776 Answers to Selected Problems Chapter 14 14.1 a) 2021.27 Hz b) H(jco c ) = 0.7071 /-45°, H(j02a> c ) = 0.981 /-11.31°, H(j5w c ) = 0.196/-78.69° c) v o ((0 c ) = 7.07cos(12,700f - 45°)V, v o (02w c ) = 9.81 cos(2540r - 11.31°) V, v 0 (5a) c ) = 1.96cos(63,500f - 78.69°) V 14.2 a) 31.42 ft b) 3419.98 Hz c) 33 ft, 5252.11 Hz 14.13 a) 5.305 kft b) 333.86 Hz 14.14 a) 150 ft b) 680ft 14.22 a) lOOkrad/s b) 15.9 kHz c) 8 d) 93.95 krad/s e) 14.95 kHz f) 106.45 krad/s g) 16.94 kHz h) 1.99 kHz 14.23 a) 5 kft, 50 mH b) 3.52 kHz, 2.88 kHz c) 636.62 Hz 14.35 a) 1 Mrad/s b) 159.15 kHz c) 15 d) 967.22 krad/s e) 153.94 kHz f) 1.03 Mrad/s g) 164.55 krad/s h) 10.61 kHz 14.36 a) 397.89ft, 3.17 mH b) 4.42 kHz, 3.62 kHz c) 800 Hz 14.46 a) 0.39H,0.10piF b) |VW1 = \V m ^\ = 0.707|V peak |, I^TTOHJ = I^Wl = 0.948|K pcak | c) [VUWHJ = 0.344|V pcak | 14.47 L = 0.225 H,C = 0.057/AF, 0.344|V r pcak | 14.48 63.7 times as large as the DTMF tones Chapter 15 15.6 a) R { = 67.16 ft, R 2 = 212.21ft b) + • »i 750 nF 67.16(1 • 212.21 ft ^^_ ^> b-"^ 15.10 a) R x = 5.10 kft, R 2 = 25.55 kft b) 5.10kn 3 ^f • VA |(1 T -VW 25.55 kH v^^ 15.15 a) 1H, IF, 0.05 ft b) 2.5 H, 250 pF, 5 kft c) 250 pF 2.5 H OVA. 5kn 15.16 a) 1/0 F (1/0* b) -, S 2 + (1/(2)5 + 1 c) 10 kft, 6.4 H, 250 pF u) # 1L e-wv-v- 250 pF 6.4 H 10 kft: e) 1562.55 s 2 + 1562.5.9 + 625 X 10 6 15.30 f cl = 38.52 Hz, f c2 = 1038.52 Hz, R L = 30.65 ft, R H = 826.43 ft 15.31 R L = 21.18 kft, R H = 1.18 kft; if Ri = 1 kft then R f = 6 kft 15.36 a) 3 b) -32.65 dB 15.38 a) First-order stage: 79.58 nF; second-order stage: 151.15 nF, 39.79 nF Answers to Selected Problems 777 b) 159.15 nF b) 2V, '77 79.58 nF lkU —VA.— 1 +2Y — H=I 1 - 4« - COS 110)()1 V lkft •—WW + I I ikn lkft 39.79 nF ;±T V V ) v in . Y i,. — + -^- sina> 0 / + 7T 2K —— T^ r cos Hunt V 77 «=2A6 1 "« 2 ~ cos (nojj) 15.39 a) First section: 3.05 kH, 20.77 kft; second section: 7.35 kil, 8.61 kil b) 3.05 kO + —IM-I(- 10 nF 10 nF : 7.35 kO HHHf- 10 nF 10 nF : 20.77 kil 8.61 kO I 15.60 a) R = 5305 £l,aR = 10,478X1, (1 - a)R = 133 ft, C = 15 nF,2C = 30 nF b) 15 nF 15 nF ^ It c) J 2 + 4 X 10V s 2 + IOOTT.? + 4 X lOV 15.61 C = 39.79 nF, |H(/w)L« = 20 -01 dB, |#07#A)I = 17.04 dB 15.62 Choose i?! = 100 kil, then R 2 = 400 kft, C, - 7.96 nF Chapter 16 v 4K 16.1 a) 1 y\ — sin nct) U t V 77 «=fe " ^, 160 _ . 320 16.2 + 20 sin aU 77 W ,,=2¾ («* - 1) 16.3 a) to (m = 31,415.93 rad/s,w„ b = 157.08 krad/s b) /„a = 5000 Hz,/ ob = 25 kHz c) a m = 0, a vb = 25 V d) % a = 0 for A: even; «>t a = -—j- sin — for fc odd; 240 bk& = 0 for A: even; b k . A = —- for k odd. irk a kb = —- sin —— V for all k; b kb = 0 for all k. irk 4 e) v a (t) = 80 £ ( 1 . nir 3 . \ — Y — sin —- cos «tt„f H— sin nwj V ""«=UAA * 2 " J /N _ 200 " (\ . nir \„ v b (t) = 25 + > j — sin —— cos nioj V 16.11 a) 27rrad/s b) yes c) no d) no 16.12 a) 62.5 Hz b) no c) yes d) yes e) yes f) a v = 0; a k = 0 for all k;b k - 0 for k even; 0.16 .kit ov = -, -, sin —— for k odd 77½ 2 4 ~ V(«TT) 2 + 4 16.22 a) 10 2 ——2 cos (nco Q t - 6 n ) A, «=1.3.5. d„ — tan" mr b) 26.23 A 778 Answers to Selected Problems 16.27 a) 41.998 cos(2000f - 0.60°) + 13.985 cos(6000f + 177.32°) + 5.984 cos( 14,0001 - 175.83°) V b) The 5th harmonic at 10 krad/s is eliminated by the bandreject filter whose center frequency is 10 krad/s. 16.28 26.83 cos(500/ + 63.43°) + 16.64 cos( 1500/ - 146.31°) + 11.14cos(2500r + 21.80°) V 16.34 a) 59.46 W b) 60 W c) -0.9041% 16.35 1.85 W 16.36 a) 74.5356 V(rms) b) 74.5306 V(rms) 16.39 a) 117.55 V(rms) b) -2.04% c) 69.2765 V(rms), -0.0081 % V V b) |C 1 < ± 1 1 o > 1 10 * f 9 8 7 6 5 4 3 2 1 i — i - • « i < i —•— i ! -3-2-10 1 2 d„(dcg) 135 90 f45 16.45 C„ = ^.C / . n ±1,±2, 6.46 a) 480 W b) 68.58 V c) -2.02% .6.49 a) \ 20 18 16 14 12 10« 8 6 4 2 f) (V) — < > - - i i > t > i —o— 1 -7-6-5-4-3-2-1 ¢12 3 4 5 6 7 -«1-1 -90 -135 0 t (dcg) 135 90 45 L 16.50 a) A a (mA) 200 150 100 50 0 180 90.56 38-2 2603 19 .1 15.4 12.73 T T f • j 0 u>„ 2<o (| 3o> (( 4o) (] 5a)„ 6a>„ 0° «0)„ 012345678 012345678 140 122.48° 120 100 80 60 40 20 0 101.98° 97 26° 90 c 90 : 0 OJ () 2(o„ 3o)„ 4o) () 5o) 0 6&>„ Answers to Selected Problems 779 b) I C, | (mA) 200 180 13.02 nc^ 19-1 6.37 7.7 9.55 • t f T T I 100 45.28 45.28 13.02 19.1 9-55 7.7 6.37 0 y.DO 7.7 6 1 T y y t -6-5-4-3-2-1 0 12 3 4 5 6 150 -6 -5 -4-3-2-10 1 2 3 4 5 6 • • • -90° -90° • -90° -97.26° -101.98°_ 12 2.48' 122^48° 101.98° 97.26° 90° • 90° 90° 1 150 Chapter 17 17.1 a) / b) 0 c) 2A COT COS(WT/2) — 2 sm(cor/2) of -20 -15 -10 10 15 20 17.2 a) co n A 4TT sin(w,//4) (co n t/4) b) 79.58 X l(T 3 ftv4 c) , 2(« z - co 1 ) 17.5 a) \ ^ « — co b) -/48fl(o ; y (a 2 + co 2 ) 4 + a 2 + (co - co a ) 2 a 2 + (co + co 0 ) 2 —ja ja a 2 + (co - co a ) 2 a 2 + (co + co 0 ) 2 e) e- JMt " sin 17.19 a) (co + COQ)T/2 sin + (&j - co () )r/2 2 (co + co 0 )(r/2) 2 (co - CO 0 )(T/2) b) F(co) -*TT[8(CO - COQ) + 8(co + co Q )) 17.20 a) 6e~ 50( u(t) mA b) Yes, check the initial conditions and final values. 17.28 a) [12.5e -/ - 7.5e" 5/ ]«(0 + 5e 5 'u(-t) V b) 5 V c) 5V d) (12.5<T' -1.5e- 5t )u(t)V e) Yes 17.30 lcos(5000f + 90°) A 17.40 a) [-24e~' + 32e~ t/2 ]u(t) + 8e'u(-t)V b) •5 -4 -3 -2 -1 0 780 Answers to Selected Problems c) -5 -4 -3 -2-10 1 2 d) 900 J e) 320 J f) 95.95% g) 99.75% Chapter 18 18.2 y n = 168.75 mS; y l2 = -156.25 mS; y 2l = -156.25 mS;y 22 = 218.75 mS 18.3 18.8 18.13 18.14 18.29 18.30 18.34 18.40 Zn = 13ft;z 12 = 12 H; 22i = 12ft;z 22 = 16 ft fl U = -4 X 10~ 4 ;a 12 = -20 ft; a 21 = -0.5fxS;a 22 = -0.02 g u = 12.5 fiSlgn = 1*5; g 2l = -250;g 22 = 50Mft a) y n =2QfiS;y u = 30nS; y 2l = 5/LIS;>?22 = 20 nS b) y n = 20MS;)/ 12 = 30nS; y 21 = 5/xS;y 22 = 20 nS 12.5 a) 28V(rms) b) 11.20mW c) 2.88 /xW 420MW 3.75 V Index a-, b-, and c-phase current and voltage, 400,419 Acceptance ratings, 367 Active filter circuits, 558-603 bandpass filters, 566-570,585-588,594 bandreject filters, 570-573,585,588-591,594 bass volume control, 559,591-593 block diagrams, 567-568 Bode plots, 560-563 broadband filters, 567-573,594 Butterworth filters, 578-586,594-595 cascading identical filters, 574-577,594-595 first-order, 560-564 higher-order, 573-591,594 high-pass filters, 562-563,585,594 low-pass filters, 560-562,578-584,594 narrowband filters, 586-591,595 op amp filters, 560-586,594 prototype filters, 561-566,594 scaling, 564-566,594 Adjoint of a matrix, 715-716 Admittance (Y), 324,347 Ammeter, 66-67,76 Amplifier circuits, 98-99,105-106,118-119,144-173, 241-244,246,289-293 analysis of, 98-99 integrating, 241-244,246,289-293 mesh-current method for, 105-106 node-voltage method for, 98-99 operational (op amp), 144-173 responses and, 241-244,246,289-293 Thevenin equivalent in, 118-119 Amplitude modulation, 659 Amplitude plots, 742-747,751-754 Amplitude spectrum, 630-632,647-648 periodic functions described using, 630-632 transition from periodic to aperiodic, 647-648 Analog meters, 67,76 Aperiodic functions, Fourier transform, 644,646-648,672 Apparent power, 370,386 Attenuation, defined, 522 Average power (P), 363-367,373-375,380-382,386, 410-411,415-417,419,623-627,635 acceptance ratings, 367 calculations of, 363-367,373-375,386 Fourier series analysis for, 623-627,635 maximum power (P max ) transfer, 380-382 measurement of, 415—417 periodic functions, calculations with, 623-627,635 power factor (pf), 365,386 sinusoidal steady-state analysis for, 363-367,373-375,386 three-phase circuits, 445-446,452-454 two-wattmeter method, 415-417,419 wye (Y) connected loads, in, 410-411,419 B Balanced three-phase circuits, 398-427 average power in, 410-411,415-417,419 circuit symbols for, 401^102,407 complex power in, 411,419 conditions for, 403 delta (A) connected loads,411—413 designing characteristics of, 398,419 electrical power, transmission and distribution of, 399, 418-419 instantaneous power in, 412,419 line current, 405 line voltage, 404-405 neutral terminal for, 401 phase current, 405 phase voltage, 405 power calculations in, 410-415 single-phase equivalent circuits and, 403-404,419 sinusoidal voltage phases for, 400-401,419 sources of three-phase voltage, 401-402 unspecified loads, 414 voltages (v), 400-405 wye (Y) connected loads, 410-411,413 wye-delta (Y-A) circuit, analysis of, 407-409 wye-wye (Y-Y) circuit, analysis of, 402-407 Bandpass filters, 525,536-545,551,566-570,585-588,594 active filter circuits, 566-570,585-588,594 bandwidth (/3), 537,539-540,551 broadband, 567-573,594 Butterworth, 585 cascading, 567-570 center (resonant) frequency (w„), 536-540,551 cutoff frequency (o» c ) 538-540 frequency response plots, 525,537 narrowband, 586-588 operational amplifier (op amp), 566-570,585,594 passive filter circuits, 525,536-545,551 qualitative analysis, 537-538 quality factor (Q), 537,540,551,586-587 quantitative analysis, 538-544 relationship between frequency and time domains, 545 series RLC circuit as, 537-545 transfer function H(s) for, 544,551 781 782 Index Bandreject filters, 525,545-549.551,570-573,585, 588-591,594 active filter circuits, 570-573,585,588-591,594 Butterworth filters, 585 cascading, 570-572 cutoff frequency (o) c ), 547-548 frequency response plots, 525,546 narrowband filters, 588-591 operational amplifier (op amp) filters, 570-573,585,594 passive filter circuits, 525,545-549,551 qualitative analysis, 546 quantitative analysis, 546-548 series RLC circuit as, 546-549 transfer function H(s) for, 549,551 Bandwidth (/3), 537,539-540,546-548,551 Bass volume control, 559,591-593 Black box, 58 Black box amplifier, 679,697 Block diagrams, 567-568 Bode plots, 560-563,741-757 amplitude, 742-747,751-754 complex poles and zeros, 749-751 correcting, 752-754 frequency response, 560-563 phase angle, 747-749,755-757 real, first order poles and zeros, 741-742 straight-line amplitude, 742-746,752-754 straight-line phase angle, 747-749 Branch, defined, 90-91 Broadband filters, 567-573,594 Butterworth filters, 578-586,594-595 bandpass, 585 bandreject, 585 design of circuits, 579-582 high-pass, 585,594-595 low-pass, 578-584,594-595 order of cascading, 582-584 transfer function H(s), 578-579 C Capacitance (C), 174,182,203 circuit parameter of, 174,182,203 displacement current, 182 terminal equations for, 182-186,203 Capacitors, 174,182-189,203,319-320,469-470,498-499,506 current (/), 182-185,203 displacement current, 182 energy (iv), 184-186,203 frequency domain, 319-320 impulse function (/) in, 498-499 Laplace transform used for, 469-470,498-499,506 power 0),183-186,203 .v-domain, 469-470,506 series-parallel combinations, 188-189,203 switching operation, 498-499 terminal equations for, 182-186,203,469-470,506 voltage (v), 183-186,203 voltage to current (v-i) relationships, 183,319-320 Cascading active filters, 567-570,574-577,594-595 bandpass, 567-570 bandreject, 570-572 Butterworth, 582-584,594-595 identical lowpass, 574-577,594-595 order of, 574-575,582-584,594-595 Cascading connections, 289-293,297 Center frequency (a>„), 536-540,546-548,551 Characteristic determinant, 706 Circuit analysis, 10-11,30-33,37-45,47^18,71-73,76, 88-143,466-521,689-693 conceptual model, 10-11 delta to wye (A to Y) transformations, 71-73,76 dependant-source circuits, 42-45 ideal circuit components, 11 impulse function K8(t) in, 498-504,506 Kirchhoff s laws, 37-41,48,471 Laplace transform in, 466-506 maximum power transfer, 120-122,129 mesh-current method, 90,99-109,129 node-voltage method, 90,93-99,106-109,129 nonplanar circuits, 90-91 Norton equivalent, 115,129 Ohm's law, 30-33,47-48,470-471 physical prototype, 11 planar circuits, 90-91 .y-domain, 470-489,506 sensitivity, resistors, 89,125-128 simultaneous equations for, 91-93 source transformations, 109-113,129 superposition, 122-125,129,482-484 terminology for, 90-91 Thevenin equivalent, 113-119,129,479-480 transfer function H(s) in, 488-497,506 two-port circuits. 689-693 Circuit theory, 6-7 Circuits, 2-87,212-305,398-427,488-489,522-603, 678-703. See also Amplifier circuits; Circuit analysis; Equivalent circuits; Ideal basic circuit elements active filter, 558-603 balanced three-phase, 398-427 constant (dc) sources, 56 current (/),11-12,37-42 dependent sources, 26-27,29,42-45 electrical engineering and, 2-8 electrical resistance, 30-33 electrical safety of, 25,46-74 energy and, 14-17 first-order, 212-263 frequency-selective, 522-603 ideal basic elements, 12-14,24-55 International System of Units (SI) for, 8-10 Kirchhoff s laws, 37-^J-l models of, 10-11,17-18,34-36 Ohm's law, 30-33 Index 783 open,35 parallel-connected (in parallel), 59-61,76 passive filter, 522-557 passive sign convention, 13 power (/;) and, 14—17 resistive, 56-87 responses of, 212-305 second-order, 264-305 series-connected (in series), 39,58,76 short. 35 time invariant, 488-489 two-port, 678-703 variables of, 2-23 voltage (v), 11-12 Coefficient of coupling, 197,204 Common mode input, 156-157 Common mode rejection ratio (CMRR), 157-159,164 Complex numbers, 725-730 arithmetic operators, 727-728 graphical representation of, 726-727 integer power of, 729 notation, 725-726 roots of, 729-730 useful identities of, 728-729 Complex poles and zeros, 749-751 Complex power (5), 370-374,386,411,419 Component values, 763 Conductance (G), 31 Constant (dc) sources, 56 Controlled sources, 27 Convergence of Fourier integral, 648-650 Convolution, 489^195,506,659-660 Fourier transform of, 659-660 integral for transfer function and, 489-495,506 Cosine function, Fourier transform of, 654-655 Cramer's method,706 Critically damped responses, 269,277-278,283,286,297-298 Current (i), 11-13,26-29,37-41,47-48,66-69,147-149, 164,177-179,182-185,203,214-216,308-309,347,400, 405,419 a-, b-, and c-phase, 400,419 capacitors, 182-185,203 defined, 11-12 displacement, 182 electric charge and, 11-13 inductors, 177-179,203 input constraints, 147-148,164 Kirchhoff s law (KCL), 37-41,48 line, 405 measurement of, 66-69 operational amplifiers (op amps), 147-149,164 RC circuits, deriving expression for, 214-216 reference direction, 13 sinusoidal source, 308-309,347 sources, 26-29,47 terminal variables, 147 Current coil, 415 Current-divider circuits, 63,76 Current-division circuit analysis, 64-66,76 Current ratios, transformers, 340-342 Current to voltage (i-v) relationships, 177-179,182-183 Cutoff frequency (to c ), 524-525,527-528,533,538-540, 547-548,550 bandpass filters, 538-540 bandreject filters, 547-548 bandwidth, relationship to, 539 center frequency, relationship to, 538-539 defined, 524-525,527,550 half-power frequency, 528 high-pass filters. 533 low-pass filters, 527-529 D Damped radian frequency ((o (i ), 274-275 Damping factor (coefficient), 275 d'Arsonval meter movement, 67-68 Decibel (dB), unit of, 739-740 Delta (A) interconnection, 71 Delta (A) connected loads, 411-413 Delta-to-wye (AtoY) equivalent circuits, 71-73,76 Delta-to-wye (A toY) transformations, 326-328 Dependent sources, 26-27,29,47,95-96,102-103 analysis of circuits with, 95-96,102-103 ideal circuit elements of, 26-27,47 interconnections of, 47 mesh-current method for, 102-103 node-voltage method for, 95-96 Derivatives of impulse functions, 435 Derived units, 9 Determinants, 706-709 Difference-amplifier circuit, 155-159,164 common mode input, 156-157 common mode rejection ratio (CMRR), 157-159,164 defined, 164 differential mode input, 156-157 Differential mode input, 156-157 Differentiation, operational transforms for, 438-439,658 Digital meters, 67,76 Direct approach, step response analysis, 281-282 Direct current (dc) sources, 56 Dirichlet's conditions, 607 Displacement current, 182 Dot convention, 190-192,195-196,204,341-342 ideal transformers, 341-342 mutual inductance (M), 190-192,195-196,204 Dual in-line package (DIP), 146 E Effective value, see Root-mean-square (rms) Electrical circuit, defined, 6 Electrical engineering, 2-8 circuit theory, 6-7 784 Index Electrical engineering (Continued) communication systems, 4 computer systems, 4 control systems, 4-5 overview of, 4-8 power systems, 5 problem solving, 3,7-8 role of, 2 signal-processing systems, 5-6 Electrical power, transmission and distribution of, 399, 418-419 Electrical safety and, 25,46-74 Energy (w), 14-17,179-181,184-186,197-200,203-204 capacitors, 184-186,203 inductors, 179-181,203 mutual inductance and storage of, 197-200,204 power and, 14-17 Equivalent circuits, 71-73,76,113-119,129,159-162, 329-332,479-480,731-738 delta to wye (A to Y), 71-73,76 frequency-domain, 329-332 ideal transformers for, 731,735-738 Laplace transform analysis of, 479-480 magnetically coupled coils and, 731-738 Norton, 115,129,329-330 operational amplifier (op amp), 159-162 7r-equivalent, 732-735 pi to tee (77 to T), 71-73 source transformations, 115-116,329-332 T-equivalent, 731-732 Thevenin, 113-119,129,329-332,479-480 transformation of, 71-73,76 Even-function symmetry, 611-612,634 Exponential form of the Fourier series, 627-629,635 F Faraday's law, 193-194 Farads (F), unit of, 182 Feedback, op amps, 147-148,164 Feedback resistors, 291-293,297 Filters, 522-523,550. See also Frequency-selective circuits Final-value theorem, Laplace transform, 455-457 First-order active filters, 560-564 First-order circuits, 212-263 defined, 212,214,246 flashing light, 213,245-246 general solutions for, 231-236,246 integrating amplifier, analysis of, 241-244,246 natural response of, 212,214-224,231-236,246 resistor-capacitor (RC), 212,214,220-224,229-236, 239,246 resistor-inductor (RL), 212,214-220,224-228, 236-238,246 sequential switching, 236-240,246 steady-state response of, 217 step response of, 212,224-236,246 unbounded response of, 240-241,246 Flashing light circuit, 213,245-246 Fourier series, 604-643 amplitude spectrum, 630-632 analysis, 607-608 application of, 619-623 average-power calculations, 623-627,635 coefficients, 607-616,634 Dirichlet's conditions, 607 exponential form of, 627-629,635 fundamental frequency, 607,634 harmonic frequency, 607,634 periodic functions and, 604-607,611-616,619-627, 630-632,634-635 phase spectrum, 630-632 root-mean-square (rms) value, 626-627,635 square wave input of sinusoids, 605,632-634 steady-state response, 619-627,635 symmetry and, 611-616,634 trigonometric form of, 617-618,634 waveforms, 604-606 Fourier transform, 644-677 amplitude spectrum transition, 647-648 aperiodic functions of, 644,646-648,672 circuit applications of, 661-663 convergence of Fourier integral, 648-650 cosine function, 654-655 defined, 646 derivation of, 646-648 elementary functions, 655 integral, 646-650 inverse, 647 Laplace transforms used to find, 650-653 mathematical properties of, 655-657 operational, 657-661 Parseval's theorem, 664-672 signum function, 653 sinusoidal steady-state response using, 663 telephone line frequency response, 645,672 time-domain functions of, 648-650,672 transient response using, 662 unit step function, 654 Frequency (/), defined, 7 Frequency domain, 312-322,329-343,441,531,545,659-660 convolution in, 660 defined, 313 delta-to-wye (A to Y) transformations, 326-328 Fourier transform and, 659-660 impedance (Z), 320,322-332,336 Kirchhoff s laws, 321-322 Laplace transform and, 441 mesh-current method, 333-334 node-voltage method, 332-333 Norton equivalent circuit, 329-330 operational transforms of, 441,659-660 passive-filter circuits and, 531,545 phasors, 312-317,347 reactance, 320 sinusoidal steady-state analysis and, 312-322,329-343 Index 785 source transformations and, 326-332 Thevenin equivalent circuit, 329-332 time domain, relationship to, 531,545 transformers, 334-343 translation in, 441,659 voltage to current (v-i) relationships, 317-320 Frequency regions, 530 Frequency response, defined, 522 Frequency response plots, 524-525,560-563 Bode plots, 560-563 magnitude plot, 524-525 phase angle plot, 524-525 Frequency scaling (kf), 564,594 Frequency-selective circuits, 522-603 active filter circuits, 558-603 attenuation, 522 circuit symbols for, 524 filters, 522-523,550 passive filter circuits, 522-557 Functional Laplace transforms, 431,436^-37,459 Fundamental frequency, Fourier series, 607,634 G Gain (A), 147,151 Graphical representation of complex numbers, 726-727 H Half-power frequency, 528 Half-wave symmetry, 613-614,634 Harmonic frequency, 607,634 Heating appliances, 361,384-385 Henrys (H), unit of, 176 High-pass filters, 524-525,532-536,551,562-563,585, 594-595 active filter circuits, 562-563,585,594 Butterworth, 585,594-595 cutoff frequency (co c ), 533 frequency response plots, 524-525 operational amplifier (op amp), 562-563,585,594 passive filter circuits, 524-525,532-536,551 qualitative analysis, 532-533 quantitative analysis, 533-536 series RC circuits, 526-532 transfer function H(s) for, 536,551 Higher-order active filters, 573-591,594 Household distribution circuit, 307,346 Hybrid parameters, two-port circuits, 683 I Ideal basic circuit elements, 12-14,24-55,174-211 active, 27 capacitors, 174,182-189,203 closed loop (path), 38,48 current (/), 26-29,37-42 dot convention, 190-192,195-196 electrical resistance, 30-33 electrical safety and, 25,46-74 inductors, 174,"l76-182,187-200,203 mutual inductance, 176,189-200,203-204 nodes, 37-38,48 passive sign convention, 13,176 passive, 27,176,203 polarity reference, 190-196 proximity switches, 175,200-202 reference direction, 13 resistors, 30-33,47-48 self-inductance, 176-182,193-194,196-197,203 series-connected (in series), 39 series-parallel combinations, 187-189,203 sources, 26-29,42-45,47 terminal equations for, 176-186,203 voltage (v), 26-29 Ideal transformers, 335,338-343,347,383-384,731-738 dot convention for, 341-342 equivalent circuits in, 735-738 frequency domain and, 334-343 impedance matching using, 343 limiting values of, 338-340 magnetically coupled coils and, 338,731-738 maximum power transfer with, 383-384 polarity of, 341-342 properties of, 338 sinusoidal steady-state analysis, 335,338-343,347 voltage and current ratios, 340-342 Identity matrix, 714-715 Ignition circuit, 265,294-296 Impedance (Z), 320,322-332,336,343,347,380-381 admittance ( Y), 324,347 combined in series and in parallel, 322-326 delta-to-wye (A to Y) transformations, 326-328 frequency domain source transformations, 326-332 matching using ideal transformers, 343 maximum power transfer with restricted, 380-381 Norton equivalent circuit, 329-330 reflected (Z,),336,347 susceptance, 324 Thevenin equivalent circuit, 329-332 voltage to current (v-i) relationships, 320 Improper rational function, 444,453^154,459 Impulse, defined, 433 Impulse function KS(t), 433-435,459,498-504,506 capacitor circuit analysis and, 498-499 circuit analysis using, 498-504,506 circuit symbols for, 498 defined, 433-434 derivatives of, 435 inductor circuit and, 499-501 Laplace transform and, 433-435,459,498-504,506 moment function (unit doublet), 435 sifting property, 434 sources, 501-504 strength (K) of, 433 switching operations for, 498-501,506 . 56 current (/),11-12,37-42 dependent sources, 26-27,29,42-45 electrical engineering and, 2-8 electrical resistance, 30-33 electrical safety of, 25,46-74 energy and, 14-17 first-order,. E Effective value, see Root-mean-square (rms) Electrical circuit, defined, 6 Electrical engineering, 2-8 circuit theory, 6-7 784 Index Electrical engineering (Continued) communication. 564,594 Frequency-selective circuits, 522-603 active filter circuits, 558-603 attenuation, 522 circuit symbols for, 524 filters, 522-523,550 passive filter circuits, 522-557 Functional