Nyquist Plot

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Jan 20, 2024 03:23 AM
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How to draw a Nyquist diagram

Drawing a Nyquist diagram, which is a graphical representation of a system's frequency response, involves plotting the complex gain of the system as a function of angular frequency . This plot is in the complex plane, with the real part of on the x-axis and the imaginary part on the y-axis. Here's a step-by-step guide to creating a Nyquist plot:

1. Obtain the Transfer Function

First, you need the transfer function of the system, where is the complex frequency variable. This function typically comes from a system's differential equations and represents the system's response in the frequency domain.

2. Substitute for

Replace with in the transfer function to get . This gives you the frequency response of the system.

3. Compute Frequency Response Over a Range of Frequencies

Calculate over a range of frequencies, typically from to . This step is often done using computational tools like MATLAB, as it can involve complex arithmetic for systems with non-trivial transfer functions.

4. Plot the Real and Imaginary Parts

For each frequency , plot the real part of on the x-axis and the imaginary part on the y-axis in the complex plane. This will create a curve in the complex plane that represents the system's frequency response.

5. Include Negative Frequencies (Optional)

For a complete Nyquist plot, repeat the process for negative frequencies, from to . However, this step can often be omitted for linear, time-invariant systems because the plot is symmetric about the real axis.

6. Analyze Stability

The Nyquist plot is often used to assess the stability of a system, especially in control systems with feedback loops. The key part of this analysis involves the Nyquist stability criterion, which relates the number of times the plot encircles the point to the number of poles of in the right half of the complex plane.

7. Interpret the Plot

Interpret the plot in terms of stability, gain margin, and phase margin. The distance from the point to the plot indicates the robustness of the system against gain variations, and the angle at the point where the plot crosses the real axis indicates the phase margin.

Example

As an example, consider a simple system with the transfer function . The Nyquist plot for this system would be a semicircle starting at (1,0) and ending at (-1,0) in the complex plane.

Tools and Software

Creating Nyquist plots by hand can be challenging for complex systems. In practice, software tools like MATLAB, Python (with libraries like NumPy and Matplotlib), or specialized control system design software are used to generate these plots accurately and efficiently.
Remember, the Nyquist plot is a powerful tool in control theory, used for analyzing the frequency response of linear systems and assessing their stability and performance in the frequency domain.

Nyquist criterion in s-plane and af-plane

The Nyquist criterion is a fundamental tool in control theory for determining the stability of a feedback system. It involves plotting the frequency response of the open-loop transfer function and analyzing its behavior in the complex plane. The criterion is typically applied in two different complex planes: the s-plane (Laplace domain) and the af-plane (Nyquist plot). Let's explore each:

Nyquist Criterion in the s-plane:

  1. s-plane (Laplace Domain):
      • The s-plane represents the complex frequency domain, where \( s = \sigma + j\omega \) is a complex number. Here, \( \sigma \) is the real part (indicating exponential growth or decay), and \( \omega \) is the imaginary part (indicating oscillation).
      • In the s-plane, poles and zeros of the system's transfer function are plotted. The location of these poles and zeros is crucial for system stability.
  1. Stability Analysis:
      • A system is generally considered stable if all poles of the closed-loop transfer function are in the left half of the s-plane (i.e., have negative real parts).
      • The Nyquist criterion in the s-plane involves checking the number and position of poles on the right half of the s-plane and their relation to the contour encircling the right half-plane.

Nyquist Criterion in the af-plane (Nyquist Plot):

  1. af-plane (Nyquist Plot):
      • The Nyquist plot is a graphical representation of the open-loop transfer function \( G(s)H(s) \) plotted over a range of frequencies. It's a plot of the imaginary part versus the real part of \( G(j\omega)H(j\omega) \) as \( \omega \) varies from \( -\infty \) to \( +\infty \).
      • This plot shows how the phase and magnitude of the open-loop transfer function change with frequency.
  1. Stability Analysis:
      • The Nyquist stability criterion involves encircling the critical point \(-1 + 0j\) in the af-plane. The number of encirclements of this point, and the direction of these encirclements, determine the stability of the closed-loop system.
      • The key principle is that the number of clockwise encirclements of the point \(-1\) must equal the number of poles of \( G(s)H(s) \) that are in the right half of the s-plane for the system to be stable.

Conclusion:

  • s-plane Analysis: Focuses on the location of poles and zeros of the transfer function in the complex frequency domain. Stability is determined by the position of poles relative to the imaginary axis.
  • af-plane Analysis (Nyquist Plot): Involves plotting the open-loop transfer function's frequency response and analyzing encirclements around the critical point \(-1 + 0j\). It provides a more visual and often more practical method of assessing stability, especially in systems where the transfer function is complex or not easily factorizable.
Both approaches are used to determine the stability of a system, but they do so from different perspectives. The s-plane analysis is more about the inherent characteristics of the system, while the Nyquist plot (af-plane) focuses on the system's response to a range of frequencies.

s-plane and af-plane

To establish connections between the s-plane (Laplace domain) and the af-plane (Nyquist plot), it's important to understand how each represents the behavior of a system and how they relate to each other in the context of control systems and stability analysis.

s-plane (Laplace Domain):

  1. Representation: The s-plane is a complex plane where the horizontal axis represents the real part (\(\sigma\)) and the vertical axis represents the imaginary part (\(j\omega\)) of the complex frequency \(s\).
  1. Poles and Zeros: In the s-plane, the system's behavior is characterized by the locations of its poles and zeros. Poles are values of \(s\) where the system's transfer function goes to infinity, and zeros are values of \(s\) where the transfer function becomes zero.
  1. Stability Analysis: Stability in the s-plane is determined by the locations of the poles. A system is stable if all poles are in the left half-plane (having negative real parts).

af-plane (Nyquist Plot):

  1. Representation: The af-plane, or Nyquist plot, is a graphical representation of the open-loop transfer function \(G(j\omega)H(j\omega)\) plotted over a range of frequencies. It plots the imaginary part versus the real part of this function.
  1. Frequency Response: The Nyquist plot shows how the phase and magnitude of the open-loop transfer function change with frequency. It provides a visual way to assess the system's response to different frequencies.
  1. Stability Analysis: The Nyquist stability criterion involves encircling the critical point \(-1 + 0j\) in the af-plane. The number of clockwise encirclements of this point, minus the number of counterclockwise encirclements, must equal the number of poles of \(G(s)H(s)\) that are in the right half of the s-plane for the system to be stable.

Connections Between s-plane and af-plane:

  1. Poles and Frequency Response: The poles in the s-plane directly influence the shape of the Nyquist plot in the af-plane. Poles on the right half of the s-plane (unstable poles) affect the Nyquist plot's encirclement of the \(-1\) point.
  1. Mapping from s-plane to af-plane: The Nyquist plot is essentially a mapping of the open-loop transfer function \(G(s)H(s)\) evaluated along the imaginary axis (\(s = j\omega\)) of the s-plane. This mapping translates the frequency response from the Laplace domain to a graphical form in the complex plane.
  1. Stability Interpretation: Both plots are used for stability analysis but from different perspectives. The s-plane directly shows the stability through the pole positions, while the Nyquist plot reveals stability through the interaction of the frequency response with the critical point \(-1 + 0j\).
  1. Complementary Tools: The s-plane is more about the inherent characteristics of the system (poles and zeros), while the Nyquist plot focuses on how the system responds to a range of frequencies. Together, they provide a comprehensive view of system stability and performance.
In summary, the s-plane and af-plane are interconnected tools used in control system analysis. The s-plane provides a fundamental understanding of system behavior through poles and zeros, while the Nyquist plot translates this behavior into a frequency response perspective, offering a practical approach to stability analysis.
 
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