Black Box Modelling of Power Transistors in the Frequency Domain

A frequency domain black box model for power transistors is proposed. It can accurately predict the behavior of the transistor for a one tone excitation with arbitrary fundamental and harmonic impedances present at the output. The model parameters can be extracted out of a limited set of “nonlinear network analyzer” measurements. Both simulated as well as measured results are given. Introduction It is not easy to design a good microwave power amplifier. Powerful CAD tools save a lot of time. It is important, however, to be aware that these simulators can only be as accurate as the mathematical models that are used. As a consequence a lot of time is spend on constructing good models for the components used. Especially constructing models that can accurately describe the large-signal hard-nonlinear behavior of power transistors is far from trivial. The state-of-the-art is to use technology dependent analytical models (e.g. Curtice Cubic, Materka, Statz, Tajima,...) or more general “small signal measurement based” models like the HP-Root model [1]. In this work another approach is proposed, based upon the use of a black-box frequency domain model. The model can simulate the behavior of a power transistor under large-signal one-tone excitation at the input, with any arbitrary impedances present at the output (fundamental and all harmonics). The model parameters are extracted based upon a relatively small set of measurements performed with a vectorial “nonlinear network” analyzer [2]. There are mainly two reasons why one can expect this approach to be more simple and accurate. Firstly one has the advantage that the model parameters are directly extracted from large signal measurements, which are the actual working conditions of the device, while the other models are based upon many small signal and DC measurements. Secondly all parasitic effects are automatically included in the black-box model, while all parasitic effects have to be explicitly identified with the other models. A drawback of the method is of course that the model will only be valid for one-tone excitation, with a frequency corresponding to the frequency used to extract the model. Theory In the following the theory of the modelling approach is explained. Suppose one has a device-under-test, called DUT, (typically one power transistor, but the theory can be applied on a whole power amplifier containing several transistors) excited by a large single-tone signal at the input and with arbitrary impedances (fundamental and harmonics) present at the output. This is depicted in Fig. 1. Conference Record of the INMMC 1996 Workshop Duisburg (Germany) Fig. 1 Schematic of the device-under-test In this figure I11 denotes the complex number representing the single-tone voltage wave incident to the input, O2i denotes the complex number representing the i th harmonic of the voltage wave scattered by the output, I1i denotes the complex number representing the i th harmonic of the voltage wave incident to the output, while ΓLi is the complex number representing the reflection factor seen by the ith harmonic at the output. For simplicity, the phases of all spectral components will be defined relative to the I11 signal, which is used as a kind of time reference signal in order to define the phases of the other components (as a consequence I11 will have an imaginary part equal to zero). Note that the voltage waves are defined in a characteristic impedance which is typically 50 Ohms. The problem is to find a model for the DUT, based upon a limited number of measurements, which allows to describe the scattered components O21,...,O2N as a function of I11 and the harmonic reflection factors ΓLi (N denotes the number of significant harmonics). This is done by identifying a black-box model which describes these scattered components O21,...,O2N as an analytical function of all incident components I11 and I21,...,I2N and their conjugates [3]. This is illustrated in Eq. 1, Eq. 1 where Fi stands for the describing function corresponding to the i th harmonic and where the superscript “+” stands for the conjugate. The solution for O21,...,O2N can then be found by solving the combined set of Eq. 1 and Eq. 2 in the 2N unknowns I21,...,I2N and O21,...,O2N, where Eq. 2 describes the reflection of the voltage waves O21,...,O2N caused by the fundamental and harmonic mismatches present at the output of the DUT. In order to simplify the identification of the describing functions, it is assumed that these functions behave linearly versus the components I22,...,I2N. These components are assumed to be relatively small signal components because of Eq. 2, the fact that the amplitude of the harmonic components O22,...,O2N is significantly smaller than O21 (the fundamental) and the amplitude of the reflection factors ΓLi is smaller than 1 (the terminations are assumed to be passive). The final mathematical model then becomes: