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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)
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