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Measurement -based modeling of vector network analyzer calibration standards and nonlinear microwave devices using artificial neural networks

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MEASUREMENT-BASED MODELING OF VECTOR NETWORK ANALYZER
CALIBRATION STANDARDS AND NONLINEAR MICROWAVE DEVICES
USING ARTIFICIAL NEURAL NETWORKS
by
JEFFREY ARENDT JARGON
B. S., University of Colorado, 1990
M. S., University of Colorado, 1996
A thesis submitted to the
Faculty of the Graduate School of the
Uni versity of Colorado in partial fulfillment
of the requirements of the degree of
Doctor of Philosophy
Department of Electrical and Computer Engineering
2003
I
_______________________
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UMI N um ber: 3087553
UMI
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This thesis for the Doctor of Philosophy degree by
Jeffrey Arendt Jargon
has been approved for the
Department of
Electrical and Computer Engineering
by
K. C. Gupta
c
Donald C. DeGroot
Date
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I
Jargon, Jeffrey Arendt (Ph.D. Electrical Engineering)
Measurement-Based Modeling of Vector Network Analyzer Calibration Standards
and Nonlinear Microwave Devices Using Artificial Neural Networks
Thesis Directed by Professor K. C. Gupta and Dr. Donald C. DeGroot
This thesis is comprised of two parts. The first segment covers artificial neural
network (ANN) modeling for improved vector network analyzer (VNA) calibrations.
Specifically, measurement-based ANNs are applied to model a variety of on-wafer
and coaxial vector network analyzer calibrations, including open-short-load-thru
(OSLT) and line-reflect-match (LRM). A sensitivity analysis of the ANNs is
performed by determining the training error as functions of the number of hidden
neurons and the number of training points. The respective accuracies of these
calibrations are then assessed using the ANN-modeled standards. As a major research
result, this doctoral thesis shows that ANN models offer a number of advantages over
using calibrated measurement data files or equivalent circuit models, namely: they do
not require the numerous details and parameters of physical models; calibration times
can be reduced because only a few training points are required to accurately model
the standards; ANN model descriptions are much more compact than large
measurement data files; ANN models, trained on only a few measurement points can
be much more accurate than direct calibrations when limited calibration data are
available; ANNs give an optimized estimate in the presence of noise; and ANN
models are able to accurately model loads with measured DC resistances slightly
outside of their training range.
in
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In the second part of this thesis, new frequency-domain models and figures of
merit for nonlinear microwave circuits are developed for sparse-tone inputs. This
section begins with a method for preserving time-invariant phase relationships when
ratios are taken between two harmonically related signals by introducing a third
signal that is used as a phase reference. Then, as another major research result, this
doctoral thesis introduces nonlinear large-signal scattering (§>) parameters, a new type
of frequency-domain mapping that relates incident and reflected signals. A general
form of nonlinear large-signal ^-parameters is presented. It is shown that they reduce
to classic ^-parameters in the absence of nonlinearities. Nonlinear large-signal
impedance (%) and admittance (|j) parameters are also introduced, and equations
relating the different representations are derived. Next, definitions of power gain,
transducer gain, and available gain are expanded by taking harmonic content into
account. An example is provided showing how the expanded definitions of gain and
nonlinear large-signal ^-parameters allow one to examine the behavior of a nonlinear
model by simply performing a harmonic balance simulation. Next, this thesis
illustrates how nonlinear large-signal ^-parameters can be used as a tool in the design
process of a nonlinear circuit, specifically a single-diode 1-2 GHz frequency-doubler.
For the case where a nonlinear model is not readily available, a method of extracting
nonlinear large-signal ^-parameters is developed using ANN models trained with
multiple measurements made by a nonlinear vector network analyzer equipped with
two sources. Finally, nonlinear large-signal ^-parameters are compared to another
form of nonlinear mapping, known as nonlinear scattering functions. The nonlinear
large-signal ^-parameters are shown to be more general.
IV
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I
DEDICATION
To my wife Soogy, and our children, Trevor and Valerie. Words can’t
possibly describe how much you all mean to me. I love you so much and appreciate
your encouragement, patience, and understanding throughout this project.
And to Mom and Dad for absolutely everything. You’re the best parents I
could ever ask for. I love you both.
I also thank the rest of my wonderful family for their love and support
throughout the years, including Jangmonim; my in-laws, nieces, and nephews in
Korea; Jon and Maria; Julie and Craig; Granddad and Rada; Grandma Boam;
Grandpa Jargon and my late Grandma Jargon.
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ACKNOWLEDGEMENTS
I thank all of the individuals who helped make this dissertation possible.
Special thanks to my thesis advisors, K. C. Gupta and Donald C. DeGroot, for their
invaluable support, guidance, and encouragement. Our last few years working
together have been very educational and enjoyable. I also thank my co-conspirator
and best friend, Michael Janezic, for making this journey together a pleasant one. I
am grateful for the support of everyone at the National Institute of Standards and
Technology, especially Robert Judish, Susie Rivera, Kate Remley, and Dennis
Friday. I am indebted to Dominique Schreurs, Alessandro Cidronali, Pete Kirby,
Huantong Zhang, Jan Verspecht, and Marc Vanden Bossche for their valuable
contributions and suggestions. I also thank Y. C. Lee, James Baker-Jarvis, and
Melinda Piket-May for serving on my committee and providing me with valuable
feedback. And last, but also least, a tip of the hat to Victor Kushmann for “keepin’ it
real.”
j
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TABLE OF CONTENTS
CHAPTER
1.
2.
I
i
INTRODUCTION
1
1.1.
ANN Modeling for ImprovedVector Network Analyzer
Calibrations
1
1.2.
Frequency-Domain Models and Figures
Nonlinear Circuits
3
1.3.
Organization of the Thesis
6
ARTIFICIAL NEURAL NETWORKS
11
of Merit
for
2.1.
Neural Network Structure
12
2.2.
Model Development
16
Model Verification
18
2.3.
PART I - ANN MODELING FOR IMPROVED VECTOR NETWORK
ANALYZER CALIBRATIONS
3.
4.
LINEAR NETWORK ANALYSIS
22
3.1.
Scattering Parameters
22
3.2.
Vector Network Analyzer Architecture
26
3.3.
Vector Network Analyzer Error Models
28
3.4.
Vector Network Analyzer Calibration Techniques
30
3.5.
ANN Modeling for Improved VNA Calibrations
32
ANN MODELING OF ON-WAFER OSLT STANDARDS
34
4.1.
Modeling the Standards
35
4.2.
Calibration Comparisons
45
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4.3.
46
Discussion
5. MODELING LOAD VARIATIONS WITH ANNS TO IMPROVE
ON-WAFER OSLT CALIBRATIONS
49
5.1.
Modeling the Standards
50
5.2.
Advantages of ANN Models
53
5.3.
Calibration Comparisons
53
5.4.
Discussion
57
6 . ANN MODELING FOR ON-WAFER LRM CALIBRATIONS
58
6.1.
Load Modeling
58
6.2.
Calibration Comparisons
60
6.3.
Discussion
62
7. ANN MODELING FOR COAXIAL LRM CALIBRATIONS
63
7.1.
Load Modeling
64
7.2.
Calibration Comparisons
67
7.3.
Discussion
68
PART II - DEVELOPING FREQUENCY-DOMAIN MODELS FOR
NONLINEAR
CIRCUITS
BASED
ON
LARGE-SIGNAL
MEASUREMENTS
8 . NONLINEAR NETWORK ANALYSIS
70
8.1.
Linear V ersus N onlinear Behavior
70
8.2.
Nonlinear Vector Network Analysis
71
8.3.
Nonlinear Vector Network Analyzers
75
8.3.1. Architecture
75
8.3.2. Calibration
77
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8.4.
9.
Developing Frequency-Domain Models for
Circuits Based on Large-Signal Measurements
Nonlinear
79
CALCULATING RATIOS OF HARMONICALLY-RELATED,
COMPLEX SIGNALS
81
9.1.
Introduction
81
9.2.
Method
80
9.3.
Example
84
9.4.
Discussion
89
10. NONLINEAR LARGE-SIGNAL SCATTERING PARAMETERS
90
10.1.
General Form
90
10.2.
Nonlinear Large-Signal Impedance Parameters
93
10.3.
Relating
94
10.4.
Nonlinear Large-Signal Admittance Parameters
97
10.5.
Relating &>and $ Parameters
98
10.6.
One-Port Network with Single-Tone Excitation
101
10.7.
Two-Port Network with Single-Tone Excitation
103
10.8.
Discussion
105
and Z Parameters
11. EXPANDING DEFINITIONS OF GAIN
HARMONIC CONTENT INTO ACCOUNT
BY
TAKING
107
11.1.
Commonly Used Definitions of Gain
107
11.2.
Expanded Definitions of Gain
109
11.3.
Expanded Definitions of Gain in Terms of ^-Parameters
111
11.4.
Discussion
113
12. USING ^-PARAMETERS AND EXPANDED DEFINITIONS OF
GAIN TO EXAMINE THE BEHAVIOR OF A PHEMT MODEL
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115
12.1.
Using ^-Parameters to Examine the Behavior of a PHEMT
Model
116
12.2.
Using Expanded Definitions of Gain to Examine the
Behavior of a PHEMT Model
121
12.3.
Discussion
125
13. USING ^-PARAMETERS TO DESIGN A DIODE FREQUENCY
DOUBLER WITH A HARMONIC-BALANCE SIMULATOR
126
13.1.
Diode Only
128
13.2.
With 1&2 GHz Filters
129
13.3.
With 1&2 GHz Filters, and Input Matching
130
13.4.
With 1&2 GHz Filters, Input and Output Matching
130
13.5.
With 1&2 GHz Filters, Input & Output Matching Optimized
131
13.6.
With 1&2,4&6 GHz Filters, In & Out Matching Optimized
131
13.7
Discussion
132
14. DETERMINING
^-PARAMETERS
FROM
ARTIFICIAL
NEURAL
NETWORK
MODELS
TRAINED
WITH
MEASUREMENT DATA
134
14.1.
Methodology
134
14.2
Sensitivity Analysis of ANN Models
142
14.3.
Results and Comparison
147
14.4.
Discussion
150
15. COMPARING
^-PARAMETERS
SCATTERING FUNCTIONS
WITH
NONLINEAR
16. SUMMARY AND FUTURE WORK
16.1.
ANN Modeling for Improved Vector Network Analyzer
Calibrations
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151
\ 58
153
16.1.1. Summary
158
16.1.2. Other Applications
160
16.1.3. Future Work
161
16.2.
Frequency-Domain Models and Figures of Merit for
Nonlinear Circuits
164
16.2.1. Summary
164
16.2.2. Future Work
167
169
BIBLIOGRAPHY
XI
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LIST OF FIGURES
Figure 2.1.
MLP3 artificial neural network architecture.
13
Figure 2.2.
Sigmoidal activation function used in hidden and output
layer neurons.
15
Figure 2.3.
Linear scaling of neural network data.
17
Figure 2.4.
Three ANN models illustrating (a) over-learning, (b) underlearning, and (c) good-learning.
19
Figure 3.1.
Determining two-port scattering parameters.
25
Figure 3.2.
Four-sampler vector network analyzer.
27
Figure 3.3.
Three-sampler vector network analyzer.
27
Figure 3.4.
Twelve-term error model for vector network analyzers.
29
Figure 3.5.
Eight-term error model for vector network analyzers.
30
Figure 3.6.
Standards measured for the OSLT, LRM, TRL, and multiline
TRL calibrations.
32
Figure 4.1.
Training error versus the number of neurons in the hidden
layer for various OSLT calibration standards.
37
Figure 4.2.
Magnitude of the ANN-modeled reflection coefficient errors
(|ASn|) for the open standard with varying numbers of
training points.
38
Figure 4.3.
Magnitude of the ANN-modeled reflection coefficient errors
(|ASn|) for the load standard with varying numbers of
training points.
38
Figure 4.4.
Comparison of magnitude and phase of the reflection
coefficients [|Sn| and Arg(Sn)] for the load standard using
an ANN model trained with 5 points, linear interpolation
with TRL using the same 5 points, and TRL with 192 points
as the reference.
40
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Figure 4.5.
Magnitude and phase of the reflection coefficients [|Sn| and
Arg(Sn)] for the open standard measured by multiline TRL
and ANN modeling.
41
Figure 4.6.
Magnitude and phase of the reflection coefficients [|Sn| and
Arg(Sn)] for the short standard measured by multiline TRL
and ANN modeling.
42
Figure 4.7.
Magnitude and phase of the reflection coefficients [|Sn| and
Arg(Sn)] for the load standard measured by multiline TRL
and ANN modeling.
43
Figure 4.8.
Magnitude and phase of the transmission coefficients [|S2i|
and Arg(S2i)] for the thru standard measured by multiline
TRL and ANN modeling.
44
Figure 4.9.
Magnitude and phase of the scattering parameters of a
calibrated 19-mm CPW transmission line.
46
Figure 4.10.
Magnitude of the scattering parameter differences (|ASij|) of a
calibrated 19-mm CPW transmission line.
47
Figure 5.1.
Real and imaginary components of Z\\ for the load standards
measured by multiline TRL and modeled by an ANN.
52
Figure 5.2.
Magnitude and phase of S21 for a calibrated 1.764-mm CPW
transmission line.
55
Figure 5.3.
Magnitude and phase of S 11 for a calibrated 1.764-mm CPW
transmission line.
56
Figure 5.4.
Magnitudes of the scattering parameter differences of a
calibrated 1.764-mm CPW transmission line.
57
Figure 6.1.
Magnitude and phase of measured and modeled scattering
parameters of the on-wafer load for the LRM calibration.
59
Figure 6.2.
Worst-case error bounds between measurements of passive
devices from on-wafer LRM and TRL calibrations and the
multiline TRL calibrations.
61
Figure 7.1.
Real and imaginary components of the measured and
modeled impedance of the coaxial load for the LRM
calibration.
66
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Figure 7.2.
Worst-case error bounds between measurements of passive
devices from coaxial LRM and TRL calibrations and the
multiline TRL calibrations.
69
Figure 8.1.
Nonlinearities transfer energy from the stimulus frequency to
products at new frequencies.
71
Figure 8.2.
Complex traveling waves at the ports of a nonlinear device
when a set of harmonically related signals is present.
72
Figure 8.3.
Flow diagram for a two-port network with a singlefrequency excitation at port 1 and a terminating load at port
2. (a) Linear case, (b) Nonlinear case considering three
harmonics.
74
Figure 8.4.
Simplified block diagram of a nonlinear vector network
analyzer.
76
Figure 9.1.
Phasor plot of the fundamental reference x\, the dividend z%,
and the divisor >’3 at the first phase reference [all phasors
identified by superscript ( 1)].
84
Figure 9.2.
Time-domain plot of the fundamental reference x\, the
dividend Z2, and the divisor y 3 at the first phase reference.
88
Figure 9.3.
Phasor plot of the fundamental reference x\, the dividend Z2,
and the divisor _y3 at the second phase reference [using
superscript (2 )].
88
Figure 9.4.
Phasor plot of the fundamental reference xi, the dividend Z2,
and the divisor y 3 at the third phase reference [using
superscript (3)].
89
Figure 12.1.
Circuit diagram of a pHEMT device operating in a two-port,
common-source configuration.
115
Figure 12.2.
Magnitude of
as a function of input power for a
nonlinear lumped element model of a 2x90 pm GaAs
pHEMT device operating at 5 GFIz and a bias of Vds - 3V
and VGS = 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V.
117
Figure 12.3.
Magnitude of # 21*1 as a function of input power for a
nonlinear lumped element model of a 2x90 pm GaAs
pHEMT device operating at 5 GHz and a bias of VDS = 3V
and Vgs - 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V.
119
xiv
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I
Figure 12.4.
Phase of £>m\ and
as a function of input power for a 120
nonlinear lumped element model of a 2x90 pm GaAs
pHEMT device operating at 5 GHz and a bias of VDS - 3V
and Vgs - 0.0, -0.4, and -1.0 V.
Figure 12.5.
Expanded power gain i as a function of input power for a
nonlinear lumped element model of a 2x90 pm GaAs
pHEMT device operating at 5 GHz and a bias of Vds - 3V
and VGS = 0.0, -0 .2 , -0.4, -0 .6 , -0 .8 , -1.0, and - 1.2 V.
121
Figure 12.6.
Expanded transducer gain C$r as a function of input power
for a nonlinear lumped element model of a 2x90 pm GaAs
pHEMT device operating at 5 GHz and a bias of VDS = 3V
and V g s = 0 .0 , -0 .2 , -0.4, -0 .6 , -0 .8 , -1.0, and - 1.2 V.
122
Figure 12.7.
The ratio ®i/(S as a function of power for a nonlinear
lumped-element model of a 2x90 pm GaAs pHEMT device
operating at 5 GHz and a bias of V d s ~ 3V and V g s = 0.0,
-0.2, -0.4, -0.6, -0.8, -1.0, and -1.2 V.
123
Figure 12.8.
The 1 dB gain compression point for a nonlinear lumped
element model of a 2x90 pm GaAs pHEMT device operating
at 5 GHz and a bias of Vds - 3V and Vgs - -0.4 V.
125
Figure 13.1.
Block diagram of a single-diode resistive doubler.
126
Figure 13.2.
Nonlinear large-signal ^-parameters used to characterize a
two-port device excited by a single-tone signal at port 1 .
127
Figure 13.3.
Final design of the single-diode resistive frequency doubler.
Electrical lengths shown are all at 1 GHz.
133
Figure 13.4.
Time-domain plots of a\ and 62 for the simulated 1-2 GHz
frequency-doubler circuit.
133
Figure 14.1.
Block diagram of a nonlinear vector network analyzer
equipped with a second source and isolators.
135
Figure 14.2.
An ANN model that maps real and imaginary values of o ’s
to b's for different real and imaginary values of amn
\{m±X)/\(n£V)\.
136
1
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1
1
Figure 14.3.
An ANN model that interpolates b’s from the measured
results for nonzero values of amn [(m^l)A(n^l)] to the
desired values for amn [(m^l)A(n^l)] equal to zero. Outputs
of the ANN model yield values of &iuti-
136
Figure 14.4.
Schottky diode in a series configuration located in the middle
of a CPW transmission line. (White area is metal.)
138
Figure 14.5.
Five hundred measurements of a2\ in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
138
Figure 14.6.
Five hundred measurements of an in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
139
Figure 14.7.
Five hundred measurements of au in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
139
Figure 14.8.
Five hundred measurements of a 22 in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
140
Figure 14.9.
Five hundred measurements of a22i in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
140
Figure 14.10. Five hundred measurements of b\\ in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
141
Figure 14.11. Five hundred measurements of b2\ in the complex plane with
the excitation from source 1 held constant and the output
from source 2 set to random phases with constant amplitude.
141
Figure 14.12. Average testing errors as functions of the number of hidden
neurons for ANN models trained to map aj and a2 to bj and
ai and a 2 to b 2. The models were developed using 250
training points and verified using 250 testing points.
146
j
j
!
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Figure 14.13. Average testing errors as functions of the number of training
points for ANN models trained to map and &2 to bi and sl\
and a 2 to b 2. The models were developed using 14 hidden
neurons and verified using 250 testing points.
146
Figure 14.14. Average testing errors as functions of the number of testing
points for ANN models trained to map ai and a 2 to bi and ai
and a2 to b 2. The models were developed using 14 hidden
neurons and 250 training points.
147
The 250 measurements of bn used for training (circles).
Values of B in r^ n were determined from the measurementbased ANN model (square) and the harmonic balance
simulation using a compact model (triangle).
149
Figure 14.16. The 250 measurements of bn used for training (circles).
Values of B>2in-tfn were determined from the measurementbased ANN model (square) and the harmonic balance
simulation using a compact model (triangle).
149
Figure 15.1.
Gkpij and Hkpy serve to map ay circles centered at zero to bkP
ellipses with variable axes also centered at zero, neglecting
FkP for illustrative purposes.
152
Figure 15.2.
If B y/ is a complex constant for a given bias and
fundamental drive level, it has the limitation that it can only
map circles into circles.
157
Figure 14.15.
X V ll
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I
LIST OF TABLES
Table 3.1.
Error terms for the twelve-term model.
28
Table 5.1.
Measured DC resistances of the five load terminations.
51
Table 9.1.
Determining the ratios of Z2 to y?, using three methods at
three different phase references.
87
Table 13.1
Simulated values for & im - §>n6i, &2111 ~ &2i6i> ®2, and
(g2/(& for each of the design stages of the diode frequency
doubler.
129
Table 14.1.
Average testing errors and correlation coefficients as
functions of the number of hidden neurons for ANN models
trained to map values from the first five harmonics of ai and
a 2 to the first five harmonics of bi. All models were
developed using 250 training points and verified using 250
testing points.
144
Table 14.2.
Average testing errors and correlation coefficients as
functions of the number of training points for ANN models
trained to map values from the first five harmonics of aj and
a 2 to the first five harmonics of bi. All models were
developed using 14 hidden neurons and verified using 250
testing points.
144
Table 14.3.
Average testing errors and correlation coefficients as
functions of the number of testing points for ANN models
trained to map values from the first five harmonics of ai and
a 2 to the first five harmonics of bi. All models were
developed using 250 training points and 14 hidden neurons.
144
Table 14.4.
Average testing errors and correlation coefficients as
functions of the number of hidden neurons for ANN models
trained to map values from the first five harmonics of ai and
a 2 to the first five harmonics of b 2. All models were
developed using 250 training points and verified using 250
testing points.
145
xvm
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Table 14.5.
Average testing errors and correlation coefficients as
functions of the number of training points for ANN models
trained to map values from the first five harmonics of ai and
a 2 to the first five harmonics of b 2- All models were
developed using 14 hidden neurons and verified using 250
testing points.
145
Table 14.6.
Average testing errors and correlation coefficients as
functions of the number of testing points for ANN models
trained to map values from the first five harmonics of ai and
a 2 to the first five harmonics of b 2. All models were
developed using 250 training points and 14 hidden neurons.
145
Table 14.7.
Differences between the measurement-based, ANN-modeled
results and the compact model simulated in commercial
harmonic-balance software.
148
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1
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I
CHAPTER 1
INTRODUCTION
This thesis is comprised of two parts. The first segment covers artificial neural
network (ANN) modeling for improved vector network analyzer (VNA) calibrations.
The second segment covers frequency-domain models and figures of merit for
nonlinear circuits based on large-signal measurements. In this chapter, an introduction
and motivation is provided for each part of the thesis. Following that, the organization
of the thesis is presented along with a brief summary of each chapter.
1.1. ANN Modeling for Improved Vector Network Analyzer Calibrations
Vector network analyzers (VNAs) are one of the most versatile instruments in
the RF and microwave industry. They can be found in calibration facilities, research
laboratories, design facilities, and on production lines. VNAs are used to measure
complex scattering parameters of devices and circuits. Engineers use them to verify
their designs, confirm proper performance, and diagnose failures. The accuracy of a
VNA measurement is highly dependent on its calibration, which accounts for
imperfections in the instrument such as impedance mismatch, loss in the cables and
connectors, the frequency response of the source and receiver, and directivity and
crosstalk due to signal leakage. There are a wide variety of calibration methods
available to VNA users, most of which can be classified into one of two categories
depending on the type of calibration standards used.
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The first category makes use of transmission lines as standards, and includes
such calibration methods as thru-reflect-line (TRL) [1-2] and multiline TRL [3-4],
Multiline TRL is the most accurate means of VNA calibration and is especially useful
for on-wafer environments, since the characteristic impedance can be calculated from
dimensional measurements of the standards, which consist simply of a number of
transmission lines of varying line lengths and a reflective termination. In an on-wafer
environment, the disadvantages of this method are that it requires a lot of real estate
on the wafer, due to the numerous long lines required for an accurate calibration, and
the different lengths of line necessitate changing the separation between probes
during the calibration process. In a coaxial environment, the disadvantages are that a
large number of expensive airline standards and numerous interconnects are required.
Consequently, a second category of VNA calibrations, which makes use of
compact, lumped-element standards, is often preferred. The most common of these
calibrations are the open-short-load-thru (OSLT) and the line-reflect-match (LRM)
[5] methods. The trade-off is that these methods tend to be less accurate, since it is
more difficult to calculate the reflection coefficients of the standards from
independent physical measurements. But, if the compact calibration kits are
characterized using a benchmark calibration, such as multiline TRL, it is possible to
perform an accurate lumped-element calibration.
Once the lumped-element standards for a given calibration kit are
characterized, one must decide whether to develop a model for each of the standards
or to directly use the measurement data obtained from the benchmark calibration. In
the first part of this thesis, ANNs are applied to improve the modeling of lumped-
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element standards in both on-wafer and coaxial environments. As a major research
result, this doctoral thesis shows that ANN models offer a number of advantages over
the use of calibrated-measurement data files and equivalent circuit models, namely,
the following: ( 1) they do not require the numerous details and parameters of physical
models; (2 ) calibration times can be reduced since only a few training points are
required to accurately model the standards; (3) ANN model descriptions are much
more compact than large measurement files; (4) ANN models, trained on only a few
measurement points, can be much more accurate than direct calibrations when limited
data are available; and (5) they give an optimized estimate in the presence of noise.
Specifically, this thesis describes how on-wafer OSLT standards have
successfully been modeled, in one case assuming the standards can be reproduced
from wafer to wafer with little variation, and in another case where the loads exhibit
significant difference among the wafers studied. This thesis also describes how load
standards have been modeled to improve both on-wafer and coaxial LRM
calibrations.
1.2. Frequency-Domain Models and Figures of Merits for Nonlinear Circuits
Although the measurement of ^-parameters using a vector network analyzer
(VNA) is invaluable to the microwave designer of linear circuits, such measurements
are oftentimes inadequate for nonlinear circuits since nonlinearities transfer energy
from the stimulus frequency to products at new frequencies. Thus, a different and
more sophisticated instrument is required to measure nonlinear devices and circuits.
A recently introduced nonlinear vector network analyzer (NVNA) is capable of
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providing accurate waveform vectors by acquiring and correcting the magnitude and
phase relationships between the fundamental and harmonic components in the
periodic signals [6-10]. An NVNA excites a nonlinear device under test (DUT) with
one or more sine-wave signals and detects the response of the DUT at its signal ports.
Assuming the DUT exhibits neither sub-harmonic nor chaotic behavior, the input and
output signals will be combinations of sine-wave signals, due to the nonlinearity of
the DUT combined with mismatches between the system and the DUT.
Even though ^-parameters cannot adequately represent nonlinear circuits,
some type of parameters relating incident and reflected signals are beneficial so that
|
the designer can “see” application-specific engineering figures of merit that are
similar to what he or she is accustomed to. Another major result of this thesis is the
proposal of definitions of such ratios that are referred to as nonlinear large-signal
scattering (# ) parameters. First, a general form of time-invariant nonlinear large-
|
signal ^-parameters is presented. Then nonlinear large-signal impedance (Z) and
j
admittance (fl) parameters are introduced, and equations for relating the different
representations are derived. It is important to note that these parameters are developed
for sparse tone inputs, rather than wideband modulation.
j
In addition to nonlinear large-signal ^-parameters, another figure of merit for
]
s
|
nonlinear circuits is introduced by expanding the definitions of power gain,
j
transducer gain, and available gain by taking harmonic content into account. Under
j
special conditions, these expanded definitions of gain can be expressed in terms of
j
\
i
!
two-port, nonlinear large-signal ^-parameters.
4
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For existing nonlinear models, nonlinear large-signal ^-parameters can
readily be generated by performing a harmonic balance simulation. For devices, with
no model available, these parameters can be extracted from artificial neural network
(ANN) models that are trained with multiple frequency-domain measurements made
on a nonlinear DUT with an NVNA.
To illustrate applications and generation of nonlinear large-signal
parameters, three examples are presented. First, nonlinear large-signal ^-parameters
and the expanded definitions of gain are shown to be useful in discovering valuable
information regarding the behavior of a nonlinear model. Specifically, a lumpedelement model of a pseudomorphic high electron mobility transistor (pHEMT) device
operating in a two-port, common-source configuration is examined. Second, this
thesis illustrates how nonlinear large-signal ^-parameters can be used as a tool in the
design process of a nonlinear circuit, specifically a single-diode 1-2 GHz frequencydoubler. And finally, a method for generating nonlinear large-signal ^-parameters
based upon ANN models trained on frequency-domain data measured using a
nonlinear vector network analyzer (NVNA) is described. A diode circuit model,
generated using this method, is compared to a harmonic balance simulation of a
commercial device model.
In both parts of this thesis, when measurement-based ANNs are utilized, not
only are demonstrations presented to show their usefulness, but specific procedures
are outlined for determining optimal models.
|
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8
1.3. Organization of the Thesis
In Chapter 2, an overview of artificial neural networks is presented since they
are common to both parts of this thesis. This chapter covers ANN structure, with
special emphasis on the multi-layer perceptron architecture, model development, and
model verification.
The first part of the thesis, which deals with ANN modeling for improved
vector network analyzer calibrations, is covered in Chapters 3-7. In Chapter 3, an
overview of linear network analysis is presented, which includes an introduction to
scattering parameters, VNA architecture and error models, calibration techniques, and
motivations for using ANNs to model calibration standards.
In Chapter 4, ANNs are applied to improve the modeling of on-wafer OSLT
standards. The ANNs are trained with measurement data obtained from a benchmark
calibration. The training errors and training times are quantified as functions of both
the number of training points and the number of neurons in the hidden layer. The
OSLT calibration using the ANN-modeled standards compares favorably (less than a
0.02 difference in magnitude) to the benchmark calibration over a 40 GHz bandwidth.
The assumption made in Chapter 4 is that the standards can be reproduced
from wafer to wafer with little variation. One study, however, found that while open,
short, and thru standards can be reproduced with minimal variance, load standards
exhibit a significant difference among wafers. So in Chapter 5, it is demonstrated that
on-wafer OSLT calibrations can be further improved by applying ANNs to model the
correlation between measured DC resistance and RF variations in load terminations.
Furthermore, ANN models are shown to offer a number of advantages over using
6
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calibrated measurement files or equivalent circuit models, including ease of use,
reduced calibration times, and compactness.
In Chapter 6 , a load is modeled using an ANN to improve an on-wafer LRM
calibration. The accuracy of the LRM calibration using the ANN-modeled load
compares favorably to a benchmark calibration with an average worst-case scattering
parameter error bound of 0.017 over a 40-GHz frequency range.
In Chapter 7, a load is modeled using an ANN to improve a coaxial LRM
calibration and improve its accuracy over an earlier reported equivalent circuit model.
This chapter shows that an ANN model allows one to develop a compact description
of the standard without having to formulate a detailed physical model.
The second part of the thesis, which deals with frequency-domain models and
figures of merit for nonlinear circuits based on large-signal measurements, is covered
in Chapters 8-15. Chapter 8 compares linear and nonlinear network analysis. A brief
overview of nonlinear vector network analyzers (NVNAs) is presented, which
includes instrument architecture, error models, and calibration techniques. Finally,
motivations are presented for introducing application-specific figures of merit such as
nonlinear large-signal scattering parameters and expanded definitions of gain.
In Chapter 9, a method is presented for preserving time-invariant phase
relationships when ratios are taken between two harmonically related, complex
signals by introducing a third signal that is used as a phase reference. This technique
shows that a reference signal must be present at the fundamental frequency in order
for time-invariant phase relationships to exist between ratios of any two harmonically
related signals. A simple example is provided to illustrate the technique. This method
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must be implemented when defining the time-invariant, nonlinear large-signal
scattering parameters in the following chapter.
In Chapter 10, the concept of nonlinear large-signal scattering parameters is
introduced. Like commonly used linear 5-parameters, nonlinear large-signal
scattering (# ) parameters can also be expressed as ratios of incident and reflected
wave variables. However, unlike linear 5-parameters, nonlinear large-signal f&parameters depend upon the signal magnitude and must take into account the
harmonic content of the input and output signals since energy can be transferred to
other frequencies in a nonlinear device. After presenting the general form of
nonlinear large-signal ^-parameters, nonlinear large-signal impedance (Z) and
admittance (|f) parameters are also presented, and equations for relating the different
representations are derived. Next, two simplifications are made, considering the cases
of a one-port network with a single-tone excitation and a two-port network with a
single-tone excitation.
In Chapter 11, the definitions of power gain, transducer gain, and available
gain are expanded by taking harmonic content into account. Under special conditions,
these expanded definitions of gain can be expressed in terms of the two-port,
nonlinear large-signal scattering parameters defined in the previous chapter.
In Chapter 12, an example is provided showing how nonlinear large-signal
scattering parameters and the expanded definitions o f gain, introduced in the previous
chapters, can be used to discover valuable information regarding the behavior of a
nonlinear model. Specifically, a lumped-element model of a pseudomorphic high
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electron mobility transistor (pHEMT) device operating in a two-port, common-source
configuration is examined.
Chapter 13 illustrates how nonlinear large-signal scattering parameters can be
used to as a tool in the design process of a simple nonlinear circuit, specifically a
single-diode 1-2 GHz frequency-doubler circuit.
In Chapter 14, an approach is described for generating ANN models for
nonlinear devices, based upon frequency-domain data measured using a nonlinear
vector network analyzer (NVNA). Specifically, models are developed for a Schottky
diode in a series configuration. This chapter demonstrate that the ANN models give
accurate descriptions of the input-output relationships of the device over the span of
the measurements. A sensitivity analysis is performed to determine how many
training points, testing points, and hidden neurons are required to adequately train the
ANN models. The models are also used to extract nonlinear large-signal scattering
parameters using appropriate ratios of wave variables measured under large-signal
conditions. An independent check is obtained by comparing the diode-circuit models,
generated by means of this methodology, to a compact model simulated in
commercial harmonic-balance software.
In Chapter 15, nonlinear large-signal scattering parameters are compared to
another form of nonlinear mapping, known as nonlinear scattering functions [11, 12].
The two formulations are shown not to be equivalent. Specifically, nonlinear largesignal scattering parameters are more general than nonlinear scattering functions,
which are useful in approximating a specific class of nonlinearity in a more compact
form.
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Finally, Chapter 16 provides a summary of the thesis with concluding remarks
and directions for future research.
10
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CHAPTER 2
ARTIFICIAL NEURAL NETWORKS
Artificial neural networks (ANNs) are neuroscience-inspired computational
tools that are trained using input-output data to generate a desired mapping from an
input stimulus to the targeted output. ANNs consist of multiple layers of processing
elements called neurons. Each neuron is linked to other neurons in neighboring layers
by varying coefficients that represent the strengths of these connections. Learning is
accomplished by adjusting these coefficients until the network provides output results
that meet prescribed values.
ANNs have been applied to diverse areas such as speech and pattern
recognition, financial and economic forecasting, telecommunications, and nuclear
power plant diagnosis, and have just recently been introduced into the area of
microwave engineering [13]. In particular, researchers have successfully used ANNs
to model microstrip vias [14], packaging and interconnects [15], spiral inductors [16],
MESFET devices [17], CPW circuit components [18], effective dielectric constant of
microstrip lines [19], and HBT amplifiers [20], to name just a few.
ANNs offer several advantages over other modeling methods [13], First, they
tend to be more accurate than polynomial regression models and allow for more
dimensions than look-up tables [21]. Second, prior knowledge about the input-output
mapping is not required to develop an ANN model. Relationships between input and
output data are developed from the training process. Third, ANNs can generalize,
which means that they can provide the correct output for inputs that have not been
11
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I
used to train the model. And fourth, ANNs have the ability to model both linear and
nonlinear input-output mappings.
2.1. Neural Network Structure
A neural network has two types of components, the processing elements
known as neurons and the connections between them known as links. Each neuron
receives stimulus from connected neighboring neurons, processes the information,
and produces an output. Input neurons receive stimulus from outside of the network,
hidden neurons receive stimuli from other neurons and send stimuli to other neurons
in the network, and output neurons produce stimuli that are used externally.
A variety of neural network architectures have been developed for numerous
applications. O f these, multi-layer perceptrons (MLP) are the most popular type in
use today. They belong to a class of structures known as feed-forward neural
networks and are capable of approximating generic classes of functions [22], In an
MLP network, neurons are grouped into layers, namely an input layer, one or more
hidden layers, and an output layer. Figure 2.1 illustrates a three-layer network
(MLP3).
One of the reasons that MLP3 networks are so widely used is due to the proof
of the universal approximation theorem for MLP networks, which in summary states
that there always exists a three-layer perceptron that can approximate an arbitrary,
continuous, multidimensional function with any desired accuracy [23-24]. From this
theorem, we can conclude that any errors in an MLP3 neural model can be attributed
12
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to too many or too few neurons, inadequate training, or the presence of a stochastic
relationship between inputs and outputs.
In an MLP3 network, each neuron, except those in the input layer, receive and
process stimuli from other neurons. Each input is multiplied by a corresponding
weight parameter w, and the resulting products are added to produce a weighted sum,
which is passed through a neuron activation function g to produce the final output of
the neuron. The output can then become a stimulus for neurons in the next layer.
z,
/
Y
1
/
Output Layer
Hidden Layer
Input Layer
w \\
W
//
*
—* g(Uj) Zj
"
11
WKi\\
Kj \ \ —►
v = t g(vK)
*
k=l,2,...,K
Figure 2.1. MLP3 artificial neural network architecture.
13
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K
Analytical functions corresponding to the network of Figure 2.1 can be written
down as follows [25], The input of the y'th hidden neuron is obtained by forming a
weighted linear combination of the I input variables x, and the corresponding weights
w(l)i to give
Uj
= X w^Xf
(2.1)
i=0
where w(1)7o is the additional bias input to the jth neuron. The output zj of the y'th
hidden neuron is obtained by transforming the linear sum of eq.(2.1)
using the
activation function of the neuron g to give
zj = g ( U j ) .
(2.2)
The network outputs are obtained in a similar manner. For each output neuron k, a
weighted linear combination of the outputs of the hidden layer neurons is formed to
give
V* = Z
•.
( 2 .3 )
7=0
The output y>k of the Mi output neuron is obtained by transforming the linear sum of
eq. (2.3) using the activation function of the neuron g to give
y k = g ( vk )•
Combining eqs. (2.1)-(2.4), the kth output of the network is
14
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(2-4)
yk=g
^
j
=0
(2)
/
T
(1)
1
2
\
w )/X i
(2.5)
Ki=0
Equation (2.5) can be written in matrix-vector form as
Y = g[W2 #g(W 1 *X>]
(2.6)
where X is the input vector, Y is the output vector, and Wi and W 2 are the weight
matrices between the input and hidden layers and between the hidden and output
layers, respectively.
The most widely used activation function is the sigmoidal function given by
g(«) =
1
1+ exp(-w)
(2.7)
and is shown in Figure 2.2. The advantages of this activation function are that it is
bounded between 0 and 1, it is continuous, monotonic, and continuously
differentiable.
1 .0 - 1
0 .8 0.6 0.4 >
r2 _
-10
Figure 2.2. Sigmoidal activation function used in hidden and output layer neurons.
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2.2. Model Development
The first step in developing a neural network is the generation of training and
testing data, which may be obtained from either simulation or measurement. Training
data are used for model development while testing data are used for model validation.
Before an ANN model is developed, the training data, both inputs and outputs,
are usually re-scaled. This is especially useful when different variables have values
that significantly vary. With linear re-scaling, each variable is assigned the same
importance for model development resulting in weights that should not be
substantially different. Linear scaling, shown in Figure 2.3 is given by
X = im in + - * ~ ^ nn_
■*max
( ; max
<
( 2 .8 )
x min
Model training is accomplished by adjusting the weights of the network by
comparing the outputs of the network to the target output variables. At the beginning
of training, the weights are initialized to small random values. The inputs are passed
through the network and the outputs are computed. The output values are compared to
target values and the derivatives of the error between them with respect to each of the
weights are calculated. Weights are then adjusted based upon the contribution to the
error such that the overall network error decreases. This procedure continues in an
iterative fashion until the overall error of the network is reduced to a prescribed level.
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I
X
max
mm
X
Xmax
Xmm
,
Figure 2.3. Linear scaling of neural network data.
One of the most popular algorithms for neural network training is back
propagation [26]. This refers to the propagation of errors backward through the
network to find the error derivatives with respect to each weight. During training, the
neural network performance is evaluated by computing the difference between
outputs of the network y* to the target values tk for all N of the training samples. The
difference E, also known as the error, is given by
e
= \ X Z b 'b . - t h , ? .
(2.9)
1 n=\k=1
Derivatives of the error with respect to each weight are computed, and the
weights w are updated along the negative direction of the gradient of E as
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I
w = w - 77^— ,
aw
(2.10)
until E becomes small enough. Here, the parameter T| is called the learning rate.
Although back propagation is the most popular training algorithm, it is
relatively slow in converging, so second-order, training algorithms such as conjugategradient and Quasi-Newton are oftentimes preferred for their increased efficiency
[13]. The Quasi-Newton approach is relatively fast due to its quadratic convergence
property, although more computer memory and implementation effort is required
since it relies on the Hessian matrix and its inverse to be calculated. The conjugategradient method is a nice compromise both in terms of memory requirements and
implementation effort, since the descent direction runs along the conjugate direction,
which can be accomplished without matrix computations.
It is important to point out that there are potential hazards when training an
ANN. One possibility is over-learning, which means that the network matches the
training data but cannot generalize well. This can occur when there are insufficient
training data or too many hidden neurons. Another possibility is under-learning,
which means the network has not learned the input-output relationship well and
cannot even match the training data. Reasons for this include insufficient hidden
neurons, inadequate training, or the training procedure being stuck on a local
minimum. Good-learning is when the network matches the training and testing data
well. Figure 2.4 illustrates the three cases.
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10 -I
6
-
42
■
□
-
4
6
Training D ata
T estin g D ata
ANN
10
8
12
(a)
lO -i
□□
6-
4-
□ □
2
4
6
■
□
Training D ata
T esting D ata
ANN
10
8
12
(b)
6-
42
■
□
-
0
2
4
6
8
Training D ata
T esting D ata
ANN
10
12
(c)
Figure 2.4. Three ANN models illustrating (a) over-learning, (b) under-learning, and
(c) good-leaming.
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I
2.3. Model Verification
Once a neural network model is trained, its quality should be tested with an
independent set of testing data. There are a number of measures that can be used to
indicate the quality of a model, a few of which are discussed below.
The relative test error S« for the Mi output of a neural model at the /th data
sample is defined as
S ., =
thl.......
tk ,m a x
k = l t ' " t K . j e Te
(2.11)
*knmin
where Te is the set of test data. The average test error 8aVg is given by
~
I sfel
_ k = lle T e
avg " Sizc(Te)K
( 2. 12)
The worst-case test error 8w-c is calculated from
K
? w -c
= max max \ou
(2.13)
k=1 /e L
Another measure that indicates the correlation between a neural model and test data is
the correlation coefficient p and it is defined as
H £ (ykl - Tavg \^kl ~ ^avg )
k=lleT„
K
K
Y Y {ykl
Y Y {fkl t'dvg)
T avg)
k - \ 1&TP
k=\l<aTP
(2.14)
where
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K
(2.15)
7av8= Siz« T e)K
and
1
' a ,6 = Sizt*Te)K {1 u -t
(2.16)
In the following portions of this thesis, ANNs will be shown to be a powerful
tool for developing measurement-based, frequency-domain models of VNA
calibration standards and nonlinear devices.
!
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3
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CHAPTER 3
LINEAR NETWORK ANALYSIS
This chapter reviews the basic concepts of linear network analysis as applied
to microwave frequencies, which in its most fundamental form involves the
measurement of incident, reflected, and transmitted waves that travel along
waveguides. Since it is difficult, if not impossible, to measure total voltage and
current at microwave frequencies, scattering parameters are usually measured. These
parameters relate to familiar concepts such as reflection coefficient, insertion loss,
and gain. Aside from the physical interpretations, scattering parameters are a
convenient way to represent linear networks for a number of reasons, namely: they
can be cascaded for multiple devices to predict system performance; they can be
converted into impedance and admittance parameters; and they are conveniently used
with computer-aided engineering circuit simulation software. Finally, scattering
parameter ratios enable measurements to be made that are independent of absolute
power. In the following sections, we briefly review scattering parameter definitions,
VNA architecture and error models, and a discuss some of the most popular types of
VNA calibration techniques.
3.1. Scattering Parameters
When an electric circuit operates at low frequencies, its size is relatively small
compared to the wavelength and thus it may be treated as an interconnection of
lumped components with unique voltages and currents defined at any point in the
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network. This leads to the formulation of impedance and admittance parameters.
However, when an electric circuit operates at microwave frequencies, its size can be
much larger than the wavelength and thus voltages and currents may not be welldefined. In this case, normalized wave variables (a and b) are more suitable for
characterizing circuits at microwave (and higher) frequencies. Relationships between
a’s and b’s for a linear circuit can expressed by a scattering matrix whose elements
are known as scattering parameters, or ^-parameters [27-30].
The number of ^-parameters for a given device is equal to the square of the
number of ports. For example, a two-port device has four 5-parameters. Each Sparameter Sy has two subscripts, i and y, where i refers to the port at which power
emerges and j refers to the port number at which power enters. When i and j are
identical, Su indicates a reflection coefficient, and when i and j are different, Sy
indicates a transmission coefficient.
Consider an JV-port network. Normalized wave variables aj and bj at the y'th
port are proportional to the incoming and outgoing waves, respectively, and may be
defined in terms of the voltages associated with these waves as follows:
aJ
V1
V7
*J =
(3.1)
Z 0J
where V'j and Vj represent voltages associated with the incoming and outgoing waves
in the transmission lines connected to the y'th port, and Z oJ is the characteristic
impedance of the line at the y'th port.
The scattering matrix S of the network expresses the relationship
between a’s and b’s at various ports through the matrix equation
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1
(3.2)
b = Sa,
where b and a are iV-element column vectors and S is an (/Vx/V)-eleinent square
matrix. For a two-port network, eq. (3.2) becomes
CN
1 1
i1 i
<>T
V "A
A. A
(3.3)
The ^-parameters along the diagonal of the scattering matrix are reflection
coefficients defined as
S jj
bJ
VJ
a J a,= 0 (i* j)
V/ '
(3.4)
Off-diagonal terms o f the scattering matrix are transmission coefficients defined as
VflJZoi
auj ai = Q { i * j )
•
(3.5)
V ;/Jz~
Sy may be determined by finding bt for a value aj (power input to port j) with incident
wave variables at all other ports equal to zero. Figure 3.1 illustrates this for a two-port
network.
I
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Forward
Transmitted
Incident
a
.
DUT
a 2 ~
Reflected
0
Reflected b, .
Su =------------ = — a, =0
Incident ax
S21 -
Transmitted b0 ,
=
a2=0
Incident
ax
Reverse
Reflected b2 ,
S22 = ------------ =
a\ Incident a2
o
Transmitted b, .
512 --------------- = —
Incident
a2
=0
Reflected
a1
b
1
DUT
Clr
Transmitted
Incident
Figure 3.1. Determining two-port scattering parameters.
It should be noted that all of the normalized wave variables, characteristic
impedances, and S-parameters are generally complex values. Making use of
magnitude and phase is required to fully characterize a linear network. Furthermore,
complex impedance must be measured in order to design efficient matching networks.
Additionally, the transformation from the frequency-domain to the time-domain
requires magnitude and phase information in order to perform an inverse Fourier
transform.
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I
3.2. Vector Network Analyzer Architecture
A vector network analyzer (VNA) is an instrument that is used to measure
complex 5-parameters [31-37], It does this by sampling the incident and reflected
waves at both ports of a device under test (DUT), and then forming ratios that are
directly related to the reflection and transmission coefficients of the device.
Frequency is swept to obtain 5-parameters over a band of interest.
Most commercial VNAs make use of either three-sampler or four-sampler
architectures. High-end VNAs often use four samplers, while less expensive VNAs
employ three samplers [4, 38].
Figure 3.2 is a simple operational schematic of a four-sampler VNA.
Directional couplers behind each test port can sample the incident and reflected
waves with the source switched to either port while the other port is terminated by a
nominally ideal load. Typically, four-sampler VNAs do not make use of all four data
channels. Rather, only the transmitted signal, and not the reflected signal, is sampled
on the terminating side. This limits the cost of the electronics and increases sweep
speed. Neglecting the reflected signal introduces additional error in the measurement.
However, this error can be corrected provided that the load reflection coefficients are
repeatable and can be measured at least once.
Figure 3.3 is a simple operational schematic of a three-sampler VNA.
Directional couplers behind each test port can sample the reflected waves with the
source switched to either port while the other port is terminated by a nominally ideal
load. A third directional coupler samples the incident wave independent of the switch
position.
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I
DUT
Figure 3.2. Four-sampler vector network analyzer.
VW
DUT
Figure 3.3. Three-sampler vector network analyzer.
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3.3. Vector Network Analyzer Error Models
When using a VNA for microwave measurements, calibration is required
regardless of the architecture of the instrument. Calibrations account for serious
systematic imperfections in the VNA such as impedance mismatch, loss in the cables
and connectors, the frequency response of the source and receiver, and directivity and
cross talk due to signal leakage. That is not to say that calibrations account for all of
the possible sources of error. Calibrations do not account for system drift,
repeatability in the switches and connectors, instrument noise, or errors in the
calibration standards.
The most popular way to describe a VNA’s systematic imperfections is the
twelve-term error model, as shown in Figure 3.4 [39-40]. There are 6 error terms for
the forward direction (Su and S21) and 6 error terms for the reverse direction (Sn and
£ 22) • These 12 terms are defined in Table 3.1.
Table 3.1. Error terms for the twelve-term model.
Error Term
eoo
en
^10^01
£30
@22
£10^32
e ’33
e ’n
e 23 e 32
£03
e ’22
e 23 e ’01
Description
Directivity
Port 1 Match
Reflection Tracking
Leakage
Port 2 Match
Transmission Tracking
Directivity
Port 1 Match
Reflection Tracking
Leakage
Port 2 Match
Transmission Tracking
28
with permission of the copyright owner. Further reproduction prohibited without permission.
F orw ard
"30
------------------------ ► -----------------------e 1 0 e 32
IP
p
1
*
^
b 2
° l 4%P
F
1
^
^00
*1
ki
^
^
II
S 21
i9
^
S 22
r 5 U
i i
^11
e 10 e 01
W
19
S n
|
9
^
i |
a 2
D U T
* 1
R everse
*
iW
^
^22
b\
a
A’,
e 23 e 32
J 21
a
a
-
e
23 e
01
U
T
22
-
b \
D U T
22
® 33 A
1
— ^ ----
S’, 2
4
^
« 2
03
Figure 3.4. Twelve-term error model for vector network analyzers.
Solving the twelve-term error model signal flow graph gives 4 equations
relating the corrected 5-parameters to the uncorrected, measured 5-parameters via the
12 error terms.
Another common error model is the eight-term model, shown in Figure 3.5. In
this model, all of the errors are accounted for by a pair of “error box” matrices on
either side of the DUT, but requires imperfect switch conditions to be satisfied. The
eight-term model and twelve-term models can be derived from one another [40], but
that discussion is beyond the scope of this chapter.
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a«
o --------► -------- \ >----------------------------------- ► -----*10
V em
eu i L
*01
V
pP
^
^
P
Sn
i
a%
IP
p
1
VI
p
*23
V e 33
i ^12
a2
*^21
^22
P^
-
DUT
^
r
14 i
b2
e22 i {
*32
O -------- M — — i i------------------------------------ M --------
Figure 3.5. Eight-term error model for vector network analyzers.
3.4. Network Analyzer Calibration Techniques
There are a wide variety of calibration methods available to VNA users, most
of which can be classified into one of two groups depending on the type of calibration
standards used. Before we discuss the various calibration methods, it is worth
mentioning that the basic procedure for calibrating a VNA is the same, regardless of
what type of calibration is performed. First, known standards are measured. Next, the
measured data is processed to determine the error coefficients. And finally, the
measured data for a DUT is corrected using the error coefficients.
The first category of VNA calibrations makes use of transmission lines as
standards, and includes such calibration methods as thru-reflect-line (TRL) [1-2] and
multiline TRL [3-4]. Multiline TRL is the most accurate means of VNA calibration
30
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and is especially useful for on-wafer environments, since the characteristic impedance
can be calculated from dimensional measurements of the standards, which consist
simply of a number of transmission lines of varying line lengths and a reflective
termination. In an on-wafer environment, the disadvantages of this method are that it
requires a lot of real estate on the wafer, due to the numerous long lines required for
an accurate calibration, and the different lengths of line necessitate changing the
separation between probes during the calibration process. In a coaxial environment,
the disadvantages are that a large number of expensive airline standards and
numerous interconnects are required.
Consequently, a second category of VNA calibrations, which makes use of
compact, lumped-element standards, is often preferred. The most common of these
calibrations are the open-short-load-thru (OSLT) and the line-reflect-match (LRM)
[5, 41] methods. The trade-off is that these methods tend to be less accurate, since it
is more difficult to calculate the reflection coefficients of the standards from
independent physical measurements. But, if the compact calibration kits are
characterized using a benchmark calibration, such as multiline TRL, it is possible to
perform an accurate lumped-element calibration.
Figure 3.6 illustrates the calibration standards that are measured for the
previously mentioned techniques of OSLT, LRM, TRL, and multiline TRL.
!
31
j
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LRM Standards
OSLT Standards
Open
Thru
Short
Reflect
Load
Match
Thru
Multiline TRL Standards
Thru
TRL Standards
|------- j
Thru
Reflect 1------ ,
Reflect
Line 1
Line
Line 2
Figure 3.6.
calibrations.
,----
Standards measured for the OSLT, LRM, TRL, and multiline TRL
3.5. ANN Modeling for Improved VNA Calibrations
Assuming a compact calibration kit is preferred, once the lumped-element
standards are characterized, we must decide whether to develop a model for each of
the standards or to directly use the measurement data obtained from the benchmark
calibration. Recently, we have applied ANNs to improve the modeling of lumpedelement standards in both on-wafer and coaxial environments [42-45]. We have
shown that ANN models offer a number of advantages over the use of calibratedmeasurement data files and equivalent circuit models, namely, the following: (1) they
do not require the numerous details and parameters of physical models; (2) calibration
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
times can be reduced since only a few training points are required to accurately model
the standards; (3) ANN model descriptions are much more compact than large
measurement files; (4) ANN models, trained on only a few measurement points, can
be much more accurate than direct calibrations when limited data are available; and
(5) they give an optimized estimate in the presence of noise.
In the following four chapters, we summarize our work in this area, describing
how we have successfully modeled on-wafer OSLT standards, in one case assuming
the standards can be reproduced from wafer to wafer with little variation (Chapter 4),
and in another case where the loads exhibit significant difference among the wafers
studied (Chapter 5). We also describe how we modeled load standards to improve
both on-wafer and coaxial LRM calibrations (Chapters 6 and 7).
33
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C H A PTE R 4
ANN MODELING OF ON-WAFER OSLT STANDARDS
The OSLT calibration [5] is one of the most widely used techniques for
calibrating VNAs. It is mainly used with devices that contain coaxial or waveguide
interfaces, but is also often applied to on-wafer environments such as microstrip and
coplanar waveguide (CPW). The calibration procedure consists of a “thru”
connection of the two VNA ports as well as the measurement (on both ports) of three
one-port standards, typically a nominal open, a nominal short, and a nominally
matched load. None of these standards needs to be ideal, but we must know their
reflection coefficients. In practice, our definition of these reflection coefficient values
is typically drawn from a model of the standard.
Manufacturers of calibration kits typically provide a description of the
standards based on equivalent circuit parameters, known as Calibration Kit
Parameters [36, 46] or Calibration Component Coefficients [37]. These parameters
assume single, real values for both load impedance and characteristic impedance and
describe the open and short circuit terminations as frequency-polynomials of
capacitance and inductance, respectively. With coaxial and waveguide standards, the
equivalent circuit approximations have worked to the satisfaction of most users, but
for on-wafer standards, a recent study [47] reported errors in scattering parameters of
up to 0.5. Considering the maximum possible value for passive devices is a
magnitude of 1, such errors are clearly unacceptable. DeGroot et al. [48] recently
documented the OSLT models and developed a general description of transmission
34
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lines to express offset reflection standards and finite-length thru standards that
accounts for lossy environments with complex impedance. Implementing this more
general description, however, still requires physical models or measurement data of
each for the individual standards.
Here, we apply ANNs to improve the modeling of on-wafer OSLT standards
used for calibrating VNAs [42]. The ANNs are trained with measurement data
obtained from a benchmark multiline TRL calibration. We assess the accuracy of an
OSLT calibration using these ANN-modeled standards and find that it compares
favorably (less than a 0.02 difference in magnitude) to the benchmark multiline TRL
calibration over a 40 GHz bandwidth. We also quantify the training errors and
training times as a function of both the number of training points and the number of
neurons in the hidden layer.
4.1. Modeling the Standards
In this study, the OSLT and multiline TRL standards and devices were
constructed of CPW transmission lines fabricated with 1.5 jam thick gold conductors
evaporated on 500 jam thick semi-insulating GaAs [49]; the gold center conductor
was 73 |am wide and separated from the ground plane by 49 |am gaps. The five line
standards included a 0.55-mm thru line and four additional lines that were 2.135, 3.2,
6.565, and 19.695 mm longer. The load standard was fabricated by terminating a 275jam section of the CPW with a single 73 jam by 73 jam nickel-chromium thin-film
resistor. All of the standards were measured using on-wafer probes. The OSLT open
circuit was defined by lifting the probe heads off the wafer, as recommended by
35
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probe manufacturers. For each standard, we measured scattering parameters at 192
frequencies from 0.5 to 40 GHz.
Once the OSLT standards had been characterized using a multiline TRL
calibration, we determined how many neurons in the hidden layer were required to
develop accurate ANN models. Our first experiment was to vary the number of
neurons in the hidden layer for each of the standards. We started with 1 hidden
neuron, noting the training error reported by the software after training was
completed, and repeated this process while incrementing the number of hidden
neurons until we reached a total of 10 neurons. We performed this experiment on
each of the standards using all 192 measured frequencies as training data. Figure 4.1
illustrates the results for S\ i of the open, short, and load, and for S21 of the thru. Each
of the standards had different errors, but no discemable improvements could be seen
for more than 5 neurons. We also measured the training time for each standard while
varying the number of hidden neurons, and found that the training time varied linearly
with the number of hidden neurons. One hidden neuron required approximately 2
seconds of training time on the computer used, while 10 hidden neurons required
about 20 seconds of training time. These training times undoubtedly vary depending
on the speed of the computer, but the values give a relative idea of how much time is
required per hidden neuron.
After we decided that 5 hidden neurons were sufficient, we studied how many
training points were required to accurately model each standard. We trained each
standard using all 192 points, and then tried smaller subsets of the measurement
points, namely 3, 5, 9, and 41 points. After the models for each of the standards were
36
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I
trained for the five sets of data, we compared them to the measurement data. To our
surprise, we found that we could achieve good accuracy with as few as 9 training
points, and that as few as 5 training points were adequate for the open and short.
Figures 4.2 and 4.3 show the magnitudes of the vector differences of Sn (|ASn|)
between the measured data and the ANN models for various numbers of training
points for the open and load standards, respectively. From these two plots, we see that
the ANN model of the open standard agrees with measurement data to within 0.015
using as few as 5 training points, and that the ANN model of the load standard agrees
with measurement data to within 0.04 for most frequencies using as few as 9 training
points.
0 . 001 -
—ir - Short S u
—▼•••• Thru S2I
Open S u
•••■»■• Load S u
1
2
3
4
5
6
7
8
9
10
Neurons in Hidden Layer
Figure 4.1. Training error versus the number of neurons in the hidden layer for
various OSLT calibration standards.
37
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0 .16 -,
0 .1 4 -
0.12
0.10
-
------ ANN M odel - 192 pts.
------ANN M odel - 41 pts.
........ ANN M odel - 9 pts.
------ ANN M odel - 5 pts.
—
ANN M odel - 3 pts.
-
0 .0 8 -
0 .0 6 -
0.04-
0 .0 2 -
0 .0 0 -
0
20
30
40
Frequency (GHz)
Figure 4.2. Magnitude of the ANN-modeled reflection coefficient errors (|ASn|) for
the open standard with varying numbers of training points.
SO -,
40-
—
0
10
20
ANN M odel - 192 pts.
ANN M o d e l-4 1 pts.
ANN M odel - 9 p ts .
ANN M odel - 5 pts.
ANN M odel - 3 p ts .
30
40
Frequency (GHz)
Figure 4.3. Magnitude of the ANN-modeled reflection coefficient errors (|ASn|) for
the load standard with varying numbers of training points.
38
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Our observation that so few training points are sufficient to model our
standards highlights an important advantage in using ANN models over calibrated
measurement data files. We found that it is possible to cut down on calibration times
by measuring only a few frequency points and developing an ANN model rather than
measuring numerous points and carrying around large data files. The ANN model,
trained on only a few measurement points, can be much more accurate than linearly
interpolating, as is commonly done in practice. For example, if one were to measure
the load standard at 5 points and perform linear interpolation between frequencies, as
shown in Figure 4.4, the maximum error would be 0.045, as opposed to only 0.016
for the ANN model trained using the same 5 points.
Next, we developed ANN models for each of the OSLT standards using 5
hidden neurons and all 192 measured points, since we already had the data on hand.
Figures 4.5-4.7 show the magnitude and phase of Su using both measured and ANN
model data for the open, short, and load standards, respectively. Figure 4.8 shows the
magnitude and phase of S21 using the measured and ANN model data for the thru
standard. Notice that the ANN models for each standard follow the trends of the
measured data, but avoid the scatter of multiline TRL calibrated measurements.
Whether or not this scatter is real, we see that ANNs follow general trends but omit
the scatter, which is usually desirable in a model as long as the scatter is less than the
repeatability of the measurements.
>
39
with permission of the copyright owner. Further reproduction prohibited without permission.
I
0.14-,
lilik
0 . 12 -
0 . 10 0 .0 8 t/5
— TRL Calibration (192 points) - Reference
— TRL Calibration (5 points)
ANN Model (5 points)
0.060.0 4 0.0 2 -
0.00
0
10
20
Frequency (GHz)
30
40
10
20
Frequency (GHz)
30
40
-l.O-i
-1 .5 -
-
2. 0 -
-2 .5 -
-3.0
0
Figure 4.4. Comparison of magnitude and phase of the reflection coefficients [|Sn|
and Arg(Sn)]for the load standard using an ANN model trained with 5 points, linear
interpolation with TRL using the same 5 points, and TRL with 192 points as the
reference.
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.020
1.015-
TRL Calibration
ANN Model
1. 010 1.0051.0000.995 0.990
0
10
20
Frequency (GHz)
30
40
TRL Calibration
ANN Model
i
•S
e
0.8
0.6
0.4
0.2
0.0
0
10
20
Frequency (GHz)
30
40
Figure 4.5. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Si
for the open standard measured by multiline TRL and ANN modeling.
41
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1.020-,
1.015
TRL Calibration
ANN Model
1. 010 1-0051.0000.995
0.990
0
10
20
Frequency (GHz)
30
40
30
40
3.16
3.14
3.12
^1 3.10
X 3.08
czT
« 3 06
%
3.04
TRL Calibration
ANN Model
3.02
3.00
0
10
20
Frequency (GHz)
Figure 4.6. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Si
for the short standard measured by multiline TRL and ANN modeling.
42
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I
0.14-1
0 . 12 0 . 10 0.0 8 0.06TRL Calibration
ANN Model
0.0 4 0.0 2 -
0.00
0
10
20
Frequency (GFIz)
30
40
30
40
0 .0 - i
-0 .5 -1 .0 -
TRL Calibration
ANN Model
-1 .5 CZ3
|
-2 .0 -2 .5 -3.0
0
10
20
Frequency (GHz)
Figure 4.7. Magnitude and phase of the reflection coefficients [|Sn| and Arg(Sn)]
for the load standard measured by multiline TRL and ANN modeling.
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
l.OOlO-i
1.0005-
TRL Calibration
ANN Model
(N 1.0000
0.9995-
0.9990
0
10
20
Frequency (GHz)
30
40
30
40
800 -i
600400-
-
200
-
200
-
TRL Calibration
ANN Model
-600-800
0
10
20
Frequency (GHz)
Figure 4.8. Magnitude and phase of the transmission coefficients [|S2i| and Arg(S 2i)]
for the thru standard measured by multiline TRL and ANN modeling.
44
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In Figures 4.5 and 4.6, the measured magnitudes of the reflection coefficient
for the open and short standards are slightly greater than 1, which is not possible for
passive devices. This discrepancy can be attributed to random errors in the TRL
calibration, which are typically as high as 0.02 at 40 GHz. Our measurements never
exceed 1 by more than this repeatability error. A similar argument can be made for
the transmission coefficients of the thru standard.
4.2. Calibration Comparisons
We performed two OSLT calibrations, one using the calibrated measurement
data of the standards and the other using the ANN models of the standards. We
calibrated a 19-mm CPW transmission line; using both OSLT calibrations, and
compared the results to measurements calibrated directly using the benchmark
multiline TRL calibration. Figure 4.9 compares the magnitudes and phases of the
scattering parameter data [|Sn|, Arg(S'n), l&il, and ArgfS^i)] for all three calibrations.
The agreement is good.
To get a more quantitative idea of the differences, we plotted the maximum
vector differences of the scattering parameters (|ASij|) for the 19-mm line between the
two OSLT calibrations and the multiline TRL calibration. Figure 4.10 illustrates the
differences. Not surprisingly, the OSLT calibration, using the calibrated measurement
data, compares better to the multiline TRL calibration, since they both make use of
the same calibration data. However, the OSLT using the ANN models still compares
well with less than a 0.02 difference in magnitude at all frequencies. The difference
here does not necessarily mean that the OSLT, using ANN models, is in error. The
45
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i
differences could be due to the presence of noise in the TRL calibration that the ANN
models avoided. Regardless of the source of error, a 0.02 difference between two onwafer calibrations spanning 40 GHz is impressive, considering that the repeatability
between two multiline TRL calibrations is usually on the order of 0.015.
4 -i
0.95 n
OSLT (TRL)
OSLT (ANN)
TRL
0 .9 0 -
Kfi
U
20-
0 .8 0 0.75
0
10
20
30
0
40
10
20
30
40
30
40
Frequency (GHz)
Frequency (GHz)
1001
6 -1
80”0
60-
200
10
20
30
0
40
10
20
Frequency (GHz)
Frequency (GHz)
Figure 4.9. Magnitude and phase of the scattering parameters of a calibrated 19-mm
CPW transmission line.
4.3 Discussion
We have successfully applied ANNs to model on-wafer OSLT standards, and
shown that such a calibration compares favorably (less than a 0.02 difference in
magnitude) to the benchmark multiline TRL calibration. In modeling these standards,
we quantified the training errors and training times as a function of both the number
of training points and the number of neurons in the hidden layer. We found that 5
46
with permission of the copyright owner. Further reproduction prohibited without permission.
2 0 -1
OSLT (TRL) vs. TRL
OSLT (ANN) vs. TRL
15-
o
» 10-
0
10
20
30
40
Frequency (GHz)
Figure 4.10. Magnitude of the scattering parameter differences (|ASij|) of a calibrated
19-mm CPW transmission line.
neurons in the hidden layer of an MLP3 architecture and that fewer than 10 training
points were sufficient to accurately model our standards.
In practice, ANN-modeled calibration standards can be easily implemented
using existing or custom software packages. In our case, we utilized MultiCal [4], a
free program developed by the National Institute of Standards and Technology, to
perform our benchmark multiline TRL calibration. The internal software on any
commercial network analyzer can also be used if the user has confidence in another
calibration method such as single-line TRL or LRM (line-reflect-match). Then, once
the OSLT standards are measured, one of a number of ANN programs may be used to
model the standards. We used software developed by Zhang et al [50] to construct our
47
with permission of the copyright owner. Further reproduction prohibited without permission.
ANN models. Finally, a program that can perform OSLT calibrations using exported
ANN models is required. We wrote custom software to perform this task, using the
equations found in references [5] and [48] to perform the OSLT calibrations.
We have shown that ANN models offer a number of advantages over using
calibrated measurement data files or equivalent circuit models, namely:
(1) They do not depend on the numerous details and parameters of physical models.
(2) Calibration times can be reduced since only a few training points are required to
accurately model the standards.
(3) ANN model descriptions are much more compact than large measurement data
files.
(4) ANN models, trained on only a few measurement points, can be much more
accurate than direct calibrations, when limited calibration data is available.
(5) They give an optimized estimate in the presence of noise.
48
I\ \
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 5
MODELING LOAD VARIATIONS WITH ANNS TO
IMPROVE ON-WAFER OSLT CALIBRATIONS
The assumption made in the previous chapter was that the standards can be
reproduced from wafer to wafer with little variation. Kirby et al. [51] studied
variations in OSLT standards from wafer to wafer on a CPW calibration set designed
for GaAs substrates, and found that open, short, and thru standards can be reproduced
with minimal variance, but that load standards exhibit a significance difference
among the wafers they studied. Furthermore, they discovered that RF variations in the
load terminations correlate directly to their measured DC resistances.
Here, we demonstrate that on-wafer OSLT calibrations of VNAs can be
further improved by applying ANNs to model the correlation between DC resistance
and RF variations in load terminations [43]. The ANNs are trained with measurement
data obtained from a benchmark multiline TRL calibration. The open, short, and thru
standards do not vary significantly from wafer to wafer, so we also model these
standards using ANNs trained with calibrated measurement data chosen from an
arbitrary wafer. We assess the accuracy of five OSLT calibrations with varying load
terminations using the ANN-modeled standards, and find that they compare favorably
(a difference of less than 0.04 in magnitude at most frequencies) to the benchmark
multiline TRL calibration over a 66 GHz frequency range.
49
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5.1. Modeling the Standards
In this study, the OSLT and multiline TRL standards and devices were
constructed of CPW transmission lines fabricated from 4.5 (im plated gold on a 625
urn thick GaAs. The load terminations were composed of TiWN (titanium tungsten
nitride) thin film resistive material [51]. The four line standards included a thru line
and three additional lines that were 0.9552, 1.239, and 1.764 mm longer. All of the
standards were measured using on-wafer probes. For each standard, we measured
scattering parameters at 165 frequencies from 1 to 67 GHz.
Since the open, short, and thru standards did not vary significantly from wafer
to wafer we modeled these standards with ANNs using calibrated measurement data
chosen from an arbitrary wafer. The ANN architecture for the open, short, and thru
standards consisted of one input (frequency) and two outputs (the real and imaginary
components) for each measured scattering parameter. Since we measured reflection
coefficients for the two terminations at both ports and all four scattering parameters
of the thru connection, we ended up with eight ANN models, excluding the load.
From our previous study in [42], we determined that 5 neurons were sufficient for the
hidden layer. We trained each model of the standards using all 165 frequencies since
we already had the data on hand.
The ANN architecture for the load standards consisted of two inputs
(frequency and DC resistance) and two outputs (the real and imaginary components)
for the impedance parameters at each port. We were unable to generate one model
that included both ports due to a systematic difference between the load
measurements at port 1 and port 2, so we settled on separate models for each port.
50
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Ten neurons were chosen for the hidden layers since the ANN models for the loads
included an additional input compared to the other standards. The measured DC
resistances for the loads are listed in Table 5.1. For each port, we trained the models
using 3 of the 5 loads. We chose loads 1, 4, and 5 since load 1 had the lowest DC
resistance, load 5 had the highest, and load 4 had an intermediate value. It is
important to train ANNs at the expected boundary values of the input parameter space
in order to ensure good performance of the model [13]. By purposely not training the
ANN with loads 2 and 3, we could test how effective the model behaved at other DC
resistances. Figure 5.1 shows the real and imaginary components of port 1
impedance, Z\\, of both measured and ANN-modeled data for the 5 load standards.
Table 5.1. Measured DC resistances of the five load terminations.
Load
1
2
3
4
5
DC Resistance (Q)
Port 1
44.73
45.85
45.20
45.38
46.45
DC Resistance (Q)
Port 2
45.01
46.13
45.27
45.64
46.71
a
51
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5 2 -i
Load 1
Load 2
Load 3
Load 4
Load 5
50-
13.
? 48'
Pi
46'
44-
_l-------------- !-------------- 1------------ 1------------ 1------------ 1------------ 1
10
20
30
40
50
60
70
50
60
70
Frequency (GHz)
8 -|
6-
53
4 -
N
2-
0-1
10
20
30
40
Frequency (GHz)
Figure 5.1. Real and imaginary components of Z\\ for the load standards measured
by multiline TRL and modeled by an ANN.
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.2. Advantages of ANN Models
One of the advantages of using ANN models as opposed to calibrated
measurement files is the compact description possible with an ANN. For example, the
ANN model we developed for the load at port 1 required 62 real-valued parameters to
generate complex S-parameters as a function of frequency and DC resistance. In
contrast, a single measurement file contains 495 real-valued numbers (165 frequency
points plus the real and imaginary components at each point). If a measurement
database of just 5 loads is utilized, the combined files would contain 2475 real-valued
numbers.
We also explored the use of ANN models for extrapolation outside the bounds
of the training data. (Generally, it is believed that ANN models are good at
interpolating but not extrapolating.) We did this by training an ANN model at port 1
using 3 of the 5 loads once again, but this time we chose loads 2, 3, and 4. By
purposely not training the ANN model with loads 1 and 5, we could test how
effective the model behaved at extrapolating. Surprisingly, both the interpolating and
extrapolating ANN models exhibited almost identical deviations between measured
and predicted values. This bodes well for the application of ANN models to our loads,
since it is conceivable that other wafers may possess DC resistances slightly outside
the range of the 5 loads we used to train the models.
5.3. Calibration Comparisons
We performed 5 OSLT calibrations, each one making use of the same ANNmodeled open, short, and thru standards as well as the ANN-modeled loads with their
53
1
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respective DC resistances. We calibrated a 1.764-mm long CPW transmission line
using each of the OSLT calibrations and compared the results to measurements
calibrated directly using the benchmark multiline TRL calibration. Figures 5.2 and
5.3 compare the magnitudes and phases of S 21 and Sn for all 6 calibrations. The
agreement is remarkably good except at a few points where the multiline TRL
calibration shows a lot of scattering.
To obtain a more quantitative idea of the differences, we plotted the maximum
magnitude of the vector differences of the scattering parameters [maxd^l)] for the
1.764-mm line for each of the OSLT calibrations and the multiline TRL calibration.
Figure 5.4 illustrates the differences. All of the OSLT calibrations using ANNmodeled standards compare favorably to the benchmark multiline TRL calibration,
with a difference of less than 0.04 in magnitude at most of the frequencies over the 66
GHz frequency range. Not surprisingly, the OSLT calibrations for loads 2 and 3 show
slightly higher differences since they were not used to train the ANN model. The
differences between the 5 OSLT calibrations and the TRL calibration do not
necessarily mean the OSLT calibrations are in error. The differences are likely due to
the presence of noise in the TRL calibration that the ANN models avoided.
Regardless of the source of error, a 0.04 difference between two on-wafer calibrations
spanning 66 GHz is impressive, considering that the repeatability between two
multiline TRL calibrations is usually on the same order.
54
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3
1 .0 4 - 1
1. 0 2 -
1. 0 0 -
0.98-
Xfl
0.96-
Multiline TRL
OSLT - LI ANN
OSLT - L2 A N N
OSLT - L3 A N N
OSLT - L4 A N N
OSLT - L5 ANN
0.94-
0.92-
0.90-
“I
1 ------------ 1------------ 1------------ 1---10
20
30
40
50
60
70
Frequency (GHz)
2-
m
S5 -1 -j
-2 -3 -
1 ------------ 1------------ 1------------ 1------------ 1------------ 1------------ 1
10
20
30
40
50
60
70
Frequency (GHz)
Figure 5.2. Magnitude and phase of S21 for a calibrated 1.764-mm CPW transmission
line.
55
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0 .1 2 —i
0. 1 0 -
0.08
Multiline TRL
OSLT - LI A N N
OSLT - L2 ANN
OSLT - L3 A NN
OSLT - L4 ANN
OSLT - L5 ANN
0 .0 4 -
0.02
0.00
10
20
30
40
50
60
70
50
60
70
Frequency (GHz)
2 -|
1
'3
2
0-
-2 -
10
20
30
40
Frequency (GHz)
Figure 5.3. Magnitude and phase of Sn for a calibrated 1.764-mm CPW transmission
line.
56
|
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80
-|
60O SLT - L I
■O SLT - L 2
O S L T -L 3
O SLT - L4
O S L T -L 5
ANN
ANN
ANN
ANN
ANN
40-
20-
~T~
10
n --------------------------r
20
30
40
~ r
50
60
70
Frequency (GHz)
Figure 5.4. Magnitudes of the scattering parameter differences of a calibrated 1.764mm CPW transmission line.
5.4 Discussion
We have successfully applied ANNs to model the correlation between DC
resistance and RF variations in load terminations and the RF performance of open,
short, and thru standards used for on-wafer OSLT calibrations of vector network
analyzers. We have shown that these modeled standards compare favorably (a
difference of less than 0.04 in magnitude at most frequencies) to the benchmark
multiline TRL calibration over a 66 GHz bandwidth. In addition, we have shown that
ANN models are able to accurately model loads with measured DC resistances
slightly outside their training range.
57
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j
CHAPTER 6
|
ANN MODELING FOR ON-WAFER LRM CALIBRATIONS
!
!
!
|
Another popular compact calibration method is LRM (line-reflect-match),
|
which requires only a short transmission-line connection, a load, and a reflection [41,
|
52]. Here, the reference impedance is set to that of the standard load. As shown
.
above, the impedance of many on-wafer loads, however, is non-ideal, which can lead
to significant error in LRM calibrations.
Here, we model a load using an ANN to improve an on-wafer LRM
]
|
calibration [44], The ANN is trained with measurement data obtained from a single-
j
line TRL calibration. Using a single-line TRL calibration enables us to build an
j
j
effective model of the load using minimal real estate on the wafer. This methodology
|
results in an LRM calibration with less overall error than by simply applying the
j
j
single-line TRL calibration. The accuracy of the LRM calibration using the ANN-
I
modeled load compares favorably to a benchmark multiline TRL calibration with an
average worst-case scattering parameter error bound of 0.017 over a 40-GHz
|
frequency range.
6.1. Load Modeling
In this study, the LRM and multiline TRL standards and devices were
|
constructed of coplanar waveguide (CPW) transmission lines fabricated from 1.5 pm
gold conductors on a layer of 500 pm thick semi-insulating GaAs [49]; the gold
1
I
I1
3
1;l
|
center conductor was 73 pm wide, and was separated from the ground plane by 49
58
i
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Lim gaps. The five line standards included a 0.55-mm thru line and four additional
lines that were 2.135, 3.2, 6.565, and 19.695 mm longer. All of the standards were
measured using on-wafer probes. For each standard, we measured scattering
parameters at 192 frequencies from 0.5 to 40 GHz.
In Figure 6.1, we plot measurements of magnitude and phase of the load’s
reflection coefficients. They were determined by a TRL calibration using only the
thru connection and the 2.135-mm line, and applying an impedance transformation to
the calibration, which yielded the measured 5-parameters referenced to 50 £2 [53].
Use of only a single line explains the inaccuracy at multiples of 26.65 GHz, where the
difference in line lengths corresponds to a multiple of half a wavelength. The figure
shows that the load deviates significantly from 50 Q.
0.20
i - -40
--6 0
Measured w/ Single-Line TRL
ANN Model
0.15 -
— 80
--1 0 0
(JQ
- -140
0 .0 5 -
- -160
0.00
-180
0
10
20
30
40
Frequency (GHz)
Figure 6.1. Magnitude and phase of measured and modeled scattering parameters of
the on-wafer load for the LRM calibration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
We then used an ANN to model the load. The ANN for the load standard
consisted of one input (frequency) and two outputs (the real and imaginary
components) for the S-parameters. Based on previous experience [42], five neurons
were chosen for the hidden layer. Figure 6.1 shows the magnitude and phase of the Sparameters of both measured and ANN-modeled data for the load standard. Notice
that the ANN model for the load standards follows the trends of the measured data,
but avoids both the spike near 26.65 GHz as well as scatter of the TRL-calibrated
measurements. Whether or not this scatter is real, we see that ANNs follow general
trends but omit noise, which is usually desirable in a model.
6.2. Calibration Comparisons
First, two consecutive multiline TRL calibrations, using all five lines, were
compared to assess the limitations on calibration repeatability caused by contact error
and instrument drift. The technique of [54] was used to determine an upper bound on
this repeatability error. Briefly, the comparison determines the upper bound for |S'ySij\ for measurements on any passive device, where S'y are the scattering parameters
of a device measured with respect to the first calibration and Sy are the scattering
parameters measured with respect to the second calibration. The bound is obtained
from a linearization, which assumes the two calibrations are similar to the first order.
The result, plotted as a solid curve in Figure 6.2, roughly indicates the minimum
deviation between any pair of calibrations. The average of the worst-case error
bounds for repeatability was 0.013.
60
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I
So
S-H
t/5
1
ffl
ti
0.01
Multiline TRL Repeatability
Single-line TRL
Simple LRM
LRM with Measured Load
LRM w ih ANN-M odeled Load
£
0.001
0
20
10
30
40
Frequency (GHz)
Figure 6.2. Worst-case error bounds between measurements of passive devices from
on-wafer LRM and TRL calibrations and the multiline TRL calibrations.
We also compared the single-line TRL calibration, which was used to develop
the ANN model, to the multiline TRL calibration. The result is plotted in Figure 6.2.
Since we used only the 2.135-mm line standard, our calibration accuracy is poor near
multiples of 26.65 GHz, where the difference in line lengths corresponds to a multiple
of half a wavelength [3]. Otherwise the single-line TRL calibration is nearly as
accurate as the multiline TRL calibration at most frequencies.
We assessed the accuracy of the LRM calibrations by comparing them to a
50-0 multiline TRL calibration. First, we compared a simple LRM calibration, where
the load is assumed to be ideal, to the multiline TRL calibration. Figure 6.2 illustrates
a large difference since the reference impedance of the LRM calibration, which is
equal to the impedance of the non-ideal load, deviates significantly from 50 Q.
61
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In order to assess the accuracy of the best LRM calibration, we compared the
multiline TRL calibration to the LRM with a fully characterized load, which involved
calibrating the load with the benchmark multiline TRL calibration and using the
calibrated measurement data file to define the load. This comparison is once again
shown in Figure 6.2. The average of the worst-case error bounds for this calibration
was 0.011. Figure 6.2 also shows the worst-case error bounds for the LRM calibration
based on the ANN-modeled load. Here, the average worst-case error bound was
0.017.
6.3. Discussion
The use of ANNs to model on-wafer LRM load standards compares favorably
to a benchmark multiline TRL calibration, with an average worst-case scatteringparameter error bound of 0.017 over a 40-GHz bandwidth.
62
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CHAPTER 7
ANN MODELING FOR COAXIAL LRM CALIBRATIONS
Multiline TRL can also be applied in coaxial environments. However, a set of
coaxial lines, some relatively long, is required to obtain a wide-band measurement.
Coaxial airlines also require considerable care to ensure a good connection without
damaging the standard. Furthermore, a set of lines can be costly. In contrast, the LRM
calibration, which requires only a thru connection, a coaxial load, and a reflection,
overcomes these limitations. Here, the reference impedance is set to that of a standard
load. The impedance of many coaxial loads, however, is non-ideal, which can lead to
significant error in LRM calibrations.
There are two approaches we can take to characterize an imperfect load. One
is to characterize it in terms of its reflection coefficient, which requires access to a
full multiline TRL calibration set. Alternatively, we can postulate a physical model of
the load and apply a minimal calibration sufficient to determine the model coefficient.
Jargon et al [55] applied this notion to coaxial lines, employing the measurement of
the load after a single-line TRL calibration to fit the parameters of an equivalent
circuit model. This provided a means for obtaining an accurate wide-band LRM
calibration with a compact coaxial standard set consisting of a reflection, a match
standard, and a line of short length.
Although the equivalent circuit model used in [55] was effective, there were a
number of difficulties with it. First, the model was specifically tailored for the load
used, so considerable time was required to develop an adequate model. The load was
63
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approximated by the impedance R+qaf+jcoL, preceded by a lossless line of
characteristic impedance
Z q,
length /, and effective permittivity er,eff. The value of R
was determined by measuring the dc resistance of the load. Zq was chosen to be 50 £2,
and er,eff was assumed to be 1. Then, L, q, and I were determined by optimizing the
model. Another disadvantage of this model is that it is not guaranteed to work for
other loads. A completely different impedance might be required to model other loads
correctly.
In an attempt to improve the accuracy of the LRM calibration, we use a
single-line TRL calibration to train an ANN model of the load. The following
sections describe our implementation of ANNs and assess the accuracy of the LRM
calibration using the ANN-modeled load, comparing it to the equivalent circuit model
and measured data [45].
7.1. Load Modeling
We used a set of commercially available GPC-7 artifacts for these
experiments. The artifacts consisted of 2.25-cm, 10-cm, and 30-cm airlines, a short
circuit, and a nominally 50 Q coaxial load. We assumed that our sexless GPC-7
connectors mated perfectly with our line, allowing a direct connection between the
two ports to serve as a thru line.
In Figure 7.1 we plot measurements of the real and imaginary parts of the load
impedance. The impedance was determined by a TRL calibration using only the thru
connection and the 2.25-cm line, and applying an impedance transformation to the
calibration, which yielded the measured 5-parameters referenced to 50 Q. The
64
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characteristic impedance of the line was determined from its capacitance and
propagation constants, allowing the reference impedance of the TRL calibration to be
accurately set to 50 Q, [53], Use of only a single line explains the inaccuracy at
multiples of 6.67 GHz, where the difference in line lengths corresponds to a multiple
of half a wavelength. The figures show that the load deviates significantly from 50 Q.
To account for the non-idealities, we developed an ANN model for the load
using 15 hidden neurons and 181 measured points. We trained our ANN model with
data taken from the single-line TRL calibration to illustrate that loads, or for that
matter almost any artifact, can be modeled using only a simple set of calibration
standards rather a large set of expensive airlines. Figure 7.1 shows that the real and
imaginary parts of the ANN-modeled load correspond closely to the measured values,
and that the model did avoid the spikes present at multiples of 6.67 GHz. Also plotted
in Figure 7.1 are the real and imaginary parts of the load as determined by the
equivalent circuit model of [55]. Not only does the ANN model match the measured
values closer than the circuit model, but it was also developed in a small fraction of
the time needed to develop and optimize the circuit model.
65
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65 -i
Measured Single-line TRL
ANN Model
Circuit Model
60-
55O
N
<L>
45-
0
5
10
Frequency (GHz)
15
20
10
Frequency (GHz)
15
20
15-i
10
-
N
5
-5-
-10
0
5
Figure 7.1. Real and imaginary components of the measured and modeled
impedance of the coaxial load for the LRM calibration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7.2. Calibration Comparisons
First, two consecutive multiline TRL calibrations, using all three airlines,
were compared to assess the limitations on calibration repeatability caused by contact
error and instrument drift. The technique of [54] was used to determine an upper
bound on this repeatability error. Briefly, the comparison determines the upper bound
for l^y - Sy\ for measurements on any passive device, where S'y are the scattering
parameters of a device measured with respect to the first calibration and Sy are the
scattering parameters measured with respect to the second calibration. The bound is
obtained from a linearization, which assumes the two calibrations are similar to the
first order. The result, plotted as a solid curve in Figure 7.2, roughly indicates the
minimum deviation between any pair of calibrations. The average worst-case error
bound for repeatability was 0.013.
We then compared the single-line TRL calibration, which was used to develop
both the ANN and equivalent-circuit models, to the mutliline TRL calibration. The
result is plotted in Figure 7.2. Since we only used the 2.25-cm line standard, our
calibration accuracy is poor near multiples of 6.67 GHz, where the difference in line
lengths corresponds to a multiple of half a wavelength [3]. Otherwise, the single-line
TRL calibration is nearly as accurate as the multiline TRL calibration at most
frequencies.
We assessed the accuracy of the LRM calibrations by comparing them to the
50 Q multiline TRL calibration. Figure 7.2 shows the maximum possible difference
|S'y - S'y] where S'y corresponds to the simple LRM calibration (load assumed to be
ideal), and Sy corresponds to the multiline TRL calibration. Here, the difference is
67
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large since the reference impedance of the LRM calibration, which is equal to the
impedance of the non-ideal load, deviates significantly from 50 Q.
To see how accurate the best LRM calibration was, we compared the multiline
I
j
TRL calibration to the LRM with a fully characterized load, which involved
j
calibrating the load with the benchmark multiline TRL calibration and using the
j
calibrated measurement data file to define the load. This comparison is once again
j
shown in Figure 7.2. The average of the worst-case error bounds for this calibration
|
\I
j
was 0.016.
Figure 7.2 also shows the worst-case error bounds for the LRM calibrations
based on both the ANN and equivalent circuit models. The average of the worst-case
error bounds for the ANN-modeled LRM calibration was 0.024, while the average for
the circuit-modeled LRM was 0.034.
7.3.
Discussion
j
1
The use of ANNs to model coaxial LRM load standards compares favorably
!
|
to a benchmark multiline TRL calibration, with an average worst-case scattering-
|
3
parameter error bound of 0.024.
j
|j
In the case of LRM calibrations,7 we have shown that ANN models offer
]
j
advantages over equivalent circuit models since they do not require detailed physical
models. Our ANN model required far less development time than our equivalent
|
circuit model and still managed to achieve higher accuracy. ANN model descriptions
are also preferred over calibrated measurement data files since they are much more
compact in size. Additionally, loads, or for that matter almost any artifact, can be
!
68
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modeled using only a simple set of calibration standards rather than being fully
characterized with a large set of expensive airlines.
St
Multiline T R L R epeatability
Single-line T R L
Sim ple LRM
LR M with M easured Load
LR M with C ircuit-M odeled Load
LR M with A N N -M odeled Load
'M
1
CO
0 . 01 -
0.001
0
2
4
6
10
8
12
14
16
18
Frequency (G H z)
Figure 7.2. Worst-case error bounds between measurements of passive devices from
coaxial LRM and TRL calibrations and the multiline TRL calibration.
69
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I
CHAPTER 8
NONLINEAR VECTOR NETW ORK ANALYSIS
In the second portion of this thesis, we focus on frequency-domain models
and figures of merit for nonlinear circuits based on large-signal measurements.
8.1. LINEAR VERSUS NONLINEAR BEHAVIOR
As mentioned in Chapter 3, vector network analyzers (VNAs) are one of the
most versatile instruments in the RF and microwave industry. They are used to
measure complex scattering parameters (^-parameters) of devices or circuits.
Engineers use them to verify their designs, confirm proper performance, and diagnose
failures. A VNA works by exciting a linear device under test (DUT) with a series of
sine wave signals, one frequency at a time, and detecting the response of the DUT at
its signal ports. Since the DUT is linear, the input and output signals are at the same
frequency as the source, and can be described by complex numbers that account for
the signals’ amplitudes and phases. The input-output relationships are described by
ratios o f complex numbers, known as ^-parameters. For a two-port network, four Sparameters completely describe the behavior of a linear DUT when excited by a sine
wave at a particular frequency.
Although the measurement of S-parameters by VNAs is invaluable to the
microwave designer for modeling and measuring linear circuits, this is oftentimes
inadequate for nonlinear circuits operating at large-signal conditions since
nonlinearities transfer energy from the stimulus frequency to products at new
70
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frequencies, as shown in Figure 8.1. Thus, conventional linear network analysis,
which relies on the assumption of superposition, must be replaced by a more general
type of analysis, which we refer to as nonlinear network analysis.
Linear Device
Time
Time
Incident
Transmitted
Frequency
I
Frequency
Nonlinear Device
Time
Time
Incident
Transmitted
| Frequency
Frequency
Figure 8.1. Nonlinearities transfer energy from the stimulus frequency to products at
new frequencies.
8.2. Nonlinear Vector Network Analysis
Nonlinear network analysis involves characterizing a nonlinear device under
realistic, large-signal operating conditions. To do this, complex traveling waves
(rather than ratios) are measured at the ports of a DUT not only at the stimulus
frequency (or frequencies), but also at other frequencies where energy may be
created. Assuming the input signals are sine-waves and the DUT exhibits neither sub­
harmonic nor chaotic behavior, the input and output signals will be combinations of
sine-wave signals, due to the nonlinearity of the DUT in conjunction with mismatches
between the system and the DUT. If a single excitation frequency is present, new
71
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frequency components will appear at harmonics of the excitation frequency, and if
multiple excitation frequencies are present, new frequency components will appear at
the intermodulation products as well as at harmonics of each of the excitation
frequencies. Figure 8.2 illustrates this concept for a set of harmonically related signals
present at the port of a DUT. In practice, there will be a limited number of significant
harmonics and intermodulation products. The set of frequencies at which energy is
present and must be measured is known as a frequency grid.
... con
©1, ©2, ... ©n
L ^
l
a,
b.
Nonlinear
a,
C0„0)2, ... con
©1, 0)2, ... ©n
L _ j_
Figure 8.2. Complex traveling waves at the ports of a nonlinear device when a set of
harmonically related signals is present.
To get a feel for the added complexity of nonlinear network analysis, as
compared to linear network analysis, we examine the flow diagram of a two-port
network excited by a source at a single frequency at port 1 and terminated with a load
at port 2. For a linear device, shown in Fig. 8.3(a), four scattering parameters relate
the incoming and outgoing waves. The outgoing wave b\ can easily be solved in
terms of a\ as
72
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where Tin is referred to as the input reflection coefficient of the network.
For a nonlinear device, the flow diagram becomes much more complicated.
Figure 8.3(b) shows a nonlinear device with only the first three harmonics considered.
In this case, we still have a single-source excitation at port 1 (a \\), but now we must
account for the fact that power may be transferred to higher harmonic frequencies.
Note that the wave variables, a and b, in the nonlinear case contain two positive
integer subscripts, the first refers to the port number and the second to the spectral
component number. Here, we have three reflection coefficients for the terminating
load and 16 nonlinear large-signal scattering parameters (&#*/) that relate the
incoming and outgoing waves, assuming a single-source excitation and no downconversion. (Nonlinear large-signal scattering parameters will be discussed in more
detail in Chapter 10.) The outgoing wave b\\ can be solved in terms of a\\ in a
similar fashion to the linear case, giving
j u - » m i + * li l' * i!‘i r ‘ -
a ll
1 *2211 1
(8-2)
Solving for bn and bu in terms of an , however, gives more involved expressions:
^12 _ g,
a \\
+ ^1221^2111^1 + ^1222^2121^2 + ^1222^2221^211 l^ l^
1—^2211^1
1—^2222-^2
0 —^2211^1 ) 0 —&2222^2)
/g 3 -v
and
_^13__^
a \\
! ^1231^2111^1 ! ^1233^2131^3 |
I - ^221 l^ i
I - 8*2233^3
^1233^2231^2111^1^3
0 _ ^2 2 1 1 ^ 1 ) 0 ~ ^ 2 2 3 3 ^ 3 )
73
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(8 4)
In this simplified case, we have not considered harmonics above 3, but if we did, eqs.
(8.4) and (8.5) would become even more involved at frequencies that are multiples of
more than one sub-harmonic. For example, the fourth harmonic is a multiple of both
the first and second harmonics.
(a)
«i
'l
' 21*1
*22kl ^
11*1
12ki
Figure 8.3. Flow diagram for a two-port network with a single-frequency excitation
at port 1 and a terminating load at port 2. (a) Linear case, (b) Nonlinear case
considering three harmonics.
74
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I
8.3. Nonlinear Vector Network Analyzers
A class of instruments known nonlinear vector network analyzers (NVNA) are
capable of providing accurate waveform vectors by acquiring and correcting the
magnitude and phase relationships between the fundamental and harmonic
components in the periodic signals [6-8]. An NVNA excites a nonlinear DUT with
one or more sine wave signals and detects the response of the DUT at its signal ports.
Assuming the DUT does not exhibit any sub-harmonic or chaotic behavior, the input
and output signals will be combinations of sine wave signals due to the nonlinearity
of the DUT in conjunction with mismatches between the system and the DUT.
With these facts in mind, the major difference between a linear VNA and an
NVNA is that a VNA measures ratios between input and output waves one frequency
at a time while an NVNA measures the actual input and output waves simultaneously
over a broad band of frequencies.
8.3.1. Architecture
Figure 8.4 illustrates a simplified block diagram of an NVNA with the RF source
connected to port 1 and bias tees located at both ports. The incident and reflected
waves at both ports of the DUT are measured through directional couplers. The
measured RF spectrum are down-converted to an IF spectrum using harmonic mixing,
which is based on the same principle as in sampling oscilloscopes. The compressed
signals are digitized using precision analog-to-digital (ADC) cards. The RF source,
the four down-converters, and the four ADC cards are timed by a common reference
clock in order to obtain a fully synchronous, phase-coherent measurement system.
75
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Synchronous
Clock
ADC
ADC
ADC
ADC
Broadband
Down
Converters
M
RF
Source
BiasT
DUT
BiasT
Figure 8.4. Simplified block diagram of a nonlinear vector network analyzer.
Despite the disadvantages of using some non-commercially available parts, a
high price tag, and a complex infrastructure that requires powerful software, the
NVNA offers a number of significant advantages over other types of measurement
systems. First, the use of four couplers and four-channel data acquisition allows the
entire spectrum of incident and reflected waves to be measured simultaneously at
both ports of a DUT, thus circumventing any phase synchronization problems.
Second, because of four couplers, all mismatches can be taken into account during
calibration without making any assumptions regarding perfect terminating loads.
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Third, data acquisition is fast. A power-bias sweep requires only a few seconds. And
fourth, the system has good dynamic range, typically around 60 dB.
8.3.2. Calibration
The calibration procedure for an NVNA is based on the assumption that the
four measured wave quantities (aM\k, aMik, bM\k, and bM2k) are linearly related to the
four physical voltage waves at the ports of the device (au, aik, b\k, and bjk) [10],
Here, a represents the incident wave, b represents the scattered wave, the first
subscript refers to the port number (1 or 2), the second subscript k refers to the
frequency index, and the superscript M refers to the measured wave quantities. The
relationship between the physical waves and the digitized quantities is characterized
by complex square matrices (one matrix for each frequency) with sixteen elements, as
given by
~<*\k
b\k
0
0
M
a \k
Pk
0
0
0
p.2k
\k
a„M
2k
0
Yik
P
->2k .
1
P \k
a2k
Y\k
0
p2k _
0
= \Ck\emCk)
M
(8.5)
Generally, the calibration procedure solves for the sixteen elements for all of
the matrices, but oftentimes as in eq. (8.5), the cross-talk terms are assumed to be
negligible and are set to zero, leaving only eight terms to be solved.
The calibration of an NVNA consists of three steps: a relative calibration that
is identical to that used in a linear VNA, an amplitude calibration that makes use of a
power meter, and a phase distortion calibration that makes use of a reference
generator. The relative calibration, such as TRL, OSLT, or LRM, accounts for
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directivity and cross-talk errors due to signal leakage, impedance mismatches and
losses in the cables and connectors. This calibration solves for the following seven
terms: pa, yik, 5u, 0C2k, P2k, Jik, and 82*. There is also a complex number C* common
to the entire matrix that must be solved for in order to achieve an absolute calibration.
The amplitude calibration solves for the magnitude of C*, and the phase calibration
solves for the phase of Q ,
The phase calibration is achieved by using a reference generator that generates
a fundamental signal and harmonics with a known phase relationship, the main
components of which are a power amplifier and a step recovery diode. The reference
generator is characterized by a sampling oscilloscope, which in turn is characterized
by a nose-to-nose calibration [56, 57], The main principle behind the nose-to-nose
calibration is that the “impulse response” of each sampler in an oscilloscope can also
be used as a pulse generator. This “kick-out” pulse occurs when an offset voltage is
applied to the hold capacitors of the sampler. It turns out that the “kick-out” pulse is
proportional to the “impulse response.” The nose-to-nose calibration refers to the
process of connecting two oscilloscopes together, and using one scope to measure the
other’s “kick-out” pulse. If the scopes are identical, the resulting waveform is the
convolution of the scope’s impulse response with the kick-out pulse. The impulse
response can then be calculated. Since no two scopes are exactly identical, three
oscilloscopes are needed to perform an accurate nose-to-nose calibration.
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j
8.4. Developing Frequency-Domain Models for Nonlinear Circuits Based on
Large-Signal Measurements
Even though 5-parameters cannot adequately represent nonlinear circuits,
some type of parameters relating incident and reflected signals are beneficial so that
the designer can “see” application-specific engineering figures of merit that are
similar to what he or she is accustomed to. In this portion of the thesis, we propose
definitions of such ratios that we refer to as nonlinear large-signal scattering (§>)
parameters. After describing a method for preserving time-invariant phase
relationships when ratios are taken between any two harmonically related, complex
signals, we present a general form of time-invariant nonlinear large-signal &parameters. Then we introduce nonlinear large-signal impedance (Z) and admittance
(H) parameters, and present equations for relating the different representations.
In addition to nonlinear large-signal ^-parameters, we introduce another
figure of merit for nonlinear circuits by expanding the definitions of power gain,
transducer gain, and available gain by taking harmonic content into account. We
show that under special conditions, these expanded definitions of gain can be
expressed in terms of two-port, nonlinear large-signal ^-parameters.
For existing nonlinear models, we can readily generate nonlinear large-signal
^-parameters by performing a harmonic balance simulation. For devices, with no
model available, we can extract these parameters from artificial neural network
(ANN) models that are trained with multiple frequency-domain measurements made
on a nonlinear DUT with an NVNA.
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To illustrate applications and generation of nonlinear large-signal 0 parameters, we present three examples. First, we show how nonlinear large-signal 0 parameters and the expanded definitions of gain can be used to discover valuable
information regarding the behavior of a nonlinear model. Specifically, we examine a
lumped-element model of a pseudomorphic high electron mobility transistor
(pHEMT) device operating in a two-port, common-source configuration. Second, we
illustrate how nonlinear large-signal ^-parameters can be used to as a tool in the
design process of a simple nonlinear circuit, specifically a single-diode 1-2 GHz
frequency-doubler circuit. And finally, we describe a method for generating nonlinear
large-signal 0-parameters based upon ANN models trained on frequency-domain
data measured using a nonlinear vector network analyzer (NVNA). We compare a
diode circuit model, generated using this method, to a harmonic balance simulation of
a commercial device model.
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CHAPTER 9
CALCULATING RATIOS OF HARMONICALLY
RELATED, COMPLEX SIGNALS
In this chapter, we describe a method for preserving time-invariant phase
relationships when ratios are taken between any two harmonically related, complex
signals [58]. We provide a simple example to illustrate our technique. Then in the
following chapter, we will show how this method is implemented when defining
time-invariant nonlinear large-signal scattering parameters.
9.1. Introduction
When two complex signals z and y exist at the same frequency w/27i, the ratio
R of the two quantities may be expressed as
(9.1)
where the phasor notation of z is represented by |z|Zcpz and that of y is represented by
[y|Z(py.
When two complex signals exist at different frequencies, obtaining a timeinvariant phase of the ratio is more involved. When ratios are taken between two
harmonically related signals, we can preserve time-invariant phase relationships by
introducing a third signal that acts as a phase reference. We show that this reference
signal must have a component at the fundamental frequency in order that the ratios of
any two harmonically related signals contain a time-invariant phase relationship. We
provide a simple example to illustrate our technique.
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9.2. Method
Consider two complex signals zk and yi that are harmonically related. Here k
and I are positive integers representing signals at the Mi and /th harmonic terms,
respectively. In phasor form,
(9.2)
Note that all phases (p considered here are in units of degrees (in terms of their
respective frequencies) and have a modulus of 360° (i.e., 0° < cp < 360°).
At first glance, a commonly assumed equation for taking the ratio of two
harmonically related complex signals zk and yi is
nt _
\Zk
K ki ~
Z
(9.3)
The factor kll serves to translate the phase from the /th harmonic of the divisor
to the Mi harmonic of the dividend, resulting in the phase of the ratio R \i given in
terms of the Mi harmonic. The superscript‘t’ is used because eq. (9.3) gives a timevariant phase. Specifically, if kll is not an integer, there will be a phase ambiguity of
360°//.
In order to try to avoid a phase ambiguity, we modify eq. (9.3) by referencing
the phases of signals zk and yi to some reference phase of a third signal x„ at the nth
harmonic, which gives
Ru
Zk
W
(9.4)
/
where
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In eq. (9.5), the kin factor serves to translate the arbitrary phase (0° < cp* < 360°) from
the nth harmonic of x„ to the kth harmonic of z* and the Hn factor serves to translate
the phase from the nth harmonic of x„ to the /th harmonic of yi. Combining eqs. (9.4)
and (9.5) gives
U
V
<t>zk
<t>x,
n
'
k ( rh - L r h
J
I
$yi
n $x.
'
(9.6)
Here, we still have the problem that if kin or Hn are not integers, eq. (9.6) gives an
inconsistent phase. Specifically, if kin or Hn are not integers, there will be up to n (n <
I) possible answers with a phase ambiguity of 360°//. If & is a multiple of / or vice
versa, there will be fewer than n possibilities, but still more than one in general.
In order to avoid any phase ambiguity, kin and lln must be integers. In order
for this to be true for all k and /, n must equal one. If the frequency of the reference
signal (xn) is set to its fundamental frequency (n - 1), then eq. (9.6) becomes
Rkl ~
z
(&* ~ k(Pxx) - y fay,
(9.7)
)
Note that eq. (9.7) is algebraically identical to eq. (9.3). It is important, however, to
leave the (p.C| terms in and perform the phase references for cpZi and
in order to
ensure that eq. (9.7) provides a time-invariant phase. Equation (9.7) can be simplified
in the case of Rk\ ( /- l) if >’i serves as both the divisor and the reference signal:
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**. =
~ f k ~ kK)-
<’ -8 >
9.3. Example
Here, we provide a simple example illustrating that eq. (9.3) gives a timevariant phase, and that eq. (9.6) gives a time-variant phase for n > 1. However, eq.
(9.6) does provide a time-invariant phase if n - 1.
In this example (see Figure 9.1), we consider signals with three phase
references, the first one being arbitrary, where the reference at the fundamental is xi =
1Z0°, the reference at the second harmonic is x2 = 1Z0°, the dividend (at the second
harmonic) is z2 = 0.7Z800, and the divisor (at the third harmonic) is yi = 0.4Z1700.
Figure 9.2 shows the time-domain representation of xi, z2, and y^. From the figure, we
1.0n Im
z\l) =0.7Z80‘
0.5-
>{l) =0.4Z170(
-
1.0
Re
-0.5
0.5
.•()
-0.5-
-1 .0 J
Figure 9.1. Phasor plot of the fundamental reference xi, the dividend z2, and the
divisor ^3 at the first phase reference [all phasors identified by superscript (1)].
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can see that the 80° phase delay in z2 corresponds to a time delay of 0.111
fundamental-unit period, and the 170° phase delay in yi corresponds to a time delay
of 0.157 fimdamental-unit period. Figure 9.2 illustrates that there is no ambiguity in
the time domain if all of the signals are synchronous with the fundamental signal.
Equation (9.7) ensures the same in the frequency domain. But if, on the other hand,
all of the signals are synchronous with a harmonic signal, portions of the waveforms
at lower frequencies will be lost, resulting in possible phase ambiguities.
At the first phase reference, we determine
definition uses
from eq. (9.3), which by
as the reference in this case:
K
- 3^3
(9.9)
Next, we calculate 7?(2)23 from eq. (9.6), using x2 as the reference, as
n(2) _ lZ2l z
"X2
23
(9.10)
where the superscript ‘(2)’ denotes the phase reference of x2. Finally, we calculate
from eq. (9.6), using xi as the reference, as
(1) _ lZ2| z
R 23
(9.11)
where the superscript ‘(1)’ denotes of phase reference of x\. At this first phase
reference, eqs. (9.9) - (9.11) give the same answer, of 1.75Z326.670, as shown in the
first column of Table 9.1.
Next, we consider a second phase reference, where the phase of the
fundamental frequency is shifted by 100°. This means that the phase at the second
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harmonic is shifted by 2 times 100°, or 200°, and the phase at the third harmonic is
shifted by 3 times 100°, or 300°. So now, the reference at the fundamental is x\ =
1Z1000, the reference at the second harmonic is X2 = 1Z2000, the dividend is z2 0.7Z2800, and the divisor is ys = 0.4Z4700 = 0.4Z110°. These values are plotted in
Figure 9.3. At this second phase reference, we again determine Rl22, R(1)23, and R(2)23
using eqs. (9.9) - (9.11). Here, R ^ - 1.75Z206.670 is inconsistent with the answer
determined at the first phase reference by 120° (360°/3). The ratio R(2)23 =
1.75Z326.670 is consistent with the answer determined at the first phase reference.
Likewise, the ratio i ?(1)23 -
1.75Z326.670 is also consistent with the answer
determined at the first phase reference. The values of all of the quantities at the
second phase reference are shown in the second column of Table 9.1.
Finally, we consider a third phase reference, where the phase of the
fundamental frequency is shifted by 200°. This means that the phase at the second
harmonic is shifted by 2 times 200°, or 400°, and the phase at the third harmonic is
shifted by 3 times 200°, or 600°. So now, the reference at the fundamental is x\ 1Z2000, the reference at the second harmonic is x2 ~ 1Z40°, the dividend is z2 =
0.7Z1200, and the divisor is yi = 0.4Z500. These values are plotted in Figure 9.4. At
this third phase reference, we again determine /?*23, Z?(1)23, and /?(2)23 using eqs. (9.9) (9.11). Here, R \ 5 = 1.75Z86.670 is inconsistent with the answers determined at the
first and second phase references by 120° (360°/3). The ratio R(2)23 = 1.75Z206.670 is
also inconsistent with the answers determined at the first and second phase
references. The ratio i ?(1)23 = 1.75Z326.670, however, is consistent with the answers
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*2
K
Quantity
1st Phase
Reference
Xi
K
0O
2nd Phase
Reference
1Z100°
0O
Table 9.1. Determining the ratios of z2 to yj using three methods at three different
phase references.
1Z 2000
Z2
0.7Z800
0.7Z2800
0.7Z120 0
T3
0.4Z1700
0.4Z110°
0.4Z500
^23 [eq. (9)]
1.75Z326.670 1.75Z206.670
3rd Phase
Reference
1Z 2000
1Z40°
1.75Z86.670
R(2)23 [eq. (10)]
1.75Z326.670 1.75Z326.670 1.75Z206.670
R(1)23 [eq. ( 11)]
1.75Z326.670 1.75Z326.670 1.75Z326.670
determined at the first and second phase references. The values of all of the quantities
at the third phase reference are shown in the third column of Table 9.1.
Examining the fifth row of Table 9.1, we see that R l23 does indeed give a timevariant phase. Since kll - 2/3 is not an integer, there is a phase ambiguity of 360° / 3,
or 120°. Examining the sixth row of Table 9.1, we see that that R(2)23 also gives a
time-variant phase since the reference signal is located at the second harmonic. Since
lln = 3/2 is not an integer, there are 2 possible answers with a phase ambiguity of
360° / 3, or 120°. Examining the seventh row of Table 9.1, we see that that R(1)23
gives a time-invariant phase since the reference signal is located at the fundamental
frequency. Thus, the only time-invariant ratio is R0)23.
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1 .0 “ i
0.5
0.5u = 0.157
0.0
0.0
-0.5
-0.5
-
1.0
-
-
0.0
0.2
0.4
0.6
Time (fundamental unit period)
1.0
0.8
0.2
Figure 9.2. Time-domain plot of the fundamental reference x\, the dividend Z2 , and
the divisor ^3 at the first phase reference.
IZKHf
+
200‘
+3001
-
1.0
Res
■0.5
0.5
+IW
-0.5-
12) =0.7Z280‘
-1.0 J
Figure 9.3. Phasor plot of the fundamental reference x\, the dividend Z2, and the
d iv iso r^ at the second phase reference [using superscript (2)].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1-O-i Im -
+400'
Re
+600°f
0.5
♦
200
-0.5
-1.0 J
Figure 9.4. Phasor plot of the fundamental reference x\, the dividend Z2 , and the
divisor J 3 at the third phase reference [using superscript (3)].
9.4. Discussion
We described a method for preserving time-invariant phase relationships
when ratios are taken between two harmonically related signals by introducing a third
signal that is used as a phase reference. We showed that a reference signal must be
present at the fundamental frequency in order for time-invariant phase relationships to
exist between ratios of any two harmonically related signals. We provided a simple
example to illustrate our technique. In the next chapter, we will implement this
method when defining time-invariant nonlinear large-signal scattering parameters.
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CHAPTER 10
NONLINEAR LARGE-SIGNAL SCATTERING PARAMETERS
In this chapter, we introduce the concept of nonlinear large-signal scattering
parameters. Like commonly used linear 5-parameters, nonlinear large-signal
scattering (# ) parameters can also be expressed as ratios of incident and reflected
wave variables. However, unlike linear 5-parameters, nonlinear large-signal U>parameters depend upon the signal magnitude and must take into account the
harmonic content of the input and output signals since energy can be transferred to
other frequencies in a nonlinear device.
After presenting the general form of nonlinear large-signal ^-parameters, we
also introduce nonlinear large-signal impedance (2) and admittance ($) parameters,
and present equations for relating the different representations. Next, we make two
simplifications, considering the cases of a one-port network with a single-tone
excitation and a two-port network with a single-tone excitation.
10.1. General Form
Consider an iV-port network. Normalized wave variables a# and bji at the y'th
port and /th harmonic are proportional to the incoming and outgoing waves,
respectively, and may be defined in terms of the voltages associated with these waves
as follows:
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a - I L
. „ - I I .
( 10. 1)
where K1}/ and Vji represent voltages associated with the incoming and outgoing
waves in the transmission lines connected to they'th port and at the /th harmonic, and
Z0j is the characteristic impedance of the line at the y'th port.
The nonlinear large-signal scattering matrix & of the network expresses the
relationship between a ’s and b’s at various ports and harmonics through the matrix
equation
b=$a.
(10.2)
where b and a are (/VxVlf)-element column vectors. Here N refers to the number of
ports and M refers to the number of harmonics being taken into account. & is an
(NxM) -element square matrix. We assume all a ’s and V s are phase referenced to a\\
to enforce time invariance, as discussed in the previous chapter.
For a two-port network with 3 harmonics, for example, eq. (10.2) becomes
[® l2 f
*1
-h _
[» 2 2 l.
J » 2 ll
a x
(10.3)
a2
where
v11
>#21
' i j 12
•'ij 13
h j2 2
h j2 3
>#31
*ij 32
>,33
(10.4)
For each nonlinear large-signal scattering parameter Bgki the index i refers to the port
number of the b wave, the index j refers to the port number of the a wave, k is the
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harmonic index of the b wave, and / is the harmonic index of the a wave. The vectors
1
1
I
'—
1
aj and bt are (M=3)-element vectors:
1
&
U>
1
II
1-cT
II
isT
aj2
(10.5)
bn
bn _
Equation (10.3) can be expanded to
*1111
^1112
^1113
^1211
^\2\2
® 1213
a \\
^12
® 1121
^1122
® 1123
^1221
^1222
^1223
a \2
^13
® 1131
® 1132
^1133
® 1231
^1232
^1233
a tt
^21
^2111
® 2112
^2113
^2211
^2212
^2213
a 2\
^22
® 2121
^2122
^2123
^2221
^2222
^2223
a 22
_^2131
^2132
^2133
^2231
^2232
® 2233_
V
_^23_
_a 23
(10.6)
Note that in each of the four sub-matrices, the diagonal elements contain the samefrequency scattering parameters, the upper right elements contain the frequency
down-conversion scattering parameters, and the lower left elements contain the
frequency up-conversion scattering parameters. If the device under consideration
contains no nonlinearities (i.e. no power is transferred to other frequencies), then eq.
(10.6) reduces to
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0
0
^1211
0
0
au
^12
®hlll
0
® 1122
0
0
^1222
0
a \2
^13
0
0
® 1133
0
0
^1233
an
^21
^2111
0
0
® 2211
0
0
a 2\
^22
0
^2122
0
0
^2222
0
a 22
0
0
^2133
0
0
V
(10.7)
o
1
^ 2 2 3 3 . _a 23
which is the matrix representation of the well-known linear 5-parameters containing
three frequencies.
10.2. Nonlinear Large-Signal Impedance Parameters
Rather than expressing the relationship between a ’s and b's in terms of a
nonlinear large-signal scattering matrix &, we can alternatively express the
relationship between voltages (F s) and currents (Ps) in terms of a nonlinear largesignal impedance matrix Z, as follows
V = Z I,
(10.8)
where V and I are (iVxM)-element column vectors. Once again N refers to the number
of ports and M refers to the number of harmonics being taken into account. Z is an
(/VxAf)2-element square matrix.
For a two-port network with 3 harmonics, for example, eq. (10.8) becomes
> r '[2„] [Z12]
JZ 2 1 ]J [Z
22]
LK
2j 13*21
L*22JJL72j
72.
where
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(10.9)
7)'l 1
7/12
-021
7y22
7y23
7/32
7y33
[ * ,] =
(10.10)
For each nonlinear large-signal impedance parameter Zy«the index i refers to the port
number of the voltage V, the index j refers to the port number of the current I, k is the
harmonic index of V, and / is the harmonic index of I. The vectors Vt and Ij are
1
■Jy\
II
y,= V t2
7*3.
i
1
1
(M=3)-element vectors:
(10.11)
*J2
J jK
Equation (10.9) can be expanded to
Vn
Vn
Vn
v21
v22
V23_
7 lll
7112
7113
7211
7212
7213
~h1
7121
7122
7123
7221
7222
7223
^12
7l31
2^1132
7133
7231
7232
7233
^13
7 lll
7 l l2
7 ll3
^2211
7212
7213
7l
7l21
7122
7l23
7221
7222
7223
72131
7 l3 2
7l33
7231
7232
7233.
122
J23
(10.12)
10.3. Relating & and Z Matrices
The & and 2 matrices can be expressed in terms of one another, if we know
how a and b relate to V and I. From eq. (10.1), we can express Iji and Vtk in terms of
dji and bik as follows:
v ik = v ik + v ik ~ 4 ^ o i (a ik + bik) >
where the subscripts refer to the /'th port and at the Mi harmonic, and
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(10.13)
(10.14)
i j , = i ; + /} = Y f o - v ; ) = - j = = ( a/, - b „ ) ,
z °j
where the subscripts refer to the /th port and at the /th harmonic.
For simplicity, we will assume for now the network under consideration
consists of two ports. Later, we can easily generalize the equations relating the & and
Z matrices for any /V-port network. If we allow the two transmission lines or
waveguides connecting the two ports to have different characteristic impedances, Z0\
and Z02, eq. (10.14) can be expressed in matrix form as
X
\ u y z 01
[0 ]
E°]
[U]/Zo2
\
\
X
ik*i
-h_
X
(10.15)
Yi.
where [U] is the identity matrix. Eq. (10.9) can be expressed as
X
+X
Y2Y Yi.
"[Z11]
[Zi2]“ X
_[Z21]
[Z 2 2 L X
(10.16)
Combining eqs. (10.15) and (10.16) gives
X
+X
Y2Y Yi.
"[Z11]
[Z 12 ]
[U]/Zol
[0]
J * 2 l]
[ZaalJ
[0]
[ u y z o2^
Vx+
vs
vs
v;
(10.17)
or
r
+X
Y2Y Y2Y
'[Z n ]
.[Z 21]
/
\
X
<N
X
[Z22 ] J V Y2Y
X
Y2Y /
where
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(10.18)
[Zi2r
■[Zl'l]
[z12r \u y z 01
_[Z21] [z22].
[°]
[0]
[U]/Zo2_
"[Z11]
.[Z2 1 ] \Z 2 2 \_
(10.19)
is the normalized impedance matrix. Eq. (10.18) can be rewritten as
r r [Z n ]
[%2] 1 ~IW
M
[Z2 2 ].
l]
_1_
.[0]
[0]~
\
> f"
VI ) Y Y
f '[Zn] [z«r
V]
[0 ]'
[Z2 2 ]. _[0]
V _[Z'21]
\
> , +"
[U]_ /
Yi.
(10.20)
and eq.(10.3) can be rewritten as
[0]
V V ^Y i
[°]
[ [ » ! ,] [*12 ]" [ u v f e ^
_[*2l] [ » 2 2 l
[0]
> f"
[U]/y[Z^_ Y i .
[0]
[ u v 4 z 7 i_
h+J
(10.21)
Combining eqs. (10.20) and (10.21) allows us to solve for & in terms of Z:
"»n]
J * 2 l]
[*12 f
[*22 ]_
[0]
[UVyfZ,o2
[0]
T
\%\\] [%2]
M
i ]
[Z22L
/r[2n] [Z,'2]' +
[Z2 1 ] [Z2 2 1
\
'[U]
[0 ]“
.[° ]
V\ J
[U]
[0]'
[0]
[U]
-1
m j z 01
[°]
(10.22)
[0 ]
[U]/ J z o2 _
If Z0\=Z02 , eq. (10.22) reduces to
hi]
►
21]
►
i2]
»22]
‘[Zn] [z;2]' + '[£/] [0]T
_[Z21] [Z22]_ _[°] ml)
"[Z[,] [z;2]' "[tf] [0]T
.[Z2 1 ] [Z2 2 ]. _[°]
96
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(10.23)
Alternatively, we can combine eqs. (10.20) and (10.21) to solve for Z in terms of 0 :
' [U]
'[Z»] [z;2r
[Z2 2 L
~[U] [0]"
+
JO] [U]_
[0]
-1
[U]
[0]
M L
>11]
>12]
^21]
^22 ]
[0]
■\lZ o2
r [u]
[0]'
LEO]
[U]\
[V]
-1
[0]
[U]
yjZo2
J
\ _1
[0]
4 Zo1
4 Z oX
1
I” 1 0
~[U]
[0]
4 Z oX
[0]
[»22lJ
Ll»2l]
[0]
[U]
&
o2
(10.24)
If Z0\-Z 02 , eq. (10.24) reduces to
'[Zn]
\%2\
\% x\
[Z 22I
/
V
/
~[U]
[0]~
_[°]
[U}_
~[U]
\ _[°]
I»12l"
_j1_
J » 2 ,l
\
[» 2 2 l. /
\~
[0 ]'
" [ » ,,]
» 1 2 ]'
lul
J » 2 ll
[»22 ]_ /
(10.25)
10.4. Nonlinear Large-Signal Admittance Parameters
We can also express the relationship between voltages (T’s) and currents (/’s)
in terms of a nonlinear large-signal admittance matrix
as follows
I
(10.26)
where $ is an (7VxA/)2-element square matrix.
97
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For a two-port network with 3 harmonics, for example, eq. (10.26) becomes
in]
. ^^
IK
fcil
1
X
l,2] Vi
122]
(10.27)
where
1,11
1,12
1,13
« , ] = 1,21
1,22
1,23
1,31
1,32
1,33
(10.28)
For each nonlinear large-signal impedance parameter %jki the index i refers to the port
number of the current 7, the index j refers to the port number of the voltage V, k is the
harmonic index of 7, and / is the harmonic index of V. The vectors V/ and 7, are once
again (M=3)-element vectors, defined in eq. (11). Equation (10.27) can be expanded
as
X
I\2
In
1 21
1
tos
1 22
lllll
I 1121
lll31
I 2 III
12121
12131
llll2
ll 122
ll 132
12112
12122
12132
ll 113
ll 123
ll 133
12113
12123
12133
ll211
ll221
ll231
I 2 2 II
12221
12231
ll212
ll222
ll232
12212
12222
12232
ll213 >11
ll223 V 12
ll233 V ,3
12213 V 2 1
12223 V22
12233 723
(10.29)
10.5. Relating 9 and $ Matrices
The & and $ matrices can also be expressed in terms of one another, using eqs.
(10.13) and (10.14) which show how a and b relate to V and 7.
Once again, for simplicity, we will assume the network under consideration
consists of two ports. If we allow the two transmission lines or waveguides
98
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connecting the two ports to have different characteristic impedances, Z0\ and Z0i, eq.
(10.14) can be expressed in matrix form as
V
~[U]/Z01
h.
[°]
f77 77 \
[U ]/Zo2_ 77 7 C _
V
)
[0]
(10.30)
where [t/] is the identity matrix. Eq. (10.27) can be expressed as
\
Vi+ + V f
H a l . V 72 . 7C_ )
H i,]
ra n
.r a n
h .
(10.31)
Combining eqs. (10.30) and (10.31) gives
7 7
7 2
.
7 7
\ u y z 01
7 2\
[0]
-i
[0]
[U]/Zo2_
\
7 + + vx~
l22] V7 2\
727 )
(10.32)
+
(10.33)
ll2]
In]
*21]
r
- ____
-
—
or
7+
W l
v x~
In]
« « ]'
727
In]
[«22]
where
\ u y z ol
Itn]
JI2 1 ] [f22]_
[°]
[0]
-1
'ran ran
[U]IZo2_ ran ran
(10.34)
is the normalized admittance matrix. Eq. (10.33) can be rewritten as
/
[0 ]'
V _ [°]
1 '[ i n ]
[in f
J in ]
[ f 2 2 ].
/
7 7
727
V
\U ]
[0 ]"
JO ]
[U]_
m
' j
flK i]
V,
[ i l '2 r
[ W ill
J
and eq.(10.3) can be rewritten as
99
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(10.35)
!
!
[U]/Jz4
[0]
r^ni
[ ^ i2] i
ivyJzT,
[0]
_[®21 ]
[^22]_
[0]
[ £ /] /,/V
Yi.
[0]
kJ
(10.36)
Combining eqs. (10.35) and (10.36) allows us to solve for & in terms of
'[» „ )
l®12 ]"
\u yjz^
; ® 2 .i
[» 2 2 l.
[°]
/
~[U]
[0]
~[U]
[0 ]'
.[0 ]
R .
[0]'
11
" [In ]
luyjz. 2 . V .[0 ] VI fl& i]
\
' [ « » ] m 2] [ u v - j z ^
flE l]
[ I 22
(10.37)
-1
[o]
l u y j z ,o l
[0 ]
/
[%2] N-1
i%2]_
If Z0\=Z02 , eq. (10.37) reduces to:
/
►l2]
h i]
►n]
hi ]
V
f
V
\
"[£/] [O f " t i n ]
+
.[0] [tf]_
"[C/]
[0]'
.[0]
lu l
[in i'
[I22 ]_
(10.38)
\
'[ i n ]
i t u i
«2l] [I22 ]_/
Alternatively, we can combine eqs. (10.35) and (10.36) to solve for $ in terms of
rm
in ]
u
122]
[t/] [0]'
[0] [tf]_
-1
[0]
4 Z oi
[0]
[U]
-A
[V]
[[Sul [»«]] VZ.l
L[»2ll [®22lJ [0]
4 Z o l_
[U]
[U]
[0]'
[0]
[U]
[0]
[ [U]
[0]
[U\
[U]
yjZo2 \^
-a
-1
K.]
+
[0]
[0]
4 z oi
^221
[0]
o2
[U]
4 Z o2
(10.39)
If Z0\=Z02 , eq. (10.39) reduces to:
100
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-[fl]
flE i]
[ i : 2r
m 2]_
/
~[U]
[» ,2 lT
[0]'
J » 2 ll
VJO]
// \U]
[0]"
XJO]
[U]_
4- 'l& lll
I » 2 ,l
[®22lJJ
(10.40)
1*12 i f
[^22 ]_y
10.6. One-Port Network with Single-Tone Excitation
For a one-port network with a single-tone excitation at the fundamental
frequency, we can extract a reflection coefficient as
'11*1
_K
\ A
k
~ kK
la11
)
a \m
~ 0 f°r V/n(/n ^ l)
(10.41)
The limitation imposed on the equation is that all other incident waves other than a\\
equal zero. Instead of simply taking the ratio of b u to an, we reference the phase of
b\k to that of a\\. To do this we must subtract k times the phase of a\\ from b\k. This
concept is identical to the simplified case presented in eq. (9.8), where a\\ serves as
both the reference and the divisor.
For a one-port network with a single-tone excitation at the fundamental
frequency, we can show that the equation relating £? and 2 reduces to the same wellknown equation as for the linear case if we assume that no energy is created in the
form of frequency down-conversion. To illustrate this, we will once take into account
M=3 harmonics for the sake of simplicity. Eq. (10.6) reduces to
101
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hi
^ n ii
^1112
^1113
an
hi =
® 1121
^1122
^1123
0
.^ 1 1 3 1
^1132
^1133_
0
hi_
(10.42)
_
for a one-port network with a single-tone excitation a\\. This matrix can be rewritten
as a set of three equations:
Z>i =
1
n iai
; bn = ^ \ \ 2 \ aw
1
> h i = ^ i u i an -
(10.43)
Likewise, eq. (10.12) reduces to
Vn
^1111
^*1112
^1113
hi
Vn =
^*1121
^1122
^1123
hi
.^ 1 1 3 1
^1132
^1133 _
^13.
(10.44)
Ju
where the voltage at the first harmonic V\\ can be expressed as
^11 = ^llllAl + ^1112^12 ^’^'1113^13 •
(10.45)
From eqs. (10.13) and (10.14), we know
^ii = y[^oiiau + ^ n ) ’
h i = ~T==iau ~ h i ) ’
\ Jo\
0
h i ~ rz— (ai2 h i ) - '
\
hi ~
o\
r r — (a \l ~ h l ) ~ ~
Jil
(10.46)
y [ z7 i’
hi
^01
■yj^oi
Combining eqs. (10.45) and (10.46) gives
y [z^ (a n +bn ) — i——[%xm (au
v
bn ) Z1112Z>12 ^ iiiih i\-
ol
Substituting eq. (10.43) into eq. (10.47) and solving for 2 im gives
102
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(10.47)
Z o l( l +
^ 1 1 1 1 ) +
*1111 =
% n 12 ^1 1 2 1
(l
+
(10.48)
^ 1 1 1 3 ^ 1 1 3 1
m)
If no energy is created in the form of frequency down-conversion (i.e.,
Ziii2=Zni3=0), then eq. (10.48) reduces to the same equation as in the linear case:
1049
(1 + S n )
( . )
Zll=Zol(l-S n )'
A similar derivation can be performed to show that
M
1050
( . )
— H i 1 2 ® 1 121 ~ H l l 3 ^ 1 1 3 1
mi
Once again, if no energy is created in the form of frequency down-conversion (i.e.,
%\\\2- § \\n - ^ ) , then eq. (10.50) reduces to the same equation as in the linear case:
1
5
r,,=— = f -15'1?.
z„ z„,(i+su)
(io. i)
10.7. Two-Port Network with Single-Tone Excitation
For a two-port network excited at port 1 by a single-tone excitation at the
fundamental frequency, we can extract an input reflection coefficient as
'm i
\K \A
k
\aii
~ kK )
a
\
/
\T
am„ = 0 for \fm\/n[(m ^ l) a (n * l)j
(10.52)
As with eq. (10.41), instead of simply taking the ratio of bu to a\\, we phase
reference to a\\. To do this we must subtract k times the phase o f an from b\k. And
the limitation once again imposed on the equation is that all other incident waves
other than a\ \ equal zero.
103
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Another valuable parameter, the forward transmission coefficient, is similarly
extracted as
_ \ b 2k \ A K ~ k K )
2141
\a.11
amn ~ ®f°r
± l) a (n *
l)]. (10.53)
This parameter provides a value of the gain or loss through a device either at the
fundamental frequency or converted to a higher harmonic frequency.
In addition to the previous two parameters, shown in eqs. (10.52) and (10.53),
an output reflection coefficient can also be useful when trying to determine the output
matching network. If a nonlinear DUT is operating under its normal drive condition
(>a\\ at some constant signal level), and a second source, excited by a small-signal
tone at frequency fk, is placed at port 2 of the DUT, one of the equations in the matrix
found in eq. (10.6) reduces to
^24 = ^2\k\a\\ + ^22kka2k •
(10.54)
If we solve eq. (10.54) for 82244, we get
v22kk
21kV 11 .
a2k
(10.55)
a2k
From eq. (10.55), it is obvious that the output reflection coefficient 82244 cannot be
determined by simply taking the ratio of 624 to a^k, since it also depends on a\\
through 8*2i4i- When we have small levels of aik, we can generate another signal A«24
that is offset slightly from the frequency of interest fk by Afk. Doing this, eq. (10.54)
becomes
104
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^2k + ^ b 2k - ^ >2 m a U + ^22kk(a 2k + ^ ° 2 k ) ’
where Aci2k «
(10.56)
<*ik and S»22m remains constant over this frequency range. Subtracting
eq. (10.54) from eq. (10.56) gives
^b>2k — ^22kk^a 2k >
(10.57)
which does not depend on &2Ui-If we solve eq. (10.57) for ^ 22kk,
_ ^b 2k
?22kk ~
A
Aa2k
(10.58)
Large a xj , Small A a 2k
Eq. (10.58) is a quasi-linear approximation of the output reflection coefficient under
normal operating conditions, and is consistent with the definition of “Hot £ 22,” which
has been used to measure the degree of mismatch at the output port of a power
amplifier at its excitation frequency.
10.8. Discussion
In this chapter, we presented the general form of nonlinear large-signal
parameters. Unlike linear A-parameters, nonlinear large-signal ^-parameters depend
upon the signal magnitude and must take into account the harmonic content of the
input and output signals since energy can be transferred to other frequencies in a
nonlinear device. We also introduced nonlinear large-signal impedance (2) and
admittance ($) parameters, and presented equations for relating the different
representations. Next, we made two simplifications, considering the cases of a oneport network with a single-tone excitation and a two-port network with a single-tone
105
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excitation. For the one-port case with a single-tone excitation at the fundamental
frequency, we showed that the equation relating % and Z reduces to the same wellknown equation as for the linear case if we assume that no energy is created in the
form of frequency down-conversion. For the two-port case excited at port 1 by a
single-tone excitation at the fundamental frequency, we extracted an input reflection
coefficient
a forward transmission coefficient l5>2ui, and a quasi-linear output
reflection coefficient: nikk-
106
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CHAPTER 11
EXPANDING DEFINITIONS OF GAIN BY TAKING
HARMONIC CONTENT INTO ACCOUNT
In this chapter, we expand the definitions of power gain, transducer gain, and
available gain by taking harmonic content into account. Furthermore, we show that
under special conditions, these expanded definitions of gain can be expressed in terms
of nonlinear large-signal scattering parameters. These expanded forms of gain and
nonlinear large-signal scattering parameters can provide us with valuable information
regarding the behavior of nonlinear models.
11.1. Commonly Used Definitions of Gain
Three types of gain are commonly used as figures of merit for two-port
networks - power gain, transducer gain, and available gain [59]. Power gain G is
defined as the ratio of the power delivered to the load Pi connected at port 2 of the
device to the input power Pm at port 1 of the device
(11.1)
where Pl and Pm can be described in terms of wave-variables referenced to some
real-valued impedance
(11.2)
and
107
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d
1 k I2
= "21 '
*1 I2
•
(11.3)
Here at and bi refer to the complex incident and reflected power-normalized waves,
respectively, where the subscript / denotes the port number. Transducer gain G j is
defined as the ratio of the power delivered to the load Pi to the power available from
the source P a ys
Gt s - A .
(11.4)
AVS
The power available from the source P ays is the maximum power that can be
delivered to the network. This occurs when the input impedance of the terminated
network is conjugate matched to the source impedance, and can be described in terms
of power-normalized waves as
Pays ~ Pin
r*s
1rin —
~1
z2
(11.5)
Available gain GA is defined as the ratio of the power available from the network
P avn to the power available from the source P ays
G = E d Z iL
(11.6)
AVS
The power available from the network P avn is the maximum power that can be
delivered to the load. This occurs when the output impedance of the terminated
network is conjugate matched to the load impedance, and can be described in terms of
power-normalized waves as
108
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AVN
~
r
(11.7)
I h l 2-
OUT
One limitation of the gain definitions described in eqs. (1), (4), and (6) is that they are
restricted to the fundamental operating frequency, and thus do not take into account
energy at higher harmonic frequencies.
11.2. Expanded Definitions of Gain
In the case of a sinusoidal input to a two-port nonlinear device, power may be
transferred to higher harmonic frequencies. Thus, we can modify the definitions of
power, described in the previous section by taking into account the harmonic
contributions and summing over all K harmonics considered. First, the input power
P in
given in eq. (11.2) can be modified to
d
_
1 v I
I2 _
l k=\
1 v
l k
Ia
(11.8)
I2
=i
Here, a,* and bik refer to the complex incident and reflected power-normalized waves,
respectively, where the subscript i denotes the port number and k denotes the spectral
component number. All waves must be taken with respect to some real-valued,
reference impedance. Similarly, we can modify the other definitions as follows:
(11-9)
I k=\
AVS
k=\
A-k=\
\k \
’
and
109
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(11.10)
Pavn —~'E\b2k\ •
(H-H)
Z- k=\
Equations (11.8-11.11) are valid under the assumption that the power in the network
is confined to a grid of frequencies that are harmonically related.
Here, the expanded power gain (S is defined as the ratio of Pl to P in and is
given by
l^ i
P
"
i2
l^ i
i2
(11.12)
— EM
Z k=\
z k=l
the expanded transducer gain 0f>7 is defined as the ratio of P l to P a vs and is given by
P
p" s
and the expanded available gain
k k f - k k l 2
2 ‘=1
2 i - ‘------- •
^Ekl
Zk=\
(11.13)
C$a
is defined as the ratio of P a v n to P A vs and is given
A
r AVN _ Z k= 1
P
1 K ,
Pavs
by
* v Ia I2
(11.14)
,2
Z k=\
110
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11.3. Expanded Definitions of Gain in Terms of ^-Parameters
Next, we show that under special conditions, the expanded definitions of gain
can be expressed in terms of the two-port, nonlinear large-signal scattering
parameters introduced in the previous chapter.
When we extract the nonlinear large-signal scattering parameters I$iui and
&2ui, described in eqs. (10.52) and (10.53), only aw is present and all other o ’s are
forced to zero. In this case, eq. (11.8) is reduced to
l k =1
z
Likewise, eq. (11.9) becomes
2 k =1
and eq. (11.10) simplifies to
Pavs
A2 k iif-
(n -17>
Equation (11.11), however, remains unchanged as
Pa w 4 S M
Zk=\
2.
(H-18)
since it contains no ay terms.
In this case, where only a\\ is present and all other a’s are forced to zero, the
expanded power gain (6 can be expressed as
ill
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1 £ |,
Z*=l
■IN
±.\ a
ii
|2
(11.19)
'*=1
i*
Dividing both the numerator and denominator of eq. (11.19) by |tfn|2, and substituting
in eqs. (10.52) and (10.53) gives the power gain in terms of the nonlinear large-signal
scattering parameters:
K
2\k\
k=\
(11.20)
K
1*=l
itti
The power gain confined to the nth harmonic frequency is
'2 1 n l
’n
k
(11.21)
*=1
The ratio (S«/(S expresses the fraction of the power gain confined to the nth harmonic
frequency compared to the power gain over all of the harmonic frequencies
considered.
Likewise, we can express transducer gain and available gain in terms of
nonlinear large-signal scattering parameters. With only an present and all other a’s
forced to zero, (Br and (§u reduce to the same expression:
= « * = £ |»21«|
*=1
(11.22)
Here, the transducer, or available, gain confined to the nth harmonic frequency is
112
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|» 2,„ |2 •
Once again, ratio
(H-23)
or C e x p r e s s e s the fraction of the transducer,
or available, gain confined to the nth harmonic frequency compared to the transducer,
or available, gain over all of the harmonic frequencies considered. Note that (8„/(S =
11.4. Discussion
In this chapter, we expanded the definitions of power gain, transducer gain,
and available gain by taking harmonic content into account. We also showed that
under special conditions, these expanded definitions of gain can be expressed in terms
of nonlinear large-signal scattering parameters.
In the following three chapters we will present three examples illustrating
applications and extraction of nonlinear large-signal ^-parameters and the expanded
definitions of gain. First, we will show how nonlinear large-signal ^-parameters and
the expanded definitions of gain can be used to discover valuable information
regarding the behavior of a nonlinear model. Specifically, we examine a lumpedelement model of a pseudomorphic high electron mobility transistor (pHEMT) device
operating in a two-port, common-source configuration. Second, we illustrate how
nonlinear large-signal ^-parameters can be used to as a tool in the design process of a
simple nonlinear circuit, specifically a single-diode 1-2 GHz frequency-doubler
circuit. And finally, we describe a method for generating nonlinear large-signal &>parameters based upon ANN models trained on frequency-domain data measured
113
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using a nonlinear vector network analyzer (NVNA). We compare a diode circuit
model, generated using this method, to a harmonic balance simulation of a
commercial device model.
114
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CHAPTER 12
USING ^-PA RA M ETERS AND EXPANDED DEFINITIONS OF GAIN TO
EXAMINE THE BEHAVIOR OF A PHEM T MODEL
In this chapter, we provide an example showing how nonlinear large-signal
scattering parameters along with the expanded definitions of gain, introduced in the
previous chapter, can be used to discover valuable information regarding the behavior
of a nonlinear model. Specifically, we examine a lumped-element model of a 2x90
pm GaAs pseudomorphic high electron mobility transistor (pHEMT) device
operating in a two-port, common-source configuration, shown in Figure 12.1. The
pHEMT model was developed by Cidronali et al [60, 61] using 5-parameter
measurements and electromagnetic simulations of the device layout.
GS
DS
ax
Figure 12.1. Circuit diagram of a pHEMT device operating in a two-port, commonsource configuration.
115
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I
j
I
j
]
|
12.1. Using 0-Parameters to Examine the Behavior of a PHEMT Model
First, we will look at 0 m i and 02 ui as a function of power for various bias
conditions, and then we will reduce the 0-parameter data set by calculating the
expanded power gain (E, the expanded transducer gain <&r, and the ratio of the power
!
gain confined to the first harmonic to that of the overall power gain (Si/(S. We
examine the model operating at a frequency of 5 GHz and powers ranging from a
j
|
small-signal level of -20 dBm to a level of 10 dBm, which is close to the maximum
|
device rating. Since the model is valid for an ^-channel device, the drain is biased
i
j
|
|
j
i
I
positively with respect to the common source. In this case, we choose Vus = 3 V. The
I
I
j
we vary
gate is biased negatively with respect to the source since this condition controls the
width of the depletion region and blocks part of the conducting channel region. Here,
V gs
from 0.0 V to -1.2 V in steps of 0.2 V.
We can easily determine the two-port, nonlinear large-signal scattering
parameters, described in eqs. (10.52) and (10.53), as a function of power and bias
using a commercial harmonic balance simulator with all a’s other than an forced to
zero. Figure 12.2 plots the magnitudes of 0 i u i for the first four harmonics (k = 1,2,
|
1
!
!
3, and 4). We see that |0 im | remains relatively flat at all bias conditions with varying
input power \au\- The value of |0 n n | decreases as Vgs decreases from 0 V to -0.2 V,
reaches a minimum of -2.325 dB at
Vgs
= -0.2 V, and then increases as
Vgs
is further
decreased from -0.2 V to -1.0 V. The parameters |0i 121(, |0i 1311, and |0n4i| generally
increase with input power at all bias conditions. This reveals that as the input power is
increased, more energy is converted to higher harmonic frequencies and appears at
port 1.
!
116
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0V as
-
20-
“ O.OV
S
'S' -2 0 •a
lin
1121
I
1121
|> -4 0 2
1131
I
mi
1141
1131
1141
-80
-20
-10
0
-20
10
-10
0
10
0
10
0
10
a n (dBm)
a xj (dBm)
V r,, - -0.4 V
m
C$
-4 0 -
s
1111
1111
1121
1121
1131
1131
1141
1141
-6 0 -80
-20
-10
0
-20
10
-10
an (dBm)
a, j (dBm)
20 -i
0
-
Va , ~ -0.8 V
CO
-a
^-o -20
3
§> -40
I
-6 0 -
-20
"O
g
1
| -4 0 I
-6 0 -
1121
1131
1141
1121
1131
1141
-80
0
-20
10
-20
-10
a, j (dBm)
a , . (dBm)
Figure 12.2. Magnitude of S i u t as a function of input power for a nonlinear lumped
element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of
VDs= 3V and VGS = 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V.
117
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 12.3 plots the magnitudes of S>2«i for the first four harmonics (k = 1, 2, 3,
and 4). At small input signals, |02in| increases as VGS decreases from 0 V to -0.4 V,
reaches a maximum of 15.481 dB at VGs = -0.4 V, and then decreases as VGS is further
decreased from -0.4 V to -1.0 V. The parameter |02m| remains relatively flat and then
gradually decreases with increasing |«n| for values of VGs from 0 V to -0.6 V. At VGs
less than -0.6 V, |02in| actually increases with increasing \au \ due to self-biasing. We
also see that |0 2 i2i|, |0 2 i3i|, and |0 2 i4i| generally increase with input power at all bias
conditions. This reveals that as the input power is increased, some of the energy
converted to higher harmonic frequencies appears at port 2.
To gain some physical insight into the pHEMT model, we can examine the
nonlinear large-signal ^-parameter data near pinch-off ( VGS = -1.0 V and Vos - 3 V)
and near Ids ~ Idss
(V gs
- 0 Y and Vos - 3 V). In these two regions, the second
harmonic content is maximum (02 121 > 02u-i for k > 2), which can be seen in Figure
12.3. Near the pinch-off region, the device draws current only for the positive part of
the gate voltage waveform, which results in a clamped waveform drain current, while
near Ids ~ Idss, the device saturates at the positive part of the gate voltage waveform
and the device draws current only for the negative portion. Thus, taking the Fourier
transform gives even harmonics of the same amplitude in both cases, but they are
180° out of phase. From Figure 12.4, we can see that near pinch-off
the phase of 02i2i is approximately 160°, while near I d s
=
(V gs
= -1.0 V),
loss ( V g s = 0 V), the phase
of 02i2i is approximately -50°. The difference in phase is about 210°, which is close
to the expected value of 180°.
118
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
20 n
OH
1o -2 0 -
-
£
si
Si
*
“
-4 0 -
rGS =o.ov
%2111
&2121
-6 0 -80-20
.g
1
1
-10
r
o
V™ = -0.2 V
-4 0 -
2
-6 0 -
&2131
&2141
"l
o“
-2 0 -
-80-20
1
10
axj (dBm)
20
0
0
-
m
T3
-20-
10
-6 0 -
-60 H
“ i-----1
-20
-20 h
M
ITO -4 0 -
-4 0 -
-80
-
T3
1
|
I
\
-10
o
a xx (dBm)
20-i
e
^
—i— 1----r
r
-10
0
axj (dBm)
-80
n
-20
10
r
~i-----1
-10
0
10
axx(dBm)
20- |
20 -i
OH
CQ
TD
^
T3
-20 h
^
”2
.n
I
1
ICg -40'
-20-
= -1.0V
-4 0 -
2111
2121
-6 0 -
-60
2131
2141
-80
-80
~_l----- 1— '----- 1— 1— I
-20
-10
0
a xj (dBm)
-20
10
-10
0
10
a x! (dBm)
Figure 12.3. Magnitude of &21/H as a function of input power for a nonlinear lumped
element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of
VDS= 3V and VGS = 0.0, -0.2, -0.4, -0.6, -0.8, and -1.0 V.
119
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
200
200
-
-
2111
100-
2121
P
03
1
<D
2131
o-
0-
2141
l-i
<
1121
-100
1131
-
—
100-
1141
-20
-15
5
-10
0
-20
CO
8
If
"O
-15
-10
5
0
a n (dBm)
# ii (dBm)
200-
200-
100-
100
—
s
i
o-
3
<
0-
2121
1131
-
100-
-20
2131
-1 0 0
1141
- 10
-
2141
0
-20
10
200-
200
100
100-
0
10
-
S3
<u
0-
o1121
fgs =-i .ov
— » 2111
2 121
2131
131
-100
-10
a n (dBm)
a xj (dBm)
U
<
2111
1121
-100
-
-
2141
1141
-20
-10
0
a n (dBm)
-20
10
-10
0
a n (dBm)
10
Figure 12.4. Phase of
and &2ui as a function of input power for a nonlinear
lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a
bias of Vds = 3V and VGS = 0.0, -0.4, and -1.0 V.
120
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12.2. Using Expanded Definitions of Gain to Examine
PHEMT Model
the Behavior of a
Rather than looking at numerous ^-parameter graphs, we can get a more
concise view of the modeled behavior of a device by reducing the nonlinear largesignal ^-parameter data set into the compact expression of power gain (S. Figure 12.5
plots the calculated values of (S using eq. (11.20) for K = 4, as a function of power for
Vqs varying from 0.0 V to -1.2 V. At small signals, the power gain increases as Vgs
decreases from 0 V to -0.4 V. The power gain reaches a maximum of 19.72 dB at
= -0.4 V, and then decreases as
V gs
V gs
is further decreased from -0.4 V to -1.2 V. This is
consistent with the fact that the trans-conductance gm, which is proportional to Y2\ and
hence £ 21, peaks at a bias of
Vqs
= -0.4 V. We can also see from Figure 12.5 that the
gain stays relatively flat and then gradually decreases with increasing \au\ for values
of
V gs
from 0 V to -0.6 V. At
Vg s
less than -0.6 V, the gain increases with increasing
\an \ due to self-biasing and harmonic production.
2 0 -i
-0.4 V
15-
-0.2 V
-0.8 V
"°-6V
vGS(S)
10-
— 0.0
-
50-
-20
-1.0V
-1.2 V
-15
0.2
— -0.4
— -
-
—
-
-
-10
5
axj (dBm)
0
5
-
0.6
0.8
1.0
- 1.2
-
10
Figure 12.5. Expanded power gain (S as a function of input power for a nonlinear
lumped element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a
bias of VDS= 3V and VGS = 0 .0 , -0 .2 , -0.4, -0 .6 , -0 .8 , - 1 .0 , and - 1.2 V.
121
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
We can also reduce the nonlinear large-signal ^-parameter data set by using the
compact expression of transducer gain d r • Figure 12.6 plots the calculated values of
d r using eq. (11.22) for K = 4, as a function of power for
Vg s
varying from 0.0 V to
-1.2 V. At small signals, the power gain increases as Vgs decreases from 0 V to
-0.4 V. The transducer gain reaches a maximum of 15.48 dB at
decreases as
V gs
Vqs
- -0.4 V, and then
is further decreased from -0.4 V to -1.2 V. And like power gain, the
transducer gain stays relatively flat and then gradually decreases with increasing |an|
for values of
Vgs
from 0 V to -0.6 V. At
Vq s
less than -0.6 V, the transducer gain
increases with increasing \an \ due to self-biasing and harmonic production.
2010
|
-
-0.4 V
-°-2 V
0-
-1 0 -
-0.6 V
-0.8 V
0.0
-1.0 V
-1.2 V
-
0.2
-0.4
-
0.6
-
1.2
-20
-20
-15
-10
5
a {j (dBm)
0
5
10
Figure 12.6. Expanded transducer gain d r as a function of input power for a
nonlinear lumped element model of a 2x90 pm GaAs pHEMT device operating at 5
GHz and a bias of Vd s ~ 3V and V q s = 0.0, -0.2, -0.4, -0.6, -0 .8 , -1.0, and - 1.2 V.
Comparing the plots of power gain and transducer gain in Figures 12.5 and 12.6,
we can see that the values of transducer gain are less than those of power gain, due to
the large values of H>iin which inflate the values of power gain. Another difference to
122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
note is that the power gain at VGS = -0.4 V is always greater than at
Vgs
- -0.2 V, but
the transducer gain at VGs = -0.4 V is only greater than at VGs - -0.2 V up to |an| = 2.6 dBm. At higher input powers, the transducer gain is slightly higher at Vgs = -0.2
V than at Vgs = -0.4 V.
Figure 12.7 plots the ratio of the power gain confined to the first harmonic to
that of the overall power gain, (&i/CS. Recall that ($|/(S = (Sri/®r- At small signals,
(&i/(S is relatively high for all values of
V g s,
but is at a maximum of 0.9958 for
-0.4 V. As |an| increases, ($i/($ remains relatively high for
At
Vgs
V gs
Vgs
-
at -0.2 V and -0.4 V.
less than -0.4 V, (Hu/d decreases dramatically with increasing |an| since much
of the energy in the device is converted to higher harmonic frequencies. In fact, at VGs
= -1.2 V, (&i/(8 drops to 0.731.
1.00
0.95
-0.4 V
-0.2 V
V:;s(V)
0 .9 0 -
0.0
- 0.2
—
— -0.4
-
-
0.6
-
-
-
0.8
- 1.0
- 1.2
—
-1.0 V
-
0 .7 5 -
-1.2 V
0.70
-20
-15
-10
5
a ]| (dBm)
0
5
10
Figure 12.7. The ratio ®i/(K as a function of input power for a nonlinear lumped
element model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of
Vds= 3V and VGS = 0 .0 , -0 .2 , -0.4, -0 .6 , -0 .8 , - 1 .0 , and - 1.2 V.
123
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
From Figures 12.5, 12.6, and 12.7, we can clearly see that the optimum gateto-source bias condition is near Vgs - -0.4 V for the device at Vos = 3 V, where d ,
d r, and d i / d are highest. Furthermore, if we look at the power gain at this particular
bias condition, shown in Figure 12.8, we find the 1 dB gain compression point to be
didB = 18.716 dB, which occurs at \aw\~ 1.40 dBm.
Also shown in Figure 12.8 is the traditional power gain G. For this example,
where a\\ is the only incident wave present, G can be expressed in terms of nonlinear
large-signal scattering parameters as:
O = l°'2l" l —
(12.1)
H & in if
We see from Figure 12.8 that at small input signals d and G are nearly identical,
but at higher powers d is greater than G since the traditional power gain does not take
into account the power generated at higher harmonic frequencies. Using G, we get the
same 1 dB gain compression point of GidB = 18.716 dB, but it occurs at |tfn| = 0.78
dBm.
124
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2 0 .0 -1
19.0-
ldB
ldB
18.0•g 17.0g
16.015.0-
—
G
14.013.0
-20
-15
-10
a
5
x, (dBm)
0
5
10
Figure 12.8. The 1 dB gain compression point for a nonlinear lumped-element
model of a 2x90 pm GaAs pHEMT device operating at 5 GHz and a bias of Vos - 3V
and VGS = -0.4 V.
12.3. Discussion
We provided an example showing how the expanded definitions of gain and
nonlinear large-signal scattering parameters can be used to examine the behavior of a
nonlinear model by simply performing a harmonic-balance simulation with all a's
other than a\\ forced to zero. Looking at the nonlinear large-signal scattering
parameters gives us an in-depth view of the modeled behavior by allowing us to
separate out the input reflection coefficients and transmission coefficients at each of
the frequency components, while reducing the nonlinear large-signal scattering
parameter data set into the compact expressions of power gain (S, transducer gain (Sr,
and the ratio (&i/(S, gives us a more concise view of the modeled behavior.
125
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CHAPTER 13
USING NONLINEAR LARGE-SIGNAL ^-PARAM ETERS
TO DESIGN A DIODE FREQUENCY-DOUBLER CIRCUIT
W ITH A HARMONIC-BALANCE SIMULATOR
Resistive frequency doublers operate on the principle that a sinusoidal
waveform is distorted by the nonlinear UV characteristic of a Schottky-barrier diode
[62]. This distortion causes powers to be generated at higher harmonic frequencies.
The design o f such doublers reduces to separating the input and output signals by
filters and determining the optimum input and output matching circuits, as illustrated
in Figure 13.1.
Although single-diode resistive doublers are not very efficient (analysis
predicts a conversion loss of at least 9 dB [63]), we choose this circuit because it is
simple enough to clearly illustrate how nonlinear large-signal ^-parameters can be
used as a design tool.
Input
Matching
Network
Output
Filtering
Network
Input
Filtering
Network
Output
Matching
Network
Figure 13.1. Block diagram of a single-diode resistive doubler.
126
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In the following sections, we describe the various steps involved in designing
a single-diode 1-2 GHz frequency-doubler circuit. Since we are using a simulator, we
can force the stimulus to consist of only |a\\\, with all other amn = 0 , where m and n
are positive integers such that m * 1 and n ^ 1. (This condition can never be
completely realized in a measurement environment.) With only an an component
present, we need only consider the parameters &n*i (eq. 10.52), a measure of the
large-signal input match at the Mi harmonic, &21« (eq- 10.53), a measure of the largesignal conversion loss or gain at the Mi harmonic, and the quasi-linear B 2222 (eq.
10.58) to determine the output matching network at the second harmonic. Figure 13.2
illustrates the setups required for determining these parameters. Determining B 2222
requires a second source at port 2 at a frequency slightly offset from CO2 •
G1
inti
Nonlinear
DUT
21k \
(a)
G2
G1
Nonlinear
DUT
2222
(b)
Figure 13.2. Nonlinear large-signal ^-parameters used to characterize a two-port
device excited by a single-tone signal at port 1.
127
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In the first step, we perform a simulation on the diode alone and use #2121 to
determine the optimum bias condition for converting power from the fundamental
frequency to the second harmonic. Second, we add filtering networks to separate the
input and output signals, and verify their proper performances by looking at &2111 and
§>1121. Third, we make use of §>1111 to determine the input matching network. Fourth,
with the input matching network in place, we place a second source at port 2 and find
the quasi-linear value of ^
2222 ,
which allows us to determine the output matching
network. Fifth, we use the optimization feature of the simulator to minimize & im b y
varying the line lengths of the input and output matching circuits. And finally, sixth,
we add 4 GHz and 6 GHz filters at the output (and re-determine the proper input and
output matching circuits) in order to reduce the values of #2141 and # 2161, which in
turn increases the value of &2121 and cleans up the output waveform.
13.1. Diode Only
In this example, we use a compact model to simulate a commercial Schottkybarrier diode. The model includes a series resistance, Rs, of 14 Q, a junction
capacitance at zero voltage, C/0, of 0.08 pF, and a reverse saturation current, Is, of
3xlO'10 A.
First, we perform a harmonic-balance simulation on the diode, sweeping the
bias voltage to determine which condition gives the highest value of &2121 for an 1.0 V. Note that in all simulations we set the generator impedance, Zg, and the load
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 13.1. Simulated values for & im - &ii 6i, &2111
of the design stages of the diode frequency doubler.
Quantity
1£ i m |
| £ l l 2 l|
1® 1 1 3 l l
l& 114l|
l& 115ll
| £ l l 6 ll
l & 211 ll
l & 212 l |
1& 2 1 3 l l
1^2141
l® 215l|
l& 216ll
(dB)
® 2/<S
<62
Diode Only
0.464
0.170
3 .2xl0'2
2.4x10‘2
1.7xl0"2
3 .9xl0'3
0.536
0.170
3.2x1 O'2
2.4xl0'2
1.7x1 O'2
3.9x10'3
-14.16
0.091
*2 1 6 1 :
($2 , and (£2/(6 for each
9.4x1 O'2
8.8x1 O'6
4 .0 x l0 '3
Diode w/
1 ,2 GHz
Filters,
Input &
Output
Match
8.7x10‘2
8.0x1 O’6
1.4x1 O'2
Diode w/
1 ,2 GHz
Filters,
Input &
Output
Match Opt.
6.0xl0'3
9.5x1 O’6
l.lx lO '2
Diode w/
1 ,2 , 4 ,6
GHz Filters,
Input &
Output
Match Opt.
2.1xl0'4
9.9x10'6
2.2x1 O'2
3.7x1 O'2
2.4x1 O'2
1.9x10'3
9.7xl0'7
4.0x10'5
0.328
1.5x10'6
5.1X10'2
2.5x10’3
3.3xl0'5
0.268
3.5x1 O'7
l . l x l O'2
1.0x1 O'6
4.0x10'5
0.326
3.3xl0'7
2.8xl0'2
2.3x10'3
l.lx lO '6
4.0xl0'5
0.331
l.lx lO '6
2.0x1 O'6
5.0x1 O'5
0.332
1.7x1 O'7
3.5xl0'2
4.5x1 O'2
4.1xl0'2
4.0x1 O'2
1.4x1 O'6
7.6x1 O'7
2.0x1 O'2
-9.73
0.978
l.lxlO"6
2.5 x l0 '2
-9.69
0.976
2.5x1 O'6
2.6x1 O'2
-9.65
0.979
2.3X10'6
2.9 x l0 '2
3.0x1 O'6
2.7x1 O'6
-9.56
0.999
Diode w/
1, 2GHz
Filters
Diode w/
1 ,2 GHz
Filters,
Input Match
0.569
1.3x1 O'5
4,9x10'3
3.5xl0'2
l.lx lO '2
1.0x1 O'6
-9.60
0.978
impedance, Zi, to 50 Q. After sweeping the voltage, we determine that the optimum
forward bias is +0.48 V.
13.2. Diode with 1 & 2 GHz Filters
With a stimulus of an = 1.0 V and a forward bias of +0.48 V, we add filtering
networks to separate the input and output signals. On the input side, we place a 2
GHz, A/4 (A/8 at 1 GHz) open-circuited stub. This creates an RF short at 2 GHz,
preventing the output power generated in the diode from traveling backward. On the
output side, we place a 1 GHz, A/4 open-circuited stub. This creates an RF short at 1
GHz, preventing any signal at 1 GHz from traveling forward.
129
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Table 13.1 lists the simulated values for 8 »im - 0n6i and ^2111 - ^ 2161, as
well as the power gain confined to the second harmonic CS2 and the ratio OS2/®, for
each of the design stages. With the 1 GHz and 2 GHz filters in place, we see that the
value of IS'iml decreases from 0.170 to 1.3xl0'5, the value of |^ 2in| decreases from
0.536 to 3.3xlO'5, and (§2 increases from -14.16 dB to -9.73 dB, when compared to
the diode-only case.
13.3. Diode with 1& 2 GHz Filters and Input Matching
Once the filters are placed in the circuit, we make use of the complex-valued
&1111 to design the input matching network with the well-known single, open-
circuited stub technique. This is possible, assuming that no energy is created in the
form of frequency down-conversion, as discussed in Section 10.6. We see in Table
13.1 that
m il reduces from 0.569 without the input matching network to 9.4x10
-2
with the input matching network in place. Likewise, (§2 increases from -9.73 dB to
-9.69 dB.
13.4. Diode with 1& 2 GHz Filters, Input and Output Matching
Whereas our input matching network is designed for 1 GHz, our output
matching network must be designed for 2 GHz. While the circuit is operating under
its normal drive condition (an = 1.0 V and a forward bias of +0.48 V) we place a
second source at port 2, excited by a small-signal tone (A«22 = 0.01 V) at a frequency
offset of 10 kHz from the desired 2 GHz, to give us the quasi-linear value of
130
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
§>2222 ,
which allows us to determine the output matching network. We make use of ^ 2 2 2 2 to
design the output matching network with the well-known single, open-circuited stub
technique. We see in Table 13.1 that with the output matching network in place, the
value of I&21211is only marginally increased from 0.326 to 0.328. This is because the
value of §>2222 is relatively low, which means the output is already almost matched to
50 Q. We also note that (82 increases from -9.69 dB to -9.65 dB.
13.5. Diode with 1& 2 GHz Filters, Input and Output Matching Optimized
With the filters and matching networks in place, we use the optimization
feature of the simulator to minimize #1111 by varying the lengths of the lines in the
input and output matching circuits. Doing this decreases the value of |&mi| from
8.7xl0 '2 to 6.0xl0 '3 while increasing the value of I&2121I from 0.328 to 0.331 and (82
from -9.65 dB to -9.60 dB.
13.6. Diode with 1& 2, 4 & 6 GHz Filters, Input and Output Matching
Optimized
From Table 13.1, we see that at the output port, |02in|> I&2131I, and I&2151I all
have values less than or equal to 4.0xl0'5, but I&2141I and |&2i6i| have noticeably
higher values (at least 2.9x10'2).
In order to clean up the output waveform, we add 4 GHz and 6 GHz filters, in
the form of A/4 open-circuited stubs, at the output. With these filters placed in the
circuit, we re-determine the proper input and output matching conditions. After
optimizing the circuit once again, the value of I&2141I decreases from 4.0x10 '2 to
131
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.4x10' and the value of |&2i6i| decreases from 2.9x10' to 2.7x10". The addition of
these filters, in turn, slightly increases I&2121I from 0.331 to 0.332 and (S2 from -9.60
dB to -9.56 dB. At this final design stage, the overall power gain is very near -9.56
dB since the ratio CI2/® = 0.999. The semi-empirical analysis of [62] predicts a
maximum gain of -9 dB. Figure 13.3 illustrates the final design of the single-diode
resistive doubler circuit. And Figure 13.4 shows the time-domain plots of a\ and bi
for the final design o f the simulated 1-2 GHz frequency-doubler circuit.
13.7. Discussion
We illustrated how nonlinear large-signal ^-parameters can be used as a tool
in the design process of a single-diode 1-2 GHz frequency-doubler. Specifically, we
used i n n to determine the input matching network, §>2222 to determine the output
matching network, and fifiui, €>21*1 (for k = 1 to 6), and
($ 2
to quantify the
performance of the circuit at each stage.
By the final stage of the design, we had created a doubler with an overall
power gain of -9.56 dB, not far from the maximum possible predicted value of -9 dB.
132
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Input
Match
4, 6 GHz
Filters
Diode
Doubler
Output
Match
+0.48 V
65.72°
-W
56.29°
-
50 Q*
vd
*T)
50 Q
0
fln=1.0V
/,=1.0 GHz
Figure 13.3. Final design of the single-diode resistive frequency doubler. Electrical
lengths shown are all at 1 GHz.
1
1
D
D
D
D
•1
D
•1
time, rtsec
time, nsec
Figure 13.4 Time-domain plots of a\ and bi for the simulated 1-2 GHz frequencydoubler circuit.
133
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CHAPTER 14
DETERMINING NONLINEAR LARGE-SIGNAL ^-PARAMETERS
FROM ARTIFICIAL NEURAL NETWORK MODELS
TRAINED WITH MEASUREMENT DATA
Although the nonlinear large-signal ^-parameters, introduced in Chapter 10,
can be easily determined for an existing model in a commercial harmonic balance
simulator by forcing all a ’s other than an to zero, they cannot be determined directly
from measurements. With currently available nonlinear vector network analyzers
(NVNAs), described in Chapter 9, the nonlinear device under test (DUT), in
conjunction with the impedance mismatches and harmonics from the system make it
impossible to set all a ’s other than an (assuming port 1 excitation) to zero. In order to
overcome this obstacle, we propose a method [64, 65] that makes use of multiple
measurements of a DUT using a second source and isolators, as shown in Figure 14.1.
This measurement set-up is similar to that introduced by Verspecht et al. [9, 10] to
generate ‘nonlinear scattering functions.’ As a side note, in the following chapter, we
will compare and contrast the ‘nonlinear scattering functions’ with our definitions of
nonlinear large-signal scattering parameters.
14.1. Methodology
To illustrate our technique of generating nonlinear large-signal ^-parameters,
let us consider the case where a DUT is excited at port 1 by a single-tone signal at
frequency f\ and signal level \an\. Utilizing a second source, multiple measurements
of a nonlinear circuit are taken for different values of amn [(m£\ )A(n^l)]. These data
134
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\
are then used to develop an ANN model that maps values of a ’s to b’s, as shown in
Figure 14.2. Once the ANN model is trained and verified, the nonlinear large-signal
^-parameters are obtained by interpolating V s from the measured results for nonzero
values of amn [(w ^I)a(r^I)] to the desired values for amn [(m^l)A(«^l)] equal to
zero, as shown in Figure 14.3. Alternatively, other conditions may be called for where
amn ^ 0 depending on the desired application-specific figure of merit.
The ANN architecture used for this modeling is the feedforward, three-layer
perceptron structure that we discussed in Chapter 2. To briefly review, this
architecture consists of an input layer, a hidden layer, and an output layer. The hidden
layer allows complex models of input-output relationships. And once again, we
utilized software developed by Zhang et al. [42] to construct the ANN models.
Precision ADC
Broadband Downconverter
Bias T
r
DUT
Bias T
s
Figure 14.1. Block diagram of a nonlinear vector network analyzer equipped with a
second source and isolators.
135
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:
\a i l l
Re,Im{<2 12}
Re,Im{6n }
Re,Im
Re,Im{a1A:}
Re,Im{«21}
Re,Im{&12}
ANN
Model
Re,Im {blk}
Re,Im{a22}
Re,Im{a2;t}
Re,Im{617J
Figure 14.2. An ANN model that maps real and imaginary values of a ’s to b’s for
different real and imaginary values of amn [(m^l)A(n^l)].
\a
Re,Im{6n }
i l l
A
'm i
Re,Im{612}
aXk{ k * \) - Q )
tf2,(a lU )= (P
ANN
M odel
'1121
Re,Im{6u }
A
'm i
R e ,I m { V }
A
1\ K \
Figure 14.3. An ANN model that interpolates h’s from the measured results for
nonzero values of amn [(m*X)A{n£\)] to the desired values for amn [{m^ 1 ) a ( « ^ 1 ) ]
equal to zero. Outputs of the ANN model yield values of
136
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To test our method of generating nonlinear large-signal ^-parameters, we
fabricated a wafer-level test circuit using a Schottky diode in a series configuration,
as shown in Figure 14.4. The two-port diode circuit was fabricated on an alumina
substrate by bonding a beam-lead diode package to the gold metalization layer with
silver epoxy. The diode was located in the middle of the coplanar waveguide (CPW)
transmission lines, with short lines connecting the diode to probe pads at both ports.
We measured the test circuit on an NVNA using an on-wafer line-reflect-reflectmatch (LRRM) VNA calibration, along with signal amplitude and phase calibrations.
This process places the reference plane at the tips of the wafer probes used to connect
with the CPW leads.
For all measurements, the first source, located at port 1, was set to a sine-wave
excitation of frequency 900 MFIz and magnitude \aw\ ~ 0.178 V ( - 5 dBm in a 50 Q
environment) at the probe tips. The second source was connected to port 2 and was
set to a sine-wave excitation of frequency 900 MHz and \ci2 \\ ~ 0.178 V. The diode
was forward-biased to +0.2 V through the probe tips. In order to obtain the nonlinear
large-signal ^-parameters, |§>im and f$2 \k\, the excitation from source 1 was held
constant, while the phase of source 2 was randomly changed for 500 different
measurements that varied slightly in magnitude. Figure 14.5 plots the resulting
measurements of an in the complex plane. The nonlinearities in the test circuit, along
with impedance mismatches, created other input components at higher harmonics, as
shown in Figures 14.6 - 14.9 for the second and third harmonics {an, an, an, and
an). These variations in ay allowed us to create an ANN model that could be used to
interpolate
Ifs
from the measured results for nonzero values of amn [(m^l ) a ( / i + 1 ) ] , as
137
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shown in Figures 14.10 and 14.11 for bn and 621, to the desired values for amn
[(m^l)A(rt^l)] equal to zero, or alternatively another desired device condition.
(i
Ci
Figure 14.4. Schottky diode in a series configuration located in the middle of a CPW
transmission line. (White area is metal.)
••
0.1-
-
0.2
-
0.2
0.1
>
-0.1-
-0 .2 -
Figure 14.5. Five hundred measurements of a 2 \ in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
138
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2 - i Im{a12} ( V ) x 103
♦y.
v ; *• ••
s . ';
*.
A Re{a12} (V) x 103
-2
- r
1
•
&
- l-
-2J
Figure 14.6. Five hundred measurements of a n in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
Im{a13} (V) x 103
1-
Figure 14.7. Five hundred measurements of ab in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
139
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2 - i Im {a22} (V) X 103
1-
Re{a22} (V) x 103
1
-2
-1
-1
-2Figure 14.8. Five hundred measurements of <222 in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
Re{a23} (V) x 103
1
-2
-1
1
2
-1-
-2J
Figure 14.9. Five hundred measurements of <223 in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
140
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0 . 2 - , Im{*u }(V )
0.1-
-
0.1
-
-0 .2 J
Figure 14.10. Five hundred measurements of bn in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
0.2-, im{&21} (V)
-
0.2
-
0.1
-
0.1
-
0.2 - 1
Figure 14.11. Five hundred measurements of bii in the complex plane with the
excitation from source 1 held constant and the output from source 2 set to random
phases with constant amplitude.
141
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14.2. Sensitivity Analysis of ANN Models
Data from the 500 measurements were used to develop two ANN models, one
for mapping values from the first five harmonics of ai and a 2 (a\\, an, ..., flis, a 2i,
an, ..., a25 ) to the first five harmonics of bi (bn, bn, ■■■, bn), and the other for
mapping values from the first five harmonics of ai and a 2 to the first five harmonics
of b2 (£>21, £>22, •••, £>25)- We performed a sensitivity analysis to determine how many
training points, testing points, and hidden neurons are required to adequately train the
two ANN models. Tables 14.1 - 14.3 summarize the results for the first model, where
we map values from the first five harmonics of ai and a 2 to the first five harmonics of
bi, and Tables 14.4 - 14.6 summarize the results for the second model, where we map
values from the first five harmonics of ai and a 2 to the first five harmonics of b 2.
First, we varied the number of hidden neurons from 1 to 20. All other
parameters were held constant. Specifically, the 500 measurements points were
divided into 250 training points and 250 testing points, and the conjugate gradient
method was used for training. Table 14.1 lists the average testing errors and
correlation coefficients for the models that map ai and a 2 to bi, and Table 14.4 lists
the average testing errors and correlation coefficients for the models that map ai and
a 2 to b 2. Both mappings show similar trends. The average testing errors decreased
with increasing numbers of hidden neurons until around 14 or 16, where the errors
were minimized. For more than 16 hidden neurons, the trend reversed and the errors
appeared to start increasing again. Figure 14.12 plots the average testing errors as a
function of the number of hidden neurons for both mappings.
142
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Next, we varied the number of training points from 5 to 250. All other
parameters were held constant. The number of hidden neurons was set to 14 since we
found that to be an ideal number from the previous analysis, and 250 testing points
were used for verification. Table 14.2 lists the average testing errors and correlation
coefficients for the models that map ai and a 2 to bi, and Table 14.5 lists the average
testing errors and correlation coefficients for the models that map ai and a 2 to b 2.
Once again, both mappings showed similar trends. The average testing errors
decreased for an increasing number of training points. However, as more and more
training points were added, diminishing returns on the testing errors were evident.
Figure 14.13 plots the average testing errors as a function of the number of training
points for both mappings.
Finally, we varied the number of testing points from 5 to 250. All other
parameters were held constant. The number of hidden neurons was once again set to
14, and the same 250 training points were used for model development. Table 14.3
lists the average testing errors and correlation coefficients for the models that map ai
and a 2 to bi, and Table 14.6 lists the average testing errors and correlation
coefficients for the models that map ai and a 2 to b 2. Both mappings showed that the
average testing errors varied little with the number of testing points. Figure 14.14
plots the average testing errors as a function of the number of testing points for both
mappings.
143
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Table 14.1. Average testing errors and correlation coefficients as functions of the
number of hidden neurons for ANN models trained to map values from the first five
harmonics of ai and si2 to the first five harmonics of bi. All models were developed
using 250 training points and verified using 250 testing points.
Hidden
Neurons
1
2
4
6
8
10
12
14
16
18
20
Average Testing
Error (%)
16.86
10.84
4.56
1.66
1.15
1.08
0.80
0.72
0.72
0.84
0.70
Correlation
Coefficient
0.94814
0.98896
0.99715
0.99971
0.99989
0.99991
0.99996
0.99997
0.99997
0.99996
0.99997
Table 14.2. Average testing errors and correlation coefficients as functions of the
number of training points for ANN models trained to map values from the first five
harmonics of ai and a to the first five harmonics of bj. All models were developed
using 14 hidden neurons and verified using 250 testing points.
2
Training
Points
5
10
25
50
125
250
Average Testing
Error (%)
20.10
9.01
3.64
1.91
0.95
0.72
Correlation
Coefficient
0.96764
0.99556
0.99891
0.99979
0.99995
0.99997
Table 14.3. Average testing errors and correlation coefficients as functions of the
number of testing points for ANN models trained to map values from the first five
harmonics of ai and a to the first five harmonics of bi. All models were developed
using 250 training points and 14 hidden neurons.
2
Testing
Points
5
10
25
50
125
250
Average Testing
Error (%)
0.80
0.74
0.68
0.68
0.72
0.72
Correlation
Coefficient
0.99998
0.99997
0.99998
0.99998
0.99997
0.99997
144
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Table 14.4. Average testing errors and correlation coefficients as functions of the
number of hidden neurons for ANN models trained to map values from the first five
harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed
using 250 training points and verified using 250 testing points.
Hidden
Neurons
1
2
4
6
8
10
12
14
16
18
20
Average Testing
Error (%)
17.88
13.22
6.48
2.04
1.43
0.90
0.82
0.78
0.73
0.78
0.99
Correlation
Coefficient
0.74320
0.91161
0.96659
0.99893
0.99951
0.99985
0.99989
0.99989
0.99992
0.99988
0.99983
Table 14.5. Average testing errors and correlation coefficients as functions of the
number of training points for ANN models trained to map values from the first five
harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed
using 14 hidden neurons and verified using 250 testing points.
Training
Points
5
10
25
50
125
250
Average Testing
Error (%)
27.08
12.99
3.72
1.75
1.09
0.78
Correlation
Coefficient
0.50237
0.91962
0.99628
0.99940
0.99978
0.99989
Table 14.6. Average testing errors and correlation coefficients as functions of the
number of testing points for ANN models trained to map values from the first five
harmonics of ai and a 2 to the first five harmonics of b 2. All models were developed
using 250 training points and 14 hidden neurons.
Testing
Points
5
10
25
50
125
250
|
I
Average Testing
Error (%)
0.87
0.84
0.81
0.80
0.81
0.78
Correlation
Coefficient
0.99995
0.99993
0.99988
0.99989
0.99988
0.99989
145
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N
0sO
20
1
15-
f
10
ce
<D
H
<u
^ Model (ap a2 -> bj)
I- Model (ap a2 -> b2)
-
.*
* *.
*
2
o
>
•
0
I
5
0
"T~
15
10
20
Number o f Hidden Neurons
Figure 14.12. Average testing errors as functions of the number of hidden neurons
for ANN models trained to map ai and a2 to bi and ai and a2 to b2. The models were
developed using 250 training points and verified using 250 testing points.
0X
s
g
30
- |
25-
Model (ap a2 -> bj)
Model (ap a2 -> b2)
20
15-
CD
H
8<D
10
f.•
•
k'
-
5-
V.
0
0
50
100
150
200
250
Number o f Training Points
Figure 14.13. Average testing errors as functions of the number of training points for
ANN models trained to map ai and a2 to bi and ai and a2 to b2. The models were
developed using 14 hidden neurons and verified using 250 testing points.
146
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1 .0 -1
go
0 .6 -
.3
Model (ap a2 -> bj)
—■* Model (ap a2 -> b2)
<i
0 .0
0
50
100
150
200
250
Number o f Testing Points
Figure 14.14. Average testing errors as functions of the number of testing points for
ANN models trained to map ai and a 2 to bi and ai and a 2 to b 2. The models were
developed using 14 hidden neurons and 250 training points.
14.3. Results and Comparison
With the sensitivity analysis complete, we decided to use 250 training points
and 250 testing points to train and verify the two ANN models. We chose to use 14
hidden neurons for mapping values from the first five harmonics of ai and a 2 to the
first five harmonics of bi and 16 hidden neurons for mapping values from the first
five harmonics of ai and a 2 to the first five harmonics of b 2. The testing error was
0.72% for the bi model and 0.73% and for the b 2 model, with respective correlation
coefficients of 0.99997 and 0.99992.
After the ANN models were developed, the nonlinear large-signal g>parameters,
and 02Ui (k = 1, 2, ..., 5), were obtained by interpolating b u and
147
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from measured results for nonzero values of an, an, ..., a n and # 21, # 22, •••> ^25 to the
desired values for <212, <213, ..., <215 and #21, 0 -22 , ■■■, a 25 equal to zero. Figure 14.15
shows the interpolated value of 611 (= Shm -an) when <212, an, ..., an and ai\,
022,
...,
<225 were set equal to zero, and Figure 14.16 shows the interpolated value of 621 (=
ftm rtfn ) when <212, an, ■■■, an and a2 \, <222,
^25 were set equal to zero.
We compared our results to a compact model provided by the manufacturer
and simulated in commercial harmonic-balance software to get an independent check
on our methodology. Our comparison was accomplished by providing the simulator
with the identical biasing conditions on the diode and a stimulus of the same
magnitude used in the measurements for a\\ and setting all other a ’s to zero.
Providing the simulated circuit with <211 of the same magnitude as the measurement
should give the same values of bn and b2k as the interpolated values of bn (=
& m r« n ) and 62/t (= &2u rtfn ) determined by the ANN models when an, an, ■■■, an
and
<221 , a 22,
■■■,
<225
are set equal to zero. Figures 14.15 and 14.16 show that the
simulated values bn and 621 agree with those determined from the measurementbased ANN models. Quantitatively, the differences between the ANN and equivalentcircuit models are shown in Table 14.7.
Table 14.7. Differences between the measurement-based, ANN-modeled results and
the compact model simulated in commercial harmonic-balance software.
Quantity
^1111
&1121
^1131
^1141
^1151
Difference
(%)
3.38
1.23
3.29
0.40
1.67
Quantity
Difference
(dBV)
-44.5
-53.3
-44.8
-63.1
-50.6
&2111
&2121
&2131
&2141
&2151
Difference
(%)
3.95
7.15
5.93
0.72
0.85
148
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Difference
(dBV)
-43.2
-38.0
-39.6
-57.9
-56.5
!
0.2-| Im{6n}(V)
• Measurements
■ ANN Model
▲ Compact Model
0 .1 -
-
0.2
-
0.1
-0 . 1 -
- 0 .2
-1
Figure 14.15. The 250 measurements of bn used for training (circles). Values of
S iiirflu were determined from the measurement-based ANN model (square) and the
harmonic balance simulation using a compact model (triangle).
• Measurements
■ ANN Model
▲ Compact Model
0.1 r
2111
-
0.2
0.2
••
-0.2-1
Figure 14.16. The 250 measurements of 621 used for training (circles). Values of
&2iirtfn were determined from the measurement-based ANN model (square) and the
harmonic balance simulation using a compact model (triangle).
149
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14.4. Discussion
We described a method of extracting nonlinear large-signal ^-parameters,
using an NVNA equipped with isolators and a second source. First, we showed how
multiple measurements of a nonlinear circuit could be used to train artificial neural
networks. Then, we extracted the desired ^-parameters by interpolating the ANN
models for all a ’s equal to zero other than a\\. We checked our approach by
comparing our results to a compact model simulated in commercial harmonic-balance
software, and showed that the two methods agree well.
We also performed a sensitivity analysis on the ANN networks, and
discovered the following: (1) The average testing error decreases for an increasing
number of training points. However, as more and more training points are added,
diminishing returns on the testing errors are evident. (2) As the number of hidden
neurons are increased, the average testing error decreases until around 14 hidden
neurons at which point more hidden neurons have no benefit and can actually lead to
increases in testing error. (3) The number of testing points does not drastically affect
the testing error. In fact, no more than 25 testing points are needed for the models
tested.
150
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CHAPTER 15
COMPARING NONLINEAR LARGE-SIGNAL 0-PARAMETERS
WITH NONLINEAR SCATTERING FUNCTIONS
Here, we compare the nonlinear large-signal ^-parameters, introduced
previously, to another form of nonlinear mapping, known as nonlinear scattering
functions, introduced by Verspecht [11, 12].
For a two-port nonlinear device, excited by a single-tone signal, and assuming
all harmonic signals are relatively small compared to the fundamental signals,
Verspecht defines nonlinear scattering functions as
^kp
where
ay
F'kp
kp
X
Z-i
i= 1,2
ijJ)
& kpij
kpij
"F
'
X
H kkpij
p ij
,) >
(15.1)
i= \, 2
j= 2 ,...,M
and bkp represent the wave variables proportional to the incoming and
outgoing waves, respectively, and M refers to the number of harmonics being taken
into account. Fkp, Gkpij, and Hkpy are functions of the fundamental components
Re(an), Re(a 2i), and Im(tf2i). The imaginary component of
an
is omitted, with the
assumption that the wave variables are phase referenced such that the phase of a n is
set to zero. F ^, Gkpij, and H ^y are assumed complex constants for a given bias and
fundamental drive condition. Note that these three terms do not depend upon the
higher harmonic signal levels. With the ay wave variables split into real and
imaginary components, Gkpij and Hkpij serve to map ay circles centered at zero to bkp
ellipses with variable axes also centered at zero, as shown in Figure 15.1. The Fkp
terms translate the ellipses about the complex plane.
151
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i
0 .5 -
-i
0.5
T—
-
1.0
-0.5
Re{«}
-h
1.0
-0.5-
•••••
b = G • Re(a) + H ■Im(a)
1
l.O-i
0.5-
-
1.0
0.5
• -0.5
-0.5-
• •••••'
-1.0J
Figure 15.1. Gkpij and Hkpij serve to map a# circles centered at zero to bkP ellipses
with variable axes also centered at zero, neglecting Fkp for illustrative purposes.
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j
For illustrative purposes, let us consider bn, taking into account the first three
harmonics. Doing this, eq. (15.1) reduces to
b n = F n +
I G,ijRe(as) + 2 «ii( Im(%)
/= 1,2
7 = 2 ,3
(15-2)
;= 1,2
y = 2 ,3
or
*11 = -*11 + ^ 1112 R e ( a 12 ) + **1 112 Im ( a 12 )
+ * -h ll3 R e ( a 1 3 ) + *^1113 I m ( a 13)
(15.3)
+ ^1122 Re(«22) + H \ \2 2 Im(<322)
+ ^1123 Re(a23) + ifn23 Im(a23) .
If we now consider the nonlinear large-signal ^-parameter representation for
bn, once again assuming a two-port network and taking into account the first three
harmonics, we have
*11 =
Z
&ijuaji
O 5 -4 )
J = 1.2
1=2,3
or
* 11 -
^ 1 1 1 1 ^ 1 1 + ^ 1 1 1 2 a 12 + ® 1 1 1 3 a 13
(15.5)
+ ^ 1 2 1 1 a 21 + ^ 1 2 1 2 ^ 2 2 + ^ 1 2 1 3 a 23 •
I
2
Here, S>ijki are functions o f all of the harmonics, not just the fundamental terms. So for
any change in any ap, a new set of &,#/ will need to be determined. Separating the real
and imaginary components of the a ’s, we can express eq. (15.5) as
153
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R e ^ 0 +%uu [Re(^i2) + J lm(an )]
+ £ n 13[Re(tz13) + j Im<>13)] + £ 121j[Re(a21) + j Im(a21)]
'11 - ^ 1 1 1 1
+
j
'1 2 1 2
(15.6)
[Re(fl22) +7M «22 )] +^1213tRe(«23) +7Im0 23)] •
The imaginary component of an is omitted once again, with the phase reference such
that the phase of an is set to zero.
We can now equate the nonlinear large-signal ^-parameters of eq. (15.6) to
the nonlinear scattering functions of eq. (15.3), with the understanding that this is
only generally valid for the special case when the nonlinear large-signal ^-parameters
are constant for a given bias and fundamental drive level, like Fkp, Gkpij, and Hkpy are
defined. Normally, however, the nonlinear large-signal ^-parameters are dependent
upon the higher harmonics as well as the bias and fundamental drive level. The
implication of this special case will be discussed shortly, after eqs. (15.3) and (15.6)
are equated. Equating the corresponding real and imaginary components of the a
wave variables in eqs. (15.3) and (15.6) gives
Fn = &H JRe(an ) + ^ \ 2 \\a 2 \ •
(15.7)
1
Additionally,
*1112
1113
_
^ 1 1 1 2
_ < A
h 3
> 7 ^ 1 1 1 2
“
# 1 1 1 2
; 7 ^ 1 1 1 3
- # 1 H
3
’
(15.8)
5
(15.9)
^1212 - ^1122 » 7 ^ 1 2 1 2 ~ # 1 1 2 2 >
and
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(15.10)
'1 2 1 3
(15.11)
_ ^1123 5 7 ^ 121 3 “ ^1123 •
Eqs. (15.8) - ( 1 5 .1 1 ) im ply
^ 1 1 1 2
=
~ j H
1112
5 ^ 1 1 1 3
=
~ j H
1113
> ^ 1 1 2 2
=
~ J H
1122
> ^ 1 1 2 3
=
~ J H
1123
’
(15.12)
which means
Re(Gkpij )= lm (H kf)ii)
; Re(Hkinj) = -lm (G kpij) .
(15.13)
Eq. (15.13) satisfies the conditions of the Cauchy-Riemann equations [66],
9 [ R e ( ^ ) ] _ 3 [Im (^ )]
a[Re(ap)]
which implies
3[Im (^)]
3[Re(^p)]
’
3[Im(ap)]
must be an analytic function of
ay.
_
d [Im (^ )]
3 [R e(^ )] ’
(15.14)
A complex-valued function is
said to be analytic on an open set W if it has a derivative at every point of W. This is
only generally true when bkp is a linear function of ay. Thus, equating the nonlinear
large-signal ^-parameters with the nonlinear scattering functions is only generally
valid in the small-signal, linear case.
As we mentioned earlier, eqs. (15.7) - (15.12) are only generally valid in the
special case when the nonlinear large-signal ^-parameters are constant for a given
bias and fundamental drive level, like Fkp, GkPy, and Hkpy are defined. Since this is not
generally true, the formulations for nonlinear large-signal ^-parameters and nonlinear
scattering functions are not equivalent.
We can make at a few important conclusions, however, after attempting to
equate the two formulations. First, if Gkpij and Hkpy are allowed to be functions of
higher harmonics, then only one of them, either Gkpy or Hkpy, or equivalently &yu, is
required since eq. (15.12) shows that they are not independent. Second, if the
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nonlinear large-signal ^-parameters are complex constants for a given bias and
fundamental drive level and are not functions of the higher harmonics, they have the
limitation that they cannot map circles into ellipses, but rather can only map circles
into circles, as shown in Figure 15.2. This is because I i s
a single, complex
constant rather than a pair of independent complex constants like G ^ j and Hkpjj. Thus,
if Biju is not dependent upon higher harmonics, it acts like a linear ^-parameter.
We have shown above that the two formulations are not equivalent. Nonlinear
large-signal ^-parameters are more general than the nonlinear scattering functions,
which are useful in approximating a specific class of nonlinearity in a more compact
form. Nonlinear large-signal ^-parameters have the advantage of being able to map
circles into any arbitrary shape, rather than being limited to ellipses.
156
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0 .5 -
-0.5
0.5
-0 .5 -
**•••«!•§•. >•••••••**
b = S ■a
I
1.0 -,
..........
0 .5 -
R e{b}
-
1.0
-0.5
0.5
1.0
-0 .5 •••
-l.O-1
Figure 15.2. If &#*/ is a complex constant for a given bias and fundamental drive
level, it has the limitation that it can only map circles into circles.
157
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CHAPTER 16
SUMMARY AND FUTURE WORK
16.1. ANN Modeling for Improved Vector Network Analyzer Calibrations
16.1.1. Summary
In the first part of this thesis, we successfully applied ANNs to model onwafer and coaxial lumped-element calibration standards. For the on-wafer OSLT
standards, we showed that a calibration using ANN-modeled standards compares
favorably (less than a 0.02 difference in magnitude) to the benchmark multiline TRL
calibration. In modeling these standards, we quantified the training errors and training
times as functions of both the number of training points and the number of neurons in
the hidden layer. We found that 5 neurons in the hidden layer of an MLP3
architecture and that fewer than 10 training points were sufficient to accurately model
our standards. We then expanded upon our method of modeling on-wafer OSLT
standards using ANNs by taking into account the load variations from wafer to wafer.
Specifically, we modeled the correlation between measured DC resistance and RF
variations in load terminations as well as the RF performances of the open, short, and
thru standards. We showed that these modeled standards compare favorably (a
difference of less than 0.04 in magnitude at most frequencies) to the benchmark
multiline TRL calibration over a 66 GHz frequency range.
We also used ANNs to successfully model on-wafer and coaxial LRM load
standards. For the on-wafer case, we showed that an LRM calibration using an ANNmodeled load compares favorably to a benchmark multiline TRL calibration, with an
average worst-case scattering parameter error bound of 0.017 over a 40 GHz
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frequency range. For the coaxial case, we found the average worst-case error bound
to be 0.024, an improvement over the equivalent circuit-modeled average worst-case
bound of 0.034.
In practice, ANN-modeled calibration standards can be easily implemented
using existing or custom software packages. In our case, we utilized MultiCal [4], a
free program developed by the National Institute of Standards and Technology, to
perform our multiline TRL and LRM calibrations. The internal software on any
commercial network analyzer can also be used if the user has confidence in another
calibration method. Then, once the lumped-element standards are measured, one of a
number of ANN programs may be used to model the standards. We used software
developed by Zhang et al. [50] to construct our ANN models. For the OSLT
experiments, we wrote custom software to perform the calibrations with exported
ANN models, using the equations found in references [5] and [48].
We have shown that ANN models offer a number of advantages over using
calibrated measurement data files or equivalent circuit models, namely:
(1) They do not require the numerous details and parameters of physical models.
(2) Calibration times can be reduced since only a few training points are required
to accurately model the standards.
(3) ANN model descriptions are much more compact than large measurement
data files.
(4) ANN models, trained on only a few measurement points, can be much more
accurate than direct calibrations, when limited calibration data are available.
(5) They give an optimized estimate in the presence of noise.
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(6) ANN models are able to accurately model loads with measured DC
resistances slightly outside their training range.
16.1.2. O ther Applications
In addition to modeling calibration standards, ANNs have been used as a
measurement-based modeling tool in a number of other RF and microwave
applications. Recently, they have been developed as a means of transforming the
measured reflection coefficients of liquids, using an open-circuited coaxial probe, to
their respective permittivity values. Tuck and Coad implemented an ANN using
training data from nine different liquids in the frequency range of 200 MHz to 16
GHz [67]. In another study, Bartley et al. implemented an ANN using training data
from eleven liquids (different mixtures of water and isopropyl alcohol) in the
frequency range of 200 MHz to 6 GHz [68], Both of these studies suggest that ANNs
can be successfully applied to determine the dielectric properties of materials from
uncalibrated reflection coefficient measurements made on an open-ended coaxial
probe.
Bartley et al. also applied an ANN to determine the moisture content in wheat
from microwave transmission measurements [69], The ANN was trained for moisture
contents between 10.6 and 19.2 %, which is referred to as “wet basis,” and bulk
densities varying from 0.72 to 0.88 g/cm3. Measurements were made from free-space
transmission-coefficient measurements on layers of wheat placed between two
antennas connected to a VNA at eight frequencies ranging from 10-18 GHz. The
ANN architecture consisted of one hidden layer with fifteen neurons. Sixteen inputs
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(the magnitudes and phases of each of the eight measured transmission coefficients)
were used for the ANN model. The one output was moisture content. For the 179
sample measurements, Bartley et al. calculated the mean squared error between the
ANN model and the measured results to be 0.028. Here, the authors point out that this
method has the potential for on-line, nondestructive moisture content measurement
for flowing grain.
16.1.3. Future Work
ANNs have many potential applications in the area of RF and microwave
measurement-based modeling [70].
For example, in the area of materials characterization, the most accurate way
of measuring the complex permittivity of a low-loss dielectric material is by using
one of a number of resonator methods [71]. Although usually limited to a single
frequency, resonators provide the required accuracy that broadband methods lack.
The disadvantage of most resonator techniques, however, is the need for accurately
machined samples of the material of interest. Recently, Krupka et al. [72] introduced
a new type of resonator that allows for nondestructive permittivity measurements,
referred to as the split-post resonator. Briefly, the permittivity of a sample is
determined from shifts in the quality factor and resonant frequency from
measurements taken with and without the sample in place. Krupka et al. utilize the
Rayleigh-Ritz method to theoretically determine the shift in resonant frequency and
quality factor from the complex permittivity and sample thickness. Using a wide
range of permittivities and sample thicknesses, they created a look-up table that is
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used to calculate the sample permittivity from measured shifts in resonant frequency
and quality factor for a given sample thickness. For values that are not explicitly
listed in the table, they interpolate between the two closest values. Researchers at the
National Institute of Standards and Technology (NIST) are currently developing an
independent mode-matching model for the split-post resonator to compare with the
method of Krupka et al. In a similar manner as described above, they will develop a
model to determine the shift in resonant frequency and quality factor from the
complex permittivity and sample thickness. But rather than creating a look-up table,
they are interested in developing an ANN model, trained by data from the modematching technique, to provide the complex permittivity for a given sample thickness
from measured shifts in resonant frequency and quality factor. The ANN will provide
an efficient model for the transformation that would otherwise be time-consuming
using the mode-matching computations each time.
Another potential application of ANNs comes in the area of power
measurement. The measurement of microwave power is a fundamental test
requirement necessary for determining output levels of signal generators, transmitters,
and radar, just to name a few. Invariably, commercial power sensors are ultimately
traceable to measurements made by primary national standards laboratories, most of
which utilize a water-bath microcalorimeter and a reference standard for coaxial and
waveguide measurements. The reference standards are usually substitution-type
bolometric power detectors, which use heat sensitive resistors terminating a
transmission line that absorbs microwave power [73].
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I
\
I
!
Not all the microwave energy incident on a power detector is absorbed by the
bolometer element, however. Some of the power is dissipated in the connector, the
transmission line, and the bolometer mounting structure, and some additional power
is lost to leakage in the mount. These factors result in a dimensionless measurement
error called the mount efficiency, which is always less than one. Additionally, the
bolometer elements are not heated identically by the same amounts of RF and DC
power. This is referred to as the RF-DC substitution error. The combination of these
two errors is defined as the effective efficiency, which is independent of mismatch
corrections.
Customers who wish their devices to be directly traceable to a national
standards laboratory submit their devices for calibration. After the measurements are
complete, the customer receives a table of effective efficiencies for the frequencies
measured. Since a typical measurement of 360 frequency points requires
approximately 48 hours due to the time required for the thermopile to stabilize at each
point, ANNs have the potential to lessen measurement times by reducing the number
of frequency points. An ANN model, could in principle, be trained to interpolate the
effective efficiency between a reduced set of measurements, especially for devices
that have been previously measured and are known to behave correctly, without
unpredictably large spikes that occur over a short frequency span.
In
addition
to
reducing
measurement
times
for
microcalorimeter
measurements without noticeably degrading accuracy, ANNs can potentially be used
in other areas where measurement times are lengthy, such as the determination of
noise parameters.
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16.2. Frequency-Domain Models and Figures of Merit for Nonlinear Circuits
16.2.1. Summary
In the second part of this thesis, we developed new frequency-domain models
and figures of merit for nonlinear microwave circuits. We began by introducing a
method for preserving time-invariant phase relationships when ratios are taken
between two harmonically related signals by introducing a third signal that is used as
a phase reference. We showed that a reference signal must be present at the
fundamental frequency in order for time-invariant phase relationships to exist
between ratios of any two harmonically related signals.
Then, we introduced nonlinear large-signal scattering parameters, a new type
of frequency-domain mapping that relates incident and reflected signals. Unlike
classical 5-parameters, nonlinear large-signal ^-parameters take harmonic content
into account and depend on the signal magnitudes. First, we presented a general form
of nonlinear large-signal ^-parameters and showed that they reduce to classic Sparameters in the absence of nonlinearities. We also introduced nonlinear large-signal
impedance (2) and admittance (|j) parameters, and presented equations for relating
the different representations. Next, we made two simplifications, considering the
cases of a one-port network and a two-port network, each with a single-tone
excitation. For the one-port network, we showed that the equation relating
and 2
reduces to the same well-known equation for the linear case assuming no power is
transferred in the form of frequency down-conversion. For the two-port case, we
extracted input reflection coefficients and forward transmission coefficients, which
can be useful for designing circuits such as amplifiers and frequency multipliers. In
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addition, we derived a quasi-linear approximation of the output reflection coefficient
under normal operating conditions. These three two-port parameters allow a designer
to “see” application-specific engineering figures of merit that are similar to what he
or she is accustomed to in the linear world.
Next, we expanded the definitions of power gain, transducer gain, and
available gain by taking harmonic content into account, and showed that under
special conditions, this generalized power gain can be expressed in terms of nonlinear
large-signal scattering parameters.
We provided an example showing how the expanded definitions of gain and
nonlinear large-signal scattering parameters allow us to examine the behavior of a
nonlinear pHEMT model by simply performing a harmonic balance simulation with
all a’s other than a\\ forced to zero. Looking at the nonlinear large-signal scattering
parameters gives us an in-depth view of the modeled behavior by letting us separate
out the input reflection coefficients and transmission coefficients at each of the
frequency components, while reducing the nonlinear large-signal scattering parameter
data set into the compact expressions of power gain (S, transducer gain (Br, and the
ratio (Si/<8, gives us a more concise view of the modeled behavior.
We also illustrated how nonlinear large-signal ^-parameters can be used as a
tool in the design process of a single-diode 1-2 GHz frequency-doubler. Specifically,
we used
111 to determine the input matching network, #2222 to determine the output
matching network, and
n iki,
(for k = 1 to 6), and ($2 to quantify the
performance of the circuit at each stage. By the final stage of the design, we had
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created a doubler with an overall power gain of -9.56 dB, not far from the maximum
possible predicted value of -9 dB.
For the case where a nonlinear model is not readily available, we described a
method of extracting nonlinear large-signal ^-parameters, using an NVNA equipped
with isolators and a second source. First, we showed how multiple measurements of a
nonlinear circuit could be used to train artificial neural networks. Then, we extracted
the desired ^-parameters by interpolating the ANN models for all o ’s equal to zero
other than a\\. We checked our approach by comparing our results to a compact
model simulated in commercial harmonic-balance software, and showed that the two
methods agree well. We also performed a sensitivity analysis on the ANN networks,
and discovered the following: (1) The average testing error decreases for an
increasing number of training points. However, as more and more training points are
added, diminishing returns on the testing errors are evident. (2) As the number of
hidden neurons are increased, the average testing error decreases until around 14
hidden neurons at which point more hidden neurons have no benefit and can actually
lead to increases in testing error. (3) The number of testing points does not drastically
affect the testing error. In fact, no more than 25 testing points are needed for the
models tested.
Finally, we compared our nonlinear large-signal ^-parameters to another form
of nonlinear mapping, known as nonlinear scattering functions. We showed that the
two formulations are not equivalent. Nonlinear large-signal ^-parameters are more
general than the nonlinear scattering functions, which are useful in approximating a
specific class of nonlinearity in a more compact form. Nonlinear large-signal &-
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parameters, however, have the advantage of being able to map circles into any
arbitrary shape, rather than being limited to ellipses.
16.2.2. Future Work
There are a number of future research topics that can be done to extend the
work presented in the second half of this thesis.
One idea is to examine whether the method of preserving time-invariant
phase relationships when ratios are taken between two harmonically related signals
can be generalized or modified to preserve consistent phase relationships when ratios
are taken between two signals not harmonically related. In this case, the third signal
would occur at a frequency that is a common factor of the first two, but may not be
readily available for use as a reference. Such a method, if discovered, could be very
useful for mixer applications.
Another topic for further consideration is to examine how well the quasilinear approximation of the output reflection coefficient, derived in Chapter 10, can
be measured using an NVNA equipped with two sources. This concept works very
well in harmonic balance simulators, but it would be interesting to see how this
parameter varies in the presence of noise and at various offset frequencies and power
levels.
To show how nonlinear large-signal scattering parameters can be used to
design nonlinear circuits, it would be useful to provide more examples. Designing a
high-efficiency amplifier, such as a Class E or Class F configuration, where harmonic
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I
terminations are important, would provide a convincing argument for the use of these
parameters.
In order to make the ANN models we derived from measurements more
useful, it would be beneficial if they could be incorporated into a commercial
harmonic balance simulator. Currently, most nonlinear models that are imported into
simulators are represented in the time-domain, but if a “hook” could be provided that
would also allow frequency-domain models, these ANN models could be placed into
the circuit of interest and used in the overall simulation, as well as in the optimization
stage of the design process.
The measurement set-up we presented for extracting nonlinear large-signal 1S>parameters was shown to be useful in diode circuits, but in the case of a transistor,
where there can be more isolation between ports 1 and 2, this may not be adequate. A
third source, placed at port 1, and possibly other additional hardware, may be required
to simultaneously cover the origins of the a\k waves, while the second source, located
at port 2, covers the origins of the au waves. Further investigation will be required to
answer this question.
These are just a few of the many research topics that can be explored in this
area. In the broader sense of measurement-based modeling, uncertainty statements on
the measured o ’s and V s will be required for the designer to fully trust and validate
his or her models.
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BIBLIOGRAPHY
1.
G. F. Engen and C. A. Hoer, Thru-reflect-line: an improved technique for
calibrating the dual six-port automatic network analyzer, IEEE Trans
Microwave Theory Tech 27 (1979), 987-993.
2.
B. Bianco, M. Parodi, S. Ridella, and F. Selvaggi, Launcher and microstrip
characterization, IEEE Trans Instrum Meas 25 (1976), 320-323.
3.
R. B. Marks, A multiline method of network analyzer calibration, IEEE Trans
Microwave Theory Tech 39 (1991), 1205-1215.
4.
D. C. DeGroot, J. A. Jargon, and R. B. Marks, Multiline TRL revealed, 60th
ARFTG Conference Digest, Washington, D.C., Dec. 2002,131-155.
5.
D. K. Rytting, Network analyzer error models and calibration methods, 52nd
ARFTG Conference, Short Course on Computer-Aided RF and Microwave
Testing and Design, Dec. 1998.
6.
M. Sipila, K. Lehtinen, and V. Porra, High-frequency periodic time-domain
waveform measurement system, IEEE Trans Microwave Theory Tech 36
(1988), 1397-1405.
7.
U. Lott, Measurement of magnitude and phase of harmonics generated in
nonlinear microwave two-ports, IEEE Trans Microwave Theory Tech 37
(1989), 1506-1511.
8.
G. Kompa and F. Van Raay, Error-corrected large-signal waveform
measurement system combining network analyzer and sampling oscilloscope
capabilities, IEEE Trans Microwave Theory Tech 38 (1990), 358-365.
9.
J. Verspecht, P. Debie, A. Barel, and L. Martens, Accurate on wafer
measurement of phase and amplitude of the spectral components of incident
and scattered voltage waves at the signal ports of a nonlinear microwave
device, 1995 IEEE MTT-S Int Microwave Symp Dig, May 1995,1029-1032.
10.
J. Verspecht, Calibration of a measurement system for high-frequency
nonlinear devices, Doctoral Dissertation, Vrije Universiteit Brussel, Belgium,
1995.
11.
J. Verspecht, D. Schreurs, A. Barel, and B. Neuwelaers, Black box modeling
of hard nonlinear behavior in the frequency domain, IEEE MTT-S Int
Microwave Symp Dig, June, 1996,1735-1738.
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12.
J. Verspecht and P. Van Esch, Accurately characterizing hard nonlinear
behavior of microwave components with the nonlinear network measurement
system: introducing ‘nonlinear scattering functions,’ Proceedings of the 5th
International Workshop on Integrated Nonlinear Microwave and
Millimeterwave Circuits, Duisburg, Germany, Oct. 1998,17-26.
13.
Q. J. Zhang and K. C. Gupta, Neural networks for RF and microwave design,
Boston, London: Artech House (2000).
14.
P. M. Watson and K. C. Gupta, EM-ANN models for microstrip vias and
interconnects in dataset circuits, IEEE Trans Microwave Theory Tech 44
(1996), 2495-2503.
15.
A. Veluswami, M. S. Nakhla, and Q. J. Zhang, The application of neural
networks to EM-based simulation and optimization of interconnects in high­
speed VLSI circuits, IEEE Trans Microwave Theory Tech 45 (1997), 712723.
16. G. L. Creech, B. J. Paul, C. D. Lesniak, T. J. Jenkins, and M. C. Calcatera,
Artificial neural networks for fast and accurate EM-CAD of microwave
circuits, IEEE Trans Microwave Theory Tech 45 (1997), 794-802.
17. F. Wang and Q. J. Zhang, Knowledge-based neural models for microwave
design, IEEE Trans Microwave Theory Tech 45 (1997), 2333-2343.
18. P. M. Watson and K. C. Gupta, Design and optimization of CPW circuits
using EM-ANN models for CPW components, IEEE Trans Microwave
Theory Tech 45 (1997), 2515-2523.
19. A. Patnaik, R. K. Mishra, G. K. Patra, and S. K. Dash, An artificial neural
network model for effective dielectric constant of microstrip line, IEEE Trans
Antennas Propagation 45 (1997), 1697.
20. M. Vai, S. Wu, B. Li, and S. Prasad, Creating neural network based
microwave circuit models for analysis and synthesis, Proc Asia Pacific
Microwave Conf, Hong Kong, Dec. 1997, 853-856.
21. Q. J. Zhang, F. Wang, and M. S. Nakhla, Optimization of high-speed VLSI
interconnects: a review, Int J RF and Microwave CAE 7 (1997), 83-107.
22. F. Scarselli and A. C. Tsoi, Universal approximation using feedforward neural
networks: a survey of some existing methods, and some new results, Neural
Networks 11(1998) 15-37.
23. G. Cybenko, Approximation by superpositions of a sigmoidal function, Math
Control Signals Systems 2 (1989) 303-314.
170
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24. K. Homik, M. Stinchcombe, and H. White, Multilayer feedforward networks
are universal approximators, Neural Networks (2) 1989, 359-366.
25. P. M. Watson, Artificial neural network modeling for computer-aided design
of microwave circuits, Doctoral Dissertation, University of Colorado at
Boulder, 1998.
26. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning internal
representations by error propagation, in Parallel Distributed Processing 1, D.
E. Rumelhart and J. L. McClelland, Editors, Cambridge, MA: MIT Press
(1986) 318-362.
27. R. H. Dicke, General microwave circuit theorems, in Principles of Microwave
Circuits, C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Editors,
Cambridge, MA: McGraw-Hill Book Company (1948) 130-161.
28. D. M. Kems and R. W. Beatty, Basic theory of waveguide junctions and
introductory microwave network analysis, Oxford: Permagon Press (1967).
29. K. C. Gupta, Microwaves, New York: John Wiley & Sons (1980).
30. G. F. Engen, Microwave circuit theory and foundations of microwave
metrology, London: Peter Peregrinus Ltd. on behalf of the Institution of
Electrical Engineers (1992).
31. R. A. Hackbom, An automatic network analyzer system, Microwave Journal,
May, 1968, 45-52.
32. S. F. Adam, A new precision automatic microwave measurement system,
IEEE Trans Instrum Meas 17 (1968), 308-313.
33. C. A. Hoer and K. C. Roe, Using an arbitrary six-port junction to measure
complex voltage ratios, IEEE Trans Microwave Theory Tech 23 (1975), 978984.
34. G. F. Engen, The six-port reflectometer: an alternative network analyzer,
IEEE MTT-S Int Microwave Symp Dig, June, 1977,44-45.
35. S. F. Adam, Microwave instrumentation: an historical perspective, IEEE
Trans Microwave Theory Tech 32 (1984), 1157-1161.
36. Hewlett-Packard, Operating and programming manual: HP 851 OB network
analyzer, Hewlett-Packard, Manual 1987.
37. Wiltron, Model 360B vector network analyzer operation manual, Wiltron
Company, Manual 1991.
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
j
I
38.
J. A. Jargon and R. B. Marks, Two-tier multiline TRL for calibration of lowcost network analyzers, 46th ARFTG Conference Digest, Scottsdale, AZ, Dec.
1995,1-8.
39.
J. Fitzpatrick, Error models for system measurement, Microwave Journal,
May, 1978,63-66.
40.
R. B. Marks, Formulations of the basic vector network analyzer error model
including switch terms, 50th ARFTG Conference Digest, Portland, OR, Dec.
1997,115-126.
41. J. T. Barr IV and M. J. Pervere, A generalized vector network analyzer
calibration technique, 34th ARFTG Conference Digest, Ft. Lauderdale, FL,
Dec. 1989, 51-60.
42.
J. A. Jargon, K. C. Gupta, and D. C. DeGroot, Artificial neural network
modeling for improved on-wafer OSLT calibration standards, Int J RF and
Microwave CAE 10 (2000), 319-328.
43.
J. A. Jargon, P. Kirby, K. C. Gupta, L. Dunleavy, and T. Weller, Modeling
load variations with artificial neural networks to improve on-wafer OSLT
calibrations, 56th ARFTG Conference Digest, Broomfield, CO, Nov. 2000, 7688.
44.
J. A. Jargon and K. C. Gupta, Artificial neural network modeling for
improved on-wafer line-reflect-match calibrations, 31st European Microwave
Conference Proceedings, London, England, Sep. 2001,229-232.
45.
J. A. Jargon and K. C. Gupta, Artificial neural network modeling for
improved coaxial line-reflect-match calibrations, Int J RF and Microwave
CAE 11 (2001), 33-37.
46.
Hewlett-Packard, Network analysis: specifying calibration standards for the
HP 8510 network analyzer, Product Note 8510-5A, 1997.
47.
R. B. Marks and D. F. Williams, Verification of commercial probe-tip
calibrations, 42nd ARFTG Conference Digest, San Jose, CA, Dec. 1993, 37-
48. D. C. DeGroot, K. L. Reed, and J. A. Jargon, Equivalent circuit models for
coaxial OSLT standards, 54th ARFTG Conference Digest, Atlanta, GA, Dec.
1999,103-115.
49. D. F. Williams, R. B. Marks, K. Phillips, and T. Miers, Progress toward
MMIC on-wafer standards, 36th ARFTG Conference Digest, Nov. 1990, 7383.
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50.
Q. J. Zhang and his neural network research team, NeuroModeler, ver. 1.2,
Department of Electronics, Carleton University, Ottawa, Canada, 1999.
51.
P. Kirby, L. Dunleavy, and T. Weller, The effect of load variations on onwafer lumped element based calibrations, 54th ARFTG Conference Digest,
Atlanta, GA, Nov. 1999, 81-90.
52.
D. F. Williams and R. B. Marks, LRM probe-tip calibrations with imperfect
resistors and lossy lines, 42nd ARFTG Conference Digest, Dec. 1993, 32-36.
53.
R. B. Marks and D. F. Williams, Characteristic impedance determination
using propagation constant measurement, IEEE Microwave Guided Lett 1
(1991), 141-143.
54.
D. F. Williams and R. B. Marks, Comparison of on-wafer calibrations, 38th
ARFTG Conference Digest, Dec. 1991,68-81.
55. J. A. Jargon, R. B. Marks, and D. F. Williams, Coaxial line-reflect-match
calibration, Asia-Pacific Microwave Conference Proceedings, Taejon, Korea,
Oct. 1995, 86-89.
56. K. Rush, S. Draving, and J. Kerley, Characterizing high-speed oscilloscopes,
IEEE Spectrum (1990), 38-39.
57. J. Verspecht and K. Rush, Individual characterization of broadband sampling
oscilloscopes with a ‘nose-to-nose’ calibration procedure, IEEE Trans Instrum
Meas 43(1994), 347-354.
58.
J. A. Jargon, D. C. DeGroot, K. C. Gupta, and A. Cidronali, Calculating ratios
of harmonically related, complex signals with application to nonlinear largesignal scattering parameters, 60th ARFTG Conference Digest, Washington,
DC, Dec. 2002,113-122.
59.
D. M. Pozar, Microwave engineering, 2nd Edition, Reading, Massachusetts:
John Wiley & Sons (1997).
60.
A. Cidronali, G. Collodi, A. Santarelli, G. Vannini, and G. Manes, Millimeterwave FET modeling using on-wafer measurements and EM simulation,” IEEE
Trans Microwave Theory Tech 50 (2002), 425-432.
61.
A. Cidronali, G. Collodi, C. Toccafondi, R. Cignani, A. Santarelli, G.
Vannini, and F. Filicori, “A distributed approach for the characterization of
parasitic networks in electron device modeling,” GAAS 2002, Milan, Italy,
Sep. 2002,149-152.
173
i
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62.
M. T. Faber, J. Chramiec, and M. E. Adamski, Microwave and millimeterwave diode frequency multipliers, Boston, London: Artech House (1995).
63.
S. A. Maas, The rf and microwave circuit design cookbook, Boston, London:
Artech House (1998).
64.
J. A. Jargon, K. C. Gupta, D. Schreurs, K. A. Remley, and D. C. DeGroot, A
method of developing frequency-domain models for nonlinear circuits based
on large-signal measurements, 58th ARFTG Conference Digest, San Diego,
CA, Nov. 2001,35-48.
65.
J. A. Jargon, K. C. Gupta, D. Schreurs, and D. C. DeGroot, Developing
frequency-domain models for nonlinear circuits based on large-signal
measurements, URSI XXVIIth General Assembly, Maastricht, the
Netherlands, Aug. 2002, CD-ROM.
66.
S. D. Fisher, Complex variables, Monterey: Brooks/Cole Publishing Company
(1986).
67.
D. Tuck and S. Coad, Neurocomputed model of open-circuited coaxial probes,
IEEE Microwave Guided Lett 5 (1995) 105-107.
68. P. G. Bartley, R. W. McClendon, and S. O. Nelson, Permittivity determination
by using an artificial neural network, IEEE Instrum and Meas Tech
Conference, 1999,27-30.
69. P. G. Bartley, S. O. Nelson, R. W. McClendon, and S. Trabelsi, Determining
moisture content of wheat with an artificial neural network from microwave
transmission measurements, IEEE Trans Instrum Meas 47 (1998), 123-126.
70.
J. A. Jargon, K. C. Gupta, and D. C. DeGroot, Applications of artificial neural
networks to RF and microwave measurements, Int J RF and Microwave CAE
1 (2002), 3-24.
71. J. Baker-Jarvis, R. G. Geyer, J. H. Grosvenor, M. D. Janezic, C. A. Jones, B.
Riddle, C. M. Weil, and J. Krupka, Dielectric characterization of low-loss
materials: a comparison of techniques, IEEE Trans Dielect Elect Insulation 5
(1998) 571-577.
72.
J. Krupka, R. G. Geyer, J. Baker-Jarvis, and J. Ceremuga, Measurements of
the complex permittivity of microwave circuit board substrates using split
dielectric resonator and reentrant cavity techniques, Seventh International
Conference on Dielectric Materials Measurements and Applications, 1996,
21-24.
174
I
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73.
F. R. Clague, A calibration service for coaxial reference standards for
microwave power, National Institute of Standards and Technology Technical
Note 1374,1995.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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