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A List of Tables Table No Title 1.1 1.2 1.3 1.4 4.1 4.2 6.1 6.2 7.1 8.1 8.2 8.3 Page No The International System of Units (SI) Derived Units in SI Standardized Prefixes to Signify Powers of 10 Interpretation of Reference Directions in Fig 1.5 Terms for Describing Circuits PSpice Sensitivity Analysis Results Terminal Equations for Ideal Inductors and Capacitors Equations for Series- and Parallel-Connected Inductors and Capacitors Value of e - t>t for t Equal to Integral Multiples of t Natural Response Parameters of the Parallel RLC Circuit The Response of a Second-Order Circuit is Overdamped, Underdamped, or Critically Damped In Determining the Natural Response of a Second-Order Circuit, We First Determine Whether it is Over-, Under-, or Critically Damped, and Then We Solve the Appropriate Equations In Determining the Step Response of a Second-Order Circuit, We Apply the Appropriate Equations Depending on the Damping Impedance and Reactance Values Admittance and Susceptance Values Impedance and Related Values Annual Energy Requirements of Electric Household Appliances Three Power Quantities and Their Units An Abbreviated List of Laplace Transform Pairs An Abbreviated List of Operational Transforms Four Useful Transform Pairs Summary of the s-Domain Equivalent Circuits Numerical Values of vo(t) Input and Output Voltage Magnitudes for Several Frequencies Normalized (so that vc = rad>s) Butterworth Polynomials up to the Eighth Order Fourier Transforms of Elementary Functions Operational Transforms Parameter Conversion Table Terminated Two-Port Equations 8.4 9.1 9.2 9.3 10.1 10.2 12.1 12.2 12.3 13.1 13.2 14.1 15.1 17.1 17.2 18.1 18.2 9 13 91 128 203 203 217 269 295 295 296 318 322 345 365 368 435 440 451 468 492 527 577 653 658 682 688 Greek Alphabet A a Alpha I i Iota P r Rho B b Beta K k Kappa © s Sigma g Gamma ả l Lambda T t Tau  d Delta M m Mu ⌼ y Upsilon E P Epsilon N n Nu £ f Phi Z z Zeta ⌶ j Xi X x Chi H h Eta O o Omicron ° c Psi ™ u Theta ß p Pi Ỉ v Omega ELECTRIC CIRCUITS TENTH EDITION This page intentionally left blank ELECTRIC CIRCUITS TENTH EDITION James W Nilsson Professor Emeritus Iowa State University Susan A Riedel Marquette University Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Vice President and Editorial Director: Marcia J Horton Executive Editor: Andrew Gilfillan Editorial Assistant: Sandra Rodriguez Marketing Manager: Tim Galligan Senior Managing Editor: Scott Disanno Production Editor: Rose Kernan Cover Design: Black Horse Designs Cover Art: Inverter 04 Oil painting by Ben Leone “TechScape” Collection www.benleone.com Manager, Cover Visual Research & Permissions: Karen Sanatar Photo Researcher: Marta Samsel Composition: Integra Publishing Services Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text Copyright © 2015, 2008, 2005 Pearson Education, Inc., publishing as Prentice Hall, One Lake Street, Upper Saddle River, New Jersey, 07458 All rights reserved Manufactured in the United States of America This publication is protected by Copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey, 07458 Library of Congress Cataloging-in-Publication Data Nilsson, James William Electric circuits / James W Nilsson, Professor Emeritus, Iowa State University, Susan A Riedel, Marquette University.—Tenth edition pages cm ISBN-13: 978-0-13-376003-3 ISBN-10: 0-13-376003-0 Electric circuits I Riedel, Susan A II Title TK545.N54 2015 621.319'2—dc23 2013037725 10 ISBN-13: 978-0-13-376003-3 ISBN-10: 0-13-376003-0 To Anna This page intentionally left blank Brief Contents Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G Appendix H List of Examples xiii Preface xvii Circuit Variables Circuit Elements 24 Simple Resistive Circuits 56 Techniques of Circuit Analysis 88 The Operational Amplifier 144 Inductance, Capacitance, and Mutual Inductance 174 Response of First-Order RL and RC Circuits 212 Natural and Step Responses of RLC Circuits 264 Sinusoidal Steady-State Analysis 304 Sinusoidal Steady-State Power Calculations 358 Balanced Three-Phase Circuits 396 Introduction to the Laplace Transform 426 The Laplace Transform in Circuit Analysis 464 Introduction to Frequency Selective Circuits 520 Active Filter Circuits 556 Fourier Series 602 The Fourier Transform 642 Two-Port Circuits 676 The Solution of Linear Simultaneous Equations 703 Complex Numbers 723 More on Magnetically Coupled Coils and Ideal Transformers 729 The Decibel 737 Bode Diagrams 739 An Abbreviated Table of Trigonometric Identities 757 An Abbreviated Table of Integrals 759 Common Standard Component Values 761 Answers to Selected Problems 763 Index 775 vii This page intentionally left blank 782 Index Inductors (Continued) switching operation, 497–499 terminal equations for, 176–181, 203, 466–467 voltage (v) in, 176–177, 202 voltage to current (v-i) relationships, 176–177, 316–317 Infinite frequency, 528 Initial-value theorem, 453–455, 458 Instantaneous power, 360–361, 384 Instantaneous real power, 362 Integer power of complex numbers, 727 Integrals, 487–493, 504, 646–648, 670, 759–760 asterisk (*) notation for, 489 convergence of Fourier transforms, 646–648, 670 convolution, 487–493, 504 equations of, 761–762 folding operation, 489–493 time domain functions, 646–648, 670 transfer function H(s) and, 487–493, 504 Integrating amplifiers, 241–244, 246, 289–293, 295 cascading connections, 289–293, 295 feedback resistors and, 291–293 first-order circuits with, 241–244, 246 response analysis of, 241–244, 289–293 second-order circuits with, 289–293, 295 sequential switching and, 243 Integration, operational transforms for, 437–438, 656 Integrodifferential equations, 440–442 Interconnection of sources, 28–29 Intermittence, 680–681 International System of Units (SI), 8–10 Inverse Fourier transform, 645 Inverse Laplace transforms, 442–452, 457 distinct complex roots of D(s), 445–447 distinct real roots of D(s), 443–444 improper rational functions, 442, 451–452 partial fraction expansion, 442–452, 457 proper rational functions, 442–443, 457 rational function F(s), 442, 457 repeated complex roots of D(s), 449–451 repeated real roots of D(s), 447–449 Inverse of a matrix, 714 Inverse phasor transform, 312–314, 345 Inverting-amplifier circuit, 150–152, 160, 164 K Kirchhoff’s laws, 37–41, 48, 58–59, 61, 319–320, 402–403, 406, 469 balanced three-phase circuits using, 402–403, 406 closed loop (path) for application of, 38, 48 current (KCL), 37–41, 48, 319–320, 406 frequency domain, 319–320 parallel-connected resistors, 59, 61 nodes, 37–38, 48 s-domain applications of, 469 series-connected resistors, 58 voltage (KVL), 38–41, 48, 319, 402–403 L Lagging power factor, 363 Laplace transform, 426–519, 648–651 applications of, 440–442, 470–482 circuit analysis using, 464–519 complete circuit response using, 427, 456–457 concept of, 428–429 convolution integral, 487–493, 504 defined, 428, 457 final-value theorem, 453–455, 458 Fourier transforms found using, 648–651 frequency domain and, 439 functional transforms, 429, 434–435, 457 improper rational functions, 442, 451–452 impulse function Kd(t), 431–433, 457, 496–502, 504 initial-value theorem, 453–455, 458 inverse, 442–452, 457 lumped-parameter circuit applications, 440–442, 457, 484 mutual inductance, analysis of a circuit with, 478–479 natural response using, 470–471 one-sided (unilateral), 428–429 operational transforms, 429, 435–440, 457 pairs, 434–435, 451 partial fraction expansion, 442–452, 457, 484–487, 504 poles (denominator polynomial), 452–453, 458, 484 proper rational functions, 442–443, 457 rational function F(s), 442–453, 457–458 s-domain, 441–455, 457–458, 466–487, 504 sinusoidal sources and, 427, 456–457 steady-state sinusoidal response, 493–495, 504 step function Ku(t), 429–430, 457 step response using, 471–476 superposition, use of in s-domain, 480–482 surge suppressors, 465, 503 Thévenin equivalent, use of in s-domain, 477–478 time domain translation in, 439–442 time-invariant circuits and, 486–487, 504 transfer function H(s), 482–495, 504 transient effects on circuits, 427, 456–457 transient response using, 473–474 unilateral (one-sided), 428–429 unit impulse function d(t), 431, 457 unit step function Ku(t), 429, 457 zeros (numerator polynomial), 452–453, 458, 484 Leading power factor, 363 Line current, 403, 406–407 Line spectrum, 628 Line voltage, 402–404 Linear transformer, 333–336, 345 Load impedance (ZL), 376–377 Loads, 62, 408–413 balanced three-phase circuits and, 408–413 defined, 62 delta ( ¢ ) connected, 409–410 Index power calculations for, 408–413 unspecified, 412 wye (Y) connected, 408–409 Loop, defined, 90–91 Low frequencies, 524 Low-pass filters, 522–530, 548, 558–560, 572–582, 592–593 active filter circuits, 558–560, 576–580, 592–593 Butterworth, 576–582, 594–595 cascading identical, 572–575, 592 cutoff frequency (vc), 526–527 defined, 522–530, 548 frequency regions, 528 frequency response plots, 522–525, 558–561 operational amplifier (op amp), 558–560, 576–580 passive filter circuits, 522–530, 548 qualitative analysis, 524–525 quantitative analysis, 526–527 relationship between frequency and time domains, 529 series RC circuits, 528–529 series RL circuits, 524–527 transfer function H(s) for, 529, 548 Lumped-parameter circuits, 440–442, 457, 484 Lumped-parameter system, 6–7 M Magnetically coupled coils, 190, 194–196, 203, 235–236, 336–338, 729–736 equivalent circuits for, 729–736 ideal transformers and, 336–338, 729–736 mutual inductance (M) of, 190, 194–196, 203 p-equivalent circuit, 730–732 step response of, 235–236 T-equivalent circuit, 729–730 Magnitude plot, 522 Magnitude scale factor (km), 562, 592 Matrices, 707–708, 712–717 adjoint of, 713–714 column, 708 identity, 712–713 inverse of, 714 partitioned, 715–717 row, 708 simultaneous equation solutions from, 707–708, 712–717 square, 708 transpose of, 712 Matrix, defined, 707 Matrix algebra for simultaneous equation solutions, 708–712, 718–722 Maximum power transfer, 120–122, 129, 376–382, 384 average power (Pmax) absorbed, 378 circuit analysis and, 120–122, 129 ideal transformer, 381–382 impedance (Z) restricted, 378–380 load impedance (ZL) for, 376–377 sinusoidal steady-state analysis and, 376–382, 384 Measurement, 8–10, 66–70, 76, 413–415, 417, 737–738 ammeter, 66–67, 76 analog meters, 67, 76 average power (P), 413–415, 417 current, 66–69 d’Arsonval meter movement, 67–68 decibel (dB) used for, 737–738 digital meters, 67, 76 International System of Units (SI), 8–10 power transmission, 737–738 resistance, 69–70, 76 two-wattmeter method, 413–415, 417 voltage, 66–69, 76 voltmeter, 66–67, 76 wattmeter, 413–415, 417 Wheatstone bridge, 69–70, 76 Memory and the weighting function, 492–493 Mesh, defined, 90–91 Mesh circuit, step response of, 474–477 Mesh current, defined, 100 Mesh-current method, 90, 99–109, 129, 190–192, 331–332 amplifier circuit analysis using, 105–106 circuit analysis using, 90, 99–109, 129 dependent sources and, 102–103 equations, 99–102, 190, 192 essential branches and, 103–104 frequency-domain circuits, 331–332 mutual inductance (M) and, 190–192 node-voltage method, comparison of, 106–109 sinusoidal steady-state analysis using, 331–332 special cases of, 103–106 supermesh, 104 Meters, see Measurement Models, 10–11, 17–18, 34–36 See also Equivalent circuits circuit, 17–18, 34–36 conception of, 10–11 construction of, 34–36 flashlight, 34–35 mathematical (circuit), 11 physical prototypes, 11 power balance, 17–18 Modulation, operational transforms for, 657 Mutual inductance (M), 176, 189–200, 203, 478–479 circuit parameters of, 189–190, 194–196, 203 coefficient of coupling and, 197, 203 dot convention for polarity, 190–193, 195–196, 203 energy (w) storage and, 197–200, 203 Laplace transform analysis and, 478–479 magnetically coupled coils and, 190, 194–196, 203 mesh-current equations for, 190–192 s-domain using, 478–479 self-inductance and, 193–194, 196–197, 203 783 784 Index N O Narrowband filters, 584–589, 593 bandpass filters, 584–586 bandreject, 586–589 quality factor (Q) and, 584–589 twin-T notch filter, 586–589 Natural response, 212, 214–224, 231–236, 246, 266–279, 285–289, 295, 470–471 critically damped voltage, 269, 277–278, 286, 295 current (i) determined for, 214–216, 286 defined, 212, 246 first-order circuits, 212, 214–224, 231–236, 246 forms of in RLC circuits, 270–279 general solution for, 231–236, 246, 266–270 Laplace transform analysis and, 470–471 method of calculating, 232 overdamped voltage, 269, 271–273, 286, 295 parallel RLC circuit, 266–279, 295 resistor-capacitor (RC) circuits, 220–224, 231–236, 246, 470–471 resistor-inductor (RL) circuits, 214–220, 231–236, 246 resistor-inductor-capacitor (RLC) circuits, 266–279, 285–289, 295 series RLC circuits, 285–289, 295 time constant (t), 216–217, 221, 246 underdamped voltage, 269, 274–277, 286–288, 295 voltage (v) determined for, 221 Negative (acb) phase voltage sequence, 398 Neper frequency (a) for, 268–269, 286 Neutral terminal, 399 Node-voltage equation (VN), 400–401 Node-voltage method, 90, 93–99, 106–109, 129, 330–331 amplifier circuit analysis using, 98–99 circuit analysis using, 90, 93–99, 106–109, 129 dependent sources and, 95–96 equations, 93–94, 129 essential nodes and, 96–97 frequency-domain circuits, 330–331 mesh-current method, comparison of, 106–109 sinusoidal steady-state analysis using, 330–331 special cases for, 96–99 supernode, 97–98 Nodes, 37–38, 48, 90–91, 93–94 circuit element, 37–38, 48 defined, 37, 90–91 essential, 91 voltage, 93–94 Noninverting-amplifier circuit, 153–154, 160–161, 164 Nonplanar circuits, 90–91 Norton equivalent circuits, 115, 129, 327 analysis of, 115, 129 defined, 115 frequency-domain source transformations, 327 impedance (Z) in, 327 source transformations used for, 115 Numerator determinant, 704–705 Octave, 744 Odd-function symmetry, 610–611, 614, 632 Ohm’s law, 30–31, 48, 59, 468–469 electrical resistance and, 30–31, 48, 59 s-domain applications of, 468–469 One-sided (unilateral) Laplace transform, 428–429 Open circuit, 35 Open-loop gain (A), 151, 164 Operational amplifiers (op amps), 144–173, 556–601 bandpass filters, 564–568, 583–586, 592–593 bandreject filters, 568–571, 583, 586–589, 592–593 bass volume control, 557, 589–591 broadband filters, 565–571, 592 Butterworth filters, 576–584, 592–593 cascading, 564–574, 592 circuit symbol for, 146 currents (i), 146–150 difference-amplifier circuit, 155–159, 164 dual in-line package (DIP), 146 equivalent (realistic model), 159–162, 164 negative feedback, 147–148, 164 filters, 556–601 first-order filters, 558–562 gain (A), 147, 151, 157–158, 164 high-pass filters, 560–561, 583, 592 higher-order filters, 571–584, 592–593 input constraints, 147–148,164 inverting-amplifier circuit, 150–152, 160, 164 low-pass filters, 558–560, 576–582, 592–593 noninverting-amplifier circuit, 153–154, 160–161, 164 realistic model (equivalent), 159–162, 164 resistance (R) and, 145, 162–163 scaling, design of using, 563–564 simplified, 146–159, 164 strain gages for, 145, 162–163 summing-amplifier circuit, 152–153, 164 terminals, 146–150 transfer characteristics, 147 voltages (v), 146–150, 164 Operational transforms, 429, 435–440, 457, 655–659 addition, 436, 656 amplitude modulation, 657 convolution, 657–658 defined, 429 differentiation, 436–437, 656 Fourier, 655–659 frequency domain functions, 439, 657–658 integration, 437–438, 656 Laplace, 431, 437–442, 459 modulation, 657 multiplication by a constant, 435, 655 scale changing, 439, 657 subtraction, 436, 656 time domain functions, 438–439, 655–659 translation, 438–439, 657 types of, 440, 658 Index Overdamped responses, 269, 271–273, 282, 286, 295–296 natural response equations, 271–273, 295 parallel RLC circuits, 269, 271–273, 282, 295 series RLC circuits, 286, 295 step response equations, 286, 296 P Pacemaker circuits responses, 213, 245–246 Parallel-connected circuits, 59–61, 75, 187–189, 203, 322–323, 692 See also Parallel RLC circuits capacitors, 188–189, 203 impedances combined in, 322–323 inductors, 187–188, 203 Kirchhoff’s law for, 59, 61 Ohm’s law for, 59 resistors, 59–61, 75 two-port circuits, 692 Parallel RLC circuits, 266–285, 295–296, 471–474 characteristic equation for, 267, 269–270, 295 circuit symbols for, 266, 280 critically damped responses, 269, 277–278, 283, 295 direct approach for, 281–282 general solution of differential equations, 266–270 indirect approach for, 281 Laplace transform analysis of, 471–474 natural response, 266–279, 295 Neper frequency (a) for, 268–269 overdamped responses, 269, 271–273, 282, 295 resonant radian frequency (v0), 268–269 s-domain, 471–474 step response, 280–285, 295–296, 471–473 transient response, 473–474 underdamped responses, 269, 274–277, 283, 295 Parallel-series two-port circuit connection, 692 Parameters of two-port circuits, 679–686, 696 Parseval’s theorem, 662–669 filter applications of, 665–666 Fourier transform and, 662–669 graphic interpretation of, 663–664 rectangular voltage pulse analysis using, 666–668 time-domain energy calculations, 662–668 Partial fraction expansion, 442–452, 457, 484–487, 504 distinct complex roots of D(s), 445–447 distinct real roots of D(s), 443–444 improper rational function, 442, 451–452 inverse Laplace transforms in, 442–452, 457 Laplace transform analysis and, 484–487, 504 Laplace transform pairs for, 451 proper rational functions, 442–443, 457 repeated complex roots of D(s), 449–451 repeated real roots of D(s), 447–449 time invariant circuits, 486–487, 504 transfer function H(s) in, 484–487, 504 Partitioned matrix, 715–717 Passband frequencies, 522–523, 548 785 Passive current elements, 27, 176, 202, 315–318 capacitors, 176, 202, 317–318 defined, 27 frequency domain, 315–318 ideal sources, 27 impedance (Z), 318 inductors, 176, 202, 316–317 reactance, 318 resistors, 315–316 voltage to current (v–i) relationships, 315–318 Passive filter circuits, 520–555 bandpass filters, 523, 534–543, 549 bandreject filters, 523, 543–547, 549 bandwidth ( b ), 535, 537, 545–546, 549 center (resonant) frequency (vo), 534–536, 545–546, 549 cutoff frequency (vc), 522–523, 525–526, 531, 533, 537, 545–546, 548 defined, 523–524 filtering capabilities of, 523–524 frequency response and, 520, frequency response plots, 522–523 high-pass filters, 522–523, 530–534, 549 low-pass filters, 522–530, 548 magnitude plot, 522 qualitative analysis, 524–525 quantitative analysis, 526–527 passband frequencies, 522–523, 548 phase angle plot, 522 pushbutton telephones, 521, 548 quality factor (Q), 535, 538, 545, 549 relationship between frequency and time domains, 529, 543 stopband frequencies, 522–523, 548 transfer function H(s) for, 522, 529, 533–534, 542, 544–545, 547–549 Passive sign convention, 13, 18 Path, defined, 90–91 Per-phase quantity (f), 403 Period, sinusoidal sources, 306 Periodic functions, 602–605, 609–614, 617–625, 628–630, 632–633, 644–646 amplitude spectrum of, 628–630 aperiodic function transition from, 644–646 average-power calculations, 621–624, 633 defined, 602, 632 Fourier coefficients and, 605, 609–614 Fourier series application of, 617–621 Fourier series representation of, 605 Fourier transforms and, 644–646 phase spectrum of, 628–630 root-mean-square (rms) value, 624–625, 633 steady-state response using, 617–621 symmetry effects, 609–614, 632 waveforms, 602–604 Phase angle (f), 306 Phase angle plots, 522, 745–474, 753–755 Phase current, 403, 405–407 Phase spectrum, 628–630 786 Index Phase voltage, 398, 402–404 Phasor diagrams, 342–344, 403, 406 Phasors, 310–315, 345 defined, 310 frequency domain and, 311, 345 inverse transform, 312–314, 345 sinusoidal steady-state analysis and, 310–315, 345 transform, 311, 345 Pi (p)-equivalent circuit, 730–732 Pi (p) interconnection, 71 Pi to tee (p to T) equivalent circuit, 71–73 Planar circuits, 90–91 Polar form of complex numbers, 723–724 Polarity, 15–16, 190–192, 195–196, 203, 339–340 dot convention for, 190–192, 195–196, 203, 341–342 ideal transformers, 339–340 mutual inductance, 190–192, 195–196, 203 power references, 15–16 voltage and current ratios, 339–340 Poles (roots), 452–453, 458, 484, 739–740, 747–749 Bode plots and, 739–740, 747–749 complex, 747–749 defined, 452 rational function F(s), 452–453, 457 real, first order, 739–740 transfer function H(s), 484 Ports, 676 Positive (abc) phase voltage sequence, 398 Potential coil, 413 Power (p), 14–18, 31–32, 48, 120–122, 129, 179–181, 183–186, 202, 359, 382–383, 397, 413–415, 417, 737–738 algebraic signs of, 15–16 balancing, model for, 17–18 capacitors, 183–186, 202 defined, 15, 18 decibel (dB) used for, 737–738 electrical, transmission and distribution of, 397, 416–417 energy (w) and, 14–17 inductors, 179–181, 202 maximum transfer, 120–122, 129 measurement of, 413–715, 417, 737–738 polarity references, 15–16 resistors and, 31–32, 48 standby (vampire), 359, 382–383 wattmeter, 413–415, 417 Power calculations, 358–395, 408–413, 417, 621–624, 633 apparent power, 368, 384 appliance ratings, 365 average (real) power (P), 361–365, 371–373, 384, 408–409, 621–624, 633 balanced three-phase circuits, 408–413, 417 balancing from an ac circuit, 374–375 capacitive circuits, 363 complex power (S), 368–372, 384, 409 delta ( ¢ ) connected loads, 409–410 equations for, 369–376 inductive circuits, 362–363 instantaneous power, 360–361, 384 maximum power transfer, 376–382, 384 parallel loads, 373–374 periodic functions and, 621–624, 633 power factor (pf), 363, 384 reactive factor (rf), 363, 384 reactive power (Q), 361–365, 371–373, 384 resistive circuits, 362 root-mean-square (rms) value, 366–367 sinusoidal steady-state analysis, 358–395 three-phase circuits, 408–413, 417 standby (vampire) power, 359, 382–383 wye (Y) connected loads, 408–409 unspecified loads, 412 Power equation, 15 Power factor (pf), 363, 384 Power systems, Primary winding, transformers, 333 Problem solving strategy, 3, 7–8 Pushbutton telephone circuits, 521, 548 Q Qualitative analysis, 524–525, 531, 535–536, 544 bandpass filters, 535–536 bandreject filters, 544 high-pass filters, 531 low-pass filters, 524–525 Quality factor (Q), 537, 539, 545, 549, 584–585 active (narrowband) filter circuits, 584–585 passive filter circuits, 537, 539, 545, 549 Quantitative analysis, 526–527, 530–531, 536–538, 544–547 bandpass filters, 536–538 bandreject filters, 544–547 high-pass filters, 530–531 low-pass filters, 526–527 Quarter-wave symmetry, 612–613, 632 R Rational function F(s), 442–453, 457–458 defined, 442 partial fraction expansion of, 442–452, 457 poles of, 452–453, 458 zeros of, 452–453, 458 RC circuits, see Resistor-capacitor (RC) circuits Reactance, impedance and, 318 Reactive factor (rf), 363, 384 Reactive power (Q), 361–365, 371–373, 384 See also Power calculations Real first-order poles and zeros, 739–740 Real models, see Equivalent circuits Real power, see Average power Reciprocal two-port circuits, 685–686, 696 Rectangular (Cartesian) form of complex numbers, 723 Reflected impedance (Zr), 334, 345 Index Resistance (R), 30–33, 48, 69–70, 113–115, 145, 162–163 conductance (G) and, 31, 60 equivalent, 58–60, 64–65 measurement of, 69–70 Ohm’s law, 30–31, 48 operational amplifiers (op amps), 145, 162–163 resistors as models of, 30–33, 48 strain gages for, 145, 162–163 Thévenin equivalent (RTh), 113–115 Resistive circuits, 56–87, 362 constant (dc) sources, 56 current-divider, 63, 75 current-division analysis, 64–66, 75–76 delta to wye ( ¢ to Y) equivalent, 71–73, 76 load, 62 measurement of voltage and current in, 66–69, 76 parallel-connected (in parallel), 59–61, 75 pi to tee (p to T) equivalent, 71–73 power calculations for, 362 resistors, 58–61, 75 series-connected (in series), 58, 75 touch screens, 57, 73–75 voltage-divider, 61–62, 75 voltage-division analysis, 64–66, 75 Wheatstone bridge, 69–70, 76 Resistor-capacitor (RC) circuits, 212, 214, 220–224, 229–236, 239, 246, 470–471, 528–534 circuit symbols for, 214, 220 cutoff frequency, 531, 533 defined, 212 frequency response plot of, 531 general solution for responses of, 231–236, 246 high-pass filters, behavior of as, 530–534 Laplace transform analysis of, 470–471 low-pass filters, behavior of as, 528–529 natural response of, 220–224, 231–236, 246, 470–471 qualitative analysis of series, 530–531 quantitative analysis of series, 531 s-domain, 470–471 sequential switching, 236, 239 step response, 229–236, 246 time constant (t), 221, 246 unbounded response, 240–241 voltage (v), deriving expression for, 221 Resistor-inductor (RL) circuits, 212, 214–220, 224–228, 231–238, 246, 524–527, 532–534 circuit symbol for, 214 current (i), deriving expression for, 214–216 cutoff frequency, 525–527, 533 defined, 212 frequency response plots of, 524–525 general solution for responses of, 231–236, 246 high-pass filters, behavior of as, 5302–534 low-pass filters, behavior of as, 524–527 natural response of, 214–220, 231–236, 246 787 qualitative analysis of series, 524–525 quantitative analysis of series, 526–527 sequential switching, 236–238 steady-state response, 217 step response, 224–228, 231–236, 246 time constant (t), 216–217, 246 transient response, 217 Resistor-inductor-capacitor (RLC) circuits, 264–303, 471–474, 535–547 bandpass filters, behavior of as, 535–543 bandreject filters, behavior of as, 543–547 characteristic equation for, 267, 269–270, 286, 295 circuit symbols for, 266, 285–286 critically damped voltage responses, 269, 277–278, 283, 286, 295–296 cutoff frequency (vc), 537, 545–546 clock for computer timing, 265, 293–294 frequency response plots, 535–536, 544 frequency selective circuits, 535–547 Laplace transform, analysis of using, 471–474 natural response of, 266–279, 285–289, 295 Neper frequency (a) for, 268–269, 286 overdamped voltage responses, 269, 271–273, 282, 286, 295–296 parallel, 266–285, 295, 471–474 qualitative analysis of series, 535–536, 544 quantitative analysis of series, 536–538, 544–547 resonant radian frequency (v0), 268–269, 286 s-domain, 471–474 series, 285–289, 295–296 step response of, 280–289, 295–296, 471–473 transfer function H(s) for, 542, 544–545, 547, 549 transient response of, 473–474 underdamped voltage response, 269, 274–277, 283, 286–288, 295–296 Resistors, 30–33, 48, 58–61, 75, 89, 125–128, 291–293, 315–316, 466, 504, 761 black box, 58 circuit symbol for, 31 component values, 761 conductance (G) and, 31 defined, 30, 48 electrical resistance and, 30–33, 47–48 feedback, 291–293 frequency domain, 315–316 integrating amplifiers with, 291–293 Laplace transforms for analysis of, 466, 504 parallel-connected (in parallel), 59–61, 75 power terminals of, 31–32, 48 resistance (R) and, 30–33, 48 s-domain, 466, 504 sensitivity analysis, 89, 125–128 series-connected (in series), 58, 75 time and frequency domain elements, 466 voltage to current (v–i) relationships, 315–316 Resonant radian frequency (vo), 268–269, 286 See also Center frequency (v0) 788 Index Response, 212–303, 309–310, 345, 427, 456–457, 470–482, 493–495, 503–504, 520, 659–661 clock for computer timing, 265, 293–294 complete, 427, 456–457 first-order circuits, 212–263 Fourier transform for, 659–661 frequency, 520 general solutions for, 231–236, 246 integrating amplifiers, 241–244, 246, 289–293, 295 Laplace transform used for, 427, 456–457, 470–482, 495–497, 503 multiple mesh circuits, 474–476 natural, 212, 214–224, 231–236, 246, 266–279, 285–289, 295, 470–471 pacemaker circuits, 213, 245–246 parallel RLC circuits, 266–285, 295–296, 471–474 resistor-capacitor (RC) circuits, 212, 214, 220–224, 229–236, 239, 246, 470–471 resistor-inductor (RL) circuits, 212, 214–220, 224–228, 231–238, 246 resistor-inductor-capacitor (RLC) circuits, 264–303, 471–474 second-order circuits, 264–303 sequential switching, 236–240, 243, 246 series RLC circuits, 285–289, 295–296 sinusoidal, 309–310, 345, 493–495, 503–504, 661 steady-state, 217, 427, 493–495, 503–504, 661 step, 212, 224–236, 246, 280–289, 295–296, 471–476 transfer function and, 493–495, 503–504, 661 transient, 217, 427, 473–474, 659–660 unbounded, 240–241, 246 RL circuits, see Resistor-inductor (RL) circuits RLC circuits, see Resistor-inductor-capacitor (RLC) circuits Root-mean-square (rms) value, 307–309, 366–367, 624–625, 633 effective value (eff) as, 366–367 periodic functions, 624–625, 633 power calculations and, 366–367 sinusoidal sources, 307–309 Roots, 443–453, 458, 727–728 complex numbers, 443–451, 727–728 distinct complex, 445–447 distinct real, 443–444 Laplace transform pairs for, 451 partial fraction expansion of D(s), 442–451, 458 repeated complex, 449–451 poles (denominator polynomial), 452–453, 458 repeated real, 447–449 zeros (numerator polynomial), 452–453, 458 Row matrix, 708 S s-domain, 441–455, 457–458, 466–502, 504 capacitor in, 467–468, 504 circuit analysis in, 468–482, 504 circuit symbols for, 466–467, 470–471, 475, 477, 479–480 final-value theorem, 453–455, 458 impulse function Kd(t) in, 496–502, 504 inductor in, 466–467, 504 initial-value theorem, 453–455, 458 inverse Laplace transform and, 442–452, 457 Kirchhoff’s laws and, 469 Laplace transform and, 441–455, 457–458, 466–487, 504 mutual inductance in, 478–479 notation of in Laplace transforms, 441–442 Ohm’s law in, 468–469 partial fraction expansion, 442–452, 457, 484–487, 504 poles (denominator polynomial), 452–453, 458 resistor in, 466, 504 responses of circuits in, 470–482 superposition, use of in, 480–482 terminal voltage-current equations in, 466–468, 504 Thévenin equivalent, use of in, 477–478 transfer function H(s) in, 482–495, 504 zeros (numerator polynomial), 452–453, 458 Scale-changing property, operational transforms, 439, 657 Scaling, 562–564, 592 frequency factor (kf), 562, 592 magnitude factor (km), 562, 592 op-amp filter design using, 563–564 Second-order circuits, 264–303 defined, 266 general solution of differential equations, 266–270 integrating amplifiers, 289–293, 295 natural response of, 266–279, 285–289, 295 resistor-inductor-capacitor (RLC), 264–303 step response of, 280–289, 295–296 Secondary winding, transformers, 333 self-impedance (Z), 333–334 Self-inductance, 193–194, 196–197, 203 Sensitivity analysis, resistors, 89, 125–128 Sequential switching, 236–240, 243, 246 first-order circuit responses and, 236–240, 243, 246 integrating amplifier with, 243 resistor-capacitor (RC) circuits, 236, 239 resistor-inductor (RL) circuits, 236–238 Series-connected (in series) circuits, 39, 48, 58, 75, 187–189, 203, 320–324, 692 See also Series RLC Circuits black box concept, 58 capacitors, 188–189, 203 impedances combined in, 320–324 inductors, 187–188, 203 Kirchhoff’s current law for, 39, 48, 58 node positions, 39, 48 resistors, 58, 75 two-port circuits, 692 Series-parallel two-port circuit connection, 692 Series RLC circuits, 285–289, 295–296 capacitor voltage in, 286–287 characteristic equation of, 286, 295 circuit symbols for, 285–286 critically damped response, 286 natural response, 285–289 Index Neper frequency (a), 286 overdamped response, 286 resonant radian frequency (v0), 286 step response, 285–289, 295–296 underdamped responses, 286–288 Short circuit, 35 Sifting property, 432 Signal-processing systems, 5–6 Signed minor, 705–706 Signum function, Fourier transform of, 651 Simultaneous equations, 91–93, 703–722 characteristic determinant, 704 circuit analysis using, 91–93 Cramer’s method, 704 determinant evaluation for, 705–708 determination of number of, 91–92 matrices for, 707–708, 712–717 matrix algebra for, 708–712, 718–722 numerator determinant, 704–705 solution of, 703–722 systematic approach using, 92–93 Sine functions, 615–616 Single-phase equivalent circuits, 402–403, 417 Sinusoidal response, 309–310, 345, 465, 495–497, 503, 506, 661 Fourier transform for, 661 Laplace transform for, 465, 495–497, 503 steady-state analysis and, 309–310, 345 steady-state component of, 310 surge suppressors, 465, 503 transfer function H(s) and, 495–497, 506 transient component of, 312 Sinusoidal sources, 306–309, 345, 427, 456–457 complete circuit response of, 427, 456–457 current (i), 306–307, 345 Laplace transforms for, 427, 456–457 period, 306 phase angle (f), 306 root-mean-square (rms) value of, 307–309 steady-state analysis, 306–309, 345 voltage (v), 306–308, 345 Sinusoidal steady-state analysis, 304–395 delta-to-wye ( ¢ -to-Y) transformations, 324–326 frequency domain, 311, 315–320, 327–341, 345 household distribution circuit, 305, 344 ideal transformer, 336–341, 345 impedance (Z), 318, 320–330, 334, 341, 345 Kirchhoff’s laws, 319–320 mesh-current method, 331–332 node-voltage method, 330–331 Norton equivalent circuit, 327 passive circuit elements, 315–318 phasor diagrams for, 342–344 phasors, 310–315, 345 power calculations, 358–395 reactance, 318 response, 309–310, 345 source transformations, 327–330 789 sources, 306–309, 345 standby (vampire) power, 359, 382–383 Thévenin equivalent circuit, 327–330 transformers, 332–341, 345 voltage to current (v-i) relationships, 315–318 Source transformations, 109–113, 115–116, 129, 327–330 circuit analysis using, 109–113, 115–116, 129 defined, 110 frequency domain, 327–330 impedance (Z) and, 327–330 Norton equivalent circuits, 115, 327 Thévenin equivalent circuit, 115–116, 327–329 Sources, 26–29, 42–45, 48, 56, 306–309, 345, 399–400, 427, 456–457, 499–502 active element of, 27 complete circuit response and, 427, 456–457 constant (dc), 56 controlled, 27 delta ( ¢ ) configurations, 399–400 dependent, 26–27, 29, 42–45, 48 ideal current, 26–29, 48 ideal voltage, 26–29, 48 impulsive, 499–502 independent, 26, 29, 48 interconnection of, 28–29 Laplace transform and, 427, 456–457, 499–502 passive element of, 27 sinusoidal, 306–309, 345, 427, 456–457 three-phase voltage, 399–400 wye (Y) configurations, 399–400 Square matrix, 708 Square wave input of sinusoids, 603, 630–632 Standby (vampire) power, 359, 382–383 Steady-state analysis, see Sinusoidal steady-state analysis Steady-state response, 217, 427, 456–457, 493–495, 503–504, 617–621, 633, 661 defined, 217 direct approach to, 619–621 first-order circuits, 217 Fourier series analysis, 617–621, 633 Fourier transform for, 661 Laplace transform analysis and, 493–495, 504 periodic functions used for, 617–621 sinusoidal, 493–495, 503–504, 661 surge suppressors and, 465, 503 transfer function H(s) and, 493–495, 504 transient effects and, 427, 456–457 Step function Ku(t), 429–430, 457 Step response, 212, 224–236, 246, 280–289, 295–296, 471–476 capacitor voltage in series RLC circuits, 286–287 critically damped voltage, 283, 295–296 defined, 212, 246 direct approach, 281–282 first-order circuits, 212, 224–236, 246 general solution for, 231–236, 246 indirect approach, 281 Laplace transform analysis and, 471–476 790 Index Step response (Continued) magnetically coupled coils and, 235–236 method of calculating, 232 multiple mesh circuit, 474–476 overdamped voltage, 282, 295–296 parallel RLC circuits, 280–285, 295–296, 471–473 resistor-capacitor (RC) circuits, 229–236, 246 resistor-inductor (RL) circuits, 224–228, 231–236, 246 resistor-inductor-capacitor (RLC) circuits, 280–289, 295–296 series RLC circuits, 285–289, 295–296 underdamped, 283, 288, 295–296 Stopband frequencies, 522–523, 548 Straight-line amplitude plots, 740–744, 750–752 Straight-line phase angle plots, 745–747 Strain gages, 145, 162–163 Summing-amplifier circuit, 152–153, 164 Supermesh, 104 Supernode, 97–98 Superposition, 122–125, 129, 480–482 circuit analysis using, 122–125, 129 defined, 122 Laplace transform analysis and, 480–482 s-domain, 480–482 Surge suppressors, 465, 503 Susceptance (B), 322 Switches, 176, 232–240, 243, 465, 496–499, 503 arcing, 176 capacitor circuits, 496–497 inductor circuits, 176, 497–499 impulse functions created by, 496–499 integrating amplifier with, 243 Laplace transform analysis and, 465, 496–499, 503 RL and RC circuits, 232–240, 246 sequential switching, 236–240, 246 surge suppressors, 465, 503 Symmetric two-port circuits, 686, 696 Symmetry, 609–614, 632 even-function, 609–610, 632 Fourier coefficients, effects on, 609–614, 632 half-wave, 611–612, 632 odd-function, 610–611, 614, 632 quarter-wave, 612–613, 632 T T-equivalent circuit, 729–730 Tee (T) interconnection, 71 Terminals, 31–32, 36, 146–150, 164, 182–186, 203, 466–468, 678 capacitor equations, 182–186, 203, 467–468 current (i) input constraints, 148–149, 164 inductor equations, 176–181, 203, 466–467 measurements for circuit construction, 36 operation amplifiers (op amps), 146–150, 164 resistor power at, 31–32 two-port circuits, 678 variable characteristics (voltage and current), 146–150, 164 voltage (v) input constraints, 147–148, 164 Terminated two-port circuits, 687–691 Thévenin equivalent circuits, 113–119, 129, 327–329, 477–478 amplifier circuit using, 118–119 analysis of, 113–119, 129 defined, 113 dependent sources, 116, 118 finding equivalent of, 114–115 frequency-domain, 327–329 impedance (Z) in, 327–329 independent sources, 117–118 Laplace transform analysis and, 477–478 resistance source (RTh), 113–115 s-domain, use of in, 477–478 source transformations used for, 115–116, 327–329 voltage source (VTh), 113–115 test source for, 118 Three-phase circuits, see Balanced three-phase circuits Time constant (t), 216–217, 221, 246 resistor-capacitor (RC) circuits, 221, 246 resistor-inductor (RL) circuits, 216–217, 246 Time domain, 311–313, 345, 438–442, 529, 543, 646–648, 655–659, 662–668 convergence of Fourier transform integral and, 646–648 convolution in, 657–658 differentiation in, 656 energy calculations, 662–668 Fourier transform and, 646–648, 655–659, 664–670 frequency domain relationships, 529, 543, 657 frequency domain transformations, 311–313, 345 integration in, 656 integrodifferential equations for, 440–442 inverse phasor transformation, 312–313, 345 Laplace transform and, 438–442 operational transforms for, 438–439, 655–659 Parseval’s theorem for, 662–668 passive-filter circuits and, 529, 543 phasor transformation, 311, 345 scale change in, 657 translation in, 438–439, 657 Time-invariant circuits, 486–487, 504 Tolerance, 62 Touch screens, 57, 73–75, 175, 200–202 capacitance of, 175, 200–202 resistive circuits of, 57, 73–75 Transducers (strain gages), 145, 162–163 Transfer function H(s), 482–495, 504, 529, 533–534, 544–545, 547–549, 569, 572–579, 583–584, 592 active-filter circuits, 569, 572–577, 584, 592 bandpass filters, 542, 549, 569, 584 bandreject filters, 544–545, 547, 549 Index Butterworth filters, 576–579, 583, 592 cascading identical filters and, 572–575 circuit analysis, 482–495, 504 convolution integral and, 487–493, 504 defined, 482 high-pass filters, 533–534, 549 Laplace transform circuit analysis and, 482–495, 504 low-pass filters, 529, 548 memory and, 492–493 partial fraction expansion, use of in, 484–487, 504 passive-filter circuits and, 529, 533–534, 542, 544–545, 547–549 poles of, 484 steady-state sinusoidal response and, 493–495, 504 time-invariant circuits, 486–487, 504 weighting function, 492–493 zeros of, 484 Transformations, see Circuit transformations Transformers, 332–341, 345, 729–736 defined, 332 dot convention for, 339–340 equivalent circuits and, 729–736 frequency domain and, 332–341, 345 ideal, 333, 336–341, 345, 729–736 self-impedance (Z), 333–334 limiting values of, 336–338 linear, 333–336, 345 winding (primary and secondary), 333 reflected impedance (Zr), 334, 345 sinusoidal steady-state analysis, 332–341, 345 voltage and current ratios, 338–340, 345 Transient effects on circuits, 427, 456–457 Transient response, 217, 473–474, 659–660 defined, 217 Laplace transform for, 473–474 Fourier transform for, 659–660 Translation, operational transforms for, 438–439, 657 Transmission parameters, 681 Transpose of a matrix, 712 Trigonometric identities, 757 Twin-T notch filter, 586–589 Two-port circuits, 676–701 analysis of, 687–691 black box amplifier, 677, 695 conversion of parameters, 684–686 hybrid parameters, 681 interconnected, 692–695 intermittence, 680–681 parameter conversion for, 682–684 parameters of, 679–686, 696 ports, 676 reciprocal, 685–686, 696 symmetric, 686, 696 terminal equations for, 678 terminated, 687–691 transmission parameters, 681 z parameters, 679–680, 687–691 791 U Unbounded response, 240–241, 246 Underdamped responses, 269, 274–277, 283, 286–288, 295–296 characteristics of, 276 damped radian frequency, 274 damping factor (coefficient), 275 natural response equations, 274–275, 295 parallel RLC circuits, 269, 274–277, 283, 295 series RLC circuits, 286–288, 295 step response equations, 286, 296 Unilateral (one-sided) Laplace transform, 428–429 Unit impulse function d(t), 431, 457 Unit step function u(t), 429, 457, 652 V Vampire (standby) power, 359, 382–383 Variable-parameter function, 431–432 Volt-amp reactive (VAR), unit of, 363, 384 Volt-amps (VA), unit of, 368, 384 Voltage (v), 11–13, 18, 26–29, 32–33, 38, 40–41, 48, 66–69, 76, 113–115, 146–150, 164, 176–179, 183–186, 188, 202–203, 221, 286–287, 306–308, 319, 338–340, 345, 398–405, 417, 466–469, 504 a-, b-, and c-phase, 398, 417 capacitors, 183–186, 202–203, 286–287, 467–468, 504 current (i) and, 11–13, 177–179, 468–469 defined, 12, 18 determination of, 176–177 electric charge and, 11–12 equivalent capacitance in, 188 frequency domain, 319, 338–340, 345 gain (A), 147 inductors, 176–179, 202, 466–467, 504 input constraints, 147–148, 164 Kirchhoff’s law (KVL), 38–41, 48, 319, 402–403 line, 402–404 measurement of, 66–69, 76 natural response and, 221 negative (acb) phase sequence, 398 node-voltage equation (VN), 400–401 operational amplifiers (op amps), 146–150, 164 phase, 398, 402–404 polarity of, 339–340 positive (abc) phase sequence, 398 power in a resistor, 32–33, 48 RC circuits, deriving expression for, 221 reference direction, 13 resistors, 466, 504 s-domain equations for, 466–469, 504 series RLC capacitor step response, 286–287 sinusoidal phase sequences, 398–399 sinusoidal source, 306–308, 345 sources, 26–29, 48, 399–400 terminal variable characteristics, 146–150, 164 792 Index Voltage (v) (Continued) Thévenin equivalent (VTh), 113–115 three-phase, 398–405, 417 transfer characteristics of, 146–147, 164 transformer ratio, 338–340, 345 wye (Y) and delta ( ¢ ) configurations, 399–400 wye-wye (Y-Y) circuit analysis and, 400–405, 417 Voltage-divider circuits, 61–62, 75 Voltage-division circuit analysis, 64–66, 75 Voltage drop, 57, 64–65 Voltage pulse analysis, 645, 666–668 Voltage to current (v–i) relationships, 176–177, 183, 315–318, 345 capacitors, 183, 317–318 frequency domain, 315–318 impedance (Z) of, 318 inductors, 176–177, 316–317 reactance of, 318 resistors, 315–316 sinusoidal steady-state analysis and, 315–316, 345 Voltmeter, 66–67, 76 W Watt (W), unit of, 363, 384 Wattmeter, 413–415, 417 Waveforms, 602–604 Weighting function, memory and, 492–493 Wheatstone bridge, 69–70, 76 Winding (primary and secondary), 333 Wye (Y) connected loads, power in, 408–409 Wye (Y) interconnection, 71 Wye (Y) source configurations, 399–400 Wye-delta (Y- ¢ ) circuit analysis, 405–407 Wye-wye (Y-Y) circuit analysis, 400–405, 417 Z z parameters, two-port circuits, 679–680, 687–691 Zero frequency, 528 Zeros (roots), 452–453, 458, 484, 739–740, 747–749 Bode plots and, 739–740, 747–749 complex, 747–749 defined, 452 rational function F(s), 452–453, 457 real, first order, 739–740 transfer function H(s), 484 This page intentionally left blank An Abbreviated List of Laplace Transform Pairs f(t) (t > – ) Type F(s) d(t) (impulse) u(t) (step) s t (ramp) s2 e - at (exponential) s + a sin vt (sine) v s2 + v2 cos vt (cosine) s s2 + v2 te - at (damped ramp) (s + a)2 e - at sin vt (damped sine) v (s + a)2 + v e - at cos vt (damped cosine) s + a (s + a)2 + v An Abbreviated List of Operational Transforms f(t) F(s) Kf(t) KF(s) f1(t) + f2(t) - f3(t) + Á F1(s) + F2(s) - F3(s) + Á df(t) dt sF(s) - f(0 - ) d2 f(t) dt2 s 2F(s) - sf(0 - ) - df(0 - ) dt n d f(t) dtn t s n F(s) - s n - f(0 - ) - s n - F(s) s L0 f(t - a)u(t - a), a > e - as F(s) e - at f(t) F(s + a) f(at), a > s Fa b a a f(x) dx tf(t) tn f(t) f(t) t - dF(s) ds (-1) n Ls dn F(s) ds n q F(u) du df(0 - ) dn - f(0 - ) df2(0 - ) Á - s n-3 dt dt2 dtn - Periodic Functions f (t) f(t) A A T 2T t ϪA Triangular Wave f (t) t T T/2 3T/2 Half-wave rectified sine 2T T T/2 3T/2 Full-wave rectified sine 2T f(t) A A T 2T t ϪA Square Wave t Fourier Series 8A np B sin a b R sin nv0t a 2 p n = 1,3,5, n q f(t) = Triangular wave 4A sin nv0t a p n = 1,3,5, n f(t) = q cos nv0t A 2A A sin v0t + a p p n = 2,4,6, n2 - Half-wave rectified sine q f(t) = Square wave f(t) - q cos nv0t 2A 4A a p p n = (4n2 - 1) Full-wave rectified sine Fourier Transforms of Elementary Functions f(t) F( V ) d(t) (impulse) A (constant) 2pAd(v) sgn(t) (signum) 2>jv u(t) (step) pd(v) + 1>jv e - atu(t)(positive-time exponential) 1>(a + jv) at e u(-t) (negative-time exponential) e - a|t| (positive- and negative-time exponential) 1>(a - jv) 2a>(a2 + v2) e jv0t (complex exponential) 2pd(v - v0) cos v0t (cosine) p[d(v + v0) + d(v - v0)] sin v0t (sine) jp[d(v + v0) - d(v - v0)] Operational Transforms f(t) F( V ) Kf(t) KF(v) f1(t) - f2(t) + f3(t) n n d f(t)>dt F1(v) - F2(v) + F3(v) ( jv)n F(v) t L- q f(x) dx F(v)>jv f(at) v F a b, a > a a f(t - a) e - jva F(v) e jv0t f(t) f(t) cos v0t q L- q x (l)h(t - l) dl F(v - v0) 1 F(v - v0) + F(v + v0) 2 X(v)H(v) q f1(t) f2(t) F (u)F2(v - u) du 2p L- q tn f(t) ( j)n dn F(v) dvn ... kg # m>s2 N#m Power watt (W) Electric charge coulomb (C) A#s Electric potential volt (V) J>C Electric resistance ohm ( Ỉ ) V>A Electric conductance siemens (S) A>V Electric capacitance farad (F)... of electrical sources An electrical source is a device that is capable of converting nonelectric energy to electric energy and vice versa A discharging battery converts chemical energy to electric. .. balancing power, simple resistive circuits, node voltage method, mesh current method, Thévenin and Norton equivalents, op amp circuits, firstorder circuits, second-order circuits, AC steady-state analysis,