134

FIRST-ORDER CIRCUITS

[CHAP. 7

Fig. 7-10

7.7

COMPLEX FIRST-ORDER RL AND RC CIRCUITS

A more complex circuit containing resistors, sources, and a single energy storage element may be converted to a The´venin or Norton equivalent as seen from the two terminals of the inductor or capacitor. This reduces the complex circuit to a simple RC or RL circuit which may be solved according to the methods described in the previous sections. If a source in the circuit is suddently switched to a dc value, the resulting currents and voltages are exponentials, sharing the same time constant with possibly diﬀerent initial and ﬁnal values. The time constant of the circuit is either RC or L=R, where R is the resistance in the The´venin equivalent of the circuit as seen by the capacitor or inductor. EXAMPLE 7.5 Find i, v, and i1 in Fig. 7-11(a).

Fig. 7-11 The The´venin equivalent of the circuit to the left of the inductor is shown in Fig. 7-11(b) with RTh ¼ 4  and vTh ¼ 3uðtÞ (V). The time constant of the circuit is  ¼ L=RTh ¼ 5ð103 Þ=4 s ¼ 1:25 ms. The initial value of the inductor current is zero. Its ﬁnal value is ið1Þ ¼

vTh 3V ¼ ¼ 0:75 A RTh 4 

Therefore, i ¼ 0:75ð1  e800t ÞuðtÞ

ðAÞ

v¼L

di ¼ 3e800t uðtÞ dt

ðVÞ

i1 ¼

9v 1 ¼ ð3  e800t ÞuðtÞ ðAÞ 12 4

v can also be derived directly from its initial value vð0þ Þ ¼ ð9  6Þ=ð12 þ 6Þ ¼ 3 V, its ﬁnal value vð1Þ ¼ 0 and the circuit’s time constant.

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