Abstract
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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> , the solution measured may not be periodic at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> . Instead, it may be quasi-periodic at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> and an oscillation frequency ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> , it may exhibit a subharmonic oscillation at ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sub> /2 or exhibit a continuous spectrum (chaos) [7], [8].
Highlights
Instability problems often faced by 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 analysis and numerical optimization algorithms
Only stable solutions can be physically observed, since only stable solutions are robust against the small perturbations that are always present in real life, coming from noise and perturbations
It is interesting to note that these eigenvalues can be obtained if we reduce number of system equations by substitution
Summary
Instability problems often faced by 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 analysis and numerical optimization algorithms. Each coexisting stable solution has its own basin of attraction or set of initial values such that the system evolves to the particular steady-state solution These initial values are conditioned by the sense of variation of Pin. a hysteresis phenomenon [5,24] is observed when increasing and decreasing Pin. Need for a complementary stability analysis Even though the mechanisms leading to the major forms of instability are generally well understood, an a priori prediction of unstable behavior in a practical microwave circuit with full models of its active and passive elements is usually impossible, unless a rigorous stability analysis is carried out. If the two-port network is not absolutely stable, input and output passive loads should be restricted to certain regions of the Smith chart, delimited by the stability circles [9]
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