Nyquist Stability Criterion and its Statement

Nyquist Criterion

Nyquist has developed a criterion to study the stability of control system in frequency domain.

Nyquist suggested to select a single valued function F(s) as 1 + G(s)H(s)

F(s) = 1 + G(s)H(s)

Configuration developed is :
Poles of 1 + G(s)H(s) = Poles of G(s)H(s) = Open loop poles of system

Zeros of 1 + G(s)H(s) = Closed loop poles of system

For stability all the zeros of 1 + G(s)H(s) should lie in left half of s-plane,none of the zeros should lie in right half of s-plane.

Nyquist suggested that , instead of finding whether all the zeros are in left half of s-plane, it is better to examine any one zero in right half of s-plane which is making the system unstable.

Active region of stability is Right half of s-plane.

Nyquist suggested to select a 𝛕(s) path which will encircle the entire right half of s-plane, instead of choosing any arbitrary path.

This path should start from s= + j ꚙ and continued till s = - j ꚙ along the imaginary axis and completed with a semicircle of ꚙ radius encircling entire right hand side of s-plane. This path is called as Nyquist path.

When, N = Number of encirclements of origin of F- plane by Nyquist plot

As per mapping theorem, N = Z – P

Z = Number of zeros of 1 + G(s)H(s) encircled by Nyquist path in s-plane.

As Nyquist path encircles only right half of s-plane

Z = Number of zeros of 1 + G(s)H(s) which are located on right half of s-plane.

P = Number of poles of 1 + G(s)H(s) which are located on right half of s-plane


For absolute stability no zero of 1+G(s)H(s) must be on right half of s-plane i.e. Z = 0 for stability.

So Nyquist criterion is obtained by substituting Z = 0 in
N = Z – P

N = - P

Statement of Nyquist Stability Criterion

Nyquist stability criterion states that,
“for absolute stability of system, the number of encirclements of new origin of F-plane by Nyquist plot must be equal to number of poles of 1+G(s)H(s) i.e. poles of 1+G(s)H(s) which are in right half of s-plane and in Clock wise direction.”

For ease of mapping ,

We consider only G(s)H(s) instead of 1 + G(s)H(s)

The stability criterion is



P = Number of poles of G(s)H(s) which are located on right half of s-plane

N = Number of encirclements of Critical point , 1 + j0 of F plane by Nyquist plot

N = - P





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