1 Black Box Modelling Of Hard Nonlinear Behavior In The Frequency Domain Jan Verspecht*, D. Schreurs*, A. Barel*, B. Nauwelaers* * Katholieke Universiteit Leuven * Hewlett-Packard NMDG B-3001 Leuven VUB-ELEC Belgium Pleinlaan 2 1050 Brussels * Vrije Universiteit Brussel Belgium fax 32-2-629.2850 Pleinlaan 2 tel. 32-2-629.2886 1050 Brussels email janv@james.belgium.hp.com Belgium Network Measurement and Description Group
2 ABSTRACT A black box model is proposed to describe nonlinear devices in the frequency domain. The approach is based upon the use of describing functions and allows a better description of hard nonlinearities than an approach based upon the Volterra theory. Simulations and experiments are described illustrating the mathematical theory. Network Measurement and Description Group
3 Frequency Domain Black Box Modelling OUTPUT device-under-test INPUT ... O 5 O 1 I 1 I 2 I 3 freq. freq. ( , , ) = Fk I 1 I 2 ... Ok For a frequency domain black box model one assumes that every spectral output component is a function of the spectral input components, no further a priori knowledge is required. Network Measurement and Description Group
4 Preexistent techniques Volterra Theory: Multiple Input Components Multiple Output Components Hard Nonlinear Is A Problem Describing Functions: One Input Component One Output Component (Fund.) Hard Nonlinear OK What is new ? New Describing Functions: Multiple Input Components Multiple Output Components Hard Nonlinear OK Network Measurement and Description Group
5 Describing Functions With Multiple Inputs with α i being the normalized frequency of the i th input component , , Ok = Fk I α 1 I α 2 ..., I α N Identifying F k is simplified by expressing that the device-under-test is time-invariant. “Delaying the input results in the same delay at the output.” This results in the following transformed mathematical formulation: ) kGk A 1 ...,A N V 1 ... VN ( ( , , , , ) Ok = VN – 1 delay input amplitudes input phase relationships e j ϕ I α i with A i and V i : Ai = I α i Pi = m 1... PN mN α 1 m 1 α NmN + ... + = 1 VN = P 1 s 1 i ... PN sNi ( ) α 1 s 1 i α NsNi ≤ ≤ Vi = P 1 + ... + = 0 1 i N – 1 Network Measurement and Description Group
6 Simple Examples Harmonic Distortion ( ) k G k A 1 O k = P 1 For an harmonic distortion analysis one input spectral component is present. The A 1 variable corresponds to the amplitude and P 1 to the phase represented as a complex number on the unit circle. The k th harmonic O k at the output can be written as a function of the input as shown above, where G k represents an arbitrary describing function. Intermodulation Products k – 1 3 P 3 – 4 ( ) ( , , ) O k = P 4 P 3 G k A 3 A 4 P 4 Suppose there are two input components with normalized frequencies 3 and 4. Every intermodulation product O k can be written as shown above. The first two arguments of the describing function G k are the amplitudes of the input components, while the third argument represents the phase relationship between the two components. Network Measurement and Description Group
7 Black Box Parametric Models Volterra Approach (VIOMAP) In practice, a parametric model for the N ∑ describing functions G k is proposed. The ) k 2 i + k ( Ok = P 1 KiA 1 parameters can be found by fitting measured data. In what follows two types of models i = 0 are investigated and compared for an harmonic distortion measurement: one approach which corresponds to the Volterra Rational Describing Function theory, and one approach based upon rational describing functions. The model k N A 1 ∑ parameters are noted K i , they are extracted ) k i ( Ok = P 1 KiA 1 - - - - - - - - - - - - - - - - by a least squares technique. k 1 + A 1 i = 0 Network Measurement and Description Group
8 VIOMAP vs. Describing Rational Ideal Compressor Characteristic (7th harmonic) The rational describing function and the Harmonic amplitude (V) VIOMAP parametric models are fitted on the 0.14 simulated 7th harmonic generation of an ideal 0.12 0.1 compressor. The number of parameters used is 0.08 in both cases equal to 5. The describing rational 0.06 does a much better job to fit the ideal curve, 0.04 0.02 with an error which is typically only one tenth 0 0 2 4 6 8 10 of the error made with the Volterra approach. Input amplitude (V) Modelling Errors (with 5 parameters used) Describing Rational VIOMAP 0.03 0.002 Model error (V) 0.02 Model error (V) 0.01 0.001 0 0 -0.01 -0.02 -0.001 -0.03 0 2 4 6 8 10 0 2 4 6 8 10 Input amplitude (V) Input amplitude (V) Network Measurement and Description Group
9 The Resistive Mixer Experiment RF 4GHz LO IF 3GHz 1GHz 7GHz ... HEMT transistor (no bias) The same approach is also tested on measured data. A resistive mixer experiment is performed for this purpose. With a resistive mixer experiment the “local oscillator” (LO) signal is applied to the gate of a microwave FET transistor, while the “radio frequency” (RF) signal is a voltage wave incident at the transistor drain. The “intermediate frequency” (IF) signals are the spectral components of the voltage wave scattered at the drain. Measurements are performed with a prototype “Vectorial Nonlinear Network Analyzer” (VNNA). Network Measurement and Description Group
10 Resistive Mixer Time Domain Waveforms Voltage waves (V) 0.15 0.1 LO/2 0.05 0 RF -0.05 -0.1 IF -0.15 0 0.2 0.4 0.6 0.8 1 Time (ns) Shown above are the measured time domain waveforms at the signal ports of the FET transistor used as a resistive mixer. The LO signal is applied to the gate. The RF signal is the voltage wave incident to the drain, the IF signal is the reflected voltage wave. While the gate voltage is high the drain represents a low impedance, such that the IF and RF are in opposite phase, while the gate voltage is low the drain represents a high impedance, such that the IF and RF are in phase. Network Measurement and Description Group
11 VIOMAP and Describing Rational HEMT Resistive Mixer: IF vs. LO (RF fixed peak amplitude of 0.2V) Intermod 1 peak amplitude (V) 0.12 0.1 0.08 : rational model 0.06 : VIOMAP model 0.04 0.02 0 0 0.2 0.4 0.6 0.8 Local oscillator peak amplitude (V) Measured data is captured sweeping the amplitude of RF and LO from 0V to 0.8V, for different RF-LO phase relationships. Two models are fit on the data (same number of parameters): a rational describing function and a VIOMAP. As an example, the first intermod (1GHz) peak amplitude is plotted versus the LO peak amplitude, with the RF having a peak amplitude of 0.2V. The rational model is smoother than the VIOMAP, corresponding better to what one physically expects. Network Measurement and Description Group
12 Conclusion The describing functions approach developed allows to construct better black box parametric models for hard nonlinear devices than an approach based upon the Volterra theory. “Vectorial Nonlinear Network Analyzer” measurements can be used in order to extract the model parameters. Network Measurement and Description Group
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