Check the Stability: Stability Analysis Methods for Microwave Circuits

The instability problems often faced by the designers of nonlinear microwave circuits are the cause of significant qualitative discrepancies between simulations and measurements, even when using powerful simulation tools based on harmonic-balance (HB) analysis and numerical optimization algorithms. Critical anomalies resulting from instability phenomena most often invalidate the prototype and demand intense investigation and resolution efforts, which may substantially increase production cycles and the final cost. Understanding instability requires awareness of two facts: 1) two or more steady-state solutions can coexist for the same values of the circuit elements and 2) stable solutions must be able to recover from the small perturbations that are always present in real life. To realize the complexity of the problem, one must take into account the fact that circuits containing nonlinear components, such as transistors and diodes, are governed by a set of nonlinear differential algebraic equations [1]-[4]. Time differentiation comes from the existence of reactive elements, involving this operation in their constitutive relationships, and nonlinearity comes from the presence of semiconductor devices, containing nonlinear functions in their intrinsic models. Nonlinear differential equation systems provide four main types of steady-state solutions: dc, periodic, quasi-periodic (having two or more fundamental frequencies with nonrational relationships), and chaotic (nonperiodic) [1], [5]-[7]. Unexpected solutions are often observed in nonlinear circuits since, in addition to the frequencies delivered by the input sources, there may be frequency components coming from the circuit self-oscillation. For instance, under a periodic excitation at ω
_in
, the solution measured may not be periodic at ω
_in
. Instead, it may be quasi-periodic at ω
_in
and an oscillation frequency ω
_o
, it may exhibit a subharmonic oscillation at ω
_in
/2 or exhibit a continuous spectrum (chaos) [7], [8].