Boldly Going Beyond S-Parameters

It’s not for nothing that RF and microwave design have long been considered close to black magic. As in many areas of expertise, a few fundamental principles apply, but a great deal of engineering experience and judgment has been required to achieve results close to those that theory predicted.

Today, thanks to powerful circuit simulation software and good device models, an engineer can easily design a small-signal RF amplifier. Scattering or S-parameters completely describe the linear behavior of semiconductor devices. Unfortunately, power amplifiers operate outside the small-signal superposition assumptions underlying the use of S-parameters. The signal levels are sufficiently large that device nonlinearities can’t be ignored, so linear design methods no longer apply.

This situation isn’t a recent revelation, and since at least the early 1990s, a few microwave test instrumentation companies and academic researchers have been investigating the problem. Their work led to the development of a small number of commercial large-signal network analyzers (LSNAs), today called nonlinear vector network analyzers (NVNAs), that provide comprehensive solutions. However, the new technology is only now becoming widely available, with the result that many large-signal circuits have suboptimal designs and performance.

During the present decade, X-parameters™, a superset of S-parameters, were developed to handle periodic nonlinear signals. X-parameters is Agilent Technologies’ trademarked name for the enhanced reflected-to-incident-wave ratios based on polyharmonic distortion (PHD) analysis.

Traditional Approaches

Most RF equipment is intended for use in a 50-? environment. But what about impedance levels within the equipment? For maximum power transfer, an amplifier’s input impedance should equal the complex conjugate of the impedance driving it. The same consideration applies at the output, where the impedance must be equal to 50 ? but the load seen by the active device may be tuned for maximum gain.

In both cases, engineers use microwave tuners to vary the complex impedance during the design stage. The procedure is called source-pull at the input and load-pull at the output. It’s straightforward to synthesize an L-C impedance-matching network that has the same effect as a tuner.

For power devices, the input matching will be established at a suitable impedance and the output optimized by varying the load-pull tuner. S-parameters eliminate this trial-and-error procedure for small-signal levels although it’s still required for large-signal operation.

Both passive and active load-pull instruments are available as well as fundamental and harmonic tuners, each with advantages and disadvantages. Automated test systems available from several manufacturers record output variables while sweeping tuner impedance through a wide range of complex values. Analysis routines then can derive contours of constant power and present the results as familiar Smith chart plots.

Passive Load-Pull
Basic passive tuners are available from Focus Microwaves and Maury Microwave (Figure 1). These devices adjust the length of a stub and the degree of coupling by positioning a conductive probe horizontally and vertically within a slotted airline. The magnitude and phase of the reflection added to the incident wave create a complex impedance load. This arrangement is adequate for characterizing amplifiers with minimal compression; however, beyond a couple of dB compression, the uncontrolled impedance at the harmonic frequencies becomes important.

Most mechanical tuners have one probe so only the impedance at the fundamental frequency can be affected. Focus Microwaves’ Model MPT Tuner, in contrast, has three probes, and the distance between them and the vertical position of each are independently programmable. This means that the impedance at the fundamental frequency and at the second and third harmonics can be set. Maury’s Automated Tuner System Software supports use of two cascaded tuners for impedance control at the fundamental and second harmonic.

Automated tuners considerably speed up and improve the accuracy of the load-pull process. A tuner calibration table is built from VNA measurement of the magnitude and phase corresponding to a number of vertical-horizontal probe-position combinations. Electromechanical tuners have virtually unlimited resolution, so the user must determine how fine the resolution needs to be vs. the time a measurement may take.

Software searches the resulting table to provide a list of probe-position combinations corresponding to the required impedances at the fundamental and second and third harmonics. In this way, large nonlinear behavior can be accommodated.

Active Load-Pull
Particularly for on-wafer probing, high loses determine the maximum reflection coefficient possible with passive electromechanical tuners. Active load-pull has the capability to overcome this problem and provide harmonic tuning at the same time.

An active load-pull instrument changes the load impedance by injecting a signal that combines with the DUT output. This is the same thing that occurs with the reflected wave from an electromechanical tuner, except that amplification is involved in an active load-pull unit so reflection coefficients larger than 1.0 are possible. This can compensate for signal loss, but, of course, oscillation also is possible.

The disadvantages include the following:
•?Two-tone or modulated signal phase errors
•?Lower measurement speed when using modulated signals
•?Power limitations and high cost of linear amplifier
•?Incompatibility with some VNAs
•?Incompatibility with source tuning
•?Parasitic oscillations

Some of the drawbacks result from impairments in the amplifier used within the active load-pull system. Focus Microwaves recommends a 10-dB backoff to ensure linearity, but this means a 300-W amplifier is needed for a 30-W test.

The problem with two-tone or modulated signals stems from the loss of phase-lock information associated with the VNA measurements. In some cases, a spectrum analyzer might be a better measuring instrument than a VNA, but it’s not nearly as fast.

A recent development undertaken by Tektronix, Cardiff University, and Mesuro is focused on applying an arbitrary waveform generator (Arb) and sampling scope to nonlinear device characterization. As described by Tektronix’s Darren McCarthy, RF market development manager, “The use of sampling scopes enables phase-coherent wideband measurements to be made. All the spectral components of multiple signals are simultaneously measured.

“The company’s Arbs enable any spectral components to be injected into the input and output of the DUT to manipulate the current and voltage waveforms. This type of completely electronic approach is fast and allows source- and load-pull solutions not only at the fundamental and harmonic frequencies but also at baseband,” he said.

As explained by Johannes Benedikt, representing both Cardiff University and Mesuro, “The measurement solution cuts down the characterization time through the simultaneous utilization of its waveform measurement and harmonic load-pull capability, a functionality that Mesuro terms waveform engineering. The design approach is based on the well-defined theoretical current and voltage waveforms for all amplifier modes of operation.

“Taking this into account allows directed changes to be made to the fundamental and harmonic load impedances so the difference between the measured and theoretical waveforms is continuously reduced. This approach eliminates the need for systematic yet time-consuming multidimensional parameter sweeps and enables the designer a more directed approach with the added benefit that the resulting device performance is close to a theoretical optimum,” he concluded.

New Instruments

NVNAs
During the 1990s, Hewlett-Packard, the Network Measurement and Description Group (NMDG) at Vrije Universiteit Brussel, and other university researchers created a few laboratory LSNAs generally based on harmonic sampling downconversion. In 1999, NMDG joined Agilent Technologies, and in 2003, Agilent licensed NVNA IP to Maury Microwave.

Agilent NMDG became independent NMDG Engineering in 2003 at the same time as the arrangement with Maury and since 2007 has been NMDG NV. The Model MT4463 LSNA sampler-based instrument that uses the Agilent IP was further developed as a commercial product by NMDG Engineering under a sublicense from Maury. The instrument was discontinued in 2008 although NMDG NV continues to support it.

As explained by Guillaume Pailloncy, an NMDG NV application engineer, “The NM4463 is a rack-and-stack system that needs to be customized to the customer’s needs. The core product consists of a sampling downconverter and a four-channel broadband data acquisition system. To perform measurements, it must be complemented with a test set consisting of couplers and bias tees, at least one synthesizer, a calibration kit, and a power meter. The NM4463, being sampler-based, simultaneously measures the fundamental, harmonics, and modulation tones in contrast to a mixer-based system that must scan through all tones of interest and reconstruct the complete signal.”

Maury’s Chief Technology Officer Gary Simpson described recent activity, “During the 2004 to 2007 period, the common industry name moved from LSNA to NVNA. In 2008, Agilent introduced the Model N5242A PNA-X VNA, a mixer-based instrument that can be configured as an NVNA (Figure 2).

“Maury has since introduced load-pull with X-parameters, using the Agilent NVNA in the system,” he continued. “When the measurement completes, an X-parameter file is saved that can be loaded directly into the Agilent advanced design system (ADS) nonlinear simulator as a large-signal PHD model. This is a major step forward for the industry. Instead of using the load-pull data to check a large-signal model, the load-pull data becomes the model.

“With Maury’s load-pull system using Agilent’s NVNA, a sweep plan that measures vs. impedance, power, bias, and frequency creates an instant large-signal model based on measurements at the actual large-signal operating conditions. Accordingly, the model is extremely accurate over the range of operation selected by the user,” Mr. Simpson concluded.

NMDG NV recently collaborated with Rohde & Schwarz (R&S) to develop the ZVxPlus Extension Kit that allows nonlinear harmonic measurements to be made with the R&S ZVA and ZVT VNAs. The kit includes attenuators, a synchronizer that enables calibration, and the ICE software platform for nonlinear HF component characterization, system configuration, absolute calibration, and measurement.

Chris Petrole, R&S campaign program manager, said, “ICE stands for integrated component characterization environment, and after calibration, ICE can be used to stimulate a component with both RF CW and DC sources. The voltage and current waveforms at the DUT ports are displayed, and derived quantities such as input and output impedance, gain, and PAE can be visualized. In the future, NMDG plans to integrate measurement-based behavioral modeling extraction tools in ICE as well as an easy interface to simulation tools.”

In 2003, Dr. Jan Verspecht, one of the key researchers at Agilent NMDG, formed his own company, Jan Verspecht b.v.b.a. He is known as the inventor of X-parameters and laid the foundations through the PHD mathematics he and others developed. Agilent continued to develop both the NVNA concept and X-parameters, culminating in the 2008 N5242A PNA-X launch.

For use as an NVNA, the PNA-X Analyzer is combined with a separate X-parameter extraction source, a measurement phase reference, and the ADS simulation software application with the X-parameter measurement option. The usual vector calibration is required together with harmonic phase and amplitude calibration.

In March 2008, Loren Betts, a research scientist at Agilent, presented a detailed discussion of NVNA technology to the IEEE Microwave Theory and Techniques Santa Clara Valley Society. In the presentation, he explained some of the characteristics of mixer-based VNAs, such as changing LO phase from sweep to sweep, that require special consideration in an NVNA. In the N5242A, this problem is overcome by referencing each A or B wave measurement to a phase-stable harmonic reference before calculating the B/A ratio:

New Parameters

Early attempts to address device nonlinearities extended linear S-parameters. The usual two-port definitions relate the reflected B waves to the incident A waves. A₁, B₁ and A₂, B₂ are the waves at ports 1 and 2, respectively.

The first number of the S-parameter two-digit subscript refers to the reflected wave port, and the second number is the incident wave port. Accordingly, S₂₁ is the ratio of output voltage at port 2 to input voltage at port 1 or voltage gain. S₂₁ and S₂₂ are of particular interest because during nonlinear operation they are not independent of the input A₁.

During 2002, while still at Agilent NMDG, Dr. Verspecht developed a presentation on Hot S₂₂ modeling. This popular approach recognizes that S₂₁ and S₂₂ are affected by a large A₁ signal so the usual S-parameter relationship

B₂ = S₂₁ A₁ + S₂₂ A₂

is modified to take this into account. The form of the equation remains the same, but S₂₁ and S₂₂ now are shown as functions of the magnitude of

A₁: B₂ = S₂₁(|A₁|) A₁ + S₂₂(|A₁|) A₂

Although this accounts for some nonlinear effects, it doesn’t address all of them. Specifically, there is a phase-related compression that requires a separate term in addition to the Hot S₂₂ equation. Through straightforward measurements, Dr. Verspecht showed that the phase of the A₂ wave relative to that of the A₁ wave matters. The extra term is linear in the conjugate of the phase-normalized A₂ wave and a key aspect of PHD modeling.

In a 2006 article, Dr. Verspecht and Agilent’s Dr. David Root described the mathematics underlying the need for the extra conjugate A₂ term. PHD refers to the class of signals consisting of a large fundamental and several smaller harmonically related signals. The assumption, proven by experiments, is that as long as the harmonics are small, they add linearly.

On the other hand, the values of the B wave to A wave ratios depend on the phases of the harmonics relative to the fundamental. As the authors explain, …”the linearization of the nonlinear system, around the simple dynamic operating point determined by the large tone, is not analytic in the sense of complex variable theory. If it were, the equations would depend only on the complex variable ? and not both ? and ?*…. Therefore, unlike linear S-parameters, the result is not independent of the phase of the small perturbation tone.”¹

The term analytic signal was introduced by Denis Gabor in his 1946 paper on communications theory. It refers to signals that have a non-zero spectrum for only positive or only negative frequencies. For real-valued time-domain signals, the Fourier transform of the positive frequency components is equal to the conjugate of the Fourier transform of the negative frequency components: F(-?) = F(?)*. From this, it follows that f_–(t) = f₊(t)* and f(t) = 2 Re f₊(t). No information is lost by considering only the spectrum for positive frequencies.²

However, this frequency-domain conjugate symmetry doesn’t apply to PHD-based analysis. The scaling factors associated with small ? and ?* perturbations about an operating point are not conjugates of each other. This result is outside the experience of circuit designers used to working with only positive frequencies and real-valued time-domain functions.

In the Verspecht-Root article, two fundamental factors were introduced. One was the nonanalytic behavior, and the other was a phasor P = e^+j?(A₁₁⁾ that represents the relative phase between the large input signal and the associated harmonics. The authors used a two-digit subscript to refer first to the number of the device port and second to the order of the harmonic. A₁₁ is the large fundamental input at port 1.

The m^th element in the set of values that represent the B wave at port p is given by

The subscripts are multiport extensions of the simple two-digit S-parameter subscripts. The digits p and q refer to the effect of the A wave incident at port q on the B wave at port p. Because distortion can produce sum and difference frequencies, m and n refer to the order of the harmonics at the p and q ports, respectively. If there were no distortion, all the terms with m ? n would equal zero.

This equation mimics the Hot S₂₂ extensions to S-parameters by expressing the basic S_pq,mn factor as a function of the magnitude of A₁₁; however, the relative phase difference between the B-wave and A-wave harmonics, m-n, is included. The second T_pq,mn factor is entirely new and the conjugate term that describes the nonanalytic behavior the authors referred to.

Agilent’s equivalent X-parameter expression is given by

Although the notation is different, equation 2 is based on the same PHD signal model as equation 1. It describes the effect of harmonic f on the B wave at port e in the presence of a large fundamental A₁₁. B_e,f is just one harmonic member in the set of values that compose B_e.

This equation sums A_gh waves weighted by the values of the ef member in each set of X^(S) and X^(T) coefficients. This corresponds to the effect on harmonic f at port e when A waves at all the harmonic frequencies h are applied to all the ports g.

The double-pair subscripts in equations 1 and 2 are similar but grouped differently. S_pq,mn refers to ports p and q and harmonics m and n. In equation 2, X_ef,gh indicates ports e and g and harmonics f and h. In both equations, the summations actually are double summations running over all combinations of q and n or g and h.

The X^(F) term has the phase of harmonic f but only de-pends on the amplitude of A₁₁, it being the only A wave involved. Because the X^(F) term indicates the direct relationship between B_e and A₁₁, the sets of X^(S) and X^(T) coefficients cannot be identical to the S_pq,mn and T_pq,mn coefficients although overall the two equations are equivalent.

Summary

If you design RF and microwave large-signal circuits, the message is clear: You can’t use linear S-parameters for large-signal circuits. And what’s even more important, you no longer need to try.

PHD or X-parameter characterization is a mathematically correct way to describe nonlinear behavior and automatically can create transportable device models. Should you happen to be involved in a top-secret snoop project, X-parameters create black-box behavioral models that convey absolutely nothing about the physics of the devices that generated the waveforms.

The best places to start learning about NVNA technology are the websites of the companies highlighted in this article. In addition, many IEEE conference papers as well as several technical articles and product datasheets are readily available.

References

1. Verspecht, J., and Root, D., “Polyharmonic Distortion Modeling,” IEEE Microwave Magazine, June 2006, pp. 44-57.
2. Kaiser, G., A Friendly Guide to Wavelets, 1994, pp. 71, 72.

April 2009