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6/01 c:/CIRC/DateDue.p65-p.15

ELECTROMAGNETIC MATERIAL CHARACTERIZATION USING A PARTIALLY
FILLED RECTANGULAR WAVEGUIDE

By

Andrew Eric Bogle

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2004

ABSTRACT

ELECTROMAGNETIC MATERIAL CHARACTERIZATION USING A PARTIALLY
FILLED RECTANGULAR WAVEGUIDE

By

Andrew Eric Bogle

Electromagnetic material characterization is the process of determining the complex
permittivity and permeability of a material. A waveguide material measurement
technique is developed for highly reflective or lossy materials. In order to extract the
complex constitutive parameters from a material, experimental reflection and
transmission scattering parameters are needed. In a traditional rectangular waveguide
material measurement the sample fills the entire waveguide cross-section, making it
difficult to obtain a significant transmission scattering parameter with highly reflective,
absorptive, or lossy materials. This thesis will demonstrate, through the use of a modal-
analysis technique, how using a partially filled rectangular waveguide cross-section
allows for better transmission responses to extract the complex constitutive parameters.
Experimental results for acrylic and magnetic radar absorbing material are compared to

stripline measurements to verify the modal-analysis technique.

To Tanqueray, why not?

iii

ACKNOWLEDGMENTS

I would like to thank Dr. Dennis Nyquist for all his help. His knowledge and patience

have been invaluable to my grth as an engineer and person.

I would like to thank Dr. Leo Kempel and Dr. Ed Rothwell for the opportunity provided
to further my education at this wonderful institution. I look forward to continuing this
journey for another degree.

Thank you to Dan, Brad, Jason, Rick, Brianna, and Greg for all their support.

Finally, Thank you to my parents, Ted and Nansi, there never ending love and support is

more than any child could ever hope for!

iv

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

LIST OF FIGURES-“- - ..... vi
CHAPTER I: INTRODUCTION- I
CHAPTER 2: M ODE-MA T CHIN G ANALYSIS 4
1. Configuration 4

2. Overview 5

3. Mode Matching Analysis 5
Transverse Fields ..................................................................................................................... 6
Boundary Conditions ............................................................................................................... 7
Testing Operators .................................................................................................................... 7
Nomenclature .......................................................................................................................... 8
Scattering Matrix ..................................................................................................................... 9
CHAPTER 3: MODES OF WAVEGUIDE REGIONS II
1. Waveguide Modes -- - ll
Cutoff Frequency ................................................................................................................... ll

2. Unloaded Waveguide Analysis 13
Geometry ............................................................................................................................... 13
Unloaded Modes .................................................................................................................... 13

3. Loaded Waveguide Analysis 16
Geometry ............................................................................................................................... l6
Loaded Modes ....................................................................................................................... 16
CHAPTER 4: NUMERICAL IMPLEMEN T A T IONS 21
l. Integral Relations 21

2. Propagation Constants 25
CHAPTER 5: RESULTS....---- 54
1. Experimental Setup <4

2. Validation 55

3. Tested Samples <7
Acrylic ................................................................................................................................... 57
MagRAM ............................................................................................................................... 57
CHAPTER 6: CONCLUSIONS 76
APPENDIX A: INTEGRAL IDEN TI TIES 77
APPENDDK B: NETWORK ANALYZER PROCEDURES - 78
1. Calibration-Kit Definition Procedure 79

2. Calibration Procedure 84

3. Measuring Data Procedure 87
APPENDIX C: SOURCE CODE - - - --------- 89
BIBLIOGRAPHY“--- -- - - - - I05

 

LIST OF FIGURES

FIGURE 2.1: CONFIGURATION ............................................................................................ 4
FIGURE 3.1: GEOMETRY OF UNLOADED WAVEGUIDE ........................................................ 13
FIGURE 3.2: GEOMETRY OF LOADED WAVEGUIDE ............................................................ 16
FIGURE 4.1: k, /k0 vs. D/A FOR MODE 1 AT 9 GHz .......................................................... 30
FIGURE 4.2: k, /k0 vs. D/A FOR MODE 1 AT 10 GHz ........................................................ 31
FIGURE 4.3: k, /k0 vs. D/A FOR MODE 1 AT 11 GHz ........................................................ 32
FIGURE 4.4: k, /k0 vs. D/A FOR MODE 2 AT 9 GHz .......................................................... 33
FIGURE 4.5: kz /k0 vs. D/A FOR MODE 2 AT 10 GHz ........................................................ 34
FIGURE 4.6: k: /k0 vs. D/A FOR MODE 2 AT 11 GHz ........................................................ 35

FIGURE 4.7: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 0.25
AT 9 GHz ................................................................................................... 36

FIGURE 4.8: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 0.5
AT 9 GHz ................................................................................................... 37

FIGURE 4.9: NORMALlZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 1 AT
9 GHZ ........................................................................................................ 38

FIGURE 4.10: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 0.25
AT 9 GHz ................................................................................................... 39

FIGURE 4.11: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, WA = 0.5
AT 9 GHz ................................................................................................... 40

FIGURE 4.12: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 1 AT
9 GHz ........................................................................................................ 41

FIGURE 4.13: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 0.25
AT 10 GHz ................................................................................................. 42

FIGURE 4.14: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 0.5
AT 10 GHz ................................................................................................. 43

FIGURE 4.15: NORMALIZED 'EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 1 AT
10 GHz ...................................................................................................... 44

Vi

FIGURE 4.16: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 0.25

AT 10 GHZ ................................................................................................. 45
FIGURE 4.17: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 0.5
AT 10 GHz ................................................................................................. 46
FIGURE 4.18: NORMALIZED Ev FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 1 AT
10 GHz ...................................................................................................... 47
FIGURE 4.19: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 0.25
AT 1 1 GHz ................................................................................................. 48
FIGURE 4.20: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 0.5
AT 1 1 GHz ................................................................................................. 49
FIGURE 4.21: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 1, D/A = 1 AT
1 1 GHz ...................................................................................................... 50
FIGURE 4.22: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 0.25
AT 1 1 GHz ................................................................................................. 51
FIGURE 4.23: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 0.5
AT 11 GHz ................................................................................................. 52
FIGURE 4.24: NORMALIZED EY FIELD vs. WAVEGUIDE LOCATION FOR MODE 2, D/A = 1 AT
1 1 GHz ...................................................................................................... 53
FIGURE 5.1: EXPERIMENTAL SETUP ................................................................................. 59
FIGURE 5.2: SAMPLE HOLDER .......................................................................................... 60

FIGURE 5.3: COMPARISON OF THEORETICAL AND EXPERIMENTAL S-PARAMETERS VALUES
................................................................................................................... 61

FIGURE 5.4: RELATIVE PERMEABILITY & PERMITTIVITY OF DATA GENERATED WITH 3
MODES THEN SOLVED WITH 2 MODES VS. FREQUENCY ............................. 62

FIGURE 5.5: RELATIVE PERMEAEILITY & PERMITTIVITY OF DATA GENERATED WITH 5
MODES THEN SOLVED WITH 4 MODES VS. FREQUENCY ............................. 63

FIGURE 5.6: RELATIVE PERMEABILITY & PERMITTIVITY OF ACRYLIC vs. FREQUENCY
USING STRIP TRANSMISSION LINE .............................................................. 64

FIGURE 5.7: RELATIVE PERMEABlLlTY & PERMITTIVITY OF ACRYLIC vs. FREQUENCY
USING WAVEGUIDE, D/A = 0.25, MODE 1 ................................................... 65

FIGURE 5.8: RELATIVE PERMEABILITY & PERMITTIVITY OF ACRYLIC vs. FREQUENCY
USING WAVEGUIDE, D/A = 0.25, MODE 2 ................................................... 66

vii

FIGURE 5.9: RELATIVE PERMEABILITY & PERMITTIVITY OF ACRYLIC VS. FREQUENCY
USING WAVEGUIDE, D/A = 0.5, MODE 1 ..................................................... 67

FIGURE 5.10: RELATIVE PERMEABILITY & PERMITTIVITY OF ACRYLIC vs. FREQUENCY
USING WAVEGUIDE, D/A = 0.5, MODE 2 ..................................................... 68

FIGURE 5.11: RELATIVE PERMEABILITY & PERMITTIVITY OF ACRYLIC vs. FREQUENCY
USING WAVEGUIDE, D/A = 1, MODE 1 ........................................................ 69

FIGURE 5.12: RELATIVE PERMEABILITY & PERMITTIVITY OF ACRYLIC VS. FREQUENCY
USING WAVEGUIDE, D/A = 1, MODE 2 ........................................................ 70

FIGURE 5.13: RELATIVE PERMEABILITY & PERMITTIVITY OF MAGRAM vs. FREQUENCY
USING STRIP TRANSMISSION LINE .............................................................. 71

FIGURE 5.14: RELATIVE PERMEABILITY & PERMITTIVITY OF MAGRAM vs. FREQUENCY
USING WAVEGUIDE, D/A = 0.09, MODE 1 ................................................... 72

FIGURE 5.15: RELATIVE PERMEABILITY & PERMITTIVITY OF MAGRAM vs. FREQUENCY
USING WAVEGUIDE, D/A = 0.09, MODE 2 ................................................... 73

FIGURE 5.16: RELATIVE PERMEABILITY & PERMITTIVITY OF MAGRAM VS. FREQUENCY
USING WAVEGUIDE, D/A = 0.09, MODE 3 ................................................... 74

FIGURE 5.17: RELATIVE PERMEABILITY & PERMITTIVITY OF MAGRAM VS. FREQUENCY
USING WAVEGUIDE, D/A = 0.09, MODE 4 ................................................... 75

viii

CHAPTER 1: INTRODUCTION

As engineering research progresses and new applications for electromagnetic materials
are found, the need to test these materials for their constitutive parameters also continues
to grow. A number of disciplines rely heavily on the material characterization proceSs
including stealth and integrated circuits. For example, in stealth technology the
constitutive parameters can describe effectively how a material can absorb incoming

radar signals.

An electromagnetic material is characterized by two constitutive parameters, permittivity
and permeability, which describe the materials susceptibility to becoming polarized or
magnetized when exposed to an impressed electric or magnetic field, respectively. Both
permittivity and permeability can be complex values, where the real parts are related to
the energy storage of the material, and the imaginary parts are related to the loss

mechanisms that convert incident electromagnetic radiation into heat.

The electromagnetic material measurement process that finds these complex constitutive
parameters usually has the same basic steps. First, a sample is machined to fit the desired
testing device. Next, the device is connected to a vector network analyzer in order to
obtain experimentally measured scattering parameters. The network analyzer works by
launching an incident wave at the test sample and then measures the signal reflected from
and transmitted through the material. Finally, numerical algorithms are used to extract
the complex constitutive parameters of the test sample from the experimental scattering

parameters.

The algorithm used most to extract the material properties is the Nicholson-Ross-Weir
(NRW) technique[1, 2]. The advantage of the NRW technique is the ability to directly
solve for the material parameters in closed form fiom the experimental scattering
parameters. For the NRW technique to be applied properly, the test samples must be
linear, homogenous, and isotropic with coplanar front and back surfaces[3, 4]. In
waveguide applications, this means the material must completely fill the waveguide

cross-section to prevent the excitation Of higher-order modes.

However, with increasing demands of industry and a number of new materials being
designed to modify signals, traditional methods used to characterize materials are
problematic. The issue in measuring these new highly reflective, absorptive, and lossy
materials is transmission coefficients that are approaching the noise floor of the network
analyzers[5-11]. This thesis offers a waveguide material measurement technique aimed
at solving this issue by only partially filling the waveguide cross-section with the testing
sample. The intention is to allow more of the incident wave to easily pass through the

sample region while still having it couple with the sample.

Chapter 2 presents the mode-matching analysis technique that will find the theoretical
scattering parameters for the complex constitutive parameters extraction process[12].
Since measurement technique being developed violates the properties of the NRW
technique, an iterative complex two-dimensional Newton’s root-searching algorithm will

be used instead[3]. Chapter 3 develops the modal field values for the loaded and

unloaded regions of the rectangular waveguide, a necessity to solve the modal-analysis
technique[13-15]. Chapter 4 discusses how to solve the integral relations developed in
Chapter 2 using the modal field values developed in Chapter 3, as well as the propagation
constant kz that is needed to solve the modal field values numerically[16, 17]. Chapter 5
shows experimental results of various sample materials using the modal-analysis

technique developed in Chapter 1. Chapter 6 presents conclusions and recommendations

for future work.

CHAPTER 2: MODE-MATCHING ANALYSIS

The first three chapters develop the theory for the problem this research addresses. After
discussing the configuration and general problem, Chapter 2 introduces the mode-

matching analysis technique.

1. Configuration

The problem at hand is one of material characterization; specifically, electromagnetic
material characterization is the process of determining the permittivity and permeability
of a sample. Rectangular waveguides are commonly used to facilitate these material
measurements since they have good operational bandwidth for single-mode propagation,
reasonably low attenuation, and good mode stability for the fimdamental mode of
propagation[18]. In addition, for various bands of operation, appropriate standard
waveguides are ubiquitous in a well-equipped electromagnetic test facility. Consider a

rectangular waveguide with a T E10 mode wave incident on a sample at z = 0 as shown in

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1. The waveguide has fixed dimensions for width
x=a/2 ‘ .. y=b
l S 2
_’ _> x=d/2 _>
4— <—>
(5441(1) (‘d'”d)
x=-d/2
—0
x=-a/2 l y—
=-d/2 x=d/2
z=0 7.=L x=-a/2 x=a/2
Top view Cross-sectional View

Figure 2.1: Configuration

-a/ 2 s x s a/ 2 and height y = b , where the width (a) is in general greater than twice the
height (b). The length (L) of the sample waveguide region (S) may vary. The sample,
which has one fixed dimension, height y = b, and two varying dimensions, width

—d/ 2 S x s d / 2 and length 2 = L , will be centered in the waveguide about x = O.

2. Overview
The overall objective of the above material measurement process is to experimentally
obtain the sample scattering parameters using a network analyzer and compare them with

their theoretical expressions. That is

thy exp _

,8 S a) —0

1(‘05 l") 11 ( ) (2.1)
O

S
thy exp
521‘"(w5,5“a#) 521 (w )=
This pair of nonlinear equations with two unknowns has a solution which is seen to

decompose into two parts. First, analytical theory 15 needed to relate S1}; y(a), a, ,u) and
Sthy ((0,632!) to the complex constitutive parameters (see mode-matching analysis).

Second, a technique is needed to accurately measure Slefp and S2“) (see calibration &

measurement). With these two parts needed for the solution, equation (2.1) can be
iteratively solved using a complex two-dimensional Newton’s method root search, giving

the desired results of the complex constitutive parameters[3].

3. Mode Matching Analysis
The mode matching technique is a powerful method for analyzing waveguides with

varying cross-sections. The concept of matching modes at a boundary has been around

for nearly forty years, but due to the limited computer power available until recently only
simple computations were possible[12]. Thus the computational emphasis was on
reducing the number of modes to the minimum so that a numerical solution could be
obtained. With the arrival of more powerful computers, the mode matching concept
could be applied to analysis of more complicated structures. This problem is not the most
computational taxing, but the steps of the mode-matching technique are followed just the

same.

Transverse Fields
The first step in this technique is to represent the generic transverse fields for the various

regions inside the waveguide as follows (see Figure 2.1),

‘

N
E} = Ele—ylz + Z Rn'éney”

 

 

"2‘ ...z < o (2.2)
fitl=h18712—nZ=:Rnhne)/nz
m N '_ ,7 N S ‘
E: = ZTné,‘:e 7"“ + Zrné'geynz
" 1 "—1 l...0<z<L (2.3)
N I _ 5,, N _. s
= ZTnhje 7"” - zrnhjeynz
n=1 n=l J
N
“2 a — :—L) i
Er :Ztn ne 7n(
"=1 >...z > L (2.4)

.. _,. _L
tnhne ”(2 )

33,
II
[V] 2

 

3
ll
—‘

J

Boundary Conditions
Next is to ensure that the tangential electric and magnetic fields are continuous along the

interfaces of the three regions. This is done with continuity boundary conditions that are
appliedon E,, H, at 2:0 and z=L.

”s “2
E, =15,

, g g (2.5)
Hf=H3

 

 

2:0 z=L
Application of these boundary conditions to the transverse fields leads to the follow

system of four algebraic equations with infinite unknowns (assuming N —+ 00)

N’ .N .N
a + Z Rnén = Z Tnjf + Z 515;: (2.6)
Hz] n=1 ":1
g N a N g N “
hl—ZRnhn=2T,,,f—Zr,,,f (2.7)
n=l n=1 n=1
N g S N s N
Z Tnége W + Z rnaanL = Ina, (2.8)
n=l n=1 n=l
N N __
Z T ”12,16 “L ’2 r nhgeyfl’ = Ztnhn (2.9)

Testing Operators
At this point testing operators are applied to equations (2.6)-(2.9) to obtain an infinite set

of equations to equal the infinite number of unknowns. Consider the following testing

operators
b/2 a/2
Jan-{wk J jém-odxdy (2.10)
CS —b/2 —a/2

* b/2 a/2 _.
III," .{}dS= j j hm -{}dxdy (2.11)
CS —b/2 —a/2
Application of (2.10) to (2.6),(2.8) and (2.11) to (2.7),(2.9) produces the following results
for (2.6)-(2.9). The integration of the equations over y yields a constant due to the y-

invariance of the problem. The fact that the modes are even with respect to x is also

used.
a/2 N (1/2
I am Eldx+z i am éndx R, =
=1
0 " 0 (2.12)
N (1/2 N (1/2
:2 3,, égdx 3+2 [5,, égdx r, m=1, ,N
n=1 0 n=1 0
a/Z‘ _. N a/Z“ -
[hm ldx—z [hm hndx Rn:
=1
0 " 0 (2.13)
N (1/2” _. N a/2_. _.
:2[ [km ridIJTn_Z jhm sdx]rn "’21, ’N
n=1 0 "=1 O
N a/2
2 am and». tn:
=1
" 0 (2.14)

Nomenclature

The following nomenclature will be used

an = gym ~éyndx (2.16)
0
a/Z _. _.
Pmn : j hxm 'hxndx (2-17)
0
a/2
Ymn = gym éjwdx (2.18)
0
a/2 .. ..
an = i h... -h;findx (2.19)
0

S S S S
Smn : Ymne—y"L 2 Umn : YmneynL 2 an = ane_ynL 2 Wmn = anernL (2'20)
to write (2.12)-(2.15) more completely. Note numerical solutions to equations (2.16)-

(2.20) will be found in a later chapter. Substituting in the nomenclature and rearranging

leads to form the more compact set of equations

N N N
2 anRn 3; YmnTn —Z Ymnrn = —M,,,1 ...m =1,...,N (2.21)
2121 n=1 n=l
N N N
Z Pman +2 anTn -2 anrn : Pml ”'m =12°"2N (2'22)
"=1 "=1 "=1
N N N
Z smnrn +2 U,,,,,r,, —2 anz,, = 0 ...m = 1,.., N (2.23)
2121 n=l 11:]
N N N
2 Van —2 W,,,,,r,, —Z Pmnr, = o ..m =1,...,N (2.24)
n=l n=1 n=l

Scattering Matrix

Equations (2.21)-(2.24) can be cast in the following partitioned-matrix form

l
l

— q—

M —Y —Y o R1
P Q —Q 0 T
0 S U —M r
_0 V —W —P t

(2.25)

ozoIg-y

 

 

 

 

 

 

r
I

Where M, Y, P, S, U, V, W are N x N sub-matrices whose elements are given by (2.16)-

(2.20), 0 is a N x N null matrix, D is a le null column vector and

R1 T1 ’1 ’1
[R]: 5 ,[r]: 3 ,[r]= 3 ,[r]: s (2.26)
RN TN rN tN
“M11 P11
[b1]: 5 ,[b2]= 5 (2.27)
‘MNI PM

It is seen that through a simple matrix inversion, equation (2.25) can be solved giving the

solutions (2.26) which are the field expansion coefficients at the boundaries between the
three regions. Knowing that Slfll'y (axe, ,u) and Sgiy ((0,841) are due to the fundamental

mode, and from the definition of our transverse fields, it is readily seen that

_ thy __ thy

lO

CHAPTER 3: MODES OF WAVEGUIDE REGIONS

To take full advantage of the mode-matching technique presented in Chapter 2, the
transverse field values must be represented with modal indexing. Hence, Chapter 3
discusses waveguide modes and develops the modal field value expressions for the

loaded and unloaded waveguide regions.

1. Waveguide Modes

In the interior of a waveguide, such as a rectangular waveguide, Maxwell’s equations can
be divided into two basic sets of solutions or modes. For one set of modes no
longitudinal or axial magnetic field component exists, these modes are called transverse
magnetic (TM) modes. The other basic set of modes has an axial magnetic field but no
axial electric field component, this set is referred to as the transverse electric (TE)
modes[18]. The TE modes are used in rectangular waveguides (with a > b), because the

T E10 mode is the dominant mode, due to the fact it has the lowest cutoff frequency. The

TE mode fields may be derived from a magnetic-type Hertzian potential having a single

component along the axis of the guide (see unloaded waveguide analysis section).

Cutoff Frequency
The cutoff frequency of a T Em” mode is given by

1
C(m2b2 + 112a2 )/2

ck
fem" = 2,? = 2.1 ‘3'"

 

where c is the speed of light in free Space, a is the width, b is the height, and m and n are

the modal values[18]. The cutoff frequency helps define the bandwidth of the

11

waveguide. The dominant mode of propagation is the mode with the largest cutoff
wavelength, the frequency corresponding to that wavelength is the low end of the
bandwidth. The first higher order mode to propagate will then define the upper bound of
the bandwidth. Most bands can be pushed if really necessary, but the closer to the cutoff

frequencies the greater the chance of error.

The transverse field for the T E10 mode is even in x (the plane of symmetry is given by
x = 0) and also y-invariant. Since this wave is incident on the sample, scattered modes
of matching properties will be excited. For T m0 modes with odd integers of m,
scattered modes are excited Since they are also even in x, about x = O, and y-invariant.
The T EmO modes with even integers of m are not excited since they are odd about x = O ,
and TEmn modes for n > O and TM ”m modes are not excited since they are both y-

dependent[19]. Hence, it is sufficient for this work to include only even symmetric

modes.

In order to solve the scattering matrix (2.25) found using the mode-matching technique,
numerical solutions to equations (2.16)-(2.20) need to be found. To get numerical
solutions to the equations (2.16)-(2.20) requires finding the modal electric and magnetic
field values for the appropriate regions. Thus, the field values must be found for two

different regions, the unloaded regions 1, 2 and the loaded region S.

12

2. Unloaded Waveguide Analysis

Geometry

For the unloaded regions 1 and 2, consider the following cross-sectional View of a
rectangular waveguide shown below. The origin is located in the center of the bottom

plate and the waveguide has dimensions of width —a/ 2 S x S a/ 2 and height y = b.

iy

 

 

 

 

y=0 — —>
I l | x
x=-a/2 x=0 x=a/2

Figure 3.1: Geometry of unloaded waveguide

Unloaded Modes

Since the TEIO mode is the generating eigenfunction for the transverse wave, the

direction of propagation has no electric component, Ez =0, this leads to the wave

function for TE modes
v31, (x, y) + 1.311, (x, y) = o (3.2)

where V,2 is the transverse Laplacian Operator. Writing the 2-D Hemholtz equation in

rectangular coordinates and letting hz (x, y) = X (x) Y (y) , gives

1 d2X(x)+ 1 d2Y(y)

=—k2 3.3
X(x) dxz Y0) dyz c ‘ ’

 

then using separation of variables, (3.3) becomes

13

 

 

X(x) (12:29) = 44;? —>X(x) = Asinkxx+Bcoskxx
1 d2Y(y) 2 . (3.4)
Y(y) dyz = “ky —>Y(y) = CSIn kyy + Dcoskyy
where k,r and ky are as yet unknown separation constants defined by
é+g=g On

which is a constraint to satisfy the Helmholtz equation. The resulting expression for
hz (x, y) is

h, (x, y) = [A sin k, (43 —x)+ Bcoskx (36 —x)][Csin kyy+ Dcoskyy] (3.6)
The partial derivatives, required for enforcing boundary conditions, are given as

41x,y>/ax=k.IAcosk.(a-xI-Bsmk.(eI-xIIICschosOI

. (37)
6h, (x,y)/ay =k, [Asinkx(% —x)+Bcoskx (%-x)][Ccoskyy-Dsinkyy]

The boundary conditions at perfect-conducting waveguide walls are

 

 

 

h ah-
3.2M =0_)A=0,—:’(—f’—y—) =0—>sinkxa=0
ax x=f9£,ye[0,b] ax x=—‘/1/§,ye[0,b]
I, , .
ah,(x,y) —0—+C=0, 5’1”) =O—)s1nkyb=0(3.8)
2y [1 a. FIJI—42%)
kx = fl, kv =E ...m,n = O,l,2,3,...
a .

Thus, the generating eigenfunction for the mn’h TE mode is given by

a

h: (x, I) = Am” cos [EU/{’2 — x):l cosl:§%z] (3.9)

The transverse field components are determined using the following relations

14

 

(3.10)

”(1)
2:27.); =¥ , y=./k§—k2 ,k=w,/gg (3.11)

Therefore, the transverse fields are given by

jaw .
ex = ———k7 Amnky cos kx (% — x)srn kyy
C

e . = 1%‘iAmnkx sin 1, (9g —x)cos k, y (3.12)

l
C

N

‘___«L" __-t.
hx— Z ,hy—Z

Where m, n = 0 is not allowed simultaneously since it leads to the trivial solution

'e', h = O. The y-invariant T EmO modes are

. . / . e
e, =L“%’—A,,,,,k,51nkx(gzxz—x)=Bsmk,(%—x) , h, =—7y (3.13)

C.
Consider the trigonometric identity

. / . "172' I . m” "I” .
srnk ‘1 —x =srn— 01/ —x =srn—cosk x—cos—srnk x
x 2 a / 2 2 x 2 x

. ...m =1,3,5,... m7: 0 ...m =1,3,5,...
3m? = , 00$—

0 ...m = 2,4,6...- 2 (-1)% ...m =2,4,6,...

 

 

(3.14)

(—1)m—21”/2 cos kxx ...m = l,3,5,...(even modes)

 

_(_1)’7‘”2 sin kxx ...m = 2.4.6,...(odd mode-2)

For T Emo even symmetric modes

.1"
ey =Bsinkx (‘fé—x)=B(—l)m /2 coskxx=Acoskxx
...m=1,3,5,... (3.15)

e, A
h =—i=——cosk x
X Z Z X

15

or on a mode-by-mode version

- I

 

 

 

_. .. e m
eym = COS kxmx 2 hxm : — Zy
. ”m >...m =1,3,5,... (3.16)
mzr jwfl 2 2
kxm :— 2 ZTEm = 2 7m = (kxm) —ko
a 7m 1

3. Loaded Waveguide Analysis

Geometry

Consider the following cross-sectional View of a waveguide shown below. A sample of
width d is centered in the waveguide, width a. The origin is placed in the center of the
bottom of the waveguide, giving an even problem about x = 0 , allowing the problem to

be solved from x=O to x=a/2.

I.

 

 

 

 

 

y=b "’ : ‘:
5 1w.) 5 law.)
I .3 II

y=0 — 1 T; —->

| | I x
x=0 x=d/2 x=a/2
Figure 3.2: Geometry of loaded waveguide
Loaded Modes

For the loaded region an appropriate choice of generating eigenfunctions is made for both
region I and II. For region I the generating eigenfunction must be even about x = O. For
region 11 the generating eigenfunction must satisfy the perfecting conducting walls of the

waveguide. The generating eigenfunctions are thus as follows

16

hz] = A] sin[k§x]

 

 

II 11 11 2 (3'17)
11;. =A cos[k,C (92—37)]
using (3.10), the transverse fields for the y-invariant T Eno modes are
. ., 1 .
631} ... Jam; 0”: = Jaw] kiA’ cos[k;x]
C C
. 1] .
II Jaw/1 511:; 1(0/11 II II . 11
e}, .—_ 12 at- : k2 ! k, A sml:kx (%—x)] (3.18)
c " c
ev
hx = ——7_
Grouping all the electric field constants, the electric fields in (3.18) can be written
6)]. = Acos [kix]
(3.19)

e9] = B sin [kg (9g — x):l

Normalizing the e]

y constant A to l and setting the fields equal to each other at the

boundary leads to the following equation

d 9, :> cos[k; 93] = Bsin [kg] (% 4%)] (3.20)

2 "2.2

 

 

From there the constant B is found to be normalized as

I
k .11
cos[——-'; 99]
B = , (3.21)

[1)]

Now, substituting in (3.21) for B into (3.19), the fields can be re-written as

17

I ,I
e). = cos[ltxx]

kid d /
COS "*2:- /(1
I]

e, _ A k” sin[k” (y, -x)]

(3.22)

3531“”)

Now, to ensure that the magnetic fields hz” and h; are continuous across there boundary

the application of a continuous field must be applied through boundary conditions at

boundary between region I and region II. This gives

 

 

 

I! ”
’—; 2:11 (3.23)
9' /

} "Ti/12% x: /2

Substituting values from (3.17) and (3.18) into (3.23) leads to the characteristic equation

Alsin[k192] A” cos[k”(/2 %)]
”912.24’coslki :12] ’“—”—~’é ki’A” sinlki’ (% W]
(12;) (kg)

Canceling like terms and multiplying by the denominators

1 ll a-d)
k’ sin k ‘12] k” cos[k ( 2 j

__X_ A cos[k’ V]sin[k” (limw (3.25)
1“] cos k; 92:] #11 sin[kl,l(afd):| ( x 2 x 2

 

 

(3.24)

 

 

 

 

gives

normalizing

l8

 

 

 

I I ’ I II
[:§:;11—sin[k3a %]sin[fi‘2—a(l—%)H—[cos[flz—a—%]cos|:kx2a(l—%):H=O(3.27)

In order to satisfy this equation, a way must be found to relate the propagation constants
k? and k; to each other, because right now there are two unknowns and only one

equation. Finding a common relationship between kg] and k; will allow equation (3.27)

to be written in terms of one unknown, thus making it solvable. Starting with the

representation for the wavenumber
1.3 + 1.9+ k3 = k2 (3.28)

and applying the y-invariance of T EmO modes

 

1,? + k3 = 1:022, 12, (3.29)
indexing for both regions a and d
(Adi-[’1 )2 + k22 = IOWA/#11,! (3-30)
rearranging
[ii/’1 = (Egg/HAIL] _kzz (3-31)

and finally, normalizing

 

2
k
[($1.10 :koa\/;[LIIUU,I —[ki) (3.32)
0

a simple relation between k9] and k,{ and k2 /k0 is achieved. Finally grouping all the

loaded transverse field values and indexing them for modal expansion gives

19

 

 

cos — A /
«I I «11 2 a
eyn = cosl:kxx] , eyn = . [A]
Sin .k} :1 (1 ...di
2 /0
511,1 2
”11,1 _ yn 11,1 _
hxn -"'——”,, 2 kx a-koa 511,1#11,1'(;z—]
T5,, 0

jaw .
211,1 = 11.1 , 11,1 2 )k
TE" 11,] ’7
722

2n

k

20

>...n =1,3,5,...

 

(3.33)

CHAPTER 4: NUMERICAL IMPLEMENTATIONS

The mode-matching technique has been introduced and the transverse field values have

be found and indexed for the T EmO modes. All that remains is to solve the integral

relations that will fill the scattering matrix (2.25). Chapter 4 solves these integral

relations and discusses the propagation constant k2.

I. Integral Relations

Now, with the modally index transverse electric and magnetic field values found in
Chapter 3, equations (2.16)-(2.20) can be solved numerically. It will be seen that only
two integrals from Chapter 3 will need to be solved, the rest of the integral relations can
be reduced to constants multiplied by these integrals. For reference the pertinent field
values are

~ __ , =1 __ I
e), — cos kxmx , eyn — cos kxnx

1 d '
cos|:kx,,a—
5].] = 20‘ _ sin[-kg1x+kg, %] (4'1)

I" '
1_d
- 1]
sm Lkxna(—~-2—-‘1]

Starting with equation (2.16) and inserting appropriate values from (4.1) the first integral

 

 

 

 

that needs to be solved is

a/2 a/2
M m” = I ”y," -e’y,,dx= I cos kxmxcos kxnxdx (4.2)
O 0

Now using the integral identity (A.2) on equation (4.2) it is seen that

21

sin [(kxm —k_,m )x] sin [(kxm +km )x] V2

4 A =0...m¢n (4.3)
2(k k

 

 

M =

mn

.I'm - x11

V
N
A
.k‘
2K
5
+
w-
31
3
v

0

and likewise, using integral identity (A.l) also on equation (4.2) gives

(1/2 (112 - %

4 a 2 x srn[2kxmx] a

an = J- ew” eyndxz I cos kmxdxz —+—— =— form =n(4.4)
0 ' 0 2 4km, 4

 

0

SO finally, it is seen that

(“2* g £...f0rm=n
an = j eym °eyndx : 4 (4'5)
0 0 f0rm¢n

where this relationship is due to mode orthogonality. The second and last integral that
must be solved is equation (2.18). Since the sample region has two different field values,

the integral must be spilt into two separate integrals Ymnl , 1’ng that can be solved

individually and added together as shown below

a/2 (NZ 1 a/Z ll
—. as —. -o -o —o
Ymn = I eym emdx = I eym -eyndx+ I eym -eyndx (4.6)
0 ‘0 51/2 I

 

 

V v

(YW'I ) (Ymnz )

Inserting the appropriate field values from (4.1), Nmn becomes

22

d/2
_ ,I
Ymn — I coskmxcoskxmxdx+

cos[k(na 2 A 2.12 (4.7)
( I sin [—ktééx + kg, %] cos kxmxdx

' 1_d/i
sin[kl.n a[- 9]]‘1/2
2
(Ymn2)

solving Ymn1 first using the integral identity (A.2)

 

%
d / 2 1 sin [(kin — km )x] sin [(k;,, + km )x]
Ymnl = j cos kmxcos km, mix: I + A I (4.8)
O 2(kxn ‘kxm) 2(kxn +kxm)

 

 

it is seen that

 

(4.9)

sin[(k(na—k xma a)/2a] sin[(kxna+kxma)%a]
)

— +
YmnI 2/ 1
2'a(k] r-na krma) %(kxna+kxma

Next solving Ym ,1 using the integral identity (A3) and making sure to keep the constant

 

] % (4.10)

2...“ -../4:, i,,,)x+rg,a,] cos[(-k,{;+k,,,)x+kg,%

2( 411-1....) ' 441,4...)

 

%

yields

23

cos[k1na gi]
Ymnz :_ (:1 '
sin[kg,a [1 0]]
2 (4.11)

GOSH/(£10 (1 _ (jg) — kxma da)2:l COS|:(kJ{{za(1— (ya) )+ kx ma d/a)/2]

+
II I]
gra(kma+k_.a\m ) %(erz a— kxma)

Finally, combining Ym"1 and YWZI gives the final result for Ymn

Y Sin[(kxn a- kxma)({2a] Sin[(k1{"a+kxma)%a] COS[k£na Zda]
"m 2/(k1a— Irma) 31'2(k,£na+kxma) sin[k§{,a ( :4”
(4.12)

emu/(gap_d,;;‘)_kx,,,a4;)11‘2] cos[(k;{,a(1_% /)+k mda /)/2]
. . ' :+

2/(k”a+k_xma) 1 /2/a(]d'{1::)—kxma)

 

 

 

 

Using the transverse relationship for fix fields, it is seen that the integrals for the

scattering matrix terms Pm” and an reduce to those of M m" and Ymn respectively, by

simply bringing the wave impedances, which have no x-dependencies, outside the

 

integrals.
a/2 _. a 1 a/ 2 1
Pm" = j hm .hmdx = j gym -éy,,dx = ZT—[an] (4.13)
0 ZTEm 2T5" 0 TE,"

1 a/ 2 ES

.. n
an= “I211 1xm htsj—Ndx — E— eym ' 3} dx (4°14)

0 TE»: 0 2TB"

1 d/2 a/2 II

an : _—T_ I é.ym 8}]‘ndx+——— IE e'ym v. eyndx (4'15)

ZTEm ZTE" 0 ZTEm IZTEn d / 2

24

67—416,". >1+——1—,,-,—116ng )1 666

TEmZTEn ZTEm T5,,
The rest of the scattering matrix terms from equation (2.20) are seen to involve the

already solved relationships for Ymn and an.

731

mne , V

_ SL _ 3L S
Smn =Ymne 7" a Umn = mn =ane 7,, 9 Wmn =ane7nL (4'17)

2. Propagation Constants

Now that the integral relations have been solved, it is seen that for a numerical

representation of Ymn , the propagation constants k? and k; must be found. The

relationship found in Chapter 2 for the propagation constants k? and k; was

 

11.1 2
kx = (#0 811,1#11,1 "1‘22 (4-13)

The common term that relates k5] and k; together is k2, which is the propagation

constant in the z-direction. Knowing that the fields must propagate in phase through

every section of the waveguide, this relationship should be expected[16]. Thus, to solve

for 1911 and k; , the appropriate value of k2 must be found, this is done with the

characteristic equation (3.27). By utilizing the relationships for [(11 and k; in terms of
k2, the characteristic equation is now in terms of one unknown and thus able to be

solved, with the value that satisfies the equation being the necessary value of kz needed

to find both k,{1 and 16,1.

25

Since kz, and thus kg and k; , is dependant on both frequency and mode, the
characteristic equation needs to be solved for each mode at every frequency point. The
most effective way to solve the characteristic equation is through the use of an iterative
solver, which will take an initial guess for k2 and slightly alter its value till the equation
is satisfied to within a specified accuracy. As with any iterative solver, the initial guess is
the critical aspect in getting the proper solution to the equation. For this problem the

value of k2 that will satisfy the characteristic equation depends on many things,

including relative permittivity and permeability, frequency, mode, and d/a (the ratio of

sample width to waveguide width). However, the equation for k2 appropriate for a

uniformly-filled guide is only dependant on relative permittivity and permeability,

frequency and mode as shown below

 

2

2
n 71'
1‘: =ko\/€//,1#11,1\/ — [ ] (4.19)

 

511,11111J koa
where the frequency is embedded in k0 and the mode is identified by n. This means that

an initial guess found with equation (4.19) will only be valid for the completely empty or
filled waveguide cases, and since this problem is to be solved for varying d/a ratios, a

method for finding initial guesses in between the empty and filled cases is needed.

Since this problem is not very computationally taxing, the method used to find these

initial guesses is a simple program. The program starts with a known initial guess for kz ,
the empty case, and using the iterative solver to satisfy the characteristic equation, k2 is

found for that d/a. The d/a ratio is then incremented in very small steps (d/a = 0.001),

26

using the previous value of k: as the initial guess for the next increment of d/a, till the
desired ratio is reached. At which point the value of kz is returned to the main program.

By incrementing in small steps of d/a and using the previous value of k2 as the initial

guess for the next point, the initial guess method is believed to be very accurate.

Figures (4.1) -— (4.3) show the normalized propagation constant k2 /k0 for four different
epsilions (1, 4, 9, 16), versus the entire range of the ratio d/a, for mode 1 at frequencies 9
GHz, 10 GHz and 11 GHz respectively. For the completely empty and filled sample
cases, that is a ratio of d/a = 0 or 1, the propagation constant has the corresponding value
found in equation (4.19), using the appropriate numbers for mode, frequency and relative
constitutive parameters. For an epsilion of unity, k2 /k0 is constant as this is just fi'ee
space in both sample and air regions. For the other permittivities, the values in between
the empty and filled cases are seen to curve in a smooth fashion with no discontinuities.

Note, that all the values of the propagation constant for mode 1 are real valued.

Figures (4.4) — (4.6) show the normalized propagation constant k2 /k0 for four different

epsilions (1, 4, 9, 16), versus the entire range of the ratio d/a, for mode 2 at frequencies 9
GHz, 10 GHz and 11 GHz respectively. Again, for the completely empty and filled
sample cases, the propagation constant has the corresponding value found in equation
(4.19), using the appropriate numbers for mode, frequency and relative constitutive

parameters. Also, for an epsilion of unity, the value of kz /k0 is constant just as in mode

1, however the value is purely imaginary. Depending on the frequency and relative

27

permittivity and permeability the value of the completely filled case can turn out to be
purely real or purely imaginary. Take epsilion 4 at d/a = 1 for example, at 9 GHz the

value is purely imaginary, but then at 10 GHz the value is purely real.

To verify that the propagation constants being found are correct, the y-oriented electric
field values were found for a few various d/a values. The ratios used are d/a = 0.25, 0.5,
and 1 for both modes 1 and 2 at frequencies 9, 10 and 11 GHz. Figures (4.7) - (4.9)
show the normalized y-oriented electric field values for five values of relative
permittivity (l, 2.5, 4, 9, 16) for mode 1 versus the waveguide location (x/a) at frequency
9 GHz for d/a = 0.25, 0.5, and 1, respectively. The waveguide location is only shown

from 0 to a/ 2 since the field is even about x = O. The field value is normalized to one at

x = 0 and goes to zero at edge due to boundary conditions.

Notice the hyperbolic shape of the curves once the field leaves the sample and enters the

air region, for these plots this takes place at half the d/a ratio. The y-oriented electric

field value assumes the sine and cosine of the k? term, which is purely imaginary for the

values of mode number, frequency and relative constitutive parameters in these figures.
Knowing that the sine or cosine of an imaginary value is hyperbolic, the shape of this
curves is justified. Note that for small enough epsilions the curve shows no hyperbolic
tendencies, and that for d/a = 1 all the curves exhibit free-space tendencies as anticipated

by transmission line theory.

28

Figures (4.10) — (4.12) show the normalized y-oriented electric field values of five
relative permittivity values (1, 2.5, 4, 9, 16) for mode 2 versus the waveguide location
(x/a) at frequency 9 GHz for d/a = 0.25, 0.5, and 1, respectively. Again, the waveguide

location is only shown from 0 to a/ 2 since the field is even about x = 0. Also, the field
value is still nomialized to one at x = 0 and goes to zero at x=a/2 due to boundary

conditions. Note that since the y-oriented electric field is even about x = 0 , that only odd

numbered modes propagate, hence mode 2 is actually TE30 which will have three peaks

across the entire waveguide width. Since these figures only show half the waveguide

width, one null and one peak should be seen in between x = O and x = a/ 2 .

Again, note the hyperbolic shape in the air region when the kg term goes purely
imaginary. Since the mode term is squared in equation (4.19), the product of the relative
permittivity and permeability must be larger before the k;’ will turn purely imaginary.
This is the reason that only the a, : 9 and 5,. =16 cases demonstrate a hyperbolic shape

in the air region. Similar results for 10 GHz and 11 GHz are seen in Figures (4.13) —

(4.24).

29

 

 

 

kzlko vs. dla, Freq 9 GHz, Mode 1

4.5 ____.__-_., fl —— A

 

 

 

- - — -epsilion 1
----- epsilion 4
- — - epsilion 9
——epsilion 16

 

 

 

 

 

 

 

0 7 T V T T f fi T T j 1 I f

0.01 0.1 0.19 0.28 0.37 0.46 0.55 0.64 0.73 0.82 0.91 1
dla

 

Figure 4.]: k2 /k0 vs. d/a for mode 1 at 9 GHz

30

 

 

 

 

kzlko vs. dla, Freq 10 GHz, Mode 1

 

 

 

 

 

 

-----epsilion1
------- epsilion 4
- -— - - epsilion 9
epsilion 16

 

 

 

 

0.01

l T T I T T T T T T

0.55 0.64 0.73 0.82 0.91
dla

0.1 0.19 0.28 0.37 0.46

 

1

 

Figure 4.2: [(2 /k0 vs. d/a for mode 1 at 10 GHz

31

 

 

 

 

kzlko

kzlko vs. dla, Freq 11 GHz, Mode 1

 

 

 

 

 

 

- - - -epstllon 1
----- epsilion 4
— - - epsilion 9
epsilion 16

 

 

 

 

0.01

0.1 0.19 0.28 0.37 0.46 0.55 0.64 0.73 0.82 0.91 1
dla

 

 

Figure 4.3: [(2 /k0 vs. d/a for mode 1 at 11 GHz

32

 

 

 

 

 

 

kzlko vs. dla, Freq 9 GHz, Mode 2

 

 

 

 

 

epsilion 1 real
— - - epsilion 1 imag
----- epsilion 4 real
— - - ~epsilion 4 imag
- - - -epsllion 9 real
—-— epsilion 9 imag
— + — epsilion 16 real
- - a- - epsilion 16 imag

 

 

 

 

 

 

 

-3 -.- ____. .
0.01 0.1 0.19 0.28 0.37 0.46 0.55 0.64 0.73 0.82 0.91 1

dla

 

 

 

 

 

Figure 4.4: [(2 /k0 vs. d/a for mode 2 at 9 GHz

33

 

 

 

kzlko vs. dla, Freq 10 GHz, Mode 2

 

 

 

 

 

 

4
3 1
epsilion 1 real
2 6,6" _, _ *' ; "T“ r ..-_ or“ — — — epsilion 1 imag
I , / ' ----- epsilion 4 real
1 , , 7 If /i 7 _7 , z. _ a - - - epsilion 4 imag
ff /' - - - - epsilion 9 real
r ‘ ..... —-— epsilion 91mag
, - + - epsilion 16 real
' - - .- - epsilion 16 imag

 

 

 

 

 

 

0.01 0.1 0.19 0.28 0.37 0.46 0.55 0.64 0.73 0.82 0.91 1
dla

 

 

 

Figure 4.5: k2 /k0 vs. d/a for mode 2 at 10 GHz

34

 

kzlko

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 _ __fi
3 4
. epsilion 1 real
2 of. ’ / . 1 ’ ' ' * ‘ - -- — epsilion 1 imag
; / ' ' ----- epsilion 4 real
‘ I _ - — . . . a
1 - f / _ _ _. epsilion4 tmag
4 g —---epstlion9real
fl ./ ,- ' +epsilion 9 imag
o ' - + - epsilion 16 real
- t- - epsilion 16 imag
-1 .
-2 _ -..._
0.01 0.1 0.19 0.28 0.37 0.46 0.55 0.64 0.73 0.82 0.91 1

dla

 

 

 

Figure 4.6: k2 /k0 vs. d/a for mode 2 at 11 GHz

35

 

 

Normalllzed Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 9 GHz,
Mode 1, dla = 0.25

1.2 a..-

 

 

 

 

 

 

0'8 “ —-——epsilion 1
— — - epsilion 2.5

0.6 . ----- epsilion 4
- - — epsilion 9
— - - -epsilion 16

0.4 ._ ~

0.2 .

 

 

 

-- Q
.‘_-._.-~-
T l T T

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

0

 

Figure 4.7 : Normalized Ey Field vs. Waveguide Location for mode 1, d/a = 0.25 at 9 GHz

36

 

 

 

Normalllzod Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 9 GHz,

 

 

 

 

 

Mode1,dla=0.5
1.2 4*—
1- , z , a 775___
\ii\~.\
‘: ~\
\‘~\
0.86 \: ‘. __,
‘\.‘\.\
\\“\
‘ . .\
\ ‘.\
0.61 \\, .-.. - - 252_.
~\\ . \
.‘\ ‘.~\\
0.4 ‘ \"\ ~ ‘1 \\ " g — —- ~— —
.\\‘ ‘\\
02-. ‘. \ i‘._\;~2 -1
' \"x‘ ‘~‘\
\ \~\ .7~\.\
4“.-‘.§:~\~
O 1 r t r i 1 i 1 I t t i .I.—-‘i—_-
0‘bibcb‘bQ3Cb’Lb‘bN‘O%\
o"; 0"“ \‘N \‘9 ’3? (if) a?" 119% “$56) '5‘“) 6'9 9’3 $96
0' 0' 0' 0' Q‘ Q‘ 0' Q 0' 0' 0' Q' 0'

Waveguide Location (xla)

 

 

epsilion 1
— - — epsilion 2.5
----- epsilion 4
- - - -epsilion 9
— - - - epsilion 16

 

 

 

 

Figure 4.8: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = 0.5 at 9 61-12

37

 

 

 

 

Normailized Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 9 GHz,

Mode 1, dla =1

 

1.2 , __

 

0.6 ~

0.2666 6 6 66 66 6 66 6

 

0 .

 

———epsilion 1
— - - epsilion 2.5
----- epsilion 4
— - - ~epsi|ion 9

 

— - - -epsilion 16

 

 

 

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49

Waveguide Location (xla)

1

T

 

Figure 4.9: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = 1 at 9 GHz

38

 

 

 

Normalllzed Ey Field

0.5- \\‘~‘\ . ,V/ , , 7 Ave

 

Normalized Ey Field vs. Waveguide Location, Freq 9 GHz,

Mode 2, dla = 0.25

 

 

 

 

epsilion 1
— — — epsilion 2.5
----- epsilion 4
- - — -epsilion 9
— - - — epsilion 16

 

 

 

\‘Q‘A

 

1 v 1 v v r 1 r v v I

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

39

Figure 4.10: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 0.25 at 9 GHz

 

 

 

 

Normaillzed Ey Field

 

Normalized Ey Field vs. Waveguide Location, Freq 9 61-12,

Mode 2, dla = 0.5

1.2 “—w

 

 

0.8 6

0.6 ..

0.4 4

0.2 1

 

 

—epsilion 1
- - - epsilion 2.5
----- epsilion 4
- - - -epsilion 9
— - - -epsilion 16

 

 

 

 

 

0 1 I ' 1 T V t r T 1 1 f

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

Figure 4.11: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 0.5 at 9 GHz

40

 

 

 

 

1.2

Normalized Ey Field vs. Waveguide Location, Freq 9 GHz,
Mode 2, dla = 1

 

0.8 4

0.6 ..

Normaillzed Ey Field

0.4 .

0.2 .

 

0

 

 

 

epsilion 1
- — — epsilion 2.5
- _________ ----- epsilion 4
— - - -epsilion 9
— - - - epsilion 16

 

 

 

 

 

0

7 F r T T T l T T T T T T i T r

0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

 

 

Figure 4.12: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 1 at 9 GHz

41

 

 

 

Normalized Ey Field vs. Waveguide Location, Freq 10 GHz,

 

 

 

 

 

Mode 1,dla=0.25
1.2
14 K‘ _ _ _ fl____ ..
§.‘.\
'6 ‘§,‘~\‘\
3 0-81 \.\.“.\\ , , c _ __ _
E ‘\ “ \
>, \. . \
“‘ ‘.\ ‘ \
0.6 1 \‘ \‘ w _
'6’ w x
z 0.4 "‘ \\ \‘ \ ;\ - __~_ ___2
‘\ \\ \\
\ \
0.2 ~ ‘\ . . \\.. .mfl
\ \ ‘. . \\
\ \ \ ~“. \
.. ........ '-.§
0 1 I Y T T Y 1 l ' 1—‘1..1-—‘--V_-.1~T.-n.n
0.00 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.30 0.34 0.37 0.41 0.45 0.49

Waveguide Location (xla)

 

 

epsilion 1
— — - epsilion 2.5
----- epsilion 4
- - - -epsilion 9
- - - epsilion 16

 

 

 

 

Figure 4.13: Normalized Ey Field vs. Waveguide Location for mode I, d/a = 0.25 at 10 GHz

42

 

 

 

Normalized Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 10 GHz,

 

Mode1,dla=0.5
1.2
1. ~~.. , L__-___

\.:}~\
\\.‘.\
: ~.\
03.. , N.‘.\ L , __,
' ‘\ \\
‘\. ~,\
.\ ‘.\
\‘ e \
05.. .\\ ‘,,\., _ --_____...
\“ “ \

\\. x \

\\ ‘ \

\ ‘ \
0.4» \6\g~ \ _ - ~.—___

\‘\ ‘ \
\ \
\ . \
\ \ ‘ \
' -0-- .5 7 \k. ...—.—
02 \‘ \‘ g ‘ \
Q \ \
\ ~ --.\
\\ \ ‘ ...\

o T T T T TIT if T 7 T ~.-l~.-?_L.

 

 

 

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.
Waveguide Location (xla)

T T T

45 0.49

 

 

epsilion 1
— - — epsilion 2.5
----- epsilion 4
— - - -epsilion 9
- - epsilion 16

 

 

 

 

Figure 4.14: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = 0.5 at 10 GHz

43

 

 

 

Normailized Ey Field

 

Normalized Ey Field vs. Waveguide Location, Freq 10 GHz,
Mode 1, dla = 1

1.2

 

 

0.8 1

0.6 1- -

0.4 ..

0.2 6

 

0

‘1'

#7

 

 

T

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.

‘

34 0.37 0.41 0.45
Waveguide Location (xla)

T l

0.49

 

—-—-epsilion 1
- - - epsilion 2.5
----- epsilion 4
- - - -epsilion 9
- - epsilion 16

 

 

 

 

 

Figure 4.15: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = l at 10 GHz

44

 

 

 

1.86
1.66
1.4 ..

1.2 .,

0.8 1

Normalized Ey Field

0.6~
0.4.- *\

0.2 ..

0

Normalized Ey Field vs. Waveguide Location, Freq 10 GHz,

Mode 2, dla = 0.25

 

 

6 if.\ ' / * * ..,___z
'. \I ‘X ' / ‘
\"/“\'/ ~ -- ~~—-—'-— .
VA" A.
' J \.

 

 

 

T T T f T f 7 T T T fl T T T 7 T r T T

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

 

 

epsilion 1
- - - epsilion 2.5
----- epsilion 4
- - - ~epsilion 9
- - - —epsilion 16

 

 

 

 

Figure 4.16: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 0.25 at 10 GHz

45

 

 

 

Normaillzed Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 10 61-12,

 

 

 

 

 

 

 

Mode 2, dla = 0.5
1.2 —«
- "-7 \
1 " r 7 N ur}/ ' ‘\*- —— “—— ————
X / . x 7. ‘.
\k '6 / . /
0-8 “*“'\‘r\.\“ f / . ,1 \ " \f ‘ ** —‘ “W“ _ ——epsuion1
'.‘\6‘\ I, / g / ‘\ \‘ - — — epsilion 2.5
0.6 1.3} .6 ,‘ .' // - - \. . -- _ _,__._- 2 ----- epsilion 4
‘\\~ '~\ 1 I. .‘ / \, \~ '. - - - ~epsilion 9
-‘\ ‘,\ .' . ,’ / ‘\ \. - - - -epsilion 16
0.4 6 {‘51 I I g / 6‘ ___
"\l' ' ; ll ‘\ \
\\ 1‘ . x ‘. ‘ -
0.2 ..t. i L L. - :13 fl i i 7 7 v FV‘ ‘__.L _._\. .
‘\ .l"'\ I \ ‘ \.
0 Y . ‘. \n. . \ . \‘
0 ‘17 ‘.’V T 7 \

 

 

O 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.
Waveguide Location (xla)

37 0.41 0.45 0.49

 

Figure 4.17: Normalized Ey Field vs. Waveguide Loc

46

ation for mode 2, d/a = 0.5 at 10 GHz

 

 

 

Normailized Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 10 GHz,

 

 

 

 

 

 

 

 

 

 

Mode 2, dla=1
1.2
1 z _ _ _ _ _ LLF__L ..
0'8 ' “ ' F * ‘ """'“""_‘ epsilion1
---epsilion 2.5
0.6 1- - — - —- —— — — — _ -..__~_._+ ----- epsilion4
----epsilion9
--*-epsilion 16
0.46 - -. . - . , v. L ____
0.2 . ,_ -- _ . _ _ ._ .. __ _,w___
o T . . . T - . . . . f. . . 4 4

 

O 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

Figure 4.18: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 1 at 10 GHz

47

 

 

 

Normalized Ey Field vs. Waveguide Location, Freq 11 GHz,
Mode 1, dla = 0.25

 

 

 

 

 

 

 

2

if 0-8 1* epsilion 1
“>1. — - - epsilion 2.5
3 0 6 ----- epsilion 4
5 - - - -epsilion 9
E - - - - epsilion 16

0.4
2

.o
N

 

 

 

 

0 fi fi 1 i fin l t 1

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

Figure 4.19: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = 0.25 at 11 GHz

48

 

 

 

Normalllzed Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 11 GHz,

Mode 1, dla = 0.5

 

 

 

 

1.2 — L _
1- , L L L L L L. L.
(t’~‘s
\\~\:\
”(‘5‘
\ 6-\
‘ ~‘\. i V W v ’ “fl: “—4
0.8T 7' \\\I.. \\ v i
\
.. \
\T. ‘~ 6.
06- ‘Q,~.\ ~—___
.\ ‘,\
. . ‘ \
\\ . ‘\
\ \
0.46 6 n . g\_ L g L g___
\ ‘ \'
i \
\\> \
02 “\ ‘s \\
. _,L . _ ._ . \LL. _ ‘LLLL L‘ LLLL L_LL_L.._q
\. \‘ ~. .\\
. \ ‘-
\. \_\ ~ ‘~\\
\‘-:’:-b.-‘.-‘:.-\.:‘:
0 7 r 1 1 T ' T T t t t i l I .7.‘i='i T

 

0 0.04 0.07 0.110.15 0.19 0.22 0.26 0.3 0.34 0.37 0.410.45 0.49

Waveguide Location (xla)

 

 

 

epsilion 1
— — - - epsilion 2.5
------- epsilion 4
- - — - - epsilion 9
- - - — - epsilion 16

 

 

 

Figure 4.20: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = 0.5 at 11 GHz

49

 

 

 

Normallized Ey Field

1.2

0.8 «

0.6 ~

0.4 .

0.2 -

0

Normalized Ey Field vs. Waveguide Location, Freq 11 GHz,

Mode 1, dla =1

 

 

 

 

 

epsilion 1
— - — epsilion 2.5
----- epsilion 4
- - -- -epsillon 9
- - epsilion 16

 

 

 

 

I T T T T T T T T T T T I I

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

Figure 4.21: Normalized Ey Field vs. Waveguide Location for mode 1, d/a = l at 11 GHz

50

 

 

 

 

Normalllzed Ey Field

 

Normalized Ey Field vs. Waveguide Location, Freq 11 GHz,

 

 

 

 

—— epsilion 1
— - - epsilion 2.5
----- epsilion 4
- - - -epsilion 9
- - - - epsilion 16

 

 

Mode 2, dla = 0.25
2.5 «it
2.- ...; L. L
/ ‘\
/ .L \
/"/ i \
15 1' - .-z..:~\ -‘ '—
.'/ r ‘\ \
/' __ .
o / // ‘\ V. \
/ / \ ‘: ‘
1. / , —\T
0.5
0 T T T T T T T T T T T T T T fiT T r

 

 

 

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

 

Figure 4.22: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 0.25 at 11 GHz

51

 

 

 

 

 

Normallized Ey Field

Normalized Ey Field vs. Waveguide Location, Freq 11 GHz,

Mode 2, dla = 0.5

1.2

 

 

 

 

epsilion 1
— - — epsilion 2.5
----- epsilion 4
— - - - epsilion 9
— - - - epsilion 16

 

 

 

 

 

0 T T T T T T T T T T T T T T T T T I' j

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

 

Figure 4.23: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 0.5 at 11 GHz

52

 

 

 

 

Normaillzed Ey Field

1.2

0.8 ~

0.6 .

0.4 -

0.2 1

0

 

Normalized Ey Field vs. Waveguide Location, Freq 11 GHz,

Mode 2, dla = 1

 

 

 

 

 

epsilion 1
- — - epsilion 2.5
----- epsilion 4
— - - -epsillon 9
- - - - epsilion 16

 

 

 

 

T . l T T T T T T T T T T T T T I T

0 0.04 0.07 0.11 0.15 0.19 0.22 0.26 0.3 0.34 0.37 0.41 0.45 0.49
Waveguide Location (xla)

 

 

Figure 4.24: Normalized Ey Field vs. Waveguide Location for mode 2, d/a = 1 at 11 GHz

53

 

 

CHAPTER 5: RESULTS

To verify the modal-analysis technique deve10ped in Chapter 2, several materials were
measured in the X-band frequency range, 8-12 GHz. While this technique is valid over
any desired frequency range, the dimensions and availability of X-band waveguide
sections made this the easiest range for a proof of concept demonstration. In this
Chapter, the experimental setup is explained, the extraction process is verified, and
experimental results will be shown for acrylic and magnetic radar absorbing material

(MagRAM). Note, mode 11 refers to all modes up to and including mode 11.

1. Experimental Setup

The experimental setup is as shown in Figure (5.1) with a WR9 waveguide (where WR
stands for “waveguide rectangular” and 9 refers to the inner waveguide width a = 0.9
inches) connected to a HP8510C Vector Network Analyzer via coaxial cables. The
sample holder used to verify the modal-analysis technique is shown in Figure (5.2). The
sample holder is connected to the waveguide with precision alignment screws to help
minimize the discontinuities across the interfaces. Great care is also taken to ensure that

the coaxial cables are spatially stabilized while calibrating and measuring samples.

To ensure that the samples, which will have varying width, are centered about the middle
of the waveguide, spacers are used. The spacers are much longer than the sample holder
region and thus can easily be removed once the sample is in place. This is an extremely
critical step in the measurement procedure, if the sample is not perfectly centered the

assumption of evenness about x = 0 is not valid.

54

Before measuring any test samples, a standard baseline was checked to further validate
the experimental setup. After performing the calibration, the two waveguide sections

where placed together and the magnitude and phase of the S-parameters was checked to
ensure that $332 z 1 -—j0 and 51632 z 0— )0. Also, an extraction was performed on the

S-parameters measured from the empty sample holder to ensure a free space baseline for

the system.

2. Validation

Before testing any samples, a few simple validation tests were done to verify the
extraction process. The first test involves reversing the extraction process to generate
scattering parameters for the special case of a completely filled waveguide cross-section.
This special case allows the generated scattering parameters to be compared with results
for well known transmission line theory and experimentally measured data. If the results

of all three scattering parameters are comparable, then based on the results for the

propagation constant k: and y-directed electric field in Chapter 4 , it may be concluded

that the theoretical scattering parameters found in the extraction process are in fact

correct.

Figure (5.3) shows the comparison the generated, experimentally measured and
theoretical transmission line scattering parameters for 9, 10 and 11 GHz, respectively.
The value of relative permittivity used to calculate the generated and theoretical

scattering parameters was based on strip transmission line data, seen in Figure (5.6),

55

which used the same acrylic measured with the waveguide setup. It is seen that the
results are very comparable for all three cases at each frequency, thus validating the

theoretical scattering parameters found in the extraction process.

The second test for validation of the extraction process involves generating fake
experimental data and then using the extraction process to find the complex constitutive
parameters. However, understanding the nature of coding, the generation and extraction
processes can not be run with the same number of modes to prove convergence, since the
same errors will be present in both cases. Consequently, a slightly altered approach is
taken to proving convergence, wherein the generation process is run with a certain
number of modes, and the extraction process is run with one fewer mode. This test
proves if the extraction process is converging to the correct complex constitutive

parameters.

Figure (5.4) shows complex constitutive parameters for a lossless, non-magnetic material
with relative permittivity of 2.5 versus a frequency range of 8 — 12 GHz. The data was
generated with 3 modes and extracted with 2 modes for a d/a = 0.25. Figure (5.5) shows
complex constitutive parameters for a lossy, magnetic material with relative permittivity
of 5 and relative permeability of 2 versus a frequency range of 8 — 12 GHz. This data
was generated with 5 modes and extracted with 4 modes for a d/a = 0.01. The results of
both cases are seen to be flawless, thus validating the convergence of the extraction

process to the correct complex constitutive parameters.

56

3. Tested Samples

Acrylic

The acrylic measured for this thesis was a clear plexi-glass material that exhibits almost
pure dielectric characteristics with very little electric loss and no magnetic properties. In
addition, due to its rigid structure, acrylic is easily machined into tight fitting strip
transmission line and waveguide samples. Based on these properties, acrylic is an

practical material to use as starting point for the modal-analysis verification.

Three different widths (d/a = 0.25, 0.5, 1) of acrylic were measured to verify the ability of
the modal-analysis technique to handle varying d/a ratios. Figures (5.7) and (5.8) show
the complex constitutive parameters of acrylic with d/a = 0.25 versus the frequency for
modes 1 and 2, respectively. The data is seen to converge in just one mode to the
baseline expected from the strip transmission line data seen in Figure (5.6). This is
verified by the mode 2 data which is the same as the mode 1 data. Since the acrylic has a
low dielectric constant, very little loss and no magnetic properties, it was expected that
very few modes would be required to have the data converge to the known complex
constitutive parameters. Similar results are seen in Figures (5.9)-(5.12) for the d/a ratios

of0.5 and 1.

MagRAM
The general concept of this material measurement technique was to better measure highly
reflective, absorptive and lossy samples, thus the ultimate verification of this technique is

to measure a test sample with those properties. MagRAM was a convenient solution in

57

that it exhibits the desired properties and can easily be cut to fit the sample holder,

comparison data was also available from the strip transmission line.

The results for the MagRAM using the strip transmission line are shown in Figure (5.13),
this will stand as the baseline for expected results using the waveguide. Figure (5.14)
shows the complex constitutive parameters of MagRAM with d/a = 0.09 versus the
frequency in gigahertz for mode 1. The complex constitutive parameters are seen to start
at the appropriate values, but then diverge as the frequency increases. Since the
MagRAM is exhibiting dielectric loss as well as magnetic properties, the use of more

modes to account for the hi gher-order modes excited in the sample region is expected.

Figure (5.15) shows the complex constitutive parameters of MagRAM with d/a = 0.09
versus the frequency for mode 2. Here, the complex constitutive parameters are seen to
follow very closely those of the strip transmission line, with just some slight deviation at
the very highest frequencies. Still, the additional modes must be included to consider the
data completely converging. Figure (5.16) shows all the modes up to mode 3 where it
can be seen that the complex constitutive parameters converge for the entire frequency
range to the expected reference data of the strip transmission line. This is verified by
Figure (5.17), where by including mode 4 demonstrates no improvement over the use

with three modes.

....

 

Figure 5.1: Experimental Setup

59

 

Figure 5.2: Sample holder

60

Frequency 9 GHz
$11 821
Data Source Real Imag Real Imag
Transmission Line Theory -0.6026 -0.1107 0.1428 -0.7774
Data Generation Program -0.6025 -0.1107 0.1428 -0.7773
Experimentally Measured -0.5942 -O.1058 0.1412 -0.7781

Frequency 10 GHz
811 821
Data Source Real Imag Real Imag
Transmission Line Theory -0.5754 -0.0212 0.0301 -0.8170
Data Generation Program 05754 -0.0212 0.0301 -0.8170
Experimentally Measured -0.5675 -0.0183 0.0261 -0.8072

Frequency 11 GHz
$11 821
Data Source Real Imag Real Imag
Transmission Line Theory -0.5387 0.0622 -0.0963 -0.8346
Data Generation Program 05387 0.0621 -0.0963 -0.8345
Experimentally Measured -0.5310 0.0628 -0.1013 -0.8300

Figure 5.3: Comparison of theoretical and experimental S-parameters values

61

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability & Permittivity of Data Generated with 3
Modes then Solved with 2 Modes vs. Frequency
3 L— L_. _L _L L...
g6 2.5 (— -------------------------------------
.?.
2
E 2 -
O
t
11615“ —--—EPReal
g ' - - — EP Imag
'8 ----- MU Real
g 1 i iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii — - — -MU Imag
0
e.
g 0.5
3
a 0 —r + “m fi' —r r “T' T r—
-0.5 —————-~--—~6--— —~- -—-—-- -————
8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GHz)

 

 

Figure 5.4: Relative Permeability & Permittivity of Data Generated with 3 Modes then Solved with 2
Modes vs. Frequency

62

 

 

Relative Permeability 8. Permittivity of Data Generated with 5
Modes then Solved with 4 Modes vs. Frequency

 

 

 

----EPReal
———Eleag
----- MUReal
2‘ —-—-MUlmag

 

 

 

....-.-o.--~-opua.~---------..--...--onu...-.....-

Relative Permeability 8. Permittivity
(a)

 

_.—-——--—-——-~o-—c—--_-—.—-—-—-_-_l—.—.—

 

 

 

 

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GHz)

 

 

 

Figure 5.5: Relative Permeability & Permittivity of Data Generated with 5 Modes then Solved with 4
Modes vs. Frequency

63

 

 

Relative Permeability & Permittivity of Acrylic vs. Frequency
using Strip Transmission Line

 

 

 

 

 

 

 

 

 

 

3 1T- —6 ‘— — —
MW
E 2 ‘ ‘ ‘ — — “‘ fl“
:
.5 15 - . i ' -L EP Real
g -——-Eleag
8 ------- MU Real
0 1.i-.-Lm.-.---.--.--.--------_----.---..-LfiLa-----Lfr-----..TH-4.~_.L._HLTL.-:.L-L
E -~—---MUImag
0
a.
. 0.5. L V _LL_L
5
S ,
g 0 'W‘C#W~IW7-WT.
-05 --LL ..L. __ LL
8.0 8.4 8.9 9.3 9.8 10.2 10.6 11.1 11.5 11.9

Frequency (GHz)

 

 

Figure 5.6: Relative Permeability & Permittivity of Acrylic vs. Frequency using Strip Transmission Line

 

 

64

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability & Permittivity of Acrylic vs. Frequency,
dla = 0.25, Mode 1
3 +__ e _ __
- _— . __—_-—:‘—'::'_"
g 2.5 ~- _
2 - i #

E
,5 EP Real

15 - ”,7- ..
g - — — - EP Imag
'3 ......... . ........ , ...................................... MU Real
. 1 —< if 2,! -. - -.-__;.._.__- - ‘4
E — - — - - MU Imag
a
O 0.5 ‘ *"—* — J
5
2
g 0 %i2;%‘%—Hfi__::~m‘:x~rfiw

-0.5 -__ —-——-- -— — -—— —

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GI-l2)

 

 

 

Figure 5.7 : Relative Permeability & Permittivity of Acrylic vs. Frequency using Waveguide, d/a = 0.25,
Mode l

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability & Permittivity of Acrylic vs. Frequency,
dla = 0.25, Mode 2
3 T—-77 7 7 77 7— 77 --7

g, 2.5 7 7 i 7777 g— 77»
E 2 -1 7 7—7 7 7 77 ———7————4
h
at
.5 EP Real

1.5 7 7 77 i ‘1
5‘ -———EP Imag
'3 ---------------------------------------------------------------------- MU Real
e 1 ~ 7 7 _ —77 7—
E —-—--MU Imag
at
, 0.5 1 e fi___
.2
fl
.2
g 0 :‘_-..:::‘_~_, :41'543L—ttrWW-rrflé

-0.5 7— i - -—- -—7 —-—— -

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (Gl-Iz)

 

 

 

Figure 5.8: Relative Permeability & Permittivity of Acrylic vs. Frequency using Waveguide, d/a = 0.25,
Mode 2

66

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability & Permittivity of Acrylic vs. Frequency,
dla = 0.5, Mode 1
3 _,_ .2 n a ‘_ _
gas- i—fflf—w7——4
e 2 7 7 7 7 7 7 7 7
E
... 15 - i _ .2 ._ EPReal
g —-—-Eleag
‘5 ............................................... MU Real
0 1 g ....... ._....__.,+M
E --—--MUlmag
E
,, 0.5 1 - A ~. ____
E
A!
g 0 77:— 5'7: :7_:_i fi?ff_‘figjfimfiw
-o.5 — ,__--_-___.__ ———.———.
8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency(Gl-lz)

 

 

Figure 5.9: Relative Permeability & Permittivity of Acrylic vs. Frequency using Waveguide, d/a = 0.5,
Mode 1

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability & Permittivity of Acrylic vs. Frequency,
dla = 0.5, Mode 2
31—“ —— __ _—
5‘ 2.5 «W , , 7——— w —
g 2 . efi____
S
a 1 5 4 i 7 7 v 7 _fi __ EP Real
5‘ —-—-EP Imag
ii} .................................................................. MU Real
. 1 7 7 7 7- — —-————.-—-—--—
E — - — - - MU Imag
a
.051 _ _ fl____
5
g
g o
-0.5 -______,_ —_ —.. __._ V.—
8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (Gl-lz)

 

 

 

Figure 5.10: Relative Permeability & Permittivity of Acrylic vs. Frequency using Waveguide, d/a = 0.5,
Mode 2

68

 

 

 

Relative Permeability 8: Permittivity of Acrylic vs. Frequency,
dla = 1, Mode 1

 

 

 

 

 

 

 

 

 

 

 

3 _ __ a , - __ _i e
g 2.5 ~ ~ - 7* e W e- -________2
E 2 1 7 7 7 __ i i
b
8
“‘15— i 7 g i g EPReaI
g ——-—-Eleag
‘8 .................................................................. MU Real
0 1 s _ . , ,, , . .,,v, ..__...._._,__-____.-2
E —-—--Mulmag
11"
. 0.5 ~ — __ _ _ -..—
,3.
2
g 0 “-“T” H“‘*—r'~"——-F—r-—‘r—*r-—=I-:=rW"-Y—'FW1‘
-o.5 -~_ — —— _—
8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12

Frequency (Gl-lz)

 

 

Figure 5.11: Relative Permeability & Permittivity of Acrylic vs. Frequency using Waveguide, d/a = 1,
Mode 1

 

69

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability 8. Permittivity of Acrylic vs. Frequency,
dla = 1, Mode 2
3 . -— — .— — —

sH—“fi 7 w
5' 25
i

2 . i,
E
.. 15 _ i ¥W EP Real
g ' —-—-Eleag
'3 ........................................................... MU Real
. 1- . ,,, , . ..... “be... .
E —-—--MU Imag
at
o 0.5 “ - —*'-_*—-
5
.2
a 0 T": ‘7 fi' T—T— r—'I- T—z-r‘r-r‘r— T—‘Ffi—Iii
-o.5 __ .— »_ - ._ -. -__‘ ---—---
8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GI-i2)

 

 

 

Figure 5.12: Relative Permeability & Permittivity of Acrylic vs. Frequency using Waveguide, d/a = 1,
Mode 2

7O

 

 

Relative Permeability 8. Permittivity of MagRAM vs. Frequency
using Strip Transmission Line

 

 

 

 

 

 

6 4W
5‘
>
.3; 5 7 __ _g
E
O
I. 4 q, __ _ W
.5 EP Real
g - - - EP imag
.. 3 ______ .4
'5 ----- MU Real
E 2 - - - -MU Imag
0 -2 We _ _ __2
D.
3
a 1 . ~—
2’ --------------------------------------------------
o 4%IY‘EW
-1

 

 

 

8.0 8.4 8.9 9.3 9.8 10.2 10.6 11.1 11.5 11.9
Frequency (GI-l2)

 

Figure 5.13: Relative Permeability & Permittivity of MagRAM vs. Frequency using Strip Transmission
Line

 

71

 

 

 

Relative Permeability & Permittivity of MagRAM vs. Frequency,
Mode 1

 

 

 

 

 

 

 

EP Real
— — - EP Imag
----- Mu Real
3 , __ __ -_._ . -_---__ — - — -Mu Imag

 

 

 

 

Relative Pemleablllty 8- Permlttivlty

 

. ! i =_h* — —
_ o *

 

 

 

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GI-l2)

 

 

Figure 5.14: Relative Permeability & Permittivity of MagRAM vs. Frequency using Waveguide, d/a =
0.09, Mode 1

 

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability & Permittivity of MagRAM vs. Frequency,
Mode 2

7 1777—7 777

6 i # d
E
>
e 5 -. _ _ __ __ _— _—
E
O
O. 4 _ ,
all EP Real
5 3-- fl ”WM -—-EP|mag
8 ----- Mu Real
E — - — ~Mu Imag
O 2.. .. , . _ ___ .-. .4
l
2
E 1 ‘
g ..................................................

o-::T—__—-_1-.— 1“?1::_‘+_?:_—T-T~j_.T—-:_7-—._r:.::

-1 -____.____ -__ -

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (Gl-lz)

 

 

 

Figure 5.15: Relative Permeability & Permittivity of MagRAM vs. Frequency using Waveguide, d/a =
0.09, Mode 2

73

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Relative Permeability 8. Permittivity of MagRAM vs. Frequency,
Mode 3

7

6 4e 2 # # 2- _ __ ._ _—
E
’ 1
g 5 ‘r 7 7
E
O
l 4 w , i 7 _ ___ 2
an EP Real
5 3 fl 2. f _ __m --—Eleag
‘8 ----- MU Real
EZ‘ _ _ _ _ A —---MUImag
8
0
:2 1~ 7 7
g ..............................................

0 TgF—E!F_i—?-T_T-?_—i_?—i_rT—~:?:J:-‘

-1 _

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GHz)

 

 

 

Figure 5.16: Relative Permeability & Permittivity of MagRAM vs. Frequency using Waveguide, d/a =
0.09, Mode 3

74

 

 

Relative Permeability & Permittivity of MagRAM vs. Frequency,
Mode 4

 

 

 

 

 

EPReal
34 f , _ —--Eleag
----- MUReal
—--—-MUImag

 

 

 

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

Relative Permeability 8. Pennittivlty

 

_D—It
-—-—-—.
-_-—.-———-—-
.—-—-
_._.-
._._—-

 

 

 

 

8 8.4 8.8 9.2 9.6 10 10.4 10.8 11.2 11.6 12
Frequency (GHz)

 

 

Figure 5.17: Relative Permeability & Permittivity of MagRAM vs. Frequency using Waveguide, d/a =
0.09, Mode 4

 

75

CHAPTER 6: CONCLUSIONS

This thesis has provided a waveguide material measurement technique for improving
transmission coefficients thru highly reflective, absorptive and lossy materials.
Specifically, a mode-matching technique was implemented to accommodate the higher-
order modes excited by the partially-filled rectangular waveguide sample region. The
sample permittivity and permeability were found by using an iterative complex two-
dimensional Newton’s root-searching algorithm to compare the theoretical S-parameters
obtained in the modal-analysis with the experimentally measured S-parameters obtained

from the network analyzer.

A couple of special case tests were done to show that the extraction process was
converging to the correct complex constitutive parameters. Then partially-filled
rectangular waveguide material characterization technique was experimentally verified,
through a comparison of acrylic and MagRam samples, with strip transmission line
measurements. These comparisons demonstrated the validity and accuracy of the

technique.

Future work for this technique should include some error analysis. Tolerances should be
checked on off-center samples, sample length, and sample thickness. A study using a
small-perturbation approximation for finding the propagation constants may also prove

useful.

76

APPENDIX A: INTEGRAL IDENTITIES

sin(2ax+b+d)
4a

 

Icos(ax+b)cos(cx+d)dx=§cos(b—d)+ forc=a (A.1)
sin[(a—c)x+b—d:|
2(a-c) +

sin[(a+c)x+b+d]
2(a+c) 7 ...forcafia

 

Icos(ax + b)cos(cx + d) dx =
(A2)

+

 

cos[(a-c)x+b—d] _
2(a—c)
_cos[(a+c)x+b+d]
2(a+c)

Isin(ax + b)cos (at + d)d.r = —

 

(A.3)

 

fare at a

77

APPENDIX B: NETWORK ANALYZER PROCEDURES

For waveguides, the simplest known method of two-port calibration is the thru-reflect-
line (TRL) method. Traditional full 2-port calibration methods typically require three
impedance standards and one transmission standard to calibrate the vector network
analyzer (VNA), whereas the TRL method relies on only one impedance standard of a
short transmission line. Although the TRL mathematical derivation is different than
conventional full 2-port calibration methods, application of the method, two sets of 2-port
measurements that differ by a short length of transmission line and two reflection
measurements, results in the same 12-term correction model. A complete mathematical

solution for the TRL calibration will not be repeated here[20-22].

TRL refers to the three basic steps in the calibration process. First is the thru, where port
1 and port 2 are connected directly to each other and transmission and reflection
measurements are taken. Next is the reflection, a short is placed on both port 1 and port 2
and reflection measurements are taken. Last is the line, where a short transmission line is
inserted between port 1 and port 2, and transmission and reflection measurements are

taken again. The length of the line section should be roughly a quarter wavelength or
900 and should not be multiples of half wavelengths of the middle frequency, the phase

shift from that point should remain within 300 and 1500. With the measurements taken
in these three steps the VNA can solve the 12-term correction model, thus giving a fill] 2-

port calibration at the interfaces between port 1, port 2 and the sample holder.

78

I. Calibration-K it Definition Procedure

This section shows step by step how to program the TRL Calibration-Kit on the HP
8510C Network Analyzer. Before starting to program the cal-kit on the 8510C the delay
of the line segment must be found. This is done by dividing the length of the line (L) by
the velocity of free space (c) as shown below

“m? = delay(s) (3.1)

611’)
/ S

 

1. Press the cal key located on the IF/Detector block of the HP 8510C Network
Analyzer. A calibration menu will appear on the display screen.

2. Select more from the available menu options.

3. Two calibration kit programs, from which the user may select, can be stored in
memory at one time. These may be standard calibration kits such as Type-N, or
they may be user defined kits. In order to begin the definition of a new non-
standard calibration kit, select modify 1 or modify 2 from the menu list.

4. The user must now define the standards needed to for the calibration kit. To
begin this process, select define standard from the display menu list.

5. Each calibration standard must be appropriately numbered, the phrase calibration
standard # will appear on the display. Enter 1 on the numeric keypad by
pressing 1 followed by X].

6. A list of standard types will appear on the display, for the TRL calibration method
one short and two delay/thru standards are needed. The short will be defined first,
which corresponds to the reflect part of the calibration name. Select short from

the menu list.

79

7.

10.

11.

12.

A list of parameters necessary to define the standard will appear on the display,
each parameter must be selected to completely define the standard. The
parameters, L0, L1, L2, and L3 belong to an inductance model that accounts for
offsets that exist in certain types of coaxial calibration standards which may cause
the reference plane to be offset slightly. Such offsets do not occur with the
waveguide short circuit terminations, therefore 0 should be entered for each of
these parameters. Select each parameter and enter 0 by pressing 0 and X] on the
numeric keypad.

Several parameters specifying the offset of the short are needed. Select specify
offset from the menu list.

Select offset delay and enter 0 s by pressing 0 and x1 on the numeric keypad.
This represents the difference in travel time the signals emanating from port 1
experience between the short circuit standard and the calibration plane which will
be established, this is assumed to be zero.

Select offset loss and enter 0 by pressing 0 and X] on the numeric keypad. This
represents the loss experienced by the waves traveling between the short circuit
standard and the calibration plane which will be established, this is also assumed
to be zero.

Select offset 20 and enter 1.0 by pressing 1 and x] on the numeric keypad. This
represents the impedance of the guiding structure which exists between port 1 and
port 2.

Next is the minimum frequency, this needs to be the cutoff frequency of the

waveguide which is found in equation (3.1). For x-band this value is 6.557 GHz.

80

13.

Select minimum frequency and enter 6.557 GHz by pressing 6.557 and G/n on
the numeric keypad.

Now the maximum frequency must be selected, this is just twice the minimum
frequency. Select maximum frequency and enter 13.114 GHz by pressing

13.114 and G/n on the numeric keypad.

14. Now the general type of guiding structure must be selected. Select waveguide

15.

16.

17.

18.

19.

20.

21.

22.

from the menu list.

Press std offset done when all parameters have been successfully entered.

The calibration standard must now be named, choosing an appropriate name will
facilitate the measurement process. Press label std and enter TRLSH using the
control wheel and select letter key. If a name already exists, use the backspace
or erase title keys to delete it before entering the new name.

When the appropriate name has been entered, press title done.

complete the process of defining the calibration standard by pressing std done
(defined).

The delay/thru calibration standards must now be defined, first will be standard
corresponding to the thru part of the calibration title. To begin this process press
define std. The phrase calibration standard # will appear on the display. Enter
2 by pressing 2 and X] on the numeric keypad.

The list of standard types will appear again on the display. Select delay/thru
from the menu list.

Select specify offset from the menu list.

Select offset delay and verify that O has been entered.

81

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

Select offset loss and verify that 0 has been entered.

Select offset Z0 and verify that 1.0 Q has been entered.

Select minimum frequency and enter 6.557 GHz by pressing 6.557 and G/n on
the numeric keypad (value for x-band, change appropriately).

Select maximum frequency and enter 13.114 GHz by pressing 13.114 and G/n
on the numeric keypad (value for x-band, change appropriately).

Verify that waveguide is select and not coax.

Press std offset done when all parameters have been successfirlly entered.

Press label std and enter TRLTI-I using the control wheel and select letter key.
If name already exists, use the backspace or erase title keys to delete it before
entering the new name.

Press title done after the name has been entered.

Press std done (defined)

Repeat steps 19-31 for the second delay/thru standard using standard # 3, offset
delay found in equation (B.1), and label std TRLLI.

Now each of the calibration standards defined above must be related to the
individual measurement classes recognized by the 8510C. To begin this process
press specify class.

Select more twice from the display menu list.

Select TRL thru and enter 2 by pressing 2 and X] on the numeric keypad.

Select TRL reflect and enter 1 by pressing 1 and X] on the numeric keypad.
Select TRL line and enter 3 by pressing 3 and X] on the numeric keypad.

Complete this process by pressing class done (spec’d).

82

39.

40.

41.

42.

43.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

Now each of the calibration classes must be labeled. To begin this process press

label class.

Select more twice from the display menu list.

Select TRL thru and enter TRLTH using the control wheel and select letter key.
Press title done.

Select TRL reflect and enter TRLSH using the control wheel and select letter

key.

. Press title done.

Select TRL line and enter TRLLI using the control wheel and select letter key.
Press title done.

Complete this process by pressing label done.

The TRL option must now be defined, to begin this process select TRL option
from the display menu list.

Verify that cal Z0 has system Z0 selected.

Verify that set ref has reflect selected.

Complete this process by selecting TRL option defined.

The new calibration kit must now be named. Select label kit and enter TRLX
using the control wheel and select letter key.

Press title done.

Press kit done modified.

Now the new calibration kit should be saved to disk so that it may be recalled at
an time. Press the disc button which is located under the auxiliary menus section

of the IF/Detector block of the 8510C.

83

56. Select store from the menu options.

57. Select cal kit from the menu options.

58. Enter 1 by pressing 1 and X] on the numeric keypad.

59. Enter the desired cal kit name, TRLX in this case using the control wheel and
select letter key.

60. Select store file from the menu options.

2. Calibration Procedure

This section shows step by step how to perform the TRL Calibration on the HP 8510C
Network Analyzer. Before beginning the calibration procedure, power up the HP 8510C
Network Analyzer and allow the device to warm up for an appropriate amount of time,
roughly an hour.

1. Start by setting up the generic parameters of the measurement. Begin this process
by selecting the cal key located on the IF/Detector block of the HP 8510C
Network Analyzer.

2. Select more from the display menu list.

3. Select Set Z0 from the display menu list and enter 1.0 (2 by pressing l and X] on
the numeric keypad.

4. Next, press the start button which located under the stimulus menu section of the
IF/Detector block of the 8510C.

5. Enter 8 GHz by pressing 8 and G/n on the numeric keypad.

6. Select the stop button and enter 12 G/n by pressing 12 and G/n on the numeric

keypad.

84

7. Select the menu button from the stimulus menu section.

8. Select number of points from the display menu list.

9. Select appropriate number of points from display menu list.

10. Select prior menu key on the numeric keypad.

11. Select step from display menu list.

12. Select dwell time from display menu list and enter 25 ms by pressing 25 and k/m
on numeric keypad.

13. Select menu button from the response menu section.

14. Select averaging on/restart from display menu list and enter 32 by pressing 32
and x] on the numeric keypad.

15. Select more from the display menu list.

16. Select Waveguide Delay from the display menu list and enter 6.557 GHz by
pressing 6.557 and G/n on the numeric keypad (value for x-band, change
appropriately).

17. Now with all the generic parameters of the 8510C setup, the calibration
measurements can be taken. Begin by pressing the cal key located on the
IF/Detector block of the HP 8510C Network Analyzer. A calibration menu will
appear on the display screen.

If the calibration kit is already shown on the display menu list, skip to step 18.

18. If the calibration kit is not shown on the display menu list, it needs to be loaded
into memory. To begin this process press the disc button which is located under
the auxiliary menu section of the IF/Detector block of the 8510C.

19. Select load from the menu options.

85

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

Select cal kit from the menu options.

Select 1 from the menu options.

Using the control wheel highlight appropriate cal kit file and select load file from
menu list.

Press the cal key located on IF/Detector block of 8510C.

Select appropriate cal kit from menu options.

Select TRL 2—port from menu options.

The display menu list now shows all the measurements that must be taken for the
12-term error model to be solved. These measurements can be done in any order,
the simplest in terms physically putting the setup together is to measure the thru,
reflect, and line in that order. For the thru measurement connect port 1 and port 2
of the waveguide to each other and select Thru. A line will appear under TRLTH
when all the measurements have been taken.

Next, place the short into port 1 and select 81] reflect, wait for the line to appear.
Now place the short into port 2 and select 822 reflect, wait for the line to appear.
Select Isolation from the display menu.

Select Omit Isolation and then Isolation done.

Now place the known line segment in between port 1 and port 2 and connect
everything together. Select Line/match and wait for the line to appear.

To complete this process select Save TRL 2-port.

Select 1 from display menu list.

The display menu list will now show correction 1 on, signifying that the 12—term

error model has been solved.

86

35. The final step to check the calibration. A good calibration will have transmission

magnitude values in the thousandths range and reflection magnitude values in the
-80dB range across the entire band. If the magnitude values are not in this range,
the calibration should be performed till they are. Otherwise the data taken from

the VNA can not be trusted

3. Measuring Data Procedure

This section shows step by step how to measure data and save it to disc on the HP 8510C

Network Analyzer.

1)

2)
3)
4)

5)

6)

7)

8)

Calibrate the HP 8510C Network Analyzer for the waveguide setup (see
Appendix C section 2).

Carefully place the desired sample in the line section of the waveguide.

Select the menu button from the stimulus menu section.

Select more from the display menu list.

Select single from the display menu list and wait for the line to appear under it.
This measures all four s-parameters simultaneously and holds them so they are
able to be saved.

Now the data needs to be saved to disk, so it can be transferred to a computer
where the extraction program can characterize the material. To begin this process
select the disc button which is located under the auxiliary menu section of the
IF/Detector block of the 8510C.

Select store from the display menu list.

Select memory all from the display menu list.

87

9) Enter the desired file name by using the control wheel and the select letter key.
The file name can only be five characters long.

10) To complete this process select store file from the display menu list.

88

APPENDIX C: SOURCE CODE

PROGRAM BOGLE

C***

C CONSIDER A PERFECTLY-CONDUCTING RECTANGULAR WAVEGUIDE CONSISTING OF
C THE FOLLOWING THREE SECTIONS. REGION A (Z<0) IS COMPRISED OF AN

C AIR-FILLED WAVEGUIDE HAVING WIDTH A AND HEIGHT B. REGION B (O<Z<L)

C IS COMPRISED OF WAVEGUIDE HAVING WIDTH A AND HEIGHT B, FILLED WITH A

C SAMPLE WIDTH D<A, HEIGHT B, AND THICKNESS L. REGION C (Z>L) IS

C COMPRISED OF AN AIR-FILLED WAVEGUIDE HAVING WIDTH A AND HEIGHT B.

C IT IS ASSUMED THAT THE CALIBRATION/REFERENCE PLANES ARE LOCATED AT

C Z=0,L. PROGRAM BOGLE COMPUTES THE RELATIVE PERMITTIVITY AND

C PERMABILITY OF THE SAMPLE MATERIAL USING A MODE-MATCHING TECHNIQUE.

C SPECIFICALLY, THE TE_10 MODE IS ASSUMED TO BE INCIDENT ON THE APERATURE
C REGION. DUE TO THE TE_10 MODE Y-INVARIANCE AND EVENNESS ABOUT X=0,

C ONLY EVEN TE_MO (M=1,3,5,...) MODES WILL BE EXCITED IN THE SURROUNDING

C WAVEGUIDE REGIONS.

THE GENERAL OVERVIEW OF THE PROGRAM CONSISTS OF THE FOLLOWING STEPS:

 

1. CALL CONST: DEFINE IMPORTANT CONSTANTS AND PARAMETERS.

2. CALL INPUT: READ IN IMPORTANT INPUT VARIABLES SUCH AS:
INITIAL FREQUENCY, FINAL FREQUENCY, WAVEGUIDE
WIDTH, D/A, SAMPLE SECTION LENGTH, OUTPUT FILENAMES

3. CALL EXPSP: READ IN EXPERIMENTAL S-PARAMETER DATA.

4. CALL CALCMUEP: FIRST, COMPUTE THEORETICAL S-PARAMETERS USING
MODE-MATCHING. NEXT, COMPARE WITH EXPERIMENTAL S-PARAMETERS
USING TWO DIMENSIONAL ROOT SEARCH AND ITERATE UNTIL DESIRED
ACCURACY HAS BEEN SATISFIED.

C 5. CALL OUTPUTMUEP: WRITE EPSILION/MU TO OUTPUT FILE.

Cit“.

00000000000000

CALL CONST

CALL INPUT

CALL EXPSP

CALL CALCMUEP
CALL OUTPUTMUEP
STOP

END

Cfliiifittfiii

Ci’i‘ttitiiii

SUBROUTINE CONST
C**fi
C SUBROUTINE CONST DEFINES FUNDAMENTAL PARAMETERS AND CONSTANTS.
C**i
IMPLICIT NONE
REAL‘8 PI, MUO, EPO, C
COMPLEX’1 6 J
COMMON/BCONST1/PI
COMMON/BCONSTZ/MUO
COMMON/BCONSTB/EPO
COMMON/BCONST4/C
COMMON/BCONSTS/J

89

PI = 4.D0*ATAN(1.DO)

MUO = 4.D-7*P|

C

EPO = 8.8540-12
= 299792508

J = DCMPLX(0.DO,1.D0)

RETURN
END

Cii‘itititfit

C**********

C***

SUBROUTINE INPUT

C SUBROUTINE INPUT READS IN ALL THE NECESSARY INPUT TERMS.

C***

IMPLICIT NONE

INTEGER*4 NoM, NP

REAL*8 Fl, FF, A, L, WL, D_A, MURre, MURim, EPRre, EPRim
COMPLEX‘16 MUR__G, EPR_G
CHARACTER*80 INFILE, OUTFILE
COMMON/BlNPUT1/A
COMMON/BINPUTZ/L
COMMON/BINPUT3/D_A
COMMON/BlNPUT4/NoM
COMMON/BlNPUTS/NP
COMMON/BINPUTS/MUR_G
COMMON/BINPUT7/EPR_G
COMMON/BlNPUTB/F|,FF
COMMON/BlNPUTQ/WL

WRITE(*,*) 'Enter 1) Initial Frequency (in GHz)

+, 2) Final Frequency (in GHz)‘

READ(*,*) Fl, FF

WRITE(*,*) 'Enter 1) A (in meters), 2) L (in meters)

+, 3) Fraction D/A, 4) Waveguide Sample Section Length (in meters)‘

READ(*,*) A, L, D_A, WL

WRITE(*,*) 'Enter 1) Number of Modes (NoM)

+, 2) Number of Frequency Points (NP)'

READ(*,*) NoM, NP

WRITE(*,*) 'Enter 1) MUR (RE,IM), 2) EPR (RE,IM)'
READ(*,*) MURre, MURim, EPRre, EPRim

MUR_G = DCMPLX(MURre,MURim)

EPR__G = DCMPLX(EPRre,EPRim)

WRITE(*,*) 'Enter Input Filename’

READ(*,80) INFILE

OPEN(10,FILE=INFILE)

WRITE(*,*) 'Enter Output Filename‘
READ(*,80) OUTFILE
OPEN(11,F|LE=OUTFILE)

80

FORMAT(A80)

90

RETURN
END

Ciiitiii’it'k

Ci*****i**t

SUBROUTINE EXPSP
Ctti
c SUBROUTINE EXPSP READS IN THE EXPERIMENTALLY MEASURED SAMPLE
c S-PARAMETERS.
C***
IMPLICIT NONE
INTEGER*4 |,NP
REAL*8 F(801),FI,FF,FS,S11M,S11P.822M,822P,821M,821P,S12M,812P,Pl
REAL*8 S11R,S11I,821R,821l,822R,822l,S12R,Si2l
REAL*8 KO,KZ,ZL,A,C,L,WL
COMPLEX‘1 6 SE(801,2,2),J
CHARACTER*80 junk
COMMON/BCONST1/Pl
COMMON/BCONST4/C
COMMON/BCONSTS/J
COMMON/BlNPUT1/A
COMMON/BlNPUTZ/L
COMMON/BINPUTS/NP
COMMON/BlNPUTB/FI,FF
COMMON/BINPUTQNVL
COMMON/BEXPSPi/F
COMMON/BEXPSPZ/SE

ZL = WL-L
FS = (FF-Fl)/(NP-1)

DO I=1,NP,1
F(|) = Fl+(I-1)*FS
ENDDO

oo l=1,9.1
READ(10,80) junk
ENDDO

oo I=1,NP,1
READ(10,*) S11R,S11|
SE(I,1,1) = DCMPLX(S11R,S1 1 I)
ENDDO

GOTOZO

DO I=1,NP,1

K0 = (2.DO*PI*F(|))/C

KZ = SQRT(K0*K0-((PI*PI)/(A*A)))

READ(10,*) S21R,821|

SE(I,2,1) = DCMPLX(SZ1R,821l)*EXP(J*KZ*ZL)
ENDDO

GOTO20

DO I=1,NP,1

91

K0 = (2.DO*P|*F(|))/C

KZ = SQRT(K0*KO-((PI*PI)/(A*A)))

READ(10,*) S12R,S12l

SE(|.1.2) = DCMPLX(S12R,S12I)*EXP(J*KZ*ZL)
ENDDO

GOTOZO

DO l=1,NP,1

K0 = (2.DO*PI*F(I))/C

KZ = SQRT(K0*K0-((PI*PI)/(A*A)))

READ(10,*) 822R,822|

SE(I,2,2) = DCMPLX(822R,822I)*EXP(J*KZ*2*ZL)
ENDDO

20 DO l=1,1o,1
READ(10,80) junk
ENDDO

80 FORMAT(A80)

RETURN
END

Cfiiti'ttitti

Ciit**i**ii

SUBROUTINE CALCMUEP
Cfiii
c SUBROUTINE CALCMUEP COMPUTES EPSILION/MU OF THE SAMPLE.
Cflii
IMPLICIT NONE
INTEGER*4 NP,|,K,NOI
COMPLEX*16 EPR_G, MUR_G, EPR_F_G, MUR_F_G, EPR_R_G, MUR_R_G
COMPLEX*16 EPR_F_S, MUR_F_S, EPR_R_S, MUR_R_S
COMPLEX*16 EPR_F(801), MUR_F(801), EPR_R(801), MU _R(801)
COMMON/BINPUT5/NP
COMMON/BlNPUTG/MUR_G
COMMON/BlNPUT7/EPR_G
COMMON/BCALCMUEPt/l
COMMON/BCALCMUEPZ/K
COMMON/BCALCMUEP3/EPR_F, MUR_F, EPR_R, MUR_R

EPR__F__G = EPR_G
MUR_F_G = MUR_G
EPR_R__G = EPR_G
MUR_R_G = MUR_G

DO l=1,NP,1

WRITE(*,100) l,NP

CALL FREQCONSTU)

K=1

CALL CTDNEWTN(EPR_F_G,MUR_F_G.EPR_F_S,MUR_F_S,1 .D-7,1 .D-3.50,N01)
EPR_F(I) = EPR_F_S
MUR_F(|) = MUR_F_S
EPR_F_G = EPR_F_S
MUR_F_G = MUR_F_S

K=2

92

CALL CTDNEWTN(EPR_R_G,MUR_R_G,EPR_R_S,MUR_R_S,1.D-7,1.D-3,50,NOI)
EPR_R(I) = EPR_R_S
MUR_R(I) = MUR_R_S
EPR_R_G = EPR_R_S
MUR_R_G = MUR_R_S
ENDDO

100 FORMAT(1X,'FREQ STEP',1X,I4,1X,'OF',1X,I4)

RETURN
END

Ciiflit‘tiii‘t

Ct‘titti’ittt

SUBROUTINE FREQCONST(1)

C***

C SUBROUTINE FREQCONST COMPUTES THE VALUES THAT CONSTANT FOR A

C SINGLE FREQUENCY.

C***
IMPLICIT NONE
INTEGER*4 l
REAL*8 A, F(801), Pl, C, OMEGA, K0, K0_A
COMMON/BlNPUT1/A
COMMON/BEXPSP1/F
COMMON/BCONST1/Pl
COMMON/BCONST4/C
COMMON/BFREQCONST1/OMEGA
COMMON/BFREQCONSTz/Ko
COMMON/BFREOCONST3/K0_A

OMEGA = 2.DO*PI*F(I)
KO = OMEGA/C

KO_A = K0*A
RETURN
END

Ciifiiii‘kiii

C**i*******

SUBROUTINE CTDNEWTN(UO,VO,UF,VF,DUV,EPS,LIMNOI,NOI)
Cifl‘.
C SUBROUTINE CTDNEWTN USES THE TWO DIMENSIONAL NEWTON'S METHOD
C TO SOLVE THE COMPLEX SOLUTION OF TWO NONLINEAR COUPLED COMPLEX
C EQUATIONS W(U,V)=0 AND Z(U,V)=0 IN TWO COMPLEX UNKNOWNS U AND V.
C THE NONLINEAR COUPLED COMPLEX FUNCTIONS W(U,V) AND Z(U,V) ARE
C SUPPLIED BY SUBROUTINE CFWZ.
Ci‘hi

IMPLICIT INTEGER’4 (I-N)

IMPLICIT REAL'8 (A-H,O-Z)

COMPLEX*16 U0,V0,UF,VF,DU,DV,W0,ZO,WUI,ZU|,WVI,ZV|

COMPLEX‘l 6 WU,WV,ZU,ZV,DET,DLU,DLV

NOI=0
10 NOI=NOI+1
IF(NOI.GT.LIMNOI) GO TO 20

DU=DUV*UO
DV=DUV*VO

93

00 00

20

CALL CFWZ(UO,V0,W0,ZO)
CALL CFWZ(UO+DU,VO,WUI,ZUI)
CALL CFWZ(UO,VO+DV,WVI,ZVI)

WU=(WUI-W0)/DU
WV=(WV|-W0)/DV
ZU=(ZUl-ZO)/DU
ZV=(ZVl-ZO)/DV

DET=WU*ZV-WV*ZU
DLU=(WV*ZO-ZV”W0)/DET
DLV=(ZU*W0-WU*ZO)/DET
UF=UO+DLU

VF=VO+DLV

write(*,*) 'before'

write(*,*) DET,DLU,DLV
T=SQRT(ABS(DLU*DLU)+ABS(DLV*DLV))

write(*,*) T

write(*,*) 'after'

|F(T.LT.EPS) RETURN
U0=UF
VO=VF
GO TO 10
WRITE(*,*) 'TOO MANY ITERATIONS IN CTDNEWTN.’

RETURN
END

C**********

Ctfiifi‘ti‘fit'fii

SUBROUTINE CFWZ(EPR,MUR,W,Z)

C***

C SUBROUTINE CFWZ READS IN THE VALUE OF EPR AND MUR
C (RELATIVE EPSILION/MU OF THE UNKNOWN SAMPLE) AND

C COMPUTES THE FUNCTIONS W AND 2

C***

IMPLICIT NONE
INTEGER'4 I,K
COMPLEX‘16 EPR,MUR,W,Z,SE(801,2,2),ST(2,2)
COMMON/BEXPSPZ/SE
COMMON/BCALCMUEP1/I
COMMON/BCALCMUEPZ/K

CALL SCATPAR(EPR,MUR,ST)
IF (K.EQ.1) THEN
w = ST(1,1)-SE(I,1,1)
z = ST(2,1)-SE(I,2,1)
ELSE
w = ST(2,2)—SE(I,2,2)
z = ST(1,2)-SE(|,1,2)
ENDIF

RETURN
END

94

Ctiitiitfitt

Ciii*iiii**

SUBROUTINE SCATPAR(EPR,MUR,ST)
C".
C SUBROUTINE SCATPAR FINDS THE THEORETICAL SCATTERING PARAMETERS
C OF THE SAMPLE USING A MODE-MATCHING TECHNIQUE.
Ctti
IMPLICIT NONE
INTEGER’4 NOM
COMPLEX'16 EPR,MUR,KML(6,20),KMUL(2,20),ST(2,2),DET
COMPLEX*16, ALLOCATABLE :: ME(:,:)
COMMON/BINPUT4/NOM

CALL UNLOADEDMODAL(KMUL)
CALL LOADEDMODAL(EPR,MUR,KML)
ALLOCATE (ME(NoM*4,NoM'4+1))
CALL MATEL(KML,KMUL,ME)
CALL CMATPAC(-1,NoM*4,NoM*4+1,ME,NoM*4,1,DET,1.D-300)

ST(1,1) = ME(1,NOM*4+1)
ST(2,1) = ME(NoM*3+1,NoM*4+1)
ST(2,2) = ST(1,1)

ST(1,2) = ST(2,1)

RETURN
END

Ciitiiititi

Ciii'i'kiittt

SUBROUTINE UNLOADEDMODAL(KMUL)
Cit"!
C SUBROUTINE UNLOADEDMODAL FINDS THE NEEDED TRANSVERSE FIELD
C VALUES, BASED ON INPUT PARAMETERS, FOR THE UNLOADED REGIONS
C OF THE WAVEGUIDE.
Cit!

IMPLICIT NONE

INTEGER’4 NOM,NUI

REAL*8 A,PI,MU0,0MEGA,K0

COMPLEX*16 J,KMUL(2,20),GM,ZMI,KXM,KXMA

COMMON/BINPUT1/A

COMMON/BINPUT4/NOM

COMMON/BCONST1/PI

COMMON/BCONSTZ/MUO

COMMON/BCONSTS/J

COMMON/BFREQCONSTl/OMEGA

COMMON/BFREQCONSTZ/KO

DO NUI=1,NOM,1
KXM = DCMPLX(((2.DO*NUI)-1.DO)*P|/A,0.D0)
GM = SQRT(KXM*KXM-K0*K0)
ZMI = GM/(J'OMEGA*MUO)
KXMA = KXM'A

KMUL(1,NUI) = KXMA

KMUL(2,NUI) = ZMI
ENDDO

95

 

RETURN
END

Ciitfifitfliti

Ci‘iiiitifiii

SUBROUTINE LOADEDMODAL(EPR,MUR,KML)
Ciii
C SUBROUTINE LOADEDMODAL FINDS INITIAL GUESSES FOR KZ/KO BASED
C ON INPUT PARAMETERS, THEN CALLS CSECANT WHICH FINDS THE ACTUAL
C VALUES KZ/KO, THEN USES THOSE VALUES OF KZ/KO IT FINDS THE NEEDED
C TRANSVERSE FIELD VALUES FOR THE LOADED WAVEGUIDE REGION.
Ciifi

IMPLICIT NONE

INTEGER'4 NoM,Nn,NI,NOI

REAL*8 L,PI,MU0,0MEGA,K0,K0_A,D_A,APPLE,APPLE2,DA

COMPLEX*16 J,EPR,MUR,KML(6,20),KZOKOI,KZOKOF,KZOKO,KZOKOA

COMPLEX*16 KXAA,KXDA,ZAI,ZDI,GNSL

COMMON/BINPUTZ/L

COMMON/BINPUTS/D_A

COMMON/BINPUT4/NOM

COMMON/BCONSTl/PI

COMMON/BCONSTZ/MUO

COMMON/BCONSTS/J

COMMON/BFREQCONSTI/OMEGA

COMMON/BFREQCONSTZ/KO

COMMON/BFREQCONSTS/K0_A

DO Nn=1,NOM,1

IF (Nn.EQ.1) THEN
APPLE = D_A*1.D1
APPLE2 = 1.01
KZOKOA = SQRT(MUR*EPR)

+ *SQRT(1 .DO—((Nn*Nn*Pl*Pl)/(K0_A*K0_A*MUR*EPR)))
KZoKO = SQRT(DCMPLX(1.DO-((Nn*Nn*PI*PI)/(K0_A*K0_A)),O.DO))
KZOKOI = KZoKO + (KZoKOA-KZoK0)*(ATAN(APPLE)/ATAN(APPLE2))
DA = D A
CALL CSECANT(KZoK0|,KZoKOF,1.D-6,1.D-7,50,NOI,EPR,MUR,DA)

ELSEIF (Nn.GT.1) THEN
NI = (2*Nn)—1
KZOKOI = DCMPLX(-1.D-5,-(NI*PI/K0_A))

DO DA=1.D-2,D_A,1.D-3
CALL CSECANT(KZoK0|,KZoKOF,1.D-6,1.D-7,50,NOI,EPR,MUR,DA)
C CALL DISPZF(KZoKOF, NOI)
KZoKOI = KZOKOF
ENDDO

ENDIF

KXAA = K0_A*SQRT(1.DO-(KZoKOF*KZoKOF))

KXDA = K0_A*SQRT((MUR*EPR)-(KZoKOF*KZoKOF))
ZAI = K0*KZoKOF/(OMEGA*MUO)

ZDI = K0*KZoKOF/(OMEGA*MUO*MUR)

GNSL = J*K0*L*KZoKOF

96

KML(1,Nn) = KZoKOF
KML(2,Nn) = KXAA
KML(3,Nn) = KXDA
KML(4,Nn) = ZAI
KML(5,Nn) = ZDI
KML(6,Nn) = GNSL

ENDDO

RETURN
END

C**********

Ci’i‘kitifiiiii

SUBROUTINE CSECANT(ZO, ZF, DFD, EPS, LlMNOl, NOI, EPR, MUR, DA)
C***
C SUBROUTINE CSECANT FINDS THE MODAL VALUE OF KZ/KO FOR A SPECFIC
FREQUENCY.
C***
IMPLICIT REAL*8 (A-H,O-Z)
lNTECER*4 NOI, LlMNOl
COMPLEX*16 ZO,ZF,F0,F,DZ,DFDZ,DLZ,21,F1,22,F2,EPR,MUR

ZF=ZO
NOl=1
F0=F(zo,EPR,MUR,DA)
C WRITE(*,10) NOI,ZO,F0
1o FORMAT(/,4X,'NOI=',l2,3X,'KZoK0|=(',E12.5,',',E12.5,')',3X,'
+F=(',E12.5,',',E12.5,')')
11 FORMAT(/,4X,'NOI=',I2,3X,' KZoK0=(',E12.5,',',E12.5,')',3X,'
+F=(',E12.5,',',E12.5,')')
IF(LIMNOI.EQ.1) RETURN
NOl=2
DZ=ZO"DFD
DFDZ=(F(ZO+DZ,EPR,MUR,DA)-FO)/DZ
DLz=-F0/DFDz
Z1=ZO+DLZ
ZF=Z1
T=ABS((Z1-ZO)/ZO)
IF(T.LT.EPS) RETURN
F1=F(Z1,EPR,MUR,DA)
C WRITE(*,11) NO|,Z1,F1
20 NOl=NOl+1
IF(NOI.GT.LIMNOI) GOTO 30
22=Zt-F1*(Zi-ZO)/(F1-F0)
ZF=Z2
T=ABS((22-z1 )/21 )
IF(T.LT.EPS) RETURN
F2=F(22,EPR,MUR,DA)
C WRITE(*,11) NOI,22,F2
zo=21
21:22
F0=F1
F1=F2
GOTO 20
30 WRITE(*,40)
4o FORMAT(/,5X,'TOO MANY ITERATIONS IN CSECANT.')

97

RETURN
END

Ciititiiit.

Ciflitl’ifitii

COMPLEX*16 FUNCTION F(KZoK0,EPR,MUR,DA)
REAL'8 KO_A, DA

COMPLEx*16 KZoKO, KX_D_A, KX_A__A, EPR, MUR
COMPLEX*16 SDX, SAX, CDX, CAX, DX, AX
COMMON/BFREQCONST3/KO_A

KX_D_A = KO_A*SQRT((MUR*EPR)-(KZoK0*KZoKO))
KX_A_A = K0__A*SQRT(1.DO-(KZoKO'KZoK0))

Dx = KX_D_A/2.DO*DA
AX = KX_A_A/2.DO*(1.DO-DA)

SDx = SIN(DX)
em = COS(DX)
SAx = SIN(AX)
CAx = COS(AX)

F = ((((KX_D_A/KX_A_A)*(1.DO/MUR))*(SDX*SAX))-(CDX*CAX))

RETURN
END

Cifiitiiitit

Ciiiiifiitii

SUBROUTINE DISPZF(ZF, NOI)
Cfifit
C SUBROUTINE DISPZF DISPLAYS THE FINAL VALUE OF ZF.
Citi’

IMPLICIT NONE

INTEGER'4 NOI

COMPLEX*16 ZF

WRITE(*,*)'KZoKOF = ',ZF,"
WRITE(*,*)'NOI = ',NOI

RETURN
END

Cii**i**i**

Ciiitii'iiii

SUBROUTINE MATEL(KML,KMUL,ME)
Ciifi
C SUBROUTINE MATEL FILLS THE SCATTERING MATRIX ME.
Ciii
IMPLICIT NONE
INTEGER*4 NoM,M,N
COMPLEX'16 KML(6,20),KMUL(2,20),ME(NOM*4,NOM*4+1)
COMPLEX*16 CM,CN,CP,CQ,CS,CU,CV,CW
COMMON/BINPUT4/NOM

DO M=1,NoM,1
DO N=1,NOM*4+1,1
IF (N.LE.NoM) THEN
ME(M,N)=CM(M,N)

98

ELSEIF (N.GE.NoM+1 .AND. N.LE.NoM*Z) THEN
ME(M,N)=-CN(M,N-NoM,KML,KMUL)
ELSEIF (N.GE.NoM*2+1 .AND. N.LE.NoM*3) THEN
ME(M,N)=-CN(M,N-NOM*2,KML,KMUL)
ELSEIF (N.GE.NOM*3+1 .AND. N.LE.NoM*4) THEN
ME(M,N)=DCMPLX(0.D0,0.DO)
ELSE
ME(M,N)=-CM(M,N-NOM*4)
ENDIF
ENDDO
ENDDO

DO M=NoM+1,NoM*2,1
DO N=1,NoM*4+1,1
IF (N.LE.NoM) THEN
ME(M,N)=CP(M-NOM,N,KMUL)
ELSEIF (N.GE.NoM+1 .AND. N.LE.NoM*Z) THEN
ME(M,N)=CQ(M-NoM,N-NOM,KML,KMUL)
ELSEIF (N.GE.NoM*2+1 .AND. N.LE.NoM*3) THEN
ME(M,N)=-CQ(M-NOM,N-NoM*2,KML,KMUL)
ELSEIF (N.GE.NoM*3+1 .AND. N.LE.NoM*4) THEN
ME(M,N)=DCMPLX(0.D0,0.DO)
ELSE
ME(M,N)=CP(M-NoM,N-NOM*4,KMUL)
ENDIF
ENDDO
ENDDO

DO M=NoM*2+1,NoM*3,1
DO N=1,NoM*4+1,1
IF (N.LE.NoM) THEN
ME(M,N)=DCMPLX(0.D0,0.DO)
ELSEIF (N.GE.NoM+1 .AND. N.LE.NoM*2) THEN
ME(M,N)=CS(M-NoM*2,N-NOM,KML,KMUL)
ELSEIF (N.GE.NoM*2+1 .AND. N.LE.NoM*3) THEN
ME(M,N)=CU(M-NOM*2,N-NoM*2,KML,KMUL)
ELSEIF (N.GE.NoM*3+1 .AND. N.LE.NoM*4) THEN
ME(M,N)=-CM(M-NoM*2,N-NOM*3)
ELSE
ME(M,N)=DCMPLX(0.D0,0.DO)
ENDIF
ENDDO
ENDDO

DO M=NoM*3+1,NoM*4,1
DO N=1,NoM*4+1,1

IF (N.LE.NoM) THEN
ME(M,N)=DCMPLX(0.D0,0.DO)

ELSEIF (N.GE.NoM+1 .AND. N.LE.NoM*2) THEN
ME(M,N)=CV(M-NOM*3,N-NoM,KML,KMUL)

ELSEIF (N.GE.NoM*2+1 .AND. N.LE.NoM*3) THEN
ME(M,N)=-CW(M-NoM*3,N-NOM*2,KML,KMUL)

ELSEIF (N.GE.NoM*3+1 .AND. N.LE.NoM*4) THEN
ME(M,N)=-CP(M-NoM*3,N-NOM*3,KMUL)

ELSE
ME(M,N)=DCMPLX(0.D0,0.DO)

99

ENDIF
ENDDO
ENDDO

RETURN
END

C'kii‘fifl'kiiii

C*****i****

COMPLEX*16 FUNCTION CM(M,N)
IMPLICIT NONE

INTEGER*4 M,N

REAL*8 A

COMMON/BINPUT1/A

IF (M.EQ.N) THEN

CM = DCMPLX(A/4.D0,0.DO)
ELSE

CM = DCMPLX(0.D0,0.DO)
ENDIF

RETURN
END

C**********

Cfltfiitiiii'k

COMPLEX’16 FUNCTION CP(M,N,KMUL)
IMPLICIT NONE
INTEGER*4 M,N

COMPLEX*16 CM,KMUL(2,20)

IF (M.EQ.N) THEN
CP = KMUL(2,M)*KMUL(2,M)*CM(M,N)
ELSE
CP = DCMPLX(0.D0,0.DO)
ENDIF

RETURN
END

C*i*t******

Ciitiiitfiiii

COMPLEX*16 FUNCTION CN(M,N,KML,KMUL)
IMPLICIT NONE

INTEGER*4 M,N
COMPLEX'16 KML(6,20),KMUL(2,20),CN1,CN2

CN = CN1(M,N,KML,KMUL) - CN2(M,N,KML,KMUL)

RETURN
END

Cfi‘tiiititfii

Ciiifii‘it’kit

COMPLEX*16 FUNCTION CQ(M,N,KML,KMUL)
IMPLICIT NONE
INTEGER*4 M,N

COMPLEX‘1 6 KML(6,20),KMUL(2,20),CP,CN1,CN2

CQ = KMUL(2,M)*KML(5,N)*CN1(M,N,KML,KMUL) -

100

+ KMUL(2,M)*KML(4,N)*CN2(M,N,KML,KMUL)

RETURN
END

Cttititi‘iit

Ct*i*****t*

COMPLEX*16 FUNCTION CS(M,N,KML,KMUL)
IMPLICIT NONE

INTEGER*4 M,N

COMPLEX*16 KML(6,20),KMUL(2,20),CN

CS=EXP(-KML(6,N))*CN(M,N,KML,KMUL)

RETURN
END

Cfititi’iiiii

C*i********

COMPLEX*16 FUNCTION CU(M,N,KML,KMUL)
IMPLICIT NONE

INTEGER*4 M,N

COMPLEX*16 KML(6,20),KMUL(2,20),CN

CU=EXP(KML(6,N))*CN(M,N,KML,KMUL)

RETURN
END

Ciififl‘ti‘ii‘ti

Citiiitifl‘ii’

COMPLEX*16 FUNCTION CV(M,N,KML,KMUL)
IMPLICIT NONE

INTEGER*4 M,N

COMPLEX*16 KML(6,20),KMUL(2,20),CQ

CV=EXP(-KML(6,N))*CQ(M,N,KML,KMUL)

RETURN
END

Cit‘i‘tii‘ti’it

C**********

COMPLEX*16 FUNCTION CW(M,N,KML,KMUL)
IMPLICIT NONE

INTEGER*4 M,N

COMPLEx*16 KML(6,20),KMUL(2,20),CQ

CW=EXP(KML(6,N))*CQ(M,N,KML,KMUL)

RETURN
END

C**********

C**********

COMPLEX*16 FUNCTION CN1(M,N,KML,KMUL)
IMPLICIT NONE

INTEGER*4 M,N

REAL*8 A,D_A,Pl,cc,dd

COMPLEX*16 KML(6,20),KMUL(2,20),aa,bb
COMMON/BlNPUT1/A

101

COMMON/BINPUT3/D_A
COMMON/BCONSTi/PI

aa = KML(3,N) - KMUL(1,M)
bb = KML(3,N) + KMUL(1,M)
cc = D_A/2

dd = N2

CN1 = dd*(S|N(aa*CC)/aa + SIN(bb‘Cc)/bb)

RETURN
END

Cii’iiiittit

Ctiitiittfii

COMPLEX*16 FUNCTION CN2(M,N,KML,KMUL)
INTEGER*4 M,N

REAL*8 A,O_A,Pl,dd

COMPLEX*16 KML(6,10),KMUL(2,10),ee,ff,gg,hh,ii,jj
COMMON/BlNPUT1/A

COMMON/BlNPUT3/D_A

COMMON/BCONST1/PI

dd = N2

ee = KML(2,N)*(1-D_A)

ff = KMUL(1,M)*O_A

99 = KML(2,N) + KMUL(1,M)
hh = KML(2,N) - KMUL(1,M)
ii = COS(KML(3,N)*D_A/2)
n=aNmmm

CN2 = (ii/jj)*dd*(COS((ee-ff)/2)/gg + COS((ee+ff)/2)/hh)

RETURN
END

Ct**§*i*i*i

Ciiitiiiitt

SUBROUTINE CMATPAC(IJOB,NR,NC,A,N,M,OET,EP)
C...
C SUBROUTINE CMATPAC IS A MATRIX-OPERATIONS SUBROUTINE PACKAGE
C FOR COMPLEX-VALUED ARRRAYS.
C...
IMPLICIT INTEGER*4(l-N)
IMPLICIT REAL*8(A—H,O-Z)
COMPLEx*16 A(NR,NC),B,DET,CONST,S

180 FORMAT(1X,'THE DETERMINANT OF THE SYSTEM EQUALS ZERO.',/,

11X,'THE PROGRAM CANNOT HANDLE THIS CASE.',//)
DET=1.
NP1=N+1
NPM=N+M
NM1=N-1
lF(lJOB) 30,10,30

10 DO 20 I=1,N

NPl=N+l
A(I,NPI)=1.
IP1=I+1

102

DO 20 J=IP1,N
NPJ=N+J
A(l,NPJ)=0.
20 A(J,NPI)=0.
30 D080.I=1,NM1
C=ABS(A(J,J))
JP1=J+1
DO 60l=JP1,N
D=ABS(A(I,J))
lF(C—O) 40.60.60
40 DET=-DET
D050 K=J,NPM
B=A(I,K)
A(|,K)=A(J,K)
so A(J,K)=B
C=D
60 CONTINUE
lF(ABS(A(J,J))-EP) 90.70.70
70 DO 80I=JP1,N
CONST=A(I,J)/A(J,J)
DO 80 K=JP1,NPM
80 A(l,K)=A(|,K)-CONST*A(J,K)
lF(ABS(A(N,N))-EP) 90,120,120
90 DET=0.
lF(lJOB) 100,100,110
100 WRITE(*,180)
WRITE(7,180)
110 RETURN
120 DO130|=1,N
130 DET=OET*A(I,I)
lF(lJOB) 140,140,110
140 DO17OI=1,N
K=N-I+1
KP1=K+1
DO 170 L=NP1,NPM
S=0.
lF(N-KP1) 170,150,150
150 D0160 J=KP1,N
160 S=S+A(K,J)*A(J,L)
170 A(K,L)=(A(K,L)-S)/A(K,K)

RETURN
END

Ci‘tttfifii‘k'kt

C**********

SUBROUTINE OUTPUTMUEP
Ctii
C SUBROUTINE OUTPUTMUEP WRITES EPSILION/MU DATA TO OUTPUT FILES
c AND THEN CLOSES THE FILES.
Ciii
IMPLICIT NONE
INTEGER*4 I,NP
REAL*8 F(801)
COMPLEX*16 EPR_F(801),MUR_F(801),EPR_R(801),MUR_R(801)
COMMON/BINPUTs/NP
COMMON/BEXPSPI/F

103

COMMON/6CALCMUEP3/EPR_F, MUR_F, EPR_R, MUR_R

WRITE(11,102)
102 FORMAT(ZX,'FREQ(GHz)',3X,'Re{EPR-F}',3X,'lm{EPR-F}',
+ 3X,'Re{MUR-F}',3X,'lm{MUR-F}',3X,'Re{EPR-R}',
+ 3X,'Im{EPR-R}',3X,'Re{MUR-R}',3X,‘Im{MUR—R}')
DO I=1,NP,1
WRITE(11,103) F(|)/1.09,EPR_F(I),MUR_F(I),EPR_R(|),MUR_R(I)
103 FORMAT(1X,9(E10.4,2X))
END DO

CLOSE (10)
CLOSE (11)

WRITE(*,*) 'Program Bogle Oone!!!‘

RETURN
END

104

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106

 

 
 

 

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