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and L 1 H, has a 100-V source applied at t 0. t > 0. Ans: 10 1 e 10t (A) Find the current for
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In Fig. 7-39, the switch is closed on position 1 at t 0, then moved to 2 at t 1 ms. Find the time at which the voltage across the resistor is zero, reversing polarity. Ans: 1:261 ms A series RL circuit, with R 100
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and L 0:2 H, has a 100-V source applied at t 0; then a second source, of 50 V with the same polarity, is switched in at t t 0 , replacing the rst source. Find t 0 such that the current is constant at 0.5 A for t > t 0 . Ans: 1:39 ms
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CHAP. 7]
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FIRST-ORDER CIRCUITS
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Fig. 7-39 7.32 The circuit of Problem 7.31 has a 50-V source of opposite polarity switched in at t 0:50 ms, replacing the rst source. Obtain the current for (a) 0 < t < 0:50 ms; b t > 0:50 ms. Ans: a 1 e 500t A ; b 0:721e 500 t 0:0005 0:50 A A voltage transient, 35e 500t (V), has the value 25 V at t1 6:73 10 4 s. Show that at t t1  the function has a value 36.8 percent of that at t1 : A transient that increases from zero toward a positive steady-state magnitude is 49.5 at t1 5:0 ms, and 120 at t2 20:0 ms. Obtain the time constant . Ans: 12:4 ms The circuit shown in Fig. 7-40 is switched to position 1 at t 0, then to position 2 at t 3. transient current i for (a) 0 < t < 3; b t > 3. Ans: a 2:5e 50 000t A ; b 1:58e 66 700 t 0:00006 (A) Find the
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Fig. 7-40 7.36
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Fig. 7-41
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An RL circuit, with R 300
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and L 1 H, has voltage v 100 cos 100t 458 (V) applied by closing a switch at t 0. [A convenient notation has been used for the phase of v, which, strictly, should be indicated as 100t =4 (rad).] Obtain the resulting current for t > 0. Ans: 0:282e 300t 0:316 cos 100t 26:68 (A)
The RC circuit shown in Fig. 7-41 has an initial charge on the capacitor Q0 25 mC, with polarity as indicated. The switch is closed at t 0, applying a voltage v 100 sin 1000t 308 (V). Obtain the current for t > 0. Ans: 153:5e 4000t 48:4 sin 1000t 1068 (mA)
What initial charge on the capacitor in Problem 7.37 would cause the current to go directly into the steady state without a transient Ans: 13:37 mC ( on top plate)
Write simultaneous di erential equations for the circuit shown in Fig. 7-42 and solve for i1 and i2 . The switch is closed at t 0 after having been open for an extended period of time. (This problem can also be solved by applying known initial and nal conditions to general solutions, as in Problem 7-17.) Ans: i1 1:67e6:67t 5 A ; i2 0:555e 6:67t 5 A
FIRST-ORDER CIRCUITS
[CHAP. 7
Fig. 7-42
Fig. 7-43
For the RL circuit shown in Fig. 7-43, nd the current iL at the following times: (a) 1 ms, (c) 0.3 ms, (d) 1. Ans: a 2:00 A; b 2:00 A; c 2:78 A; d 3:00 A
(b) 0 ,
A series RC circuit, with R 2 k
and C 40 mF, has two voltage sources in series with each other, v1 50 V and v2 100u t (V). Find (a) the capacitor voltage at t , (b) the time at which the capacitor voltage is zero and reversing polarity. Ans: a 13:2 V; b 55:5 ms Find the unit-impulse response of the circuit of Fig. 7-44; i.e., v2 for v1  t (a unit-area narrow voltage pulse).   R 1 t= R1 C Ans: v2 2  t u t e R1 R1 C
Fig. 7-44 7.43 In the circuits of Fig. 7-45, RC 5 10 7 and v1 t 10 cos 1000t 3 cos 2000t . Find v2 t . Assume tan  %  when  < 18. Ans: v2 t % 10 cos 1000 t 10 6 3 cos 2000 t 10 6 v1 t 10 6
Fig. 7-45
CHAP. 7]
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