Modern Control Engineering

448 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

jv

j 2 s Plane

j 1

0

• j 1


• j 2


• 2 – 121 3

A B

C

D

s

jv

j 2

j 1

0

• j 1


• j 2


• 121 3

A B

D C

s

jv

j 2

0

• j 2


• 1 1 3

AB

D C

s

jv

• 2 – 12031

H G

E F

D C

A B

s

Im

2 F(s) Plane

1

0

• 1

–2

12

A

A

D

D

A

C

B

B

DC

E F G

B

D C

C

B

Re

Im

2

1

0

• 1

–2

(^12) Re
Im
2
1
0

• 1

–2

– (^121) Re
Im
2
1
0

• 1

–2

– (^1213) Re

• 1 3


• 2 – 1 3


• 2 2

j 1

• j 1

j 2

j 1

• j 1
  
  
  • j 2
  
  
  
  

3

H

A

(a)

(b)

(c)

(d)

Figure 7–46

Closed contours in the s

plane and their

corresponding closed curves

in the F(s)plane, where

F(s)=(s+1)/(s-1).

Mapping Theorem. LetF(s)be a ratio of two polynomials in s. Let Pbe the num-

ber of poles and Zbe the number of zeros of F(s)that lie inside some closed contour in

thesplane, with multiplicity of poles and zeros accounted for. Let the contour be such

that it does not pass through any poles or zeros of F(s). This closed contour in the splane

is then mapped into the F(s)plane as a closed curve. The total number Nof clockwise

encirclements of the origin of the F(s)plane, as a representative point straces out the

entire contour in the clockwise direction, is equal to Z-P. (Note that by this mapping

theorem, the numbers of zeros and of poles cannot be found—only their difference.)

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