Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 37

Table 4.1 Fundamental
features of Laplace transform

y(t) Y(s)

x ð t Þ exp ð at Þ X(s - a)

X ð s Þ e st 0 x ð t t 0 Þ

dx ð t Þ
d t

sX(s) - x(0)

R

t 1

s X ð s Þ

x ð s Þ ds

1

x ð t Þ g ð t Þ X(s)G(s)

Table 4.2 Important time
functions and their Laplace
and Z transforms

x(t) X(s) X(z)

kd ð t Þ k k

z
z 1

1 ð t Þ 1

s

zT S
z 1

t1 ð t Þ 1

s 2

Þ 2

ð

1
s 3

t 2
2 1 ð t Þ

z ð z þ 1 Þ T S
z 1 Þ 2

ð

z
z exp ð aT S Þ

exp ð at Þ 1 ð t Þ

1
s þ a

1
s þ a

t exp ð at Þ 1 ð t Þ zT S exp ð aT S Þ

½ z exp ð aT S Þ 2

Þ 2 ð

b
s 2 þ b 2

sin ð bt Þ 1 ð t Þ z sin ð bT S Þ

z 2 2z cos ð bT S Þþ 1

s
s 2 þ b 2

cos ð bt Þ 1 ð t Þ z ½ z cos ð bT S Þ

z 2 2z cos ð bT S Þþ 1

1 exp ð sT S Þ
s

z 1
z

X ð s Þ
s

X ð s Þ

Z

Solutions

(a) From the definition ( 4.8 ) one obtains

s e st

X ð s Þ¼ Z 1 ð t Þ e st dt ¼ Z

¼ 1

1 1

1

e st dt ¼ 1 ¼ 0 1

s s ::

0

1 1

(b) Expanding the cos function and using the features from Table 4.1 one may

derive:

¼ 1
2

¼

1 ð t Þ

1
2

1
s jx þ

1
s þ jx

s
s 2 þ x 2 :

e jxt þ e jxt X ð s Þ¼ L