# Nyquist Plot : Graph, Stability, Example Problems & Its Applications

The Bode plot & the Nyquist plots are very popular plots, especially for Electrochemical Impedance Spectroscopy or EIS data among electrochemists. So, Nyquist Plot is named after a Swedish-American namely “Harry Nyquist”. He is an electric engineer & developed this plot for electronics purposes in the year 1932. During an EIS, a lot of information is collected & this gathered information needs to be presented. So, a picture gives more information than hundred words. So a graphical representation like a Nyquist plot is used to show an Electrochemical Impedance Spectroscopy. This article provides information on **Nyquist plot** – working, advantages & its disadvantages.

## Nyquist Plot Definition

The graphical representation which is widely used for transfer functions is known as the Nyquist plot. This is a frequency response plot that is used to assess the control system with feedback stability. It is a parametric plot for the real & imaginary part of a transfer function within the complex plane because the frequency parameter sweeps throughout a specified interval. In Cartesian coordinates, the nyquist plot transfer function’s real part is plotted on the X-axis whereas the imaginary part of the transfer function is plotted on the Y-axis.

Nyquist Plot is used in automatic control as well as signal processing for analysis of stability because anyone can instantly verify whether a loop with negative feedback meets Nyquist’s stability principle. If the Nyquist plot of the open loop control system covers approximately the point over the real axis afterward the equivalent closed loop system is unstable.

### Nyquist Plot Graph

The Nyquist plot graphs are the extension of polar plots used mainly for finding the closed-loop control systems stability by simply changing ‘ω’ from −∞ to ∞.that means, these plots are mostly utilized for drawing the open-loop transfer function’s total frequency response. The Nyquist plot simply evaluates the control system’s stability with feedback. So, in a Cartesian coordinate system, the transfer function’s real par is simply plotted over the X-axis whereas the imaginary part is simply plotted over the Y-axis.

The similar Nyquist plot can be explained simply with polar coordinates, where the transfer function’s gain is the radial coordinate, and the transfer function’s phase is the equivalent angular coordinate.

The Nyquist plot can be understood by knowing some of the terminologies used. In Nyquist plot, a closed path within a complex plane is known as a contour.

#### Nyquist Path

The Nyquist path or Nyquist Contour is a closed contour within the s-plane that encloses totally the complete right-hand side of the s-plane. To enclose the plane’s total RHS, a large semicircle lane is drawn by a diameter along the ‘jω’ axis & center at the source. The semicircle radius is simply treated as Nyquist Encirclement.

#### Nyquist Encirclement

A point is known to be encircled by a line if it is found in the curve.

#### Nyquist Mapping

The procedure by which a point within the s-plane is changed into a point within the F(s) plane is known as mapping & F(s) is known as the function of mapping.

The stability analysis of the feedback control system mainly depends on recognizing the location roots for the characteristic equation above the s-plane.

Thus, if the root on the s-plane lies on the left face then the controls system is stable. So, the relative stability of the system can be determined through different frequency response techniques like the Nyquist plot, Bode plot & Nichols plot.

### Nyquist Stability Criterion

The Nyquist stability criterion is mainly used to recognize the existence of roots for a characteristic equation in the S-plane’s particular region. Nyquist stability criterion like N = Z – P simply says that. ‘N’ is the total number of encirclements regarding the origin, ‘P’ is the number of poles & ‘Z’ is the total number of zeros.

In Case 1: When N = 0 (no encirclement), thus Z = P = 0 & Z = P.

If N = 0, P should be ‘0’ so the system is stable.

In Case 2: When N is greater than 0 (clockwise encirclement), thus P = 0, Z ≠0 & Z > P

In these two cases, the system is unstable.

In Case 3: When N is less than 0 (counter-clockwise encirclement), thus Z = 0, P ≠0 & P > Z

Thus, the system is stable.

#### How to Draw Nyquist Plot?

There are many steps involved in drawing nyquist plot that is discussed below.

- In step 1: Need to check the poles for an open loop transfer function like G(s)H(s) within ‘s’ plane.
- In step 2: Choose the correct Nyquist contour by including the whole right side of the s-plane by simply drawing a semicircle of radius ‘R’ where R tends to infinity.
- In step 3: Recognize different segments on the outline with location to the Nyquist path.
- In step 4: The mapping segment needs to perform through the segment by simply substituting the respective segment equation in the mapping function. Generally, we have to draw the polar plots for the particular segment.
- In step 5: Generally, the segment mapping are reflect images of mapping for the particular path of the positive imaginary axis.
- In step 6: The semicircular lane that covers the right half of the plane normally maps into a point within the G(s) H(s) plane.
- In step 7: Interconnect all the various mapping segments to yield the necessary Nyquist diagram.
- In step 8: Note the no. of clockwise encirclements about (-1, 0) & decide stability through N = Z – P.

Once the Nyquist plot is drawn, we can discover the closed-loop control system’s stability with the Nyquist stability criterion. So, if the critical point (-1+j0) lies at the outside of the encirclement, then the closed-loop control system is completely stable.

The open loop transfer function is the G(S)H(S) = N(S)/D(S).

The closed-loop transfer function is the G(S)/1+ G(S)H(S).

N(s) = zero is the open loop zero & D(s) is the open loop pole.

From a stability viewpoint, no closed loop poles must lie on the RH face of the s-plane. The characteristics equation like 1 + G(s) H(s) equal to zero signifies closed-loop poles.

When 1 + G(s) H(s) is equal to zero thus q(s) must be zero.

So, from the stability viewpoint, zeroes of q(s) shouldn’t lie within the Right-Hand Plane of the s-plane.

To describe the strength, the whole RHP needs to be considered. So we imagine a semicircle that includes all points within the RHP by considering the semicircle radius ‘R’ that tends to infinity.

### Stability Analysis with Nyquist Plot

From the Nyquist plot, we can recognize whether the control system is stable, unstable, or marginally stable depending on the parameter values.

- Gain cross-over frequency & phase cross-over frequency.
- Gain margin & phase margin.

#### Phase Cross-over Frequency.

The frequency at which point the Nyquist plot meets the negative real axis is called the phase cross-over frequency and it is denoted with ωpc.

#### Gain Cross over Frequency

The frequency at which point the Nyquist plot has one magnitude is called the gain cross-over frequency and it is denoted with ωgc.

The control system stability based on the main relationship between the two frequencies like phase cross-over as well as gain cross over is discussed below.

- If the ωpc is higher as compared to the ωgc then the control system is stable.
- If the ωpc is equivalent to the ωgc then the control system is slightly stable.
- If the ωpc is less as compared to ωgc then the control system is not stable.

#### Gain Margin

The gain margin is equivalent to the reciprocal of the Nyquist plot’s magnitude at the phase cross-over frequency.

Gain margin (GM) =1/Mpc

Where ‘Mpc’ is the magnitude within normal scale at the ωpc or phase cross-over frequency

#### Phase Margin

The phase margin is equivalent to the sum of 180 degrees & the phase angle at the ωgc or gain cross-over frequency.

PM = 1800 + ϕgc

Where ϕgc is the phase angle at the gain cross-over frequency (ωgc).

The control system’s stability depends on the main relationship between the two margins like the gain margin & the phase margin given below.

If the gain margin is higher than one & the phase margin is positive, then the control system is stable.

If the gain margin is equivalent to one & the phase margin is ‘0’degrees, then the control system is slightly stable.

If the gain margin is low than one & the phase margin is negative, then the control system is not stable.

### Nyquist Plot Example Problems

**Ex1:** If the Nyquist plot cuts the negative real axis at the 0.6 distance then what is the system gain margin?

We know that the gain margin of the system can be defined as the amount of change required within open loop gain to make a closed loop system unstable is

Gain margin or GM = 1/|G| wpc

Where, the system’s gain is |G| and wpc is the phase crossover frequency.

The phase crossover frequency can be defined as; the frequency at which point the system gain is ‘0’.

Gm = 1/0.6 = 1.66

**Ex2:** The open loop system transfer function of unity gain negative feedback system can be given as G(s) = 1/S(S+1). The Nyquist curve within the S-plane includes the whole right side plane & small area around the origin on the left side shown in the following graph. The no. of encirclements of the (-1+ j0) point through the G(S) Nyquist plot, equivalent to the Nyquist contour which is indicated as ‘N’ then ‘N’ equivalent to?

The no. of encirclements for the (-1+ j0) significant point is given through N = P-Z.

Where ‘N’ is the number of encirclements of this critical point in the anti-clockwise direction.

‘P’ is the number of open-loop poles within the right side of the S-plane.

‘Z’ is the number of closed-loop poles within the right side of the S-plane.

N = P for stability Z = 0.

The above-given formula is only valid once the Nyquist curve is defined for the right side of the S-plane & the poles are excluded at the source. The curve rotation should be clockwise & the critical point’s encirclement is in the direction anti-clockwise.

**G(s) = 1/S(S+1).**

The open-loop poles are present at S = 0,-1

The transfer function of closed-loop = 1/S^2+S+1

The number of the closed pole over the right side is zero.

But the Nyquist contour is defined for the total half side of the S-plane & contains the pole at the origin also.

Thus, at S=0 the open-loop pole is considered as the pole within the right side of the S-plane.

N = P-Z =>1-0 =>1

### Advantages and Disadvantages

The **advantages of Nyquist plot** include the following.

- The Nyquist plot is an extremely helpful tool in determining system stability.
- It has many advantages over the Routh-Horwitz & root locus as it simply manages time delays.
- But, it is most helpful because it gives us a method to utilize the Bode plot to decide stability.
- By using this, control system stability can be decided.
- An open-loop transfer function is found by simply measuring its frequency response.
- It is better as compared to the root locus in terms of time delay which means that the Nyquist plot can simply manage the time delay within the system.
- It can locate the open-loop transfer function’s frequency response.
- It finds the no. of poles available poles on the right face of the s-plane.
- It finds the system’s relative stability/

The **disadvantages of Nyquist plot** include the following.

- Nyquist plot utilizes some difficult mathematical methods.
- It cannot resolve the complete strength of the system.
- It does not give precise information about the available poles on the right face of the s-plane.

### Nyquist Plot Applications

The applications of the Nyquist plot include the following.

- The Nyquist plot is used to establish the system stability through a graphical process within the frequency domain.
- A Nyquist plot or a frequency response plot is mainly used in control engineering & signal processing.
- These are the extension for polar plots, used to find the closed-loop control system stability.
- It is an extremely useful tool in determining system stability.
- Using a Nyquist plot, we can monitor the distance between the two points (–1, 0) & the point where the curve crosses the negative real axis.

**How is Nyquist Plot used to Determine Stability?**

Stability can be determined by using Nyquist Plot by simply looking at the no. of encirclements of the point (−1, 0). The variety of gains on which the system will be steady can be determined by looking at the real axis crossings. This plot provides some data regarding the transfer function’s shape.

**What are Nyquist Criteria for Sampling?**

The Nyquist criteria need that the sampling frequency is a minimum of two times the maximum frequency contained within the signal. If the sampling frequency is low than twice the highest analog signal frequency, then a phenomenon called aliasing will happen.

**What is used for Nyquist Plot?**

An open loop transfer function is used for Nyquist Plot.

**What is the Nyquist Rule?**

Nyquist’s rule simply states that a periodic signal should be sampled at above twice the signal’s maximum frequency component. In fact, because the time available is limited, a sample rate is somewhat higher than it requires.

**What is Nyquist Bit Rate Formula for Noiseless?**

Nyquist simply states that in a bandwidth ‘B’ channel, you can transmit up to 2B orthogonal signals for each second thus, Rp ≤ 2B, wherever ‘Rp’ is the pulse rate.

**What does Nyquist’s Plot Represent?**

The Nyquist plot represents some information regarding the form of the transfer function. So, for example; this plot gives information on the variation between the no. of poles & zeros of the transfer function through the angle at which point the curve reaches the origin.

Thus, this is an overview of the Nyquist plot – advantages, disadvantages & its applications. Nyquist plots are used for analyzing the properties of the control system like stability, phase margin, & gain margin. **Nyquist Plot using Matlab** assists us in making a Nyquist plot graph, related to frequency response generated through a dynamic model. Here is a question for you, what is a bode plot?