Robox's Multi-AC source - Loop gain / return ratio PUBLIC

CircuitLab can calculate "return ratio" or "loop gain" to show when a dependent source can create an unstable system.

In reply to this forum post.

The original circuit presented here:

and in CircuitLab format here:

is poorly formed and unstable for transient analysis. If we turn off the independent voltage and current sources, and look only at the feedback loop created by the dependent source and the remainder of the passive network, we get the circuit shown here. This feedback loop is from the dependent source, through the network, to the measured control quantity, and back to the dependent source.

Injecting a current at port I1, what current is returned through the capacitor? If you run the frequency domain simulation, you'll find a peak of magnitude 0.8 with zero phase. This means that at that frequency, 1 unit of current injected from the dependent source causes 0.8 units of current through the capacitor. Because the dependent source has a gain of 2, this new 0.8 units of current through the capacitor causes 1.6 units of current to be injected from the dependent source. This new 1.6 causes 1.6*0.8=1.28 new current in the capacitor, which itself causes 2.56 new current in the dependent source -- and so on and so on, to infinity.

Because this 1.6 is greater than 1, the system is unstable. Any small signal injected at this frequency is amplified literally to infinity -- this system is unstable. (There are other more accurate ways of explaining this in terms of phase margin etc., but I hope this qualitative picture is broadly helpful!)

The original system is therefore also unstable, and won't simulate "properly" in transient mode.

https://en.wikipedia.org/wiki/Return_ratio