Stability Analysis Using Nyquist Plot
The nyquist plot helps in analysis of the stability of a control system. It tells whether the system is stable, unstable or marginally stable. The stability is analyzed using various parameters which are as follows:
- Gain cross-over frequency
- Phase cross-over frequency
- Gain margin
- Phase margin
Phase Cross Over Frequency
It is the frequency at which point the Nyquist plot crosses the negative real axis is called the phase cross-over frequency and it is denoted with ‘ωpc‘.
The stability of the control system based on the relationships between the two frequencies i.e., phase cross-over and gain cross-over is given below:
ωpc > ωgc ->System is stable
ωpc < ωgc ->System is unstable
ωpc = ωgc ->System is marginally stable
Phase Margin
The phase margin indicates how much more phase shift we may put in the open loop transfer function before our system becomes unstable. It can be calculated from the phase at the gain cross-over frequency.
Phase Margin (PM) = 180∘+∠G(jω)H(jω)∣ω=ωgc
The stability of the control system based on the relationships between the two margins i.e., gain margin and phase margin is given below:
GM > 1 and PM is positive -> System is stable
GM < 1 and PM is negative ->System is unstable
GM = 1 and PM is 0o ->System is marginally stable
Gain Crossover Frequency
It is the frequency at which the Nyquist plot has unity magnitude. It is denoted by ‘ωgc‘.
Gain Margin
The gain margin is the amount of open loop gain that can be increased before our system becomes unstable. It can be calculated from the gain at the phase cross-over frequency.
Gain Margin (GM):
Let us understand the above concept with the help of an example.
Question: Find the gain crossover frequency and phase margin of the given transfer function
Solution
PM = 180∘+∠G(jω)H(jω)∣ω=ωgc
ωgc = Frequency at which magnitude of G(s)H(s) is equal to 1.
G(jω)H(jω) =
|G(jω)H(jω)| = 1
= 1
On solving the quadratic equation we will get:
ωgc = (we will neglect the negative term inside the root)
ωgc will now become =
ωgc = 0.786 rad/sec
Now finding the phase at gain crossover frequency
∠G(jω)H(jω)∣ω=ωgc = – tan-1 (ωgc)
∠G(jω)H(jω)∣ω=ωgc = – tan-1 (0.786)
∠G(jω)H(jω)∣ω=ωgc = -90o – 38.16o
∠G(jω)H(jω)∣ω=ωgc = -128.16
PM = 180 – 128.16
PM = 51.84o
Nyquist Plot
A Nyquist plot is a graphical representation used in control engineering. It is used to analyze the stability and frequency response of a system. The plot represents the complex transfer function of a system in a complex plane. The x-axis represents the real part of the complex numbers and the y-axis represents the imaginary part. Each point on the Nyquist plot reflects the complex value of the transfer function at that frequency.
- Nyquist Stability Criteria
- Important Terminologies of Nyquist Plot
- How to draw Nyquist Plot?
- Stability analysis using Nyquist Plot
- Solved Example of Nyquist Plot
- Advantages of Nyquist Plot
- Disadvantages of Nyquist Plot
- Application of Nyquist Plot
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