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
- 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
]
- 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
]
- 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
⎤
⎦
- 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
]