Stabilizing a Potentially Unstable Two-Port

Criteria for evaluating the stability of a two-port (primarily a single active device) based on the [v,i] terminal parameter sets have been known for years [1]. More recently, Bodway [2] expressed the stability criteria in terms of the S-parameter matrix. Smith developed his chart and published a useful and thorough treatment of his work [3]. However, Smith's rectangular coordinate system origin does not coincide with the center of the chart. White [4] presents the equations for the general contours of the Smith Chart in a easily understood translated coordinate system with its origin at the center of the unity radius chart. This paper brings together the prior work [2,4] and presents a set of equations for finding the real (r and g) circles in the stable operation portion of the Smith Chart and the necessary and sufficient conditions for their existence. A seemingly complicated problem is reduced to simple algebra by a novel change of variables which also highly simplifies the complexity of the expressions manipulated to achieve the desired results.