# Frequency Response Analysis

 Question 1
Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path, where the gain $K=2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ has no poles on the closed right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is _______

 A 0 B 1 C 2 D 3
GATE EC 2022   Control Systems
Question 1 Explanation:

For K = 2, point A will be -0.8x2 = -1.6
Hence N = -2, P = 0
(By default Nyquist contoure is considered in clockwise direction)
P - N = 2
Number of closed loop pole in right side of the complex plane.
 Question 2
The complete Nyquist plot of the open-loop transfer function G(s)H(s) of a feedback control system in the figure.

If G(s)H(s) has one zero in the right-half of the s-plane, the number of poles that the closed-loop system will have in the right-half of the s-plane is
 A 0 B 1 C 4 D 3
GATE EC 2021   Control Systems
Question 2 Explanation:
The given Nyquist plot is not matched according to the data given in the question.

 Question 3
The pole-zero map of a rational function G(s) is shown below. When the closed counter $\Gamma$ is mapped into the G(s)-plane, then the mapping encircles.
 A the origin of the G(s)-plane once in the counter-clockwise direction. B the origin of the G(s)-plane once in the clockwise direction. C the point -1+j0 of the G(s)-plane once in the counter-clockwise direction. D the point -1+j0 of the G(s)-plane once in the clockwise direction.
GATE EC 2020   Control Systems
Question 3 Explanation:
s-plane contour is encircling 2-poles and 3-zeros in clockwise direction hence the corresponding G(s) plane contour encircles origin 2-times in anti-clockwise direction and 3-times in clockwise direction.
Therefore, Effectively once in clockwise direction.
 Question 4
For an LTI system, the Bode plot for its gain is as illustrated in the figure shown. The number of system poles $N_p$ and the number of system zeros $N_z$ in the frequency range $1Hz\leq f\leq 10^7Hz$ is
 A $N_p=5,N_z=2$ B $N_p=6,N_z=3$ C $N_p=7,N_z=4$ D $N_p=4,N_z=2$
GATE EC 2019   Control Systems
Question 4 Explanation:

Number of poles $(N_{P})$= 6
Number of zeros $(N_{Z})$ = 3
 Question 5
The figure below shows the Bode magnitude and phase plots of a stable transfer function $G(s)=\frac{n_{o}}{s^{2}+d_{2}s^{2}+d_{1}+d_{0}}$
Consider the negative unity feedback configuration with gain k in the feedforward path. The closed loop is stable for $k \lt k_{o}$. The maximum value of $k_{o}$ is ______.
 A 0.1 B 0.2 C 0.3 D 0.4
GATE EC 2018   Control Systems
Question 5 Explanation:
For G(s)
$M_{\mathrm{dB}}\left(\omega_{p c}\right)=20 \mathrm{dB}$
When cascaded with k ,
\begin{aligned} \mathrm{GM}_{\mathrm{dB}} &=-20 \mathrm{dB}-20 \log _{10}(k) \gt 0 \mathrm{dB} \\ 20+20 \log _{10}(k) & \lt 0 \\ 20 \log _{10}(k) & \lt -20 \\ k & \lt 10^{-1}=0.10\\ \text{So,}\quad k_{0}&=0.10 \end{aligned}

There are 5 questions to complete.