# Gain margin using Root locus plot

How can I find Gain margin of any control system given its polar plot In schaum's series I have read that Gain Margin=Ku/Ki but in given figure the system never becomes unstable so shouldn't Gain margin be infinity.

Also should the gain margin of the above system be infinite

• @Chu Plz correct me ..... but I have read that the gain margin is additional gain in forward path necessary to bring a system on verge of instability. And I have mentioned Initial gain in forward path Ki so what do you mean by -1 point – SUNITA GUPTA Jul 23 at 8:48
• A 2nd order system with positive denominator coefficients (as is this system) cannot be unstable, so GM is infinite. – Chu Jul 23 at 8:54
• This is correct - Never used root locust, only found the Gm through bode plots. So if you generate a bode plot, find the point where the phase plot crosses -180 deg now run straight up through that line into your gain plot, the difference from that point to 0db is your gain margin. You can see an example here se.mathworks.com/help/control/ref/margin.html sorry for not knowing about the root locust, i was taught this way, maybe you could use it as an alternative – Sorenp Jul 23 at 8:55
• @Chu Yes but can't second order system can be marginally stable like in above figure.I am just getting confused by wordings "on verge of instability" {kuo} because above system goes to just become marginally stable and then comes back to stability So should i consider the point as the system was on verge of instability. – SUNITA GUPTA Jul 23 at 9:07
• Yes it’s critically stable at that point, and the loop gain is infinite. – Chu Jul 23 at 10:59