Phase Margin

The questions below are due on Thursday April 06, 2023; 11:59:00 PM.

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Consider again the feedback system below,

where H(s) has the three-pole transfer function

H(s) = \frac{10*1000 * 1000}{(s+10)(s+1000)(s+1000)}

and K(s) = K is a constant gain factor. From Black's formula, the closed loop transfer function is

G(s) = \frac{KH(s)}{1+KH(s)}.

You may find it easiest to do this problem by trial and error using MATLAB's bode and/or margin plots, though you should be able to intelligently tune K. Also, when looking at overshoot, you can right click the step response in Matlab and have it display the peak amplitude and the final steady-state (settled) value.

Determine a value for K so that KH(s) has a phase margin that is close (within 1 degree) to 90 degrees.

What is the peak overshoot, delta above steady-state, in the unit step response of the closed loop system when its loop transfer function has 90 degrees of phase margin? Give an answer in the form of (peak value - final value). If there is no overshoot, enter zero.

Where are the poles of the closed loop system with 90 degrees of phase margin?

Left-half of s-plane

Right-half of s-plane

On the s=jw axis

Now set K to yield about 60 degrees of phase margin. What is the peak overshoot, delta above steady-state, in the unit step response of the closed loop system when its loop transfer function has 60 degrees of phase margin? Give an answer in the form of (peak value - final value). If there is no overshoot, enter zero.

Where are the poles of the closed loop system with 60 degrees of phase margin?

Left-half of s-plane

Right-half of s-plane

On the s=jw axis

Now set K to yield 30 degrees of phase margin. What is the peak overshoot, delta above steady-state, in the unit step response of the closed loop system when its loop transfer function has 30 degrees of phase margin? Give an answer in the form of (peak value - final value). If there is no overshoot, enter zero.

Where are the poles of the closed loop system with 30 degrees of phase margin?

Left-half of s-plane

Right-half of s-plane

On the s=jw axis

Now set K to yield 0 degrees of phase margin. One pole will be in the left-half of the s-plane. Where are the remaining conjugate pair poles of the closed loop system?

Left-half of s-plane

Right-half of s-plane

On the s=jw axis

Now set K to yield -60 degrees of phase margin (phase drops below 180 before gain drops below 1). Where are the poles of the closed loop system? Solve numerically for the exact locations (for your own learning) and then indicate generally where they are below.

Left-half of s-plane

Right-half of s-plane

On the s=jw axis

In the case with -60 degrees of phase margin, is the system stable or unstable?

Stable

Unstable