Black Box Modelling Of Hard Nonlinear Behavior In The Frequency - - PowerPoint PPT Presentation

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Network Measurement and Description Group

Black Box Modelling Of Hard Nonlinear Behavior In The Frequency Domain

Jan Verspecht*, D. Schreurs*, A. Barel*, B. Nauwelaers* *

Hewlett-Packard NMDG VUB-ELEC Pleinlaan 2 1050 Brussels Belgium fax 32-2-629.2850

• tel. 32-2-629.2886

email janv@james.belgium.hp.com

* Katholieke Universiteit Leuven

B-3001 Leuven Belgium

* Vrije Universiteit Brussel

Pleinlaan 2 1050 Brussels Belgium

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Network Measurement and Description Group

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.

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Network Measurement and Description Group

I1 I2 I3 O1 O5

...

INPUT OUTPUT freq. freq. device-under-test Ok Fk I1 I2 ... , , ( ) = Frequency Domain Black Box Modelling

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.

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What is new ?

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 New Describing Functions: Multiple Input Components Multiple Output Components Hard Nonlinear OK

Preexistent techniques

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Network Measurement and Description Group Ok VN ( )kGk A1 ...,AN V1 ... VN 1 – , , , , ( ) = Pi e jϕ Iαi     = Ai Iαi = Ok Fk Iα1 Iα2 ...,IαN , ,     = α1s1i ... αNsNi + + = VN P1 m1...PN mN = Vi P1 s1i...PN sNi = α1m1 ... αNmN + + 1 = 1 i N 1 – ≤ ≤

( )

Describing Functions With Multiple Inputs

Identifying Fk 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: with αi being the normalized frequency of the ith input component with Ai and Vi : delay input amplitudes input phase relationships

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Harmonic Distortion

Simple Examples

Intermodulation Products

Ok P1

kGk A1

( ) = Ok P4P3

1 –

( )

k

Gk A3 A4 P4

3P3 4 –

, , ( ) =

For an harmonic distortion analysis one input spectral component is present. The A1 variable corresponds to the amplitude and P1 to the phase represented as a complex number on the unit circle. The kth harmonic Ok at the output can be written as a function

• f the input as shown above, where Gk represents an arbitrary describing function.

Suppose there are two input components with normalized frequencies 3 and 4. Every intermodulation product Ok can be written as shown above. The first two arguments of the describing function Gk are the amplitudes of the input components, while the third argument represents the phase relationship between the two components.

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Volterra Approach (VIOMAP) Rational Describing Function Ok P1 ( )k KiA1 2i k + i = N

∑

        = Ok P1 ( )k KiA1 i i = N

∑

        A1 k 1 A1 k +

• 

       =

Black Box Parametric Models

In practice, a parametric model for the describing functions Gk is proposed. The parameters can be found by fitting measured

• data. In what follows two types of models

are investigated and compared for an harmonic distortion measurement:

• ne

approach which corresponds to the Volterra theory, and

• ne

approach based upon rational describing functions. The model parameters are noted Ki, they are extracted by a least squares technique.

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2 4 6 8 10

• 0.001

0.001 0.002

Input amplitude (V) Model error (V)

2 4 6 8 10

• 0.03
• 0.02
• 0.01

0.01 0.02 0.03

Input amplitude (V) Model error (V)

2 4 6 8 10 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Input amplitude (V) Harmonic amplitude (V)

VIOMAP

VIOMAP vs. Describing Rational

Describing Rational Ideal Compressor Characteristic (7th harmonic)

Modelling Errors (with 5 parameters used)

The rational describing function and the VIOMAP parametric models are fitted on the simulated 7th harmonic generation of an ideal

• compressor. The number of parameters used is

in both cases equal to 5. The describing rational does a much better job to fit the ideal curve, with an error which is typically only one tenth

• f the error made with the Volterra approach.

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The Resistive Mixer Experiment

HEMT transistor (no bias) LO 3GHz RF 4GHz IF 1GHz 7GHz

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).

...

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Voltage waves (V) Time (ns)

Resistive Mixer Time Domain Waveforms

LO/2 RF IF

0.2 0.4 0.6 0.8 1

• 0.15
• 0.1
• 0.05

0.05 0.1 0.15

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.

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Network Measurement and Description Group

0.2 0.4 0.6 0.8 0.02 0.04 0.06 0.08 0.1 0.12

Local oscillator peak amplitude (V) Intermod 1 peak amplitude (V) : rational model : VIOMAP model

VIOMAP and Describing Rational

HEMT Resistive Mixer: IF vs. LO

(RF fixed peak amplitude of 0.2V) 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.

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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.