Instability as seen in time domain

Question

It is my understanding that an unstable closed loop in a minimum-phase system occurs when the loop gain (A*B of the characteristic equation) encircles the -1 point on the Nyquist chart. In https://www.analog.com/media/en/technical-documentation/application-notes/an149fa.pdf in Fig.1 and Fig.3 instability is shown, which looks like relatively small disturbances on the output. By instability do they mean an encirclement of the -1 point or just coming close to it (i.e. low gain/phase/modulus margin)?

Would an encirclement not mean that the output starts oscillating and the oscillations exponentially diverge to some physical limit such as an operational amplifier rail? And a borderline situation would be when the loop gain exactly passes through the -1 point, which would create stable oscillations of a fairly high value?

I am basically trying to find out how a loop instability can demonstrate itself in the time domain and if there are any special cases.

Answer 1

Run a time domain analysis (=transient analysis) of any unstable circuit in a circuit analyzer program and see, how the oscillation starts. It can be DC - the output voltage drifts to the minimum or maximum, it can occur at a single frequency or it can be complex when the oscillations can happen in numerous frequencies. Non-inearity can even make the oscillation in such cases chaotic (=no repeating pattern).

Some starting impulse is needed, but in common transistor oscillator circuits starting the analysis is generally enough, because the capacitors start to get charged. Using Matlab or another simulation system with theoretical noiseless blocks needs a starting impulse, a precharged state variable or a noise injection.

To see clearly the growth of the amplitude use a sinewave oscillator and have long enough analysis duration. An example:

You can see how the circuit searches a while the operating point and finally the oscillation escalates. The amplitude is limited by the non-linearity of the transistor. The amplitude stops to the value where the gain of the transistor is reduced so much that further amplitude growth would make the oscillation impossible. The oscillation is not pure sine because the transistor amp distorts badly.

With more advanced analysis programs (Matlab for ex.) you can draw the Nyqvist plot of the loop gain and adjust some parameter to see what's the difference a) as Nyqvist plots b) in time domain and between systems which are

• stable
• stable, but nearly oscillating
• unstable, but so near the stability that the amplitude grows only slowly or stays the same, if started somehow
• strongly unstable, the amplitude grows to the clipping limit during the first oscillation cycle.

Using a circuit analyzer is not especially good for this, because the loop gain can be affected by the loading much when the loop is closed for the time domain analysis. Low frequency opamp circuits only could be recommended. It's better to use ideal amplifier, integrator and summing blocks + an amplitude limiter. At least MicroCAP has them. Matlab is the perfect one.