Closed-form solutions to large-signal PA problems: Wirtinger calculus applied to X-parameter

A powerful new technique is presented and used to derive compact closed-form expressions for the optimum output reflection coefficient for maximum power transfer to the load of a PA under large-signal input drive. Wirtinger calculus is applied to a non-analytic X-parameter expression to formulate the extremal problem in a form that can be solved exactly, by hand, in explicit form. Compact formulas for the exact solution are valid for small and large input drive levels and small-to-moderate output mismatch, and reduce to the well-known S-parameter result in the small-signal limit. The results are validated numerically and against measured load-pull data on a commercial power amplifier. The method enables a fast, prescriptive, X-parameter based production test for a key large-signal PA figure of merit, beyond what is possible from S-parameters, and without the need for searching the measured performance space.