Loop transfer function of a feedback system is \(G\left( s \right)H\left( s \right) = \frac{{s + 3}}{{{s^2}\left( {s - 3} \right)}}\;.\) Take the Nyquist contour in the clockwise direction. Then, the Nyquist plot of G(s) H(s) encircles \(- 1 + j0\)
once in clockwise direction
Nyquist Plot Analysis of the Given Transfer Function
Open-Loop Transfer Function Details
The open-loop transfer function provided is:
\( G\left( s \right)H\left( s \right) = \frac{{s + 3}}{{{s^2}\left( {s - 3} \right)}} \)
Identifying Poles and Zeros
To determine the behavior of the Nyquist plot, we first identify the poles and zeros of the transfer function:
- Poles: The poles are the values of s for which the denominator is zero.
- \( s = 0 \) (a pole of order 2)
- \( s = 3 \)
- Zeros: The zeros are the values of s for which the numerator is zero.
- \( s = -3 \)
Applying the Nyquist Stability Criterion
The Nyquist stability criterion relates the number of encirclements of the critical point \( (-1 + j0) \) on the polar plane to the number of poles and zeros in the right-half of the s-plane (RHP).
The criterion is given by the formula:
\( N = P - Z \)
Where:
- \( N \) is the number of clockwise encirclements of the \( (-1 + j0) \) point.
- \( P \) is the number of poles of \( G(s)H(s) \) in the RHP.
- \( Z \) is the number of zeros of \( G(s)H(s) \) in the RHP.
Determining RHP Poles (P)
We examine the poles of \( G(s)H(s) \):
- The pole at \( s = 3 \) is located in the right-half of the s-plane.
- The poles at \( s = 0 \) are located on the imaginary axis, not in the RHP.
Therefore, the number of poles in the RHP is \( P = 1 \).
Determining RHP Zeros (Z)
We examine the zeros of \( G(s)H(s) \):
- The zero at \( s = -3 \) is located in the left-half of the s-plane.
There are no zeros in the RHP. Therefore, the number of zeros in the RHP is \( Z = 0 \).
Calculating the Number of Encirclements (N)
Using the Nyquist criterion formula:
\( N = P - Z = 1 - 0 = 1 \)
The result \( N = 1 \) indicates that the Nyquist plot encircles the \( (-1 + j0) \) point once.
Direction of Encirclement
A positive value of \( N \) signifies that the encirclements are in the clockwise direction. The Nyquist contour specified is taken in the clockwise direction, which is consistent with the standard application of the \( P - Z \) rule yielding a positive \( N \) for clockwise encirclements.
The presence of poles at the origin (like \( s^2 \)) requires indenting the Nyquist contour around these poles. When the contour is indented into the RHP (clockwise), the mapping of this small semicircle contributes to the overall phase change. However, the formula \( N = P - Z \) correctly accounts for the net number of encirclements of the \( (-1 + j0) \) point.
Conclusion
Based on the analysis using the Nyquist stability criterion, where \( P=1 \) and \( Z=0 \), the Nyquist plot of the given transfer function \( G(s)H(s) \) encircles the point \( (-1 + j0) \) once in the clockwise direction.
Important Questions from Nyquist Plot
In Nyquist plot of a system on adding a pole at s = 0, then plot will -
The Nyquist plot of the transfer function \(G\left( s \right) = \frac{K}{{\left( {{s^2} + 2s + 2} \right)\left( {s + 2} \right)}}\)
Does not encircle the point (–1 + j0) for K = 10 but does encircle the point (-1 + j0) for K = 100 . Then the closed-loop system (having unity gain feedback) is
A closed-loop control system is stable if the Nyquist plot of the corresponding open-loop transfer function
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:
Consider a negative unity feedback system with forward path transfer function \(G\left( s \right) = \frac{K}{{\left( {s + a} \right)\left( {s - b} \right)\left( {s + c} \right)}}\), where K, a, b, c are positive real numbers. For a Nyquist path enclosing the entire imaginary axis and right half of the s-plane is the clockwise direction, the Nyquist plot of (1 + G(s)), encircles the origin (1 + G(s)) –plane once in the clockwise direction and never passes through this origin for a certain value of K. then, the number of poles of \(\frac{{G\left( s \right)}}{{1 + G\left( s \right)}}\) lying in the open right half of the s-plane is ______.
