Electric Circuits, 9th Edition P82 potx

786 Index Impulse function K8(t) (Continued) unit 5(0,433-434,459 variable-parameter function, 433-434 Impulsive sources, 501-504 Independent sources, 26,29,47 Indirect approach, step response analysis, 281 Inductance (L), 174,176-182,189-200,203-204 circuit parameter of, 174,176,203 Faraday's law, 193-194 mutual, 176,189-200,203-204 self, 176-182,193-194,203 Inductors, 174,176-182,187-200,203,318-319,468-469, 499-501,506 behavior of, 174 circuit symbol for, 176 current (i) in, 111-119,203 energy (vv) in, 179-181,203 frequency domain, 318-319 impulse function (I) in, 499-501 Laplace transform used for, 468-469,499-501,506 power (p) in, 179-181,203 i-domain, 468-469,506 series-parallel combinations, 187-188,203 switching operation, 499-501 terminal equations for, 176-181,203,468-469,506 voltage (v) in, 176-177,203 voltage to current (v-i) relationships, 176-177, 318-319 Infinite frequency, 530 Initial-value theorem, Laplace transform, 455-457 Input constraints, 147-148,165 Instantaneous power, 362-363,386,412,419 Integer power of complex numbers, 729 Integrals, 646-650,761-762 Integrating amplifiers, 241-244,246,289-293,297 cascading connections, 289-293,297 feedback resistors and, 291-293,297 first-order circuits with, 241-246 response analysis of, 241-244,246 second-order circuits with, 289-293,297 sequential switching and, 243 Integration, operational transforms for, 439-440,658 Integrodifferential equations, 442-443 Interconnected two-port circuits, 694-697 International System of Units (SI), 8-10 Inverse Fourier transform, 647 Inverse Laplace transforms, 444-454,459 distinct complex roots of D(s), 447-449 distinct real roots of £)(5), 445-446 improper rational function, 444,453-454 partial fraction expansion, 444-454 proper rational functions, 444-453,459 rational function F(s), 444,459 repeated complex roots of D(s), 451-453 repeated real roots of D(s), 449^451 Inverse of a matrix, 716 Inverse phasor transform, 314-316,347 Inverting-amplifier circuit, 150-152,160,164 K Kirchhoff s laws, 37-41,48,321-322,471 closed loop (path) for application of, 38 current (KCL), 37-41,48,321-322 frequency domain, 321-322 nodes, 37-38 s-domain applications of, 471 voltage (KVL), 38,40-41,48,321 L Laplace transform, 428-521,650-653 applications of, 442-444,472-484 circuit analysis and, 466-521 complete circuit response using, 429,458^159 convolution integral, 489-495,506 defined, 430,459 final-value theorem, 455^4-57 Fourier transforms found using, 650-653 frequency domain and, 441 functional transforms, 431,436-437,459 impulse function K8(t), 433-435,459,498-504,506 initial-value theorem, 455-457 inverse, 444-454,459 lumped-parameter circuit applications, 442-444,459 mutual inductance, analysis of a circuit with, 480-481 natural response using, 472-473 one-sided, 430-431 operational transforms, 431,437^442,459 pairs, 436-437 partial fraction expansion, 444-454,459,486-489,506 poles (roots), 454-455,486 rational functions F(s), 444-454,459 s-domain, 443-455,468-489,506 sinusoidal sources and, 429,458-459 steady-state sinusoidal response, 495-497,505-506 step function Ku(t), 431-432,459 step response using, 473-478 superposition, use of in .s-domain, 482-484 surge suppressors, 467,505 Thevenin equivalent, use of in s-domain, 479-480 time domain translation in, 440-443 time-invariant circuits and, 488,506 transfer function H(s), 484^197,506 transient response using, 475—476,506 unilateral, 430-431 unit impulse function 5(0,433-434,459 unit step function KK(0,431,459 zeros (roots), 454-455,486 Line current, 405 Line spectra, 630 Line voltage, 404-405 Linear simultaneous equations, 705-724 Cramer's method,706 determinants, 706-709 matrices, 709-724 solution of, 705-724 Index 787 Loads, 62,410-414 balanced three-phase circuits and, 410-414 defined, 62 delta (A) connected,411-413 power calculations and, 410-414 unspecified, 414 wye (Y) connected, 410-411,413 Loop, 38,48,90-91 closed, 38,48 defined, 90-91 Low-pass filters, 524-532,550,560-562,574-584,594-595 active filter circuits, 560-562,578-584,594 Butterworth, 578-584,594-595 cascading identical, 574-577,594-595 cutoff frequency (<w c ), 527-529 frequency regions, 530 frequency response plots, 524-525 operational amplifier (op amp), 560-562,578-584,594 passive filter circuits, 524-532,550 qualitative analysis, 526-527 quantitative analysis, 528-529 relationship between frequency and time domains, 531 series RC circuits, 530-531 series RL circuits, 526-529 transfer function H(s) for, 531,550 Lumped-parameter system, 6-7,442-444,459 M Magnetic coupling, see Mutual inductance Magnetically coupled coils, 235-236,338,731-738 equivalent circuits for, 731,735-738 ideal transformers and, 338,731-738 ^-equivalent circuit, 732-735 step response of, 235-236 T-equivalent circuit, 731-732 Magnitude plot, 524 Magnitude scaling (k m ), 564,594 Matrices, 709-724 Maximum power transfer, 120-122,129,378-384,386 average power (P ms . x ) absorbed, 380-382 circuit analysis and, 120-122,129 ideal transformer, 383-384 impedance (Z) restricted, 380-381 sinusoidal steady-state analysis and, 378-384,386 Measurement, 8-10,66-70,76 ammeter, 66-67,76 analog meters, 67,76 current, 66-69 d'Arsonval meter movement, 67-68 digital meters, 67,76 International System of Units (SI), 8-10 resistance, 69-70 voltage, 66-69,76 voltmeter, 66-67,76 Wheats tone bridge, 69-70,76 Memory, transfer function concept of, 494-495 Mesh, defined, 90-91 Mesh circuit, step response of, 476-478 Mesh-current method, 90,99-109,129,190-192,333-334 amplifier circuit analysis using, 105-106 circuit analysis using, 90,99-109,129 defined, 100 dependent sources and, 102-103 equations, 99-102,190,192 frequency-domain circuits, 333-334 mutual inductance (M) and, 190-192 node-voltage method, comparison of, 106-109 sinusoidal steady-state analysis using, 333-334 special cases of, 103-104 supermesh, 104 Meters, see Measurement mho, unit of, 31 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 Moment function (unit doublet), 435 Mutual inductance (M), 176,189-200,203-204,480-481 Laplace transform analysis of a circuit with, 480-481 circuit parameters of, 189-190,194-196,203-204 coefficient of coupling and, 197,204 dot convention for polarity, 190-192,195-196,204 energy (w) storage and, 197-200,204 mesh-current equations for, 190-192 self-inductance and, 196-197,204 N Narrowband filters, 586-591,595 bandpass filters, 586-588 bandreject, 588-591 quality factor (Q) and, 586-587 twin-T notch, 588-589 Natural response, 212,214-224,231-236,246,266-279, 285-289,297,472-473 critically damped voltage, 269,277-278,286,297 defined, 212,246 forms of RLC circuit, 270-279 general solution for, 231-236 Laplace transform, using, 472-473 method of calculating, 232 overdamped voltage, 269,271-273,286,297 parallel RLC circuit, 266-279 resistor-capacitor (RC) circuits, 220-224,231-236,246, 472-473 resistor-inductor (RL) circuits, 214-220,231-236,246 resistor-inductor-capacitor (RLC) circuits, 266-279 series RLC circuits, 285-289 time constant, 216-217,221-222,246 underdamped voltage, 269,274-277,286-288,297 Negative (acb) phase sequence, 400 Negative feedback, op amps, 147-148 788 Index Neper frequency (a), 268,286 Neutral terminal, 401 Nodes, 37-38,48,90-91 circuit description by, 37-38,48 defined, 90-91 essential, 91 Node-voltage equation (V N ), 402-403 Node-voltage method, 90,93-99,106-109,129,332-333 amplifier circuit analysis using, 98-99 circuit analysis using, 90,93-99,106-109,129 defined, 93 dependent sources and, 95-96 equations, 93-94 frequency-domain circuits, 332-333 mesh-current method, comparison of, 106-109 sinusoidal steady-state analysis using, 332-333 special cases for, 96-97 supernode, 97-98 Noninverting-amplifier circuit, 153-154,160-161,164 Nonplanar circuits, 90-91 Norton equivalent circuits, 115,129,329-330 circuit analysis of, 115,129 defined, 115 frequency-domain, 329-330 impedance (Z) in, 329-330 source transformations used for, 115,329-330 Numerator determinant, 706-707 0 Odd-function symmetry, 612-613,616,634 Ohm (O), unit of, 31 Ohm's law, 30-33,47-48,470-471 electrical resistance and, 30-33,47-48 s-domain applications of, 470^471 Ohmmeter, 145 One-sided Laplace transform, 430-431 Open circuit, 35 Operational amplifiers (op amps), 144-173,560-586, 594-595 bandpass filters, 566-570,585,594 bandreject filters, 570-573,585,594 broadband filters, 567-573,594 Butterworth filters, 578-586,594-595 cascading identical filters, 574-577,594-595 circuit symbol for, 146 currents (/),146-150 difference-amplifier circuit, 155-159,164 dual in-line package (DIP), 146 equivalent (realistic model), 159-162,164 feedback, 147-148,164 filters, 560-577,594 first-order filters, 560-564 gain (/1), 147,151 high-pass filters, 562-563,585,594 higher-order filters, 573-591,594 input constraints of, 147-148 inverting-amplifier circuit, 150-152,160,164 low-pass filters, 560-562,578-584,594 noninverting-amplifier circuit, 153-154,160-161,164 resistance, 145,162-163 scaling, design of using, 565-566 simplified, 146-159,164 strain gages for, 145,162-163 summing-amplifier circuit, 152-153,164 terminals, 146-150 transfer characteristics of, 146-147,165 voltages (v), 146-150,164 Operational transforms, 431,437-442,459,657-661 addition, 438,658 amplitude modulation, 659 convolution, 659-660 defined, 431 differentiation, 438-439,658 Fourier, 657-661 frequency domain functions, 441,659-660 integration, 439-440,658 Laplace, 431,437-442,459 multiplication by a constant, 437,657 scale changing, 441,659 subtraction, 438,658 time domain functions, 440-441,659-660 translation, 440-441,659 types of, 442,660 Overdamped responses, 269,271-273,282,286,297-298 P Parallel-connected circuits, 59-61,76,187-189,203 capacitors, 188-189,203 inductors, 187-188,203 resistors, 59-61,76 Parallel RLC circuits, 266-285,297-298 characteristic equation for, 267,269-270,297 circuit symbols for, 266,280 critically damped responses, 269,277-278,283,297-298 direct approach for, 281-282 indirect approach for, 281 natural response, 266-279,297 Neper frequency (a) for, 268 overdamped responses, 269,271-273,282,297-298 resonant radian frequency (wo), 268-269 step response, 280-285,298 underdamped responses, 269,274-277,283,297-298 Parameters of two-port circuits, 680-688,698 Parseval's theorem, 664-672 filter applications of, 667-668 Fourier transform and, 664-672 rectangular voltage pulse application of, 668-670 time-domain energy and, 664-672 Partial fraction expansion, 444^154,459,486-489,506 distinct complex roots of D(s), AA1-AA9 distinct real roots of D(s), 445-446 improper rational function, 444,453-454 inverse Laplace transforms in, 444-454,459 repeated complex roots of D(s), 451-453 Index 789 repeated real roots of D(s), 449-451 time invariant circuits, 488-489 transfer function H(s) in, 486-489,506 Partitioned matrices, 717-719 Passband frequencies, 524-525,550 Passive circuit elements, 27,176,203,317-320 frequency domain, 317-320 ideal sources, 27 inductors and capacitors as, 176,203 Passive filter circuits, 522-557 bandpass filters, 525,536-545,551 bandreject filters, 525,545-549,551 bandwidth (0), 537,539-540,546-548,551 center (resonant) frequency (o) a ), 536-540,546-548,551 cutoff frequency (<u c ), 524-525,527-528,538-540,550 defined, 525-526 filtering capabilities of, 525-526 frequency response and, 522 frequency response plots, 524-525 high-pass filters, 524-525,532-536,551 low-pass filters, 524-532,550 magnitude plot, 524 passband, 524-525,550 phase angle plot, 524 pushbutton telephones, 523,550 quality factor (Q), 537,540,546-548,551 relationship between frequency and time domains, 531,545 stopband, 524-525,550 transfer function H(s) for, 531,536,544,549,551 Passive sign convention, 13,176 Path, 38,48,90-91 closed, 38,48 defined, 90-91 Period, sinusoidal sources, 308 Periodic function, 604-607,611-616,619-627,630-632, 634-635 amplitude of, 630-632 average-power calculations with, 623-627,635 defined, 604,607 excitation, 619 Fourier coefficients in, 607,611-627 Fourier series application of, 619-623 phase spectra of, 630-632 root-mean-square (rms) value of, 626-627,635 steady-state response using, 619-627 symmetry, 611-616,634 waveforms, 604-606 Periodic waveforms, 604-606 Phase angle (</>), 308 Phase angle plots, 524,747-749,755-757 Phase spectrum, Fourier series and, 630-632 Phasor diagrams, 344-346 Phasors, 312-317,347 defined, 311 inverse transform, 314-316,347 transform, 313-317,347 7t-equivalent circuit, 732-735 Pi (77-) interconnection, 71 Pi to tee (77 to T) equivalent circuits, 71-73 Planar circuits, 90-91 Polarity, 15-16,190-192,195-196,204,341-342 algebraic references, 15-16 dot convention for, 190-192,195-196,204,341-342 ideal transformers, 341-342 mutual inductance, 190-192,195-196,204 voltage and current ratios, 341-342 Poles (roots), 454-455,486,741-742,749-751 Bode plots and, 741-742,749-751 complex, 749-751 rational function F(s), 454-455 real, first order, 741-742 transfer function H(s), 486 Ports, defined, 678 Positive (abc) phase sequence, 400 Potential coil, 415 Power (/?), 14-17,31-32,48,179-181,183-186,203, 410-415,419 algebraic signs of, 15-16 average (P), 410-411,415-417,419 balanced three-phase circuit calculations for, 410-415 capacitors, 183-186,203 complex (S), 411,419 delta (A) loads and, 411-413 energy and, 14-17 inductors, 179-181,203 measurement of, 415-417,419 polarity references, 15-16 resistors and, 31-32,48 wattmeter, 415-417,419 wye (Y) loads and, 410-411,413 Power calculations, 360-397,445^146,452-454,623-627,635 apparent power, 370,386 average (real) power (P), 363-367,373-375,386, 623-627,635 capacitive circuits, 365 complex power (5"), 370-374,386 heating appliances, 361,384-385 inductive circuits, 364-365 instantaneous power, 362-363,386 maximum power transfer, 378-384,386 periodic functions, 623-627,635 power factor (pf), 365,386 reactive factor (rf), 365,386 reactive power (£?), 363-367,373-375,386 resistive circuits, 364 root-mean-square (rms) value, 368-369 sinusoidal steady-state analysis, 360-397 three-phase circuits, 445-446,452-454 Power factor (pf), 365,386 Power systems, see Balanced three-phase circuits Primary winder, transformers, 335 Proper rational functions, 444-453,459 Prototype filters, 561-566,594 Proximity switches, 175,200-202 Pushbutton telephone circuits, 523,550 790 Index Q Quality factor (£>), 537,540,546-548,551,586-587 active (narrowband) filter circuits, 586-587 passive filter circuits, 537,540,546-548,551 Quarter-wave symmetry, 614-615,634 R Rational functions F(s), 444-455,459. See also Partial fraction expansion defined, 444 improper, 444,453-454,459 inverse Laplace transform using, 444-455,459 partial fraction expansion of, 444-454,459 poles of, 454-455 proper, 444-453,459 zeros of, 454-455 RC circuits, see Resistor-capacitor (RC) circuits Reactance, 320 Reactive factor (rf), 365,386 Reactive power (0,363-367,373-375,386 Real models, see Equivalent circuits Real power, see Average power Real, first order poles and zeros, 741-742 Reciprocal two-port circuits, 687-688,698 Rectangular-pulse waveforms, 604,606 Rectangular voltage pulse, 668-670 Reflected impedance (Z,.), 336,347 Resistance (R), 30-33,47-48,69-70,113-115,145,162-163 conductance (G) and, 31 measurement of, 69-70 Ohm's law, 30-31,47 operational amplifiers (op amps), 145,162-163 resistors as models of, 30-33,47-48 strain gages for, 145,162-163 Thevenin equivalent (Rjh), 113-115 Resistive circuits, 56-87 constant (dc) sources, 56 current-divider, 63,76 current-division analysis, 64-66,76 delta to wye (A to Y) equivalent, 71-73,76 load. 62 measurement of voltage and current in, 66-69 parallel-connected (in parallel), 59-61,76 pi to tee (IT to T) equivalent, 71-73 rear window defroster grid, 57,73-75 resistors, 58-61 series-connected (in series), 39,58,76 voltage-divider, 61-62,76 voltage division analysis, 64-66,76 Wheatstone bridge, 69-70,76 Resistor-capacitor (RC) circuits, 212,214,220-224, 229-236,239,246,472-473,530-536 circuit symbol for, 214 cutoff frequency, 533 defined, 212 general solution for responses of, 231-236,246 high-pass filters, behavior of as, 532-536 Laplace transform, using for analysis of, 472-473 low-pass filters, behavior of as, 530-531 natural response of, 220-224,231-236,246,472^173 qualitative analysis of series, 532-533 quantitative analysis of series, 533-536 s-domain, 472-473 sequential switching, 236,239 step response, 229-236,246 time constant (T), 221-222,246 unbounded response, 240-241 voltage (v), deriving expression for, 221-222 Resistor-inductor (RL) circuits, 212,214-220,224-228, 236-238,246,526-529 circuit symbol for, 214 current (/), deriving expression for, 214-216 cutoff frequency, 527-529 defined, 212 general solution for responses of, 231-236,246 low-pass filters, behavior of as, 526-529 natural response of, 214-220,231-236,246 qualitative analysis of series, 526-527 quantitative analysis of series, 528-529 sequential switching, 236-238 steady-state response, 217 step response, 224-228,231-236,246 time constant (r), 216-217,246 transient response, 217 Resistor-inductor-capacitor (RLC) circuits, 264-305, 473-476,537-549 bandpass filters, behavior of as, 537-545 bandreject filters, analysis of as, 546-549 characteristic equation for, 267,269-270,297 critically damped voltage response, 269,277-278 cutoff frequency, 538-539,547-548 ignition circuit, 265,294-296 Laplace transform, analysis of using, 473-476 natural response of, 266-279,285-289,297 overdamped voltage response, 269,271-273,297 parallel, 266-279,473-476 qualitative analysis of series, 537-538,546 quantitative analysis of series, 538-544,546-549 .v-domain, 473-476 series, 285-289 step response of, 287-289,297-298,473-475 transient response of, 475-476 underdamped voltage response, 269,274-277,297 Resistors, 30-33,47^18,58-61,76,89,125-128,291-293, 297,317-318,468,506 black box, 58 circuit symbol for, 30 electrical resistance and ,30-33,47-48 feedback, 291-293,297 frequency domain, 317-318 integrating amplifiers with, 291-293,297 Laplace transform used for, 468,506 parallel-connected (in parallel), 59-61,76 power terminals of, 31-32,48 Index 791 s-domain, 468,506 sensitivity analysis, 89,125-128 series-connected (in series), 58,76 voltage to current (v-i) relationships, 317-318 Resonant frequency, see Center frequency (o) 0 ) Resonant radian frequency (co () ), 268-269,286 Response, 212-305,429,458-459,472-478,495-497, 505-506,522,661-663 complete, 429,458-459 first-order circuits, 212-263 flashing light circuit, 213,245-246 Fourier transform for, 661-663 frequency, 522 general solutions for, 231-236,246 ignition circuit, 265,294-296 integrating amplifiers, 241-244,246,289-293,297 Laplace transform used for, 429,458-459,472-478, 495-497,505-506 multiple mesh circuits, 476-478 natural, 212,214-224,231-236,246,266-279,285-289, 472^173 parallel RLC circuits, 266-279,473^176 resistor-capacitor (RC) circuits, 212,214,220-224, 229-236,239,246,472-473 resistor-inductor (RL) circuits, 212,214-220,224-228, 236-238,246 resistor-inductor-capacitor (RLC) circuits, 264-305, 473^175 second-order circuits, 264-305 sequential switching, 236-240,246 series RLC circuits, 285-289 sinusoidal, 495-497,505-506,663 steady-state, 217,495^197,505-506,663 step, 212,224-236,246,280-289,297-298,473^178 transfer function and, 495^197,505-506 transient, 217,475-476,506,662 unbounded, 240-241,246 RL circuits, see Resistor-inductor (RL) circuits RLC circuits, .see Resistor-inductor-capacitor (RLC) circuits Root-mean-square (rms) value, 309-311,368-369, 626-627,635 periodic functions, 626-627,635 power calculations and, 368-369 sinusoidal sources, 309-311 Roots of complex numbers, 729-730. See also Poles; Zeros S .y-domain, 443-455,468-489,506 capacitor in, 469-470,506 circuit analysis in, 470-489,506 circuit symbols for, 468^-70 inductor in, 468-469,506 inverse Laplace transform and, 443-444,506 Kirchhoff s laws and, 471 Laplace transform and, 443-444,468^184,506 mutual inductance in,480-481 Ohm's law in, 470-471 partial fraction expansion in, 444-454,486-489,506 poles (roots) in, 454-455,486 resistor in, 468,506 responses of circuits in, 472-484 superposition, use of in, 482-484 terminal voltage-current equations in, 468-470,506 Thevenin equivalent, use of in, 479-480 transfer function H(s) in, 484-489,506 zeros (roots) in, 454-455,486 Safety, electrical circuits, 25,46-74 Scale-change property, operational transforms, 441,659 Scaling, 564-566,594 frequency (k f ), 564,594 magnitude (k, n ), 564,594 op-amp filter design using, 565-566 Second-order circuits, 264-305 characteristic equation for, 267,269-270,286,297 critically damped responses, 269,277-278,283,286, 297-298 defined, 266, general solution for responses of, 266-270 ignition, example of, 265,294-296 integrating amplifiers and, 289-293,297 natural response of, 266-279,285-289,297 overdamped responses, 269,271-273,282,286,297-298 parallel RLC, 266-285,297 resistor-inductor-capacitor (RLC) circuits, 264-305 second-order differential equation for, 266-270 series RLC, 285-289,297-298 step response of, 281-289,297 underdamped responses, 269,274-277,283,286-288, 297-298 Second-order differential equations, 266-270 Secondary winder, transformers, 335 Self-inductance, 176-182,193-194,204 Sensitivity analysis, resistors, 89,125-128 Sequential switching, 236-240,246 resistor-capacitor (RC) circuits, 236,239 resistor-inductor (RL) circuits, 236-238 Series-connected (in series) circuits, 37-39,48,58,76, 187-189,203 capacitors, 188-189,203 inductors, 187,203 nodes, 37-39,48 resistors, 58,76 Series RLC circuits, 285-289 characteristic equation of, 286 circuit symbols for, 285-286 critically damped response, 286 natural response, 285-289 Neper frequency (a), 286 overdamped response, 286 resonant radian frequency (w 0 ), 286 step response, 285-289,297-298 underdamped responses, 286-288 Short circuit, 35 Siemens (S), unit of, 31 Sifting property, 434 792 Index Signum function, Fourier transform of, 653 Simultaneous equations, 91-93 current analysis using, 92-93 determination of number of, 91-92 systematic approach using, 92-93 Single-phase equivalent circuits, 403-404,419 Sinusoidal response, 311-312,347,495-497,505-506,663 Fourier transform for, 663 steady-state analysis and, 311-312,347 steady-state component of, 31 surge suppressors, 505 transfer function H(s) and, 495-497,506 transient component of, 312 Sinusoidal sources, 308-311,347,429,458-459 complete circuit response and, 429,458-459 current, 308-309 period, 308 phase angle (<£), 308 root-mean-square (rms) value of, 309-311 sources, 308-311,347 steady-state analysis, 308-311,347 voltage, 308-310 Sinusoidal steady-state analysis, 306-359,360-397 delta-to-wye (A-to-Y) transformations, 326-328 frequency domain, 312-322,329-343 heating appliances, 361,384-385 household distribution circuit, 307,346 ideal transformer, 338-343,347 impedance (Z), 320,322-332,336,343,347 Kirchhoff's laws, 321-322 mesh-current method, 333-334 node-voltage method, 332-333 Norton equivalent circuit, 329-330 passive circuit elements, 317-320 phasor diagrams for, 344-346 phasors, 312-317,347 power calculations, 360-397 reactance, 320 response, 311-312,347 source transformations, 329-332 sources, 308-311,347 Thevenin equivalent circuit, 329-332 transformers, 334-343,347 voltage-to-current (V-I) relationships, 317-320 Sinusoidal voltage phases, 400-401,419 Source transformations, 71-73,109-116,129,326-332 circuit analysis using, 109-116,129 defined, 110 delta-to-wye (A to Y), 326-328 equivalent circuits, 71-73 frequency domain, 326-332 impedance (Z) and, 326-332 Norton equivalent circuits, 115,329-330 Thevenin equivalent circuit, 115-116,329-332 Sources, 26-29,42-45,47,56,401^102,429,458-459,501-504 active element of, 27 complete circuit response and, 429,458-459 constant (dc),56 controlled, 27 dependent, 26-27,29,42^15,47 ideal current, 26-29,47 ideal voltage, 26-29,47 impulsive, 501-504 independent, 26,29,47 interconnection of, 28-29 Laplace transform and, 429,458-459,501-504 passive element of, 27 sinusoidal, 429,458-459 three-phase voltage, 401-402 Square waves, 604-606,632-634 formation of, 604,606 sinusoid input as, 605,632-634 Steady-state analysis, see Sinusoidal steady-state analysis Steady-state response, 217,467,495-*97,505-506,619-627. 635,663 direct approach to, 621-623 first-order circuits, 217 Fourier series analysis, 619-627,635 Fourier transform for, 663 Laplace transform analysis and, 467,495-497,505-506 periodic functions used for, 619-627 sinusoidal, 495-497,505-506,663 surge suppressors and, 467,505 transfer function H(s) and, 495-497 Step function Ku(t), Laplace transform, 431-432,459 Step response, 212,224-236,246,297-289,297-298,473^78 critically damped, 283,298 defined, 212,246 direct approach, 281-282 general solution for, 231-236,246 indirect approach, 281 Laplace transform, analysis of using, 473^178 magnetically coupled coils and, 235-236 method of calculating, 232-235 multiple mesh circuit, 476^478 overdamped, 282,298 parallel RLC circuits, 280-285,473^75 resistor-capacitor (RC) circuits, 229-236,246 resistor-inductor (RL) circuits, 224-228,231-236,246 resistor-inductor-capacitor (RLC) circuits, 280-285 series RLC circuits, 285-289 underdamped, 283,288,298 Stopband frequencies, 524-525,550 Summing-amplifier circuit, 152-153,164 Supermesh, 104 Supernode, 97-98 Superposition, 122-125,129,482^184 circuit analysis using, 122-125,129 defined, 122 Laplace transform and, 482-484 Surge suppressors, 467,505 Susceptance, 324 Switches, 175,200-202,236-240,246,467,498-505 arcing, 176 Index 793 capacitor circuit operation, 498^99 first-order circuits, 236-240,246 impulse functions created by, 498-504 impulsive sources for, 501-504 inductor circuit operation, 499-501 Laplace transform and, 467,498-505 proximity, 175,200-202 sequential switching, 236-240,246 surge suppressors, 467,505 Symmetry, 611-616,634,688,698 even-function, 611-612,634 Fourier coefficients, effects of on, 611-616,634 half-wave, 613-614,634 odd-function, 612-613,616,634 quarter-wave, 614-615,634 reciprocal two-port circuits, 688,698 T T-equivalent circuit, 731-732 Tee (T) interconnection, 71 Telephone line frequency response, 645,672 Terminals, 30-33,35-36,146-150,164,680 circuit connections, 35-36 current (/) input constraints, 147-148,164 operation amplifiers (op amps), 146-150,164 resistor power and, 31-32 two-port circuits, 680 variables (voltage and current), 146-147 voltage (v) input constraints, 147-148,165 Terminated two-port circuits, 689-693 Thevenin equivalent circuits, 113-119,129,329-332,479-480 amplifier circuit using, 118-119 circuit analysis of, 113-119,129 defined, 113 finding equivalent of, 114-115 frequency-domain, 329-332 impedance (Z) in, 329-332 independent sources used for, 117-118 Laplace transform, analysis of using, 479^180 resistance source (/?jh), 113-115 s-domain, use of in, 479-480 source transformations used for, 115-116,329-332 voltage source (Vn,), 113-115 Three-phase circuits, see Balanced three-phase circuits Time constant (T), 216-217,221-222,246 resistor-capacitor (RC) circuits, 221-222,246 resistor-inductor (RL) circuits, 216-217,246 Time domain, 440-443,531,545,648-650,659-660,664-672 convolution in, 659-660 energy, 664-671 Fourier transform and, 648-650,659-660,664-672 frequency domain, relationship to, 531,545 integral for, 648-650,672 integrodifferential equations for, 442-443 Laplace transform and, 440-443 operational transforms, 440-441,659-660 Parseval's theorem for, 664-672 passive-filter circuits and, 531,545 translation in, 440-441,659 Time invariant circuits, 488-489,506 Transducers, 145 Transfer function H(s), 484-497,506,531,536,544,549, 551,578-579 active-filter circuits, 578-579 Butterworth filters, 578-579 circuit analysis, 488^197,506 convolution integral and, 489-495,506 defined, 484 Laplace transform circuit analysis and, 484-497,506 memory and, 494-495 partial fraction expansion, use of in, 486-489,506 passive-filter circuits and, 531,536,544,549,551 poles and zeros of, 486 steady-state sinusoidal response and, 495^-97,506 time invariant circuits, 488-489,506 weighting function, 494-495 Transformation, see Source transformation Transformers, 334-343,347,731-738 defined, 335 dot convention for, 341-342 equivalent circuits and, 731-738 frequency domain and, 334-343 ideal, 335,338-343,347,731-738 limiting values of, 338-340 linear, 335-338,347 primary and secondary winders for, 335 reflected impedance (Z r ), 336,347 sinusoidal steady-state analysis, 334-343,347 voltage and current ratios. 340-342 Transient response, 217,475-476,506,662 circuit analysis of, 212 Fourier transform for, 662 Laplace transform for, 475-476,506 Translation, operational transforms for, 440-441,659 Transmission parameters, two-port circuits, 683 Triangular waves, 604,606 Trigonometric form of Fourier series, 617-618,634 Trigonometric identities, 759 Twin-T notch filter, 588-589 Two-port circuits, 678-703 analysis of, 689-693 black box amplifier, 679,697 conversion of parameters, 684-686 hybrid parameters, 683 interconnected, 694-697 parameters of, 680-688,698 reciprocal, 687-688,698 symmetric, 688,698 terminal equations for, 680 terminated, 689-693 transmission parameters, 683 z parameters, 681,689-692 Two-wattmeter method, three-phase circuits, 415^117,419 794 Index U Unbounded response, 240-241,246 Underdamped responses, 269,274-277,283,286-288, 297-298 parallel RLC circuits, 269,274-277,283,297-298 series RLC circuits, 286-288 Unilateral Laplace transform, 430-431 Unit doublet (moment function), 435 Unit impulse function 8(/), 433-434,459 Unit step function Ku(t), 431,459,654 Laplace transform of, 431,459,654 Fourier transform of, 654 V Variable-parameter function, 433-434 Voltage (v), 11-13,26-29,38,40-41,47,66-69,113-115, 146-150,164,176-177,184-186,203,221-222, 308-310,347,400-405,419,468-470,506 a-, b-, and c-phase, 400,419 capacitors, 183-186,203,469-470,506 defined, 11-12 electric charge and, 11-13 gain (,4), 147 inductors, 176-177,203,468-469,506 input constraints, 147-148,165 Kirchhoff s law (KVL), 38,40-41 line, 404-405 measurement of, 66-69 negative (acb) phase sequence, 400 node-voltage equation (V N ), 402^103 operational amplifiers (op amps), 146-150,164 positive (abc) phase sequence, 400 RC circuits, deriving expression for, 221-222 reference direction, 13 resistors, 468,506 s-domain equations for, 468-470,506 sinusoidal phases, 400-401,419 sinusoidal source, 308-310,347 sources, 26-29,47,401-402 terminal variables, 146 Thevenin equivalent (^ Th ), 113-115 three-phase, 401^105 transfer characteristics of, 146-147,164 Voltage-divider circuits, 61-62,76 Voltage division circuit analysis, 64-66,76 Voltage ratios, transformers, 340-342 Voltage to current (v-i) relationships, 176-177,183,317-320 capacitors, 183,319-320 frequency domain, 317-320 impedance (Z) of, 320 inductors, 176-177,318-319 reactance of, 320 resistors, 317-318 Volt-amp reactive (VAR), unit of, 365,386 Voltmeter, 66-67,76 W Watt (W), unit of, 365,386 Wattmeter, defined, 415,419 Waveforms, 604-606 Wavelength (y), defined, 7 Weighting function, transfer function concept of, 494-495 Wheatstone bridge, 69-70,76 Wye (Y) connected loads, 410^111,413,419 Wye (Y) interconnection, 71 Wye-delta (Y-A) circuit, analysis of, 407-409 Wye-wye (Y-Y) circuit, analysis of, 402^407 Z z parameters, two-port circuits, 681,689-692 Zero frequency, 530 Zeros (roots), 454-455,486,741-742,749-751 Bode plots and, 741-742,749-751 complex, 749-751 rational function F(s), 454-455 real, first order, 741-742 transfer function H(s), 486 Periodic Functions /(0 A - Triangular Wave 7/2 7 37/2 Half-wave rectified sine /(0 /i " 0 A T 27 Square Wave 7/2 7 37/2 Full-wave rectified sine 27 Fourier Series /(o = — S 17 «=1.3,5. Triangular wave sin nojQt A A . 2A ^, cos /10),,( /(0 = — + T sm w <>' ^ ~J—r 77 * ^ /»=2.4,6. " ~~ 1 Half-wave rectified sine /YA 4/1*1, /(0 = — 2i r, sm IH0 " 1 /(0 2 A «=1.3,5, 4/1 ^ cos no) 0 t Square wave TT TT f=\ (4/r - 1) Full-wave rectified sine . resistor-capacitor (RC) circuits, 220-224,231-236,246, 472-473 resistor-inductor (RL) circuits, 214-220,231-236,246 resistor-inductor-capacitor (RLC) circuits, 266-279 series RLC circuits, 285-289. (RL) circuits, 212,214-220,224-228, 236-238,246 resistor-inductor-capacitor (RLC) circuits, 264-305, 473^175 second-order circuits, 264-305 sequential switching, 236-240,246 series RLC circuits,. 282,298 parallel RLC circuits, 280-285,473^75 resistor-capacitor (RC) circuits, 229-236,246 resistor-inductor (RL) circuits, 224-228,231-236,246 resistor-inductor-capacitor (RLC) circuits, 280-285