Page 131 - Circuit Analysis II with MATLAB Applications

The Laplace Transform of Common Functions of Time

TABLE 4.1 Summary of Laplace Transform Properties and Theorems

Property/Theorem Time Domain Complex Frequency Domain
1 Linearity c f t + c f t c F s + c F s
1 1
2 2
2
2
1
1
+ } + c f t + } + c F s
n n
n n
2 Time Shifting ft – a u t – 0 a e – as Fs

3 Frequency Shifting – as Fs + a

e ft
4 Time Scaling f at 1 s
§·
-
--F -- -
a
a ©¹
5 Time Differentiation d
See also (4.18) through (4.20) ----- ft sF s – f0
dt
6 Frequency Differentiation tf t – d
See also (4.22) -----Fs
ds
7 Time Integration t
³ f W dW Fs f0 ------------
----------- +
– f s s
8 Frequency Integration f t f
-------- Fs s
d
t ³
s
9 Time Periodicity f t + nT T – st

³ ft e dt
0
--------------------------------
1 – e – sT

10 Initial Value Theorem lim ft
t o 0 lim sF s = f 0
s o f
11 Final Value Theorem lim ft lim sF s = f f
t o f s o 0

12 Time Convolution f t *f t F s F s
2
1
2
1
13 Frequency Convolution f t f t 1
1
2
-------- F s *F s
2Sj 1 2

Circuit Analysis II with MATLAB Applications 4-13
Orchard Publications