þ 0:7 VÞ
In order to obtain Eq. (17.21), we have again applied KVL to the baseemitter loop in Fig. 17.19b. For the particular component values shown in Fig. 17.19b and β 100, the output voltage is V C 8:44 V. It is not far from VCC/2, which is the desired value. Thus, both circuits approximately satisfy the condition for the maximum output voltage swing given by Eq. (17.20). The next step is to check the βdependency. XVII874
Chapter 17
Section 17.2: LargeSignal Circuit Models of a BJT Table 17.3. Performance of the bias circuit in Fig. 17.22 for different β. β 50 100 250
IC 1.82 mA 1.86 mA 1.89 mA
VC 7.35 V 7.18 V 7.07 V
Other bias circuits exist, in particular those which use the current sources along with the voltage sources. The current sources are constructed using transistors as well. Thus, often one transistor circuit is used to bias another.
XVII879
Chapter 17
Bipolar Junction Transistor and BJT Circuits
Section 17.3 Practical BJT Circuits at DC The transistor circuits of this section behave very similar to the transistor bias circuits considered previously. However, they serve a completely different purpose. Namely, they are used as drivers and switches for different (power) loads such as LEDs, automotive lights, solenoids, etc.
17.3.1 ConstantCurrent Sources: Active Region of Operation General Since the BJT behaves as a dependent current source, it can be used to control the current through many other devices. For example, if a load requires a constant current, a BJT can be connected as shown in Fig. 17.23 with a constant reference voltage connected to the base of a BJT through a ﬁxed resistance. Doing so establishes a constant current in the base, which is ampliﬁed in the collector (by the current gain β) and ﬂows through the load. The transistor circuits so constructed are called constantcurrent BJT sources. b)
+
+
a)
Load
VL
IC

C
B VBE
E

IB VREF
C
B
VCC
+
+
VCC
VBE
VREF
E

RB
IC
Load
VL
RE
IE
Fig. 17.23. Simple BJT conﬁguration as constantcurrent sources.
In the current sources, the BJT is operating in the active region. The collector current is determined using the largesignal DC model. In Fig. 17.23a, b, one has, respectively IC IE
β
V REF
V REF
0:7 V RB 0:7 V
RE
) IC
ð17:25aÞ β V REF 0:7 V βþ1 RE
ð17:25bÞ
Those equations are the same as Eqs. (17.19) and (17.21) of the previous sections established for the topologically equivalent bias circuits. Although Eq. (17.25a) provides a means of setting the collector current to a particular value, β varies widely due to variations in transistor dopant concentrations, operating temperature, and current levels. Therefore, this design is not very practical since it is too βdependent. An improved
XVII880
Chapter 17
Bipolar Junction Transistor and BJT Circuits
occurs when the solenoid is turned off. The procedure to determine the value of RB is similar to the previous example; however, much higher current values are used: 1. If the transistor were a perfect switch, the maximum load current would be 12 V=3 Ω 4 A. 2. Given that β1 100 and β2 25, the overall current gain of the Darlington pair is 2625 see Eq. (17.31). This is sometimes referred to as superbeta. 3. Therefore, the minimum amount of base current required to saturate the Darlington pair is I B 4 A=2625 1:52 mA. Note that the saturation voltage for the Darlington pair is approximately 0.9 V since only one transistor (the ﬁrst one) goes into saturation and the other remains in the active mode, i.e., 0.2 V + 0.7 V 0.9 V. This is a major drawback of the Darlington pair since it substantially increases its power dissipation and heat sink requirements. 4. To overdrive the Darlington pair, double the base current, i.e., choose I B 2 1:52 mA 3:04 mA. 5. KVL equation around the baseemitter loop yields V TH I B ðRTH þ RB Þþ V BE . Note that there are two 0.7 V voltage drops from base to emitter for the Darlington pair. 6. Solving for RB ﬁnally yields RB
V TH
2 0:7 V IB
RTH
5 1:4 0:00304
100
1184 Ω
ð17:32Þ
XVII888
Chapter 17
Bipolar Junction Transistor and BJT Circuits
17.4.2 Simpliﬁed Model of the BJT CommonEmitter Ampliﬁer The simpliﬁed model of the BJT commonemitter ampliﬁer introduced below ignores a smallsignal resistance/impedance of the EBJ. However, it correctly describes the major modeling steps and it is mathematically consistent. The starting point is the ﬁxedbase transistor bias circuit from Fig. 17.19a. We replace the DC bias source VBB by a variable input voltage υIN(t). The DC collector voltage VC will now become the variable output voltage υOUT(t). The corresponding circuit is shown in Fig. 17.33a. We do not yet connect a load resistance RL. a)
b) VCC
vCC
vOUT(t), V Slope is voltage gain A V0
RC vOUT(t)
Q  DC operating point (bias point)
VOUT
B
+
vIN(t) RB
0.7 V
0.2 V sat.
0 0.5
0.7
vIN(t), V 0.9
VIN
Fig. 17.33. BJT commonemitter ampliﬁer and its simpliﬁed voltage transfer characteristic.
Voltage Transfer Characteristic To grasp the key ampliﬁer concept, we will simplify the circuit analysis. We will use the largesignal DC circuit model but assume that input/output voltages and currents are now functions of time. The corresponding solution has been given by Eq. (17.20), that is, υOUT ðt Þ
V CC
β
RC ðυIN ðt Þ RB
0:7 VÞ
ð17:35Þ
Equation (17.35) is plotted by a thick in Fig. 17.33b. This curve is known as the voltage transfer characteristic of the ampliﬁer. The input voltage in Fig. 17.33a is a combination of a certain DC voltage plus a very small input AC signal to be ampliﬁed. Then, the output voltage will be a combination of a certain DC voltage plus an ampliﬁed (but still relatively small) replica of the AC signal; see Fig. 17.33b. This is the ampliﬁer concept.
Linear Expansion of Circuit Variables and Quiescent Point In transistor ampliﬁers, the smallsignal ampliﬁcation is superimposed onto the DC solution. Mathematically, the separation of large DC and small (at least for now) AC quantities is done in the form (lowercase indexes are used for small AC signals):
XVII890
Chapter 17
Bipolar Junction Transistor and BJT Circuits
17.4.3 SmallSignal BJT Analysis and Superposition BaseEmitter SmallSignal Resistance The simpliﬁed ampliﬁer model developed previously assumes exactly zero smallsignal resistance between base and emitter. This would be true if the v i dependence for the baseemitter junction were inﬁnitely steep, as is indeed imposed by the largesignal DC model. As a matter of fact, the exponential v i Shockley dependence given by Eq. (17.7) υBE υBE iB I SB exp 1 I SB exp ð17:40Þ VT VT has a very steep, yet ﬁnite, slope in the active region. Its accurate consideration will give us a ﬁnite smallsignal baseemitter resistance rπ at quiescent point Q as illustrated in Fig. 17.34. This ﬁgure shows the “zoomed in” area around the Qpoint. The value of rπ is just the inverse slope at that point. To ﬁnd rπ, we insert the expansion (17.36) for the base current iB I B þ ib and the corresponding expansion for the baseemitter voltage υBE V BE þ υbe into the Shockley equation (17.40) and obtain using Taylor series V BE þ υbe υbe υbe υbe I B exp IB þ IB IB þ ð17:41aÞ I B þ ib I SB exp VT VT VT rπ iB Q  DC operating point (bias point)
tangent at Q
IB
1/r 0.55
VBE
vBE, V
0.75
Fig. 17.34. Finding the smallsignal baseemitter ampliﬁer resistance rπ.
given that jυbe j=V T 1
ð17:41bÞ
which is the critical smallsignal approximation. From Eq. (17.41a), one has rπ
υbe ib
VT IB
βV T IC
ð17:42Þ
XVII892
Chapter 17
Bipolar Junction Transistor and BJT Circuits V mout /V ms, dB
Bode plot
5 MHz
Avo=25.2 dB
25 20 15 10 5 0 5 0
10
1
10
10
2
3
4
5
10 10 10 frequency of input voltage, Hz
10
6
10
7
10
8
Fig. 17.40. Bode plot for the ampliﬁer gain of the commonemitter ampliﬁer.
XVII900
Chapter 17
Summary
Summary Largesignal exponential ﬁrstorder transistor model
υBE iC ¼ I S exp 1 , VT IS υBE 1 , iC ¼ β iB exp iB ¼ β VT Largesignal DC circuit model of the npn transistor
Largesignal DC circuit model of the pnp transistor
Transistor bias circuit with base resistance (basebias) in active region (βdependent, common emitter conﬁguration) V BB 0:7 V RB V B ¼ 0:7 V, V E ¼ 0 V
V C ¼ V CC βRC I B , I B ¼
Used as a constantcurrent source Used as a smallsignal commonemitter ampliﬁer
(continued)
XVII901
Chapter 17
Bipolar Junction Transistor and BJT Circuits
Transistor bias circuit with emitter resistance in active region (βindependent) V C ¼ V CC βRC I B , I B ¼
V BB 0:7 V ðβ þ 1ÞRE
V C V CC ðRC =RE ÞðV BB 0:7 VÞ V B ¼ V BB , V E ¼ V BB 0:7 V Used as a constantcurrent source Used as a smallsignal commonemitter ampliﬁer Transistor bias circuit with voltage divider (fourresistor bias circuit) in active region (βindependent) V C ¼ V CC βRC I B , I B ¼ RTH ¼
V T H 0:7 V RT H þ ðβ þ 1ÞRE
R1 R2 R2 , V TH ¼ V CC R1 þ R2 R1 þ R2
V C V CC ðRC =RE ÞðV T H 0:7 VÞ V B ¼ V E þ 0:7 V, V E ¼ ðβ þ 1ÞRE I B Used as a constantcurrent source Used as a smallsignal commonemitter ampliﬁer Transistor bias circuit with emitter resistance and dual power supply in active region (βindependent) V C ¼ V CC β RC I B , I B ¼
V CC 0:7 V RB þ ðβ þ 1ÞRE
V C V CC ðRC =RE ÞðV CC 0:7 VÞ V B ¼ V E þ 0:7 V, V E ¼ ðβ þ 1ÞRE I B V CC Used as a smallsignal commonemitter ampliﬁer
Transistor constantcurrent sources and constantvoltage sources operating in active region (similar to DC bias circuits) Load current is controlled by VREF (βdependent) IL ¼ β
V REF 0:7 V RB
(continued)
XVII902
Chapter 17
Summary
Load current is controlled by VREF (βindependent) IL
V REF 0:7 V RE
Constantcurrent source—the 5 V—LED drive of 20 mA (any color LED, βindependent) V T H 0:7 V R2 , V TH ¼ V CC R1 þ R2 RE
I LED
Constantvoltage source (voltage follower or voltage buffer, βindependent) in common collector or emitter follower conﬁguration V L ¼ V TH RTH ¼
I L RT H 0:7 V V TH 0:7 V βþ1
R1 R2 R2 , V TH ¼ V CC R1 þ R2 R1 þ R2
Transistor switch operating in saturation region RB ¼
V T H 0:7 V V CC RTH , I B 2 β RL IB (design selection)
Darlington pair Darlington pair (superbeta transistor) I C ¼ ððβ1 þ 1Þðβ2 þ 1Þ 1ÞI B I E ¼ ðβ1 þ 1Þðβ2 þ 1ÞI B βeff β2 for equal transistors (continued)
XVII903
Chapter 17
Bipolar Junction Transistor and BJT Circuits
General characteristics of voltage ampliﬁer Opencircuit voltage gain Aυ0: Aυ0
υin Input resistance Rin : Rin
iin Output resistance Rout
υout υin RL ¼1
υtest : Rout
itest υin ¼0
Overall volt. gain: Gυ
υout Rin RL ¼ Aυ0 υS Rin þ RS RL þ Rout
Smallsignal transistor model (hybridπ model) at Qpoint Smallsignal EBJ resistance: rπ
υbe V T ¼ ib IB
Smallsignal transconductance: g m
ic β ¼ υbe rπ
Smallsignal basebias ampliﬁer circuit—common emitter ampliﬁer (no load) (βdependent)—Class A
VT V BB 0:7 V , V C ¼ V CC β RC I B , I B ¼ IB RB RC Aυ0 ¼ β , Rin ¼ RB þ rπ , Rout ¼ RC RB þ rπ
rπ ¼
Capacitive coupling eliminates the DC bias at the output (continued)
XVII904
Chapter 17
Summary
Smallsignal ampliﬁer circuit with emitter resistance—common emitter ampliﬁer (no load) (βindependent)—Class A
rπ ¼
VT V BB 0:7 V , V C ¼ V CC β RC I B , I B ¼ IB ðβ þ 1ÞRE
Aυ0
RC , Rin ¼ rπ þ ðβ þ 1ÞRE , Rout ¼ RC RE
Capacitive coupling eliminates the DC bias at the output Smallsignal fourresistor bias ampliﬁer circuit—common emitter ampliﬁer (no load) (βindependent)—Class A
VT V TH 0:7 V , V C ¼ V CC β RC I B , I B ¼ IB RTH þ ðβ þ 1ÞRE RC Aυ0 , Rin ¼ R1 R2 ðrπ þ ðβ þ 1ÞRE Þ, RE Rout ¼ RC
rπ ¼
Capacitive coupling eliminates the DC bias at the output Gain enhancement of the prior design with shunt capacitance The Qpoint parameters remain the same The smallsignal parameters: RC Aυ0 ¼ β , Rin ¼ R1 R2 ðrπ þ ðβ þ 1ÞRE Þ, rπ Rout ¼ RC
XVII905
Chapter 17 iC iB
Bipolar Junction Transistor and BJT Circuits
iE þ iB βiB þ iB ð1 þ βÞiB υBE 1 I SB exp VT
ð2Þ all unknown currents close to each wire in this circuit topology.
ð3Þ E
are valid for the pnp BJT in the active region. Consider each equation separately. Problem 17.15. Shown in the ﬁgure below are four BJT transistors. The input base current is always equal to 10 μA. Each BJT has a commonemitter current gain β of 50 and is properly biased to operate in the active region. 1. Redraw the circuit shown in the ﬁgure. 2. Indicate collector, base, and emitter, and denote the transistor type (npn or pnp) for each device. 3. Indicate the current directions and determine the values of all other currents in the circuit. a) iB
c) iB
b) iB
d) iB
Problem 17.16. Shown in the ﬁgure below is the Sziklai pair (George Clifford Sziklai (1909–1998) was an electrical engineer at Lockheed Martin), which includes the npn and the pnp BJTs. The npn BJT base current is 10 μA. Given that (1) the npn BJT has the current gain β of 10 and VBE of +0.7 V and (2) the pnp BJT has current gain β of 5 and VBE of 0.7 V, show the values and the directions of
B C C B iB
E
Problem 17.17. Repeat the previous problem if υBE of the pnp transistor is 0.7 V. Problem 17.18. A. Who invented the ﬁrst npn transistor, and in what year? B. Who were two major collaborators (and rivals) of W. Shockley? Hint: See the reference in this subsection and the article devoted to W. Shockley from Wikipedia.
17.2 LargeSignal Circuit Models of a BJT 17.2.1 Equivalent LargeSignal Circuit Model of a BJT Problem 17.19. Draw the largesignal equivalent BJT model with: A. Voltagecontrolled current source B. Currentcontrolled current source Problem 17.20. For the transistor circuit shown in the ﬁgure below: A. Draw the equivalent circuit diagram using the largesignal BJT model. B. Solve the circuit for unknown voltages υBE, υ, and υCE at room temperature of 25 C. Assume I S ¼ 10 14 A for the EBJ Shockley diode.
XVII908
Chapter 17
Bipolar Junction Transistor and BJT Circuits
A. Basetoemitter voltage in the saturation region (“BaseEmitter Saturation Voltage” in the datasheet) B. Collectortoemitter voltage in the saturation region (“CollectorEmitter Saturation Voltage” in the datasheet)
Problem 17.29. Repeat Problem 17.28 for A. I B ¼ 0:1 mA and V CE ¼ 1 V B. V CE ¼ 1 V and V BE ¼ 0:4 V C. I C ¼ 2 mA and I B ¼ 20 μA D. V CE ¼ 0 V
for this popular transistor make.

IE
A. B. C. D.
For I B ¼ 0:1 mA and V CE ¼0:4 V For V CE ¼ 0:3 V and V BE ¼0:4 V For I C ¼ 1 mA and I B ¼ 20 μA For V CE ¼ 5 V
Problem 17.27. Repeat Problem 17.26 for β ¼ 100 and A. I B ¼ 0:1 mA and V CE ¼ 0:6 V B. V CE ¼ 0:7 V and V BE ¼ 0:4 V C. I C ¼ 2 mA and I B ¼ 20 μA D. V CE ¼ 0 V Problem 17.28. Determine the region of operation for a pnp BJT with β ¼ 100 shown in the ﬁgure using the largesignal DC circuit model.
IB

B
E
+
VEB
VEC
VCC=20 V
V
500 k =RB
4.7 k =RC C
B VBE
VCE
=100
E
Problem 17.31. For threetransistor circuit shown in the ﬁgure, ﬁnd the collector current IC and collectortoemitter voltage VCE A. For β ¼ 100 B. For β ¼ 300 Use the largesignal DC circuit model of the transistor.

+
IE
VCC=20 V
+
E
+
VBE
+
+ VCE
+ IB
A. Solve the transistor circuit shown in the ﬁgure that follows by determining unknown voltages VBE, V, and VCE using the largesignal DC BJT model and the method of assumed states. B. Solve the same circuit using the largesignal BJT model. Assume room temperature of 25 C and I S ¼ 10 14 A for the EBJ Shockley diode. C. Compare both solutions to each other.

B
Problem 17.30.

IC
17.2.4 Transistor Circuit Analysis Using the Method of Assumed States

Problem 17.26. Determine the region of operation for an npn BJT with β ¼ 200 shown in the ﬁgure using the largesignal DC circuit model.
C IC
XVII910
Chapter 17 VCC=20 V
+10V

C
B
100 k =RB B
VCE E
C
VCE
VBE


VBE
15 k =RC
+
VB
+
IB
IC
3.5 k
V
+
500 k
+
+
IC
vBB
IB

+10V
E

a)
Problems
0V +10V
+5V IC
+
b)
5k
V

VB
C
B
VCE
+
IB
+
930 k

E

VBE
VBB 0V 1V 2V 3V 4V
IB
IC
VCE
Region
0V
5k
V

VB
C
B
VCE
+
IB
E

VBE
+
930 k
5V
IC
15k =RC C
B
VCE
+
Problem 17.32. A. For the BJT circuit shown below, determine unknown parameters listed in the table that follows Assume β ¼ 100. Use the largesignal DC circuit model of the transistor. B. Perform the corresponding laboratory experiment with a 2N3904 smallsignal npn BJT and ﬁll out a similar table. Note: This conﬁguration is sometimes referred to as an NPN inverter with resistive load.
VCC=15 V
VBE
vBB
IB
IE
VBB 1V 2V 3V 4V 0V
IE
IC
+
+
IC
Problem 17.33. A. For the BJT circuit shown below, determine unknown parameters listed in the table that follows. Assume β ¼ 100. Use the largesignal DC circuit model of the transistor. B. Perform the corresponding laboratory experiment with a 2N3904 smallsignal npn BJT and ﬁll out a similar table.

+5V
E

+10V

c)
1k =RE
VCE
Region
XVII911
Chapter 17
Bipolar Junction Transistor and BJT Circuits
Problem 17.36. For the circuit shown in the 17.2.5 DC Transistor Bias Circuits 17.2.6 βIndependent Biasing and ﬁgure, determine (the npn BJT has β ¼ 100): A. Collector current IC Negative Feedback B. Emitter current IiE 17.2.7 Common DiscreteCircuit Bias Use the largesignal DC circuit model of the Arrangement transistor. 17.2.8 Other Bias Circuits Problem 17.34. For the circuit shown in the ﬁgure, ﬁll the table that follows. Use the largesignal DC circuit model of the transistor.
+10V
+10V IC
50k
200
VCC=20 V
C
+
B
V
4.7 k =RC
62 k =RB
E C
100
IE
+
B
VC
β 50 100 250
VBE
0V
E

VB
VBB=2 V
IC
VC
Problem 17.37. For the circuit shown in the ﬁgure, determine the transistor current gain β. Use the largesignal DC circuit model of the transistor. +10V
+10V IC=9mA
1k
Problem 17.35. For the circuit shown in the ﬁgure, ﬁll the table that follows. Use the largesignal DC circuit model of the transistor.
51 C B E
VCC=20 V
+
1k
V
0V
7.5 k =RC VC

C
Problem 17.38. A. Sketch the fourresistor bias circuit. Label two base resistances, collector resistance, and emitter resistance. B. What purpose could this circuit have?
+
B VBE

VB
VBB=2 V
E VE 1 k =RE
β 50 100 250
IC
VC
Problem 17.39. For the circuit shown below, determine the emitter current, iE, using the method of Thévenin equivalent when R1 ¼ 43 kΩ, R2 ¼ 3 kΩ, RC ¼ 20 kΩ, RE ¼ 1 kΩ:
XVII912
Bipolar Junction Transistor and BJT Circuits
17.3.3 BJT Switches: Saturation Region 17.3.4 Application Example: Automotive BJT Dome Light Switch 17.3.5 Application Example: Door Lock BJT Switch and Darlington Pair Problem 17.47. For the circuit of Fig. 17.28, V REF ¼ 3 V, V CC ¼ 24 V, and the load resistance is 10 Ω. Assume the minimum value β ¼ 100. A. Find the maximum possible load current (if the transistor were a perfect switch). B. Find the minimum base current required to saturate the transistor. C. Determine a value for RB to overdrive the transistor by approximately a factor of 2.
IC
VL RB
C
B
VCC VBE
IB
E
Problem 17.49. For the circuit shown in the ﬁgure below, V CC ¼ 12 V, and the bulb turn on resistance is 40 Ω. Assume the minimum value β ¼ 30. A. Find the maximum possible load current (if the transistor were a perfect switch). B. Find the minimum base current required to saturate the transistor. C. Determine a value for RB to overdrive the transistor by approximately a factor of 2.
IC
+
Problem 17.46. For the circuits of Fig. 17.24b, V CC ¼ 5 V, the emitter current is 30 mA and RE ¼ 15 Ω. The transistor has β ¼ 50. A. What is the voltage across the emitter resistance? B. What is the LED voltage in Fig. 17.24b?
+
Problem 17.45. For the circuit of Fig. 17.25b, R1 ¼ 200 Ω, R2 ¼ 300 Ω, and V CC ¼ 12 V. A. Determine Thévenin voltage and resistance for the voltage divider. B. Determine the voltage drop that results for a load current of 30 mA. Assume β ¼ 100. Repeat for a load current of 100 mA. C. Determine the load voltage for two load currents from part B.
+
Problem 17.44. For the circuits of Fig. 17.24a, the required LED current is 30 mA and RE ¼ 15 Ω. What is the largest LED voltage that can be tolerated before the transistor is pushed into saturation?
RB

Problem 17.43. Solve task A of the previous problem for the circuit in Fig. 17.24a given β ¼ 50.
Problem 17.48. For the circuit shown in the following ﬁgure, V CC ¼ 12 V, and the bulb turn on resistance is 40 Ω. Assume the minimum value β ¼ 50. A. Find the maximum possible load current (if the transistor were a perfect switch). B. Find the minimum base current required to saturate the transistor. C. Determine a value for RB to overdrive the transistor by approximately a factor of 2.

C. Given V CC ¼ 12 V, what is the range of possible load voltages?

Chapter 17
B
VEB
E
VCC
C
IB
XVII914
Chapter 17
Bipolar Junction Transistor and BJT Circuits
A. Determine DC bias parameters VB, VC, VE, IB, IC. B. Determine smallsignal ampliﬁer parameters rπ, Aυ0, Rin, and Rout. C. Write the expression for υOUT(t).
D. Sketch υin(t) and υOUT(t) for two cycles to scale on the same plot. Note: Use the exact expressions for circuit parameters without simpliﬁcations resulting from the condition β 1.
XVII918
Chapter 18
Chapter 18: MOS FieldEffect Transistor (MOSFET) Overview Prerequisites:  Knowledge of basic circuit analysis  Exposure to theory of the pnjunction (optional)  Exposure to BJT circuit analysis and amplifiers (Chapter 17, optional) Objectives of Section 18.1:  Learn physical composition of the fieldeffect transistor, four and threeterminal configurations  Understand principle of operation of the MOSFET  Realize the origin and understand the value of MOSFET threshold voltage  Be able to estimate threshold voltage based on MOSFET’s physical composition Objectives of Section 18.2:  Learn MOSFET test circuits  Become familiar with the dynamics of channel inversion and quantify the underlying mechanism  Derive MOSFET equations (largesignal model) for three regions of operation from first principles  Pay special attention to largesignal MOSFET model in saturation  Become familiar with vi dependencies for the NMOS and PMOS transistors Objectives of Section 18.3:  Learn the resistorswitch model of the MOSFET for switching applications  Apply the resistorswitch model of the MOSFET to logic gates  Understand the value of the triode and cutoff regions for switching applications  Become fluent with the method of assumed states for MOSFET DC circuit analysis  Use the loadline method, either graphically or analytically  Solve in basic MOSFET DC bias circuits
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_18
XVIII919
Chapter 18
MOS FieldEffect Transistor (MOSFET)
Objectives of Section 18.4:  Learn circuit topology of the commonsource MOSFET amplifier  Analyze and characterize the voltage transfer characteristic of the commonsource amplifier  Understand the value of the saturation region for amplifier applications  Be able to properly select the quiescent (bias) point  Formulate the smallsignal MOSFET model and solve in the commonsource amplifier circuit Application Examples:  Output resistance of digital logic gates  Basic MOSFET switching actuator
Keywords: Fieldeffect transistor (FET), Metaloxide semiconductor FET (MOSFET), Enhancementmode MOSFET, EMOSFET, Depletionmode MOSFET, nchannel MOSFET, pchannel MOSFET, NMOS transistor, PMOS transistor, MOSFET drain terminal, MOSFET source terminal, MOSFET gate terminal, MOSFET body terminal, MOSFET substrate terminal, Fourterminal MOSFET, Threeterminal MOSFET, MOSFET channel, MOSFET threshold voltage, MOS capacitor, Ideal MOS capacitor model, Surface spacecharge region of MOS capacitor, Surface voltage of MOS capacitor, Surface potential of MOS capacitor, Onesided pnjunction approximation, Strong inversion in MOS capacitor, Inversion layer of MOS capacitor, Flatband voltage, Work function difference, Junction FET (JFET), Metal–semiconductor FET (MESFET), Triode region of a MOSFET, Saturation region of a MOSFET, Cutoff region of a MOSFET, Process transconductance parameter, MOSFET transconductance parameter, MOSFET lumped process parameter, MOSFET turnon resistance, Channel pinchoff, Saturation current, Saturation velocity, Velocity saturation region, Early effect, Channel modulation effect, Transconductance curve, Largesignal MOSFET model in saturation, MOSFET parameter extraction, MOSFET onstate resistance, MOSFET resistorswitch model, CMOS logic gates, CMOS NOT gate (inverter), CMOS NAND gate, CMOS NOR gate, Gate output resistance, Method of assumed states, Gatebias (ﬁxedgate) MOSFET circuit, Diodeconnected MOSFET circuit, Loadline analysis, Load line, Basic MOSFET actuator device, Commonsource MOSFET ampliﬁer, Voltage transfer characteristic, Quiescent point of NMOS ampliﬁer, Smallsignal MOSFET model, Opencircuit smallsignal voltage gain, Smallsignal MOSFET transconductance, Smallsignal ground
XVIII920
Chapter 18
Section 18.1: Principle of Operation and Threshold Voltage
Section 18.1 Principle of Operation and Threshold Voltage In this chapter we study the ﬁeldeffect transistor (FET). The most important member of the FET family is the metaloxidesemiconductor FET or MOSFET. Similar to the npn and pnp BJT transistors, MOSFETs are subdivided into nchannel MOSFETs and pchannel MOSFETs, also known as NMOS and PMOS transistors. The abbreviation CMOS, or complementary MOS, implies an integrated circuit which incorporates both of these types of transistors on the same substrate. It is the CMOS transistor that allows highdensity chip integration as part of microelectronic analog and digital circuits. MOSFETs are used in both logic gates and in memory cells. Discrete power MOSFETs are also deployed in many power engineering applications. We will concentrate on the enhancementmode MOSFET (or EMOSFET), which relies on a positive gatetosource threshold voltage. Other MOSFET types (depletionmode MOSFET ) may have either negative or nearzero threshold voltages.
18.1.1 Physical Structure: Terminal Voltages and Currents An enhancementmode nchannel MOSFET (NMOS transistor) is a fourterminal semiconductor device. The NMOS transistor shown in Fig. 18.1 consists of a pdoped substrate (the Si wafer) into which two n (or rather heavily doped n+) regions, the source and the drain, are formed through ion implantation. The gate electrode (source of the control voltage) used to be a metal ﬁlm, but nowadays it is a heavily doped polysilicon. The gate length L, also known as channel length, can be as small as 30 nm. The gate isolation, necessary to form a capacitor, is a SiO2 dielectric. It is formed directly from the Si substrate by thermal oxidation of Si. There are four metal electrodes corresponding to four transistor terminals: the gate terminal (G), the source terminal (S), the drain terminal (D), and the body (or substrate) terminal (B). The basic geometrical device parameters are the channel length, L, in horizontal direction, and the channel width, W, in vertical direction in Fig. 18.1. The channel length is the distance between the two pn+ junctions; the channel width characterizes the region of electron carrier ﬂow between the drain and the source. Typical substrate acceptor concentrations (pdoping) are in the range of N A 1016 1017 cm 3 . The doping of the two n+ domains (donor doping) is large. For example, N D 1019 cm 3 for a power MOSFET.
XVIII921
Chapter 18
MOS FieldEffect Transistor (MOSFET)
When strong inversion takes place, the depletion layer width Wd no longer increases because the inversion layer starts blocking the electric ﬁeld. Its maximum value is given by Eq. (18.3) with V S 2φF . A critical distinction between the NMOS capacitor and the NMOS transistor is the channel formation time. While for an NMOS capacitor it can take minutes to collect the necessary electrons from the pdoped semiconductor with few free electrons, the inversion electrons for the transistor are readily available from two nearby n+ regions the source and the drain.
18.1.6 Threshold Voltage The threshold voltage VTh of an NMOS transistor is deﬁned as the gatesource voltage (18.2) at the onset of strong inversion when V S 2φB according to Eq. (18.11). VOX still follows Eq. (18.5). Therefore, p ð18:12Þ V Th V FB þ 2φB þ 2εqN A ð2φB Þ=C OX The new extra term VFB on the righthand side of Eq. (18.12) is called the ﬂatband voltage of the MOS capacitor. This term is needed for two reasons. The somewhat less important one is the presence of charges in the oxide layer due to ionic contamination. The second, important reason is the builtin voltage or potential of a boundary between two materials. This effect is similar to the builtin potential or voltage of the pnjunction. The builtin voltages of the metaloxide boundary and of the semiconductoroxide boundary do not cancel each other; the corresponding voltage difference is known as a work function difference ψ GS between the gate and the semiconductor; it appears across the oxide layer (Fig. 18.6). Without going into further details, we may assume V FB ψ GS and write NA NA ½V, ψ GS jnþpoly 0:7 0:03ln ½V ψ GS jAl 0:66 0:03ln 1013 1013 ð18:13Þ
0
GS ,
V
Al/n Si n+ poly/n Si
0.4
Al/p Si
0.8 n+ poly/p Si 1.2 13 10
10
14
10
15
10
16
10
17
NA , cm
10
18
3
Fig. 18.6. Work function difference ψ GS as a function of body doping for gate electrodes of polysilicon and aluminum, respectively, on a pSi body of an NMOS transistor.
XVIII928
Chapter 18
MOS FieldEffect Transistor (MOSFET)
Section 18.2 Theoretical Model of a MOSFET 18.2.1 Test Circuit and Operating Regions Figure 18.11 shows the test circuit for the threeterminal NMOS transistor. Gatesource voltage υGS and drainsource voltage υDS are varied. The drain current iD is measured at every particular voltage combination. Our goal is to derive analytical expressions for the drain current iD. The analytical models described below rely on the corresponding measured data, which have been obtained with circuits similar to that in Fig. 18.11. D iD G
+ A 
+ vDS
+ vGS
S

0V
Fig. 18.11. Schematic diagram of the NMOS transistor test circuit.
The NMOS (and PMOS) transistor has three operating regions: triode, saturation, and cutoff listed in Table 18.2. All three regions are used. Most common MOSFET switching circuits like logic gates utilize the cutoff and triode regions in order to characterize two binary steady states logic 0 and 1. However, during the fast transition between the states, the transistors enter the saturation region. MOSFET ampliﬁer circuits solely utilize the saturation region of operation. Table 18.2. The three operating regions of an NMOS transistor. Region Triode Saturation Cutoff
Condition on υGS υGS > V Th υGS > V Th υGS V Th
Condition on υDS υDS < υGS V Th ¼ υOV υDS > υGS V Th ¼ υOV Immaterial
The regions of operation are determined by the value of υDS as compared to the control voltage υGS. The triode region starts with a linear (or ohmic) subregion. Table 18.2 indicates one more useful voltage parameter: the overdrive voltage υOV υGS V Th . The NMOS transistor typically operates at large overdrive voltages.
XVIII932
Chapter 18
MOS FieldEffect Transistor (MOSFET)
μns is the electron surface mobility; μns 450 cm2 =ðV sÞ or less. The drain current that ﬂows from drain to source is thus given by (W is the channel width) iD
W QINV υ
W 0 k ðυGS L n
V Th ÞυDS ,
0
kn
C OX μns
ð18:16Þ
0
The constant kn with units of A/V2 (more often mA/V2) is called the process transconductance parameter. The name implies that it is determined by the particular fabrication 0 technology. The constant k n ðW =LÞk n with the same units is the MOSFET transconductance parameter (also called the lumped process parameter); it also includes information about the gate dimensions. Typically, kn is on order of 1 mA per V2 or less for smallsignal MOSFET transistors. For power MOSFETs, however, it can be much larger: on the order of 100 mA per V2. Equation (18.16) states that at small positive υDS the NMOS transistor behaves like a linear resistance rDS, which is controlled by the gatesource voltage, iD
υDS , rDS
rDS
1 k n ðυGS
ð18:17Þ
V Th Þ
The resistance rDS can be measured in the laboratory. It is also called the turnon resistance. This resistance of a MOSFET is a key parameter that is typically speciﬁed in the manufacturer’s datasheets (in contrast to kn).
18.2.3 Nonlinear Subregion of Triode Region at Strong Inversion When υDS increases (but still remains less than υGS V Th ), the situation depicted in Fig. 18.12b is observed. Close to the source region, the gate still “sees” the absolute source voltage (0 V in this case) as the terminal voltage. However, close to the drain, the gate does not “see” 0 V, but sees the drain voltage as the terminal voltage. The resulting voltage becomes υGD υGS υDS . Therefore, a variable gatesource voltage υGS(x) is effectively applied across the channel. The tip of the inversion layer becomes thinner, which is schematically shown in Fig. 18.12b. Introducing an asyet unknown channel voltage proﬁle y(x), we have 0 x 0 ð18:18aÞ υGS ðxÞ υGS yðxÞυDS , yðxÞ 1 x L Consequently, the charge of the inversion layer given by Eq. (18.15) also becomes a function of x as illustrated in Fig. 18.12b: QINV ðxÞ
C OX ðυGS ðxÞ
V Th Þ
C OX ðV OV
yðxÞυDS Þ
ð18:18bÞ
The vertical potential electric ﬁeld in the channel is now variable too, that is, XVIII934
Chapter 18
Section 18.2: Theoretical Model of a MOSFET
dυGS ðxÞ dx
E ðx Þ
υDS
dy dx
ð18:18cÞ
Next, the current along the inversion layer, iD iD
0
W QIN V μns EðxÞ
W k n υDS ðV OV
W QINV υ, becomes
yðxÞυDS Þ
dyðxÞ dx
ð18:18dÞ
By KCL, the current along the inversion layer must remain constant. If this condition is enforced, Eq. (18.18c) becomes a nonlinear ODE augmented with the boundary conditions Eq. (18.18a). It allows us to ﬁnd the voltage proﬁle y(x) along the channel analytically. The corresponding solution has the form (the proof is suggested as one of the homework problems) yðxÞ
m
r m2
ð2m
x 1Þ , L
dy dx
1 m Lm
0:5 , yðxÞ
m
υOV υDS
ð18:18eÞ
The proﬁle y(x) is quite linear ( x=L) everywhere in the channel at small υDS and close to the source for any υDS, but it becomes steeper when approaching the drain at large υDS. Since all channel parameters are now deﬁned, the transistor current can be calculated by picking up any point along the channel. An alternative and more common approach is to integrate Eq. (18.18d) from x to L and use boundary conditions Eq. (18.18a) along with the constantcurrent condition. Either method gives the simple ﬁnal expression for the drain current in the form: iD
k n υGS
1 υDS 2
V Th υDS
ð18:19Þ
Equation (18.19) reveals a nonlinear (parabolic) dependence of iD on υDS; this is an exact result. We could still use Eq. (18.17) too. However, rDS is no longer constant; it becomes voltage dependent, i.e., rDS
k n υGS
1 1 2υDS
V Th
ð18:20Þ
and increases with increasing υDS. When the drainsource voltage υDS is small compared to υGS V Th , the nonlinear MOSFET model is reduced to a linear one. Equation (18.19) becomes asymptotically equivalent to Eq. (18.16), and Eq. (18.20) reduces to Eq. (18.17).
18.2.4 Saturation Region As υDS continues to increase and eventually reach υGS V Th , the tip of the inversion layer in Fig. 18.12b becomes inﬁnitely thin since the inversion layer charge in Eq. (18.18b) is XVIII935
Chapter 18
Section 18.2: Theoretical Model of a MOSFET
Table 18.5. Minimum channel lengths, oxide capacitances, and surface mobilities for some CMOS processes. Note that 1 f F=μm2 ¼ 10 7 F=cm2 , εOX ¼ 3:45 1013 F=cm (SiO2 oxide). Parameter OX C OX ¼ εtOX ðf F=μm2 Þ
0.5μm process NMOS PMOS 3.8 3.8
0.25μm process NMOS PMOS 5.8 5.8
0.18μm process NMOS PMOS 8.6 8.6
0.13μm process NMOS PMOS 12.8 12.8
μns cm2 =ðV sÞ or μps cm2 =ðV sÞ
500
180
460
160
450
100
400
100
VTh (V)
0.7
0.8
0.5
0.6
0.5
0.5
0.4
0.4
XVIII941
Chapter 18
MOS FieldEffect Transistor (MOSFET)
index assuming that the DC parameters already correspond to the desired operating point (i.e., are the quiescentpoint parameters). The quiescentpoint parameters may denote the corresponding DC bias sources, for example, V IN V GS . The equations in (18.33) are applicable to all MOSFET smallsignal ampliﬁer models. The DC parameters must satisfy the largesignal DC circuit model separately, that is, V OUT
V DD
kn RD ðV IN 2
V Th Þ2
ð18:34Þ
18.4.4 MOSFET Biasing for Ampliﬁer Operation In order to use the MOSFET as an ampliﬁer, it is necessary to bias the output to a point Q in the saturation region where the transfer characteristic is fairly linear while at the same time providing enough dynamic range for the output voltage to swing in both a positive and negative direction. In the case of Fig. 18.28, this point has been chosen to be V Th
2 V < V IN
2:3 V < ð1 þ sÞV Th
2:42 V
ð18:35Þ
to establish V OUT 5:14 V according to Eq. (18.34). This way, a small variation in VIN will cause a large variation in VOUT, realizing the desired ampliﬁcation; see Fig. 18.28. The amount of ampliﬁcation or the opencircuit smallsignal voltage gain Aυ0 is simply the slope of the voltage transfer characteristic at this point, approximately 30 V/V in Fig. 18.28. The negative sign indicates that the output voltage swing will be inverted with respect to the input.
18.4.5 SmallSignal MOSFET Model and Superposition Our goal is to solve the circuit in Fig. 18.27. In order to do so, we will introduce a smallsignal MOSFET model. The concept of the linear expansion (18.33) allows us to split the largesignal MOSFET model used previously and depicted in Fig. 18.29a into two parts. Both of them are shown in Fig. 18.29b and c respectively. The DC solution is still described by the nonlinear largesignal circuit model. At the same time, the AC solution is described by a linear smallsignal MOSFET model in Fig. 18.29c. In the smallsignal model, the change in output current id is equal to a gain factor gm multiplied by the change in input voltage υgs. The gain factor gm is known as the smallsignal MOSFET transconductance. In the smallsignal model, all constant DC bias sources are replaced by ground (the smallsignal ground condition) since their voltages do not change with time; the AC signal is thus shorted out. According to Fig. 18.29, we solve the MOSFET ampliﬁer circuit twice: the ﬁrst time at DC using the largesignal DC model and the second time at AC using the smallsignal model. The DC solution will provide the necessary information for the AC solution. The complete solution is found as the sum of the DC and AC solutions, respectively. In other words, the superposition principle applies. This is a remarkable fact given that the DC model is inherently nonlinear. XVIII956
Chapter 18
MOS FieldEffect Transistor (MOSFET) vIN, vOUT, V 9
vOUT(t)
5
vIN(t) 1
0
1
t/T
2
Fig. 18.31. Input and output voltages for the commonsource ampliﬁer from Example 18.10.
XVIII960
Chapter 18
Summary
Summary MOSFET physical characteristics VTh—threshold voltage (0.4–4 V for enhancementmode nchannel MOSFET) [V] k n ¼ ðW =LÞC OX μns —MOSFET transconductance par. (lumped process par.) [mA/V2] MOSFET modeling NMOS test circuit and regions of operations (largesignal model) Triode ½ υGS > V Th , υDS < υGS V T h : 1 2 iD ¼ k n ðυGS V T h ÞυDS υDS 2 Saturation ½υGS > V T h , υDS υGS V T h : 1 iD ¼ k n ðυGS V Th Þ2 2 Cutoff ½υGS V T h : iD ¼ 0 PMOS test circuit and regions of operations (largesignal model) Triode ½ υSG > jV Th j, υSD < υSG jV Th j: 1 iD ¼ k p ðυSG jV Th jÞυSD υ2SD 2 Saturation ½υSG > jV Th j, υSD υSG jV Th j: 1 iD ¼ k p ðυSG jV Th jÞ2 2 Cutoff ½υSG jV Th j: iD ¼ 0 Resistorswitch model in triode region υDS 1 NMOS: iD ¼ , rDS ¼ rDS k n ðυGS V Th Þ PMOS: iD ¼
υSD 1 , rDS ¼ rDS k p ðυSG jV Th jÞ
rDS may be as small as 0.3–0.5 Ω for discrete (smallsignal or power) MOSFETS MOSFET circuits at DC CMOS Logic gates (two stable states correspond to triode and cutoff regions) NOT (inverter) Most basic and most important digital circuit
For alternative drawing see Chap. 13 (continued)
XVIII961
Chapter 18
MOS FieldEffect Transistor (MOSFET) NAND
For alternative drawing see Chap. 13 NOR
For alternative drawing see Chap. 13 AND (OR is constructed similarly) AND = NAND + NOT OR = NOR + NOT
For alternative drawing see Chap. 15 Method of assumed states/Load line method
(continued)
XVIII962
Chapter 18
Summary Gatebias circuit (all three regions are possible) The load line: iD ¼ ðV DD υDS Þ=RD If V GS V Th then cutoff k n RD If V DD > 1 þ ðV GS V Th Þ then saturation 2 k n RD ðV GS V Th Þ then triode If V DD 1 þ 2
Gatebias circuit with voltage divider (all three regions are possible) Eliminates the need in the second voltage supply (VGS) Equivalent to the previous circuit when V GS ¼
R2 V GS V DD or R2 ¼ R1 R1 þ R2 V DD V GS
May be used as a bias circuit for the ampliﬁer or as a sensor switch when R1 (or R2) is variable and RD is the load (light bulb, motor, etc.) Diodeconnected MOSFETS (current sources, always in saturation or cutoff) If V DD V Th , then iD ¼ 0 NMOS 1 2 If V DD > V Th , then iD ¼ k n ðV DD V Th Þ NMOS 2 PMOS If V DD jV Th j, then iD ¼ 0 1 If V DD > jV Th j, then iD ¼ k n ðV DD V Th Þ2 PMOS 2 Diodeconnected MOSFET with resistance (always in saturation or cutoff) If V DD V Th , then iD ¼ 0 If V DD > V Th , then V DD V Th 1 þ iD ¼ RD
q
1 þ 2k n R2D ðV DD V Th Þ k n R2D
Used to estimate threshold voltage: If RD is sufﬁciently large (iD is small) then V Th υGS MOSFET commonsource ampliﬁer – Commonsource ampliﬁer = smallsignal voltage ampliﬁer – Commonsource ampliﬁer is similar to the BJT commonemitter ampliﬁer – In a MOSFET ampliﬁer, the input resistance can be made inﬁnitely large (or kept ﬁnite if necessary) – An analog of the BJT emitter follower is the MOSFET source follower (continued)
XVIII963
Chapter 18
MOS FieldEffect Transistor (MOSFET) Voltage transfer function and Qpoint
0 υIN V Th
υOUT ¼ V DD
cutoff ðsmall υIN , not usedÞ kn V Th < υIN ð1 þ sÞV Th υOUT ¼ V DD RD ðυIN V Th Þ2 saturation ðυIN V Th Þ 2 υOUT > V Th , not usedÞ υIN > ð1 þ sÞV Th r p 1 þ 2k n RD V DD 1 V DD 1 > 0, V GS V Th þ s¼ , V DS V DD k n RD k n RD V th 2 Ampliﬁer circuit analysis, transconductance, and opencircuit smallsignal gain
Transconductance: gm ¼ k n ðV GS V Th Þ ½A=V Smallsignal gain: Aυ0 ¼ g m RD ½V=V Input resistance: Rin ¼ 1 ½Ω Output resistance: Rout ¼ RD ½Ω Smallsignal output: υout ðt Þ ¼ A0υ υin ðt Þ Input with DC bias: υIN ðt Þ ¼ V GS þ υin ðt Þ Output with DC bias: υOUT ðt Þ ¼ V DS A0υ υin ðt Þ
XVIII964
Chapter 18
Problems
Problems 18.1 Principle of Operation and Threshold Voltage
Problem 18.7. Repeat the previous problem when the NMOS body is GaAs and the insulating layer is Al2O3. Problem 18.8. Estimate surface voltage of the
18.1.1 Physical Structure: Terminal semiconductor body at the onset of strong channel inversion at room temperature of 25 C given Voltages and Currents 17 3 18.1.2 Simpliﬁed Principle of Operation N A ¼ 5 10 cm . The NMOS body is Si. Problem 18.1. What do the abbreviations FET, MOSFET, and CMOS stand for?
Problem 18.9. Repeat the previous problem when the body material is GaAs.
Problem 18.2. Draw circuit symbols of: A. Fourterminal symmetric NMOS transistor B. Threeterminal asymmetric NMOS transistor with the body tied to the source and C. The same but simpliﬁed symbol
Problem 18.10. Estimate threshold voltage VTh for a Si NMOS transistor with n+ polysilicon gate, N A ¼ 2 1016 cm 3 , and the SiO2 oxide layer with the thickness of 20 nm at room temperature of 25 C.
For B and C, label transistor currents and transistor voltages.
Problem 18.11. Repeat the previous problem for an aluminum gate.
Problem 18.3. Repeat the previous problem for the PMOS transistor.
Problem 18.12. Repeat Problem 18.10 for the PMOS transistor with the ndoped body of the same doping concentration.
Problem 18.4. An NMOS transistor has A. The gatedrain voltage of 2 V and gatesource voltage of 1 V B. The source current of 1 mA What is the drainsource voltage? What is the drain current? Problem 18.5. Repeat the previous problem for the PMOS transistor.
Problem 18.13. Estimate threshold voltage VTh for Si NMOS transistors used in analog ICs and fabricated in four CMOS processes listed in Table 18.1. Every transistor has the n+ polysilicon gate and the SiO2 oxide layer. The corresponding doping concentrations are N A ¼ ½3:5, 4:5, 7:5, 10:5 1016 cm 3 . Keep at least two signiﬁcant digits. Compare your solutions with the typical values reported elsewhere: V Th ¼ ½0:7, 0:5, 0:5, 0:4 V. Assume room temperature of 25 C.
18.1.3 NMOS Capacitor 18.1.4 Voltage Across the Oxide Layer 18.1.5 Voltage Across the Semiconductor 18.2 Theoretical Model of a Body 18.1.6 Threshold Voltage MOSFET 18.1.7 PMOS Transistor 18.2.1 Test Circuit and Operating 18.1.8 Oxide Thicknesses and Capaci Regions tances in CMOS Processes 18.2.2 Linear Subregion of Triode region Problem 18.6. Given the semiconductor surface potential ϕ ¼ 2 V and the uniform accep at Strong Inversion S
tor concentration 5 1016 cm 3 for the pbody of the NMOS transistor, estimate the voltage across the SiO2 oxide layer with the thickness of 10 nm. The NMOS body is Si.
Problem 18.14 A. Draw the schematic test circuit for the NMOS transistor. Label transistor terminals. When the gatesource bias voltage
XVIII965
Chapter 18
MOS FieldEffect Transistor (MOSFET)
is 0 V, which region of operation is encountered? B. Repeat the same tasks for the PMOS transistor. Problem 18.15. Determine the total charge (show units) stored in the inversion layer of the MOSFET transistor with L ¼ 0.8 μm, W ¼ 16 μm, C OX ¼ 2:3 f F=μm2 . The overdrive voltage is 2 V; the drainsource voltage is 0 V. How many electrons are stored in the inversion layer? Problem 18.16. For a CMOS process of MOSFET fabrication, L ¼ 0.8 μm, and W ¼ 16 μm. Furthermore, the electron surface mobility is μn ¼ 550 cm2 =V s and the oxide capacitance is C OX ¼ 2:3 f F=μm2 . Determine the MOSFET transconductance parameter and show units. Problem 18.17. An NMOS transistor in the linear subregion of the triode region operates at υOV ¼ 4 V. Given rDS ¼ 100 Ω determine the 0 lumped process parameter kn and show units. Problem 18.18 A. Determine the MOSFET transconductance parameter (show units) for NMOS transistors used in analog ICs and fabricated in four CMOS processes listed in Table 18.5. Use the given channel length and the channel width ten times greater than the length. B. Determine turnon resistances rDS in every case given that the overdrive voltage is equal to the threshold voltage. Problem 18.19. Repeat the previous problem for the PMOS transistor with parameters listed in Table 18.5. The corresponding resistance is given by rDS ¼ k p ðυSG 1 jV Th jÞ, υOV ¼ υSG jV Th j.
charge behavior, it is suggested to simplify Eq. (18.18). Namely, the inversion layer of the MOSFET at nonzero drainsource voltages would be described by a straightforward linear voltage dependence υGS ðxÞ ¼ υGS Lx υDS , x 2 ½0, L present in some undergraduate texts. Do you see one critical contradiction of this model? Problem 18.21. Plot variable gatesource voltage υGS(X) to scale over the interval X ¼ Lx 2 ½0, 1 given that V Th ¼ 1 V, V OV ¼ 1 V, and A. υDS ¼ 0:1V OV : B. υDS ¼ 0:5V OV : C. υDS ¼ 1:0V OV : Use the vertical scale from 1 V to 2 V for every ﬁgure. Problem 18.22. Repeat the previous problem for the normalized charge, QINV(X)/QINV(0), of the inversion layer. Problem 18.23. Show that the solution for the channel voltage proﬁle given by Eq. (18.18e) guarantees the condition of the constant current along the channel. Problem 18.24. An NMOS transistor has k n ¼ 2 mA=V2 and V Th ¼ 0:7 V. The gatesource voltage is 3 V. A. At which value of υDS does the transistor enter the saturation? B. What value of iD is obtained in saturation? Problem 18.25. An NMOS transistor has (the 0.25μm Si CMOS process) L ¼ 0:25 μm, t OX ¼ 6 nm, μn ¼ 460 cm2 =ðV sÞ, and V Th ¼ 0:5 V. Given W ¼ 10 μm ﬁnd υGS, which assures the operation in the saturation region with the transistor current of 1 mA.
18.2.5 The vi Dependencies 18.2.6 PMOS Transistor 18.2.3 Nonlinear Subregion of Triode 18.2.7 LargeSignal MOSFET Model in Region at Strong Inversion Saturation 18.2.4 Saturation Region Problem 18.26. A nchannel power MOSFET Problem 18.20. To avoid complications caused by a nonlinear channel voltage and inversion
has the following parameters: V Th ¼ 3 V and k n ¼ 100 mA=V2 .
XVIII966
Chapter 18
Problems
A. Plot the drain current for sourcedrain voltages from the interval υDS ¼ ½09 V and at three values of the gatesource voltage, υGS ¼ 3, 4, and 5 V on the same ﬁgure. B. Plot the boundary between the triode region and the saturation region. 300
iD, mA
20
iD, mA
NMOS transistor
16
VGS=5 V
12 8 4
n channel MOSFET
0
0
2
4
6
8 vDS, V
200
100
0
0
2
4
6
8
vDS, V
Problem 18.27. Repeat the previous problem for υGS ¼ 6, 7, and 8 V. iD, mA
Problem 18.29. A PMOS transistor has the following parameters: jV Th j ¼ 3 V and k p ¼ 100 mA=V2 . A. Plot the drain current for sourcedrain voltages from the interval υSD ¼ ½09 V and at three values of the gatesource voltage υGS ¼ 4, 5, and 6 V on the same ﬁgure. B. Plot the boundary between the triode region and the saturation region.
n channel MOSFET iD, mA
p channel MOSFET
1200
1200 800
800 400
400
0
2
4
6
8
vDS, V
Problem 18.28 A. Based on Table 18.3, construct the similar table for the power dissipated by an NMOS transistor. B. In the ﬁgure that follows, show graphically power dissipated by the transistor at two values of υDS: 2 V and 5 V. C. Which region, triode or saturation, leads to the smallest power dissipation?
0
2
4
6
8
vSG, V
Problem 18.30. For the circuit shown in the ﬁgure that follows, determine the region of MOSFET operation as well as the drain current iD for each set of conditions given. Assume k n ¼ 100 mA=V2 and V Th ¼ 3 V A. υGS ¼ 3 V, υDS ¼ 3 V: B. υGS ¼ 10 V, υDS ¼ 8 V: C. υGS ¼ 5 V, υDS ¼ 1 V:
XVIII967
Chapter 18
MOS FieldEffect Transistor (MOSFET)
+ A
D iD
+
G
vDS
+

S
vGS
B. Plot the boundary between the triode region and the saturation region and indicate the slope 1/rDS in the ﬁgure. C. Plot rDS as a function of υGS to scale. iD, mA 1000

800
0V 600
Problem 18.31. For the circuit shown in the ﬁgure that follows, determine the region of MOSFET operation as well as the drain current iD for each set of conditions given. Assume k p ¼ 100 mA=V2 and V Th ¼ 2 V. A. υGS ¼ 3 V, υDS ¼ 5 V: B. υGS ¼ 10 V, υDS ¼ 8 V: C. υGS ¼ 5 V, υDS ¼ 1 V:
400 200 0 0
2
4
6
8 vDS, V
2
4
6
8 vGS, V
rDS, 1000
100
0V
+ vSG

S
10
G
+ iD
vSD

1 0
D + A 
18.3 MOSFET Switching and Bias Circuits 18.3.1 Triode Region for Switching Circuits. Device Parameter Extraction 18.3.2 ResistorSwitch Model in Triode Region Problem 18.32. A power MOSFET has the lumped process parameter k n ¼ 130 mA=V2 and the threshold voltage of 2.0 V. A. Plot the drain current for drainsource voltages from the interval υDS ¼ ½09 V to scale and determine MOSFET turnon resistance rDS given the gatesource voltage of 5 V.
Problem 18.33. Repeat the previous problem when the threshold voltage changes to 3 V. Problem 18.34. A measurement curve for a certain NMOS transistor is shown in the ﬁgure below. Approximately determine the threshold voltage VTh and the (MOSFET) lumped process parameter kn (show units). 1000
rDS,
100
10
1 0
2
4
6
8
vGS, V
XVIII968
Chapter 18
Problems
Problem 18.35. Repeat the previous problem for the measurement curve shown in the ﬁgure that follows.
VDD VA PMOS M3
rDS, 20
VB PMOS M4 Vout
NMOS M2
2 0
2
4
6
NMOS M1
8 vGS, V
Problem 18.36. The datasheet for an IRF510 enhancedmode nchannel power MOSFET from Fairchild reports the average drainsource onstate resistance rDS ¼ υDS =iD ¼ 0:4 Ω for iD ¼ 3:4 A and υGS ¼ 10 V. Assuming threshold voltage to be 2.0 V, sketch rDS as a function of υGS to scale.
Problem 18.38. For a logic gate shown in the ﬁgure that follows, construct the truth table in the form of Table 18.7 and determine the output gate resistance for every input voltage combination. Assume V DD > V Th for the NMOS transistor and V DD > jV Th j for the PMOS transistor. Label turnon resistances of M1,2,3,4,5,6 as rDS1,2,3,4,5,6.
rDS,
VDD
20 10
M3
M4 M6
Vout 1
VA 0
5
vGS, V
M1
M5
10
VB
M2
18.3.3 Application Example: Output Resistance of Digital Logic Gates Problem 18.37. For a logic gate shown in the ﬁgure that follows, construct the truth table in the form of Table 18.7 and determine the output gate resistance for every input voltage combination. Assume V DD > V Th for the NMOS transistor and V DD > jV Th j for the PMOS transistor. Label turnon resistances of M1,2,3,4 as rDS1,2,3,4.
18.3.4 MOSFET Circuit Analysis at DC 18.3.5 Application Example: Basic MOSFET Actuator
Problem 18.39. In a ﬁxedgate NMOS transistor circuit, the NMOS transistor has the lumped process parameter k n ¼ 1:0 mA=V2 and the threshold voltage of 1.0 V. Furthermore, V GS ¼ 7 V, V DD ¼ 10 V, and RD ¼ 1 kΩ.
XVIII969
Chapter 18
Problems VDD iD D G
S
Problem 18.43. In the circuit shown in the ﬁgure that follows, the PMOS transistor has the lumped process parameter kp and the threshold voltage V Th < 0. Derive the analytical expression for the drain current iD as a function of VDD valid for any values of VDD. VDD
Problem 18.46. In a ﬁxedgate NMOS transistor circuit shown in the ﬁgure, the NMOS transistor has the lumped process parameter k n ¼ 100 mA=V2 and the threshold voltage of 2.0 V. Furthermore, V DD ¼ 20 V, R2 ¼ 1 kΩ, and RD ¼ 40 kΩ. Determine υGS, the region of operation of the transistor, and the solution for the drainsource voltage υDS and drain current iD when A. R1 ¼ 9 kΩ: B. R1 ¼ 4 kΩ: C. R1 ¼ 2:33 kΩ: D. R1 ¼ 1:5 kΩ: At which value of the resistance R1 is the load power (power into RD) maximized? At which value of the resistance R1 is the MOSFET power loss minimized?
iD VDD
S
VDD
G
RD
+
D
iD D
R1 G
VDS
+

VDD
VGS R2

Problem 18.44. In the circuit shown in the ﬁgure that follows, the NMOS transistor has the lumped process parameter kn and the threshold voltage VTh. Derive an analytical expression for the drain current iD as a function of VDD and RD valid for any values of VDD, RD.
S
Problem 18.47. Repeat the previous problem when the lumped process parameter changes to 200 mA/V2.
iD
RD D
18.4 MOSFET Amplifier
G
18.4.1 MOSFET CommonSource Ampliﬁer S 18.4.2 Voltage Transfer Characteristic 18.4.3 Principle of Operation and Qpoint Problem 18.45. In the circuit shown in the 18.4.4 MOSFET Biasing for Ampliﬁer ﬁgure to the previous problem, a sufﬁciently Operation
large resistance RD has been chosen so that the drain current is very small. Measured υDS is 2.5 V. What is approximately the threshold voltage VTh of the transistor?
Problem 18.48. In the commonsource ampliﬁer circuit in Fig. 18.27, the MOSFET has the lumped process parameter k n ¼ 100 mA=V2
XVIII971
Chapter 18
MOS FieldEffect Transistor (MOSFET)
and the threshold voltage of 2 V. The source voltage is V DD ¼ 20 V. A. Identify all values of υIN corresponding to the saturation region when
RD
1 kΩ:
RD
500 Ω:
RD
100 Ω:
B. Identify the DC Qpoint voltage VIN within the saturation region corresponding to V OUT ¼ V DD =2 when
RD
1 kΩ:
RD
500 Ω:
RD
100 Ω:
Problem 18.49. Plot to scale the voltage transfer characteristic for the commonsource ampliﬁer circuit in Fig. 18.27 built with the generalpurpose 2 N7000 NMOS transistor from Fairchild, which has the lumped process parameter k n ¼ 90 mA=V2 and the threshold voltage of 2.0 V. Indicate the saturation region. The source voltage is V DD ¼ 10 V and RD ¼ 90 Ω. Assume that the output voltage is constant in the triode region for simplicity. v OUT, V 1.2V DD
0.6V DD
0
0
0.5V DD
v IN, V
1.0V DD
18.4.5 SmallSignal MOSFET Model and Superposition 18.4.6 MOSFET Transconductance 18.4.7 Analysis of CommonSource MOSFET Ampliﬁer Problem 18.50. In a commonsource MOSFET ampliﬁer in Fig. 18.30a, the transistor has k n ¼ 100 mA=V2 and V Th ¼ 3 V. Furthermore, V DD ¼ 20 V. For A. RD ¼ 5 kΩ B. RD ¼ 1 kΩ C. RD ¼ 100 Ω design the ampliﬁer by performing the following steps: 1. Identify all values of VGS corresponding to the Qpoint in the saturation region. 2. Determine VGS which assures that the Qpoint (the DC operating point) is in saturation region and there is enough dynamic range for the output voltage to swing in both a positive and negative direction, i.e., V DS ¼ V DD =2. 3. Determine the transistor’s transconductance at the Qpoint. 4. Determine the opencircuit smallsignal voltage gain Aυ0 of the ampliﬁer. Problem 18.51 A. Repeat the previous problem when RD ¼ 120 Ω. B. Plot input and output voltages υIN ðtÞ ¼ V GS þ υin ðt Þ, υOUT ðt Þ ¼ V DS þ υout ðt Þ to scale over two periods given that υin ðt Þ ¼ 0:1 cos ðωt Þ ½V. Problem 18.52 A. Repeat Problem 18.50 when RD ¼ 120 Ω and V DD ¼ 10 V. B. Plot input and output voltages υIN ðtÞ ¼ V GS þ υin ðt Þ, υOUT ðt Þ ¼ V DS þ υout ðt Þ to scale over two periods given that υin ðt Þ ¼ 0:2 cos ðωt Þ ½V.
XVIII972
ERRATUM TO
Practical Electrical Engineering Sergey N. Makarov, Reinhold Ludwig, Stephen J. Bitar
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2
DOI 10.1007/978 3 319 21173 2
The original version of this volume contained incorrect page numbers associated with the index entries. We have updated these with the correct page numbers in the current volume.
The updated original online version for this book can be found at DOI 10.1007/978 3 319 21173 2
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2_19
E1
Index
A abc phase sequence, 547, 573 Absolute voltages in circuit, 72 73 acb phase sequence, 548, 554, 573 AC circuit analysis KVL, KCL and equivalent impedances, 411 412 Norton equivalent circuits, 415 417 phasor diagram, 412 413 single frequency, 417 source transformation, 413 415 superposition theorem, 417 418 Thévenin equivalent circuits, 415 417 AC coupled ampliﬁer, 208 AC direct micro hydropower system, 79 AC fuse, 528, 529 AC power distribution system automotive alternator, 550 line currents, 553 554 line voltages, 551 553 neutral conductor, 546 polyphase distribution systems, 546 phase voltages, 551 553 residential household, 551 single phase three wire, 546 single phase two wire, 546, 547 synchronous three phase AC generator, 550 synchronous three phase AC motor, 550 551 three phase four wire, 546 electronic circuits, 525 load voltage amplitude, 525 maximum power efﬁciency, 542 543, 571, 572 maximum power transfer average power, 544 equality, 544 impedance matching, 545 load impedance, 545 radio frequency and communication circuits, 544 Ohm’s law, 525
rms voltage and AC frequencies, 528 529 and rms current, 526 528 time averaging, 526, 567 Active differentiator, 292 Active ﬁlters, 290 Active one port networks, 178 Active reference conﬁguration, 52 AC voltage source, 68 Air core coil, 275 Alternating current (AC) circuits (see AC circuit analysis) power (See AC power) American Wire Gauge (AWG), 39, 78 Ampere’s law, 583 584, 631 Ampliﬁer circuit analysis (see Ampliﬁer circuit analysis) circuit model, 195 197 differential gain and common mode gain, 223 225 differential input signal, 222 223 dynamic circuit elements, 283 284, 290 291 ideal ampliﬁer model, 197 198 input/output resistances, 198 instrumentation, 226 229 load cell and uses, 229 output current, 198 summing point constraint, 197 198 Ampliﬁer circuit analysis cascading ampliﬁer stages, 215 217 current ﬂow, 206 207 DC imperfections, 217 220 discrete resistance values, 210 gain tolerance, 210 211 input bias and offset currents, 219 220 input load bridging, 212 214 input load matching, 214 215 input offset voltage, 217 218 input/output resistances, 211 212 inverting ampliﬁer, 203 204
The original version of the index was revised. An erratum can be found at DOI 10.1007/978 3 319 21173 2_19
© Springer International Publishing Switzerland 2016 S.N. Makarov et al., Practical Electrical Engineering, DOI 10.1007/978 3 319 21173 2
973
Index
Ampliﬁer circuit analysis (cont.) multiple input, 207 208 negative feedback, 211 non inverting ampliﬁer, 201 203 output offset voltage, 218 219 potentiometers, 210 resistance values, 209 210 summing point constraints, 240 voltage follower/buffer ampliﬁer, 204 205 voltage vs. matched ampliﬁer, 212 215 Amplitude envelope, 824 Amplitude modulated signal, 479 Amplitude transfer function See Filter circuits Analog computers, 293, 698 699 Analog ﬁlter amplitudes, 436 DC circuit, 436 load connection, effect, 439 440 qualitative analysis, 436 437 RC voltage divider circuit, 435, 436, 473 real valued voltages, 436 transient circuits, 436 two port network, 437 439 Analog output voltage, 692 Analog pulse counter, 290 Analog sinusoidal voltage, 717 718 Analog to digital converter (ADC) conversion time, 727 728 and DAC circuit, 691 692 equation, 726 727 ﬂash, 724 725 quantization error, 727 speed and throughput rate, 727 728 successive approximation circuit, 728 concept, 728 DAC formula, 730 logic control block, 729 operation, 730 AND gate, 664 667 Apparent power, 535 Arbitrary linear networks, 155 157 Arduino applications, 757 compiling and uploading code, 761 IDE, 757 759 language and compiler, 759 loop() function, 760 open source platform, 757 setup() function, 760 sketch, 760 translation, 761 Arduino programming arrays, 777 779 conditional statements, 771 773 interrupts, 780 782 loops, 774 777 PWM generation, 782 783
974
serial communication, 779 780 square wave generation, 782 783 strings, 777 779 switch statements, 773 774 Arduino syntax arithmetic operations, 764 765 assignment statements and features, 763 764 data types, 762 functions, 769 interfacing, IO pins, 769 770 library, 767 objects, 768 769 Arithmetic operations characters and relation, C language, 765 ﬂoating point and integer, 764 IDE, 765 typecasting, 764 Arrays element, 777 indexing, 777 linear, 777 ASCII codes, 702 704, 735 Assignment statements declaration, 763 ﬂoating points, 763 MOTOR_PIN, 763 typecasting, 764 Astable multivibrator, 330 Asynchronous transmission, 698 Autotransformer, 598 599, 635 Average generator voltage, 60 Average power, 532 533, 569
B Balanced delta connected load, 560, 575 Balanced delta connected source, 560 563, 575 Balanced phase voltages, 578, 552 Balanced three phase load, 549 Balanced three phase source, 549 Balanced three phase systems apparent power, 556, 558 average power, 556, 558 constant torque, 557 line currents, 557 material consumption, 558 560 phase voltages, 557 reactive power, 556, 558 total instantaneous load power, 556 wye wye conﬁguration, 556 Balanced Y and Δ networks, 121 Bandlimited spectrum, 459, 461, 463 Bandwidth, parallel resonant circuit, 495 Base emitter voltage, 854, 855, 861, 867, 871, 872, 892 Battery capacity, 61, 62 Battery energy storage, 63 Battery pack, 100 Battery voltage, 61
Index
Bias circuit, dual polarity power supply, 878 Bidirectional switch, 50 Binary counter, 208, 709 Binary numbers analog computers, 699 ASCII codes, 702 704 conversion, 698 700 decimal fraction, 700 digital voltage, 692 693 MATLAB, 700 Binary weighted input DAC, 707 708 Bipolar junction transistor (BJT), 20 assumed states, 870 automotive dome light application, 886 887 base collector (bias) voltage, 853 base collector voltage, 854, 861 base current, 853 base emitter (bias) voltage, 853 bias circuits, 878 879 collector current, 853 base current, 858 859 qualitative description, 857 Shockley equation, 857 858 collector emitter (bias) voltage, 853 constant current sources circuit limitations, 881 collector current, 880, 881 color LEDs, 881, 882 ﬁxed resistance, 880 LED driver, 881 882 DC transistor bias circuits, 873 874 dome light switch, 886 887 door lock BJT switch and Darlington pair, 887 888 early effect, 863 early voltage, 863, 907 EBJ and CBJ, 853 emitter current, 853 ground side switches, 884 independent biasing and negative feedback, 874 875 large signal DC circuit model, 868 870 maximum load current, 886 negative side and positive side, 884 npn BJT switch, 884 885 npn junction transistor, 853, 854 operating regions, 856 overdriving, transistor base, 886 pnp transistor, 863 865 principles built in electric potential, 855 common emitter conﬁguration, 854 doping proﬁles, 854, 855 electron controlling device, 856 intrinsic Fermi energy level of electrons, 855 potential boundary, 855 qualitative description base current, 859, 860 collector current, 859, 860 diffusion electron motion, 859 Ebers Moll model, 860
975
emitter current, 859, 860 forced common emitter current gain, 860 forward biased pn junctions, 859 reverse common emitter current gain, 860 transistor v i dependencies, 861 863 voltage controlled switches, 884 voltage follower circuit comparison, 882, 883 circuit limitations, 883 884 constant voltage, 882 emitter follower conﬁguration, 882 Thévenin equivalents, 883 Bistable ampliﬁer circuit See Switching RC oscillator Bit rate, 695 696 Blocking capacitor, 289 Body capacitance, 263 Body control module, 886 Boolean algebra, 668, 669, 670 Boolean expressions, 669, 673, 686 Branch currents, 93 Branch voltages, 93 Branches of electric network, 91 93 Break frequency ampliﬁer gain, 452 3 dB frequency, 443 half power frequency, 440 high pass ﬁlter, 440 441, 447 mirror reﬂected, 444 Built in voltage, pn junction, 797, 799 Butterworth response, 499 501 Bypass capacitor, 287 288
C Capacitance conductors, 261 dynamic behavior, 278 286 equal conductors separated, large distances, 262 ground, 261 262 Capacitive coupling of an ampliﬁer, 209 Capacitive reactance, 534 Capacitive touch screens, 265 Cascading ampliﬁer stages, 215 217 Center tapped transformer 180 power divider, 602 180 power splitter, 602 Central processing unit (CPU) brain, 748 functions, 748 instruction, 748 instruction set, 748, 749 opcode, 748 operands, 748 software, 749 Channel pinch off, 936 Charge separation principle, 58 Chassis ground, 71, 72 Circuit analysis methods, 159 163, 183 Circuit components
Index
Circuit components (cont.) bias, 46 vs. circuit elements, 31 semiconductor, 20 Circuit elements absolute voltage and voltage drop, 72 73 vs. circuit components, 31 current ampliﬁer, 65 ideal diode, 49 independent ideal voltage source, 52 linear passive, 33 nonlinear passive, 47 48 transconductance ampliﬁer, 65 transresistance ampliﬁer, 66 υ i characteristic, 34 35 Clamp on ammeter, 602 Clock frequency, 696, 698 Closed loop AC gain, 453 455 Closed loop conﬁguration, 192, 231 Closed loop gain, 202, 230 232 CMOS NAND gate, 946 947 CMOS NOT gate (inverter), 945 Collector base junction (CBJ), 853, 856 Collector emitter voltage, 854, 855, 861, 862, 867, 871, 873 Common base conﬁguration, 854 Common base current gain, 858, 862, 906 Common collector conﬁguration, 854 Common emitter ampliﬁer ﬁxed base transistor bias circuit, 890 linear expansion, 890 891 small signal ground, 891 small signal resistance/impedance, 890 voltage transfer characteristic, 890 Common ground, dual polarity power supply, 101 Common mode ampliﬁer circuit gain, 224 Common mode input signal, 197 Common mode rejection ratio (CMRR), 224 Common mode voltage of sensor, 222, 224, 226 Common (neutral) terminal (ground), 71 Common source MOSFET ampliﬁer DC bias solution, 958 input/output voltages, 959 large signal/small signal, 959 saturation region, 959 small signal model, 959 voltage controlled current source, 953 Comparator, 194 195 Compensated Miller integrator, 291 292 Complementary MOS (CMOS) circuits H bridge, 658 logic inverter/NOT gate, 660 NAND logic gate, 665 NOR logic gate, 662 Complementary transistors, 646 Complex load power, balanced three phase system, 558 Complex power, 535
976
Complex transfer function cascading ﬁlter circuits, 445 447, 476 complex expression, 445 frequency response, 445 high pass ﬁlter, 445 low pass ﬁlter, 445, 446 next stage ﬁlter load, 447 448 Compliance, 19 Conditional statements function, 772 logic circuit, 772 logical expression, 772 nested if statements, 773 state programming, 771 structure, 771 Conductance Ohm’s law, 33 transconductance, 65 Conduction angle, 819 Conservative ﬁeld, 5 Constant voltage drop model assumed states, 808 diode current, 809 diode voltage, 809 voltage supply, 808 Contour integral, 5 Control switching circuits full H bridge switch, 650 half H bridge, 650 motor speed controller, 650 one quadrant switch, 650 Corner frequency, 443, 444 Coulomb force, 3, 7, 37, 94 Coupled inductors mutual inductances, 615 N coupled, 615 phasor form, 613, 614 T network, 615 616 two coupled inductors, 616, 638 Coupling coefﬁcient energy, 616 ideal transformer, 616 617 mutual inductance, 617 CPU See Central processing unit (CPU) Current ampliﬁer current controlled current source, 65 and transconductance, 233 using op amp, 209 Current controlled current source, 65 Current controlled voltage source, 65 66 Current divider circuit, 112 113 Current division rule, 112 Current limiter, 111 112 Current limiting resistor, 111, 134 Current reference directions, 587, 613 Current transformer, 602 Cutoff frequency, 496, 497
Index
D DAC See Digital to analog converter (DAC) DAC scaling voltage factor, 708 Damping coefﬁcient, 348, 352, 353, 358, 381 Data bus, 707 Data storage methods, 755 Data types, 762 3 dB frequency, 443 DC coupled ampliﬁer, 208 DC coupled single supply ampliﬁer, 220 221 DC imperfections, 217 220 DC operating point, 810, 811, 842, 843 DC restorer circuit, 827, 828 DC steady state, 284 285 Decoupling capacitor, 289 Decoupling inductor, 289 Decrement operator, 775 Delta delta distribution system, 561 Δ to Y transformation, 120, 121 De Morgan’s laws, 669 670 Dependent sources vs. independent sources, 64 Depletion enhancement mode MOSFET, 921 Depletion mode MOSFET, 921 Device under test (DUT), 264 DFT See Discrete Fourier transform (DFT) Differential ampliﬁer circuit gain, 224 Differential input signal, 197, 222 223 Differential input voltage, 192, 200, 224 Differential resistance, 49 Differential sensor, 223 Differentiator ampliﬁer circuit, 292 293 gain at very high frequencies, 292 Digital logic gates CMOS NOT gate, 945 logic inverter, 945 output resistance, 945 947 Digital memory element, 332 Digital output voltage, 694 Digital repeater, 194 Digital signal processing (DSP), 464, 691, 707 Digital switching circuits AND gate, 665 667 combinational logic circuits, 668 669 latch, 673 logic circuit analysis motor state, 671 672 truth table, 670 671 logic circuit synthesis, 672 673 logic gate, 660 NAND gate, 664 665 NOR gate, 661 663 NOT gate/logic inverter, 660 661 OR gate, 663 664 universal property, NAND gates, 669 670 Digital to analog converter (DAC) binary weighted input, 707 708 equation, 711
977
math voltage, 708 709 operations, 707 output voltage range, 711 PWM, 716 quantization levels, 711 R/2R ladder, 714 716 relative accuracy, 712 714, resolution, 712 resolution voltage, 709 710 Thévenin equivalent, 715 716 Digital voltage ADC, 691 692 vs. analog voltage, 692 ASCII codes and binary words, 702 binary numbers, 692, 693, 698 700 bit rate, 695 696 clock frequency, 696 697 hexadecimal numbers, 700 702 parallel vs. serial representation, 693, 694 timing diagram, 697, 698 tri state, 704 705 Digital to analog converter (DAC), 207, 247 Diode, 20 bridge rectiﬁer, 820, 821 clamper circuit, 827 828, 847 electronic (see Electronic diode) voltage doubler, 847 multiplier, 847 quadrupler circuit, 829 Discrete circuit transistor ampliﬁers base emitter loop, 874 four resistor bias circuit, 876 integrated circuit transistor ampliﬁers, 876 Thévenin equivalent, 876, 877 voltage supply, 876 Discrete Fourier transform (DFT) applications, 464 deﬁnition, 462 463 FFT, 461 fundamental frequency, 461 IFFT, 461 implementation, 479 480 sampling interval, 461 sampling points, 461 spectrum, 463 Dotted terminals, 582, 587, 600, 612, 635 Double clipper, 831 DRAM See Dynamic random access memory (DRAM) DSP See Digital signal processing (DSP) Dual in line (DIP N) package, 191 Dual polarity power supply, 101 Dynamic circuit elements capacitance (see Capacitance) instantaneous energy and power, 283 284 inductance (see Inductance) Dynamic random access memory (DRAM), 317 Dynamic resistance, 49
Index
E Earth ground, 71, 72 EAS See Electronic article surveillance (EAS) Electric circuit, 39, 53, 57, 69, 70, 71, 84 Electric current density, 11 Electric ﬁeld intensity, 3 Electric network description, 92 topology, 92 95 Electric permittivity, 9 Electric transformer AC power transfer, 579 Ampere’s law, 583 584 application, 634 635 autotransformer, 598 599 center tapped, 600 602, 636 current, 602 603, 636 ﬁxed load voltage, 595 596 ﬁxed source voltage, 596 597 function, 579, 580 high frequency transformer model, 609 610 magnetic circuit, 579 mechanical analogies, 589 multiwinding, 599 600 phasor form, 590, 591, 633 referred load impedance, primary side, 592 593, 633 referred source network, secondary side, 590 592, 633 transformer currents, 583 transformer efﬁciency, 610, 637 usage, 579 voltage regulation, 609 610, 637 Electromagnetic material processing electromagnetic forming, 316 317 self induced Lorentz force, 317 Electronic article surveillance (EAS), 506 Electronic diode anode and cathode, 797 automotive battery charging system application, 845 delta/wye connection, 821 diode circuit transformation, 822 three phase diode rectiﬁer, 822 823 vehicle, 821 breakdown region, 797 constant voltage drop diode model, 815, 830 diode clipper circuits, 830 diode limiter circuits, 830 double diode clipper circuit, 831 envelope/peak detector circuit amplitude modulated signal, 824 demodulation, 823 and operation, 824 826 square law region, 826 ESD protection, 831 forward bias region, 797, 809 forward bias voltage regulator, 814 iterative method, 811, 842 843 LED, 802
978
load voltage and diode current, 814 load line analysis, 810, 842 mechanical analogy, 798 negative diode clipper circuit, 831 1N4148 Si switching diode, 797, 798 photodiode, 802 PIN diode, 802 positive diode clipper circuit, 830 power electronics, 814 rectiﬁers and regulators, 814 reverse bias region, 797 room temperature, 798 800 Schottky diode, 801 802 Shockley’s ideal diode equation, 798 799 shunt voltage regulator, 814 switching diode, 797 symbol, 797 terminal voltage, 797 thermal voltage, 799, 800 varactor diode, 937, 977 voltage reference circuit, 814 voltage regulator, 814 wave shaping circuits clamper and multiplier circuit operation, 827 diode clamper circuit/DC restorer, 827 828 diode voltage doubler and multiplier, 828 830 positive, negative and double clipper, 830 832 power electronics, 827 transfer characteristics, 832 833 Zener breakdown voltage, 797 Zener diode, 801 Electronic ignition system, 324, 328 Electronic switch, 50 51, 643, 645 Electrostatic discharge (ESD) ESDS device, 263 ICs, 263 prediction of, 263 Electrostatics, 3 10 conductor, 7 8 Coulomb’s Law, 9 10 electric charge, 3 electric ﬁeld deﬁnition, 3 laboratory power source, 4 electric voltage and electric potential, 4 5 vs. ground, 5 7 Element’s polarity, 81 Encapsulation, 767 Energy spectral density, 479 Energy stored in capacitance, 261 Energy stored in inductance, 282, 283 Envelope detector circuit, 823 Equipotential lines, 6 Equipotential surface, 6, 8 Equivalent circuit element, 103 104 Equivalent electric circuits, 99 Equivalent electric networks, 99
Index
Equivalent resistance, 102 104, 120 Error signal, 230 232 Essential mesh, 92 Exciting current, 581 Exponential diode model, 809
F Faraday’s law current and inductance, 581 dot convention, 582 ideal open circuited transformer, 630 631 induction, 59, 60, 276 primary winding, 580 single lossless inductor, 580 transformer analysis, 579 transformer voltages, 581 582 Fast Fourier transform (FFT) digital signal processing, 464 input sinusoidal signal, 464 inverse Fourier transform, 465 monotonic frequency data, 465 numerical differentiation, 465 466, 480 Feedback factor, 230 Feedback gain, 230 Feedback loop, 192, 199 Fermi potential, 927 Filter circuits Bode plot, 441 445, 474 476 decibel (dB), 441 444, 474 476 ﬁrst order ﬁlters, 435 high pass, 440 441 low pass, 440 phase transfer function, 444 445, 475 476 roll off, 441 444, 474 476 Filter termination, 439 First order high pass ﬁlter, 441, 448, 466, 480 First order low pass ﬁlter, 443, 448, 449, 455, 480 First summing point constraint, 197 198 Fixed resistors, 41 42 Flash ADC circuit, 724, 725 operation, 725 Flash memory, 756 Flat band voltage, 928 Fluid analogies See Hydraulic analogiesFor loop, 774 776 Forbidden state, 655, 670, 672 Forced beta, 860 Forced current gain, 861 Forward current, 71 Fourier transform bandlimited, 459 converting computational electromagnetic solution, 467 472 deﬁnition, 478 479 DFT, 461 462 direct, 458 Fourier spectrum, 458 input pulse signal, 466 467, 480
979
inverse, 458 mathematical properties, 460 461, 479 rectangular pulse, 459 reversal property, 458 sampling theorem, 463 464 voltage/current signal, 458 Four resistor bias conﬁguration, 894 Four terminal NMOS transistor BJT, 922 integrated circuits, 922 Frequency band bandwidth, 451 bode plot, 447 passband, 443 Full H bridge DC motor, 655 656, 681 682 transistor switch, 655 Full scale output voltage range, 711 Full wave rectiﬁer, 819 820 Functions, Arduino language, 765 767 Function header, 766 Fundamental frequency, 461
G Gain tolerance, ampliﬁer, 210 211 Gauge factor (GF), 44 Gauss’ theorem, 7 Generic solar cell application, 170 171, 179, 188 maximum power extraction, 185 187 nonlinear circuits, 167 168, 187 188 solar panels, 179 Ground reference, 5 6 Ground side switch, 644 646, 651, 660, 679
H Half H bridge DC motor, 653 654 motor functions, 654 switching states, NMOS transistors, 682 Half power bandwidth, 491, 492, 497, 503, 520 Half power frequency See Filter circuits Half wave rectiﬁer AC power supply, sinusoidal voltage, 817 current limiting resistor, 818 diode rectiﬁer, 818 dual supply, 845 electronic DC power supplies, 817 ideal diode rectiﬁer, 817, 818 plot source and load voltage, 818 819 rectiﬁed voltage vs. source voltage, 819, 820 Hard limiter, 832 Hardware description language (HDL), 672 Harmonic voltage and current AC power supplies, 391 AC voltage, 391 angular frequency, 391 392
Index
Harmonic voltage and current (cont.) beneﬁt, 391 complex exponent, 404 deﬁnition, phasor, 396 398 electric circuit, 410 electrode, 410 frequency, phase and amplitude, 391 impedance, 405 407 leading and lagging AC signal, 395 amplitude and phase, 395 frequency, 394 phase shifted AC voltages, 394 power electronics, 393 zero phase cosine, 393 magnitude impedance, 408 410 human body, 410 Maxwell’s theory, 392 measurements, 396 operations, phasors and phasor diagram, 401 404 and parameters, 391, 392 phasors to real signals, 399 polar and rectangular forms, 399 401 real signals to phasors, 398 399 resistance, 405 signals, 391 steady state AC circuits, 391 steady state AC voltage, 391 steady state alternating current, 391 Hexadecimal numbers, 700 702 High voltage side, 587 Howland current source (Howland current pump), 255 Hydraulic analogies AC circuits, 19 DC steady state, 18, 19 semiconductor circuit components, 20
I υ i characteristics circuit elements, 31 current source, 55 deﬁnition, 31 c plot, 53 ideal (Shockley) diode, 47 ideal diode, 48 ideal switch, 35 inverse slop, 49 Ohm’s law, 47 ohmic conductor, 47 passive circuit elements, 47 48 practical current source, 58 practical voltage source, 55 resistances 34 35 solid line, 58 static resistance, 48 transfer characteristic, 66 67
980
two terminal switch, 50 voltage source, 52 53 Ideal ammeter, 69 Ideal ampliﬁer model, 197 198, 217, 252 Ideal diode model, 31, 47, 79 assumed states, 804, 805, 838 840 forward bias region, 804 OFF diode, 804 806 ON diode, 804 806 OR logic gate, 806 807 output voltage, 807 v i characteristics, 804 Ideal loaded transformer circuit symbol, 585 contour, 584 linear interface, 585 power conservation, 585 power rating, 587 588 vs. real transformer, 586 587, 631 632 stored energy, 586 terminologies, 587 Ideal MOS capacitor model, 925 Ideal open circuited transformer See Faraday’s Law Ideal transformer equations, 579, 602 Ideal voltmeter and ammeter, 69 70, 85 IDFT See Inverse discrete Fourier transform (IDFT) Impedance matching, 545 Increment operator, 775 Incremental resistance, 49 Independent ideal current source active reference conﬁguration, 55 deﬁnition, 55 symbols, 56 57 υ i characteristic, 56 Independent ideal voltage source, 52 55 Inductance dynamic behavior, 281 283 self inductance, 272 273 series and parallel, 275 276 solenoid with and without magnetic core, 273 275 Inductor choke, 289 fringing effect, 274 ideal, 275 1 mH, 276 277 physical, 272 Input and output (I/O) devices, 750 Input bias current, 219 220 Instantaneous generator voltage, 60 Instruction set, 748 Instrumentation ampliﬁer, 222 229 Instrumentation transformers, 579, 603 Integrated circuit (IC), 191, 229 Integrated development environment (IDE) edit menu, 759 features, 758 ﬁle menu, 759 installation, 758
Index
license agreement, 758 sketch menu, 758 tools menu, 759 USB connection, 757 visual representation, 757 windows, 758 Interface analogRead function, 769 Arduino programs, 770 IO pins, 769 pinMode function, 769 PWM signal, 770 Internal compensation, 451 Internal source resistance, 54, 57 Interrupt service routine (ISR), 781 Interrupts debounced, 781 ISR, 781 microprocessor, 781 polling, 780 queue, 781 trigger state, 781 volatile, 782 Inverse discrete Fourier transform (IDFT), 461 Inversion layer of MOS capacitor, 928, 935, 936 Inverting ampliﬁer, 203 204 Inverting input, 191, 201, 207 Inverting Schmitt trigger, 332, 376 Isolation transformers, 579 ISR See Interrupt service routine (ISR) Iterative method for nonlinear circuits deﬁnition of, 169 explicit iterative scheme, 169 implicit iterative scheme, 169
J Junction FET (JFET), 931
K Karnaugh maps, 673 Kirchhoff’s current law (KCL), 93 95 Kirchhoff’s voltage law (KVL), 14, 95 98
L Lagging power factor, 536 Large signal BJT circuit model, 866 Large signal circuit model base bias transistor circuit, 866 equivalent circuit model, 866 ﬁrst order circuit, 866 linear current controlled current source, 866 nonlinear voltage controlled current source, 866 Large signal MOSFET model in saturation digital switching circuits, 939
981
transconductance curve, 940 Latch inverters, 673 SRAM, 688 stable states, 673 Laws of Boolean algebra, 668 Leading power factor, 537 Least signiﬁcant bit (LSB), 707 LED See Light emitting diode (LED) Lenz’s law, 630 Library, 767 Light emitting diode (LED), 802 Line currents, 553 Line integral, 5 Linear circuit (deﬁnition of homogeneity additivity superposition), 116 Linear feedback system, 230 Linear load, 47, 607 Linear oscillators, 330 Linear passive circuit element components, 31 ﬁxed resistors, 41 42 Ohm’s law, 33 Ohmic conductors, 36 39 passive reference conﬁguration, 32 33 power delivered, 35 36 resistive sensors, 42 45 symbols and terminals, 31 32 variable resistors (potentiometers), 42 voltage across, 32 υ i characteristic (see υ i characteristic) Lines of force, 4 Line to line voltages, 552 Line to neutral voltages, 547, 552, 557, 575 Load cell, 229 Load impedances per phase, 549, 573 Load line (deﬁnition, method of), 167 168 Load reﬂections, 592 Load line analysis actuator device, 951, 952 ampliﬁer (see MOSFET ampliﬁer) current mirror, 950 diode connected, 950 lumped process parameter, 950 Ohm’s law, 948 quadratic equation, 949 Load line method, 810 811, 842 Logic inverter, 660 661, 683 Loops electric network, 92 93 for loop, 774 776 while loop, 776, 777 Lorentz force, 59, 60, 315, 317 Low frequency asymptotes, 444, 449 Low voltage side, 588, 632 LSB See Least signiﬁcant bit (LSB) Lumped process parameter, 934
Index
M Magnetic ﬁeld, 271, 272, 299 Magnetic ﬂux density, 272, 273, 299, 315 Magnetic induction, 272, 299 Magnetic permeability of vacuum, 274 Magnetic radiators application, 638 coil array, 622 tuned radiators, 623 624 two identical, 622 623 uncoupled inductors, 623 wireless inductive power transfer, 622, 638 Magnetizing inductance, 581 Magnetizing reactance, 606, 637 Magnetostatics Ampere’s law, 14 16 current carrying conductor, 12 13 current ﬂow model vs. electrostatics, 11 12 electric circuit, 13 14 electric current, 11 electric power transfer, 16 17 Matched load, 161 Matched switching transistors, 646 Matching circuit arbitrary complex impedances, 595 real valued impedances, 593 594 transformer, 634 Material conductivity, 37, 44 Material resistivity, 37 Maximum available source current, 54 Maximum available source power, 54 Maximum operating frequency, 800 Maximum power efﬁciency, 162 163, 184 185 Maximum power theorem (principle of maximum power transfer), 161, 171 Maximum power transfer for AC circuits, principle of, 544 545, 572 Maximum sensitivity of voltage divider circuit, 108 109 Mechanical analogy, 332 Memory binary representation, 754 bytes, 749 data storage methods, 754 755 ﬂash, 756 organization, 753 755 program execution, 749 storage, 881 types, 755 756 Mesh analysis (mesh current analysis), 142 linear circuits, 146 147 Meshes of electric network, 91 93 Metal oxide semiconductor (MOS) transistors complementary MOS circuits, 645 ground side switch, 645 power side switch, 645 Metal semiconductor FET (MESFET), 931 1 mH inductor, 276 277 mho, 33
982
Microcontrollers architecture, 747, 748 buses, 850 851 CPU, 748 749 feature, 747 hierarchy, 748 I/O devices, 747, 850 memory, 747, 849 timers, 850 USART, 851 852 Miller integrator ampliﬁer circuits, dynamic elements, 290 291 compensated, 291 292 Mobility of charge carriers, 37 Modulation depth, 824 MOS ﬁeld effect transistor (MOSFET) ampliﬁer linear small signal, 956 957 Q point, 955 956 transconductance, 957 958 voltage transfer characteristic, 953 954 channel, 924 CMOS process, 930 common source ampliﬁer (see Common source MOSFET ampliﬁer) DC circuit, 947 JFET, 931 linear/ohmic subregion, 932 load line method (see Load line analysis) MESFET, 931 n and p channel, 921 NMOS capacitor, 924 925 and PMOS transistor, 932 on state resistance, 944 oxide layer pn junction depletion layer, 926 SiO2, 926 parameter extraction, 943 944, PMOS transistor, 929 930, 939 process transconductance parameter, 934 resistor switch model, 944 saturation current, 936 saturation region, 935 937 semiconductor body Fermi potential, 927 inversion layer, 927, 928 mass action law, 927 simpliﬁer transistor circuit, 933 terminal voltages and currents drain source voltage, 923 gate drain voltage, 923 gate source voltage, 923 threshold voltage, , 921 933 transconductance parameter curve, 940 lumped process parameter, 934 transistors, 646
Index
turn on resistance, 934, 942 944, 968 v i dependencies, 937 938 work function difference, 928, 929 Motor control states (forward mode), 650, 652 Motor speed controller, 650 Motor switching forward mode, 650 free run to stop, 650 motor brake, 650 reverse mode, 650 Multiwinding transformer, 599 600, 636 Mutual inductance, 272 273
N NAND gate vs. input voltages, 662 symbol, 663 three inputs, 664 n channel metal oxide semiconductor ﬁeld effect (NMOS) transistor, 20 n channel MOSFET, 653 Near ﬁeld communication (NFC), 506 Near ﬁeld wireless link, 506, 510 511, 522 Negative clipper, 831 Negative equivalent (Thévenin) resistance, 182 construction and use, 158 dependent source, 158 voltage controlled voltage source, 158 Negative feedback ampliﬁer’s stability, 199 idea of, 199 summing point constraints, 199 201 Negative phase sequence, 524, 573 Negative temperature coefﬁcient (NTC), 43 Neper (Np), 348, 358 Neutral wire, 553 555, 573 576 NFC See Near ﬁeld communication (NFC) NMOS transistor channel inversion, 924 threshold voltage, 928 Nodal analysis, 142, 176 177 circuit current I, 176 linear circuits, 146 148 supply current I, 176 Nodes of electric network, 92 94 Noise signals, 530 Nonideal low frequency transformer, 617 Nonideal transformer model harmonic source and linear load, 607 impedance load, 607 input voltage, 609 load current, 607 load voltage, 607 power factor, 609 rms values, 606 Non inverting ampliﬁer, 201 203
983
Non inverting input, 192, 193 Non inverting Schmitt trigger, 332, 376 Nonlinear circuits, 167 168 deﬁnition of linearization dynamic/small signal resistance, 116 Nonlinear passive circuit elements, load, 46 Non ohmic circuit elements, 46 Non reference, 141 Nonvolatile memory, 755 NOR gate, 661 vs. input voltages, 660 three inputs, 664 NOT gate, 660 661, 663, 665, 666 n type/n channel MOS transistor (NMOS) control voltage, 646 ground side switch, 645 intrinsic threshold voltage, 648 load current, 647 metal electrodes, 647 threshold voltage, 647 Nyquist rate minimum acceptable sampling rate, 723 modulation theory, 723 Nyquist Shannon sampling theorem, 723 oversampling and undersampling, 723 sampling frequency, 723 Nyquist Shannon sampling theorem, 723
O Objects, 768, 769 Octave, 443, 474 Offset null terminals, 192, 218 Ohm’s law, 35, 36, 39, 47, 49, 67 resistance and conductance, 33 Ohmic conductor, 36 39 One port network, 99 Open circuit, 34 small signal voltage gain, 956, 958 source voltage, 54, 57, 60, 65, 67, 83 voltage gain, 192, 232, 889, 891, 895 Open loop AC gain, 453 454, 477 Open loop conﬁguration, 192, 199 Open loop voltage gain, 192 Operating point, 49, 50 Operational ampliﬁer application, 194 195 bandwidth, 455 456, 477 478 closed loop AC gain, 454 455, 477 deﬁnition, 191 frequency bandwidth, 451, 456 457, 478 open circuit/open loop voltage gain, 192 open loop AC gain, 453 454, 477 open loop ampliﬁer gain, 451, 477 open loop gain behavior, 451 452 power rails and voltage transfer characteristic, 193, 194 symbol and terminals, 191 192
Index
Operational ampliﬁer (cont.) unity gain bandwidth vs. gain bandwidth product, 452 453, 477 OR gate, 661 662, 669 Output resistance, 863, 873, 889, 891, 895, 896 Output terminal, 192, 196
P Parallel battery bank, 101 Parallel connection, 103 Parallel data transmission, 751 Parallel digital output voltage, 693 Parallel plate capacitor application, 267 269 capacitive touchscreens, 270 271 capacitor marking, 269 270 ceramic capacitors, 269 circuit symbol, 266 267, 275 276 dielectric permittivity of vacuum, 265 electrolytic capacitors, 269 fringing effect, 265 inductor marking, 277 mutual capacitance method, 271 273 opposite charge Q, 264 self capacitance method, 270 271 solenoid with and without magnetic core, 273 275 static capacitance, 266 total charge +Q, 264 touch screens, 265 Parallel resonant RLC circuit amplitude, 493 circuit voltage, 494 duality, 495, 519, 520 parallel RLC tank circuit, 493 partial time constants, 493 phasor representation, 493 Q factor, 494 resonance condition, 493 resonant frequency, 493 494 resonant phasor currents, 495 second order RLC ﬁlters, 504 505 Parallel vs. serial representation, 693, 694 Parseval’s theorem, 479 Passive linear circuit elements, 278 Passive reference conﬁguration, 32 33 p channel MOSFET, 929, 944 Peak detector circuit, 823 Peltier Seebeck effect, 44 Peripherals, 750 Per phase solution, 553 PFC capacitor, 539 541, 543 Phase impedances, 549, 557, 575 Phase voltages, 547 549, 572, 573 Photocell, 43 Photoresistor, 44, 45 Piecewise linear diode model, 801, 816, 817
984
PMOS transistor, 929 930 Polarity AC source, 67 active reference conﬁguration, 52, 56 independent, 52 voltage, 32, 34 Polling, 780 Positive feedback, 330, 335, 372 Positive phase sequence, 548, 549, 551, 556, 560 Postﬁlter, 710 Potential transformers, 603 Potentiometer, 42 Potentiometric position sensor, 44 45 Power angle, 531 532, 569 Power conservation law for electric networks, 99 Power factor correction application, 542 average load power, 541 capacitance value, 540 capacitor, 539 current amplitude, 541 impedance, 540 load impedance, 542 modiﬁed load, 539 power angle, 539 reactive load power, 540 RLC circuit, 539 shunt capacitor, 539, 540 Power rails, 193 Power related networking theorems, 98 99 Power side switch, 644 646, 679, 884 Power switching circuits DC motor, 651 652 full H bridge, 655 657 half H bridge, 653 654 one quadrant switch, 652 653 PWM motor controller, 657 659 resistive load, 651 switching quadrants, 650 651 Power terminals, 230 Power triangle, 570 571 inductive and capacitive load, 535, 536 whip antenna, 537 Poynting vector, 17 Practical current source deﬁnition, 58 open circuit voltage, 57 short circuit current, 57 υ i characteristic, 58 Practical voltage source, 54 55 Primitive data types, 762 Process transconductance parameter, 934 Product of sums approach, 672 673 Proximity sensors, 511 516, 520, 521 P type/p channel MOS transistor (PMOS), 646 control voltage, 646 intrinsic threshold voltage, 648
Index
output voltages, 648 649 power side, 653 power side switch, 679 threshold voltage, 646 Pull down switch See Ground side switch Pull up switch See Power side switch Pulse width modulation (PWM), 716, 768, 782, 783 average supply voltage, 657 commercial controllers, 658 duty cycle, 657 motor controller, 657 659, 682 realization, 658, 659 supply voltage, motor, 657 voltage form, 657 659 PWM See Pulse width modulation (PWM)
Q Quality factor constant, 489 deﬁnition, resonant frequency, 488 inductor and capacitor, 489 maximum amplitude, 489 mechanical resonator, 489 series resonant LC circuit, 490 series resonant RLC circuit, 491 Quality factor of series resonant RLC circuit, 488 Quantization error, 727 Quiescent point (Q point), 49, 810, 812 NMOS ampliﬁer, 955 parameters, 891
R Radiation resistance, 47 Radio frequency (RF) inductor choke, 289 power, 823 Rail to rail ampliﬁers, 194 Reactive power, 535 Real transformer, 604 equivalent circuit values, 606 model parameters and extraction, 604 606 nonideal low frequency core loss resistance, 604 leakage resistance, 604 leakage inductance, 604 ohmic resistance, 604 Steinmetz model, 604 Receiver circuit, 522 Faraday’s law, 508 open circuit voltage, 509, 510 operating frequency, 510 quality factor, nonideal inductor, 510 voltage multiplier, 510 Reconstruction ﬁlter, 710 Rectangle rule, 461 Reduction of resistive networks, 104 105
985
Relative magnetic permeability, 275 Replacing a node by a loop, 120 Resistive sensors, 42 45 Resistor, 40 ECE laboratory kit, 77 ﬁxed, 41 42 photoresistor, 44 thermal, 42 variable, 42 Resolution voltage, 708 710 Return current and ground, 71 Reversal property, 458, 479 Riemann sum approximation, 461 RL ﬁlter circuits, 448 450, 476 rms voltage AC signals, 567 application, 567 DC voltage, 530 root mean square value, 530 sawtooth/triangular wave, 530 single frequency voltage signals, 530 Rotating magnetic ﬁeld, 550 Rotor, 550 R/2R ladder DAC, 714 715 R 2R ladder network, 157
T T and Π networks, 121 122 Telephone hybrid circuit, 600 Tellegen’s theorem, 98 Thermistor constant, 43 Thermistor equation, 43 Thermocouple, 43 Thévenin equivalent, 153, 155, 158, 162, 167, 181 Thévenin’s and Norton’s theorems, 153 182 Three terminal MOSFET, 922, 923, 932 Three terminal networks, 119 Threshold voltages, 332 Timing diagram, 697, 698 T network, 122, 132 T pad, 122 Transconductance, 66, 67 ampliﬁer, 65, 233 curve, 940 Transfer characteristic, 66 67 Transfer resistor, 854, 862 Transient circuit, 284 Transient RC circuit capacitor, 310 digital memory cell, 317 318 electromagnetic railgun, 314 316, 369 energy accumulating capacitor circuit forced response, 319 resistor voltage, 319 source voltage, 320 Thévenin resistance, 318 energy consideration, 313
Index
Transient RC circuit (cont.) energy release capacitor circuit electromagnetic material processing, 312 load resistor, 310 time/relaxation constant, 311 ﬂuid mechanics analogy, 314 static/dynamic circuit, 310 time constant, 312 313 Transient RL circuit energy accumulating capacitor circuit charging, 325 inductor/resistor voltages, 326 inhomogeneous, 325 mH inductor, 326 supply current, 327 voltages and currents, 321 zero inductor current, 326 energy release inductor circuit charged, 321 current source, 321 dynamic circuit, 321 inductor, 321, 322 magnetic ﬁeld energy, 321 supply current, 322 time/relaxation constant, 322 inductor current continuous function, 324 ﬁnite magnetic ﬁeld energy, 324 ﬂuid mechanics analogy, 324, 325 independent function, 324 inductor inertia, 324 laboratory ignition circuit, 328 329 static/dynamic circuit, 321 voltage supply, 327 Transistor circuit analysis collector and base resistances, 871 current controlled current source model, 872, 873 Transistor test circuits, 856, 866, 871 Transistor threshold voltage digital circuits, 647 switching behavior, 646 Transmission line, 14 Transmitter circuit, 522 magnetic ﬁeld, 507 operation frequency, 507 phasor method, 507 quality factor, 507 series capacitor, 507 Transmitting antenna, 163 Transresistance ampliﬁer, 65, 233 Triode region, MOSFET electron surface mobility, 934 lumped process parameter, 934 process transconductance parameter, 934 turn on resistance, 934 Tri state buffer, 704 705 Tri state digital voltage buffer, 704 components, 705
986
True power, 544 Truth table AND gate, 666 logic circuit, 661, 670 671 NAND gate, 665 OR gate, 666 Tuning diode, 800 Two coupled inductors circuit analysis, 607 circuit symbol, 613 Two port networks, 121 122 Two terminal network (deﬁnition of input port output port), 119 Two terminal switch, 50 Typecasting, 764
U Undamped resonant frequency, 348, 352, 381, 384, 385 Underdamped circuit, 357, 360 Unidirectional switch, 50 Unity common mode gain stage, 227 Universal synchronous/asynchronous receiver/ transmitter (USART) CPU and external serial devices, 752 parallel data transmission, 751 serial data transmission, 751 USART See Universal synchronous/asynchronous receiver/transmitter (USART)
V Variable resistor, 42 Velocity saturation region, 936 Very high frequencies, 285 Virtual ground circuit, 220 221 Virtual ground of dual polarity power supply, 101 Virtual ground (integrated) circuit, 221 Volatile memory, 755 Voltage ampliﬁer, 65 Voltage controlled current source, 65 66 Voltage controlled voltage source, 65 Voltage difference, 5, 32, 52, 61 Voltage divider circuit, 105 107 Voltage division rule, 107 Voltage drop, 5, 32, 37, 56 absolute voltage, 72 74 Voltage follower (buffer) ampliﬁer, 204 205 Voltage polarity, 31, 32, 55, 582, 613 Voltage transfer characteristic, 193, 953, 954 Volt amperes (VA), 535 Volt amperes reactive (VAR), 535
W Wattmeter, 538 Wattmeter current coil, 538 Wheatstone bridge (deﬁnition of difference signal difference voltage balanced), 113 115
Index
While loop, 776, 777 Wireless inductive power transfer coaxial coupled inductors, 619 coil conﬁguration, 618 data transfer, 618 ﬁnite magnetic core, 621 magnetic near ﬁeld calculations, 620 MATLAB script, 621 mutual inductance, 620 RFID, 619 transmission frequency, 620 Work function difference, 928 wye connected load, 549, 560 Wye connected source, 549, 560, 561 Wye wye distribution system, 552 Wye/Y conﬁguration, 549
987
X XOR gate, 670
Y Y and Δ Networks, 119, 121 Y to Δ transformation, 121
Z Zener diode piecewise linear, 816 817 shunt voltage regulator, 815 817 voltage regulator, 843, 844 Zero level detector, 195, 240