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Analysis, modelling, and design of microwave planar ferrite devices and antenna circuits using spectral domain method of moments

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ANALYSIS, MODELLING, AND DESIGN OF MICROWAVE PLANAR
FERRITE DEVICES AND ANTENNA CIRCUITS USING SPECTRAL DOMAIN
METHOD OF MOMENTS
by
Tarief M. Fawzy Elshafiey
A Dissertation Presented in Partial Fulfillm ent
of the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
December 1996
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UMI Number: 9710349
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ANALYSIS, MODELLING, AND DESIGN OF MICROWAVE PLANAR
FERRITE DEVICES AND ANTENNA CIRCUITS USING SPECTRAL DOMAIN
METHOD OF MOMENTS
by
Tarief M. Fawzy Elshafiey
has been approved
November 1996
APPROVED:
. Chairperson
5
■
-
8-
Lo-fi*
Supervisory Committee
:pa:trnO$t Chairperson
Dean, G raduate College
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ABSTRACT
T he objective of this research is to develop the theoretical and numerical models,
based on the method of moments, for the analysis of various ferrite microwave devices
and antenna circuits. The transmission m atrix for a normally biased ferrite layer is
derived in a closed form. The Green’s function is formulated using the transmission
m atrix. A 2-D model is implemented to analyze the edge-guided mode microstrip
isolator. The resistive film term ination over one edge of the microstrip, to absorb the
backward wave, is considered in this work. The closed form transmission m atrix for
any arbitrarily biased ferrite slab is also derived to formulate a very general Green’s
function for multi-layer a rb itra rily magnetized ferrite structures. Two numerical
models are implemented: phase shifter model and a m agnetostatic surface wave
model. Good agreement with the previously published results is achieved. A novel
approach to analyze scattering from arbitrarily shaped patches on arbitrarily biased
ferrite substrates is presented. In th at approach, the excitation vectors for various
single and multi-layer antenna structures are derived in a closed form using the
transm ission matrix. A 3-D model is implemented to study the Radar Cross Section
(RCS) for several ferrite antennas. A novel cross patch antenna is proposed which
results in significant RCS reduction.
111
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roT* k ^ok
^
a
(Over all Endued with knowledge is One, the All-Knowing)
/d L lz ll ASal^. 0A. *111 ^f»V) U i t )
(Those truly fear Allah, Among His Servants who have knowledge)
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To th e spirit of m y father
and
To m y sons, M ahm oud, Abdelrahm an and Ahmed
V
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ACKNOWLEDGMENTS
I would like to express my deep gratitude to my advisor, Dr. James T. Aberle
for his friendship, encouragement, guidance and advice throughout the duration of
this research. When you work with someone like Dr. Aberle you feel that you are
working with a friend.
A special thanks to Dr. Samir M. El-Ghazaly for his sincerity and his brother­
hood. I do not like to say much about his encouragement and solving many personal
problems th at I faced, but I ask the All-mighty ALLAH to bless him and his family.
I am also grateful to the members of my committee: Dr. R. Renaut, Dr. R. Grondin,
and Dr. B. Welfert who unfortunately was on leave when this work was completed
and defended.
Deep appreciation is due to my friends and colleagues at the Telecommunications
Research Center, in particular, Dr. David Kokotoff for his time discussing MoM and
ferrites and Chris Bishop for providing his mesh generator that I used to generate
the Cross-Patch mesh, and for his help in revising the manuscript of this work.
I can not forget two of my sincere friends, Dr. Osman Ibrahiem and Dr. Khalid
Shehata of the Postgraduate Navy School at Monterey, CA. Their constant support
and advice especially after I faced many difficulties gave me much strength and
confidence.
I would like also to take the opportunity to thank many Brothers for their com­
pany and encouragement, Hussein Mahmoud, Bahader Yildirim, Samir Hammadi,
and Sohel Imtiaz. Their friendship and humor has made this process bearable.
I wish to express my sincere appreciation to the Egyptian Government who gave
me this opportunity and partially supported this program. My appreciation also
extends to Gen. Dr. Essam Ibrahiem, one of the very active members of the adm in­
istration of this program. His understanding and consideration for the difficulties
th at I faced gave me the chance to complete this work.
vi
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I would especially like to acknowledge the supplication of my mother and contri­
bution of my wife. Their enduring love, patience and understanding has extended
fax above what could be expected from any person.
vii
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TABLE OF CONTENTS
Page
LIST OF T A B L E S .........................................................................................................
xii
LIST OF F I G U R E S ......................................................................................................... xiii
CHAPTER
1
IN T R O D U C T IO N .........................................................................................
1.1
M otiv atio n .............................................................................................
1
1.2
Previous Work
...................................................................................
4
1.3
M e th o d s ................................................................................................
11
1.4
2
1
1.3.1
Introduction
.....................................................................
11
1.3.2
M ethod of M o m e n ts ........................................................
14
Research O b je c tiv e s .........................................................................
15
GREEN’S FUNCTION FORMULATION FOR A NORMALLY BI­
ASED FE R R IT E S L A B ...............................................................................
18
2.1
In tro d u c tio n .........................................................................................
18
2.2
Transmission M atrix F o rm u la tio n ..................................................
19
2.3
Discussion on the Difficulty of Deriving Normally Biased Trans­
2.4
mission M a t r i x ..................................................................................
25
Green’s Function F o rm u latio n .........................................................
26
viii
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CHAPTER
2.5
3
Page
2.4.1
Green’s Function for a Single Ferrite Substrate
2.4.2
Green’s Function for a Ferrite-Dielectric Substrate
.
28
2.4.3
Green’s Function for a General Multi-layer Structure
30
2.4.4
Circuit Model Interpretation of the Green’s Function
26
for a Multi-layer S tru c tu re ..............................................
30
Results and Conclusion......................................................................
31
2-D FULL-WAVE ANALYSIS OF AN EDGE-GUIDED MODE ISO­
LATOR ............................................................................................................
39
3.1
In tro d u c tio n ..........................................................................................
39
3.2
Full Wave F o rm u latio n .......................................................................
40
3.2.1
Green’s Function F o rm u latio n ........................................
41
3.2.2
Basis Functions and Resistive Region Tr eat ment . . .
41
3.2.3
Numerical Considerations in the Evaluation of the
Spectral Domain In teg ratio n ..........................................
44
Complex Root S earching.................................................
46
Numerical Results and C onclusion..................................................
46
3.2.4
3.3
4
...
ANALYSIS OF PHASE SHIFTERS AND TRANSDUCERS USING
A GENERAL GREEN’S FUNCTION
...................................................
65
4.1
In tro d u c tio n .........................................................................................
65
4.2
Transmission Matrix for an Arbitrarily-Biased Ferrite Slab . .
65
ix
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CH A PTER
4.3
Green’s Function F o rm u la tio n .........................................................
81
4.4
Varying the Magnetization A n g l e ..................................................
82
4.5
Planar Phase S h ifters.........................................................................
83
4.6
4.7
5
Page
4.5.1
Introduction
......................................................................
83
4.5.2
Slot Line Phase S h if te r s ..................................................
84
4.5.3
Microstrip Phase S h i f t e r s ...............................................
88
Magnetic Surface Wave T ransducers...............................................
93
4.6.1
Introduction
93
4.6.2
Full-Wave Analysis of the Microstrip Phase Shifters .
......................................................................
Results and D iscussion......................................................................
93
94
3-D ANALYSIS OF RADAR CROSS SECTION OF A FERRITE
PATCH A N T E N N A ........................................................................................114
5.1
In tro d u ctio n ............................................................................................ 114
5.2
Plane Wave Propagation in a Ferrite Medium: an Introduction
to the Excitation Vector F o rm u la tio n ..............................................116
5.3
5.4
T h e o ry ......................................................................................................122
5.3.1
Full Wave F orm ulation.........................................................122
5.3.2
Excitation V ecto r.................................................................. 127
5.3.3
Green’s Function F o rm u latio n ........................................... 132
Impedance Matrix In te rp o latio n ........................................................ 134
x
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CHAPTER
5.5
6
Page
Results and C o n clu sio n s............................................................... 137
C O N CLU SIO N ...................................................................................................166
R E F E R E N C E S ............................................................................................................. 168
APPENDIX
A
THE TRANSMISSION MATRIX OF A NORMALLY BIASED FER­
RITE S L A B ................................................................................................... 174
B
THE TRANSMISSION MATRIX OF A DIELECTRIC SLAB . . . .
185
C
THE SEM I-INFINITE SPACE GREEN’S FU N C T IO N ........................... 187
D
RESISTIVE MATRIX ELEMENTS IN SPATIAL D O M A IN
E
GREEN’S FUNCTION FOR A SLOT LINE WITH A FER RITE
190
S U B S T R A T E ................................................................................................ 192
E .l
G reen’s Function for a Single Slot Line Ferrite Substrate . . . 193
E.2
G reen’s Function for a Multilayer Slot Line Ferrite Substrate .
xi
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195
LIST OF TABLES
Table
3.1
Page
Percentage of the impedance m atrix f il lin g ..............................................
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42
LIST OF FIGURES
Figure
Page
1.1 Symbolic representation of phase shifter, isolator and circulator with
ideal properties...................................................................................
2
1.2 3-D edge-guided isolator with transverse slot l o a d i n g ...........................
5
1.3 3-D edge-guided isolator with resistive film loading..................................
6
1.4 3-D edge-guided isolator with shorted load.................................................
7
1.5 Microstrip patch antenna on a normally biased ferrite substrate. . . .
9
1.6 Current basis functions for (a) Longitudinal current (b) Transverse
c u rre n t.................................................................................................
14
2.1 Geometry of single layer isolator structure....................................
26
2.2 Geometry of double layer isolator structure...................................
32
2.3 Geometry of drop-in element isolator structure............................
32
2.4 Circuit representation for a single ferrite layer structure............
33
2.5 Comparison of the computed Green’s function versus Pozar’s(Imag(Gxx))
{d = 7.62 x 10~4m, e, = 12.0, 4irMs = 2100.0G, H dc = 700.00e,
A H = O.OOe, R 3 = 0.00, W = 1.016 x 10~2m, / = 3.6GHz,
K x = (110.0, —10.0))..........................................................................
33
2.6 Comparison of the computed Green’s function versus Pozar’s(Real(Gxx)).
The param eters are the same as in Fig. 2.5..................................
34
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Figure
2.7
Page
Comparison of the computed Green’s function versus Pozar’s (Imag(Gxy)).
The param eters axe the same as in Fig. 2.5......................................
2.8
Comparison of the computed Green’s function versus Pozax’s (Real(Gxy)).
The param eters axe the same as in Fig. 2.5......................................
2.9
34
35
Comparison of the computed Green’s function versus Aberle’s(Imag(Gxx)).
(dd = 0.1 x 10-2 m, e<f = 30.0, dj = 0.762 x 10-3 m, e/ = 12.0,
K x = 110.0ra d /m , f = 2.0G H z).........................................................
35
2.10 Comparison of the computed Green’s function versus Aberle’s(Imag(Gyy)).
The param eters axe the same as in Fig. 2.9......................................
36
2.11 Comparison of the computed Green’s function versus Aberle’s (Imag(Gxy)).
The param eters axe the same as in Fig. 2.9......................................
36
2.12 Comparison of the computed Green’s function versus Aberle’s (Imag(Gxx)).
(dd = 0.1 x 10-2m , td = 9.8, df = 0.762 x 10-3m, e/ = 12.0,
da = 0.1 x 10_2m, ea = 1.0, K x = 110rad/m , f = 2.0G H z ) .........
37
2.13 Compaxison of the computed Green’s function versus Aberle’s (Imag(Gyy)).
The param eters axe the same as in Fig. 2.12....................................
37
2.14 Compaxison of the computed Green’s function versus Aberle’s (Imag(Gxy)).
The param eters are the same as in Fig. 2.12....................................
38
3.1 Edge-guided isolator with resistive film loading..........................................
39
3.2 Geometry of single layer structure.................................................................
40
3.3 Geometry of double layer structure...............................................................
40
3.4 Geometry of drop-in element structure.........................................................
41
xiv
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Figure
Page
3.5 Current basis functions for (a) Longitudinal current (b) Transverse
c u r r e n t...............................................................................................................
43
3.6 Integration contour in the complex plane of either kx or ky.....................
46
3.7 Symmetric current distribution over dielectric m icrostrip........................
50
3.8 Asymmetric current over ferrite microstrip in forward and backward
directions for R 3 = 0.0ft..................................................................................
50
3.9 Asymmetric current over ferrite microstrip in forward and backward
directions for R a = 100.0ft..............................................................................
51
3.10 The phase constants of forward and backward waves (d = 7.62 x
10-4m, t f = 12.0, 4v M . = 1750.0G, Hdc = 800.00e, A H = 80.00e,
R , = 100.0ft. W = 1.016 x l0 -2m )...............................................................
51
3.11 Computed isolation and insertion loss [d = 7.62 x 10-4m, e/ = 12.0,
4 ttM s = 1750.0G, Hdc = 800.00e, A H = 80.0Oe, R 3 = 100.0ft,
W = 1.016 x 10-2m )........................................................................................
52
3.12 Comparison of the insertion loss for three isolator structures (47tA/s =
1750.0G, Hdc = 800.OOe, A H = 80.0Oe, R 3 = 100.0ft, W = 1.016 x
10~2m. For th e single-layer: dj = 7.62 x 10-4m , ej = 12.0. For
the double-layer: dd = 2.62 x 10_4m, ed = 3.0. For the triple-layer:
dd = 4.0 x 10_4m, ed = 8.9 da = 5.0 x 10-3m, ea = 1.0)..........................
52
3.13 Comparison of the isolation for three isolator structures. The same
param eters are as in Fig. 3.12........................................................................
53
3.14 The effect of the dielectric thickness on the insertion loss (dj = 7.62 x
10-4m, e/ = 12.0, 4;rM. = 2100.0G, Hdc = 700.0Oe, A H = 80.0Oe,
R s = 100.0ft, W = 1.016 x 10"2m, td = 30.0)............................................
xv
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53
Figure
Page
3.15 The effect of the dielectric thickness on the isolation (<// = 7.62 x
10-4m, e/ = 12.0, 471 M s = 2100.0G, Hdc = 700.0Oe, A H = 80.00e,
R, = lOO.Ofi, W = 1.016 x 10-2m, ed = 30.0).......................................
54
3.16 The effect of the dielectric constant of the dielectric layer on the
isolation and the insertion loss, (dj = 7.62 x 10~4m, e/ = 12.0,
At M s = 2100.0G, H dc = 700.00e, A H = 80.0Oe, R s = lOO.Ofi,
W = 1.016 x 10_2m , Dd = 0.381 x 10-3m, Freq.= 6.0 GHz)............
55
3.17 The effect of the external DC bias on the isolation, (d = 7.62 x 10_4m,
£f = 12.0, AirM3 = 2100.0G, A H = 80.0Oe, R s = 100.0D, W =
1.016 x 10-2m )...........................................................................................
56
3.18 The effect of the external DC bias on the insertion loss, (d = 7.62 x
10_4m, ef = 12.0, 4ttM s = 2100.0G, A H = 80.0Oe, R s = 100.0D,
W = 1.016 x 10-2m )..................................................................................
57
3.19 The frequency behavior of /xe/ / , (47tM„ = 2100.0G, Hdc = 800.OGe). .
58
3.20 The effect of the film resistance on the insertion loss and the isolation,
(d = 7.62 x 10-4m, ef = 12.0, ArM s = 1750.0G, Hdc = 800.00e,
A H = 80.Ge, Freq.= 5.0 GHz, W = 1.016 x10-2 m )...............................
59
3.21 The effect of the resistive film width on the insertion loss, (d = 7.62 x
10"4m, ef = 12.0, 4ttM s = 2000.0G, Hdc = 700.0Ge, A H = 80.0Ge,
R , = 100.0n, W = 1.016 x 10-2m )..............................................................
60
3.22 The effect of the resistive film width on the isolation, (d = 7.62 x
10-4m, ef = 12.0, 4ttM s = 2000.0G, Hdc = 700.0Ge, A H = 80.0Ge,
R a = lOO.OD, W = 1.016 x 10_2m )..............................................................
61
3.23 3-D edge-guided isolator with resistive film loading..................................
62
xvi
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Figure
Page
3.24 Compaxison between the numerical and experimental insertion loss
(d = 7.62 x 10-4m, ef = 12.0, 4;rM , = 2100.0G, Hdc = 700.0Oe,
A H = 80.00e, R s = 100.0ft, W = 1.016 x 10~4m )..................................
63
3.25 Comparison between the numerical and experimental isolation (d =
7.62 x 10-4m , £/ = 12.0, 47rMs = 2100.0G, H dc = 700.00e, A H =
80.0Oe, R s = lOO.Ofi, W = 1.016 x 10_4m )...............................................
64
4.1
Geometry of single layer structure...............................................................
66
4.2
Magnetization angle (a) an approximate picture (b) a real picture. . .
83
4.3
A hysteresis curve for a ferrite sample........................................................
85
4.4
Cross-section of basic slot line single-layer ferrite planar phase shifter.
85
4.5
n = 0 Chebychev basis function of the electric field in the slot in
transverse directions.........................................................................................
4.6
86
n = 1 Chebychev basis function of the electric field in the slot in
longitudinal directions.....................................................................................
87
4.7
Cross-section of basic microstrip single-layer ferrite planar phase shifter. 89
4.8
Cross-section of the phase shifter using oppositely-magnetized ferrite
layers...................................................................................................................
89
The odd mode of the dual structure...........................................................
90
4.10 The odd mode representation of the dual structure................................
90
4.9
4.11 n=0 Chebychev basis function of the electric current on the strip in
longitudinal directions.....................................................................................
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92
Figure
Page
4.12 n = l Chebychev basis function of the electric current on the strip in
transverse directions........................................................................................
92
4.13 Geometry of MSSW transducers with microstrip embedded between
dielectric ferrite structure...............................................................................
93
4.14 Geometry of MSSW transducers
in
multilayer practical structure. . .93
4.15 Geometry of MSSW transducers
in
two-layerstructure.............
94
4.16 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Imag(Gxx)). (d = 7.62 x 10-4m , ej = 12.0,
4irMs = 2100.0G, Hdc = 700.0Oe, A H = 0.Oe, R , = Oft, W =
1.016 x 10-2m , / = 3.6GHz, K x = (110.0, -1 0 .0 )).................................
98
4.17 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Real(Gxx)). The param eters are the same as in
Fig. 4 .1 6 ............................................................................................................
98
4.18 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Imag(Gyy)). The param eters are the same as
in Fig. 4 . 1 6 .....................................................................................................
99
4.19 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Real(Gyy)). The param eters are the same as in
Fig. 4 .1 6 ............................................................................................................
99
4.20 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Imag(Gxy)). The param eters are the same as
in Fig. 4 . 1 6 ........................................................................................................ 100
xviii
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Figure
Page
4.21 Compaxison of the computed Green’s function versus Pozar’s for the
normally biased slab (Real(Gxy)). The parameters are the same as in
Fig. 4 .1 6 ............................................................................................................... 100
4.22 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Imag(Gyx)). The param eters are the same as
in Fig. 4 . 1 6 ........................................................................................................ 101
4.23 Comparison of the computed Green’s function versus Pozar’s for the
normally biased slab (Real(Gyx)). The param eters are the same as in
Fig. 4 .1 6 ............................................................................................................... 101
4.24 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Imag(Gxx)) (d = 7.62 x 10-4m, t j = 12.0,
iirM s = 2100.0G, Hdc = 700.00e, A tf = 0.Oe, R 3 = 0Q, W =
1.016 x 10-2m, / = 3.6GHz, K x = (110.0, -1 0 .0 )).................................... 102
4.25 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Real(Gxx)). The parameters are the same
as in Fig.4 .2 4 ..................................................................................................... 102
4.26 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Imag(Gyy)). The parameters are the same
as in Fig.4 .2 4 ..................................................................................................... 103
4.27 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Real(Gyy)). The parameters are the same
as in Fig.4 .2 4 .....................................................................................................103
4.28 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Imag(Gxy)). The parameters are the same
as in
Fig.4 .2 4 ..................................................................................................... 104
xix
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Figure
Page
4.29 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Real(Gxy)). The param eters are the same
as in Fig. 4 .2 4 ..................................................................................................... 104
4.30 Compaxison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Imag(Gyx)). The param eters are the same
as in Fig. 4 .2 4 ..................................................................................................... 105
4.31 Comparison of the computed Green’s function versus Elsharawy’s for
transversely biased slab (Real(Gyx)). The param eters are the same
as in Fig. 4 .2 4 ..................................................................................................... 105
4.32 Comparison of the calculated differential phase shift versus theoretical
and experim ental results for a microstrip single layer phase shifter
{df = 0.635 x 10-3m, e, = 12.9, 4ttM s = 2300.0G,
= 150.0Oe,
S = 0.45 x 10~3m, a = 1.27 x 10“2m, / = 1.52 x 10"2, 9 = 90°, (f>= 90°). 106
4.33 Comparison of the calculated differential phase shift versus theoretical
and experim ental results for a microstrip single layer phase shifter
{df = 0.635 x 10-3m , e, = 12.9, 4jtM, = 2300.0G, Hdc = 150.0Oe,
5 = 0.45 x 10-3m, a = 1.27 x 10~2m, I = 1.52 x 10-2 , 9 = 90°, <p = 80°). 107
4.34 Comparison of the normalized propagation constants for dual strip
phase shifters (d f = 1.0 x 10-3m, t j — 17.5, 47rM , = 1500.0G, Hdc =
0Oe, S = 1.0 x 10 3m , a = 1.0 x 10 2m )........................................................ 107
4.35 Comparison of
the calculated differential phase shift for dual strip
phase shifters (d f = 1.0 x 10-3m, ef = 17.5, 4 itM 3 = 1500.0G, Hdc =
0Oe, S = 1.0 x 10~3m, a = 1.0 x 10_2m, 9 = 90°, <f>= 80°)........................108
xx
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Figure
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4.36 Compaxison of the calculated symmetric propagation constant for the
two layer transducer (dd = 1.27 x 10_3m , d f = 2.03 x 10-3m, e/ =
17.5, t d = 10.2, 4irM3 = 2267.0G, Hdc = 144.0Oe, AH = 300.00e,
S = 0.3 x 10-3m ).................................................................................................108
4.37 Comparison of the calculated asymmetric propagation constant for
the two layer transducer (dd = 1.27 x 10-3m, df = 2.03 x 10-3m, t f =
17.5, ed = 10.2, 4ttM s = 2267.0G, Hdc = 144.0Oe, A H = 300.00e,
S = 0.3 x 10-3m )................................................................................................ 109
4.38 Comparison of the calculated attenuation constant for the three layer
transducer (d\d = 2.5 x 10-4m, d2d = 2.5 x 10-4m, df = 0.5 x 10-4m,
t f = 15.0, eld = 9.8, eu = 10.0, 4ttM s = 1780G, Hic = 600.0Oe,
A H = 45.0Oe, S = 0.5
x
10-4m ).....................................................109
4.39 Comparison of the calculated phase constant for the three layer trans­
ducer ( d u = 2.5 x 10-4m, d2d = 2.5 x 10-4m, df = 0.5 x 10-4m,
t f = 15.0, eu = 9.8, eld = 10.0, 4?rM s = 1780G, Hdc = 600.0Oe,
A H = 45.0Oe, S = 0.5
x
10~4m ).....................................................110
4.40 The effect of number of basis functions on the attenuation constant
4.41 The effect of number of basis functions on the phase constant
. . . .
. Ill
Ill
4.42 Comparison of the insertion loss along a MSSW transducer (dd =
1.27 x 10_3m, d f = 2.03 x 10_3m, ef = 17.5, ed = 10.2, 4irMa =
2267.0G, H dc = 144.0Oe, A H = 490.00e, 5 = 0.3 x 10"3m, I =
12.7 x 1 0 '3m )..................................................................................................... 112
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Figure
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4.43 The effect of the magnetization angles on the insertion loss along a
MSSW transducer (dd = 1.27 x 10-3 m , d f = 2.03 x 10~3m , cj = 17.5,
t d = 10.2, 4ttM s = 2267.0G, Hdc = lU.OOe, A H = 44O.O0e, S =
0.3 x 10_3m , I = 12.7 x 10~3m )........................................................................112
4.44 The effect of the 3-dB line width on the insertion loss along a MSSW
transducer (dd = 1.27 x 10-3m, d f = 2.03 x 10-3m, e/ = 17.5, ed =
10.2, 4:irMs = 2267.0G, Hdc = 144.O0e, S = 0.3 x 10~3m, I = 12.7 x
1 0 '3m )....................................................................................................................113
5.1
Incident wave on a normally biased single ferrite layer............................... 119
5.2
Geometry of single ferrite layer with a general incident wave.................... 122
5.3
Microstrip patch antenna on a normally biased ferrite substrate
5.4
RWG basis function and corresponding edge connectivity....................... 126
5.5
Geometry of a patch antenna with a biased ferrite as a cover layer . . 141
5.6
Fields and currents of a patch antenna with a biased ferrite as a cover
. . . 123
l a y e r ......................................................................................................................141
5.7
Compaxison of RCS for a microstrip patch antenna (Ms = H0 = 0,
eT = 13.0, d = 1.3 x 10-3m, L = W = 1.3 x 10"2m, 0{ = 30°, <?,• = 45°) 142
5.8
Comparison of RCS for a microstrip patch antenna (M3 = Ho = 0,
= 13.0, d = 1.3 x 10-3m, L = W = 1.3 x 10-2m, 0t- = 30°, fa = 45°) 142
5.9
Comparison of RCS for a microstrip patch antenna (Ms = H0 = 0,
er = 4.0, d = 3.0 x 10~4m, L = 1.25 x 10"2m, W = 2.5 x 10_2m,
Qi = 45°, fa = 0 ° ) ...............................................................................................143
XXI1
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Figure
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5.10 Comparison of RCS for a microstrip patch antenna (Ma = H q = 0,
er = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10-2m, W = 0.4 x 10~2m,
0{ = 60°, fa = 45°)
............................................................................................143
5.11 Comparison of RCS for a microstrip patch antenna (Ms = Ho = 0,
tr = 12.8, d = 6.0 x 10_4m, L = 0.55 x 10_2m, W = 0.4 x 10~2m,
0i = 60°, fa = 45°)
............................................................................................144
5.12 Comparison of RCS for a microstrip patch antenna, the bias field
is in the y-direction, (47rMj = 1780.0G, Ho = 360.00e, tr = 12.8,
d = 6.0 x 10-4m, L = 0.55 x 10-2m, W = 0.4 x 10~2m, 0,- = 60°,
<f>i = 4 5 ° )...............................................................................................................144
5.13 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (47rMs = 1780.0G, Ho = 360.00e, ^ = 12.8,
d = 6.0 x 10_4m, L = 0.55 x 10_2m, W = 0.4 x 10_2m, 6; = 60°,
4>i = 4 5 ° )...............................................................................................................145
5.14 Comparison of RCS for a microstrip patch antenna, the bias field is
in th e x-direction, (47rM a = 1780.0G, H0 = 300.00e,
= 40.0G,
Crd = 2.2, eTf = 13.0 dd = 1.3 x 10-3m, d j = 0.5 x 10-3m L —
4.0 x 10-2m, W = 3.0 x 10-2m, 9{ = 30°, fa = 4 5 ° ) .................................. 145
5.15 Comparison of RCS for a microstrip patch antenna, the bias field is
in the x-direction, (47rM s = 1780.0G, H 0 = 300.OGe, AH = 40.0G,
Crd = 2.2, trf = 13.0 dd = 1.3 x 10_3m, dj = 0.5 x 10~3m L —
3.0 x 10“2m, W = 4.0 x
10"2m,0{ = 30°, fa = 4 5 ° ) .................................. 146
xxiii
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Figure
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5.16 Comparison of RCS for a microstrip patch antenna, the bias field
is in th e y-direction, (4t M 3 = 1780.0G, Ho = 360.OOe, er =
d = 6.0 x 10-4m, L
=
0.55
x 10~2m,
W
12.8,
=
0.4
x
10"2m,
x
10-2m,
x
10-2m,
x
10_2m,
x
10~2m,
<t>i = 4 5 ° ) ...............................................................................................................146
5.17 Comparison of RCS for a microstrip patch antenna, the bias field
is in the y-direction, (47rMa = 1780.0G, Ho = 360.OOe, tr = 12.8,
d = 6.0 x 10_4m, L
=
0.55
x 10-2m,
W
=
0.4
4>i = 4 5 ° ) ...............................................................................................................147
5.18 Comparison of RCS for a microstrip patch antenna, the bias field
is in the y-direction, (4irMs = 1780.0G, H0 = 36O.O0e, er = 12.8,
d = 6.0 x 10-4m, L
=
0.55
x 10_2m,
W
=
0.4
fa = 4 5 ° ) ...............................................................................................................147
5.19 Comparison of RCS for a microstrip patch antenna, the bias field
is in the r-direction, (4ttM s = 1780.0G, Ho = 360.OOe, tr = 12.8,
d = 6.0 x 10_4m, L
=
0.55
x 10-2m,
W
—
0.4
(f>i = 4 5 ° ) ...............................................................................................................148
5.20 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (47tM, = 1780.0G, H0 = 360.00e, er =
d = 6.0 x 10-4m, L
=
0.55
x 10-2m,
W
12.8,
=
0.4
<f>i = 4 5 ° ) ...............................................................................................................148
5.21 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (AirMa = 1780.0G, Ho = 360.00e, er = 12.8,
d — 6.0 x 10_4m, L = 0.55 x 10_2m, W = 0.4 x 10~2m, 0,- = 60°,
<f>i = 4 5 ° ) ...............................................................................................................149
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5.22 Comparison of RCS for a microstrip patch antenna, the bias field is
in th e x-direction, (A tM s = 1780.0G,
H
q
= 300.OOe, A H = 40.0G,
trd = 2.2, trf = 13.0 dd = 1.3 x 10_3m, df = 0.5 x 10-3m L =
4.0 x 10~2m, W = 3.0 x 10"2m, 0,- = 30°, fa = 4 5 ° ............................. 150
5.23 Comparison of RCS for a microstrip patch antenna, the bias field is
in th e x-direction, (47rM s = 1780.0G, H0 = 300.OOe, A H = 40.00,
eTd = 2.2, Crf = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10_3m L =
4.0 x 10-2m, W = 3.0 x 10-2m, d{ = 30°, ^, = 4 5 ° ............................. 151
5.24 Comparison of RCS for a microstrip patch antenna, the bias field is
in the x-direction, (4ttM 3 = 1780.0G, H0 = 300.OOe, A H = 40.00,
erd — 2.2, erj = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10-3m L =
4.0 x 10~2m, W = 3.0 x 10_2m, 9{ = 30°, & = 4 5 ° ............................. 152
5.25 Comparison of RCS for a microstrip patch antenna, the bias field is
in the x-direction, (47tM 3 = 1780.0G,
H
q
= 300.OOe, A H = 40.00,
erd = 2.2, 6rf = 13.0 dd = 1.3 x 10-3m, df = 0.5 x 10-3m L =
3.0 x 10~2m, W = 4.0 x 10_2m, 9{ = 30°, fa = 4 5 ° ............................153
5.26 Comparison of RCS for a microstrip
in the x-direction, (47tM s = 1780.0G,
patch antenna,
H
q
the bias
fieldis
= 300.OOe, A H = 40.00,
trd = 2.2, Crf = 13.0 dd = 1.3 x 10~3m, dj = 0.5 x 10~3m L =
3.0 x 10-2m, W = 4.0 x 10-2m, 0,- = 30°, <f>i = 4 5 ° ............................154
5.27 Comparison of RCS for a microstrip
in the x-direction, (47rMa = 1780.OG,
patch antenna,
H
q
the bias
= 300.OOe, A H = 40.0G,
erd = 2.2, trf = 13.0 dd = 1.3 x 10-3m, dj = 0.5 x 10-3m L =
3.0 x 10-2m, W = 4.0 x 10-2m, 0,- = 30°, fa = 4 5 ° ............................155
xxv
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fieldis
Figure
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5.28 Comparison of RCS for a microstrip patch, antenna, the bias field is
in the x-direction, (4xM s = 1780.0G,
H
q
= 300.00e, A H = 40.0G,
6rd = 2.2, er/ = 13.0 dj = 1.3 x 10~3 m, dj = 0.5 x
3.0 x 10- 2m, W = 4.0 x 10- 2m,
1 0 ~3m
L =
= 30°, fa = 4 5 ° ........................ 156
0,-
5.29 Comparison of RCS for a microstrip patch antenna, the bias field is
in the x-direction, (4irM3 = 1780.0(7,
H
q
= 300.00e, A H = 40.0(7,
eTd = 2.2, trf = 13.0 dd = 1.3 x 10_3 m, df = 0.5 x 10~3m L =
3.0 x 10“ 2m, W = 4.0 x 10“ 2 m,
0t-
= 30°, fa = 4 5 ° ........................ 157
5.30 Comparison of RCS for a microstrip patch antenna, the bias field is
in the x-direction, (47rMs = 1780.0G, H0 = 300.00e, A H = 40.0(7,
trd = 2.2, tr/ = 13.0 dd = 1.3 x 10- 3 m, df = 0.5 x
3.0 x 10- 2m, W = 4.0 x 10_ 2 m,
0,-
1 0 ~3m
L =
= 30°, ^, = 4 5 ° ........................ 158
5.31 Geometry and dimensions of a cross patch compared to the full patch
antenna...................................................................................................................158
5.32 Comparison of RCS for a microstrip patch antenna (M s =
H
eT = 12.8, d = 6.0 x 10_4 m, L = 0.55 x 10- 2 m, W = 0.4 x
q
= 0,
1 0 - 2 m,
0,- = 60°, fa = 4 5 ° ............................................................................................... 159
5.33 Comparison of RCS for a microstrip patch antenna (M3 = Ho = 0,
tr = 12.8, d = 6.0 x 10- 4 m, L = 0.55 x 10- 2 m, W = 0.4 x 10- 2m,
0{ = 60°, fa = 4 5 ° ............................................................................................... 159
5.34 Comparison of RCS for a microstrip patch antenna (M„ = H0 = 0,
tr = 12.8, d = 6.0 x
1 0 _4 m,
L = 0.55 x 10_ 2m, W = 0.4 x 10- 2m,
0,- = 60°, fa = 4 5 ° ............................................................................................... 160
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Figure
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5.35 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (4irMs = 1780.0G, Ho = 360.00e, tr = 12.8,
d = 6.0 x 10- 4 m , L = 0.55 x 10_2m, W = 0.4 x 10- 2m , 0,- = 60°,
4>i = 4 5 ° ...............................................................................................................160
5.36 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (4xiVfa = 1780.0G, H q = 360.00e, tr = 12.8,
d = 6.0 x
1 0 _4 m ,
L = 0.55 x 10_2m, W = 0.4 x 10_ 2m , 0,- = 60°,
<f>i = 4 5 ° ............................................................................................................... 161
5.37 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (4irMa = 1780.0G, H q = 360.OOe, tr = 12.8,
d = 6.0 x
1 0 _4 m ,
L = 0.55 x 10“ 2m, W = 0.4 x 10- 2m, 0,- = 60°,
(f>i = 4 5 ° ............................................................................................................... 162
5.38 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (47rMa = 1780.00, H q = 360.OOe, tr = 12.8,
d = 6.0 x 10- 4 m , L = 0.55 x
1 0 ~2m,
W = 0.4 x 10- 2m , 0; = 60°,
<f>i = 4 5 ° ............................................................................................................... 163
5.39 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, ( 4 ^ ^ , = 1780.0G, Ho = 360.OOe, tT = 12.8,
d = 6.0 x
1 0 - 4m ,
L = 0.55 x
1 0 ~2m,
W = 0.4 x 10- 2m , 0,- = 60°,
<f>i = 4 5 ° ............................................................................................................... 164
5.40 Comparison of RCS for a microstrip patch antenna, the bias field
is in the x-direction, (4 ^ ^ ^ = 1780.OG, H0 = 360.00e, tr = 12.8,
d = 6.0 x 10_4m , L = 0.55 x 10"2m, W = 0.4 x 10~2m , 0; = 60°,
(pi = 4 5 ° ............................................................................................................... 165
A .l Geometry of single layer structure................................................................... 175
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C .l
Geometry of single layer isolator structure................................................... 188
D .l
Current basis functions for (a) Longitudinal current (b) Transverse
c u rre n t.................................................................................................................. 191
E .l
Geometry of single layer slot line structure.................................................. 193
E.2
Geometry of single layer slot line structure.................................................. 195
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CHAPTER
1
INTRODUCTION
1.1
M otivation
T he objective of this research is the development of numerical models suitable for
the analysis of a variety of microwave ferrite devices such as isolators, phase shifters,
M agnetostatic Surface-Wave (MSSW) transducers and circulators. The main reason
behind this work is the extensive progress in the epitaxial growth of ferrite substrates.
M icrostrip is one of the most widely used transmission lines in the design of microwave
integrated circuits (MIC’s). Although microstrip circuits have relatively high line
losses and low power capability, they tire easy to fabricate, have small footprint and
weight, exhibit large bandwidth, facilitate realization of passive circuits, and allow for
good integration of chips, ferrites, and lumped elements. Ferrite materials are widely
used in conjunction with planar structures for many microwave applications. Their
high resistivity enables an electromagnetic wave to penetrate the m aterial so that
the m agnetic field component of the wave can interact with the magnetic moment
of th e ferrite. The interaction is distinguished by the remarkable behavior of the
microwave permeability of ferrite. The permeability shows a clear resonance at a
frequency which is simply related to the strength of the applied magnetic field within
the ferrite. Another useful property of the microwave perm eability of a magnetized
ferrite is th a t it causes cross coupling of linearly, circularly or elliptically polarized
waves which would remain uncoupled in an isotropic dielectric medium. Because of
this coupling, the propagation of electromagnetic waves through a ferrite medium
can be nonreciprocal. Many microwave devices have been implemented based on the
nonreciprocal phenomena. Among the most im portant of these devices are phase
shifters, isolators, MSSW transducers and circulators. These devices have found an
im portant place in microwave receivers, transm itters and duplexers. The symbolic
representations of a phase shifter, an isolator and a circulator are shown in Fig. 1.1.
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9
(c)
fl»
(■)
-A A A A
Isolator
Gyrator
Circulator
Fig. 1.1: Symbolic representation of phase shifter, isolator and circulator with ideal prop­
erties.
In these devices, the microwave signal level is kept low such th at nonlinear effects
in the ferrite axe negligible. Thus, a linear (small signal) analysis can be used to
sim ulate most ferrite devices.
One of the most useful microwave ferrite devices is the isolator, which is a twoport device having unidirectional transmission characteristics. The scattering m atrix
for an ideal isolator has the form [1 ]
o o
0
1
1
( 1.1)
1
[s
indicating that both ports are matched, but transmission occurs only in the direction
from port 1 to port 2. Since th e S m atrix is not unitary, the isolator m ust be lossy.
And, of course, [S] is not symmetric, since an isolator is a nonreciprocal component.
A common application uses an isolator between a high-power source and a load to
prevent possible reflections from damaging the source. An isolator can be used in
place of a matching or tuning network, but any power reflected from the load will be
absorbed by the isolator as opposed to being reflected back to the load, as the case
when a matching network is used. Although there are several types of ferrite isolators,
we concentrate here on the field displacement isolator. Field displacement isolators
are simple and compact microwave ferrite devices with good electrical performance.
The principle of operation of these devices is based on the field displacement effect;
i.e, the microwave field configurations of the forward and backward propagating
waves are different from each other. If an absorbing resistive film is placed at the
regions of low electric field of forward propagating waves and high electric field of
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backward waves, then different attenuation of these two waves occurs, and an isolator
is realized.
T he circulator is another im portant ferrite device. It is a three-port device that
can be lossless and matched at all ports. The scattering m atrix of such an ideal
device can be w ritten as [1].
[s].
0 0 1
1 0 0
0 1 0
The S-m atrix shows th at power flow can occur from ports
1,
( 1.2 )
1
to 2, 2 to 3, and 3 to
but not in the opposite direction. Thus, complete transmission between adjacent
arms takes place in one sense of circulation only. A circulator with any number of
arms can be built.
M agnetostatic surface waves (MSSW) axe potentially im portant for carrying out
signal processing directly at microwave frequencies because of their low propagation
loss, ease of excitation, and possibility for electrically variable delay. Operations of
MSSW delay lines have been dem onstrated from
1
to 15 GHz [2]. MSSW transducers
have potential applications in several signal processing devices, such as delay lines,
filters, resonators, and oscillators. The magnetostatic waves are efficiently excited
by simple microstrip transducers.
Ferrite materials have also found wide applications in a class of microwave devices
called phase shifters. These are two-port devices that perm it the passage of guided
waves with very little attenuation but with a variable phase delay controlled by
changing the bias field of the ferrite. There are many types of ferrite phase shifters,
both reciprocal (same phase shift in either direction) and nonreciprocal, and they find
use in a variety of laboratory test equipment. The most significant use is in phased
array antennas where the antenna beam can be steered in space by electronically
controlled phase shifters.
This work comprises investigations of the edge - guided Wave microwave devices
including isolators and MSSW transducers and phase shifters, and in addition, the
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4
radax cross section (RCS) of ferrite patch antennas. The spectral domain m ethod of
moments (MoM) is used in our work.
1.2
Previous Work
Over th e past twenty years, more than one hundred papers have been published on
the theory and application of edge-guided wave (EGW) devices. During this period,
EGW multi-octave isolators [3, 4, 5, 6 ], circulators [7, 8 ] and phase shifters [9, 10]
were constructed and used in operational systems. All of the previous work reported
in this area has comprised either experimental implementations or highly simplified
analyses. To the best of our knowledge, the work presented here represents the first
full-wave analysis for the EGW devices. In this section, we summarize m ajor aspects
of previous work in this area.
Kane [11], in his experimental work on edge-guided isolators, introduced a trans­
verse slot discontinuity located at one edge of the upper conductor as shown in
Fig. 1.2 to attenuate the backward wave.
Hines [4] presented an approximate analysis and a physical description of wave
propagation in a wide microstrip line printed on a normally magnetized ferrite sub­
strate. He divided the microstrip line into three zones and then solved Maxwell’s
equations for the propagation constant under some approximations. Hines concluded
th at th e dominant mode resembles the TEM mode, except that there is a strong
transverse field displacement causing wave energy to be concentrated along one edge
of the line. He also reported experimental work to show the performance of a lossy
term inated isolator. In his experiment, Hines constructed a device consisting of a
normally magnetized ferrite slab over a ground plane and under a wide microstrip
conductor as shown in Fig. 1.3. Waves incident from the port
1
are guided by the
straight edge of the microstrip and propagate with relatively low attenuation. Waves
incident from port
2
are guided toward the other edge of the conductor where they
suffer high attenuation in the resistive film. Hence, nonreciprocal behavior is ob-
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S22
S12
S21
Ho
S li
Fig. 1.2: 3-D edge-guided isolator with transverse slot loading
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6
tained.
e,^i
S21
S12
Ha
n X/4
S ll
Fig. 1.3: 3-D edge-guided isolator with resistive film loading.
Hines increased the bandwidth and improved the performance of the device by
adding capacitance to the low loss edge to compensate for the inductive susceptance
of the fringing fields. Similar devices have been reported which operate in other
frequency bands [5,
6,
12], Cortucci [13] introduced magnetic losses into Hines’
model and accounted for the finite curvature of the guiding edges. He performed
an experim ent to check the accuracy of his theoretical results and found th a t the
overall behavior of the experimental results are in accordance with the theoretical
results. However, he found that the theoretical value of the spacing between adjacent
resonances of the EG wave structure is greater than the experimental value by a factor
of 1 .6 .
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De Santis [14], using an equivalent model, evaluated the fringing field effects in
edge-guided waves propagating along ferrite microstrip devices. He used the fringingfield param eter to evaluate the ratio between the reactive power stored in the fringing
fields and th e RF power in the ferrite under the strip conductor in a disc resonator.
Araki [15] investigated the transmission characteristics of a ferrite substrate
stripline in comparison with an ordinary dielectric substrate stripline. His studies
included reflection problems of ferrite striplines with one edge shorted to the ground
(short-end) and wide striplines with a transverse slot (open-end) using the eigenmode
expansion m ethod. He confirmed his analytical results by constructing an EG mode
isolator with short-end as shown in Fig. 1.4. He showed experimental results for the
device for which the insertion loss is about 2.5 dB, isolation is about 35 dB, and the
usable frequency bandwidth is about 0.6 GHz. However, the characteristics of this
isolator are not as good as another type of EG mode isolator previously described
by Araki [16].
A
Shorted to the gound plane
TOP VIEW
Conductor
Ferrite Slab
Ho
Ground Plane
Fig. 1.4: 3-D edge-guided isolator with shorted load.
In a very good review, De Santis [17] covered the basic concepts of EGW, includ­
ing the partially magnetized state of the ferrite substrate and higher order modes of
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8
propagation th at m ay exist.
The transverse field displacement effect (TFDE) modes as well as dispersion
diagram of the directional waves in various geometries have been widely reported [4],
[18] - [20]. In 1976, Courtois et al [21] introduced the first very high frequency (VHF)
edge-mode isolator. They used a high applied dc field to construct a wide bandwidth
isolator in the VHF range. An isolator with an operating range of 225-400 MHz is
described.
Lin [22] suggested a new method to enhance the performance of field-displacement
isolators by putting th e ferrite slab of m oderate thickness along the axis of the
rectangular waveguide. She verified her theoretical analysis by experimental results,
indicating that the bandwidth may be increased to 20 percent. Dydyk [23, 24], in a
two-part series, examined the propagation characteristics of the edge-guide isolator.
Shively [25], investigated the effects of resistive materials on microstrip antennas
as well as other microstrip structures. We shall use resistive film loading of one of
the microstrip conductor edges to simulate the performance of the EG mode isolator.
Recently, researchers have begun to use the spectral domain approach for analysis
of a variety of ferrite geometries with dc magnetic bias. Yang [26] studied the mi­
crostrip open-end discontinuity on a nonreciprocal ferrite substrate using a full-wave
MoM analysis. In his work, an exact Green’s function and a careful numerical inte­
gration scheme axe used to determine the properties of an open-end for three biasing
directions: longitudinal bias, transverse bias and normal bias. Pozar [27] described
the radiation and scattering characteristics of microstrip antennas on normally bi­
ased ferrite substrates using spectral domain method of moments. He concluded
that the extra degree of freedom offered by the biased ferrite can be used to obtain a
number of novel characteristics, including switchable and tunable circularly polarized
radiation from a microstrip antenna having a single feed point, dynamic wide-angle
impedance matching for phased arrays of microstrip antennas, and a switchable radar
cross section (RCS) reduction technique for microstrip antennas. Fig.
1.5
shows the
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9
geometry considered by Pozar.
d
Fig. 1.5: Microstrip patch antenna on a normally biased ferrite substrate.
Alexopoulos et al [28] also studied microstrip patches on ferrite substrates. They
compared the RCS for the case of an arbitrary biased ferrite to the case of an unbiased
ferrite. They found that the resonant frequencies of a patch vary significantly with the
change of the bias field except for those resonant modes with a dominant magneticfield component in the direction of the bias field. They also found that the magnetic
loss affects RCS at resonance significantly. Their analysis is based on a full-wave
integral equation formulation in conjunction with the method of moments. In spite of
a reasonable effort and many publications about the scattering from printed antennas
on a biased ferrite substrates, an efficient and versatile formulation of the RCS is
not available yet. W ith the exception of Yang [29], multi-layer structures are not
yet discussed. However, neither Yang nor anyone else show the excitation vector
expression or a detailed derivation of the RCS formulation.
It is very im portant for anyone working in the area of ferrites to deeply under­
stand the physics of magnetostatic surface and volume wave mode in ferrite m aterial.
There are two types of modes th at may exist in a ferrite sample. The first type is
the pure electromagnetic mode or the dynamic mode. This mode does not interact
appreciably with the electron spin in the ferrite material. It acts as if the ferrite ma­
terial is isotropic with an effective permeability. The second type is the extraordinary
mode, also called the magnetostatic wave (MSW) mode. Magnetostatic modes are
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10
a class of wave phenomena that appear in ferrimagnetic m aterials and are situated
between the electromagnetic waves at the long wavelength end of the spectrum and
the spin waves at the short wavelength end. In contrast to norm al electromagnetic
waves, this decrease in the wavelength does not require a proportional increase in the
frequency of the wave, but can be obtained by a change in the value of the DC field
th at magnetizes the ferrite. Auld [30] demonstrated that in m agnetostatic waves, the
electric field components become less and less im portant for determ ining the propa­
gation, and thus, Amperes law simplifies in the magnetostatic lim it to V x H = 0. In
specimens w ith finite dimensions, there are also surface dipoles, so th a t the propaga­
tion can be quite different from th at in the infinite medium. Walker [31] has devel­
oped a general theory of magnetostatic modes in ellipsoidal samples. Experimental
verification of this theory was first made by W hite and Solt [32] with absorption
experiments. Auld showed th at these modes can exist under certain conditions in
specimens of any size, and th at they gradually merge with the electromagnetic waves.
Damon and Eshbach [33, 34] extended Walker’s theory to slabs and studied the mag­
netostatic modes in ferrite materials. Their studies established the existence of three
types of m agnetostatic waves (MSW): magnetostatic surface waves (MSSW’s), mag­
netostatic forward volume waves (MSFVW’s) and m agnetostatic backward volume
waves (MSBVW ’s). The orientation of the internal bias field relative to the ferrite
structure and to the propagation direction determine which particular wave type can
exist. M SFVW ’s propagate perpendicularly to the applied m agnetic field which is
itself normal to the ferrite substrate. MSBVW’s propagate perpendicularly to the
applied magnetic field which is in the longitudinal direction of the substrate plane.
MSSW propagates in the same direction as the applied magnetic field. Both are in
the longitudinal direction of the substrate plane.
In our analysis for a normally biased ferrite structure, we deal prim arily with
m agnetostatic forward volume waves. However, the surface wave mode may also be
excited depending on the operating frequency range as described by Pozar [27].
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11
The last subject of this review concerns units. In the Gaussian system of units,
the magnetization is measured in Gauss (lG au ss = 10“ 4 W e6er/m 2), and the field
strength is measured in Oersteds (4tt x 10-3 O ersfed =
1A / m ) .
Thus, fio = 1G/Oe
in Gaussian units, implying th a t B and H have the same numerical values in a
nonmagnetic m aterial. Saturation magnetization is usually expressed as AirM3 Gauss.
In Gaussian system of units, the lamor frequency can be expressed as fo = u m/2ir =
Ho^ H o/ I k = (2.8MHz/Oersted) (H0Oersted), and f m = wm/2 tt = fi0j M s/2ir =
(2.8M H z (Oersted), (AirMsGauss).
On the other hand, In the MKS systems of
units, the permeability for free space fig is equal to 4?r x 10~7F / m and the dielectric
constant for free space e0 = ( 3^ ) x lO~9H / m .
It follows that ^ = = is equal to
3 x 108 m /s , the velocity of light in free space, and ^/no^o — 377fl is th e impedance
of free space. The conversion factors from Gaussian to MKS units are given by [35],
B {G ) x 10“ 4 =
1.3
1.3.1
B ( W b / m 2)
4trM{G) x 79.5 =
M (A/m )
H(Oe) x 79.5 =
H(A/m)
Methods
Introduction
The most widely used method for analyzing microstrip and printed line structures
is the spectral domain approach. This method is essentially a Fourier-transformed
version of the spatial domain integral equation method. The method was first intro­
duced by Yamashita and M ittra [36] for com putation of the characteristic impedance
and the phase velocity of a microstrip line based on a quasi-TEM approximation.
As the operating frequency is increased, dispersion characteristics of the microstrip
become im portant for an accurate design. This requirement has led to the full wave
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12
analysis of microstrip lines, represented by the work of Denlinger [37], who solved
the integral equations using a Fourier transform technique. His technique is strongly
dependent on the assumed current distribution. To avoid this difficulty, another
method was introduced by Itoh and M ittra [38], now commonly called the spectral
domain approach (SDA). In SDA, Galerkin’s method is used to form ulate a homo­
geneous system of equations to determine the propagation constant and the current
distribution from which the characteristic impedance is derived. In this m ethod the
Fourier transform is taken along the direction parallel to the substrate and perpendic­
ularly to the strip. The use of the Fourier transform domain analysis and Galerkin’s
m ethod leads to several im portant features in SDA [39]:
• Easy formulation in the form of a pair of algebraic equations
• Variational nature in determ ination of the propagation constant
• Ability to identify the physical nature of the modes corresponding to each
solution
The SDA method is computationally efficient due to significant analytical pre­
processing. This method has the following limitations:
• SDA assumes infinitesimal thickness for the strip conductor
• It is difficult to treat a strip having finite conductivity
• The substrate is assumed to be infinite in the transverse direction
In spite of these limitations, SDA is a very popular and widely used numeri­
cal technique. However, the formulation of a Green’s function represents a major
complexity in th at approach. Solving for a Green’s function as a boundary condi­
tion problem is quite difficult for magnetic substrates and becomes more difficult
for multi-layer structures. In addition, Green’s functions determ ined using such a
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13
procedure axe useful only for the specific structures for which they are derived. A
more flexible approach is to find the transmission m atrix of the medium and then
use this m atrix to find the Green’s function. The transmission m atrix of a medium
does not depend on the geometry of the total structure. Thus, it is possible to form
a “software” library of transmission matrices for several m aterial layers and to use
this library for th e analysis of different multi-layer structures. In this work, we follow
the approach used by El-Sharawy [40] to derive the transmission m atrix for a lon­
gitudinally biased ferrite slab. W ith this approach, the transmission m atrix for any
material, ferrite or dielectric, can be derived in closed form. Thus, a considerable
improvement in the flexibility of the numerical technique is achieved.
Dyadic Green’s functions axe concise representations of vector-input, vectoroutput systems. For our analysis, they axe the relationships between vector elec­
tromagnetic fields and vector current sources. For instance, Gxy is the electric field
in the x direction due to a current source in y direction. In a m atrix representation
of the dyadic G reen’s function, Gxy is just one of four elements in a 2 x 2 matrix.
The tangential current vector is related to the vector electric or magnetic field by
the dyadic Green’s function.
To expand th e electric current densities in the strip, piecewise linear basis func­
tions axe used to model both longitudinal and transverse currents as shown in Fig. 1.6.
To apply these basis functions in the spectral domain, the Fourier transform is found
analytically. Once the Green’s function is obtained and the basis functions are se­
lected, Galerkin’s technique is used to fill the impedance m atrix. The complex prop­
agation constant is the variable that forces the determinant of the impedance matrix
to zero. The isolation and the insertion loss axe calculated using the attenuation
constant. The current basis function coefficients correspond to the eigenvector of
the impedance m atrix for a given propagation constant. There axe many numerical
considerations which are investigated in detail including integration limits, integra­
tion intervals, num ber of integration points, and the integration path modification
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14
to avoid the m agnetostatic wave poles.
ft)
Fig. 1.6: Current basis functions for (a) Longitudinal current (b) Transverse current
1.3.2
Method of Moments
The Method of Moments (MoM) is a powerful tool for solving linear integral-differential
equations, such as the determ inistic problem
£ ( /) = «
(1.3)
where L is a linear operator, / is the unknown function, and g is the function resulted
from the application of L on / . In the moment method, the
unknown function /
(in our analysis, the current distribution function) is replaced by an approximate
function f a which is a linear combination of a series of known functions (basis), the
coefficients of which are to be determined in the process
/ “ = £<■ „/„
(i-4)
n
Here, the moment m ethod uses a finite number of terms (functions) to represent an
unknown function. Thus th e original problem becomes
6 + J 2 anL(fn) = g
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(1.5)
15
where 6 is the error (residual) due to the introduction of / “. Another series of
functions, called testing or weighting functions, wm, in the range of L, is used to
take the inner product with Equation 1.5. The moment methods lets the inner
product < wm,6 > be zero. Comparing equation 1.5 with the original problem
Equation 1.3, it is noted th at by letting inner product < wm,S > be zero, MoM
approximates the exact solution / with / “ in a sense that equates the projections of
L ( f ) and L ( f a) on
where C{wm) is the space spanned by the wm. Since the
error 6 is orthogonal to the projections (inner product is zero), it is of second order,
and the moment method solution minimizes the error 6 [41]. If the dimensions of
the linearly independent basis functions is equal to or greater than the dimensions
of the solution space, the moment method is exact.
The moment method is not only used in the spectral domain, but also widely
used in th e spatial domain, where a wider variety of basis functions can be used [42].
1.4
Research Objectives
This research attem pts to address some of the current issues involving numerical
modeling and design optimization of microwave ferrite devices and antennas. The
impetus behind the work is to develop theoretical and numerical techniques which
allow rigorous analysis of various microwave ferrite devices with different m agnetiza­
tion directions. Software tools such as
Maple V are extensively utilized. Maple V
is an interactive computer algebra system which is useful in evaluating complex alge­
braic expressions such as the derivation of the transmission m atrix for normally and
arbitrarily biased ferrite slab. In addition, a variety of data visualization programs
such as
GNUPLOT and GLE are utilized during our work.
This is the first effort to present a full-wave analysis of edge-mode microstrip
isolator. This also is the first attem pt to study the effect of the magnetization angle
on the performance of microwave ferrite devices, phase shifters and m agnetostatic
surface wave transducers. The effect of the magnetization angle cannot be stud­
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16
ied without the formulation of the general Green’s function that has two degree of
freedom (6 and <f>).
The generalization of exsisting methods is the main contribution of this work.
This work provides a cleax and versatile formulation for the scattering from a biased
ferrite antenna. This formulation includes general plane wave incident angles, general
biasing directions, and multi-layer ferrite and dielectric structures. The Green’s
function is formulated using the closed form transmission m atrix. The excitation
vectors axe also derived in closed form. The expression of the excitation vectors is
general for any incident angle, any number of layers, and any magnetization angle.
This study of ferrite microwave devices and antennas addresses various topics.
In chapter 2, the Green’s function for a normally biased ferrite slab is evaluated.
A validation of the Green’s function is also presented by comparison with dielectric
Green’s function w ith a Green’s function for a single ferrite slab. Chapter 3 presents
the full-wave analysis of the edge-guided microstrip isolator using the spectral domain
approach, the moment method, and microwave circuit theory. In this analysis, we
present the figures of merit for three isolator structures, discuss the effect of adding
a dielectric layer underneath the ferrite layer, and determine the optimum value
and location of the resistive layer that is added to absorb the backward wave. We
compare our results with published experimental results. Chapter 4 presents the
derivation of the general Green’s function for arbitrarily biased ferrite structures.
Two special cases (the transversely biased Green’s function and the normally biased
Green’s function) dem onstrate the validity of the Green’s function. We also present
a versatile model using the general Green’s function to analyze various microwave
ferrite devices. As an example, we present the results for differential phase shifters
and magnetostatic surface wave transducers. A compaxison with the available results
in the literature shows the efficiency and the versatility of the proposed model. In
chapter 5, we study th e scattering from arbitrarily biased ferrite patch antennas. We
present a unique and versatile approach for evaluating the excitation vectors using
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17
th e generalized transmission m atrix th at we derive in chapter 4. Our results agree
well w ith many previously published results. In chapter 5, we also propose a novel
cross-patch antenna and describe its advantages over the rectangular patch antenna.
Finally, Chapter
6
summarizes the dissertation and gives recommendations for future
work.
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CH APTER 2
GREEN’S FUNCTION FORMULATION FOR A NORMALLY BIASED
FER RITE SLAB
2.1
Introduction
Many people use ferrite materials extensively in microwave devices such as phase
shifters, isolators, circulators, and antennas. The most popular technique for analyz­
ing planar structures is the spectral domain approach, which can be adopted for a
wide variety of geometries. However, the spectral Green’s function for a biased ferrite
substrate does not have a simple analytic form. Solving for a Green’s function as a
boundary condition problem is quite difficult for magnetic substrates and becomes
more difficult for multilayer structures. In addition, Green’s functions developed us­
ing conventional methods axe useful only for the particular configuration for which
they were derived. A more flexible approach is to find the transmission m atrix of the
medium and then use this m atrix to obtain the Green’s function numerically. The
transmission matrices of different layers are combined to form the Green’s function
for a multilayered structure. The transmission m atrix approach used in our paper was
developed by El-Sharawy [43]. This approach yields the Green’s function for general
multi-layered structures in a more efficient and versatile way than other approaches.
For instance, Krowne’s method [44], [45] is based on eigenvector matrices th at have
to be constructed to find the transmission m atrix. As a result, in contrast to our
work, closed form transmission matrices are not generally available. The approach
of Morgan et al [46] and Berman [47], like Krowne’s, requires numerically evaluated
matrices and is lim ited to the treatm ent of propagation through a single anisotropic
layer bounded either on both sides by isotropic media or an isotropic medium on
one side and a ground plane (PEC) on the other. The work of Tsalamengas [48]
is based on solving a boundary value problem. The eigenvectors are determined
numerically, then the Green’s function is formulated through a lengthy procedure.
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19
Thus, Tsalamengas’ approach is much less versatile than our approach. While the
Green’s function for a multilayer structure is derived in closed-form by Hsia and
Alexopoulos [49], their analysis is lim ited to one particular structure. In contrast,
our approach can be used for any multi-layer structure. Lee and Harackiewicz [50]
treated the case of in-plane biased ferrite substrate using the transmission m atrix
approach by El-Sharawy [43]. In [50], the authors refer to the work of Pozar [51]
and Yang, et al. [52]. In these papers, the Green’s function is valid only for a single
grounded ferrite layer. Also in [50], the capability to handle multilayer structures is
not dem onstrated.
It should also be noted th at the in-plane biased case is much easier to treat than
the normally biased ferrite, as may be readily deduced by examining the permeabil­
ity tensor for each case. To the best of our knowledge, the closed form transmission
m atrix for a normally biased ferrite layer has not been presented in the literature.
The derivation of this m atrix would probably be intractable without the use of a
com puter algebra system. While the scope of this chapter is limited to the normally
biased case, this case has numerous applications for antennas and microwave devices.
We hope th at this work will represent a useful reference for those attem pting to de­
velop efficient numerical models of general multilayer microwave and antenna ferrite
circuits.
2.2
Transmission Matrix Formulation
The transmission m atrix T for a m aterial layer is a 4 x 4 matrix w ritten as [43]
_ __
E2 - f
.
.
.
____
' =E
Ei
t
^ .
—™
=T z
Ei
=T =J
.
Y
T
.
=E =T = r - j
where T , Z , Y , T are 2 x 2 submatrices of T, Ei, E? are the tangential electric
field at the boundaries of the layer, J \ and J i are the tangential surface currents
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20
defined by J n = z x H n, where H n is the tangential magnetic field at the n th surface
of the layer, and
denotes the spatial Fourier transform defined as
E ( K X, K y) = f
I E ( x , y)e~iKzXe~iKyy dx dy
J —COJ —oo
(2.2)
The transmission m atrix for a ferrite layer is derived in Appendix A. Results for
a dielectric slab can be found easily by taking the limit of the ferrite formulation
when th e magnetization is set to zero and can be found in Appendix B [40].
The elements of the the transmission m atrix for a ferrite layer are given by
T\i
— ------ r [(n 2 + n i n 2 2(n2 —n i)
=
(—ni —n xn 2 +
T12 =
1
1
- n x) cosh(Rxd) +
+ n 2) cosh(i? 2^)]
f f 2 = —— ------ -[(n 2 + n in 2 +
i[n2 — ni)
1
+ nx) cosh(Rxc/) +
(—ni — riin2 — 1 —n 2) cosh(/? 2d)]
Tii
=
ZT = -3[(Tii + l ) ( D s - D e ) s i n h ( R i d ) +
(« 2
7 i4
=
+ l)(Ds — D 7) sinh(R2^)]
Z 12 = -^-[(ni + 1 )(Z?5 + Dq) sinh(Rxd) +
(n2 + 1)(Z?8 + D7) sinh(i? 2 ^)]
T2i
=
T2i = ——
r[(—n 2 + n xn 2 +
Z{n2 —ni)
1
—n x) cosh(Rxd) +
(n x —n xn 2 — 1 + n 2) cosh(i? 2d)]
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21
?22
ff2=
=
tt,
— r [ ( ~ n 2 + n in 2 - 1 + ni) cosh (Rid) +
Z{ri2 — ni)
(nx —riiTi 2 + 1 — n 2) cosh(i22<0]
T 23 =
Z 21 =
+ ! ) ( A - A>) sinh(/?1rf) +
(—n2 + l)(Ds —Dt) sinh(i?2^)]
T24 = Z 22 = ^ [(- n i +
1 )(^ 5
+ A>) sinh(Rid) +
(—n 2 + 1)(A* + D j) sinh(i?2</)]
fzx
= 2{ri2 - n i ) [(n2 ~ 1){Dl ~ ° 2) sinh^ 1<f) +
=
(ni - 1)(Z)3 —D 4) sinh.(R2d)]
fz2
=
? i 2
=
2
('n 2
_
n
)
[(n 2
+
! ) ( A
~
D )sm h(Rid)
2
+
(«i + 1)(Z)3 — Z?4)sinh(i?2^)]
^33
=
f 1Jl = ^[(Dl - D
2 )(D5 -
D 6)cosh(Rl d) +
(D4 —Dz)(D8 —D7) cosh.(R2d)]
T34 = f
^2
= -J [ ( D i - D 2 )(D5 + D6)cosh(R1d) +
(D 4 - D 3 )(D 8 + D 7) cosh(i?2cO]
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22
= 2 K - m ) [(n2 ~ 1)(£>1 + ° 2) siah(<R ^
?41 =
+
(—n i + 1)(£>4 + ^ 3 ) sinh(i? 2<i)]
?42
=
^22 =
2 { n 2
■—
M l) ^ n 2
—
(nj + 1)(^4 + ^ 3 ) sinh(i?2^)]
£3
=
Tj1 = ^ - [ ( D l + D2)(D5 - D 6)cosh(R1d) +
(D4 + D 3){D s - Dr) cosh (R2d)}
T4A = f ^ = ^[(Dl + D 2)(D5 + D6)cosh(Rl d) +
(D4 + D3)(D8 + Dr) cosh(i? 2 ^)]
where
Dl
=
n
=
5
‘•"2
we
K+KA r fi2(- i —
2
*
if2
+ " 2- f )
uj2e2R 2R i ( n 2 — n \ ) ^ Z
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23
u>2e2R.2Ri(n2 —ni)
u 262R 2Ri(n2 — ni)
-------------D2
u 2e2R 2 Ri{n 2 —n x)
9i +
9
i
+
R
2
9 2
2k2er (fi2 91
_
92
- K-K+(n +
2 (fi + k )
k 2)
k
+ 1)
I<1(H + K — 1)
2{n + k )
k ±2
= K 2 + K 2 — u)2hqc([j, =F k )
Ax
= u 2fio en (K l + K \ - u 2n 0e - ----- — )
fi
K+ =
K.
K s+ jK t
= K x - jK,
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24
and
2k$er([l2 — K2) — K - K + (fi + K + 1)
^ =
2(fi -f- /c)
2k$6r([i2 -
_
h
1
~
K2)
k
K +
2kler(fi2 -
I)
k
k)
- K ^K +(fi -
K2)
K
K)
K \ K t ( n — k — 1) ( f i +
4(/z — K)(fi + k )
where
2(n -
+
- K - K +(fi +
2 {fi +
2kQ6r(fi2 — K2) — K - K + ( f l — k + 1)
2
(h -
k
K +
I)
)
]
+ 1)
and fi are the elements of the permeability tensor and will be defined later.
Transmission matrices have the following useful properties
f(o) = 7
T { a + b)
=
T{a)T{b)
f {-d)
=
f
(2.3)
\d)
where / is the identity m atrix. The first property means th a t for zero thickness,
the field and current components axe equal satisfying the boundary conditions. VVe
can think of the second property as dividing a single ferrite slab into two sub-layers
with the same ferrite param eters but with different thickness. We can prove easily
th at the transmission m atrix for the slab is the multiplication of the transmission
matrices of the sub-layers. The third property simply means th at when we reverse
the reference axis from one side of the slab to the other side weget the
the transmission m atrix.
inverse of
In general, the three properties are useful forchecking the
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25
computer routines written to com pute the transmission m atrix. We have shown that
the transmission m atrix defined, by equations (2.3) satisfies these identities.
2.3
Discussion on the Difficulty of Deriving Normally Biased Transmission Matrix
To understand the reason of the difficulty of deriving the transmission m atrix for a
normally biased ferrite slab compared to the other transmission matrices of in-plane
biased ferrite slab, either in th e x or y direction, we can refer to the permeability
tensor for each case. For a ferrite magnetized in the 2 -direction, the permeability
tensor is given by
f* =
fl
- jk
jk
ft
0
0
0
0
1
(2.4)
For a ferrite magnetized in the y-direction, the permeability tensor is given by
fl
0
0
1
0
JK
-
jk
0
(2.5)
fl
For a ferrite magnetized in the x-direction, the permeability tensor is given by
'1 0
fi =
O '
0
fi
0
JK
-
jk
( 2 .6 )
fl
We can notice th at the top left 2 x 2 submatrix in each tensor is responsible for
defining the relation between the field components in x and y directions. Because
of the zeroes in the submatrix, th e field components in the x and y are not coupled.
Therfore, the wave equation in the case of in-plane biasing is a second order differ­
ential equation, while the wave equation in the case of normal bias is a fourth order
differential equation.
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26
2.4
Green’s Function Formulation
Using the transmission m atrix, Green’s functions can be form ulated in the spectral
domain for single and multi-layer structures. The Green’s function relates the tan­
gential electric field on one surface to the surface currents on the same or another
surface. This relation has the form,
E 3(kx, k y) = G (kx,ky)Js(kx, k y)
(2.7)
where
GXX Gly
G =
2.4.1
Gyx
( 2 .8 )
Gyy
Green’s Function for a Single Ferrite Substrate
E J
Air
Ground
Planes
1
I
E J
Fig. 2.1: Geometry of single layer isolator structure.
The Green’s function is formulated at the plane of the source, which is the ferrite-air
interface of Fig. 2.1. The electric surface current at this plane, J s, is split into two
-= +
equivalent currents J
— —
and J
as follows
13 = zx(T-T)
= T +7
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( 2.9)
where (-) denotes the tangential fields just below the plane of source (on the ferrite
side of the interface) and (+ ) denotes the tangential fields just above the plane of
source (on the air side of the interface).
From the transmission m atrix equation, we can write the following equation for
the upper air region,
-----1- 1
E
_
T
= Ta
" -=ru ’
E
J
.
' =E =T ' * -=ru "
E
T a z na
-=u
=T =J
J
Ya f ° .
( 2 . 10 )
In this equation, E and J represent the field and the current at a distance dj_ from
the air-ferrite interface. We force E = 0 by placing a ground plane at distance dj
from the source plane. This results in the following equation
E
=
=T=J~1_ +
Z af a J
=
gT
(2.11)
where the superscript (—1), here and throughout this work, means the m atrix inver­
sion.
Gais a semispace Green’s function, which is calculated by taking the limit of
the dielectric Green’s function when the distance ddgoes to infinity and the dielectric
constant goes to unity, see Appendix C.
Ga = lim Gd
dd—oo
Ed-1
(2.12)
and Gd is formed using the dielectric transmission m atrix derived by El-Sharawy
[40].
For the lower region, we have
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28
.T
’ =E =T ' — -|
E
E
-=i =
TJ r ! f
-=i
. J .
. J .
L *1 T , .
•
r
infs'
II
— _
E
.
i
(2.13)
-=i
where E and J represent the field and current at the other side of the ferrite slab.
We force E = 0 by placing a ground plane at distance dj from the source plane.
This results in the following equation
_ _ _____ =t =j ~1__
E
= Z jT f J
=
(2.14)
G SJ
where G j is th e ferrite region Green’s function, which can be calculated using the
derived ferrite transmission m atrix. From equations (2.9), (2.11) and (2.14) we obtain
J, =
where E s, E
and E
T +J
= (Go.
+GS W
(2.15)
are equal due to the continuity condition of the electric field
in the plane of source. The total Green’s function, which is the sum of the upper
and lower semispace components, is given by
Es =
=
2.4.2
=J =T~l = -1
(Tf Z f
+ G a )-V ,
=MS_
G J,
(2.16)
Green’s Function for a Ferrite-Dielectric Substrate
In this section, we present the Green’s function for two additional structures. The
first structure is similar to that depicted in Fig. 2.1 except for the addition of another
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29
dielectric layer under the ferrite layer as shown in Fig. 2.2. The only difference in
the derivation of the Green’s function of this structure from the previous one is the
transm ission m atrix of the lower region. We modify the transmission m atrix of the
lower region to include the dielectric region as follows
=
(2.17)
[?A [h
The Green’s function is found to be,
=
where Z
= -1
=J=T~l
G= (T Z
+ Ga T 1
(2.18)
=j
and T are the elements of [Tnew\.
The second structure, called a “drop-in element,” is a structure compatible with
Monolithic Microwave Integrated Circuits (MMIC). In this structure, a dielectric
substrate with relative perm ittivity of 9.8 is used. To form an EG isolator, we place
a piece of ferrite with a resistive thin film on top of the dielectric strip as shown
in Fig. 2.3. In this structure, the source is at the ferrite-dielectric interface. We
divide th e structure into two regions. The upper region includes the ferrite and air
layers. The lower region is the dielectric region. We can consider the total adm ittance
Green’s function as the sum of two parallel admittances, one for the upper region and
another for the lower region. The two semispace Green’s functions are formulated
using the transmission m atrix as previously described. The transmission m atrix for
the upper region is given by
[ ? J = (?/][?•]
(2.19)
where the air transmission m atrix is the dielectric transmission m atrix with er= l.
The final form of the Green’s function can be found as
=
= J= T
G = (f,Z „
~1
=J=T ~l
+ f dZd
)-■
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(2.20)
30
2.4.3
Green’s Function for a General Multi-layer Structure
For a general planar structure with any combination of ferrite and dielectric layers
and any arbitrary location of the source, the Green’s function can be easily formu­
lated following the procedures described in the preceding sections.
2.4.4
Circuit Model Interpretation of the Green’s Function for a Multi-layer Struc­
ture
Consider the structure shown in Fig. 2.1. In th at structure, the surface current J3
at th e plane of source can be split into two currents, J + and J - , which implies th at
the two layers, ferrite in the lower region and air in the upper region, are in parallel.
Thus, we can use the circuit model shown in Fig. 2.4 as an analogy to the structure.
From the definition, the Green’s function of a microstrip structure has the units of
an impedance. The total impedance of the analogous circuit is given by the following
equation,
Z to ta l
=
{ -y 6 f
+
( 2 .2 1 )
6a
where Z a is given in equation 2.11 and Z j is given in equation 2.14 for the air and
ferrite layer, respectively. Notice the sim ilarity between equation 2.21 and equa­
tion 2.16. For th e multilayer structure shown in Fig. 2.2, we can simply apply the
same procedure. By investigating equation 2.18, we conclude th at we added two
adm ittance term s for the upper and lower regions and then inverted to get the total
impedance of the structure, and hence the Green’s function. Generally speaking, the
adm ittance of th e region above the plane of the source is added to the adm ittance of
the region underneath the plane of the source, and the total summation is inverted
to get an expression for the total impedance.
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31
2.5
Results and Conclusion
We compared the Green’s function of the structure shown in Fig. 2.1 derived us­
ing the transm ission m atrix approach with the Green’s function derived by Pozar
[27] using th e boundary condition method. Excellent agreement is achieved as is
clear from Figures 2.5-2.8. No Green’s function is available in the literature for the
dielectric-ferrite structure shown in Fig. 2.2. However, we replaced the ferrite with
dielectric layer to form a multi-layer dielectric structure and compared its Green’s
function with th a t formulated by Aberle [53]. Excellent agreement is clear from
Figures 2.9-2.11. We did a similar comparison with Aberle’s results for the drop-in
element structure shown in Fig. 2.3. Again, excellent agreement is clear from Figures
2.12-2.14. In comparing our results with Pozar, we demonstrate the validity of the
ferrite transmission m atrix. In comparing our results with Aberle, we dem onstrate
our ability to obtain Green’s functions for multi-layer structures.
The case of normally magnetized ferrite slab is considered to be significantly more
difficult than the cases of transversely or longitudinally magnetized ferrite slab. One
can reach this conclusion simply by comparing the permeability tensors for each case
[54]. As a result of this complication, no closed form for the transmission m atrix
of a normally magnetized ferrite medium has been available to date. The Green’s
function is th e core of the spectral domain MoM and is used extensively to analyze
planar microwave structures as well as antenna problems. Normally biased ferrite
medium is widely used in many microwave devices such as isolators, circulators, and
phase shifters, and more recently in antenna applications [27], [28]. Our technique is
based on the exact derivation of the transmission m atrix, and no approximation has
been made in the Green’s function formulation. In addition, the Green’s function is
formulated in a way to increase the numerical efficiency in the MoM applications.
Our next step is to generalize the derivation of the normally biased ferrite slab
transmission m atrix to the arbitrary biased ferrite slab.
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t 1
E J
Fig. 2.2: Geometry of double layer isolator structure.
Ground
Planes
E 1J 1
Fig. 2.3: Geometry of drop-in element isolator structure.
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33
J,
TTTT
7777"
Fig. 2.4: Circuit representation, for a single ferrite layer structure.
— Thi* w o rk '
— Polar
3060
B 2550
O
CQ
c 2040
S
ao
6
1530
1020
-280
-140
0
140
280
Ky/k0
Fig. 2.5: Comparison of the computed Green’s function versus Pozar’s (Imag(Gxx)) (d =
7.62 x 10-4 m, ef = 12.0, 4k M s = 2100.0G, Hdc = 700.00e, A H = 0.0Oe, R 3 = 0.00,
W = 1.016 x 10-2m, / = 3.6GHz, K x = (110.0, -10.0)).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
400
—
This work
Pozar
350
(§ 2 5 0
200
150
-280
-140
0
Ky/kO
140
280
Fig. 2.6: Comparison of the computed Green’s function versus Pozar’s (Real(Gxx)). The
parameters are the same as in Fig. 2.5.
— This work
— Pozar
2480
-1240
-2480
-280
-140
0
Ky/kO
140
280
Fig. 2.7: Comparison of the computed Green’s function versus Pozar’s (Imag(Gxy)). The
parameters are the same as in Fig. 2.5.
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35
240
120
-120
-240
-280
-140
0
Ky/kO
140
280
Fig. 2.8: Comparison of the computed Green’s function versus Pozar’s (Real(Gxy)). The
parameters are the same as in Fig. 2.5.
This work
Aberle
o.o
-4.5
0
S'
as
c
'5b
a
c
-9.0
-13.5
-18.0
-480
-240
0
240
480
Ky/kO
Fig. 2.9: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxx)).
(.dd = 0.1 x 10"2 m, ed = 30.0, df = 0.762 x 10~3 m, ef = 12.0, K x = HO.Orad/m.
/ = 2.0 GHz).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
18000
Thii work.
Aberle /
14400
&
10800
S ' 7200
3600
-480
-240
0
Ky/kO
240
480
Fig. 2.10: Comparison of the computed Green’s function versus AberLe’s (Imag(Gyy)). The
parameters are the same as in Fig. 2.9.
Thw work
Aberle
I
S3
c
’&
S3
s
I—I
-40
-80
-480
-240
0
Ky/kO
240
480
Fig. 2.11: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxy)). The
parameters are the same as in Fig. 2.9.
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37
This work
Aberle
-480
-240
0
240
480
Ky/kO
Fig. 2.12: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxx)).
(dd = 0.1 x 10_2m, ed = 9.8, dj = 0.762 x 10~3m, e/ = 12.0, da = 0.1 x 10~2m,
€a = 1.0, K x = 110r a d / m , f = 2.0G\ff2).
12000
This work
— Aberle
9600
<3 7200
« 4800
2400
0
-480
-240
0
Ky/kO
240
480
Fig. 2.13: Comparison of the computed Green’s function versus Aberle’s (Imag(Gyy)). The
parameters are the same as in Fig. 2.12.
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38
This work
Aberle
Fig. 2.14: Comparison of the computed Green’s function versus Aberle’s (Imag(Gxy)). The
parameters are the same as in Fig. 2.12.
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39
CHAPTER 3
2-D FULL-WAVE ANALYSIS OF AN EDGE-GUIDED MODE ISOLATOR
3.1
Introduction
The isolator is one of the most widely used magnetic devices. The principle of
operation of these devices is based on the field displacement effect; i.e, the microwave
field configurations of the forward and backward propagating waves are different. If
an absorbing resistive film is placed at one edge of the conductor, then different
attenuations of these two waves occur and an isolator is realized. Fig. 3.1 shows
the geometry of an isolator with a resistive thin film. Experimental work on this
type of isolator has been widely reported in the literature [4] - [11]. Approximate
theoretical analyses have also been reported [3] - [6]. However, no full-wave analysis
of this structure has been reported to date.
C on d u ctor Strip
R esistiv e F ilm
F errite S u h strate
d
G round P lan e
Fig. 3.1: Edge-guided isolator with resistive film loading.
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40
3.2
Full Wave Formulation
We investigated three different isolator structures. The first structure comprises a
single normally magnetized ferrite substrate as shown in Fig. 3.2. In the second
structure, we added another dielectric layer underneath the ferrite layer as shown in
Fig. 3.3. The th ird structure, called “drop-in elem ent”, is an isolator structure com­
patible with Monolithic Microwave Integrated Circuits (MMIC). In this structure, a
dielectric substrate w ith relative perm ittivity equal to 9.8 is used. To form an EG
isolator, we place a piece of ferrite with a resistive thin film on top of the dielectric
as shown in Fig. 3.4.
«
«
E I
d4
Ground
Plane
dt
Fig. 3.2: Geometry of single layer structure.
mimmimmmmmmmmm
r
t
B
J
E
J
f -
Fertile
Fig. 3.3: Geometry of double layer structure.
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41
E J
Ground
Plane
E J
Fig. 3.4: Geometry of drop-in element structure.
3.2.1
Green’s Function Formulation
As outlined in Chapter 2, the Green’s function for the three isolator structures are
formulated using the transmission m atrix approach. The details of the derivation
and the Green’s function expressions are mentioned in the previous chapter and will
not be repeated here.
3.2.2
Basis Functions and Resistive Region Treatm ent
One very im portant step in any MoM numerical solution is the selection of basis
functions. In general, one chooses the set of basis functions that has the ability to
accurately represent the unknown physical quantity while minimizing the com puta­
tional effort required to employ it. Theoretically, there are many possible basis sets.
However, only a lim ited number are used in practice. These sets may be divided
into two general classes, the entire domain basis functions and the subdomain basis
functions. Using th e subdomain basis functions to represent the current distributions
in solving the problem on hand is necessary, in order to properly treat the resistive
region.
Due to the edge effect, we cannot use piecewise constant (pulse) basis functions
in the y direction. Instead, we can combine piecewise linear basis functions in the
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42
Table 3.1: Percentage of the impedance matrix filling
nx
3
5
7
9
ny
1
3
5
7
n = (nx + ny)
4
8
12
16
Pulse-Tri angle
75
43.75
30.5
23.4
Triangle-Triangle
93.75
73.43
54.8
43.35
x direction with a half triangle at the edges. An alternative is to use piecewise
linear basis functions in both x and y directions. Differentiating between the two
alternatives, we find th at the number of non-zero elements in the impedance m atrix
which utilizes triangles in x and pulses in y to be ( N x N —( N —2)2). For the case
of triangles in x and y, the number of elements is ( N x N — ( N — 4)2) — 1. The
following table gives us a feeling for the difference between the two choices in terms
of the percentage of the impedance m atrix we have to fill.
Although th e filling percentage of the impedance m atrix using pulse-triangle com­
bination is less than the filling percentage of triangle-triangle combination, the nu­
merical effort required to evaluate the latter is less than the numerical effort required
to evaluate th e former (the Fourier transform of pulse-triangle contains sine while
the the Fourier transform of triangle-triangle contains sine2). We selected piece-wise
linear basis functions to represent the current on the microstrip as shown in Fig. 3.5.
The current in the y-direction (transverse) is zero at the edges of the conductor and
the current in th e x-direction (longitudinal) is maximum at the edges to satisfy the
edge conditions.
Five basis functions in the y-direction and seven basis functions in x-direction are
found to be sufficient for convergence of the solution. In m atrix notation, the system
m atrix can be w ritten as
' Er' _ /
J y . " l
Z xx
Zyx
Zxy
Zyy .
+
Rxx
0
0
Ryy
\
‘ Jr '
J J ,.
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43
Resistive Layer
R*
2L
(»)
Conductor
2L
(b)
Fig. 3.5: Current basis functions for (a) Longitudinal current (b) Transverse current
We separate the resistive m atrix from the impedance matrix, and integrate the
resistive m atrix in the spatial domain in closed form. We also exploit the block
Toeplitz symm etry of the impedance m atrix. This means that only the first two
rows and the first two columns of each subm atrix must be calculated. The remaining
term s of each subm atrix can be filled by using the terms of the first two rows and
first two columns, thus reducing the time needed to calculate the impedance m atrix.
The elements of the resistive m atrix can be derived in a closed form in the spatial
domain. As an example,
r2d
2d
/ 1 . J 1J/ = JR
Jo
3
Ryy( 1,1)
= R,
Ryy( 1,2)
= Rs
#yy(l,3)
= 0.0
d
J x - J 2d l = R sJo
6
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44
R a j “ j 2 . J 2dl = R ^
R *,( l ,l )
=
Ra
/ Ji ■J xdl =
*■3
to
R * * ( 1,2)
=
Rs
f 2d
d
j J\ • J2dl = fl>6
to
R * x ( 1,3)
=
0.0
R x z i 2,2)
=
R,
o
2d
It
r2d
f t .
=
10s
Ryy{ 2,2)
-
r -t
where 2d is the width of each basis elements. The entire resistive m atrix for the
case of five basis functions in the x-direction and three basis functions in the ydirection is given in Appendix D. A similar m atrix can simply be derived for different
combinations of basis functions in the x and y directions and for different resistive
region widths.
3.2.3
Numerical Considerations in the Evaluation of the Spectral Domain Integra­
tion
The spectral domain integrals required in the evaluation of the impedance m atrix
elements are evaluated numerically using sixteen-point Gaussian quadrature. The
numerical integration must be carried out to a sufficiently large value of (3 for the
integral to converge. As the subdomain size becomes smaller, this upper lim it must
increase due to the spectral properties of the Fourier transform of the subdomain
basis functions. The numerical integration must be performed with enough intervals
to adequately model any rapid changes of the integrand. As the two subdomain basis
functions in the integrand become physically farther apart, the integrand becomes
more oscillatory requiring more intervals to assure accurate results. The integrand
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45
has poles corresponding to the propagation constants of magnetostatic modes. A
pole extraction m ethod was used [56], [57], [58]- [60] in the past, which in addition
to the numerical integration, requires the calculation of residues and the application
of the Cauchy principle at the singularity points. For a multilayer or anisotropic
structure, the location of the singularities usually introduces more complexity, if not
more difficulty, into the problems. A nice way to avoid this singularity problem is
to deform the integration path in a flexible m anner to account for a different range
of m agnetostatic modes. The deformed path should be applied to both propagation
directions, kx and ky. Fig. 3.6 shows the deformed integration path. Although the
deformed integration path can be arbitrarily selected, care must be taken in choosing
the proper p ath for numerical integration. The integrand increases exponentially as
the distance from the real axis increases; therefore, if the contour is too far from the
real axis, numerical problems can occur. Also, if the path is too close to the real
axis (where th e singularities are), the integrand is not a smooth function [61]. T he
following numerical expressions are used in the course of our simulation,
• Maximum integration limit
• Number of integration points
N { = 1 + [C2A/3{S max\
where
C i ,2 — numerical constants determined experimentally
lmin =
S mcLX =
minimum edge length in the discretization
maximum dimension of the structure
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46
A/3,- =
w idth of the if ft. region
Im (kx or Icy)
■\
Real(kxorky)
\
\
/
/
Fig. 3.6: Integration contour in the complex plane of either kx or k y.
3.2.4
Complex Root Searching
The propagation constant is a complex value th at makes the determ inant of the
impedance m atrix equal to zero. There is no input or excitation to the problem, i.e.
the governing equation is homogeneous. The solution is found when the determ inant
of the coefficient m atrix is zero or less than a given small value. This condition can
be achieved by an iterative root searching process. If there is no loss in the ferrite
layer or the strip, a simple Newton’s method or interval-halving should be sufficient;
however, a complex root searching algorithm should be employed to compute the
phase velocity and the attenuation for the device. It is found th at M uller’s threepoint method is an efficient root searching algorithm. It can be used for complex
roots, and it converges relatively quickly. Usually a convergence can be reached
within seven iterations.
3.3
Numerical Results and Conclusion
We compared our Green’s function for the single layer structure shown in Fig. 3.2 with
the Green’s function derived by Pozar [27] using the boundary condition method. An
excellent agreement between these two methods was achieved.
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47
Since the Green’s function of a multi-layer structure which includes a normally bi­
ased ferrite substrate is not available in the literature, we compared the limiting case
of the ferrite with th e Green’s function derived by Aberle for multi-layer dielectric
structures [53]. Again excellent agreement was achieved.
We constructed a 2-D MoM code for simulating an EG mode isolator with resistive
loading as shown in Fig. 3.1. First, we examined the limiting case of ferrite with
zero ferrite parameters, which is essentially the dielectric case, and compared our
results to the widely published results for dielectric microstrip. The computed current
distribution in the longitudinal direction over the conductor is shown in Fig. 3.7. As
expected, the current is symmetric. In addition, very good agreement with the
dielectric case is obtained for the propagation constant.
For the ferrite case, Fig. 3.8 shows the asymmetric longitudinal current distribu­
tion over the conductor for no surface resistance, and Fig. 3.9 shows the longitudinal
current distribution for a surface resistance equal to 100 Q over half of the strip.
The phase constants for forward and backward waves are shown in Fig. 3.10. The
computed insertion loss and isolation axe given in Fig. 3.11
A preliminary analysis of the three isolator structures indicates th at the best
electrical performance is given by the double-layer structure shown in Fig. 3.3. While
the performance of the triple-layer structure shown in Fig. 3.4 is not as good as the
other two structures, its advantage is that it can be compatible with MMIC. Fig. 3.12
compares the insertion loss of the three isolator structures and Fig. 3.13 compares
the isolation of the three structures.
For a single layer ferrite isolator, the field ellipticities at the upper and the lower
boundaries with air counteract each other. If the air at one of these boundaries
is replaced by a dielectric layer, as in Figures 3.3 and 3.4, one of the counteract­
ing ellipticities is replaced by a co-acting ellipticity which leads to increase in the
nonreciprocity and the isolation as well [40]-- [63].
The effect of the dielectric thickness of the ferrite-dielectric structure shown in
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48
Fig. 3.3 is studied, Figures 3.14 and 3.15 give the results. We note th at decreasing
the dielectric thickness will increase the isolation and decrease the insertion loss
which results in an improvement in the isolator performance. It is worthwhile to
note th a t the isolation and the insertion loss do not seem to depend significantly on
the dielectric constant of the dielectric layer as shown in Fig. 3.16.
T h e normally biased ferrite structure excites the magnetostatic forward volume
wave which has the frequency range / & < / < / #
where H 0, £irMs, and
7
[33, 34], where
h
=
7 H0
fa
=
7 y/H oiffo+ teM .)
axe the applied DC magnetic field, the magnetization of
the ferrite, and the gyro-magnetic ratio, respectively. It is shown from Figures 3.17
and 3.18 th at the peaks move forward when we increased the external DC bias, H a,
following the limits of the volume wave. The isolation begins after the cut-off lim it of
the volume wave, f a - The frequency range for negative fj,ef / is
u <
7
H0(H0 + At M 3) <
(H 0 + 4irM3) as shown in Fig. 3.19. The upper limit of the isolation is when
Heff = 0, which is the second peak in Fig. 3.11 and it occures at frequency
47rMs).
7
(H0 -j-
That peaks limit the operating range (the band width) of an isolator has
been pointed out and experimentally demonstrated in [1 1 ].
T he optim um resistance of the film is determined from Fig. 3.20. We note th at
both th e isolation and the insertion loss are equal when the resistance is zero, which
is physically true. When the resistance is around twenty times the characteristic
im pedance of the line, we find that the isolation returns to its value when th e resis­
tance is zero. A possible explanation is th at the high resistance acts like an open
circuit and the backward wave will pass through the resistance free region. T he inser­
tion loss is not a strong function of the resistance since the forward wave propagates
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49
m ainly in the resistance free region.
T he optim um width of the resistive film may be deduced from Figures 3.21 and
3.22. Increasing the width of the resistive film increases both insertion loss and
isolation. When we covere the entire conductor with a resistive film the isolation
and insertion loss become equal and the nonreciprocity vanishes.
Finally, we compared our results for the insertion loss and the isolation for the
2-D structure shown in Fig. 3.1 with the experimental results published in [11] for
the structure shown in Fig. 3.23. Fair agreement is clear from Figures
3.24 and
3.25. The difference between the numerical and the experimental structures is the
difference between Fig. 3.1 and Fig. 3.23. In Fig. 3.1 we assumed m atched ports
while in Fig. 3.23 matched ports are assumed only at the center frequency. Thus,
the best agreement with the experimental results is in the middle of the frequency
range. In addition, in our analysis, the external bias field is assumed to be exactly
perpendicular to the substrate which is not necessarily true in the practical case.
This problem suggests it is essential to have a flexible tool to be able to handle ar­
bitrary biased ferrite slab, the topic of the next chapter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
1.0
D ttkctricC j
0.96
uh
0.92
0.9
0.88
g 0.86
Number of the Basis Function on the Conductor
Fig. 3.7: Symmetric current distribution over dielectric microstrip.
1.0
—
Forward Wava, RasO.O Ohm*
Backward Wave* RasO.O Ohm*
.5 0.8
0.6
O 0.4
0.0
Number of the Basis Function on the Conductor
Fig. 3.8: Asymmetric current over ferrite microstrip in forward and backward directions
for R s = 0.0ft.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
1.0
Forward Wat*, Ra*100 Oho*
—— Backward Watt, Ras 100 Ohna
•S 0.8
0.6
O 0.4
0.0
Number of the Basis Function on the Conductor
Fig. 3.9: Asymmetric current over ferrite microstrip in forward and backward directions
for R s = 100.Oft.
14.00
Forward Wave ----Backward Wave ----12.00
10.00
o
*
o
O
I
8.00
6.00
4.00
2.00
0.00
2J0Q
3.00
4.00
5.00
6.00
7.00
Frequency GHz
9.00
11.00
Fig. 3.10: The phase constants of forward and backward waves (d = 7.62 x 10-4 m, €/ =
12.0, 4t M s = 1750.0G, H dc = 800.00e, A H = 80.0Oe, R s = lOO.Ofi, W = 1.016 x
10_2m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.00
Insertion Loss
Isolation
-5.00
-15.00
a■a
a©
So
- 20.00
-25.00
-35.00
-40.00
too
3.00
4.00
6.00
7.00
Frequency GHz
9.00
11.00
Fig. 3.11: Computed isolation and insertion loss (d = 7.62 x 10-4 m, e/ = 12.0, 4 irM3 =
1750.0G, H dc = 800.00e, A H = 80.00e, R , = 100.0ft, W = 1.016 x 10"2m).
0.00
•5.00
a
-
10.00
J
o
■g
8
5
-15.00
- 20.00
-25.00
200
3.00
4.00
6.00
7.00
Frequency GHz
9.00
11.00
Fig. 3.12: Comparison of the insertion loss for three isolator structures (4ttM3 = 1750.0G,
Hdc — 8OO.O0e, A H = 8O.O0e, R3 = 100.0ft, W = 1.016 x 10-2 m. For the single-layer:
dj = 7.62 x 10-4 m, c/ = 12.0. For the double-layer: dd = 2.62 x 10-4 m, ed = 3.0. For
the triple-layer: dd = 4.0 x 10_4m, ed = 8.9 da = 5.0 x 10~3m, ea = 1.0).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
0.00
Single-layer -----
-
10.00
-
20.00
Double-layer ----Triple-layer-----
w
-30.00
om
•50.00
•60.00
100
3.00
6.00
4.00
7.00
Frequency GHz
11.00
9.00
Fig. 3.13: Comparison of the isolation for three isolator structures. The same parameters
are as in Fig. 3.12.
i
1
r
Dd=0.381d-3 m
Dd=0.635d-3m
Dd=1.143d-3 m
oj-2.4
5.4
7.2
9.0
10.8
Frequency GHz
Fig. 3.14: The effect of the dielectric thickness on the insertion loss (dj = 7.62 x 10-4 m,
ef = 12.0, 4wMa = 2100.0G, Hdc = 700.00e, A H = 80.0Oe, R s = lOO.Ofi, W =
1.016 x 10-2 m, ed = 30.0).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
Dd=0 J8 1 d -3 m
Dd=0.635d-3 m
Dd=1.143d-3 m
5.4
7.2
Frequency GHz
Fig. 3.15: The effect of the dielectric thickness on the isolation (df = 7.62 x 10-4 m, ej =
12.0, 471-Jlf, = 2100.0G, H dc = 700.0Oe, A H = 80.00e, R s = 100.0(2, W = 1.016 x
10~2m, ed = 30.0).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i
0.0
i
i----------------1
i---------------- 1----------------1
i---------------- 1—
Insertion LossIsolation
-4.2
PQ
T3
-
-8.4
12.6
-16.8
e'
I
_____ !_____ i_____ I_____ I_____ I_____ I_____ !_____ I_____ L
9.6
19.2
28.8
38.4
48.0
Dielectric Constant
Fig. 3.16: The effect of the dielectric constant of the dielectric layer on the isolation and
the insertion loss, (df = 7.62 x 10~4m, ej = 12.0, 4 t M s = 2100.0G, H dc = 700.00e,
A H = 8O.O0e, R a = lOO.Ofi, W = 1.016 x 10"2m, D d = 0.381 x 10~3m, Freq.= 6.0
GHz).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
Ho=700 Oe
----- Ho=800 Oe
Ho=900 Oe
-19
-a
S-38
-76
Ha
3.6
5.4
7.2
9.0
10.8
Frequency GHz
Fig. 3.17: The effect of the external DC bias on the isolation, (d = 7.62 x 10_4m, ej = 12.0,
4ttM 3 = 2100.0G, A H = 80.00e, R s = 100.0D, W = 1.016 x 10“ 2m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
o.o
Ho=700 Oe
Ho=800 Oe
Ho=900 Oe
-3.6
n
CO
§ -7.2
C
o
"S -10.8
0)
09
ai
i—
-14.4
-18.0
3.6
5.4
7.2
9.0
10.8
Frequency GHz
Fig. 3.18: The effect of the external DC bias on the insertion loss, (d = 7.62 x 10-4 m,
ef = 12.0, 4ttM 3 = 2100.0G, A H = 80.0Oe, R a = 100.0ft, W = 1.016 x 10"2m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
8
^eff
4
0
-4
8
2
3
4
5
6
7
8
9
10
11
Frequency GHz
Fig. 3.19: The frequency behavior of Me//> (4 tMs = 2100.0G, Htc = 800.00e).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
Insertion Loss
Isolation
0.0
-5.4
-16.2
-
21.6
0
40
80
120
160
200
Ohms
Fig. 3.20: The effect of the film resistance on the insertion loss and the isolation, (d =
7.62 x 10-4m, e/ = 12.0, 4ttM3 = 1750.0G, Hdc = 800.0Ge, AH = 80.Oe, Freq.= 5.0
GHz, W = 1.016 x 10-2m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
25
Rs width =0.25 w
Rs width =0.50 w
Rs width =0.75 w
Rs width=1.00 w
0
2-25
_ J _________ I_________ I_________ 1_________ !_________ 1_________ 1_________ i_________ !_
3.6
5.4
7.2
9.0
10.8
Frequency GHz
Fig. 3.21: The effect of the resistive film width on the insertion loss, (d = 7.62 x 10-4 m,
€/ = 12.0, 4x114, = 2000.0G, Hdc = 700.00e, A H = 80.00e, R s = 100.0Q, W =
1.016 x 10-2m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
—
Rs width =0.25 w~
Rs width =0.50 w
Rs width =0.75 w
Rs width=1.00 w
-21
C -42
-84
3.6
5.4
7.2
9.0
10.8
Frequency GHz
Fig. 3.22: The effect of the resistive film width on the isolation, (d = 7.62 x 10-4m, cj —
12.0, 4xMa = 2000.0G, Hdc = 700.00e, AE = 80.00e, Ra = 100.0ft, W = 1.016 x
10-2 m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 3.23: 3-D edge-guided isolator with resistive film loading.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
63
Numerical
Experimental"
-14
-28
2-42
-56
3.6
7.2
Frequency GHz
5.4
9.0
10.8
Fig. 3.24: Comparison between the numerical and experimental insertion loss (d = 7.62 x
10-4 m, ef = 12.0, 4ttM5 = 2100.0G, H dc = 700.00e, A H = 80.00e, R 3 = lOO.Ofi,
W = 1.016 x 10"4m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
64
0.00
-5.00
-10.00
-15.00
§
§
-20.00
I
-25.00
a
-30.00
-35.00
-40.00
-45.00
240
3.00
4.00
540
6.00
7.00
Frequency GHz
9.00
U.00
Fig. 3.25: Comparison, between the numerical and experimental isolation (d = 7.62 x
10-4 m, e/ = 12.0, At M s = 2100.0G, Hdc = 700.0Oe, A H = 80.0Oe. R s = lOO.Ofi,
W = 1.016 x 10~4m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
CH APTER 4
ANALYSIS OF PHASE SHIFTERS AND TRANSDUCERS USING A GENERAL
GREEN’S FUNCTION
4.1
Introduction
A rigorous analysis of multilayer planar ferrite structures can best be carried out
using the spectral domain method of moments. To formulate the integral equation,
the Green’s function for the structure is needed. Over the past few years, several
techniques have been employed to formulate the Green’s function for a single ferrite
slab with a given bias direction [43], [27], [64] and [65]. In addition, Green’s func­
tions for multilayer structures comprising both ferrite and dielectric layers have been
investigated [66], [67], [68], [69] [70] and [71]. In this paper, we present a Green’s
function which overcomes the lim itations of previously presented Green’s functions
and allows us to treat structures which include any number of dielectric and ferrite
layers (possibly with different bias directions).
Previous spectral domain analyses for anisotropic materials have assumed that
the bias field is along one of the principal axes of the ferrite slab. However, in many
practical applications, the bias axis may not be perfectly aligned along the principle
axis due to misalignment of the bias magnet. This deviation in the bias angle may
alter th e electromagnetic characteristics of the structure. Using the derived Green’s
function, we can study the effect of bias angle deviation on circuit performance.
4.2
Transmission M atrix for an Arbitrarily-Biased Ferrite Slab
In this section, the transmission m atrix is formulated in the spectral domain for a
m agnetic substrate of thickness d as shown in Fig. 4.1. The transmission m atrix T
is a 4 x 4 m atrix written as [40]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where
T
e
,
Z
t
,
Y
t
,
T
are 2 x 2 submatrices of T,
j
denotes the spatial Fourier
transform defined as
E ( kx, k y) =
r
f°° E ( x , y ) e - jk**e-jkyydkxdky
(4.2)
J — OO «/—OO
E i , E 2 axe the tangential electric fields at the boundaries of the layers and J 1 and
J 2 axe the tangential surface currents defined by «/,- = z x H n where H n is the
tangential magnetic field at th e n th surface of the layer.
z
A
z= 0
h
e 2
z '
'
y
d
Fig. 4.1: Geometry of single layer structure.
To find the transmission m atrix for a ferrite slab, we start from Maxwell’s equa­
tions
—j u p H
=
V x E
V
- ( [ J -
■H ) =
0
(4.3)
jueE
=
V x. H
eV • E
=
0
where ^ is the permeability tensor of the ferrite. For an arbitrary magnetized ferrite
slab, the permeability tensor is given by [72]
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67
Mil
M=
M21
_ M31
Ml2 Ml3
M22 M23
M32 M33
where
Mu =
M+ (Mo —y)smOcas <f>
Ml2 =
(Mo
M13 =
Mo —M
— - — sin2 0 cos (p —jKsmd sin <p
/X21 =
Mo —M • 2
— - — sin 0 sin 2 <?i —jk cos 0
M22 =
M+ (Mo —y)sm 2 9sm2<p
M23 =
Mo —M
— „— sin20 sin <p + j k sin 6 cos <p
/131
=
Mo —M
— - — sin20 cos <p + jk sin 9 sin <p
/132
=
Mo —M
— - — sin 20 sin <f>— jk sin 0 cos <p
M33 =
^^sin 2gsin 2 ^ -(- j/c cos 0
Mo ~ (Mo ~ M)sin2^
/ 1 . u o w7n \
M= ( l + -T2----- 7 )
U/o - w ‘
K=
-2
w0
- U20
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.4)
68
7
Um =
U= U)0 + —, U0 = ~fH0
(jJq is the precession frequency, Ho is a z-directed impressed DC magnetic field,
the gyromagnetic ratio, T =
7
is
is the relaxation tim e, and AH is the 3-dB line
width.
From Maxwell’s equations, we can write
Vx(/i
-V x E) — k%erE = 0
(4 .5 )
where k0 = Uy/eo. M anipulation of (4.5) yields three scalar equations
{ j K y d”
( J^y
d
f*5 q z 2
^ ^ E y ~Qz
d2
{J.gKyKx 4- fl4 Q^ 2
d*
d
~
) Ey 4 -
d
d ~
(- f i 7 K 2 4- fi&KyKx - jii 4 K y — + jfisK x ^ j ) E z= 0
d
( - j f i SK x- ^ - fl9 K XKy -
d
^m i K x ~dz +
d2
~
d2
d
~
- j ^ K y - ^ ) Ex +
d
+J/i3KxdZ ~
Ey +
d
d
(,(i7 K xK y - nsl<l + JUiK y-^ - j/jl2 K x — ) E z= 0
r\
O
^m s K x d z + H K x K y ~
~
Ex +
— f*6 K 2 + JUlKy— + flZK x Ky) Ey +
( - f i 4 K y K x + fi 5 I<l + yLXK j2 - jjL2 K xKy - kleT) E z= 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
Eliminating E z from the three equations and rearranging we obtain
d 2E x
dEx
~
d 2E v
dEv
~
w i - q^ T + W2~&T + Wz x +W4~ d z ^ + W5~ d f + We y=
d 2 Ex
dEx
W7~q~T + Wa~ dT +
~
109
d2E v
dEv
+Wl01 h 2’ + Wll~ d T + Wl2
Wi
= fi5Cl + K 2 (fi 2 fi4 - MiMs)
w 2
= jK y in e P i - M7 M2 ~ M4M3 + MsMi) +
, v
^4’6^
0
^4'7^
E y~ 0
j K xK l { - n e f i 2 + M7Hs - MsM4 + HsHz) - jCiKy{fi6 + Ms)
wz
=
K l K : ( f j 9fi5 ~ PaVe) + K xK 2(fi8fi3 - ( j , 9fi4 - [ i 9fi 2 - H7iie)
(M9Mi — M7 M3 ) ~ C i K 2 (fi9 + mi) + C \ K xK y{y. 4 + M2 ) — P2 C 1 K
w4
= - K xK y(fi4 (i2 - msMi) - M4 C 1
w5
= ~ j K l K y ( f l 7 f l s + M5 M3 ~ M8M4 — M6M2)
2
—j K x K 2 { —f i 7f i 2 — M4M3 + M sM i + M6M1) + J f r C i K y + j f i e C i K x
w6
= - K 2 K l ( / i s f i z + M7 Me -
M9M2 -
M9M4) -
K x K y ( f i 9fx 1 -
y. 7 li z )
—K ^ K y ( —fisfi 6 + M9 M5 ) + fi9 C \ K XKy
IV7
= M 2C 1 +
K XK y { f l 2 fi 4 -
M1M5)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
+ C\
70
ws = ]K xK * ( - i i 7fi2 +
MsMi + MiMe - M 3 M4 ) +
jK lK y { n 7fi5 - H 2 M6
~ M s M 4 + M sM s)
= —W6
Wg
ww
=
- K l ( n 2 H4
^11
=
- j K l K y(flIfle - fi7f*2 - f*4ll3 + HaHl) -
-
M1 M5 ) -
P\C\
j K x i - f * 8M4 - M2 M6 + M7M5 + M5M3 ) + 3CiKx {fiz + fl7)
W12
= ~ K y K x ( —/J.7fl3 + M9Ml) —K x K y(—fJ.gfl4— flgfi2 + M8M3 + M7M6) +
-^r(M8M6 —^ 9 /^5 ) + ClKl(fig + ^ 5 ) —C \ K xK y(n4 + fl2) + fl\CiKy — C\
Cl
=
UJ2Cgtr
Ml
=
(M22M33 — M 2 3 M 3 2 ) / A / 1
M2
=
M3
=
(M12M23 — M l 3 M 2 2 ) / A / l
M4
=
( ~ M21M33 + M 2 3 M 3 l ) / A / l
Ms
=
(M 11 M33 — M i 3 M 3 i ) / A m
M6
=
( — M llM 23 + M i 3 M 2 i ) / A m
M7
=
(M21M32 — M 2 2 M 3 l ) / A M
( —M12M33 + M l 3 M 3 2 ) / A / Z
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-jfisC iK
71
A
fl8
= (—/*11^32 + Ml2/*3l)/A^
P9
= (l*11P22 — /*12/*2l)/AM
f l
=
f l l l f l 2 2 [ i 3 3
~
f & l l 1 * 2 3 ^ 3 2
~
f c l l t U
f t S S
+
f * 2 l f i l 3 f 1 3 2
+
^ 3 1 ^ 1 2 ^ 2 3
~
H 3 1 ^ 1 3 ^ 2 2
Rewrite equations (4.6) and (4.7) in the following form
w i D 2 E x + \V2 D E x + w 3 Ex + wliD 2 E y + W 5 D E y + w 6Ey
=
0
(4.8)
W7 D 2 E x + w gDEx + wqEx + w\qD 2 Ey + w \ \ D E y + w\ 2 E y =
0
(4-9)
or
(w \D 2 + W2 D + 11)3 ) E x + (w\D 2 -f-11)5 D + wo)Ey =
( 1V7 D 2 + w$D + iv^)Ex + (w\qD2 + W\\D + w\ 2 )Ey
0
(4.10)
— 0
(4.11)
where D 2 = Jp- and D = J j. Equating the operational determ inant of the coupled
equations (4.11) and (4.11) to zero, we obtain the following fourth order equation
C 01D 4 + C02 D 3 + C03 D 2 + C04 D + Cos = 0
where
C 01 =
wiW\q —UJ7UJ4
C02
Wi Wu + W2 Wiq — WjWs — w8 w 4
=
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.12)
72
C q3
=
W1W12 + W 2W n — W 7W 6 — W sW s — w s w 4
C04
=
W
Co5
=
W 3 W i 2 — W qW q
2
W
12 +
W3Wn
— W 8W6 — W 9W 5
By investigating equation (4.12), we found th at solving for the four roots analytically
is quite difficult due to the fact th at this equation contains the full coefficients. At
this step and knowing the five coefficients, we find the four roots numerically. As a
check, the four roots have to satisfy the following relations
R iR 2R3R 4 =
R \R 2R3 + R1R1R4 + R \R 3Rj4 -f- R 2R 3R 4 =
R i R 2 + R \R 3 + R1R4 + R 2R z + R 2R4 + R3R4 =
i?i + R\ + R\ + R\
=
C04
Cqi
Cp3
Cqi
Cq2
The solution of the coupled partial differential equations (4.6) and (4.7) is given by
Ex = Crie x p (i? i 2 ) + C2 exp(R 2 z) + C3 exp(R 3 z) + C ^ e x p ^ z )
(4.13)
Ey = Di exp(i?iz) + D2 exp(iZ2-j) + D3 exp(R 3 z ) + D4 exp(R 4 z)
(4.14)
At this point we appear to have eight arbitrary constants in the solutions (4.13) and
(4.14). But it follows from [73] that the general solution of a system of two second
order equations involves only four arbitrary constants. That is also clear from the
order of the operational determ inant, which is four. There must be some relation
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
73
between the eight constants. We can discover it by substituting the solutions in (4.13)
and (4.14) into either of the original equations in (4.6) and (4.7). On substituting in
the first equation, we get
Dx =
N XCX
X?2 =
N 2 C2
D3
=
N3C3
D4
— N4C4
where
N
_
jy
_
2
_
WXR% + W2 R 2 + W3
W4 RI + W3 R 2 + w6
w xR l + w2R 3 + w3
W4 RI + W5 R 3 + w6
3
_
4
wxR\ + w2R x + w3
w 4R x + w5R x + w6
wxR\ + W2 R 4 + w3
W4R I + W5R4 + we
Again, the solution in (4.13) and (4.14) can be written as
Ex = Cx exp(Rxz) + C2 exp(R 2 z) + C3 exp(i?3z) + C4 exp(R^z) (4.15)
E y = N XCXexp{Rxz) + N 2 C 2 exp(R 2 z) + N 3 C3 exp(R 3 z) + N 4 C4 exp(R 4 z) (4.16)
From Maxwell’s equations, we can write the relationship between the components of
the magnetic field and the electric field in matrix form as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
74
Hx
Hy
Hz
uj2eAh
h u h\2
h 21 h.22 h 23
^31 ^32 ^33
(4.17)
u e ( K yE x — K x E y)
where
^11
— f^33-HxK y + UJ2t(fl33lil2 — ^ 13 ^ 3 2 )
hi 2 — —fl3 3 K^ + U2e(fl23fl22 ~ ^23/^32)
/ll3
— —K x (lliz Kx + fl 23Ky) + u 2e(fli3fj,22 — ^ 23/^32)
^21
— (^33-Hy — W2e([lufl33 ~ ^31^13)
^22 — P33KXK y + U)2e(ll33fi21 ~ H23f*3l)
h 23 — —K y(fli 3 K x + H23Ky) ~ W2e(/il3/*21 —/*ll/*23)
^31 — —Ky(fi32Ky + fi3i K x ) + u)2e(finfi32 —^ 31^ 12)
^32 — E x (fi 32K y + fi 3l K x ) — U)2e(fJ,2X^32 ~ (*31^22)
^33 — ^i(/^12-^y + /^ll^Cr) + ^y(/^21^Cr + /i22-^y) + w2^(^21^12 ~/^ll/^22)
Ah
— K x Ky(fl33fli2 — ^13^32 + /*2l/*33 ~ ^3X^23)
+^y(/*33/*22 ~ ^23^32) + K 2(llfl33 — fi3Xf*X3)
+o;2e(^u (//23M32 ~ ^ 33/^22) + ^ 12(^ 21/^33 — V3 1 H2 3 ) + ^ 13(^ 22/^31 —H21 H3 2 ))
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
Appling (4.17) to the solutions (4.15) and (4.16) and puting z — 0, we can evaluate
--- 1
1
the constants C \, C 2 , C 3 and C4 in m atrix form as follows
1
1 ' \ c x]
1
1
Nx n 2 iV3 N 4
c2
Dx d 2 Dz D 4
C3
. D s De d 7 Ds . lc<.
'
Eyi
HX1
.
(4.18)
The inverse of equation (4.18) can be written as
E Xl
c 2
E y i
II
o '
f—
<
1
1 C X 1
c 3
. C
4
(4.19)
K
.
.
K
.
where
r Mxx Mx 2
A/ 14 ‘
M 2l M 22 M 23 A/24
M 31 M 32 A/33 A/34
M 42 A/43
M
1
A
m
x
3
a x
--- 1
£
. M
and
D\
=
<jJ€{jh.nRi + j h i z R i N i + h i3 ( K y — N i K x))
D2
— ide(jhuR2 +
D3
=
u e ( ]h u R 3 + J^i27?3^3 + h 1 3 ( K y — N 3 K X))
D4
=
w e(j^ n i ?4 + jh w R i N ^ + hi3(Ky — N^ K x))
D5
=
w€.(]h2iR\ + ]h-22 R \N \ + h,23(Ky —N \ K X))
Dq =
a;e(j/i2ii?2 + ^ ^ 22 ^ 2 ^ 2 + ^23(7vy — iV2/ f x))
+ hi3(Ky — N 2 K 2;))
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76
D?
=
u e { j h 2\ R z + ]h 2i R z N z + h,2z ( K y — N z K x ))
Ds
=
ue(]fi2iR4 + jh22R-AN^ + h.2a{Ky — N ^ x))
and
— N2^D zD s — D4D7) + N z^DqD^ — 7 ) 2 75s ) + -/V4 ( 7 ) 2 D7 —
Mu
M 12 =
7)3( 75$ — 7)8) + 7) 2 ( 7 )8 —D 7) + 7) 4(757 — 75$)
•A7i 3 =
^ 2 ( 7 ) 7 — Ds) + Nz{Ds — Ds) + N ^ D s — D7)
Mi 4
—( ^ 2 ( 7)3 —7) 4 ) + ^ 3 ( 7)4 —7) 2 ) + -^4 (7)2 — 7) 3 ))
=
M 21
M 22
D qD^)
= - ( ^ ( 7 ) 3 7 ) 8 - 7 ) 47 ) 7 ) + iV3( 7 ) 5 7 ) 4 - 7 ) 17)8) + ^ 4 ( 7 ) 47 ) 7 - 7 )5 7 ) 3 ))
=
—(7)i(7 ) 8 —7 ) 7 ) + 7)3(755 —7)8) -F 7) 4(757 — 7) 5 ))
M 23 =
—( N i ( D 7 —7)8) + TV3 (7)s —7 ) 5 ) + N^(Ds — D 7))
M 24. = Ni(Ds — 7 )4 ) + ^ 3 ( 7)4 —7)i) + ^ 4 ( 7)4 —7) 3 )
M 31
=
TVX( 7 ) 2 7)8 — D 4De) + ^ { D s D ^ — DiD s) + N ^ (D i Ds — D5D2)
M 32
=
D i ( D g —75$) + 7 ) 2 (7 5 5 —7)8) + 7 )4 ( 7 5 $ — 7 ) 5 )
-AT33
=
Vl(7)$
■M34 =
7)8) + TV2 (7)8 —7 ) 5 ) + ^ 4 ( 7 ) 5 —75$)
—(Vx(7)2 —7) 4 ) + TV2 (7)4 —7)i) + iV4(Di — 7) 2 ))
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77
M 44 =
—(Ni(D2D7 — D3Dq) + N2{D$D3 — D \D t) -+• N^^DiDs — D 5 D 2 ))
M 42 =
Di(Dq — D 7 ) + ^
M 43 =
N i ( D 7 — D&) + N 2 {Ds — D 7 ) + Ns(Ds — D 5 )
M 44 =
—{N \( D 3 — D 2 ) + - ^ (D i — D3) + Nz(D 2 — Di))
Ajvr =
N \ ( D 2(Ds — D 7 ) + D3(Ds — Ds) + D 4 (D 7 — 7^6))
2 (^ 7
—D 5 ) -f- D3{Ds — D&)
+iV2(Di(£>7 — 7}g) + D z( D s — D$) + D 4(D$ — £?7))
+jV3(£ M £>8 — -^6) + D 3{Dg — Ds) + D 4(Dq — £?s))
+7V4(£)i(£)8 — £^7) + £>2(£>r — £^5) + D 3(D$ — Ds))
Following the same procedure for z = —d, we get an expression relating the field
components at the second surface of the slab to the arbitrary constants th a t appear
in the solution.
'
'
Eyi
HX2
.H y i
.
rcii
c2
= M (-d)
c3
.CA.
(4.20)
where
exp (—Rid)
Ni e x p ( - R i d )
M {-d) =
D\ exp( Rid)
. £ )5 exp(—Rid)
exp(—R 2d)
exp (—R 3d)
exp(R4d)
N 2e x p ( - R 2d) N 3 exp(—R 3d) N 4exp(—R 4d)
D2 exp(—R 2d) D3 e x p ( - R 3d) D4e x p ( - R 4d)
£>6 e x p ( - R 2d) D 7e x p ( - R 3d) D4 exp(—R 4d)
Convert the m agnetic fields to the electric currents at both surfaces of the slab using
the following relations
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78
’ E Xl '
E X1
E yi
H Xl
. K
Eyi
= [5]
. 4
.
(4.21)
Jx l
.
E X2
Ey 2
H X2
.Hy>
Eyi
= [5]
.
.
4
4
(4.22)
.
where
i ■■
o
[5] =
o
0
0 1
0 0
0
' l
0
0 0
0 1
1 0
(4.23)
Using equations (4.19), (4.20), (4.21) and (4.22) we can write the final expression
th at relate the electric field and current components at both sides of the ferrite slab
is given by
E:
E 3/1
I
E *2
:
i
I X2
I 3/2
= [5 ]- 1 [M (-d)][M (0)]- 1 [5]
J
L
4
(4.24)
j
From equation (4.24) we conclude the transmission m atrix for arbitrary biased ferrite
slab
f ( d ) = [ 5 ] - 1[M (-d)][M (0)]" 1 [5]
where the elements of the transmission m atrix are given by
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(4.25)
79
Tn = T ^
=
M u e x p ( -R id ) + M 2i exp( - R 2d) +
A/31 exp(—R3d) + iV/41 exp(—iLjc?)
/\2
= f f 2 = Afi2 exp( —
+ A/22 exp(—R 2d) +
A/32 exp(—R3d) + A/42 exp(—/?4 (f)
r 13 = Z ^
=
—M u exp (—Rid) —A/ 24 exp(—/?2</) —
A/34 exp(—i?3 </) —A/44 exp(—/?4 cZ)
/\4
=
=
A/13 exp(—/?!</) + A/ 23 exp(—R 2 d) +
A/33 exp(—/?3 rf) + A/43 exp(—i?4 </)
/2 1
= T2l =
N i M u exp {—Rid) + JV2A/21 exp(—R 2d) +
NzMzi exp(—R 3d) + A 4 M 41 exp(—R Ad)
T 22 = T22 =
iViAfi2 exp(—i?ic/) + A 2A/22 exp(—/?2</) +
A/3 A/ 32 exp(—i?3 c/) + N 4 M 42 exp(—R 4 d)
T23 = Z 21 =
—N i M u exp(—Rid) — N 2M 24 exp(—R 2 d) —
N 3 M 34 exp(—R 3 d) — N 4 M 44 exp(—R 4 d)
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80
f 24 = Z22 = N i M i 3 e x p ( - R i d ) — N 2 M 23 exp(—R 2 d) —
N 3 A/33 exp( - R 3 d) —N 4 M 43 e x p ( - R 4 d)
T31 =
=
—D 5 M u exp {—Rid) —DqM2i exp (—R 2 d) —
D 7 M 31 exp(—R 3 d) — D 8 M 4i exp(—R 4 d)
Z32 = ?X2 =
—D 5 M l 2 exp(—R i d ) —D 6 M 2 2 e x p ( - R 2 d) -
D 7 M 3 2 exp(—R 3 d) — DSM 42 exp(—R 4 d)
T33 = T/x =
D 5 M 14 exp(—/?xcf) + D 6 M 24 exp(—R 2 d) +
D 7 M 3 4 exp(—R 3 d) + D 8 M 4 4 exp(—R 4 d)
Tm = f
(2
— - D 5 M 13 e x p ( - R i d ) - D 6 M 23 e x p ( - R 2 d) —
D rM^e xjpi—Rsd) — D 8 M 43 exp(—R 4 d)
T41 = F2T =
A M u exp(—i?x<f) + D 2 M 2i e x p ( ~ R 2 d) +
D 3 M 3i exp(—R 3 d) + D 4 M 41 exp(—R 4 d)
r 42 =
=
D \ M x2 e x p ( - R i d ) + D 2 M 22 exp(—R 2d) +
D 3 M 32 exp(—R 3 d) + D 4 M 42 exp(—i?4t/)
T43 = f/x
=
—D 5Mx4exp(-i?x<f) — D 6M24ex p (-/? 2 ^ )—
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81
6 xp(—R^d)
—D 8 M 44 exp(—
T 44 = T22 — D\M \z exp (~ R \ d) + Di M 23 exp (—
+
D 3 M 33 exp(—i ?3 J) + D 4 M 4 3 exp(—iltd)
Transmission matrices have the following properties
f(o) = 7
T ( a + b) =
f ( a ) f ( 6)
(4.26)
f (-d) = f (d)
where I is the identity m atrix. These properties make transmission m atrices ex­
tremely convenient for deriving Green’s functions for multilayered geometries.
4.3
Green’s Function Formulation
Using the transmission m atrix, Green’s functions can be formulated in the spectral
domain for single and multi-layer structures. The Green’s function relates the tan­
gential electric field on one surface to the surface currents on the same or another
surface. This relation has the form,
E 3( k x , k y )
= G (kx , k y ) J s(kr ,ky)
(4.27)
where
G =
Gx
Gyx
Gyy
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(4.2S)
82
Details of the Green’s function formulation is given in chapter 2. The Green’s func­
tion for the grounded ferrite slab as shown in Fig. 2.1 is given by
=
m s
G
= T
=J
= t-i
= (Tf Zf
= -1
+Ga T 1
(4.29)
=J
where Z j and T f are the elements of the derived ferrite transmission m atrix and
Ga is a semispace Green’s function, which is calculated by taking the lim it of the
dielectric Green’s function when the distance dj, goes to infinity and the dielectric
constant goes to unity. The Green’s function for the multilayer structure shown in
Fig. 2.3 is given by
=
=J =T~l
- r - 1
G = (T UZU + T i Z i
)-'
(4.30)
Following the procedures outlined in [65], the Green’s function for any general
multilayer structure can be easily constructed.
4.4
Varying the Magnetization Angle
Most researchers assume that the field bias is along one of the cartesian principle axes
of the ferrite sample. These bias directions yield permeability tensors th at contain
four zero elements out of nine, which simplifies the analysis. However, in practical
devices, the DC magnet used to bias the ferrite slab may not be exactly aligned
along the required axes. This situation is depicted in Fig. 4.2. This variation in the
magnetization angle can change the electrical characteristics of the ferrite device. In
addition to treating unintentional deviation of the bias angle from the principle axis
of the ferrite sample, our approach allows us to discover interesting behaviors th at
may occur when th e ferrite sample is magnetized along oblique angles.
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83
1
Ho
(a)
(b)
Fig. 4.2: Magnetization angle (a) an approximate picture (b) a real picture.
4.5
Planar Phase Shifters
4.5.1
Introduction
Ferrimagnetic materials have found wide application in a class of microwave com­
ponents called phase shifters. Phase shifters are two-port devices that perm it the
passage of a guided wave with very little attenuation but with a variable phase delay
controlled by the external bias field of the ferrite. Phase shifters generally operate
in two states, denoted collectively by M , and M = ± M r, where M r is the rem nant
m agnetization. The difference between the phase shifts which occur in these two
states is called the differential phase shift. To reduce power requirements associated
w ith m aintaining an external magnetic field, ferrite devices are commonly operated
in a rem nant state. When the ferrite is initially demagnetized and the bias field is
off, both M and H 0 are zero. An external magnetic field in the desired direction is
applied long enough to magnetize the ferrite to near-saturation. When this external
field is removed, the ferrite magnetization drops to a remnant value, and the ferrite
remains magnetized in that direction until an external field is reapplied. Fig. 4.3
shows a typical hysteresis curve. This figure illustrates the variation in m agnetiza­
tion, Af, with bias field, H0.
The two most important figures of m erit used to describe ferrite phase shifters
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84
axe related to the total differential phase shift, A <£, and the bandwidth, A /. The
first figure of m erit, F , can be defined in terms of the total differential shift per unit
length by [74]
F = A <j>(deg/cm).
(4.31)
The bandw idth of a phase shifter, A /, is defined to be the range of operating fre­
quencies over which the insertion loss is less than some specified value (typically 3
dB). T he bandwidth, when expressed as a percentage, is found from A/ and the
center frequency, f a, using
B W = ^ r - x 100
(4.32)
Jo
4.5.2
A.
Slot Line Phase Shifters
Introduction
The direction of magnetization of the ferrite substrate for a phase shifter is given by
9 = 90° and <f>= 90°. The case analyzed here is shown in Fig. 4.4. This structure
features a single-layer transversely magnetized ferrite slot line.
B.
Full-Wave Analysis of the Slot Line Phase Shifter
The propagation constant of an infinitely long slot line structure is found using a
full-wave spectral domain analysis based on Galerkin’s method of moments [75] [74]. This m ethod uses the fact th at the tangential electric field on the conducting
plane and the surface current in the slot are zero. Following the approach of [40],
the unknown electric field components in the slot, E x(y) and E y(y), are expanded in
the slot using entire domain basis functions as
Ny
Fy(y) =
(4.33)
71=0
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85
Ms
Mr
Ho
-Mr
-Ms
Fig. 4.3: A hysteresis curve for a ferrite sample.
a
W
,
♦
■
. s\
ss-’\sss
Ferrite
«.%
df
PF.C
Fig. 4.4: Cross-section of basic slot line single-layer ferrite planar phase shifter.
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86
M y) = £
‘W -
n=0
The expansion functions fyn and f m are given by
/„ = (-
(4.34)
/= . = ( - 1 ) “£ « § ? ) y ^ l - ( ^ ) 2
(4.35)
where W is the width of th e slot and the functions, Tn and Un, are Chebychev
polynomials of the first and second kinds, respectively. We select n to be odd or
even according to
the physical behavior of the electric fields in the slot.Thus, only
the even values of
n
are used for f yn and only the odd valuesare used for f m .
Ey(y)
o
-W/2
W/2
Fig. 4.5: n = 0 Chebychev basis function of the electric field in the slot in transverse
directions.
Applying Galerkin’s approach yields an adm ittance m atrix of the form
Y=
J- x x
x xy
Y
*yx
Y
x yy
with elements given by
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(4.36)
87
Ex(y)
0
-W/2
W/2
Fig. 4.6: n = 1 Chebychev basis function of the electric field in the slot in longitudinal
directions.
Y ™= 1.
£
^ k ti) G „ ( - l 3 , k ^ ) F ^ ( k yi)
(4.37)
{=—oo,even
The indices (p,q) refer to the subm atrix of Y , and the indices (l,m) refer to the
testing and expansion functions, respectively [74]. The functions Fpi(ky) are the
Fourier transforms of the functions in (4.34) and (4.35), and they are available in
closed form as
Fyn
= 3n^ J n ( ^ Y ~ )
(4.38)
- ^ ( »
(4.39,
+
For enclosed structures like the one on hand, ky takes the discrete values ky{ =
Y, where i is an even integer [43] and a is the spacing between the conducting
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88
sidewalls. The propagation constant /3 is the value of kx th at makes the determ inant
of th e adm ittance m atrix zero. Reversing the magnetic bias, we get the propagation
constant in th e reverse direction. The difference between the propagation constants
of the forwaxd and reverse waves is used to find the nonreciprocal phase shift per
unit length, given by
A <£ = £ / - 0r
(4.40)
We can control the total differential phase shift by adjusting the length of the phase
shifter.
4.5.3
A.
Microstrip Phase Shifters
Introduction
Microstrip phase shifters, shown in Fig. 4.7 for a single ferrite layer structure and in
Fig. 4.8 for a double ferrite layer structure, axe compatible with coaxial connectors.
Dual ferrite oppositely-magnetized layers have been found to increase nonreciprocity
[74]. Thus, the nonreciprocal phase shift of the structure shown in Fig. 4.8 is ex­
pected to be higher than that of the structure shown in Fig. 4.7. In practical cases,
a thin dielectric layer can be inserted between the two ferrite layers shown in Fig. 4.8
to prevent m agnetic leakage from one ferrite layer to the other. In addition, the di­
electric layer param eters give us some control over the bandwidth and nonreciprocity
of the structure.
The two strips of Fig. 4.8 can be excited independently. Thus, two mode distri­
butions axe possible, the even and the odd modes. However, in this paper, we shall
limit our analysis to the odd mode shown in Fig. 4.9 and compare our results to ones
available in the literature.
Due to sym m etry of the odd mode of th e structure shown in Fig. 4.9 with respect
to the x-y plane, we can simplify the analysis by placing a perfect electric conductor
(PEC) between the ferrite layers [74]. The equivalent structure shown in Fig. 4.10 is
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89
PEC
Fig. 4.7: Cross-section of basic microstrip single-layer ferrite planar phase shifter.
PEC
Fig. 4.8: Cross-section of the phase shifter using oppositely-magnetized ferrite layers.
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90
Fig. 4.9: The odd mode of the dual structure.
then easier to analyze.
v.., ; >jci
M M i
PEC
Fig. 4.10: The odd mode representation of the dual structure.
B.
Full Wave Analysis
The analytical procedure used for the microstrip is very similar to the procedure
applied in the previous subsection for the slot line. For the microstrip, the unknown
is the electric current on the strip instead of the electric field in the slot. In addi-
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91
tion, an impedance m atrix formulation is used for the microstrip in contrast to the
adm ittance m atrix formulation for the slot line. The electric current on the strip in
both directions is expanded using Chebyshev polynomials as
Nx
Jx{y) = ' ^ c n hxn
(4.41)
n=0
Ny
Jy{y)
~
dnhyn
n=0
where the expansion functions hm and hyn are defined as
hrn = ( - i r n t f ) / J 1 - ( ^ ) 2
(4.42)
hm = ( - 1 ) " ( / „ ( f ) y i - ( | ) ’
(4.43)
where S is the w idth of the strip. Based on the physical behavior of the electric
currents on the strip, we use only even values of n with hxn and only odd values of
n with hyn.
Galerkin’s approach yields an impedance m atrix of the form
Z=
ZXX
Zxy
Zyx
Zyy
(4.44)
with elements given by
(4.45)
i=—oo,odd
The solution for th e propagation constant is the value of (3 that forces the determ inant
of the impedance m atrix to zero.
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92
o
-W/2
W/2
Fig. 4.11: n=0 Chebychev basis function of the electric current on the strip in longitudinal
directions.
Jy(y)
o
-W/2
W/2
Fig. 4.12: n = l Chebychev basis function of the electric current on the strip in transverse
directions.
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93
4.6
4.6.1
Magnetic Surface Wave Transducers
Introduction
M agnetostatic surface wave (MSSW) transducers are another class of microwave
ferrite devices. They are widely used in delay lines. MSSWs exist when we magnetize
the ferrite slab in the plane of the slab, and the direction of wave propagation is
along th e magnetization direction [76]. We investigate this case by setting
6
= 90°
and <f> = 180°. In our analysis, Galerkin’s technique is used to find the complex
propagation constant of an infinitely long microstrip transducer in various multilayer
structures shown in Figures 4.13, 4.14 and 4.15.
The results are verified where
possible by comparison to previously published ones. The effect of the m agnetization
angle on the predicted results is also investigated.
Fig. 4.13: Geometry of MSSW transducers with microstrip embedded between dielectric
ferrite structure.
D2d
or
m sm m
G
JB«
3
fariu-
Did
Fig. 4.14: Geometry of MSSW transducers in multilayer practical structure.
4.6.2
Full-Wave Analysis of the M icrostrip Phase Shifters
The MSSW tranducers can be analyzed by following an approach similar to the one
outlined in section 4.5.3. We note th at the expression for the impedance m atrix is
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Fig. 4.15: Geometry of MSSW transducers in two-layer structure.
no longer a discrete sum m ation b ut rather an infinite integral given by
4.7
Results and Discussion
To validate our approach, we obtained the Green’s function of the single layer ferrite
structure shown in Fig. 2.1 for two special cases - normally magnetized
and transversely magnetized
(6
(6
= 0°)
= 90°, <j> = 90°). In the normally magnetized case,
we compared our Green’s function to the one derived by Pozar using the boundary
condition method [27]. Excellent agreement is achieved as is clear from Figures 4.164.23. We notice that in our results there axe some noise for laxge values of K y. In our
expressions for the transmission m atrix we have an exponential terms th at may cause
this noise for large arguments. Using a normalization method, this noise is simply
avoided. In the transversely magnetized case, we compared our Green’s function
with the Green’s function derived by Elsharawy [40]. Again, excellent agreement is
achieved as shown in Figures 4.24-4.31.
To verify the present theory for the single layer microstrip phase shifter shown in
Fig. 4.7, we compared our predicted differential phase shift A <f>with the numerical
result of Elsharawy [40] and the experimental result of Riches et. al. [77]. Excellent
agreement with [40] and fair agreement with [77] is shown in Fig. 4.32. The discrep­
ancy between the numerical and experimental results is attributed to the fact that
the m agnet’s fringing field is probably not strong enough to magnetize the entire
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95
length of the ferrite substrate in the experimental case. We also examined the effect
of magnetization angle deviation by changing the magnetization angle from <f>= 90°
to <f>= 80°. The result is shown in Fig. 4.33. While this change in magnetization
angle decreased slightly the discrepancy between our theoretical result and the ex­
perim ent, it does not seem th at the magnetization angle deviation entirely explains
the discrepancy between theory and experiment in this case.
Further validation for our theory is obtained by comparing our results for the
forward and backward propagation constants and differential phase shift of the struc­
ture shown in Fig. 4.8 with those of Koza [74]. Excellent agreement is achieved as
shown in Fig. 4.34 for the normalized propagation constants and in Fig. 4.35 for the
differential phase shift.
In our analysis of the MSSW transducers in Figures 4.13, 4.14 and 4.15, we used
two different kinds of basis functions, piecewise linear subdomain basis functions and
entire domain basis functions (Chebyshev polynomials). Both basis sets are conver­
gent with a reasonable number of basis functions. In this dissertation, the presented
results are obtained using the entire domain basis set. The complex propagation
constant is computed and compared to the results given in [40] for both symmetric
and asymmetric cases. In the symmetric case, we represent the electric current dis­
tribution by even symmetric basis functions in the longitudinal direction and odd
symm etric basis functions in the transverse direction. For MSSW transducers, this
assum ption is considered to be questionable. A better assumption is to use both even
and odd basis functions to represent the current in both directions. This is referred
to as the asymmetric case. Fig. 4.36 indicates good agreement with Elsharawy’s
symm etric case, and Fig. 4.37 shows fair agreement with his asymmetric case.
We also analyzed the three layer structure shown in Fig. 4.14. This structure is
considered to be a practical one, in which a layer of ferrite is deposited on the top of
an alum ina dielectric with dielectric constant 9.8. It has the advantage of compatibil­
ity with MMIC circuits. Comparison of the attenuation constant computed using our
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96
approach with the result of [40] is given in Fig. 4.38 for both symmetric and asym ­
m etric current distributions. Comparison of the phase constants is given by Fig. 4.39.
MSSW can exist over the range of frequencies given by
+ u>0u>m < uj < u Q+ ~um.
The first peak in Fig. 4.38 corresponds to the second limit of MSSW frequency range,
4.172 GHz. For the symmetric case in Fig. 4.38, there is only one attenuation m ax­
im um at 4.18 GHz. However, for the asymmetric case, there are two maxim a, the
first one at 4.18 GHz and the second one at 4.609 GHz, which is due to the coupling
to th e MSSW. We varied the number of the basis functions from 5 to 13. Figures
4.40 and 4.41 show th at the results are almost the same after 9 basis functions are
used.
The structure shown in Fig. 4.13 is used to model the experiment carried out by
Elsharawy [40]. The comparison of the symmetric and asymmetric theoretical cases
with th e experimental result is shown in Fig. 4.42. A peak of S 21 = 23 dB occurs in
the measured curve at 4.34 GHz while a peak of about 20.3 dB at 4.11 GHz occurs
in Elsharawy’s theoretical result. The peak in our curve is about 21.4 dB at 4.35
GHz.
The bandwidth of the device can be defined as the frequency range over which
the insertion loss is within 3-dB of its peak value. The bandwidth of Elsharawy’s
theoretical result is about 0.94 GHz, while the bandwidth of our theoretical result
is about 0.70 GHz. The experimental result exhibits a bandwidth of 0.76 GHz.
Elsharawy attributed the discrepancy between the measured and calculated results
to the lack of accurate knowledge of magnetization magnitude and angle and to the
line width at the surface. Using our Green’s function, we studied the effect of the
m agnetization angle, and the results are shown in Figures 4.43 and 4.44 for several
different values of 0 and <£, respectively. Varying 9 has very slight effect on the
results, while varying <f> has a clear effect on the results and improvement in the
agreement with the experiment is observed. We studied another factor th a t may
also has an impact on the results, the 3-dB line width. Wheu we decreased the
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97
3-dB line width, a better agreement with the experim ental results is achieved at
A H = 420.0Oe as shown in Fig. 4.45. The measured 3-dB line width for the ferrite
m aterial under the experiment was 490.OOe ± 2% a t 9.0G H z. Since the 3-dB line
width depends on the operating frequency, using A H = 420.0Oe for the comparison
in the frequency range of 2.5 : 5.0G H z is acceptable. We noticed also th at increasing
the material loss decreases th e attenuation which agrees with what Elsharawy stated
in his dissertation.
This work
Pozar
3000
J
2400
aB*
a
a
1800
s
1200
600
-320
-160
0
Ky/k0
160
320
Fig. 4.16: Comparison of the computed Green’s function versus Pozar’s for the normally bi­
ased slab (Imag(Gxx)). (d = 7.62x 10-4m, £/ = 12.0,4 k M s = 2100.0G, Hdc = 700.0Ge,
A H = O.Oe, R 3 = Ofi, W = 1.016 x 10~2m, / = 3.6GHz, K x = (110.0, -10.0)).
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98
350
This work
Pozar
300
250
150
100
-320
-160
160
320
Ky/kO
Fig. 4.17: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Real(Gxx)). The parameters are the same as in Fig. 4.16
12000
— This work
•— Pozar
.
9600
&
a
7200
4800
2400
-320
-160
0
Ky/kO
160
320
Fig. 4.18: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Imag(Gyy)). The parameters are the same as in Fig. 4.16
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99
os m
15
d ' i2°
-180
-240
-320
-160
0
Ky/kO
160
320
Fig. 4.19: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Real(Gyy)). The parameters are the same as in Fig. 4.16
This work
Pozar
3200
^
1600
CJ
b
2
0
S
d
CO
S
|- < -1600
-3200
-320
-160
0
Ky/kO
160
320
Fig. 4.20: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Imag(Gxy)). The parameters are the same as in Fig. 4.16
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100
This work
— Pozar
320
160
CQ
£
-160
-320
-320
-160
0
Ky/kO
160
320
Fig. 4.21: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Real(Gxy)). The parameters are the same as in Fig. 4.16
— This work
Pozar
3200
*
1600
I
O
c
o
-1600
-3200
-320
-160
0
Ky/kO
160
320
Fig. 4.22: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Imag(Gyx)). The parameters are the same as in Fig. 4.16
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101
This work
Pozar
320
160
I
-160
-320
-320
-160
0
Ky/kO
160
320
Fig. 4.23: Comparison of the computed Green’s function versus Pozar’s for the normally
biased slab (Real(Gyx)). The parameters are the same as in Fig. 4.16
This work
Elsharawy
3000
2400
'So 1800
1200
600
-320
-160
0
Ky/kO
160
320
Fig. 4.24: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Imag(Gxx)) (d = 7.62 x 10-4 m, e/ = 12.0, 4 irMs = 2100.0G’,
H dc = 700.00e, A H = O.Oe, R s = 0Q, W = 1.016 x 10~2m, / = 3.6 G H z .
K s = (110.0,-10.0)).
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102
This work
Elsharawy'
288
180
144
-320
-160
0
Ky/kO
160
320
Fig. 4.25: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Real(Gxx)). The parameters are the same as in Fig. 4.24
This work
Elsharawy
12000
9000
3000
-320
-160
0
Ky/kO
160
320
Fig. 4.26: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Imag(Gyy)). The parameters are the same as in Fig. 4.24
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103
-32
« -64
-96
-128
-320
-160
0
Ky/kO
160
320
Fig. 4.27: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Real(Gyy)). The parameters are the same as in Fig. 4.24
3200
1600
I
IC
*5
C3
S
0
-1600
-3200
-320
-160
0
Ky/kO
160
320
Fig. 4.28: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Imag(Gxy)). The parameters are the same as in Fig. 4.24
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104
This work
Elsharawy
252
126
I
CQ
£
-126
-252
-320
-160
0
Ky/kO
160
320
Fig. 4.29: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Real(Gxy)). The parameters are the same as in Fig. 4.24
This work
Elsharawy .
3200
1600
I
£
e
•N
bo
es
g
-1600
-3200
-320
-160
0
Ky/kO
160
320
Fig. 4.30: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Imag(Gyx)). The parameters are the same as in Fig. 4.24
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105
260
130
-130
-260
-320
-160
0
Ky/kO
160
320
Fig. 4.31: Comparison of the computed Green’s function versus Elsharawy’s for trans­
versely biased slab (Real(Gyx)). The parameters are the same as in Fig. 4/24
This Work-90-90
Riches-Experimeot.
ELaharawy-Theo ry
7
8
9
10
Frequency GHz
11
12
Fig. 4.32: Comparison of the calculated differential phase shift versus theoretical and ex­
perimental results for a microstrip single layer phase shifter (df = 0.635 x 10-3 m.
ff = 12.9, 4x M a = 2300.0G, Hdc — 150.0Oe, S = 0.45 x 10_3m, a = 1.27 x 10-2 m.
/ = 1.52 x 10"2, 9 = 90°, <f>= 90°).
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106
Thi# Woric-90-80
Richei-Experiment..
E I*harawy-Theo ry
cu 20
7
8
9
10
11
12
Frequency GHz
Fig. 4.33: Comparison of the calculated differential phase shift versus theoretical and ex­
perimental results for a microstrip single layer phase shifter (df = 0.635 x 10-3m,
€/ = 12.9, 4irMs = 2300.0G, Hdc — 150.00e, 5 = 0.45 x 10-3m, a = 1.27 x 10-2m.
I = 1.52 x 10"2, 9 = 90°, <t>= 80°).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
i
*
i
This Wort-Forward
Roza-Forward
This Wort-Backward
Koza-Backward
-2
7.2
9.6
12.0
14.4
16.8
Frequency GHz
Fig. 4.34: Comparison of the normalized propagation constants for dual strip phase shifters
(df = 1.0 x 10-3 m, e/ = 17.5, 47rMs = 1500.0G, Hdc = 0Oe, S = 1.0 x 10"3m,
a = 1.0 x 10~2m).
40
— This Wort
— Roza
35
O 30
-S-25
20
15
7.2
9.6
12.0
14.4
16.8
Frequency GHz
Fig. 4.35: Comparison of the calculated differential phase shift for dual strip phase shifters
(df = 1.0 x 10-3m, ef = 17.5, 4 xM s = 1500.0G, H dc = 0Oe, 5 = 1.0 x 10-3m.
a = 1.0 x 10-2 m, 0 = 90°, <p = 80°).
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108
• This Work-Real
Elsharawy*ReaI
- This Wbrk-Imag.
EUharawy-lmag.
6.4
S . 3.2
es
<2
- o.o
bo
|- 3 .2
-6.4
r
2.4
3.2
4.0
4.8
5.6
Frequency GHz
Fig. 4.36: Comparison of the calculated symmetric propagation constant for the two layer
transducer (dd = 1.27 x 10-3m, df = 2.03 x 10-3m, e/ = 17.5, q = 10.2, 4ttM s =
2267.0G, Hdc = 144.0Oe, A H = 300.00e, S = 0.3 x 10"3m).
This Work-Real
Elsharawy*Real
— This Wdrk-Imag.
EUharawy-lmag.
6.4
$5_ 3.2
73
,- c 0.0
a -3.2
r
-6.4
2.4
3.2
4.0
Frequency GHz
4.8
5.6
Fig. 4.37: Comparison of the calculated asymmetric propagation constant for the two layer
transducer (dd = 1.27 x 1 0 '3m, df = 2.03 x 10_3m, ef = 17.5, e<* = 10.2. 4-iV/s =
2267.0G, Hic = 144.0Oe, A H = 300.00e, S = 0.3 x 10-3m).
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109
60
-i
1----- 1----- 1----- s----- r
“T
—
4.0
4.25
This Work-Symm.
Elshajawy-Symm.
This Work-Asym.
Elsharawy-Asym.
4.5
Frequency GHz
Fig. 4.38: Comparison of the calculated attenuation constant for the three layer transducer
(i n = 2.5 x 10-4m, d.2d = 2.5 x 10-4m, df = 0.5 x 10-4 m, ej = 15.0, eu = 9.8,
exd = 10.0, 4trMs = 1780G, Hdc = 600.00e, A H = 45.0Oe, S = 0.5 x 10~4m).
8.0
—
—
—
-
6.4
3
This Work-Symm.
Elsharawy-Symm..
This Work*Asym.
EIsharawy-Asym.
3.2
1.6
0.0
3.5
3.75
4.0
4.25
4.5
4.75
5.0
Frequency GHz
Fig. 4.39: Comparison of the calculated phase constant for the three layer transducer
(d u = 2.5 x 10_4m, did = 2.5 x 10_4m, df = 0.5 x 10-4 m, e/ = 15.0, eu = 9.8.
eld = 10.0, 4ttM s = 1780G, Hdc = 600.0Ge, AH = 45.0Oe, S = 0.5 x 10_4m).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
110
45
u
3.5
3.75
4.0
4.25
4.5
4.75
5.0
Frequency GHz
Fig. 4.40: The effect of number of basis functions on the attenuation constant
8
This Work-BF=5
This Work-BF=9
This Work-BF=13
6
/-—
V
4
2
0
3.5
3.75
4.0
4.25
4.5
4.75
5.0
Frequency GHz
Fig. 4.41: The effect of number of basis functions on the phase constant
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
I
Thu Work-Symm.
Elsharawy-Symm
This Work*Asym.
Elsharawy-Asym.
EUharawy-Exp,
3.5
4.0
Frequency GHz
Fig. 4.42: Comparison of the insertion loss along a MSSW transducer (dd = 1.27 x 10~3m.
df = 2.03 x 10“ 3m, ef = 17.5, ed = 10.2, A vM . = 2267.0G, Hdc = 144.0Oe, A H =
490.0Oe, S = 0.3 x 10"3m, I = 12.7 x 10"3m).
30
1------
rThis Work*fe90f £=00
This Work*$=95, £=00
This Work-JblOO, £=00
Elshar&wy-Exp.
24
ffl 18
CM
— !2
2.5
3.0
3.5
4.0
4.5
5.0
Frequency GHz
Fig. 4.43: The effect of the magnetization angle, 9, on the insertion loss along a MSSW
transducer (dd = 1.27 x 10-3 m, df = 2.03 x 10- 3 m, e/ = 17.5, ed = 10.2, 4 ~ M S =
2267.0G, H dc = 144.0Oe, A H = 440.00e, 5 = 0.3 x 10~3m, / = 12.7 x 10- 3 ,m
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
112
30
24
-i
r
This Work-#=90, *=00
This Work-#=90, *s05
This Work fa DO, *slO
Elsharawy-Exp.
-I
PQ 1 8
-a
—
12
2.5
3.0
3.5
4.0
4.5
5.0
Frequency GHz
Fig. 4.44: The effect of the magnetization angle, o, on the insertion loss along a MSSW
transducer (dd = 1.27 x 10-3 m, df = 2.03 x 10-3 m, ej = 17.5, e<* = 10.2, 4 ttM 3 =
2267.0G, Hdc = 144.0Oe, A H = 44O.O0e, S = 0.3 x 10"3m, I = 12.7 x 10“ 3m).
30
This Work*AH=490 Oe
This Work-&H=450 Oe
This \Vork-AH=420 Oe
—
24
—
Elsharawy-Exp.
!2
2.5
3.0
3.5
4.0
4.5
5.0
Frequency GHz
Fig. 4.45: The effect of the 3-dB line width on the insertion loss along a MSSW transducer
(idd = 1.27 x 10~3m, df = 2.03 x 10~3m, ef = 17.5, ed = 10.2, 4;rA/a = 2267.0G.
H dc = 144.0Oe, S = 0.3 x 10_3m, I = 12.7 x 10_3m).
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113
CHAPTER 5
3-D ANALYSIS OF RADAR CROSS SECTION OF A FERRITE PATCH
ANTENNA
5.1
Introduction
The basic configuration of a microstrip antenna consists of a sandwich of two parallel
conducting layers separated by a single thin dielectric substrate. The lower conductor
functions as a ground plane, and the upper conductor acts as a radiating element
which may be a patch of regular shape, a dipole, or a monolithically printed array of
patches or dipoles and the associated feed network. The patch conductor typically
has some regular shape, for example, rectangular, circular or elliptical. The feed is
often a coaxial probe or a microstrip transmission line. Microstrip antennas exhibit
all the advantages of the microstrip devices: a) they are light weight, have small
size, and exhibit low profile planar configurations which can be made conformal;
b) they are inexpensive to build and ideally suited for large scale production by
printed circuit techniques; c) they are compatible with modular designs (solid state
devices such as oscillators, amplifiers, phase shifters, etc., can be added directly to
the antenna substrate board); d) their feed lines and matching networks can be
fabricated simultaneously with the antenna structure so th at discontinuities due to
connectors can be minimized. All these advantages compensate, at least in part,
some of the disadvatages: a) simple microstrip antennas have narrow bandwidths;
b) their gain is low; c) they have a small power handling capability; d) their dielectric
losses reduce th e radiation efficiency; e) unwanted surface waves may cause spurious
radiation at th e edges of the microstrip patch [78].
Recently, there has been great interest in incorporating ferrites in patch antenna
designs to incorporate phase shifting, impedance matching, frequency tuning, and
nonreciprocal effects in the operation of antennas. It has been reported th at by
including ferrite materials, the main beam from an antenna array can be scanned.
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114
and th a t the radiation frequency of a microstrip antenna can be tuned by varying the
dc bias. Ferrite patch antennas, similar to the dielectric antennas, consist of metalic
patches deposited on ferrite substrates. The substrates are electrically grounded by
placing them on a m etal surface.
There are m any possibilities and combinations for using ferrite materials in printed
antenna systems. Bias fields can be applied in different directions; ferrite mate­
rials can be used as single substrates, in multilayer substrate configurations with
dielectrics, or as cover layers for printed antennas; ferrites can also be used inhomogeneously with dielectric materials [27].
In spite of a reasonable effort and a large number of publications concerning the
scattering from the printed antennas on a biased ferrite substrates, an efficient and
versatile formulation of the RCS is not available yet. In addition, the multilayer
structure is not discussed yet except by Yang [29]. However, neither Yang nor any
one else has presented the excitation vector expression or detailed derivation.
This work presents a clear and versatile formulation of such problems. The gen­
eralization is th e m ain contribution of this chapter. The Green’s function is formu­
lated using the closed form transmission matrix. The excitation vectors are given in
a closed form too. T he expression of the excitation vectors is general for any incident
angle, any num ber of layers, and any magnetization angle.
The I E E E dictionary of electrical and electronics terms defines R C S as a mea­
sure of reflective strength of a target defined as 4tt times the ratio of the power per
unit solid angle scattered in a specified direction to the incident power per solid an­
gle. More precisely, it is the limit of that ratio as the distance from the scatterer to
the point where the scattered power is measured approaches infinity [79]:
\E s c a t\2
a = lim 47rr2T -r—oo
\Etnc 2
(5.1)
where E scat is the scattered electric field and E 'nc is the field incident to the target.
Three cases are distinguished: monostatic or backscatter, forward scattering, and
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115
bistatic scattering. The units for radar cross section are square meters. T he crosssectional area does not relate directly to the physical size of a target. Due to the
large dynamic range of RCS, a logarithmic power scale is most often used with the
reference value of aref = lm 2.
crdBsm = crd B m 2 = 10logl0^ ~ = 10 lo g x f i f e f
(5.2)
1
Two unit notations axe used. The dB sm notation is very common w ithin the aca­
demic, government, and industrial organizations. The d B m 2 notation is less used,
although it is sometimes seen in radar system design literature.
5.2
Plane Wave Propagation in a Ferrite Medium: an Introduction to the Excitation
Vector Formulation
The first step in our analysis is to derive the propagation constant of a wave propa­
gating at an arbitrary angle ( 9 \ <f>1) to the direction of the dc magnetization in ferrite.
W ithout loss of generality, we can consider the incident wave to be traveling in the
2 -direction
in an arbitrary magnetized ferrite medium.
The full perm eability tensor of an arbitrarily magnetized ferrite medium is given
by [72]
ft =
ftn
ft\ 2
ft\3
ft2 l
ft22
ft2Z
. ftzi
ft32
ft33 .
where
[*n
=
f t 12
=
(ti3
=
ft +
( f t o — f t ) s i n 29 c o s 2 <£
s i n 2 0 s in 2 < £ +
jk cos
9
(to — ft
— -— sin20 cos 6 —j k sin 9 sin 6
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116
uo — U
9
f i 2i
=
— -— sin
H22
=
y + {yo
Uo
o
0
sin 2 6
— jk cos 0
—y ) sin 20 sin V
—u
y.22 =
— ~— sin 2 0 sin 6 + JK sin 0 cos 6
y.31 =
u0 — a
— -— sin 20 cos 6 + JK sin » sin 6
yz 2 =
— -z— sin 20 sin 6 ~ JK sin 0 cos 6
y&
=
yo
—y
y o - {yo - y ) sin 20
U0 U}m
(* = (! + ^Ll]2 ----—r.j*7 )
K =
um
UJUJm
w2 —W2
= 747riV/s, w= w0 + ^, w0 = 7 Ho
wo is the precession frequency,
#0
is a z-directed impressed dc magnetic field,
7
is
the gyromagnetic ratio, T = —
^27? is the relaxation time, and AH is the 3-dB line
-fAH
width.
The four Maxwell’s equations are given by
—j u
y -H
ju e E
=
=
V x E
=
V x H
V • ( y -H)
_
eV • E
=
0
(5.4)
= 0
If the wave is propagating in the z-direction with a propagation constant j3 and
its angular frequency, w, and if the conditions for a plane wave ^
=
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0
are
117
substituted into equation (5.4), that equation becomes
ueEx
(5.5)
= u;eEx
(5.6)
= Ez
(5.7)
j(3Ey
= —jujB x
(5.S)
-jfiE x
= -ju jB y
(5.9)
0
= -j0 B z
(5.10)
/3Hy =
fiHy
0
Equation (5.4) becomes
It is interesting to note that, because it is a plane wave, the longitudinal component
of the m agnetic flux density, fiBz, is zero, but there still may be a finite longitudinal
(normal) component of the magnetic field intensity, H z.
After substituting equation (5.3) into Maxwell’s equations and after some ma­
nipulation we find
Bi
P
P2
^
=
T
±
\
/
w2e
2
V -4 r - 0
(5-ID
p
(5. 12)
where
=
M33
Q
=
( 5 . 13)
^33
Ml
—
~M22M33 +
M23M32
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(5-14)
118
=
/^3
= ^21/^33 —^23^31
(5.16)
=
(5-17)
P
4
~
1 1 ^3 3 +
(5-15)
(J -2
f i
H
13/^31
^ 1 2 ^ 3 3 — ^1 3 /^ 3 2
As a validation, we set the parameters in equation 5.11 to the case of wave normally
incident on a norm ally biased ferrite slab, 0 = 0°, <f>= 0°. We obtain the following
expression for th e propagation constant for this case,
P±
=
u
y
/
e
(
p
±
K
(5.18)
)
The resulted propagation constant given by equation 5.18 is exactly the result reached
by Fuller for this case [80].
Ground plane
Fig. 5.1: Incident wave on a normally biased single ferrite layer.
For a single grounded ferrite substrate which is normally biased, the field com­
ponents outside and inside the ferrite region as seen in Fig. 5.1 can be written as
e]k°yv' ejk°' cos s‘
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The field components inside the ferrite slab consist of right and left-handed circularly
polarized partial waves propagating along positive and negative directions, and are
expressed as
^
=
[ A + eJ^ +xut e-,/3+yv' eJ/3+' cos e ‘ +
B + e -jP + x u ' e ~]0+yv ‘e - j
0 + - c o s 8‘ j
(a* - ja#) +
[A_eJ/3-xut ej(3~yvCejf3~~ cos e‘ + 5 _ e~j/3~zu‘e
t ~ji3- - cos e‘]
{ a e + ja t)
(a t+ ja g ) +
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120
} _ ^ _ e j P - x u c e J 0 - y v c e J p - z c o s S c _|_
e - j P - x u c g - J 0 - y v c g - j 0 - z cos5 ' j
n~
(a t-ja g )
E[
[A+eJ0+xv,t e]l3+yvt e~3t3+zcos6t + B +e~jt3+xu' e~JP+yv‘eJ*3+zcoset].
=
(5.19)
(—ag + ja,j,) +
^ _
e J 0 - x u ‘ £ j 0 - y v c e ~ ] 0 - z c o s 9 C _|_ g _ e - j 0 - x u ‘ e ~ ] 0 - y v ' & j 0 - z c o s S ' j
(a g + j a t )
1TX =
— |r a r x E[
V
_
L
_
(5.20)
e ] 0 + x u ‘ e j/3+ y v ‘ e ~ ] 0 + : c o s 8 ‘ _|_ Q ^ e - ] 0 + x u ( e ~ ] 0 + y v ‘ £ j 0 + z c o s 6 ‘ -l
T] +
(-it-jag)
+
+
_ ^ _ [ A _ e 3 0 -x v ‘ e J 0 -y v l e -tf-z c o s O *
g ^ e - ] 0 - x u ‘ e - j ( 3 - y v c e j ( 3 - z c o s e ‘ -l
v~
( - a t + jag)
The generalized Snell’s law, which comes from matching the phases of E and H at
the air-ferrite interface, leads to the following expression for the transmission angle
.
Sm
k 0 sin 6 l
P±
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(5'21)
121
The total field in each region is the sum of the incident and reflected components.
First, we apply the continuity of the electric and magnetic fields at the interface,
z = d. Then, we solve for the vector reflection coefficients, Rg and R ^ . This procedure
results in a tedious and complex expression for single normally biased ferrite layer.
The procedure for an arbitrarily biased ferrite layer are even more complex, and for
an multi-layer arbitrarily biased imulti-layer structure, they may be impossible to
derive.
5.3
Theory
5.3.1 Full Wave Formulation
First we consider the single ferrite layer given in Fig. 5.2 with a general incident wave
with either soft or hard polarization.
m
e
'
j
’
1
1
E 2
J 2
—
2
Fig. 5.2: Geometry of single ferrite layer with a general incident wave.
The transmission m atrix T for a ferrite layer is a 4 x 4 matrix written as [43]
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122
-
II'Fh
II
J l
= E
= T
= T
.
=
-
=
h
-
^2
-
=J
where T , Z , Y , T
'
-i
=E
=T
f
z
= T
=J
Y
T
'
e
.
h
2
(5.22)
.
=
axe 2 x 2 submatrices of T , E l , E 2 axe the tangential electric
field at the boundaries of the layer, J x and J 2 are the tangential surface currents
defined by J n = z x H n, where H n is the tangential magnetic field at the n th surface
of the layer, and
denotes the spatial Fourier transform defined as
E {K x, K y) = f 0
E { x ,y ) e - jK*xe - jKyy dx dy
J —OOJ —CO
(5.23)
Ferrite Substrate
d
\
G round Plane
Fig. 5.3: Microstrip patch antenna on a normally biased ferrite substrate
The structure, shown in Fig. 5.3, is illuminated by an incident wave, E l. In the
absence of the conducting patches, E T is reflected into the air region by the ground
plane and the ferrite substrate. The total field E° in this case is the summ ation of
these two components. W ith the presence of the patch, a surface current is induced
on the patch conductor which causes a scattered field, E s. The total field in the air
region is the vector sum of the incident, the reflected, and the scattered fields,
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123
El
= E ' + E r + E3
Hl
=
W + H' + H3
(5.24)
(5.25)
Rewriting equations (5.24) and (5.25), we get
E l = E° + E 3
=
H° + H 3
(5.26)
(5.27)
The integral equation is a statem ent of the boundary conditions on the patch that
the total tangential field on the surface of the conducting patch must be zero, that
is
a~ x E 3 = —dz x (E °)
(5.28)
Equation (5.28) represents an integral equation since E 3 is given by
=
J G ' J . ds
(5.29)
where J s is the surface current density on the patch and G is the Green’s function
derived in our previous works [65] and [81].
The method of moments is essentially a technique for transforming integral equa­
tions into m atrix algebric equations th at can be solved numerically using a computer.
Applying the Galerkin’s approach to equation (5.29), using the basis functions as
weighting functions leads to a system of linear algebric equations which can be solved
for the unknown coefficients of the basis functions. In a m atrix form we can write
[Zij][Ij] = N
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(5.30)
124
where [Z\ is known as the impedance m atrix with elements
^
Z{j
—
~
ro o
2
J
ro o —
x
J k y ) ’ G" [ k x j
z|z ) • J
—
k y ' j
d k x dky
(o.31)
and [e,-] is the excitation vector defined by
< = I E ° J ’J s
(5-32)
where u is either z or y and E° is the tangential total electric field at the air-ferrite
interface with the absence of the conducting patch.
The patch currents in the integral equations (5.28) and (5.29) are expanded in
term s of a
set of knownfunctions. Analysis of arbitrarily shaped patches requires
the use of subdomain basis functions. Entire domain basis functions are difficult to
implement on a complex structure and may take many modes to describe fine electric
current details accurately. A convenient choice is a linearly-varying basis function
with triangular support. This set of basis functions ensure current continuity from
triangle to triangle, and it has been introduced by Rao, Wilton and Giisson (RWG)
to evaluate the electromagnetic scattering by surfaces of an arbitrary shape using
the EFIE formulation [82].
Assuming th a t a suitable triangulation has been defined to approximate the mi­
crostrip antenna, we note th at each basis function is associated with an interior
(nonboundaxy) edge and vanishes everywhere except in the two triangles attached
to th at edge. Thus, within the triangle X) shown in Fig. 5.4, the surface current can
be approximated as
3
___
J *{x >y ) = J L h Jj ( x ’ y )
( 5 -3 3 )
i= i
where Ij is a coefficient to be determined via MoM and Jj is the piece of the basis
function on the triangle J2 associated with j th edge. The Jj are given by
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125
J j (r) = ± 2 A~ ( r ~ ri)
where A ^ is the area of triangle
(5-34)
lj is the magnitude of the j th edge vector given
by
(j
^'mo<f(j-(-l,3)+l
T 'm o d .^ j,
(5.35)
3 )+ l
r is the position vector to an arbitrary point in the triangle
and r j is the position
vector to node j . The sign multiplying lj is determined by the connectivity of the
mesh. The Fourier transform of the piece of the basis function on triangle J2 along
the j th edge is given by
7 j$ ) = j
j J ] ( r ) e - * TdA
(5.36)
where
k = xkx + yky
(5.37)
The Fourier transform of this basis functions has been evaluated in a closed form by
M clnturff and Simon [83].
(x4,]T4)
<*i
Fig. 5.4: RWG basis function and corresponding edge connectivity
The impedance m atrix elements in the spectral domain MoM solution of an arbi­
trary microstrip structure using RWG basis functions th at has x and y-components
for each set may be written as
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126
J
Z {j
—
4-ti^ j
to o
J
"hJ r t (
^ -n
“f*J yt (
^xi
" I”
J
yi (
to o
—
zi
^“*5
xi(
—
^ y ) ' G x x (.^ x 7
^*y) ’ J x j i j ^ x i k y )
(5.38)
k y ) ' G x y {,^xi k y ) ’ J y j ^ x t k y )
) ‘
G y x ( ^ n ^-y) * J x j (
j ky )
G yy ( kx 7 ^ y
^ y ) }
^*277 f c y ) •
) *
*JIjj ( ^X 7
dfcr dl\*y
Notice the difference between the two im pedance equations, (5.31) and (5.39). Equa­
tion (5.39) is a special case of (5.31) where we use one set of basis functions having
both x and y components, which is the case of RWG basis functions.
5.3.2
A.
Excitation Vector
Soft Polarized Incident Wave
As mentioned above, the total tangential electric field in the absence of the conduct­
ing patch is E°, which is given in equation 5.26 as the summation of the incident
and reflected waves at the air-ferrite interface. Applying the boundary conditions on
both sides of the ferrite layer shown in Fig. 5.2, we can write
E+
=
E[ + E[
(5.39)
E*
=
Ef
(5.40)
E2
=
0
(5.41)
Since there is no current source at surface 1, J =
+ ./f = 0, hence, ./f = —J* .
T he current above surface 1 in terms of the magnetic fields is given by
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127
J+
(5.42)
= zxHf
=
z x ( H i + H[)
=
z x (— —an x H{ + — an x H[)
f]0
TjO
where an is th e normal vector of a general incident wave which has the following
form
On =
oxu' -f ayv' + az cos 9l
(5.43)
ul =
sin 9' cos 4>l
(5.44)
vl =
sin 9l sin <f>*
(5.45)
After m anipulation, equation 5.43 becomes
cos 9l J t = -------(E\ - E l)
lo
(5.46)
Rewriting equation 5.22 we get
—r
'
El + El
^
( e x- X )
-E
=T
f
=T
z
'
_
_
0
=J
Y
f
(5.47)
J 2
Solving equation 5.47 for E ' x and E Ti we can write
=
& i =
J
= T ~ l
cos 9'I + rjoT Z
= J = T ~ l
cos 9'I — Tj0 T Z
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(5 .4 8 )
12S
The total tangential electric field at the ferrite-air interface is the sum of the incident
and the reflected electric field components which leads to the following expression
for E + 1
2 cos 6 XI
=J = T~l
(5.49)
E\
. cos O'I + t\qT Z
In general, we can write equation 5.49 in matrix form as
C0
[£ + i] =
r ”t>
ri4> ri<t>
11
°21
u 12
(5.50)
[&l]
'-'22 J
The general incident soft-polarized plane wave can be expressed as
E[ = a^ejkQXU' ejhzu' e3kaZCOsd'
(5.51)
Substituting equation 5.51 into equation 5.50 and the result into equation 5.32, the
excitation vector elements are found to be
=
( - C f i sin <?+ Cf 2 COS <l>)e3k°dcos6' j * { - k Qu \ - k 0 v l)
(5.52)
ey =
(—C2i sin <j>+ C%i cos <f>)ej k o d c o s O 1 Jq(—kQu \ —kov')
(5.53)
<
where J* and Jq are th e Fourier transforms of the basis functions components.
The same procedures are used to find the excitation vector for the double layer
structure shown in Fig. 5.5, a conducting patch on the top of a dielectric grounded
plane and a ferrite cover on the top of both. The total tangential electric field in
terms of the soft-polaxized incident field is found to be
= T
-J
= T ~ l =E
£ += - ( Y , - T f Z,
fj)
f j i ,
Ci +
u
)+ —
no
Ci -
u
) E \
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(5.54)
129
where subscript f stands for ferrite, the superscript (—1) throughout this work means
m atrix inversion, / is the unity matix, and
contains the reflection coefficients is
given from equation 5.48 as
= j
cos
I+ j]0 T
U -
Z
Zj V t-i
COS 0 ‘ 7 — 7/0
=J
=T~l
{o.zz)
T Z
=T
T and Z
are the submatrices of the new transmission m atrix which results from the
multiplication of the ferrite times the dielectric transmission matrices. In a m atrix
form, equation 5.54 can be w ritten as
[£>] =
Wl
11
cti
C<
f> [£'u]
11
(5.56)
In final form, the excitation vectors axe given by
exq
= ( - C f x sin 0 + C ?2 cos Q)ejho{d'i+d' ]cosa'J~*{-k 0 u \ - k Qv l)
(5.57)
e*
= ( - C ^ s m t + C ^ c o s ^ e ^ ^ + ^ ^ ' J Z i - k o u ^ - k o v 1)
(5.58)
where dd and dj are the thicknesses of the dielectric and the ferrite layers, respectively
as seen in Fig 5.6.
B.
Hard Polarized Incident Wave
For the single ferrite layer shown in Fig. 5.2, the difference between the soft and the
hard-polarization case is the expression of the incident plane wave which is given by
E[ = dg ejk°xu' ejk°xu' ejh°' cos 8'
(5.59)
Following the same procedures as the soft-polarization case, we can prove the follwing
relations
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130
2/
E+1
c o s O'
=
E\
=J = T~l
(5.60)
sfsr + ^
^
In a m atrix form, equation 5.60 can be writen as
.
C8
no
11
12
riO riB
U21 0 22
(5.61)
The excitation vector elements for a hard polarized plane wave incident on a single
ferrite layer axe given by
ef =
(Cf, cos 9 + CI,J sin
. / H - i o u '. - t o i '1)
(5.62)
(5.63)
For the double layer structure shown in Fig. 5.5, the final expression is found to
be
=
t
—j
= r~ 1 =
E+ = - ( Y f - Tf Zf
e
=J =T~l
f f ) Tf Zj
( I + T g) + —
s - ( / - r () E \
(5.64)
COS t / l T]o
In a m atrix form, equation 5.64 can be written as
[E+] =
CO
11
O
C21
r'Q
°12
(5.65)
r^O
u 22
The reflection coefficent Fg is found to be
=j = t - i
f•L.e
i :_ d F + r>o T
Z
=J =T~
1
TF-nof z
cos 8
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(5 .6 6 )
131
=T
=J
where T
and Z
are the submatrices of the new transmission m atrix resulting from
the multiplication of the ferrite transmission m atrix by the dielectric transmission
m atrix.
The final form of the excitation vectors is
=
{C9
n cos <f>+ CU sin <j>)ejko(dd+df ) cosB' J = ( - k Qu \ - k Qv l)
(5.67)
eyq =
{Ce2l cos d>+ C e22 sin ^ )eJM«*«+<*/)cos6' J $ ( - k 0 u \ - k Qv')
(5.68)
<
5.3.3
A.
Green’s Function Formulation
General
The detailed derivation of the Green’s functions for a single ferrite layer and for
some multi-layer dielectric- normally biased ferrite structures using the transmission
m atrix approach are outlined in chapter 2 and for arbitrary biased ferrite structures
chapter 4. We will summarize the Green’s functions for the structures analysed in
this work.
For the single structure shown in Fig. 5.2, the Green’s function is found to be
=J=T ~l
Es =
{Tf Z f
-
- i
_
+ G a ) " 1J S
= MS_
=
G
Js
(5.69)
Notice th at the Green’s function given in equation 5.69 is not limited for certain bias
direction. The elements Z j and T j are elements of any transmission m atrix derived
for any bias direction. Ga is a semispace Green’s function, which is calculated by
taking the limit of the dielectric Green’s function when the thickness goes to infinity
and the dielectric constant goes to unity.
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B.
Derivation of the Green’s Function for a Two-Layer Structure Patch Antenna
The dielectric layer in Fig. 5.6, the lower region, is represented by the following
relation
E
= Td
T
Ei
Ji
=E
—T
Td
Zd
= T
=J
Ya
Ei
(5.70)
Ji
f,
We set E[ = 0 by placing a ground plane at distance dd from the source plane. This
results in the following equation
=
J
= J = T ~1__
T dZ d E
(5.71)
For the ferrite layer, the cover region, we have
(5.72)
Applying the boundary conditions at the air-ferrite interface leads to the following
relation
_+
E
=
. J
h K
Ju
.
- =E
■
r
-1
If If
—
.
—T
j f
El U
(5.73)
X.
f,_
After some manipulations we get
= T
J
= j = -l
=E
=T
=
-1
= ( Y f + T f Ga ) ~i \/ T f + Z s Ga )E
(5.74)
The total current at the source plane, the conducting patch, is the sum of the two
current components above and below the surface, J
J3
=
J
and J
+J
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(5.75)
133
'
=T
=J = -l
=E
=T
= - 1
= J = T~l
( Y f + T f Ga ) ( Tf + Z f Ga )~1 + f dZ d
E3
(5.76)
From, the definition, the Green’s function relates the current to the electric field,
hence the final expression of the Green’s function is obtained in the following form
=T
'
E,
= J = -1
=E
=T
= - l
=J = T~l
=
( Y f + T f Ga ) ( Tf + Z f Ga )~ 1 + T dZ d
=
= \rs_
G J3
-l
J3
(5-77)
(5.78)
Following similar procedures, the Green’s function for any structure is simply
obtained.
»
5.4
Impedance M atrix Interpolation
To improve the computational efficiency of the MoM code, we follow the same tech­
nique outlined in Aberle’s notes [84], where a cubic spline algorithm is used to inter­
polate the impedance m atrix at intermediate frequencies. A so-called natural spline
is used. The cubic spline algorithm used here differs from conventional cubic spline
algorithms because we are interpolating matrices and so do not wish to store them
in memory.
The MoM code computes the impedance matrix at a set of discrete frequencies
given by
fm
—
f st art 4" (m
1)
771 = 1, ..., iVy
(o. f9)
where
A/
=
f “° [ ~
(5.80)
IS j — 1
and N j is the number of frequency points at which the impedance matrix is calcu­
lated. We denote the impedance m atrix at frequency f m by Zm The first task in
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134
the development of the cubic spline interpolation algorithm is to compute the second
derivative matrices at each f m. We denote the second derivative m atrix at frequency
fm by Z"m. We have
Z'[ =
0
(5.81)
Nf - 1
Z"m =
£
(5.82)
n=2
Z N,
=
(5.83)
0
where T is m atrix whose inverse is an N j — 2 by N j —2 tridiagonal m atrix given by
T -i
=
a =
6 =
b
a
0
a 0 ••• 0 0
b a • ■■ 0 0
a b ••• 0 0
0
0 0 ••• a b
( A /) 2
(5.84)
(5.85)
2(A f f
(5.86)
and J3n is a m atrix given by
-Sn — Zn+x — 2Zn + Zn_!
(5.87)
To avoid storing more than a few Zn’s in memory at any one time, the impedance
matrices are read from disk as needed, and the second derivative matrices are written
to disk as computed. The algorithm for computing the B n matrices is based on
writing B n as
Bn
—
Qn
Q n—
1
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(5.8S)
135
where
Qn =
Zn+l - Zn
(5.89)
Thus, the algorithm for computing the B n is
Qi
— Z i — Z\
n = 2,..., Nf —1
for
Bn — Zn+1
Zn
Qn—1
Qn — B n + Q n-1
end
(5.90)
The algorithm can be implemented by storing three matrices at a given tim e (Q n- 1 ,
Zni 3*nd Bn)-
Once the Z"m have been obtained the cubic spline interpolated value of the
impedance m atrix at frequency / can be obtained as
Z ( /)
a Z m+ f3Zm+1 + 7Zm +
=
8 Zm+l
(5.91)
where
a
=
~ i f
1
(S-92)
/3
= I-a
(5.93)
7
= ^ Q ( a 2 - l) (A / ) 2
(5.94)
s
= i/j(/32 - l ) ( A I f
(.5.95)
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136
m
=
f - f st art
A/
+ 1
(5.96)
and [x] is the largest integer less than or equal to x, i.e.,
fm < f < fm+l
5.5
(5-97)
Results and Conclusions
To validate the results of our approach, we compared the numerical values of the
RCS components, for the dielectric to those in [85], [86], and [87]. The dielectric
case is the limiting case of ferrite when we set the ferrite parameters to zero. Figures
5.7, 5.8, 5.9, 5.10 and 5.11 indicate the results are identical.
For the in-plane biased cases, x-biased and y-biased, we compared our results of
<70# for the two cases against those in [52], and very good agreement is clear from
Figures 5.12 and 5.13. To make sure th at our approach is also correct and valid
for the multi-layer structure, we compared our results for the case of a ferrite cover
on the top of a dielectric patch against those in [29], the only case available in the
literature for a multi-later strucure. Figures 5.14 and 5.15 show good agreement.
In reference [52], the thickness of the ferrite cover is not mentioned; therefore, we
assumed it to be 0.5mm. If we knew the exact thickness, we might obtain a better
agreement.
Figures 5.16, 5.17 and 5.18 show the other elements of the RCS with and without
the bias field for the case given in [52], a patch on a ferrite substrate. In these figures,
the bias field as well as the propagation direction is the ^-direction. It is clear that
the first resonance peak is the same for the biased and unbiased cases. The invari­
ant location of the first resonant peak can be explained by investigating Maxwell’s
equation for the T M \ q cavity mode. The longitudinal magnetic field component, Hy,
and th e transverse electric field component, E z are almost unaffected by the bias
field. Therefore, the first patch resonance does not depend on the bias field [52].
This concept of analysis is also true for the fourth resonance, which is considered the
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137
resonance of T M 20 mode. In general, the resonances of T Mm0 modes are unaffected
by the bias field. For the resonance of the other modes, the bias field increases their
resonant frequency, but the RCS levels rem ain almost the same.
Figures 5.19, 5.20 and 5.21 show the other elements of the RCS with and with­
out the bias field for th at case given in [52]. In these figures, the bias field is the
x-direction while the propagation direction is still in y-direction. W ith the same ar­
gument as for the previous case, those resonances corresponding to the T M 0n modes
axe unaffected by the bias field. Yang et al. also explains this phenomenon [52].
We can notice also th at the RCS level is below —60 d B s m for frequencies less than
8 GHz .
Figures 5.22, 5.23 and 5.24 show the elements of the RCS with and without a
ferrite cover on the top as shown in Fig. 5.5 for the case given in [29]. The bias field
is assumed in the x-direction. Using the cavity model concept, the first resonance
is assumed to be for the T M \ 0 mode. W ith the ferrite cover, the first resonance is
eliminated. The third resonance has less RCS level compared with no ferrite cover
case. Under the assumption th a t the bias field and the propagation direction are
parallel, that structure can excite m agnetostatic surface wave in the frequency range
when fief / is negative. The frequency range for the negative fie/ / is give by [29]
\JUJq{u1q + UJm) < U! < UJq + U!m
(5.98)
In our case, the frequency range of the m agnetostatic surface wave is in the range
of 2.2G H z < f < 5.8GHz.
As a m atter of fact, the eliminated first resonance
frequencyi occurs in the cut-off region. In the frequency range of the magnetostatic
surface wave, the RCS level is decreasing with increasing the frequency [29].
Figures 5.25, 5.26 and Fig. 5.27 show elements of RCS with and without a ferrite
cover as seen in Fig. 5.5. The param eters are the same as in the prevous case except
the patch dimensions are interchanged, L = 3.0cm and W = 4.0cm. We can look for
this case from another view, keep the patch dimensions as it was before and change
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138
the bias field direction, from x to y. Using this argum ent, we can better understand
the new mode configurations. As in the case of the single biased ferrite patch, when
the bias field is along the propagation direction, the first resonance is barely affected
by the bias field. The surface wave has a strong effect on the second resonance.
Therefore, the second resonance is eliminated by adding the ferrite cover.
Using the phenomenon of RCS elimination due to the property of the magneto­
static surface wave, Yang in [29] suggests to design a switchable antenna. At one
bias level, the surface wave is a propagating wave, the antenna is “on” near the res­
onant frequency. At the other biased level, the m agnetostatic surface wave is highly
attenuating, and the antenna is “off”.
We studied the previous case, but the structure is inverted, ferrite underneath
the patch and dielectric layer as a cover. To define the effect of that reversal, we use
the same material param eters as in the previous case. Because of the fact that the
resonant frequencies of a patch are mainly determined by the patch’s size and the
m aterial underneath (not above) it, the RCS changes. Figures 5.28, 5.29, and 5.30
show plots of the RCS due to different polarizations.
As a novel antenna structure, we propose the cross patch shown in Fig. 5.31.
Using the linearly varying basis functions enables us to analyze such configurations.
The cross-patch dimensions are kept constant for all the results, Lc = Wc = 0.05cm.
The comparisons of the RCS of the cross patch for the dielectric case versus the RCS
of the full patch are shown for the co-polarized case in Figures 5.32 and 5.33 and for
the cross-polarized case in Fig. 5.34. The same comparison is repeated for the ferrite
case, the results are shown in Figures 5.35, 5.36 and 5.37. The effect of the bias field
is insignificant as seen from Figures 5.38, 5.39 and 5.40. In general, we notice that
the first resonance is almost at the same frequency for both patches, the cross and
the full. However, the RCS level is reduced. The cross patch excites less number of
modes than the case of full patch. That this is true can be seen by the reduction in
RCS resonances when the cross patch is used. The fact th at the RCS decreases may
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139
be due to the smaller physical area of the cross patch.
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140
Fertile layer
Dielectric layer
Df
^
Ground plane
Fig. 5.5: Geometry of a patch antenna with a biased ferrite as a cover layer
Free space
Df
Ferrite layer
Dd
Dielectric layer
Ground plane
Fig. 5.6: Fields and currents of a patch antenna with a biased ferrite as a cover layer
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141
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This work
Abe tie
Pozar
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S3 -40
-60
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-80
Frequency (GHz)
Fig. 5.7: Comparison of RCS for a microstrip patch antenna (M s =
d = 1.3 x 10-3 m, L = W = 1.3 x 10-2m, 0,- = 30°, <pt- = 45°)
H
q
= 0, er = L3.0,
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Thia work
Aberle
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-80
2
3
4
5
6
7
8
Frequency (GHz)
Fig. 5.8: Comparison of RCS for a microstrip patch antenna ( M
d = 1.3 x 10“ 3m, L = W = 1.3 x 10“ 2m,
= 30°, & = 45°)
3
=
H
q
= 0, er = 13.0,
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142
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2.9
3.0
3.1
3.2
3.3
Frequency (GHz)
Fig. 5.9: Compaxison of RCS for a microstrip patch, antenna (Ms = Hq = 0, eT = 4.0,
d = 3.0 x 10- 4 m, L = 1.25 x 10 - 2m, W = 2.5 x 10 - 2 m, 0,- = 45°, <?,- = 0°)
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ThU work
Aberle
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-80
4
8
12
16
20
Frequency (GHz)
Fig. 5.10: Comparison of RCS for a microstrip patch antenna (Ms = Hq — 0 , er — 12.8,
d = 6.0 x 10- 4 m, L = 0.55 x 10- 2 m, W = 0.4 x 10- 2 m, 0; = 60°, <?>,• = 45°)
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143
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This wort
Aberle
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-80
4
8
12
20
16
Frequency (GHz)
Fig. 5.11: Comparison of RCS for a microstrip patch antenna ( M s = H q = 0, er = 1‘2.8,
d = 6.0 x 10_ 4 m, L = 0.55 x 10- 2m, W = 0.4 x 10"2 m, 0,- = 60°, 4>{ = 45°)
-20
Thi* work
Yang et al.
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CO
05 -60
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-80
4
8
12
Frequency (GHz)
16
20
Fig. 5.12: Comparison of RCS for a microstrip patch antenna, the bias field is in the
y-direction, (4ttM 3 = 1780.0G, H0 = 360.0Oe, er = 12.8, d = 6.0 x 10- 4 m, L =
0.55 x 10_ 2 m, W = 0.4 x 10_ 2 m, 0,- = 60°, <f>i = 45°)
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144
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O S
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Frequency (GHz)
Fig. 5.13: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (47rM, = 1780.0G, Ho = 360.00e, er = 12.8, d = 6.0 x 10 - 4 m, L =
0.55 x 10“ 2m, W = 0.4 x 10"2 m, 0{ = 60°, & = 45°)
-10
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-30
CQ -40
-o
-60
-70
-80
1
2
3
4
5
Frequency (GHz)
Fig. 5.14: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4itM 3 = 1780.0G, H q = 300.00e, AH = 40.0G, er(f = 2.2, er/ = 13.0
d& — 1.3 x 10_ 3 m, dj = 0.5 x 10 -3m L = 4.0 x 10 - 2m, W = 3.0 x 10- 2 m. 0,• = 30°,
4>i = 45°)
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145
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Thi* work with a biued ferrite cover
Yang with a biued ferrite cover
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1
2
3
4
5
Frequency (GHz)
Fig. 5.15: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (AwMs = 1780.0G, Ho = 300.00e, AH = 40.0G, erd = 2.2, er/ = 13.0
dd = 1.3 x 10- 3 m, df = 0.5 x 10-3m L = 3.0 x 10~2m, W = 4.0 x 10- 2 m, 8 { = 30°.
4>i = 45°)
-20
Biased Ferrite
Unbiased Ferrite •
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Ias -60
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-80
4
8
12
16
20
Frequency (GHz)
Fig. 5.16: Comparison of RCS for a microstrip patch antenna, the bias field is in the
y-direction, (4ttMs = 1780.0G, H0 = 360.0Ge, eT = 12.8, d = 6.0 x 10_ 4 m, L =
0.55 x 10- 2m, W = 0.4 x 10~2 m, 0; = 60°, fa = 45°)
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146
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Biased Ferrite
Unbiased Ferrite *
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CQ -60
-80
-90
-100
Frequency (GHz)
Fig. 5.17: Comparison of RCS for a microstrip patch antenna, the bias field is in the
y-direction, (4xM s = 1780.0G, Ho = 360.OOe, er = 12.8, d = 6.0 x 10-4m, L =
0.55 x 10-2m, W = 0.4 x 10-2m, 0; = 60°, <p,- = 45°)
-30
Biased Ferrite
— Unbiased Ferrite •
-40
-60
BJ -70
-80
-90
4
8
12
Frequency (GHz)
16
20
Fig. 5.18: Comparison of RCS for a microstrip patch antenna, the bias field is in the
y-direction, (4irMs = 1780.0G, H0 = 36O.O0e, er = 12.8, d = 6.0 x 10_4m, L =
0.55 x 10-2m, W = 0.4 x 10_2m, 0; = 60°, o,- = 45°)
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147
Biaxed Ferrite
Unbiased Ferrite
os -60
Frequency (GHz)
Fig. 5.19: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (4 t M s = 1780.0(7, So = 360.0(7e, er = 12.8, d = 6.0 x 10-4m, L =
0.55 x 10_2m, W = 0 A x 10~2m, = 60°, fa = 45°)
-30
Biased Ferrite
Unbiased Ferrite
-40
«
-60
4
8
12
Frequency (GHz)
Fig. 5.20: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (4irM 3 = 1780.0(7, S 0 = 360.00e, er = 12.8, d = 6.0 x 10_4m, L =
0.55 x 10-27n, W = 0.4 x 10~2m, 6 { = 60°, <j>i = 45°)
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148
— Biased Ferrite
— Unbiased Ferrite
-40
03
QQ -60
Frequency (GHz)
Fig. 5.21: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (47nV/a = 1780.0(7, Ho = 360.00e, er = 12.8, d = 6.0 x 10-4 m, L =
0.55 x 10-2 m, W = 0.4 x 10_2m, 0,- = 60°, <£,- = 45°)
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149
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with a biased ferrite cover
without a ferrite cover
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1
2
3
4
5
Frequency (GHz)
Fig. 5.22: Comparison, of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4ttM 3 = 1780.0G, H 0 = 300.0Oe, A H = 40.0G, erd = 2.2, er/ = 13.0
dj, = 1.3 x lO-3 m, df = 0.5 x 10_3m L = 4.0 x 10_2m, W = 3.0 x 10-2 m, 9{ = 30°.
<Pi = 45°
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150
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with a biased ferrite cover
without &ferrite cover
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1
2
3
4
5
Frequency (GHz)
Fig. 5.23: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4irM3 = 1780.0G, Ho = 300.00e, AH = 40.0(7, erd = 2.2, er/ = 13.0
dd = 1.3 x 10_377i, df = 0.5 x 10-3m L = 4.0 x 10"2m, W = 3.0 x 10“2m, 9{ = 30°,
4>i = 45°
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151
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vrith a biased ferrite cover
without &ferrite cover
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Frequency (GHz)
Fig. 5.24: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4tcM„ = 1780.0G, H q = 300.00e, AH = 40.0G, er(* = 2.2, erj = 13.0
dd = 1.3 x 10-3m, dj = 0.5 x 10-3m L = 4.0 x 10-2m, W = 3.0 x 10-2m, 0,- = 30°.
4>i = 45°
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152
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with a biased ferrite cover
without a biased ferrite cover
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ID
3
&
r r ? -5 0
CQ -40
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-80
Frequency (GHz)
Fig. 5.25: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, ( 4 t M a = 1780.0G, H0 = 300.00e, A H = 40.0G, ertf = 2.2, erf = 13.0
di = 1.3 x 10- 3 m, d f = 0.5 x 10-3 m L = 3.0 x 10-2 m, W = 4.0 x 10_2m, 0,- — 30°,
4>i = 45°
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153
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with a biased ferrite cover
without a biased ferrite cover'
-20
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CQ -40
-60
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-80
Frequency (GHz)
Fig. 5.26: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4ttMs = 1780.0(7, Ho = 300.0(?e, A H = 40.0(7, er(f = 2.2, er/ = 13.0
di = 1.3 x 10-3m, dj = 0.5 x 10-3m L = 3.0 x 10-2m, 17 = 4.0 x 10-2m. 6 { = 30°,
4>i = 45°
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154
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with a biased ferrite cover
without a biased ferrite cover'
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-30
CQ -40
-60
-70
-80
Frequency (GHz)
Fig. 5.27: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4 ttM s = 1780.0C?, H q = 300.00e, A H = 40.0G, erd = 2.2, er/ = 13.0
dd = 1.3 x 10-3 m, dj = 0.5 x 10_3m L = 3.0 x 10~2m, W = 4.0 x 10- 2 m, 0,- = 30°.
4>i = 45°
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155
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s?
cn
O
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1
2
3
4
5
Frequency (GHz)
Fig. 5.28: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (AkM, = 1780.0(2, H0 = 300.00e, AH = 40.0G, erj = 2.2, ery = 13.0
di = 1.3 x 10_3m, dj = 0.5 x 10-3m L = 3.0 x 10-2 m, W = 4.0 x 10“2m, 0{ = 30°,
<f>i = 45°
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156
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3
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«
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Frequency (GHz)
Fig. 5.29: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4 kM s = 1780.0G, H q = 3OO.O0e, A H = 40.0(7, €rd = 2.2, er/ = 13.0
dd = 1.3 x 10-3 m, d f = 0.5 x 10_3m L = 3.0 x 10-2 m, W = 4.0 x 10_2m, 0,- = 30°,
= 45°
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157
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— Ferrite-Dielectric cover
•— Dielectric-Ferrite cover '
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CO -40
■o
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2
3
4
5
Frequency (GHz)
Fig. 5.30: Comparison of RCS for a microstrip patch antenna, the bias field is in the xdirection, (4wMs = 1780.0G, Ho = 300.00e, A H = 40.0G, erd = 2.2, er/ = 13.0
dd = 1.3 x 10-3m, df = 0.5 x 10_3m L = 3.0 x 10~2m, IF = 4.0 x 10"2m, 6 { = 30°,
< P i = 45°
Ferrite ___________;
:................................................................
/
r ^_____ Ground plane
Fig. 5.31: Geometry and dimensions of a cross patch compared to the full patch antenna.
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158
Unbiased Ferrite-Cross patch
Unbiased Ferrite-Full patch
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CQ -50
4
8
12
20
16
Frequency (GHz)
Fig. 5.32: Comparison of RCS for a microstrip patch antenna ( M s = H q = 0, er = 12.8.
<f = 6.0x 10-4 m, L = 0.55 x 10“ 2m, W = 0.4 x 10~2m, 0{ = 60°, & = 45°
)
Unbiased Ferrite-Cross patch
Unbiased Ferrite-Full patch
-40
co
CQ -60
4
8
12
16
20
Frequency (GHz)
Fig. 5.33: Comparison of RCS for a microstrip patch antenna (Ms = Hq = 0, er = 12.8,
d = 6.0 x 10-4 m, L = 0.55 X 10-2m, W = 0.4 x 10~2m, 9; = 60°, 4 > i = 45°
)
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159
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Unbiased Ferrite-Cross patch
Unbiased Ferrite-Full patch
-40
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-80
-90
-100
Frequency (GHz)
Fig. 5.34: Comparison of RCS for a microstrip patch antenna (Ms = Ho = 0, er = 1‘2.8,
d = 6.0 x 10_4m, L = 0.55 x 10-2m, W = 0.4 x 10-2m, 8 { = 60°, <pi = 45°
)
-20
Biued Ferrite-Cross patch
Biased Ferrite'FuIl patch
-30
-40
-50
CO
a s
-60
-70
-80
Frequency (GHz)
Fig. 5.35: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (47rMs = 1780.0G, Hq = 360.00e, er = 12.8, d = 6.0 x 10-4m, L =
0.55 x 10_2m, W = 0.4 x 10_2m, 9{ = 60°, <?>,• = 45°
)
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-30
— Biased Ferrite-Cross patch
— Biased Ferrite-Full patch
-40
-50
«
-60
-80
-90
-100
4
8
12
Frequency (GHz)
16
20
Fig. 5.36: Comparison of RCS for a microstrip patch antenna, the bias field is in the
ar-direction, (4t M s = 1780.0(7, Ho = 360.0d?e, er = 12.8, d = 6.0 x 10-4 m, L =
0.55 x 10~2m, W = 0.4 x 10" 2m, 9{ = 60°, & = 45°
)
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161
-30
Biased Ferrite-Cross patch
Biased Ferrite-Full patch
-40
-50
CQ
•o
-60
-70
-80
-90
-100
Frequency (GHz)
Fig. 5.37: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (4icMs = 1780.0G, H q = 360.00e, er = 1*2.8, d = 6.0 x 10~4m, L =
0.55 x 10-2m, W = 0.4 x 10~2m, 0{ = 60°, <?,- = 45°
)
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162
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Biased Ferrite-Cross patch
Unbiased Ferrite-Cross patch -
-40
-50
cn
-60
02
05 -70
-80
-90
Frequency (GHz)
Fig. 5.38: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (47rMj = 1780.0G, H q = 360.00e, er = 12.8, d = 6.0 x 10-4 m, L =
0.55 x 10-2m, W = 0.4 x 10"2m, 6 { = 60°, <p{ = 45°
)
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163
-40
Biased Ferrite-Cross patch
Unbiased Ferrite-Cross patch.
-50
-60
-80
-90
Frequency (GHz)
Fig. 5.39: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (4irMs = 1780.0G, H q = 360.00e, er = 12.8, d = 6.0 x 10~4m, L =
0.55 x 10-2m, W = 0.4 x 10-2m, 0,- = 60°, <f>i = 45°
)
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164
-30
Biased Ferrite-Cross patch
Unbiased Ferrite-Crass patch -
-40
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-60
OS -70
-80
-90
4
8
12
16
20
Frequency (GHz)
Fig. 5.40: Comparison of RCS for a microstrip patch antenna, the bias field is in the
x-direction, (4xM3 = 1780.0(7, H q = 360.00e, er = 12.8, d = 6.0 x 10“4m, L =
0.55 x 10_2m, W = 0.4 x 10-2m, 0,- = 60°, fc = 45°
)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
165
CHAPTER 6
CONCLUSION
A num ber of issues involving planar structures printed on biased ferrite are inves­
tigated in this dissertation. First, the transmission m atrix of the normally biased
ferrite substrate is derived in a closed form. Using the transmission m atrix, the
G reen’s functions for normally biased ferrite structures is formulated. The perfor­
m ance of three edge-guided microstrip isolators is studied. A full-wave MoM based
on Galerkin’s technique is utilized in this analysis. The resistive film termination on
one side of the microstrip to absorb the backward wave is considered in this work.
The effects of the resistive film, resistance value, and resistance width on the inser­
tion loss and isolation are studied. The insertion loss does not depend strongly on
the resistance since the forward wave propagates mainly in the resistance free region.
Second, the transmission m atrix for an arbitrarily biased ferrite substrate is also
derived in closed form. The generalized Green’s function for a multi-layer arbitrarily
biased ferrite structures is formulated. As an application of the generalized Green’s
function, two microwave ferrite devices are studied, a phase shifter biased in the
transverse direction and a magnetostatic surface wave transducer, biased in the lon­
gitudinal direction. Excellent agreement with previously published numerical and
experim ental results is achieved. The effect of magnetization angle is also studied.
T hird, the RCS of a biased ferrite antenna is evaluated. A 3-D full-wave MoM
based on Galerkin’s technique is utilized. The RCS formulation is done in a very
efficient and versatile manner. The transmission m atrix of the arbitrarily biased
ferrite substrate is used to derive the excitation vectors in a closed form. This novel
approach enabled us to analyze a variety of complex antenna geometries.
Last, a novel cross-patch antenna is presented. The RCS is computed and com­
pared to th at of the full-patch antenna. A reduction in both the number of resonant
peaks and RCS values is noted. In conclusion, this work makes five major contribu­
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166
tions to the study of microwave ferrite devices
1. Development of a closed form transmission m atrix for a normally biased ferrite
substrate
2. Full wave analysis of edge-guided mode microstrip isolator
3. Development of a closed form for arbitrarily biased ferrite substrate
4. RCS formulation for arbitrarily-shaped patches on arbitrarily-biased ferrite
sunstrates using the trasmission matrix approach
5. Analysis of a novel cross-patch antenna.
The future work includes but is not limited to the following areas,
• analysis of more complex microwave devices such as three-dimensional circula­
tors, isolators, phase shifters, and transducers
• analysis of antenna radiation patterns and input impedance.
• analysis the plasma materials that has a perm ittivity tensor instead of the
permeability tensor with the transmission m atrix technique.
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REFERENCES
[1] D. M. Pozar, Microwave Engineering. Addison-Wesley, 1990.
[2] K. G. Achintya and C. W. Denis, “Microstrip excitation of magnetostatic surface
waves: Theory and experiment,” IEEE Transactions on Microwave Theory and Tech­
niques, vol. MTT-23, pp. 998-1006, Dec. 1975.
[3] Peripheral Mode Isolator Operates From 3.5 to 11 GHz. Microwaves, Apr. 1969.
[4] M. E. Hines, “Reciprocal and nonreciprocal modes of propagation in ferrite stripline
and microstip devices,” IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 442451, May 1971.
[5] B. Chiron and G. Forterre, “Emploi des ondes de surace electromagnetiques pour la
realisation de dispositifs gyromagnetiques a tres grande largeur de band,” in Dig., Int.
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[6 ] M. Blanc, L. Dusson, and J. Guidevoux, “Etude de la fonction isolation a tres large
band utilisant des materioux ferrites,” in Dig., Int. Semin. Microwave Ferrite Devices.
1972.
[7] P. D. Santis and F. Pucci, “The edge guided wave circulator,” IEEE Trans. Microwave
Theory Tech., vol. MTT-23, pp. 516-519, June 1975.
[8 ] P. D. Santis, “Symmetrical four-port edge-guided wave circulators,” IEEE Trans.
Microwave Theory Tech., vol. MTT-24, pp. 10-18, Jan. 1976.
[9] M. E. Hines, “Ferrite phase shifters and multiport circulators in microstrip and
stripline,” in IEEE G-MTT International Microwave Symposium, pp. 108-109, 1971.
[10] M. E. Hines, “A new microstrip isolator and its application to distributed diode am­
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[11 ] R. C. Kane and T. Wong, “An edge-guided mode microstrip isolator with transverse
slot discontinuity,” in IEEE MTT-S Digest, pp. 1007-1010, 1990.
[12] M. E. Hines, “Ferrite transmission devices using the edge-guided mode of propaga­
tion,” in IEEE G-MTT Int. Microwave Symp. Dig., 1972.
[13] G. Cortucci and P. De Santis, “Edge-guided waves in lossy ferrite microstrips,” in
Proc. European Microwave Conference, 1973.
[14] P. De Santis, “Fringing-field effects in edge-guided wave devices,” IEEE Trans. Mi­
crowave Theory Tech., vol. MTT-24, pp. 409-415, July 1976.
[15] K. Araki, T. Koyama, and Y. Naito, “Reflection problem in a ferrite stripline,” IEEE
Trans. Microwave Theory Tech., vol. MTT-24, pp. 491-498, Aug. 1976.
[16] K. Araki, T. Koyama, and Y. Naito, “A new type of isolator using the edge-guided
mode,” IEEE Trans. Microwave Theory Tech.. vol. MTT-23 (Letter), p. 321. March
1975.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
168
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APPENDIX A
TH E TRANSMISSION MATRIX OF A NORMALLY BIASED FER R ITE SLAB
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
174
Jl
Et
z=0
z=-d
Fig. A.l: Geometry of single layer structure.
In this Appendix, the transmission m atrix is derived in the spectral domain for a
magnetic substrate of thickness d as shown in Fig. A.I. The symbolic manipulator
of Maple Vis extensively utilized to alleviate the huge algebraic burden and to obtain
the solution of a fourth order differential equation. To find the t r ansmission m atrix
for a ferrite slab, we start from Maxwell’s equations
—jupo ft -H
=
V x E
V • (pQft ■H )
=
0
(A.l)
juieE
=
V x H
eV • E
= 0
where ft is the permeability tensor of the ferrite. For a ferrite magnetized in the
z-direction, the permeability tensor is given by
fl
f1 = Pa
—jk 0
JK
ft
0
0
0
1
where
K
=
Wm =
tZ)2
o —up
74 7TiV/s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A.2)
175
u
=
Uo =
u 0 +. -3
7 Ho
wo is the precession frequency, Ho is a z-directed impressed DC magnetic field,
the gyromagnetic ratio, T =
7
is
is the relaxation time, and AH is the 3-dB line
width. To simplify the formulation, the wave equation is expressed in term s of fields
th at axe perpendicular to the direction of magnetization.
From Maxwell’s equations, we can write
' V x £ ) - k ltTE = 0
Vx(/i
(A.3)
where ko = u}y/fi0 e0. Manipulation of (A.3) yields three scalar equations
r\2
r\ 2
A
a
~ k 0 ^ A ) E* + { - K x K y - JK— ) Ey +(-K l<y— + j f i Kx — ) E z= 0
(A.4)
Q 2
^
a2
^
A
A
( - K xK v + j k — ) E x + (I< 1 -n — - k le r & ) E y +{]nKy— + nK x — ) E z= 0 (A.5)
d
^
d
d
(jI<xf i + K yK— ) Ex +(-I<xK— + j K yn — ) Ey + (/i/^ + /x /^ -fc o £ r A) E z= 0 (A. 6 )
Eliminating E z from the three equations and rearranging such th at
and
are separated, we obtain
d 2 Ex
.
W9 q z 2 + (w 7 w 2 — w3 ws)Ex + (W7 W4 — w3 w 2 )Ey = 0
(A.7)
d '2E
— w9~q^T + ( ^ 5 ^ 4 - wiw 2 )E y + {w5 w 2 - wiw 6 )Ex = 0
(A.8 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
176
where
u>i
= K x K y([i2 — k 2) — jC ik .
w2
= CxK xK y - iiK x K y( K 2x + K 2)
wz
= C\n —K x2{fi2 — k 2 )
wA = fiKKKl + K D - ^ K l + K D + C ^ - K l )
w5
= Cin — K y2(fi2 — k 2)
we
= fiK y{K l + K 2) —
w7
= K xK y(/j.2 — k2) + jClK
Wg
=
Cl
= k%erA
A
=
+ K 2) + Cx{Ci - I<1)
W 7W \ — W 2 W 5
(J2 — k 2
Next, wedefine elliptical polarization components as
E+ =
Ex + j E y
(A.9)
E~
Ex —jE y
(A .10)
=
Noting th at Ex = \{ E + + E~) and E y = -^-(E+ —E ~ ). we can rewrite (A.7) and
(A .8) as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Decoupling equations (A .11) and (A.12) generates the following fourth order differ­
ential equations
dAE +
^
-
d2E +
+ A
d zA
^
-
+ A
^ - o
(A .13)
= 0
dz2
(A .14)
where
, _ 2kltr{n 2 - K2) - K_K +(fl + K + 1)
1
f
2Arger (/i2 - K2) — A ./v + (ft — K + 1)
2 ( / i + /c)
=
. 2k%£r(fl2- K2) -
2
K - K + (fl + K + 1)
2 ( f i -(-
K 2+ K 2_ { p -
2(p -
k
k
)
2fc026r ( ^ 2 - K2)- K - K + j f l - K + 1)
)
2{y. —
k
)
K - l){fl + K + I)
4 (fJL — K,)(fl + K)
K+ = K x + jK y
(A .15)
K - = K x — jA 'y
(A .16)
The solutions of (A .13) and (A. 14) can be w ritten as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
178
E + = A \s in h (R \z ) + B ico sh(R iz) + A 2 sin h (R 2 Z) + B 2 Cosh(R2 z)
(A .17)
E
(A .18)
= A zsinh(R xz) + B zcosh(R \z) + A 4 sin h (R 2 z) + B \cosh(R 2 z)
where the constants A’s and B’s are determ ined from the boundary conditions, and
R i and i ?2 are the roots of the differential equations given by
Ri
Ri
=
~
=
\-
2
W
+ V%
£
(A.19)
~
-
h
(A.20)
a
From Maxwell’s equations, we can write the relationship between the elliptical com­
ponents of the magnetic field and the electric field in m atrix form as
'
H+
tr ­
fC=K±
ue
'
=
a
T
_ k -2
KL
2
■Z= £ + k+*
'
a£+
'
L
dj:
8E~
dz
J
(A.21)
where
k±2 =
Ai
K 2 + K 2 - u } 2fi0e { fi^ K )
— uj2 fi0ef i ( K2 + K 2 — u)2floe- ---------
Substitution of (A.17) and (A.18) into (A.21) yields after some manipulation
H + = D i[A icosh(R iz) + B isin h (R iz)] + D 2 [A2 cosh{R 2 z) + B 2 sin h (R 2 z)] (A.22)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
179
H = D 2 [AiCosh(Riz) + B \s in h { R \z )] + D 3 [A2 Cosh(R 2 z) +
At z
B 2 sin h (R 2 Z)] (A.23)
= 0,we can evaluate the constants Ax, A 2 , Bx and B 2 as follows
Ax =
D 5 H + + D 6 H~
(A.24)
A2 =
DsH + + D 7 H~
(A.25)
Bx =
B2 =
U2
- E + + ----- — E-
77.2 — 711
712 — ^1
---- — E+ + ---- 1-----1
712 —^1
^2 —
(A.27)
where
1 =
u e „ ,K + K a T * ' (^
—
we „ , K 2
„
d 2
=
D3 =
, ,
,
K l,
+ "'f )
K+K-
—
,
V ~ + k ' ) ni )
^ R 2( - ^ + ( ~ K ± ^ + k+2 )n2)
4 =
u e n ,K + K 2(^
. _2
5
uj2e2R2Ri(n2 —rii )
(A -26)
/fL
+ n2_2 ^
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ISO
________ Ax________
Dr =
u 2 e2 R 2 R i(n 2 — ni)
-A x
Do
u 2 e2 R 2 R i{n 2 —n x)
D* =
T li
Di
9i + -^1
=
92
9i + R 2
n2 =
92
9\
=
92
=
2k ltr (n 2 — K2) — K - K + ( f l + k + 1)
2 {fi + k)
K+(p + K — 1)
2 (n -f K)
Equations (A.25)-(A.27) can be w ritten in m atrix form as follows
-Ai
a2
= M -1 (0)
Bi
. b2 .
'
E+
'
bt
(A.28)
Ht
where
0
0) =
_n2_
Tl2— Tli
0
0
-i
f l 2 “ Tlx
1__
„
712
7 1 2 — TU
Ds
Ds
0
0
Dr
0
0
Following the same procedure for z = —d, we obtain
El
b2
Ht
.h 2
'
'
.
A\
a2
= M (-d )
Bi
. b2
(A.29)
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LSI
where
—sinh(R id)
—nisin h (R id )
Dicosh(Rxd)
D 2 cosh(R\d)
M {-d ) =
cosh(Rid)
cosh(R 2 d)
n 2 cosh(R 2 d)
■riicosh(Rid)
—D isinh(R id) —D \sin h {R 2 d)
—D 2 sin h (R id ) —D zsinh{R 2 d)
—sin h (R 2 d)
—n 2 sin h (R 2 d)
D icosh(R 2 d)
Dzcosh{R 2 d)
From equations (A.28) and (A.29) we can write
'
Et
e2
Ht
h2
'
E f
a2
Ef
= M (- d )
= M ( - d ) M ~ l {0)
B!
Ht
. b2_
.H r
Et
El
{d)
Ht
. nr
'
'
=f
(A.30)
where
=±
(A.31)
We use the following relations between th e elliptical and cartesian field and current
components at both sides of the slab
' E f
Er
= [5]
Ht
.n r .
E X2
-
Ejn
Jx 2
Jy? .
(A.32)
' E f
Eyi
Er
= [5]
Jxi
Ht
.n r .
. Jyi .
'
(A.33)
where
[5] =
J
1 -J
0
0
0
0
0
1
o
1
0
0
-J
J
1
1
(A.34)
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
182
Using equations (A.30), (A.32) and (A.33) we can write
EXl
Eyi
JXx
’K '
Tyi
= [S]-1^ (d)[S]
J 12
. 4
(A.35)
. Jyi .
.
From th e above equation we can write the transmission m atrix for normally biased
ferrite slab
=±
T(d) = [S]-l T (<f)[S]
(A.36)
where th e elements of the transmission m atrix are given by
T \\
=
T-jf = —---------- -[(n2 + nxn2 — 1 —ni)cosh(Rid) + (—ni —/ix/i2 + 1 + ri2 )cosh(Rul)]
Z{n2 — ni)
2(n2 T\z
[(n2 + 7ixn2 + 1 + ni)cosh(Rid) + (—Tix —n \n 2 — 1 —n 2 )cosh{R 2 d)\
T lx )
—D s)sinh(R id) + (n 2 + 1)(Z?8 — D 7 )sin h (R 2 d)\
=
T \4 =
Z l2 = ~^ [(tix + l)(f ^5 + D o)sinh(Rid) + (n 2 + 1)(-D8 + D 7 )sin h (R 2 d)]
T2i
T* =
=
T 22 =
T 23
=
2( ti2 - Tlx)
[(—712 + 71x712 + 1 —ni)cosh(R id) + (nx —riin 2 — 1 + n 2 )cosh{R 2 d)\
-1
T 22 = ^- n _ n j [(—:n 2
Z 2l
=
T 24 — Z 22 —
- [ ( - n x
+
1 )(A >
+ 71x 712
-
— 1 + ni)cosh(R id) + (rii —n xn 2 + 1 - n 2 )cosh(R 2 d)\
D e)sinh(R id) + ( - n
2 +
1
)(D 8 - D 7 )sin h (R 2d ) ]
Tix + l)(-^5 + D 6 )sin h (R ld) + (—n2 + 1)(-D8 + D 7 )sinh(R 2 d)]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
183
^31
= Yu = 2(n 2 — Tii) ^
^32
= Yyi = ^
~ l ^ Dl ~ Di) s in h (R i d) + (n i “ 1)(-°3 - D 4 )sin h (R 2 d)]
n ”) [(” 2
—^ 2 )sinh(R id) + (ni -f-1 )(.0 3 —D 4 )sin h (R 2 d)]
f 1Jl = ^[ ( D 1 - D 2 ){D 5 - D 6 )cosh(Rl d) + (D 4 - D
faa =
T34
=f
241
=
^42 =
n2
(2 =
5^2
3 )(Ds
- D 7 )cosh(R 2 d)}
J-[{D t - D 2 ){D 5 + D 6 )cosh(Rl d) + (D 4 - D 3 )(DS + D 7 )cosh(R 2 d)}
=
2^
--- n "")^772 _
=
777---------- r [(n 2
Z{n2 — n 4)
+
D 2 )sinh(R id) + (—ni -f- 1 )(Z?4 + D 3 )sin h (R 2 d)]
+ D 2 )sinh(R id) — (ni + 1)(-O4 -t- D 3 )sin h (R 2 d)]
Taz = T i = y [(£>! + D 2 )(D 5 - De)cosh(Rl d) + (D 4 + D 3 )(D 8 - D 7 )cosh(R 2 d)}
T44 =
T 22 —
+ D 2 ){D$ + Ds)cosh(Rid) + (D 4 + D 3 )(D 8 + Dr)cosh(R 2 d)]
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APPENDIX B
THE TRANSMISSION MATRIX OF A DIELECTRIC SLAB
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185
The transmission m atrix of a dielectric slab can be w ritten as [40],
=E ~ J
T — T — cosh(kd)
Z
1 0
0 1
(B .l)
= yr^-sinh(kd)
kkj.
kl ~ kd
k*ky
kxky
ky - k\
= Y^-sinh{kd)
rCKfl
kl ~ kd
(B.2)
Y
Y
k x ky
kx ky
kl — kj
(B.3)
where,
zc
=
y=
(B.4)
kd
=
x /^ o
(B.5)
k
=
+ fc»2 - k i
(B.6)
where d, is the dielectric slab thickness, er , is the dielectric constant, rj is the freespace intrinsic im pedance and to, is the free-space wave number.
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APPENDIX C
TH E SEMI-INFINITE SPACE GREEN’S FUNCTION
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187
E
J
Air
Ground
Planes
Ferrite
t
I
E
J
Fig. C.l: Geometry of single layer isolator structure.
___ +
E
.
T
*ZU
= T
—
-1 a
J
.
' =E
=
fa
- T
= T '
■
=J
^
■
E
SUL
J
(C .l)
We force E = 0 by placing a ground plane at distance dj, from the source plane, as
shown in Fig. C .l. This results in the following equation
= T = J ~ l
!_
E
=
ZaTa
=
GaJ
J
(C
.2 )
In the above equation, Ga has units of impedance. Most of the semi-infnite G reen’s
functions has the units of admittance. Therfore, the admitance Green’s function can
be w ritten as
= J —T ~ l
—+
J
=
T aZ a
E
(C.3)
Using the elements of the dielectric trasmission m atrix, we can write
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1SS
= -1
G„ =
JYC
kkdtanh{kd)
kj - kj
- k xky
- k xky
kj-kj
|t£—>00
(C.4)
After taking th e limit, the adm ittance Green’s function has the following form
r
a
kkd
kj~kj
- k xky
~ k Xky
k j - kj
(C.o)
where,
Yc v
(C.6)
kd =
IjJy/JtQt
(C.7)
y/kj + kj - kj
(C.S)
k =
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APPENDIX D
RESISTIVE MATRIX ELEMENTS IN SPATIAL DOMAIN
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190
I
2
[3
4
5
Conductor
(»)
!y
Resistive L ay er Rs
(b)
Fig. D.l: Current basis functions for (a) Longitudinal current (b) Transverse current
We selected the piece-wise linear subdomain basis function as shown in Fig. D .l.
The conductor is subdivided into N equal overlapped segments. The elements of the
submatrices of th e total resistance m atrix given in equation 3.1. R xx and Ryy, are
found to be
i
f
R y y — R$d>
00
3
1
6
0
1
6
1
3
0
0
0
2
3
1
6
0
1
6
1
3
0
0 0
Similar matrices can be easily derived for a different number of basis function
and for a different location of the resistance.
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APPENDIX E
GREEN’S FUNCTION FO R A SLOT LINE W ITH A FERRITE SUBSTRATE
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Fig. E.l: Geometry of single layer slot line structure.
E .l
Green’s Function for a Single Slot Line Ferrite Substrate
The Green’s function for a ferrite slot line relates the slot currents to the slot fields
as
Jx = Gs l E x
(E .l)
The electric field is continous on surface (1) and (2)
Et
= E~=EX
E+ =
E~ = E 2
(E.2)
(E.3)
(E.4)
where E xand E 2 are the tangential electric fields at the boundaries of the layer, J *
and J } are the currents due to the electric fields in the upper and lower semispaces
(above surface (1) and below surface (2), respectively), these currents are defined as
J+
=
G 1 E+ = Gl E l
(E.5)
J <2
=
G 2 E 2 = G2 E 2
(E.6)
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193
(E.T)
where G\ and G 2 are semi-infinite space Green’s functions above surface (1) and
below surface (2), respectively.
The total surface currents at both sides of the slab can be written as J x and J 2
Jx =
aB x ( f f + - f f + ) = J + - J f
(E.8)
J2 =
dn x { H t - H t ) = J t ~ J 2
(E.9)
(E.10)
Note the an is +ve in the z-direction for surface (1) and -ve in the z-direction for
surface (2).
The surface current at surface (1) is due to the slot electric field while there is no
current source at surface (2) hence, J 2 = 0
J2
— —J 2 — —G 2 E 2
Jx
=
J t —J\ — G \E \ — J\
(E .ll)
(E.12)
The transmission m atrix relates the electric fields and the electric currents just inside
the slab on both sides.
r
____
e
.
J
^
2
" 2
.
'
=E
-T
f
z
=T
=J
Y
T
'
_ —_
Ex
From equations E.12, E.12 and E.13 we can write
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194
= E
= T
=E
__
=T
_____ = T _
E 2 = T E x ~h Z J\ = T E \ + Z G \E \ — Z J \
=
J2
=
t
_
=
j
___
=
t
__
=
j
(E.14)
_____ - J _
- G 2 E 2 = Y E 1 + T J x = Y E i + T G \E \ - T
Jx
(E.15)
S ubstitute by E 2 from equation E.15 to E.15 and after some manipulation we get
-T
J,
= [Gx + {G2Z
=
E.2
- J
_
-E
+ T )\--l/\ G" 2T
=T
+Y)]El =
Gs l E x
(E.16)
(E.17)
Green’s Function for a Multilayer Slot Line Ferrite Substrate
Jl.
Fig. E.2: Geometry of single layer slot line structure.
Before the second layer is added, we have that the current and the electric field below
surface
2
are related by
J 2 = G2 E 2
(E.1S)
After adding the second layer, the relation between J 2 and E 2 is no longer simple.
At surface (3), there is no current source so we can write J 3 = 0 and
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195
(E-19)
J 3 = G3E 3
The transmission m atrix of the new added layer can be w ritten as
—_
e
2
' =E
—
. ^2 .
T
=T
Z
= T
=J
Y
f
'
_
e3
—+
(E.20)
. ^3 .
From equation E.19 and E.20 we can find the new relation between J 2 and E 2 as
J2 =
G2 E 2
_
=r = j_
= e
= t_
G 2 = (Y + T G3)(T + Z G 3 ) - 1
(E.21)
(E.22)
The above expression for G 2 replaces the G 3 in the main equation E.17 and G 3
becomes the new semi-infinite free space Green’s function below the newlayer num ber
(3).
Noticealso th at
the elements of the new layer transmsition m atrix are quite
different than the transmission matrix of the m ain ferrite layer, but because of the
simplicity we used the same symbols without any extra notation.
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