CHAPTER 4 Problems and Solutions 不要用看的,要動手! Problem 4 How long will it take the following waveform to pass through 10 cycles? ω = 2,000 rad/s = 2π/T T = 2 π ÷ 2000 rad/s = 3.142 ms 走10個cycles 需要 10× T = 31.42 ms Problem 5 At what instant of time will the following waveform be 6 V? Associate t = 0s with θ of sinθ equal to 0°. v 12 sin( 200 30) 6 12 sin 30 t0 Problem 6 a. At what angle θ (closest to 0°) will the following waveform reach 3 mA? i = 8.6 × 10-3 sin 500t b. At what time t will it occur? v 12 sin( 200 30) 3 i 8.6 10 sin 500 t 3 10 3 3 8.6 10 sin 500 t sin 500 t 0.349 500t sin t 1 0.349 20.426 0.3565rad 0.3565rad 500rad / s 0.713ms Problem 7 What is the phase relationship between the following pairs of waveforms? 相角-72° 相角-16° a. 電流領先電壓 56° b. 電流領先電壓140° Problem 8 Write the sinusoidal expression for a current i that has a peak value of 6 μA and leads the following voltage by 40° i的 peak value 為6 μA ,且領先電壓40° i (t) = 6×10-6A sin(1000t+46°) Problem 9 Write the sinusoidal expression for a voltage V that has a peak value of 48 mV and lags the following current I by 60° v的 peak value 為48 mV ,且落後電流60° v (t) = 48×10-3V sin(ωt-90°) Problem 10 Determine the effective value of each of the following. a.Veff b.I eff v peak 45.25V 2 i peak 4.243mA 2 c.Veff Vdc 20V Problem 11 Write the sinusoidal expression for each quantity using the information provided a. Ieff = 36 mA, f = 1 kHz, phase angle = 60° b. Veff = 8V, f =60 Hz, phase angle = -10° a.i( t ) 2 36mA sin( 2f 60) 3 50.9 10 A sin( 6,283.2 t 60) b.v ( t ) 2 8V sin( 2f 10) 11.31V sin( 377 t 10) Problem 12 Determine the average value of the following. Problem 13 Determine the average value of the waveform in Fig. 4.80. v av A 1 A BC D 8ms ( 2 ms )( 4 V ) B (1ms )( 2 V ) 2 C 1 (1ms )( 2 V ) 2 v av 0.25V D 1 2 (1ms )( 2 V ) Problem 14 Determine the average value of the waveform in Fig. 4.81. 求出交點 v av A BC D E 8ms v av 0.5mV Problem 15 Determine the average value of i2 from θ= 0 to π if i = 6 sin θ (integral calculus required). i av (6 sin ) d 2 0 18A 2 Problem 16 a. Determine the sinusoidal expression for the voltage drop across a 1.2-kΩ resistor if the current iR is 8×10-3 sin 200t. b. Find the power delivered to the resistor. c. What is the power factor of the load? a.v peak i peak R 9.6V b.I eff i peak 2 5.657mA v R ( t ) 9.6V sin 200t PI 2 eff R 38.4mW Problem 17 a. Find the sinusoidal expression for the current through a 2.2-kΩ resistor if the power delivered to the resistor is 3.6 W at a frequency of 1000 Hz. b. Find the sinusoidal expression for the voltage across the resistor. a.P I 2 eff R 3.6 W i peak I eff 40.45mA 2I eff 57.2mA i( t ) 57.2mA sin( 6.283 10 t ) 3 b.v peak i peak R 125.84V v( t ) 125.84V sin( 6.283 10 t ) 3 Problem 18 a. Find the sinusoidal expression for the voltage drop across a 20-mH coil if the current iL is 4 sin(500t + 60°) b. Find the power delivered to the coil. c. What is the power factor of the load? a.X L L (500rad / s) 20mH 10 v p i L X L ( 4A )(10) 40V v( t ) 40V sin( 500 t 150) 電壓領先電流90º b.P 0W c.Power factor FP 0 Problem 19 Determine the sinusoidal expression for the current ic of a 10-μF capacitor if the vo1tage across the capacitor is Vc = 20× 10-3 sin(2000t + 30°) XC ip 1 C v peak XC 1 6 ( 2000rad / s)(10 10 F) 20mV 50 50 0.4mA i C ( t ) 0.4mA sin( 2000 t 120) 電流領先電壓90º Problem 20 a. For the following pairs determine whether the element is a resistor, inductor, or capacitor. b. Determine the resistance, inductance, or capacitance. 1. v = 16 sin(200t + 80° ) i = 0.04 sin(200t-10° ) 2. v = 0.12 sin(1000t + 10° ) i = 6×10-3cos(1000t + 10° ) v = 16 sin(200t + 80° ) i = 0.04 sin(200t-10° ) 電流落後電壓90º,為電感。 XL L v peak i peak XL 16V 400 L 0.04A 400 200rad / s 2H v = 0.12 sin(1000t + 10°) i = 6×10-3cos(1000t + 10° ) = 6×10-3sin(1000t+100°) 電流領先電壓90º,為電容。 XC C v peak i peak 1 X C 0.12V 20 6mA 1 C 1 (1000rad / s)( 20) 50F Problem 21 a. For the following pairs determine the power delivered to the load. b. Find the power factor and indicate whether it is inductive or capacitive. 1. v = 1600 sin(377t + 360° ) i = 0.8 sin(377t + 60° ) 2. v = 100 sin(106t- 10° ) i = 0.2 sin(106t - 40° ) a.P1 (1600V )(0.8A ) P2 cos 60 320 W 2 (100V )( 0.2A ) cos 30 8.66 W 2 b.FP1 cos 60 0.5 leading FP 2 cos 30 0.866 lagging Problem 22 Convert the following to the other domain. Problem 23 Perform the following operations. State your answer in polar form. Problem 24 Using phasor notation, determine the vo1tage (in the time domain) across a 2.2-kΩ resistor if the current through the resistor is i = 20× 10-3 sin (400t + 30°). i( t ) I( j) I( j) (0.707)( 20mA )30 14.142mA 30 V ( j) I( j) Z R (14.142mA 30)( 2.2k0) 31.11V30 time domain v( t ) 2 31.11V sin( 400 t 30) 44V sin( 400 t 30) I( j) i( t ) (0.707)( 20mA )30 14.142mA 30 v( t ) V( j) 2 31.11sin( 400t 30) 44V sin( 400t 30) Problem 25 Using phasor notation, determine the current (in the time domain) through a 20-mH coil if the voltage across the coil is vL = 4 sin(1000t + 10°). I( j) i( t ) v( t ) V ( j) V ( j) (0.707)( 4V )10 2.828V10 v( t ) V( j) X L L (1000rad / s)( 20mH ) 20 I( j) V( j) ZL 2.828V10 2090 141.4mA 80 time domain i( t ) 2 141.4mA sin( 1000t 80) 0.2A sin( 1000 t 80) Problem 26 Using phasor notation, determine the voltage (in the time domain) across a 10-μF capacitor if the current ic = 40×10-3 sin(10t + 40°). i( t ) I( j) I( j) (0.707)( 40mA )40 28.284mA 40 XC 1 C ZC 1 C 90 10k 90 V ( j) I( j) Z C 282.84V 50 time domain v( t ) 2 282.84V sin( 10 t 50) 400A sin( 10 t 50) Problem 27 For the system in Fig. 4.82, determine the vo1tage v1 in the time domain. V1 ( j) E ( j) V2 ( j) (0.707)(100V)0 (0.707)( 20V )60 (70.71 j0) (7.071 j12.247) 63.639 j12.247 64.807 10.893 time domain v1 ( t ) 2 (64.807 V) sin( 377 t 10.893) 91.64V sin( 377 t 10.893) 再度提醒 V1 ( j) E( j) V2 ( j) (0.707)(100V)0 (0.707)( 20V)60 (70.71 j0) (7.071 j12.247) 63.639 j12.247 64.807 10.893 2 (64.807 V) sin( 377 t 10.893) 91.64V sin( 377 t 10.893) Problem 28 For the system in Fig. 4.83, determine the current i in the time domain. I( j) I1 ( j) I 2 ( j) ... 1.273A 60 time domain i( t ) ... 1.8A sin( 400 t 60) Problem 29 For the series ac network in Fig. 4.84, determine: a. The reactance of the capacitor. b. The total impedance and the impedance diagram. c. The current I d. The voltages VR and VC using Ohm's law. e. The voltages VR and VC using the voltage-divider rule. f. The power to R g. The power supplied by the voltage source e. h. The phasor diagram. i. The Fp of the network. j. The current and voltages in the time domain. a. The reactance of the capacitor. XC 1 C 1 (1000rad / s)(0.1F) 10k b. The total impedance and the impedance diagram. ZT ZR ZC 2k j10k 10.198k 78.69 c. The current I I( j) E( j) ZT 1200 10.198k 78.69 11.767 mA 78.69 d. The voltages VR and VC using Ohm's law. e. The voltages VR and VC using the voltage-divider rule. VC I( j) ZC (11.767mA 78.69)(10k 90) 117.67V 11.31 VC E ( j) ZC ZT f. The power to R P I R (11.767mA ) (2k) 276.925mW 2 2 g. The power supplied by the voltage source e. S EI (11.767mA )(120V) 1.412VA i. The Fp of the network. FP ZR 0.196 leading ZT j.The current and voltages in the time domain. Problem 30 Repeat Problem 29 for the network in Fig. 4.85 , after making the appropriate changes in parts (a), (d) and (e). Problem 31 Determine the voltage vL (in the time domain) for the network in Fig. 4.86 using the voltage-divider rule. Problem 32 For the series RLC network in Fig. 4.87, determine: a. ZT b. I. c. VR, VL, VC using Ohm's law. d. VL using the voltage-divider rule. e. The power to R. f. The Fp. g. The phasor and impedance diagrams. Problem 33 Determine the voltage VC for the network in Fig. 4.88 using the voltage-divider rule. Problem 34 For the parallel RC network in Fig. 4.89, determine: a. The admittance diagram b. YT, ZT c. I. d. IR, IC using Ohm's law. e. The total power delivered to the network. f. The power factor of the network. g. The admittance diagram Problem 35 Repeat Prob1em 34 for the network in Fig. 4.90, replacing IC with IL in part (d). Problem 36 Find the currents I1 and I2 in Fig. 4.91 using the current divider rule. If necessary, review Section 2.10. I1 ( j) I I 2 ( j) I ZL ZL ZR ZR ZL ZR 2A0 2A0 0.8k90 0.8k90 1.2k0 1.2k0 0.8k90 1.2k0 1.11A56.31 1.66A 33.69 Problem 37 For the parallel RLC network in Fig. 4.92, determine: a. The admittance diagram. b. YT, ZT c. I, IR, IL, and IC. d. The total delivered power. e. The power factor of the network. f. The sinusoidal format of I, IR, IL, and IC. g. The phase relationship between e and iL. Y T Y R Y L Y C 1mS 0 1mS 90 0 . 5 mS 90 1mS j0 . 5 mS 1 . 118 mS 26 . 565 ZT 1 YT 0 . 894 k 26 . 565 E I ... 55 . 93 mA 3 . 435 ZT IR IC E 50 mA 0 ZR E 25 mA 120 ZC FP cos 26 . 565 0 . 894 IL E 50 mA 60 ZL 2 P I R R 2 .5 W lagging Problem 38 For the network in Fig. 4.93, determine: a. The short-circuit currents I1 and I2 , b. The voltages V1 and V2 . c. The source current I. a.I1 I b.V1 0 c.I E R2 E 5A R2 V2 20 5A I2 0 Problem 39 Determine the current I and the voltage V for the network in Fig. 4.94 I E R1 R 2 V VR 2 10A E R2 R1 R 2 100V