Question

How do you solve for phase margin for both an open and closed loop transfer function without Matlab?

Answer 1

Phase margin tells about the stability of closed loop system. If we know the closed loop transfer function we can define it for open loop system but generally it is measured from open loop transfer system.

Now we can take here example of unity feedback system whose closed loop transfer function is

Instability occurs when characteristic equation becomes

To calculate phase margin we need gain cross over frequency

It is the frequency at which

This frequency is critical point if

then system is unstable

The phase margin is relative to 180 degree where magnitude is unity.

is known as phase margin frequency.

Note: To calculate phase margin from open loop or closed loop transfer function, the only thing which you need is characteristic equation which you can calculate from both the transfer functions. Phase margin tells only about closed loop system stability.