404

16.7

THE LAPLACE TRANSFORM METHOD

[CHAP. 16

CIRCUITS IN THE s-DOMAIN

In Chapter 8 we introduced and utilized the concept of generalized impedance, admittance, and transfer functions as functions of the complex frequency s. In this section, we extend the use of the complex frequency to transform an RLC circuit, containing sources and initial conditions, from the time domain to the s-domain. Table 16-2 Time Domain

s-Domain Voltage Term

s-Domain

RIðsÞ

sLIðsÞ þ Lið0þ Þ

sLIðsÞ þ Lið0þ Þ

IðsÞ V0 þ sC s

IðsÞ V0  sc s

Table 16-2 exhibits the elements needed to construct the s-domain image of a given time-domain circuit. The ﬁrst three lines of the table were in eﬀect developed in Example 16.1. As for the capacitor, we have, for t > 0, vC ðtÞ ¼ V0 þ

1 C

ðt iðÞ d 0

so that, from Table 16-1, VC ðsÞ ¼

V0 IðsÞ þ Cs s

EXAMPLE 16.5 In the circuit shown in Fig. 16-4(a) an initial current i1 is established while the switch is in position 1. At t ¼ 0, it is moved to position 2, introducing both a capacitor with initial charge Q0 and a constant-voltage source V2 . The s-domain circuit is shown in Fig. 16-4(b). The s-domain equation is

RIðsÞ þ sLIðsÞ  Lið0þ Þ þ in which V0 ¼ Q0 =C and ið0þ Þ ¼ i1 ¼ V1 =R.

IðsÞ V0 V2 þ ¼ sC sC s

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An