# How to reduce gain margin to finite value?

I am trying to observe bode plots for a system but i am getting infite gain margin. I know that positive phase margin and/or positive gain margin represent stable system while negative phase margin and/or negative gain margin represent unstable system

But what about infite gain margin and/or infinite phase margin, What it represents?

How can i reduce infinite gain margin to a finite value?

My MATLAB code and output snap are given below:

clc;clear;close all
num=[1]
den=[0.1 1]
sys=tf(num,den)
margin(sys)

I would prefer to call the gain margin in that plot "undefined" rather than "infinite".

When gain is measured as Vout/Vin the gain margin is defined as the factor by which you have to multiply the gain, at the phase cross-over frequency (phase = -180 degrees), to get a gain equal to 1. When gain is measured in dBs the gain margin is defined as the number of dBs which must be added to the gain, at the phase cross-over frequency (phase = -180 degrees), to get a gain of 0 dB (gain of 1).

So to have an infinite gain margin implies that the gain is 0 (equal to negitive infinity dBs) at the phase crossover but in your plot you don't have a phase cross-over frequency where the phase gets up to -180 degrees so I would prefer to say that the gain margin is "undefined" rather than "infinite".

To be able to determine an actual (not undefined) value for gain margin, the phase must have a lag of -180 degrees at some frequency. A phase cross-over frequency then exists where the gain margin also exists and can be measured.

The infinite gain margin becomes obvious by looking at the bode plots. The phase of the system lies between 0° and -90°, therefore the 'critical' phase of +/-180° is never reached. Speaking in terms of the nyquist plot, the open loop transfer function will never touch or envelope the critical point -1+0j (for positve gain). The phase margin is -180° because |G(jw)| takes on the value of 1 only for w = 0. Again, taking a look at the nyquist plot reveals that a rotation of 180° is required for the open loop transfer function to touch the cricital point for w = 0 (marginal stability).

Finally, the gain margin becomes finite if the phase of the open loop transferfunction takes on values of +/-180° (maybe shifted by multiples of 360° = full rotations). I recommend looking at the definition of gain and phase margins via the nyquist plot to get a better graphical understanding of this quantities.

• Yes - I fully agree to the above explanation. However, it should be mentioned that the shown example (Bode plot) represents an idealized system only! Due to parasitics and/or finite gain-frequency characteristics of amplifiers such a first-order function does not exist in the physical reality.
– LvW
Nov 29, 2023 at 9:11
• Do you mean that practically in reality, gain margin or phase margin can be never infinity?
– ATJ
Nov 30, 2023 at 9:39