Nyquist Plot Overview | Control Systems - Electrical Engineering (EE) PDF Download

 Page 1


Introduction
• Nyquist plots are the continuation of polar
plots for finding the stability of the closed loop
control systems by varying ? from -8 to 8.
• Nyquist plots are used to draw the complete
frequency response of the open loop transfer
function.
• The Nyquist stability criterion determines the
stability of a closed-loop system from its open-
loop frequency response and open-loop poles.
Page 2


Introduction
• Nyquist plots are the continuation of polar
plots for finding the stability of the closed loop
control systems by varying ? from -8 to 8.
• Nyquist plots are used to draw the complete
frequency response of the open loop transfer
function.
• The Nyquist stability criterion determines the
stability of a closed-loop system from its open-
loop frequency response and open-loop poles.
The Nyquist Criterion can be expressed as,
Z = P + N
where Z = number of zeros of 1 + G(s)H(s) on the
right-half s-plane
N = net encirclements around the point (-1+j0).
(clockwise encirclements are taken as
positive and anticlockwise encirclements
are negative)
P = number of poles of G(s)H(s) in the
right-half of s-plane
Page 3


Introduction
• Nyquist plots are the continuation of polar
plots for finding the stability of the closed loop
control systems by varying ? from -8 to 8.
• Nyquist plots are used to draw the complete
frequency response of the open loop transfer
function.
• The Nyquist stability criterion determines the
stability of a closed-loop system from its open-
loop frequency response and open-loop poles.
The Nyquist Criterion can be expressed as,
Z = P + N
where Z = number of zeros of 1 + G(s)H(s) on the
right-half s-plane
N = net encirclements around the point (-1+j0).
(clockwise encirclements are taken as
positive and anticlockwise encirclements
are negative)
P = number of poles of G(s)H(s) in the
right-half of s-plane
The stability of linear control systems using the Nyquist 
stability criterion, three possibilities can occur:
1. There is no encirclement of the (–1+j0) point. This
implies that the system is stable if there are no poles
of G(s)H(s) in the right-half of s- plane; otherwise, the
system is unstable.
2. There are one or more counterclockwise
encirclements of the (–1+j0) point. In this case the
system is stable if the number of counterclockwise
encirclements is the same as the number of poles of
G(s)H(s) in the right-half of s- plane; otherwise, the
system is unstable.
3. There are one or more clockwise encirclements of the
(–1+j0) point. In this case the system is unstable.
Page 4


Introduction
• Nyquist plots are the continuation of polar
plots for finding the stability of the closed loop
control systems by varying ? from -8 to 8.
• Nyquist plots are used to draw the complete
frequency response of the open loop transfer
function.
• The Nyquist stability criterion determines the
stability of a closed-loop system from its open-
loop frequency response and open-loop poles.
The Nyquist Criterion can be expressed as,
Z = P + N
where Z = number of zeros of 1 + G(s)H(s) on the
right-half s-plane
N = net encirclements around the point (-1+j0).
(clockwise encirclements are taken as
positive and anticlockwise encirclements
are negative)
P = number of poles of G(s)H(s) in the
right-half of s-plane
The stability of linear control systems using the Nyquist 
stability criterion, three possibilities can occur:
1. There is no encirclement of the (–1+j0) point. This
implies that the system is stable if there are no poles
of G(s)H(s) in the right-half of s- plane; otherwise, the
system is unstable.
2. There are one or more counterclockwise
encirclements of the (–1+j0) point. In this case the
system is stable if the number of counterclockwise
encirclements is the same as the number of poles of
G(s)H(s) in the right-half of s- plane; otherwise, the
system is unstable.
3. There are one or more clockwise encirclements of the
(–1+j0) point. In this case the system is unstable.
Consider the closed-loop system shown in Fig. 
The closed-loop transfer function is
C(s)
R(s)
= 
G(s)
1 + G(s)H(s)
The characteristic equation is 1 + G(s)H(s) = 0
For stability, all roots of the characteristic 
equation must lie in the left-half of s- plane. 
Page 5


Introduction
• Nyquist plots are the continuation of polar
plots for finding the stability of the closed loop
control systems by varying ? from -8 to 8.
• Nyquist plots are used to draw the complete
frequency response of the open loop transfer
function.
• The Nyquist stability criterion determines the
stability of a closed-loop system from its open-
loop frequency response and open-loop poles.
The Nyquist Criterion can be expressed as,
Z = P + N
where Z = number of zeros of 1 + G(s)H(s) on the
right-half s-plane
N = net encirclements around the point (-1+j0).
(clockwise encirclements are taken as
positive and anticlockwise encirclements
are negative)
P = number of poles of G(s)H(s) in the
right-half of s-plane
The stability of linear control systems using the Nyquist 
stability criterion, three possibilities can occur:
1. There is no encirclement of the (–1+j0) point. This
implies that the system is stable if there are no poles
of G(s)H(s) in the right-half of s- plane; otherwise, the
system is unstable.
2. There are one or more counterclockwise
encirclements of the (–1+j0) point. In this case the
system is stable if the number of counterclockwise
encirclements is the same as the number of poles of
G(s)H(s) in the right-half of s- plane; otherwise, the
system is unstable.
3. There are one or more clockwise encirclements of the
(–1+j0) point. In this case the system is unstable.
Consider the closed-loop system shown in Fig. 
The closed-loop transfer function is
C(s)
R(s)
= 
G(s)
1 + G(s)H(s)
The characteristic equation is 1 + G(s)H(s) = 0
For stability, all roots of the characteristic 
equation must lie in the left-half of s- plane. 
Example 1
G(s)H(s) = 
(?? +1)(?? +2)
?? (?? +3)
Open loop zeros: -1, -2
Open loop poles:  0, -3
The characteristic equation is 1 + G(s)H(s) = 0
1+ 
(s+1)(s+2)
s(s+3)
= 0
(?? +0.38)(?? +2.62)
?? (?? +3)
= 0  
Roots of the system:  - 0.38, - 2.62
Closed-loop system  T.F =  
G(s)
1 + G(s)H(s)
= 
(?? +1)(?? +2)
(?? +0.38)(?? +2.62)
Closed-loop poles: - 0.38, - 2.62  [zeros of 1 + G(s)H(s)]