HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
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 0 and nonzero, the result is an exponential decay function. In Table 8-1, several functions are given with corresponding values of s for the expression Aest . Table 8-1 f ðtÞ 5t
10e 5 cos ð500t þ 308Þ 2e3t cos ð100t 458Þ 100:0
5 þ j 0 þ j500 3 þ j100 0 þ j0
10 5 2 100.0
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 Vm (not shown). If ! ¼ 0, the function is an exponential decay with an initial value Vm . And ﬁnally, with both ! and nonzero, the damped cosine is the result.
GENERALIZED IMPEDANCE (R; L; C) IN s-DOMAIN
A driving voltage of the form v ¼ Vm est applied to a passive network will result in branch currents and voltages across the elements, each having the same time dependence est ; e.g., Ia e j est , and Vb e j est . 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).
A series RL circuit with an applied voltage v ¼ Vm e j est will result in a current i ¼ Im e j est ¼ Im est , which, substituted in the nodal equation Ri þ L
di ¼ Vm e j est dt
will result in RIm est ¼ sLIm est ¼ Vm e j est
Vm e j R þ sL
Note that in the s-domain the impedance of the series RL circuit is R þ sL. fore has an s-domain impedance sL.
The inductance there-