Nyquist Stability Criterion

On the basis of argument, the Nyquist stability criterion operates. The corresponding G(s)H(s) plane must go around the origin P-Z times if there are P poles and Z zeros are enclosed by the closed path of the ‘s’ plane, according to this statement. Therefore, we can express the number N of encirclements as,

N=P−Z

• The direction of the encirclement in the G(s)H(s) plane will be the opposite of the direction of the enclosed closed path in the ‘s’ plane if the enclosed closed path in the ‘s’ plane only consists of poles.
• The encirclement in the G(s)H(s) plane will be in the same direction as the enclosed closed path in the ‘s’ plane if the enclosed closed path in the ‘s’ plane only contains zeros.

Let’s now choose the entire right half of the ‘s’ plane as a closed path and apply the argumentation principle to it. The Nyquist contour is the name of this chosen path.

If all of the closed loop transfer function’s poles are located in the left half of the ‘s’ plane, the closed loop control system is said to be stable. Therefore, the characteristic equation’s roots are the same as the poles of the closed loop transfer function. Finding the roots becomes harder as the characteristic equation’s order rises. Let’s compare these roots of the characteristic equation using the following correlation.

• The poles of the characteristic equation and the open loop transfer function share the same polarities.
• The poles of the closed loop transfer function and the characteristic equation’s zeros are identical.

If there isn’t an open loop pole in the right half of the ‘s’ plane, the open loop control system is stable.

i.e., P=0⇒N=−Z

If there isn’t a closed loop pole in the right half of the ‘s’ plane, the closed loop control system is stable.

i.e., Z=0⇒N=P

According to the Nyquist stability criterion, the quantity of encirclements around the critical point (1+j0) must equal the poles of the characteristic equation, which are the poles of the open loop transfer function in the right half of the ‘s’ plane. The recognizable equation plane results from moving the origin to (1+j0).

Rules for Drawing Nyquist Plot

Follow these rules for plotting the Nyquist plots.

• Locate the poles and zeros of open loop transfer function G(s)H(s) in ‘s’ plane.
• Draw the polar plot by varying ω from zero to infinity. If pole or zero present at s = 0, then varying ω from 0+ to infinity for drawing polar plot.
• Draw the mirror image of above polar plot for values of ω ranging from −∞ to zero (0⁻ if any pole or zero present at s=0).
• The number of poles or zeros at the origin will be equal to the number of half circles with infinite radius. The point where the mirror image of the polar plot ends will be where the infinite radius half circle begins. And this half circle with infinite radius will come to an end at the beginning of the polar plot.

We can use the Nyquist stability criterion to determine the stability of the closed loop control system after creating the Nyquist plot. The closed loop control system is utterly stable when the critical point (-1+j0) is outside of the encirclement.

Stability Analysis using Nyquist Plot

Based on the values of these parameters, we can determine from the Nyquist plots whether the control system is stable, marginally stable, or unstable.

• Gain cross over frequency and phase cross over frequency
• Gain margin and phase margin

Phase Cross over Frequency

The phase cross over frequency is the frequency at which the Nyquist plot crosses the negative real axis (phase angle is 180). Its symbol is ωpc.

Gain Cross over Frequency

The gain cross over frequency is the frequency at which the Nyquist plot has a magnitude of one. Its symbol is ωgc.

As determined by the relationship between phase cross over frequency and gain cross over frequency, the stability of the control system is listed below.

• The control system is stable. If the phase cross over frequency (ωpc) is higher than the gain cross over frequency (ωgc).
• The control system is only marginally stable. If the phase cross over frequency (ωpc) and the gain cross over frequency (ωgc) are equal.
• The control system is unstable if phase cross over frequency (ωpc) is lower than gain cross over frequency (ωgc).

Gain Margin

The gain margin GM is equal to the Nyquist plot’s magnitude at the phase cross over frequency divided by its reciprocal.

GM=1/Mpc

Where, Mpc is the magnitude in normal scale at the phase cross over frequency.

Phase Margin

The sum of 180 and the phase angle at the gain cross over frequency is the phase margin PM.

PM=180+ϕgc

Where, ϕgc is the phase angle at the gain cross over frequency.

Below is a list of the control system’s stability. Based on the relationship between the gain margin and the phase margin.

• The control system is stable if the gain margin (GM) exceeds one and the phase margin (PM) is positive.
• The control system is only marginally stable. If the gain margin (GM) is equal to one and the phase margin (PM) is zero degrees.
• The control system is unstable. If the gain margin (GM) is less than one and/or the phase margin (PM) is negative.