UNIVERSITY OF NOTTINGHAM School of Electrical and Electronic Engineering

Signal Processing and Control Engineering H62SPC

Laplace Transforms

2.1 Convert the following into partial fractions:

a. 2

1 2 3s

s s s( )( )( )+ + +

b. 1

1 2 32( )( )s s s+ + +

c. 2 66 2 2s

s s s+

+ +( )( )

2.2 Obtain the time functions corresponding to the Laplace transforms:

a. F ss

s s s( )

( )( )= ++ +

32 4 2

b. F ss

s s s s( )

( )( )= ++ + +

2 510 2 52

For b. find the damping ratio and the natural frequency for the oscillatory term.

2.3 The output y(t) of a system is defined by the following differential equation, with the initial conditions specified. Determine Y(s) and hence the solution for y(t). a. ( )D D y e t2 2 5 8+ + = at t = 0 y = 1, &y = 3 b. ( )D D D y t3 25 12 8 16+ + + = at t = 0 y = 0 , &y = 1, &&y = 5

2.4

v

R=2ohm L=1H

C

i

For the circuit above at t = 0, i = 50A and v = 100V. C is an unknown capacitance. Show :

V ss

c

s sc

( ) =+ +

+ +

100 212

212

Answers:

2.1 a. -1,4,-3 b.. 12 ,- 12 s,- 12 c. 14 , 116 , 516 , 14 2.2 0.447, 2.24rads/sec

2.3 a. ( )( )Y s ss s s( ) = ++ + +2

2

71 2 5

, ( ) ( )( )y t e t tt( ) cos sin= 2 2 2 b. ( )( )Y s

ss s s s

( ) = ++ + +3

2 2

161 4 8

, ( )y t t e e tt t( ) sin= + + + 3 2 3 22