Radiofrequency Stability Analysis

Abstract

This article will provide essential concepts for a basic understanding of the nonlinear dynamics of microwave circuits, so that a designer may distinguish between different types of steady‐state solutions, identify instability problems, and comprehend mechanisms for instability. The aim is to increase a designer's awareness and knowledge on what can go on in a nonlinear circuit. The harmonic‐balance method will be briefly outlined, with emphasis on the need to combine this method with a stability analysis to ensure the physical existence of any solution obtained. The article will cover the stability analysis of microwave circuits in small‐ and large‐signal regimes. The main approaches for stability analysis will be presented and compared, from the most familiar ones, Rollet factor and stability circles, to very recent and efficient techniques, such as pole‐zero identification. Another problem covered will be the abrupt change in the solution observed when a parameter, such as a bias voltage or input power, is modified continuously.

Check the Stability: Stability Analysis Methods for Microwave Circuits

“…Taking into account the high number of passive and/or active elements in most practical nonlinear circuits, the fact that the circuit can be unstable in small signal and/or large signal and the variety of instability phenomena, one can conclude that the only way to accurately predict undesired instability phenomena is to complement the frequency-domain simulation with a rigorous stability analysis. The difficulties of the widely used harmonic-balance method for the analysis/prediction of self-oscillations [5][6][7]20] are briefly explained in the following. Harmonic balance only provides steady-state solutions, represented with a Fourier series, usually in terms of one or two fundamental frequencies.…”

“…Assume that a small-signal input current source In at the frequency  is introduced into the circuit. Due to the small amplitude of the current generator, the nonlinear devices can be linearized about the dc solution, as in the case of the stability analysis based on (7). This provides the linear system:…”

“…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 admit four main types of steady-state solutions: dc, periodic, quasi-periodic (having two or more fundamental frequencies with non-rational relationships) or chaotic (non-periodic) [1,[5][6][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.…”

“…Instead, it may be quasiperiodic at in and an oscillation frequency o, it may exhibit a subharmonic oscillation at in/2 or exhibit a continuous spectrum (chaos) [7][8].…”

“…When particularizing the analysis to a given type of waveform, as done with scattering parameters or with harmonic balance, the solution obtained will be mathematically valid, but it might not be stable (or physical) [7][8]. The only way to verify the physical existence of a given steady state solution is to perform a stability analysis [12][13].…”