M2S parameters: a Multi-Tone Multi-Harmonic Measurement Approach for the Characterization of Nonlinear Networks

Abstract

A multi-tone multi-harmonic scattering (M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> S) parameter measurement approach, suitable for multi-port nonlinear network characterization, is proposed. The M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> S parameters, based on the concept of incommensurate excitation frequencies, provide a generalization to the multi-input multi-harmonic case of the well-known single-input describing functions representation using only the fundamental frequency. The M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> S parameters are directly measurable quantities featuring a mild nonlinear dependency on the incident wave amplitudes only, with the advantage of a much larger information content with respect to single-tone periodic measurements. The validity of the proposed approach is demonstrated for RF transistor and power amplifier load-pull applications, where good accuracy can be achieved with a smaller number of measurements in comparison with conventional load-pull techniques, since the M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> S parameters do not depend on the incident wave phases.