CHAP. 7]

7.10

FIRST-ORDER CIRCUITS

139

RESPONSE OF FIRST-ORDER CIRCUITS TO A PULSE

In this section we will derive the response of a ﬁrst-order circuit to a rectangular pulse. The derivation applies to RC or RL circuits where the input can be a current or a voltage. As an example, we use the series RC circuit in Fig. 7-17(a) with the voltage source delivering a pulse of duration T and height V0 . For t < 0, v and i are zero. For the duration of the pulse, we use (6b) and (6c) in Section 7.3: v ¼ V0 ð1  et=RC Þ V i ¼ 0 et=RC R

ð0 < t < TÞ

ð14aÞ

ð0 < t < TÞ

ð14bÞ

When the pulse ceases, the circuit is source-free with the capacitor at an initial voltage VT . VT ¼ V0 ð1  eT=RC Þ

ð14cÞ

Using (3) and (4) in Section 7.2, and taking into account the time shift T, we have v ¼ VT eðtTÞ=RC i ¼ ðVT =RÞe

ðtTÞ=RC

ðt > TÞ

ð15aÞ

ðt > TÞ

ð15bÞ

The capacitor voltage and current are plotted in Figs. 7-17(b) and (c).

Fig. 7-17

EXAMPLE 7.11 In the circuit of Fig. 7-17(a), let R ¼ 1 k and C ¼ 1 mF and let the voltage source be a pulse of height V0 and duration T. Find i and v for (a) V0 ¼ 1 V and T ¼ 1 ms, (b) V0 ¼ 10 V and T ¼ 0:1 ms, and (c) V0 ¼ 100 V and T ¼ 0:01 ms. We use (14) and (15) with the time constant of  ¼ RC ¼ 1 ms. For convenience, time will be expressed in ms, voltages in V, and currents in mA. We also use the approximation et ¼ 1  t when t  1. (a) V0 ¼ 1 V, T ¼ 1 ms. For 0 < t < 1 ms, v ¼ ð1  et Þ; i ¼ et , and VT ¼ ð1  e1 Þ ¼ 0:632 V

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An