def: encircled point a is encircled in the counter in ccw by ...iesl.yolasite.com/resources/nyquist...
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Nyquist Stability Criterion : Fundamentals
Def: Encircled
point A is encircled in the counter in CCW by closed path .
point B is NOT encircled by closed path .
Nyquist Stability Criterion : Fundamentals
Def: Enclosed
(a) point A is not enclosed by . point B is enclosed by .
A point or region is said to be enclosed by a closed path if the point or
region lies to the right of the path when the path is traversed in
any prescribed direction.
(b) point A is enclosed by . point B is not enclosed by .
Nyquist Stability Criterion : Fundamentals
Def: # of encirclement
1Consider the phasor from point A to .s
the net angle traversed by the phasor = 2 (rad)N
# of encirclement = N
(a)A : # of encirclement = +1
B : # of encirclement = +2
(b)A : # of encirclement = -1B : # of encirclement = -2
Nyquist Stability Criterion : Fundamentals
Determination of N
Nyquist Stability Criterion : Fundamentals
Consider single-valued function ( ) : .s C C
s
closed path does not go through any pole of ( ) is closed path.s s
Nyquist Counter Nyquist plot
Nyquist Stability Criterion : Fundamentals
Nyquist Counter
Therefore, we choose a contour s, say, Nyquist path, in the s-plane that encloses the entire right-half s-plane. The contour consists of the entire j axis ( varies from to +) and a semicircular path of infinite radius in the right-half s plane.
Nyquist contour
r 0
S
j
s-plane
As a result, the Nyquist path encloses all the zeros and poles of F(s)=1+G(s)H(s) that have positive real parts.
Nyquist Stability Criterion : Fundamentals
Principle of the Argument:
# of encirclements of the origin by N
Sketch of PF:
N Z P
# of zeros of ( ) encircled by sZ L s
# of poles of ( ) encircled by sP L s
1
1 2
( )( )( )( )
K s zL ss p s p
11 1 2
1 2
( ) ( ) ( )
[ ( ) ( ) ( )]
L s L s L s
K s zs z s p s p
s p s p
0K
( ) ( ) ( )L s G s H s
Nyquist Stability Criterion : Fundamentals
1 11
1 1 1 2
( )( )( )( )
K s zL ss p s p
1-z
2z
1p2p 0
s1
2
2 1
j
0
F
)(sFu
jv
S
F
If, for example, F(s) has two zeros and one pole are enclosed by s, it can be deduced from the above analysis that F encircles the origin of F plane one time in the clockwise direction as s traces out s.
Nyquist Stability Criterion : Fundamentals
# of encirclement of the (-1+0j) by Nyquist plot of ( )N s
= # of poles of ( ) encircled by (RHP poles of ( ))sP L s L s
N Z P
= # of zeros of ( ) encircled by (RHP zeros of ( ))sZ s s
stable 0 ZN P
Nyquist Criterion :
( ) 1 ( )s L s
But most of systems have P =0, Therefore stability, N = 0
