CHAP. 7]

143

FIRST-ORDER CIRCUITS

RESPONSE OF RC AND RL CIRCUITS TO SUDDEN SINUSOIDAL EXCITATIONS

7.14

When a series RL circuit is connected to a sudden ac voltage vs ¼ V0 cos !t (Fig. 7-19), the equation of interest is Ri þ L

di ¼ V0 ðcos !tÞuðtÞ dt

ð18Þ

The solution is iðtÞ ¼ ih þ ip

where

ih ðtÞ ¼ AeRt=L

and

ip ðtÞ ¼ I0 cos ð!t  Þ

By inserting ip in (18), we ﬁnd I0 : V0 ﬃ I0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R2 þ L2 !2 Then

and

 ¼ tan1

iðtÞ ¼ AeRt=L þ I0 cos ð!t  Þ

L! R

t>0

From ið0þ Þ ¼ 0, we get A ¼ I0 cos . Therefore, iðtÞ ¼ I0 ½cos ð!t  Þ  cos ðeRt=L Þ

Fig. 7-19

7.15

SUMMARY OF FORCED RESPONSE IN FIRST-ORDER CIRCUITS Consider the following diﬀerential equation: dv ðtÞ þ avðtÞ ¼ f ðtÞ dt

ð19Þ

The forced response vp ðtÞ depends on the forcing function f ðtÞ. Several examples were given in the previous sections. Table 7-2 summarizes some useful pairs of the forcing function and what should be guessed for vp ðtÞ. The responses are obtained by substitution in the diﬀerential equation. By weighted linear combination of the entries in Table 7-2 and their time delay, the forced response to new functions may be deduced.

7.16

FIRST-ORDER ACTIVE CIRCUITS

Active circuits containing op amps are less susceptible to loading eﬀects when interconnected with other circuits. In addition, they oﬀer a wider range of capabilities with more ease of realization than passive circuits. In our present analysis of linear active circuits we assume ideal op amps; that is; (1) the current drawn by the op amp input terminals is zero and (2) the voltage diﬀerence between the inverting and noninverting terminals of the op amp is negligible (see Chapter 5). The usual methods of analysis are then applied to the circuit as illustrated in the following examples.

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An