Page 121 - Circuit Analysis II with MATLAB Applications
P. 121
Properties of the Laplace Transform
F s F s } F s
n
2
1
respectively, and
c c } c 1 2 n
are arbitrary constants, then,
c f t + c f t + } + c f c F s + c F s + } + c F (4.11)
s
t
2 2
1
1 1
2
n n
1
2
n n
Proof:
f
L c f t + c f t + } + c f t ` = ³ > c f t + c f t + } + c f t dt
@
^
n n
1 1
2 2
n n
2 2
1 1
t 0
f f f
= c 1³ f t e – st dt + c 2³ f t e – st dt + } + c n³ f t e – st dt
n
1
2
t 0 t 0 t 0
= c F s + c F s + } + c F s
n
1
2
1
2
n
Note 1:
It is desirable to multiply f t by u t to eliminate any unwanted non-zero values of f t for t . 0
0
2. Time Shifting Property
a
The time shifting property states that a right shift in the time domain by units, corresponds to mul-
tiplication by e – as in the complex frequency domain. Thus,
ft – a u t – a e – as Fs (4.12)
0
Proof:
a f
L ft – a u t – 0 a ` = ³ 0e – st dt + ³ f t – a e – st dt (4.13)
^
0 a
Now, we let t – a = W ; then, t = W + a and dt = dW . With these substitutions, the second integral
on the right side of (4.13) becomes
f – s W + a – as f – sW – as
³ f W e dW = e ³ f W e dW = e Fs
0 0
3. Frequency Shifting Property
The frequency shifting property states that if we multiply some time domain function f t by an
– at
exponential function e where a is an arbitrary positive constant, this multiplication will produce a
shift of the s variable in the complex frequency domain by units. Thus,a
Circuit Analysis II with MATLAB Applications 4-3
Orchard Publications
