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‘ a LIBRARY
2M Michigan State
University

This is to certify that the
thesis entitled

Preliminary Investigation of Negative Impedance Converters
with Microstrip Lines

presented by

Rodolph Sanon, Jr.

has been accepted towards fulfillment
of the requirements for the

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5/08 K‘IProlecc8PrelelRC/DateDue.indd

Preliminary Investigation of Negative Impedance Converters
with Microstrip Lines

By

Rodolph Sanon, Jr.

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2008

ABSTRACT

Preliminary Investigation of Negative Impedance Converters with
Microstrip Lines
By
Rodolph Sanon, Jr.

The purpose of this thesis is to design a negative impedance converter (N IC) such
that it functions as a load to a microstrip transmission line. Like other transmis-
sion lines, the performance of the microstrip depends on its load impedance which
determines the reflection of a wave traveling along the line at the load. The wave
characteristics of the microstrip for a traditional — positive — load impedance, a. short
circuit, and an open circuit are already known. However, the predictability of the
performance of a microstrip with negative load impedance is not well understood.

In order to observe the behavior of a microstrip with a negative load impedance,
an electronic device must be designed, built, implemented, and integrated to the
microstrip. In this thesis, a theoretical model of a NIC will be simulated with a
microstrip such that it behaves as its load impedance. Since microstrip lines are
generally observed at high frequencies, the NIC must be designed to optimally perform
at the desired frequency bandwidth of interest, in this case C-band (6-8 GHZ).

A technique called de-embedding will be used to attach the N IC to a microstrip
line. This process will detail how the NIC affects the microstrip’s performance. The
means of how it will be done is through the Thm-Reflect-Line (TRL) calibration
technique. The microstrip will be modeled such that it. will be de-embedded us—
ing the calibration technique. In addition, a parameter study will be performed to

characterize the behavior of the microstrip with a NIC.

Copyright by
Rodolph Sanon, Jr.
2008

For my family and my late godmother J alene Crawford

ACKNOWLEDGMENTS

A lot. of people have been involved directly and indirectly for their contributions
for the work displayed here in this thesis and many other things during my time at.
Michigan State. I would first. like to thank my advisor Dr. Lee Kempel for helping
me in all phases in conducting this work. I am thankful to work as your student.
and to find a project as stimulating and challenging as this one. You also gave me
a valuable perspective in exploring opportunities in industry and academia and will
be a valuable resource in my future endeavors. To Dr. Edward J. Rothwell, my
surrogate adviser, who helped me understand what it takes to be a graduate student
and the daily struggles that come with it. To Dr. Shanker B. for helping me establish
a means to think critically, even though my head split. in two in the process. To the
students of the Electromagnetics Research Group for making my time as a. graduate
special. We had some fun times at the Engineering Library.

Thanks to Dr. Percy Pierre and Dr. Barbara O’Kelly for their support through
the Sloan Fellowship Program. To Sheryl Hulet, Vanessa l\-"Iitchner, and members of
the staff of the Department. of Electrical and Computer Engineering for their support.
To Michael Archbold for providing an opportunity to go to graduate school when no
opportunity existed due to unfortunate circumstances.

A special thank you to friends and family around the country for their love and
support during my time in graduate school at Michigan State. A personal thanks to
the family of my late godmother J alene Crawford who helped me establish discipline

when my parents were watching over me in my youth.

‘7

TABLE OF CONTENTS

LIST OF TABLES ................................. viii

LIST OF FIGURES ................................ ix

KEY TO SYMBOLS AND ABBREVIATIONS ................. xiv
CHAPTER 1

Introduction ..................................... 1
CHAPTER 2

Negative Impedance Converters .......................... 3

2.1 Foster’s Reactance Theorem ....................... 3

2.1.1 The Complex Poynting Theorem ................ 4

2.1.2 Microwave Network Analysis ................... 6

2.1.3 Verifying Foster’s Reactance Theorem .............. 9

2.2 Operation of Negative Impedance Converters .............. 12

2.2.1 Voltage Inversion NICs vs. Current Inversion NICs ...... 13

2.2.2 Grounded NICs vs. Floating NICs ............... 15

2.2.3 Open-circuit. Stable vs. Short-circuit Stable .......... 17

2.2.4 BJT vs. FET ........................... 19
CHAPTER 3

De-Embedding Test Fixtures and l\=’Iicrostrip Circuits .............. 29

3.1 Microstrip Circuits ............................ 31

3.1.1 Transmission Line Theory .................... 32

3.1.1.1 Arbitrary Load Impedance ............... 34

3.1.1.2 Arbitrary Characteristic Impedance .......... 38

3.1.1.3 Open-circuited Load Impedance ............ 39

3.1.1.4 Short-circuited Load Impedance ............ 40

3.1.2 Microstrip ............................. 42

3.2 TRL Calibration ............................. 45

3.2.1 Scattering, Transmission, and ABCD Parameters ....... 50

3.2.1.1 The Transmission Matrix ............... 53

3.2.1.2 The Scattering Matrix ................. 55

3.2.1.3 The ABCD Transmission Matrix ........... 56

3.2.1.4 Transformation between Parameters ......... 57

3.2.2 Deriving the Thru-Reflect-Line Calibration Method ...... 59

3.2.2.1 The Thm Measurement ................ 60

3.2.2.2 The Reflect Measurement ............... 61

3.2.2.3 The Line Measurement ................ 64

3.2.2.4 Integrating the TRL Standards Together ....... 68

vi

CHAPTER 4

Designing and Modeling the l\<licrostrip and the NIC .............. 73
4.1 Microstrip Design ............................. 73
4.2 Negative Impedance Converter Circuit Design ............. 75
4.3 De-embedding the Test Port Cables and the Microstrip ........ 88

4.3.1 The T hm Measurement Revisited ................ 89
4.3.2 The Reflect h’Ieasurement Revisited ............... 89
4.3.3 The Line Measurement Revisited ................ 90
4.4 Modeling the Test Fixture, Microstrip and NIC ............ 92
4.4.1 Modeling the Test Fixture .................... 93
4.4.2 Modeling the Microstrip ..................... 104
4.4.3 Modeling the Negative Impedance Converter .......... 115
4.5 Comparing Computational Results with Simulated Results ...... 119

CHAPTER 5

Conclusions and Future Work ........................... 164

BIBLIOGRAPHY ................................. 167

vii

LIST OF TABLES

Table 4.1 Values of Zm for negative capacitor with load capacitance of 10 pF
in C-band ............................... 117

Table 4.2 Values of Zm for negative inductor with load inductance of 0.1 nH
in C-band ............................... 118

viii

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 2.12
Figure 2.13
Figure 2.14
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10
Figure 3.11
Figure 3.12

Figure 3.13
Figure 3.14
Figure 3.15

LIST OF FIGURES

Two-port network hybrid parameter model [1] ........... 13
Hybrid parameter model for VINIC [1] ............... 14
Hybrid parameter model for CINIC [1] ............... 15
GNIC circuit [7] ........................... 16
FNIC circuit [7] ............................ 16
Open-circuit stable N IC [7] ..................... 17
Short—circuit stable N IC [7] ..................... 18
N IC terminated at both its ports [7] ................ 19
npn and pnp BJT [10] ........................ 20
The BJT in active mode viewed as two diodes [12] ........ 22
The Early effect [11] ......................... 23
Voltage-current relationship for both BJTs [10] .......... 24
n-channel and p-channel MOSFET [12] .............. 26
Voltage-current relationship for n-channel MOSFET [11] ..... 27
Test fixture diagram locating the measurement and device planes 30
Transmission line model [5] ..................... 31
Transmission line terminated with load impedance [5] ....... 35
Wave reflection and transmission at intersection of transmission

lines with different characteristic impedances [5] .......... 38
Transmission line terminated with an open circuit [5] ....... 40
Transmission line terminated with a short circuit [5] ....... 41
l\=’Iicrostrip geometry [5, 20] ..................... 42
Field lines due to microstrip [5] ................... 43
Flow chart describing the de—embedding process [19] ....... 47
Two-port S-parameter network [18] ................. 48
Two—port S-parameter network represented as a signal flow graph [5] 48
Signal flow graph representing the test fixture halves and the DUT

[16] ................................... 50
Schematic representing the test. fixture halves and the DUT [5] . . 54
Two—port ABCD-parameter network for series impedance [5] . . . 57
Schematic of Thru Standard Connection [5, 18] .......... 60

ix

Figure 3.16
Figure 3.17
Figure 3.18
Figure 3.19
Figure 3.20
Figure 4.1

Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9

Figure 4.10
Figure 4.11
Figure 4.12
Figure 4.13
Figure 4.14
Figure 4.15
Figure 4.16
Figure 4.17

Figure 4.18
Figure 4.19

Figure 4.20
Figure 4.21

Figure 4.22
Figure 4.23

Signal flow graph of Tina Standard Connection [5] ........ 61

Schematic of Reflect Standard Connection [5, 18] ......... 62
Signal flow of Reflect Standard Connection [5] ........... 63
Schematic of Line Standard Connection [5, 18] .......... 65
Schematic of Line Standard Connection [5] ............. 65

Negative impedance converter with capacitively terminated load
[7, 9] .................................. 75

Negative impedance converter with inductively terminated load [7, 9] 76

High frequency small-signal equivalent circuit of a BJT [10] . . . 77
High frequency small-signal equivalent circuit of a MOSFET [10] 77
Typical frequency response of a BJT [11] .............. 78
Ebers-Moll model of npn BJT [10] ................. 80
Basic common-emitter amplifier [24] ................ 81

Input voltage-current. relationship of common-emitter amplifier [24] 83

Output. voltage-current relationship of common-emitter amplifier

[24] ................................... 83
Transfer characteristic of common-emitter amplifier [11] ...... 84
Basic common-base amplifier [24] .................. 84
Input voltage-current relationship of common-base amplifier [24] . 86

Output voltage-current relationship of common-base amplifier [24] 86

Transfer characteristic of common-base amplifier [11] ....... 87
Test fixture modeled as a lossy transmission line [16] ....... 93
Test fixture modeled as a lossless transmission line [16] ...... 94

Magnitude of transmission coefficient for Thru standard of the test
fixture ................................. 98

Phase of transmission coefficient for Thm standard of the test fixture 99

Magnitude of reflection coefficient for Reflect standard of the test
fixture ................................. 100

Phase of reflection coefficient for Reflect standard of the test fixture 101

Magnitude of transmission coefficient for Line standard of the test
fixture ................................. 102

Phase of transmission coefficient for Line standard of the test fixture.103

Magnitude of reflection coefficient for Thru standard of the mi-
crostrip (Sonnet vs. Ansoft Designer) ................ 105

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.28

Figure 4.29

Figure 4.30

Figure 4.31

Figure 4.32

Figure 4.33
Figure 4.34
Figure 4.35

Figure 4.36

Figure 4.37

Figure 4.38

Figure 4.39

Figure 4.40

Figure 4.41

Figure 4.42

Phase of reflection coefficient for Thr'u standard of the microstrip
(Sonnet vs. Ansoft Designer)

Magnitude of reflection coefficient for Reflect standard of the mi-
crostrip (Sonnet vs. Ansoft Designer) ................

Phase of reflection coefficient for Reflect standard of the microstrip
(Sonnet vs. Ansoft Designer)

Magnitude of reflection coefficient for Line standard of the mi-
crostrip (Sonnet. vs. Ansoft Designer) ................

Phase of reflection coefficient for Line standard of the microstrip
(Sonnet vs. Ansoft Designer)

oooooooooooooooooooo

Magnitude of transmission coefficient for Thru standard of the mi-
crostrip (Sonnet vs. Ansoft Designer) ................

Phase of transmission coefficient for Thru standard of the mi-
crostrip (Sonnet vs. Ansoft Designer) ................

Magnitude of transmission coefficient for Line standard of the mi-
crostrip (Sonnet vs. Ansoft Designer) ................

Phase of transmission coefficient for Line standard of the mi-
crostrip (Sonnet vs. Ansoft Designer) ................

Negative impedance converter with capacitively terminated load .
Negative impedance converter with inductively terminated load

Magnitude of reflection coefficient of full-length microstrip vs. cas-
caded half-length microstrips (Sonnet vs. Ansoft Designer) . . . .

Phase of reflection coefficient of full-length microstrip vs. cascaded
half-length microstrips (Sonnet vs. Ansoft Designer)

Magnitude of transmission coefficient of full-length microstrip vs.
cascaded half-length microstrips (Sonnet vs. Ansoft Designer)

Phase of transmission coefficient of full-length microstrip vs. cas-
caded half—length microstrips (Sonnet vs. Ansoft Designer) . . . .

Magnitude of reflection coefficient of cascaded half-length mi-

106

107

108

109

110

111

112

113

114
116
116

121

122

123

124

crostrips with 50-9 resistor in between (Sonnet vs. Ansoft Designer) 127

Phase of reflection coefficient of cascaded half-length microstrips
with 50-9 resistor in between (Sonnet vs. Ansoft Designer) . . . .

Magnitude of transmission coefficient of cascaded half-length mi-

128

crostrips with 50-9 resistor in between (Sonnet vs. Ansoft Designer) 129

Phase of transmission coefficient of cascaded half-length mi-

crostrips with 50-0 resistor in between (Sonnet vs. Ansoft Designer) 130

xi

Figure 4.43

Figure 4.44

Figure 4.45

Figure 4.46

Figure 4.47

Figure 4.48

Figure 4.49

Figure 4.50

Figure 4.51

Figure 4.52

Figure 4.53

Figure 4.54

Figure 4.55

Figure 4.56

Figure 4.57

Figure 4.58

Magnitude of reflection coefficient of cascaded half-length mi-
crostrips with 75-9 resistor in between (Sonnet vs. Ansoft Designer) 131

Phase of reflection coefficient of cascaded half-length microstrips

with 75-9 resistor in between (Sonnet vs. Ansoft Designer) . . . . 132

Magnitude of transmission coeflicient of cascaded half-length mi-
crostrips with 75-9 resistor in between (Sonnet vs. Ansoft Designer) 133

Phase of transmission coefficient of cascaded half-length mi-
crostrips with 75-9 resistor in between (Sonnet vs. Ansoft Designer) 134

Magnitude of reflection coefficient of cascaded half-length mi-
crostrips with 100-9 resistor in between (Sonnet vs. Ansoft Designer) 135

Phase of reflection coefficient. of cascaded half-length microstrips

with 100-9 resistor in between (Sonnet. vs. Ansoft Designer) . . . 136

Magnitude of transmission coefficient of cascaded half-length mi-
crostrips with 100-9 resistor in between (Sonnet vs. Ansoft Designer) 137

Phase of transmission coefficient of cascaded half-length mi-
crostrips with 100-9 resistor in between (Sonnet vs. Ansoft Designer) 138

Magnitude of reflection coefficient of cascaded half-length mi-
crostrips with 10-pF capacitor in between (Sonnet. vs. Ansoft De-

signer) ................................. 139

Phase of reflection coefficient of cascaded half-length microstrips

with 10—pF capacitor in between (Sonnet vs. Ansoft Designer) . . 140

Magnitude of transmission coefficient of cascaded half-length mi-
crostrips with 10-pF capacitor in between (Sonnet vs. Ansoft De-

signer) ................................. 141

Phase of transmission coefficient of cascaded half—length mi—
crostrips with 10-pF capacitor in between (Sonnet vs. Ansoft De—

signer) ................................. 142

Magnitude of reflection coefficient of cascaded half-length mi-
crostrips with 0.1-nH inductor in between (Sonnet vs. Ansoft De-

signer) ................................. 143

Phase of reflection coefficient of cascaded half-length microstrips

with 0.1-nH inductor in between (Sonnet vs. Ansoft Designer) . . 144

Magnitude of transmission coefficient of cascaded half-length mi-
crostrips with 0.1-nH inductor in between (Sonnet vs. Ansoft De-

signer) ................................. 145

Phase of transmission coefficient of cascaded half-length mi-
crostrips with 0.1-nH inductor in between (Sonnet vs. Ansoft. De-

signer) ................................. 146

xii

Figure 4.59

Figure 4.60

Figure 4.61

Figure 4.62

Figure 4.63

Figure 4.64

Figure 4.65

Figure 4.66

Figure 4.67

Figure 4.68

Figure 4.69

Figure 4.70

Figure 4.71

Figure 4.72

Figure 4.73

Figure 4.74

Magnitude of reflection coefficient of cascaded half—length mi-
crostrips with —10-pF capacitor in between (Sonnet vs. Ansoft
Designer)

Phase of reflection coeflicient of cascaded half-length microstrips
with —10-pF capacitor in between (Sonnet vs. Ansoft Designer) .

Magnitude of transmission coefficient of cascaded half-length mi-
crostrips with —10-pF capacitor in between (Sonnet vs. Ansoft
Designer)

ooooooooooooooooooooooooooooooo

Phase of transmission coefficient of cascaded half—length mi-
crostrips with —10-pF capacitor in between (Sonnet vs. Ansoft
Designer)

ooooooooooooooooooooooooooooooo

Magnitude of reflection coefficient. of cascaded half-length mi-
crostrips with —0.1-nH inductor in between (Sonnet vs. Ansoft
Designer) ...............................

Phase of reflection coefficient of cascaded half-length microstrips
with —0.1-nH inductor in between (Sonnet vs. Ansoft Designer) .

Magnitude of transmission coefficient of cascaded half-length mi-
crostrips with ——0.1—nH inductor in between (Sonnet vs. Ansoft
Designer) ...............................

Phase of transmission coeflicient of cascaded half-length mi-
crostrips with -0.1—nH inductor in between (Sonnet vs. Ansoft
Designer)

ooooooooooooooooooooooooooooooo

Magnitude of reflection coefficient of parameter study of cascaded
system with capacitor varied from 10 pF to —10 pF ........

Phase of reflection coefficient of parameter study of cascaded sys-
tem with capacitor varied from 10 pF to —10 pF

Magnitude of transmission coefficient of parameter study of ca.-
caded system with capacitor varied from 10 pF to —10 pF

Phase of transmission coefficient of parameter study of cascaded
system with capacitor varied from 10 pF to -—10 pF ........

Magnitude of reflection coefficient of parameter study of cascaded
system with inductor varied from 0.1 nH to —0.1 nH

Phase of reflection coefficient of parameter study of cascaded sys-
tem with inductor varied from 0.1 nH to —0.1 nH .........

Magnitude of transmission coefficient of parameter study of cas-
caded system with inductor varied from 0.1 nH to —0.1 nH . . . .

Phase of transmission coefficient of parameter study of cascaded
system with inductor varied from 0.1 nH to —0.1 nH

xiii

147

148

149

150

151

152

153

154

156

157

158

159

161

162

KEY TO SYMBOLS AND ABBREVIATIONS

AC: Alternating Current

BJ T: Bipolar-Junction Transistor

CINIC: Current Inversion Negative Impedance Converter

DC: Direct Current

DUT: Device Under Test

EM: Electromagnetic

FET: Field-Effect Transistor

FNIC: Floating Negative Impedance Converter

GNIC: Grounded Negative Impedance Converter

JFET: Junction Field-Effect Transistor

MESFET: Met3.1—Semiconductor Field-Effect Transistor

MOSF ET: l\/Ietal—Oxide Semiconductor Field-Effect Transistor

NIC: Negative Impedance Converter

PCB: Printed Circuit Board

RF: Radio Frequency

RMS: Root-Mean—Square

xiv

SMT: Surface Mount Technology

SWR: Standing Wave Ratio

TE: Transverse Electric

TEM: Transverse Electromagnetic

TM: Transverse Magnetic

TRL: Thru-Reflect-Line

VINIC: Voltage Inversion Negative Impedance Converter

VN A: Vector Network Analyzer

VSWR: Voltage Standing Wave Ratio

XV

CHAPTER 1

INTRODUCTION

In recent decades, the electronic industry has manufactured and sold products that
use high frequency technology. More importantly, radio-frequency (RF) and circuit
design engineers have designed, built, and tested devices that can be used for mi-
crowave and RF applications. One type of high frequency device that is commonly
integrated with electrical circuit components is the microstrip transmission line. In
this thesis, a theoretical negative impedance converter will be connected to a mi-
crostrip transmission line such that the negative impedance converter acts as a load
to the microstrip. Its characteristics will be observed in C-band or the frequency
range of 6 to 8 Gigahertz (GHz).

Like other transmission lines, the performance of the microstrip is dependent on
its load impedance. The load impedance of the microstrip impacts how the a wave
traveling along the line will reflect at the load. Transmission line theory has shown
how a traditional — positive — load impedance, a short circuit, and an open circuit will
affect the wave characteristic of the microstrip. These wave characteristics include the
input impedance and the dimensions of the microstrip. However, the predictability of
the performance of a microstrip with negative load impedance is not well understood.

When conducting microstrip analysis with a negative load impedance, an elec-
tronic device must be designed, built, implemented, and integrated to the microstrip.
In this case the negative load impedance will be realized through a negative impedance
converter (NIC). Constructing a negative impedance converter requires knowledge of
Foster’s reactance theorem, its relation to NICs, the types of negative impedance con-
verters that exist, and the active and passive circuit elements needed to implement

them. Chapter 2 covers these concepts.

Chapter 3 describes the theory of designing a microstrip circuit. Topics also
included are transmission line theory and microwave network analysis. In addition,
it also shows the reader how to integrate the microstrip line with the NIC. The
process which it take is called de-embedding and it will be done by using Thru-
Reflect-Lz’ne (TRL) calibration. This is important because although microstrip lines
can be integrated with electrical circuit components, the high frequency properties if
the NIC must be inferred from measurements that include a device—under—test (DUT)
and a test fixture (e.g. the microstrip line).

Chapter 4 displays the design and operation of the type of NIC that will be used for
testing. It also has a design procedure for performing the TRL calibration standards
using a HP 8510 vector network analyzer (VN A). In addition, data will be collected
to compare computational results using circuit design software simulation programs.

These results explain how the NIC affects the microstrip’s performance.

CHAPTER 2

NEGATIVE IMPEDANCE CONVERTERS

In order to know how a. NIC works, one must know the concept of Foster’s reactance
theorem. The relation between Foster’s reactance theorem and NICs is that nega-
tive impedance converters are composed of non-Foster circuit elements. Aberle and
Loepsinger-Romak [1] state that. NICs are circuits that do not obey Foster’s reac—
tance theorem; hence, these type of circuits are classified as non-Foster circuits [1].
This chapter will describe Foster’s reactance theorem, how NICs are implemented
and operated, different models of N ICs, and the types of classifications of NICs that

exist.

2.1 Foster’s Reactance Theorem

Comprehending Foster’s reactance theorem requires deriving the complex Poynting
theorem by applying l\’Iaxwell’s equations in the frequency (phasor) domain and using
microwave network analysis. Rothwell [2] does point out, however, that if Maxwell’s
equations are applied to obtain Poynting’s theorem, it must be assumed that the
fields are linked to an energy flux that propagates at the speed of light and the
region of space that is observed is symmetric [2]. In addition, it. is postulated that
the electromagnetic (EM) fields are time-harmonic [3]. Using the complex Poynting
vector, T9) = éERFE" x 3*], where the asterisk (*) denotes the complex conjugate of
the vector field, the complex Poynting theorem can be evaluated. The % term arises
from the fact that the time-harmonic fields are the root-mean-square (RMS) values
of the instantaneous fields (e. g., peak 2 7‘ms\/2); the instantaneous fields correspond

to peak values of the EM fields [4].

2.1.1 The Complex Poynting Theorem

Using F araday’s Law and Ampere’s Law,

V x E : —J.t,u,—I-_f (21a)
V X ff = 7 +1.06? (2.1b)

the complex average power density can be evaluated.
. —->* . -—> _ -—> —> —>
By dotting (2.1a) by H and the comugate of (2.1b) by E and usmg J = J i+ JC
where J ,- 18 the unpressed current. density due to an external source and J C = (IE

is the conduction current density, the following equation is obtained

—-> ——-> -—> ——>
H*-(VXE)=—jwhH-H* (2.2a)
E-(Vxfi*)=E-7;+aE-E*+jweE-E (2.2b)

Subtracting (2.2a) from (2.2b) gives

——> —> —>

E-(VxH*)—H*-(Vx E): ~+0E-E*—jweE- *+jw,u.—ff-Ff* (2.3)

e1
5}
e1

Using the vector identity
v-(Xx§)=§.(vX74’)—X-(vx’§) (2.4)

and dividing by 2, (2.3) is reduced to

§[V-(H* x 13)] = gE' J¥+§UIE|2+5M1|H12— -Jw€|E| (M)
or
1 ——> -—>* 1—> —>* 1 —>2 _ 1 —+2 ——>2

Equation (2.5) characterizes energy conservation in point (differential) form. To
express (2.5) in integral form, it has to be integrated over a volume region of space
V and the divergence theorem needs to be applied to the left side of (2.5) to apply

the energy conservation equation to an entire region

—]]]V%[E x H*]=%§(EXH*)-fi(18=%]JI(E 7:)(11/

 

(2.6)
1 —* 2 . 1 —* 2 "* 2
_ . / ., _ _ , ,
+ 2]]fa|E|di+123]]]4[H|H| €|E| ]dv
V V
or
1 —’ 1 _’ —->* A
—§]J](E ”2;wa )-ndS
1 _+ 2 1 _) 2 _) 2 (2.7)
_ 9 _ , _
+ 2]]]0|E| dV+J-.a]]]4[H|H| e|E| [W
V V
which can be written as
P3 = Pf + Pd + jw(ll"‘m — m) where
1 —-> ——> -—+.
P3 = —5 J] (E - J 3‘ )dV 2 complex power supplied by J " (2.8a)
V
1 —-> ——>
Pf = —2- @(E x H*) - 53113 = complex power exiting V (2.8b)
S
1 —->
Pd 2 — [J10] E [2dV = real power dissipated in lossy medium (2.8c)
2 V
_ 1 —>
le = 4 [if le [QdV = time-average magnetic energy (2.8d)
_ 1 —>
I *‘e = 21— ]JI E] E [2dV = time—average electric energy (2.8c)

Equation (2.7) is referred to as the c0771ple11: Poynting theorem and is valid only

for lossless (non-dissipative) materials. The real part. of (2.7) or (2.8) characterizes

a time-average power balance while the imaginary part. of the (2.7) or (2.8) refers to

reactive power [2, 3].
2.1.2 Microwave Network Analysis

Microwave network analysis must. be applied to (2.7) or (2.8) for a N-port network
to derive Foster’s reactance theorem. For this particular case, a two-port network
will be observed to derive Foster’s reactance theorem. It is assumed that the two-
port network is bounded within a perfect electric conductor except at the terminal
port cross sections. It is also postulated that only the principal waveguide mode
propagates in each terminal due to the fact that the terminals are far away from
the network resulting in all higher-order waveguide modes decaying to an amplitude
that is negligible at the reference planes. Describing the parameters for the two-port
network will be described later.

A

Using (2.7) or (2.8) and introducing mode vectors a = —fi. x R and fi = n. x 5,
mode voltage V, and mode current I (all of which are real), the transverse electric and
magnetic fields can be expressed in terms of modal functions, voltages, and currents

by modal expansion.

N

it = 251V, (2.93)
i=1

——) N A

H, = 2 22,1, (2%)
i=1

Equation (2.9) represents modal expansions of the general electric and magnetic
fields of the waveguide terminal. By establishing orthogonality and using normal-
ization for the modal fields, (2.9) is dot multiplied by arbitrary mode vectors and

integrated over the cross section to get

if 52' ' €de = ffim '3de = 527' (2-10)
05 s

and

_.
f] E, . €,:dS = V,- (2.11a)
] 197, 3,115 = I,- (2.11b)
CS

where V,- and I,- are the modal expansion coefficients of the microwave network, CS

is designated as the closed surface of the terminal, and

- 0 ifi 74 j,
0771‘ = (2.12)

1 ifz' = j.
is the Kronecker delta function. Using (2.7), (2.10), (2.11), and the vector identity
——+ -—> ——+ ——> —> -—> —~> —> -—>
A-(BXC)=B-(CXA)=C°(AXB) (2.13)
the modal power transport for ports 1 and 2 is

Pm = ’Pf (2.14)

:2

The reason why there is no PS term in (2.14) is because there the microwave net-

work that is being observed is a source-free region. Now knowing the power entering

ports 1 and 2, the input impedance and admittance can be computed.

Pin. _ Pd +J‘2wifi771—1 _ m)

 

 

 

 

2n +J [I[2 [I]? ( )
PT P — at; W. _ “w
Yin = G+jB : m _ d J ( m. e) (2.16)

[V]:2 — IV]2
By examining equations (2.14), (2.15), and (2.16), the following observations can

be made:

1. There is no dissipated power in a lossless network resulting in no input resistance

and no input. conductance (R = 0, G = 0 and Pd 2 0).
2. The input. resistance R (or conductance G) is always positive for a lossy network.
3. At resonance, the input reactance X and input susceptance B are zero.

4. The input resistance (or conductance) is an even function of a) while the input
reactance (or susceptance) is an odd function of a; where they both depend on

frequency.

The relation for frequency is w = 27r f where w is the radial frequency and f is the
more commonly used temporal frequency.

Since the lossless case is only being considered, (2.14) will be imaginary making
V out of phase with 1* by 90 degrees (90°). The boundary conditions (ii x E> = 0
and ii X ff 2 7 for a perfect electric conductor) and the source-free Maxwell curl
equations where 7 = 0 from (2.1) associated with the microwave network are only
satisfied when E is real and ff is imaginary over the surface area S' and within the
volume region of space V. The uniqueness theorem requires this particular solution
to be the only solution and making the EM fields E and Ff in phase quadrature

within the network.

2.1.3 Verifying Foster’s Reactance Theorem

Theorem 2.1 (Foster’s Reactance Theorem) The slope of the reactance or sus-

ceptance with respect to frequency of a lossless n-port network is always positive. In

other words,

(LX (13

Proving Foster’s reactance theorem requires examination how the source-free

Maxwell curl equations change with frequency. Taking the derivative of (2.1) with

—)
respect to a) and setting J to zero, (2.1) becomes

—>

M _l a}?
——=—-' _,-,_ 2.1-.
Vx (9w JuH jay 3w ( 8a)
—> E
V X (9—11 = -—jeE —jwe—a,— (2.18b)
0a) 0w

—)
—> 8E
Dot multiplying (2.18a) by H * and the conjugate of (2.18b) by $— and subtract

the two results in

E ——> —-> —+ —> E
V (%JXH*) ——ju|H|2-—jw)uH* ~8—+Jtue * @33—
61? at?
V- —><E* =—]6[E|2—jwu—— E*+]wH*-+
8a) (9a)
v [93? x H* — 5: x E*] = —ju|H[2 —je|E|2 (2.19)

Integrating (2.19) throughout the region of space, in this case the microwave net-

work, and applying the divergence theorem to the left-hand side of (2.19) creates the

following expression
Iffv [— X H“ —:7)—:7 X E‘*]dV= @387: x fi*_§_fi_‘ x E‘*] .fidS
= ‘1' ill

V

where the right—hand side of (2.20) is proportional to the total EM energy enclosed

 

#171)? + EIEF] dV (2.20)

in the network. Performing modal expansion using (2.9) to (2.20) and using (2.13)

 

 

and the vector identity E X h = —h x 5 creates the following equation.
‘9 N A N(
8E ‘_‘)* (9H —‘** A 8( (719 1) *)
S S 221
’V A N]
.-I
_ (Z 8(577) x :(EjVJ :10] ndS
i=1 W j=1
N N
(9V ,, 01 ,, A A A
= 2:]:8wilj + 5:19] fiw X h)-ndS
i=1j=1 S
N N
31/2 a: (912'. * A A A
= :Z[8w Ij+ dw'V f][e - (—n x e)]dS
i=1j=1 ‘ CS
N N
8V2 * 012 t A A
imam—w Ilse-MS
i=1]=1 ‘ CS

 

".2: 5514‘s; 2' ('2'
8

(2.21)

Substituting (2.21) into (2.20),

N
2]]:(‘3w _Va+ V27_0£:[— _ —j Iffllle|2 + 5] El“ r7ldV_ — ]2(l—_lm " f—e) (2.22)

i=1

10

V 1
Since 8‘[Z,-n] = 3 [7] = X = E7 it can be concluded that the frequency derivatives

are

 

 

 

 

 

(1X 1 0V 4 .
a: 75.; 2W717""W€) >0
J I =constant
dB 1 <91 4 —, _,
I J w V=c0nstant
and from (2.15)
4w , —
X : mfg-(7177” — life) (2.24)
and
4w —.— .
B : W<Lie — tbyn) (2.25)

validating Foster’s reactance theorem with (2.23). The effect that results from (2.23)

is that all poles and zeros of the reactance or susceptance function are simple and

dX X B B

must alternate with respect to w [1, 3]. In addition, :1;- > Z and g: > 2; since [3]
_ |I|2 ”(IX X“ M2 ‘dB 3‘

W=——————— =——— — >0 2.26

e 8 idw w, 8 _dw + w_ ( )
_,_ IIIQ 'dX X“ M2 rdB B"

W =— —— — =——— ——— >0 2.27

m 8 de+w_ 8 _dw wd ( 7

 

 

 

 

If a network device had an impedance and admittance function that does not
obey (2.23), it is classified as a “non-Foster” element. Typical non-Foster circuit ele-
ments contain active circuit components such as capacitors and inductors to produce

negative inductors and negative capacitors, respectively. The negative capacitor has

 

an input impedance function Zm = — and the negative inductor has an input

ijL
impedance function Zm = —ijL. The negative capacitor has a load capacitance
value of C L and the negative inductor has a load inductance value of L L- An appli-

cation that is best used to visualize how negative capacitors, negative inductors and

11

other non-Foster circuits operate are negative impedance converters.

2.2 Operation of Negative Impedance Converters

A practical application for the theory behind non—Foster circuits are NICs. Ideally
it is defined as a two-port device where the input impedance of the network is the
negative impedance of the load impedance attached to the network, i.e., a capacitor or
inductor, that is usually scaled by a constant creating a negative element [7]. Negative
impedance converters were originally going to be used for reduction of resistive loss in
electrical circuits 80 years ago [1, 6]. In the 1950s. NICs were used to develop a new
type of telephone repeater that. was cost—effective [6]. Recently, NICs have been used
to eliminate resistive loss in amplifier-speaker systems and for impedance matching
for electrically small antennas [1, 9]. Although the means of how NICs are modeled
has been revolutionized, building NICs that are stable, lossless, and broadband have
been quite a challenge.

In order to model an NIC, examine Figure 2.1 where Zm = —kZL with k > 0. If

‘, = 1, the following parameters must. be set

hll = 0 (2.28)
hgg = 0 (2.29)
llvlg - hgl = I (2.30)

The types of NICs that exist fall into several classifications: the type of inversion
of the NIC (voltage inversion vs. current inversion), the reference level (grounded
vs. floating), the stability of the NIC (open-circuit. stable vs. short-circuit stable),
and the type of transistor models that. exist [bipolar-junction transistor (BJT) vs.

field-effect transistor (FET)].

12

 

 

 

 

‘ h11

+
V1 h12v2 ® 172171 O h22 v2

 

 

 

 

 

 

 

 

 

 

Figure 2.1. Two-port network hybrid parameter model [1]

2.2.1 Voltage Inversion NICs vs. Current Inversion NICs

Revisiting Figure 2.1, if 1112 = fig 2 —1, the NIC is categorized as a voltage inversion
NIC (VINIC) while the NIC is a current inversion NIC (CINIC) if h12 = h21 = 1.
Analyzing the hybrid parameter models for the VINIC shown in Figure 2.2 and the
CINIC displayed in Figure 2.3, the following relationships are established for the
VINIC

12m = v1 = —'v2 = ——*v L (2.31a)

int = i1 = —’i2 = 7L (2.31b)

Zin 2 77—777“ = ——.77L 2 —ZL (2.3IC)
2in 2L

13

and the CINIC

 

 

 

 

 

 

 

I’m 2 'U1 = U2 2 “UL (2.323)
2:,” :11 = 2:2 = —iL (2.32b)
T." ”U
Zi‘n. = fl 2 —IL‘ 2 ‘ZL. (2.320)
7in "7L
11 =11)? l2 :lout
+ +
VI — I)!" V2 11 ZL v2 V0”!
+

 

 

Figure 2.2. Hybrid parameter model for VINIC [1]

l4

 

 

 

 

i1 Z 1.in 1.2 : _lout
+ +
+
V] : vm v2 i1 ZL v2 : out

 

 

 

 

 

Figure 2.3. Hybrid parameter model for CINIC [1]

where 1:22,, is the input voltage, im is the input current, u L is the load voltage, and i L
is the load current of the NIC network. In the VIN IC, the input voltage is inverted
while the input current remains unchanged at the load. In the CINIC, the opposite
takes place. VINICs and CINICS are also known as series NICs and shunt NICs,
respectively [6].

2.2.2 Grounded NICs vs. Floating NICS

If the load impedance of the NIC network is connected to ground, it is called a
grounded NIC (GNIC). Otherwise, it is called a floating NIC (FNIC). Figure 2.4
displays the GNIC while Figure 2.5 displays both the FNIC [1].

15

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4. GNIC— circuit [7]

 

 

 

 

 

 

 

 

 

 

 

Figure 2.5. FNIC circuit. [7]

16

 

 

2.2.3 Open-circuit Stable vs. Short-circuit Stable

One of the difficult things in observing a NIC is that the computational analysis
does not. always correlate with the physical characteristics of the negative impedance
converter. The cause of this is that. NICs are only stable provided that certain con-
ditions are satisfied. Another term for this is conditional stability [1]. Even though
the predictability of implementing a stable NIC can be complex at times, one thing
is common among all NICs: they are open-circuit stable at one port and short-circuit
stable at the other port [7].

Examining Figure 2.6, if a load impedance Z L is attached to port 2, the NIC
network is stable if port 1 is an open circuit. The restriction for the NIC being

open-circuit stable is that. [Zn] 2 |Z,-,,,1|.

 

 

 

 

 

1 NIC 2 2L

 

 

 

 

 

 

 

 

 

 

 

Figure 2.6. Open-circuit. stable NIC [7]

17

Analyzing Figure 2.7, if a. load impedance is attached to port 1, the NIC network
is stable if port. 2 is short-circuited. The constraint for the NIC being short-circuit

stable is that [2L2] S [Zing].

 

 

 

 

 

 

 

ZL 1 NIC 2

 

 

 

 

 

 

 

 

 

 

 

Figure 2.7. Short—circuit stable NIC [7]

The limitations which the magnitude of the load impedance with relation to the
magnitude of the input impedance is the result of the inherent conditional stability of
NICs. The load impedances will determine how large or how the small it will be with
respect to the input impedance of the network. Figure 2.8 depicts a NIC terminated
with load impedances at both ports to help visualize the relationships of conditional

stability for both cases.

18

 

 

 

 

 

ZL 1 NIC 2 2L

 

 

 

 

 

 

 

 

 

 

 

 

 

2'1 2'2

m m

Figure 2.8. N IC terminated at both its ports [7]

2.2.4 BJT vs. FET

Negative impedance converters can be built using operational amplifiers (op—amps),
BJTs, and F ETs. However, since the NIC that will be built will operate in C—band,
op-amps will not be considered because they are generally used for low frequency
and audio frequency applications. Most NICs that are transistor-based are VINICs
because the transistors operate as dependent current sources resulting in the input
current of the N IC network to not invert at the load.

The B] T is a three terminal device (base, collector, and emitter) that is controlled
by the current. flowing into one terminal. Two types of BJTS exist: the npn transistor
and the pnp transistor; the voltage-current characteristics of each type of BJT are

the same except that the polarities for each type of transistor are different. Typically

19

npn transistors are used more often than pnp transistors due to better performance
in most. circuit applications. As a result, only npn transistors will be considered

from this point forward. Figure 2.9 displays circuit symbols for both npn and pnp

transistors.

 

Figure 2.9. npn and pnp BJT [10]

The voltage relationship for each junction of the BJ T is

VBE = VBC + VCE (npn) (2.33a)

VEB = VCB + VEC (Imp) (2331))

20

while the current relationship for each terminal of the BJ T is

IE=IB+IC (2.34)

Three circuit configurations are most common with BJTs: common-emitter,
common-base, and common-collector. Most BJT circuits are of the common-emitter
configuration where the input voltage is 7B E (base-emitter voltage) and the output.
voltage is VCE (collector-emitter voltage). The common-base configuration is used
for special applications where the input voltage is VB E and the output voltage is VBC
(base-collector voltage). The common-collector configuration is rarely, if ever, used
where the input voltage is VBC and the output voltage is VCE-

There are four modes of operation for the BJT: cutoff, inversion (reverse-active),
saturation, and active. In cutofir mode, the base-emitter and base-collector junctions
are reverse biased. N 0 current is flowing into the input terminal of the transistor no
matter what the output voltage is. In reverse-active mode, the base-emitter junction
is reverse biased and the base-collector is forward biased. There is still no current
flowing from the input terminal of the BJT. The BJT undergoes saturation when
the base-emitter junction is forward biased and the base-collector junction is reverse
biased. The output voltage is small enough for the BJT to function and the output
terminal is almost but not entirely grounded since typical voltages in this mode of
operation are typically around Vsat = 0.2 volts (V) where Vsat corresponds to the
saturation voltage of the BJT. The BJT is active if the base—emitter junction is
forward biased and the base-collector is reverse biased. In active mode, the BJT can
be visualized as two diodes sharing an anode (npn) or a cathode (pnp) as shown in
Figure 2.10 and VBE z 0.7 volts for the npn BJT and VEB z 0.7 volts for the pnp
BJT.

21

 

 

Figure 2.10. The BJT in active mode viewed as two diodes [12]

In addition, the output current goes through a small change with the base—emitter
voltage for a known input current where the output current increases with input
current. This is due to the Early effect. The slope of the voltage-current relationship

due to the Early effect shown in Figure 2.11 is

 

 

d'iout 2out
= 2.35
dvout VA ( )

22

Increasing 13

 

I
I
’
II I
I I
I ”
’ a
’ ’ '
I I v
’ ’ ”
I”’ ” v“
”"a ‘—

 

 

 

Figure 2.11. The Early effect [11]

where VA is the Early voltage of the transistor for a BJT operating in the common-
emitter configuration. Values of (2.35) typically fall anywhere between 0.01 to 0.05

milliamperes per volt (mA/V). The reciprocal of (2.35) is known as the output re—
d' 1
70777 = —). Another mode of operation that the BJT can
dvout 7‘0

undergo is avalanche breakdown. This mode will not be considered too much since

sistance of the BJT (

 

the application that will be needed in making a NIC does not. require a very large
applied voltage [10, 11]. The voltage—current relationship for each the npn and pnp

BJT is shown in Figure 2.12

23

Increasing lB
Saturationl/ Active ‘

 

 

 

 

 

 

 

 

 

 

 

 

Invert [7
Active /
’ Saturation]

 

Figure 2.12. Voltage-current relationship for both BJTs [10]

using the relation

10 = [00(eVBE/77VT — 1) z t3 F] B common-emitter configuration (2.36a)
IC = 100(eVBE/77VT — 1) z a F1 E common—base configuration (2.36b)
0’F 10. . . . , ,
where [)7 F = (—1————) z T— 18 the common-emitter dlrect current (DC) gain whlch
_ 07F B

has a typical value of approximately 100, a F is the common-base DC gain, 100 =
OFIEO where I E0 is the is the saturation current. of the base—emitter junction, 77 is
the emission coefficient that has a value of 1 or 2 depending on the physics of the
junction, VT = E} is the thermal voltage with k designated as Boltzman’s constant

(1.381 X 10‘23 Joules per degree Kelvin (J / Kl), T corresponding to the temperature in

24

degrees Kelvin, and q being the electric charge (1.602 X 1019 Coulombs) [VT z 25 mV
at 300 K (room temperature)]. The first equalities of (2.36) is better known as the
Ebers-Moll equation.

The BJT is sometimes viewed as a current-controlled current. device as is shown
in the second equalities of (2.36). However, this is not an accurate way of depicting
how a BJT operates. Modeling BJT circuits based on the common emitter current
DC gain makes it a bad circuit due to the fact that the value of 13 F can vary from 20
to 1000 depending on what. type of BJT that the designer uses, the collector current,
the collector-emitter voltage, and the temperature of the transistor [12]. Looking
at the first equality of (2.36), it can be shown that. the BJT is better modeled as
a. transconductance device. Details of this will be explained later, but it should be
noted that the BJT can be visualized better as a transconductance device rather than
a current controlled current device from (2.36) where the input voltage determines
the output current. Examples of N ICs composed of BJTs are shown in [1], [7], [8],
[9], and [13].

The FET is a four-terminal device (body, gate, drain, source) that is controlled
by the amount of voltage applied to it. Generally the body is connected directly to
the substrate. Three common types of F ETs exist: the junction field-effect transistor
(JFET), the metal-oxide semiconductor field-effect transistor (MOSF ET), and the
metal semiconductor field-effect transistor (MESFET). Most NICs that have been
built using FETs have been made from MOSFETs. Circuit symbols for each F ET
are shown in figures Figure 2.13. The NICs that are made from FETs use MOSFETS

and examples are shown in [14] and [15].

25

l'° , Jl'o

 

 

 

 

G I. G av
+ I B VDS - j ——H B :0
VGS.—II IS VSG + '8

S S
n-channel MOSFET p-channel MOSFET

Figure 2.13. n-channel and p-channel MOSFET [12]

Two types of MOSFETs exist: n-channel and p-channel MOSFETs. Both types
of MOSFETS can be classified as either enhancement-mode or depletion-mode. Unlike
BJTs, no current flows into the gate terminal of the MOSFET. The voltage current
characteristics of the n-channel and p-channel MOSFETs are the same except that
the polarities are different and the enhancement—mode and depletion-mode MOSFETS
have the same voltage-current relationship except. that the enhancement-mode MOS-
FET has a positive (n-channel) or negative (p-channel) threshold voltage VT R while
the depletion-mode MOSF ET has the opposite threshold voltage with respect to the
enhancement-mode MOSFEET. The voltage-current relationship for the n-channel

enhancement-mode MOSFET is shown in Figure 2.14.

26

i _ _ lncreasinvaS
D VDS _ VGS VTR

Linear / Saturation
I

 

 

 

 

 

 

 

 

Figure 2.14. Voltage-current relationship for n-channel MOSF ET [11]

Although the MOSFET and other F ETs can be considered as a. voltage controlled
current device, like the BJT, it is better to consider them as transconductance de-
vices. There are four modes of operation for the MOSFET: cutofi, subthreshold, linear

(ohmic), and saturation where the voltage—current relationship is defined as

f

 

0 cutoff (for VGS << VTR),
KeVGS_VTR subthreshold (for VG S < VTR),
ID = {
K[2(VGS — VTRlvDS — V1235] linear (for VGS > VTR and 0 < VDS < VGS - VTR),
K(VGS — VTRl2 saturation (for VGS > VTR and VDS 2 VGS — VTR)
t
(2.37)
. [in 60:1; VI” [Ln I’I’f _ .
where K = ———- = —C0;E— Is the conductance parameter With a. value of
2 tom L 2 L

27

around 0.5 mA/V2, pm is the electron mobility, C01- : E91 is the gate oxide-layer
OI

capacitance per unit area, cm; is the dielectric constant. of the oxide material within

the MOSFET, VGS — VT R = VDS, sat is the voltage at which the MOSF ET operates

in saturation, and the gate dimensions tar, W, and L correspond to the gate thickness,

width, and length, respectively. VG S is the gate-to—source voltage, and VDS is the

drain-to—source voltage and Figure 2.14 displays where each mode of operation is

located.

28

CHAPTER 3

DE—EMBEDDIN G TEST FIXTURES AND MICROSTRIP CIRCUITS

Attaching the N IC to a microstrip transmission line such that it acts as its load is
not as simple as one thinks. Recording measurements of the NIC will not be accurate
because the instrument will account for the microstrip, the NIC, and the coaxial
cables connected from the equipment to the system. This is a problem since most RF
circuits use printed circuit board (PCB or PC board) or surface mount technologies
(SMT). A test fixture will be needed to connect the cables to the microstrip which
is not a coaxial transmission line. Essentially, the instrument will account for the
coaxial cables, the test fixture, the microstrip, and the NIC.

Other errors will occur due to improper calibration of the instrument and other
external or internal sources that are unpredictable. Errors due to improper calibration
can be fixed and removed using proper calibration techniques. Random errors are
harder to eliminate due to their unpredictability and can result from unidentified
sources of noise within the system, equipment testing the system, or the environment
of where the data will be taken. Such errors form an important component in the
uncertainty of the measurement.

One solution that removes the effects of the coaxial cables connected from the
equipment measuring the system, the test fixture, and the microstrip from the features
of the N IC is to perform a step-by-step process called de-embedding. This process will

undergo two stages [16]:

1. In the first stage, the instrument measuring the composite system of the mi-
crostrip and the NIC moves the reference plane from the coaxial interface to

the input port of the composite system producing accurate data.
2. In the second stage, the instrument measuring the device under test (DUT)

29

moves the reference plane from the microstrip interface to the input port of the

DUT — the N IC — producing accurate data.

The type of de-embedding that. will be conducted is called two-stage de—embedding.
Figure 3.1 gives a visualization of where the measurement and device planes are lo—
cated on the overall system. The means of how the test fixture and microstrip will
be de-embedded is through T RL calibration. This calibration technique will manip—
ulate scattering parameter (S—parameter) and transmission parameter (T—parameter)

matrices to allow the user to obtain the measurements of the NIC.

 

Measurement Measurement
Plane Device Planes Plane
I I I l
I I

DUT

 

 

l
[ Coax Cable I Microstrip

 

 

I
Microstrip ] Coax Cable I
Test Fixture

 

 

 

 

Figure 3.1. Test fixture diagram locating the measurement and device planes

This chapter will describe the operation of a microstrip transmission line through a

brief overview of transmission line theory and microwave network analysis contribut—

30

ing to the microstrip. It will also show the user how to de—embed the test fixture
and microstrip using TRL calibration. Once the test fixture and microstrip are de-
embedded, the NIC will be able to be attached to the line and measurements can
be taken without the microstrip, test fixture, cables connected to the circuit, or any

other external sources of interference impeding accurate measurements of the NIC.

3.1 Microstrip Circuits

The best way to illustrate how a microstrip works is to introduce transmission line
theory. The figure shown below displays a lumped circuit model of a transmission
line. The wave that propagates along the line is a transverse—electromagnetic (TEM)
wave in the E-direction; in other words, the field components along the E—direction

are zero [5].

 

i(z,t) RAz LAz i(z+Az,t)
——> A A A nmn ——>
+ +
1
v(z, t) CAz__ GAz v(z + Az,t)

 

 

 

Figure 3.2. Transmission line model [5]

31

3.1.1 Transmission Line Theory

The quantity R (series resistance per unit length) characterizes the series resistance
due to the finite conductivity of the conducting wall of the line, C (shunt conductance
per unit length) cmresponds to the shunt. conductance due to the dielectric loss of the
material between the conducting wall, L (series inductance per unit length) signifies
the total self-inductance of the conducting wall, and C (shunt capacitance per unit
length) is the capacitance due to the conducting wall.

Constructing a loop for Figure 3.2 and applying Kirchhoff "s voltage and current

laws yields the following equations.

 

‘ Z,t
u(:, t) — iv(:: + Az. t) = RAzi(z,t) + LAzalgt‘ ) (3.1a)
i(:, t) — i(z + A2, 1‘) = Gsz(z, t) + CAz—(Egfl (3.1b)

Dividing (3.1) by A2 and taking the limit Az ——> 0 obtains the following equations.

 

 

(77—22:) = —Ri(z,t) — L237) (3.2a)
0i(z,t) _ ,. ~ v(z.t)
dz — G'z.(.'., t) C (it (3.2b)

Equation 3.2a represents the transmission line or telegraph equations in the time
domain. If the assumption is taken that the TEM waves are time-harmonic, v(z, t)

and i(z, t) can be expressed in phasor form

m, t) = n[V(z)ej“’7]

i(z, t) = §R[I(z)ejwf]

and (3.2a) reduces from a partial differential equation to an ordinary differential

32

equation such that

 

 

(l::l) = —(R + ij)I(3) (3.4a)
(12:3) 2 —(G +ij)V(3) (3.41))

 

If the propagation constant 7 = a + jl3 = \/(R + ij)(G + ij) is introduced
where a is the attenuation constant and 173 is the phase constant, the equations of

(3.4) can be solved simultaneously to acquire the wave equations for V(::) and 1(2).

 

 

' dz, = 72%;) (3.5a)
(1‘31 2 ,
(2 l = 721(2) (3.51))
The solutions for (3.5) are
V(z) = V7777: + V—el’: (3.6a)
1(2) 2 1+e—lz + I—elz (3.6b)

where e“7z characterizes a TEM wave propagating in the +3 direction while e777
characterizes a TEM wave propagating in the ——3 direction. Substituting (3.6) into

(3.4a) and solving for [(3) gives

’3:[—V+e—lz + V_e7‘7] = —(R +ij)I(z)

77 +—'z —er.-:
I: =-————-V 7 —V .' 3.7
<> “Mi e e l < >

33

reducing (3.6) into only one unknown

V(::) = V+e_lz + V_e7lz (3.8a)
1 ,- .-

1(3) = —[V+e—'i~ — V‘el‘] (3.8b)
Zn

and comparing the equations of (3.6) defines the characteristic impedance Z0 as

R +ij /R +ij W v-
0 7 0+ ij 1+ 1- 7 9)

If the transmission line is lossless, R = G = 0, '7 2 3'3 = jwv LC, (1 = 0, l3 2 (UV LC,
L

and Z = —.
0 c

If a load impedance is attached to one end of the transmission line, the T EM

wave propagating along the line will reflect back. How much the wave reflects back is

determined by what type of load impedance is at the end of the line. Three cases will

be considered: an arbitrary load, an open circuit, and a short circuit. For convenience,

it is assumed that the transmission line is lossless.
3.1.1.1 Arbitrary Load Impedance

For an arbitrary load where Z L 7E Z0 shown in Figure 3.3, the impedance at. the load
will be Z L and the voltage and current of the line will be the sum of the incident
and reflected waves expressed by (3.8) with ”y = jl3 [5]. Using this relation, the load

impedance can be found at z = 0 where

20 (3.10)

34

V<z> 1(2) 1,

 

 

 

 

 

 

 

 

 

 

Figure 3.3. Transmission line terminated with load impedance [5]

Using (3.10), the reflection coefficient F L can be computed when solving for V‘.

v— = ————ZL _ Z‘) W
ZL + Z()
V_ ZL — ZO
r = ——- = —— 3.11
L V+ ZL + ZO ( )
and (3.8) for the lossless case can be expressed as
V(z) = V+[e_jt7’: + I‘Lejfiz] (3.12a)
v+ ~ ~
1(3) = Z—[e-Jrh — FLeJ’7‘] (3.121))
0

It can be concluded that from looking at (3.12) that the waves propagating along

35

the transmission line are standing waves if an arbitrary load is attached one end of
the transmission line. If Z L = 20, the load impedance is matched and F L = 0 by
(3.11). Physically, no reflection occurs anywhere along the line.

Using (3.12), the time—average power flow along the point z anywhere along the

line will be

1 .‘ s- - .-
= gen — Pie—W + rLe~723~ — |FL|2]

1iv+l2
= 27T(1_IFL|2) (3-13)

since Pie-7237' + I‘LeflfJ73 = j2%[FLejQ-‘7z] which is purely imaginary and does not
contribute to (3.13) since Pay is purely real and constant [5]. The total power delivered

to the load is
|V+|2 _ [Vl2IFL12

P ,2
a" 220 220

(3.14)

Maximum power is delivered to the load if it is a. matched load since F = 0 no power
is delivered to the load for a short—circuited and open-circuited transmission line since
IFLI = 1-

Evaluating the standing wave ratio (SVVR) of the line requires first taking the

magnitude of (3.12a)

|V(z)| = |V+||1+ 11672777

 

= |V+||1+ |rL|ej(9—2l")| (3.15)

where l = —2 represents the distance measured from the load impedance at z = 0 and

0 is the phase of the reflection coefficient I‘ L = |FL|679. From (3.15), the maximum

(6—251) (6—231) = _1

voltage occurs when e3 = 1 and minimum voltage occurs when 6]

36

where

Vmam = IV+|(1 + [FLD (3.168.)

Vmin 2“ lV+|(1_ [FLll (3'16b)

and the SVVR. can be found from (3.16).

. ’7 a 1 + [FLI
SII'I/ R : mar :
Vmin 1 "‘ [FL]

 

 

(3.17)

Since —1 < FL < 1, the voltage SW'R (VSVVR) is a real number such that 1 S
S lVR S 00 where S l-VR = 1 corresponds to a matched load.
Referring back to (3.12) and applying (3.11) for anywhere along the line, the

reflection coefficient can be calculated at any point l.

- —J'.A3l . ,
V 6 €_Jle

FL(ll=—_f—=FLI

V+el .31 (3.18)

The input impedance of the transmission line is found by analyzing the line at
distance 2. = —l by applying (3.11).
V(—l) v+[€—j,3l + FL6_7"77[]

— —— z .. .,
I(—l) 0V+[e-Jdl _ rLe—M]
ejt’il + ZL " ZOB—jfll

 

 

 

e"jt37 + fie—7’57 _ 7 ZL + Z0
06—] 77 — FLe—j’jl .31 L 20 —J,3z
— e
ZL + 20

)647’771+ZL( _ZO)e—j,3l
0(ZL + Zole 737- (ZL - Zole 7‘77
OZL cos(/3l) + jZO sin(,z3l)
0Z0 cos(i31) + JZL sin(,<3l)
0ZL +J'Z0 tan(,8l)
0Z0 +JZL tan(l3[)

 

 

(3.19)

 

37

. . . . A .
If the length of the transmlssion lme 1s half of a wavelength (1 = 277 the Input

impedance is equal to the load impedance from (3.19), i.e., Z,” = Z L- If the length

. . . . /\ 2n/\ + 1 ,
of the transmissmn lme lS quarter of a wavelength (1 = — or T), the mput
impedance is inversely proportional to the load impedance of the transmission line.

2
To be more precise, Zm = ZQ and the transmission line is known as a quarter-wave

transformer [5].
3.1.1.2 Arbitrary Characteristic Impedance

Looking at Figure 3.4, if transmission line of characteristic impedance Z0 is feeding
a line of an arbitrary characteristic impedance Zarb 76 Z0, the input impedance seen

by the feed line is Zarb, i.e., Zm =2 Zarb and (3.11) is now expressed as

Zarb "" 20

r = ——
L Zarb + ZO

(3.20)

 

 

Figure 3.4. Wave reflection and transmission at intersection of transmission lines with

different characteristic impedances [5]

38

The incident voltage of the feed line is expressed from (3.12) for z < 0. Assuming

that no reflections are present in the feed line, the transmitted voltage of the line is

V(z) = V+Te_jflz where
2Zarb

T21 F =———
+ L Zarb+ZO

2 transmission cocfificient of the line (3.21)

The transmission and reflection coefficients are often expressed in decibels (dB)

as the insertion loss and return loss, respectively where

[L = —2Olog10(|T|) = insertion loss (3.22a)

RL 2 —2010g10(|FL]) 2 return loss (3.22b)

with the insertion loss relating to mismatched characteristic impedances of a transmis-
sion line and the return loss relating to mismatched loads at the end of a transmission
line.

3.1.1.3 Open-circuited Load Impedance

For an open—circuited load impedance, Z L = 00 as shown in Figure 3.5, the following

relation from (3.11) occurs.

r r=—=1 3%
233100 L ZL ( )

The voltage and current of the line from (3.12) become

V(z) = V+[e_j"7: + 87772] = 2V+ cos(,.~i3.:) (3.24a)
+ w M —W+ .
[(2) = ij—J’j" _ 613"] = —]—-— sin(,£3:) (3.24b)
Zn 20

and the input impedance is

Zm = —jZ() Got-(til) (3.25)

39

The standing wave ratio for an open circuit without the transmission line varies over

the line.

 

 

 

 

O‘_'—

Figure 3.5. Transmission line terminated with an open circuit [5]

3.1.1.4 Short-circuited Load Impedance
For a short—circuited load impedance, Z L = 0 as shown in Figure 3.6, the following

relation from (3.11) occurs.

1'. F =—=—1 3.26
Zimo L ( l

40

 

O
,_J
+

 

 

 

 

Figure 3.6. Transmission line terminated with a short circuit [5]

The voltage and current of the line from (3.12) become

 

 

V(z) = V+[e—fl7‘: — 67777:] = —j2V+ sin(,d::) (3.27a)
v+ A, ,~ 2v+
1(2) = [e’Jtfi‘ + 67’5"] = cos(,r3z) (3.27b)
20 Zn
and the input impedance is
Zn 2 J'Zota11(t3l) (3.28)

The standing wave ratio for a. short circuit by itself without a transmission line is

SIVR = 00 from (3.17).

41

3.1 .2 Microstrip

A microstrip is best defined as a rectangular waveguide consisting of a metallic strip
of width W and thickness t mounted on top of a dielectric substrate of height h with
a relative permittivity er sitting on a conducting plate operating in the microwave
spectrum of a frequency range of 100 MHz to 300 GHz [20]. Figure 3.7 and Figure
3.8 show a diagram of the geometry of the microstrip with the fields distributed

accordingly.

 

Figure 3.7. Microstrip geometry [5, 20]

42

 

Figure 3.8. Field lines due to microstrip [5]

Due to the presence of the dielectric substrate, the fields due to the microstrip are not
TEM waves but hybrid transverse electric-transverse magnetic (TE—TM or HE—HM)
mode because most of the fields are concentrated within the substrate between the
conducting strip and the ground plane although these are fringing fields. The phase
velocity at which the waves propagate in the substrate is less than c z 3 X 108 meters
per second (III/S), which is the phase velocity of the wave in the air [5]. Since the
thickness of the dielectric substrate is very thin and much smaller than a wavelength
A (h < A) at any frequency f, the hybrid mode can be considered as a quasi-TEM

wave allowing for very accurate approximations of the phase velocity up, propagation

43

constant 1’3, and characteristic impedance Z0 where for a given aspect ratio T
I

 

1 w
c = = — 3.29a

,/,1.1.06() 3 ( )
2 ,

k0 _ 3- = a 11050 = 5 (3.2%)
A c
A

,. z _ 2 _.__C (3.29c)

.13 = k0 , /€eff (3.2911)

where ur is the relative permeability (for this work, non-magnetic materials will be
assumed), [10 z 47r X 10‘7 Henrys per meter (H / m) is the free-space permeability,

60 m 8.854 X 10.172 F arads per meter (F / m) is the free-space permittivity,

 

 

 

. 1 1
(eff = 6": [1+ ———I—] (3.30)
l.
1 12—
+ W
and
60 8h W W
In ——, + — for — S 1
‘/6€ff VI/ 4}}. h
20 = 1207? W (331)
W W for 7? Z 1
‘/€€’ff [F +1393 + 0667111<7 +1.444):|

where eeff is the effective dielectric constant of the microstrip and 1 < 1?fo < ET. The
effective dielectric constant can be viewed as the dielectric constant of a. homogeneous

medium consisting of both the dielectric and air regions of the microstrip [5]. If a

44

specific characteristic impedance (Z0) is required, then

 

 

&4 IV
IV_ eQA—‘Z for7<2
_‘ 2 ,—1 :1 W
h — B—l—ln(2B—1)+€r [ln(B—1)+0.39—Ob H for—>2
7r er 6,— h

 

am)

where

20 (7' +1 fr _1 0.11
A = —- 0.23

60 2 + Er +1( + 6T )
_3Wn
— 2Z0\/37

 

 

 

B

can be used to design the microstrip line. Provided that the microstrip is postulated
to be a quasi-TEM line, the attenuation due to dielectric loss and conductor loss are,

respectively,

. _ hoe-,«tandfilfeff — 1)
ad _ 2 e(,ff(67~ — 1)
a - RS
7'nw

 

(3.33a)

 

mam

lwpo , . . .
where R,- = —2— is the conductor surface reslstlvlty.
a

3.2 TRL Calibration

It was previously mentioned that taking measurements of the NIC attached to the
microstrip for a calibrated instrument will not. be fairly easy because it cannot discern
the measurements due to the test fixture, the microstrip, and the DUT as a complete
system and the components individually [19]. S-parameters and T-parameters are
circuit representations in matrix form that will be considered for the de-embedding

process. In this process, the S—parameters of the NIC can be found — or de-embedded

— from the overall system. Rhonda Franklin states that the method where the test
fixture is illustrated by evaluating its S-parameters and T-parameters separately is
called unterminating [19].

Two types of de-embedding exist: one-tier de-embedding and two-tier de-
embedding. If two—tier de-embedding is used, the equipment needs to be calibrated by
standards for the VNA. The test fixture and all other external devices with respect to
the DUT are described by using the calibration standards of the DUT. The two-tier
de-embedding method uses data transfer of standard measruements, test fixture de-
scription, and DUT de—embedding. If one-tier de—embedding is used, the calibration
standards of the DUT can be found by taking measurements of the instrument after
calibration. One-tier de—embedding is preferred over two-tier de-embedding for three

reasons:

1. Propagation of measurement discrepancies are reduced creating more accurate

data of the device being tested.
2. The VN A offers de—embedded S-parameters without extra data processing.

3. The de-embedded data. can be seen directly via a better graphical illustration

on the VNA.

The only problem is that data can only be shown for one data point of the DUT [19].
The only case where two—tier de—embedding is preferred over one-tier de-embedding is
if measurements need to be repeated over and over again therefore saving time when
different test fixtures are used. Figure 3.9 shows a flow chart of how to de-embed the

test fixture and all other external devices of the DUT.

46

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

System
calibration with Test Fixture
coaxial "" S-Parameters
standards with standards
Test Fixture Characterization Measure Test Fixture
| Outputs: __, With DUT
nputs: - g
- S-Parameter eff l .
TRL Calibration _ 1—~ De-Embed Test Fixture Effects
Standards L Inputs:
- Physical - S-Parameters - DUT S- Outputs:
Length of Line of Test Fixture Parameters - De-Embedded
Connectors - Test Fixture DUT S-
Characterization Parameters
Measurements

 

 

 

 

Figure 3.9. Flow chart describing the de—embedding process [19]

With the information provided, it can be concluded that the type of de—embedding
that will be used will be one-tier de—embedding. The type of one-tier de-embedding
that will be used is called TRL calibration. This calibration technique has been used
for nearly thirty years to de-embed the DUT from the test fixture and other external
devices using T-parameters [18, 21, 22]. A two—port network can be represented by a

schematic or a signal flow graph as shown in Figure 3.10 and Figure 3.11.

47

 

 

 

 

 

V" +
1 Two-port V2
V1— Network , v2—

 

 

 

Figure 3.10. Two—port S-parameter network [18]

 

 

 

 

 

SiiY 4322
4 t a i t
v1— 812 V2—

Figure 3.11. Two—port S-parameter network represented as a signal flow graph [5]

48

Each component of the composite system can be represented as a two-port net-
work. The total network is composed of a DUT placed in between connectors A and B
where each connector is viewed as a two-port error—boa: placed in between the actual
measurement plane and the device plane of the DUT. Since two-stage de-embedding
will be used to measure the S-parameters of the DUT, connectors A and B are des-
i gnated as the test port cables of the instrument analyzing the composite system in

the first stage and characterized as two segments of the microstrip divided evenly in

the second stage. If a network is constructed as such, the TRL calibration technique

Will identify the characteristics of each connector before any measurements are taken
then it will relocate the reference planes from the test cable ports of the VN A to the
boundaries of the DUT [5, 16, 17, 18].

Figure 3.12 depicts how the cascaded system would look like using a series of
signal flow graphs. Data that needs to be taken using this calibration method are the
three standards of TRL calibration and the measurements of the composite system.

These standards are:

1. Thru — this standard takes measurements where connectors A and B are con-

nected together with no device placed in between them.

2. Reflect — this standard collects data where either connector A or connector B

is terminated with a short circuit or open circuit.

3. Line — this standard records measurements where an empty transmission line

of a given, specified length is connected in between connectors A and B.

Before implementing this method for the necessary application, it is best to under-

St and the concept of S-parameters, T-parameters, and ABCD parameters.

49

 

 

 

 

Connector A Connector B

Figure 3.12. Signal flow graph representing the test fixture halves and the DUT [16]

3.2.1 Scattering, Transmission, and ABCD Parameters

Referring back to Section 2.1.2 for a N -port network and applying it to a two-port
network, it was assumed that for a microwave network, ii X E7 = 0 over the surface
of the enclosed network except- over the cross sections of the terminal ports. If the
transverse fields are known at each port of the network, then all modal voltages
or currents of the network will be known [3]. Specified 1, or V2: at each port leads
to a specified transverse magnetic or electric field fit or Ft, respectively. As a
consequence, mixed tangential fields Etang and fitang are specified over the enclosed

network surface S and the EM fields within the network volume V established by the

50

uniqueness theorem. Since the microwave network is linear, the impedance matrix [2]

for a N—port network is defined by

 

 

 

 

 

 

N
Vlzzzijlj fori=1,2,--- ,N (3.343)
j=1
Vi
Zij = -I— (3.341))
1 1,:0, he)
' ' ' e ' 1
Vi 211 312 31N I1
V2 321 2’22 Z N 12
= ‘7 (3.34c)
_I/.N" _3Nl 3N2 ZNN_ _1N_
[V] = [:][I] (3.34d)

where V,- and I, are the mode voltage and mode current in the i777 port, zij is the
open—circuit transfer impedance, and 2,,- is the open-circuit input impedance seen

looking into port i. The inverse of (3.34) is the admittance matrix defined by

 

 

 

 

 

 

N
Ii = Zyijl/j fori=1, 2, - - ' , JV (3.353.)
j=1
I.
yij = V7 (3.35b)
J szo, kyéj
_ . - , _ -
11 gm 912 yiN V1
12 @121 3122 yN V2
= 2 (3.35c)
_IN‘ _ZN1 yN2 yNN, _V1V"
[1 l = [yllVl (335(1)

51

where [y] = [3]“1, yij is the short—circuit transfer admittance, and 9117 is the short—
circuit input admittance. To show that an isotropic network is reciprocal, the Lorentz

. . . ‘—'> "’1, “—7, “'"t . . .
reczproczty theorem Will be used. For fields (Ea, H ) and (E , H a) in a linear and

isotropic bounded medium

fifexfib—befia)-ad3=/aid-7”—fib-7g,+Eb-7a+7i7“-7$’n)dl/
s V
=j£[(?z.><Ba)-B7b—(iiXEb)-B7a]d3=0
S

(3.36)

—) A —1) .
since a source-free region is observed and ii X E a = n X E 77 = O on the conductmg
surface of the network.

If (2.9) is substituted into (3.36) and the second term is moved to the right side

of the equation

N N
Z V313 / (a, x in) - fidS = Z V513, / (a, x h.,,) . fidS (3.37a)
n=1 CS" 11:1 n
N N
Z V5215; = Z V313 (3.37b)
n=1 n=1

Substituting (3.34a) into (3.37a) verifies that isotropic networks are reciprocal

N

N N N
Z 177; Z 371111132, = Z 1% Z Zninlybn
n=1 m=1

n=1 m=1

N N N N
= Z 131 2: 37227213 = Z 1% Z znmlfi, (338a)
n=1 m=1

m=1 7221
N N
2: (3mm — 271771)].72113 : 0 (338b)

m=1 n=1

where (3.38b) is true only when zmn = an- Similarly, substituting (3.35a) into

(3.37a) yields gm." 2 ynm. If the network is lossless, zmn and ymn are purely imagi-

nary [5] For a two-port network, N = 2.

While it is good to use these matrices to derive microwave networks analysis for N—
port networks, it is better to use matrices that. describe the waveguide characteristics
for each terminal port [3]. The voltages and currents of each port can expressed as a
form of (3.8) or (3.12) for lossless networks where the first is the incident wave and

the second term is the reflected wave.

”n = VJ + Vn‘ (3.39a)

In 2 I; + I; = (V; - V7?) (3,39b)

Z};

3.2.1.1 The Transmission Matrix

The T—parameter matrix is derived from (3.39) with respect to traveling waves. These
matrices are best used for cascaded networks. For TRL calibration, T—parameter
matrices are used for the composite system consisting of both connectors A and B
and the DUT. This also applies to two-stage de-embedding. Using (3.39) for a two-

port network

V2. = T11 T12 W (3 40)
V; T21 T22 V1—
where
T11 = —2+— (3.413)
V1 V;=0
V—
T12 = -2—_ (3.41b)
1 Vl+___0
v+
T21 = V21 (3.41c)
1 ‘/'1—:0
v+
T22 = -2—_ (3.41d)
1 v1+:0

 

53

Examining Figure 3.13, the incident and reflected waves at the input of the DUT
are the reflected and incident. waves, respectively at the output of connector A. The
same applies for connector B with respect to the DUT. In general, the incident and
reflected waves at. the output of network N are the reflected and incident waves,
respectively at the input of network N + 1 [3]. For a. cascaded network, the T-

parameter matrix is the product of each individual network, i.e.,

 

 

 

 

 

[T] = [TNllTN—il ' ' ' [T2llT1l (3-42)
I I
+ ' L +
V1—" . —""V2
Connector A [ Connector B
_. I _

 

 

 

 

 

 

 

Reference Planes for DUT

Figure 3.13. Schematic representing the test fixture halves and the DUT [5]

3.2.1.2 The Scattering Matrix

The S—parameter matrix: is also derived from (3.39) with respect. to traveling waves.

S-parameters are the reported quantity from a VNA. In the case of TRL calibra-

tion, S-parameters are evaluated from the T-parameter matrices for each individual

component of the composite system. Using (3.39) for a two-port network

Vf __ S11 512
V; 521 522
where
5 V1—
11 : ‘_"
V+ ,
1 V2 :0
V1—
S12 = ‘7:
2 V,+_—_0
V—
2
S21 = F
1 V2+:0
V—
2
522 = V:
2 V,+=0
and the general expression for a N —port network is
V2.—
J V320, kg

 

(3.43)

(3.44a)

(3.44b)

(3.44c)

(3.443)

(3.45)

with 511 being the port 1 reflection coefficient with port 2 matched, 512 designated as

the transmission coefficient from port 2 to port 1, 821 corresponding to the transmis-

sion coefficient from port 1 to port 2, 822 representing the port 2 reflection coefficient

with port 1 matched, Si,- known as the port 2' reflection coefficient with all other

ports are terminated with matched loads, and Sij being the transmission coefiicient

from port j to port 3' and all other ports are terminated with matched loads [5]. If

55

t

the microwave network is symmetric, then [S] = [S], where [S]t is the transposed

S-parameter matrix. If the network is lossless, [S]t[S]* = [U] where [S]*

is the unitary matrix and [U] is the identity matrir where

10- 0

01
[U]:

0 0

_00- 1.

 

 

3.2.1.3 The ABCD Transmission Matrix

([Sl‘T1

(3.46)

The AB CD-parameter transmission matria: is closely related to (3.12) with respect to

total voltages and currents. Like T—parameter matrices, these matrices are best used

for cascaded networks. ABCD-parameter matrices will not be needed for the TRL

calibration method, but they will be needed to visualize the NIC connected to the

microstrip such that the parameter Z shown in Figure 3.14 is the series impedance of

the two-port network corresponding to the input impedance of the NIC. Using (3.39)

for a two-port network

V1 A B V2

11 CD 12

where

 

 

V2 [2:0 V1

B = fl — V1 = Z
12 V2_0 Vl/Z
I

C: 71 =0

D_a _a=.
12 V2:0 11

(3.47)

(3.483.)

(3.48b)

(3.48c)

(3.48d)

 

 

 

 

 

 

 

Figure 3.14. Two-port ABCD-parameter network for series impedance [5]

In order to analyze two-port networks based on ABCD-parameters, the current is
flowing out of port 2 where Figure 3.10 has current flowing into port 2. In general,
the input and output voltages and currents at the output of network N are the input
and output voltages and currents waves at the input of network N + 1. As was
done for T—parameters, the ABCD-parameter matrix for a cascaded network is the
product of the individual networks of the system being analyzed. It can be shown

that AD - BC 2 1 for a. reciprocal network [5].
3.2.1.4 Transformation between Parameters

Expressing S-parameters in terms of impedance parameters yields

[S] = [z — 2:0][3 + 20]—1 (3.49)

where

0 :02

where 301 and 302 are the characteristic impedances for ports 1 and 2, respectively,
and for a two-port network

S11 512 1 (211 — Zo)(322 + 20) - 212321 2212Z0

— AZ
S21 S22 232120 (311 + Zo)(222 - Zo) — 212321
(3.50)
where AZ 2 (:11 + Z0)(::22 + Z0) — 212221. Expressing T-parameters in terms of

S-parameters for a two—port network yields

T11 T12 1 S12521-511522 511

[T] = = —S—— ‘ (3.51)
T21 T22 21 —322 1
and performing the inverse relationship gives
S11 S12 1 T12 T11T22 — T12T21
521 522 22 1 —T21
Using (3.52), reciprocity for a two—port network holds when
. . 2502
T11T22 - T12T21 = a = 1 or 812 = 521 (3.53)
Expressing ABCD-parameters in terms of S-parameters yields
A B _ 1 (1 +511)(1 —S12)+512521 Zo[(1+311)(1+512) —3125211
C D 2521 [(1— 511)(1— 512)/ZO - 512321] (1 — 511)(1- S12) + 312521
(3.54)

58

and performing the inverse relationship gives

 

A+—ZIi—CZO—D 2(AD—BC)
0
B
311 512 _ 2 —A+Z—CZO+D
— B
521 522 A+—+CZO+D

Zn

3.2.2 Deriving the Thru—Reflect-Line Calibration Method

With the knowledge obtained for the scattering, transmission, and the ABCD pa-

rameters, signal flow graphs can be used to derive a set of equations for each TRL

calibration standard. This technique has been used by Barba. via l\’la.tthews and Song

and Pozar [5, 18, 21]. This section of the thesis is essentially a recasting of material

in [18] and [21]. Using (3.51) and (3.52), the scattering and transmission parameters

can be evaluated for each individual component of the composite system using the

following formula.

[Tcompl = [TA] ' [TDUTl ' [TB]

where

Tcompll Tmmpl2 , ,
[Tcomp] = = T-matrix of composue system

Team [)2 1 Tcom p22

_

-

TA11 TA12
[T A] = = T-matrix of connector A

LTA‘ZI 7:422

p

T811 T812
[T3] = = T-matrix of connector B

LTBQI T822

TDUTn TDUT12
[TDUTl =

 

TDUT21 TDUT22

: [TA]—1 - [Tcomp] - [TB]_1 2 T—matrix of the DUT

59

(3.56)

(3.57a)

(3.57b)

(3.57c)

(3.57d)

3.2.2.1 The Thru Measurement

The Thru measurement is realized by having connectors A and B directly connected
to each other as shown by a. schematic in Figure 3.15 and a signal flow graph in Figure

3.16. Using (3.51), the T-parameter matrix for the Thra standard is

T T
[TTl = T11 T12 =[TAl'lTTHRUl'lTBl= [TAl'lTBl (3-58)
TT21 TT22

where the ideal transmission matrix for the Thru standard is the identity matrix
given by

1 0
[TTHRUI = (3.59)
0 1

 

 

 

 

 

 

 

 

 

 

I

I
+ - +
_ Connector A : Connector B _

Reference Plane for DUT

Figure 3.15. Schematic of Thra Standard Connection [5, 18]

60

TH RU

 

 

 

 

 

 

 

 

+ _ 821 1 812 +
_ S11" 0822 V822 A 311 _
v < t < < < v
1 S12 1 $21 2
Connector A Connector B

Figure 3.16. Signal flow graph of Thru Standard Connection [5]

3.2.2.2 The Reflect Measurement

The Reflect measurement is realized by placing a short circuit. or open circuit at one
end of connector A or B as shown by a schematic in Figure 3.17 and a signal flow

graph in Figure 3.18.

61

 

 

I r
+ i i +
1.. ConnectorA EFL { Connector 8 2_
V "_" I
1 i FL '

 

 

 

 

 

Reference Planes for DUT

Figure 3.17. Schematic of Reflect Standard Connection [5, 18]

The reflection coefficient seen by the V NA will be a function of the reflection coefficient

F L and the S—parameters of the connector by the following equation

T.412 TAll ‘FL
SAl‘ZSAQIFL _ TA22 TA22

 

 

 

S = S + — 3.60
R11 A11 1_ 5A22FL 1+ TA21 TL ( )
TA22
"T
and using (3.60), the ratio All is found.
A22
SRll _ m

TAll : _1_ . TA22 (3 61)

 

 

62

REFLECT

 

 

 

 

 

 

 

 

 

v: 83‘ 842 = V;
_ S11 81:2[SQL Q" "522 AS11 _
V = < < v
1 821 2
ConnectorA Connector B

Figure 3.18. Signal flow of Reflect Standard Connection [5]

The same method applies to connector B using the same reflection coefficient F L-

 

T T
B21 + 1311,1111

 

 

53125321FL _TB‘22 T322
R22 322 1-5311PL 1 TB__1_2 FL ( )
7322

 

T
Using (3. 62) to find the ratio 811

 

TB22
TB21
T 1 8322 + T—
Bll _ B22 4
. _ _. (3.63)
T822 FL 1 + —-—812 SRO‘)
T311 "

63

then taking the product of (3.61) and (3.63) yields

T412) ( T312 >
s — ‘ 1+——= -s
TAU _TBll _ ( R11 TA22 T811 R22

TA22 T322 _ TB21 T312
51222 + - ——
TB22 T322

3.2.2.3 The Line Measurement

 

 

 

(3.64)

 

The Line measurement is realized by having an empty transmission line of known
length 6 placed in between connectors A and B as shown by a schematic and in
Figure 3.19 and a signal flow graph in Figure 3.20.. The S-parameters for this system

will be measured and using (3.51), the T-parameter matrix for the Line standard is

T T
[TLl= L11 L12 =lTAl'lTLINEl'lTBl (3-65)

TL21 TL22

where the ideal transmission matrix for the Line standard is a phase-shifted identity

matrix given by

e"7€ 0
lTLINEl = . (3.66)
0 eV
and (3.65) becomes
[TL] = [TAl ° [TLINEl ' [TAl—l ° [TTl (3-67)

64

 

 

 

 

 

 

 

 

 

 

. +
Vl+—'J I 5 —"V2
ConnectorA : 6‘11: ConnectorB _
V1 i 3 ._v2
I

Reference Planes for DUT

Figure 3.19. Schematic of Line Standard Connection [5, 18]

 

 

 

 

 

 

 

 

 

LINE
8 ‘1’ s
V:- = >21 e: >12 > V;
_ 311' “322 V S22 A 811 __
V < < < t < ‘V
l 812 e” . 821 .2
Connector A Connector B

Figure 3.20. Schematic of Line Standard Connection [5]

65

If [TLT] is defined by

_ TLT11 TLT12
[TLTl = lTLl ' lTTl 1 =
TLT21 TLT22

_ 1 TL11TT22 — TL12TT21 TL12TT11 — TLllTT12 (3 68)

T
l Tl TL‘21TT'2‘2 — TL22TT21 TLQQTTll — TLQITTIQ

where lTTl is the determinant of [TT]. Using (3.68), (3.67) can now be expressed as

lTLTl ' [TAl = [TAl ' lTLINEl (3-69)

and expressing each component of (3.69) in linear form,

TLT11TAn + TL:r12TA21 = Time—75 (3-703)
TLT11TAl2 + TL:r12TA22 = T141128“ (3-70b)
TLT21TA11 + TLT22TA21 = 721216—76 (3700)
TLT21TA12 + TLT22TA22 = 73122676 (370d)

then taking the ratios of (3.70a) to (3.70c) and (3.70b) to (3.70d), a. pair of quadratic

66

equations are evaluated such that they have the same solutions.

TA1 2 TA11
TLT11(—"T 1 ) + (TLT22 — TLT11)_T — TLT12 = 0
A21 A21
(3.71a)
. T 12 2 TA12
TLT21(—TA ) + (TLT22 - TLrnl—T - TLT12 = 0
A22 A22
(3.71b)

 

TA11TA12= 1
TA21’TA22 2TLT21

 

 

[TLTn - TL:r22 i «Tum — TLTn)2 + 4(TLT21TLT12ll

(3.71c)

The roots of (3.71c) are determined by choosing a value of FL. If a metallic

plate is placed at the load of the microstrip and the experimentalist is aware that the

- . - . T
reflection coefficrent seen by the VNA is S A11 2 TA12

 

, it can be shown that this root
has a larger magnitude and thus is assigned the reflection coefficient.

Referring back to (3.58) and solving for [TB]
[TBl = lTAl—l ' [TTl (3-72)
expanding (3.72) in matrix form

T311 T312 _ 1 TT11TA22-TT21TA12 TTIZTA22“TT22TA12 (3 738)

T .
TB21 TB22 I Al TT21TA11_TT11TA21 TT22TA11—TT12TA21

67

and taking the ratios of T321 to T322 and T312 to T311 yielding

 

 

 

 

 

 

 

. TT21 " ——1‘ TIM
T1321 2 TA11 (3 74a)
T862 TA21 '
‘ T:r22 — —T 'TT12
A11
TA12
T312 _ TT12 __ TA22 TT22 3 74b
T15311 _ TAl? ( i )
TT11 — #77— ’TT21
A22
and taking the products of (3.74a) and (3.74b) to get
T
TA TB TTll — TAT—Z ° TT21
11 _ 11 _ A22 (375)

TA22 TB21 TT22 _ T_A21 'TT12
TA11

3.2.2.4 Integrating the TRL Standards Together

Once all the measurements for each standard of the TRL calibration technique have
been computed and defined, the results are put together and the S-parameters of

the DUT are evaluated from (3.57d). Using both (3.64) and (3.75) creates a new

 

 

 

T
expression for A11
A22
TA12 TA12 T1312
T (TTII — T— 'TT21) (31211 — ‘77— 1 + 7“— ' 31222
A11 2 i A22 . A22 B11

(3.76)

 

T T T T
A22 \ (TT22 — T—AQ-l 'TT12) (51122 + 54%) (1+ 751—21 ' 31211)
A11 B22 A11

The sign of the ratio of (3.76) is selected with the condition that. the value of F L
produces the same numerical value for (3.61) and (3.76). Another relation found from

(3.76) is
TA12
T- — — -T -1
TB11 T11 TA22 T21 (TAU)
T _ TA21 T
B22 TT22 _ §;__ . TT12 A22
A11

 

(3.77)

68

and using (3.51) to produce the S-parameters of connector A with the reflection

coefficients being

 

 

 

 

 

T
5.411 = ”“2 (3.78a)
TA22
T T . T _
TA22 TA11 TA22
and referring to (3.52) and calculating the product S A125 A21
TA11 [ T412 TA‘H]
5, 5 = 1 _ ‘ . 7 3.79
A12 A21 TA22 TA22 TAn ( )

For connector B. (3.51) and (3.73) can be used to find its reflection coefficients at for

ports 1 and 2

 

 

 

 

 

T _ TA12 TA11 -T
T T12 T ' T T22
51311 = ——-B 12 = A11 A22 (3.80a)
T322 TAll . TT22 __ TA21 'TT12
TA22 TA22
TA11 TA21
TAO? °TT21 — m ' TT11
S = “ 3.80b
822 TA11 'TT _ TA21,T ( )
TA22 22 TA22 T12

Calculating the determinant of lSBl and using (3.80) yields the product 53125321

_TBl2ST21 _ T811 T3125121
33225322 TB22 SB22SB22

T
33123321 = 54% + SBIISB‘Z‘Z (3-81b)

‘-

 

lSBl = 313115322 - 331231321 = (53-813)

W’ hen separating the off-diagonal elements of [S Al and [33], two postulates are
made: (1) The determinants of the T-parameter matrices for the Tina and Line

standards are equal to each other and (2) the determinants for the T-parameter

69

matrices of the connectors A and B are equal to each other

 

 

 

5.412 S812
A21 B21
lTAl = ITBI (3-83)
. _. . . SA12
Now usmg the assumptions made from above v1a (3.82) and (3.83), the ratios S
A21
and 8312 are found
5321

5 412 51312 ST12
—‘ = —“ : —-—- 3.84
SA21 5321 ST21 ( )

and evaluating 5,412, 5,421, S312, and 5321 yield

 

S
SA12= SA12'SA21 2% (385a)

ST21

S _ SA12SA21

S
SBi2 = S1.312 ' 3321‘] TE? (3-850)

SB 2 S3125321
1 S812

(3.85b)

 

(3.85d)

Now everything needed to evaluate the elements of [TDUTl from (3.57) can be

expressed using (3.51) and (3.52) and applying it to [S Al and [83] where the T-

70

parameters of the DUT are

TA22(TcompllTB22 - TcomplQTBQl) — TA12(Tcomp21TB22 - Tcomp22TB21)

 

 

 

 

TDUT11 =
(TA11TA22 — TA12TA21)(T311TB22 — TBi2TB21)
(3.86a)
TDUT12 = TA22(T('0mp12TBll “ TcompllTBl2) “ TAl2lemp22TB11 _‘ Tcomp‘ZlTB12)
(TA11TA22 - TA12TA21)(TB11TB22 - T312T321)
(3.86b)
_ TA11(Tc0-mp21TB22 - Tcomp22TB21) " TA21(Tcomp11TB22 — Tcomp12TB21)
TDUT21 — .
(TA11TA22 — TA12TA21)(TB11T322 — T312TB21)
(3.86c)
T _ TA11(Tcomp22TBll " Tcomp21TB12) "‘ TA21(Tc0mp12TBQ2 " TcompllTBl2)
DUT22 —

(TA11TA22 — TA12TA21)(T311T322 — T312TB21)
(3.86d)

and the S-parameters of the DUT are

SB11(SA11 Scomp22 — IScompl) + (Scampll - SA11)|SB|

 

 

 

 

s =
DUT“ 8311 . (Scampgz - lSAl — 5,422 . Iscompn + (Scampusm — ISAl) - ISBI
(3.8721)
SD ___ ’Scomp12SA2ISB21
UT12 SBll ‘ (Sc0mp22 ' ISAI _ 8A2? ' lScompl) + (500772.111131422 — ISAI) ' ISBI
(3.87b)
SDUT21 = ‘Scomp213A12SBIQ
SBll ' (Scamp22 ' ISAI _ SA22 ' lScompl) + (ScampllsA22 — lSAl) ' ISBI
(3.87c)
SDUT22 = 5322(5A22Scomp11 - ISAl) + Scamp22|SAl — 5A22|Scomp|
SBll ‘ (5007711922 ' lSAl "’ 5/122 ‘lScompl)+(Sc0mp118A22 " ISAI) ' ISBl

(3.87d)

Considering that the de-embedding process will have two stages, here is how the

process will be conducted for the test. fixture and the microstrip [16]:

1. Construct a simulated model of the test. fixture using S-parameter and T-

71

parameter matrices to represent connectors A and B of the test fixture.

. Collect the data for the S-parameters of the composite system of the test fixture,

microstrip. and NIC and convert them to T—parameters.

. Perform TRL calibration and apply (3.337(1) and (3.86) to the measured T-
parameters and convert, them to S-parameters using (3.87) to remove the effects
of the S—parameters due to the test fixture to obtain the S—parameters of the

integrated system of the Illicrostrip and the NIC.

. Construct a simulated model of the microstrip using S-parameter and T—

parameter matrices to represent. two segments of the microstrip divided evenly.

. Collect the data for the S—parameters of the integrated system of the microstrip

and N IC and convert them to T-parameters.

. Perform TRL calibration and apply (3.57d) and (3.86) to the measured T-
parameters and convert them to S~parameters using (3.87) to remove the effects

of the S-parameters due to the microstrip to obtain the S-parameters of the NIC

(DUT).

72

CHAPTER 4

DESIGNING AND MODELING THE MICROSTRIP AND THE NIC

All the parameters have been gathered to construct the components of the composite
system for two—stage de-embedding. These parameters must be chosen such that
the most optimal results are produced. The microstrip will be constructed by the
properties of a Rogers RT / Duroid 5870 high frequency laminate. The N IC that will be
chosen for this particular project is a grounded, open—circuit stable VINIC consisting
of BJTs. The type of transistor that will be used is the RF npn silicon-germanium
(Si-Ge) NESG204619 BJ T. Unfortunately there is no way to create a simulated model
of the N IC using this transistor with the software available. Therefore, the RF npn
silicon (Si) NESG2107M33 transistor will be considered.

This chapter depicts the dimensions of the microstrip, analyzes the NIC circuit,
and shows how to perform one—tier de—embedding using the TRL calibration technique
given the parameters of the system that will be observed for both stages of the de-
embedding process. It also compares and contrasts simulated designs of the NIC with

computational analysis the composite system and the DUT via TRL calibration.

4.1 Microstrip Design

Referring back to Section 3.1.2, for a Rogers RT / Duroid 5870 high frequency laminate
with a known characteristic impedance Z0 2 50 Ohms (S2) and a dielectric constant
6r 2 2.33, the aspect ratio can be calculated. The initial guess is to use the case

W W
for L:- < 2. From (3.32), A = 1.18601 and —h— : 3.00401. Therefore 71— > 2,

B = 7.75913 and g = 2.97028 with the effective dielectric constant eeff = 2.40665.
If the thickness of the dielectric substrate is 0.7874 millimeters (0.031 inches), the

width of the conducting strip is 2.3388 millimeters (mm).

73

The electrical length. 6, of the microstrip that is a quarter of a wavelength long

operating at 7 GHZ is

27r A 7r
=36=k ,. €=QOO=———=— 4.1'
95 . 0%fo A 4 2 ( <1)
2 2
k0 2 3‘7: 2 w p.060 2 fl = 146.709 radians per meter (rad/1n) (4.11))
c
90°——7r O
E 2 J9— : 6.90171 mm (4.1c)

A70, “Eff

These measurements are only estimates of the what the dimensions of the mi-
crostrip should be. Using a program called LineGauge by Zeland Software, more
precise dimensions of the microstrip can be made using h. = 0.7874 mm, t = 0.004
mm, Z0 2 90 Q, o = 900, Er = 2.33, f = 7 GHZ, W = 2.33227 nun, E = 7.6354
mm, Eeff = 1.96908, and A = 30.5213 mm. With the given width and thickness
of the microstrip, the aspect ratio is now % = 2.96199. The reason for choosing
the operating frequency to be 7 GHZ is that this frequency lies in the middle of the
frequency bandwidth of interest.

W hile these parameters produce the most optimal results, one should consider
making the microstrip longer. In this case, the length of the microstrip will be 150
mm (6 = 150 mm) making the microstrip nearly two and a half wavelengths long. It
is necessary to have a microstrip long enough for use of practical applications as the
length of f = 7.6354 mm is too small.

The attenuation constants due to dielectric and conductor loss can also be calcu—
lated knowing that the loss tangent. of the laminate is tan(6) = 0.0012 and the surface
resistivity is RS 2 2 x 108 megaohms (Mil), (3.33a) and (3.33b) will have values of

(rd = 0.126018 nepers per meter (Np/In) and ac = 1.71501 X 1015 Np/m, respectively.

4.2 Negative Impedance Converter Circuit Design

A schematic of the NIC is displayed in Figure 4.1 and Figure 4.2. In this design,

the NIC is composed of BJTs. For high frequency applications, BJTs are preferred

20m . . .
—, which is proportional to
in

the unity-gain frequency fT, is significantly much larger for BJTs than MOSFETS

over MOSFETS because the transconductance gm =

making BJTs more useful for high frequency amplifier design [12]. For this thesis,
it is necessary to have a higher unity-gain frequency to have the BJT to maintain
its expected performance when operating in the active region and the MOSFET to
maintain its anticipated performance when operating in saturation. Otherwise, the

transistors will not function properly.

 

 

 

 

 

 

 

Figure 4.1. Negative impedance converter with capacitively terminated load [7, 9]

 

Q1 ’”

 

 

 

 

 

 

 

 

Figure 4.2. Negative impedance converter with inductively terminated load [7, 9]

The unity-gain frequency

9 m

 

BJT
fr 2 27réCBC + CBE) (4.2)
_m_ y T
QWCOX IOSFE

is the frequency where the small-signal current gain is equal to one or

gm 7‘7r

3 u} z ,
A“) 1+J’W'7'7rchE'l‘CBC)

 

= 1 (4.3)

where C BC is the base-collector junction capacitance, CB E is the base-emitter ca-
pacitance, and CO X = Cm: - Areagate is the gate oxide-layer capacitance [10, 11, 12].
These parameters are found using alternating current (AC) small-signal equivalent
circuit analysis. The small-signal models for the BJT and MOSFET are shown in

Figure 4.3 and Figure 4.4.

76

 

 

 

 

Ea CE

Figure 4.3. High frequency small-signal equivalent circuit of a BJT [10]

 

G. F H 10

CG :: gd
ngBE
3 F O 1 S

 

 

Figure 4.4. High frequency small-signal equivalent circuit. of a MOSF ET [10]

77

The unity-gain frequency represents the largest frequency that. a transistor can
operate to produce gain [11]. The given specifications require that the unity-gain
frequency be much larger than 8 GHz. Since BJTs will be used to construct the NIC,
,8F(w) >> 1 in C—Band. It should also noted that frequency bandwidth of interest must
be greater than the break frequency 1123 (e.g., f B < C-Band < fT). This frequency is
the point where the small-signal current gain falls by 3 dB. Figure 4.5 illustrates the

trend for a transistor as the frequency increases.

18F (50)]0 -31dB
gmrzr T

-20 dB/dec

 

(03 ca} 0)

 

Figure 4.5. Typical frequency response of a BJT [11]

The transconductance for a BJT operating in the active region is

__ (910 813 N 81C ~ qIC 10
gm. — OVBE (“)VCE , N aVBE N kT VT
VCE VBE VCE

, =c0nstant =c0nstant

 

 

=c0n.stant

78

and the transconductance for a MOSF ET operating in saturation is

BID

9771 = —.
()V
GS VD 3 :con stant

= 21\’(VGS - VTR) = QVKID (4.5)

It was 1')1‘e\~’iously mentioned in Section 2.2.4 that NICs composed of BJTs are
VlNICs because BJTS are trai'isconductance devices. In other words, transistors are
dependent current sources whose the input voltage controls the output current. It
should also be noted that the NIC is grounded and open-circuit stable. Achieving
the required specifications of the N IC make it necessary to favor the GNIC over the
FNIC since most NIC designs are GNICs and this type of classification of NIC will
be considered for this particular design. The NIC is open-circuit stable because the
port that is not terminated with the load impedance is an open—circuit.

Revisiting Figure 4.1 and Figure 4.2, circuit. analysis will show that the NIC is
a VINIC. The transistors Q1 and Q2 have different configurations. Transistor Q1
is operating as a common-base current follower and transistor Q2 is operating as a
common-emitter inverter. The load impedance Z L is connected at the output of Q1
and is connected in between the base and emitter resistor R1 of Q2. It will be shown
that Vout = VZL :2 —V.,-,,, the voltage at the base of Q1 is in phase with Vm, and the
input impedance is equal to the negative of the load impedance, i.e, Zm = —ZL.

Looking back to (2.36), the Ebers—Moll equations will be expanded to observe all

modes of operation of the BJT mentioned in Section 2.2.4 using Figure 4.6

IE = IEofeng/m’} — 1) — O'RlcofevB‘i/"VT — 1) (4-68)
10 = aFIEoeVBE/"VT — 1) — 100(6VBC/"VT — 1) (4.61)
13 = (1 — aF)IEo<eVBE/"VT — 1) + <1 — amlcoeVBc/"VT — 1) (41%)

79

 

 

 

 

Figure 4.6. Ebers-Moll model of npn BJT [10]

where I E0 is the fomrard saturation current of the base—emitter junction, 1C0 =
(11:11.30 is the reverse saturation current of the base-collector junction, up is

the forward common-base DC gain, a R is the reverse common—base DC gain,

, a I , . . .
(3F 2 ———F——— x —-Q is the fomzard-actwe common—emitter DC gain, and ,8}; =
(1 — 0F) I B
OR I E . . . . .
——-— z —— is the reverse-active common-emitter direct. current (DC) gain
(1 - 0’1?) I B

[10, 11]. The transconductance characteristics can be observed for the common—base

current follower Q1

Ii-nQi = IEQ1 = IE0(6VBEQl/'M — 1) — Q’RICO(€VBCQ‘/T]VT - 1) (47a)

Iothl = ICQI = O‘FIEO — (1 — aFanflcofevBCQl/"VT - 1) (“W

80

and the common—emitter inverter Q2 [10]

111102 = IBQ2
= (1 — O'FllEofi’IVBEQ?MVT — 1) + (1 — 0'12)Ico(€(VBCQ2IVCEQMWT — 1)

(4.8a)

Ioth2 : ICQ-z

_ (O'FIEO — [COG—VCEQz/"W)lIBQ2 +(1— OFVEO + (1 — G'RNCOl I I

_ —v /er + co—GF E0
(1 — aFllEo + (1 - (112)1006 C592 7 T

= 100(1 — e—W’wz/"VT)[IBQ. + (1 — OFNEO + (1 — G‘Rflcol

(1 — O‘FVEO + (1 — aRflCOe—VCEQJ’WT

 

 

(4.8b)

Since the active mode of the BJT is only being considered, the input-output
transfer characteristics must be observed for all modes of operation to determine the

range of which transistors Q1 and (22 need to operate in the active region.

 

 

out

in

 

Figure 4.7. Basic common—emitter amplifier [24]

If only Q2 is observed, then it is an inverter with an emitter resistor R1 which is a

common-emitter amplifier as shown in Figure 4.7. Considering AC sources. if Q2 is

81

operating in the active region,

. 'v 1 t - VBE .
2E0, = 0“ R1 Q2 (4.9a)

 

"out _ Z'EQ2 _ ’Uout — VBEQ2

 

 

(BQZ, = iCQ1 — igut = Q'Fim — 7; — 5F + 1 — (BF + URI (4.91))
ico2 = firinoz = Q'F'iEQQ = 3%(1’0111 - VBEQQ) (4-90)
113-an = Pout (49d)
1’0th, = Verso.2 = VCQ.2 — iCQle
= VCQ2 —- (1’31: +1)iBQ2R1
= VBQl - (vout - VBEQQ) = —vout + (VBQ1 + VBEQ2) (4-96)

while transfer characteristics for the saturation region are

vothQJni-n : VCEQ2,sat Z VBQI _ (18F + lfiBQZmzale

= VBQ1 — ("voutflnar _ VBEQZ) = ‘L’outjnaa: + (VBQl + VBEQz) (4-103')

 

vout,ma:i: = (VBQ1 + VBEQZ) ‘ VCEQ2,sat (4.10b)
. . Uoutnnaa: _ VB E .
zBQz.sat = 'Z'Bszanar = (23F + 1)R1 Q2 (410C)
. . , . aF
ZCQ2,sat : zCannar : HFZBQ2.SU,t = Efvoutrmar _ VBEQ2) (4-10d)

The inverter is in cutofir mode when ”out < VBEQ2 and 730th2 = VBQ 1. The voltage-
current relationship at the input and output of the inverter is shown in Figure 4.8 and
Figure 4.9, respectively. The input-output voltage relationship of the inverter is shown
in Figure 4.10 where the applied input and output. voltage ranges needed for Q2 to
operate in the active region are VBEQ2 < 17,-"Q2 = "00m < (VBQ1 +VB EQ 2) —VC EQQ. sat

and —U0ut,ma$ + (VBQ1 + VBEQZ) < ’00.th2 = VCEQ2 < VBQ,» respectively.

VCE > VBE (on)

n-
I

VBE (on) VBE

 

Figure 4.8. Input vcfltage-current relationship of common-emitter amplifier [24]

Increasing (B
It

 

 

 

 

 

 

 

 

Cutoff IS VCE

Figure 4.9. Output voltage-current. relationship of common—emitter amplifier [24]

83

 

 

V011! A
__ Cutoff
CC
V
slope = ——0“’
v.
Active m
Saturation
vsat -— , . A
VBE vmax Vin

 

Figure 4.10. Transfer characteristic of common-emitter amplifier [11]

If Q2 and R1 are removed from the NIC circuit, the circuit will be a current

 

. . . . 1
follower With a base reSistor R2 and a load capamtor CL of impedance Z L = , C

.70" L
or a load inductor L L of impedance Z L = ij L which is a common-base amplifier as

shown in Figure 4.11.

 

Figure 4.11. Basic common-base amplifier [24]

84

Considering AC sources, if the BJT is operating in the active region,

 

 

iEQl = iin (4.1161)
. I'm . . . VBQ1

'. = :1, a z '. 4.11b
’BQi ,3F +1 10a + 1.12, Ice. + R2 ( )

. 1’3}? , vout _ VBEQ 'Uout
CQI flF + 1 in F in BQ2 out (1‘3F + URI ZL

 

(4.11c)

vino. = m = V11. - VBEQi = 1122112 - VBEQ. = (1862. - iCQ2132 — VBEQ1

 

 

 

 

fin OF
= —- —— 1 — V R — V
13F + 1 R1(Lou.t BEQQ] 2 BEQ,
011732 73172.32 aFRz
= — . . —,_———- V — V 1 4.11d
R1 0111+ 13F+1+ R1 BEQ2] BEQ, ( )
"vow, = Pout = VBCQ1 = VCEQ, — VBEQ, = ioutZL (4116)

while transfer characteristics for the saturation region are

vout,min : VCEQ1,sat " VBEQI : —aFZLiin,sat (4123')

2in,sat Z Zin,m.a;i: (4'12b)

and that = 0 in cutofi mode. The voltage-current relationship at the input and
output of the current follower is shown in Figure 4.12 and Figure 4.13, respectively,
where the input current range is 0 < i217, < iin,sat and the output voltage range needed

for Q2 to operate in the active region is —0’FZLi-zin,sat < 1.10,,th = VBCQ, < 0.

85

 

Figure 4.12. Input voltage-current relationship of common-base amplifier [24]

10 Increasing
0

IE

 

 

 

 

 

 

 

Figure 4.13. Output voltage-current relationship of common-base amplifier [24]

86

  
 

Active Slope : —aFZL

‘

—V Y Saturation if
B m

A_vsa E

t

 

Figure 4.14. Transfer characteristic of common-base amplifier [11]

Equation (4.11d) can be used to find um in terms of vout only

 

 

 

OFRQ iz‘nRz aF112 ~ OFR2
[bl-7?. : ——-E-—Uout + [BF—I '1' R1 VBEQ2] _ VBEQl ~ _ R1 vout

R2 R2, R2 . .

% —Evout = ‘R‘IloutZL = -§1-(ICQ1 - ZBQ2)ZL

= _—[i_2_ 0‘ i- — UOUt—VBEQz ~ __Ii2_a ? Z
R1 F "in (18F +1)R1 L R1 F “tn L
R2 .

z ’E'linZL (4.13)

since the last three terms are negligible compared to the first term by order of mag-

nitude, a F as 1, and )3]: >> 1 in the first. and last expressions of (4.13), respectively.

. . . . . . u- , R
UtiliZing this information, it can be shown from (4.13) that Zm = —l-n— = —§2Z L
2in 1
R . . .
where k = —2-. It 18 also shown from (4.9) that the voltage at the base of Q1 is in

El
phase with the input voltage um and that ’Uout = —u,-n from (4.13).

To simplify things. here is how the NIC works:

87

1. An input current flows out of the voltage source 1.1.," and through the common—
base current follower Q1 generating an output voltage 210m 2 —ZLi,-n at load

impedance Z L-

2. The output voltage produces feedback through the collector—emitter stage via
the base of the common—emitter inverter Q2 inverting Pout at the base of Q1

resulting in the voltage at the base of Q1 to be in phase with 11m.

3. Since 'l'Ouf = —2.1,,,, the input impedance of the NIC is the negative of the load
impedance of the NIC multiplied by a scaling factor, i.e., Z,” = —kZ L where k.
is the scaling factor defined by how the NIC is implemented. For this particular

5’2
" N‘t.k=— 9.
(II‘CIII R1 []

Even though all the design parameters have been computed for the N IC design,

there are a few precautions that need to be taken when operating this circuit [12]:

1. As the collector current 10 increases by a. factor of 10, the base—emitter voltage
VB E increases 60 millivolts (mV). The general relationship between the change

. I
of collector current and base-emitter voltage is AVE E = VT ln(%).
Cl

2. As the temperature T of the transistor increases, the base-emitter VB E decreases

2.1 mV per degree Celsius (mV/OC).

4.3 De-embedding the Test Port Cables and the Microstrip

The test port cables and microstrip will be de-einbedded using TRL calibration.
Connectors A and B will be designated as the test port. cables of the VNA for the
first stage of twostage de-embedding. It will be assumed that the test port cables will
have equal length. In the second stage of the de—embedding process, the microstrip
will be cut evenly in two, i.e., the lengths of connectors A and B are half the electrical

length of the microstrip and they are equal to each other. If this is the case for both

88

stages of the de-einbeding proces, then from (3.56) and (3.57)

[TAl=lTBl and [SAl=lSBl (4-14)

Referring back to Sections 3.2.2.1, 3.2.2.2, and 3.2.2.3, the measurcmei'its for each

standard of both stages of the de—embedding process will be conducted.
4.3.1 The Thru Measurement Revisited

From (3.58). [TT] can be evaluated as

lTTl = [TA] ° [TB] = [TA] ' lTAl (4-15)

4.3.2 The Reflect Measurement Revisited

In the first stage of two-stage de-embedding, it will be assumed that the test port
cables have low loss and the characteristic impedance of the test fixture connectors
are matched with the microstrip. Details will be explained in Section 4.4.1. In the
second stage of two—stage de—enibedding, it will be assumed that the microstrip is
lossless. If the connectors are terminated with a short. circuit, S11 = 8'22, 521 2 S12,

and F L = —1 making new expressions for equations (3.60) to (3.64)

 

 

 

 

2 TA12 _ TAii
S T T 9
S = S __ .421 2 A22 A2“ 4.16
R11 A11 1+SA11 1_ TA21 ( )
TA22
where the ratio 1 is expressed in (3.61). For connector B, (3.62) becomes
A22
SBIQSB21
S. = S = S - ————————
R22 R11 B22 1+ 5811
2 _TB:21 _ T1311
S
2 5311 _ A = T322 T822 (417)
1 + 3311 1 + T812
T322

89

and the ratio expressed in (3.63) becomes

 

 

 

TA21
, s + ——
TBii _ T.411 _ 11 TA22
_ _ — r (4.18)
T322 TAQ‘) 1 A12 ,SRH
TAi

The product of (3.61) and (3.63) is now

TA11_TBll = TAii .TAll = TA11 2
TA22 T822 TA22 TA22 TA22

TA12 T312
51711 — —) (1+ —— ° 51322
= ( TA22 T311

T321) ( T312)
SR2? + — 1 - —
< T322 T322

= 11 (4.19)

 

 

 

 

4.3.3 The Line Measurement Revisited

To produce the best results for this standard, the length of the empty transmission
line in the first. stage and the electrical length of the empty, lossless microstrip in
the second stage, 6, must be a quarter of a wavelength (e.g., E = 2) [17]. This
makes ’7' = jS in (3.66) where ,13t_l,-,,e = w\/L_C from Section 3.1.1 for the test port
cables and 13.,mjcmstrip = BEM from (3.29) for the microstrip. Equation (3.72) is
converted to

[TB] = [TA] = lTAl—l ' WI] (420)

90

and is expanded in the same way shown in (3.73) and the ratios of (3.74) which

remain the same

 

 

 

 

 

T421
T 9 _ __‘__ . T
TB” = T421 = m T411 T11 (4 21a)
T T 1 TA21 '
B” A22 TT22 - m ° TT12
T412
r — —‘ -T
T1912 = T.412 = T12 T.422 T22 (4 21b)
:r :r _ T412 '
B11 .411 TTH _ TA” 'TT21
and (3.75) is now
TA12
T ’TT21

, T11 — —
T.411 _ T311 _ TAii .TAll _ T422

T ‘9 T 0 — T 9 T — TA21
.42.. B.1 A-2 A21 TT‘Z‘Z _ T , TT12
A11

 

 

 

(4.22)

The S-paraineters of the DUT are recalculated from (3.57d). Using both (3.64)

 

T
and (3.75) creates a new expression for All using (4.14)
A22

T412 T412 TA12
T (TTll — —T 'TT21) (51211 ‘- _T 1 + —T “31211
All = :1: A22 A22 A11

T TA21 TA21 T.421
A22 \ (TT2‘2 — —T ' TT12) (31211 + —T 1 + _T ' S11211
A11 A22 A11

 

 

(4.23)

 

Following the same procedure to evaluate the S—parameters the DUT as shown in

(3.87) through equations (3.77) to (3.85) obtains new expressions for (3.86) and (3.87)

91

where the T-parameters of the DUT are

T.422(T(.'o-mp11TA22 - Tcompl2TA21) — TA12(71?07‘121)21TA22 — Tcomp22r421)

 

 

 

TDUTii =
(TAiiTA22 — T412TA21)(TAiiTA22 — T141213121)
(4.24a)
TDUT 0 _ T.422(Tmmpi2T.4ii - TcompllTA12) — T412(Tcomp22TAii — Tcomp21TAi2)
1.. — .
(721171422 — TA12TA21)(T.411TA‘22 — T111273421)
(4.24b)
TDUT _ TAllchomp2lTA22 _ Tcomp22TA2l) _ TA21(TC0771[)11TA22 — 7107121212TA21)
21 - . .
(TAiiTA22 — TA12TA21)(TA11TA22 — TAi2TA21)
(4.24c)
T _ TAllchon'ip22TAll "‘ Tcomp21TA12) '" TAQlchomp12TA22 ‘" TcompllTA12)
DUT22 —

 

(TAiiTA22 — TA12TA21)(TA11TA22 — T412111.421)
(4.24d)

and the S—parameters of the DUT are

SAllfsAllScompll _ lScompl) ‘1’ (Scompll — SA11)lSAl

 

 

 

 

SD T11 =
U 8.411 ' (Scampll ' ISA] " SAll ' lscompl) +(ScompllsA11 _ ISAI) ' lSAl
(4.25a)
SDUT : _Scomp21312421
12 SAM ' (Scompll ' ISA] — 5.411 ° IScompl) '1' (ScampllsAll _ lSAl) ' lSAl
(4.251))
SDUT2 = —Scomp218’2412
1 SAii ' (Scompll ' |SA| — 5.411 ' lscompl)+(scomp115.411 - lSAI) ° lSAl
(4.25c)
SDUT22 : 5.411(SAIIScomp11 “ ISAl) '1' ScmnplllSAl _ SAlllScompl
SA11'(Scomp11 ' lSAl _ SAll'lscompl)+(ScompIISA11_lSAl)'lSAl

(4.25d)

4.4 Modeling the Test Fixture, Microstrip and NIC

With all the necessary measurements needed, the test fixture, microstrip, and NIC can

be modeled using equations (4.14) to (4.25). Each component. will be modeled accord-

92

ing to their physical parameters. It must be noted that the S-parameter, T-paran‘ieter,
and ABCD-parameter matrices have to be represented as complex numbers.
4.4.1 Modeling the Test Fixture

The type of test fixture that will be used is an Anritsu 3680-20 Universal Test. Fixture

and will be modeled according to Figure 4.15.

 

 

 

 

 

 

 

 

#— f A__) !+— f B ——L
. _ i I _ T
IgA—‘flgAl IgB—‘IBEBI
I l I l
a1 : DUT : “B
I I I I
IZA¢ZOI 'ZB¢Z0'
i i i i
Measurement Device Device Measurement
Plane Plane Plane Plane

Figure 4.15. Test fixture modeled as a lossy transmission line [16]

The ideal case would be to model each side of the test fixture as a lossless transmission

line with a characteristic impedance of 50 Q as shown in Figure 4.16.

93

 

 

 

1‘— 5 A—J

i i

: 6A = :83 A :

l l

Port1| l

I

T 3
Measurement Device
Plane Plane

 

DUT

-1;
.T.

 

 

 

 

l I

: 63 = 18613 :

I I Port 2

l I

l l

i 7
Device Measurement
Plane Plane

Figure 4.16. Test. fixture modeled as a lossless trai‘isn‘iission line [16]

from (3.221))

lI‘Ll:1ORL/20

94

However, since an accurate model needs to account for insertion loss and return loss, it
will be first assumed that the test fixture is a lossy transmission line with an arbitrary
characteristic impedance Zarb and a return loss of —30 dB (II‘LI = 1030/20 = 10’1'5 =
0.0316228) [16, 23]. In this case, the magnitude of the reflection coefficient was found

(4.26)

and the magnitude of the transmission coefficient. can be found from (3.22a)

(4.27)

For the first stage of the de-embedding process, the measurement plane is located
at the ends of the cables while the device plane is located at. the ends of integrated
system of the microstrip and the NIC. The distance between these two planes will be
designated as [A for connector A and 6 B for connector B of the test fixture. Since
the signal sent from the V NA to the integrated system via. the test port cables will
encounter dispersion (the signal sent from the cables will be distorted as it propagates

down the line). the propagation constant. expressed in Section 3.1.1 is expanded to [5]

 

4,: a +1.3 = \/(R + June + No)

R G
2. 'wL-'u;.. 1 — 1 —
J J C\/( wit +2120)

= jwm\/1—j(£ + G ) RG (428)

 

 

 

wL LIE —w2LC

Suppose now that return loss of the cables is small. This goes under the condition
that R << wL and G << wC making the conductor and the dielectric loss of the cables

very small. This makes RC << wQLC and

 

:zsjw\/L—C]1—%(—R— + 9—)] (4.29.1)

wL 100
1 C /L 1 R
1’3 2 a; LC (4.29c)
Z — M m x/LC (4.29d)

0—G+ij

If the signal propagating along the line is not dispersive, the ideal case is presented

95

where g = g and

C

 

= RV % +jwv LC = (1 +113 (4.30a)

C R

{3 :: WV LC (4.30C)

Using this information, it can be concluded that the conductor and dielectric losses
for connectors A and B will be chiefly due to the test fixture. For simplicity, since
characteristic impedance of the test port cables and the test fixture cannot be directly
obtained without measuring the characteristic impedances of the cables and the test
fixture, it will be presumed from (4.29) that the test port cables are lossless and the
input impedance at the device plane (20 = 50 Q) is matched with the characteristic
impedance of the cables (ZO = 50 Q) because the loss of the transmission line is small.
This makes the reflection coefficient between the test port cables and the connectors
of the test fixture zero. Since the distance between the two planes for each connector
are equal and 3.5 mm long (i A = f B = 3.5 mm) as specified by [23], equations
(4.16) to (4.25) can be applied to the test fixture to evaluate the S-parameters of the
composite system of the microstrip and the NIC where they are defined by [16] for

the C-band frequency bandwidth

SAll 5.412 0 e‘jflfA 0 e—j(27rf/C)€A
8A2, 5422 6136.4 0 [Jew/()5, 0

(4.31)

96

w 27r .
where .13 = — = f and Figures 4.17 to 4.22 model the S-parameters of the test
c c

 

port. cables for each TRL calibration standard. The TRL calculations were performed
using MATLAB developed by The MathWorks. It is shown from these figures that.
the coax cables and the test fixture connectors have low loss. For purposes here,
it will be assumed that the coaxial cables and test fixture connectors are lossless.
Therefore, performing de—embedding in the first stage is not. necessary. However, if
the characteristic impedances of the cables and the test fixture connectors are not

the same, then this stage of the de-embedding process needs to be performed.

97

SZ1-Magnitude (THRU - Test Fixture)

 

 

821 (dB)

 

 

 

6:2 6:4 6:6 618 i 72 7:4 7:6 7:8 8
Frequency (GHz)

Figure 4.17. Magnitude of transmission coefficient. for Thru standard of the test

fixture

98

821-Phase (THRU - Test Fixture)

 

 

 

 

"50).,

-521

-54t 1
Eng—56
:83—58 1
9-60 1
31’ —62-
.1“: -11

-66~ 1

-68- i

-70 1 1 1 1 1 1 1 m 1

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.18. Phase of transmission coefficient for Thru standard of the test fixture

99

 

S11-Magnitude (REFLECT - Test Fixture)
10 r 1 r 1 r 1 . 1 1

 

311 (dB)

 

 

”106 62 64 66 68 7 72 74 76 78 8

Frequency (GHz)

 

Figure 4.19. Magnitude of reflection coefficient for Reflect standard of the test fixture

100

S11-Phase (REFLECT - Test Fixture)
130,\1 e 1 a! .11 1

128-
1261
13124 _ t , _ V
$3122- , , f w , J
9120 '
8116

1:“

a. 116
114-
112-
110 1 1 1 1 1 m 1 1 1

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

 

F

 

 

 

Figure 4.20. Phase of reflection coefficient for Reflect standard of the test fixture

101

$21-Magnitude (LINE - Test Fixture)
10 4 1 1 1 1 1 1 1 1

 

 

CID-#0303

321 (dB)

 

 

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.21. Magnitude of transmission coefficient for Line standard of the test fixture

102

821—Phase (LINE - Test Fixture)
200 IF 1 1 1 r 1 1 1

l

 

150*

 

1oo~ f ‘> . ‘w r.

 

 

 

 

 

Phase (Degrees)
I
8 c?

 

 

 

 

 

 

 

7

—100 ~ 1 ~ I *2 ,

 

-150r

 

 

 

 

 

 

 

 

 

 

 

 

_2012 . 1 1 . 1 1 * 1 . 1_
06 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.22. Phase of transmission coefficient. for Line standard of the test. fixture

103

4.4.2 Modeling the Microstrip

Referring back to Section 4.1, the S-parameters of the microstrip will be modeled
for the second stage of the de—embedding process. The S-parameters for a lossless
microstrip of a length E = 150 mm and a phase constant analyzed in the frequency

band of 6 CH2 to 8 GHz in 50 MHz increments can be modeled from [16]

5411 S1412 0 6’13“ 0 e-JIQWf/CMA

[3.4] = [SB] = .
31421 5A9”) e_jd£4 0 e-j(27rf/C)£A O

(4.32)
where e‘j (27111 / CV14 with (1’ A = If B = g = 75 mm. A model can be made to visu-
alize how the microstrip will behave. However, an EM simulator called Sonnet by
Sonnet Software models more precise data of the microstrip since it is a full-wave
l\"laxwell’s equation solver. Another simulation program called Ansoft Designer by
Ansoft Corporation can also be used to describe how the microstrip will behave using
TRL calibration. Figures 4.23 to 4.32 model the S-parameters of the microstrip for
each T RL calibration standard using the S-parameters of the half-length microstrip
modeled in Sonnet and Ansoft Designer. The TRL calculations for each simulation

program was performed using MATLAB.

104

S11-Magnitude (THRU - Microstrip)

 

 

 

 

 

 

 

 

_30 1 _ fl 4 f
. _ Sonnet
-351 ~ , ~ .' » -D- Ansoft Designer
_4&Ifla-4 EBBGD ab; Q _.
iii 11 ‘ p U 13 13
1‘ 1. c1 1 1 P a)
_45 _ , [:1 [1].», Ci 1... L73 1 -
1 y , [j \\ \ P
A :3 I [a 111‘ f] /
m _ _ l 1' , . 9] a
U 50 , , 1 1 1
v [ [p \ I , 1" I
‘— \ . f I C] 1 d]
F — — ‘ 111 "1 1:1 1 d
U) 55 a: 1 l 1 l 1 I
I ~ I l p l
l 1 1 l ,
-60 c , , 1‘ 1 1‘ I _
l l ‘ , , I
1, 1‘ , , l
-65 1’ ' {111) ,l l : F
I] 1
l [l
-70— [5 g, —
_75 1 1 i g 1 i I 4 m
6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.23. IV’Iagnitude of reflection coefficient for Thru standard of the microstrip

(Sonnet vs. Ansoft Designer)

S11-Phase (THRU - Microstrip)

 

 

 

 

 

 

 

 

 

 

 

200 r w r r I 1 I I I
K ‘ ”K. ’ --—e—Sonnet
1501* \2 , _ +Ansoft Designer_
I \g R ~ . '
‘51? \R
1004
§ 50
E: I
(D 1
e \
a: i '* HE.
(D
2 —50 .
0- 1
-1001 g "
-150. \\fi
_20 ‘ 1 1 1 1 1 1 1 1 1
6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.24. Phase of reflection coefficient. for Thr‘u standard of the microstrip (Sonnet

vs. Ansoft Designer)

106

S11-Magnitude (REFLECT - Microstrip)

 

 

 

 

 

_O.1 T 1 J
- Sonnet
, f -D- Ansoft Designer»
A —0.2 -*
m
E
‘—
‘—
‘D -0.25- ~
11:BB
BU
B1}
DEBBCHEBBDBB
911mb
-O.3~ ‘ #1235888 _
CLa
15831388ij
-O.35

 

 

 

6 61.2 614 6:6 618 r 7.3 714 7:6 71.8 8
Frequency (GHz)

Figure 4.25. Magnitude of reflection coefficient for Reflect standard of the microstrip

(Sonnet. vs. Ansoft Designer)

107

S11-Phase (REFLECT - Microstrip)

 

 

 

 

 

 

 

200 , , w —
fig -3? Sonnet
150_ ‘ f—Ansoft Designer H
W? N A
\e
’.\
1 003k 3}: _
a; - 44x

7,; as? as,
g 501 “g at; ~
L e\ w:-
3 a 531K
El 0 X is —
(D R\ RE
0) "‘\ SE
2 -501 “a; W
a x

‘100* Rs; ‘

E
-151 x _
-20

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.26. Phase of reflection coefficient. for Reflect standard of the microstrip

(Sonnet vs. Ansoft Designer)

108

S11-Magnitude (LINE - Microstrip)

 

 

 

 

 

 

 

 

"2 1 1 I 1 fi j I 1 I
-13.. Sonnet
—3(]1 ‘1 , -13- Ansoft gesrgner
. {£18 BEBE] B GD‘CLD- j  , /":.' game B BR
-401- 1:1 - - ' EU . .-B _; t . . ‘ID
11:: h a
[3
_501_ ‘ t1 : I’D u .1
a [:3 Val-P :7
3 _ \ 111 ‘3 _
601
‘— [D l
\— 1 ,
a) 1 ,
-701 1 a ‘
, 11] ’1
1
-80)— 7 1 1! _
‘ 1
‘ 1
—90t— , y‘ '1 a
‘1
_1O ‘ 1 1 1 1 1 ¥ 1 1 1
6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.27. Magnitude of reflection coefficient for Line standard of the microstrip
(Sonnet vs. Ansoft. Designer)

109

S11-Phase (LINE - Microstrip)

200 I T f I I I_ I I I
-—='r- Sonnet

+Ansoft Designer

 

 

 

 

150*

 

100‘?-~

i

50

 

Phase (Degrees)
9

-501
21K? KY
-1001 ‘ , \a ~

T

 

 

 

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.28. Phase of reflection coefficient for Line standard of the microstrip (Sonnet

vs. Ansoft Designer)

110

SZ1-Magnitude (THRU - Microstrip)

 

 

 

 

 

—O'1 T I I T i T f l
- e—Sonnet
-D--Ansoft Designer i
-O.15*~ " x 1 4
-O.2L ~ —

821 (dB)

*3

 

 

 

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.29. Magnitude of transmission coefficient for Thr‘u standard of the microstrip

(Sonnet vs. Ansoft Designer)

111

S21-Phase (THRU — Microstrip)
200 r . r w w

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r; ‘ =--Sonnet
150_ i “~ +Ansoft Designer a
~\‘2: S,
h e»
‘ix, . .1'
375. V
100 — ”-Igh - , ‘ . i _
‘05
\m
A .‘k‘
w 3:.
GD 50 ~ 33:. ~
9 :2
a) 3:
(D \s
D O — a - d
g _50~ . Y a A 1 . a
n. 731‘
{xv ~ )1;
—1oor~~xfi at» 1
s. x
\‘x ' Ex ‘
-1 50 _ Ex»: , N; ,1 _
at; \ . - “Ad:
-200

 

 

6 6:2 6:4 sis 618 % 7:2 7:4 7:6 7:8 8
Frequency (GHz)

Figure 4.30. Phase of transmission coefficient for Thm standard of the microstrip

(Sonnet vs. Ansoft. Designer)

112

SZ1-Magnitude (LINE - Microstrip)

 

 

 

 

 

 

 

 

—O.1 ' I 1 T 1
*-—*-Sonnet
1»~— k ~D- Ansoft Designer 4!
-015- “wt“ _F .. a. _
A -O.2~ —
co
'3
E
‘0 -o.25+ —
EEBBBD
. QDQESBBCBSB
_0 3_ B‘BQEEGB B a
BEBBfifl
_0'356 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.31. Magnitude of transmission coefficient. for Line standard of the microstrip

(Sonnet. vs. Ansoft Designer)

H3

S21-Phase (LINE - Microstrip)

 

 

 

 

 

  

200 I I F I 1 1 f 1
I ' ----e—Sonnet

150 ~ +Ansoii>esigner 1

100— PR , a
A K
m +— 35%; _
g 50 “9:11
C) "‘37;
‘D 1
9, 0 i
(D
U)
2 -501 -
o.

-100r 1

-15Qg _

 

 

 

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.32. Phase of transmission coefficient for Line standard of the microstrip

(Sonnet vs. Ansoft. Designer)

114

4.4.3 Modeling the Negative Impedance Converter

Referring back to Section 3.2.1.3, the S-parameters of the NIC will be modeled accord-
ing to the ABCD—parameters of the composite system of the microstrip and the NIC
such that the parameter Z shown in Figure 3.14 represents the series impedance
of the two—port. network corresponding to the input impedance of the NIC, i.e.,
Z = Zinc-VIC = —kZL = —%§ZL. The reason of doing this is that the compos-
ite system is a cascaded network of two-port networks. The S-parameters of the
composite system can be found using (3.48) and (3.55) for the second stage of the

de—embedding process

 

 

 

 

Z;
‘51? 2
2 EE Zin 100
Z - 100 Z. 100
[Scompl = 2 + 2:? = 2711-50 lnztn (4.33)
R2 18 . . . . .
where k. = —, w = 27r f , Zm cap = —.——— for the negative capac1tor shown in F igure
R1 ’ ij'
4.33, and Ziand = —jkwL for the negative inductor shown in Figure 4.34. For

simplicity, the values of R2 and R1 will be the same (e.g., R2 = R1 = 1. kiloohm

(kQ)) for both the negative capacitor and negative inductor making k. = 1. This
1

— ij

picofarads (pF) and the value of the inductor is 0.1 nanohenrys (nH). Table 4.1 gives

 

will make Zhwap = and Zin,ind = —ij. The value of the capacitor is 10
values of Z22” for the negative capacitor while Table 4.2 gives values of Zm for the
negative inductor. All tables and figures are observed in the C-band frequency range

in 50 MHz increments.

 

 

 

 

 

 

 

 

Figure 4.33. Negative impedance converter with capacitively terminated load

 

 

 

 

in

——’ 0.1nH

 

 

 

 

Figure 4.34. Negative impedance converter with inductively terminated load

116

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency (GHz) Zin = fljwlCL (9) Frequency (GHz) Zm = — 3:110: (Q)

6 12.65258 7 32.27364
6.05 j2.63()66 7.05 j2.25752
6.1 j2.6()91 7.1 j2.24162
6.15 j2.58789 7.15 j2.22594
6.2 j2.56702 7.2 j2.21049
6.25 j2.54648 7.25 j2.19524
6.3 j2.52627 7.3 j2.1802
6.35 j2.50638 7.35 j2.16537
6.4 j2.4868 7.4 12.15074
6.45 j2.46752 7.45 j2.13631
6.5 j2.44854 7.5 32.12207
6.55 j2.42985 7.55 j2.10801
6.6 j2.41144 7.6 j2.09414
6.65 j2.39331 7.65 j2.08046
6.7 j2.37545 7.7 j2.06695
6.75 j2.35785 7.75 j2.05361
6.8 j2.34051 7.8 j2.04045
6.85 j2.32343 7.85 j2.02745
6.9 j2.30659 7.9 j2.01462
6.95 j2.29 7.95 j2.00195

7 j2.27364 8 j1.98944

 

Table 4.1. Values of Zm for negative capacitor with load capacitance of 10 pF in

C-band

117

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency (GHZ) Zin = —ijL (Q) Frequency (GHZ) Zm = —ijL (Q)
6 -j3.769911 7 -j4.39823
6.05 43.801327 7.05 44.429646
6.1 43.832743 7.1 44.461062
6.15 -j3.864159 7.15 -j4.492477
6.2 43.895575 7.2 44.523893
6.25 -j3.926991 7.25 -j4.555309
6.3 43.958407 7.3 44.586725
6.35 -j3.989823 7.35 -j4.618141
6.4 -j4.021239 7.4 —j4.649557
6.45 44.052655 7.45 44.680973
6.5 —j4.08407 7.5 -j4.712389
6.55 -j4.115486 7.55 -j4.743805
6.6 44.146902 7.6 44.775221
6.65 44.178318 7.65 44.806637
6.7 44.209734 7.7 44.838053
6.75 -j4.24115 7.75 -j4.869469
6.8 -j4.272566 7.8 -j4.900885
6.85 44.303982 7.85 44.9323
6.9 44.335398 7.9 44.963716
6.95 -j4.366814 7.95 -j4.995132
7 —j4.39823 8 -j5.026548

 

Table 4.2. Values of Z21” for negative inductor with load inductance of 0.1 nH in

C-band

118

 

4.5 Comparing Computational Results with Simulated Results

The TRL calibration technique will be conducted for the first stage of the de-
embedding process by substituting (4.14) into (3.56) and (4.31) into (3.51) to obtain

the following relation

[Tcompl = [TA] ' [TDUT] ' [TA] (434)

However, since the DUT of the first stage of de-embedding is the composite system
in the second stage, the test fixture connectors are modeled using (4.31), and the
measurements of the composite system are given by (4.33), the S-parameters do not.
need to be found since computational analysis produces the same result. Remember,
the first stage of de—embedding moves the reference plane from the ends of the test
port cables to the edge of the composite system.

The second stage of de-embedding undergo TRL calibration as well. This time
connectors A and B (the half-length microstrips) are modeled by equation (4.32) or
using Sonnet for more accurate analysis. Referring back to microstrip simulations
displayed in Figures 4.23 to 4.42, the S-parameters of the microstrip will be used to
conduct the TRL c.a.lil:)ration technique for the second stage. The NIC is represented
by (4.33). Following the procedure detailed in Section 3.2.2 and simplified in Section
4.3, the S—parameters of the NIC can be evaluated using (4.25). The figures and
tables mentioned in Section 4.4.1 model the S—parameters for the negative capacitor
and negative inductor examined in the C-band frequency range in 50 MHz increments.

In order to check if the results are accurate, a parameter study is performed in
MATLAB comparing simulated data of the half—length microstrips done in Sonnet
and Ansoft Designer. The simulated data will then be put into the theoretical model
obtained in (4.33). First the S-parameters of a full-length microstrip (5 = 150 mm)

were compared to the S—parameters of two half-length microstrips (€44 = 75 mm)

119

cascaded together in Sonnet and Ansoft Designer. It was concluded that the full-
length microstrip produced the same results as the cascaded half—length microstrips
for both simulation programs. It also shows that the S-parameters of the cascaded
system using the simulated data from Sonnet. and Ansoft designer produce similar
results when they are compared against each other. This is shown in Figures 4.35 to

4.38.

120

S11-Magnitude (Cascaded Half-Length vs. Full Length Microstrip)

_30 I ‘ I T I 1 I j I
Half (Sonnet)
d [TUBE a a a an —a—- Full (Sonnet)
._ a ‘4 ~-+— Half (Ansoft DeSIgner)
40qu -' * m. +Fu|I (Ansoft Designer) F“
K i] .- ' V.\ . l , ' f- d
\\ .‘ .,‘ 34‘ m by X3 :5 I i?
it I 4'6 - ‘ ‘ r , “ ‘ v
A “ \I f I -\\ Id/
55-505 z I"
T- ‘ f I”
f— I
U) I '
, I I
-609 / 6] " ' F
-70~ ‘

 

 

 

 

 

 

 

 

 

 

Frequency (GHz)

6 6:2 6:4 616 6.8 7 7:2 7:4 7:6 7:8 8

Figure 4.35. Magnitude of reflection coefficient of full-length microstrip vs. cascaded

half-length microstrips (Sonnet vs. Ansoft Designer)

121

S11- Phase (Cascaded Half-Length vs. Full Length Microstrip)

 

 

 

 

 

 

200
1x 7.; - - Half (Sonnet)
150- y "I? » K 4:. Full (Sonnet) y
I} \ 1 a 7' - Half (Ansoft Designer)
6 I +Full (Ansoft Desi ner)
100”i ' e ) ~ 9 _
I r . ~

I

Phase (Degrees)
O

 

 

 

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.36. Phase of reflection coefficient of full-length microstrip vs. cascaded

half-length microstrips (Sonnet vs. Ansoft Designer)

122

S21- -Magnitude (Cascaded Half-Length vs. Full Length Microstrip)

 

 

 

 

 

 

 

-0. 1 . . I ‘ P
Half (Sonnet)
a -5- Full (Sonnet)
-O 1 _ DEB a 330% Half (Ansoft Designer) L
- 3:08 0 e s B a" —*~?— Full (Ansoft Designer) .3
A —O.2— P
m
E
2?:
‘0 -0.25r _
wag?
$Mflfi
'0 3r ‘7 NWWRW d
' 'TTWV
-O.3

Figure 4.37.

caded half-l

 

" 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 78 8

Frequency (GHz)

Magnitude of transmission coefficient of full-length microstrip vs. cas-

ength microstrips (Sonnet. vs. Ansoft Designer)

123

S21- Phase (Cascaded Half-Length vs. Full Length Microstrip)
200 a 4

 

 

 

 

 

 

 

 

5 fl Half (Sonnet)
150 _ 93 -5- ~ Full (Sonnet) 4
BE . ~~— Half (Ansoft Designer) 134
100L 8% _, +FulI(Ansoft Designer)1
TD‘ , (Tia . ‘ f
a) is
é: 0_ 5355 _
(D _
5’3 -50. ' Xe   ’   "
4003‘»... 85 ' ‘ 2
8h '
‘9... a
—150* mg ‘ V , f \Eéi "
‘2006 6.2 6.4 616 6:8 7 7.2 714 7:6 7:8 8
Frequency (GHZ)

Figure 4.38. Phase of transmission coefficient. of full-length microstrip vs. cascaded

half—length microstrips (Sonnet vs. Ansoft Designer)

124

Using (4.15), the half-lengths modeled in Sonnet were created to produce a cas-
caded network of a half-length microstrip, a zero-length line. and another half-length
microstrip. The computed S-parameters were compared to the cascaded network of
a half-length microstrip, a zero-length reactive component, and another half-length
microstrip modeled in Ansoft Designer. Examining Figures 4.23 to 4.32, it is shown
that the ranges that the S-parameters for the TRL calibrations for both types of
models (Sonnet vs. Ansoft Designer) are nearly identical despite that fact that the
trends of the transmission coefficients being different. The reflection coefficients for
both models do have similar ranges and trends.

Finally, a parameter study of components in the structure (half-length microstrip,
active component, half-length microstrip) were performed. Several components were
used: a 50-9 resistor. a 75-9 resistor, a 100-Q resistor. a 10—pF capacitor, and a
0.1-nH inductor. Then, the reactive component values were varied from positive to
negative (e. g. the capacitor was varied from —10 pF to 10pF in 4 pF increments and
the inductor was varied from —0.1 nH to 0.1 nH in 0.04 nH increments). The follow-
ing figures display the behavior of the cascaded system using resistive and reactive
components.

Figures 4.39 to 4.50 display the S—parameters of the cascaded network with the
50—0, 75—9, and IOO-Q resistors using the half—length microstrips modeled in Sonnet
and Ansoft Designer. Figures 4.51 to 4.54 display the S—parameters of the cascaded
network with the 10—pF capacitor and Figures 4.55 to 4.58 display the S-parameters
of the cascaded network with the 0.1-nH inductor using the half-length microstrips
modeled in Sonnet. and Ansoft Designer. Figures 4.59 to 4.62 display the S-parameters
of the cascaded network with the -—10-pF capacitor and Figures 4.63 to 4.66 display
the S-parameters of the cascaded network with the ——0.1-nH inductor using the half-
length microstrips modeled in Sonnet and Ansoft. Designer using (4.33) in MATLAB

to model the NIC.

The purpose of conducting the parameter study is to analyze how the cascaded
system behaves as the active component values vary from impedance to positive
reactance to negative reactance. Another thing it does is distinguish a negative
capacitor with a positive inductor and a negative inductor with a positive capacitor.

Equations (4.35) and (4.36) establish the relationshi1.)s for each of them.

Z, r . . = --.-—
IV [Crap ij

Zind = jw‘L (4.35)

ZNIC,2'nd = —J'w‘L
1

—— 4.
ij ( 36)

anp =

The key difference between the negative capacitor and the positive inductor is that the
frequency relationships of the impedance functions are inverses of each other (e. g., the
frequency of the negative capacitor is inversely proportional to its input impedance
while the frequency of the positive inductor is proportional to its impedance). Simi-
larly, the frequency relationships of the impedance functions for the negative inductor
and positive capacitor are opposite when they are compared to each other. This is in

spite of the fact. that the phase for each case is the same.

126

S11—Magnitude (50-Ohm Resistor - Cascaded System)
—9.55 . I .

 

 

 

 

 

 

 

 

' ~9— Sonnet
-5- Ansoft Designer
-9.65r EDD/I EU . ~ —
E \D‘ ;; [5.435513
if U , : ,4 DD

3 ‘9]- C1 2 "121.» E]

E D )1

‘— o 13 .

5 -9.75~ an me _
-9.8— _
.99 ‘1 1 m 1 I I I I I

6 6.2 6.4 6.6 8.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.39. Magnitude of reflection coefficient of cascaded half-length microstrips

with 50-9 resistor in between (Sonnet vs. Ansoft Designer)

127

 

I I

S11-Phase (50-Ohm Resistor - Cascaded System)
200 e 4. fl . . .

 

~— Sonnet
. +Ansoft Designe .

x
\

 

 

150

 

1.
I "J

100 -

50

Phase (Degrees)
O

-150

 

 

 

l

6 6:2 654 61.6 618 7 72 7.4 716 7:8 8

 

-200

Freouencv (GHz)

Figure 4.40. Phase of reflection coefficient of cascaded half-length microstrips with

50-9 resistor in between (Sonnet vs. Ansoft Designer)

128

SZ1-Magnitude (50-Ohm Resistor - Cascaded System)
-3.' . . . I 4 .

 

I

 

-- ~‘— Sonnet
-5- Ansoft Designer

r/I‘

 

 

 

S21 (dB)
do
\.It

. BBBGDEBEJ
6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

 

 

 

-3.

Figure 4.41. l\~'Iagnitude of transmission coefficient of cascaded half-length microstrips

with 50-0 resistor in between (Sonnet vs. Ansoft Designer)

129

SZ1-Phase (50-Ohm Resistor - Cascaded System)
200 F I I I I I

 

 

 

 

 

 

 

 

 

 

1 “- Sonnet
“S .
150_ 3% +Ansoft DeSIgne‘. -
8g 13‘s..
:3 ‘ . 3'.
.4. .
100— 58.44 * * -~
a) 2‘?
an 50— *8. ~
9 7;;
a) “K
a) 0 .3:
(I) 95.x“
cu ..
.c ‘507 \g * " » 1
o. R
(“L \X
1%.
34
400—3.. *4 * ~
n-3{3 )s:\_
"a 45‘
-150- K . Kg _
a: ‘84

 

 

’2006 6:2 6:4 6:6 6:8 7 72 7.4 7.6 71.8 8

Frequency (GHz)

 

Figure 4.42. Phase of transmission coefficient of cascaded half-length microstrips with

50-9 resistor in between (Sonnet vs. Ansoft Designer)

130

S11-Magnitude (75-Ohm Resistor - Cascaded System)
-7.45 I I I l I

 

 

 

 

 

 

 

 

66.4 "0— Sonnet
GP 3.4 s -D- AnsoftDeSIgner
’75ij3 7:1 .- ' If; ctr-DEB ‘14
ch” 1 :3 . 1;. tr . E.
* .3 - 4
)3 [IE .1
A "7.55” D ID _
m CI
3 5 44‘
‘- t3 E1 13 {33/13
‘0 -7.6~ -
—7.65~ _
—7.7 . . . A . . . . .
6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.43. Magnitude of reflection coefficient. of cascaded half-length microstrips

with 75-9 resistor in between (Sonnet vs. Ansoft Designer)

131

S11-Phase (75-Ohm Resistor - Cascaded System)
200 r

 

j

i l I J I

we Sonnet

" +Ansoft Designer _
xx - 3%

 

 

 

150—

 

)7
v

100r

Phase (Degrees)
‘?

 

/,

-150

L

 

 

 

_2006 6:2 6:4 6:6 68 i 712 71.4 7:6 7:8 8
Frequency (GHz)

Figure 4.44. Phase of reflection coefficient. of cascaded half-length microstrips with

75—9 resistor in between (Sonnet vs. Ansoft Designer)

132

SZ1-Magnitude (75-Ohm Resistor - Cascaded System)

 

 

 

 

 

_4- T F i T T I l L I
~- w" Sonnet
-D- Ansoft Designer
-4.9* , ’ —
EB _ i * ~ a
3 5r
F
N
a)
—5.1— ~
Lk[:1 3.0438 E 31:13
EH38 B a 3 DE
-5.2L ‘DEEBBDDQB til-E T
E] B B
; Game 8 C“

 

 

 

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.45. Magnitude of transmission coefficient. of cascaded half-length microstrips

with 75-0 resistor in between (Sonnet vs. Ansoft Designer)

133

SZ1-Phase (75-Ohm Resistor - Cascaded System)

 

 

 

 

 

 

 

 

 

 

 

 

200 r , f , t r 1 L .
Fer-Sonnet
1 50 +Ansoft Designe; H
. , 4%
100 _
A X
8 50 ~— _
22 s
C) 4'
é: 0 \is *
.31
8 is
co _ rs: _ q
& 5° X
”X
—100 Ks ' s
X .
_2006 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.46. Phase of transmission coefficient of cascaded half-length microstrips with

75—9 resistor in between (Sonnet vs. Ansoft Designer)

134

S11-Magnitude (100-Ohm Resistor - Cascaded System)
-6.1 I I ,

 

 

 

 

 

 

 

 

—©- Sonnet
{F Ansoft Designer
—6.15— DEE 8 ° 1
[:1 D”
,8] \D C i 5838.
i? D gap [13
A _6.2_.“ at} [Ziml ; _
g: D 121/D
‘- [Km [21.
"’ —6.25~ " {ma _
—6.3L ‘1 tr
_6'356 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.47. i\r-‘Iagnitude of reflection coefficient of cascaded half-length microstrips

with IOU-Q resistor in between (Sonnet vs. Ansoft Designer)

135

S11-Phase (100-Ohm Resistor - Cascaded System)

 

 

 

 

 

   

 

 

 

200 F i r i r
‘ s e~Sonnet

150 _ . RX? +Ansoft Designe; H

100~ l a
A i
8 50— -
92
8’
9, 0" “
Q)
U)
2 -5o— —
D.

-2006 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.48. Phase of reflection coefficient of cascaded half-length microstrips with

100-Q resistor in between (Sonnet vs. Ansoft Designer)

136

SZ1-Magnitude (100-Ohm Resistor - Cascaded System)

 

 

 

 

 

._- Sonnet
-6r -' -a- Ansoft Designert
' 7,47‘ \ /,7 .
-6.1~ / , —
E 1,.
B
‘_ —6.2— ~
N
(I)
-6.3t ' -
CAD» ’BBEDBBBBB‘
n a E! CLD
D a
:1) a DD
\88 B {385” 11
EU
_6.4_ \DE] B [33:14:15]

 

 

 

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.49. Magnitude of transmission coefficient of cascaded half—length microstrips

with 100-Q resistor in between (Sonnet. vs. Ansoft Designer)

137

O-Ohm Resistor - Cascaded System)

 

200

SZ1-Phase (1O

 

 

 

 

 

 

 

 

 

 

 

 

 

’ ' ~e— Sonnet
15o +Ansoft DesigneR
‘33»; . ; .
100 ' F - -
F
71? FE¥
g 50 ‘ SF “
c» ‘ka
CD
g 0 F; —
g FE»:
-50 ~X 1 j V. r
if. 3g
i L
-100 Kgak 4
“\w
—2006 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.50. Phase of transmission coefficient of cascaded half-length microstrips with

1009 resistor in between (Sonnet. vs. Ansoft Designer)

138

S11-Magnitude (10-pF Capacitor - Cascaded System)

 

 

 

 

 

 

 

 

-2 f 1 r 1
_.:,_ Sonnet
~CJ-vAnsoft Designer
-28r . .._. r,-,..h.,_
[13+] :1 .
Dr:
-30i \Q '. ‘i
a i‘ [ETD—DB _‘
ZJja . D
3 .
a -32 C; 1:]; Q. _
U I’D ‘ I\
:j a a
27) -34r - ca B Q —
9 1:1 pi ‘ F]
I; Q Ci ) \Q
—38 ~ b V. . .
‘ \D‘Dflfi ,» E]\
5P
-38. ¥ A
_4 ‘ 1 1 1 1 1 1 L — 1 1
6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.51. Magnitude of reflection coefficient of cascaded half-length microstrips

with 10-pF capacitor in between (Sonnet. vs. Ansoft Designer)

139

S11-Phase (10-pF Capacitor - Cascaded System)
200 I r ‘. f '

 

I I

: _ ~e— Sonnet
Sf .

i F V +Ansoft Desngnerj

i

 

 

 

150

 

(<1

7X:

 

100*

r

50

Phase (Degrees)
O

 

 

 

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.52. Phase of reflection coefficient of cascaded half-length microstrips with

10-pF capacitor in between (Sonnet vs. Ansoft Designer)

140

SZ1-Magnitude (10-pF Capacitor - Cascaded System)
-0.1 . . . X F .

 

f I

 

 

 

 

 

 

 

Sonnet
-D--Ansoft Designer
A -O.2— _
m
E
‘—
8i
-025L ; . . . . _ - , a
CH; E] St}
DU .
EBBBD‘UBBBG
304.:
—O.3H E] S Bgmflfl F
. BBB ,
. DEB E] BBD‘UB B
' E3
-O.3

 

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.53. Magnitude of transmission coefficient of cascaded half—length microstrips

with lO—pF capacitor in between (Sonnet vs. Ansoft. Designer)

141

 

S21-Phase (10-pF Capacitor - Cascaded System)
200 . T r

l— _T *1 f L

 

— +2—— Sonnet
  . +Ansoft Designer
x 1 ‘in
n . . l. "‘X

 

 

 

150—

100-

50*

Phase (Degrees)
I
2:3 <9

HL
IJT
1‘.

I
A
01
Q

T

 

 

 

 

‘ 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.54. Phase of transmission coefficient of cascaded half-length microstrips with

10—pF capacitor in between (Sonnet vs. Ansoft Designer)

142

S11-Magnitude (0.1-nH Inductor - Cascaded System)

 

 

 

 

 

 

 

 

‘24 f i i ' F ‘ l
‘ ~©~ Sonnet
-D-Ansoft Designer CF
-26- Egasga._ . ”g H g_i;;a. r
p/ aq BF
D [1 [Z] .,
‘3 . d
-28- D 88. ,4 flfl/ r
p EBBUD
5:: DP
:: -30~ ~
:35
-32_. _
-34F fl .
_36 1 1 1 1 1 1 1 1 1
6 612 6!i 613 613 7 712 7J1 715 713 8
Frequency (GHz)

Figure 4.55. Magnitude of reflection coefficient of cascaded half—length microstrips

with 0.1-nH inductor in between (Sonnet vs. Ansoft Designer)

143

Phase (Degrees)

S11-Phase (0.1-nH Inductor

- Cascaded System)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

200 . . , , L . J L
.7 »—-;~—s nnet
150~ iFt. 148—Ansoft Designer
100'- F\?\ .4
50' ' ‘ Fit: ~ 4
0X\ ‘ F. _
-50 F ' ‘ Fix X . . . _
\ s. _
—100L- + 1 FR \ -
’150’ ' FX * is
F.
i:
-2006 6.2 8.4 6.6 8.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.56. Phase of reflection coeflicient of cascaded half-length microstrips with

0.1-11H inductor in between (Sonnet vs. Ansoft Designer)

144

SZ1-Magnitude (0.1-nH Inductor - Cascaded System)

 

 

 

 

 

 

 

 

"0.1 n I I m I L r I I
--—~—Sonnet
—e- Ansoft Designer
_0_15i_"-~~ .. i.“ ,1 """ , -
A —O.2~ -
m
E
8
-O.25- _
"03* B flames—518. _
, V . EESB$
'0'358 8.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.57. Magnitude of transmission coefficient of cascaded half—length microstrips

with 0.1-nH inductor in between (Sonnet vs. Ansoft Designer)

145

SZ1-Phase (0.1-nH Inductor - Cascaded System)
200 e . . . v I . T .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'f -e— Sonnet
7 3% f .1 f +Ansoft Designer L
150 F F. .
F ‘ ' F
”a
100 - - ' a: ' - » ~
A H ER”? - i I
a”) 50 _ . FF . . . . , , . _
CD >3
L. -.~\
99
e O ,_ E51“ .. _
8 , F: x .
cu . . .
E ’50“ ‘ “Ft , . . _ . . .. _
. , , 1 Fc . .
3g 3:
100 F; . FF .
Fag _
WF— FF:
‘1 50 * F}; p ' ~ - ' . FF») —
"Ff - ’ ‘_:
-200

 

 

 

8 6:2 6:4 6T8 6:8 7 7:2 7:4 7:8 7:8 8
Frequency (GHz)

Figure 4.58. Phase of transmission coefficient of cascaded half-length microstrips with

0.1-nH inductor in between (Sonnet vs. Ansoft Designer)

146

S11-Magnitude (-10-pF Capacitor - Cascaded System)

 

 

 

 

 

~ Sonnet
. ., «fl (9 _D_. '
—3o«L ‘ = games a a a A??? pes'gneW
" Q B [3, t W .0fo . 4:3
3:: Ella :UD/EVC]
r. E1 [3 Si 'D/il'
Ci 3 1'; »— U’U "
x (3’
3 [3 GET
_40 a _
ES
3
\—
‘—
(I)
“50 F f
-60 _ ..

 

 

l

8 6:2 84 6:6 6:8 7 72 7:4 7i8 7:8 8
Frequency (GHz)

 

Figure 4.59. Magnitude of reflection coefficient. of cascaded half-length microstrips

with —10—pF capacitor in between (Sonnet vs. Ansoft Designer)

147

S11—Phase (-10-pF Capacitor - Cascaded System)

 

 

 

 

 

 

 

 

 

200 T I r l I T I I I—
T; I +Sonnet
150- " Ky *i +Ansoft Designer“
' \w . i :
J "\T g
100- ' - » > §e\. . . . . .d
8 50 r fix .i _
9.) 9R g
8 (a I m at _
I; \s
("3 XX j
a “50* x ° '
-100L \ _
\a
-150_ U f
‘2006 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.60. Phase of reflection coefficient of cascaded half-length microstrips with

—10-pF capacitor in between (Sonnet. vs. Ansoft Designer)

148

SZ1-Magnitude (-10-pF Capacitor — Cascaded System)
-O.1 I I I 1 I i

 

 

 

 

 

 

 

 

““— Sonnet

"D‘ Ansoft Designer .

-o,15_“‘ W X WM, ‘ _

A -O.2” ‘
m
B
:9;

-O.25— fl

mag B
-0 3* flBBBBBDfiflBBB ‘
Baflflfi E] B
‘33:]ij
£356 6-2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.61. h‘Iagnitude of transmission coefficient of cascaded half-length microstrips

with ——10—pF capacitor in between (Sonnet. vs. Ansoft Designer)

149

SZ1-Phase (-10-pF Capacitor - Cascaded System)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

200 1 I N I I I
’ N‘NSonnet
150 , . +Ansoft Designer
N : : Ni
a ' .
”5'
TNT '
A E‘s,
3’3 50 A N —
2 :\_
on ‘1;
5, 0 \N -
3 NW
(D 3% 1
i _50 .. FEE 1
SK .
-100 . x: _
-150 W _. _
-2006 6.6 6.8 7 7.2 7.4 7.6 7.8 8

 

Frequency (GHz)

Figure 4.62. Phase of transmission coefficient. of cascaded half-length microstrips with

—10-pF capacitor in between (Sonnet. vs. Ansoft. Designer)

150

811 (dB)

 

S11-Magnitude (-O.1-nH Inductor - Cascaded System)

-24

I

 

I I r 1

 

-«>- Sonnet
-D- Ansoft Designer

23

 

 

If".

 

 

1

 

6

6:2 6:4 6:6 6:8 7 7:2 7:4 7:6 7:8 8
Frequency (GHz)

Figure 4.63. h‘Iagnitude of reflection coefficient, of cascaded half-length microstrips

with —0.1—nH inductor in between (Sonnet vs. Ansoft Designer)

151

S11-Phase (-0.1-nH Inductor - Cascaded System)
200 I 1 1 ,

 

 

 

 

 

 

 

 

 

E -+N-Sonnet
150_' NR +Ansoft Desrgner
! ”N i \
1 X A . , \. _
N N
A g R
8 50” XX \Kfi
0’ TR ‘3:
a) \‘i\
g 0: , “N :
GD \vg
U)
2 -50 ‘ 1
“- 1
-1oo,»’ —
—150- ~
_2006 6:2 64 8.8 8.8 7 7:2 7:4 7:6 7.8 8
Frequency (GHz)

Figure 4.64. Phase of reflection coefficient. of cascaded half-length microstrips with

—O.1-11H inductor in between (Sonnet vs. Ansoft Designer)

152

SZ1-Magnitude (-0.1-nH Inductor - Cascaded System)

-O.1 n I I I I I f f T

-'--— Sonnet
-D-~Ansoft Designer

 

 

 

 

 

 

l ‘ , , 1
:7 , _. 7 ' ~ —‘
-o 15 ‘ ~
I A“ ‘ x. ,4’ V
I , I

821 (dB)

I
9
N
on

 

 

 

CBS 83337585
—o.3t- I 8889:3813? J
SBBD‘DfiB ‘
BBB—@8518
{334:3
'0'358 612 6.4 8.8 68 7 7:2 7.4 716 7:8 8
Frequency (GHz)

Figure 4.65. Magnitude of transmission coefficient of cascaded half-length microstrips

with —0.1-nH inductor in between (Sonnet vs. Ansoft Designer)

153

SZ1-Phase (-O.1-nH Inductor - Cascaded System)

 

 

 

 

 

 

 

 

 

 

200 I I N . I a I I L
_ * .1,— Sonnet
150_ I I K +Ansoft DesIgne;
I I \3’ Rx}:
100— 5 I ~
8 50— :
9
8’
e OI 1
(I)
g
.c -50- 7
d
4009):: I ~
1% I
“150“ 'i II _
_2006 6:2 614 8.8 8.8 7 7.2 714 7.8 7:8 8
Frequency (GHz)

Figure 4.66. Phase of transmission coefficient of cascaded half-length microstrips with

—O.1-nH inductor in between (Sonnet vs. Ansoft Designer)

154

The trends for the cascaded networks including the resistor, capacitor and inductor
are similar using the half-length microstrips modeled in Sonnet and Ansoft Designer.
The general trend for the resistor is that the reflection coefficient increases as the
resistance increases. The opposite trend is true for the transmission coefficient as
the resistance increases. For the reactive components (e.g., capacitor and inductor),
there is little reflection while there is nearly complete transmission.

Figures 4.67 to 4.74 shows a parameter study of reactive components in the cas-
caded network. The reactive component used for Figures 4.67 to 4.70 and 4.71 to
4.74 is a capacitor and an inductor, respectively. The values of the reactive compo—
nents were varied from positive to negative and the trends of the S~parameters as the
reactance transitions from positive to negative is displayed in these figures.

For the following figures for the capacitor, the reflection coefficient increases in
magnitude as the capacitance decreases from its nominal value of 10 pF to O pF and
decreases in magnitude as the capacitance decreases from O to —10 pF. The transmis—
sion coefficient decreases in magnitude as the capacitance decreases from its nominal
value of 10 pF to 0 pF and increases in magnitude as the capacitance decreases from
() to ——10 pF. For the following figures for the inductor, the reflection coefficient de-
creases in magnitude as the inductance decreases from its nominal value of 0.1 nH
to 0 nH and increases in magnitude as the inductance decreases from 0 to —0.1 nH.
The transmission coefficient increases in magnitude as the inductance decreases from
its nominal value of 0.1 nH to 0 HH and decreases in magnitude as the inductance
decreases from O to —0.1 nH. It should be noted that 0 pF denotes an open circuit and
O nH represents a short circuit. The phase of the reflection coefficient and transmis-
sion coefficient relatively remains the same for both the capacitor and inductor as it
transitions from positive to negative reactance. The half-length microstrips modeled
in Sonnet were used in this parameter study because it produces results that will be

similar to experimental data.

155

S11-Magnitude (Negative Capacitor - Cascaded System)

 

 

    

 

 

 

 

 

 

'15 I a I
91:88 a a 89%,.-- N» 1 -10 pFL
_ fw‘” Fifi . t , 1341+}? BB. 3‘3 _ :
20 a E] B B B‘EHZI-E] E1 *3 B 8 EH} TMTFW + g p: C3
‘—Cl—' _ p
4% I “I:
i . , ’ %
L
A ' P ' \fi
CD N.
"C
v ....
‘_
(73
-45 ~ w
--50 - I.
l
-55- j
-60 _ _
-85 , . .
6 6.5 7 7.5 8

Frequency (GHz)

Figure 4.67. Magnitude of reflection coefficient of parameter study of cascaded system

with capacitor varied from 10 pF to —10 pF

156

 

S11-Phase (Negative Capacitor - Cascaded System)

200

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I , VF l“---1IOpF
. ‘éfii+_6 pF h
150 _D_._2 pl:
~-—*—2pF
100 +6pF
A +10pF
? 50 . \
8 i
a mi . . .1
m E
(D \ _
2 .504 fix __ _
o. g ,
A a
-100~* ‘ , \g‘fi . , _
‘ EEK
-15o-1 - a;
i
‘2006 6:2 6:4 6:6 6:8 ‘7 73 7:4 7:6 7:8 8

Frequency (GHz)

Figure 4.68. Phase of reflection coefficient of parameter study of cascaded system

with capacitor varied from 10 pF to —10 pF

157

821 (dB)

 

 

 

 

 

 

 

 

 

SZ1-Magnitude (Negative Capacitor - Cascaded System)
-O.12 I I , I T I r , L
-~ -10 pFfi
v ’*_ ‘6 pF 57
“0.14 _D_ _2 pl: F;
“ -~— 2 pF
__ + 6 pF £3
0.16 + 10 pF
as
, r**--**:r~x
—O.18 , Ep/ , x x-
-0.2 . q
-o.2z _
—o.24— ’ _
_0'266 62 6:4 6:6 618 t 712 7:4 7i6 7:8 8

Frequency (GHz)

Figure 4.69. Magnitude of transmission coefficient of parameter study of cascaded

system with capacitor varied from 10 pF to —10 pF

158

 

 

 

 

 

 

 

 

SZ1-Phase (Negative Capacitor - Cascaded System)
200 f ' T F I I I I L
[R in," -10 pF
_ : ... , , , +-6 9F _
150 l 1‘ ‘D“"2 PF t
i 1; fl~2pF
100 — ‘ [I y L" \.. * + 6 pF
A I : ' +10 pF
am) 50 _ I V i _
e I .v’ ;
C) i ' "I '
a) l “ a: :
e 0* . " I ‘
ax
ff“ —50i- Ii \t: i —
—10 i I: ' i d
i fix: i
—1 5 3 i -
? *" t. i

  

 

 

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.70. Phase of transmission coefficient of parameter study of cascaded system

with capacitor varied from 10 pF to —10 pF

159

 

S11-Magnitude (Negative Inductor - Cascaded System)
-20 1 1 I r

 

 

 

 

 

 

 

 

, -0- -0.1 nH m
—*— '0.06 NHLw
i i : ~ ~~ . I -—a—--o.02 nH’
401%»: ~~ a g ., \ +002 nH a:
h \ +0.06 nH
I +0.1 nH
\_ / A
A ‘40 >63 ‘ ‘ ‘
m ”.
‘— / ‘
‘— [j i
CD ~ %
_50 iii, i 3.-
. I 11 i
U i. I} 1 i
\q‘ [p l“ l1
\t 1' 1, i
1: i
1 if
i . i
_7 a 1 i 1 1 1 1 1 1
6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.71. Magnitude of reflection coefficient of parameter study of cascaded system

with inductor varied from 0.1 nH to —0.1 nH

160

S11-Phase (Negative Inductor - Cascaded System)

 

 

   
   
    
   

200 r a a a. a I , I I
.1 ' ' -o.1 nH
-—+——-0.06 nH
15° -o.02 nH
—-~—o.02 nH
100' +0.06 nH *

 

 

 

50 “\a

 

Phase (Degrees)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6
Frequency (GHz)

Figure 4.72. Phase of reflection coefficient of parameter study of cascaded system

with inductor varied from 0.1 nH to —0.1 nH

161

321 (dB)

$21-M

agnitude (Negative Inductor - Cascaded System)

 

 

 

 

 

 

 

 

-O.12 r I I
-O.1 nH
’—*— -0.06 nH
-O.13~ -D. -0.02 nH
Gig? ‘ “P“ 0.02 nH
—o.14ia:%a t; ‘52:ng +006 “H i
V regain ‘ ~ ~ +1 + 0.1 nH
53R c j . a0
_O.15 ” h \f t -- -
-0.16 — 1» _
-O.17F _
-0.18- -
_O.19 1 1 i 1 1 n 1 1 i
6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8

Frequency (GHz)

Figure 4.73. Magnitude of transmission coefficient of parameter study of cascaded

system with inductor varied from 0.1 nH to —0.1 nH

162

SZ1-Phase (Negative Inductor - Cascaded System)

 

 

   

 

 

 

 

 

 

 

 

 

 

 

200 1 r e . w e a l 1
w “a” -0.1 nH
~1—--O.06 nH
150 —a-. -o.02 nH
—~~«o.02 nH it
100* +0.06 nH
A +0.1nH
8 50— ~ A- -
e
U)
(D
e 0* *
(D
(D
E —501 ~
0.
-10 _
-15 a

 

 

  

 

 

6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Frequency (GHz)

Figure 4.74. Phase of transmission coeflicient of parameter study of cascaded system

with inductor varied from 0.1 nH to —0.1 nH

163

CHAPTER 5

CONCLUSIONS AND FUTURE WORK

The purpose of this thesis is to analyze the behavior of a negative impedance converter
attached to a microstrip transmission line in the C-band frequency bandwidth. This
required modeling the NIC such that it operated in this frequency range. Using
microwave network analysis, Fosters reactance theorem, and circuit analysis, non-
Foster circuits can be realized with NICs. The non-Foster circuit that was observed
was a grounded, voltage inversion, open—circuit stable NIC. This N IC was modeled
using bipolar-junction transistors in such a way that it would be able to operate in the
frequency bandwidth of interest and have it produce at optimal performance. In this
particular case, the main transistor operated in the common-base configuration to
reduce instabilities inherent for BJTs. Another transistor operating in the common-
emitter configuration was present in the non-Foster circuit, but its purpose was to
bias the voltage at its output to be in phase with the input voltage of the overall
circuit.

The two—stage de-embedding process by the way of Thru—Re ect—Line calibration
was needed to measure the S-parameters of the NIC without any effects from the
microstrip, test fixture, and test port cables. The first stage removed the measure-
ments of the test fixture and recorded only the data of the composite system of the
NIC attached to the microstrip. The second stage removed the measurements of
the microstrip and tabulated information of only the NIC. Transmission line the-
ory, microstrip circuit analysis, and microwave network analysis involving scattering
and transmission parameters were required to perform the TRL calibration technique
standards.

Once all the individual components of the entire system have been modeled in

164

Sonnet and Ansoft Designer, computational analysis was performed to analyze how
the NICS would behave at. high frequencies. The NICS that were observed were
a negative capacitor and a negative inductor. A parameter study of components
embedded within the half-length microstrips designed in Sonnet was performed to
observe how the cascaded network would perform. A 50-9 resistor, a 75-9 resistor,
a IOU-Q resistor, a 10-pF capacitor, and a lO-nH inductor were used to conduct this
study. Then, the reactive component values were varied from positive to negative
values to see the trends of the performance of the cascaded network. This helped
distinguish a negative capacitor from a positive inductor and a. negative capacitor
from a positive capacitor.

Significant work still needs to be done with this project. Future work to consider
is use additional circuit. simulation software programs such as Agilent ADS (Advanced
Design System) or Ansoft HFSS (High Frequency Structural Simulator) to analyze
how the NIC circuit will interact with the half-length microstrips. This will help the
experiment alist observe how the NIC will perform using silicon—germanium transistors
as they produce lower noise, higher amplification, and need less power to achieve
maximum performance at. high frequencies compared to silicon transistors.

The circuit will be built and tested using the TRL calibration procedure outlined in
[19] comparing all simulated models and the computational information evaluated in
MATLAB. In addition, the characteristic impedances of the Anritsu 3680-20 Universal
Test Fixture and the test port cables of the VN A (in this case the HP 8510 vector
network analyzer) need to be measured to construct a better model of connectors A
and B for the calibration method. Also, the TRL calibration procedure needs to be

performed on the VNA outlined in [19].

165

BIBLIOGRAPHY

166

 

[1]

[10]

[111

[12]

BIBLIOGRAPHY

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C.A. Balanis, Advanced Engineering Electromagnetics, lst edition, W'iley, N.Y.,
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D.1\I. Pozar, Microwave Engineering, 3rd Edition, Wiley, N.Y., 2005.

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“Agilent Network Analysis Applying the 8510 TRL Calibration for Non-Coaxial
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P. Barba, “Numerical S~parameter and characterization of Inhomogeneously
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R. Franklin, “IVIicrowave Test FIxture Characterization De—embedding Tech-
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P. J. Matthews and J. J. Song, “RF Impedance Measurement Calibration,” LS
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