Circuit stability theory for non-Foster circuits

The successful design of non-Foster circuits requires tests for stability that are accurate and reliable. It is widely known by designers of amplifiers and active circuits that popular tests for the stability of 2-port and n-port circuits can give incorrect results. Despite periodic debate over the past 50 years, the validity of Rollett's K-factor test has never been proved mathematically; it was argued only on physical grounds. We present linear non-Foster circuits that serve as explicit counterexamples to Rollett's test. One counterexample circuit is given that is unconditionally stable according to the full statement of Rollett's test including Rollett's proviso. Yet the circuit is provably unstable with simple resistive termination, in direct contradiction of Rollett. A second counterexample circuit is shown that gives the stronger result that Rollett's test cannot be repaired, and indeed many other stability tests are defective too. Armed with these results, a different approach to the design of stable circuits is discussed that has enabled us to build stable non-Foster antenna impedance matching circuits. An example of a stable non-Foster broadband impedance matching circuit for a small loop antenna is given. The design of the circuit was made possible by using a new capability available in certain circuit EDA CAD tools.