Page 127 - Circuit Analysis II with MATLAB Applications

Properties of the Laplace Transform

Proof:

The Laplace transform of a periodic function can be expressed as

f T 2T 3T
^
L ft ` = ³ ft e – st dt = ³ f t e – st dt + ³ f t e – st dt + ³ f t e – st dt + }
0 0 T 2T
In the first integral of the right side, we let t = W , in the second t = W + T , in the third t = W + 2T ,
and so on. The areas under each period of f t are equal, and thus the upper and lower limits of
integration are the same for each integral. Then,
T T T

+
L ft ` = ³ f W e – sW dW ³ f W + T e – s W + T dW + ³ f W + 2T e – s W + 2T d + } (4.29)
^

W
0 0 0
Since the function is periodic, i.e., f W = f W + T = f W + 2T = } = f W + nT , we can write

(4.29) as
T
– sT – 2sT – sW
L f W ` = 1 + e + e + } ³ f W e dW (4.30)
^
0
By application of the binomial theorem, that is,
1
2
3
a
1 + + a + a + } = ------------ (4.31)
1 – a
we find that expression (4.30) reduces to
T – sW
³ f W e dW
0
^
L f W ` = ----------------------------------
W e – sT
–
10. Initial Value Theorem

The initial value theorem states that the initial value f 0 of the time function f t can be found
s
from its Laplace transform multiplied by and letting s o f .That is,
lim ft = lim sF s = f 0 (4.32)
t o 0 s o f
Proof:
From the time domain differentiation property,

d
----- ft œ sF s – f0
dt
or

Circuit Analysis II with MATLAB Applications 4-9
Orchard Publications