Analysis of Conditional Stability and Unconditional Stability and Instability for Microwave 3-Ports

Abstract

A full and rigorous treatment of conditional and unconditional stability and instability for an active 3-port microwave circuit, represented at a given frequency by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula> -parameters, is carried out. By terminating a given port with a variable termination one may study at the input and the output of the resulting 2-port circuit the occurrence of unconditional stability and instability, as well as of special cases of conditional stability, defined as reactive unconditional stability or instability. Such conditions are especially relevant in microwave oscillator design and were exhaustively identified for the first time and treated in previous analysis by the authors. Drawing from its conclusions, one obtains in the plane of the variable termination and with a simple algorithm, a single parametric representation, providing at once, in a closed and direct form, the entire collection of boundaries of the regions, wherein the listed conditions occur at ports of the resulting 2-port circuit. Guidelines for the design of microwave amplifiers and oscillators, both in series and shunt configuration, are sketched. Numerical examples, validated by simulations, are also presented.