# How to find the stability margin of a transfer function?

In the control there are two kinds of margin

• Phase Margin
• Gain Margin

There are plenty of examples available to find about it.

Now here is my doubt.

What is stability margin? I was studying this pdf. On page-13, just before the Ziegler-Nichols' Tuning Methods topic, there is a paragraph which contains the stability margin values. But it does not explain how to find these stability margin values. Can anyone explain the steps to find the stability margin?

The easiest way to visualize both margins is to plot the openloop transfer function, also called a Nyquist plot.

Now the gain margin is the gain by which you can multiply the transfer function such that it will cross the minus one point. This can be found by finding where the transfer function crosses the negative x-axis. Since applying a gain to the transfer function scales the entire plot uniformly, therefore the gain margin is one over the distance between the negative x-axis crossing and the origin. On a side note: it is possible that the gain margin is infinite if the transfer function does not cross the negative x-axis (excluding the origin itself). This concept can also be translated to a bode plot of the open loop transfer function, but now you have to find the point(s) where the phase is -180° (plus or minus a hole number of times 360°), since that translates to the negative x-axis. The gain margin itself can be found by taking one over the modulus (transforming the dB to normal gain) at those frequencies.

The phase margin is the angle by which the Nyquist plot can be rotated clockwise around the origin before the transfer function would cross the minus one point, or in other words the angle between a point which crosses the unit circle (relative to the origin), the origin and the minus one point. In a bode plot of the transfer function the unit circle translates to the 0 dB line of the magnitude plot, so the phase margin can also be found by looking at the phase at the frequencies of the 0 dB crossings relative to the -180° (plus or minus a hole number of times 360°), since that translates to the phase of the minus one point.

You can actually define a third margin, namely the modulus margin, which is the smallest distance between the transfer function and the minus one point in the Nyquist plot. This also translates to one over the highest value of the sensitivity function.

• Thanks @fibonatic for your reply. Is there any example available about the sensitivity function? – Ashutosh Kumar Aug 31 '16 at 7:26
• @AshutoshKumar You mean something like this this also gives some additional explanation of the two margins. – fibonatic Aug 31 '16 at 10:58
• Thanks @fibonatic for that video link. Your first link explains me lot about the sensitivity margin and second one,video link, give me a proper glimpse about it. I am making a humble request to you. Is there any example available so that I can cross check myself ? Thanks in Advance – Ashutosh Kumar Sep 1 '16 at 6:11

There are some typos on Page 13 which may add to the confusion. In figure 8.9, the right figure, the k graph is labelled ki, and the ki graph is labelled k. In the text, the stability margins have been misassigned: 0.67 should go with 0.18, 0.60 with 0.30, 0.55 with 0.36, 0.48 with 0.58.