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Supplementary Problems
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8.22 In the RLC circuit of Fig. 8-30, the capacitor is initially charged to V0 200 V. after the switch is closed at t 0. Ans: 2e 1000t sin 1000t A Find the current transient
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A series RLC circuit, with R 200
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, L 0:1 H, and C 100 mF, has a voltage source of 200 V applied at t 0. Find the current transient, assuming zero initial charge on the capacitor. Ans: 1:055 e 52t e 1948t A What value of capacitance, in place of the 100 mF in Problem 8.23, results in the critically damped case Ans: 10 mF
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CHAP. 8]
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HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
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Fig. 8-30
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Fig. 8-31
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Find the natural resonant frequency, j j, of a series RLC circuit with R 200
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, L 0:1 H, C 5 mF. Ans: 1000 rad/s A voltage of 10 V is applied at t 0 to a series RLC circuit with R 5
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, L 0:1 H, C 500 mF. Find the transient voltage across the resistance. Ans: 3:60e 25t sin 139t V In the two-mesh circuit shown in Fig. 8-31, the switch is closed at t 0. Find i1 and i2 , for t > 0. Ans: i1 0:101e 100t 9:899e 9950t A ; i2 5:05e 100t 5:00 0:05e 9950t A A voltage has the s-domain representation 100 308 V. Express the time function for (a) s 2 Np/s, (b) s 1 j5 s 1 . Ans: a 86:6 e 2t V ; b 100 e t cos 5t 308 V Give the complex frequencies associated with the current i t 5:0 10e 3t cos 50t 908 Ans: 0; 3 j50 s 1 A .
A phasor current 25 408 A has complex frequency s 2 j3 s 1 . What is the magnitude of i t at t 0:2 s Ans: 4:51 A Calculate the impedance Z s for the circuit shown in Fig. 8-32, at (a) s 0; b s j1 rad/s, (c) s j2 rad/s, (d) jsj 1. Ans: a 1
; b 1:58 18:438
; c 1:84 12:538
; d 2
Fig. 8-32
Fig. 8-33
The voltage source in the s-domain circuit shown in Fig. 8-33 has the time-domain expression vi t 10e t cos 2t Obtain io t . Ans: 7:07e cos 2t 98:138
In the time domain, a series circuit of R, L, and C has an applied voltage vi and element voltages vR , vL , and vC . Obtain the voltage transfer functions (a) VR s =Vi s , (b) VC s =Vi s :
HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
[CHAP. 8
Ans:
Rs=L ; R 1 s2 s L LC
1=LC R 1 s2 s L LC The response is the voltage Vi s .
Obtain the network function H s for the circuit shown in Fig. 8-34. Ans: s 7 j2:65 s 7 j2:65 s 2 s 4
Fig. 8-34 8.35 Construct the s-plane plot for the transfer function of Problem 8.34. Ans: See Fig. 8-35. 7:02 9:0 2:86 38:918 3:50 51:418
3:61 5:0 56:318 36:878 Evaluate H j3 from the plot.
Fig. 8-35 8.36 Obtain H s Vi s =Ii s for the circuit shown in Fig. 8-36 and construct the pole-zero plot. s s2 1:5 : s2 1
Ans:
H s
See Fig. 8-37.
Write the transfer function H s whose pole-zero plot is given in Fig. 8-38. s2 50s 400 s2 40s 2000
Ans:
H s k
CHAP. 8]
HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
Fig. 8-36 8.38
Fig. 8-37 Use the geometrical
The pole-zero plot in Fig. 8-39 shows a pole at s 0 and zeros at s 50 j50. method to evaluate the transfer function at the test point j100. Ans: H j100 223:6 26:578
Fig. 8-38
Fig. 8-39
A two-branch parallel circuit has a resistance of 20
in one branch and the series combination of R 10
and L 0:1 H in the other. First, apply an excitation, Ii s , and obtain the natural frequency from the denominator of the network function. Try di erent locations for applying the current source. Second, insert a voltage source, Vi s , and obtain the natural frequency. Ans: 300 Np/s in all cases
In the network shown in Fig. 8-40, the switch is closed at t 0. di 25 A=s dt
At t 0 , i 0 and
Obtain the natural frequencies and the complete current, i in if . Ans: 8:5 Np/s, 23:5 Np/s; i 2:25e 8:5t 0:25e 23:5t 2:5 A
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