### Introduction

Due to the demands of today’s wireless communication systems, it has become ever more important to improve the speed and accuracy of the RF component models used to simulate the system. Complex, nonlinear interactions between RF components in the system are difficult or impossible to simulate without more rigorous models at the component level. This paper will discuss various approaches to RF component level models for the system level designer and detail some of the issues related to model extraction, simulation speed, ease of use, accuracy, intellectual property protection, and model limitations. I will compare and contrast three primary RF component models, full nonlinear circuit IP encoded models, simple low-pass behavioral models, and X-parameter Poly Harmonic Distortion (PHD) models.

### Full Nonlinear IP Encoded Circuit Model

In terms of accuracy and flexibility, the full, nonlinear circuit model description of an RF component has the fewest limitations (with some important caveats I will discuss below…). Generally this is supplied as an IP encoded “black-box” model (such as Advanced Design System’s IP encoder) where any underling intellectual property of the RF component supplier is fully protected. First, this type of model can incorporate all aspects of the RF component with the fewest limitations. Take an RF amplifier component for example. The IP encoded model allows system simulation over bias, temperature, and frequency range, source and load mismatch, etc. There are also few limitations when it comes to simulating stability (or predicting instability) when inserting the component into a system simulation. However, (now for those caveats…) the model can be very time consuming to create and verify. This is not a problem for the system designer, but it is for the component supplier. The model can be very complex, thus slow to simulate at the system level (and this is a big problem for the system designer). Generally, the full nonlinear circuit model offers the slowest system level simulation speed due to the complexity of the underlying model (2 to 3 orders of magnitude slower than behavioral models). The model may also cause simulator convergence issues, again decreasing system simulation speed. Also, custom design kits (including custom compiled component supplier models etc.) may be needed by the system designer to run the model. The IP encoded models are very easy to use, but are not compatible with all system level simulation tools. Component models suitable for other system level simulator tools can be generated from the IP encoded model however. In summary, the IP encoded RF component model is easy to use and very accurate, but offers the slowest overall system level simulation speed.

Figure 1 shows an example of an ADS IP encoded nonlinear SP6T switch model embedded in a GSM RF transmit system chain consisting of an amplifier, low-pass filter, transmission line, and switch. Figure 2 shows a system level simulation of the 3rd harmonic versus frequency using the IP encoded nonlinear SP6T model.

Figure 1: GSM RF Transmit System Chain

Figure 2: Simulated 3rd Harmonic of GSM RF Transmit System Chain

From this simulation, it is clear that the low-pass filter’s match at the harmonic frequency has a significant impact on system level harmonics transmitted to the antenna.

Low-Pass Behavioral Models

In terms of simplicity and simulation speed, the behavioral model is a very good choice for narrow-band system simulation. José C. Pedro and S. A. Maas give a very good overall summary of various behavioral models in their 2005 MTT transactions paper [1]. Here I will discuss two of the more popular types, the low-pass memory-less behavioral model and the behavioral model parameterized in frequency [2]. The simplest low-pass memory-less behavioral model assumes that the signal modulation bandwidth is small as compared to the RF carrier frequency and the amplitude and phase transfer characteristics are invariant with frequency. The model is extracted from either measured or simulated single-tone (or two-tone depending on the type) complex voltage gain versus input power (voltage amplitude) at the RF carrier frequency. This AM-AM (amplitude to amplitude) and AM-PM (amplitude to phase) characteristic is then fit using a polynomial or a more complex function as in [2]. Using this modeled characteristic, the component’s output response (in the time domain) is constructed by multiplying the complex input envelope by the modeled complex envelope gain (for the corresponding input magnitude) at that moment in time. The result can then be converted back into the frequency domain (if desired) using the Discrete or Fast Fourier Transform to calculate useful parameters such as Adjacent Channel Power Ratio ACPR.

If the component is driven by a sufficiently wide bandwidth signal, or the RF gain response is narrow due to matching/filtering components, the narrow-band assumption of this simple behavioral model breaks down and accuracy degrades. One example of this is at a band-edge for a WLAN power amplifier where the gain and phase characteristics can change rapidly versus frequency. In this situation, the assumption of frequency invariance in the AM-AM and AM-PM response is no longer valid. To address these and other similar issues, behavioral models have been developed to overcome the deficiencies inherent to the memory-less narrowband approximation assumed by the low-pass equivalent AM–AM and AM–PM models. The idea behind these models is that the bandwidth of the input is no longer so small compared to the one of the system that a CW signal ceases to be a reasonable representation. The solution to this problem consists of sensing that bandwidth in all possible frequency points. This leads to models that are typically AM–AM and AM–PM memory-less models parameterized in frequency [2].

The benefits of the behavioral models discussed above are many. They are simple to extract/create from measured or simulated single-tone or two-tone input power sweeps. Due to their mathematical simplicity, they lead to very fast system level simulations (2-3 orders of magnitude faster that full non-linear component models). They are also “black-box” models, so component supplier intellectual property is fully protected. Predictions of “near-in” nonlinearities such as IP3, EVM, ACPR, etc. from narrow-band modulated signals are in general very accurately predicted. Many system level simulators such as Matlab, Ptolemey, etc. support these types of behavioral models.

There are, however, a number of disadvantages to the behavioral models discussed above. If the RF component has significant nonlinear memory, model accuracy will degrade. Behavioral models that can simulate nonlinear memory do exist [3-4], but the model extraction is more complex. Behavioral models also have limitations with respect to stability. The model can become invalid if the component is unstable and generates undesired frequencies due to the system level environment. For example, due to RF or bias supply feedback, the component may oscillate and generate frequencies that will not be captured using the simple “black-box” behavioral model. Lastly, the simple behavioral model is valid for one set of source and load impedances (those used to extract the model). If the RF component model is loaded differently in the system, model accuracy degrades.

This brings us to X-parameter models…

### X-parameter (PHD) Models

X-parameters are the mathematically correct supersets of S-parameters valid for nonlinear (and linear) components under large-signal (and small-signal) conditions [5]. The X-parameter PHD model is a “black-box” behavioral model whereby the model theory derives from a multi-harmonic linearization around a periodic steady state determined by a large-amplitude single input tone. For this reason, the model is known as the PHD model [6]. The assumption is that the system to be modeled can depend in a strongly nonlinear way on its large-signal drive, but nevertheless responds linearly to additional signal components at the harmonic frequencies considered as “small” perturbations around the time-varying system state. This is referred to as the “harmonic superposition” principle [7].

X-parameter models have many of the same benefits as the simple behavioral models already discussed (because the are simply another form of a behavioral model). They preserve component supplier intellectual property. They are simple to create from either measurements using a Nonlinear Vector Network Analyzer (NVNA) or from simulations using Advanced Design System (ADS). They are accurate and very fast to simulate in a system simulator. In terms of speed, for narrow band modulation, I think the low-pass memory-less behavioral model is still faster, but not by much. The model is more transportable like S-parameters (it’s simply a parameterized data file). One of the most useful features of X-parameter models is how they maintain accuracy when inserted into a mismatched system. Unlike the low-pass behavioral models, they can correctly predict performance even under strong load mismatch [5]. Now for some of the downsides…

X-parameter models are very accurate assuming your system simulation stays within the parameter space of the model. For some RF components, this means that the X-parameters can take a very long time to extract from many measurements or simulations. Depending on the number of ports, frequencies, biases, powers, etc. the X-parameter files can become very large (Gbyte range). This can slow down system simulation when the X-parameter file needs to be loaded. I have found that when using ADS to generate an X-parameter model, single-tone simulation based models are very fast to extract and file sizes are small. Two-tone simulation based models become cumbersome and file sizes are moderate to large, and three or more tone models take a very long time to generate and the file sizes can be enormous. If the RF component is very complex, stick with two-tones or less.

### Summary

I have discussed the relative merits of three RF component model types suitable for RF system designers. I believe that all three have their appropriate place in the system designer’s tool kit. Remember, the more tools you give to engineers, the more damage they can do…