Higher Engineering Mathematics, Sixth Edition

Inverse Laplace transforms 599

Problem 12. Determine the poles and zeros for

the function:F(s)=

(s+ 3 )(s− 2 )

(s+ 4 )(s^2 + 2 s+ 2 )

and plot them on a pole-zero map.

For the numerator to be zero,(s+ 3 )=0and(s− 2 )=0,
hencezeros occur ats=− 3 and ats=+ 2 Poles occur
when the denominator is zero, i.e. when(s+ 4 )=0, i.e.
s=− 4 ,andwhens^2 + 2 s+ 2 =0,

i.e.s=

− 2 ±

√

22 − 4 ( 1 )( 2 )

2

=

− 2 ±

√

− 4

2

=

− 2 ±j 2

2

=(− 1 +j)or(− 1 −j)

The poles and zeros are shown on the pole-zero map of
F(s)in Figure 63.2.

2 j

j

j

24 23 22 21 0 123

Figure 63.2

It is seen from these problems that poles and zeros
are always real or complex conjugate.

Now try the following exercise

Exercise 225 Further problemson poles

and zeros

1. Determine for the transfer function:

R(s)=

50 (s+ 4 )

s(s+ 2 )(s^2 − 8 s+ 25 )

(a) the zero and (b) the poles. Show the poles

and zeros on a pole-zero diagram.

[

(a)s=−4(b)s= 0 ,s=− 2 ,

s= 4 +j 3 ,s= 4 −j 3

]

2. Determinethepolesandzerosforthefunction:
   F(s)=

(s− 1 )(s+ 2 )

(s+ 3 )(s^2 − 2 s+ 5 )

and plot them on

a pole-zero map.

[

poles ats=− 3 ,s= 1 +j 2 ,s= 1 −j 2 ,

zeros ats=+ 1 ,s=− 2

]

3. For the functionG(s)=

s− 1

(s+ 2 )(s^2 + 2 s+ 5 )

determine the poles and zeros and show them

on a pole-zero diagram.

⎡

⎣

poles ats=− 2 ,s=− 1 +j 2 ,

s=− 1 −j 2 ,

zero ats= 1

⎤

⎦

4. Find the poles and zeros for the transfer func-

tion:H(s)=

s^2 − 5 s− 6

s(s^2 + 4 )

and plottheresults in

the s-plane.

[

poles ats= 0 ,s=+j 2 ,s=−j 2 ,

zeros ats=− 1 ,s= 6

]