Page 184 - Schaum's Outline of Theory and Problems of Electric Circuits

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
CHAP. 8]

Fig. 8-15 173

The result implies that, in the time domain, iðtÞ¼ 0:248vðtÞ, so that both voltage and current become
1t
infinite according to the function e . For most practical cases, must be either negative or zero.
The above geometrical method does not seem to require knowledge of the analytic expression for
HðsÞ as a rational function. It is clear, however, that the expression can be written, to within the
constant factor k, from the known poles and zeros of HðsÞ in the pole-zero plot. See Problem 8.37.

8.9 THE NATURAL RESPONSE
This chapter has focused on the forced or steady-state response, and it is in obtaining that response
that the complex-frequency method is most helpful. However, the natural frequencies, which charac-
terize the transient response, are easily obtained. They are the poles of the network function.

EXAMPLE 8.10 The same network as in Example 8.8 is shown in Fig. 8-16. Obtain the natural response when a
source VðsÞ is inserted at xx . 0

Fig. 8-16

The network function is the same as in Example 8.8:

2
s þ 12
HðsÞ¼ ð0:4Þ
ðs þ 2Þðs þ 6Þ
The natural frequencies are then 2 Np/s and 6 Np/s. Hence, in the time domain, the natural or transient current
is of the form

i n ¼ A 1 e 2t þ A 2 e 6t