How could we predict the transient response of the multi-loop circuit?

Question

Typically, when analysis the stability of a single loop circuit, we can use the open-loop method, just to break the loop and get the loop gain. Then, using the Bode diagram, we can easily get the PM(Phase Margin) information, from that we can predict how the transient response will look like.

However, when analysis the stability of a multi-loop circuit, if we also use the open-loop method, we will get two different loop gain from different break point.

Theoretically, we can still get the same absolute stability information from both two loop gain, but the two loop gain show the different information about PM.

So, my question is -- How could we predict the transient response of the multi-loop circuit?

For example, if one of the loop gain is 90°, and the another loop gain is 45°, will the circuit’s transient response look like overdamped or underdamped?

Hope to get an answer, thanks.

Answer 1

So, my question is -- How could we predict the transient response of the multi-loop circuit?

I wouldn't hesitate to use a free simulation tool like micro-cap (or quite possibly LTspice). In microcap you can create H(s) Laplace blocks of any reasonable complexity and, you can give the input a transient step change and look at how the output responds in the time domain.

Or, do it the mathematical way by combining H1(s), H2(s) and B2 then solve by incorporating B2 to get a formulaic solution.