Nyquist Plot

In order to determine the stability of closed loop control systems by varying ω a value from −∞ to ∞ , Nyquist plot are a continuation of polar plots. In other words, Nyquist plots are used to depict the open loop transfer function’s entire frequency response.

Nyquist Stability Criterion

The argumentation principle underlies the Nyquist stability criterion’s operation. It states that 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. As a result, we can express N as the number of encirclements.

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 direction of the encirclement in the G(s)H(s) plane will be the same as that of the enclosed closed path in the ‘s’ plane if the enclosed closed path in the ‘s’ plane is made up entirely of 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. So Therefore, the characteristic equation’s roots are the same as the poles of the closed loop transfer function. And So Finding the roots becomes harder as the characteristic equation’s order rises. So Let’s compare these roots of the characteristic equation using the following correlation.

• The Poles of the characteristic equation are same as that of the poles of the open loop transfer function.
• The zeros of the characteristic equation are same as that of the poles of the closed loop transfer function.

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. So 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.
• To create the polar plot, change between zero and infinity. Drawing a polar plot requires varying values between 0+ and infinity if a pole or zero is present at s = 0.
• 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. So 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 list below.

• If the phase cross over frequency ωpc is greater than the gain cross over frequency ωgc, then the control system is stable.
• If the phase cross over frequency ωpc is equal to the gain cross over frequency ωgc, then the control system is marginally stable.
• So If phase cross over frequency ωpc is less than gain cross over frequency ωgc, then the control system is unstable.

Gain Margin

The gain margin GM is determined by multiplying the magnitude of the Nyquist plot at the phase cross over frequency by its reciprocal.

GM=1Mpc

So When the phase cross over frequency is use, Mpc is the magnitude in normal scale.

Phase Margin

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

PM=1800+ϕgc

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

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

• So The control system is stable if the gain margin (GM) exceeds one and the phase margin (PM) is positive.
• So 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.