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This is to certify that the
thesis entitled

Error Analysis of Material Characterization Using a Wave
Guide Applicator

presented by

Jason Jeffrey Meeusen

has been accepted towards fulfillment
of the requirements for the

 

 

 

MS. degree in Electrical Engineering
MaTo} Professor’s Signature
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To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

AUG 2, (l 2005

 

 

 

 

 

 

 

 

 

 

 

 

 

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Error Analysis of Material Characterization Using a Wave
Guide Applicator

By

Jason Jeffrey Meeusen

A THESIS

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

MASTER OF SCIENCE
Electrical and Computer Engineering

2004

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ABSTRACT

Error Analysis of Material Characterization Using 3 Wave Guide
Applicator
By

Jason Jeffrey Meeusen

Error analysis is the process of finding the amount of estimated error in a final
calculated result, given the uncertainty of all the variables within the calculation.
This thesis will cover an error analysis method applied to the characterization of
electromagnetic materials, which uses the error propagation equation to estimate the
final uncertainty.

The thesis will first discuss the extraction of the electromagnetic properties of a
material using an X-band wave guide system. Next, the identification of errors such
as uncertainty in network analyzer measurements, sample thickness, and waveguide
dimensions will be discussed. Finally, the errors must be quantified so they can be
used to obtain an uncertainty on the complex permeability and permittivity of the
material sample by using the error propagation equation.

Results will show that each contributing factor to the uncertainty of the complex
permeability and permittivity is largely dependant on the electromagnetic properties

of the material sample.

For My Loving Wife and Supportive Family

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ACKNOWLEDGMENTS

First and foremost I would like to thank my wife, Katherine, for her endless
patience and support through my journey in graduate school. Without her, I would
not have had anything but school in my life. I only hope that I can now repay her
for what she has done for me.

A sincere thank you to my graduate adviser, Dr. Rothwell. The writing of this
thesis would not have been possible without his generous support and knowledge. He
has an unparalleled sense of humor and he took me in when I needed help. Words
cannot express my gratitude.

I would also like to thank Lydell Frasch and Garrett Heil at Boeing for their
generous financial support and help pertaining to the research involved with this
thesis.

Thanks to Brian Wright and Roxanne Peacock for all of their advice and inter-
esting conversations throughout my time here at MSU. I thank them also for their
generous support by letting me work for them through most of my college career.

Finally, I would like to thank my parents. Without them, college would have never

been an option. Thank you for believing in my potential.

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TABLE OF CONTENTS

LIST OF TABLES ................................. vii
LIST OF FIGURES ................................ viii
CHAPTER 1
Introduction ..................................... 1
CHAPTER 2
Material Characterization ............................. 3
2.1 Methods of Material Characterization .................. 3
2.1.1 Wave Guide Applicator ...................... 4
2.2 Practical Uses ............................... 5
CHAPTER 3
Derivation of Equations .............................. 7
3.1 Waveguide Electromagnetic Theory ................... 7
3.2 Forward Relationship ........................... 11
3.2.1 Forward Relationship Results .................. 19
3.3 Inverse Relationship ........................... 21
3.3.1 Inverse Relationship Results ................... 27
3.4 Compensation Within Inverse Relationship ............... 29
3.4.1 Compensated Inverse Relationship Results ........... 31
CHAPTER 4
Error Analysis of Material Characterization ................... 44
4.1 Measurement of Errors .......................... 44
4.1.1 Systematic Errors ......................... 45
4.1.2 Random Errors .......................... 46
4.1.3 Standard Deviation ........................ 46
4.2 Error Analysis ............................... 47
4.2.1 Propagation of Errors ...................... 47
4.2.2 Computer Calculation of Uncertainties ............. 50
4.3 Spline Method ............................... 52
4.4 Error Analysis Results .......................... 54
CHAPTER 5
Conclusion ..................................... 125
APPENDIX A
Fortran Code of Forward Problem ......................... 128

 

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APPENDIX B

Fortran Code of Inverse Problem ......................... 131
APPENDIX C
Fortran Code of Inverse Problem with Compensation .............. 134
APPENDIX D
Fortran Code of Inverse Problem with Error Analysis .............. 138
APPENDIX E
Matlab Code for Graphing Fortran Output With MagRAM .......... 160
APPENDIX F
Matlab Code for Graphing Fortran Output With Rexolite ........... 162
BIBLIOGRAPHY ................................. 165

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Table 2.1
Table 2.2

Table 3.1

Table 3.2
Table 3.3
Table 3.4
Table 3.5
Table 3.6
Table 3.7
Table 3.8

Table 3.9

Table 3.10

Table 4.1

Table 4.2
Table 4.3

Table 4.4
Table 4.5

Table 4.6
Table 4.7

LIST OF TABLES

Information for the waveguide used by The Boeing Company . . . 4

Information for the measurement equipment used by The Boeing

Company ............................... 5
Material properties for MagRAM as provided by The Boeing Com-

pany .................................. 19
Calculated S-parameters for MagRAM using Forward program . . 20
Material properties for Rexolite as provided by The Boeing Company 20
Calculated S-parameters for Rexolite using Forward program . . . 21
Calculated p and c for MagRAM using Inverse program ..... 28
Calculated p and c for Rexolite using Inverse program ...... 29
Sample data for MagRAM provided by The Boeing Company . . 32
Calculated material parameters for MagRAM using Compensated

Inverse program ............................ 33
Sample data for Rexolite provided by The Boeing Company . . . 34
Calculated material parameters for Rexolite using Compensated

Inverse program ............................ 35
Input arguments used for independent error analysis of Rexolite

and MagRAM using Error Analysis program ............ 54
Comparison of resultant uncertainties for Rexolite ......... 55
Different cables used for error analysis using setup #1 for the com-

parison ................................. 56
Different setups used for error analysis as shown from Table 2.2 . 58
Input arguments used for independent error analysis of Rexolite

and MagRAM using Error Analysis program ............ 58
Comparison of resultant uncertainties for MagRAM ........ 59
Comparison of resultant uncertainties for Rexolite ......... 60

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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 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

LIST OF FIGURES

Waveguide with one material layer

ooooooooooooooooo

Waveguide with one material layer

Waveguide with multiple material layers ..............

Illustration of phase compensation for a piece of material in a
waveguide ...............................

The Boeing Company results vs. the Compensated Inverse pro-
gram results for permeability of MagRAM .............

The Boeing Company results vs. the Compensated Inverse pro-
gram results for permittivity of MagRAM

The Boeing Company results vs. the Compensated Inverse pro-
gram results for permeability of Rexolite ..............

The Boeing Company results vs. the Compensated Inverse pro-
gram results for permittivity of Rexolite ..............

Worst case error for SH and setup #1 using “HP 8510 Specifica-
tions & Performance Verification Program”

Worst case error for S22 and setup #1 using “HP 8510 Specifica-
tions & Performance Verification Program”

oooooooooooo

Worst case error for S12 and setup #1 using “HP 8510 Specifica-
tions & Performance Verification Program”

Worst case error for S21 and setup #1 using “HP 8510 Specifica-
tions & Performance Verification Program”

Error analysis of permeability for 0.3% tolerance in material thick-

ness for MagRAM

Error analysis of permittivity for 0.3% tolerance in material thick—
ness for MagRAM

Error analysis of permeability for 0.3% tolerance in material thick-
ness for Rexolite

oooooooooooooooooooooooooo

...........................

Error analysis of permittivity for 0.3% tolerance in material thick-
ness for Rexolite

ooooooooooooooooooooooooooo

Error analysis of permeability for 1.0% tolerance in material thick-
ness for MagRAM

Error analysis of permittivity for 1.0% tolerance in material thick-
ness for MagRAM

Error analysis of permeability for 1.0% tolerance in material thick-
ness for Rexolite

oooooooooooooooooooooooooo

Error analysis of permittivity for 1.0% tolerance in material thick-
ness for Rexolite

ooooooooooooooooooooooooooo

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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

Figure 4.24

Figure 4.25

Figure 4.26

Figure 4.27

Figure 4.28

Figure 4.29

Figure 4.30

Figure 4.31

Error analysis of permeability for 3.0% tolerance in material thick-
ness for MagRAM

Error analysis of permittivity for 3.0% tolerance in material thick-
ness for MagRAM

oooooooooooooooooooooooooo

oooooooooooooooooooooooooo

Error analysis of permeability for 3.0% tolerance in material thick-
ness for Rexolite

Error analysis of permittivity for 3.0% tolerance in material thick-
ness for Rexolite

ooooooooooooooooooooooooooo

Error analysis of permeability for 0.3% tolerance in waveguide
width for MagRAM ..........................

Error analysis of permittivity for 0.3% tolerance in waveguide

width for MagRAM ..........................
0.3%

Error analysis of permeability for tolerance in

width for Rexolite

waveguide

Error analysis of permittivity for
width for Rexolite

Error analysis of permeability for 1.0% tolerance in waveguide

width for MagRAM ..........................

Error analysis of permittivity for 1.0% tolerance in waveguide

width for MagRAM ..........................
1.0%

Error analysis of permeability for tolerance in

width for Rexolite

waveguide

Error analysis of permittivity for
width for Rexolite

oooooooooooooooooooooooooo

Error analysis of permeability for 3.0% tolerance in waveguide

width for MagRAM ..........................

Error analysis of permittivity for 3.0% tolerance in waveguide
width for MagRAM ..........................

3.0%

oooooooooooooooooooooooooo

Error analysis of permeability for tolerance in

width for Rexolite

waveguide

Error analysis of permittivity for
width for Rexolite

oooooooooooooooooooooooooo

Error analysis of permeability using test setup #1 with cable model
HP-85132C and MagRAM ......................

Error analysis of permittivity using test setup #1 with cable model
HP-85132C and MagRAM ......................

Error analysis of permeability using test setup #1 with cable model
HP-85132C and Rexolite .......................

ix

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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

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

Error analysis of permittivity using test setup #1 with cable model
HP-85132C and Rexolite .......................

Error analysis of permeability using test setup #1 with cable model
HP-85132D and MagRAM ......................

Error analysis of permittivity using test setup #1 with cable model
HP-85132D and MagRAM ......................

Error analysis of permeability using test setup #1 with cable model
HP-85132D and Rexolite .......................

Error analysis of permittivity using test setup #1 with cable model
HP-85132D and Rexolite .......................

Error analysis of permeability using test setup #1 with cable model
HP-85132E and MagRAM ......................

Error analysis of permittivity using test setup #1 with cable model
HP-85132E and MagRAM ......................

Error analysis of permeability using test setup #1 with cable model
HP-85132E and Rexolite .......................

Error analysis of permittivity using test setup #1 with cable model

HP-85132E and Rexolite .......................

Error analysis of permeability using test setup #1 with cable model
HP-85132F and MagRAM ......................

Error analysis of permittivity using test setup #1 with cable model
HP—85132F and MagRAM ......................

Error analysis of permeability using test setup #1 with cable model
HP-85132F and Rexolite .......................

Error analysis of permittivity using test setup #1 with cable model
HP-85132F and Rexolite .......................

Error analysis of permeability using test setup #1 with no cable

and MagRAM .............................

Error analysis of permittivity using test setup #1 with no cable
and MagRAM .............................

Error analysis of permeability using test setup #1 with no cable
and Rexolite

.............................

Error analysis of permittivity using test setup #1 with no cable
and Rexolite

ooooooooooooooooooooooooooooo

Error analysis of permeability using test setup #2 and MagRAM

Error analysis of permittivity using test setup #2 and MagRAM .

Error analysis of permeability using test setup #2 and Rexolite
Error analysis of permittivity using test setup #2 and Rexolite

Error analysis of permeability using test setup #3 and MagRAM

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

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Figure 4.55
Figure 4.56
Figure 4.57

Figure 4.58

Figure 4.59

Figure 4.60

Figure 4.61

Figure 4.62

Figure 4.63

Figure 4.64

Error analysis of permittivity using test setup #3 and MagRAM .
Error analysis of permeability using test setup #3 and Rexolite
Error analysis of permittivity using test setup #3 and Rexolite

Error analysis of permeability using test setup #1 with 0.3% ma-
terial thickness tolerance and 0.3% waveguide width tolerance for

MagRAM ...............................

Error analysis of permittivity using test setup #1 with 0.3% ma-
terial thickness tolerance and 0.3% waveguide width tolerance for
MagRAM ...............................

Error analysis of permeability using test setup #1 with 0.3% ma-
terial thickness tolerance and 0.3% waveguide width tolerance for
Rexolite ................................

Error analysis of permittivity using test setup #1 with 0.3% ma-
terial thickness tolerance and 0.3% waveguide width tolerance for
Rexolite ................................

Error analysis of permeability using test setup #1 with 3.0% ma-
terial thickness tolerance and 3.0% waveguide width tolerance for
MagRAM ...............................

Error analysis of permittivity using test setup #1 with 3.0% ma-
terial thickness tolerance and 3.0% waveguide width tolerance for
MagRAM ...............................

Error analysis of permeability using test setup #1 with 3.0% ma-
terial thickness tolerance and 3.0% waveguide width tolerance for
Rexolite ................................

Error analysis of permittivity using test setup #1 with 3.0% ma-
terial thickness tolerance and 3.0% waveguide width tolerance for
Rexolite ................................

xi

114
115
116

117

118

119

120

121

122

123

124

CHAPTER 1

INTRODUCTION

When working with a material, engineers often need the electromagnetic properties
(constitutive relations) of the material in question. Reasons for needing the material
properties can vary greatly. An engineer might be interested in the material’s ability
to absorb radar energy to avoid detection. Another engineer might be trying to shrink
the physical size of an antenna by surrounding it with a dielectric while keeping the
antenna’s electrical size constant. In both cases, the engineers need to know the
constitutive relations of the material and the frequencies at which the values are
valid. Without an accurate way to determine the constitutive relations of materials,
both tasks would be very difficult guessing games.

To understand how a material will react in the presence of an electromagnetic
wave, four things are needed [1]. First, you need Maxwell’s equations. Second, you
need to know the electromagnetic properties of the material in question. Third,
you need to know the incident electromagnetic fields. Finally, you need to know the
boundary conditions. Knowing these four things enables one to calculate the reflected
and transmitted waves associated with that medium and its incident waves [2].

It is also possible to restate the problem. Instead of finding the transmitted
and reflected waves, we could calculate the permeability (,u) and permittivity (c) of
an unknown material by measuring the transmitted and reflected fields [3]. Since
the boundary conditions, incident, transmitted and reflected waves are known, the
electromagnetic properties can be calculated, within a certain tolerance, for a piece

of material.

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Tolerance is the key word. When dealing with the accuracy of calculating a and
6, many factors come into play. One will notice that there are many other things
that need to be measured other than the incident and reflected waves. For example,
the material thickness, the dimensions of the waveguide and the accuracy of the
equipment used to do the measurements need to be represented in some way. Every
measurement has an error or tolerance associated with it that effects the final result
[4, 5]. Using methods discussed in this thesis, the uncertainty of e and ,u can be
obtained.

This thesis is divided into 5 chapters. Chapter 2 will discuss material properties,
why they are needed and what methods can be used to obtain them. Chapter 3 will
discuss the method and calculations used to obtain the material properties and how
to verify the results. The error analysis will be covered in Chapter 4 and includes
types of errors and their inclusion into the material property calculations. Lastly,

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CHAPTER 2

MATERIAL CHARACTERIZATION

For the purpose of this thesis, the materials are assumed to be isotropic and homoge-
neous to make the derivations easier. In order to describe how the material responds

to an incident wave, constitutive relations must be formed [1]. The constitutive rela-

(7‘11

tions for the derivations in this thesis are B = riff and D = c .

When the constitutive relations, or electromagnetic properties, of a material are
known, the transmitted and reflected waves can be calculated or simulated to aid in
the development of a product. Not all the energy from an incident wave is transmitted
or reflected. Some of the energy is absorbed into the material itself and dissipated as
heat. In order to compensate for the losses associated with a medium, it is necessary
to describe p. and 6 as complex values: it = u’ —jp", c = 6, —j6” [1]. Describing p, and

c in complex form also eliminates the need to include conductivity (0) in Maxwell’s

equations since the effects of 0 can be included in complex 6 [6].

2.1 Methods of Material Characterization

There are several methods that exist for the measurement of the permittivity and
permeability of a given material. Generally, a sample is placed in the path of a
traveling electromagnetic wave, either in free space, inside a propagating structure or
resonator cavity. Since the reflection and transmission coefficients of the experimental
device are directly related to electromagnetic properties of the material, they are
measured using a network analyzer and used to find the permittivity and permeability

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space as well as bounded free space measurements.
2.1.1 Wave Guide Applicator

The method of material characterization explored in this thesis uses a propagating
structure or waveguide. This waveguide is owned and operated by The Boeing Com-
pany and is built to run in the X-band of frequencies. Listed below in Table 2.1 is

what is known about the waveguide used for the material measurements.

 

 

 

 

Waveguide
Dimensions 0.02286 x 0.01016
(Inches)
Frequency Range 8.2-12.4
(GHz)

 

 

 

Table 2.1. Information for the waveguide used by The Boeing Company

A detailed equipment list is also provided by The Boeing Company which is es—

sential in the error analysis portion of the thesis. This list is shown in Table 2.2.

 

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Measurement Equipment
Name Part Number

Setup #1 Network Analyzer HP-8510C (85103C & 85102B)
Test Set HP-8514B (45 MHz - 20 GHz)
Source HP-834IB (10 MHz - 20 GHz)
Calibration Kit HP-85050C (7mm Precision Cal Kit)
Calibration Technique Thru-Reflect-Line (TRL)
Test Port Cables HP-85132E (3.5mm - 7.0mm)

Setup #2 Network Analyzer HP—8510C (85101C & 851028)
Test Set HP-8517B (45 MHz - 50 GHz)
Source HP-83651A (10 MHz - 50 GHz)
Calibration Kit HP-85050C (7mm Precision Cal Kit)
Calibration Technique Thru-Refiect-Line (TRL)
Test Port Cables HP-85132E (3.5mm - 7.0mm)

Setup #3 Network Analyzer HP-8510C (85101C & 85102B)
Test Set HP-8515A (45 MHz - 26.5 GHz)
Source HP-834IB (10 MHz - 20 GHz)
Calibration Kit HP-85050C (7mm Precision Cal Kit)
Calibration Technique Thru—Reflect-Line (TRL)
Test Port Cables HP-85132E (3.5mm - 7.0mm)

Setup #4 Network Analyzer HP—8510C (85101C & 85102A)
Test Set HP-8515A (45 MHz - 26.5 GHz)
Source HP-8341A (10 MHz - 20 GHz)
Calibration Kit HP-85050C (7mm Precision Cal Kit)
Calibration Technique Thru-Reflect-Line (TRL)
Test Port Cables HP-85132E (3.5mm - 7.0mm)

 

 

 

 

Table 2.2. Information for the measurement equipment used by The Boeing Company

2.2 Practical Uses

Material response is an integral part of electromagnetics and cannot be ignored. Ma-
terial characterization is useful for many different areas of Electrical Engineering. An
example would be an antenna engineer using a dielectric to shrink the physical size
of an antenna while keeping the same effective electrical size to fit a small enclosure
for a GPS system or cellular phone.

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erties of the dielectric need to be obtained so the engineers can determine the correct
radius required for the desired propagating impedance.

Lastly, an aircraft manufacturer could use the electromagnetic properties of a
material to make the skin of their aircraft absorbent to a specific range of frequencies.
It is, therefor, necessary to measure material properties as accurately as possible and

understand the limitations of such a measured result.

 

 

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CHAPTER 3

DERIVATION OF EQUATIONS

Although the goal of this thesis is to determine the uncertainty of the relative per-
meability and permittivity of a medium, the derivation of the equations used for
measuring the material properties is necessary in understanding the whole procedure.
The first calculations to be derived will establish the relationship between the trans-
mission and reflection coefficients of a guided wave incident on a flat, parallel piece
of material with predetermined ,u and c. This relationship between material property
and the incident wave will be referred to as the “forward relationship”.

The transmission and reflection coefficients in this thesis will be referred to as
7' and p respectively. When referring to the measured quantities at. the waveguide
ports, the transmitted and reflected waves will be referred to as their “S-parameter”
equivalents or S21 and 811 respectively.

Once the forward relationship is established, the relationship in the inverse direc-
tion can be derived. A relationship in the inverse direction would determine the u
and e of a material given 7' and p. The inverse relationship is essential to the error

analysis of the measured )1 and c.

3.1 Waveguide Electromagnetic Theory

The calculations used and derived in this section are cited largely from [1] and [2].
For the rectangular waveguide used in experiments reported in this thesis, the
electromagnetic fields are traveling in the 2 direction with the lowest mode of prop-

agation to ensure that there is only one propagating wave to measure. The lowest

 

 

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mode of propagation for The Boeing Company’s waveguide is the TE10 mode, there-
for, everything will be derived for this mode.

The setup used for the experiments reported in this thesis can be seen in Figure
3.1 where the waves that exist outside the material can be represented as incident,
transmitted and reflected waves. The waves propagating inside of the material are
represented as a and fl. Since Em, Em”, and a are propagating in the +2 direction
and are assumed to be polarized in the +i direction, they can be represented by the
exponential function

Ea; = Ae—jkzz (3.1)

Also in Figure 3.1 the waves Eref and B are traveling in the —23 direction and can be

represented by the exponential function

E; = Bejkzz (3.2)

Inside each exponential, the term k2 is called the axial wavenumber and it relates
the velocity of the traveling wave in the Z direction with its angular frequency. The

general form of the wavenumber is written as

k=f=wfin an

'1)

where

k2 = k3, + k5 + kg (3.4)

In order to find kz, it is necessary to find the cutoff wavenumber (kc), or the wavenum-

 

 

 

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1332
1.3 = k2 -— k3, = k2 (1— 75%) (3.7)

substituting (3.3) and (3.5) into (3.7) results in

2
a, = k-‘/1—§C§ (3.8)

where fC is the cutoff frequency inside the waveguide while propagating in a particular
medium.

In order to obtain the cutoff frequency, the eigenvalues of the waveguide need to
be calculated. The eigenvalues are the transverse wavenumbers kg; and ky determined
from the boundary conditions on the waveguide walls. The eigenvalues in this problem

are calculated as

kw = 175 m=0,1,2... (3.9)
0
kg = % n=0,1,2... (3.10)

where a and b are the dimensions of the waveguide and the integers m and n. are used

to represent the mode that the waveguide is operating in (TEmn). The eigenvalues

 

 

can be "

H? 'r

-l 5"1 A

I "—
“.9

F.
if
The ‘

 

can be substituted into the equation for kc

 

kc = wig“; z l/(ZEIY‘L (1)92

Solving for fC by equating (3.5) and (3.11) gives

 

+

arm-(H

fc -_— 27r\1/ZE\/(:ndz)2

(ii-T

 

 

+
A
slii
v

to

or since the wave is propagating in the TE10 mode

1

fc=2a\/;L—€-

 

where 11 and 6 depend on the medium in which the wave is propagating.

The I? fields in the waveguide can be calculated by using Faraday’s law

-o

VxE=—jwpff

For the forward propagating waves, substituting (3.1) into (3.15) gives

Ae—jkzz
Hy = T

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

where C refers to the wave impedance inside a waveguide for f > fC and is written as

 

10

(3.17)

 

 

If“ '
...~ eq;
The ‘

and are 1

,.
E‘II'
1".

 

Similarly for the backward propagating waves, substituting (3.2) into (3.15) gives

Bejk: Z

Hy: C

 

Using (3.3) and (3.8), equation (3.17)is rewritten as

 

um
C _
2
w,/;1€ — f2-
and then simplified to
I p.
C = —
2 6
_ J;
f2

This equation for C will be used in the rest of the thesis.

(3.18)

(3.19)

(3.20)

The transmission (T) and reflection (p) coefficients are ratios of the electric fields

and are defined as

 

7’ : Etran
Einr'
[1 _ Ercf
Einc

3.2 Forward Relationship

(3.21)

(3.22)

The calculations used and derived in this section are cited largely from [1] and [2]

The “Forward Relationship” derived here will relate the transmission and refiec-

tion coefficients to the relative permeability and permittivity (pmr and (Wu) within

a material slab given umr ° #0 = um and cmr - 60 = em, as depicted in Figure 3.1.

11

 

9.24 <13.

4‘.

,
Since IE1.

Vim; ‘
\l)

Hymn!

”17‘er r

J [1"]

 

 

 

In order to start this derivation, two new variables E,- and E f will be introduced as

shown in Figure 3.2 and are defined as

Er = Einc 'I' Eref (3'23)

Ef : Etran (3.24)
and similarly

Hi : Hinc "I" Href (325)

Hf = Htran (326)

Since these variables represent the transverse components of the electric and magnetic
fields, vector notation is not necessary. Using (3.1), (3.2), (3.16) and (3.18), the above

magnetic field equations can be found to equal

[‘1' : mc _ 76f 327
1 C0 C0 ( )
Etrrm
H = — 3.28
f (0 ( )

where (0 is the wave impedance for bounded free space. Using (3.20), (0 is found to

1
C0 = —f2 g (3.29)
1‘ T

equal

where fd) is

(3.30)

12

 

“IF-n: p
. .
. l

AIfi‘Jh is t

AS in

 

31

VI.

 

which is the cutoff frequency inside bounded free space derived from (3.14).

The waveguide problem will be further complicated by introducing more than one
layer of material inside, as depicted in Figure 3.3. It can be seen from Figure 3.3 that
there are two waves propagating in each layer of material. These waves are labeled
am and 6m respectively.

Inside the material, the a and I3 waves can be combined into a single expression
and written as

E17 = Ante—jkznlz + B1716‘jkzmz (3.31)

where the first exponential represents a wave traveling in the +2 direction and the
second exponential represents a wave traveling in the —z direction. Since (3.31) is

traveling in a material layer, the km, term is expressed from (3.8) as

 

2
w .
kzm : ’C‘Vflmrcmr 1 “ f}? (332)
where c is the speed of light and fan is
f — ——3——— (3 33)
cm 2‘1 \/ Hmr € 1711‘ I
which is the cutoff frequency inside the material derived from (3.14).
As in section 3.1, the If field can be calculated from (3.15) to be
A ‘jkzmz _ , .779sz
Hy = me Ema (3.34)

 

CT",

where (m is the wave impedance inside of the material layer. (m can be expressed

13

from

where '

I311-

szlvai .

 

([1

 

from (3.20) as

 

1 1. 1. .
Cm = ———‘ ] ‘l—O‘ ] I mr (3.35)
60 6mr

1_ fgm
f2

 

where fan is given in (3.33).
The E and If field equations on the first boundary of a material layer can be

solved in terms of the second boundary by first letting

 

 

Ex," = E,;(;.~ = zm) = A,,,e—jk3m2m + B,,,ef"zmzm (3.36)
A, fjkzmzm _ B_ {jkzmzm
Ham : Hy(z : 27") : rue rne (3.37)
' Cm
E;I.‘m—l = EI(Z = gin—1) z Ame—jkzmzm—l + Brrte‘jkfzmzm—1 (338)
A ,e-J'kzmzm—l _ B ejkszm—l
CT".
Solving equations (3.36) and (3.37) in terms of Am, and Bm gives
Ex," +cmHym = 2Ameri’vmzm (3.40)
Ell/‘1” — C1,),Hym : ZBynejkzmzm (3.41)
SO

1 H eikszm 3 42
Am = '2‘ [Exm'l‘Cm ym] , ( - )

I _.- .
Bm = 5 [Esrm " CmHym] 8 szm~m (3.43)

14

ff)

.__.

wt we

"',-‘ P
. A
HAAAAJAL“ 1

 

EQUa
Blind (if:

titration:

.L

Previoug

D?

 

Substituting (3.42) and (3.43) into (3.38) and (3.39) yields

 

 

1 . . _ - . ,
EIm—l : 5 [Earm 'i' CmHym] ejkzmqne Jklmzm—l
I _: .~ , ,' -~ 4
+5 [Exm — (ml—131m] 8 Jk“m~m€-’k"m‘m-l
ejkzmtm _+_ e-jkzmtm ejkzmtm _ e—jkzmtm
= E1171 2 + Canym 2
: Exm COS (kzmtm) + jCynHym Sin (kzyntm) (3.44)

where tm = 2m — zm_1

Similarly,

Sin (kzmtm)

Hym—l = Hym C05 (kzmtm) + jEzm Cm

(3.45)

Equations (3.44) and (3.45) are written so that the E and If fields on the m — 1
boundary is completely represented in terms of the mth boundary. By arranging the
equations in this order, each material layer in Figure 3.3 can be represented by the
previous and next layer’s electric and magnetic field. Arranging (3.44) and (3.45) into

matrix form gives

Exm_1 C05 (kzmt'm) ij Sin (kzmtm) El'm
= _ ' (k t ) (3.46)
.7 sm ‘ .
Hym-l 2m m COS (k-zmtm) H 31m
Cm.
or in a more compressed form
E1; _ E1
m 1 = M m (3.47)
Hym—l Hym

15

_ L

 

where

Be
layer a:

‘
T2 mi'f‘l ‘

or

NC" 111::

(K “E

A.‘ “I. Orig) 1

 

where the wave matrix is given by

1” cos (kzmtm) ij Sin (kzmtm) ( )
, = 3.48
‘. 4 . k7 t ’
M COS (kzmtm)

(771

Because of the continuity of El. and Hy at each interface, the fields exiting one
layer are equal to the fields incident on the next layer as seen in Figure 3.3. So, for

n layers of material, the matrices can be cascaded as

E, _ E
Tm‘ =MIMQ...M,, 1"" (3.49)
Hym—l Hym
or
E, _ n E,
1'" 1 = I] Mm I'" (3.50)
Hym~l m=l Hym

Since all the la ers of material have been re resented l) the cascaded 1W matrices,
y P y
Exm_1, Hym—l’ Egg", and Hym are equal to the bounded free space equations found

in (3.23), (3.24), (3.27) and (3.28) respectively. So, the matrix in (3.50) becomes

E, n 13,
Hi "1:1 Hf

No matter how many layers of material are in the system, as long as there is

only one unknown layer, the matrix equation can be reduced into three layers for

16

 

{hf

1.1K.- '-

 

 

OT it: rv .

4.1'.

mathematical convenience

E,- E f
= Alleft ' Munknoum ' Alright (3-52)
H, H f
where A116 ft and [Wright are the contraction of all the known layers on the left and

right sides of the unknown layer. If the system happens to only consist of one layer,

A118 ft and Mrz'ght can be replaced with the identity matrix [6 (1)] so that

A! total = A116 ft ' Alunkno-wn ' [wright = A’Iunknown (3°53)

The transmission and reflection coefficients will now be found in terms of the
unknown matrix entries. Given (3.48), (3.52) and (3.53), a single layered system can

be written as

 

Ei COS (kzmtm) ij Sill (kmntm) Ef
— .. (k t ) (3-54)
Sin 7
H i J Aim m COS (Alzmtm) H f
Cm
or to make the derivation easier
E- 4111 4112 E
’ = f (3.55)
H7; M21 A122 Hf
Using (3.55), E,- and H,- are
Ez' = M11Ef+Mlng (3.56)
Hi = A/IglEf -+- [”2ng (3.57)

17

 

TEEN] 5"

Factor (

('11th a4

 

!_'

Then by substituting (3.23), (3.24), (3.27) and (3.28) into (3.56) and (3.57) gives

1
Einc + Eref = AJIIEtran + A112Z3Etran (3'58)
1 1 1
—Ei1zc " _Eref = A421Etran + A122—Etran (3-59)
C0 C0 (0

Equation (3.58) is then rearranged to give
1
Eref : A“IllEtr‘arn. + A112C_OEtran - Einc (3-60)
Then substituting into (3.59) and multiplying both sides by (0 results in

1
Einc " AillEt'ran + A'112'C‘6Etran — Einc = COAI21Etran + A’I22Etran (3'61)

Factor out Em", and apply (3.21) to find the expression for the transmission coeffi-

cient as

7' = 2 (3.62)

A41
M11 + $12 + MziCo + M22

 

where the terms A111, 4112, Mm and A122 are given in (3.48).
The reflection coefficient can be derived in the same fashion by substituting (3.58)

and (3.59) into (3.22) and applying some algebra to get

 

M
Ma + 713 — M21C0 —- M22
Mil + 7— + M21C0 + M22

0

18

 

3.2.1
A p.“ _
the P"
FUN}:
used ;:
IIiPrb‘li

the ra:

[my
V‘;

sh!- -
p‘ie I‘ll".

 

" r
pd‘rflnw.

3.2.1 Forward Relationship Results

A program was written in FORTRAN to compute the S-parameter coefficients given
the permeability and permittivity of a material layer inside the waveguide. The
Forward program is shown in Appendix A for reference. Two sets of data were
used in the testing of the Forward calculations. The first set of data consists of the

measured a and e of lV’IagRAM provided by The Boeing Company at 201 points in

the range of 8-12 GHz. Representative points are shown in Table 3.1.

 

 

 

 

 

 

 

 

 

 

 

Relative Relative Relative Relative

Frequency Permittivity Permittivity Permeability Permeability

(GHz) (Real) (Imaginary) (Real) (Imaginary)
8.32600 20.61915 -0.45778 1.61163 —2.15671
9.04000 20.50717 -0.50155 1.47979 -2.08967
9.77500 20.37240 ~0.49106 1.35299 -2.02382
10.61500 20.40467 -0.40625 1.21399 -1 .93652
11.87500 20.41216 -0.36015 1.04721 -1.80428

 

 

Table 3.1. Material properties for MagRAM as provided by The Boeing Company

Using the Forward program, the input arguments in Table 3.1 were used to com-
pute the S-parameters located at the boundaries of the material layer. These S-
parameters will not correspond to real life measurements because the proper com-
pensation is not accounted for. The results will however accurately depict the S-
parameters at the boundaries of the medium in the waveguide. The resulting S-
parameters are shown in Table 3.2 for each corresponding frequency. The values in

Table 3.2 will be verified with the Inverse program in section 3.3.1.

19

 

Frequency S 1 1 S I 1 321 521
(GHz) (6113) (D69) (dB) (Deg)
8.32600 -3.54334 -167.69275 -20.59249 134.90907
9.04000 -3.80197 -166.39375 —21.50234 124.98930
9.77500 -3.95370 -165.44933 -22.58655 115.41341
10.61500 -4.00873 —164.76779 -23.90743 104.60103
11.87500 -4.01569 —164.24632 -25.85478 89.90709

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.2. Calculated S-parameters for MagRAM using Forward program

The second set of data consists of the measured u and e of Rexolite provided by
The Boeing Company at 201 points in the range of 8-12 GHz. Representative points

are shown in Table 3.3.

 

 

 

 

 

 

 

Relative Relative Relative Relative

Frequency Permittivity Permittivity Permeability Permeability

(G112) (Real) (Imaginary) (Real) (Imaginary)
8.32600 2.52555 -0.00317 1.00098 0.00214
9.04000 2.52327 -0.00102 1.00282 0.00007
9.77500 2.52133 -0.00260 1.00247 -0.00152
10.61500 2.52196 —0.00300 1.00200 —0.00172
11.87500 2.52593 —0.00206 0.99806 -0.00154

 

 

 

 

 

 

 

Table 3.3. Material properties for Rexolite as provided by The Boeing Company

Using the Forward program, the input arguments in Table 3.3 were used to com—
pute the S-parameters located at the boundaries of the material layer. As men-
tioned with MagRAM, these are the material boundary S-parameters and will not
coincide with waveguide measurements unless properly compensated. The resulting
S-parameters are shown in Table 3.4 for each corresponding frequency. The values in

Table 3.4 will be verified with the Inverse program in section 3.3.1.

20

 

3.3 I

Tht‘ (r;

In \.
given i
niat).
' 1'1 (1,.

7‘ v
FxliL

 

 

Frequency 511 $11 $21 521
(GHz) (dB) (Deg) (43) (Deg)
8.32600 -5.03803 -147.09491 -I .63356 -56.98653
9.04000 -5.40875 -150.64702 4.47723 -60.61267
9.77500 -5.61971 45480183 -1.40905 -64.74479
10.61500 -5.74420 45979144 -1.36612 -69.74192
11.87500 -5.84244 -167.33481 -1.32812 -77.33021

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.4. Calculated S-parameters for Rexolite using Forward program

3.3 Inverse Relationship

The calculations used and derived in this section are cited largely from [3].

In section 3.2 the equations were derived that allow the calculation of p and r
given emr and umr. In this section, the reverse will be accomplished. Given the
reflection and transmission coefficients, the relative complex material properties em,-
and mm will be calculated.

From (3.48) the matrix entries for the material layer can be rewritten as

a. (ejkzmtm + e‘jkzmtm) % (ejkzmtm _ e—jkzmtm)

M = (3.64)

Cm
1-1— ( .lkzmtm _ efilkzmtm) l (ejkzmtm + e—jkzmtm)
2 Cm 2
using Euler’s Relation. To make the inverse derivation easier, the following substitu-

tion variables will be defined

u = ejkzmtm (3.65)
Cr = ‘7’; (3.66)
r—l

21

‘ D
As...) .

 

 

l u
5) ":
Ali‘n (.4

Re ..
.5 0n ht

Solving (3.67) for (1. results in

Also from (3.29), (3.35) and (3.66), (T can be written as

($54 a
4,: f it" (3.69)

rein

72—

 

1—

Using (3.32) and (3.65), the natural log of u can be calculated as

1n“ = jkzmtm

2
,w
J‘C—thHmrEmr‘) 1 — f;? (3.70)

The material property um,- can now be solved by taking the product of (3.69) and

 

(3.70) as shown

 

 

,/ _ [.39
f 2 , .w 2
((1)011 u) #mTJ—tmviumremr 1 _ fCT2n
/ fr2m 6mr C f
1 _ __'__
f2

clnu
Mmr = C" 2 (3.71)
thm 1 _ Lg

 

When calculating 1n u, care has to be taken because u is a complex number. Depend-

ing on how thick the material slab is in comparison to the wavelength of the electric

22

 

 

“1’"! . .-

where

r‘ r

,

x r .
\hebd \J‘u

 

J .
Lift . V)..“‘1i

9

.,, .
~‘4.}\"q

Ff'v'II
rip:
r“ 41%: :.

{0 r7

wimp

field, lnu is calculated as
1n u = ln lul + jarg(u) — j27rn (3.72)

where it starts at 0 for tm < /\/4 and is increased by 1 for every /\/4 increase in
thickness.
The material property 61,”. can be calculated by first dividing (3.69) by (3.70) to

get

2
l J—tm5mr (1 _ £07711)
3 = C f (3.73)

Cr 2

 

then by substituting

ckc

fem = —_—""
27!") l-hnrémr

(3.74)

derived from (3.5) and substituting it back into (3.73) results in

 

lnu _w c2k2. 1
‘_ = JFt-m (fmr “ ( C ) (3'75)

Cr 277)2#/mr 1 _ 129
V f2

Substituting kc = g from (3.11) into (3.75) and solving for emr yields

( fEO lnu 02
6,: l——-,w +— (3.76)
mr f2 J—C'thr 4‘12f2/‘mr

From (3.71) and (3.76), the only unknowns in the equations for the material

 

properties are (T, which from (3.68) can be described in terms of p and In ii. In order

to complete the task of deriving the inverse relationship between T, p, ,u. and e, the

23

 

 

 

P by l1:

unknowns p and u have to be described in terms of r and p or the transmission and

reflection coefficients respectively. From (3.64) and (3.65) the following substitutions

can be made

Mil = :-
Mm = g
4&1: g
M22 = %

(u+u-1)

a—u4)

C

‘Q
V
~

l H

9—44)

A
3

u+u—1)

A

(3.77)
(3.78)
(3.79)

(3.80)

Using (3.21), (3.66) and (3.77) through (3.80), r can be written in terms of u. and

p by first solving

‘Ilw

AI
6111+ —C1—2 + 6’121C0 + (”22
0

1 _1 1
5W + u )+ §Cr(

u+

1
l —
+2

u—

1 1 1 _
u — u_1) + ——(u — u_1) + 5(u + u 1)

2Cr

(13 + -:——p) (u — u—l)

l—p

 

 

24

(3.81)

[her

an 11‘ .

 

 

III (‘1:

 

then solving (3.81) for r results in

_ u(1 - 292)
r — u2 _ p2 (3.82)

To get the reflection coefficient in terms of u and 1), equation (3.63) has to be

analyzed in the same fashion as was used to determine (3.81). For the numerator

1W 1 _ 1 _
NIH-F-C—ég-CoilWQl—Afgg = -2-(u+u 1)+§Cr(u—u 1)

11 1
72r—(u — u—1)— §(u+u-1)

_1,,_1 1+p l-p
- 2 u 1—p 1+p

 

 

 

1 2
= (u _ Z) 1 _pp2 (3.83)

Note that the denominator of (3.62) is the same as the denominator in (3.63). So
obtaining p is accomplished by multiplying the result in (3.83) by the reciprocal of
(3.81) which results in

p(u2 — 1)

= __ 3.84
p u2_p2 ( )

In order to complete the derivation of the “Inverse Relationship”, u and 19 need

25

 

 

 

 

 

 

V'-
l..-

‘-.

t

its.

to be found in terms of p and 7'. Two new variables will be defined and are given as

g = p—T (3.85)

6 = p+r (3.86)

Substituting (3.82) and (3.84) into (3.85) and (3.86) gives

 

—1

g z p“ (3.87)
u—p
, 1

5 = Bu—Jr— (3.88)
u+p

Multiplying g and 6 and adding 1 to each side of the resulting equation yields

(1)2142 - 1) + (u2 — 102)
(u2 - p2)

 

€5+1=p2—T2+I= (3.89)

Solving (3.84) in terms of (u2 — p2) and substituting it into the denominator of (3.89)

and rearranging the numerator results in

p2u2 — p2 + u2 — 1

 

 

2 2 _
p —T +1—p p(u2—1) (3.90)
Next, factor out u2 — 1 which gives
2 2 2
— 1
p T +1 : p + (3.91)

26

Rearranging (3.91) results in a quadratic expression given as

2 (p2—72+1
p — —-——

p )p+1=0 (3%)

which has a solution of

2 2 2 2 2
— +1 -— 1
p = L_T__ :l: \/(_p___:__—+_—__) _1 (393)

 

2p 2p

In practice, the — sign is chosen in the above expression.

Once p is known, u can be found by solving

 

 

 

.UO-pU
p7 _ 1’ u2_p2
p—p _pW’-U
u2_p2
_ ul-pU
_ 1—fi
”r = u @9n
p—p

3.3.1 Inverse Relationship Results

As in 3.2.1, a program was written in FORTRAN to compute the material properties
)1 and 6 given the S-parameters at the boundaries of the material layer inside the
waveguide. The Inverse program is shown in Appendix B for reference. Two sets
of data were used in the testing of the Inverse calculations. The first set of data
consists of the S-parameters given in Table 3.2 for MagRAM which are the result of
the Forward program shown in Appendix A.

Using the Inverse program, the input arguments in Table 3.2 were used to compute

27

u and e of the material layer. The resulting material properties are shown in Table

3.5 for each corresponding frequency.

 

 

 

 

 

 

 

 

 

 

 

 

Relative Relative Relative Relative

Fiequency Permittivity Permittivity Permeability Permeability

(GI-12) (Real) (Imaginary) (Real) (Imaginary)
8.32600 20.61913 -0.45779 1.61163 -2.15671
9.04000 20.50718 -0.50154 1.47979 -2.08967
9.77500 20.37239 -0.49107 1.35299 -2.02382
10.61500 20.40467 -0.40625 1.21399 -1.93652
11.87500 20.41216 -0.36015 1.04721 -1.80428

 

Table 3.5. Calculated p and e for MagRAM using Inverse program

 

It can be seen by comparing Table 3.1 and Table 3.5 that the precision between
the Forward and Inverse programs is fairly close.

The second set of data consists of the S—parameters given in Table 3.4 for Rexolite
which are the result of the Forward program in Appendix A.

Using the Inverse program, the input arguments in Table 3.4 were used to compute
p and e of the material layer. The resulting material properties are shown in Table

3.6 for each corresponding frequency.

28

 

 

 

 

 

 

Relative Relative Relative Relative

Frequency Permittivity Permittivity Permeability Permeability

(6712) (Real) (Imaginary) (Real) (Imaginary)
8.32600 2.52555 -0.00317 1.00098 0.00214
9.04000 2.52327 -0.00102 1.00282 0.00007
9.77500 2.52133 -0.00260 1.00247 -0.00152
10.61500 2.52196 -0.00300 1.00200 -0.00172
11.87500 2.52593 -0.00206 0.99806 -0.00154

 

 

 

 

 

 

 

 

 

Table 3.6. Calculated u and e for Rexolite using Inverse program

 

It can be seen by comparing Table 3.3 and Table 3.6 that the precision between

the Forward and Inverse programs is fairly close.

3.4 Compensation Within Inverse Relationship

The calculations used and derived in this section are cited largely from [3].

The equations in section 3.3 all assume that the unknown slab of material is
placed at the calibration plane of the waveguide. When measuring the transmission
and reflection coefficients of a material slab in a waveguide using a network analyzer,
it is critical to be able to determine the location of the material slab to adjust the

S-parameters accordingly.

Figure 3.4 depicts the phase relationships in the waveguide where
(pr - Reference phase of the waveguide
(1311, (2522, (1521 - Actual reflection and transmission phases

(911m, $22,”, $211,, - Measured reflection and transmission phases

Phase of the material thickness in air

¢t

7,251 and (152 Phase of the air space to the left and right of the material

29

Assuming the material and measurement system is homogeneous and isotropic,

the following equations can be written from Figure 3.4

¢r = ¢1+¢t+¢2 (3-95)
¢2im = ¢1+<f>2+¢21 (3.96)
9511"). = 2°¢1+¢11 (3-97)
0522-"). = 2°¢2+¢22 (3-98)
9511 = $22 (3.99)

Algebraic manipulation of the above equations yield

 

_ $11m—¢22m
9511 —

2 - 9r + Cbt (3.100)

((521 = 9521171 - (M + (it (3.101)

which gives the phase relationships at. the material boundaries given the phase mea-
surements at the waveguide ports. a, is the reference phase measured during the

calibration of the network analyzer and (fit is the phase of the material calculated by
9: = km. - tm (3.102)

where kzm is found in (3.32) and tm is the thickness of the material.

Using (3.100) and (3.101), it is no longer necessary to know the position of the
material in the waveguide. By measuring S11 and S22, it is possible to correct the
phases in the 311 and S21 measurements to remove the uncertainty in the position

of the material relative to the calibration plane of the waveguide.

30

3.4.1 Compensated Inverse Relationship Results

The Boeing Company has provided measured S-parameter and material property
data for the development of the error analysis software. The data includes values
for MagRAM and Rexolite. Modifying the Inverse program in Appendix B with the
compensation methods derived in Section 3.4, resulted in the Compensated Inverse
program and is shown in Appendix C for reference.

The MagRAM data used to test the Compensated Inverse program is shown in
Table 3.7. The S-parameters are used as inputs for the program, while the perme-
ability and permittivity values are used as a reference to check whether the program

yields valid results.

31

 

Sample Thickness = 124.50 mil

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency 8.32600 9.04000 9. 77500 10.61500 11.87500
(GHZ)
811 (dB) -3.54352 ~3.80211 —3.95380 -4.00880 -4.01574
Sll (Deg) -169.33398 —168.47723 —167.77812 -167.45778 -167.29394
S22 (dB) —3.62212 -3.86602 -3.99953 -4.09377 -4.17685
S22 (Deg) -127.08378 -117.07008 -108.07798 -98.73757 -86.03702
S21 (dB) -20.59144 -21.50134 -22.58553 -23.90637 -25.85370
S21 (Deg) 154.40008 148.61645 142.94182 136.27872 127.49588
S12 (dB) -20.59534 —21.49905 -22.58432 -23.90067 -25.85262
Sl2 (Deg) _ 154.42669 148.68650 142.93125 136.27909 127.55798
Reference
821 (dB) -0.00053 -0.00027 0.00265 0.00212 0.00000
Reference
S21 (Deg) 0.00350 -0.00699 -0.00350 -0.02447 -0.01049
Relative
Permittivity 20.61915 20.50717 20.37240 20.40467 20.41216
(real)
Relative
Permittivity -0.45778 -0.50155 -0.49106 -0.40625 -0.36015
(imaginary)
Relative
Permeability 1.61163 1.47979 1.35299 1 .21399 1.04721
(real)
Relative
Permeability -2. 15671 -2.08967 -2.02382 -1.93652 -1.80428
(imaginary)

 

 

 

Table 3.7. Sample data for MagRAM provided by The Boeing Company

32

 

 

Given the S-parameters from Table 3.7, the Compensated Inverse program yields

the results shown in Table 3.8 which compare very closely with Boeing’s data.

 

Frequency 8.32600 9.04000 9.77500 10.61500 11.87500
(GHZ)
Relative
Permittivity 20.61928 20.50754 20.37291 20.40534 20.41297
(real)
Relative
Permittivity -0.45979 -0.50345 -0.49285 -0.40801 —0.36191
(imaginary)
Relative
Permeability 1.61192 1.48004 1.35321 1.21419 1.04739
(real)
Relative
Permeability -2.15636 —2.08933 -2.02350 -1.93620 ~1.80399
(imaginary)

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.8. Calculated material parameters for MagRAM using Compensated Inverse
program

In addition, all 201 data points for permeability and permittivity provided by The
Boeing Company from 8-12 GHz were plotted with the 201 data points calculated by
the Compensated Inverse program. The results can be seen in Figure 3.5 and Figure
3.6.

The Rexolite data used to test the Compensated Inverse program is shown in

Table 3.9.

33

 

Sample Thickness = 139. 50 mil

 

 

 

 

 

 

 

Frequency 8.32600 9.04000 9.77500 10.61500 11.87500
(GHZ)
811 (dB) -5.03857 -5.40918 -5.62008 -5.74449 -5.84262
Sll (Deg) -147.04711 -150.62139 -154.75536 -159.74187 -166.98874
S22 (dB) -5.08363 -5.43419 -5.61501 -5.74029 -5.89618
S22 (Deg) —103.47583 -97.73404 -93.16832 -88.85808 -83.45555
S21 (dB) -1.63333 -1.47705 -1.40894 -1.36598 -l.32803
S21 (Deg) -35.15038 -34.14061 -33.90210 -34.24785 ~35.21472
812 (dB) -1.63423 -1.47861 -1.40818 -1.36501 -1.32737

 

Sl2 (Deg) -35.12359 -34.06997 -33.87642 -34.20636 ~35.16429

Reference

 

 

 

 

S21 (dB) -0.00053 -0.00027 0.00265 0.00212 0.00000

Reference

S21 (Deg) 0.00350 -0.00699 -0.00350 -0.02447 -0.01049
Relative

Permittivity 2.52555 2.52327 2.52133 2.52196 2.52593
(real)
Relative
Permittivity -0.00317 —0.00102 -0.00260 —0.00300 -0.00206
(imaginary)
Relative
Permeability 1.00098 1.00282 1.00247 1.00200 0.99806
(real)
Relative
Permeability 0.00214 0.00007 -0.00152 -0.00172 -0.00154
(imaginary)

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.9. Sample data for Rexolite provided by The Boeing Company

34

 

Given the S-parameters from Table 3.9, the Compensated Inverse program yields

the results shown in Table 3.10 which compare very closely with Boeing’s data.

 

Frequency 8.32600 9.04000 9.77500 10.61500 11.87500
(GHZ)
Relative
Permittivity 2.52540 2.52316 2.52125 2.52189 2.52588
(real)
Relative
Permittivity -0.00325 -0.00109 -0.00267 -0.00306 -0.00212
(imaginary)
Relative
Permeability 1.00104 1.00286 1.00250 1.00203 0.99807
(real)
Relative
Permeability 0.00219 0.00012 -0.00149 -0.00169 -0.00151
(imaginary)

 

 

 

 

 

 

 

 

 

 

 

Table 3.10. Calculated material parameters for Rexolite using Compensated Inverse
program

In addition, all 201 data points for permeability and permittivity provided by The
Boeing Company from 8-12 GHz were plotted with the 201 data points calculated by
the Compensated Inverse program. These data points can be seen in Figure 3.7 and
Figure 3.8.

As seen from the above tables and figures, the results from the Compensated
Inverse program compare very closely with the measurements given by The Boeing
Company and are not distinguishable in the figures without zooming in to the third

or fourth decimal place.

35

 

 

Bounded Material Bounded
Freespace Layer Freeepace

 

 

Einc Bum

 

 

 

 

 

Figure 3.1. Waveguide with one material layer

36

 

 

 

Bounded Material Bounded

 

 

 

 

Freespace lgyer Freespace
12,
E, > Br ’
'13..
in» so ”an a... no, 6..

 

 

 

Figure 3.2. Waveguide with one material layer

37

 

 

 

 

Bounded Material ________________________ Material Bounded
Freespace Layer 1 Layer m+1 Freespace
0. 0.
1 I z I “ml an»: I
E, ------- E
_____’ __f+
'15: '32 Ba Bm+1
“111+!
"02 8. ”I, 6I "'29 82 ”In, all! 8m” “09 so

 

 

 

 

 

 

 

Figure 3.3. Waveguide with multiple material layers

38

 

 

 

 

 

<—¢1—> 4—0.—> <—¢z—->

 

‘— ¢21m

$5“. —'> «011—,

/ \ 6%"
on / \mn

Figure 3.4. Illustration of phase compensation for a piece of material in a waveguide

 

 

 

 

 

39

Permeability vs. Frequency

 

I-

'6

I-

I-

U-

 

       

—----4------fi---cud-Oc-nnhu--o-‘co-ud

h-QOI-d-----—fi------‘---—--fi----ufl-uouuuh-uuundcu—od

t----

J
I
I
I
I
1

6.6.1

'61

"'.I

II.‘

.I.I.'I.l

8.5

 

 

1.e----

1.2---
1----

u

—
5
4|

as: Essa-Ema

Freq (GHz)

 

         

'''''''''''''''''''

"" ""l

" 'J|I'-'

lllllllll

...................

I---L

I
I
I
I
I

 

 

 

1 -l-‘ "6'J6' '6 I--- I'III
.
u
.
a
—
I 66" IIIILI'I I '6'- I'l'l
. . u u a
- n c u .
. - - u u
n - n P F
4 4 o- 0- 0-

3-2-5 Ease-ma

8.5

Freq (GHz)

the Compensated Inverse program

Figure 3.5. The Boeing Company results vs.
results for permeability of MagRAM

40

Permittivity vs. Frequency

 

vir—

 

      
   
 

- u-d

-----------‘

IIIIIIIIIIIIIII

""""""""""

IIIIIIIIIIIIIII

1
Freq (GHz)

8.5

 

 

22
21.5

20---

n
5
U

21
19.5----

2
saw: gags-can.

 

"-

1"

III

v!"

III

Y'l

 

III;

L-----J----d

I'l

L-----J------

6-1

"1

'61

"l

8.5

 

 

U
1----

Base 522.83

-2 t---

Freq (GHz)

the Compensated Inverse program

Figure 3.6. The Boeing Company results vs.

results for permittivity of MagRAM

41

 

 

 

 

 

d d I d

c n - u

g n - -

g c u u

. . . . 1

- - o .

c - u n

p g u c

. . . . 5
""L'-"r 'I'b-"l- 6666 n

. . . . 1|

- a a u 1

. - c a

u — a -

c n a -
"I'h'666-1 """" “I---" 6666 1

. . . . 1

c u c .

u u n n

u u u u 5. 1.2
'-"‘-"'--r ....... ”'6'-" ..... U H

. . . . 4|. G

u u u n (1"

n u n n W

.u u. q u 1 F

- u o c

n u o n

c - - -
'-"L--"'.r ...... h'-"”l"ll 5

u u u u 9

- . . u

u - — u

q u u o
....... 9

n a o u

- - u a

- - o o

- u o -
j--'L-|"I-r '''''' ”I66'” 6666 5-

" u n u 8

c n o -

p p n h

1 5 1 5 9 5

1. D 9. U 8.

4| 0 U

cams Ego-mean.

 

 

......

J:v--

__J____-_J______L_____
v V VV V V V V V

"""""""""""""

 

 

 

0.05
-005

amp-e gage-ma

10 10.5 11 11.5 12

9.5

8.5

Freq (GHz)

the Compensated Inverse program

Figure 3.7. The Boeing Company results vs.

results for permeability of Rexolite

42

Permittivity 113. Frequency

 

 

   

 

 

" u u u
. . . .
. . . .
... ....... 1. ...... .- ........ n-.. 2
. . . . 1
. . . .
. . . .
. . . .
. . . . 5
TL lllllll L lllll L IIIIIIII .ll .
. . . . 4|
. . . . 1|
. . . .
. . . .
o c - -
FJ. ..... Ll--- 'J— ........ "'1 1
. . . . 1|
. . . .
. . . .
n u n u 5.?)
.- ....... .fl ...... .n ........ "i U H
. . . . 1| 0
. . . . II\
n u u n W
I. 1111111 . rrrrrr . 11111111 .11 0 rl
.u .n .u n 1F
. . . .
. . . .
. . . .
1. ....... 1” ...... .. ........ 5
u _ u - 9
. . . .
. . . .
. . . .
I" ....... J- ...... .“ ........ "1 9
. . . .
. . . .
. . . .
. . . .
L- ....... .“ ...... .h ........ n- 5
u u u L 8
. . . .
n — h n
8. 5 5 5
2 5 2 4.
2 2

one regs-ma

 

'''''

.....

‘‘‘‘‘

IIIII

IIIII

IIIII

 

" u

. .

. .

.1 .............. 1. ...... 2

. . 1

. .

. .

. .

. . 5

.r lllllllllllll u" IIIII i1

. . 1

. .

. .

. .

. . 1

1 llllllllllllll J llllll

. . 1

. .

. .

u u 542.

.r '''''''''''''' I” ..... 10H

. . 1G

. . [ls

" u w

. 11111111111111 . 111111 or

u. .n 1F

u n

. .

n. .............. .H ...... .5.

n u g

. .

. .

. .

v .............. .n ...... 9

n u

. .

. .

.r .............. .u ...... 5
8

 

 

0.1

D

-01

38.5 rage-ma

the Compensated Inverse program

Figure 3.8. The Boeing Company results vs.

results for permittivity of Rexolite

43

CHAPTER 4

ERROR ANALYSIS OF MATERIAL CHARACTERIZATION

When calculating the permeability and permittivity of a medium using measured
quantities, it is important to remember that the values calculated are not absolute.
All of the data used during the calculations have some kind of uncertainty or error
associated with them. Some uncertainties can be the result of the measurement
equipment having a specific tolerance or error in the system. There can also be
uncertainty associated with the expanding and contracting of the material and the
waveguide due to changes in temperature or maybe human error when reading a
measurement device. For these reasons, it is important to understand how the final
values behave given certain changes in the input variables.

In this chapter, the above topics will be discussed with relevance to this thesis
and then a method to analytically and graphically interpret these tolerances will be

shown.

4.1 Measurement of Errors

Errors can effect the accuracy and precision of an experiment. Accuracy is the mea-
sure of how close the calculated result is to the true value. Precision, on the other
hand, is usually referred to as the uncertainty of an experiment, and it effects the
repeatability of the result. If an experiment has high repeatability, then each consec-
utive experiment will give results very close in value to one another.

It is possible to have high precision with low accuracy. However, having high

accuracy will generally give better precision. Before the errors can be analyzed or

44

compensated for, they first need to be understood.
4.1.1 Systematic Errors

Different types of errors have different effects on precision and accuracy. The sys-
tematic error is not easy to detect and not easily studied by statistical analysis [4].
This type of error is caused by human bias or faulty calibration of equipment. For
example, a micrometer my have a zeroing error, reading 0.2mil when the jaws are
closed. Suppose five measurements of the width, tm, of a material layer were 124.6,
124.6, 124.7, 124.8 and 124.8mil. The mean of these—124.7mil—would obviously be an
incorrect ‘best estimate’ (see Section 4.1.3) of tm. For better accuracy, the zero error
should be subtracted from each measurement or from the average to give 124.5mil
[5]. Moreover, it might have happened that the temperature was fluctuating as the
measurements proceeded. If the measurements are critical or vary largely with tem-
perature changes, the coefficient of temperature expansion may be needed to correct
the measurements accordingly.

Systematic errors usually do not effect precision (assuming everything else is con-
stant) and give low accuracy because they have constant or slowly variant offset from
the real value. The best way in dealing with this type of error is to make sure that
all of the equipment used for the experiment is calibrated correctly. Sometimes there
is more than one method for calibrating a piece of equipment such as a network an-
alyzer. It is then advisable to find the best method given the frequency range of the

experiment.

45

4. 1.2 Random Errors

When all systematic errors have either been eliminated or corrected for, ‘true’ mea-
surements usually are not obtained because of variations called random errors. Ran-
dom errors may arise from ambiguities or uncertainties in the process of measurement,
or from fluctuations which are too irregular or fast to be observed in detail [5]. These
types of errors will yield different results for each run of an experiment. The problem
of reducing random errors is essentially one of improving the experimental method
and refining the techniques, as well as simply repeating the experiment [4]. In order
to better quantify random errors, it is better to run a large number of experiments

to find the standard deviation for use in some kind of error analysis.
4.1.3 Standard Deviation

Given 11 measurements 11,12, . . . ,1?” and assuming n is greater than 1, the mean is

calculated as

The deviation from each measurement is then calculated as

61: $1_ X", 62 = 1:2 "' X", . . . 6n = 1‘" "" X" (4.2)

Given the number of measurements and the deviation from each measurement, the

mean square deviation is

0,2,: (6¥+6g+-~+6g)/n (4.3)

In general, the mean is considered to be the best estimate of the ‘true’ value

46

under the prevailing experimental conditions. The variance 0,2, and the standard
deviation an characterize the uncertainties associated with the experimental attempts
to determine the ‘true’ values. The standard deviation, on, has the limiting value a
as n increases, and provides a reasonable best estimate for the precision [4].

When asymmetrical distributions occur, the standard deviation is less a guide to
an assessment of the experiment. Only the complete curve can really be trusted with
non-symmetric deviation curves. Nevertheless, if a single number has to be given,

there is usually no better measure of precision than the standard deviation [5].

4.2 Error Analysis

Error analysis is the method of combining uncertainties in separate measurements to

find the total uncertainty for the calculated measurements.
4.2.1 Propagation of Errors

If the final value or result is a function of one or more measured variables, the un-
certainty in each measured variable will propagate into the final value. Therefore, a
method must be determined to estimate the amount of error which makes its way
into the final result.

Given a variable W, which is dependant on variables A, B and C, the estimation

of error can be found for W by expanding it about A, B and C in a Taylor series

[4, 7].
(9W 3W
W 2: W0 + (A A0) (—) + (B - Bo) (—) +
8A 8000 BB AOCO
a 1 2 82W
+(C' — C0) (—) — {(A — A0) ( ) + + (4.4)
00 A030 2' 6A2 BOCO

47

Knowing the actual errors, AA = A — A0 and so forth, the first term in the Taylor

expansion would give

W 2 W0 + AA (96%),; C + AB (5E)A + AC (%g) (4.5)
0 '0 000 A030
where the terms in parenthesis are the partial derivatives of W with respect to each
measured variable, given the other variables are held fixed as indicated by the sub-
scripts.

From (4.5), the error in W can be found from AW 2 W — W0. One will notice
that this approximation neglects the higher order terms within the Taylor expansion.
However, if there is no covariance between the measured variables, the higher order
terms can be neglected [4]. If the errors are large, the second partial derivatives
(82VV/8A2, etc.) and partial cross derivatives (OZW/BA BB, etc.) should be included
[5].

There is one fundamental problem with the Taylor approximation in (4.5), the
actual errors are generally not known for each variable. However, the standard devia—
tion is usually obtainable such as shown in Section 4.1.3. Given that W is a function
of A, B and C,

W = f(A, B, C), (4.6)

the most probable value for W is given by
We = W = NEE-C) (4-7)

The uncertainty in the resulting value of W can be analyzed by taking each indi-

48

vidual measurement Ai, Bi, . .. into individual results W, such as

W, = f(A,-,B,-,...) (4.8)

The variance can then be found from (4.7) and (4.8) as

Next, the Taylor expansion from (4.5) can be rewritten as

W,—W2(A,-—Z) (633/7) +(B,-—‘§) (98%) +... (4.10)

and then substituted into (4.9) to give

0%,, 2 lim NZ(A1“A)(%LX)+(B'_§)(%%)+"']2

2 lim %Z:(A— A)2 (ii—ZYJF +(-—B —B)2 (fig/)2 +...](4.11)

 

The variance of W can now be written in terms of the variances of A, B and C by

substituting (4.9), with respect to each variable, into (4.11) which results in

aw 2 aw a__W
02 ~ __

Since the variance is simply the square of the standard deviation, the standard devi-

49

ation can be found by taking the square root of (4.12) as

2 aw2 2 aw2 2 aw2
Orv”: 0A '54— +08 E}— +0C 30— +... (4.13)

4.2.2 Computer Calculation of Uncertainties

 

An analytical approach for finding a solutions to (4.13) would begin by creating a

function for W in terms of each measured variable, such as

H7 = function(A, B, C, . . .) (4.14)

Then, the partial derivatives are calculated using a one-sided or central difference

method. Using a central difference method results in

functi0n(A + dA, B, C, . . .) — functi0n(A — dA, B, C, . . .)

 

 

 

1V 2
d A 2dA ’
leB : functzon(A, B + (18, C, . . .) — functzon(A, B — dB, C, . . . ),
2dB
leC : functzon(/l, B, C + dC, . . .) — functzon(A, B, C — dC, . . . ),
2dC
etc' (4.15)

where dA, dB, dC, and so forth are the standard deviations 0A: 03, 00 respectively.

From (4.12) and (4.15), the variance of W can be written as

aw? 2 dA2 (an/A)2 + (132 (dWB)2 + dC2 (dWC)2 + . .. (4.16)

50

Since the standard deviations in the denominator of (4.15) cancel with the ones

outside the parenthesis in (4.16), (4.16) is easier written as

 

 

 

le2 ~ functi0n(A + (1A, B, C, . . .) — fu-n.cti0n(A — (1A, B, C, . . .) 2
I — 2
function.(A, B + dB, C, . . .) — functi0n(A, B — dB, C, . . .) 2
2
+ function(A, B,C + dC, . . .) — function(A, B, C — dC, . . .) 2
2
+ (4-17)

which simplifies to

aw? 2 (fdA)2+(de)2+(de)2+... (4.18)

 

 

 

where
fdA —— function.(A + (1A, B, C, . . .) — function(A __ dA, 83 C, . . .)
_ 2 ,
de _ fanct’i0n(A, B + (18, C, . . . ) — functi0n(A, B — dB, C, . . .)
_ 2 ,
(1C _ functi0n(A,B,C+dC,...)-—function(A,B,C—dC,...)
f — 2 ,
em (4.19)

51

The standard deviation of the entire system can then be calculated from (4.18) as
aw 2 sqrt[(fdA)2 + (de)2 + (de)2 + . . l (4.20)

4.3 Spline Method

The calculations used and derived in this section are cited largely from [8].

When working with the HP8510 network analyzer, a program is available that
calculates the uncertainties of measurements given that specific equipment was used
with a specific calibration process. This program, called “HP 8510 Specifications 86
Performance Verification Program”, can display a table of the worst case error for
amplitude and phase for all of the S-parameters.

Example plots using setup #1 from Table 2.2 and the HP program are displayed
in Figure 4.1 through Figure 4.4 using 21 points for each output. Each graph shows
the amount of error for setup #1, given the measured amplitude of each S—parameter.
In order to use this information in the Error Analysis Program (Appendix D), an
approximation has to be made for all the error values that are not explicitly given in
the table provided by the HP program. An approximation is achieved by interpolating
the desired value from the values that are given.

The method of interpolation used in the Error Analysis Program is called ‘Cubic
Spline Interpolation’ [8]. Given a tabulated function y, = y(:r,~), where 2' = 1...N,
focus is concentrated on a particular interval between CL‘J‘ and ivy-+1, and the linear

interpolation of that interval is found as

y = (lyj + byj+1 (4-21)

52

where

a = 4.4-Li (4.22)
rj+1 _ ‘rj
and
:13 — xj
b=1—a=—— (4.23)
$j+1 ‘ 333'

The purpose of the cubic spline method, however, is to have an interpolation
formula that is smooth in the first derivative and continuous in the second derivative.
So, if the tabulated second derivative is provided in addition to the above information,

the interpolation is given as

II II
3/ = my + byj+l + 0313' + dyj+1 (4.24)
where
c- -1-(a3—a)(;z:- —:1:-)2 (4 25)
and
1
a = 6(b3 —— lam-+1 — raj)? (4.26)

In order to obtain a tabulated second derivative for this task, the function ‘spline’
from Numerical Recipes [8] is called at the beginning of the program that calculates
the second derivative of each data point with respect to the data surrounding it. This
second derivative is then used to calculate the interpolated values on the fly using the

function called ‘splint’ from Numerical Recipes [8].

53

4.4 Error Analysis Results

The error analysis program was first run with the standard deviations of the waveguide
width and the material thickness independently from one another to see how each one
contributed to the total error. The standard deviations used for each test are shown
in Table 4.1 along with references to the figures for each case. Every figure contains
three lines. The center line refers to the calculated permeability or permeability of
the material. The lines above and below the center line refer to the maximum and
minimum permittivity or permeability possible, based on the amount of uncertainty

introduced into the calculations.

 

 

 

 

 

 

 

 

 

Input Argument Error Figure References ]
Material 0.3% Figure 4.5 through Figure 4.8
Thickness 1.0% Figure 4.9 through Figure 4.12
(tm) 3.0% Figure 4.13 through Figure 4.16
Waveguide 0.3% Figure 4.17 through Figure 4.20
Width 1.0% Figure 4.21 through Figure 4.24
(a) 3.0% Figure 4.25 through Figure 4.28

 

 

 

 

Table 4.1. Input arguments used for independent error analysis of Rexolite and Ma-
gRAM using Error Analysis program

Results from figures in Table 4.1 are shown in Table 4.2 for simplicity.

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Resultant Resultant
Uncertainty Uncertainty
Type of From From
Material Constitutive Percent Material Waveguide
Tested Parameter Error Thickness Width
MagRAM Permeability 0.3% 0.03-0.2% 0.03-0.04%
(real) 3.0% 04-23% 0.36—0.44%
Permeability 0.3% 0.6-0. 75% 048-0. 13%
(imaginary) 3.0% 6.0-7.5% 5.5-1.4%
Permittivity 0.3% 0.2-0.06% 0.38-0.1%
(real) 3.0% 1.8—0.6% 4.0-1.09%
Permittivity 0.3% 8.5-17.3% 7.7-3.3%
L (imaginary) 3.0% 85-174% 75-33%
Rexolite Permeability 0.3% 0.03-0.08% 0.027—0.014%
(real) 3.0% 0.34-0.82% 0.3-0.15%
Permeability 0.3% 0.56-0.0% 0.56—0.0%
(imaginary) 3.0% 6.0-7.5% {3.5-1.4%
Permittivity 0.3% 0.17-0.12% 0.19-0.05%
(real) 3.0% 1.68-1.18% 200.5%
Permittivity 0.3% 0% 0%
(imaginary) 3.0% 2.52% 3.2-1.6%

 

Table 4.2 shows the amount of uncertainty introduced in the calculation of per-
meability and permittivity given that either the material thickness has uncertainty or
the waveguide width has uncertainty. Each entry in the table has a range for the final
contributed uncertainty. For instance, the 3.0% row and waveguide width column
indicate a 7.7-3.3% uncertainty for the imaginary part of permittivity. This means it
has a 7.7% final contributed uncertainty at. the lowest frequency tested and a 3.3%
final contributed uncertainty at the highest frequency tested.

The amount of final uncertainty contributed from each test seems to be consistent
with the input uncertainty amount. In other words, a ten fold increase in uncertainty

(0.3% to 3.0%) yields roughly a ten fold increase in the final uncertainty contribution

Table 4.2. Comparison of resultant uncertainties for Rexolite

55

 

(060.75% to 60-75%).

Next, the error numbers obtained from the “HP 8510 Specifications & Performance
Verification Program” mentioned in Section 4.3 were tested in the Error Analysis
program. Setup #1, as described in Table 2.2, was chosen with five different cable
selections to determine if The Boeing Company was using the best suited cables for

the specified frequency range. Table 4.3 shows the different cable selections that were

 

 

 

 

 

 

tested.
Cables Figures ]
HP85132C Figure 4.29 through Figure 4.32
Single Long Cable (3.5mm - 7.0mm)
HP85132D Figure 4.33 through Figure 4.36
Pair Short Cables (3.5mm - 7.0mm)
HP85132E Figure 4.37 through Figure 4.40
Single Long Cable (3.5mm - 7.0mm)
HP85132F Figure 4.41 through Figure 4.44
Pair Short Cables (3.5mm - 7.0mm)
No Cables Figure 4.45 through Figure 4.48

 

 

 

 

Table 4.3. Different cables used for error analysis using setup #1 for the comparison

56

From Figure 4.29 through Figure 4.44 it can be seen that cable HP-85132E is the
best cable from the four that were tested. Cable HP-85132E has roughly 7% error
for permeability and 3.5% error for permittivity. The effects of uncertainty on the
imaginary part of permittivity is quite high for all four cable options. By running the
error propagation program with the “no cable” option, it is shown that in most cases,
the equipment has more influence on the uncertainty than the material thickness and
waveguide width. It should be stressed that the HP Specifications & Performance
Program was set up to output the “worst case” error for each setup since there was
no configuration to output the standard deviation of error.

Using cable HP-85132E as specified by The Boeing Company, three other network
analyzer setups were simulated. The additional network analyzer setups are shown
in Table 2.2.

All four setups from Table 2.2 were tested in the error analysis program without.
additional error from the material thickness or waveguide width. Table 4.4 summa-
rizes which figures represent the output from each system. According to the “HP
8510 Specifications & Performance Verification Program”, the error associated with
setup #3 and #4 are identical. Since the two systems perform identically, additional
plots will not be created for system #4 and the reader will be referred to the system
#3 plots instead. The error contribution from systems #1, #3, and #4 are identi-
cal to each other. System #2 has slightly more error (roughly 0.2%) and would be

considered the least preferred setup of the four.

57

 

 

 

 

 

 

 

Analyzer Setup ] Figures

Setup #1 Figure 4.37 through Figure 4.40
Setup #2 Figure 4.49 through Figure 4.52
Setup #3 Figure 4.53 through Figure 4.56
Setup #4 Figure 4.53 through Figure 4.56

 

 

 

 

Table 4.4. Different setups used for error analysis as shown from Table 2.2

Finally, the total amount of error was tested with combinations of material thick-
ness standard deviations, waveguide width standard deviations and network analyzer
errors from setup #1. The results should be typical of a worst case measurement if

the system is properly calibrated. The error combinations are shown in Table 4.5.

 

 

 

 

 

 

 

Measurement Waveguide Material Figure References
Equipment Width Thickness

Setup #1 0.3% 0.3% Figure 4.57 through Figure 4.60

Setup #1 3.0% 3.0% Figure 4.61 through Figure 4.64

 

 

Table 4.5. Input arguments used for independent error analysis of Rexolite and Ma-
gRAM using Error Analysis program

From the figures stated above, tables were created to show the correlation between
the uncertainty in the measurements, and the uncertainty in the complex a and c for
MagRAM and Rexolite. The data which shows the relationship for MagRAM is
shown in Table 4.6. The data which shows the relationship for Rexolite is shown in

Table 4.7.

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Resultant Resultant
Uncertainty Uncertainty
From From Uncertainty

Constitutive Percent Material Waveguide From Combined

Parameter Error Thickness Width System #1 Uncertainty
Permeability 0.3% 0.03-0.2% 0.03-0.04% 7.6-7.2% 7.6-7.2%
(real) 3.0% 04-23% 0.360.44% 7.68-7.67%
Permeability 0.3% 0.6-0.75% 048-0. 13% 8.9-6.49% 8.9-6.5%
(imaginary) 3.0% 60-75% 55-14% 13.2-10.48%
Permittivity 0.3% 0.2-0.06% 038-0. 1% 3.8-3.4% 3.87-3.4%
(real) 3.0% 1.8-0.6% 401.1% 533.56%
Permittivity 0.3% 8.5-17.3% 7.7-3.3% 292-280% 293-281%
(imaginary) 3.0% 85-174% 75-33% 323-340%

 

Table 4.6. Comparison of resultant uncertainties for MagRAM

By examining Table 4.6 it can be seen that the total amount of error is largely
influenced by the system uncertainties for both a 0.3% amount of uncertainty and
for a 3.0% amount of uncertainty in the measurements. The material thickness and
waveguide width had high contributions for the imaginary permeability and the real
permittivity at the 3.0% level, but the rest of the uncertainty is due to the measuring

system itself.

59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Resultant Resultant
Uncertainty Uncertainty
From From Uncertainty
Constitutive Percent Material Waveguide From Combined
Parameter Error Thickness Width System #1 Uncertainty
Permeability 0.3% 0.03—0.08% 0.027-0.014% 64-26% 64-26%
(real) 3.0% 0.34-0.82% 0.3-0.15% 64-28%
Permeability 0.3% 0.56-0.0% 0.56-0.0% Large Large
(imaginary) 3.0% 3.3-9.5% 3.37-0.0% Large
Permittivity 0.3% 0.17-0.12% 0.19-0.05% .31-1.49% .36-1.49%
(real) 3.0% 1.68-1.18% 2.0-0.5% 2.0-1.8%
Permittivity 0.3% 0% 0% Large Large
(imaginary) 3.0% 2.5-2.8% 3.2-1.6% Large

 

Table 4.7. Comparison of resultant uncertainties for Rexolite

By examining Table 4.7 it can be seen that the total amount of error is largely

influenced by the system uncertainties for both a 0.3% amount of uncertainty and for

a 3.0% amount of uncertainty in the measurements. The waveguide width contributed

the most for permittivity at the 3.0% level, but the rest of the uncertainty is due to

the measuring system itself.

60

 

 

 

 

 

 

 

 

 

 

 

 

0.8

0.6
Measured S11 Reflection Coefficient

0.4

0.2

q . q _ 1 q a q
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
m ..... .. ..... .15.. ..... 8. t ....... u ...... .. ...... .. ...... .
. . . . D n . . .
WI . . . . .m . . .
. n n . c . . .
ms. 0 . . H mm H n u
S . . . . e . . .
. . . . 0 . . .
h . . . . C . . .
t . . . . . . .
c' "0'-.. 0000 .0000'0" ''''' n """" l 5- n ..... u """"" " '0'0'00" IIII J]
W n n . . U 0 . . .
vl . . . . .l . . .
O . . . . a . . .
N u u u u e u n n
E . . . . w . . .
e u u u “ Du ” u n
S . . . . . . .
a fi'.".- .... “'00000" '''' n ''''' I 4. H lllll O lllllll " l-|-'l'" ''''' r
C u . . . 0 H . .
MN . . . . S . . .
. . . . . .
m u m u n m n n u
. . . . 3 . . .
W n u u u a u n H
Al . . . . 2 e . . .
1 ..... .. uuuuu pill... ..... r...- .M ....... . uuuuuu L cccccc .. nnnnnn 1
S u u u " Jn. " u n
u n n u n u n
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. . . . q .
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b 0 h I
1 8 PO 4 2 U U 5 U 5 0

0.00.0

Btw mmmcn. Sm
came; Sew ammo .285 Sm

Figure 4.1. Worst case error for Sll and setup #1 using “HP 8510 Specifications &

Performance Verification Program”

61

822 Worst Case Error With Setup #1

 

 

 

 

 

 

 

 

 

d u u d

n n u u

u u u u

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0. D 0. 0. 0.

U U U 0

came; Sum 33 $52) mmm

0.8

0.6

Measured S22 Reflection Coefficient

0.4

 

 

I---

'0'-

IIIIIII

uni

C------J----------q

000000

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.-‘--------r--------

 

 

5
0

Btu $ch mmm

0.8

0.6
Measured S22 Reflection Coefficient

0.4

0.2

Figure 4.2. Worst case error for 822 and setup #1 using “HP 8510 Specifications &

Performance Verification Program”

62

812 Worst Case Error With Setup #1

 

 

   

 

 

 

 

 

 

 

 

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1
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EB .25 $8 .295 Stm $sz Em

-90

-70

-50
Measured 812 Level (dB from ref)

63

-10

 

10
Figure 4.3. Worst case error for 812 and setup #1 using “HP 8510 Specifications &

Performance Verification Program”

 

 

   
          
  

 

 

 

 

 

 

 

 

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EB .25 ...me .985 33m mmmzn. Km

-90

-70
Measured 821 Level (dB from ref)

64

-10

 

' 10
Figure 4.4. Worst case error for 821 and setup #1 using “HP 8510 Specifications &

Performance Verification Program”

Error Analysis of Permeability vs. Frequency

 

----+

IIIIIIIII

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10 10.5 11 11.5

9.5

Freq (GHz)

 

I

T T I T

U
L-----4------L-----J--_---L-----J---_--L-----J--—-.i

 

 

 

 

. c

. u

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p n
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4

325. ..___._$ean_

12

11.5

Freq (GHz)

Figure 4.5. Error analysis of permeability for 0.3% tolerance in material thickness for

MagRAM

65

Error Analysis of Permittivity vs. Frequency

 

 

 

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11 11.5 12

10.5
Freq (GHz)

Figure 4.6. Error analysis of permittivity for 0.3% tolerance in material thickness for

MagRAM

66

Error Analysis of Permeability vs. Frequency

 

    

 

 

 

d

. o
- -
. -

. .
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n c
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11 11.5 12

10.5
Freq (GHz)

 

IIIIIIIIIIIII

IIIIIIIIIIIIIIIIIIIIIIIIIII

""""""""""

Freq (GH'z)

 

 

0.05

 

BEE. ..EBmemn.

 

Figure 4.7. Error analysis of permeability for 0.3% tolerance in material thickness for

Rexolite

67

Error Analysis of Permittivity vs. Frequency

 

 

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10.5 11 11.5
Freq (GHz)

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9.5

8.5

 

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11.5

10.5 11
Freq (GHz)

9.5

8.5

Figure 4.8. Error analysis of permittivity for 0.3% tolerance in material thickness for

Rexolite

68

Error Analysis of Permeability vs. Frequency

 

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10

9.5

8.5

Freq (GHz)

Figure 4.9. Error analysis of permeability for 1.0% tolerance in material thickness for

MagRAM

69

Error Analysis of Permittivity vs. Frequency

 

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Freq (GHz)

Figure 4.10. Error analysis of permittivity for 1.0% tolerance in material thickness

for MagRAM

70

 

 
 
  

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10.5
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9.5

Figure 4.11. Error analysis of permeability for 1.0% tolerance in material thickness

for Rexolite

71

Error Analysis of Permittivity vs. Frequency

 

 

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12

10.5 11 11.5
Freq (GHz)

10

9.5

8.5

 

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Figure 4.12. Error analysis of permittivity for 1.0% tolerance in material thickness

for Rexolite

72

Error Analysis of Permeability vs. Frequency

 

 

    

     
  
 

 

 

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10.5
Freq (GHz)

 

 

 

        

 

 

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11.5

Freq (GHz)

Figure 4.13. Error analysis of permeability for 3.0% tolerance in material thickness

for MagRAM

73

Error Analysis of Permittivity vs. Frequency

 

 

10.5

8.5

 

 

 

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2

11.5 1

11

10

9.5

(GHz)

Freq

 

   

IIIIIIIIIIIIIIIIIIIIIIII

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Freq (GHz)

Figure 4.14. Error analysis of permittivity for 3.0% tolerance in material thickness

for MagRAM

74

 

 

 

 

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11 11.5 12

10.5
Freq (GHz)

8.5

 

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12

11.5

10.5 11
Freq (GHz)

9.5

8.5

Figure 4.15. Error analysis of permeability for 3.0% tolerance in material thickness

for Rexolite

75

Error Analysis of Permittivity vs. Frequency

 

    

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11

10.5

10

9.5

8.5

Freq (GHz)

 

 

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11.5

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10.5
Freq (GHz)

9.5

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Figure 4.16. Error analysis of permittivity for 3.0% tolerance in material thickness

for Rexolite

76

 

         

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83.5 3.33635

Freq (GHz)

Figure 4.17. Error analysis of permeability for 0.3% tolerance in waveguide width for
77

MagRAM

Error Analysis of Permittivity vs. Frequency

 

 

       

 

 

 

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11.5

11

10.5

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9.5

8.5

Freq (GHz)

 

 

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Freq (GHz)

Figure 4.18. Error analysis of permittivity for 0.3% tolerance in waveguide width for

MagRAM

78

ErrorAnalysis of Permeability vs. Frequency

 

 

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I

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11.5

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Freq (GHz)

Figure 4.19. Error analysis of permeability for 0.3% tolerance in waveguide width for

Rexolite

79

Error Analysis of Permittivity vs. Frequency

 

 

 

     

   

 

 

 

 

 

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10.5 11 11.5 12

10

9.5

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8.5

Freq (GHZ)

Figure 4.20. Error analysis of permittivity for 0.3% tolerance in waveguide width for

Rexolite

80

 

 

    
  

11

10.5

10

Freq (GHz)

81

    

   
  

 

 

 

 

 

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Figure 4.21. Error analysis of permeability for 1.0% tolerance in waveguide width for

MagRAM

Error Analysis of Permittivity vs. Frequency

 

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9.5 10 10.5 11 11.5 12
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Figure 4.22. Error analysis of permittivity for 1.0% tolerance in waveguide width for
82

MagRAM

Error Analysis of Permeability vs. Frequency

 

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11.5

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Freq (GHz)

 

 

 

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Figure 4.23. Error analysis of permeability for 1.0% tolerance in waveguide width for

Rexolite

83

Error Analysis of Permittivity vs. Frequency

 

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11.5

11

10.5
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Freq (GHz)

Figure 4.24. Error analysis of permittivity for 1.0% tolerance in waveguide width for

Rexolite

84

 

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Error Analysis of Permittivity vs. Frequency

 

 

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Figure 4.27. Error analysis of permeability for 3.0% tolerance in waveguide width for

Rexolite

87

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Figure 4.28. Error analysis of permittivity for 3.0% tolerance in waveguide width for

Rexolite

88

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89

Figure 4.29. Error analysis of permeability using test setup #1 with cable model

HP-85132C and MagRAM

Error Analysis of Permittivity vs. Frequency

 

 

   

    

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9.5

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Figure 4.30. Error analysis of permittivity using test setup #1 with cable model

HP-85132C and MagRAM

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Figure 4.31. Error analysis of permeability using test setup #1 with cable model

HP-85132C and Rexolite

91

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Figure 4.32. Error analysis of permittivity using test setup #1 with cable model

HP-85132C and Rexolite

92

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Figure 4.33. Error analysis of permeability using test setup #1 with cable model

HP-85132D and MagRAM

93

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HP-85132D and MagRAM

94

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Figure 4.35. Error analysis of permeability using test setup #1 with cable model

HP-85132D and Rexolite

95

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HP—85132D and Rexolite

96

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Figure 4.37. Error analysis of permeability using test setup #1 with cable model

HP-85132E and MagRAM

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Error analysis of permittivity using test setup #1 with cable model

HP-85132E and MagRAM

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Figure 4.39. Error analysis of permeability using test setup #1 with cable model

HP-85132E and Rexolite

99

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Figure 4.40. Error analysis of permittivity using test setup #1 with cable model

HP—85132E and Rexolite

100

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thickness tolerance and 0.3% waveguide width tolerance for MagRAM

118

 

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124

CHAPTER 5

CONCLUSION

Given the particular method of error analysis used within this thesis, it is safe to
assume that the limiting factor of material characterization is the cables which are
used while measuring the S-parameters from the propagating wave structure. It
should be taken into account that the error analysis done with the cables was based
on a “worst case” scenario. To gain a better understanding of the system, it would
be advisable to measure the standard deviation of errors for the system and run that
data through the error analysis program.

From Figure 4.57 through Figure 4.64 it was shown that the material thickness
and waveguide thickness did not contribute to much of the final error as long as the
standard deviation of error was within reasonable limits.

Figure 4.29 through Figure 4.48 show the dramatic difference between the theo-
retical non-cable test setup versus the four cables that were tested. It is concluded
that although the measuring instruments themselves contribute to more uncertainty
than the material thickness and waveguide width, they do not contribute much of the
final uncertainty encountered in this thesis.

Given that all the instruments are properly calibrated and all the input parameters
are properly measured, the number one contributing factor to the final results, within
the scope of this thesis, is the cable that is used on the test instruments.

Even though this thesis included much of the obvious sources of error, significant
improvements can still be realized in terms of the accuracy of the program. It was

mentioned in references [2, 3] that the angle the material is to the normal of wave

125

propagation is a large contributing factor to the source of error. This was not explored
in this thesis but the Error Analysis program could be expanded to allow for this extra
variable if the Inverse program was rewritten to allow for this input parameter.

A much more difficult addition to the program would involve the amount of er-
ror given the amount of gap between the material layer and the waveguide walls.
This analysis would require a different inverse calculation altogether, but would yield

interesting results.

126

 

APPENDICES

127

1XFH?EHVI)IXZIX

FIJIUFILAJV CXDI)E3()F‘FT)FUWDAI{I)IPFUDHBIJEBJ

This appendix contains the code written from the derivations in Section 3.2 which
calculate the S-parameters of a slab of material inside a waveguide given the permit-
tivity and permeability of the medium. The input and output files were written in
Microsoft Excel and saved as comma separated values or .csv files.

!Forward problem version 2.0 ”‘
I******IIIIII***************************************************
real pi, mu_o, eps_o, a, b, f, w, tm, etran, zo

complex mu_m, eps_m, mu, eps, f_c, f_co, var_1, k, km ‘

 

complex 1, 2m, m11, m12, m21, m22, eref, einc, rho, tau
real freq,eps_r,eps_i,mu_r,mu_i

real db_mag,angle,slldb,s21db,slla,s21a

real db_delta,ang1e_delta,tm_delta

i = cmp1x(0.,1.)
pi = 4.*atan(1.)

! tm

tm

.0035433 !Rexolite material thickness
.0031623 !MagRAM material thickness
mu_o = 4*pi*1.e-7

eps_o = 8.854e-12

a = 0.02286 !width of waveguide

open(unit=1, status=’old’, form=’formatted’, action=’read’, &
&defau1tfile=’/Documents and Settings/meeusenl/My Documents/&
&fortran/’, &

&file=’forward-input.csv’) !open this file in this directory

open(unit=2, status=’replace’, form=’formatted’, &
&action=’readwrite’, defaultfile=’/Documents and Settings/&
&meeusen1/My Documents/fortranl’, &

&file=’forward_output.csv’) !open this file in this directory

128

do

read(1,*,iostat=ios) freq,eps_r,eps_i,mu_r,mu_i
if(ios .gt. 0.0) cycle !if error, goto next input
if(ios /= 0) exit !If end-of-file, exit do loop

if (isnan(coefficient)) cycle !if file entry is not a

!number, goto next input

eps_m = eps_r + (i*eps_i)

mu_m = mu_r + (i*mu_i)

mu = mu_o*mu_m !static permeability

eps = eps_o*eps_m !static permittivity

f = freq*1.e9 !add 9 zeros to frequency to make GHz

f_c = 1/(2*a*sqrt(mu*eps)) !material cutoff frequency

f_co = 1/(2*a*sqrt(mu_o*eps_o)) !waveguide cutoff

!frequency

w = 2*pi*f

var_1 = sqrt(1-(f_c/f)**2) !variable for intermediate
!calculations

k = w*sqrt(mu*eps) !free space wavenumber

km = k*var_1 !material wavenumber

!material wave impedance

zm = (1/var_1)*sqrt(mu_o/eps_o)*sqrt(mu_m/eps_m)
!free space wave impedance

20 = (1/sqrt(1-(f_co/f)**2))*sqrt(mu_o/eps_o)

m11 = cos(km*tm) !matrix terms

m12 = i*zm*sin(km*tm)

m21 = i*(sin(km*tm)/zm)

m22 = cos(km*tm)

eref = m11+(m12/zo)-m21*zo-m22

einc = m11+(m12/zo)+m21*zo+m22

etran = 2

rho = eref/einc !reflection coefficient
tau = etran/einc !transmission coefficient

129

slldb db_mag(rho) !s-parameters
s21db db_mag(tau)
sila = angle(rho)

s21a = angle(tau)

write(2,"(f8.5,’,’,f10.5,’,’,f10.5,’,’,f10.5,’,’&
&,f10.5,’,’,f8.5)") &
&freq,s11db,slia,s21db,s21a,tm

and do

close(unit=1, status=’keep’) !close opened file

write(*,*) ’Finished ***@@@***’

close(unit=2, status=’keep’) !close opened file
end

!***********************************************************
!convert complex number to magnitude in dB
real function db_mag(input)
real mag
complex input
mag = abs(input)
db_mag = 20.*1og10(mag)

end function

!***********************************************************
!convert complex number to phase
real function angle(input)
use global
complex input
angle = atan2d(aimag(input),real(input))

end function

130

APPENDIX B

FORTRAN CODE OF INVERSE PROBLEM

This appendix contains the code written from the derivations in Section 3.3 which
calculate the constitutive parameters of a slab of material inside a waveguide given the
S—parameters at the boundaries. The input and output files were written in Microsoft
Excel and saved as comma separated values or .csv files.

! inverse research problem 4.0

!it************************************************************

module global !global variables
real pi, c, a, mu_o, eps_o, f_co, kc
real freq
complex 1

end module global
!***************t*********************************************
module values !only used by mu_r and eps_r
real sllm, s21m, slli, s21i
real w, ko, f, mu_m, eps_m, lamda, quartwave, n
real 1amda_g
complex rho, tau, var2, p, u, zr, lnu, lnu_test
end module values
y*************************************************************
program main !main program declaration
use global
complex mu_r, eps-r, find_mu_r, find_eps_r
real slldb, s21db, 811a, s21a, tm
pi = 4*atan(1.)
i = cmplx(0,1)
a = 0.02286 !width of waveguide
mu_o = 4*pi*1.e-7
eps_o = 8.854e-12

c = 1/sqrt(mu_o*eps_o)

!cutoff frequency in air
f_co = 1./(2.*a*sqrt(mu_o*eps_o))
kc = pi/a

open(unit=1, status=’old’, form=’formatted’, &

&action=’read’, defaultfile=’/Documents and Settings/&

131

&meeusen1/My Documents/fortran/’, file=’forward_output.csv’)
!open this file in this directory

open(unit=2, status=’rep1ace’, form=’formatted’, &
&action=’readwrite’, defaultfile=’/Documents and Settings/&
&meeusen1/My Documents/fortran/’, &

&file=’reverse_output.csv’) !open this file in this directory

do
read(1,*,iostat=ios) freq,slldb,s11a,s21db,s21a,tm
if(ios .gt. 0.0) cycle !if error, goto next input
if(ios /= 0) exit !If end-of-file, exit do loop
if (isnan(coefficient)) cycle
!if file entry is not a number, goto next input
mu_r = find_mu_r(811db, s21db, 311a, s21a, tm)
!note, freq is a global variable
eps_r = find_eps_r(slldb, s21db, 811a, s21a, tm)
write(2,"(f8.5,’,’,f8.5,’,’,f8.5,’,’,f8.5,’,’,f8.5)") &
&freq,eps_r,mu_r

end do

close(unit=1, status=’keep’) !close opened file

write(*,*) ’Finished ***@@@***’

close(unit=2, status=’keep’) !close opened file
end program main

!****************’1‘********************************************

!find complex permeability of material given s-parameters

complex function find_mu_r(sl1db, s21db, 311a, s21a, tm)
use global
use values
real slldb, s21db, sila, s21a, tm

slim = 10.**(slldb/20.) !convert from db
s21m = 10.**(s21db/20.)

8111 = 311m*sind(811a) !find imaginary part
s21i = s21m*sind(s21a)

sllr = sllm*cosd(slla) !find real part

132

s21r = s21m*cosd(321a)

rho s11r + $11i*i !combine real and imaginary parts
tau = s21r + s21i*i
f = freq*1.e9 !add 9 zeros to frequency to make GHz

2.*pi*f

8
"

var2 = (rho**2 - tau**2 + 1)/(rho*2)
p = var2-sqrt(var2**2 - 1)

u = (tau*p)/(p-rho)

ko = w/c

zr = (1+p)/(1-p)
n=0
do
lnu = clog(u) - i*2*pi*n
write(*,*) ’n =’, n
find_mu_r = (zr*lnu)/((i*ko*tm)*(sqrt(1.-(f_co/f)**2)))
if (rea1(find_mu-r) > 0) exit !help track imaginary
n=n+1 !component of Inn to
end do !make sure it doesn’t

!wrap around.
end function

!*************************************************************

!find complex permittivity of material given s-parameters

complex function find_eps_r(slldb, s21db, slia, s21a, tm)
use global
use values
real slldb, s21db, 311a, s21a, tm

complex find_mu_r, mu_r

!required for permittivity calculation
mu_r = find_mu_r(811db, s21db, sila, s21a, tm)

find-eps_r = ((1nu*sqrt(1-(f_co/f)**2))/(i*ko*tm*zr)+&
&((c**2 * kC**2)/((2*pi)**2 * mu_r * f**2)))
end function

!*************************************************************

133

APPENDIX C

FORTRAN CODE OF INVERSE PROBLEM WITH COMPENSATION

This appendix contains the code written from the derivations in Section 3.4 which
calculate the constitutive parameters of a slab of material inside a waveguide given the
S-parameters at the boundaries. The input and output files were written in Microsoft

Excel and saved as comma separated values or .csv files.

! inverse research problem 5.0

!*************************************************************

module global !global variables
real pi, c, a, mu_o, eps_o, f_co, kc
real f,w
complex 1

end module global
!*************************************************************
module values lonly used by mu_r and eps_r
real silm, lem, 3111, 8211
real ko, mu_m, eps_m, lamda, quartwave, n
real lamda_g
complex rho, tau, var2, p, u, zr, lnu, lnu_test
end module values
!****************t***************************#****************
! The main program calls funstions to compute epsilon and mu, then
it calls funstions ! to computer the standard deviation of error
for the real part of epsilon and mu ! (eps-r_p and mu_r_p) and the
comlex part of epsilon and mu (eps_r_pp and mu_r_pp)

program main !main program declaration
use global
complex mu_r, eps_r, find_mu_r, find_eps_r
real freq,811db,811am,s22db,s22am,s21db,s21am,si2db,&
&sl2am,amp_r,phi_r
real 311a,s21a,tm,slldbm,s21dbm,s22dbm,sl2dbm
real beta, ko

pi = 4*atan(1.)

i = cmplx(0,1)

a = 0.02286 lwidth of x-band waveguide
mu_o = 4*pi*1.e-7

134

 

eps-o = 8.854e-12
c = 1/sqrt(eps_o*mu_o)

f_co = 1./(2.*a*sqrt(mu_o*eps_o))

kc = pi/a

!choose the correct thickness for the material sample
tm .0031623 !thickness of MagRAM material
tm=.0035433 !thickness of Rexolite material

open(unit=1, status=’old’, form=’formatted’, &
&action=’read’, defaultfile=’/Documents and Settings/&
&meeusen1/My Documents/fortran/’, &

&file=’rexolite.csv’) !open this file in this directory

open(unit=2, status=’replace’, form=’formatted’, &
&action=’readwrite’, defaultfile=’/Documents and Settings/&
&meeusen1/My Documents/fortran/’, &

&file=’result.csv’) !open this file in this directory

do

read(1,*,iostat=ios) freq,empty,slldbm,sllam,s22dbm,&
&s22am,s21dbm,s21am,sl2dbm,sl2am,empty,amp_r,phi_r

if(ios .gt. 0.0) cycle !if error, goto next input
if(ios /= 0) exit !If end-of-file, exit do loop
if (isnan(freq)) cycle

!if file entry is not a number, goto next input

freq*1.e9 !add 9 zeros to frequency to make GHz
2*pi*f

ko = f*2*pi/c

beta = ko*sqrt(1-(f_co/f)**2)

phi_t = -(beta*tm)*180./pi

w

!Compensation for the position of the material within
!the waveguide. Requires addition s-parameters from
!network analyzer.

811a = ((311am+s22am)/2)-phi_r+phi_t

135

 

lea = s21am-phi_r+phi_t
slldb slldbm
s21db s21dbm

mu_r = find_mu_rfslldb, s21db, 811a, $21a, tm)
!note, freq is a global variable
eps_r = find_eps_r(811db, s21db, slla, s21a, tm)

write(2,"(f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5)") &
&freq,eps_r,mu_r

end do

close(unit=1, status=’keep’) !close opened file

write(*,*) ’Finished ***@@@***’

close(unit=2, status=’keep’) !close Opened file

 

end program main

!*********10!!ka*************************************************

! This function computes mu in its raw complex form.

complex function find_mu_r(slldb, s21db, slla, s21a, tm)
use global
use values
real slldb, s21db, slla, s21a, tm

811m = 10.**(811db/20.) !convert from db
s21m = 10.**(821db/20.)

sili = 811m*sind(slla) !find imaginary part
8211 = s21m*sind(s21a)

sllr = sllm*cosd(slia) !find real part

321r = $21m*cosd(s21a)

rho = silr + 311i*i !combine real and imaginary parts
tau = s21r + s21i*i
n=0

var2 = (rho**2 - tau**2 + 1)/(rho*2)

p = var2-sqrt(var2**2 - 1)
u = (tau*p)/(p-rho)
ko = w/c

136

zr = (1+p)/(1-p)

do
lnu = clog(u) - i*2*pi*n
find_mu_r = (zr*1nu)/((i*ko*tm)*(sqrt(1.-(f_co/f)**2)))
if (real(find_mu_r) > 0) exit !help track imaginary
n=n+1 !component of lnu to

end do lmake sure it doesn’t

!wrap around.
! write(*,*) ’1nu=’, imag(1nu) !visual verification of

!complex log n working correctly

end function
!*******III*****************************************************

! This function computes epsilon in its raw complex form. This
function calls ! find_mu_r to simplify coding so it is less

redundant.

complex function find_eps_r(slldb, s21db, 811a, s21a, tm)
use global
use values
real slldb, s21db, s11a, s21a, tm

complex find_mu_r, mu_r
mu_r = find_mu_r(slldb, s21db, s11a, s21a, tm)
find_eps_r = ((lnu*sqrt(1-(f_co/f)**2))/(i*ko*tm*zr)+&

&((c**2 * kc**2)/((2*pi)**2 * mu_r * f**2)))

end function
g*************************************************************

137

APPENDIX D

FORTRAN CODE OF INVERSE PROBLEM WITH ERROR
ANALYSIS

This appendix contains the code written from the derivations in Chapter 4 which
calculate the standard deviation of the final error tolerance given the error tolerance
of all the input variables. The input and output files were written in Microsoft Excel

and saved as comma separated values or .csv files.

! inverse research problem 8.0

!31K****************************************************************

module global !global variables
real pi, c, mu_o, eps_o
real f,w
complex i

end module global
!*****************************************************************
module values !only used by mu_r and eps_r
real sllr, s21r, slli, s21i
real ko, mu_m, eps_m, lamda, quartwave, n
real lamda_g
complex rho, tau, var2, p, u, zr, lnu, lnu_test
end module values
!*****************************************************************
! The main program calls funstions to compute epsilon and mu, then
it calls funstions ! to computer the standard deviation of error
for the real part of epsilon and mu ! (eps_r_p and mu-r_p) and the
comlex part of epsilon and mu (eps_r_pp and mu_r_pp)

program main !main program declaration
use global
complex mu_r, eps-r, find_mu_r, find_eps_r
real mu_r_p-delta,eps_r_p_delta,mu_r_p_error,eps_r_p_error
real mu_r_pp_de1ta,eps_r_pp_delta
real mu_r_pp_error,eps_r_pp_error
real freq,sllam,322am,s21am,sl2am,amp_r,phi_r
real tm,slldbm,sllm,s21dbm,s22dbm,s22m,sl2dbm,a
real ko
real sillin_delta,slla_de1ta
real s21db_delta,s21a_delta,tm_delta

138

 

80

real s22lin_delta,s22a_de1ta

real sl2db_delta,sl2a_de1ta,a_de1ta

real mu_real_upper,mu_rea1_lower

real mu_imag_upper,mu_imag_lower

real eps_rea1_upper,eps_real_lower

real eps_imag_upper,eps_imag_lower

real refl,level,y_sil_lin,y_sll_ang,y_s21_db,y_s21_ang
real y_sl2_db,y_sl2_ang,y_s22_lin,y_s22_ang
integer n

real, dimension(50): xa1,ya_sll_lin,ya_sll_ang
!largest anticipated array sizes

real, dimension(50)::ya_s21_db,ya_s21_ang
real, dimension(50)::Y2_sll_lin,Y2_511_ang
real, dimension(50)::Y2_s21_db,Y2_s21_ang
real, dimension(50)::xa2,ya_s22_lin,ya_s22_ang
real, dimension(50)::ya_sl2_db,ya_sl2_ang
real, dimension(50)::Y2_s22_lin,Y2_s22_ang
real, dimension(50)::Y2_sl2_db,Y2_sl2_ang
rea1(8) YP1,YPN

character*80 filenamel, filename2

real yesno, percentl, percent2

!Initial values
percent1=0.0
percent2=0.0

!Input name of file which contains s-parameters
write(*,*) ’Name of the material filez’
read(*,80) filenamel

format(A80)

!Enter the thickness of the material in inches
write(*,*) ’Thickness of material in Inches:’
read(*,*) tm

!Percent of uncertainty in material thickness ie. 3.0,
!0.1,...
write(*,*) ’Percent of error for material thickness:’

read(*,*) percentl

139

!Percent of uncertainty in waveguide width ie. 3.0,
!O.1,...

write(*,*) ’Percent of error for waveguide width ’
read(*,*) percent2

write(*,*) ’Do you have a file for’

write(*,*) ’the Network analyzer error?’

write(*,*) ’0 - no’

write(*,*) ’1 - yes’

read(*,*) yesno

!enter name of file containing uncertainty of the network
lanalyzer setup.
if (yesno .gt. 0) then
write(*,*) ’Name of the network analyzer error file:’
read(*,80) filename2

endif

pi = 4*atan(1.)

i = cmplx(0,1)

mu_o = 4*pi*1.e-7

eps_o = 8 8546-12

c = 1/sqrt(eps_o*mu_o)

a = 0.02286 !width of x-band waveguide
!convert percent into inches
tm_delta=percent1*tm/100.
a_de1ta=percent2*a/100.

open(unit=1, status=’old’, form=’formatted’, &
&action=’read’, defaultfile=’/Documents and Settings/&
&meeusen1/My Documents/fortran/’, file=filename1)
!open this file in this directory

open(unit=2, status=’replace’, form=’formatted’, &
&action=’readwrite’, &

&defaultfile=’/Documents and Settings/&
&meeusen1/My Documents/fortran/’, file=’resu1t.csv’)

!open this file in this directory

140

 

rad.

 

“

 

open(unit=3, status=’replace’, form=’formatted’, &

Raction=’readwrite’, &

&defaultfile=’/Documents and Settings/&

&meeusen1/My Documents/fortran/’, &

&file=’resu1t_error.csv’)

!open this file in this directory

!Don’t run this code if answer is no

if (yesno .gt. 0) then

open(unit=4, status=’old’, form=’formatted’, &

&action=’read’, defaultfi1e=’/Documents and Settings/&

&meeusen1/My Documents/fortranI’, file=filename2)

!open this file in this directory

n=0

!creating arrays with the data from file

read(4,*,iostat=ios) refl,y_sli_lin,y_311_ang,&
&y_s22_lin,y_s22_ang,empty,level,y_sl2_db,&
&y_sl2_ang,y_s21_db,y_s21_ang

if(ios .gt. 0.0) cycle !if error, goto next input
if(ios /= 0) exit !If end-of-file, exit do loop
if (isnan(coefficient) .or. &

&y_811_lin .eq. 0.0) cycle

!if file entry is not a number, goto next input

!Input uncertainty data into arrays
n=n+1

xa1(n)=refl
ya_s11_lin(n)=y_sll_lin
ya_s11_ang(n)=y_sll_ang
ya_s22_lin(n)=y_s22_lin
ya_s22_ang(n)=y_s22_ang
xa2(n)=level
ya_s21_db(n)=y_s21_db
ya_s21_ang(n)=y_s21_ang
ya_s12_db(n)=y_sl2_db
ya_sl2_ang(n)=y_sl2_ang

enddo

141

!set to a high number to make a "natural spline"
YP1=1E30
!set to a high number to make a "natural spline"
YPN=1E30

!calculate second derivatives of arrays for
!interpolation.

call SPLINE(xa1,ya_811_lin,n,YP1,YPN,Y2_sll-lin)
call SPLINE(xa1,ya_slI_ang,n,YP1,YPN,Y2_s11_ang)
call SPLINE(xa1,ya_s22_lin,n,YP1,YPN,Y2_s22_lin)
call SPLINE(xa1,ya_s22_ang,n,YP1,YPN,Y2_s22_ang)
call SPLINE(xa2,ya_s21_db,n,YP1,YPN,Y2_s21_db)
call SPLINE(xa2,ya_s21_ang,n,YP1,YPN,Y2_s21-ang)
call SPLINE(xa2,ya_sl2-db,n,YP1,YPN,Y2_sl2_db)
call SPLINE(xa2,ya_sl2_ang,n,YP1,YPN,Y2_sl2_ang)
!Spline only has to be called once to calculate
!the second derivatives

endif

do
read(1,*,iostat=ios) freq,empty,sl1dbm,s11am,&
&s22dbm,322am,s21dbm,s21am,sl2dbm,s12am,empty,&
&amp_r,phi_r

if(ios .gt. 0.0) cycle !if error, goto next input
if(ios /= 0) exit !If end-of-file, exit do loop
if (isnan(freq) .or. freq .eq. 0.0) cycle

!if file entry is not a number, goto next input

freq*1.e9 !add 9 zeros to frequency to make GHz
2*pi*f
k0 = f*2*pi/c

V

10.**(811dbm/20.)
10.**(s22dbm/20.)

$11m
s22m

!If answer to question is no, set all analyzer

142

!uncertainties to zero. Else, find uncertainties by
!interpolating existing data.
if (yesno .gt. 0) then
call SPLINT(xa1,ya_sll_lin,&
&Y2_311_lin,n,Slim,slllin_delta)
call SPLINT(xa1,ya_sl1_ang,&
&Y2_sll_ang,n,811m,511a_de1ta)
call SPLINT(xa1,ya_s22_lin,&
&Y2_s22_1in,n,s22m,s22lin_de1ta)
call SPLINT(xa1,ya_822_ang,&
&Y2_s22_ang,n,s22m,s22a_delta)
call SPLINT(xa2,ya_sl2_db,&
&Y2_sl2_db,n,812dbm,812db_delta)
call SPLINT(xa2,ya_812_ang,&
&Y2_sl2_ang,n,sl2dbm,si2a_delta)
call SPLINT(xa2,ya_s21_db,&
&Y2_s21-db,n,s21dbm,s21db_delta)
call SPLINT(xa2,ya_s21_ang,&
&Y2_s21_ang,n,s21dbm,s21a_delta)

 

else

slllin_de1ta=0.0
slla_delta=0.0
8221in_de1ta=0.0
s22a_delta=0.0
sl2db_delta=0.0
sl2a_delta=0.0
s21db_delta=0.0
s21a_de1ta=0.0

endif

!Find complex permeability
mu_r = find_mu_r(sllm,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)

!note, freq is a global variable
!Find complex permittivity

eps_r = find_eps_r(s11m,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)

143

!Find uncertainty in real part of complex permeability

mu_r_p_delta = mu_r_p_error(811m,s22m,s21dbm,sl2dbm,&
&sllam,322am,s21am,sl2am,amp_r,phi_r,tm,a,&
&8111in_delta,811a_delta,s12db_de1ta,sl2a_delta,&
&a_delta,s21db_delta,s21a_delta,s22lin_delta,&
&s22a_delta,tm_delta)

!Find uncertainty in real part of complex permittivity

eps_r_p_delta = eps_r_p_error(sllm,s22m,s21dbm,s12dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a,&
&slllin_delta,$11a_delta,sl2db_delta,sl2a_delta,&
&a_delta,s21db_delta,s21a_delta,s22lin_delta,&
&s22a_delta,tm_delta)

!Find uncertainty in imaginary part of complex permeability
mu_r_pp_delta = mu_r_pp_error(811m,s22m,s21dbm,sl2dbm,&
&311am,322am,s21am,sl2am,amp_r,phi_r,tm,a,&
&slllin_delta,811a_delta,sl2db_de1ta,sl2a_delta,&
&a_delta,s21db_delta,s21a-de1ta,s22lin_delta,&
&s22a_delta,tm_delta)

!Find uncertainty in imaginary part of complex

!permittivity.

eps_r_pp_delta = eps_r_pp_error(sllm,s22m,s21dbm,sl2dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a,&
&slllin_delta,slla_delta,sl2db_de1ta,s12a_delta,&
&a_de1ta,s21db_delta,s21a_delta,s22lin_delta,&
&s22a_de1ta,tm_de1ta)

!Find upper and lower uncertainty bars.
mu_real_upper = rea1(mu_r)+mu_r_p_delta

mu-rea1_lower = real(mu_r)-mu_r-p-de1ta

eps_real_upper real(eps_r)+eps_r_p_delta

eps_real_lower real(eps-r)-eps_r_p_delta
mu_imag_upper = imag(mu_r)+mu_r_pp_delta
mu_imag_lower = imag(mu_r)-mu_r_pp_delta
eps_imag_upper = imag(eps_r)+eps_r_pp_delta

eps_imag_lower = imag(eps_r)-eps_r_pp_delta

144

!0utput file with complex permeability, permittivity, and

!all associated uncertainties.

write(2,"(f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5&
&,’,’,’,’,f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5)") &
&freq,mu_r,eps_r,mu_r_p_delta,eps_r_p_de1ta,&
&mu_r_pp_delta,eps_r-pp_delta

!0utput file with complex permeability, permittivity, and

!all upper and lower uncertainty bars.

write(3,"(f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5,’,’,f10.5,’,’,&
&f9.5,’,’,f9.5,’,’,f9.5,’,’,f9.5)“) &
&freq,mu_real_upper,mu_rea1-lower,eps_real_upper,&
&eps_rea1_lower,mu_imag_upper,mu_imag_lower,&
&eps_imag_upper,eps_imag-lower

end do

close(unit=1, status=’keep’) !close opened file

close(unit=2, status=’keep’) !close opened file

close(unit=3, status=’keep’) !close opened file

close(unit=4, status=’keep’) !close opened file

write(*,*) ”

write(*,*) ’Finished...’

write(*,*) ”

end program main

!III*********IIHI!*****************************************************
! This part of the program does the actual computation of the
!total error of the real part of mu (mu_r_p_error) based on the
!results of individual contributions of errors (deltal,
!delta2,...)

real function mu_r_p_error(sllm,s22m,s21dbm,sl2dbm,sllam,&

&s22am,s21am,sl2am,amp_r,phi_r,tm,a,&
&s11db_delta,slla_delta,s12db_delta,&
&sl2a_delta,a_de1ta,s21db_delta,s21a_delta,&
&s22db_delta,s22a_delta,tm_delta)

use global

use values

real 311m,s22m,s21dbm,s12dbm,sllam

145

 

real
real
real
real
real

real

deltal

s22am,s21am,sl2am,amp_r,phi_r,tm,a
sl11in_de1ta,811a_delta,s21db_delta,s21a_de1ta
s22lin_delta,s22a_de1ta,sl2db-delta,s12a_de1ta
deltal,delta2,delta3,delta4,delta5,tm_delta
delta6,delta7,delta8,delta9,delta10,a_de1ta
find_mu_r_p

(find_mu_r_p(sl1m+8111in_delta,s22m,s21dbm,&
k812dbm,sllam,s22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find_mu_r_p(sllm-slllin_de1ta,s22m,&
&s21dbm,s12dbm,sllam,s22am,s21am,sl2am,&
&amp_r,phi_r,tm,a))/2

delta2 = (find_mu_r_p(s11m,s22m+s22lin_de1ta,s21dbm,&

&sl2dbm,811am,s22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find_mu_r_p(311m,s22m-s22lin_delta,&
&s21dbm,sl2dbm,811am,s22am,s21am,sl2am,amp_r,&
&phi_r,tm,a))/2

delta3 = (find_mu_r_p(811m,s22m,s21dbm+s21db_de1ta,&

asl2dbm,811am,$22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find_mu_r_p(811m,s22m,&
&s21dbm-s21db_de1ta,812dbm,&

&811am,s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta4 = (find_mu_r_p(311m,s22m,s21dbm,&

&sl2dbm+s12db_de1ta,sllam,s22am,s21am,sl2am,&
&amp_r,phi_r,tm,a)-find-mu_r_p(sllm,s22m,&
&s21dbm,sl2dbm-sl2db_de1ta,&
&sllam,s22am,821am,sl2am,amp_r,phi_r,tm,a))/2

deltas = (find-mu_r_p(s11m,s22m,s21dbm,sl2dbm,&

&sllam+811a_delta,s22am,s21am,sl2am,amp_r,&
&phi_r,tm,a)-find_mu_r_p(sl1m,s22m,s21dbm,&
&sl2dbm,sllam-s11a_delta,s22am,s21am,sl2am,&
&amp_r,phi_r,tm,a))/2

delta6 = (find_mu_r_p(311m,s22dbm,s21dbm,s12dbm,&

146

&sllam,s22am+s22a_delta,s21am,sI2am,amp-r,&
&phi_r,tm,a)-find_mu_r_p(811m,s22dbm,s21dbm,&
&sl2dbm,sllam,s22am-s22a_de1ta,s21am,sl2am,&
&amp_r,phi_r,tm,a))/2

delta? = (find_mu_r_pCslIm,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am+s21a_delta,sl2am,amp_r,phi_r,tm,a)-&
&find_mu_r_p(811m,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am-s21a_delta,sl2am,amp_r,phi_r,tm,a))/2

delta8 = (find_mu_r_p(811m,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am,sl2am+sl2a_delta,amp_r,phi_r,tm,a)-&
&find_mu_r_p(811m,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sl2am-s12a_delta,amp_r,phi_r,tm,a))/2

delta9 = (find_mu_r_p(sllm,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm+tm_de1ta,a)-&
&find_mu-r-p(silm,s22m,s21dbm,812dbm,311am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm-tm_delta,a))/2

delta10 = (find_mu_r_p(sllm,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm+tm_delta,a)-&
&find_mu_r-p(811m,s22m,s21dbm,si2dbm,sllam,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a-a_delta))/2

mu_r_p_error = sqrt(de1ta1**2 + delta2**2 + delta3**2 +&
&delta4**2 + delta5**2 + delta6**2 +&
&delta7**2 + delta8**2 + delta9**2 +&
&delta10**2)

end function

!***IIIItIIIIIUIIII:*II!III************II!*************************IIHII************
! This part of the program does the actual computation of the
!total error of the complex part of mu (mu_r_pp_error) based on
!the results of individual contributions of errors (deltal,
!delta2,...)

147

 

real function mu_r_pp_error(sllm,s22m,s21dbm,s12dbm,811am,&

&s22am,s21am,sl2am,amp_r,phi_r,tm,a,&
&slldb_de1ta,811a_delta,sl2db_delta,&
&sl2a_de1ta,a-delta,s21db_delta,s21a_delta,&
&s22db_delta,s22a_delta,tm_delta)

use global

use values

real 311m,s22m,s21dbm,sl2dbm,sllam

real s22am,s21am,sl2am,amp_r,phi_r,tm,a

real slllin_delta,$11a_delta,s21db_delta,s21a_delta

real s22lin_delta,s22a_de1ta,sl2db_de1ta,812a_delta

real deltal,delta2,delta3,delta4,delta5,tm_delta

real delta6,delta7,delta8,delta9,delta10,a_delta

real find_mu_r_pp

 

delta1 = (find_mu_r-pp(sl1m+s1llin_delta,s22m,s21dbm,&
&s12dbm,sllam,s22am,s21am,leam,amp_r,phi_r,&
&tm,a)-find_mu_r_pp(sl1m-3111in_delta,s22m,&
&s21dbm,sl2dbm,sl1am,s22am,s21am,sl2am,amp_r,&
&phi_r,tm,a))/2

delta2 = (find_mu_r_pp(811m,s22m+s22lin-delta,s21dbm,&
&sl2dbm,sllam,s22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find_mu_r_pp(311m,s22m-s22lin_delta,&
&s21dbm,sl2dbm,311am,$22am,s21am,sl2am,amp_r,&
&phi_r,tm,a))/2

delta3 = (find_mu_r-pp(sllm,s22m,s21dbm+s21db_delta,&
&sl2dbm,811am,s22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find-mu_r_pp(sllm,s22m,&
&s21dbm-s21db_delta,sl2dbm,&
&311am,s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta4 = (find_mu_r_pp(s11m,s22m,s21dbm,&
&sl2dbm+312db_de1ta,sllam,s22am,s21am,sl2am,&
&amp_r,phi_r,tm,a)-find_mu_r_pp(811m,s22m,&
&s21dbm,sl2dbm-812db_delta,&
&811am,s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

148

deltas = (find_mu_r_pp(811m,s22m,s21dbm,s12dbm,&
&sllam+sila_de1ta,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)-&
&find_mu_r_pp(311m,s22m,s21dbm,sl2dbm,&
&s11am-slla_delta,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta6 = (find_mu_r_ppCsllm,s22dbm,s21dbm,sl2dbm,sllam,&
&s22am+s22a,delta,s21am,sl2am,amp_r,phi-r,tm,a)-&
&find_mu_r_pp(811m,s22dbm,s21dbm,sl2dbm,sllam,&
&sZ2am-s22a_delta,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta7 = (find_mu_r_pp(sllm,s22m,s21dbm,sI2dbm,sllam,&
&s22am,s21am+s21a_delta,sl2am,amp_r,phi_r,tm,a)-&
&find_mu_r_pp(sllm,522m,s21dbm,sl2dbm,sliam,&
&s22am,s21am-s21a_delta,s12am,amp_r,phi_r,tm,a))/2

delta8 = (find-mu_r_pp(311m,s22m,s21dbm,sl2dbm,s11am,&
&s22am,s21am,sl2am+312a_delta,amp_r,phi_r,tm,a)-&
&find_mu_r_pp(sllm,s22m,s21dbm,sl2dbm,s11am,&
&s22am,s21am,sl2am-sl2a_de1ta,amp_r,phi_r,tm,a))/2

delta9 = (find_mu_r_pp(811m,s22m,s21dbm,312dbm,811am,&
&322am,s21am,sl2am,amp_r,phi_r,tm+tm_delta,a)-&
&find_mu_r-pp(811m,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm-tm_delta,a))/2

delta10 = (find_mu-r_pp(311m,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm+tm_delta,a)-&
&find_mu_r_pp(811m,s22m,521dbm,812dbm,811am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a-a_delta))/2

mu_r_pp_error = sqrt(de1ta1**2 + delta2**2 + delta3**2 +&
&de1ta4**2 + delta5**2 + delta6**2 +&
&de1ta7**2 + delta8**2 + delta9**2 +&
&delta10**2)

end function

149

 

l*****************************************************************

! This part of the program does the actual computation of the

!total error of the real part of epsilon (eps_r_p_error) based on

!the results of individual contributions of errors (deltal,

!delta2, ...)

real function eps_r-p_error(811m,s22m,321dbm,sl2dbm,sllam,&

use global

use values

&s22am,s21am,s12am,amp_r,phi_r,tm,a,&
&811db_delta,slla_delta,sl2db_de1ta,&
&s12a_delta,a_de1ta,s21db_de1ta,s21a_de1ta,&
&822db_de1ta,s22a_de1ta,tm_delta)

real $11m,s22m,s21dbm,s12dbm,sl1am

real s22am,s21am,sl2am,amp_r,phi_r,tm,a

real slllin_delta,311a_delta,s21db_delta,s21a_de1ta
real s22lin_delta,s22a_de1ta,s12db_delta,sl2a_delta
real deltal,delta2,delta3,de1ta4,delta5,tm_de1ta
real delta6,delta7,delta8,delta9,delta10,a_delta
real find_eps_r_p

deltal = (find_eps_r_p(s11m+slllin_delta,s22m,s21dbm,&
&312dbm,811am,s22am,s21am,312am,amp_r,phi_r,&
&tm,a)-find_eps_r_p(sl1m-sIllin_de1ta,s22m,&
&s21dbm,sl2dbm,311am,s22am,s21am,sl2am,amp_r,&
&phi_r,tm,a))/2

delta2 = (find_eps_r_p(311m,s22m+s22lin_delta,s21dbm,&
&s12dbm,sllam,s22am,s21am,812am,amp_r,phi_r,&
&tm,a)-find_eps_r_p(811m,s22m-s22lin_delta,&
&s21dbm,sl2dbm,s11am,s22am,s21am,812am,amp_r,&
&phi_r,tm,a))/2

delta3 = (find_eps_r_p(sllm,s22m,s21dbm+s21db_delta,&
&sl2dbm,811am,s22am,s21am,sl2am,amp_r,phi_r,&

&tm,a)-find_eps_r_p(811m,s22m,&
&s21dbm-s21db_delta,s12dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

150

delta4 = (find_eps_r_p(811m,322m,s21dbm,&
&s12dbm+s12db_delta,sllam,s22am,s21am,sl2am,&
&amp_r,phi_r,tm,a)-find_eps_r_p(sllm,s22m,&
&s21dbm,sl2dbm-sl2db_delta,&

&s11am,s22am,s21am,sl2am,amp-r,phi_r,tm,a))/2

deltas = (find_eps_r_p(sllm,s22m,s21dbm,312dbm,&
&sllam+slla_delta,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)-&
&find_eps_r_p(811m,s22m,s21dbm,sl2dbm,&
&311am-slla_de1ta,&
&s22am,s21am,s12am,amp_r,phi_r,tm,a))/2

delta6 = (find_eps_r_p(sllm,s22dbm,s21dbm,sl2dbm,sllam,&
&s22am+s22a_delta,s21am,812am,amp_r,phi_r,tm,a)-&
&find_eps_r_p(811m,s22dbm,s21dbm,sl2dbm,811am,&
&s22am-s22a_de1ta,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta? = (find_eps_r_p(811m,s22m,s21dbm,sl2dbm,sllam,&
&s22am,521am+s21a_de1ta,si2am,amp_r,phi_r,tm,a)-&
&find_eps_r_p(sllm,s22m,s21dbm,sl2dbm,s11am,&
&s22am,s21am-s21a_delta,sl2am,amp_r,phi_r,tm,a))/2

delta8 = (find_eps_r_p(311m,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sl2am+si2a_delta,amp_r,phi_r,tm,a)-&
&find_eps_r_p(s11m,s22m,s21dbm,s12dbm,811am,&
&s22am,s21am,s12am-sl2a_de1ta,amp,r,phi-r,tm,a))/2

delta9 = (find_eps_r_p(sllm,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm+tm_delta,a)-&
&find_eps_r_p(311m,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am,s12am,amp_r,phi_r,tm-tm_delta,a))/2

delta10 = (find_eps_r_p(811m,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sl2am,amp_r,phi_r,tm+tm_delta,a)-&
&find_eps_r_p(slim,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a-a_de1ta))/2

151

 

eps_r_p_error = sqrt(de1ta1**2 + delta2**2 + delta3**2 +&
&delta4**2 + delta5**2 + delta6**2 +&
&delta7**2 + delta8**2 + delta9**2 +&
&de1ta10**2)

end function

g****************************************************************
! This part of the program does the actual computation of the
!total error of the complex part of epsilon (eps_r_pp_error) based
!on the results of individual contributions of errors (deltal,
!delta2, ...)

real function eps_r_pp_error(sllm,s22m,s21dbm,sl2dbm,s11am,&
&s22am,521am,sl2am,amp_r,phi_r,tm,a,&
&311db_de1ta,311a_delta,sl2db_delta,&
&sl2a_de1ta,a_de1ta,s21db_delta,s21a_delta,&
&s22db_delta,s22a_de1ta,tm_delta)

 

use global

use values

real 311m,s22m,s21dbm,sl2dbm,811am

real s22am,s21am,s12am,amp_r,phi_r,tm,a

real s1111n_delta,sl1a_de1ta,s21db_delta,s21a_delta
real s22lin_delta,s22a_delta,sl2db_de1ta,sl2a_delta
real delta1,delta2,delta3,delta4,delta5,tm_de1ta
real delta6,delta7,delta8,de1ta9,delta10,a_de1ta
real find_eps_r_pp

deltal = (find_eps_r_pp(sl1m+slllin_delta,s22m,s21dbm,&
&sl2dbm,sllam,s22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find_eps_r_pp(sllm-slllin_delta,s22m,&
&s21dbm,sl2dbm,sllam,s22am,s21am,sl2am,amp_r,&
&phi_r,tm,a))/2

delta2 = (find_eps_r_pp(311m,s22m+s22lin_delta,s21dbm,&
&s12dbm,811am,s22am,s21am,sl2am,amp_r,phi_r,&
&tm,a)-find_eps_r-pp(sllm,s22m-s22lin_delta,&
&s21dbm,sl2dbm,311am,822am,821am,s12am,amp_r,&
&phi_r,tm,a))/2

152

delta3 = (find_eps_r_pp(sllm,s22m,s21dbm+s21db_de1ta,&
&sl2dbm,sllam,$22am,s21am,sl2am,amp-r,phi_r,&
&tm,a)-find_eps_r_pp(sllm,$22m,&
&s21dbm-s21db-delta,sl2dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta4 = (find_eps_r_pp(811m,s22m,s21dbm,&
&sl2dbm+sl2db_delta,s11am,s22am,s21am,sl2am,&
&amp_r,phi_r,tm,a)-find_eps_r_pp(811m,s22m,&
&S21dbm,$12dbm-812db_delta,&

&311am,s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

deltas = (find_eps_r_pp(s11m,s22m,s21dbm,sl2dbm,&
&sllam+s11a_delta,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)-&
&find_eps_r-pp(sllm,s22m,s21dbm,sl2dbm,&
&sllam-811a_delta,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta6 = (find_eps_r_pp(s11m,s22m,$21dbm,sl2dbm,sllam,&
&s22am+s22a_delta,s21am,sl2am,amp_r,phi_r,tm,a)-&
&find_eps_r_pp(sl1m,s22dbm,s21dbm,sl2dbm,sllam,&
&s22am-s22a_delta,s21am,sl2am,amp_r,phi_r,tm,a))/2

delta? = (find_eps_r_pp(811m,s22m,s21dbm,812dbm,311am,&
&s22am,s21am+s21a_delta,sl2am,amp_r,phi_r,tm,a)-&
&find_eps_r_pp(sllm,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am-s21a_delta,sl2am,amp_r,phi_r,tm,a))/2

delta8 = (find_eps_r_pp(sl1m,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am,sl2am+sl2a_delta,amp_r,phi_r,tm,a)-&
&find_eps_r_pp(811m,s22m,s21dbm,sl2dbm,811am,&
&s22am,s21am,sl2am-sl2a_de1ta,amp_r,phi-r,tm,a))/2

delta9 = (find-eps_r_pp(811m,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm+tm_de1ta,a)-&
&find_eps-r_pp(sl1m,s22m,s21dbm,s12dbm,sllam,&
&s22am,s21am,sl2am,amp-r,phi_r,tm-tm_delta,a))/2

153

 

delta10 = (find_eps_r_pp(sllm,s22m,s21dbm,sl2dbm,s11am,&
&s22am,s21am,$12am,amp_r,phi_r,tm+tm-delta,a)-&
&find_eps_r_pp(811m,s22m,s21dbm,sl2dbm,silam,&
&s22am,s21am,si2am,amp_r,phi_r,tm,a-a_delta))/2

eps_r_pp_error = sqrt(delta1**2 + delta2**2 + delta3**2 +&
&delta4**2 + delta5**2 + delta6**2 +&
&delta?**2 + delta8**2 + delta9**2 +&
&delta10**2)
end function
!*****************************************************************
! This function returns the real result of the calculated mu and

!names it mu_p or mu prime.

real function find_mu_r_p(sllm,s22m,s21dbm,sl2dbm,sllam,&
&s22am,821am,sl2am,amp_r,phi_r,tm,a)
real 811m,s22m,s21dbm,sl2dbm,sllam
real s22am,s21am,sl2am,amp_r,phi_r,tm,a
complex find_mu_r

find_mu_r_p = real(find_mu_r(sllm,s22m,s21dbm,s12dbm,&
&811am,s22am,s21am,sl2am,amp_r,phi_r,tm,a))

end function
!***********a*****************************************************
! This function returns the complex result of the calculated mu

!and names it mu_pp or mu double prime.

real function find_mu_r-pp(sl1m,s22m,s21dbm,s12dbm,s11am,&
&s22am,s21am,s12am,amp_r,phi_r,tm,a)
real sllm,s22m,s21dbm,s12dbm,811am
real s22am,s21am,s12am,amp_r,phi_r,tm,a
complex find_mu_r

find_mu_r_pp = imag(find_mu_r(811m,s22m,s21dbm,sl2dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a))

end function

154

!*****************************************************************

! This function returns the real result of the calculated epsilon

!and names it eps_p or epsilon prime.

real function find_eps_r_p(811m,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,si2am,amp_r,phi_r,tm,a)
real 811m,s22m,s21dbm,sl2dbm,s11am
real s22am,s21am,sl2am,amp_r,phi_r,tm,a

complex find_eps_r

find_eps_r_p = real(find_eps_r(811m,822m,s21dbm,sl2dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a))

end function
!***************a*************************************************
! This function returns the complex result of the calculated

!epsilon and names it eps_p or epsilon double prime.

real function find_eps_r_pp(sllm,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)
real $11m,s22m,s21dbm,s12dbm,sllam
real s22am,s21am,sl2am,amp_r,phi_r,tm,a
complex find_eps_r

find_eps_r_pp = imag(find_eps_r(sllm,s22m,s21dbm,sl2dbm,&
&sllam,s22am,s21am,sl2am,amp_r,phi_r,tm,a))

end function

!*****************************************************************

! This function computes mu in its raw complex form.

complex function find_mu_r(sllm,s22m,s21dbm,sl2dbm,311am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)
use global
use values
real 811m,s22m,s21dbm,sl2dbm,sllam
real s22am,s21am,leam,amp_r,phi_r,tm,a
real f_co

155

 

f_co = 1./(2.*a*sqrt(mu_o*eps_o))
ko = w/c

beta = ko*sqrt(1-(f_co/f)**2)
phi_t = -(beta*tm)*180./pi

s11a = ((silam+s22am)/2)-phi_r+phi_t
$21a = s21am-phi_r+phi_t

s21m = 10.**(s21dbm/20.) !convert from dB to linear

3111 = sllm*sind(811a) !find imaginary part
s21i = s21m*sind(s21a)

sllr = $11m*cosd(811a) !find real part
s21r = s21m*cosd(s21a)

 

rho = sllr + slli*i !combine real and imaginary parts
tau = s21r + s2li*i
n = 0

var2 = (rho**2 - tau**2 + 1)/(rho*2)
p = var2-sqrt(var2**2 - 1)
u = (tau*p)/(p-rho)

zr = (1+p)/(1-p)

do
lnu = clog(u) - i*2*pi*n
write(*,*) ’n =’, n
find_mu_r=(zr*lnu)/((i*ko*tm)*(sqrt(1.-(f_co/f)**2)))
if (real(find_mu_r) > 0) exit !help track imaginary
n=n+1 !component of lnu to-
end do !make sure it doesn’t-
write(*,*) ’lnu=’, imag(1nu) !wrap around

end function

156

l*****************************************************************
! This function computes epsilon in its raw complex form. This
function calls ! find_mu_r to simplify coding so it is less

redundant.

complex function find_eps_r(sllm,s22m,s21dbm,sl2dbm,sllam,&
&s22am,s21am,sI2am,amp_r,phi_r,tm,a)
use global
use values
real sllm,s22m,s21dbm,s12dbm,sllam
real s22am,s21am,sl2am,amp_r,phi_r,tm,a
real f_co, kc

complex find_mu_r, mu_r

f_co = 1./(2.*a*sqrt(mu_o*eps_o))

kc = pi/a

mu_r = find_mu_r(311m,s22m,s21dbm,s12dbm,s11am,&
&s22am,s21am,sl2am,amp_r,phi_r,tm,a)

find_eps_r = ((1nu*sqrt(1-(f_co/f)**2))/(i*ko*tm*zr)+&
&((c**2 * kc**2)/((2*pi)**2 * mu_r * f**2)))
end function

l*****************************************************************

!cubic spline routine (numerical recipes in fortran)

!William H. Press, Saul A. Teukolsky, etc..., Numerical Recipes in
!FORTRAN, Cambridge University Press, New York: New York, 1992

SUBROUTINE SPLINE(X,Y,N,YP1,YPN,Y2)
integer N,NMAX,I,K
PARAMETER (NMAX=100)
real(8) YP1, YPN
real(4) X(N),Y(N),Y2(N),U(NMAX)
real P,QN,SIG,UN
IF (YP1.GT..99£30) THEN
Y2(1)=0.
U(1)=0.
ELSE
Y2(1)=-0.5

157

 

U(1)=(3./(X(2)-X(1)))*((Y(2)-Y(1))/(X(2)-X(1))-YP1)
ENDIF
D0 I=2,N-1
SIG=(X(I)-X(I-1))/(X(I+1)-X(I-1))
P=SIG*Y2(I-1)+2.
Y2(I)=(SIG-1.)/P
U(I)=(6.*((Y(I+1)-Y(I))/(X(I+1)-X(I))-(Y(I)-Y(I-1))&
&/(X(I)-X(I-1)))/(X(I+1)-X(I-1))-SIG*U(I-1))/P
ENDDO
IF (YPN.GT..99E30) THEN
QN=0.
UN=0.
ELSE
QN=0.5
UN=(3./(X(N)-X(N—1)))*(YPN-(Y(N)-Y(N-1))/(X(N)-X(N-1)))
ENDIF
Y2(N)=(UN-QN*U(N-1))/(QN*Y2(N-1)+1.)
D0 K=N-1,1,-1
Y2(K)=Y2(K)*Y2(K+1)+U(K)
ENDDO
RETURN
END

 

!**********************4!******************************************

!cubic spline routine (numerical recipes in fortran)

!William H. Press, Saul A. Teukolsky, etc..., Numerical Recipes in
!FORTRAN, Cambridge University Press, New York: New York, 1992

!Modified by: Jason Meeusen, March 2004, In order to allow spline
!program to accept negative input arguments.

SUBROUTINE SPLINT(XA,YA,Y2A,N,X,Y)
DIMENSION XACN),YA(N),Y2A(N)
KLO=1
KHI=N
DO WHILE(KHI-KLO.GT.1)
K=(KHI+KLD)/2
IF(X.GE.0)THEN !Original splint program

158

IF(XACK).GT.X)THEN !did not work with
KHI=K !negative ’X’ values
ELSE
KLO=K
ENDIF
ELSE !This inclusion solves the
IF(XA(K).LT.X)THEN !negative ’X’ input problem
KHI=K
ELSE
KLO=K
ENDIF
ENDIF
ENDDD
H=XA<KHI)-XA(KLO)
IF (H.EQ.0.) PAUSE ’Bad XA input.’
A=(XA(KHI)~X)/H
B=(X-XA(KLO))/H
Y=A*YA(KLO)+B*YA(KHI)+&
&((A**3-A)*Y2A(KLO)+(B**3-B)*Y2A(KHI))*(H**2)/6.
RETURN
END

159

AJ?PHJNUDIXIIE

DAAUFIuAEB(ZCHDEZINDIIISILAJTHHIJC}INDIUFILAII(DITFFWJTTVVITTI
NIAJSILAJVI

Zversion 1.0
clear all

P=csvread(’C:\Documents and Settings\meeusen1\My Documents...
\fortran\proper-magram.csv’,4,0);%for displaying magram

M=csvread(’C:\Documents and Settings\meeusen1\My Documents...
\fortran\resu1t.csv’); Zfor getting mu and eps

E=csvread(’C:\Documents and Settings\meeusen1\My Documents...

\fortran\resu1t_error.csv’); Xfor getting upper and lower

freq=M(:,1); mu_m=M(:,2); eps_m=M(:,4); mu_ma=M(:,3);
eps_ma=M(:,5); mu=P(:,5); mua=P(:,6); eps=P(:,3); epsa=P(:,4);
mu_rea1_upper=E(:,2); mu_real_lower=E( ,3); eps_real_upper=E(:,4);
eps-rea1_lower=E(:,5); mu_imag_upper=E(:,6); mu_imag_lower=E(:,?);
eps_imag_upper=E( ,8); eps_imag_lower=E(:,9);

figure(l) subplot(2,1,1)
plot(freq,mu_rea1_upper,’k’,freq,mu_real_lower,’k’,...
freq,mu_m,’k’,freq,mu,’k’) axis tight

ylim([.8,1.9]) Zfor displaying magram

grid ylabel(’Permeability (rea1)’) x1abe1(’Freq (GHz)’)

title(’Permeability vs. Frequency’)

subplot(2,1,2)
plot(freq,mu_imag_upper,’k’,freq,mu_imag_lower,’k’,...
freq,mu_ma,’k’,freq,mua,’k’) axis tight
ylim([-2.6,-1.55]) Zfor displaying magram

grid ylabel(’Permeability (imag)’) x1abel(’Freq (GH2)’)

figure(2) subplot(2,1,1)
plot(freq,eps_real_upper,’k’,freq,eps_real_lower,’k’,...
freq,eps_m,’k’,freq,eps,’k’) axis tight

ylim([19.1,22]) Zfor displaying magram

grid ylabel(’Permittivity (rea1)’) x1abe1(’Freq (GHz)’)
title(’Permeability vs. Frequency’)

160

 

subplot(2,1,2)
plot(freq,eps_imag_upper,’k’,freq,eps_imag_lower,’k’,...
freq,eps_ma,’k’,freq,epsa,’k’) axis tight
ylim([-2.4,1.4]) Xfor displaying magram

grid ylabel(’Permittivity (imag)’) x1abel(’Freq (GHz)’)

161

 

. ,4
‘33...
me“-

1AFHPEHNI)LXLF‘

DAIVFIJAEI(DCHDEIFWDFLISILAJWHHIIC}INDIUFILAII(DIFPFWJTTVVITTI
REXOLITE

Zversion 1.0
clear all

P=csvread(’C:\Documents and Settings\meeusen1\My Documents...
\fortran\proper_rexolite.csv’,4,0); Zfor displaying rexolite

M=csvread(’C:\Documents and Settings\meeusen1\My Documents...
\fortran\resu1t.csv’); Zfor getting mu and eps

E=csvread(’C:\Documents and Settings\meeusen1\My Documents...

\fortran\resu1t_error.csv’); Xfor getting upper and lower

freq=M(:,1); mu_m=M(:,2); eps_m=M(:,4); mu_ma=M(:,3);
eps_ma=M(:,5); mu=P(:,5); mua=P(:,6); eps=P(:,3); epsa=P(:,4);
mu_real_upper=E(:,2); mu_real_lower=E(:,3); eps_real_upper=E(:,4);
eps_real_lower=E(:,5); mu_imag_upper=E(:,6); mu_imag_lower=E( ,7);
eps_imag_upper=E( ,8); eps_imag_lower=E(:,9);

figure(l) subplot(2,1,1)
plot(freq,mu_real_upper,’k’,freq,mu_real_lower,’k’,...
freq,mu_m,’k’,freq,mu,’k’) axis tight

ylim([ 85,1.15]) Xfor displaying rexolite

grid ylabe1(’Permeabi1ity (rea1)’) x1abe1(’Freq (GHz)’)
title(’Permeability vs. Frequency’)

subplot(2,1,2)
plot(freq,mu-imag_upper,’k’,freq,mu_imag_lower,’k’,...
freq,mu_ma,’k’,freq,mua,’k’) axis tight
ylim([-.11,.11]) Zfor displaying rexolite

grid ylabel(’Permeability (imag)’) x1abe1(’Freq (GHz)’)

figure(2) subplot(2,1,1)
plot(freq,eps_real_upper,’k’,freq,eps_rea1_lower,’k’,...
freq,eps_m,’k’,freq,eps,’k’) axis tight
ylim([2.44,2.61]) Zfor displaying rexolite

grid ylabe1(’Permittivity (real)’) x1abel(’Freq (GHz)’)
title(’Permeability vs. Frequency’)

162

subplot(2,1,2)
plot(freq,eps_imag_upper,’k’,freq,eps_imag_lower,’k’,...
freq,eps_ma,’k’,freq,epsa,’k’) axis tight
ylim([-.1?,.16]) Zfor displaying rexolite
grid ylabe1(’Permittivity (imag)’) x1abel(’Freq (GHz)’)

163

 

BIBLIOGRAPHY

 

164

 

BIBLIOGRAPHY

[1] Balanis, Constantine A., Advanced Engineering Electromagnetics, John Wiley
& Sons, 1989

[2] The Fundamentals of Plane Wave Electromagnetic Calculations, BOEING-STL
99X0016

[3] A Method for Determining Permittivity and Permeability from Guided Wave
S-Parameter Measurements, BOEING-STL 99X0011

[4] Bevington, Philip R., and D. Keith Robinson, Data Reduction and Error Anal-
ysis for the Physical Sciences, McGraw-Hill, New York: New York, 1992

[5] Barford, N. C., Experimental Measurements: Precision, Error and Truth, John
Wiley & Sons, New York: Chichester, 1985

[6] Rothwell, Edward. J ., and Micheal J. Cloud, Electromagnetics, CRC Press LLC,
Florida: Boca Raton, 2001

[7] Fogiel, Dr. M., and Staff of Research and Education Association, Handbook of
Mathematical Formulas, Tables, Functions, Graphs, Transforms, Research and
Education Association, New York: New York, 1983

[8] William H. Press, Saul A. Teukolsky, etc..., Numerical Recipes in FORTRAN,
Cambridge University Press, New York: New York, 1992

[9] Digital Fortran Language Reference Manual, Digital Equipment Corporation,
Massachusetts: Maynard, 1997

[10] Hanselman, Duane, and Bruce Littlefield, Mastering Matlab 6, Prentice Hall,
New Jersey: Upper Saddle River, 2001

165

 

 

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