Page 180 - Schaum's Outline of Theory and Problems of Electric Circuits

169
HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
CHAP. 8]
nonzero, the function is a damped cosine. Only negative values of are considered. If and ! are zero,
the result is a constant. And finally, with ! ¼ 0and nonzero, the result is an exponential decay
st
function. In Table 8-1, several functions are given with corresponding values of s for the expression Ae .
Table 8-1

f ðtÞ s A
10e 5t 5 þ j 10
5 cos ð500t þ 308Þ 0 þ j500 5
2e 3t cos ð100t 458Þ 3 þ j100 2
100:0 0 þ j0 100.0

When Fig. 8-10 is examined for various values of s, the three cases are evident. If ¼ 0, there is no
damping and the result is a cosine function with maximum values of V m (not shown). If ! ¼ 0, the
function is an exponential decay with an initial value V m . And finally, with both ! and nonzero, the
damped cosine is the result.

8.6 GENERALIZED IMPEDANCE (R; L; C) IN s-DOMAIN
st
A driving voltage of the form v ¼ V m e applied to a passive network will result in branch currents
j st
j st
st
and voltages across the elements, each having the same time dependence e ; e.g., I a e e , and V b e e .
Consequently, only the magnitudes of currents and voltages and the phase angles need be determined
(this will also be the case in sinusoidal circuit analysis in Chapter 9). We are thus led to consider the
network in the s-domain (see Fig. 8-11).

Fig. 8-11
j st
st
j st
A series RL circuit with an applied voltage v ¼ V m e e will result in a current i ¼ I m e e ¼ I m e ,
which, substituted in the nodal equation
di j st
Ri þ L ¼ V m e e
dt
will result in
V e j
m
st
st
j st
RI m e ¼ sLI m e ¼ V m e e from which I m ¼
R þ sL
Note that in the s-domain the impedance of the series RL circuit is R þ sL. The inductance there-
fore has an s-domain impedance sL.