How to Draw a Nyquist Plot: Problem 6 in Frequency Response Analysis for Control Systems

TL;DR

Draw the Nyquist plot by substituting s = jω into the open-loop transfer function, calculating magnitude and phase for different frequency values, and plotting the resulting points on a polar graph. For this problem, the gain K is assumed to be 1, and closed-loop stability is examined by checking encirclements of the point −1 + j0. Read on for the calculation steps and stability test.

Transcript

hello friends in this video we are going to solve a problem on how to draw the nyquist plot of a system whose transfer function is given to us so let's try to solve the problem so our problem is we have to draw the nyquist plot and examine the stability of the closed loop control system which is having the open loop transfer function as gshs equals... Read More

Key Insights

• 🎮 The Nyquist plot helps in understanding the stability characteristics of a control system.
• ❓ By analyzing the encirclements made by the plot, the stability of the system can be determined.
• 💁 Calculating the magnitude and phase angle of the transfer function provides valuable information about the system response at different frequencies.
• 👻 The Nyquist plot is a mirror image of the polar plot, allowing for easier analysis and interpretation.
• 🤩 The open loop transfer function plays a key role in drawing the Nyquist plot and examining system stability.
• 😥 The number of encirclements around the point -1+j0 indicates the overall system stability.
• 🔺 The magnitude and phase angle equations are used to calculate the values for different frequencies.

Questions & Answers

Q: How do you draw the Nyquist plot in Problem 6 for the control system?

First substitute s = jω into the given open-loop transfer function to obtain G(jω)H(jω). Then derive its magnitude and phase equations, calculate both for different values of ω, and plot the resulting points on a polar graph to construct the Nyquist plot.

Q: What is the first step in solving this Nyquist plot problem?

The first step is to put s = jω into the open-loop transfer function. This converts it into the frequency-domain expression G(jω)H(jω), which can then be separated into real and imaginary parts.

Q: What gain value is assumed in the calculation?

The calculation assumes that the gain K equals 1. The magnitude and phase expressions used for the plot are then developed with this assumed gain.

Q: Why is the transfer function separated into real and imaginary parts?

Separating the expression into real and imaginary parts makes its magnitude and phase angle easier to calculate. These two quantities are needed at different frequency values to generate the Nyquist plot.

Q: How is the magnitude of G(jω)H(jω) calculated?

For each complex term, take the square root of the real part squared plus the imaginary part squared. In the denominator shown, the real part is 8 and the imaginary part is 4ω − ω³, so its magnitude contains the square root of 64 + (4ω − ω³)².

Q: How is the phase angle of the open-loop transfer function calculated?

The phase of a complex expression is found using tan inverse of the imaginary part divided by the real part. Numerator phase angles are added, while the denominator phase angle is subtracted; the derivation also uses tan⁻¹(∞) = 90 degrees.

Q: Why are multiple frequency values used when drawing the Nyquist plot?

Both the magnitude and phase equations contain ω, so their values change with frequency. A table is created using different values of ω, and the calculated magnitude and phase pairs provide the points needed for the plot.

Q: How is closed-loop stability determined from the Nyquist plot?

Stability is examined by counting the plot’s encirclements around the point −1 + j0. According to the page summary, no encirclements indicate that the system is stable.

Summary & Key Takeaways

• The video discusses the process of drawing the Nyquist plot for a control system using the open loop transfer function.

• The magnitude and phase angle of the transfer function are calculated for different values of frequency (omega).

• A table of magnitude and phase angle values is created, which is then used to create the Nyquist plot by plotting the points on a polar graph.

• The stability of the system is determined by counting the number of encirclements made by the plot around the point -1+j0. If there are no encirclements, the system is stable.