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(a) Explain how the Nyquist stability criterion can be used to determine the stability of a closed-loop feedback system. Formulate the criterion for a system with no right-half plane open-loop poles. (b) Consider the Nyquist plot of the frequency response of the system G(jw) = 500 shown in Figure Q4 below. (jw+1)(jw+10)2 i) Explain how the gain margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. What is the maximum value of an additional gain K for which the closed loop would still be stable?

(a) Explain how the Nyquist stability criterion can be used to determine the stability of a closed-loop feedback system. Formulate the criterion for a system with no right-half plane open-loop poles. (b) Consider the Nyquist plot of the frequency response of the system G(jw) = 500 shown in Figure Q4 below. (jw+1)(jw+10)2 i) Explain how the gain margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. What is the maximum value of an additional gain K for which the closed loop would still be stable?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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CALCULATE Q 4 PART A B i, ii

Q4 (a) Explain how the Nyquist stability criterion can be used to determine the stability of a closed-loop feedback system. Formulate the criterion for a system with no right-half plane open-loop poles. (b) Consider the Nyquist plot of the frequency response of the system G(jw) = 500 shown in Figure Q4 below. (jw+1)(jw+10)2 i) Explain how the gain margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. What is the maximum value of an additional gain K for which the closed loop would still be stable? 0.5 -0.5 -1 -1 -0.5 0.5 1 Real Axis Figure Q4 ii) Explain how the phase margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. Assuming that the corresponding point on the plot occurs at 4 rad/s, what is the maximum delay which can be added to the system before the closed loop would become unstable? Sketch the approximate Bode diagram for the system in part (b) of the question, clearly showing the asymptotes and corner frequencies. Indicate the gain and phase margins in your diagram. (d) Explain how the bode plot changes for the system in part (b) when an additional gain of K = 10 is added. Sketch the Bode diagram of the modified system and comment on the stability of the closed loop in this case. Imaginary Axis
Transcribed Image Text:Q4 (a) Explain how the Nyquist stability criterion can be used to determine the stability of a closed-loop feedback system. Formulate the criterion for a system with no right-half plane open-loop poles. (b) Consider the Nyquist plot of the frequency response of the system G(jw) = 500 shown in Figure Q4 below. (jw+1)(jw+10)2 i) Explain how the gain margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. What is the maximum value of an additional gain K for which the closed loop would still be stable? 0.5 -0.5 -1 -1 -0.5 0.5 1 Real Axis Figure Q4 ii) Explain how the phase margin can be obtained from the Nyquist plot and determine its approximate value from Figure Q4. Assuming that the corresponding point on the plot occurs at 4 rad/s, what is the maximum delay which can be added to the system before the closed loop would become unstable? Sketch the approximate Bode diagram for the system in part (b) of the question, clearly showing the asymptotes and corner frequencies. Indicate the gain and phase margins in your diagram. (d) Explain how the bode plot changes for the system in part (b) when an additional gain of K = 10 is added. Sketch the Bode diagram of the modified system and comment on the stability of the closed loop in this case. Imaginary Axis
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