Page 168 - Schaum's Outline of Theory and Problems of Electric Circuits

FIRST-ORDER CIRCUITS
CHAP. 7]

Fig. 7-39 157

7.32 The circuit of Problem 7.31 has a 50-V source of opposite polarity switched in at t ¼ 0:50 ms, replacing the
first source. Obtain the current for (a)0 < t < 0:50 ms; ðbÞ t > 0:50 ms.
Ans: ðaÞ 1 e 500t ðAÞ; ðbÞ 0:721e 500ðt 0:0005Þ 0:50 ðAÞ

7.33 A voltage transient, 35e 500t (V), has the value 25 V at t 1 ¼ 6:73 10 4 s. Show that at t ¼ t 1 þ the
function has a value 36.8 percent of that at t 1 :

7.34 A transient that increases from zero toward a positive steady-state magnitude is 49.5 at t 1 ¼ 5:0 ms, and 120
at t 2 ¼ 20:0 ms. Obtain the time constant . Ans: 12:4ms

7.35 The circuit shown in Fig. 7-40 is switched to position 1 at t ¼ 0, then to position 2 at t ¼ 3 . Find the
transient current i for (a)0 < t < 3 ; ðbÞ t > 3 .
Ans: ðaÞ 2:5e 50 000t ðAÞ; ðbÞ 1:58e 66 700ðt 0:00006Þ (A)

Fig. 7-40 Fig. 7-41

7.36 An RL circuit, with R ¼ 300
and L ¼ 1 H, has voltage v ¼ 100 cos ð100t þ 458Þ (V) applied by closing a
switch at t ¼ 0. [A convenient notation has been used for the phase of v, which, strictly, should be indicated
as 100t þð =4Þ (rad).] Obtain the resulting current for t > 0.
Ans: 0:282e 300t þ 0:316 cos ð100t þ 26:68Þ (A)

7.37 The RC circuit shown in Fig. 7-41 has an initial charge on the capacitor Q 0 ¼ 25 mC, with polarity as
indicated. The switch is closed at t ¼ 0, applying a voltage v ¼ 100 sin ð1000t þ 308Þ (V). Obtain the
current for t > 0. Ans: 153:5e 4000t þ 48:4 sin ð1000t þ 1068Þ (mA)

7.38 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(þ on top plate)

7.39 Write simultaneous differential equations for the circuit shown in Fig. 7-42 and solve for i 1 and i 2 . The
switch is closed at t ¼ 0 after having been open for an extended period of time. (This problem can also be
solved by applying known initial and final conditions to general solutions, as in Problem 7-17.)
Ans: i 1 ¼ 1:67e 6:67t þ 5 ðAÞ; i 2 ¼ 0:555e 6:67t þ 5 ðAÞ