Understanding New Methods for Nonlinear Device Characterization
By Malcolm Edwards, AWR Corporation

Developing compact analytical and empirical models has always presented a frustrating challenge for high-frequency circuit designers. Model development is time consuming, costly, can potentially expose a device manufacturer’s intellectual property, and the quality of the resulting model varies considerably from manufacturer to manufacturer. In addition, since most compact model parameters are extracted from linear 50-ohm S-parameters and DC IV static and pulsed data, their ability to predict behavior under extreme nonlinear conditions or non-50 ohm terminations is questionable.

Fortunately, recent developments in measurement and modeling technology are focusing on technology-independent, measurement-based black box models. The concept is being actively embraced by both software developers and test equipment manufacturers. To understand this important trend, this article examines the different nonlinear models and measurement systems available today and how they can be used with AWR’s Microwave Office high-frequency design software.

Linear and nonlinear device models are the building blocks of most RF and microwave designs, and S-parameters are used to represent linear devices. As “black-box” models, they can easily be obtained using a vector network analyzer and distributed for simulation. They use superposition to equate the linear relationship between incident and reflected waves at all of the device’s ports. Conversely, nonlinear devices distort waveforms so that their behavior cannot be represented through superposition or S-parameters.

Nonlinear models describe the behavior of transistors including the large-signal region in which power amplifiers and mixers operate. Large-signal device models are continually evolving to keep up with changes in semiconductor technology. To standardize the model parameters used in different simulators, an industry working group of semiconductor vendor companies and EDA vendor companies called the Compact Model Council (CMC) is working to choose, maintain, and promote their use. An elusive but extremely valuable goal of these efforts is to predict next-generation circuit performance and identify a direction for developing models to support it.

This requires selecting operating conditions that define nonlinear device characteristics, replicate this behavior, and extract the parameters required for these model equations. An alternative to using standard or evolving compact models for the next-generation of devices is to use the measured data directly as is the case for S-parameters and linear devices. This is the underlying concept embodied in expanding linear S-parameters into a more general form that can portray nonlinear behavior. It departs from a table of scalable parameters used by the compact model’s intrinsic nonlinear equations, instead favoring a data set directly based on measured device behavior using a specific stimulus and set of terminal impedances.

This measurement-based, black-box model is the concept behind the current breed of commercial offerings known in the general sense as non-linear behavioral models or more specifically as X-parameters™, S-functions, the Cardiff model, and others. X-parameters and S-functions are extensions of the polyharmonic distortion (PHD) modeling approach developed by Verspecht, et al., and relate the spectra found at the device terminals to a given set of stimuli and termination impedances. The Cardiff model developed by Tasker et al., is a similar table-based model that relates IV waveform data at the device terminals for a given stimuli and set of load/source impedances.

These evolving measurement-based nonlinear models and the measurement techniques required to extract them represent nearly 20 years of research and development conducted by various commercial and academic organizations. Test equipment vendors such as Agilent, Anritsu, Rohde & Schwarz, and Tektronix are offering systems targeting microwave nonlinear characterization systems. The Tektronix system is based on a sampling oscilloscope approach, and the others use nonlinear vector network analyzer techniques. Anritsu, Rohde & Schwarz, and Tektronix have aligned themselves with specialized technologists (HFE Sagl, NMDG, and Mesuro) to develop their commercial nonlinear measurement and measurement-based model technologies.

Understanding Nonlinear Behavior
A periodic signal (CW or modulated) can be represented in the time or frequency domain. When such a signal drives a device into its nonlinear region, the shape of the IV waveform is distorted so that it cannot be described simply by applying a scaling factor to the input signal. In the frequency domain, this behavior can be represented by changes to the harmonic and intermodulation spectral components as functions of the changing stimuli and terminal impedances. The importance of having the nonlinear model replicate this behavior for each stimulus and terminal impedance cannot be overstated.

The polyharmonic distortion approach to modeling is based on frequency-domain measurements and is identified from the responses of a device under test (DUT) stimulated by a set of harmonically-related discrete tones. The fundamental tone is dominant and the harmonically-related tones are relatively small, so that the principle of harmonic superposition can be accurately applied. This principle asserts that the magnitude of the small test signals is such that the perturbation can be viewed as a linear process. This is analogous to mixer theory, which dictates that only the local oscillator (LO) signal is large enough to bring a nonlinear device into a time-dependent linear operating mode when the injected small-signal tones undergo multiplication in the time domain (i.e. frequency shifting).

The harmonic superposition principle is represented graphically in Figure 1. The fundamental tone that drives the DUT into a nonlinear operating mode is represented by the black tone in the figure. At the output port, the generated harmonic components (all black) are visible. The first small-signal test signal, a second harmonic tone (blue), is injected into Port 1 and results in perturbation of the four tones. The next small-signal test signal, the third harmonic (green), is injected into Port 1 and again results in perturbation of the four tones. This process continues until all harmonics and ports have been accounted for.

The DUT is connected to a large-signal network analyzer (LSNA) and a model is automatically extracted that accurately describes all aspects of nonlinear behavior, such as the amplitude and phase of harmonics, compression characteristics, AM-PM spectral regrowth, amplitude-dependent input, and output match. A benefit of this approach is that it provides much more than figures of merit such as Psat and two-tone third-order intercept point. The PHD model can be used in an EDA environment to consistently describe many nonlinear characteristics and in the design and optimization of circuits utilizing the nonlinear device.

S-functions and X-parameters are commercial implementations of the PHD model and have been grouped together because they share the same origin and generally target the same devices and subsystems. The two approaches currently have some major differences that may become less pronounced in the future as the techniques mature and converge.

S-functions
S-functions are an extension of S-parameters for nonlinear components, offering a simpler way to accelerate system design using nonlinear components by providing more complete system-level models. S-functions can predict harmonic and modulation behavior of nonlinear devices under different mismatch conditions. As with S-parameters, S-functions can be cascaded to predict nonlinear behavior of circuits and systems. S-functions are easily determined with the modeling option of the NMDG VNAPlus extension kits that extend Rohde & Schwarz network analyzers, for example, using additional hardware and software to characterize nonlinear behavior.

The characterization is performed in the frequency domain and can be converted into the time domain under real-life conditions for any terminal impedance by way of load/source-pull measurements. Microwave Office software can import this behavioral model to directly design larger circuits using the measured data or to provide more detailed data sheets.

The Cardiff Model
Unlike S-functions and X-parameters that capture the amplitude and phase information of a device’s spectral response, the Cardiff measurement system and associated model obtains the incident and reflected time domain current/voltage waveforms at the ports of the DUT. The test set-up is similar to a vector network analyzer but uses a sampling oscilloscope rather than harmonic mixing or sampling in the time domain. The resulting model uses four table-based nonlinear functions representing the corrected device currents and voltages to represent device behavior for a given input stimulus, bias, and terminating impedance. The system can employ single- or multiple-tone large-signal measurements including harmonic load-pull.

By controlling the load terminations at all harmonic frequencies while being able to view the voltage and current waveforms at the device’s current-generating plane in real-time, the designer can shape the waveform to match the theoretical values that will produce optimum results. The resulting behavioral model, obtained under the load condition that provides the optimum performance, can be extracted and invoked within Microwave Office software. As a result, modeling and design engineers can fully characterize their devices or power amplifiers for any signal level and impedance environment. For the same set of environmental conditions (drive, bias, and terminating impedance), such a model should be a more accurate representation of device behavior compared to a compact model extracted outside of these operating parameters.

In their paper “Highly-Efficient Operation Modes in GaN Power Transistors Delivering Upwards of 81% Efficiency and 12 W Output Power,” Wright, Heikh, Roff, Tasker, and Benedikt demonstrated how waveform engineering was used to optimize an inverse Class-F power amplifier to achieve drain efficiency greater than 81%, 12-W output power at 900 and 2100 GHz using a wide-bandgap gallium nitride (GaN) high electron mobility transistor (HEMT) as shown in Figure 2. This capability lets designers understand what performance their device can achieve with the right matching networks.

To design these networks or cascade devices into multi-stage amplifiers, a simulation environment is required, which can be demonstrated by showing how Microwave Office software works with these new classes of models. The Cardiff model is incorporated into a Microwave Office simulation using a netlist-based component (Figure 3) that is linked to the Cardiff current and voltage data table via the microwave data interchange format or MDIF file (Figure 4).

Multi-Tone Intermodulation Analysis
To ensure a problem can be solved, the intermodulation frequency set must be truncated to limit the number of tones required to solve the HB algorithm. All this is accomplished by the user in the Microwave Office simulator settings window. In current implementations of the nonlinear behavioral models there is no explicit support for a multi-tone FDD description in the intermodulation sense. This means there is no direct way to investigate multiple tones around the main drive tone.

However, other indirect methods (complex envelope or circuit envelope solvers) can be used to investigate intermodulation, and each one implies certain assumptions. Both methods use the AM-AM and AM-PM information in the model and assume that spectral widening is such that intermodulation effects are narrowband and centered on the carrier. This is not perceived as a limitation that prevents their use for this task. All models have limitations, and these are in their first generation, with continuous development in the areas of measurement, data extraction, support in simulations, and application scope.

All parties involved in PHD models (measurement, extraction, and simulation) are investigating additions to the modeling process that support intermodulation within a steady-state solver without the need to invoke the complex envelope and circuit envelope solvers. Several approaches are also being investigated to allow the capture of memory effects.

Summary
AWR’s Microwave Office software can simulate measurement-based, extracted nonlinear models as well as the current table-based S-function, X-parameter, and Cardiff models. Model data stored in an MDIF file is referenced through a netlist-based component within the software. Spectral-based models such as S-functions and X-parameters are addressed as frequency-domain devices in the simulator using spectral mapping. These model types eliminate the need to solve the nonlinear devices in the time domain as well as the need to harmonically balance the linear and nonlinear branches of the simulation network.

AWR CORPORATION
www.awrcorp.com
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