0195136047.pdf

144 TIME-DEPENDENT CIRCUIT ANALYSIS

becomes an inherent part of the final total solution. For cases with zero initial condition, it can be

seen that multiplication bysin the frequency domain corresponds to differentiation in the time

domain, and dividing bysin the frequency domain corresponds to integration in the time domain.

Some other properties of the Laplace transform are listed in Table 3.3.2.

From the entries of Table 3.3.2, observe the time–frequency dualism regarding frequency

differentiation, frequency integration, and frequency shifting. The initial-value and final-value

theorems also display the dualism of the time and frequency domains. As seen later, these theorems

will be effectively applied in the network solutions.

In order to use the tabulated transform pairs of Table 3.3.1, algebraic manipulations will

become necessary to makeF(s) correspond to one of the tabulated entries. While the process may

be simple in some cases, in other casesF(s) may have to be rearranged in a systematic way as

a sum of component functions whose inverse transforms are tabulated. A formalized approach

to resolveF(s) into a summation of simple factors is known as the method ofpartial-fraction

expansion(Heaviside expansion theorem).

Let us consider a rational function (i.e., one that can be expressed as a ratio of two

polynomials),

F(s)=

N(s)

D(s)

(3.3.5)

whereN(s) denotes the numerator polynomial andD(s) denotes the denominator polynomial. As

a first step in the expansion of the quotientN (s)/D(s), we check to see that the degree of the

polynomialNis less than that ofD. If this condition is not satisfied, divide the numerator by the

denominator to obtain an expansion in the form

N(s)

D(s)

=B 0 +B 1 s+B 2 s^2 +···+Bm−nsm−n+

N 1 (s)

D(s)

(3.3.6)

wheremis the degree of the numerator andnis the degree of the denominator.

TABLE 3.3.2Some Properties of the Laplace Transform

Property Time Domain Frequency Domain

Linearity af(t)±bg(t) aF(s)±bG(s)

Time delay of shift f(t−T )u(t−T) e−sTF(s)

Periodic functionf(t)=f(t+nT ) f (t), 0 ≤t≤T F(s)

1 −e−Ts

where

F(s)=

∫T

0 f(t)e−stdt

Time scaling f(at)^1

a

F

(s

a

)

Frequency differentiation tf (t) −dF(s)

(multiplication byt) ds

Frequency integration f(t)

t

∫∞

s

F(s)ds

(division byt)

Frequency shifting f(t)e−at F(s+a)

(exponential translation)

Initial-value theorem lim

t→ 0

f(t)=f( 0 +) slim→∞sF (s)

Final-value theorem t→∞limf(t)=f(∞)where limit exists slim→ 0 sF (s)