Page 126 - Circuit Analysis II with MATLAB Applications

Chapter 4 The Laplace Transformation

8. Integration in Complex Frequency Domain

This property states that integration in complex frequency domain with respect to corresponds to
s
t
--------
division of a time function f t by the variable , provided that the limit lim ft exists. Thus,
t o 0 t
ft f
d
-------- œ ³ Fs s (4.27)
t s
Proof:
f
Fs = ³ ft e – st dt
0
Integrating both sides from to , we get
s
f
f f f – st
d
³ Fs s = ³ ³ ft e dt d s
s s 0
Next, we interchange the order of integration, i.e.,

f f f – st
d
d
³ Fs s = ³ ³ e s d f t t
s 0 s
and performing the inner integration on the right side integral with respect to , we get
s

f f 1 – st f f ft – st ­ ft ½
-
d
d
³ Fs s = ³ – --e s ft t = ³ --------e t d = L ® -------- ¾
t
t
t
s 0 0 ¯ ¿
9. Time Periodicity
The time periodicity property states that a periodic function of time with period corresponds toT
T
the integral ³ ft e – st dt divided by 1 – e – sT in the complex frequency domain. Thus, if we let f t

0
T
be a periodic function with period , that is, f t = ft + nT , for n = 1 2 3 } we get the trans-

form pair
T – st
³ ft e dt
0
ft + nT œ ----------------------------- (4.28)

1 – e – sT

4-8 Circuit Analysis II with MATLAB Applications

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