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Nyquist Path Or Nyquist Contour

The overall transfer of a system is given by


The characteristic equation is 1 + G(s) H(s) = 0

The main purpose in study the stability of the closed loop system is to determine whether the characteristic equation has any root in the right half of s-plane i.e. whether C(s)/R(s) has any pole in right half of s-plane.

For this purpose, we use a contour in s-plane which encloses the entire right half plane. This contour having the encirclement in clockwise direction and radius ‘R’ approaches infinity. This path or contour is known as Nyquist contour.

Nyquist stability criterion as follows:

A feedback system or closed loop system is stable if the contour ΓGH of the open loop transfer function G(s) H(s) corresponding to the Nyquist contour in the s-plane encircles the point (–1 + j0) in counter clockwise direction and the number of counterclockwise encirclements about the (–1 + j0) equals the number of poles of G(s) H(s) in the right half of s-plane i.e. with positive real parts.

In common case of open loop stable system, the closed loop system is stable if the contour ΓGH of G(s) H(s) does not pass through or does not encircle (–1 + j0) point, i.e. net encirclement is zero.

General Construction Rules of the Nyquist Path


  • Step 1: Check G(s) for poles on jω axis and at the origin
  • Step 2: Using eqn (i) to eqn (iii) sketch the image of the path a – d in the G(s)-plane. If there are no poles on jω axis equation (ii) need not be employed.
  • Step 3: Draw the mirror image about the real axis of the sketch resulting from step 2.
  • Step 4: Use eqn (iv) plot the image of path def. This path at infinity usually plot into a point in the G(s) - plane.
  • Step 5: Use eqn (viii) plot the image of path ija (pole at origin)
  • Step 6: Connect all curves drawn into the previous steps.


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