# 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 â€˜Râ€™ 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 Î“_{GH} of the open loop transfer function G(s) H(s) corresponding to the Nyquist contour in the s-plane encircles the point (â€“1 + j0) in counter clockwise direction and the number of counterclockwise encirclements about the (â€“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 Î“_{GH} of G(s) H(s) does not pass through or does not encircle (â€“1 + j0) point, i.e. net encirclement is zero.

General Construction Rules of the Nyquist Path

** **

**Step 1:**Check G(s) for poles on jÏ‰ axis and at the origin**Step 2:**Using eq^{n}(*i*) to eq^{n}(*iii*) sketch the image of the path a â€“ d in the G(s)-plane. If there are no poles on jÏ‰ 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 eq^{n}(*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.