Models based on X-parameters can provide insights into the linear and nonlinear behavior of key components in wireless systems, including power amplifiers and mixers.

The most common method to accurately characterize RF/microwave components under linear conditions has been through the use of S-parameters. However, modeling nonlinear behavior of certain components, such as amplifiers and mixers, is challenging because S-parameters cannot be applied effectively and accurately under large-signal conditions. Approximation techniques have been used for modeling nonlinear behaviorwith partial success by complementing linear S-parameters with nonlinear component specs typically found in datasheets such as 1-dB gain compression point, two-tone third-order intercept point, etc. A much more accurate and comprehensive approach to model nonlinear behavior of RF/microwave components is through the use of X-parameters, which were developed to represent both linear and nonlinear characteristics.

X-parameters were developed by Agilent Technologies to describe the behavior of both linear and nonlinear components in response to large-signal conditions. X-parameters reduce exactly to Sparameters in the small-signal limit and have the same simple use model as S-parameters. Because they contain information on all the harmonics and intermodulation spectra generated in response to large signals, they are much more powerful than S-parameters and any other nonlinear models available in the industry. X-parameters correctly characterize impedance mismatches and frequency mixing behavior to allow accurate, much faster simulation of cascaded nonlinear Xparameter blocks (e.g., amplifiers and mixers) in design.

X-parameters can be obtained in one of two ways: generated from a circuit-level design in Agilent's Advanced Design System (ADS) software or measured using the Nonlinear Vector Network Analyzer (NVNA) software running inside the Agilent PNA-X network analyzer. When generated from a circuitlevel design, they offer a simple means of quickly and accurately capturing a component's nonlinear behavior and saving it as transportable RF intellectual property (IP) models that can be used for circuit or system designs. X-parameter models can be used to share design performance without revealing design topology.

Agilent has published the equations underlying the X-parameter theory and the Xparameter files are in an open, non-encrypted format. Agilent has taken these steps to enable broad industry adoption and to encourage others to join in the development of the technology. To gain a better understanding of how circuit-level designers can easily generate fast and accurate, transportable X-parameter models, consider the example of a two-stage MMIC power amplifier (PA) designed in ADS for 3GPP Long Term Evolution (LTE) applications (* Fig. 1*). The goal is to generate a 50-O X-parameter model of the component. The same process outlined in this article can be used to generate accurate X-parameter models for mixers and other nonlinear components.

The first step in creating an Xparameter model is to generate the component's X-parameters. In ADS, this can be done by inserting the circuit-level design into a schematic page, attaching it to an X-parameter source, load, and bias, and clicking the "Simulate" button. In seconds, an X-parameter model is generated that can be e-mailed to the system integrator for immediate use.

To validate the accuracy of the generated model and compare it with the actual circuit-level MMIC PA, both the X-parameter model and MMIC PA design are inserted into a nonlinear simulation setup and nonlinear simulation and analysis are performed. * Figure 2* shows the magnitude and phase of the fundamental, as well as the second and third harmonics of both results. This comparison clearly demonstrates that the X-parameter model has the same accuracy as that of the circuit level design and, therefore, a system integrator can insert the MMIC PA model into an LTE uplink transmit system design and use it as if it were the actual circuit-level PA.

The MMIC PA model was generated assuming a 50-O load and works well within a system matched to 50 O, accurate within about a 2.0:1 VSWR. If non-50-O modules are used in the system, a designer must be able to sweep the entire load over the Smith chart and generate a model that would work with any load impedance, not just in the 50-O region.

* Figure 3* helps show the importance of load-dependent models. It shows the MMIC PA connected to a duplexer and antenna. If the load impedance on the PA is unknown, an impedance mismatch in magnitude and phase could result at both fundamental and harmonic frequencies. The only way to accurately predict the behavior of the PA in the system under any load impedance is a load-dependent X-parameter model that contains accurate information on the magnitude and phase of the fundamental frequency and all the harmonics.

An example of a design problem would be where the gamma load of the second harmonic on the PA creates distortion that degrades cell phone performance and possibly even PA efficiency and shortens battery life. To correct the problem, the exact magnitude and phase content of the second harmonic tone must be known in order to filter the unwanted harmonic signals. Unlike other available industry models that capture nonlinear behavior only on the fundamental frequency, the X-parameter model accurately captures the behavior on all the harmonics. By providing complete information on the magnitude and phase of the second harmonics, the model allows designers to filter out this unwanted second harmonic and improve the overall design and performance of the cell phone.

Generating a load-dependent model is simple and follows the process previously outlined, with the exception that a load sweep must be added to the design. A designer simply inserts the circuit-level PA design into a template in ADS, clicks the Simulate button and a model is automatically generated. This newly generated load-dependent X-parameter model is fully IP-protected and is automatically stored in the project's data set folder and can be immediately shared with the system integrator for accurate higher up simulation and tradeoff analysis on matched or mismatched cascaded modules.

* Figure 4* shows simulation results from both the load-dependent model and the circuit-level PA with a gamma of 0.7 and phase between -180 and +180 deg. With these criteria, the generated model is accurate with any load impedance within 70 percent of the Smith chart. The overlaid power and power-added-efficiency (PAE) contours of the model and the circuit level PA demonstrate the accuracy of the X-parameter model to the circuitlevel PA.

To further evaluate the X-parameter model under mismatch conditions, it will be used to represent two cascaded PAs with mismatch between them. Individually, the output return loss of the PA (S_{22}) is excellent when it is driven hard. But if the PA is driven with a small signal, S_{22} naturally degrades and moves away from 50 O because the output FET capacitance and resistance change as a function of drive level. Cascading two of these PAs will therefore result in mismatch between them. The source impedance of the second PA is no longer 50 O. Rather, it is now the degraded S_{22} of the first PA since it is driven with a small signal. This scenario offers a good test case for the model. * Figure 5* shows the simulation results for the cascaded PAs and the cascaded models. Again, the overlaid results demonstrate the high accuracy of the model under any load impedance and with cascaded mismatch conditions.

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Circuit designers and system engineers often pay a great deal of attention to intermodulation distortion (IMD) products and third-order intercept (TOI) terms, especially in receiving systems. Such terms are usually introduced in a system by components such as mixers and amplifiers. A twotone IMD test is commonly used to determine the TOI distortion product on such components. The test generates two tones of the same power, but separated by a very small frequency offset or spacing. These two tones mix together and generate higherorder distortion products at different mixing frequencies. While some of the second-order and third-order distortion products are generated far away from the passband signal and can be easily filtered out, other third-order terms (e.g., 2f_{1} - f_{2} and 2f_{2} - f_{1}) fall very close to the system's passband signal and cannot be filtered out. These terms can be problematic to the system and therefore, must not only be accounted for, but kept at lower power levels as compared to the main receive signal in the passband.

A commonly used method of determining the third-order intercept point (TOI) is to extrapolate the linear input power (P_{in}) and output power (P_{out}) lines of both the fundamental and third-order tones from their small-signal linear range until they intersect at the point where they are equal. That point is the TOI point. The two-tone X-parameter models can also be used to accurately determine IMD products and TOI with much faster simulation speed. X-parameters models with more than two tones can also be generated. To generate the model, a designer simply inserts the circuit in the ADS template, clicks on the "Source" button to enter two tones that are equal in power, and then performs the simulation. A two-tone model is generated and automatically placed in the project's data folder.

As an example, a two-tone X-parameter model was generated for the two-stage MMIC PA using the ADS harmonic balance simulator. The Xparameter model was simulated in 3 sfour times faster than simulating the actual circuit-level PA. Notably, simulation speed using the Xparameter model could actually be as high as 100 times the speed using the circuit-level design as the size of the circuit gets larger and more complex. The resulting model is highly accurate and exactly matches the fundamental tones and third-order terms of the circuit-level PA. * Figure 6* demonstrates the accuracy of overlaid two-tone analysis results from the circuit level PA and the X-parameter model.

One of the key uses of X-parameter models is in wireless system verification. The TOI point, for example, is a figure of merit that determines a system's linearity and dynamic range when excited with sinusoidal sources. To accurately determine the linearity of a system or circuit excited with digitally modulated sources, wireless verification tests with fully modulated waveforms are commonly used. Trajectory, constellation, and spectral displays, along with parameters such as adjacent-channel power ratio (ACPR), EVM, and PAE are used in such an analysis. Determining these wireless specifications in ADS is straightforward. A designer simply generates an IP-protected Xparameter model and then hands it off to the system integrator where it can be used to perform system-level wireless verification and tradeoff analysis with accuracy and much faster simulation speed.

Consider the evaluation of EVM on a transmitter PA as an example. A designer must insert either the circuit-level-PA or the X-parameter model into the test bench as shown (* Fig. 7*). The proper parameters can then be set and the "Simulate" button clicked. The test bench shown in

*has a realistic LTE transmit source that is identical to Agilent's N7624B/ N7625B Signal Studio for 3GPP LTE application (commonly used in the laboratory). This source is used to excite the DUT with the proper waveforms defined by a specific wireless specification. With the test bench, a designer can either simulate the Xparameter model by deactivating the circuit-level PA or simulate the circuitlevel PA by deactivating the model.*

**Fig. 7**To maintain linearity, the PA is tested for EVM at an input power of -2 dBm, which is slightly below its 1-dB compression point. For this example, the Agilent Ptolemy system simulator was used along with ADS. It co-simulates with the ADS Circuit Envelope simulator to perform the EVM verification on the PA design at the circuit level.

The envelope subcircuit-level setup in the top-level block of * Fig. 7* includes a duplexer and an antenna at its output. An engineer at a PA design house would model the duplexer with a filter and a coupler, followed by a 50-O termination for the antenna. The system integrator, on the other hand, with access to the circuit-level duplexer and antenna from other design houses, could use these, along with the PA's X-parameter model, to evaluate system EVM and immediately feed it back to the PA design house, should there be any needed modification to improve the PA performance (

*).*

**Fig. 8** * Figure 9* shows the EVM results using the X-parameter model. The accurate model enables a simulation speed that is 25 times faster than with a circuit-level PA. Since it is faster, the designer can quickly simulate and generate an EVM plot at many input power points (

*).*

**Fig. 10** X-parameters are applicable to both large-signal and small-signal conditions, and can be used for linear and nonlinear components. While they can be generated from both simulation and measurement for faster design development, the true benefit to circuit-level designers and system designers/integrators comes from using X-parameters generated from circuit-level designs. Such models provide high accuracy, fast simulation and full IP protection. For more information on X-parameters, visit: * www.agilent.com/find/eesof-x-parameters*.