New simple proofs of the two-port stability criterium in terms of the single stability parameter μ/sub 1/ (μ/sub 2/)

Abstract

The classical scattering-parameter stability criterium for a linear two-port makes use of two conditions involving the Rollet parameter plus one additional parameter. A new stability criterium was developed by Edwards and Sinksky on the basis of a condition on a single parameter, i.e., 1 or 2 . This paper presents a new, simpler, and more straightforward set of proofs of the single-parameter stability criterium for a linear two-port. The first proof is algebraic and shows the equivalence of the conditions 1, 1 with the condition 1 ( = 1 2). The second proof, which is geometrical, relies only on the classical stability circle concepts in an improved way with respect to the treatment by Edwards and Sinksky.

On driving non‐passive macromodels to instability

“…This example shows that, in addition and as a generalization of existing techniques [15][16][17][18][19][20][21], the structured singular value can be adopted as a single metric for the robust stability assessment of any type of device, including active elements, provided that the specific coupling structure of its terminations is predefined.…”

Simple derivation and proof of geometrical stability criteria for linear two‐ports

“…Moreover, the geometrical implications and proofs presented in [10,12] have been made more comprehensible here, because through (29)-(30), one has reverted to the familiar geometrical concepts dictated by (11)-(14). For collation, we note that the combination of one geometrical constraint on the stability circle in the source, load, input plane, or output plane, together with one auxiliary condition, will lead us to a simple derivation of the geometrical stability parameters.…”

“…These conditions have been used to discuss the geometrical implications of Ј-parameters [10], qw well as to aid in providing the geometrical proofs of unconditional stability [10,12]. In reality, they can also be utilized to derive the geometrical stability parameters, provided further constraints are asserted for sufficiency.…”

“…These remaining constraints become immediately identifiable as Eqs. (12) and (14) if one recognizes, correspondingly,…”

“…However, the derivation and proof of the geometrical criteria are somewhat involved for demanding careful and elaborated analysis of stability conditions for many different cases, for example, ͉S 22 ͉ 2 Ϫ ͉⌬͉ 2 ѥ 0, 1 Ϫ ͉S 22 ͉ 2 ѥ 0, and 1 Ϫ ͉S 22 ͉ 2 ѥ ͉S 12 S 21 ͉ for 1 Ϫ ͉S 22 ͉ 2 Ͼ 0. Following [10], several authors reviewed and provided more straightforward proofs of the single-parameter criteria, for example, [11,12]. A slight modification of the proof has also been attempted to deduce the stability parameters in a more direct way [11].…”