Page 124 - Circuit Analysis II with MATLAB Applications

Chapter 4 The Laplace Transformation

6. Differentiation in Complex Frequency Domain

This property states that differentiation in complex frequency domain and multiplication by minus one,
t
corresponds to multiplication of f t by in the time domain. In other words,
d
tf t œ – -----Fs (4.21)
ds

Proof:

f
^
L ft ` = Fs = ³ ft e – st dt
0
*
Differentiating with respect to s, and applying Leibnitz’s rule for differentiation under the integral, we
get

d d f – st f w – st f – st
-----Fs = ----- ³ ft e dt = ³ e ft dt = ³ t – e ft dt
ds ds s w
0 0 0
f
>–
= ³ >– tf t e – st dt = L tf t @
@
0
In general,
n
n n d

t ft œ – 1 --------Fs (4.22)
ds n
The proof for n t 2 follows by taking the second and higher-order derivatives of Fs with respect
to . s
7. Integration in Time Domain

s
This property states that integration in time domain corresponds to Fs divided by plus the initial
s
value of f t at t = 0 , also divided by . That is,

t Fs f0
³ f W dW œ ----------- + ------------ (4.23)
s
s
– f

b
* This rule states that if a function of a parameter is defined by the equation F D = ³ fx D x where f is some
d

D
a
known function of integration x and the parameter , a and b are constants independent of x and , and the par-
D
D
dF
x D

w

tial derivative fw D w e exists and it is continuous, then ------- = ³ b -----------------d . x
D
dD a w
4-6 Circuit Analysis II with MATLAB Applications
Orchard Publications