Consider the Bode and Nyquist plot of the return ratio, L (s), of an unknown sys

Question

Consider the Bode and Nyquist plot of the return ratio, L (s), of an unknown system shown in Figure 3 (next page). We know that there is one unstable pole in the return ratio, L (s). (a) What is the Gain Margin and Phase Margin of L (s)? (b) Is the system stable? If not, how many unstable closed loop poles are there? (c) Find the range of gain K, if any, for which the system is unstable and give the number of unstable closed loop poles for each range. (d) Approximately, what gain K, if any. would give you a phase margin of 30 degree? (e) What is the approximate Gain Margin when the gain in part (d) is applied to the system? Is this system stable with this gain?

Explanation / Answer

a) Phase margin is the (phase at which gain is 0) + 180

In the given Bode plot, gain is zero for a corresponding phase of -180 at 0.1 rad/sec. Hence phase margin is 0

Gain margin is gain at frequency at which the ohase is -180.

from the diagram this happens twice - first at 0.1 rad/sec when gain is 0, second at approximately 8 rad/sec when gain is -20 dB

Hencegain margin is 0- aforementioned gain = 0-(-20) = 20 dB.

b) Systems with positive margins are stable. Hence the given system is stable

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