Accurately Characterizing Hard Nonlinear Behavior of Microwave Components with the Nonlinear Network Measurement System: Introducing `Nonlinear Scattering Functions'

This paper describes an original way of dealing with the measuring and modeling of microwave transistor nonlinear behavior. Although generalizations are possible, the method described in this particular paper deals with transistor behavior under a large signal one-tone excitation, with arbitrary impedance terminations for the fundamental and the harmonics. First the mathematical theory of the “nonlinear scattering functions” is described. Next the measurement set-up and the actual extraction of the model parameters is highlighted. Finally the model is implemented in a commercial harmonic balance simulator. Using the simulator, model verification is performed by comparing measured and modeled behavior. INTRODUCTION There is a growing number of applications relying heavily on microwave technology: GSM, CDMA cellular phone, Local Multipoint Distribution Service, Japanese PCS and PDC, WCDMA, Wireless Local Loop,... The design of the power amplifiers used in these systems is often one of the toughest problems to solve. Powerful CAD tools potentially 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]. Despite the fact that a lot of effort goes into building models for power transistors, the result is often not accurate enough to satisfy the designer. In this case time consuming loadpull measurements are being used (often these loadpull measurements need to be iterated a few times during the design cycle). In this work another modeling approach is proposed, based upon the use of a black-box frequency domain model. The method is called “nonlinear scattering functions” and can be considered as an extension of “scattering parameters” into hard nonlinear behavior. The model can accurately 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 an experimental loadpull set-up build around a “Nonlinear Network Measurement System” [2]. These measurements are actually a combination of passive and active harmonic loadpull measurements. 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 loadpull measurements, which are very close to the actual working conditions of the device. This implies that the behavior of the model in the simulator will be consistent with the measured harmonic loadpull behavior. Classical modeling approaches are based upon many small signal and DC measurements, which are far from the actual operating conditions. This often results in inconsistency between measured and modeled loadpull behavior. 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. This makes the “nonlinear scattering functions” really technology independent. 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. MATHEMATICAL THEORY Mathematical notations Extending the concept of “scattering-parameters” [3] in order to describe nonlinear behavior is not trivial. One needs to go back to the basics. “Scattering parameters” are called this way because they relate incident and reflected (or scattered) travelling voltage waves at the signal ports, thereby completely describing the behavior of a linear microwave device. The “nonlinear scattering functions” have the same purpose: relating incident and reflected travelling voltage waves. Some necessary mathematical notations and concepts are introduced in the following. Incident voltage waves will be denoted by the symbol “a” and reflected voltage waves will be denoted by the symbol “b”. In many cases travelling voltage waves (and the corresponding sparameters) are defined in a characteristic impedance of 50 Ohm. When dealing with nonlinear behavior, however, it may be convenient to use different characteristic impedances. The characteristic impedance used for the definition of the waves will be indicated between brackets as a subscript. When the impedance is not indicated it is assumed to be 50 Ohm, or that the value is irrelevant for the given formula. The relationship between voltage “v”, current “i” (defined as being positive when flowing into the device-under-test, called DUT, signal port) and the travelling voltage waves is given by . (1) As mentioned in the introduction, we will describe the device behavior under sinusoidal (onetone) excitation. Excluding subharmonic and chaotic behavior, all signals appearing at the DUT ports will have the same periodicity (= the reciprocal of the fundamental frequency). The periodic signals will be described by their complex Fourier series coefficients, which are commonly called the “spectral components” of the signal. Each “spectral component” has an associated “harmonic index”, which denotes the ratio between the associated frequency and the fundamental. The “harmonic index” will be indicated by the last subscript. An “harmonic index” equal to zero corresponds to DC. The first subscript indicates the respective DUT signal port. Port 1 will typically correspond to the input (gate, base) and port 2 with the output (drain, collector) of the DUT. Some examples: • a(10)21 refers to fundamental of the incident voltage wave at port 2, defined with a characteristic impedance of 10 Ohm. • b13 refers to the third harmonic (with frequency equal to 3 times the fundamental frequency) at port 1, defined with a characteristic impedance of 50 Ohm. a Z ( ) v Zi + 2 -------------= b Z ( ) v Zi – 2 -------------= The black-box model Conceptually, writing down the black-box model equation is trivial. First thing to do is to “phase normalize” all signals by applying a time delay (applying a phase shift to the spectral components proportional to the harmonic index) such that a11 has zero phase. This way all spectral component phases are uniquely defined. This is done as follows: , with superscript “NN” denoting “not normalized”. (2) One can then simply write . (3) This equation simply states that the scattered voltage wave spectral components are complex functions of the real and imaginary parts of all the incident voltage wave spectral components. The functions “Skp” are called the “nonlinear scattering functions”. Note that Im(a11) does not appear in the equation since this value is always zero because of the phase normalization. The modeling problem is now transformed in identifying the functions Skp, for all scattered spectral components. In practice it will be sufficient to consider a limited number of harmonics (including up to the 4th harmonic has been enough for all practical cases investigated until now). Generally speaking identifying the functions Skp would imply the identification of a set of multidimensional nonlinear functions, which is practically very hard to do. In many cases, however, signal conditions are such that the functions Skp can be simplified. For a power amplifier with one dominant tone at the input, all harmonic signals will be relatively small compared to the fundamental signals. It is then possible to expand the functions Skp into a MacLaurin series [7] for all harmonic components (excluding the fundamental components at both signal ports). This results in . (4) In this equation N represents the highest harmonic index considered. Fkp, Gkpij and Hkpij are functions of Re(a11), Re(a21) and Im(a21) (= fundamental components) ,they are given by