Dr. David E. Root received BS degrees in physics and mathematics, and, in 1986, his PhD degree in physics, all from MIT. He joined Hewlett-Packard Co. (now Agilent Technologies) in 1985 where he has held both technical and management positions. He is presently Principal Research Scientist and Modeling Architect at Agilent Technologies’ High Frequency Technology Center in Santa Rosa, CA. His current responsibilities include nonlinear behavioral and device modeling, large-signal simulation, and nonlinear measurements for new technical capabilities and business opportunities for Agilent. David was elected IEEE Fellow in 2002 “for contributions to nonlinear modeling of active semiconductor devices.”

He was 2006-2008 IEEE MTT-S Distinguished Microwave Lecturer. David received the 2007 IEEE ARFTG Technology Award “for contributions to nonlinear RF and microwave device measurement and behavioral modeling.” He is Chair of the IEEE Working Committee on CAD (MTT-1). He has been a member of the Technical Program Committee of the International Microwave Symposium since 1995. Dr. Root was Visiting Scholar and Lecturer at the University of California, San Diego Fall, 2005. He co-edited the recent book, Fundamentals of Nonlinear Behavioral Modeling for RF and Microwave Design, **Artech House**, 2005. In 2008, Dr. Root and seven collaborators on a cross-functional team shared Agilent Technologies’ Barnholt Award for the company’s top innovation: X-parameters and the Agilent Nonlinear Vector Network Analyzer.

**Vye:** You describe X-parameters* as a superset of S-parameters.

**Root:** Yes. X-parameters are a rigorous superset of S-parameters. X-parameters apply to nonlinear (and linear) components under large-signal (and small-signal) conditions. X-parameters reduce exactly to S-parameters in the small-signal limit. X-parameters have the same simple use model as S-parameters, but they are much more powerful because they contain information about harmonic and intermodulation spectra generated by the nonlinear component in response to large-signals, which of course S-parameters can not describe. Moreover, X-parameters enable the design of nonlinear circuits and systems from knowledge only of the X-parameters of the constituent components, including the effects of mismatch at the fundamental and harmonic (and even intermodulation) frequencies. X-parameters represent intrinsic properties of the nonlinear component in a way that completely protects the IP, promoting sharing and re-use. X-parameters can be measured with simple automated measurements on a nonlinear vector network analyzer (NVNA) or independently generated from a detailed circuit model in the ADS simulator, and immediately used for the design of a nonlinear circuit or system using ADS.

**Vye:** Are X-parameters based on the scattering functions theory developed by Jan Verspecht and others back in the early 1990s?

**Root:** Yes. Jan Verspecht developed the foundations for what we now call X-parameters in 1995 as part of his Ph.D. “annex” while employed at our parent company, Hewlett-Packard. Since 2004, Dr. Verspecht has worked together with our multi-divisional Agilent team, under contract with Agilent, to significantly enhance the capabilities and extend the applicability of X-parameters in both the measurement and modeling realms.

**Vye:** Are X-parameters like S-parameters with harmonic indices for the travelling and scattered waves?

**Root:** Like S-parameters, X-parameters describe the mappings of incident signals at input ports to scattered signals at output ports. Unlike S-parameters, however, X-parameters also inherently include frequency conversion terms where signals at one frequency at an incident port cause outputs at other ports at different frequencies. Thus terms like XS21,32 and XT21,32 relate scattered waves at port 2 at the fundamental frequency to incident waves at port 3 at the second harmonic. The XS terms are related to “hot S-parameters”, that by themselves provide an incomplete description of scattering by a nonlinear system excited by large signals. The XT terms are new and have no analogue in S-parameter theory, but can be proved are required to be taken into account under strongly nonlinear conditions. X-parameters of both types depend nonlinearly on signal conditions that determine a “large-signal operating point” such as the bias, power, and load.

**Vye:** And X-parameters are frequency domain models as opposed to time domain?

**Root:** X-parameters actually represent intrinsic properties of the device, just like S-parameters. Are they models or measurements? The same question applies to S-parameters. Whether you consider X-parameters to be “models” or fully calibrated (for mismatch at the fundamental or harmonic frequencies) “measurements” is really a matter of perspective. X-parameters are most naturally formulated in the frequency domain, or more generally in the complex envelope domain. They represent the mappings between incident time-varying spectral components (complex amplitudes) and scattered spectral components for a nonlinear component.

**Vye:** Are they only for two-port devices? If not what are the practical port limitations.

**Root:** X-parameters are completely scalable with respect to the number of ports, just like S-parameters. X-parameters have been demonstrated (in simulation) for mixers, for example, which are three-port devices. In ADS2009 Update 1, customers will be able to generate X-parameters from nonlinear circuit schematics with an arbitrary number of ports. On the instrumentation side, the Agilent Nonlinear Vector Network Analyzer (NVNA) systems are presently available only for two-port devices. However, extensions of the NVNA for multi-port (>2) devices are being developed and their commercialization is planned as part of the Agilent Nonlinear Technology roadmap.

**Vye:** How does a frequency domain-based X-parameter affect simulation technology such as harmonic balance, where time domain-based nonlinear models traditionally represent active components and frequency-domain models represent linear models and FFT algorithms are used to resolve the parsed linear/non-linear network?

**Root:** This is an interesting and important question. X-parameters are the natural formulation of the nonlinear component’s nonlinear properties from the perspective of their external terminals in a way that is actually most natural for solution by the harmonic balance simulation algorithm. Harmonic balance solves a set of nonlinear algebraic equations in the frequency domain. This algorithm is much more efficient for solving large-signal steady-state nonlinear problems than solutions of time-domain differential equations provided that the excitation signals consist of several discrete tones. The X-parameters directly express the nonlinear characteristics of the component in the frequency domain, with a one-one correspondence to specific terms in the reduced harmonic spectral Jacobian expressed in travelling wave space. With an X-parameter component representation, there is therefore no need for the simulator to convert a time-domain nonlinear model to the frequency domain! Moreover, a frequency domain (or envelope domain) description of a (linear or nonlinear) microwave component can be much more accurate than a low-order time domain system of ordinary differential equations such as define standard “compact” device models. In this respect, the nonlinear simulation technologies of harmonic balance (HB) and circuit envelope (CE) were actually far ahead of the measurement and modeling technologies.

**The NVNA and X-parameters essentially do in measurements and modeling what the HB and CE algorithms do in simulation.**

One might call X-parameters “experimental harmonic balance” or “measurement-based circuit envelope.” The philosophy for us at Agilent was to develop measurements on new instruments and to formulate nonlinear component models in the mathematical language native to the simulator algorithms that solve optimally the class of problems of most importance to our customers.

**Vye:** Aren’t nonlinear responses more efficiently represented in the time domain?

**Root:** Not necessarily. Certainly lumped nonlinear systems can be efficiently described in the time domain. But even a purely linear transmission line can not be efficiently described by a low-order system of ordinary differential equations, let alone a nonlinear dispersive component. Actual RF and microwave components such as amplifiers and integrated circuits are often highly dispersive and are therefore much more accurately described in the frequency domain. It should be noted that X-parameters are extremely accurate in the frequency domain, but are not limited to mild nonlinearities like the historical frequency domain approaches such as Volterra series. X-parameters apply even for highly nonlinear conditions.

**Vye:** It’s my understanding that S-parameters were born out of necessity that is, since the technology did not exist to measure high frequency currents and voltages 40+ years ago, measuring the ratio of reflected and transmitted voltages was the only practical way to characterize passive devices. Would you agree?

**Root:** Although I’m not a historian, I agree with your statement. S-parameters contain no more and no less information than conventional Y or Z-parameters. However, S-parameters are much easier to measure at high frequencies because the components do not need to be terminated with shorts or open circuit boundary conditions, which are hard to achieve at high frequencies and can cause damage to the device. Voltage and current sources at high frequencies are also impractical, whereas microwave sources based on traveling waves are ubiquitous. X-parameters retain the property of being easy to measure at high frequencies, but they require a more powerful instrument than a linear VNA – in particular an NVNA – to provide the necessary information.

**Vye:** The network analyzer was “instrumental” in popularizing the use of S-parameters by making them easy to obtain. EEsof’s touchstone popularized their use in design software. S-parameters established the link between high frequency device characterization, test equipment and design CAE. Is the thinking to expand this relationship between network analyzers and design software to include nonlinear devices?

**Root:** You have identified the precise objective. The S-parameter paradigm was so powerful over the past 40 years largely because of the interoperability among measurement, modeling, and design. Vector network analyzers (VNAs) could easily measure S-parameters, linear S-parameter blocks could represent - or faithfully “model” the linear component - and then a simulator could be used to design linear systems with complete predictive certainty from knowledge only of the individual S-parameters of the constitutive components. The same use model applies for X-parameters, but we can finally drop the severe “linear” limitation. X-parameters can now be measured on a new type of instrument, the Nonlinear Vector Network Analyzer (NVNA). An X-parameter component in ADS encapsulates the X-parameter representation of the component nonlinear behavior. Nonlinear circuits and systems can immediately be designed using conventional nonlinear harmonic balance or circuit envelope simulators simply by dragging and dropping the X-parameter component into the ADS design schematic. Agilent has worked to develop and commercialize all these nonlinear pieces of the puzzle and designed them to fit together, seamlessly.

**Vye:** If measurement technology has advanced to the point of being able to measure high frequency voltages and currents real-time, why characterize non-linear devices in the frequency domain?

**Root:** There are many ways to measure high-frequency voltages and currents. Some involve high-frequency scopes, for example, which measure in the time domain. However, to completely characterize a multi-port device, it is necessary to employ an instrument with many channels for input and output at multiple ports, and develop a way of calibrating the instrument at the device reference planes. For the highest dynamic range, such as can be required for designs of highly linear amplifiers, a mixer-based instrument operating in the frequency domain is the preferred solution. It is also natural to use the HW and leverage part of the vector calibration methods from linear VNAs (of course an absolute calibration as well as cross-frequency phase calibration is also needed to get all the nonlinear information necessary for X-parameters and is a key part of the NVNA). Moreover, the frequency-domain X-parameters of a device are the direct intrinsic nonlinear properties of the device that fit most naturally with the harmonic balance or circuit envelope analysis algorithms used by the simulator to most efficiently solve the design problems. The time-domain large-signal waveforms can be easily and accurately reconstructed from the X-parameters by an FFT since the magnitudes and cross-frequency phases are all accounted for.

**Vye:** Behavioral models are currently the mainstay of addressing nonlinear devices in system simulations, and accuracy depends on increasing the behavioral description with more parameters such as linear gain or S-parameters, IP3, P1db, noise figure, etc. Will an X-parameter model always be more accurate than the most fully defined behavioral model?

**Root:** For multi-tone, steady-state nonlinear behavior in the frequency domain, X-parameters are the complete behavioral description from which many of these other “nonlinear figures of merit (FOM)” are derivable. In fact, X-parameters enable, for the first time, measurements such as 3rd order harmonic magnitude and phase, to be properly calibrated for source harmonics and mismatch at fundamental and harmonic frequencies. That is, X-parameters serve to extract the “true” nonlinear FOM of the device, defined as the response to a prescribed ideal (spectrally pure) source with the device terminated at precise loads (often specified as perfectly matched) at the fundamental and harmonic frequencies. Previous industry standard harmonic measurements, including those done now with VNAs and spectrum analyzers, do not correct for source harmonics or harmonic mismatch. X-parameters fundamentally address and solve this issue. X-parameters systematically contain more nonlinear information according to the complexity of the large signals used to stimulate the device. Sometime in the near future X-parameter corrected IP3 will be the standard way to specify this FOM, by separating the device intrinsic property from the influence of the measurement system. X-parameters will continue to evolve in many dimensions, including memory effects and noise. As these capabilities are added systematically to the instrument and X-parameter component in ADS, X-parameters will be the most complete behavioral representation of components for these types of applications.

**Vye:** Another advantage would seem to be the fewer required measurements to produce an X-parameter model versus a behavioral one (requiring network analyzer, inter-modulation measurements, power measurements, etc.)?

**Root:** A great advantage of X-parameters is that there is no need to configure multiple and often disparate instruments to obtain a comprehensive behavioral model. For example, we have demonstrated at Agilent, and we are now working with industry-leading companies to replace several different test stations with an NVNA for X-parameter measurements from which the behavioral model is constructed. The NVNA is endowed with the ability to control auxiliary instruments, such as bias monitors and supplies, external sources, and load tuners, to capture X-parameters over the requisite set of independent variables (bias, frequency, load, etc.) to provide as complete a behavioral model as desired. In one customer case, the complete set of data required to create full X-parameter behavioral models with the NVNA was acquired 30 times faster than using multiple conventional instruments. It was fully automated, so the engineering and setup time was dramatically reduced as well. Moreover, the X-parameter model was much more powerful than any alternative, allowing the system integrator to account properly for the output mismatch at the second harmonic when the GSM amplifier was integrated into the cell phone. “We didn’t even think this was possible” was a quote from a Ph.D. expert designer at a key system integrator company who evaluated an early version of X-parameters.

**Vye:** Does the X-parameter file format support noise parameters?

**Root:** X-parameters are compatible with parameters related to noise and will evolve to support noise. At present, however, the commercial solution from Agilent does not include noise.

**Vye:** Were X-parameters waiting for vector network analyzers to be able to measure non-linear performance before they could be adopted for use in design software?

**Root:** X-parameters could be used in design software independently of the existence of an NVNA to provide them from actual component data. Just like S-parameters can be computed from a circuit using an SP analysis, X-parameters can be computed for a circuit schematic in an “XP analysis (new simulator) in ADS. Techniques for doing this have been published in the literature for some time, and used internally at Agilent. This capability of generating X-parameters from circuit models will be unveiled by Agilent for the external market at the International Microwave Symposium in June and will be available in ADS2009 Update 1. The ability to simulate with X-parameters, computed from simulation or other analysis, was therefore also possible independent of the existence of the NVNA. An enhanced implementation of the X-parameter simulation capability will also be available in ADS2009 Update 1.

Of course one of the greatest values of X-parameters is the ability to directly and immediately use fully calibrated nonlinear component data directly for the design of nonlinear circuits and systems. Without an instrument to easily measure X-parameters, this value was not realizable by the industry. Agilent’s NVNA, first released in 2008 with significant new enhancements this year, is still the only available instrument to automatically measure X-parameters and enabling them to be simulated immediately in nonlinear design software (ADS).

**Vye:** Can you tell us about the adoption rate of integrated device manufacture (IDM) to characterize their parts with X-parameters and make them available to designers?

**Root:** We can’t disclose sales figures, but we are delighted to say that the adoption rate is proceeding even faster than our original projections, and despite the economic downturn. It is even becoming a “lock-out” spec with some system integration customers, who are now requiring their component vendors to supply X-parameters for their parts as a condition for doing business.

**Vye:** Does Agilent have any programs in place to help both the IDMs and the engineering community with this adoption?

**Root:** Agilent is devoting many resources to training and application seminars all over the world, including workshops at the International Microwave Symposium and other conferences. This is for both the augmented X-parameter simulation and new X-parameter generation functionality in ADS and new NVNA capabilities – including load-dependent X-parameters that now unify load-pull as well as S-parameters for complete two-port nonlinear functional block modeling from nonlinear X-parameter data. This breakthrough interoperable technology satisfies the long-expressed but heretofore unmet industry need for a rigorous yet practical generalization of S-parameters to the nonlinear realm while maintaining the same simple and powerful use model.

**Vye:** Are there any special considerations about the file format? Are they fairly easy to work with in terms of size?

**Root:** The file sizes are generally quite manageable, typically ranging from as small as 10kB to 10 MB depending on how many independent settings of bias, frequency, power, and load desired. The degree of automation and accuracy, and the overall speed of the solution, generally does not impose significant constraints on the file size.

**Vye:** Do you see designers generating X-parameters from simulation and sharing the information with system designers?

**Root:** We are already seeing signs of this in our early work with customers, even before the release of the first X-parameter generation product in ADS 2009 Update 1 (to be introduced in June). We expect this capability to provide key early design wins, even before some of the component hardware is fabricated. Later in the design cycle, when actual component HW is available, measured X-parameters can be substituted for those initially computed from the circuit models for final bottom up verification. We are using this capability for Agilent internal proprietary designs, also as a means of speeding up simulations by replacing complicated nonlinear circuits with simpler but accurate X-parameter representations.

In the technical literature, much of this work is also known as the “Poly-Harmonic Distortion (PHD) model.” Agilent is rebranding “PHD” with “X-parameters.” “X-parameters” refers to the generic technology and science behind it. The simulator component that reads X-parameter data-files is called the “X-parameter simulation component” The capability to generate X-parameters from a detailed schematic in the simulator is called “X-parameter generation.”. This capability is achieved using a new analysis mode – new simulator – called “XP analysis” in analogy with the familiar “SP-analysis” for S-parameters.

* X-parameters is a trademark of Agilent Technologies