Page 168

CHAP. 7]

157

FIRST-ORDER CIRCUITS

Fig. 7-39 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  e500t ðAÞ; ðbÞ 0:721e500ðt0:0005Þ  0:50 ðAÞ

7.33

A voltage transient, 35e500t (V), has the value 25 V at t1 ¼ 6:73  104 s. Show that at t ¼ t1 þ  the function has a value 36.8 percent of that at t1 :

7.34

A transient that increases from zero toward a positive steady-state magnitude is 49.5 at t1 ¼ 5:0 ms, and 120 at t2 ¼ 20:0 ms. Obtain the time constant . Ans: 12:4 ms

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. transient current i for (a) 0 < t < 3; ðbÞ t > 3. Ans: ðaÞ 2:5e50 000t ðAÞ; ðbÞ  1:58e66 700ðt0:00006Þ (A)

Fig. 7-40

Find the

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 þ ð=4Þ (rad).] Obtain the resulting current for t > 0. Ans:  0:282e300t 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 Q0 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: 153:5e4000t 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 i1 and i2 . 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: i1 ¼ 1:67e6:67t þ 5 ðAÞ; i2 ¼ 0:555e6:67t þ 5 ðAÞ

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