Page 137 - Circuit Analysis II with MATLAB Applications

The Laplace Transform of Common Functions of Time

Thus, we have the transform pair

G t – a œ e – as (4.54)

for V ! . 0
Example 4.6

– at
Find L e ^ u t `
0
Solution:

f f
–
– at – at – st s + a t
L e ^ u t ` = ³ e e dt = ³ e dt
0
0 0
f
1
–
1 ·
= § – ----------- e s + a t = -----------
© s + a ¹ s + a
0
Thus, we have the transform pair
– at 1
e u t œ ----------- (4.55)
0
s +
a
for V ! – . a
Example 4.7

­ n – at ½
Find L t e u t ¾ 0
®
¯ ¿
Solution:

For this example, we will use the transform pair of (4.52), i.e.,

n n!
t u t œ ----------- (4.56)
0
s n + 1
and the frequency shifting property of (4.14), that is,

e – at ft œ Fs + a (4.57)

Then, replacing with s + a in (4.56), we get the transform pair
s

Circuit Analysis II with MATLAB Applications 4-19
Orchard Publications