This is the definitive guide to X-parameters, written by the original inventors and developers of this powerful new paradigm for nonlinear RF and microwave components and systems. Learn how to use X-parameters to overcome intricate problems in nonlinear RF and microwave engineering. The general theory behind X-parameters is carefully and intuitively introduced, and then simplified down to specific, practical cases, providing you with useful approximations that will greatly reduce the complexity of measuring, modeling and designing for nonlinear regimes of operation. Containing real-world case studies, definitions of standard symbols and notation, detailed derivations within the appendices, and exercises with solutions, this is the definitive stand-alone reference for researchers, engineers, scientists and students looking to remain on the cutting-edge of RF and microwave engineering.

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One of the key features that led to the wide adoption of S-parameters was the availability of hardware and calibration techniques capable of making quick, accurate, and repeatable S-parameter measurements. S-parameters can also be easily extracted in simulation from device or circuit models. In either case, the resulting S-parameters can immediately be used in simulation or design tools. In order to achieve similar success in the nonlinear domain, X-parameters must be easily measured and also easily extracted from simulation.

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This chapter presents a concise treatment of S-parameters, meant primarily as an introduction to the more general formalism of X-parameters. The concepts of time invariance and spectral maps are introduced at this stage to enable an easier generalization to X-parameters in the ensuing chapters. The interpretations of S-parameters as calibrated measurements, intrinsic properties of the device under test (DUT), IP-secure component behavioral models, and composition rules for linear system design are presented. The cascade of two linear S-parameter components is considered as an example to be generalized to the nonlinear case later. The calculation of S-parameters for a transistor from a simple nonlinear device model is used as an example to introduce the concepts of (static) operating point and small-signal conditions, both of which must be generalized for the treatment of X-parameters.

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The considerations used for the definition of X-parameters apply to the steady-state behavior of time-invariant nonlinear components with incident (and hence also scattered) waves on a harmonic frequency grid. The generality of the formalism comes at the cost of considerable complexity. Each spectral map is a nonlinear function of every applied DC-bias condition and all the magnitudes and phases of each spectral component of every signal at every port. Sampling such behavior in all variables for many ports and harmonics would be prohibitive in terms of data acquisition time, data file size, and model simulation speed.

Arbitrary-load-dependent X-parameters, automatically measured with a load-tuner working with an NVNA, are used to characterize and model a packaged 10W GaN transistor. A full nonlinear two-port functional block model for PA and other circuit design is immediately available for nonlinear simulation. It is demonstrated that the model predicts well the independent effects of harmonic load tuning without having to independently control harmonic loads during characterization. The nonlinear model can be used effectively to obtain optimal fundamental and harmonic impedances for device operation, as well as predict, accurately, other nonlinear FOMs including PAE and harmonic distortion. Source-pull is shown to be unnecessary except for efficient power transfer, yet the model is fully capable of predicting correct device response when embedded in any source and load impedance at the fundamental and harmonics.

Zusammenfassung Zur Beschreibung nichtlinearer hochfrequenter Bauelemente wie Mischer oder Verstärker reichen herkömmliche Größen wie S-Parameter nicht aus. Wir zeigen, wie sich das Großsignalverhalten solcher Bauelemente, die Arbeitspunktabhängigkeit ihrer Eigenschaften und ihre Erzeugung neuer Frequenzanteile mit Hilfe so-genannter X-Parameter wiedergeben lassen. Das Verfahren wird an mehreren Beispielen demonstriert.

Non-linear RF devices such as mixers or amplifiers cannot be described in sufficient detail by common quantities such as S-parameters. We show how the large-signal characteristics of such devices, the influence of the operating point on their characteristics, and the generation of new frequency components can be modeled by the so-called X-parameters. The approach is demonstrated by way of several examples.