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## Stability determination

We take the input stability circle on plane as the example. There are six possible ways for the stability circle drawn on the plane as the shown in Figure 17. Figure 17: Six possible ways for the input stability circle

We consider only the case where and . The circle represents the location where on the plane. Note that at the origin of the plane, , hence a stable region. Therefore, the area including the origin of the plane up to the circle is the stable region. We can find that for all values of in Figure 17 (a) and 17 (d).

We re-examine the value of  Case (i)  this corresponds to Figure 17 (d), (e), (f). But Figure 17 (d) is unconditionally stable, where Because , we can re-write the condition as Case (ii)  this corresponds to Figure 17 (a), (b), (c). But Figure 17 (a) is unconditionally stable, where Because , we can re-write the condition as Therefore, a necessary condition for unconditionally stability is If we define then K>1 is a necessary condition for unconditionally stability.

If we examine the Figure 17 (b), (c), (e), (f) again, we find that

• Figure 17 (b), (e), (f) have K < 1 and
• Figure 17 (c) has K >1.

Therefore, K > 1 is not the sufficient condition for stability. We must find a way to distinguish Figure 17 (a) from Figure 17 (c). The difference between two Figures is that in Figure 17 (a). Therefore Combine with the inequality  Because for Figure 17 (a), (c). Finally, the necessary and sufficient conditions for unconditionally stability for the input port are Therefore, the necessary and sufficient conditions for unconditionally stable two-port network are

1. and ,
2. and , and
3. Example In a 50 system, a transistor has the following S-parameter at 1.3 GHz. Examine the stability of the transistor.  Solution   The transistor is unconditionally stable.

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