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So there is a well known condition which relates the phase margin with the damping ratio for a unity feedback system:

$$\Phi_m = \tan^{-1} \frac{2 \zeta}{\sqrt{-2 \zeta^2 + \sqrt{1+4\zeta^4}}}$$

This equation assumes that the closed-loop transfer function is a damped second order function: $$T(s) = \frac{L(s)}{1+L(s)} = \frac{\omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2}$$

where L(s) is the open-loop transfer function of a unity-feedback system (with L(s) = G(s)). The derivation for the phase margin above is done by assuming that $$L(s) = G(s) = \frac{\omega_n^2}{s(s+2\zeta \omega_n)}$$ My question is, how would the phase margin equation above change if the system was non-unity feedback (i.e., if the open-loop transfer function was L(s) = G(s)H(s), where G(s) is in the feedforward path, and H(s) is in the feedback path)? I could not find any book or paper that provides this derivation.

Your help is much appreciated!

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  • \$\begingroup\$ Please provide a link to a site/paper/document that defines that well known condition. \$\endgroup\$ – Andy aka May 10 '20 at 13:21
  • \$\begingroup\$ Control Systems Engineering (Norman Nise, 6th edition) \$\endgroup\$ – Johnny Que May 10 '20 at 13:57
  • \$\begingroup\$ I don't think that your 2nd formula is correct - shouldn't it be \$2\omega_n^2\$ in the denominator? \$\endgroup\$ – Andy aka May 10 '20 at 13:58
  • \$\begingroup\$ The was a square in the formula for G(s), which I corrected (should not have been a square) \$\endgroup\$ – Johnny Que May 10 '20 at 14:03
  • \$\begingroup\$ I still don't think your 2nd formula is correct - maybe you can photograph the page of the book and post to your question. \$\endgroup\$ – Andy aka May 10 '20 at 14:46
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Johnny Que - may I ask you WHY do you think that the shown relation between the damping factor and the phase margin (for a second-order sysytem) would be valid for unity feedback only?

I rather think - better: I am convinced - that it applies to all 2nd-order systems.

That means: For the loop gain expression L(s)=G(s)H(s) .

Reference: Robert C. Dorf: "Modern Control Systems", 6th Edition, Addison Wesley, 1992

Comment: The formula is derived in the referenced book.

More than that, it can be shown that for PM < 65 deg and damping < 0.707 this expression can be approximated with good accuracy by PM=damping/0.01.

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  • \$\begingroup\$ So the derivation of the phase margin assumes that $$L(s) = G(s) = \frac{\omega_n^2}{s(s+2\zeta \omega_n)}$$...With this expression (and with unity feedback), you can get the typical second order response T(s) that I wrote in my original post. However, with non-unity feedback, you can't assume $$L(s) = G(s)H(s) = \frac{\omega_n^2}{s(s+2\zeta \omega_n)}$$ because then you won't get the second order function T(s), since in the non-unity feedback case, $$T(s) = \frac{G(s)}{1+G(s)H(s)}$$ \$\endgroup\$ – Johnny Que May 10 '20 at 14:01
  • \$\begingroup\$ @JohnnyQue You seem to be missing the point that phase margin is an open-loop phenomenon so you just calculate PM for the open loop case. \$\endgroup\$ – Andy aka May 10 '20 at 14:14
  • \$\begingroup\$ @Andy aka The point of the phase margin equation I posted in my original post is to relate the phase margin with the damping of the closed-loop response. So the PM (which is calculated using open-loop data) can be used to relate the closed-loop behavior...it's not just an "open-loop phenomenon." \$\endgroup\$ – Johnny Que May 10 '20 at 14:22
  • \$\begingroup\$ @JohnnyQue - Phase margin is defined as the amount of change in open-loop phase needed to make a closed-loop system unstable. In other words it's an open-loop phenomenon that can be used to predict closed-loop behaviour. \$\endgroup\$ – Andy aka May 10 '20 at 14:43
  • \$\begingroup\$ I think - The phase margin is basically a "closed-loop phenomenon". It is true that it is defined - very often - for the open loop (loop gain) because this allows a simple measurement technique. But it originates from the closed-loop system: The phase margin is the additional phase which must be inserted into the closed loop in order to shift the closed-loop poles to the imaginary axis (oscillation). Hence, the formula under discussion involves closed-loop parameters \$\endgroup\$ – LvW May 10 '20 at 14:59
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Taking a 2-pole amplifier, the open loop frequency response of which is shown below.

O.L Frequency response.

From this open loop frequency response, and assuming a H(s) = 1/2, I generated the loop transfer function. Then substituting s=jw I entered various values of w into the transfer function until I converged upon the value of w which resulted in a loop gain of 1, thereby also obtaining the phase at this frequency which reveals the phase margin.

Next I generated the closed loop transfer function, G(s)/(1+GH(s)), from the denominator of which I obtained the value of zeta.

I found that for a dc closed loop gain of 2 the value of zeta is 0.707 and the phase margin is 65.5 degrees.

I repeated the above analysis for a closed loop gain of 4, H(s)=1/4, and found the value of zeta to be 1 and the phase margin to be 76.3 degrees.

These results for dc closed loop gains of 2 & 4 (zeta = 0.707 & 1 respectively) agree almost exactly with the results given by your phase margin equation in your question.

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