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# Automated matching control system using load estimation and microwave characterization

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AUTOMATED MATCHING CONTROL SYSTEM USING LOAD ESTIMATION AND
MICROWAVE CHARACTERIZATION
By
JAESEOK KIM
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2008
1
UMI Number: 3370928
INFORMATION TO USERS
The quality of this reproduction is dependent upon the quality of the copy
submitted. Broken or indistinct print, colored or poor quality illustrations and
photographs, print bleed-through, substandard margins, and improper
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
______________________________________________________________
UMI Microform 3370928
unauthorized copying under Title 17, United States Code.
_______________________________________________________________
ProQuest LLC
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106-1346
c 2008 Jaeseok Kim
?
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To my family
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ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my advisor, Professor William R.
not have been possible without his guidance and support. My deep recognition goes to
Professor Kenneth O, Professor John G. Harris, and Professor Gloria J. Wiens for serving
on my supervisory committee and for their valuable suggestions. Many thanks go to Mr.
Larry Luce from Freescale Semiconductor for their valuable input and generous funding
for this research. Thanks also go to my colleagues in the Electronic Circuits Laboratory
(ECL) for their discussion of ideas and years of friendship. Last but not least, I owe a
special debt of gratitude to my family. Without their sel?ess love and support, I cannot
imagine what I would have achieved.
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page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER
1
2
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.1
1.2
1.3
1.4
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15
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BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.1
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AUTOMATIC MATCHING CONTROL . . . . . . . . . . . . . . . . . . . . . .
34
3.1
3.2
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2.2
2.3
3
3.3
3.4
Why Automatic Impedance Matching? . . . . . . . .
Maximum Power Transfer on Impedance Matching .
Challenges of Automatic Impedance Matching . . . .
Proposed Solution: Automatic Matching Control with
Impedance Matching Networks with Lumped
2.1.1 L-type Matching Network . . . . . . .
2.1.2 ?-type Matching Network . . . . . .
2.1.3 T-type Matching Network . . . . . .
2.1.4 Bandwidth of Matching Network . . .
Pattern Recognition . . . . . . . . . . . . .
2.2.1 The k -Nearest Neighbor Classi?er . .
2.2.2 Arti?cial Neural Network (ANN) . . .
2.2.3 Principal Component Analysis (PCA)
Multi-Port Re?ectometers . . . . . . . . . .
2.3.1 Six-Port Re?ectometer . . . . . . . .
2.3.2 Four-Port Multistate Re?ectometer .
Elements
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Overview . . . . . . . . . . . . . . . . . . . . . . .
System Overview . . . . . . . . . . . . . . . . . . .
3.2.1 Impedance Matching Tuner . . . . . . . . . .
3.2.2 Controller . . . . . . . . . . . . . . . . . . .
3.2.3 Search algorithm . . . . . . . . . . . . . . .
Experimental Results . . . . . . . . . . . . . . . . .
3.3.1 Characterization of Automatic Tuner System
3.3.2 Measurement Results . . . . . . . . . . . . .
Conclusion and Discussion . . . . . . . . . . . . . .
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4
LOAD IMPEDANCE ESTIMATION . . . . . . . . . . . . . . . . . . . . . . . .
52
4.1
4.2
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EFLECTOMETER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.3
4.4
5
5.1
5.2
5.3
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AUTOMATIC MATCHING CONTROL USING LOAD ESTIMATION . . . . .
94
Overview . . . . . . . . . . . . . . . .
Matching Control Procedures . . . . .
Characterization of Matching Network
Bias Search for Impedance Matching .
Characterization Results . . . . . . . .
Impedance Matching Results . . . . . .
Conclusion . . . . . . . . . . . . . . . .
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80
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6.1
6.2
6.3
6.4
6.5
6.6
Overview . . . . . . . . . . . . . . . . . . . .
Multistate Re?ectometers . . . . . . . . . . .
Tunable Matching Network . . . . . . . . . .
Experimental Results . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . .
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Estimation
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80
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Overview . . . . . . . . . . . . . . . . . . . . . . . .
System Overview . . . . . . . . . . . . . . . . . . . .
High Impedance Probe . . . . . . . . . . . . . . . . .
5.3.1 Least Square Fitting . . . . . . . . . . . . . .
5.3.2 Arti?cial Neural Network . . . . . . . . . . . .
Load Estimation Methods . . . . . . . . . . . . . . .
5.4.1 Radical Center . . . . . . . . . . . . . . . . . .
5.4.2 Least Square Fitting . . . . . . . . . . . . . .
Experimental Results . . . . . . . . . . . . . . . . . .
5.5.1 High Impedance Probe Estimation . . . . . . .
5.5.2 Load Estimation . . . . . . . . . . . . . . . . .
5.5.3 Comparison of Coupler and Coupler-Free Load
Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
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THREE PORT AND FOUR PORT REFLECTOMETERS . . . . . . . . . . . .
5.6
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5.5
7
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(AMC)
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5.4
6
Overview . . . . . . . . . . . . . .
System Overview . . . . . . . . . .
4.2.1 Automatic Matching Control
4.2.2 Load Estimation Method . .
Experimental Results . . . . . . . .
Conclusion . . . . . . . . . . . . . .
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94
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CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
APPENDIX
6
A
DERIVATION OF LOAD IMPEDANCE CIRCLE EQUATION USING INPUT
IMPEDANCE MAGNITUDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
B
THREE-PORT AND FOUR-PORT MULTISTATE REFLECTOMETERS . . . 121
C
STANDARD AND MIXED-MODE S-PARAMETER TRANSFORMATION . . 124
D
DERIVATION OF DIFFERENTIAL INPUT REFLECTION COEFFICIENT . 126
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7
LIST OF TABLES
Table
page
3-1 Speci?cation of the matching tuner . . . . . . . . . . . . . . . . . . . . . . . . .
37
3-2 Mismatched load speci?cation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3-3 Comparison of brute-force, greedy, and single-step algorithms . . . . . . . . . .
46
3-4 Percentage of catastrophic case for brute-force, greedy, and single-step algorithms 47
3-5 Comparison of brute-force, greedy, and single-step algorithms avoiding catastrophic
cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4-1 Summary of mismatched load estimation results . . . . . . . . . . . . . . . . . .
57
5-1 Summary of heavily and slightly mismatched loads . . . . . . . . . . . . . . . .
73
5-2 Comparison of coupler and coupler-free load estimation in term of mean square
error (MSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6-1 Summary of mismatched loads . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
7-1 Mean square error (MSE) of neural network ?tting models using training and
testing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7-2 Average error of closed-form models . . . . . . . . . . . . . . . . . . . . . . . . . 106
7-3 Mapping table between bias voltage and a mismatched load to be matched . . . 107
7-4 Inverse mapping table between a mismatched load to be matched and bias voltage107
7-5 Impedance matching results using coupler-free load estimation and S-parameter
measurement data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7-6 Impedance matching results using three-port re?ectometer load estimation and
S-parameter measurement data . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7-7 Impedance matching results using coupler-free load estimation and S-parameters
estimated by neural network models . . . . . . . . . . . . . . . . . . . . . . . . 109
7-8 Impedance matching results using three-port re?ectometer load estimation and
S-parameters estimated by neural network models . . . . . . . . . . . . . . . . . 110
8
LIST OF FIGURES
Figure
page
1-1 Network matching an arbitrary load impedance to a transmission line . . . . . .
14
1-2 Maximum power delivered when source and load impedances are matched. . . .
14
1-3 Automatic matching control (AMC) . . . . . . . . . . . . . . . . . . . . . . . .
16
2-1 The L section matching networks . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2-2 The ? section matching network . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2-3 The T section matching network . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2-4 Typical pattern recognitions system diagram . . . . . . . . . . . . . . . . . . . .
22
2-5 Nearest-neighbor rule leads to a partitioning of the input space into Voronoi cells 23
2-6 Multilayer perceptron
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2-7 Arti?cial neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-8 Sigmoidal nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-9 Principal component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2-10 Six-port re?ectometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2-11 Determination of ?L from the radical center of three circles . . . . . . . . . . . .
31
2-12 Four-port re?ectometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3-1 Automatic matching control (AMC) system diagram . . . . . . . . . . . . . . .
35
3-2 Recon?gurable ?ve-stub matching tuner . . . . . . . . . . . . . . . . . . . . . .
37
3-3 Comparison of simulation and measurement of matching tuner. The matching
tuner is tuned with typical bias (2.56 2.56 2.56) and matched load . . . . . . . .
38
3-4 Mismatched load 1 measurement (S11 = 0.11 + j0.09 at 3.5 GHz). Motor positions
are (16725, 2262, 5000). Re?ection coe?cient |S11 |=0.14. . . . . . . . . . . . . . 40
3-5 Mismatched load 2 measurement (S11 = 0.19 + j0.40 at 3.5 GHz). Motor positions
are (17105, 1424, 5000). Re?ection coe?cient |S11 |=0.45. . . . . . . . . . . . . . 41
3-6 Mismatched load 3 measurement (S11 = -0.11 + j0.22 at 3.5 GHz). Motor positions
are (17722, 2005, 5000). Re?ection coe?cient |S11 |=0.24. . . . . . . . . . . . . . 41
3-7 Matching capability of ?ve-stub matching tuner at 3.5 GHz
9
. . . . . . . . . . .
42
3-8 Matching tuner measurement with matched load. Typical and automated biases
are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) . . . . . . . . . . . . . . . . .
44
3-9 Matching tuner measurement with mismatched load 1. Typical and automated
biases are (2.56V 2.56V 2.56V) and (3.2V 3.86V 3.52V) . . . . . . . . . . . . . .
45
3-10 Matching tuner measurement with mismatched load 2. Typical and automated
biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) . . . . . . . . . . . . . .
49
3-11 Matching tuner measurement with mismatched load 3. Typical and automated
biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V) . . . . . . . . . . . . . .
50
3-12 Optimization surface of greedy algorithm . . . . . . . . . . . . . . . . . . . . . .
51
4-1 System diagram of load impedance estimation for automatic impedance matching 53
4-2 Radical center of three circles, ?L1 , ?L2 , and ?L3 . . . . . . . . . . . . . . . . . .
56
4-3 Impedance matching procedure using estimated load impedance. . . . . . . . . .
56
5-1 Automatic matching control (AMC) . . . . . . . . . . . . . . . . . . . . . . . .
60
5-2 Three port re?ectometer integrated with a high impedance probe A) System
diagram with schematic B) Fabrication on FR4 board . . . . . . . . . . . . . . .
61
5-3 Varactor SMV1405 capacitance versus reverse voltage . . . . . . . . . . . . . . .
62
5-4 Varactor SPICE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5-5 Radical center of three circles, ?L1 , ?L2 , and ?L3 . . . . . . . . . . . . . . . . . .
67
5-6 Least square nonlinear ?tting of high impedance probe model . . . . . . . . . .
70
5-7 Arti?cial neural network of high impedance probe model . . . . . . . . . . . . .
71
5-8 High impedance probe estimation error distribution . . . . . . . . . . . . . . . .
72
5-9 Load estimation using estimated S11 from high impedance probe . . . . . . . . .
74
5-10 Coupler and coupler-free load estimation with mismatched #1 . . . . . . . . . .
75
5-11 Coupler and coupler-free load estimation with mismatched #2 . . . . . . . . . .
76
5-12 Coupler and coupler-free load estimation with mismatched #3 . . . . . . . . . .
77
5-13 Coupler and coupler-free load estimation with mismatched #4 . . . . . . . . . .
78
6-1 Re?ectometers A) Three-port B) Four-port . . . . . . . . . . . . . . . . . . . . .
81
6-2 Schematic of a lumped power divider. Port 3 is a coupled port, which can be
used as a reference port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
10
6-3 Recon?gurable three-port matching network A) Schematic B) Implementation
on FR4 board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
6-4 Tunable element impedance with a bias from 0 V to 10 V. A varactor is in parallel
and in series with an inductor. и and О denote S11 and S21 , respectively. A) In
series with 3.3 nH B) In parallel with 1.8 nH . . . . . . . . . . . . . . . . . . . . 86
6-5 Matching capability of three-port re?ectometer at 2.4 GHz . . . . . . . . . . . .
91
6-6 The q-point distribution of three-port re?ectometer at 2.4 GHz . . . . . . . . . .
92
6-7 Multistate re?ectometer estimation using estimated S11 from high impedance
probe A) Small mismatch (MSE=0.09) B) Large mismatch (MSE=0.80) . . . .
93
7-1 Automatic matching control supports load estimation. . . . . . . . . . . . . . .
97
7-2 Neural network models for a 2-port matching network were trained by backpropagation.
A) S11 B) S21 C) S12 D) S22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7-3 Neural network models for a 2-port matching network were tested by 10% of
measurement data. A) S11 B) S21 C) S12 D) S22 . . . . . . . . . . . . . . . . . . 104
7-4 Closed form models for a 2-port matching network were trained by nonlinear
least square ?tting. A) S11 B) S21 C) S12 D) S22 . . . . . . . . . . . . . . . . . . 105
7-5 S-parameters from vector measurement and neural network model were compared
in terms of coupler-free load estimation assisted matching control. A) Vector
measurement by large mismatch B) Neural network model by large mismatch
C) Vector measurement by small mismatch D) Neural network model by small
mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7-6 S-parameters from vector measurement and neural network model were compared
in terms of re?ectometer load estimation assisted matching control. A) Vector
measurement by large mismatch B) Neural network model by large mismatch
C) Vector measurement by small mismatch D) Neural network model by small
mismatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
11
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Ful?llment of the
Requirements for the Degree of Doctor of Philosophy
AUTOMATED MATCHING CONTROL SYSTEM USING LOAD ESTIMATION AND
MICROWAVE CHARACTERIZATION
By
Jaeseok Kim
December 2008
Major: Electrical and Computer Engineering
The automation of the impedance matching of radio frequency (RF) ports enables the
test engineer to compensate the undesired e?ects, which are not uncommon in RF systems
and make the impedance matching iterative, time-consuming, and an empirical process.
Numerous recon?gurable matching networks have been presented for automated matching.
However, the automation still relies on an iterative control to achieve a matching goal,
because it lacks the knowledge of the RF target and the matching network. Our goals
were to develop an automatic matching control system that uses this knowledge to set the
impedance matching in a non-iterative fashion and to develop a method to extract circuit
parameters systematically while keeping the additional necessary parts to a minimum.
To achieve this goal, we use the principles of a re?ectometer to extract knowledge of
the RF target and various microwave modeling methods to characterize the matching
network. Our results demonstrate the proposed ideas and include an automatic matching
control using a tunable microstrip bandpass ?lter, a load estimation technique using
the microstrip ?lter, a new lumped matching network for the automatic matching in
embedded RF testing, and a new matching control algorithm using the load estimation.
12
CHAPTER 1
INTRODUCTION
1.1
Why Automatic Impedance Matching?
The design process for radio frequency (RF) systems often starts with the de?nition
of the interface between the smaller components, where the impedance matching is critical
for the following reasons.
?
?
?
?
Deliver maximum power between components
Improve the signal-to-noise ratio (SNR)
Reduce amplitude and phase errors
Provide isolation between circuit stages
However, the actual implementation of the impedance matching is impaired easily by a
deviation from the typical component value used during the design process. It is critical to
be able to evaluate this deviation when selling a component commercially, so techniques to
create and evaluate matching are important for testing RF production ICs.
For example, the impedance of radio frequency (RF) ports in a device under test
(DUT) to a loadboard is poorly de?ned due to unwanted e?ects, such as pogo pin
connections, socket parasitics, and manufacturing variation. In addition, the input
impedance of a handheld device antenna, especially the reactance part of the impedance,
is varying as the environment around the device is changing. The unwanted e?ects and
varying environment are not uncommon in RF systems and the matching of a poorly
de?ned impedance is di?cult, slow, and expensive, during part test.
A recon?gurable matching network can be used to set the impedance matching of RF
match targets against the deviation caused by the unwanted e?ects. The recon?guration
is conducted by a closed-loop feedback control, a so-called automatic matching control,
which senses the impedance mismatch and determines proper component values for the
impedance matching.
13
Z0
Matching
network
ZL
Figure 1-1. Network matching an arbitrary load impedance to a transmission line
ZS
VS
I
ZL
VL
Figure 1-2. Maximum power delivered when source and load impedances are matched.
In this thesis, we will develop a recon?gurable matching network and a matching
control algorithm for an automatic matching control (AMC) system, which automatically
sets the impedance matching of RF match targets.
1.2
Maximum Power Transfer on Impedance Matching
The basic idea of the impedance matching is illustrated in Figure 1-1. A two-port
network is placed between an arbitrary load (ZL ) and a transmission line (Z0 ). The load
impedance ZL is converted to be matched with the source impedance Z0 . Impedance
matching enables the delivery of maximum power to the load, although there are multiple
re?ections between the matching network and the load.
Maximum power delivery is explained as follows. In Figure 1-2, AC power is being
transferred from the source, with phasor magnitude voltage |VS | (peak voltage) and ?xed
source impedance ZS , to a load with impedance ZL , resulting in a phasor magnitude
current |I|. |I| is simply the source voltage divided by the total circuit impedance.
|I| =
|VS |
|ZS + ZL |
(1?1)
The average power PL dissipated in the load is the square of the current multiplied by
the resistive portion (the real part) RL of the load impedance.
PL =
2
Irms
RL
1
1
= |I|2 RL =
2
2
(
|VS |
|ZS + ZL |
)2
14
RL =
1
|VS |2 RL
2 (RS + RL )2 + (XS + XL )2
(1?2)
The denominator is minimized by making
XL = ?XS
(1?3)
The power equation is reduced to
PL =
1 |VS |2 RL
2 (RS + RL )2
(1?4)
Similarly, the power is maximized by making
RL = RS
(1?5)
Therefore, two conditions, Equation 1?3 and 1?5, are combined and written as complex
conjugate matching condition.
ZL = ZS?
(1?6)
where * denotes a complex conjugate. The complex conjugate matching is the matching
goal of the proposed automatic matching control.
1.3
Challenges of Automatic Impedance Matching
Recon?gurable matching networks have been proposed for the automatic matching
control (AMC). The proposed networks were designed using various tuning elements such
as varactors [1][2][3][4], CMOS switches [5], p-i-n diodes [6][7], or MEMS switches [8].
The recon?gurable matching network is adapted by a matching control algorithm
which determines component values by the iteration of a closed-loop feedback. When RF
match targets are not precisely de?ned, ?nding necessary matching components becomes
highly iterative, time consuming, and an empirical process. Even if they are found for a
certain RF match target, they seldom provide the impedance matching to other targets.
An improvement on the impedance matching control algorithm has been achieved
by using heuristic approaches [6]. Although the presented algorithms can converge faster
than a brute-force approach, they may still get trapped in a local minimum which is one
of typical optimization problems. Further, the iterative nature of the algorithms makes
15
Source
Z0
Coupler
Reconfigurable
matching network
Reflection
coefficient
detector
Match decision
and biasing circuit
ZL
Figure 1-3. Automatic matching control (AMC)
convergence time to achieve the impedance matching increasing proportionally to the
complexity of the matching network.
For embedded RF testing, lumped elements are preferred to distributed elements due
to the compact size at the frequency equal to or less than 2.4 GHz. Most matching control
assumes that a degree of mismatch is measured through a directional coupler, which is too
large to be embedded.
So far the challenging problems of the automatic matching control have been
described. Next section we will propose solutions to them.
1.4
Proposed Solution: Automatic Matching Control with Load Estimation
A proposed solution to the challenges of the impedance matching is an automatic
matching control (AMC) supported by a load impedance estimation. As shown in Figure
1-3, the automatic matching control recon?gures the matching network through the
closed-loop feedback consisting of a coupler, a re?ection coe?cient detector, and a match
decision circuit. An estimate of a load re?ection coe?cient helps the matching control to
achieve an impedance match without trying di?erent biases iteratively. This thesis focuses
on developing recon?gurable matching networks and matching control algorithms with
load estimation especially for embedded RF testing.
16
CHAPTER 2
BACKGROUND
2.1
Impedance Matching Networks with Lumped Elements
When circuit dimensions are not small relative to the wavelength, the phase change
from one point to another in the circuit is not negligible. In this case, the equivalent
voltage and current waves along a transmission line are expressed as the sum of the
incident and re?ective waves, given by
V (z) = V + e???z + V ? e+??z
(2?1)
V + ???z V ? +??z
?
e
e
Z0
Z0
(2?2)
I(z) =
where Z0 and ? are the characteristic impedance and the propagation constant of the
transmission line, respectively.
If the transmission line is terminated with other than its characteristic impedance Z0 ,
the re?ection happens as a result of discontinuities. The re?ection coe?cient ? is de?ned
as the ratio of the incident to the re?ected wave along a transmission line, given by
V ? +?2?z
V ? e+??z
?(z) = + ???z = + e
V e
V
(2?3)
If the transmission line is terminated with a load impedance ZL , the load re?ection
coe?cient and the load impedance are written as
?L = ?(z)
V (z) ZL =
I(z) = Z0
z=0
=
z=0
V?
V+
V++V?
1 + ?L
= Z0
+
?
V ?V
1 ? ?L
(2?4)
(2?5)
The load re?ection coe?cient can be also expressed in terms of ZL and Z0 as
?L =
ZL ? Z0
ZL + Z 0
17
(2?6)
The re?ection coe?cient ? is a complex number that describes both the magnitude and
the phase shift of the re?ection. The simplest cases, when the imaginary part of ? is zero,
are
?
?
?
? = ?1
?= 0
? = +1
Maximum negative re?ection, when short-circuited.
No re?ection, when perfectly matched.
Maximum positive re?ection, when open-circuited.
The voltage standing wave ratio (VSWR), which represents the degree of re?ection
in another way, is de?ned as the ratio of the maximum to the minimum magnitude of the
voltage wave. The voltage standing wave ratio can be expressed in terms of the re?ection
coe?cient given by
VSWR =
|V (z)|max
1 + |?L |
|V + ||e???z + ?L e+??z |max
=
= + ???z
+??z
+ ?L e
|min
|V (z)|min
|V ||e
1 ? |?L |
(2?7)
Conversely, the re?ection coe?cient can be obtained from the measurement of voltage
standing wave ratio along the transmission line.
|?L | =
VSWR ? 1
VSWR + 1
(2?8)
In a traditional vector network analyzer, a complex re?ection coe?cient is calculated
from the measurement of incident and re?ected wave powers. If the phase of a re?ection
coe?cient is not necessary, e.g., the degree of mismatch, the magnitude of a re?ection
coe?cient can be measured by a VSWR detector.
2.1.1
L-type Matching Network
The simplest matching network is a L-type network with two lumped elements. The
values of the lumped elements can be found through the analytic solution as follows. The
impedance seen looking into the matching network shown in Figure 2-1A should be equal
to Z0 for impedance matching:
Z0 = ?X +
1
?B + 1/(RL + ?XL )
18
(2?9)
?X
?X
Z0
ZL
?B
Z0
A
?B
ZL
B
Figure 2-1. The L section matching networks
where the load impedance ZL = RL + ?XL . Rearranging and separating into real and
imaginary parts gives two equations given by
B(XRL ? XL Z0 ) = RL ? Z0
(2?10)
X(1 ? BXL ) = BZ0 RL ? XL
(2?11)
The solution for B and X are given by
?
?
XL ▒ RL /Z0 RL2 + XL2 ? Z0 RL
B=
RL2 + XL2
X=
1
XL Z0
Z0
+
?
B
RL
BRL
(2?12)
(2?13)
Note that the solutions exist only when RL > Z0 .
Similarly, the solution for Figure 2-1B is given by
?
(Z0 ? RL )/RL
B=▒
Z0
?
X = ▒ RL (Z0 ? RL ) ? XL
(2?14)
(2?15)
Note that the solutions exist only when RL < Z0 .
2.1.2
?-type Matching Network
The L-type network cannot provide an impedance match of every point on the Smith
chart. By adding the third element, the whole Smith chart can be covered. The third
element added to the L-type networks results in either ?- or T-type network. In this
section, the analysis of the ?-type network is presented as follows.
19
?X
Z0
?B1
?B2
ZL
Figure 2-2. The ? section matching network
?X1
?X2
?B
Z0
ZL
Figure 2-3. The T section matching network
The impedance seen looking into the matching network shown in Figure 2-2 should be
Z0 .
1
= ?B1 +
Z0
1
1
?X +
?B2 + 1/(RL + ?XL )
(2?16)
Rearranging and separating into real and imaginary parts gives two equations given
by
B1 Z0 (X + XL ) ? B1 Z0 B2 XXL ? B2 (XRL ? XL Z0 ) = Z0 ? RL
(2?17)
X(1 ? B2 XL ) = (B1 + B2 )Z0 RL ? XL ? B1 B2 XZ0 RL
(2?18)
There are three variables, X, B1 , and B2 , de?ned by two equations. Hence, there
exist multiple solutions satisfying the impedance matching. For a uniquely determined
solution, the quality factor, which determines the bandwidth of matching networks, can be
used as one of design speci?cations.
2.1.3
T-type Matching Network
Similar to the ?-type network, a T-type network is analyzed as follows. The
impedance seen looking into the matching network shown in Figure 2-3 should be Z0 .
Z0 = ?X1 +
1
?B + 1/(RL + ?(X2 + XL ))
20
(2?19)
Rearranging and separating into real and imaginary parts gives two equations given
by
2.1.4
BZ0 (X2 + XL ) ? BX1 RL = Z0 ? RL
(2?20)
X1 (1 ? B(X2 + XL )) = BZ0 RL ? (X2 + XL )
(2?21)
Bandwidth of Matching Network
In a resonant circuit, the ratio of its resonant frequency f0 to its bandwidth is known
as the loaded Q of the circuit.
QL =
?0
f0
=
Bandwidth in Hertz
(2?22)
The matching networks are used for an impedance match at a certain frequency. The
frequency response of the networks can be classi?ed as either a low-pass or a high-pass
?lter. At each node of the networks, there is an equivalent series input impedance,
denoted by Rs + ?Xs . Hence, a circuit node Q, denoted by Qn , can be de?ned at each node
as
Qn =
|Xs |
Rs
(2?23)
If the equivalent parallel input admittance at the node is Gp + ?Bp , the circuit node Q can
be expressed in the form
Qn =
|Bp |
Gp
(2?24)
For a narrowband range of frequencies around f0 , a matching network can be viewed
as a bandpass ?lter. Hence, the loaded Q of the bandpass ?lter is given by
QL =
?0
|BT |
RT
= ?0 RT CT =
=
BW
GT
|XT |
(2?25)
If the matching network holds the complex conjugate matching at each node, Rs1 =
Rs2 and Xs1 = ?Xs2 . Rs1 and Rs2 are the resistance seen looking into a source and a load
at the node. Xs1 and Xs2 are the reactance seen looking into a source and a load at the
21
node. The loaded Q can be written as the ratio of the resistance to the reactance.
QL =
Rs1 ||Rs2
Qn
Rs1 ||Rs2
=
=
|Xs1 |
|Xs2 |
2
(2?26)
If multiple internal nodes exist such as T or ? networks, the QL of the matching
circuits is the half value of the highest Qn in the circuits. In general, higher values of QL
can be obtained using matching circuits with more elements. For example, adding one
element to a L network results in either a T network or a ? network, which has higher
values of QL than a L network.
When a bandwidth of matching networks is a design consideration, L networks are
not suitable because the QL of the networks is ?xed with the complex conjugate matching
condition. Instead, the higher order ladder networks can have either higher or lower QL
depending on the highest Qn of the networks.
2.2
Pattern Recognition
Most pattern recognition systems are partitioned into four components such as
preprocessing, feature extraction, classi?cation, and postprocessing, as shown in Figure
2-4. Preprocessing simpli?es subsequent operations without losing relevant information,
feature extraction measures useful properties for classi?cation, classi?cation assigns the
extract feature into a category, and postprocessing uses the classi?cation results to decide
on the recommended action.
Preprocessing
Feature Extraction
Classification
Postprocessing
Figure 2-4. Typical pattern recognitions system diagram
22
Voronoi edge
Voronoi vertex
v
Figure 2-5. Nearest-neighbor rule leads to a partitioning of the input space into Voronoi
cells
The microwave components have been successfully modeled by the pattern recognition
techniques thanks to their adaptability to complex phenomena [9]. Applying the pattern
recognition techniques to the proposed matching control becomes a new research
opportunity for the advancement of the automated RF system. In this section, two
classi?ers, a nearest neighbor classi?er and a neural network, and principal component
analysis (PCA), one of popular preprocessing methods, are introduced and described.
2.2.1
The k -Nearest Neighbor Classifier
The task of the classi?er is to use the feature vector provided by the feature extractor
to assign the object to a category. The nearest-neighbor rule for classifying a test point x
is to assign it the label associated with the training point x? nearest to it. This rule allows
us to partition the feature space into cells consisting of all points closer to a given training
point x? ? a so-called Voronoi tessellation of the space, as shown in Figure 2-5
An obvious extension to of the nearest-neighbor rule is the k-nearest-neighbor rule. A
decision is made by examining the labels on the k nearest neighbors and taking a vote.
23
P
P
Input Layer
P
f ()
P
f ()
P
f ()
f ()
f ()
Hidden Layer
Output Layer
Figure 2-6. Multilayer perceptron
2.2.2
Artificial Neural Network (ANN)
Arti?cial neural networks (ANNs) are distributed, adaptive, generally nonlinear
learning machines built from many di?erent processing elements (PEs) [10]. As a
nonlinear statistical data modeling tool, neural networks are used to model complex
relationships between inputs and outputs or to ?nd patterns in data.
One of the most popular and powerful neural network designs is known as a multilayer perceptron. A multilayer perceptron consists of a set of neurons interconnected by
weighted connections, as illustrated in Figure 2-6. There are an input layer, one or more
hidden layers, and an output layer. As shown in Figure 2-7, each neuron has weighted
coe?cients wl,j,m that are adjusted to train the algorithm, which are linearly combined,
then passed through a nonlinear activation function f . The multilayer perceptron can be
represented as
yk = f
(
?
(
w2,j,k f
j
?
i
24
))
w1,i,j xi
(2?27)
wl,j,0
wl,j,1
wl,j,2
P
wl,j,3
vl,j
yl,j
f (и)
wl,j,ml?1
Layer l, Neuron j
Figure 2-7. Arti?cial neuron
1
0.8
f(x)
0.6
0.4
0.2
0
-10
-5
0
x
5
10
Figure 2-8. Sigmoidal nonlinearity
where f is the activation function (typically, nonlinear sigmoidal function), wl,j,k is a
weight from the kth node of the (l-1)th layer to the jth node of the lth layer, xi and yk
denote the ith input and the kth output node, respectively.
The activation function f () provides the nonlinearity necessary for the neural
network. In general, any monotonically increasing function can be used. One commonly
used function is the sigmoidal nonlinearity, de?ned by
f (x) =
1
1 + e?x
25
(2?28)
and plotted in Figure 2-8. This function has the property that the derivative is easy to
compute.
f ? (x) = f (x)(1 ? f (x))
(2?29)
In training the multilayer perceptron, a supervised training algorithm is used, in
which a set of known input/output data combinations are presented to the network. Using
a backpropagation algorithm, the network is trained so that the network output matches as
closely as possible the desired output, for each input data point.
2.2.3
Principal Component Analysis (PCA)
Tight correlation among features makes di?cult the training of neural networks.
Principal component analysis is used to decorrelate features by representing data along the
direction with the largest variance and is often used together with neural networks. How
feature vectors can be represented in terms of principal components is described in this
section.
The problem begins with a idea of how a single vector x0 represents all vectors in
a set of n d-dimensional samples x1 , x2 , и и и , xn . One of solutions is to ?nd a vector x0
such that the sum of the squared distances between x0 and the various xk is as small as
possible. The squared-error criterion function J0 (x0 ) is de?ned as
J0 (x0 ) =
n
?
||x0 ? xk ||2
(2?30)
k=1
and is used to seek the value of x0 to minimize J0 . The solution is the sample mean m,
de?ned as
1?
xk
m=
n k=1
n
(2?31)
However, the sample mean is zero-dimensional representation of the data set. It is more
interesting to get one-dimensional representation by projecting the data onto a line
running through the sample mean. The line equation is written as
x = m + ae
26
(2?32)
where e is a unit vector in the direction of line and the scalar a can take any real value.
If xk is represented by m + ak e, then an optimal set of coe?cient ak can be obtained by
minimizing the squared-error criterion function.
The squared-error criterion function is written as
J1 (x) = J1 (a1 , a2 , и и и , an , e) =
n
?
||(m + ak e) ? xk ||2
k=1
=
n
?
a2k ||e||2 ? 2
n
?
k=1
ak eT (xk ? m) +
k=1
n
?
||xk ? m||2
(2?33)
k=1
The derivative of J1 with respect to ak is set to zero and the optimal set of coe?cient
ak is given by
n
?
?J1
=2
(ak ||e||2 ? eT (xk ? m)) = 0
?ak
k=1
(2?34)
ak = eT (xk ? m)
(2?35)
where the unity vector ||e||2 = 1.
By substituting the solution ak into J1 , J1 is written as
J1 (x) = J1 (e) = ?
n
?
[e (xk ? m)] +
T
2
=?
||xk ? m||2
(2?36)
k=1
k=1
n
?
n
?
e (xk ? m)(xk ? m) e +
T
T
n
?
||xk ? m||2
(2?37)
k=1
k=1
= ?eT Se +
n
?
||xk ? m||2
|k=1
{z
(2?38)
}
independent of e
where the scatter matrix is de?ned as
S=
n
?
(xk ? m)(xk ? m)T
(2?39)
k=1
Using the method of Lagrange multipliers, the vector e minimizing J1 becomes an
eigenvector of the scatter matrix as follows.
Se = ?e
27
(2?40)
15
10
Principal Component 1
Dimension 2
5
0
?5
Principal Component 2
?10
?15
?15
?10
?5
0
Dimension 1
5
10
15
Figure 2-9. Principal component analysis
? eT Se = ??eeT = ??
(2?41)
In summary, ?nding the best one-dimensional projection of the data (best in
least-sum-of-squared-error sense) is equivalent to projecting the data onto a line through
the sample mean in the direction of the eigenvector of the scatter matrix having the
largest eigenvalue. The coe?cients ak are called the principal components. Principal
component analysis reduces the dimensionality of feature space by limiting directions
along which the scattering (variance) is the greatest. Those directions are geometrically
illustrated in Figure 2-9.
2.3
2.3.1
Multi-Port Reflectometers
Six-Port Reflectometer
A six-port re?ectometer was proposed as an alternative network analyzer [11]. The
basic structure of the six-port re?ectometer is illustrated in Figure 2-10. The port 1 (P1)
is connected to a signal source, the port 2 (P2) is terminated with a device under test
28
a1
a2
Six-port Reflectometer
ZL
P5
a6
a4
b6
P4
P3
b5
b4
b3
a5
b2
a3
b1
?2
P6
Figure 2-10. Six-port re?ectometer
(DUT) having an impedance ZL , and the port 3, 4, 5, and 6 (P3, P4, P5, and P6) are
connected to power detectors. The six-port junction is characterized by 12 complex waves
ai and bi , i = 1, и и и , 6 and the scattering equations
? ? ?
b
S S иии
? 1 ? ? 11 12
? ? ?
?b2 ? ?S21 S22 и и и
? ? ?
?.?=? .
.. . .
?.? ? .
.
.
?.? ? .
? ? ?
b6
S61 S62 и и и
or
bi =
6
?
de?ned as
?? ?
S16
a
? ? 1?
?? ?
? ?
S26 ?
? ?a2 ?
? ?
.. ?
?? . ?
. ? ? .. ?
?? ?
S66
a6
(2?42)
i = 1, и и и , 6
Sij aj ,
(2?43)
j=1
Moreover, the detectors are terminated with de?ned loads. The terminations are described
i = 3, и и и , 6
a i = b i ?i
(2?44)
where ?i is the input re?ection coe?cient of the ith power detector. Equation 2?43 and
2?44 can be solved in terms of a2 and b2 . Then, the incident waves on the detectors are
written as
(
bi = Ai a2 + Bi b2 = b2 Ai
a2 Bi
+
b2
Ai
)
= b2 Ai (?2 ? qi )
29
i = 3, и и и , 6
(2?45)
where Ai and Bi are functions of Sij and ?j
The power detector yields only amplitude or power response, therefore the output of
the ith detector is written as
Pi = |bi |2 = |b2 |2 |Ai |2 |?2 ? qi |2
(2?46)
With an analogy with a six-port re?ectometer analysis, one of port is a reference port
(P4), where A4 is ideally zero. The power detected in the reference port 4 is written as
P4 = |b4 |2 = |A4 a2 + B4 b2 |2 = |B4 |2 |b2 |2
(2?47)
The normalized power by the reference port is written as
Pi
|Ai |2
=
|?2 ? qi |2
P4
|B4 |2
i = 3, 5, 6
(2?48)
The re?ection coe?cient of the DUT can be expressed in terms of the detector power as
|?2 ? qi |2 =
|B4 |2 Pi
|Ai |2 P4
i = 3, 5, 6
(2?49)
The equation represents circles on the Smith chart and the re?ection coe?cient of a DUT
?2 is the cross point of the circles.
Although these circles should be intersected in a single point ideally, it may not
happen in reality due to measurement errors. In this case, the re?ection coe?cient can be
determined by the radical center of three circles, as shown in Figure 2-11.
2.3.2
Four-Port Multistate Reflectometer
A six-port re?ectometer has been paid attention by many researcher due to its simple
structure. The fundamentals of the six-port re?ectometer were extended to the design of a
multistate re?ectometer and a four-port multistate re?ectometer with only two detectors
was proposed [12][13][14]. The diagram of the multistate re?ectometer is illustrated in
Figure 2-12
30
?L6
6
RL
q6
?L3
3
RL
R L5
q3
q5
?L5
Figure 2-11. Determination of ?L from the radical center of three circles
a1
a2
Four-port Reflectometer
ZL
P3
?2
b4
b3
a4
b2
a3
b1
P4
Figure 2-12. Four-port re?ectometer
31
With an analogy with the six-port re?ectometer, the operation of a four-port
re?ectometer is described by the following equations,
(
bi = Ai a2 + Bi b2 = b2 Ai
a2 Bi
+
b2
Ai
)
= b2 Ai (?L ? qi )
i = 3, 4
?2 ? q3 2 A4 2 P3
?2 ? q4 = A3 P4
(2?50)
(2?51)
where the system parameters A3 and A4 are given (Refer to Appendix B for derivation) by
A3 =
S21 S32 ? S22 S31
S21
q3 =
S31
S22 S31 ? S21 S32
(2?52)
A4 =
S21 S42 ? S22 S41
S21
q4 =
S41
S22 S41 ? S21 S42
(2?53)
and
The multistate re?ectometer is equipped with the facility to operate in di?erent states
k (k = 1, 2, 3) in order to provide enough information for the determination of both the
magnitude and the phase of the re?ection coe?cient of a device-under-test (DUT).
The equations are written with superscript k, which denotes a state.
? ? q (k) 2 A(k) 2 P (k)
2
4 3
= (k)
3
(k)
A3 P4(k)
? 2 ? q4 (2?54)
The optimum-performance criteria for the multistate re?ectometer are as follows.
?
?
?
The centers q (k) should be equally spaced around the origin (i.e., 120? angular
separation).
The centers q (k) should be equidistant from the origin (i.e., equal magnitudes).
The system parameters B (k) should have zero magnitudes.
Similar to the six-port re?ectometer, the reference port, which depends only on b2 to
set A4 to zero, is assigned to port 4. The equation can be written as
2 B (k) 2 P (k)
(k) ?2 ? q3 = 4(k) 3(k)
A3 P4
32
(2?55)
The power measurement with three di?erent network states results in three circles de?ned
by the equation. The complex re?ection coe?cient of a device-under-test (DUT) can be
determined in the same way as the six-port re?ectometer.
33
CHAPTER 3
AUTOMATIC MATCHING CONTROL
3.1
Overview
The ability to test RF devices on the loadboard requires a good broadband match.
Moreover, the bandwidth of many wireless devices is governed by the input impedance of
an antenna. However, the input impedance is one of the parameters that varies most in
di?erent environments on the loadboard and next to an antenna and the input reactance
varies with frequency more than does the input resistance of the antenna [15][16].
Various automatic matching tuners and control algorithms have been proposed. A
phase detector was used to correct the reactive part of the antenna mismatch in [17].
General matching network design and tuning strategies were studied in [18]. A generic
algorithm has been widely used as a tuning algorithm [18][19][20] and heuristic search
algorithms were studied in [6]. In addition, various narrowband techniques at di?erent
frequencies have been studied for CMOS switched capacitors at 2.4 GHz [5] and a p-i-n
diode switched capacitors at 390 MHz [6].
Prior to this work, a broadband recon?gurable matching network was proposed, but
a supporting matching control system and algorithms were not developed [1]. In this
work, a novel automatic broadband matching control was developed for a broadband
recon?gurable matching network from 2.5 to 4.5 GHz. The matching network consists
of a ?ve-stub microstrip ?lter and three varactors. A matching algorithm was developed
using a greedy search algorithm to determine the varactor bias for impedance match over
the large fractional bandwidth (> 50%). The work demonstrates the feasibility of the
automatic matching control circuit over 2.5 GHz to 4.5 GHz.
The rest of this chapter is organized as follows: Section 3.2 gives an overview of the
proposed system. Section 3.3 presents the experimental results. Section 3.4 concludes and
describes the future direction.
34
Magnitude/Phase
Measurement Chip
Coupler
12-Channel
Microprocessor
Core
Serial
Interface
4-Channel
12-Bit DAC
Lookup
Table
for
Varactor
Bias
R S 232C
Reconfigurable Tuner
Magnitude
Phase
Network Analyzer
(Agilent E8358A)
300k ~ 9GHz
GPIB
Computer
(LabVIEW)
Choke Coil
Figure 3-1. Automatic matching control (AMC) system diagram
3.2
System Overview
As shown in Figure 3-1, the proposed automatic matching control system consists of
a recon?gurable tuner, a network analyzer, a host computer, and a microcontroller. In the
current implementation, the tuner is directly connected to the network analyzer. Various
antenna mismatches were simulated by an automatic load-pull system.
The automatic matching is performed by a closed-loop feedback operation as follows.
First, the network analyzer measures the re?ected wave power (S11 parameter). Second,
the host computer calculates the available bandwidth. Last, the algorithm searches for the
varactor bias for broadband impedance match and the microcontroller sets the varactor
bias. The above procedures are repeated until the algorithm search converges to the
optimal bias, which minimizes the impedance mismatch. This technique can be applied to
an industrial automatic test equipment (ATE) system and a loadboard for conducting RF
part test.
35
3.2.1
Impedance Matching Tuner
As shown in Figure 3-2, the tuner, manufactured on Rogers Corporation Duroid 6006
board by the University of Arizona, has ?ve stubs along a microstrip line. Three stubs are
connected to a reverse-biased varactor diode MPV1965, whereas two stubs have no tuning
elements. The e?ective length of the tunable stub is controlled by the capacitance of the
varactor. The varactor reverse bias and capacitance range from 0 to 5.12 V and 4.5 to 1
pF, respectively.
The tuner is designed to provide wide bandwidth (up to 2.5 GHz) at the frequency
of 3.5 GHz, as shown in Figure 3-3. The fractional bandwidth is 71% = 2.5 / 3.5. Agilent
E8358A PNA network analyzer was used to measure S-parameters and the bandwidth
was measured below 10 dB for the return loss. The insertion loss is as low as 2 dB. The
dimensions of the tuner are shown in Table 3-1.
The matching capability of the matching tuner can be represented by the load
re?ection coe?cient ?L to be matched by the tuner. The input re?ection coe?cient ?in is
written as
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(3?1)
The matching capability is derived from the load re?ection coe?cient by setting the input
re?ection coe?cient to zero.
?L ?in =0
=
S11
S11 S22 ? S12 S21
(3?2)
The distribution of the matching capability with the full range of varactor bias voltage
illustrates the coverage of mismatched loads that can be compensated by the matching
tuner.
3.2.2
Controller
The controller consists of a host computer, a microcontroller, and a digital-to-analog
converter (DAC). The computer provides interfaces with measurement instruments
and executes search algorithm. The microcontroller consists of a 12-channel 12-bit
36
Choke Inductor
DC Block Capacitor
Stub #2
Stub #1
Varactor 1
Stub #4
Stub #3
Varactor 2
Stub #5
Varactor 3
Figure 3-2. Recon?gurable ?ve-stub matching tuner
Table 3-1. Speci?cation of the matching tuner
Stub / Interconnect
Width (mils)
Length (mils)
Stub #1 & #5
Between #1 and #2
Stub #2 & #4
Between #2 and #3
Stub #3
27
39
22
36
34
416
342
429
481
384
analog-to-digital converter (ADC), a 4-channel 12-bit DAC, a universal asynchronous
receiver/transmitter (UART), and digital input/outputs. To provide a su?cient
number of voltage biases, an additional octal 12-bit DAC (AD5328) was soldered on
the microcontroller evaluation board. Microcontroller ?rmware was developed in the
embedded-C language and stored in the internal ?ash memory. The ?rmware supports the
controller?s communication with both the host computer and the DAC through RS-232C
and I2 C. It can interpret and execute GPIB-like commands, such as *IDN?, from the host
computer.
37
0
Magnitude (decibels)
-10
-20
-30
-40
S11
S21
S12
S22
-50
-60
0
1
2
3
4
5
Freq [GHz]
6
7
8
9
6
7
8
9
A Simulation
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
B Measurement
Figure 3-3. Comparison of simulation and measurement of matching tuner. The matching
tuner is tuned with typical bias (2.56 2.56 2.56) and matched load
38
The host computer and other instruments were logically connected through the
NI-VISA standard provided by MATLAB. A dedicated interface program was developed
to automate the control of the automatic load-pull system. The program supports the
communication between the host and the load-pull system and is normally called by
MATLAB.
3.2.3
Search algorithm
A greedy algorithm was used to ?nd the varactor bias for impedance match. The
algorithm searches for the locally optimal bias per single varactor at a time and can be
expressed as
BW(vi ),
v?i = argmax
vi
i = 1, и и и , d,
vmax
vi = 0, r
, и и и , vmax ,
2
?
1
{z
}
|
(3?3)
2r
where BW() is the measured bandwidth, vmax is the maximum reverse bias voltage applied
to varactors, d is the number of varactors, r is the number of DAC bits.
The algorithm has a computational complexity of O(n), compared with O(n3 )
of brute-force approach (dimension = # of varactors = 3). Although it may fail to
converge to the globally optimal bias, it can reduce the searching time signi?cantly. For
example, suppose that a DAC resolution is 6 bits. Two algorithms? searching time are
3
dimension О resolution = 3 О 26 = 192 and resolutiondimension = 26 = 262144,
respectively. The greedy algorithm is faster by the factor of 2r
3.3
d?1
3?1
/d = 26
/3 ? 1365.
Experimental Results
A control program was developed using MATLAB to perform all testing procedures
including instrument initialization, mismatch load setup, S-parameter measurement, and
search for the varactor bias.
3.3.1
Characterization of Automatic Tuner System
We used Maury Microwave?s automatic tuner system (ATS) and mechanical load-pull
tuners for various mismatched loads. The S-parameters of the load-pull tuner are
39
Type
S11
|S11 |
Motor position
Matched
Mismatched
Mismatched
Mismatched
Mismatched
1
2
3
4
0 + j0
0.11 + ?0.09
0.19 + ?0.40
?0.11 + ?0.22
0.14 ? ?0.01
0
0.14
0.45
0.24
0.14
(100, 5000, 5000)
(16725, 2262, 5000)
(17105, 1424, 5000)
(17722, 2005, 5000)
(20464, 2228, 5000)
Mismatched
Mismatched
Mismatched
Mismatched
5
6
7
8
0.35 + ?0
0.55 + ?0
0.65 + ?0
0.75 + ?0
0.35
0.55
0.65
0.75
Simulated
Simulated
Simulated
Simulated
+j1.0
0
+j2.0 S11
S21
S12
S22
+j0.5
-5
+j0.2
5.0
2.0
0.0
1.0
-20
+j5.0
0.5
-15
0.2
Magnitude (decibels)
-10
\infty
-25
-30
-j0.2
-35
S11
S21
S12
S22
-40
-45
0
1
-j5.0
-j0.5
2
3
4
5
Freq [GHz]
6
7
8
9
-j2.0
-j1.0
Figure 3-4. Mismatched load 1 measurement (S11 = 0.11 + j0.09 at 3.5 GHz). Motor
positions are (16725, 2262, 5000). Re?ection coe?cient |S11 |=0.14.
determined by three motors and have to be measured over the frequency range of interest
because they are unknown and frequency-dependent.
The characterization was performed by Maury Microwave?s ATS software MT993
and the mapping ?le, so-called tuner file, was created. Each line of the tuner ?le contains
motor positions, S-parameters, and operating frequency.
From the tuner ?le, one matched load and three mismatched loads were selected. The
measurements of three mismatched loads are shown in Figure 3-4 to 3-6. The S11 at 3.5
GHz and motor positions are summarized in Table 3-2.
40
+j1.0
0
+j2.0 S11
S21
S12
S22
+j0.5
-5
+j0.2
5.0
2.0
1.0
0.0
0.5
-15
+j5.0
0.2
Magnitude (decibels)
-10
\infty
-20
-j0.2
-25
S11
S21
S12
S22
-30
-35
0
1
-j5.0
-j0.5
2
3
4
5
Freq [GHz]
6
7
8
-j2.0
-j1.0
9
Figure 3-5. Mismatched load 2 measurement (S11 = 0.19 + j0.40 at 3.5 GHz). Motor
positions are (17105, 1424, 5000). Re?ection coe?cient |S11 |=0.45.
+j1.0
0
+j2.0 S11
S21
S12
S22
+j0.5
-5
+j0.2
+j5.0
5.0
2.0
0.0
1.0
-20
0.5
-15
0.2
Magnitude (decibels)
-10
\infty
-25
-j0.2
-j5.0
-30
S11
S21
S12
S22
-35
-40
0
1
-j0.5
2
3
4
5
Freq [GHz]
6
7
8
9
-j2.0
-j1.0
Figure 3-6. Mismatched load 3 measurement (S11 = -0.11 + j0.22 at 3.5 GHz). Motor
positions are (17722, 2005, 5000). Re?ection coe?cient |S11 |=0.24.
Three mismatched loads used in this work show the almost-constant return and
insertion loss from 2.5 to 4.5 GHz. As described in Table 3-2, three mismatched loads have
|S11 | at 3.5 GHz = 0.14, 0.45, and 0.24, respectively.
3.3.2
Measurement Results
The matching capability S11 /(S11 S22 ? S12 S21 ) at the center frequency of 3.5 GHz was
calculated from the S-parameter measurement results, where the varactor bias ranges from
0 V to 5.12 V by 0.16 V step in 32 voltage levels. As shown in Figure 3-7, the matching
capability can cover a small portion on the Smith chart, because the tuner was designed
41
+j1.0
+j2.0
+j0.5
+j5.0
5.0
2.0
1.0
0.5
0.0
0.2
+j0.2
\infty
?j5.0
?j0.2
?j2.0
?j0.5
?j1.0
Figure 3-7. Matching capability of ?ve-stub matching tuner at 3.5 GHz
can be evaluated by the available bandwidth, which has been used as the performance
metric through this work.
The S-parameters of the matching tuner were measured with typical and optimal
bias. 2.56 V was set as the typical bias for the varactor. The optimal bias was found by
the search algorithm as described before.
The search algorithm found the optimal bias for the 50 ? matched load and the
mismatched load 1, whereas it failed to ?nd for the more severely mismatched load 2 and
3. For the mismatched load 2 and 3, the optimal bias found for the 50 ? case was used.
42
As shown in Figure 3-8 and Figure 3-9, the optimal bias for the 50 ? matched load
increased the available bandwidth from 1 GHz to 2 GHz. However, the optimal bias does
not increase the bandwidth for other mismatched loads, because the matching tuner
together with the larger mismatch (? = 0.24, 0.45) did not respond to the change in
varactor capacitance.
The comparison of brute-force, single-step proposed in [6], and the greedy algorithms
is presented in Table 3-3, where the performance metric for the cost function of three
algorithms is the 10 dB available bandwidth and the number of trials represents how many
biases each algorithm has evaluated until it converges. 100 di?erent initial states were
randomly generated and used for each experiment and the results were averaged over 100
experiments. The mismatched loads used in this experiment are summarized in Table
3-2. The degree of mismatch covers from 0 to 0.75. As shown in Table 3-3, the greedy
algorithm outperforms brute-force and single-step algorithms in terms of the number of
trials and the available bandwidth.
For severely mismatched loads, the single-step algorithm often failed to have available
bandwidth. In this catastrophic case, it is not fair to compare the performance by taking
the average of experimental results. Instead, the catastrophic case was analyzed and
excluded from the result. First, we de?ne 10% of the center frequency (0.35 GHz) as
the bandwidth limit for the catastrophic case. During the experiment, the number of
the catastrophic cases were counted to show how often the catastrophic case happens
for each algorithm. As summarized in Table 3-4, the single-step algorithm su?ered from
the catastrophic case even for modestly mismatched loads, e.g. ?L = 0.24 and all three
algorithms failed to optimize for the severely mismatched load, e.g. ?L = 0.75. This
result showed the similar trend as the broadband matching bandwidth for di?erent load
impedance presented by [1], which limited the real and imaginary part of mismatched load
impedance to 25 to 100 ? and ?50 to 50 ?, respectively. Second, the catastrophic case
was excluded from Table 3-3. The experimental result avoiding the catastrophic case is
43
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
6
7
8
9
6
7
8
9
A Typical bias
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
B Automated bias
Figure 3-8. Matching tuner measurement with matched load. Typical and automated
biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V)
44
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
6
7
8
9
6
7
8
9
A Typical bias
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
B Automated bias
Figure 3-9. Matching tuner measurement with mismatched load 1. Typical and automated
biases are (2.56V 2.56V 2.56V) and (3.2V 3.86V 3.52V)
45
Table 3-3. Comparison of brute-force, greedy, and single-step algorithms
Type
# of Trials
Available BW (GHz)
brute
greedy
single
brute
greedy single
Matched (Measured)
Mismatched 1 (Measured)
Mismatched 2 (Measured)
Mismatched 3 (Measured)
Mismatched 4 (Measured)
32768
32768
32768
32768
32768
273.60
272.64
340.80
332.16
281.28
358.29
414.99
454.95
420.39
353.70
1.28
1.24
0.76
1.24
1.24
1.22
1.21
0.68
1.16
1.20
1.17
1.09
0.25
0.63
1.04
Mismatched
Mismatched
Mismatched
Mismatched
32768
32768
32768
32768
323.52
312.00
281.28
284.16
473.85
550.53
642.87
642.87
1.16
0.72
0.36
0.32
1.10
0.51
0.36
0.32
0.40
0.06
0.00
0.00
32768
300.16
479.16
0.92
0.86
0.52
5
6
7
8
(Simulated)
(Simulated)
(Simulated)
(Simulated)
Average
summarized in Table 3-5. Provided that the catastrophic case can be always avoided by
some technique, the performance di?erence between the greedy and other two algorithms
is not as large as when the catastrophic case happens. However, the technique to avoid the
There are two reasons to make the greedy algorithm outperform other algorithms.
First, the greedy algorithm is less dependent on an initial state compared with the
single-step, because the greedy algorithm searches for the suboptimal solution for each
varactor. Second, the suboptimal solution of the greedy algorithm turned out to be close
to the global optimal solution of the brute-force. In the other word, the greedy algorithm
managed to reach close to the global optimal solution, whereas the single-step may get
trapped in local minima.
The optimization surface during an experiment on the greedy algorithm is illustrated
in Figure 3-12. The z-axis represents the available bandwidth in GHz and the x- and
y-axes represent the 32 varactor voltage levels from 0 V to 5.12 V by 0.16 V step. The
other varactor bias is set to 16th voltage level. The surface showed the steep threshold,
which can be modeled by either sigmoid or round functions. The ?at surface above the
46
Table 3-4. Percentage of catastrophic case for brute-force, greedy, and single-step
algorithms
|?L |
Type
Percentage of catastrophic cases (%)
brute-force
greedy
single-step
Matched (Measured)
Mismatched 1 (Measured)
Mismatched 2 (Measured)
Mismatched 3 (Measured)
Mismatched 4 (Measured)
0.01
0.14
0.45
0.24
0.14
0
0
0
0
0
0
0
0
0
0
0
3
65
40
1
Mismatched
Mismatched
Mismatched
Mismatched
0.35
0.55
0.65
0.75
0
0
0
100
0
39
0
100
57
93
100
100
5
6
7
8
(Simulated)
(Simulated)
(Simulated)
(Simulated)
Table 3-5. Comparison of brute-force, greedy, and single-step algorithms avoiding
catastrophic cases
Type
# of Trials
Available BW (GHz)
brute
greedy
single
brute
greedy single
Matched (Measured)
Mismatched 1 (Measured)
Mismatched 2 (Measured)
Mismatched 3 (Measured)
Mismatched 4 (Measured)
32768
32768
32768
32768
32768
273.60
272.64
340.80
332.16
281.28
358.29
424.48
348.69
287.55
356.45
1.28
1.24
0.76
1.24
1.24
1.22
1.21
0.68
1.16
1.20
1.17
1.12
0.59
1.05
1.05
Mismatched
Mismatched
Mismatched
Mismatched
32768
32768
32768
N/A
323.52
388.72
281.28
N/A
345.98
366.43
N/A
N/A
1.16
0.72
0.36
N/A
1.10
0.72
0.36
N/A
0.91
0.48
N/A
N/A
32768
311.75
355.41
1.00
0.96
0.91
5
6
7
8
(Simulated)
(Simulated)
(Simulated)
(Simulated)
Average
threshold explains why the suboptimal solution of greedy algorithm is pretty close to the
global optimal solution.
3.4
Conclusion and Discussion
microstrip matching tuner, which was provided by the University of Arizona. The
proposed matching control demonstrated the feasibility of the automatic broadband
matching control using the closed-loop feedback and the greedy search algorithm. The
47
measurement results showed that the greedy search algorithm could ?nd the optimal bias
for the matching tuner, but the matching tuner did not show the good tunability against
the large mismatches, |?| > 0.14. Currently we are working on improving the tuner?s
tunability against the large mismatch.
48
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
6
7
8
9
6
7
8
9
A Typical bias
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
B Automated bias
Figure 3-10. Matching tuner measurement with mismatched load 2. Typical and
automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V)
49
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
6
7
8
9
6
7
8
9
A Typical bias
0
-10
Magnitude (decibels)
-20
-30
-40
-50
S11
S21
S12
S22
-60
-70
0
1
2
3
4
5
Freq [GHz]
B Automated bias
Figure 3-11. Matching tuner measurement with mismatched load 3. Typical and
automated biases are (2.56V 2.56V 2.56V) and (4.8V 3.84V 4.64V)
50
1.4
1.2
1
0.8
0.6
0.4
0.2
0
40
35
30
30
25
20
20
15
10
10
0
5
0
Figure 3-12. Optimization surface of greedy algorithm
51
CHAPTER 4
4.1
Overview
The overall goal of the proposed research is to develop a compact automatic matching
control (AMC) circuit that sets the impedance match for radio frequency (RF) ports of
a socketed device under test (DUT) under automatic test equipment (ATE) test [21].
An AMC provides match setting and examination capability for RF test frequencies to
assist in a variety of RF test protocols. In this chapter, a new load impedance estimation
technique is presented that facilitates the AMC system. A load impedance was not able to
be determined in an AMC system prior to this work.
Various search methods for the optimal state of a tunable matching network have
been proposed [6][17][22]. However, the exhaustive search methods require millions of
measurements depending on the number of the matching network states and become a
limiting factor to the design of the matching network.
In prior work, most search methods try to reduce searching time using heuristic
approach based on exhaustive search, but their searching time increases proportionally to
the matching network complexity. If the load impedance ZL is known to the AMC system,
then the search is no longer necessary. Instead, the AMC system can set a matching
network topology and component values through direct analysis. Thus, the AMC system
needs to search only a few times to achieve the impedance match.
The proposed impedance estimation technique is based on the principle of a
multi-state re?ectometer. As an alternative network analyzer, measurement methods
based on a multi-state re?ectometer were presented using various approaches, such as a
scalar network analyzer [23], a four-port junction [13][14], and a tunable microstrip ?lter
[24]. Similar to the prior work, the proposed method measures a single port multiple
times instead of the measurement of multiple ports in a six-port re?ectometer [11], then
a complex re?ection coe?cient is determined by some mathematical manipulation. The
52
Zin
Tunable Impedance Matching Network
Z0
ZL
Power Detector
Control varactors
Automatic Matching Controller
Figure 4-1. System diagram of load impedance estimation for automatic impedance
matching
measured power represents the magnitude of a re?ection coe?cient and is depicted as a
circle on the Smith chart. Because the measured power reading results from the di?erent
networks and the same load, the intersect point of these circles becomes the load re?ection
coe?cient. In this work, we demonstrate that the tunable matching network designed for
impedance match can perform load estimation.
The rest of this chapter is organized as follows: Section 4.2 gives an overview of the
AMC and the proposed method. Section 4.3 presents the experimental results. Section 4.4
concludes and describes the future direction.
4.2
4.2.1
System Overview
Automatic Matching Control (AMC)
As shown in Figure 4-1, an automatic matching control system consists of a tunable
matching network, a power detector, and a controller. The automatic matching is
performed by a closed-loop feedback operation as follows. First, the power detector
53
measures the re?ected wave power. Second, the controller calculates the degree of
mismatch (S11 parameter). Last, the control algorithm searches for the varactor bias
for impedance match and the controller sets the varactor bias. The above procedures are
repeated until the control algorithm achieves a speci?ed matching goal. The matching
control system presented in [25], which was not able to estimate a load, was used in this
work.
4.2.2
An input re?ection coe?cient is expressed in terms of a load re?ection coe?cient and
two-port S-parameters as
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(4?1)
A complex re?ection coe?cient, such as ?in and ?L , is determined by both magnitude
and phase of a re?ected wave. However, a phase detector is not as compact and accurate
as a power detector. For this reason, an embedded test circuit often measures only wave
magnitude using power detectors and the phase information of the re?ection coe?cient
becomes unavailable. In this case, a complex input re?ection coe?cients is also measured
only in magnitude and Equation 4?1 can be written as
S12 S21 ?L |?in | = S11 +
1 ? S22 ?L (4?2)
The load re?ection coe?cient derived from the magnitude of the input re?ection
coe?cient is depicted as a circle on the Smith chart. By manipulating Equation 4?2 in the
same way as stability circle equation derivation [26], the circle equation is given by
2
? ?
(S
|?
|
?
?S
)
11 ?L ? 22 2 in 2
|S22 | |?in | ? |?|2 S
S
12
21
|?in |
=
|S22 |2 |?in |2 ? |?|2 where ? = S11 S22 ? S12 S21 and * denotes a complex conjugate.
54
(4?3)
The load re?ection coe?cient ?L is expressed as a circle whose center and radius are
? ?
(S22 |?in |2 ? ?S11
)
CL =
|S22 |2 |?in |2 ? |?|2
S
S
12
21
|?in |
RL = |S22 |2 |?in |2 ? |?|2 (center)
(4?4)
(4?5)
(Refer to Appendix A for the derivation.) The relationship between input and load
re?ection coe?cients was also analyzed as a circle equation in [8]. Their result looks
di?erent but is merely a special case of the general equation presented in this work.
Based on the derived equations, an estimation method measures a re?ected wave
power |?in | of an input port multiple times. The measured power represents only the
magnitude of a re?ection coe?cient and is depicted as a circle on the Smith chart.
Because the measured power reading results from the di?erent network and the same load,
the intersect point of these circles becomes the load re?ection coe?cient. We performed
the load impedance estimation as follows.
1.
2.
3.
Measure power in the input port of a matching network with three di?erent sets of
biases
Calculate the center and radius of the circle of the load re?ection coe?cient
Calculate the radical center of three circles
The radical center is the approximation of the center of three circles? overlapped region, as
illustrated in Figure 4-2. The coordinates of the radical center are given by
2
2
RL1
? RL2
+ x22
2x2
2
2
R ? RL3 + x23 + y32 ? 2xx3
y = L1
2y3
x=
(4?6)
(4?7)
Note that the least-squares method developed for the six-port measurement can be used
for higher accuracy [27].
Assuming the load is found through the proposed estimation method, the control
algorithm can set impedance matching using S-parameters of the matching network and
55
?L3
3
RL
(x3 , y3 )
?L1
1
RL
R L2
(0,0)
(x2 , 0)
?L2
Figure 4-2. Radical center of three circles, ?L1 , ?L2 , and ?L3 .
Start
Measure reflected power
Calculate bias for matching
Reflected power = 0
No
Yes
End
Figure 4-3. Impedance matching procedure using estimated load impedance.
56
Table 4-1. Summary of mismatched load estimation results
Type
Mismatched
Mismatched
Mismatched
Mismatched
#1
#2
#3
#4
|S11 |
Mean squared error
(MSE)
0.14
0.45
0.24
0.14
0.596300
7.982729
0.066334
0.148351
cL | > 1
MSE excluding |?
0.011048
0.080232
0.066334
0.031391
allow quick matching control of the AMC. The revised matching control procedure is
illustrated in Figure 4-3.
4.3
Experimental Results
In our experiments, four di?erent loads were used to evaluate the proposed estimation
methods. Maury Microwave?s Automatic Tuner System (ATS) generated the speci?ed
loads precisely every measurement and Agilent E8538 PNA network analyzer measured
the input re?ection coe?cient |?in |. The re?ection coe?cient magnitude of the loads are
|?L1 | ? 0.14, |?L2 | ? 0.45, |?L3 | ? 0.24, |?L4 | ? 0.14 over the range of frequency from
2.5 GHz to 4.5 GHz. Three sets of varactor biases were arbitrarily selected to set the
matching network to di?erent states. These values are (2.56 4.8 4.8), (4.8 3.84 4.64), and
(4.8 4.8 2.56).
The center and radius of the load re?ection coe?cient were calculated using Equation
cL was estimated using the radical center
4?4 and 4?5 and the load re?ection coe?cient ?
equations Equation 4?6 and 4?7.
When the centers of two or more of three circles are close to each other, the
estimation is not accurate because the radical center is often not inside the overlapped
area of three circles. Sometimes, the estimated load re?ection coe?cient goes out of a unit
circle on the Smith chart, which is de?nitely incorrect for a passive load. The incorrect
estimates were excluded from the accuracy statistics.
The estimation accuracy is evaluated by the mean squared error (MSE) between
cL |2 }. The proposed method
measured and estimated load re?ection coe?cients, E{|?L ? ?
57
achieved the mean squared errors (MSE), 0.011, 0.080, 0.066, and 0.031, respectively. The
experimental result is summarized in Table 4-1.
4.4
Conclusion
The automatic matching control (AMC) system provides impedance match and
examination capability for radio frequency (RF) ports by the closed-loop feedback.
However, the AMC system prior to this work was not able to estimate a load impedance.
In this work, we demonstrated that the measured power provides not only the degree of
mismatch but also the estimate of an unknown load through the multistate re?ectometer
measurement. The proposed load estimation allows the quick control of the AMC system
by achieving a matching goal without an exhaustive iteration.
Although the varactor biases for estimation were arbitrarily selected, selecting
varactor biases should be based on the careful arrangement of q-points according to
the six-port re?ectometer theory. The ideal arrangement of q-points is known as the
same magnitude and 120? phase di?erence [11]. As future work, we are working on new
matching network design and bias selection to account for the e?ect of q-points.
58
CHAPTER 5
EFLECTOMETER
5.1
Overview
Mobile devices equipped with radio frequency (RF) subsystems are widely used
and drive emerging technologies for the better performance at low power consumption.
Impedance matching between the RF subsystems plays a critical role in improving the
low-power system e?ciency by maximizing power transfer and signal-to-noise ratio (SNR).
However, RF impedance matching is often highly iterative and time-consuming, when the
RF port between subsystems is poorly de?ned.
As an e?ort to automate the RF impedance matching, an automatic matching control
(AMC) circuit has been developed for cellular phone antenna [6] and loadboard testing
[21]. The prior research work demonstrated that an unknown impedance mismatch can be
resolved by a tunable matching network and an iterative search method.
The disadvantage of the automatic matching control is that the iterative search
may slow down the automation process and make impedance matching in real-time very
challenging. To improve slow response time, a load impedance estimation method was
developed to discover an unknown load and to enable immediate impedance matching [25].
However, the load estimation method, as well as the automatic matching control, measures
the degree of mismatch through a distributed directional coupler. When the system area
or volume matters, for example, on-chip embedded RF testing or compact loadboard
testing, the coupler is preferred to be removed or replaced with other equivalent parts. As
alternatives to the directional coupler, a lumped-element coupler [28][29] and an active
coupler [30] have been proposed.
We propose a novel coupler-free load estimation method for an automatic matching
control (AMC) circuit using a lumped-element three-port re?ectometer. The proposed
three-port re?ectometer consists of a two-port lumped-element ?-type matching network
and a high impedance probing port attached to the input port of the matching network.
59
Source
Z0
Coupler
Reconfigurable
matching network
Reflection
coefficient
detector
Match decision
and biasing circuit
ZL
Figure 5-1. Automatic matching control (AMC)
The re?ectometer can be recon?gured by three varactor-based tunable capacitors and used
to set impedance matching within the unit circle on the Smith chart at 2.4 GHz.
The coupler-free load estimation can estimate an unknown load from the combined
power of incident and re?ected waves through a high-impedance probe. The use of
the high-impedance probe and coupler-free estimation can eliminate the need of the
directional coupler and make compact fabrication of impedance measurement on a chip
feasible.
The rest of this chapter is organized as follows: Section 5.2 gives an overview of the
automatic matching control and the coupler-free re?ectometer. Section 5.3 shows the
high impedance probe model and ?tting methods. Section 5.4 describes radical center
and least-square load estimation methods. Section 5.5 presents the experimental results.
Section 5.6 concludes and describes the future direction.
5.2
System Overview
Figure 5-1 illustrates the system diagram of an automatic matching control
(AMC). The automatic matching control recon?gures the matching network through
the closed-loop feedback consisting of a coupler, a re?ection coe?cient detector, and a
match decision circuit. The closed-loop feedback enables the match decision circuit to
search for varactor biases for minimizing the RF signal re?ection in a trial-and-error
process. In this work, the coupler has been replaced with a high impedance probe, which
enables the automatic matching control to perform the coupler-free load estimation.
60
Tunable Impedance Matching Network
Zin
3.3nH
Z0
5Z0
Power Detector
1.8nH
1.8nH
ZL
Control varactors
Automatic Matching Controller
A
B
Figure 5-2. Three port re?ectometer integrated with a high impedance probe A) System
diagram with schematic B) Fabrication on FR4 board
61
Capacitance (pF)
100.0
SMV1413
10.0
1.0
SMV1405
SMV1408
0.1
0
5
10
15
20
25
30
Reverse Voltage (V)
Figure 5-3. Varactor SMV1405 capacitance versus reverse voltage
Cathode
Anode
Figure 5-4. Varactor SPICE model
The proposed tunable matching network is based on the bandpass ? topology, which
is known to provide impedance matching with any point on the Smith chart [3][31]. As
shown in Figure 5-2, the matching network is a ?-type band-pass ?lter with lumped
inductors and tunable capacitors and was fabricated on 6.3 О 4.8 mm FR4 printed circuit
board (PCB). The network was designed to set impedance matching at 2.4 GHz and has
three ports for input, output, and probe.
Skyworks SMV1405-074LF was used as the tunable capacitor. It contains two
hyper-junction varactor diodes in a single package, where two varactors are connected in
common cathode. The capacitance of a single varactor ranges from 1 pF (bias = -8 V)
to 2.7 pF (bias = 0 V), as shown in Figure 5-3. The common cathode pin is connected
62
to a bias voltage supply through a DC blocking choke inductor. The SPICE model of
SMV1405, as shown in Figure 5-4, was used for preliminary simulation.
A high impedance probe was implemented with a 250?(= 5Z0 ) chip resistor and a
transmission line. The high impedance probe was used to measure the input re?ection
coe?cient of the matching network. The relationship model between the high impedance
probe and the input re?ection coe?cient will be presented in next Section.
5.3
High Impedance Probe
As shown in Figure 5-2, a high impedance probe consists of a 250? chip resistor at
the measurement node and a transmission line terminated with a matched port, which is
connected to a network analyzer. For an embedded test system, the high impedance probe
and the network analyzer can be replaced by an on-chip bipolar power detector presented
in [32].
The following are worthy to note in this demonstration as shown in Figure 5-2.
1.
Series chip resistor (250?) is attached to a vector network analyzer through a
transmission line to emulate the operation of a high-impedance probe.
2.
Used a vector network analyzer to measure and manipulate full S-parameter data
3.
Derived |?in | from S-parameters measured by a vector network analyzer to emulate
the operation of a power detector.
4.
Used |1 + ?in | from a vector network analyzer to see the system response with a
power detector and to verify Equation 5?1.
The high impedance probe has no directivity and measures the summation of both
incident and re?ected waves. The measurement at the high impedance probe can be
written as
an + bn = an + ?n an = an (1 + ?n )
(5?1)
where an , bn , ?n are normalized incident and re?ected waves and a re?ection coe?cient
of port n. The network analyzer port is matched with impedance Z0 and connected in
series with the chip resistor 5Z0 . The network analyzer measures Z0 /(Z0 + 5Z0 ) of a
63
probe node and acts as a voltage divider. In other words, the output response of the high
impedance probe is scaled down by a voltage divider ratio Z0 /(Z0 + 5Z0 ) and followed by
the phase-delay (e?? ) of a transmission line. Assuming that the input port of the matching
network is port 1 and the high impedance probe is port 3, the measured power in the high
impedance probe is written as
2
Z0
??
P3 = |b3 | = |S31 a1 | = e S11 a1 (1 + ?in ) ,
5Z0 + Z0
2
2
(5?2)
where an and bn are normalized incident and re?ected waves of port n. The ideal
relationship described by Equation 5?2 does not exactly ?t the measurement result
due to non-ideal circuit e?ects, such as ?nite impedance of the probe and transmission line
loss. Instead of the ideal model, a relationship model with parasitics and a neural network
approximation model were used to take the non-ideal e?ects into account.
The node measured by the high impedance probe is connected to the port 1 through
a transmission line. The measurement of the node is deteriorated due to the non-ideal
e?ects, which are included in a modi?ed relationship model. The relationship between the
measured S-parameters from the input port 1 and the high impedance probe is modi?ed
and written as
S11 = (S31 r1 e??1 ? 1)r2 e??2
(5?3)
where r1 , r2 , ?1 , ?2 are magnitude and phase ?tting parameters. The relationship
model can be trained by a least square nonlinear ?tting. An arti?cial neural network,
a well-known nonlinear approximation model, was also used to represent the relationship.
The training procedure for two models will be described in more detail.
5.3.1
Least Square Fitting
Least square nonlinear ?tting was applied to train the relationship model. The ?tting
parameters, r1 , r2 , ?1 , ?2 , are obtained by minimizing the least square equation written as
(
)
?
?
?
S11 (V) ? (S31 (V)r1 e??1 ? 1)r2 e??2 2
r?1 , r?2 , ?1 , ?2 = arg min
V
64
(5?4)
where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. The ?tting algorithm
is the large-scale algorithm, a subspace trust region method based on the interior-re?ective
Newton method described in [33][34].
5.3.2
Artificial Neural Network
An arti?cial neural network is well known for the approximation of a nonlinear
model. The nonlinear relationship between S11 and the high impedance probe, described
by Equation 5?3, is approximated by an arti?cial neural network. The input and
output of the neural network are the measured S31 and the estimate of S11 . The used
neural network has 15 perceptrons in the hidden layer and was trained by traditional
backpropagation algorithm.
5.4
The equation for coupler-free load estimation is based on the stability circle derivation
as presented in [26, Chapter 11] and [25]. According to the stability circle derivation, the
center and radius of the circle were given as
? ?
(S22 |?in |2 ? ?S11
)
CL =
|S22 |2 |?in |2 ? |?|2
S
S
12
21
|?in |
RL = |S22 |2 |?in |2 ? |?|2 (center)
(5?5)
(5?6)
When the incident and re?ected waves are combined under the absence of a coupler,
the measured power of the input port is expressed as
pin = |a1 + b1 |2 = |a1 |2 |1 + ?in |2
(5?7)
Therefore, |1 + ?in | instead of |?in | can be used to estimate a load impedance. The
manipulated S-parameter data from a network analyzer, |1 + ?in |, were used to verify the
power detector equation rewritten as
S
S
?
12
21
L
|1 + ?in | = 1 + S11 +
1 ? S22 ?L 65
(5?8)
1 + S11 ? (1 + S11 )S22 ?L + S12 S21 ?L |1 + ?in | = 1 ? S22 ?L
(5?9)
By taking the square of both sides, the equation is written as
|1 ? S22 ?L |2 |1 + ?in |2 = |1 + S11 ? (S22 + ?)?L |2
(5?10)
? = S11 S22 ? S12 S21
(5?11)
where
The next derivation follows the stability circle derivation presented in [25]. Then, the
center and radius of the circle for coupler-free load estimation are given as
?
(S22 |1 + ?in |2 ? (S22 + ?)(1 + S11
))?
|S22 |2 |1 + ?in |2 ? |S22 + ?|2
S12 S21
|1 + ?in |
RL = |S22 |2 |1 + ?in |2 ? |S22 + ?|2 CL =
(center)
(5?12)
(5?13)
where ? = S11 S22 ? S12 S21 and * denotes a complex conjugate. The detail in derivation is
given in Appendix A.
The center and radius of three circles were used to estimate an unknown load. The
estimation is based on radical center and least square ?tting widely used for a six-port
re?ectometer. The radical center of three circles is the simplest geometric method and the
least square ?tting is more accurate and a statistical approach by minimizing the sum of
squared distance from three circles.
5.4.1
The radical center is the approximation of the center of three circles? overlapped
region, as illustrated in Figure 5-5. The coordinates of the radical center are given as
2
2
RL1
? RL2
+ x22
2x2
2
2
R ? RL3 + x23 + y32 ? 2xx3
y = L1
2y3
x=
(5?14)
(5?15)
Note that all mismatched loads are passive and should be within a unit circle. If the
magnitude of the estimated re?ection coe?cient is larger than one, the large error can
66
?L3
3
RL
(x3 , y3 )
?L1
1
RL
R L2
(0,0)
(x2 , 0)
?L2
Figure 5-5. Radical center of three circles, ?L1 , ?L2 , and ?L3 .
dominate the overall mean square error to lead to wrong performance statistics. In this
case, the magnitude is set to one (on the unit circle) while keeping the phase information.
5.4.2
Least Square Fitting
Least square nonlinear ?tting was applied to estimate a load and the radical center
was used as an initial parameter, as described in [27]. The least square ?tting can provide
more accurate results in sacri?ce of high computation cost. The same ?tting algorithm
used in the high impedance probe was used here. The ?tting parameter is the re?ection
coe?cient of the unknown load, a complex number on the Smith chart, and is obtained by
minimizing the least square equation written as
??L = arg min
?
|Circle(V) ? ?L |2
V
67
(5?16)
where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. Note that Circle(V)
is described by Equation 5?12 and 5?13 and the distance between a circle and a point is
de?ned as a distance between a tangential line to the circle and the point.
5.5
Experimental Results
A vector network analyzer was used to measure the three-port S-parameters of the
network at 2.4 GHz. The varactor bias voltage ranges from 0 to 5.12 V by 0.32 V step,
16 voltage levels and the number of states of the three-varactor matching network is
163 = 4096. Therefore, the total number of three-port S-parameter is 163 О 32 = 36, 864.
The following experimental results come from the 36,864 S-parameter data of 4096 states
of the matching network.
First, ?tting the measurement data from the high impedance probe to the input port
S-parameter will be presented. The ?tting were performed by the least square ?tting of
the relationship model with parasitics and the backpropagation training of the neural
network approximation model. The error distribution of both methods is also presented.
Second, a coupler-free load estimation is performed by radical center and least square
and the results using two methods are compared in terms of mean square error (MSE).
The input data to the load estimation is the estimated data from the high impedance
probe.
Last, for a fair comparison of coupler and coupler-free load estimation methods, the
estimation performance is evaluated with the same measurement data used in [25].
5.5.1
High Impedance Probe Estimation
Two models based on least square ?tting and arti?cial neural network were used for
the high impedance probe estimation. The ?tting parameters of least square are r1 , r2 , ?1 ,
and ?2 , magnitude and phase of two complex numbers representing non-ideal e?ects.
Least square ?tting was trained by a large-scale algorithm, a subspace trust region method
based on the interior-re?ective Newton method. The high impedance probe S-parameters
68
S31 were converted by the relationship model and four trained ?tting parameters and
compared with the input re?ection coe?cient S11 .
The neural network used in this work has two input nodes representing the real and
imaginary parts of the high impedance probe S-parameter S31 , 15 perceptrons in the
hidden layer, and two output nodes representing the real and imaginary parts of the input
re?ection coe?cient S11 . The traditional backpropagation algorithm was used to train the
neural network.
Figure 5-6 and 5-7 show estimation results of the input re?ection coe?cient S11
from the high impedance probe S-parameter S31 using least square ?tting and arti?cial
neural network, respectively. As explained in Section 5.3, the distribution of the high
impedance probe S-parameters is the downscale of the input re?ection coe?cient through
voltage divider. The experiment results showed that the input re?ection coe?cient can be
successfully estimated through the high impedance probe without disturbing the matching
network.
Mean square error (MSE) for least square ?tting and the trained neural network are
0.000732 and 0.000366. Although the neural network achieved lower estimation error, least
square ?tting also has advantage of faster training and simpler model representation over
the neural network. The magnitude of error distribution of both methods are shown in
Figure 5-8. As expected, the neural network showed its peak of error distribution closer to
zero than least square ?tting.
5.5.2
The input re?ection coe?cients estimated from the high impedance probe were
used to estimate an unknown load using the coupler-free load estimation. One matched
mismatched loads for the experiments. They were classi?ed into two groups, slightlymismatched and heavily-mismatched loads and both groups include the matched load as
69
+j1.0
+j1.0
+j2.0
5.0
?j0.2
?j5.0
?j0.5
2.0
0.0
+j5.0
1.0
+j0.2
Ц
?j0.2
+j2.0
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
+j0.5
0.2
+j0.5
?j2.0
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
A Input re?ection coe?cient S11
B High impedance probe S-parameter S31
+j1.0
Fitting from S31
+j2.0
S11
+j0.5
5.0
2.0
1.0
0.5
+j5.0
0.2
+j0.2
0.0
?j0.2
Ц
?j5.0
?j0.5
?j2.0
?j1.0
C Fitting S31 to S11
Figure 5-6. Least square nonlinear ?tting of high impedance probe model
70
Ц
+j1.0
+j1.0
+j2.0
5.0
?j0.2
?j5.0
?j0.5
2.0
0.0
+j5.0
1.0
+j0.2
Ц
?j0.2
+j2.0
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
+j0.5
0.2
+j0.5
?j2.0
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
A Input re?ection coe?cient S11
B High impedance probe S-parameter S31
+j1.0
Fitting from S31
+j2.0
S11
+j0.5
5.0
2.0
1.0
0.5
+j5.0
0.2
+j0.2
0.0
?j0.2
Ц
?j5.0
?j0.5
?j2.0
?j1.0
C Fitting S31 to S11
Figure 5-7. Arti?cial neural network of high impedance probe model
71
Ц
1400
1200
1000
800
600
400
200
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
A Least square
1400
1200
1000
800
600
400
200
0
0
0.01
0.02
0.03
0.04
0.05
0.06
B Neural network
Figure 5-8. High impedance probe estimation error distribution
72
0.07
0.08
Table 5-1. Summary of heavily and slightly mismatched loads
|S11 | at 2.4 GHz
Tuner motor position
Matched
0.01
(100, 5000, 5000)
Slightly mismatched #1
Slightly mismatched #2
Slightly mismatched #3
0.13
0.38
0.20
(16725, 2262, 5000)
(17105, 1424, 5000)
(20464, 2228, 5000)
Heavily mismatched #1
Heavily mismatched #2
Heavily mismatched #3
0.78
0.76
0.69
(15781, 526, 5000)
(17835, 624, 5000)
(20572, 804, 5000)
a reference. All used loads are carefully generated by Maury Microwave?s load-pull system
(MT986A and MT982B01) and their speci?cation is summarized in Table 5-1.
The load estimation was performed separately for slightly mismatched and heavily
mismatched groups. During the experiment, a di?erent set of three states of matching
network were manually chosen to give better estimation results. The chosen voltage biases
for three di?erent states for slightly mismatched and heavily mismatched groups are
((0.32, 2.56, 4.48), (4.48, 0, 2.88), (4.8, 0, 0)) and ((0.32, 4.16, 3.52), (1.92, 0, 3.84), (3.2,
0.32, 0)), respectively.
As shown in Figure 5-9, the estimation on slightly mismatched loads showed smaller
error than heavily mismatched loads. Although the accuracy is not good enough to replace
an high-precision instrument, the rough estimate of magnitude and phase of the load
re?ection coe?cient is very useful for the automatic matching control to set immediate
impedance matching.
5.5.3
Comparison of Coupler and Coupler-Free Load Estimation
For the comparison of coupler and coupler-free load estimation methods, the
measurement data used in the prior research work [25] were used to evaluate the
estimation performance.
Two load estimation methods were evaluated with radical center and least square
?tting described in Section 5.4. The four mismatched loads used in the prior research
73
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
?j0.2
?j5.0
?j0.5
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.0
0.5
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
?j2.0
Ц
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
A Small mismatch (MSE=0.21)
B Large mismatch (MSE=0.86)
Figure 5-9. Load estimation using estimated S11 from high impedance probe
Table 5-2. Comparison of coupler and coupler-free load estimation in term of mean square
error (MSE)
|S11 |
Mismatched
Mismatched
Mismatched
Mismatched
Mismatched
#1
#2
#3
#4
0.14
0.45
0.24
0.14
Least square
Coupler
Coupler-free Coupler
Coupler-free
0.056840
0.120909
0.035607
0.054044
0.050865
0.134596
0.069265
0.056333
0.003665
0.061899
0.009934
0.035334
0.013576
0.034983
0.009930
0.018138
work were also used for fair comparison. The experimental results are summarized in
Table 5-2 in terms of mean square error (MSE).
Although mean square error (MSE) of the coupler-free load estimation is slightly
higher than MSE of coupler load estimation, the estimation accuracy of two methods are
comparable for both radical center and least square ?tting algorithms, as shown in Figure
5-10 to 5-13. When the coupler-free estimation is employed to an automatic matching
control system, the system dimension will be dramatically reduced without compromising
the system performance.
74
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
5.0
2.0
0.0
+j5.0
1.0
Ц
?j0.2
?j5.0
?j0.5
?j2.0
?j0.5
?j2.0
?j1.0
+j1.0
+j1.0
True
+j2.0 Estimate
?j5.0
?j0.5
5.0
2.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
0.5
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.2
+j0.2
?j0.2
?j2.0
?j1.0
?j1.0
C Least square with coupler
D Least square without coupler
Figure 5-10. Coupler and coupler-free load estimation with mismatched #1
75
Ц
?j5.0
?j0.5
?j2.0
Ц
?j5.0
?j1.0
+j0.5
0.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
5.0
2.0
0.0
+j5.0
1.0
Ц
?j0.2
?j5.0
?j0.5
?j2.0
?j0.5
?j2.0
?j1.0
+j1.0
+j1.0
True
+j2.0 Estimate
?j5.0
?j0.5
5.0
2.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
0.5
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.2
+j0.2
?j0.2
?j2.0
?j1.0
?j1.0
C Least square with coupler
D Least square without coupler
Figure 5-11. Coupler and coupler-free load estimation with mismatched #2
76
Ц
?j5.0
?j0.5
?j2.0
Ц
?j5.0
?j1.0
+j0.5
0.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
5.0
2.0
0.0
+j5.0
1.0
Ц
?j0.2
?j5.0
?j0.5
?j2.0
?j0.5
?j2.0
?j1.0
+j1.0
+j1.0
True
+j2.0 Estimate
?j5.0
?j0.5
5.0
2.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
0.5
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.2
+j0.2
?j0.2
?j2.0
?j1.0
?j1.0
C Least square with coupler
D Least square without coupler
Figure 5-12. Coupler and coupler-free load estimation with mismatched #3
77
Ц
?j5.0
?j0.5
?j2.0
Ц
?j5.0
?j1.0
+j0.5
0.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
5.0
2.0
0.0
+j5.0
1.0
Ц
?j0.2
?j5.0
?j0.5
?j2.0
?j0.5
?j2.0
?j1.0
+j1.0
+j1.0
True
+j2.0 Estimate
?j5.0
?j0.5
5.0
2.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
0.5
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.2
+j0.2
?j0.2
?j2.0
?j1.0
?j1.0
C Least square with coupler
D Least square without coupler
Figure 5-13. Coupler and coupler-free load estimation with mismatched #4
78
Ц
?j5.0
?j0.5
?j2.0
Ц
?j5.0
?j1.0
+j0.5
0.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
5.6
Conclusion
We presented a novel coupler-free load estimation and a lumped-element re?ectometer
integrated with a high impedance probe. The estimate of an unknown mismatched load
can help an automatic matching control (AMC) to set impedance matching on the load
without iterative search.
A high impedance probe was presented for the integrated on-chip RF testing. Two
relationship models together with estimation methods, least square and arti?cial neural
network, achieved mean square error 0.000732 and 0.000366, respectively. The lease square
?tting has only four ?tting parameters compared with about 20 parameters of the neural
network model and enables faster training without compromising the accuracy. The
estimated results were also used as an input to the coupler-free load estimation, which
discovered an unknown load for the automatic matching control.
A novel coupler-free load estimation was proposed with a lumped-element re?ectometer.
The proposed method demonstrated that it could discover an unknown mismatched load
and provide with the high accuracy enough for an automatic matching control to exploit.
The coupler-free load estimation removed the need of any distributed components and
achieved compact system dimension without compromising the estimation accuracy.
The experimental results showed that the proposed coupler-free estimation achieved the
comparable accuracy as the prior research work adopting a coupler. The proposed load
estimation can be integrated with a non-iterative automatic matching control system
without extra components or performance degradation.
79
CHAPTER 6
THREE PORT AND FOUR PORT REFLECTOMETERS
6.1
Overview
due to their potential to overcome the uncertainty of RF systems such as parasitics and
manufacturing variation. Recently, an automatic matching control (AMC) was proposed
to provide the impedance matching of the radio frequency (RF) ports of a device under
test (DUT) or a cellular phone antenna [21][6]. Also, load estimation was proposed to
overcome the iterative nature of the automatic matching control system [25].
Recon?gurable matching networks in the prior work were not designed for the
load estimation [1][5][6]. Even the load estimation proposed in [25] used the arbitrarily
recon?gured matching networks, whose design was not targeted for load estimation
purpose. In this work, three-port and four-port re?ectometers were designed to support
both the automatic matching control and the load estimation using lumped elements and
compact power detectors for size reduction and embedded RF testing.
In the network analyzer research community, a new method using detector power
readings and some mathematical manipulation was introduced to measure a complex
re?ection coe?cient of a device under test (DUT). The new method, so-called a six-port
re?ectometer, was ?rst proposed as an alternative of a conventional network analyzer [11].
Later, four-port multistate re?ectometers were presented to reduce the required number of
ports and parts [12][13][14]. Recently, a lumped-element structure was also presented for
the integration on a chip [35].
We propose three-port and four-port lumped-element multistate re?ectometers for
both a recon?gurable matching network and load estimation. Carefully chosen q-points
can improve the accuracy of the load estimation and it can still recon?gure a matching
network for the automatic matching control.
80
a1
a2
Three-port Reflectometer
ZL
b2
?2
b3
a3
b1
P3
A
a1
a2
Four-port Reflectometer
ZL
?2
b4
b3
P3
a4
b2
a3
b1
P4
B
Figure 6-1. Re?ectometers A) Three-port B) Four-port
6.2
Multistate Reflectometers
As shown in Figure 6-1, port 1 is fed with a signal source with impedance Z0 , port
2 is connected with a device under test (DUT) having a complex re?ection coe?cient ?2
(sometimes load re?ection coe?cient denoted by ?L ) to be measured, and port 3 and 4 are
power detector ports.
Similar to a six-port re?ectometer analysis, power readings can be expressed in
terms of incident and re?ected wave powers of port 2. The power readings of an n-port
re?ectometer is written as
bi = Ai a2 + Bi b2 ,
81
i = 3, 4, и и и , n
(6?1)
Pi = |bi |2 = |b2 |2 |Ai |2 |?2 ? qi |2 ,
i = 3, 4, и и и , n
(6?2)
where
?2 =
a2
,
b2
qi = ?
Bi
Ai
(6?3)
The complex re?ection coe?cient of a DUT can be calculated by power detector readings
and some mathematical manipulation, which is explained in more detail in [11].
By adding a network state k, the operation of multistate re?ectometers can be
described by the following equation,
(k)
|?2 ?
(k)
qi |2
=
Pi
(k)
(k)
|b2 |2 |Ai |2
i = 3, 4, и и и , n,
,
k = 1, 2, 3
(6?4)
where k is a network state (typically, set by di?erent biases) and three network states are
needed in order to determine the complex re?ection coe?cient of a DUT.
Especially for a four-port case, port 3 is assigned as a reference port which depends
only on b2 (in other words, b3 = B3 b2 or A3 = 0). The equation can be written as
B (k) 2 P (k)
(k)
|?2 ? q4 |2 = 3(k) 4(k) ,
k = 1, 2, 3
A4 P3
(6?5)
If all power detectors are perfectly matched to Z0 , then the calibration constants, Ai ,
Bi , and qi , are expressed in terms of S-parameters and given by (refer to Appendix B for
the derivation)
(k)
(k)
Ai
=
(k)
(k)
(k)
S21
(k)
(k)
S21 Si2 ? S22 Si1
,
(k)
Bi
=
Si1
(k)
S21
(k)
,
(k)
qi
=
Si1
(k)
(k)
(k)
(k)
S22 Si1 ? S21 Si2
(6?6)
The above calibration constants can be obtained by either a direct measurement or
modeling.
The reference port can be realized by a lumped power divider, which was presented
for the integration on a chip in [35], as shown in Figure 6-2. In prior work, a directional
coupler was widely used for a reference port, but the compact lumped power divider is
preferred for size reduction in embedded test. The scattering matrix of the divider with
82
2
Za
Z0
1
2
0
20
1
1
3
30
Zc
10
20
3
30
Figure 6-2. Schematic of a lumped power divider. Port 3 is a coupled port, which can be
used as a reference port.
respect to Z0 is given by
?
?
1
1
? ?
? ?
Za
Zc ?
? 0
1 + Z0 1 + Z0 ? a 1
?b1 ? ?
?
??
? ? ?
?
1
??
?b ? = ?
?
?
0
0
?
?
a
? 2 ? ? 1 + Za
? 2?
?
Z0
? ? ?
?? ?
? 1
? b3
b3
0
0
1 + ZZ0c
(6?7)
A three-port re?ectometer has the same basic structure as a four-port one except
that a reference port and a power divider do not exist. Because a reference port does not
exist, an input signal power has to be known to a re?ectometer for the normalization of
wave powers. In other words, a reference port can be replaced by a prede?ned input signal
power in some cases, e.g., testing RF parts under an automatic test equipment (ATE).
Hence, the wave power |b2 | can be estimated through the relationship between |b2 | and |a1 |
if an input power |a1 | is precisely de?ned. The relationship is described by the equation
given by
b2 = S21 a1 + S22 a2 = S21 a1 + S22 ?L b2
b2 =
S21
a1
1 ? S22 ?L
In summary, the three-port re?ectometer can be used if the signal source is precisely
de?ned and the four-port re?ectometer is applicable to other general cases.
83
(6?8)
(6?9)
6.3
Tunable Matching Network
The simplest lumped-element matching network is L-type, but cannot set impedance
matching of all points on the Smith chart. T - and ?-types are known for matching
capability with any point on a Smith chart [31]. Although T - and ?-type topologies are
equivalent in terms of matching capability, the ?-type is preferred for impedance matching
purpose because it has a smaller number of tunable elements along with a signal path,
which result in lower insertion loss than T -type.
Each branch of the matching network has an inductor connected with a varactor
either in parallel or series. As shown in Figure 6-3, the basic structure is ?-type band pass
?lter tuned at the center frequency of 2.4 GHz, where capacitors are replaced with tunable
varactors.
The tunable capacitor used in the matching network is Skyworks SMV1405-074LF,
which contains two common cathode diode varactors in a single package. The common
cathode port is connected to a bias voltage supply through a choke coil. The typical
capacitance of each varactor with a bias 1 V is 1.21(min) to 1.45(max) pF and the
capacitance ranges from 2.1 pF (0.5 V) to 0.95 pF (10 V). The capacitance versus reverse
voltage is shown in Figure 5-3. The SPICE model used in simulations is shown in Figure
5-4.
Agilent ADS was used to perform S-parameter simulations. According to the
simulation results, the varactor in series with an inductance 3.3 nH and in parallel with
1.8 nH shows the largest tuning range at 2.4 GHz as shown in Figure 6-4. The inductance
value of 3.3 nH and 1.8 nH was used for the design of the matching network.
A noninvasive measurement is mandatory not to disturb the original design of a
matching network. Similar to the prior work [35], a high impedance power detector
measures an internal node. The noninvasive power detector is emulated by inserting high
resistance in series with a measurement port and deembedding the e?ect of the resistance.
As explained earlier, the four-port re?ectometer needs a power reading that depends only
84
P3
High input impedance
power detector
Z0
ZL
A
B
Figure 6-3. Recon?gurable three-port matching network A) Schematic B) Implementation
on FR4 board
85
S(2,1)
S(1,1)
freq (2.400GHz to 2.400GHz)
S(2,1)
S(1,1)
A
freq (2.400GHz to 2.400GHz)
B
Figure 6-4. Tunable element impedance with a bias from 0 V to 10 V. A varactor is in
parallel and in series with an inductor. и and О denote S11 and S21 ,
respectively. A) In series with 3.3 nH B) In parallel with 1.8 nH
86
on the re?ected wave power from a DUT. The reference port can be easily realized using a
lumped power divider.
6.4
The matching capability of a tunable matching network can be represented by the
load re?ection coe?cient ?L to be matched by the matching network. The matching
capability is derived as follows. The input re?ection coe?cient ?in is written as
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(6?10)
The equation can be rewritten in terms of the load re?ection coe?cient as
S11 ? ?in
S11 S22 ? S12 S21 ? S22 ?in
?L =
(6?11)
The matching capability is derived from the load re?ection coe?cient by setting the input
re?ection coe?cient to zero.
?L ?in =0
=
S11
S11 S22 ? S12 S21
(6?12)
The coverage on the Smith chart speci?ed the matching capability illustrates the
distribution of the load re?ection coe?cient to be matched.
According to the six-port re?ectometer principle, an unknown load is the same as the
point intersected by three circles, speci?ed by a center, so-called q-point, and a radius.
The circle is represented by calibration constants de?ned by Equation 6?6. Note that the
q-point does not change even if a mismatched load varies, whereas the load estimation
method proposed in [25] has changed the circle center as a mismatched load varies. The
constant q-point enables to keep the optimum-performance criteria for the multistate
In reality, the three circles represented by calibration constants seldom intersect at
a point due to the non-ideal e?ects. The geometric center of the overlap of the circles is
87
estimated by the radical center and least square estimation methods widely used for the
six-port re?ectometer [27].
The radical center is the approximation of the center of three circles? overlapped
region. The coordinates of the radical center are given as
2
2
RL1
? RL2
+ x22
2x2
2
2
R ? RL3 + x23 + y32 ? 2xx3
y = L1
2y3
x=
(6?13)
(6?14)
Least square ?tting can enhance the accuracy of load estimation especially when three
circles failed to meet the optimum-performance criteria. The load re?ection coe?cient can
be obtained by the least square equation as
??L = arg min
?
|Circle(V) ? ?L |2
(6?15)
V
where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. Note that Circle(V)
is represented by calibration constants de?ned by Equation 6?6 and the distance between
a circle and a point is de?ned as a distance between a tangential line to the circle and the
point.
6.5
Experimental Results
The matching capability of the proposed three-port re?ectometer, de?ned as
S11 /(S11 S22 ? S12 S21 ), is shown in Figure 6-5. The matching capability was measured by
changing each varactor bias from 0 V to 5.12 V in 16 levels by 0.32 V step. The matching
capability covers a unit circle on the Smith chart completely, showing its capability on any
One of important calibration constants is q-point given by the equation
qi =
Si1
S22 Si1 ? S21 Si2
(6?16)
The q-points were measured with respect to the same bias range as used for the matching
capability. The measured q-points are distributed along the unit circle as shown in Figure
88
Table 6-1. Summary of mismatched loads
|S11 | at 2.4 GHz
Tuner motor position
Matched
0.01
(100, 5000, 5000)
Slightly mismatched #1
Slightly mismatched #2
Slightly mismatched #3
0.13
0.38
0.20
(16725, 2262, 5000)
(17105, 1424, 5000)
(20464, 2228, 5000)
Heavily mismatched #1
Heavily mismatched #2
Heavily mismatched #3
0.78
0.76
0.69
(15781, 526, 5000)
(17835, 624, 5000)
(20572, 804, 5000)
6-6. Some set of q-points can be selected from the distribution in order to satisfy the
optimum-performance criteria for the multistate re?ectometer. A set of q-points was
carefully selected for better estimation performance.
The load estimation using three-port re?ectometer was performed separately on
slightly and heavily mismatched loads. The speci?cation of mismatched loads is given
in Table 6-1. First, the S-parameter of the input port was converted from the high
impedance port emulating high impedance power detector. The calibration constants
were obtained from the S-parameters through direct measurement of the three-port
re?ectometer and conversion using the high impedance power detector. The q-points were
chosen to achieve higher estimation accuracy for two separate experiments. Then, radical
center estimation was applied to estimate an unknown load re?ection coe?cient. When
the magnitude of the estimated re?ection coe?cient is larger than one, it is incorrect for
passive mismatched loads. In this case, the magnitude was set to one with keeping the
phase.
As shown in Figure 6-7, estimation of slightly mismatched loads showed much smaller
estimation error than heavily mismatched loads. The mean square error for slightly and
heavily mismatched loads are 0.09 and 0.80, respectively. Due to the larger estimation
error, the estimated load re?ection coe?cient of the heavily mismatched loads often go
beyond a unit circle. As described, the magnitude was set to 1 and only the phase was
kept for impedance matching. However, the estimated phase is still quite useful because
89
the estimated magnitude larger than one often results from heavily mismatched load,
whose magnitude is close to one. The next chapter will demonstrate that an automatic
matching control can achieve impedance matching using the estimated phase information.
6.6
Conclusion
We proposed a three-port lumped-element re?ectometer for both load estimation and
impedance matching. The proposed re?ectometer can be easily extended to a four-port
demonstrated that the tunable multistate re?ectometer can help the automatic matching
control (AMC) to estimate a load re?ection coe?cient as well as to set impedance
matching. The high impedance power detector replaced the distributed coupler and
realized the dramatic size reduction of an automatic matching control system without
compromising the load estimation and matching capability. The matching capability
covered completely the unit circle on the Smith chart. Although the load estimation
result is not accurate to be used as an high-precision instrument, the estimated phase
information can still enable the automatic matching control to achieve faster impedance
matching on heavily mismatched loads. We are working toward the integration of the
proposed load estimation and a novel automatic matching control system capable of an
immediate impedance matching.
90
+j1.0
+j0.5
+j2.0
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
?j0.2
\infty
?j5.0
?j0.5
?j2.0
?j1.0
Figure 6-5. Matching capability of three-port re?ectometer at 2.4 GHz
91
+j1.0
+j2.0
+j0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
?j0.2
\infty
?j5.0
?j2.0
?j0.5
?j1.0
Figure 6-6. The q-point distribution of three-port re?ectometer at 2.4 GHz
92
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
?j5.0
?j0.5
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
?j0.2
?j5.0
?j0.5
?j2.0
Ц
?j2.0
?j1.0
?j1.0
A
B
Figure 6-7. Multistate re?ectometer estimation using estimated S11 from high impedance
probe A) Small mismatch (MSE=0.09) B) Large mismatch (MSE=0.80)
93
CHAPTER 7
AUTOMATIC MATCHING CONTROL USING LOAD ESTIMATION
7.1
Overview
An automatic matching control (AMC) system has been developed to automate
time-consuming impedance matching procedure [5][6][21]. The impedance matching of the
automatic matching control was performed by recon?guring a tunable matching network
until the lowest mismatch is achieved. The recon?guration was controlled by heuristic
iterative methods, which showed a good trade-o? between system response and impedance
matching accuracy. Also, load estimation reusing the existing tunable matching network
of the automatic matching control system was proposed to facilitate the automation of
impedance matching [25]. In this work, the load estimation technique was integrated with
the existing automatic matching control system to achieve immediate impedance matching
without compromising matching accuracy.
Traditional automatic matching control systems achieved impedance matching of
unknown or even varying mismatched loads by the feedback loop of a tunable matching
network, a mismatch detector, and match control circuit [21]. The feedback loop is
controlled by iterative methods of a match control circuit, which searches for the value
of tuning elements in a trial-and-error process. However, the trial-and-error approach
slowed down the system response and various heuristic approaches have been developed to
improve the system response without compromising matching capability. Nevertheless, the
system response of the heuristic approaches is still proportional to the complexity of the
matching network and gets slower as more tuning elements and levels are added.
We will demonstrate that an estimated load can be used for a matching control
circuit to achieve immediate impedance matching without using heuristic approaches.
The proposed matching control can ?nd the value of tuning elements by examining the
characterization table of a matching network. Therefore, the precise characterization as
94
well as the load estimation play an important role in this immediate impedance matching
approach.
Various characterization methods for a microwave device have been reported to
reduce the microwave design complexity. A neural network has been widely used to
characterize microwave devices, such as the approximation of S-parameters of BJTs [36]
and modeling parameters of microwave components [9]. Also, closed form equation was
also presented for S-parameters of BJTs [37]. In this work, neural network models and the
closed form equation were used to approximate measured S-parameters and the accuracy
of the characterization methods was evaluated in terms of mean squared error (MSE)
between true and estimated values.
The proposed matching control consists of two tasks. First, a characterization table in
terms of tuning elements was built from the direct measurement of the matching network
or approximation models such as a neural network and closed form equations. Next, the
value of the tuning elements was found by minimizing the degree of mismatch. The degree
of mismatch was calculated from the magnitude of the input re?ection coe?cient. The
experimental results of the immediate impedance matching approach will be presented.
7.2
Matching Control Procedures
The same lumped-element tunable matching network that is used for load estimation
was used to develop matching control procedures supporting load estimation presented
in Chapter 5 and 6. The matching network has a ?-type bandpass ?lter topology and
three varactor diodes as tuning elements. The recon?guration of the matching network
was performed by changing the varactor bias voltages. Its matching capability covers all
re?ection coe?cients within the unit circle on the Smith chart at 2.4 GHz.
The load re?ection coe?cient ?L of a device under test (DUT) is assumed to be
estimated by load estimation techniques presented in Chapter 5 and 6. When the DUT is
connected to the port 2 of a tunable matching network, the input re?ection coe?cient ?in
95
looking into the port 1 of the matching network is written as follows.
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(7?1)
where Sij is the S-parameter from port j to port i of the matching network. Note that the
S-parameters are the function of a bias voltage vector, denoted by v. Therefore, the input
re?ection coe?cient can be explicitly written as the function of v.
?in (v) = S11 (v) +
S12 (v)S21 (v)?L
,
1 ? S22 (v)?L
v = (v1 , v2 , и и и , vn )T
(7?2)
where T denotes a transpose and vn is the nth bias voltage. The load re?ection coe?cient
of the DUT can be derived from the input re?ection coe?cient.
?L =
S11 (v) ? ?in (v)
S11 (v)S22 (v) ? S12 (v)S21 (v) ? S22 (v)?in (v)
(7?3)
The mismatched load to be matched by the matching network set by a bias voltage v,
denoted by ?M , is the load re?ection coe?cient that makes the input re?ection coe?cient
zero.
?M (v) = ?L =
?in =0
S11 (v)
S11 (v)S22 (v) ? S12 (v)S21 (v)
(7?4)
b, as a bias voltage
Now, let us de?ne an optimal bias voltage vector, denoted by v
vector that minimizes the magnitude of the input re?ection coe?cient ?in , the degree
of mismatch.
b = arg min |?in (v)|
v
v
(7?5)
If the bias voltage, that minimizes the input re?ection coe?cient to zero, can be found for
b can be expressed
all possible load re?ection coe?cients, the optimal bias voltage vector v
using ?M as follows.
}
} { { b = v?in (v) = 0 = v?M (v) = ?L
v
Finding the optimal bias voltage can be expressed as ?nding a bias voltage whose ?M
is equal to ?L . Therefore, the mapping table between ?M and v should be calculated
96
(7?6)
Start
Characterization
Measure power three-times
Calculate bias for matching
End
Figure 7-1. Automatic matching control supports load estimation.
to perform the bias search and the mapping table can be converted from the matching
network characterized by the S-parameters. This procedure is based on Equation 7?4. The
S-parameters can be obtained from direct measurement using a vector network analyzer
or a neural network ?tting model. The characterization methods will be introduced in the
next Section.
7.3
Characterization of Matching Network
The characterization of a matching network is a procedure to discover the S-parameter functions to be used to calculate an input re?ection coe?cient or a mismatched
load to be matched. The S-parameters of the matching network were measured using
a vector network analyzer while changing the bias voltage. The measurement points
were determined by the number of varactors and bias voltage levels. Although the more
voltage levels can produce more accurate characterization results, 16 voltage levels were
chosen as good trade-o? between measurement time and characterization accuracy. The
97
characterization results were converted to a form of a mapping table, ?M (v), for easy
access and searches.
Unknown S-parameters between measurement points were approximated by a
multivariate linear interpolation. The multivariate linear interpolation is interpolating a
function of multiple variables on a regular grid, as an extension of a linear interpolation.
It performs linear interpolation ?rst on one direction, then again in the other direction.
Suppose we want to interpolate a value of an unknown function f at the point (x, y).
The value of the function f at four neighbor points on a regular grid, f (x1 , y1 ), f (x1 , y2 ),
f (x2 , y1 ), and f (x2 , y2 ), are assumed to be known, then the interpolation of the function f
at the point (x, y) can be written as follows.
f (x1 , y1 )
(x2 ? x)(y2 ? y)
(x2 ? x1 )(y2 ? y1 )
f (x2 , y1 )
+
(x ? x1 )(y2 ? y)
(x2 ? x1 )(y2 ? y1 )
f (x1 , y2 )
(x2 ? x)(y ? y1 )
+
(x2 ? x1 )(y2 ? y1 )
f (x2 , y2 )
+
(x ? x1 )(y ? y1 )
(x2 ? x1 )(y2 ? y1 )
f (x, y) ?
(7?7)
If the unknown function to interpolate has a smooth surface over neighbor points, the
linear interpolation reduces the number of measurement points signi?cantly without the
loss of the characterization detail.
The other method to estimate unknown S-parameters between measurement points
is an approximation ?tting function to the measured S-parameters. As described, the
measured S-parameters are the functions of the bias voltage vector v, given as follows.
?
?
?S11 (v) S12 (v)?
(7?8)
S(v) = ?
?
S21 (v) S22 (v)
The S-parameter functions were approximated by a curve ?tting model, such as a closed
form and a neural network. First, an arti?cial neural network was used to approximate
98
the S-parameter functions. Four independent neural network models approximate two-port
S-parameter functions, S11 (v), S12 (v), S21 (v), and S22 (v). The input and output of the
neural network are bias voltage vector and real and imaginary parts of the S-parameter,
respectively. The feed-forward topology was used with 15 perceptrons in the hidden layer.
The well-known backpropagation algorithm was used to train the neural networks.
The cascaded network is easily represented by ABCD-parameters and the matching
network was decomposed into basic components, such as a transmission line, series
impedance, and shunt admittance. The representation of the ABCD-parameters was used
as closed-form equations. The ABCD-parameters for the basic components are given as
follows.
?
?
? cos(2??) ?Z0 sin(2??)?
?
?
?Y0 sin(2??) cos(2??)
?
?
?1 Z ?
?
?
0 1
?
?
? 1 0?
?
?
Y 1
(transmission line)
(7?9)
(series impedance)
(7?10)
(7?11)
The closed-form ABCD-parameters were derived from the cascaded network of a
transmission line (?1 ), a ? network (Y1 , Z3 , and Y2 ), and a transmission line (?2 ). The
varactor capacitance was calculated from the varactor SPICE model used in Chapter 5
and 6.
ABCD11 =(cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 )
+? (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 )
(7?12)
ABCD12 =? (cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 )
+ (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) cos(2??2 )
99
(7?13)
ABCD21 =(?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 )
+? (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 )
(7?14)
ABCD22 =? (?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 )
+ (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) cos(2??2 )
(7?15)
The admittance Y1 , Y2 and impedance Z3 are the function of bias voltage which determines
the varactor capacitance. The transmission line delay ?1 , ?2 , parasitic parameters, ?tting
parameters of the varactor SPICE model are trained by a nonlinear least square ?tting
algorithm. The ABCD-parameters based ?tting model was converted to S-parameters for
fair comparison with direct measurement and neural network model.
7.4
Bias Search for Impedance Matching
The goal of bias search is to ?nd the optimal bias voltage vector given by
{ } { }
b = v?in (v) = 0 = v?M (v) = ?L
v
(7?16)
Due to the discrete measurement data of a mapping table ?M (v), it is not always possible
to ?nd the optimal bias voltage vector. Instead, we choose the bias voltage vector closest
to the optimal bias voltage vector and this procedure can be written as follows.
b = arg min |?M (v) ? ?L |
v
v
(7?17)
From now on, the optimal bias voltage vector is rede?ned as the bias minimizing the
magnitude between ?M and ?L .
The bias search consists of two steps, coarse and ?ne search. The coarse search was
performed on the mapping table converted from the direct measurement of S-parameters.
100
Then, the ?ne search was performed again on the linear interpolation of a voxel1 of the
bias voltage vector found in the coarse search.
The cost function for both coarse and ?ne search is mean square error (MSE) of
magnitude between ?M of eight bias voltage vectors of a voxel and an estimated load
?L . Note that a voxel for 3D data has eight vertexes, similar to a cubic. For example, a
unit voxel consists of eight points, (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0),
and (1,1,1). Eight points of the voxel for the coarse search were selected from the voltage
bias used for the direct measurement. Once the voxel is found by the coarse search, the
mapping table of the voxel is interpolated by multivariate linear interpolation with higher
resolution. Then, the ?ne search is performed on the interpolated data. The proposed
two-step search reduces the required number of measurement points and the memory
usage by the partial interpolation. In addition, the bias search can be made faster using
binary search algorithm on sorted data.
7.5
Characterization Results
The S-parameters of a tunable matching network are the function of bias voltage.
The S-parameters and the corresponding bias voltages should be measured together.
During the measurement, the bias voltage of each varactor was set by a 12 bit digitalto-analog converter (DAC) from 0 to 5 V by 0.32 V step in 16 levels. The S-parameters
were measured using a vector network analyzer at 2.4 GHz. All measurement procedures
were automated by an instrument control program written in MATLAB. The program
running in a host computer communicated with a microcontroller and a DAC to set bias
voltage, then sent a GPIB command for the network analyzer to measure S-parameters.
Note that the microcontroller will eventually replace the host computer and implement
all control programs. Due to the slow response of the network analyzer, the network
analyzer failed to measure correct S-parameters immediately after changing bias voltage.
1
A voxel is a basic unit cell for 3D data, similar to a pixel for 2D picture.
101
The control program paused for a few seconds after bias setting such that the network
analyzer could get into steady state and measure correct S-parameters. The bias voltage
of three varactors was changed in 16 levels, therefore the total number of measurements is
16 О 16 О 16 = 4096.
Four neural network models approximated S11 , S12 , S21 , and S22 , respectively
and were trained by backpropagation algorithm. The input, hidden, and output layers
consist of 3 bias voltages, 15 hidden perceptrons, and 2 (real and imaginary) parts of
S-parameters, respectively. Because the matching network is a reciprocal passive network,
the neural network models for S12 and S21 should generate similar outputs and can be
merged into a single model. During training the neural networks, 10% of the S-parameter
data were used for validation and 10% for testing purposes.
The estimation results using the trained neural network showed good agreement
with the measurement data, as shown in Figure 7-2. As expected, the results for S12 and
S21 were close enough to merge into a single model. To check the over?tting problem of
neural networks, the trained networks were compared using training and testing data.
The compared estimation results, as shown in Figure 7-3, demonstrated the comparable
error for both data, therefore the trained networks have no over?tting problem and can
approximate unknown S-parameter data.
Four closed-form models representing S11 , S12 , S21 , and S22 , were trained by a
nonlinear least square algorithm. The estimation results using the trained closed-form
models were compared with the S-parameter measurement data. The estimation error was
higher than the neural network models by a few orders of magnitude, as shown in Figure
7-4. To improve the closed-form models, more parasitic e?ects and ?tting parameters can
be added, but more parameters may cause optimization problems such as initial parameter
setting and local minima. In this work, only the neural network models were used to
evaluate the automatic matching control. The estimation statistics of neural network and
closed-form models are summarized in Table 7-1 and 7-2.
102
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.5
A
B
+j1.0
+j1.0
?j5.0
?j0.5
5.0
5.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.5
?j2.0
?j1.0
True
+j2.0 Estimate
?j0.2
?j2.0
Ц
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
C
D
Figure 7-2. Neural network models for a 2-port matching network were trained by
backpropagation. A) S11 B) S21 C) S12 D) S22
103
Ц
?j5.0
?j1.0
+j0.2
2.0
?j0.2
?j2.0
+j0.5
0.2
1.0
0.0
?j5.0
?j0.5
0.0
Ц
+j5.0
2.0
?j0.2
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.5
A
B
+j1.0
+j1.0
?j5.0
?j0.5
5.0
5.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.5
?j2.0
?j1.0
True
+j2.0 Estimate
?j0.2
?j2.0
Ц
?j5.0
?j1.0
+j0.2
2.0
?j0.2
?j2.0
+j0.5
0.2
1.0
0.0
?j5.0
?j0.5
0.0
Ц
+j5.0
2.0
?j0.2
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
Ц
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
C
D
Figure 7-3. Neural network models for a 2-port matching network were tested by 10% of
measurement data. A) S11 B) S21 C) S12 D) S22
104
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.5
A
B
+j1.0
+j1.0
?j5.0
?j0.5
5.0
5.0
0.0
1.0
Ц
+j5.0
0.5
+j0.2
0.2
5.0
2.0
1.0
?j0.2
True
+j2.0 Estimate
+j0.5
+j5.0
0.5
?j2.0
?j1.0
True
+j2.0 Estimate
?j0.2
?j2.0
Ц
?j5.0
?j1.0
+j0.2
2.0
?j0.2
?j2.0
+j0.5
0.2
1.0
0.0
?j5.0
?j0.5
0.0
Ц
+j5.0
2.0
?j0.2
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
Ц
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
C
D
Figure 7-4. Closed form models for a 2-port matching network were trained by nonlinear
least square ?tting. A) S11 B) S21 C) S12 D) S22
105
Table 7-1. Mean square error (MSE) of neural network ?tting models using training and
testing data
Type
Training data
Testing data
S11
S21
S12
S22
1.90 О 10?4
1.50 О 10?4
1.79 О 10?4
2.49 О 10?4
2.25 О 10?4
1.69 О 10?4
2.06 О 10?4
2.80 О 10?4
Table 7-2. Average error of closed-form models
Type
Mean square error
Average error
S11
S21
S12
S22
1.20 О 10?2
5.99 О 10?3
6.00 О 10?3
7.95 О 10?3
1.10 О 10?1
7.74 О 10?2
7.75 О 10?2
8.92 О 10?2
The S-parameters were converted to a mapping table, which is used for bias search
and matching control, and the conversion procedure is described as follows. Two di?erent
forms of the mapping table were used in this work. The ?rst form of the mapping
table, as shown in Table 7-3, consists of a bias voltage vector and the corresponding
mismatched load to be matched by a matching network. The mapping table was obtained
by Equation 7?4 and S-parameter data. However, the magnitude of the mismatched load
to be matched may be out of the unit circle on the Smith chart and the value is incorrect
for a passive network. In this case, the corresponding bias voltage cannot be chosen for
impedance matching of any mismatched load and can be removed from the mapping table.
The second form of the mapping table, as shown in Table 7-4, is an inverse mapping
table, where mismatched loads to be matched are evenly distributed on the Smith chart
and the corresponding optimal bias voltage vectors were calculated through the two-step
search.
7.6
Impedance Matching Results
Impedance matching was performed by matching control using bias search on the
106
Table 7-3. Mapping table between bias voltage and a mismatched load to be matched
Bias voltage (V)
to be matched by matching network
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
+0.54 ? ?0.99
+0.36 ? ?1.08
+0.13 ? ?1.16
?0.18 ? ?1.22
?0.60 ? ?1.18
?1.11 ? ?0.91
?1.54 ? ?0.32
?1.58 + ?0.55
?1.13 + ?1.24
?0.50 + ?1.53
..
.
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
0.00,
..
.
0.00
0.31
0.63
0.94
1.26
1.57
1.89
2.20
2.52
2.83
Table 7-4. Inverse mapping table between a mismatched load to be matched and bias
voltage
to be matched by matching network
Bias voltage (V)
+0.00
+0.50
+0.25 + ?0.43
?0.25 + ?0.43
?0.50 + ?0.00
?0.25 ? ?0.43
+0.25 ? ?0.43
+1.00
+0.87 + ?0.50
+0.50 + ?0.87
+0.00 + ?1.00
?0.50 + ?0.87
?0.87 + ?0.50
?1.00
?0.87 ? ?0.50
?0.50 ? ?0.87
?0.00 ? ?1.00
+0.50 ? ?0.87
+0.87 ? ?0.50
(4.22,
(2.61,
(4.72,
(2.30,
(1.89,
(1.57,
(0.31,
(1.35,
(0.00,
(2.64,
(0.50,
(0.09,
(3.15,
(4.38,
(2.36,
(3.59,
(4.72,
(2.30,
(0.72,
107
4.12,
2.01,
3.46,
2.20,
1.67,
0.00,
0.85,
0.13,
0.00,
1.98,
0.09,
0.09,
3.78,
3.18,
4.72,
2.86,
3.27,
2.11,
0.00,
2.83)
4.72)
2.99)
2.01)
1.64)
0.31)
4.63)
0.06)
4.72)
0.82)
0.35)
0.00)
4.60)
4.12)
3.15)
3.49)
2.99)
1.79)
4.72)
estimation methods. The mapping table was converted from the characterization results
and examined by the two-step search for an optimal bias voltage. The optimal bias
voltage was applied to the matching network and the degree of mismatch, the magnitude
of the input re?ection coe?cient, was measured to evaluate the capability of impedance
matching. The slightly and heavily mismatched loads used for load estimation were
con?gured as the same way used in Chapter 5 and 6.
The experimental results demonstrated that the matching control could achieve
immediate impedance matching through the load estimation and the mapping table. The
optimal bias voltage and the degree of mismatch were summarized in Table 7-5, 7-6, 7-7,
and 7-8.
Because the re?ectometer load estimation method achieved lower estimation error
than the coupler-free load estimation, the impedance matching by the re?ectometer
resulted in the lower degree of mismatch. In addition, S-parameter data estimated by the
neural network models were so close to measurement data that the bias search and the
corresponding impedance matching were almost identical considering the measurement
error.
The impedance matching results demonstrated that the neural network model can
be used for the automatic matching control. The number of coe?cients of each neural
network models is as low as 20, because the input, hidden, and output layers have 3, 15,
and 2 nodes. Compared with the large size of S-parameter measurement data, 163 = 4096,
the neural network model is more appropriate for compact microcontroller system with
limited available memory.
The estimate of the mismatched loads and the load re?ection coe?cient matched
by the matching network were plotted in Figure 7-5 and 7-6, respectively. Due to the
small estimation error, a few hundredth, of neural network models, the mapping table
converted from the neural network models was almost identical to that of measurement
108
Table 7-5. Impedance matching results using coupler-free load estimation and S-parameter
measurement data
?L
Optimal bias voltage
(V)
Input re?ection
coe?cient ?in
Degree of
mismatch |?in |
+0.14 ? ?0.77
+0.64 + ?0.40
?0.69 + ?0.04
(2.03, 0.17, 4.39)
(4.71, 0.02, 4.71)
(0.02, 4.71, 2.38)
+0.04 + ?0.16
+0.22 + ?0.09
+0.05 ? ?0.06
0.16
0.24
0.08
?0.01 + ?0.01
+0.12 ? ?0.01
+0.38 + ?0.06
?0.20 ? ?0.01
(4.49,
(4.05,
(3.70,
(4.45,
+0.03 + ?0.03
+0.02 ? ?0.01
+0.03 + ?0.01
+0.01 + ?0.03
0.04
0.02
0.03
0.03
4.64,
1.05,
0.08,
4.30,
4.45)
1.59)
4.45)
4.64)
0.09
Average
Table 7-6. Impedance matching results using three-port re?ectometer load estimation and
S-parameter measurement data
?L
Optimal bias voltage
(V)
Input re?ection
coe?cient ?in
Degree of
mismatch |?in |
+0.14 ? ?0.77
+0.64 + ?0.40
?0.69 + ?0.04
(1.68, 0.87, 4.42)
(2.44, 0.14, 4.68)
(0.02, 4.01, 2.28)
+0.05 + ?0.17
?0.05 ? ?0.05
?0.13 + ?0.03
0.18
0.07
0.13
?0.01 + ?0.01
+0.12 ? ?0.01
+0.38 + ?0.06
?0.20 ? ?0.01
(4.23,
(4.11,
(3.20,
(4.49,
+0.01 + ?0.00
?0.02 ? ?0.01
+0.01 ? ?0.01
+0.01 + ?0.02
0.01
0.02
0.01
0.02
4.30,
0.87,
0.02,
4.20,
4.20)
1.68)
4.64)
4.71)
Average
0.06
Table 7-7. Impedance matching results using coupler-free load estimation and
S-parameters estimated by neural network models
?L
Optimal bias voltage
(V)
Input re?ection
coe?cient ?in
Degree of
mismatch |?in |
+0.14 ? ?0.77
+0.64 + ?0.40
?0.69 + ?0.04
(1.28, 4.30, 0.02)
(4.71, 4.71, 4.14)
(1.94, 0.11, 2.28)
+0.01 + ?0.15
+0.21 + ?0.08
+0.01 ? ?0.06
0.15
0.22
0.06
?0.01 + ?0.01
+0.12 ? ?0.01
+0.38 + ?0.06
?0.20 ? ?0.01
(4.61,
(4.61,
(4.52,
(4.11,
+0.02 + ?0.05
+0.02 ? ?0.00
+0.03 + ?0.01
+0.01 + ?0.03
0.05
0.02
0.03
0.03
1.34,
1.46,
3.60,
0.08,
4.49)
4.23)
4.23)
4.71)
Average
0.08
109
Table 7-8. Impedance matching results using three-port re?ectometer load estimation and
S-parameters estimated by neural network models
?L
Optimal bias voltage
(V)
Input re?ection
coe?cient ?in
Degree of
mismatch |?in |
+0.14 ? ?0.77
+0.64 + ?0.40
?0.69 + ?0.04
(1.28, 4.42, 0.02)
(2.69, 2.41, 3.89)
(1.68, 0.05, 2.35)
+0.05 + ?0.17
?0.04 ? ?0.05
?0.15 + ?0.03
0.18
0.06
0.15
?0.01 + ?0.01
+0.12 ? ?0.01
+0.38 + ?0.06
?0.20 ? ?0.01
(4.20,
(4.42,
(4.33,
(4.17,
+0.01 + ?0.00
?0.01 ? ?0.00
+0.02 ? ?0.02
+0.02 + ?0.02
0.01
0.01
0.03
0.03
0.74,
1.90,
3.07,
0.02,
4.45)
4.55)
4.11)
4.71)
Average
0.07
data. Therefore, the impedance matching results from both mapping tables were also
almost identical.
The matching control achieved the low degree of mismatch, -26 dB or less, for
the slightly mismatched loads. The larger estimation error for the heavily mismatched
loads results in the larger degree of mismatch, around -12 dB or less, depending on the
accuracy of load estimation. The matching control can be integrated with one of the load
estimation methods. The coupler-free load estimation can be implemented in a compact
size, whereas the re?ectometer load estimation can achieve better impedance matching for
7.7
Conclusion
We demonstrated that an automatic matching control system could achieve
an immediate impedance matching by utilizing load estimation techniques and the
characterization of a matching network. The matching network was characterized by
neural network models and closed-form equations as well as S-parameter measurement
data. The neural network models achieved the comparable characterization accuracy with
much fewer coe?cients than measurement data. The smaller number of coe?cients can be
easily stored and implemented by a compact microcontroller with built-in memory.
110
+j1.0
+j1.0
True
+j2.0
Estimate
Matched by AMC
?j0.2
?j5.0
?j0.5
?j0.2
?j0.5
?j2.0
?j2.0
?j1.0
?j1.0
A
B
+j1.0
True
+j2.0
Estimate
Matched by AMC
?j0.2
?j5.0
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.5
+j5.0
0.2
+j0.2
True
+j2.0
Estimate
Matched by AMC
+j0.5
0.2
+j0.5
?j0.5
Ц
?j5.0
+j1.0
0.0
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0
Estimate
Matched by AMC
+j0.5
0.2
+j0.5
?j0.2
?j2.0
Ц
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
C
D
Figure 7-5. S-parameters from vector measurement and neural network model were
compared in terms of coupler-free load estimation assisted matching control.
A) Vector measurement by large mismatch B) Neural network model by large
mismatch C) Vector measurement by small mismatch D) Neural network
model by small mismatch
111
+j1.0
+j1.0
True
+j2.0
Estimate
Matched by AMC
?j0.2
?j5.0
?j0.5
?j0.2
?j0.5
?j2.0
?j2.0
?j1.0
?j1.0
A
B
+j1.0
True
+j2.0
Estimate
Matched by AMC
?j0.2
?j5.0
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.5
+j5.0
0.2
+j0.2
True
+j2.0
Estimate
Matched by AMC
+j0.5
0.2
+j0.5
?j0.5
Ц
?j5.0
+j1.0
0.0
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0
Estimate
Matched by AMC
+j0.5
0.2
+j0.5
?j0.2
?j2.0
Ц
?j5.0
?j0.5
?j2.0
?j1.0
?j1.0
C
D
Figure 7-6. S-parameters from vector measurement and neural network model were
compared in terms of re?ectometer load estimation assisted matching control.
A) Vector measurement by large mismatch B) Neural network model by large
mismatch C) Vector measurement by small mismatch D) Neural network
model by small mismatch
112
The bias search was performed on a mapping table between the mismatched load
and bias voltage, which approximated the inverse function of the mismatched load to be
matched. The two-step coarse and ?ne search algorithm could achieve the same degree
of mismatch with a fewer number of measurement data points than one-step full search.
The experimental results showed that the degree of mismatch could be as low as -12 dB
for heavily mismatched loads and -26 dB for slightly mismatched loads. Note that the bias
search can be made faster by using binary search and the sorted mapping table.
Although the matching control and matching network in this work were developed for
on-chip embedded test systems, they can be applied to di?erent matching networks, such
will get more attention with a rise of automated RF systems, because the automatic
matching control system utilizing load estimation needs to know the characteristics of a
matching network.
113
CHAPTER 8
CONCLUSION
The impedance matching of radio frequency (RF) ports of a socketed device under
test (DUT), which usually su?ers from a pogo pin connection, poorly de?ned RF
tolerance, and manufacturing variation, is di?cult, time-consuming, and empirical process.
In this work, a recon?gurable matching network works with adaptive matching control
to achieve the impedance matching and to reduce the undesired e?ects. In addition,
the matching control is assisted by the characterization and microwave modeling of the
matching network.
Background theory presented in Chapter 2 provides the basic of a lumped impedance
matching, various nonlinear microwave modeling techniques, and the fundamentals of
six-port and four-port re?ectometers.
An automatic matching control (AMC) presented in Chapter 3 facilitates the
impedance matching over a broadband frequency using a microstrip bandpass ?lter.
The microstrip ?lter consists of ?ve stubs and three varactors connected to the end of
the stub. The center frequency and the bandwidth are 3.5 GHz and 2 GHz, respectively
and the insertion loss over the passband is as low as 2 dB. The proposed system employs
a greedy search algorithm to determine the varactor biases for impedance match over a
large fractional bandwidth 71% = 2.5 / 3.5. The greedy algorithm outperforms brute-force
and single-step algorithms in terms of the number of trials and the available bandwidth,
respectively. The work demonstrates the feasibility of the automatic matching control
Load estimation techniques presented in Chapter 4 exploits the principle of a six-port
re?ectometer to calculate the complex re?ection coe?cient of a DUT from three wave
power readings. Three wave power detectors are realized by changing varactor biases
and measuring re?ected wave power using a scalar network analyzer. The performance of
the estimation is evaluated by the mean square error (MSE) between true and estimated
114
re?ection coe?cients. It demonstrates that tunable matching networks and compact
power detectors can be used to estimate the load re?ection coe?cient of a DUT for the
automatic matching control (AMC) system.
A coupler-free load estimation presented in Chapter 5 demonstrates the dramatic
reduction of the automatic matching control system size by replacing a distributed coupler
with a high impedance probe. The coupler-free load estimation discovers an unknown
load from the power of the combined incident and re?ected waves measured by a high
impedance probe. The experimental results show that the coupler and coupler-free load
estimation methods are comparable in terms of estimation performance.
The lumped-element re?ectometer presented in Chapter 6 is designed not only
for an automatic matching control, but also used as a multistate re?ectometer for load
estimation. A lossless ? section network is used as a network topology for the proposed
three-port. The four-port re?ectometer can be extended by adding a reference port
to the three-port re?ectometer and the reference port can be realized with a resistive
power divider. The three-port matching network was fabricated on a FR4 printed circuit
board (PCB) using lumped chip inductors and varactors. The re?ectometer analysis and
experimental result showed that the matching capability could cover a unit circle on the
Smith chart and provide the optimum performance criteria of the matching network.
The novel matching control and characterization methods presented in Chapter 7
are developed to support load estimation techniques. A tunable matching network is
characterized by vector measurement or neural network models of S-parameters. The
S-parameter data are converted to a mapping table of a load re?ection to be matched by
the matching control. The equation of the mapping table is derived from the analysis of
a bias voltage vector, an input re?ection coe?cient, and a load re?ection coe?cient. The
optimal bias minimizing the degree of mismatch is discovered by two-step bias search.
The two-step search performs coarse search on the mapping table followed by ?ne search
on the multivariate linear interpolation of the mapping table. The experimental results
115
show that the neural network ?tting model achieves comparable accuracy as vector
measurement and that the impedance matching by the proposed matching control is as
low as -12 and -26 dB for heavily and slightly mismatched loads, respectively.
One of the most challenging tasks in implementing the automatic matching control
system with load estimation is the full vector measurement of the matching network.
In case of the embedded RF test system where the embedded matching network has no
external node, it is often very di?cult, even if not impossible, to measure using a regular
vector network analyzer. The development of a measurement procedure using only existing
power detectors will be valuable. Some researchers have proposed a calibration method
using only power detectors [38] and the calibration of multistate perturbation-two-port
(PTP) [39]. The measurement of the embedded matching network can be implemented in
similar way. In addition, although single-ended RF system was assumed in this work, the
proposed work can be extended to a di?erential RF system. The preliminary analysis of
an input re?ection coe?cient of di?erential RF system is provided (Appendix C and D).
116
APPENDIX A
DERIVATION OF LOAD IMPEDANCE CIRCLE EQUATION USING INPUT
IMPEDANCE MAGNITUDE
A complex input re?ection coe?cient is written in terms of two-port S-parameters Sij
and a load re?ection coe?cient ?L as
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(A?1)
The magnitude of the input re?ection coe?cient becomes
S
S
?
12
21
L
|?in | = S11 +
1 ? S22 ?L S11 ? S11 S22 ?L + S12 S21 ?L |?in | = 1 ? S22 ?L
(A?2)
(A?3)
By taking the square of both sides, the equation is written as
|1 ? S22 ?L |2 |?in |2 = |S11 ? ??L |2
(A?4)
? = S11 S22 ? S12 S21
(A?5)
where
The equation is manipulated as follows.
? ?
|?in |2 + |S22 |2 |?L |2 |?in |2 ? S22 ?L |?in |2 ? S22
?L |?in |2
?
= |S11 |2 + |?|2 |?L |2 ? ??L S11
? ?? ??L S11
(A?6)
where * denotes a complex conjugate.
?
)?L
(|S22 |2 |?in |2 ? |?|2 )?L ??L ? (S22 |?in |2 ? ?S11
?
|?in |2 ? ?? S11 )??L = |S11 |2 ? |?in |2
?(S22
?L ??L ?
?
?
|?in |2 ? ?? S11 )??L
)?L + (S22
(S22 |?in |2 ? ?S11
|S22 |2 |?in |2 ? |?|2
|S11 |2 ? |?in |2
=
|S22 |2 |?in |2 ? |?|2
117
(A?7)
(A?8)
2
? ? 2
)
(S
|?
|
?
?S
22
in
11
?L ?
|S22 |2 |?in |2 ? |?|2 ? 2
S22 |?in |2 ? ?S11
|S11 |2 ? |?in |2
+
=
|S22 |2 |?in |2 ? |?|2 |S22 |2 |?in |2 ? |?|2 |S11 |2 |S22 |2 |?in |2 ? |?|2 |S11 |2 ? |S22 |2 |?in |4 + |?|2 |?in |2
=
||S22 |2 |?in |2 ? |?|2 |2
? ?
|S22 |2 |?in |4 + |?|2 |S11 |2 ? ?? S11 S22 |?in |2 ? ?S11
S22 |?in |2
+
||S22 |2 |?in |2 ? |?|2 |2
=
? ?
|S11 |2 |S22 |2 + |?|2 ? ?? S11 S22 ? ?S11
S22
|?in |2
2
2
2
2
||S22 | |?in | ? |?| |
(A?9)
(A?10)
(A?11)
2
S11 S22 ? ?
|?in |2
=
2
2
2
|S22 | |?in | ? |?| (A?12)
2
S12 S21
|?in |2
=
|S22 |2 |?in |2 ? |?|2 (A?13)
Now, the equation represents a circle on the Smith chart.
2
? ?
?L ? (S22 |?in | ? ?S11 ) |S22 |2 |?in |2 ? |?|2 S12 S21
|?in |
=
2
2
2
|S22 | |?in | ? |?| (A?14)
The center and radius of the circle are given by
? ?
(S22 |?in |2 ? ?S11
)
CL =
2
2
|S22 | |?in | ? |?|2
S12 S21
|?in |
RL = 2
2
2
|S22 | |?in | ? |?| (center)
(A?15)
(A?16)
When the incident and re?ected waves are combined under the absence of a coupler,
the measured power of the input port (P1) is expressed as
pin = |a1 + b1 |2 = |a1 |2 |1 + ?in |2
118
(A?17)
Therefore, |1 + ?in | instead of |?in | can be used to estimate a load impedance. The
complex input re?ection coe?cient is rewritten as
1 + ?in = 1 + S11 +
S12 S21 ?L
1 ? S22 ?L
(A?18)
The magnitude of the input re?ection coe?cient becomes
S
S
?
12
21
L
|1 + ?in | = 1 + S11 +
1 ? S22 ?L 1 + S11 ? (1 + S11 )S22 ?L + S12 S21 ?L |1 + ?in | = 1 ? S22 ?L
(A?19)
(A?20)
By taking the square of both sides, the equation is written as
|1 ? S22 ?L |2 |1 + ?in |2 = |1 + S11 ? (S22 + ?)?L |2
(A?21)
? = S11 S22 ? S12 S21
(A?22)
where
The equation is manipulated as follows.
? ?
|1 + ?in |2 + |S22 |2 |?L |2 |1 + ?in |2 ? S22 ?L |1 + ?in |2 ? S22
?L |1 + ?in |2
?
?
= |1 + S11 |2 + |S22 + ?|2 |?L |2 ? (S22 + ?)?L (1 + S11
) ? (S22
+ ?? )??L (1 + S11 )(A?23)
where * denotes a complex conjugate.
?
(|S22 |2 |1 + ?in |2 ? |S22 + ?|2 )?L ??L ? (S22 |1 + ?in |2 ? (S22 + ?)(1 + S11
))?L
?
?
?(S22
|1 + ?in |2 ? (S22
+ ?? )(1 + S11 ))??L = |1 + S11 |2 ? |1 + ?in |2
?L ??L ?
(A?24)
?
?
?
+ ?? )(1 + S11 ))??L
|1 + ?in |2 ? (S22
))?L + (S22
(S22 |1 + ?in |2 ? (S22 + ?)(1 + S11
|S22 |2 |1 + ?in |2 ? |S22 + ?|2
|1 + S11 |2 ? |1 + ?in |2
=
(A?25)
|S22 |2 |1 + ?in |2 ? |S22 + ?|2
119
2
?
? 2
(S
|1
+
?
|
?
(S
+
?)(1
+
S
))
22
in
22
11
?L ?
|S22 |2 |1 + ?in |2 ? |S22 + ?|2
? 2
S22 |1 + ?in |2 ? (S22 + ?)(1 + S11
|1 + S11 |2 ? |1 + ?in |2
)
+
=
|S22 |2 |1 + ?in |2 ? |S22 + ?|2 |S22 |2 |1 + ?in |2 ? |S22 + ?|2 (A?26)
|1 + S11 |2 |S22 |2 |1 + ?in |2 ? |S22 + ?|2 |1 + S11 |2 ? |S22 |2 |1 + ?in |4 + |S22 + ?|2 |1 + ?in |2
=
||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2
|S22 |2 |1 + ?in |4 + |S22 + ?|2 |1 + S11 |2
+
||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2
?
?
(S ? + ?? )(1 + S11 )S22 |1 + ?in |2 + (S22 + ?)(1 + S11
)S22
|1 + ?in |2
? 22
(A?27)
||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2
|1 + S11 |2 |S22 |2 + |S22 + ?|2
2
2 |1 + ?in |
2
2
2
||S22 | |1 + ?in | ? |S22 + ?| |
?
?
?
?
(S + ? )(1 + S11 )S22 + (S22 + ?)(1 + S11
)S22
|1 + ?in |2
? 22
||S22 |2 |1 + ?in |2 ? |S22 + ?|2 |2
=
(A?28)
(1 + S11 )S22 ? (S22 + ?) 2
|1 + ?in |2
=
|S22 |2 |1 + ?in |2 ? |S22 + ?|2 (A?29)
2
S12 S21
|1 + ?in |2
=
|S22 |2 |1 + ?in |2 ? |S22 + ?|2 (A?30)
Now, the equation represents a circle on the Smith chart.
2
?
?
?L ? (S22 |1 + ?in | ? (S22 + ?)(1 + S11 )) |S22 |2 |1 + ?in |2 ? |S22 + ?|2
S12 S21
|1 + ?in |
= 2
2
2
|S22 | |1 + ?in | ? |S22 + ?| (A?31)
The center and radius of the circle are given by
?
))?
(S22 |1 + ?in |2 ? (S22 + ?)(1 + S11
|S22 |2 |1 + ?in |2 ? |S22 + ?|2
S12 S21
|1 + ?in |
RL = 2
2
2
|S22 | |1 + ?in | ? |S22 + ?| CL =
120
(center)
(A?32)
(A?33)
APPENDIX B
THREE-PORT AND FOUR-PORT MULTISTATE REFLECTOMETERS
Port 3 is terminated with a detector whose load re?ection coe?cient is ?3 . The
incident wave power a1 is written as
a 3 = ?3 b 3
(B?1)
Therefore, three-port S-parameter equations can be expressed in a matrix form as
? ? ?
?? ?
?b1 ? ?S11 S12 S13 ?3 ? ?a1 ?
? ? ?
?? ?
?b ? = ?S S S ? ? ?a ?
? 2 ? ? 21 22 23 3 ? ? 2 ?
? ? ?
?? ?
S31 S32 S33 ?3
b3
b3
(B?2)
The incident power wave into a load terminating the port 3, b3 is written as
b3 = S31 a1 + S32 a2 + S33 ?3 b3
b3 =
(B?3)
1
(S31 a1 + S32 a2 )
1 ? S33 ?3
(B?4)
Also, the incident power wave into the port 1 is given by
a1 =
1
(?S22 a2 + b2 ? S23 ?3 b3 )
S21
(B?5)
By plugging Equation B?5 into Equation B?4, the equation becomes
1
b3 =
1 ? S33 ?3
(
(
S23 S31 ?3
1+
S21 (1 ? S33 ?3 )
(
S31
)
1
S23 ?3
S22
?
a2 +
b2 ?
b3
S21
S21
S21
1
b3 =
1 ? S33 ?3
((
)
S22 S31
S32 ?
S21
)
+ S32 a2
)
S31
a2 +
b2
S21
(B?6)
)
(B?7)
Now, the re?ected power wave b3 is written as
(
)
S21 (1 ? S33 ?3 )
S21 S32 ? S22 S31
a2
S31
b3 =
b2
?
S21 (1 ? S33 ?3 ) + S23 S31 ?3 S21 (1 ? S33 ?3 )
b2
S22 S31 ? S21 S32
S21 S32 ? S22 S31
=
b2 (?L ? q3 )
S21 (1 ? S33 ?3 ) + S23 S31 ?3
121
(B?8)
where
?L =
a2
b2
q3 =
S31
S22 S31 ? S21 S32
(B?9)
To simplify the above equations, suppose that the detector port is matched (?3 = 0).
The equations is simpli?ed as
b3 =
?3 =0
S21 S32 ? S22 S31
b2 (?L ? q3 )
S21
(B?10)
and also expressed in terms of A3 and B3 as
b3 = A3 a2 + B3 b2
(B?11)
where
A3 =
S21 S32 ? S22 S31
S21
B3 =
S31
S21
(B?12)
Also, the re?ected wave power b2 is simpli?ed as
b2 ?3 =0
= S21 a1 + S22 ?L b2
b2 =
S21
a1
1 ? S22 ?L
(B?13)
(B?14)
By plugging Equation B?14 into Equation B?10, b3 can be simpli?ed as
S21 S32 ? S22 S31
b2 (?L ? q3 )
S21
S21 S32 ? S22 S31
S21
a1 (?L ? q3 )
=
S21
1 ? S22 ?L
S21 S32 ? S22 S31
=
a1 (?L ? q3 )
1 ? S22 ?L
b3 =
(B?15)
Similar to the three-port re?ectometer, a four-port re?ectometer with matched
detector ports (?3 = ?4 = 0) is governed by the following equation as
S21 S42 ? S22 S41
b2 (?L ? q4 )
S21
S21 S42 ? S22 S41
=
a1 (?L ? q4 )
1 ? S22 ?L
b4 =
122
(B?16)
where
q4 =
S41
S22 S41 ? S21 S42
(B?17)
The equation is also expressed in terms of a2 and b2 as
b4 = A4 a2 + B4 b2
(B?18)
where
A4 =
S21 S42 ? S22 S41
S21
123
B4 =
S41
S21
(B?19)
APPENDIX C
STANDARD AND MIXED-MODE S-PARAMETER TRANSFORMATION
Common- and di?erential-mode normalized power waves are de?ned by
1
ac1 = ? (a1 + a2 )
2
1
ad1 = ? (a1 ? a2 )
2
(C?1)
1
bc1 = ? (b1 + b2 )
2
1
bd1 = ? (b1 ? b2 )
2
(C?2)
The common- and di?erential-mode power waves are written in a compact matrix
?
form as
bmm
?
b
?1
? ? ? c1 ?
?
? ?
?bd1 ?
?b1
1 ?
? ? ? ?
?1
?
=? ?=? ?=
?
?b ?
2?
?b2
? c2 ?
?0
? ?
?
0
bd2
?
1 0
?1 0
0 1
0 1
?? ?
0 ? ?b1 ?
?? ?
? ?
0?
? ?b2 ?
? ? ? = Mbstd
? ?
1?
? ?b3 ?
?? ?
b4
?1
(C?3)
The mixed- and standard-mode normalized power waves are converted to each other
by using the following relationship as
bmm = Mbstd
M = M?1
?
1
?
?
1 ?
?1
?
=
?
2?
?0
?
0
(C?4)
?
1 0 0
?
?
?1 0 0 ?
?
?
?
0 1 1?
?
0 1 ?1
(C?5)
Now, mixed-mode S-parameter equations are written in a matrix form as
? ? ?
?? ?
S
S
S
S
b
a
? c1 ? ? cc11 cd11 cc12 cd12 ? ? c1 ? ?
?? ?
? ? ?
?? ?
?bd1 ? ?Sdc11 Sdd11 Sdc12 Sdd12 ? ?ad1 ?
? ? ?
? ? ? ?S11 S12 ? ??a1 ?
mm mm
bmm = ? ? = ?
?? ? = ?
?? ? = S a
? ? ?
?? ?
S21 S22
?a2
?bc2 ? ?Scc21 Scd21 Scc22 Scd22 ? ?ac2 ?
? ? ?
?? ?
bd2
Sdc21 Sdd21 Sdc22 Sdd22
(C?6)
124
The equation is manipulated as follows.
bmm = Smm amm
(C?7)
Mbstd = Smm Mastd
(C?8)
bstd = M?1 Smm Mastd
(C?9)
Therefore, the transformation between mixed- and standard-mode S-parameters are
given by
Sstd = M?1 Smm M
(C?10)
Smm = MSstd M?1
(C?11)
and
125
APPENDIX D
DERIVATION OF DIFFERENTIAL INPUT REFLECTION COEFFICIENT
A network with one di?erential port can be written as
?b1 = Smm?a1
11
where mm denotes mixed-mode and
? ?
? ?
bc1 ?
?ac1 ?
?b1 = ?
?a1 = ? ? ,
? ?,
bd1
(D?1)
?
?
?Scc11 Scd11 ?
Smm
?
11 = ?
Sdc11 Sdd11
It can be easily extended to two di?erential port network as follows.
? ? ?
?? ?
mm
mm
?
?b1 ? ?S11 S12 ? ??a1 ?
? ?=?
?? ?
mm
?b2
Smm
S
?a2
21
22
(D?2)
(D?3)
Suppose that a di?erential load is connected to the di?erential port 2. Then, there is
the following relationship between incident and re?ected waves.
?
?a2 = ?mm
L b2
where
?
(D?4)
?
??Lcc ?Lcd ?
?mm
=
?
?
L
?Ldc ?Ldd
Therefore, the s-parameter equation can be written as
?
? ? ?
??
mm
mm
?
?b1 ? ?S11 S12 ? ? ?a1 ?
=
?
? ? ?
??
?
?b2
Smm
Smm
?mm
21
22
L b2
(D?5)
(D?6)
?b2 can be written as
?b2 = Smm?a1 + Smm ?mm?b2
21
22
L
(D?7)
mm
mm ?
a1
(I ? Smm
22 ?L )b2 = S21 ?
(D?8)
It can be solved as
126
?b2 = (I ? Smm ?mm )?1 Smm?a1
22
L
21
(D?9)
From the S-parameter equation and the above equation, ?b1 can be written as
?b1 = Smm?a1 + Smm ?mm?b2 = Smm?a1 + Smm ?mm (I ? Smm ?mm )?1 Smm?a1
11
12
L
11
12
L
22
L
21
(D?10)
The input re?ection matrix is de?ned as
?b1 = ?mm?a1
in
?
where
(D?11)
?
??incc ?incd ?
?mm
?
in = ?
?indc ?indd
(D?12)
By the comparison of Equation D?10 and D?11, the input re?ection matrix is
mm
mm mm
mm mm ?1 mm
?mm
in = S11 + S12 ?L (I ? S22 ?L ) S21
(D?13)
The mixed-mode re?ection matrix can be converted to standard-mode as follows.
?1 mm
?std
in = M ?in M
?1 mm
?1 mm
?1
mm mm ?1
?1 mm
= M?1 Smm
11 M + M S12 MM ?L MM (I ? S22 ?L ) MM S21 M
(D?14)
(D?15)
The standard-mode input re?ection coe?cient matrix is written as
std
std std
std std ?1 std
?std
in = S11 + S12 ?L (I ? S22 ?L ) S21
where
?std
in
?
?
??in11 ?in12 ?
=?
?,
?in21 ?in22
Sstd
11
?
?
?S11 S12 ?
=?
?,
S21 S22
?std
L
Sstd
22
127
?
?
??L55 ?L56 ?
=?
?
?L65 ?L66
?
?
?S33 S34 ?
=?
?
S43 S44
(D?16)
(D?17)
(D?18)
Sstd
12
?
?
?S13 S14 ?
=?
?,
S23 S24
Sstd
21
128
?
?
?S31 S32 ?
=?
?
S41 S42
(D?19)
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132
BIOGRAPHICAL SKETCH
Jaeseok Kim was born in Seoul, Korea. He received the B.S. degree in 1994, from
Inha University, majoring in electronic engineering. He received the M.S. degree in 2002
from the University of Florida, majoring in electrical and computer engineering. Since
2006, he has been with the Electronic Circuit Laboratory (ECL) at the University of
Florida, pursuing his Ph.D. degree. His research interests include RF impedance matching
control algorithm and system, machine intelligence of RF systems, and embedded RF
on-chip testing.
133
e 6-4. The inductance
value of 3.3 nH and 1.8 nH was used for the design of the matching network.
A noninvasive measurement is mandatory not to disturb the original design of a
matching network. Similar to the prior work [35], a high impedance power detector
measures an internal node. The noninvasive power detector is emulated by inserting high
resistance in series with a measurement port and deembedding the e?ect of the resistance.
As explained earlier, the four-port re?ectometer needs a power reading that depends only
84
P3
High input impedance
power detector
Z0
ZL
A
B
Figure 6-3. Recon?gurable three-port matching network A) Schematic B) Implementation
on FR4 board
85
S(2,1)
S(1,1)
freq (2.400GHz to 2.400GHz)
S(2,1)
S(1,1)
A
freq (2.400GHz to 2.400GHz)
B
Figure 6-4. Tunable element impedance with a bias from 0 V to 10 V. A varactor is in
parallel and in series with an inductor. и and О denote S11 and S21 ,
respectively. A) In series with 3.3 nH B) In parallel with 1.8 nH
86
on the re?ected wave power from a DUT. The reference port can be easily realized using a
lumped power divider.
6.4
The matching capability of a tunable matching network can be represented by the
load re?ection coe?cient ?L to be matched by the matching network. The matching
capability is derived as follows. The input re?ection coe?cient ?in is written as
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(6?10)
The equation can be rewritten in terms of the load re?ection coe?cient as
S11 ? ?in
S11 S22 ? S12 S21 ? S22 ?in
?L =
(6?11)
The matching capability is derived from the load re?ection coe?cient by setting the input
re?ection coe?cient to zero.
?L ?in =0
=
S11
S11 S22 ? S12 S21
(6?12)
The coverage on the Smith chart speci?ed the matching capability illustrates the
distribution of the load re?ection coe?cient to be matched.
According to the six-port re?ectometer principle, an unknown load is the same as the
point intersected by three circles, speci?ed by a center, so-called q-point, and a radius.
The circle is represented by calibration constants de?ned by Equation 6?6. Note that the
q-point does not change even if a mismatched load varies, whereas the load estimation
method proposed in [25] has changed the circle center as a mismatched load varies. The
constant q-point enables to keep the optimum-performance criteria for the multistate
In reality, the three circles represented by calibration constants seldom intersect at
a point due to the non-ideal e?ects. The geometric center of the overlap of the circles is
87
estimated by the radical center and least square estimation methods widely used for the
six-port re?ectometer [27].
The radical center is the approximation of the center of three circles? overlapped
region. The coordinates of the radical center are given as
2
2
RL1
? RL2
+ x22
2x2
2
2
R ? RL3 + x23 + y32 ? 2xx3
y = L1
2y3
x=
(6?13)
(6?14)
Least square ?tting can enhance the accuracy of load estimation especially when three
circles failed to meet the optimum-performance criteria. The load re?ection coe?cient can
be obtained by the least square equation as
??L = arg min
?
|Circle(V) ? ?L |2
(6?15)
V
where V = (V1 , V2 , V3 )T is a bias voltage vector for three varactors. Note that Circle(V)
is represented by calibration constants de?ned by Equation 6?6 and the distance between
a circle and a point is de?ned as a distance between a tangential line to the circle and the
point.
6.5
Experimental Results
The matching capability of the proposed three-port re?ectometer, de?ned as
S11 /(S11 S22 ? S12 S21 ), is shown in Figure 6-5. The matching capability was measured by
changing each varactor bias from 0 V to 5.12 V in 16 levels by 0.32 V step. The matching
capability covers a unit circle on the Smith chart completely, showing its capability on any
One of important calibration constants is q-point given by the equation
qi =
Si1
S22 Si1 ? S21 Si2
(6?16)
The q-points were measured with respect to the same bias range as used for the matching
capability. The measured q-points are distributed along the unit circle as shown in Figure
88
Table 6-1. Summary of mismatched loads
|S11 | at 2.4 GHz
Tuner motor position
Matched
0.01
(100, 5000, 5000)
Slightly mismatched #1
Slightly mismatched #2
Slightly mismatched #3
0.13
0.38
0.20
(16725, 2262, 5000)
(17105, 1424, 5000)
(20464, 2228, 5000)
Heavily mismatched #1
Heavily mismatched #2
Heavily mismatched #3
0.78
0.76
0.69
(15781, 526, 5000)
(17835, 624, 5000)
(20572, 804, 5000)
6-6. Some set of q-points can be selected from the distribution in order to satisfy the
optimum-performance criteria for the multistate re?ectometer. A set of q-points was
carefully selected for better estimation performance.
The load estimation using three-port re?ectometer was performed separately on
slightly and heavily mismatched loads. The speci?cation of mismatched loads is given
in Table 6-1. First, the S-parameter of the input port was converted from the high
impedance port emulating high impedance power detector. The calibration constants
were obtained from the S-parameters through direct measurement of the three-port
re?ectometer and conversion using the high impedance power detector. The q-points were
chosen to achieve higher estimation accuracy for two separate experiments. Then, radical
center estimation was applied to estimate an unknown load re?ection coe?cient. When
the magnitude of the estimated re?ection coe?cient is larger than one, it is incorrect for
passive mismatched loads. In this case, the magnitude was set to one with keeping the
phase.
As shown in Figure 6-7, estimation of slightly mismatched loads showed much smaller
estimation error than heavily mismatched loads. The mean square error for slightly and
heavily mismatched loads are 0.09 and 0.80, respectively. Due to the larger estimation
error, the estimated load re?ection coe?cient of the heavily mismatched loads often go
beyond a unit circle. As described, the magnitude was set to 1 and only the phase was
kept for impedance matching. However, the estimated phase is still quite useful because
89
the estimated magnitude larger than one often results from heavily mismatched load,
whose magnitude is close to one. The next chapter will demonstrate that an automatic
matching control can achieve impedance matching using the estimated phase information.
6.6
Conclusion
We proposed a three-port lumped-element re?ectometer for both load estimation and
impedance matching. The proposed re?ectometer can be easily extended to a four-port
demonstrated that the tunable multistate re?ectometer can help the automatic matching
control (AMC) to estimate a load re?ection coe?cient as well as to set impedance
matching. The high impedance power detector replaced the distributed coupler and
realized the dramatic size reduction of an automatic matching control system without
compromising the load estimation and matching capability. The matching capability
covered completely the unit circle on the Smith chart. Although the load estimation
result is not accurate to be used as an high-precision instrument, the estimated phase
information can still enable the automatic matching control to achieve faster impedance
matching on heavily mismatched loads. We are working toward the integration of the
proposed load estimation and a novel automatic matching control system capable of an
immediate impedance matching.
90
+j1.0
+j0.5
+j2.0
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
?j0.2
\infty
?j5.0
?j0.5
?j2.0
?j1.0
Figure 6-5. Matching capability of three-port re?ectometer at 2.4 GHz
91
+j1.0
+j2.0
+j0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
?j0.2
\infty
?j5.0
?j2.0
?j0.5
?j1.0
Figure 6-6. The q-point distribution of three-port re?ectometer at 2.4 GHz
92
+j1.0
+j1.0
True
+j2.0 Estimate
?j0.2
?j5.0
?j0.5
5.0
0.0
2.0
Ц
+j5.0
1.0
+j0.2
0.5
5.0
2.0
1.0
0.5
0.0
+j5.0
0.2
+j0.2
True
+j2.0 Estimate
+j0.5
0.2
+j0.5
?j0.2
?j5.0
?j0.5
?j2.0
Ц
?j2.0
?j1.0
?j1.0
A
B
Figure 6-7. Multistate re?ectometer estimation using estimated S11 from high impedance
probe A) Small mismatch (MSE=0.09) B) Large mismatch (MSE=0.80)
93
CHAPTER 7
AUTOMATIC MATCHING CONTROL USING LOAD ESTIMATION
7.1
Overview
An automatic matching control (AMC) system has been developed to automate
time-consuming impedance matching procedure [5][6][21]. The impedance matching of the
automatic matching control was performed by recon?guring a tunable matching network
until the lowest mismatch is achieved. The recon?guration was controlled by heuristic
iterative methods, which showed a good trade-o? between system response and impedance
matching accuracy. Also, load estimation reusing the existing tunable matching network
of the automatic matching control system was proposed to facilitate the automation of
impedance matching [25]. In this work, the load estimation technique was integrated with
the existing automatic matching control system to achieve immediate impedance matching
without compromising matching accuracy.
Traditional automatic matching control systems achieved impedance matching of
unknown or even varying mismatched loads by the feedback loop of a tunable matching
network, a mismatch detector, and match control circuit [21]. The feedback loop is
controlled by iterative methods of a match control circuit, which searches for the value
of tuning elements in a trial-and-error process. However, the trial-and-error approach
slowed down the system response and various heuristic approaches have been developed to
improve the system response without compromising matching capability. Nevertheless, the
system response of the heuristic approaches is still proportional to the complexity of the
matching network and gets slower as more tuning elements and levels are added.
We will demonstrate that an estimated load can be used for a matching control
circuit to achieve immediate impedance matching without using heuristic approaches.
The proposed matching control can ?nd the value of tuning elements by examining the
characterization table of a matching network. Therefore, the precise characterization as
94
well as the load estimation play an important role in this immediate impedance matching
approach.
Various characterization methods for a microwave device have been reported to
reduce the microwave design complexity. A neural network has been widely used to
characterize microwave devices, such as the approximation of S-parameters of BJTs [36]
and modeling parameters of microwave components [9]. Also, closed form equation was
also presented for S-parameters of BJTs [37]. In this work, neural network models and the
closed form equation were used to approximate measured S-parameters and the accuracy
of the characterization methods was evaluated in terms of mean squared error (MSE)
between true and estimated values.
The proposed matching control consists of two tasks. First, a characterization table in
terms of tuning elements was built from the direct measurement of the matching network
or approximation models such as a neural network and closed form equations. Next, the
value of the tuning elements was found by minimizing the degree of mismatch. The degree
of mismatch was calculated from the magnitude of the input re?ection coe?cient. The
experimental results of the immediate impedance matching approach will be presented.
7.2
Matching Control Procedures
The same lumped-element tunable matching network that is used for load estimation
was used to develop matching control procedures supporting load estimation presented
in Chapter 5 and 6. The matching network has a ?-type bandpass ?lter topology and
three varactor diodes as tuning elements. The recon?guration of the matching network
was performed by changing the varactor bias voltages. Its matching capability covers all
re?ection coe?cients within the unit circle on the Smith chart at 2.4 GHz.
The load re?ection coe?cient ?L of a device under test (DUT) is assumed to be
estimated by load estimation techniques presented in Chapter 5 and 6. When the DUT is
connected to the port 2 of a tunable matching network, the input re?ection coe?cient ?in
95
looking into the port 1 of the matching network is written as follows.
?in = S11 +
S12 S21 ?L
1 ? S22 ?L
(7?1)
where Sij is the S-parameter from port j to port i of the matching network. Note that the
S-parameters are the function of a bias voltage vector, denoted by v. Therefore, the input
re?ection coe?cient can be explicitly written as the function of v.
?in (v) = S11 (v) +
S12 (v)S21 (v)?L
,
1 ? S22 (v)?L
v = (v1 , v2 , и и и , vn )T
(7?2)
where T denotes a transpose and vn is the nth bias voltage. The load re?ection coe?cient
of the DUT can be derived from the input re?ection coe?cient.
?L =
S11 (v) ? ?in (v)
S11 (v)S22 (v) ? S12 (v)S21 (v) ? S22 (v)?in (v)
(7?3)
The mismatched load to be matched by the matching network set by a bias voltage v,
denoted by ?M , is the load re?ection coe?cient that makes the input re?ection coe?cient
zero.
?M (v) = ?L =
?in =0
S11 (v)
S11 (v)S22 (v) ? S12 (v)S21 (v)
(7?4)
b, as a bias voltage
Now, let us de?ne an optimal bias voltage vector, denoted by v
vector that minimizes the magnitude of the input re?ection coe?cient ?in , the degree
of mismatch.
b = arg min |?in (v)|
v
v
(7?5)
If the bias voltage, that minimizes the input re?ection coe?cient to zero, can be found for
b can be expressed
all possible load re?ection coe?cients, the optimal bias voltage vector v
using ?M as follows.
}
} { { b = v?in (v) = 0 = v?M (v) = ?L
v
Finding the optimal bias voltage can be expressed as ?nding a bias voltage whose ?M
is equal to ?L . Therefore, the mapping table between ?M and v should be calculated
96
(7?6)
Start
Characterization
Measure power three-times
Calculate bias for matching
End
Figure 7-1. Automatic matching control supports load estimation.
to perform the bias search and the mapping table can be converted from the matching
network characterized by the S-parameters. This procedure is based on Equation 7?4. The
S-parameters can be obtained from direct measurement using a vector network analyzer
or a neural network ?tting model. The characterization methods will be introduced in the
next Section.
7.3
Characterization of Matching Network
The characterization of a matching network is a procedure to discover the S-parameter functions to be used to calculate an input re?ection coe?cient or a mismatched
load to be matched. The S-parameters of the matching network were measured using
a vector network analyzer while changing the bias voltage. The measurement points
were determined by the number of varactors and bias voltage levels. Although the more
voltage levels can produce more accurate characterization results, 16 voltage levels were
chosen as good trade-o? between measurement time and characterization accuracy. The
97
characterization results were converted to a form of a mapping table, ?M (v), for easy
access and searches.
Unknown S-parameters between measurement points were approximated by a
multivariate linear interpolation. The multivariate linear interpolation is interpolating a
function of multiple variables on a regular grid, as an extension of a linear interpolation.
It performs linear interpolation ?rst on one direction, then again in the other direction.
Suppose we want to interpolate a value of an unknown function f at the point (x, y).
The value of the function f at four neighbor points on a regular grid, f (x1 , y1 ), f (x1 , y2 ),
f (x2 , y1 ), and f (x2 , y2 ), are assumed to be known, then the interpolation of the function f
at the point (x, y) can be written as follows.
f (x1 , y1 )
(x2 ? x)(y2 ? y)
(x2 ? x1 )(y2 ? y1 )
f (x2 , y1 )
+
(x ? x1 )(y2 ? y)
(x2 ? x1 )(y2 ? y1 )
f (x1 , y2 )
(x2 ? x)(y ? y1 )
+
(x2 ? x1 )(y2 ? y1 )
f (x2 , y2 )
+
(x ? x1 )(y ? y1 )
(x2 ? x1 )(y2 ? y1 )
f (x, y) ?
(7?7)
If the unknown function to interpolate has a smooth surface over neighbor points, the
linear interpolation reduces the number of measurement points signi?cantly without the
loss of the characterization detail.
The other method to estimate unknown S-parameters between measurement points
is an approximation ?tting function to the measured S-parameters. As described, the
measured S-parameters are the functions of the bias voltage vector v, given as follows.
?
?
?S11 (v) S12 (v)?
(7?8)
S(v) = ?
?
S21 (v) S22 (v)
The S-parameter functions were approximated by a curve ?tting model, such as a closed
form and a neural network. First, an arti?cial neural network was used to approximate
98
the S-parameter functions. Four independent neural network models approximate two-port
S-parameter functions, S11 (v), S12 (v), S21 (v), and S22 (v). The input and output of the
neural network are bias voltage vector and real and imaginary parts of the S-parameter,
respectively. The feed-forward topology was used with 15 perceptrons in the hidden layer.
The well-known backpropagation algorithm was used to train the neural networks.
The cascaded network is easily represented by ABCD-parameters and the matching
network was decomposed into basic components, such as a transmission line, series
impedance, and shunt admittance. The representation of the ABCD-parameters was used
as closed-form equations. The ABCD-parameters for the basic components are given as
follows.
?
?
? cos(2??) ?Z0 sin(2??)?
?
?
?Y0 sin(2??) cos(2??)
?
?
?1 Z ?
?
?
0 1
?
?
? 1 0?
?
?
Y 1
(transmission line)
(7?9)
(series impedance)
(7?10)
(7?11)
The closed-form ABCD-parameters were derived from the cascaded network of a
transmission line (?1 ), a ? network (Y1 , Z3 , and Y2 ), and a transmission line (?2 ). The
varactor capacitance was calculated from the varactor SPICE model used in Chapter 5
and 6.
ABCD11 =(cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 )
+? (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 )
(7?12)
ABCD12 =? (cos(2??1 )(1 + Y2 Z3 ) + ?Z0 sin(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 )
+ (cos(2??1 )Z3 + ?Z0 sin(2??1 )(1 + Y1 Z3 )) cos(2??2 )
99
(7?13)
ABCD21 =(?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) cos(2??2 )
+? (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) Y0 sin(2??2 )
(7?14)
ABCD22 =? (?Y0 sin(2??1 )(1 + Y2 Z3 ) + cos(2??1 )(Y1 + Y2 + Y1 Y2 Z3 )) Z0 sin(2??2 )
+ (?Y0 sin(2??1 )Z3 + cos(2??1 )(1 + Y1 Z3 )) cos(2??2 )
(7?15)
The admittance Y1 , Y2 and impedance Z3 are the function of bias voltage which determines
the varactor capacitance. The transmission line delay ?1 , ?2 , parasitic parameters, ?tting
parameters of the varactor SPICE model are trained by a nonlinear least square ?tting
algorithm. The ABCD-parameters based ?tting model was converted to S-parameters for
fair comparison with direct measurement and neural network model.
7.4
Bias Search for Impedance Matching
The goal of bias search is to ?nd the optimal bias voltage vector given by
{ } { }
b = v?in (v) = 0 = v?M (v) = ?L
v
(7?16)
Due to the discrete measurement data of a mapping table ?M (v), it is not always possible
to ?nd the optimal bias voltage vector. Instead, we choose the bias voltage vector closest
to the optimal bias voltage vector and this procedure can be written as follows.
b = arg min |?M (v) ? ?L |
v
v
(7?17)
From now on, the optimal bias voltage vector is rede?ned as the bias minimizing the
magnitude between ?M and ?L .
The bias search consists of two steps, coarse and ?ne search. The coarse search was
performed on the mapping table converted from the direct measurement of S-parameters.
100
Then, the ?ne search was performed again on the linear interpolation of a voxel1 of the
bias voltage vector found in the coarse search.
The cost function for both coarse and ?ne search is mean square error (MSE) of
magnitude between ?M of eight bias voltage vectors of a voxel and an estimated load
?L . Note that a voxel for 3D data has eight vertexes, similar to a cubic. For example, a
unit voxel consists of eight points, (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0),
and (1,1,1). Eight points of the voxel for the coarse search were selected from the voltage
bias used for the direct measurement. Once the voxel is found by the coarse search, the
mapping table of the voxel is interpolated by multivariate linear interpolation with higher
resolution. Then, the ?ne search is performed on the interpolated data. The proposed
two-step search reduces the required number of measurement points and the memory
usage by the partial interpolation. In addition, the bias search can be made faster using
binary search algorithm
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