# Intuitive understanding of relation between phase margin and peak overshoot and/or damping factor [closed]

Second, percent overshoot is reduced by increasing the phase margin, and the speed of the response is increased by increasing the bandwidth.

"Control Systems Engineering, Norman S. Nise"

The mathematical proof of this statement is available in the book but I am unable to understand it in a physical sense.

• Have a play with a simulator ... or (safe) real system.
– user16324
Commented Jun 5, 2022 at 11:16

It may become more understandable if you relate it to the open loop gain, k.

With regard to a simple 2nd order transfer function with 2 poles.....

Increasing k moves the closed loop system closer to the point of instability. Increasing k increases percentage overshoot and reduces rise time (faster response). Damping ratio, zeta is reduced.

So, increasing k moves the system closer to the point of instability and so phase margin and gain margin are reduced.

Interestingly, increasing k reduces the damping ratio, zeta but increases the undamped natural frequency,Wn and so changing k has no effect on settling time.

EDIT

The behavior of a closed loop system with just 2 poles (2nd order system) is totally determined by the combination of the values of zeta, the damping ratio and Wn, the undamped natural angular frequency. These two variables determine the position of the poles on the S-plane and so you can also say that the behavior of a 2nd order system is determined by the position of the poles on the S-plane.

Well, you might say, I've already said that closed loop system behavior changes with the value of K, the open loop gain. That is true and it is true because changing the value of K in the open loop transfer function alters the values of zeta and Wn in the closed loop transfer function. So ultimately, the behavior of the closed loop system depends solely on the values of zeta and Wn in the closed loop transfer function.

Considering a closed loop system with a low value of open loop gain,K such that the value of zeta is greater than one (over damping). The two poles on the S-plane will be real and spaced apart on the real axis. As the open loop gain,K is increased the poles will move towards each other (converge) until they meet, still on the real axis. At this value of K, the poles have the same value. Zeta has a value of 1. This is critical damping. If the open loop gain is increased further, zeta will become less than one (underdamping), and the two poles will diverge away from each other moving away from each other at right angles to the real axis and parallel to the imaginary axis. The two poles now have an imaginary component, they have become complex conjugate poles and the system is now oscillatory with some overshoot in response to a step input in the time domain. Oscillatory does not mean unstable, it means that, in response to a step input, the closed loop system will oscillate a bit before the output settles down to a steady state value.

This tracing of the movement of the poles of a closed loop system in response to the increase of some system parameter, such as open loop gain, from 0 to infinity is the basis of the Root Locus technique.

As K is increased and the poles move away from each other parallel to the S-plane's imaginary axis we can say that the distance from the origin to the pole is equal to Wn the undamped natural angular frequency and the cosine of the angle which that line makes with the real axis is equal to zeta, the damping ratio and so we can see that, as K increases, the poles move further from the real axis causing a reduction in damping ratio (angle gets closer to 90 degrees so cosine of the angle and hence zeta reduces) and the distance from the origin increases, increasing Wn.

Hence we can see how the the position of the poles are determined by the values of zeta and Wn.

Incidentally, the distance from the real axis to the pole, the distance along the imaginary axis, is equal to Wd, the damped angular frequency which is the frequency that the closed loop system will oscillate at in response to a step input before settling down to its steady state value.

There is a way to link the open-loop phase margin $$\\phi_m\$$ with the closed-loop quality factor $$\Q_c\$$. The exercise involves the compensated open-loop gain $$\T(s)\$$, observed in the vicinity of the crossover frequency. For the purpose of a simple analysis, the response is approximated as a two-pole system. This works for a second-order system without zeroes and practical experiment - at least for switching converters - show a good correlation between theory and experiments. You can see my APEC 2009 seminar (start slide 30) for the mathematical treatment of this expression.

The below picture is excerpted from my last APEC 2021 seminar and shows how the quality factor depends on the phase margin you choose for your compensation strategy:

You see that for having coincident poles ($$\Q\$$=0.5, fast response and no ringing), then the phase margin should be 76°. Actually, the answer to the question "what phase margin do you select?" depends on the transient response you want. If your customer or your project cannot tolerate any overshoot and you want a rock-solid design even it has a sluggish response, then a large phase margin 70-90° can be adopted. On the opposite, if you want a nervous system with a fast transient response but can tolerate a reasonable overshoot, then a reduced phase margin like 50-60° makes more sense. Finally, the very minimum value below which the response becomes unacceptably ringing is 30° and it is the value given by the modulus margin condition together with a 6-dB gain margin.

The below response shows the impact of different phase margins while the crossover frequency was kept constant at 5 kHz:

The phase margin truly affects the recovery speed of your system as you can see.

For the response speed, the answer is simple: no gain, no feedback. It means that your system will react as long as it has gain in the considered frequency domain. Before the 0-dB crossover, the system can reject perturbation and operates in closed-loop in dc and ac but, beyond crossover, the control system operates in ac open loop. Any perturbation of frequencies beyond crossover won't be rejected by the loop and the system will do what it can. Also, pushing crossover in the goal of making the system responsive up to high frequencies is not a good idea since by pushing crossover, the control system becomes more susceptible to noise. Also, the plant transfer function may include resonance or right-half-plane zeroes which will limit your ambitions in terms of crossover frequencies.

Below shows the response of a switching converter whose phase margin is kept constant but crossover frequency has been increased. You can see that the recovery time is almost constant but the point at which the converter recovers its capability becomes smaller as crossover increases. At a certain value, the dropout is fixed by the capacitor ESR and it would not help pushing crossover farther:

For your "intuitive understanding" it is helpful to realize that your question touches the relation between the time domain (overshoot) and the frequency domain (stability margin, damping factor=1/pole quality).

In detail: The step response of a system is the inverse Laplace transform of the transfer function H(s) divided by the complex frequency variable "s".

The denominator of the 2nd-order complex transfer function H(s) is identical with the characteristic polynom which gives the solution of the corresponding differential equation.

More than that, the transfer functions poles (zeroes of the transfer functions denominator, when plotted in the complex frequency plane) have a real part (sigma) which determines the decreasing amplitude in the step response [exp(-sigma*t)].

The position of this pole pair allows the definition of the pole frequency wp (magnitude of the pointer between the origin and the pole) and a quality factor (pole-Q): Qp=wp/(2*sigma).

For sigma=0 (Qp infinite) we have: pole frequency wp=wn (natural frequency) which is identical to the oscillation condition (zero phase margin). For smaller (finite) Qp-values the step response shows oscillations with decreasing amplitudes - for Qp-values in the range of Qp=1 (and smaller) the oscillations merge into a single overshoot, which disappears completely for Qp<0.5.

Formula overshoot g=f(Qp) (with g given in %):

g=100*exp[-Pi/SQRT(4Qp²-1)].