Page 154 - Schaum's Outline of Theory and Problems of Electric Circuits

FIRST-ORDER CIRCUITS
143
CHAP. 7]
7.14 RESPONSE OF RC AND RL CIRCUITS TO SUDDEN SINUSOIDAL EXCITATIONS
When a series RL circuit is connected to a sudden ac voltage v ¼ V cos !t (Fig. 7-19), the equation
0
s
of interest is
di
Ri þ L ¼ V 0 ðcos !tÞuðtÞ ð18Þ
dt
The solution is
Rt=L
where i h ðtÞ¼ Ae and i p ðtÞ¼ I 0 cos ð!t Þ
iðtÞ¼ i h þ i p
By inserting i p in (18), we find I 0 :
L!
V 0 1
I 0 ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and ¼ tan
2 2
2
R þ L ! R
Rt=L
Then iðtÞ¼ Ae þ I 0 cos ð!t Þ t > 0
þ
From ið0 Þ¼ 0, we get A ¼ I 0 cos . Therefore,
iðtÞ¼ I 0 ½cos ð!t Þ cos ðe Rt=L ފ

Fig. 7-19

7.15 SUMMARY OF FORCED RESPONSE IN FIRST-ORDER CIRCUITS
Consider the following differential equation:

dv
ðtÞþ avðtÞ¼ f ðtÞ ð19Þ
dt

The forced response v ðtÞ depends on the forcing function f ðtÞ. Several examples were given in the
p
previous sections. Table 7-2 summarizes some useful pairs of the forcing function and what should be
guessed for v p 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.