Rollett stability factor - derivation and numerical meaning

Question

For two-port networks, a value called the Rollett stability factor, K, can be derived, whose value can then be used to assess one of the criteria for the network to be unconditionally stable (K > 1).

I suppose that only holds whenever both of the stability circles are entirely outside of the unit circle on in the Γ domain - is that correct?

Hence, I wonder:

1. How can it be derived from the expressions for radii and centers of the stability circles (r_L, c_L, r_S, c_S)?
2. What meaning does the value of K have, e.g. in the expression for G_max? (from here) - can it be visualized directly?

Answer 1

The Rollett stability factor is only applicable for two-port networks and is used to assess one of the criteria for unconditional stability (K > 1) when both stability circles are entirely outside the unit circle in the Γ domain.

To derive the Rollett stability factor, you need the expressions for the radii and centers of the stability circles (rL, cL, rS, cS). The Rollett stability factor is then defined as K = min(rL, rS) / max(rL, rS). This value represents the ratio of the smaller to the larger of the two stability circles, and it provides an indication of the network's stability. If K > 1, the network is considered to be unconditionally stable.

The Rollett stability factor can also be visualized directly as the ratio of the smallest to largest stability circles. The value of K has a direct relationship to the expression for the maximum gain of a two-port network. The formula for the maximum gain, Gmax, is defined as Gmax = 1 / (1 - K^2). The maximum gain represents the highest possible power gain that can be achieved by the two-port network without causing instability. If K > 1, Gmax will always be finite, indicating that the network is unconditionally stable.