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Finite difference time domain analysis of microwave ferrite devices and mobile antenna systems

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FIN ITE DIFFERENCE TIM E DOMAIN ANALYSIS OF
MICROWAVE FERRITE DEVICES AND MOBILE ANTENNA SYSTEMS
by
Bahadir Suleyman Yildirim
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
May 1998
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
UMI Number: 9821688
Copyright 1998 by
Yildirim, Bahadir Suleyman
All rights reserved.
UMI Microform 9821(88
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FINITE DIFFERENCE TIME DOMAIN ANALYSIS OF
MICROWAVE FERRITE DEVICES AND MOBILE ANTENNA SYSTEMS
by
Bahadir Suleyman Yildirim
has been approved
December 1997
APPROVED:
5
7
t E L L ^ 1
7W -
. Chair
7
Z
C eJL
2
ACCEPTED:
Department Chair
Dean, G raduate College
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ABSTRACT
This dissertation presents analysis and design of shielded mobile antenna systems
and microwave ferrite devices using a finite-difference time-domain method. Novel
shielded antenna structures suitable for cellular communications have been analyzed
and designed with emphasize on reducing excessive radiated energy absorbed in user’s
head and hand, while keeping the antenna performance at its peak in the presence of
user. These novel antennas include a magnetically shielded antenna, a dual-resonance
shielded antenna and, a shorted and truncated microstrip antenna. The effect of mag­
netic coating on the performance of a shielded monopole antenna is studied exten­
sively. A param etric study is performed to analyze the dual-resonance phenomenon
observed in the dual-resonance shielded antenna, optimize the antenna design within
the cellular c o m m u n ic a tio n s band, and improve the antenna performance. Input
impedance, near and far fields of the dual-resonance shielded antenna are calculated
using the finite-difference time-domain method. Experimental validation is also pre­
sented. In addition, performance of a shorted and truncated microstrip antenna has
been investigated over a wide range of substrate parameters and dimensions.
Objectives of the research work also include development of a finite-difference
time-domain technique to accurately model magnetically anisotropic media, includ­
ing the effect of non-uniform magnetization within the finite-size ferrite material due
to demagnetizing fields. A slow wave thin film isolator and a stripline disc junc­
tion circulator are analyzed. An extensive param etric study calculates wide-band
frequency-dependent parameters of these devices for various device dimensions and
material parameters. Finally, a ferrite-filled stripline configuration is analyzed to
study the non-linear behaviour of ferrite by introducing a modified damping factor.
iii
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I especially dedicate this dissertation to my m other Firdevs Yildirim, my father
Mustafa Yildirim, and my wife Maura Yildirim for their deep love and continuous
support.
iv
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TABLE OF CONTENTS
Page
LIST OF F I G U R E S .................................................................................................
ix
CHAPTER
1
IN T R O D U C T IO N ......................................................................................
1
2
FINITE-DIFFERENCE TIME-DOMAIN M E T H O D ..........................
5
2.1
Introduction.......................................................................................
5
2.2
FDTD M e th o d .................................................................................
6
2.3
2.4
2.2.1
FDTD Updating E q u a tio n s ..........................................
6
2.2.2
Stability C onsiderations.................................................
11
2.2.3
Source of Excitation and Feed M o d elin g .....................
12
2.2.4
Numerical Dispersion.......................................................
13
2.2.5
Frequency Dependent P aram eters.................................
14
Absorbing B o u n d a rie s....................................................................
15
2.3.1
Introduction
....................................................................
15
2.3.2
Mur A B C ..........................................................................
16
2.3.3
Berenger’s Perfectly Matched Layer( P M L ) ................
17
A Non-Uniform FDTD G r i d ...........................................................
22
v
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CHAPTER
2.5
3
Page
Summary and Conclusions..............................................................
24
SHIELDED ANTENNAS FOR MOBILE COMMUNICATIONS . . .
25
3.1
In tro d u ctio n .......................................................................................
25
3.2
Near-to-Far Zone T ransform ation.................................................
27
3.2.1
I n tr o d u c tio n ...................................................................
27
3.2.2
Surface Equivalence T h e o r e m .......................................
28
3.2.3
Phasor Field Q u a n titie s .................................................
30
3.2.4
Far Zone Radiated F i e l d s .............................................
31
3.2.5
Far Fields of a Linear Dipole A n t e n n a .......................
36
A Magnetically Shielded Monopole A n te n n a ..............................
37
3.3.1
Theory of Magnetic S h ie ld in g .......................................
37
3.3.2
Source of Excitation and Feed M o d e lin g ....................
40
3.3.3
FDTD A nalysis.................................................................
41
A Dual Resonance Shielded Cellular Phone A n t e n n a ................
49
3.3
3.4
3.4.1
Theory of Dual R esonance..............................................
49
3.4.2
Param etric Study of an Initial Antenna Configuration
51
3.4.3
Calculation of Input Impedance and Experimental Ver­
ification ............................................................................
VI
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55
CHAPTER
3.5
3.6
4
Page
3.4.4
Near and Far Field Measurements
.............................
62
3.4.5
FDTD Calculation of Near and Far F i e ld s ................
64
A Shorted and Truncated Microstrip A n te n n a ..........................
73
...................................................................
73
3.5.1
Introduction
3.5.2
Calculation of Input Impedance and Near Fields
. .
75
Summary and C onclusions.............................................................
78
TIME DOMAIN ANALYSIS OF MICROWAVE FERRITE DEVICES 82
4.1
In tro d u ctio n .......................................................................................
82
4.2
Theory of Magnetized F e r r i t e .......................................................
83
4.2.1
Maxwell's Equations in Anisotropic M e d ia ................
83
4.2.2
T he Equation of Motion of the Magnetization Vector
84
4.2.3
Small Signal Analysis of Ferrite Biased in X Direction
86
4.2.4
Small Signed Analysis of Ferrite Biased in Z Direction
90
4.3
4.4
A Finite-Difference Time-Domain Algorithm for Ferrite . . . .
91
4.3.1
Introduction
...................................................................
91
4.3.2
Derivation of Updating Equations for Ferrite Media .
94
A Slow Wave T hin Film Isolator...................................................
97
4.4.1
97
Introduction
...................................................................
vii
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CHAPTER
4.5
4.6
Page
4.4.2
Theory of Isolation Using M agnetostatic Surface Waves 98
4.4.3
Isolator Measurements and FDTD A n a ly s is .............
99
A Stripline Disc Junction C ir c u la to r .........................................
109
4.5.1
Introduction
...................................................................
109
4.5.2
Frequency Dependent Parameters of the Circulator .
110
4.5.3
Non-Uniform M agnetization.....................................
114
4.5.4
FDTD Analysis and V a lid a tio n ..............................
116
The Non-Linear Phenomena in F e r r i t e ......................................
127
4.6.1
Introduction
127
4.6.2
Modified Damping Factor and FDTD Algorithm
4.6.3
FDTD Analysis of a Ferrite-Filled S t r i p l i n e ........
...................................................................
. .
128
130
Summary and C onclusions............................................................
133
CONCLUSION............................................................................................
135
R E F E R E N C E S ............................................................................................................
138
4.7
5
138
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LIST OF FIGURES
Figure
Page
2.1 Electric and magnetic field vectors on Yee’s cell.......................................
6
2.2 Gaussian and Rayleigh pulses in time domain, /? = 64 and A t = 5-10-12
seconds............................................................................................................
13
2.3 Frequency spectrums of Gaussian and Rayleigh pulses............................
14
2.4 Implementation of Berenger’s PML ABC on a 2-D FDTD grid
21
2.5 A simple non-uniform mesh.........................................................................
23
3.1 Evolution from (a) a classical monopole to (b) the magnetically shielded
antenna, and finally (c) the dual-resonance shielded antenna................
26
3.2 Surface equivalence theorem.........................................................................
29
3.3 Vectors used in far zone field analysis........................................................
33
3.4 Transient input current of the 0.5A dipole antenna..................................
37
3.5 Azimuth plane directive gain in polar coordinates. Directivity of the
0.5A dipole antenna is calculated as 2.21 dB............................................
38
3.6 Comparison of elevation plane normalized directive gains of the 0.5A
dipole calculated by FDTD and theory (under zero wire diameter as­
sumption).......................................................................................................
ix
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39
Figure
3.7
Page
A shielded monopole antenna mounted on a finite size perfectly con­
ducting ground plane....................................................................................
3.8
The four electric field components are driven by a source voltage to
model physical coaxial cable connection to the monopole antenna. . .
3.9
40
42
Cross sections of the actual magnetically shielded antenna (left) and
its rectangular implementation used in FDTD simulations (right). . .
44
3.10 Comparison of input resistance for unshielded, shielded and magneti­
cally shielded antennas. Coating m aterial is 4.00mm thick...................
46
3.11 Comparison of input reactance for unshielded, shielded and magneti­
cally shielded antennas. Coating m aterial is 4.00mm thick...................
46
3.12 Comparison of input resistance for unshielded, shielded and magneti­
cally shielded antennas, <r* = 28425 D /m ..................................................
47
3.13 Comparison of input reactance for unshielded, shielded and magneti­
cally shielded antennas, a* = 28425 D /m ..................................................
47
3.14 Normalized electric field magnitude along a straight line. Each case is
normalized to its own maxima. The monopole is located at 0 mm, and
the shield is located at +8 mm....................................................................
48
3.15 Electric field magnitude is normalized with respect to the monopole
antenna along a straight line. The monopole is located at 0 mm, and
the shield is located at +8 mm....................................................................
48
3.16 Resonances of shielded, shielded and shorted, and unshielded monopole
antennas..........................................................................................................
x
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51
Figure
Page
3.17 Cross section of the (a) actual dual-resonance shielded antenna and (b)
staircase approximation used in FDTD simulations.................................
52
3.18 Dual-resonance shielded antenna on a finite ground plane......................
52
3.19 Delta-gap feed model (a) for a wire represented by a single cell and (b)
for a thick wire resolved by many cells.......................................................
53
3.20 Comparison of delta-gap and coaxial cable feed models. Ll=30, L2=64,
L3=70, D l= 8 , D2=8, H=112 (mm)............................................................
3.21 Effect of L l over input resistance.
54
L2=64, L3=70, D l= 8, D2=8,
H=112 (m m )...................................................................................................
55
3.22 Effect of L l over input reactance. L2=64, L3=70, D l=8, D2=8, H=112
(mm).................................................................................................................
3.23 Effect of L2 over input resistance.
56
L l=30, L3=70, D l= 8, D2=8,
H=112 (m m )...................................................................................................
56
3.24 Effect of L2 over input reactance. Ll=30, L3=70, D l=8, D2=8, H=112
(mm).................................................................................................................
3.25 Effect of L3 over input resistance.
57
L l=30, L2=64, D l=8, D2=8,
H=112 (m m )...................................................................................................
57
3.26 Effect of L3 over input reactance. Ll=30, L2=64, Dl=8, D2=8, H=112
(mm).................................................................................................................
58
3.27 Effect of D2 over input resistance. L l=30, L2=64, L3=70, D l= 8 ,
H=112 (m m )...................................................................................................
xi
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58
Figure
Page
3.28 Effect of D2 over input reactance.
L l= 30, L2=64, L3=70, D l= 8,
H=112 (m m )...................................................................................................
59
3.29 A non-uniform mesh in x-y plane is used in FDTD simulations of the
dual-resonance shielded cellular phone antenna.......................................
61
3.30 Calculated and measured input resistance of the dual-resonance shielded
cellular phone antenna..................................................................................
62
3.31 Calculated and measured reactance of the dual-resonance shielded cel­
lular phone antenna.......................................................................................
63
3.32 H-plane far field measurements of the dual-resonance shielded antenna
in the absence of user....................................................................................
65
3.33 E-plane far field measurements of the dual-resonance shielded antenna
in the absence of user....................................................................................
66
3.34 H-plane fan: field measurements of the dual-resonance shielded antenna
in the presence of user...................................................................................
67
3.35 Normalized near field measurements of the dual-resonance shielded an­
tenna in the absence of user........................................................................
68
3.36 Calculated H-plane far field pattern of the dual-resonance shielded an­
tenna in the absence of user........................................................................
69
3.37 Calculated E-plane far field pattern of the dual-resonance shielded an­
tenna in the absence of user........................................................................
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70
Figure
Page
3.38 Near electric field distribution of the dual-resonance shielded antenna
in x-y plane in the absence of user..............................................................
71
3.39 Near electric field distribution of the dual-resonance shielded antenna
in y-z plane in the absence of user..............................................................
72
3.40 A classical open microstrip antenna (left), ground plane and resonant
patch have the same dimensions (right).....................................................
74
3.41 A shorted and truncated microstripantenna..............................................
75
3.42 Input resistance of the shorted and truncated microstrip antenna for
different ground plane sizes. L=30 mm, W =30 mm, H=5 mm, er = 10.2. 76
3.43 Effect of substrate permittivity over the resonance of the shorted and
truncated microstrip antenna. L=30 mm, W =30 mm, H=5 mm. . . .
76
3.44 Effect of substrate thickness over the resonance of the shorted and
truncated microstrip antenna. L=30 mm, W =30 mm, er = 10.2. . . .
77
3.45 Near electric field distribution of the shorted and truncated microstrip
antenna in x-y plane.....................................................................................
78
3.46 Near electric field distribution of the shorted and truncated microstri
p antenna in y-z plane..................................................................................
4.1
79
(a) The regular FDTD algorithm and, (b) the modified FDTD algo­
rithm in the presence of ferrite....................................................................
93
4.2
Cross section of the thin film isolator........................................................
100
4.3
Top view of the th in film isolator...............................................................
100
xiii
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Figure
4.4
Page
A non-uniform mesh is applied in z direction to accurately model ferrite
substrate.........................................................................................................
4.5
Various material levels of the thin film isolator. Higher level is the gold
conductor level, lower level is the YIG substrate......................................
4.6
102
103
Comparison of FDTD calculations and measurements of insertion loss
and isolation. #0=1600 Oe, Afa=1780 G, hp = 400 fim, hp = 48 fim,
Ha = 500 fim,
4.7
wq
= 250 fim, w? = 30 fim, ws = 40 fj.ni,wp = 100 fim . 104
Effect of wq over insertion loss. # o=1600 Oe, M,=1780 G, hp = 400
fim, hp = 48 fim ,
= 500 fim,
wt
= 30 fim, ws = 40 fim,
we
fim ...................................................................................................
4.8
106
Effect of wq over isolation. #o=1600 Oe, M,=1780 G, hp = 400 fim,
hp = 4 8 fim, h& = 500 fim,
4.9
= 100
wt
= 30 fim, ws = 40 fim , wp = 100 fim .
106
Effect of the dc bias field strength over insertion loss. M,=1780 G,
hp = 400 fim, hp = 48 fim, hA = 500 fim,
wq
= 250 fim, w? = 30
fim, ws — 40 fim , wp = 100 fim .................................................
107
4.10 Effect of the dc bias field strength over isolation. M,=1780 G, hp =
400 fim, hp = 48 fim, hA = 500 fim,
wq
= 250 fim,
wt
= 30 fim,
ws = 40 fim, w p = 100 fim .........................................................
107
4.11 Comparison of insertion loss and isolation for open and shorted isolator
structures. #o=1600 Oe, M,=1780 G, hp = 400 fim , hp = 48 fim,
hA = 500 fim,
wq
= 250 fim, w? = 30 fim, ws = 40 fim, wp = 100 fim . 108
xiv
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4.12 Comparison of insertion loss and isolation for truncated YIG substrate.
#0=1600 Oe, M 3= 1780 G, hp = 400 fim, hp = 48 fim , hA = 500 fim,
wq
= 250 fim, wq- = 30 fim , w s = 40 fim, wp = 100 f i m ......................
108
4.13 Comparison of insertion loss and isolation for different a values. #o=1600
Oe, M,=1780 G, hp = 400 fim, hp = 48 fim , hA = 500 fim,
fim ,
wq
= 250
= 30 fim , w$ = 40 fim , wp = 100 fim .........................................
109
4.14 A stripline disc junction circulator (a) top view (b) cross section. . . .
Ill
w t
4.15 The spatial variation of the demagnetizing factor
at z=0 as func­
tion of the radial distance from the center of the disk for different values
of q = L /a ........................................................................................................
4.16 The spatial variation of the demagnetizing factor
115
at r=0 along
the central axis of the disk for different values of q = L /a ...........................116
4.17 A three dimensionalmesh view of the stripline junction disc circulator. 118
4.18 Characteristic impedance Z c of the stripline. The experimental value
is 48.06 ohms...................................................................................................
119
4.19 Insertion loss of the circulator for different values of A H . Hext = 2550
Oe, # z=0.95....................................................................................................
120
4.20 Isolation of the circulator for different values of A H . Hext = 2550 Oe,
jVz=0.95...........................................................................................................
121
4.21 Return loss of the circulator for different values of A H . H ^ t = 2550
Oe, #2=0.95....................................................................................................
xv
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121
Figure
Page
4.22 Insertion loss of the circulator for different values of the externally
applied dc field H ^ A H = 45 Oe, Nz=0.95................................................122
4.23 Isolation of the circulator for different values of the externally applied
dc field Ha t. A H = 45 Oe, N z=0.95.........................................................
123
4.24 Return loss of the circulator for different values of the externally ap­
plied dc field Hext- A H = 45 Oe, 1VI= 0.95................................................
123
4.25 Insertion loss of the circulator for different values of the metal disk
diameter. Hext = 2550 Oe, A H = 45 Oe, 1V*=0.95...................................... 124
4.26 Isolation of the circulator for different values of the meteil disk diameter.
Hext = 2550 Oe, A H = 45 Oe, Nz= 0.95....................................................
124
4.27 Return losses of the circulator for different values of the meted disk
diameter. Hext = 2550 Oe, A H = 45 Oe, N z=0.95...................................... 125
4.28 Insertion losses of the circulator for uniform emd non-uniform magne­
tization cases. Hext = 2550 Oe, A H = 45 Oe................................................125
4.29 Isolation of the circulator for uniform and non-uniform magnetization
ceises. H at = 2550 Oe, A H = 45 Oe..........................................................
126
4.30 Return losses of the circulator for uniform emd non-uniform magneti­
zation cases. Hext = 2550 Oe, A H = 45 Oe.............................................
126
4.31 A ferrite-filled stripline cross section. Stripline is 31.85 mm long. . . .
131
4.32 Insertion loss of a ferrite-filled stripline for different source pulse am­
plitudes. hc = 0.266 Oe................................................................................
xvi
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132
Figure
4.33 Insertion loss of a ferrite-filled stripline for different source pulse am­
plitudes. hc = 0.133 Oe................................................................................
xvii
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CHAPTER 1
INTRODUCTION
Progress in cellular technology in 1990s and widespread availability of cellular
phones generated serious concerns in the antenna research community to investigate
possible electromagnetic (EM) energy exposure of the user when a cellular phone
is operated closely. It has been shown th at almost half of the radiated energy of a
classical cellular antenna (in the absence of any shielding mechanism) is absorbed by
user’s head and hand [l], [2]. The amount of absorbed energy can be hazardous at high
power output levels. Besides, antenna characteristics such as the radiation pattern,
gain and input impedance are greatly influenced by the presence of humans. The
EM energy absorbtion in human tissues due to cellular phone antennas are beyond
the scope of the research presented in this dissertation. On the other hand, there has
been no significant research in the analysis and design of new and improved antenna
structures to minimize the radiated energy towards user, while keeping the antenna
performance at its peak.
The research presented in this dissertation initiated from the development of a
cellular phone antenna which aimed to reduce excessive EM radiation towards user.
Meanwhile, antenna characteristics are aimed to be maintained almost unaltered by
the presence of user. As an initial step of solving this problem, a magnetically shielded
antenna was proposed. In this study, an ordinary monopole antenna mounted on a
finite ground plane and a metallic shield were located very close (« 8 mm) to each
other. The shield interior was coated by an isotropic magnetic material. It has been
shown th a t such an antenna configuration reduces EM radiation in the direction of
user considerably. The possibility of using ferrite as an alternative coating material
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2
led to the development of a ferrite code by deriving special FDTD updating equations
to solve electric and magnetic fields inside the ferrite medium. However, magnetic
materials for cellular phone applications sire found to be heavy and expensive. Conse­
quently, even though the magnetically shielded antenna performed well, the need for
a cheap and easy to manufacture antenna arises. For this purpose, a dual-resonance
shielded antenna is introduced.
The research in this dissertation is split into two branches. T he first branch
takes its root from the magnetically shielded antenna and evolves through the dual­
resonance shielded and, shorted and truncated microstrip antennas. The second
branch aims the validation of initially developed ferrite code. For this purpose, var­
ious microwave ferrite devices are analyzed. These include a thin film slow wave
isolator and a stripline disc junction circulator. The ferrite code is expanded in abil­
ity by combining some advanced techniques such as the non-uniform computational
grid (as implemented in thin film isolator) and the non-uniform magnetization (as
implemented in stripline circulator).
Chapter 2 of this dissertation deals with the implementation of a finite-difference
time-domain (FDTD) method to obtain electric and magnetic field solutions in a
finite computational domain. In section 2.2, electric and magnetic field updating
equations are derived in the presence of a media whose material content can be very
complex (different dielectrics and/or metals). Stability of the FDTD method, source
of excitation, numerical dispersion are reviewed in subsections. Absorbing boundary
conditions including Mur and Berenger’s perfectly matched layer are handled in sec­
tion 2.3. Implementation of an expanding non-uniform grid architecture to handle
fine geometrical details of the structure under investigation is discussed in section 2.4.
The analysis and design of various shielded mobile antennas are presented in
Chapter 3. A near-to-far zone transformation, a necessary numerical tool required
to obtain far field radiation patterns of any antenna structure from its transient near
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3
fields, is presented in section 3.2. Input impedance of a magnetically shielded antenna
is analyzed and compared to an unshielded monopole in section 3.3. A parametric
study is performed to investigate the effect of magnetic loss and thickness of the coat­
ing material on the input impedance. The dual-resonance shielded cellular antenna
is presented in section 3.4 including a brief theory of operation. The effect of various
geometrical features over the input impedance of an initial design are parametrically
investigated in section 3.4.2. A proto-type of the dual-resonance shielded antenna has
been built and tested at Arizona State University’s Anechoic Chamber. Measured
antenna param eters, such as input impedance and far fields are compared with FDTD
calculations. An excellent agreement has been achieved between measurements and
theory. In section 3.5, a shorted and truncated microstrip antenna is introduced
as an internal alternative of the dual resonance shielded antenna. The bandwidth of
this antenna is found to be narrow for cellular phone applications. Further research is
planned as future work to study stacked microstrip antenna configurations to increase
the bandwidth and enhance the overall performance.
Chapter 4 deeds with the analysis and design of microwave ferrite devices using
the FDTD method. An introductory level theory of magnetized ferrites, including
the equation of motion of the magnetization vector, magnetic loss mechanisms in
ferrite and small signal analysis, are given in section 4.2. An algorithm is developed
to solve electric emd magnetic fields inside the ferrite and its implementation is dis­
cussed in section 4.3. Special updating equations are derived for the magnetic field
components perpendicular to the direction of the dc bias field. Since the ferrite algo­
rithm is derived from the linearized equation of motion of the magnetization vector,
the analysis is limited to linear ferrite devices. A slow wave th in film isolator whose
operating mechanism depends on magnetostatic surface waves is analyzed in section
4.4. A param etric study calculates insertion loss and isolation over a wide range of
device dimensions and substrate parameters. Good agreement has been achieved be­
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4
tween measurements and FDTD calculations. In section 4.5, a stripline disc junction
circulator is analyzed. Frequency dependent circulator parameters are calculated for
both uniform and non-uniform magnetization cases. A very good agreement has been
achieved between measurements and theory. The non-linear behaviour of ferrite is
briefly touched in section 4.6. The linear ferrite algorithm is extended to non-linear
case by modifying the regular damping factor. A simple ferrite-filled stripline struc­
ture is analyzed. Initial results are presented.
This dissertation is concluded in Chapter 5. Possible future extensions of the
current research are briefly touched. These include stacked microstrip antennas for
wireless communications, further investigation of the non-linear phenomena in ferrite,
and FDTD analysis of various non-linear microwave ferrite devices such as frequency
selective limiters and signal-to-noise enhancers.
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CHAPTER 2
FINITE-DIFFERENCE TIME-DOMAIN METHOD
2.1
Introduction
The Finite-Difference Time-Domain (FDTD) method basicaUy solves Maxwell’s
curl equations in discrete time emd space within a finite computational domain trun­
cated by proper boundary conditions. The numerical solutions of partial differential
equations using finite differences and stability requirements can be found in [3] and
[4] in detail. The FDTD method was first introduced by Yee in 1966 [5] for the solu­
tion of electromagnetic (EM) boundary value problems. Yee replaced Maxwell’s curl
equations by a set of finite difference equations and analyzed diffraction of an EM
pulse by a perfectly conducting square in a two-dimensioned FDTD grid. In order to
advance field solutions in time, Yee used a unique cell architecture to enable Faraday
and Ampere’s Law contours. Detailed information regarding Yee’s cell and FDTD
time stepping algorithm will be given in the next section.
The FDTD method was initially used to calculate scattering from and penetration
into arbitrary-shaped structures having complex material contents illuminated by
plane waves [6]-[7]. This method was applied to solve several other EM problems such
as the solution of frequency domain response of passive planar microstrip circuits [8][9], microwave circuits having both passive and active elements (solid-state devices)
[10]-[11], analysis and design of next generation cellular phone antennas [12], [13], [14].
The theoretical details of the FDTD method and its application to EM modeling of
various structures can also be found in [15] and [16].
Section 2.2 of this C hapter deals with the solution of Maxwell’s curl equations in
time domain using finite differences emd derivation of the FDTD updating equations.
Stability, source of excitation, numerical dispersion, and frequency dependent param­
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6
eters are touched in consequent subsections. Section 2.3 presents different types of
absorbing boundary conditions used in this dissertation and their implementation
within the FDTD computational domain. A non-uniform computational grid which
is used to model fine geometrical features of certain structures is also discussed in
section 2.4.
(i,j,k+l)
- 3 ^
E x(ij,k)
(i+1 j,k)
Fig. 2.1: Electric and magnetic field vectors on Yee’s cell.
2.2
FDTD M ethod
2.2.1
FDTD Updating Equations
The FDTD method solves Maxwell’s curl equations for electric and magnetic
vector fields in discrete time and space within a finite computational domain truncated
by proper boundary conditions known as Absorbing Boundary Conditions (ABC).
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7
The theoretical details of different type of ABCs used in the research work presented
in this dissertation will be given in section 2.3. In the presence of electric and magnetic
losses, Maxwell's curl equations in differential form for a source free and isotropic
medium are
V x E = - p . ™ - <T'it
V xF =
eft
(2.1)
+ (tE
(2.2)
In (2.1) and (2.2), a is the electric conductivity in S/m and a* is the magnetic
resistivity in Cl/m. Therefore, <
j E and a*H terms stand for electric and magnetic
losses which may exist inside the medium. In cartesian coordinates, (2.1) and (2.2)
yield the following scalar equations
8HX
=
fj. \ d z
dt
dy
(2.3)
)
dH,
dt k
(2 ' 4)
dH.
K
i
r
dEx = i ( d H z _ dH;
=
dt
e\dy
dz
(26)
8E,_1(8H.
d H„
\
dE._l(dH,
d H.
\
i t
- 7 U
r _
( 25)
*j
Following Yee’s original notation [5], a space point in a rectangular grid is
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(
)
8
(2.9)
(*. h k ) - (*Ax, jA y , k A z)
Let F denote any function of discrete space and time
F (zAx, j Ay, kA z, n A t) = F ^k
(2-10)
where Ax, A y and A z are, respectively the grid space increments in x, y and z
directions. A t is the tim e increment.
Using a central finite-difference approximation, space and time derivatives of F
can be written as
dF
dt
+ O (Ax)2
Ax
dx
n + l/2
F ij,k
—F
At
(2.11)
,n—1 /2
+ O (A t)2
(2.12)
[n (2.11) O (Ax)2 is the error term which represents the all remaining terms in
Taylor series expansion and equation (2.11) is known as second-order accurate centred
finite-difference scheme in space. Similarly, (2.12) is second-order accurate in time.
Applying Yee’s finite-difference scheme to (2.3),
+1/ 2 - n r 1' 2
At
i
pa
ritj-hl/2,k
gitj - l / 2 , t
(2.13)
Ay
The H™.. k field component in (2.13) is evaluated at time step n. Since the value
of HXi. k at time step n is not available, the following semi-implicit approximation
[15] is used
=
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(2.14)
9
By substituting (2.14) in (2.13), leaving
on
^
hand si<*e
passing
the all remaining term s to the right, the finite-difference update equation for Hx field
component can be w ritten as
1 T i n —1 /2
2 P i- i.k
1+
,
r j ^ t j a *ij* +
_PTI
JPTl
At
En
2l* i,j,k /
\
sij+ l/2 ,k
1 +1
\
y>.j.k - l / 2
y>.j.*-H/2
m.i.k
'
(2.15)
: i . j ' —l./2»k
2f*i,i,k
The finite-difference update equations for other magnetic and electric field compo­
nents can be derived from (2.4)-(2.8) in a similar manner
/
H " -1/2 +
fp+1/2 _
11+ w
M
- )
-AS— K'i‘
V
\ /
I
+
'
f.
H*+
1/ 2 =
z t . j .k
i
}
V + 2WjVk /
^ _
cm+1
________I z j . i . k
I
cm
,
1 +
.i + %2e>./,fc
s S r./
2 <«.>.*
<7i.,.fcAt
/
At
I
e i.J .k
wB+l/2
'
1 /2 . t
(2.17)
—
si . j —l / 2 ,k
(2.18)
Ay
_ H-n-f-l/2
Hj.j.t+1/2
^
»«.j.fc-l/2
Ax
At
£ Vi.j.k
n ~+
/
p r n + l/ 2
t . j . k —l / 2
i+ l/2 .j.k
i~ l/2 . j.fc
At
...A z
r r f t+ l / 2
n-n+l/2
1+
^
1
2eitJ*tfe
n O + l/2
a *i+l /2. j. k
2Ci ,j .k
1 + £LL*At
(2.19)
gn+l/2
jyi*+l/2
l + ZLi*A t
«».».*A t '
_ a t+ l/2
i.j.t+ 1 /2
Az
\ ~
2 ei,j\Jfc /
1
> « - l/2 .j.f c
n *i.i+l/2.fc
rrn+I/Z
1 +
=
\
Ax
<Ti.i.fcA t
g n + l
V i.j.k
)
* ■ + 1 /2 ./ .*
"■i.i.fcAt
x i .j.k
£
Ay
Zi >k +
g '.i.t A t \
*
(2.16)
j+1/2. k ^ -
2M-i* I fin-1/2 ,
i + ^
,
-i'
B"
"i—k / 2 . j . k
Az
rM-n/2 1
„ « + l/2
t.j—1/2.J1
\
'
(2 .20)
Ay
As can be seen from (2.15)-(2.20) th at the field components on the right hand
side ctre weighted by some constitutive parameters and as well as A t. These weighing
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10
factors are referred as Yee’s updating coefficients. Electric field updating coefficients
can be written as follows
-
4
-
^
)
/
(
1+!^
<2-21)
r )
c‘''-=(dy/((1+^ )
(2'22)
and magnetic field updating coefficients are
d
^ *
d
^
=
(
( ' - W
-
£
^
) / {
)
/
{
i
i
+ W
+ W
)
( 2 -23)
)
(2 -2 4 )
where p can be x, y or z, and Ap is the cell size in p direction.
Assuming that the structure under investigation contains different types of ma­
terials (dielectric and/or magnetic), electric and magnetic field updating coefficients
can easily be calculated from (2.21)-(2.23) before the FDTD time stepping algorithm
starts. If cubical FDTD cells are used, the finite-difference update equations given
by (2.15)-(2.20) can be further simplified as follows
in + l/2
TT
H x\i,j,k
p.
=
D *,Hz
I
TT
|»j,ifc # *
i n —1 /2
\ij,k
.
+ L'b,H:
i
(
E y
, p
I
\
■ n - 1 /2
in + l/2
VI
H z
=
A ,
IS /2= a
Hy l«j\*
■n - 1 / 2
lij.fc
+ d kh.
l » j ,f c + l / 2 ~ E y lij .f c —1 /2 \
_ p
in
I
ltj- l/2 ,J b
^
E z l i V i / 2 j,fc - E x
+ E x \ij,k-l/2
E * l i j ‘+ l / 2 ,fc
+ E V l ? - l / 2 j,f c
- E x
/o
(4.40)
l« j+ 1 /2 ,fc /
ir-i/2o
l*J,fc+l/2
/2 /
n
i j - 1 /2!,Jfc
E V lr + l/2j
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)
/
(2.26)
(2.27)
11
rr
(
tt
in + l/2
' i f i 1/? ’4
' “ tf
in + l/2
rr
(
1 /2
T « z li- l/2 j,J t
(
rr
x
in + l/2
' S t l k 4*
\
‘
(2-28)
" » l«\i,Jfc+l/2 /
rr
' » $ /2
|» + l/2
in + l/2
\
J
<2-29)
n z lt+ l/2 j,f c
/
tt
\
in + l/2
r r i ^ V / V 'M
(2-30)
+ i l * l » j —l/2 ,fc ~ n = l » j + l / 2 , f c /
The electric and magnetic field components are located half cell away from each
other in Yee’s cell as shown in Fig. 2.1. This unique configuration of the fields enables
the realization of Ampere’s and Faraday Law contours. After determining the spatial
resolutions based on the geometrical features and the operating frequency, a suitable
time step size should be chosen by preserving the stability condition.
2.2.2 Stability Considerations
Stability of any numerical finite-difference scheme requires th a t the numerical
solution of the differential equation should not grow in future tim e steps in an uncon­
trollable manner. Once the proper spatial resolutions are chosen, stability is achieved
using Courant-Levy-Friedrichs (CFL) stability condition, which sets a relation be­
tween the time and space increments. T he stability condition for a three-dimensional,
spatially homogeneous FDTD grid is [15]
i/At < A t / y/jle <
. i
V A x2
- -1A t/2
(2.31)
A z2
where v is the maximum wave velocity in the medium. Unless the whole FDTD
computational domain is filled by a dielectric material, u is equal to the speed of light
c. Assum ing that cubical FDTD cells whose cell size is A u me used, (2.31) becomes
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12
uA t <
2.2.3
Au
(2.32)
Source of Excitation and Feed Modeling
In order to obtain wideband frequency domain results, structure under investiga­
tion should be excited by a pulsed voltage source using a proper feed model. Every
structure may require a unique feed model and this subject will be touched in detail in
the analysis of different structures. Pulses having Gaussian or Rayleigh distributions
are frequently used in FDTD simulations and are expressed in discrete-time form as
VG(t) = A0e -a^ - ^ t)2
VR{t) = A ie~a^
(2.33)
At)2 v ^ ( t - /3At)
(2.34)
where A q and A i are the amplitudes of the Gaussian and Rayleigh pulses, respectively,
t is simply the discrete time t = nA t where n is a positive integer. 2(3 is the pulse
width from zero to truncation.
The Gaussian pulse exists from r = 0 to r = 2(3A t and at r = /3At has a
maximum, while the Rayleigh pulse is zero at r = (3A t. It can be seen from (2.34)
th at the Rayleigh pulse mathematically is the time derivativeof the Gaussian pulse.
The amplitude of the pulse at truncation is determined by a.
A practiced selection
of a is given in [16] as
“ =
(2-35)
Fig. 2.2 shows Gaussian and Rayleigh pulses in time domain with /? = 64 and
a time step size of A t = 5 • 10-12 seconds. Both pulses lasts for 640 ps and pulse
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13
amplitude is zeroed after this time. The frequency spectrum s of both pulses are shown
in Fig. 2.3. As can be seen, Gaussian pulse has a peak at dc, while Rayleigh pulse
has zero spectral energy at dc. The isolation of dc and very low frequencies from the
frequency spectrum can be useful in the analysis of specific devices.
Gaussian Pulse
Rayleigh Pulse ~
1.0
0.8
TJ 0.6
S
o. 0.4
I “
- a o.o
®
:=
-0-2
"5
g -0.4
j g - 0.6
-
0.8
-L 0
50
100
125
150
175
200
Time Steps
Fig. 2.2: Gaussian and Rayleigh pulses in time domain, /? = 64 and A t = 5-10” 12 seconds.
2.2.4
Numerical Dispersion
Physical dispersion is the relation between the wavenumber k and the frequency /
of the propagating EM wave mode. Numerical dispersion can be explained as follows.
The numerical wave inside the FDTD computational domain will have a wavenumber
k which differs than the wavenumber k of the physical wave, k is a function of the
grid resolution and the angle of incidence of the propagating wave [15]. Assuming
th at an excitation source in the form of a Gaussian pulse is applied inside the FDTD
computational domain. Gaussian pulse source will have several spectral components
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14
-20
Gaussian Pulse
Rayleigh Pulse .
-40
-60
-60
3 -120
u -140
-160
-180
Frequency (GHz)
Fig. 2.3: Frequency spectrums o f Gaussian and Rayleigh pulses.
with different wavelengths in frequency domain. Numerical dispersion states that the
longer wavelengths (lower frequencies) will have higher phase velocities than that of
the shorter wavelengths. Since the longer wavelengths will see a higher resolution,
their phase velocities will also be higher. Poor resolution of the FDTD grid can cause
higher modes to be cut-off. This phenomenon is completely unphysical, and can
also be considered as a low-pass filtering effect which is inherent to FDTD algorithm.
Consequently, the cell size should be at least A/10 for the highest frequency of interest
to reduce numerical dispersion. Time step size A t should also be close to the stability
limit in order to reduce numerical dispersion further.
2.2.5
Frequency Dependent Parameters
The frequency response of the structure under investigation can be obtained by
applying a Feist Fourier Transform (FFT) to the time domain data. This data can
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15
be the electric field, magnetic field, voltage or current. The frequency dependent
parameters which are used most in this dissertation are input impedance of an antenna
or characteristic impedance of a stripline transmission line, insertion loss, return loss
and isolation of various microwave ferrite devices.
Input impedance of an antenna can be found by dividing the frequency domain
terminal voltage by current. Term in al or input voltage depends on the applied feed
technique. For delta-gap feed model, input voltage is the applied source pulse itself.
Input current is found by applying Ampere’s Law to the circulating magnetic fields
around the conductor. Characteristic impedance of a stripline transmission line can
be found from the ratio of the frequency domain voltage between the signal and
ground conductors, and the current which is found from the circulating magnetic fields
around the signal conductor. Return loss is a measure of how well a transmission fine
is matched to another and should be high for better matching. Insertion loss is a
measure of low loss power transmission and should be small for b etter transmission.
Isolation is clearly the opposite of insertion loss and a measure of power attenuation
in required direction. Frequency dependent quantities are widely used in Chapter 3
and Chapter 4.
2.3
Absorbing Boundaries
2.3.1
Introduction
In some electromagnetic modeling problems like the scattering and radiation anal­
ysis, infinitely long transmission lines and waveguiding structures, or the realization of
matching for multiport microwave devices, termination of the FDTD computational
domain is required. This requirement arises due to limited computer resources in
terms of both memory and run time, in most cases. During the FDTD time-stepping
algorithm, some neighboring field components are needed to update a specific field
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16
component. In a finite computational domain, a field component located at the outer­
most boundary will not be updated correctly, since some of the required neighboring
fields will be in outer space and their values are not known. This numerical problem
brings reflections from the boundary to computational domain and eventually cor­
rupts the useful data. This problem is solved by implementing absorbing boundary
conditions (ABC) on the outermost boundary to truncate the FDTD grid. Ideally,
absorbing boundaries behave like as if the FDTD grid is unbounded and fields propa­
gate to infinity without any numerical reflections. Regardless of the physics involved
in the design of an ABC, all ABCs should introduce matched mediums to th at of the
interior of the FDTD grid.
There are two main ABCs considered in this dissertation and are implemented.
Mur ABC [17] is based on Enquist and Majda’s [18] one-way wave equation which
allows numerical wave to propagate only in one direction and is classified as an an­
alytical ABC. The local and globed errors associated with Mur ABC was shown to
be reduced with the implementation of Mei-Fang’s super-absorbtion technique [19].
Super-absorbtion technique itself is not an ABC, but an error-elimination method to
improve the performance of an existing ABC. In 1994, Berenger [20] introduced a
revolutionary ABC called Perfectly Matched Layer (PML). This ABC is based on a
highly lossy and matched non-physical absorber medium which surrounds the FDTD
problem space.
2.3.2
Mur ABC
Mur ABC [17] was developed from the one-way wave equation which allows the
wave propagation only in one direction. This particular one-way equation is derived
from the wave equation by using an appropriate factorization scheme [21]. A threedimensional wave equation in cartesian coordinates can be written as
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17
(£+£+£-?£)*<*•»•*•*>=»
<2-36>
where U is a scalar wave function and c is the speed of light (assuming the wave
propagation is in free space). Let L denote a partial differential operator [21] such
that
L=
where Dx2 represents ^
=
? D*2
(237)
and so on. Using (2.37), the wave equation (2.36) can be
factored in the following simple form
LU = L +L~U = 0
(2.38)
where L+ and L~ are the one-way wave operators associated with the waves traveling
in opposite directions.
It is shown in [18] that the application of the one-way wave operator to a scalar
wave function U will exactly absorb a plane wave traveling toward the boundary at
any angle of incidence. However, numerical errors are introduced due to discretization
scheme and this will cause numerical reflections from the boundary.
2.3.3
Berenger’s Perfectly Matched Layer (PML)
Perfectly Matched Layer (PML) ABC was introduced by Berenger [20] in 1994 to
achieve high-accuracy FDTD simulations. PML layers are based on a highly lossy non­
physical absorbing medium which surrounds the FDTD computational domain. PML
medium finally ends with perfectly conducting conditions as described in Berenger’s
publication. The key of the PML technique is to split any scalar field component into
two sub-components. As an example Ex will be written as the sum of E xy and Exz.
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18
We gain an extra degree of freedom in specifying the material parameters of PML
medium by splitting the scalar field components. Following Berenger’s split-form
notation for electric and magnetic fields, (2.3)-(2.8) can be written as
dH xy
IM,- g T +
8 H XZ
.
_
tt
d ^ E y x + E y z )
,
^ ° ~ d t~
dHyz
dt
.
«rr
_
9 (E „
« r r
+
E Xz )
Wz--------
_
+ a 'y H zy
+
d ( E Xy +
E xz)
di
„
_
d E ,
d ( H y x + H y Z)
d jH z y +
_
_
—
1 411
431
,n
(2 ' 44)
_ d i H ^ ± J h y )_
y xy ~
dy
e0—d5T“
+ VzEyz ~
t
dt
.
,n
Eyz)
€o~ d T + azExz ~ --------- dz-------
QEyx
,
(2.42)
d ( E yx +
_
*rr
dH zy
d E xz
j
+ a*xHyx = d (EzxJ~
x yx
dx
d H zx
dt
.
.
(2'39l
~ -------- dz-------
m ~ d T + -----------
Vo-^
&{Ezx + E„j)
^
H zz)
dz
d jH z x + H z y )
dx
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(2.45)
(2.46)
(2.47)
(2.48)
19
(2.49)
(2.50)
where <r and a* denote electric conductivity and magnetic resistivity, respectively. A
reflectionless PML medium will be true, if the following condition is satisfied
* = £
(2.51)
The above medium in which the EM wave propagates will have the same wave
impedance as the free space does. Therefore, a wave propagating through such a
medium will not be reflected back due to impedance mismatch. The wave impedance
of the PML medium is found to be independent from the angle of incidence and the
frequency of outgoing waves [20]. The electric and magnetic losses should increase
gradually as a function of the depth of the PML layer. At the PML-FDTD space
boundary, both a and a* should be zero. W ith increasing depth, both losses should
also increase and achieve <rmax and
values next to perfectly conducting walls.
The electric loss as function of the depth is
(2.52)
where a can be crx, cry or az. 8 is the PML layer thickness. amax is the maximum
value of the electric conductivity and pe is the depth of a. n is an integer and chosen
to be 2 for a parabolic conductivity variation.
The magnetic loss variation inside the PML medium is similar to th a t of the
electric losses and defined as
(2.53)
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20
where a* can be a*, a* or a*, pn accounts for the depth.
It should be emphasized th a t since the electric and magnetic field components are
located half-cell apart, there should also be a half-cell interval between pe and phThe amount of the reflection from the perfectly conducting wall (PEC) at normal
incidence is given as [20]
R (0) = e(~2/{n+l))ama*s/eoC
(2.54)
where c is the speed of light.
It is reported in [22] th a t Z?(Q) < 10-5 yields the optim al PML performance.
Using this value of the reflection coefficient, <rmax cam be found from (2.54). For a
reflectionless medium,
is determined from (2.51). A PML layer thickness of 4
to 8 cells provide excellent results in the reduction of the reflection without causing a
heavy computational burden. It is also reported [22] th a t PML ABC is found to be
at least 40 dB more accurate than the second-order M ur ABC.
Fig. 2.4 shows the application of PML ABC in a two-dimensional FDTD grid. The
FDTD computational domain is enclosed by a PML medium which may have a thick­
ness of severeil cells. PML medium is finally ended by perfect electric wall conditions.
PML layers at the left and right sides have only ax and a* present satisfying (2.51).
By Berenger’s original notation, these layers are indicated as PML(crI , cr*,0,0). At
the bottom and top sides only cry and a* are present and PM L(0,0, ay, <r’ ). At the
corners all losses are present and PML(<tx,<7*, ay,<r*). T he EM waves generated by a
wave source will propagate toward the PML-FDTD space boundary. The wave sub­
component traveling in the normal direction to the PML medium will be subject to
am intense attenuation due to the presence of a and a* in th a t direction. Meemwhile
the tcingential wave sub-component will penetrate into PML medium without any
reflections due to impedcince mismatch, since a amd a* au:e zero in the direction of
the tangentiad wave sub-component. This wave sub-component will be attenuated at
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21
PMUO.O. ffy . o ’ )
PMU ox ,o * ,Oy ,Oy )
Outgoing waves
vacuum
PML(
PML( a. .a '0.0)
Wave
Source
.a* ,0.0)
PML(0.0. Oy ,o * )
Perfect conductor
Fig. 2.4: Implementation of Berenger’s PML ABC on a 2-D FDTD grid.
the comer regions. Due to rapid increase in losses, exponential updating coefficients
rather than Yee’s regular updating coefficients sure recommended to be used inside
the PML medium [20], [23]. These exponential updating coefficients are
k ) ,i = ( e ^
Cb
—
Da i , , =
(2.55)
)
1- e
(2.56)
(2.57)
( .^ )
1 —e
—
<r*
.
Iiit
iii*
\\
TiJ,k
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(2.58)
22
PML ABC is applied to term inate PEC parallel plate and asymmetric dielectric
slab optical waveguides in 2-D by Reuters [24]. It has been shown in [24] that ultrawide band excitation which contains significant energy below cut-off frequency(ies)
of each waveguiding structure was absorbed w ith reflections of less than -75 dB,
which proves th at PML ABC is very effective in the existence of evanescent energy
and multimode operation. Chen [25] modified PML (MPML) ABC to enhance absorbtion rates of evanescent waves (modes below cut-off frequencies) by introducing
additional degree of freedom in specifying the relative permittivity er and permeabil­
ity fir. MPML and PML ABCs showed the same performance for the propagating
modes as reported in [25]. However, MPML outperformed PML in the absorbtion of
evanescent waves and enabled a smellier white space between the structure and ABC
walls.
2.4
A Non-Uniform FDTD Grid
In some of the research work presented in this dissertation, a non uniform grid
architecture is applied in FDTD simulations of specific structures having small ge­
ometrical features compared to the rest of the structure. For example, the metallic
shield of the dual-resonance shielded antenna was analyzed using a very fine mesh
in the shield region and a coarser mesh elsewhere. Using a very fine mesh over the
entire FDTD computational domain could result in tremendous waste of computer
resources. An expanding grid algorithm was previously introduced by Gao et al. [26]
to analyze RF energy deposition in human tissues at frequencies higher them 915
MHz. Heinrich et al. [27] reported that the overall accuracy of non-uniform mesh is
better th an the uniform one. This behaviour is explained in the better performance of
non-uniform mesh near the singularities, even though the non-uniform mesh distorts
the second-order accurate nature of the finite-difference equations to the first-order.
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23
A non-uniform grid can be generated by some expansion functions. Selection
of these functions can vary as long as the condition 0.5Am±i < Am < 2Am±i is
maintained to avoid large local errors [15]. Here m is the spatial index variable ij, or
k. A is the cell size in x, y or z directions. Following function is used to generate an
expanding grid in one dimension
Am = A m- i ( m - l ) q
(2.59)
where q is the expansion coefficient and a real number.
Hx(i,j,k)
( i + l j jc )
Fig. 2.5: A simple non-uniform mesh.
Fig. 2.5 shows a simple non-uniform mesh applied on four neighboring cells. For
example, the spatial derivative in y direction is required for Ex and Hx field compo­
nents. Spatial derivative of E x will be taken in the sam e way as in the case of uniform
mesh. While the spatial derivative of Hx will differ due to the varying spatial distance
between the two consecutive Hx field components and should be redefined. Let Ah
to be th e new spatial distance,
can be written
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24
rrn+1/2 _ fjn+l/2
o rr
O tt-x
js, a x j . j + u k
dy ~
x i.j.k
Ah
A* = Aj +
(2.60)
Similar modifications should be done for all other magnetic field components.
Since a very fine mesh is introduced associated with the non-uniform grid ar­
chitecture, the stability condition of the FDTD algorithm should also be modified
as
^
5
/ I +
V
V Ax2
mm •
where Axmin, A ymin and A 2min. are
\ 4 .
Aw
2ir-e
"m
- 1 -
Az2m m•
(2J51>
the minimum cell sizes in a three-dimensional
FDTD space.
2.5
Summary and Conclusions
In this chapter, the basics of the FDTD method are touched. These include the
solution of Maxwell's curl equations using finite differences, derivation of FDTD up­
dating equations for electric and magnetic field components in a 3-D computational
domain, stability condition, source of excitation and numerical dispersion. In addi­
tion, information on implementing of Mur and PML absorbing boundary conditions
is given. Finally, an expanding non-uniform grid architecture is presented to handle
fine geometrical features of the structure of interest. It can be concluded th a t am
FDTD ailgorithm equipped with suitable absorbing boundaries is capable of solving
elecromagnetic fields inside the computational domaun which may include arbitrairilyshaped structures having complex material! contents.
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CHA PTER 3
SHIELDED ANTENNAS FOR MOBILE COMMUNICATIONS
3.1
Introduction
This research work initiated from, the development of a cellular phone antenna
which aimed to reduce excessive electromagnetic (EM) radiation towards user. Effect
of a person on the performance of a classical quarter-wave long monopole antenna was
investigated by Toftgard [1] using homogeneous head and hand models. Jensen [2]
used heterogeneous head and hand models to accurately simulate EM wave interaction
of actual human tissues, and analyzed the performance of various antennas. It has
been showed th a t almost half of the radiated energy is absorbed by user’s head and
hand [1] , [2]. T he amount of absorbed energy can be hazardous at high power output
levels. Besides, antenna characteristics such as the far field radiation patterns, gain
and input impedance are greatly influenced by the presence of humans. Absorbtion
of EM energy by human head modeling was also studied in [28],[29] and found th at
complex head models (more anatomical information is provided) yield lower specific
absorbtion rates (SAR) than the ones calculated using a homogeneous sphere to
account for user’s head. However, the absorbtion of EM energy in human tissues due
to cellular phone antennas are beyond the scope of this research and, analysis and
design of next generation cellular phone antennas will be presented and emphasized
only.
Recent progress in cellular technology and widespread availability of cellular phones
has generated serious concerns in antenna research community to develop a b etter
antenna to be used in cellular systems. Practically, such an antenna should have
minimized radiation towards user and antenna characteristics should not be altered
significantly by the presence of user. As an initial step of solving the ideal cellular
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26
antenna problem, a magnetically shielded antenna is proposed [12]. In this study,
a quarter-wave long wire antenna m ounted on a finite ground plane and a metallic
shield are located very close to each other. The shield is coated by a magnetic ma­
terial. It has been shown that such an antenna configuration reduces the near EM
field radiation towards user considerably. Besides the magnetically shielded antenna,
a new antenna is introduced which does not require any coating material other than
copper; m aking the new antenna inexpensive and easy to manufacture [13]. The new
antenna is named as the dual-resonance shielded antenna since it uses the metallic
shielding concept and its operating region lies in the valley between the first two
resonances. The evolution from an ordinary monopole to the dual-resonance shielded
antenna is shown in Fig. 3.1.
(a)
(b)
(c)
Fig. 3.1: Evolution from (a) a classical monopole to (b) the magnetically shielded antenna,
and finally (c) the dual-resonance shielded antenna.
A proto-type of the dual-resonance shielded cellular antenna has been built and
tested at Arizona State University’s Anechoic Chamber. Measured antenna parame­
ters, such as input impedance and far fields, are compared with FDTD calculations
and excellent agreement has been achieved. It has been shown th a t the near field
radiation towards user is reduced by about 90% and approximately a 1.5 dB higher
gain than that of the monopole antenna is obtained in the far zone. Moreover, the
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27
new antenna is inherently matched to 50 12 transmission lines and provides an almost
omnidirectional far zone pattern in the presence of user [13]. More features of this an­
tenna are presented in section 3.4. Furthermore, a shorted and truncated microstrip
antenna is presented in section 3.5, which can be well suited as an internal antenna
in wireless communications systems [14].
In all FDTD simulations in this Chapter, PML absorbing boundaries are used
with a PML reflection coefficient of f?(0) = 10-5. PML layer thickness is selected
to be at least 6 cells in all simulations. Calculation of antenna far fields from its
transient near fields is described in the next section.
3.2
Near-to-Far Zone Transformation
3.2.1
Introduction
Yee’s finite-difference time-domain algorithm numerically solves Maxwell’s curl
equations and the result is time domain electric and magnetic fields which exist at
all discretized space locations inside the FDTD computational domain. These time
domain known fields can be used to find voltages and currents at specific locations
depending on the type of application. In the analysis of antennas, various character­
istics of an antenna are desired. These characteristics are input impedance, gain, near
and far fields. Input impedance calculations require input voltage and current at the
antenna input terminals. Input current can easily be calculated from the circulating
magnetic fields around the antenna simply by applying Ampere’s Law. However,
finding far fields require a more complicated procedure as will be explained in this
subsection.
Previously, Umashankar and Taflove analyzed electromagnetic scattering of com­
plex objects [7] using FDTD method. They found scattered far field pattern and
radar cross section (RCS) of conducting and dielectric cylinder structures illuminated
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28
by plane waves. Their technique requires single frequency sinusoidal excitation for
FDTD calculations, and hence, wide band scattering results are not available. Mal­
oney [30] applied FDTD method to find transient near zone and far zone radiation
from simple cylindirical and conical monopole antennas driven by a coaxial trans­
mission line. He also calculated the far zone transient electric field using equivalent
current sources defined on a virtual surface away from the actual antenna surface.
Luebbers [31] calculated far zone transient fields by applying a pulsed source and
then transformed these results to frequency domain by applying a Fast Frequency
Transform (FFT). The technique presented in this section is quite similar to [7]. Far
fields are found at a single frequency point at each (9, <f>) location.
3.2.2
Surface Equivalence Theorem
Electromagnetic fields radiated by electric and magnetic current sources J and M ,
respectively, in a volume V, are found via a two step procedure [32]. The first step
is to determine auxiliary vector potentials A and F , in the existence of these current
sources, which are
p - £ / / / , »
m
where k = uiy/JIe and R is the distance to observation point from any point in the
source. T he second step is to find electric and magnetic fields due to auxiliary vector
potentials as follows
E = —jw A -
- A) - j V x F
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(3.3)
29
(3.4)
H = - V x A - j u F - j — V (V - F)
jj,
uifxe
ES,H S
Ee ,H
Ec , H
B
No sources
and zero fields
>eqv
>eqv
Virtual surface
(a)
(b)
Fig. 3.2: Surface equivalence theorem.
Careful examination of (3.1)-(3.4) shows that in order to find radiated far fields,
current sources J and M should be well-defined over a volume V. In antenna appli­
cations, surface electric and magnetic current sources induced on the surface of an
antenna need to be determined. However, when the problem involves of finding far
fields of arbitrary-shaped antenna structures, defining these current sources might
not be an easy task. To avoid complexity, the structure of interest is enclosed by
a virtual surface. The virtual surface is physically away from the actual antenna
surface but is still within the near field region of the antenna. The next step is to
find electric and magnetic surface current sources existing on this virtual surface,
and from these, far fields of the antenna. Surface currents can easily be found using
Surface Equivalence Theorem [33]. Actual electromagnetic sources generate Js and
Ma currents on the surface of an arbitrary-shaped structure as shown in Fig. 3.2(a).
These surface currents are used to find E a and Ha fields at the boundary of regions
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30
A and B, separated by a virtual surface S a. Our aim is to define equivalent current
a#
sources Ja
^
and M a
^
on Sa such th a t we can obtain the same E s, Ha fields in region
B. The equivalence theorem states th a t all sources and fields are zeroed in region A,
as shown in Fig. 3.2(b), and the equivalent surface currents are given as
M’. . , . = - n X £ .
= h y .H ,
(3.5)
(3.6)
Here E a and Ha are the total tangential electric and magnetic fields in phasor form
on the virtual surface, n is the unit vector normal to and outward from the virtual
surface. Next section deals with finding the phasor form electric and magnetic fields
from the transient field quantities.
3.2.3
Phasor Field Quantities
Electric and magnetic fields inside th e FDTD computational domain are time do­
main quantities. These time domain fields should be transformed into phasor form to
find far zone fields. This is done using a Discrete Fourier Transform (DFT) algorithm.
Let fij,k(n$t) denote any field component at grid location (i , j , k ) inside the compu­
tational space. Once the antenna is driven by a single frequency sinusoidal source
voltage, all field components within the computational domain will have transient
field values until they reachsteady state. All field componentsshow pure sinusoidal
variation during steady state and phasor form field quantities can now be obtained
using the exponential Fourier series
2 Ar_1
cfc = a? H fijk(n8t) exp(—jfcu/0tn)
iV n = 0
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(3-7)
31
In (3.7), St is the time step size, N is the number of time samples, k harmonic
number, u/0 angular frequency of operation and tn = nSt. Since the applied voltage
has only one frequency component, k is 1. However, transients in time domain data
(before steady state) will cause several harmonics in the frequency domain. In order to
find the correct amplitude information, the number of samples should start somewhere
in steady state region.
3.2.4
Fair Zone Radiated Fields
Regardless of the complexity of the antenna structure, any antenna is enclosed
by a virtual surface located a couple of cells away from the surface of the antenna.
Phasor form tangential electric and magnetic fields on this surface are used to find
equivalent surface currents JSeqv and M a<.qu. By its nature, a three-dimensioned FDTD
computational domain has six outer walls which are in complete fit with the cartesian
coordinate system. There will be also six virtual surfaces th a t enclose the structure
of interest. The important point is to locate E and H fields on the virtual surface.
This is because electric and magnetic field components are always half cell offset from
each other. Assuming that the tangential electric fields exist exactly on the surface,
then, the tangential magnetic fields are located 1/2 cell inward, toward the center of
the computational domain.
The effect of each virtual surface on a specific far-zone point (9, <f>) is calculated and
then overall contribution is found by adding contributions coming from six surfaces.
The procedure of finding far-zone fields is outlined as follows
1. Find phasor form electric and magnetic fields on each virtual surface.
2. Find equivalent surface currents using the field equivalence theorem.
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32
3. Find radiation integrals Lg, L$, Ng and N# from the equivalent surface currents
and repeat this procedure for each virtual surface.
4. Combine the overall contributions of six virtual surfaces to find final values of
the radiation integrals at a specific (0, <j>) location.
5. Find far fields using the radiation integrals.
From the field equivalence theorem, equivalent surface currents are given by (3.5)(3.6). The tangential electric and magnetic fields are spatially averaged and centered
on the unit surface 8s. Once the M s and Ja are calculated, radiation integrals can be
found from the following expressions [34]
Ng = J J (Jx cos 9 cos <f>+ Jy cos 9 sin <f) — Jz sin 9) exp(jk r • dr)ds'
N <ft = j
sin <f>+ Jy cos <f>) exp ( j k r
•
L# =
(3.9)
ar)ds'
Lg — j j (Mx cos 9 cos (f>+ My cos 9 sin <j) — M z sin 9) exp ( jk r
(3.8)
•
ar)ds
sin 4>+ My cos <j>)exp ( j k r *ar)ds
(3.10)
(3-11)
A reference point is needed to keep track of the correct time delay. This point
can be the (1,1,1) cell or the center of the computational domain. Fig. 3.3 shows the
vector orientations for far zone field calculations, r is basically the vector from the
reference cell to the source point of integration (center of the unit surface element).
f is the vector in the direction of the far field observation point. dr is the unit vector
^
in the direction of r. Therefore, the product r • dr represents the difference in paths
from source to the observation point and is given as
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33
Far zone point (0,<|>)
Unit surface
element 5s
Virtual surface
x
Fig. 3.3: Vectors used in far zone field analysis.
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,he copydph, „ wner. Further reproduction prohibited w«hou« permission.
34
[
J
f
r - ar = (axx + Oy-y -+- azz ) - (ax sin 9 cos <f>+ Sty sin 9 sin <f>+ ar cos 9)
(3-12)
Far zone electric and m agnetic fields in spherical coordinates [32] are given as
Er~0
E^ _ i k^
E^
(3.13)
H k r)
47rr
Na)
+ ,2 « |H f c r )
47rr
tfr~0
(3.16)
i,
As can
47rr
77
ikm X -jkr)
L*
47rr
77
be seenfrom these equations, the
(317)
(
far zone radiated fields do not have
dependence of radial distance from the antenna and angular fieldspredominate. The
radiated power associated with far zone fields is found from the power density vector
(Poynting vector). In consideration of radiated power, real and time averaged power
is understood. Reactive power that appears from the imaginary part of E x H has no
contribution in the far zone since the reactive power component vanishes for fcr
1.
Therefore, the time-average real power density is
Wav = \R e { E x H •}
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(3.19)
35
Since the radial field components are negligible compared to 9 and <j>components,
Wav becomes
Wav =
= ar i R e{E 9H ; -
e +h
;}
(3.20)
Radiated power is calculated by integrating Wav over a closed spherical surface which
encloses the antenna, that is
Prad — i T l o Wav r2sin 9 de d<t>
(s -21)
Radiation intensity U is defined as the radiated power per unit solid angle and given
by the following expression
U = r2Wav
(3.22)
Using (3.22), directive gain of an antenna can be written as
-
D{9, cf>) = 4tr - 2t ^ W ' f )
f i x £U{9,<f>)sin9ddd<f>
(3.23)
K
‘
Directive gain does not include losses due to conduction and/or mismatch. If the
power input to the antenna is known, gain can easily be calculated from
<?(»,*) =
(3.24)
in
where P;n is the input power. In term s of phasor form terminal current and voltage,
Pin can be written as
Pin = \ v x r
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(3.25)
36
3.2.5
Far Fields of a Linear Dipole Antenna
After constructing a near-to-far zone transformation code based on the theory
outlined in the previous subsections, a 0.5A dipole is analyzed for validation. The
dipole is assumed to be very thin, therefore its normalized radiation pattern can be
found using simple analytical formulas given in most antenna books. The dipole is
center-fed, and excited by a single frequency sinuzoidal source at 1.0 GHz by simply
driving the E z field component at the feed gap. During steady state, phasor form
tangential electric and magnetic fields on selected virtual surfaces are found using a
simple DFT algorithm implemented within the FDTD time stepping loop. A near-tofar zone transformation based on the surface equivalence theorem is applied to phasor
form field quantities to find the far fields of the antenna. Azimuth and elevation plane
radiation patterns, directive gains, and directivity are calculated.
For validation
purposes, elevation plane normalized directive gain pattern calculated by FDTD is
compared with an analytically obtained curve [34]. This curve is found from the
following expression
/ cos ( |r c o s 0) —cos ( t ) \
F («•»> = (
' d e ------ “ )
(3'26)
where 9 is the elevation angle in radian, k is the wavenumber, and I is the length of
the dipole in terms of wavelength. It should be noted th at (3.26) is derived under the
assumption of negligible dipole diameter (ideally zero) [34].
0.5A linear dipole is analyzed using cubical FDTD cells with a cell size of 3.0 mm.
FDTD computational space contains 28 x 28 x 76 cells (excluding the PML layers).
Virtual surface is located
6 cells away from
the PML ABC. Far field patterns are
calculated with an angle increment of 1 degrees. Time step size is chosen to be 95% of
the stability limit. T he transient input current of the dipole achieves steady state after
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37
several cycles of the applied sinusoidal source as shown in Figure 3.4. Analytically,
directivity of the 0.5A dipole is found to be 2.16 dB [34]. The calculated directivity is
about 2.21 dB and the error is less th an 2.5%. Azimuth plane directive gain pattern
is shown in Figure 3.5. For comparison purposes, elevation plane normalized far field
p attern calculated by FDTD m ethod is compared with the analytical curve obtained
from (3.26). An excellent agreement has been achieved in the calculation of far zone
field patterns of the half-wave long linear dipole antenna.
0.01
0.008
0.006
0.004
0.002
-
0.002
-0.004
-0.006
-0.008
-
0.01
1000
2000
3000
Time steps
4000
5000
6000
Fig. 3.4: Transient input current of the 0.5A dipole antenna.
3.3
A Magnetically Shielded Monopole Antenna
3.3.1
Theory of Magnetic Shielding
Current shielding technology uses metallic reflectors. However, when a metallic
shield is placed closely behind the antenna, the antenna performance will be ruined
in term s of far fields and input impedance. First of all, input resistance will drop
to alm o st zero level, which makes matching very difficult. This phenomenon occurs
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38
270
Fig. 3.5: Azimuth plane directive gain in polar coordinates. Directivity o f the 0.5A dipole
antenna is calculated as 2.21 dB.
because of the excessive capacitance between the antenna and the shield, which shorts
out the antenna. For matching purposes, input resistance of the antenna should be
raised. Second problem is a consequence of the image theory. The metallic shield
introduces an image which counteracts with the antenna when the electrical distance
between the antenna and the shield is very short. This counteraction will degrade
far fields significantly. A monopole antenna, mounted and centered on a perfectly
conducting ground plane, as shown in Fig. 3.7, is analyzed to prove the concept of
magnetic shielding. A metallic shield is located behind the antenna at a distance
of
8 mm.
The shield has been implemented simply with three PEC walls rising in
z direction. The distance between the antenna and the shield is very short. The
disadvantages associated with such a short separation can be overcome by coating
the shield interior with a thin layer of magnetic material. Following equations are,
respectively, propagation constant and characteristic impedance of such a coating
medium.
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39
Fig. 3.6: Comparison o f elevation plane normalized directive gains of the 0.5A dipole
calculated by FDTD and theory (under zero wire diameter assumption).
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40
Monopole
Metallic shield
\
Ground plane
Fig. 3.7: A shielded monopole antenna mounted on a finite size perfectly conducting
ground plane.
k = u/(/ze)1/2
(3.27)
) ‘/2
(3.28)
- ,= ( f
It can easily be seenfrom (3.27) and (3.28) that, if a dielectric material is used for
coating, electrical distance between the antenna and the shield will increase. However,
intrinsic impedance of the medium will be reduced. The only way to increase both
the electrical distance and the input impedance is to use a magnetic material.
3.3.2
Source of Excitation and Feed Modeling
An actual cellular phone antenna is fed by a coaxial cable. Coaxial feed can be
modeled by considering the fact th a t the electric field inside a coaxial cable is radially
dependent. The electric field at any point between the inner and outer conductors of
a coaxial cable is given by [35]
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41
(3 2 9 )
where p is the radial position of a particular point between the two conductors, a
and b are the radii of inner and outer conductors, respectively. V accounts for the
source voltage which can be either pulsed or sinusoidal. In impedance calculations,
a pulsed source voltage is used to obtain wideband impedance results. In calculation
of the near field magnitude, a sinusoidal source is used.
A monopole antenna mounted on a perfectly conducting ground plane is driven
by radially directed four electric field components as shown in Fig. 3.8 as proposed
in [36]. The analogy between the radially directed electric field components in the
FDTD grid and the electric field distribution inside a coaxial cable can be established
in the following manner. Let a denote r 0, the wire antenna radius, and b denote Ax.
This is because the inner conductor of coax is connected to the monopole and outer
conductor is connected to four neighboring cells. The electric field is located half cell
away from the monopole, therefore p = A x/2. Using (3.29), these four electric field
components can now be written as
B U i,j,k ) = _ £ £ ( < _
3.3.3
=
(3.30)
=
(3.31)
FDTD Analysis
The monopole antenna is m ounted and centered on a PEC ground plane whose
dimensions are 48 x 48 (mm). The antenna is 42 mm long and about A/4 at 1.8 GHz.
This is not the exact cellular frequency. However, the aim of this study is to investigate
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42
Monopole
z
y
EyOJJc) /
x
EjcOj'J c)
Fig. 3.8: The four electric field components axe driven by a source voltage to model phys­
ical coaxial cable connection to the monopole antenna.
the magnetic shielding concept with reasonable amount of computer memory and run
time available when this research was conducted. The structure of interest is shown
in Fig. 3.7. The distance between the metallic shield and the monopole is
Shield transverse dimensions are L = 16 mm and W =
8 mm.
8 mm.
Shield is 42 mm high
and rises in z direction parallel to the antenna. Cross sections of the actual shielded
antenna and its rectangular implementation used in FDTD simulations are shown in
Fig. 3.9. PML ABC is used to truncate FDTD computational space with a PML layer
thickness of 8cells. T he magnetically shielded antenna is analyzed by considering the
input impedance and the near field magnitude of following cases;
• Ordinary (unshielded) monopole.
• Shielded monopole.
• Magnetically shielded monopole. Different magnetic loss terms are used for the
coating material.
Cubical Yee cells sure used in all cases. The cell size is chosen to be 2.00 mm for the
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43
unshielded and shielded monopole antenna cases, and 1.33 mm for the magnetically
shielded antenna case. Magnetic loss of the coating material is introduced in the
form of complex permeability. Assuming that EM fields exhibit sinusoidal variation,
§1 —►juty (2.1) takes the following form
V x £ = —j uj f i H
fM = HQn'r = flo I fJLr V
where
ijl
u//zo/
j
(3.32)
is the complex permeability.
The imaginary term on the right side of (3.32) accounts for the magnetic loss.
Then, the magnetic resistivity of a* can be found easily. Constitutive parameters of
the magnetic coating material are set to er = 10.0, a = 0.0 and fir = 10.0 —j X .
Here r subscript denotes the relative permeability. Three different loss terms are
investigated for X = l.0,2.0 and 5.0. For instance, from (3.32), the magnetic resistivity
corresponding to X=1.0 is cr* = 14212.0 Q/m a t 1.8 GHz. The cell size of 1.33 mm
used in the magnetically shielded case corresponds to a cell resolution of A/10 at 2.25
GHz where A is the wavelength in the coating medium. In impedance calculations,
Rayleigh pulse is used to obtain broadband frequency domain results. Meanwhile, a
single frequency sinusoidal source is used in the calculation of near fields.
Input impedance of the antenna is found from the frequency domain input voltage
and current at the antenna input terminals. The time domain input voltage is given
by
=
where Vr is the Rayleigh pulse source voltage and scaled by
(3'33>
• This is due to the
distribution of radially directed electric fields inside the coaxial cable. Transient input
current is found by applying Ampere’s Law around the feed point of the antenna,
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44
Perfectly conducting shield
Magnetic coating
Monopole
Fig. 3.9: Cross sections of the actual magnetically shielded antenna (left) and its rectan­
gular implementation used in FD T D simulations (right).
I = jc H d l
(3.34)
It is stated in. (3.34) th a t the circulating magnetic fields along a closed loop will
induce a current that is perpendicular to the plane of the magnetic fields. Since
the electric and magnetic fields inside the FDTD computational domain are discrete
quantities, interpreting (3.34) to our problem will result in the following equation
I = (Hx( i , j - 1, k) - H x(i,j, k ) ) A x + (Hy( i J , k) - Hy(i - 1,;, k)) A y
(3.35)
Here, antenna islocated at (ij,k )th cell and all fields are evaluated at time step
n + |A t.
Frequency transformation of the time domain voltage and the current is done by
applying a simple FFT algorithm. Frequency domain voltage is divided by current
at each frequency point to obtain the input impedance.
rr r t \ — FFT{Vm(t)}
Z“ ( / ) - F f T { « t ) }
(3 36)
t3'36)
where FFT{. } denotes the Fast Fourier Transform operator.
Input impedance of the unshielded, shielded, and magnetically shielded monopole
antennas are calculated and compared. Fig. 3.10 and Fig. 3.11 shows the input
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
resistances and reactances, respectively. Unless the antenna is shielded, it will be
referred simply as the monopole for convenience. The monopole antenna has an
input resistance of about 25 ohms at 1.8 GHz on a 48 x 48 mm PE C ground plane.
When it is shielded, input resistance drops to a few ohms (< 5 ohms) due to excessive
capacitive coupling between the antenna and the shield. Since this capacitance shorts
out the antenna, matching will be a major problem. Input resistance should be raised
to match th e antenna to the feed line. It can be seen in Fig. 3.10 th a t using a lossy
magnetic coating material inside the shield will increase the input resistance. The
thickness of the magnetic coating material is set to 4.00 mm for both a * = 28425
Clfm and a* = 71062 Cl/m cases. Since the shield dimensions are
8x
16 (mm), this
material fills about half of the shield. It is clear from Fig. 3.10 th a t increasing the
magnetic loss will also increase the input resistance of the antenna. Input reactances
of the unshielded and shielded antennas are found to be close around 1.8 GHz. On
the other hand, using a lossy magnetic coating material will significantly increase the
input reactance. This is due to the high perm ittivity material being used.
The effect of thickness of the coating m aterial is also investigated as shown in
Fig. 3.12. 4.00 mm, 2.67 mm and 1.33 mm thick magnetic coatings are compared for
a fixed loss term of <r* = 28425 Cl/m. It is observed th at thicker coating materials will
result in higher input resistances. 1.33 mm thick coating yields an input resistance
which agrees excellent with th at of the unshielded monopole in the frequency range of
1.0-2.0 GHz. Input reactances for different coating thicknesses are shown in Fig. 3.13.
It can be concluded th a t the magnetically shielded antenna can be well matched to
the feed line if the thickness and the constitutive parameters of the magnetic material
are well adjusted.
Second step is the calculation of near fields. T he antenna is excited by a single
frequency sinusoidal source at 1.8 GHz using the coaxial feed model. Once all the field
components inside the FDTD computational domain achieve steady state, magnitudes
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
46
300
250
Monopole
Shielded monopole,
Magnetic coating it>=28425
Magnetic coating tr =71061
200
3 150
100
1.0
1.1
L2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz)
Fig. 3.10: Comparison of input resistance for unshielded, shielded and magnetically
shielded antennas. Coating material is 4.00 mm thick.
300
200
Monopole
Shielded monopole,
Magnetic coating tr[=28425
Magnetic coatingtr =71061
i 100
uC® a o
•o
aa.
J5 -loo
-200
-300
1.0
1.1
L2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz)
Fig. 3.11: Comparison o f input reactance for unshielded, shielded and magnetically
shielded antennas. Coating material is 4.00 mm thick.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
300
250
Monopole
4.00 mm coating
2.67 mm coating
1.33 mm coating
g 200
u
'O
§ 150
100
1.0
1.1
L2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz)
Fig. 3.12: Comparison of input resistance for unshielded, shielded and magnetically
shielded antennas, a* = 28425 0 /m .
300
Monopole
4.00 mm coating
2.67 mm coating
1.33 mm coating
200
I 100
©
§ o
T3
aa>
M-100
-200
-300
1.0
LI
1.2
1.3
L4
L5
1.6
1.7
1.8
1.9
2.0
Frequency (GHz)
Fig. 3.13: Comparison of input reactance for unshielded, shielded and magnetically
shielded antennas, a * = 28425 Q /m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
LI
—
1.0
Monopole
Shielded monopole,
4.00 mm <
coating <r=28425
L33 mm coating <x*=28425
08 0.4
0.1
■SS
0.0
Distance (mm)
Fig. 3.14: Normalized electric field magnitude along a straight line. Each case is normalized
to its own m axim a. The monopole is located at 0 mm, and the shield is located at + 8
mm.
1.6
1.4
Monopole
Shielded monopole,
4.00 mm coating </=28425
L33 mm coating <r*s28425
'S 1-2
"O0.8
«
0.6
® 0.4
0.0
Distance (mm)
Fig. 3.15: Electric field magnitude is normalized with respect to the monopole antenna
along a straight line. The monopole is located at 0 mm, and the shield is located at + 8
mm.
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49
of the electric field components can be found easily. In early work of determining the
signal magnitude, slope variation of the sinusoidal signal was observed. When the
slope changes from positive to negative values, a positive amplitude is picked up.
A slope change from negative to positive values yields a negative amplitude. The
difference between the positive and negative cycle amplitudes is the peak-to-peak
signal amplitude [12]. Alternatively, a simple DFT algorithm could also be used as
explained in section 3.2.3. In any case, phase information is not needed.
The magnitude of the electric field is found along a straight line in y direction.
This line passes through the middle of the shield and is one cell off from the antenna in
x direction. Fig. 3.14 shows the electric field distribution of the unshielded, shielded
and magnetically shielded antennas. The electric field magnitude is plotted from -38
mm to +38 mm. T he antenna is located at zero reference point. The horizontal
axis of Fig. 3.14 shows the relative distance away from the antenna. In each case,
magnitude of the electric field is normalized to its own maxima. Field distribution
is found to be symmetric when there is no shield. The PEC shield is located at
+8
mm and the field m agnitude exhibits a sharp drop at this location. The electric field
magnitudes of above cases are normalized to th a t of the monopole antenna as shown
in Fig. 3.15. The near electric field distribution of 1.33 mm thick magnetic coating is
found to be quite similar to that of the monopole. Except, near fields drop sharply
in the direction of the shield. Clearly, shielding of the antenna reduces EM radiation
in one direction which can be the user direction.
3.4
A Dual Resonance Shielded Cellular Phone Antenna
3.4.1
Theory of Dual Resonance
The magnetically shielded antenna presented in section 3.3 requires a thin mag­
netic coating m aterial to eliminate the unwanted effects of a closely located metallic
R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission.
50
shield. Magnetic materials for cellular phone applications are heavy and expensive.
Consequently, even though the magnetically shielded antenna performed well, the
need for a cheap and easy to manufacture antenna arises. For this purpose, a dual­
resonance shielded antenna suitable for cellular applications is introduced.
The theory behind the dual-resonance shielded cellphone antenna can be explained
best by looking at Fig. 3.16. Input resistances of the shielded, shielded and shorted,
and unshielded monopole antennas m ounted on a 48 x 48 (mm) PEC ground plane
sure shown. The monopole and the shield are both 42 mm long which corresponds
to A/4 at about
1.8 GHz.
This antenna and shield configuration is as same as the
one used in the study of magnetic shielding, except the magnetic coating material is
excluded. The unshielded antenna exhibits a resonance around 3.1 GHz. When the
monopole is shielded, a very high and narrow resonance peak is observed around 3.3
GHz. The off-resonance input resistance is only a few ohms in the neighborhood of
1.8 GHz. The drop in input resistance is due to the excessive capacitance between
the antenna and the metallic shield as explained in detail in section 3.3.1. Our aim is
to move this resonance frequency down while keeping the antenna dimensions same.
To do so, the monopole is shorted to the metallic shield at the open end. As a result,
the resonance frequency shifted down to 1.5 GHz. Input resistance is still very high
(> 5KQ) at resonance and very low off-resonance. If an open wire is attached to the
shorted one in parallel, a second resonance appears whose frequency is close to the
first one. This will lead to the dual-resonance mechanism.
The new design shielded antenna on a finite size ground plane is shown in Fig. 3.18.
Due to the open and shorted configuration of the wires, multiple resonances occur.
First two resonances and the frequency bandw idth between them can be adjusted by
selecting proper antenna dimensions such th a t an almost 50 ohms input resistance
between 800-900 MHz band can be obtained. The input reactance of the antenna can
be m inim ized as well. The new shielded antenna operates in the valley between the
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51
first two resonances [13]-
8000
— Monopole
— Shielded monopole
— Shielded and shorted monopole
6000
3
43
w
§ 4000
43
a
Si
i—
2000
01.0
1.5
2.0
2 .5
3.0
3.5
4.0
Frequency (GHz)
Fig. 3.16: Resonances o f shielded, shielded and shorted, and unshielded monopole anten­
nas.
3.4.2
Parametric Study of am Initial Antenna Configuration
In order to get an idea about the behaviour of the proposed dual-resonance
shielded antenna and guide the research towards the aimed direction, a paramet­
ric study is performed for the structure shown in Fig. 3.18. The shield is rectangular
in shape to avoid complexity during the mesh generation process and mounted on a
50 x 50 (mm) ground plane. The open and shorted wires, the shield and the ground
plane are assigned an electric conductivity of a = 2.564 • 107 S/m. This value of
conductivity corresponds to that of brass. Initially selected antenna dimensions sure
Ll=30, L2=64, L3=70, D l= 8, D2=8, and H=112 (mm). This configuration is inves­
tigated for different LI, L2, L3 and D2 values. Cell sizes are Ax = Ay = 1.0 mm and
Az = 2.0 mm. The FDTD computational domain has
66 x 66 x 70 cells, excluding
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52
Metallic shield
Shorts
Wire looldng up and
shorted at end
Wire looking down and open at end
(a)
(b)
Fig. 3.17: Cross section of the (a) actual dual-resonance shielded antenna and (b) staircase
approximation used in FD T D simulations.
Metallic shield
LI
L3
D2
L2
Ground plane
L4
Fig. 3.18: Dual-resonance shielded antenna on a finite ground plane.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
the PML region. T he antenna is excited by driving the E z field component at the
bottom of the shorted wire with a Rayleigh voltage pulse. This feed model is shown
in Fig. 3.19 and analogous to delta-gap source model used in MoM calculations. This
type of feed model is not physical, since there is always one cell gap between the
antenna and the cellular box as shown in Fig. 3.19.
Thick wire
Thin wire
Top of the cellular box
Ground plane
(a)
(b)
Fig. 3.19: Delta-gap feed model (a) for a wire represented by a single cell and (b) for a
thick wire resolved by many cells.
For parametric stu d y simulations, a coaxial cable analog feed model could be used
as described in section 3.3.2. However, it will be shown in the following subsections
th at the proto-type of the dual resonance shielded antenna has a relatively thicker
diameter. This diam eter should be resolved by as many as 16 cells and the imple­
mentation of delta-gap feed model will be much easier than the coaxial feed model.
Both models are supposed to yield almost equally accurate results. Fig. 3.20 shows
the comparison of input resistances of the dual-resonance shielded antenna obtained
using coaxial and delta-gap feed models. The agreement is found to be excellent. The
antenna parameters are L l=30, L2=64, L3=70, D l= 8, D2= 8, and H=112 (mm) for
this case.
From Fig. 3.21, decreasing L l will shift the first resonance to a higher frequency,
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54
1000
Delta gap feed
Coaxial feed
800
Si 400
200
0.5
0.6
0.7
0.8
0.9
1.0
LI
Frequency (GHz)
Fig. 3.20: Comparison of delta-gap and coaxial cable feed models. L l= 30, L2=64, L3=70,
D l= 8 , D 2= 8, H =112 (mm).
while the second resonance is not affected. This initially reveals that the first reso­
nance is due to the shorted wire and the second resonance is due to the open wire.
A third and weak resonance is also visible in Fig. 3.21 especially when L l=24 mm.
This is due to the mutual coupling between the wires which increase when the two
resonances get close to each other. Reviewing the input reactance in Fig. 3.22 indi­
cates that the reactance is almost zero for Ll=30 mm at about 840 MHz and L2=24
mm at about 870 MHz. This is a quite desirable feature especially for matching
purposes. Decreasing L2 from 64 mm to 52 mm will move the second resonance to a
higher frequency while the first resonance is also shifted upward slightly. Clearly, the
mutual coupling is lower for L=52 mm case. Refer to Fig. 3.23 and Fig. 3.24 for the
effect of L2 over the input impedance. In Fig. 3.25, L3 has been increased from 70
mm to 76 mm. The first resonance shifted down by about 20 MHz, and the second
one by about 35 MHz. From Fig. 3.23 and Fig. 3.25 it can be said that the secondary
resonance is a strong function of L2 and weakly related to L3. It is also observed th a t
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55
further separation of the two resonances results in lower input resistance in the valley
region. The effect of wire separation is also studied by changing D2 from
8 mm to 4
mm. By bringing the open and shorted wires even closer, we expect a higher m utual
coupling between the wires. However, Fig. 3.27 reveals th at lowering D2 will shift the
secondary resonance by about 100 MHz and a lower resistance is obtained between
the two resonances.
500
400
^ 300
a. 200
100
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Frequency (GHz)
Fig. 3.21: Effect of Ll over input resistance. L2=64, L3=70, D l= 8, D2=8, H=112 (mm).
3.4.3
Calculation of Input Impedance and Experimented Verification
A proto-type of the dual-resonance shielded antenna wets built and tested at Ari­
zona State University’s Anechoic Chamber. Input impedance, near and far fields of
the proto-type antenna were measured. Fig. 3.1(c) shows the new antenna mounted
on a cellular box. The cellular box was made from brass and covered by a 2.3 mm
thick lossless dielectric material whose permittivity is er = 2.5 except the top of the
box. The cellular box dimensions are 64.8 x 34.8 x 102.4 (mm) in x, y, and z directions,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
1000
800
600
~
400
200
o. -200
-400
-600
-800
-1000
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Frequency (GHz)
Fig. 3.22: Effect o f L l over input reactance. L2=64, L3=70, D l= 8 , D 2=8, H =112 (mm).
500
400
300
-a
a>
0.200
100
0.5
0.6
0.7
0.9
1.0
1.1
Fig. 3.23: Effect of L2 over input resistance. L l= 3 0 , L3=70, D l= 8 , D 2=8, H =112 (mm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
1000
800
600
400
200
-400
-600
-800
-1000
0.5
0.6
0.7
0.8
0.9
1.0
1-1
Frequency (GHz)
Fig. 3.24: Effect of L2 over input reactance. L l= 3 0 , L3=70, D l= 8 , D 2= 8, H =112 (mm).
500
400
300
0 .2 0 0
100
0.5
0.6
0.7
0.9
1.0
1.1
Fig. 3.25: Effect of L3 over input resistance. L l= 3 0 , L2=64, D l= 8 , D 2 = 8 , H = 112 (mm).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
1000
800
600
400
200
0.-200
-400
-600
-800
-1000
0.5
0.6
0.7
0.8
0.9
L0
1.1
1.2
Frequency (GHz)
Fig. 3.26: Effect o f L3 over input reactance. L l= 3 0 , L2=64, D l= 8 , D 2 = 8 , H =112 (mm).
500
400
u
300
0.200
100
0.5
0.6
0.7
0.9
L0
1.1
Fig. 3.27: Effect o f D2 over input resistance. L l= 3 0 , L2=64, L3=70, D l= 8 , H =112 (mm).
R e p ro du ced with permission o f the copyright owner. Further reproduction prohibited without permission.
59
1000
D2>8 mm
D2-4 mm
800
600
a. -200
s
~
-400
•800
-1000
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Frequency (GHz)
Fig. 3.28: Effect of D2 over input reactance. L l= 30, L2=64, L3=70, D l= 8 , H =112 (mm).
respectively. The antenna is located centrally on the cellular box. For fine details of
the antenna and the shield configuration Fig. 3.17 should be referred. The cellular
box, antenna and the shield are assigned a conductivity of 2.564 • 107 (S/m ) during
the FDTD simulations. Metallic shield is approximated to a curve-like shape using
staircased cells. Wires have square cross sections rather than circular. The shield has
a radius of 5.08 mm and the wire diam eter is 2.33 mm. The separation between the
open and shorted wires is 3.92 mm. The dimensions of the proto-type antenna are
given in the following table and the corresponding lengths can also be referred from
Fig. 3.18.
The 2.33 mm thick wire is subject to skin depth phenomenon and need to be
modeled precisely. For this reason, FDTD analysis is carried out using a non-uniform
grid architecture in x-y plane as shown in Fig. 3.29. The FDTD computational
domain in x-y plane is divided into two regions, a high resolution space formed by
uniform high resolution cells and an expanding grid region to handle large objects
compared to the dimensions of the antenna. On the other hand, the cell size in z
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60
direction is kept uniform and Az = 2.54 mm. It is clear that without the application
of a non-uniform mesh, handling large objects like the cellular phone box would cost
a huge memory requirement and an extremely long computational time. The cell size
in high resolution x-y plane is chosen as Ax = Ay = 0.565 mm. The antenna is
centrally located in this region. Away from the antenna cell size expands in size such
th at the physical dimensions of the cellular box and the thickness of the dielectric
coating material are approximated in best. Non-uniform cells are generated using the
expansion function given by (2.59).
Dimensions of the
proto-type antenna (mm)
H
127.00
LI
25.40
L2
77.50
L3
80.00
L4
2.54
Dl
5.08
D2
3.92
The thick wire is resolved by 16 cells in a 4 x 4 matrix resolution. Consequently, 16
E z field components at the bottom of the shorted wire are used to drive the antenna
as shown in Fig. 3.19(b). Input impedance of the antenna is desired at wide range
of frequencies. For this purpose, each E z field component is loaded by a Rayleigh
source voltage pulse. This feed technique is completely similar to the one used in
parametric study cases. Time domain input current of the antenna is found using the
circulating magnetic field components around the cells used for antenna feed. The
input voltage is the applied Rayleigh pulse itself. Frequency domain input voltage
and current are obtained by applying a simple FFT algorithm to time domain field
quantities.
Dividing voltage by current at each frequency point yields the input
impedance. Calculated input impedance of the dual-resonance shielded cellphone
antenna is compared with impedance measurement as shown in Fig. 3.30 and Fig. 3.31.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
90 r
20
-
10 r
0i— — —l - l i i- L-Li
0
20
11hi
40
60
80
Distance in x direction (mm)
u
j — — — ------ >
100
120
Fig. 3.29: A non-uniform mesh in x-y plane is used in FDTD simulations o f the dualresonance shielded cellular phone antenna.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
62
An input resistance of almost 50 ohms is obtained within a bandwidth of 824-849
MHz. This band is known as the transmit band and a minimum voltage standing
wave ratio (VSWR) is required due to power limitations of the cellular phone. The
869-894 MHz band is the receive band and VSWR is better than 2.0. Input reactance
is about zero ohms in the vicinity of 870 MHz. Clearly, an excellent agreement has
been achieved in the calculation of input impedance.
300
250
a 150
100
50
0.5
0.6
0.7
0.8
Frequency (GHz)
L0
1.1
Fig. 3.30: Calculated and measured input resistance of the dual-resonance shielded cellular
phone antenna.
3.4.4
Near and Far Field Measurements
In H-plane far field measurements, a 1.5 dB more gain than the conventional
monopole antenna is observed in the absence of user as shown in Fig. 3.32. E-plane
far field measurement of the monopole, Fig. 3.33, shows that the lobes are tilted to
the ground which is not a desirable feature for cellular communications. Measured Eplane far field pattern of the dual-resonance shielded antenna shows th a t less energy
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63
300
200
-100
-200
-300
0.5
0.6
0.7
0.8
Frequency (GHz)
1.0
1.1
L2
Fig. 3.31: Calculated and measured reactance of the dual-resonance shielded cellular phone
antenna.
is radiated into ground than the monopole antenna. T he dual-resonance shielded
antenna has also higher gain in the upper hemisphere.
In the presence of user, far field gain of the monopole dropped by about 2.6 dB
down while the dual resonance shielded antenna’s gain dropped by about
1dB which
is not significant. This implies that in the presence of user radiation pattern is not
altered significantly. H-plane pattern in the presence of user is shown in Fig. 3.34.
Human head is modeled simply by a dielectric block of er ~ 50 to account for the
presence of user. Near field measurements are shown in Fig. 3.35. The normalized
near field for the monopole antenna lies on the circumference and is symmetric in the
absence of user. T he larger pattern within the graph shows the near field distribution
of the 10.16 mm diam eter shielded antenna and the near field radiation towards user is
blocked by about 90%. The smelliest pattern is the near field of the 12.7 mm diam eter
shielded antenna and the near field towards user is blocked by about 97%. It is clear
th at the near field radiation towards user is reduced by about 90% using a 10.16 mm
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64
diameter antenna without significantly changing the omnidirectional characteristics
required for cellular phone devices.
3.4.5
FDTD Calculation of Near and Far Fields
A near-to-far zone transformation technique explained earlier in this chapter is
implemented with slight revisions to find the far fields of the dual-resonance shielded
cellular phone antenna. These revisions were necessary due to the application of nonuniform mesh inside the FDTD computational domain. It is pointed out in section 3.2
that the vectors addressing the electric and magnetic fields on the virtual surfaces and
far zone observation points should be calculated. In the presence of a non-uniform
mesh, these vectors are calculated simply by summing the physical length of each cell
referenced to (1,1,1) cell. This brings an extra computational effort within the far
zone transformation code.
A sinusoidal source at 870 MHz is used to drive the antenna by implementing the
same feed model used in impedance calculations. The tangential electric and magnetic
fields located on virtual surfaces are transformed to phasor field quantities by applying
a DFT algorithm within the FDTD time stepping loop after steady state. Far fields
of the antenna are found from these phasor form near fields. Radiation patterns in
E and H planes are calculated by FDTD with an tingle increment of
1 degree both
in 9 and <j>. H and E plane radiation pattern measurements in the absence of user
are compared with FDTD calculations as shown in Fig. 3.36 and Fig. 3.37. The
agreement is found to be excellent in both cases.
Near field distribution of the dual-resonance shielded antenna is found on selected
x-y and y-z planes. A simple DFT algorithm is implemented within the time stepping
loop to find phasor field quantities from the time varying fields on these planes. Then,
phasor form electric fields me properly averaged at (ij,k) location of each cell, since
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
they are located half cell apart. Fig. 3.38 shows the contour plot near field distribution
in x-y plane. The shield and the wires are highly visible. The electric field is very
intense inside the shield and energy is directed in the opposite direction of the shield.
Fig. 3.39 shows the near electric field distribution in y-z plane. The EM fields do
penetrate insignificantly behind the shield, mostly by way of coupling through the
shield edges. Most of the EM energy is confined in the region between the antenna
and the shield, and directed in the opposite direction of the user.
S h i e l d e d a n t e n n a on
a L G 0 2 0 I .r e o
pnone
REGUENCY = 3FC -H z
FREQUENCY = 8 2 0 tlHz
120
/*\
130
-30
Fig. 3.32: H-plane far field measurements of the dual-resonance shielded antenna in the
absence of user.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
66
Sh i a i d e d a n t e n n a an
moncDO :e on
FREQUENCY = 0 7 0 HKz
anone . c -o ic n e
FREQUENCY = 8 7 0 “Hz
120
r
1
50
180
-150
Fig. 3.33: E-plane far field measurements of the dual-resonance shielded antenna in the
absence of user.
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67
; n3 ne
J/
1 3C *
G. - 2 0 . - 3 0 . -2 0
Fig. 3.34: H-plane far field measurements of the dual-resonance shielded antenna in the
presence of user.
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Fig. 3.35: Norm alized near field measurements o f the dual-resonance shielded antenna
the absence o f user.
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69
Fig. 3.36: C alculated H-plane far field pattern o f the dual-resonance shielded antenna in
the absence o f user.
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70
Fig. 3.37: Calculated E-plane far field pattern o f the dual-resonance shielded antenna in
the absence o f user.
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71
70
60
50
c
JO
| 40
T3
>-
= 30
9
o
20
10
10
20
30
40
50
Cells in X direction
60
70
80
Fig. 3.38: Near electric field distribution of the dual-resonance shielded antenna in x-y
plane in the absence of user.
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72
t--------------------1------------------- 1------------------- 1--------------------r
_______i________i________i_______ i________i------------ 1-----------10
20
30
40
50
60
70
Cells in Y direction
Fig. 3.39: Near electric field distribution of the dual-resonance shielded antenna in y-z
plane in the absence o f user.
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73
3.5
A Shorted and Truncated Microstrip Antenna
3.5.1
Introduction
A shorted and truncated microstrip antenna is presented to be used in mobile
communications systems. The shorted microstrip antenna can be well suited as an
internal antenna in cellular systems due to its small size and low profile. Furthermore,
microstrip antennas are flexible in a way th at several individual antennas can be
mounted on a single substrate, or in the form of layers within the same substrate to
form antenna arrays. The beam of such antenna arrays can be directed electronically.
Under these considerations, a properly designed microstrip antenna can substitute
the dual-resonance shielded antenna as an internal antenna alternative.
A brief information will be given on regular (open) microstrip antennas before
going through th e shorted and truncated one. A microstrip antenna is basically
a resonant structure having a dielectric substrate sandwiched between two metallic
layers on top and bottom. The metallization on top acts as the radiating patch,
while the bottom one is the ground plane. In a classical microstrip antenna, the
ground plane is much larger than the radiating patch as shown in Fig. 3.40. The
radiating (resonant) patch dimensions should be about A/2 for an efficient radiation,
where A is the wavelength in the substrate. At a frequency of 850 MHz, a microstrip
antenna having a relative dielectric perm ittivity of er =
2.2, should have a dimension
of 11.9 cm. This dimension is quite large (larger than the cellular phone) for an
antenna designed for cellular applications. Increasing er can help reducing the antenna
dimensions at th e expense of narrower bandwidth and lower gain.
The ground plane and the dielectric substrate of the new microstrip antenna are
truncated to th e size of the radiating patch. The radiating patch is shorted to the
ground plane by simply shorting th e edge of the structure.
Previously, effect of
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74
a finite ground plane on the radiation patterns of an open microstrip antenna was
investigated [37]. For a completely truncated ground plane, surface waves propagating
parallel to air-dielectric interface will be partially reflected back from the edges, and
react with the main radiator. This will alter the input impedance of the microstrip
antenna considerably [38]. Sanad [39] reported that when the size of the ground plane
is closely reduced to the size of the radiating patch of an open microstrip antenna,
radiation patterns show good isotropic and polarization independence characteristics.
Shorting the microstrip antenna causes radiating patch to radiate at the same
resonance frequency of an open microstrip antenna with twice the resonant dimensions
[40]. It is reported in [41] that, instead of wrapping the edge of the microstrip by a
copper strip, shorting posts can be used to realize shorted microstrip antenna from the
manufacturing point of view. However, the design presented here will have a shorted
edge rather than the shorting posts. By shorting the antenna, the same resonance
frequency of 850 MHz can be obtained using a dimension of 5.95 cm, with a dielectric
substrate of er = 2.2. Based on above, a shorted and truncated microstrip antenna is
designed and its input impedance is calculated using the FDTD method [42].
Resonant patch
[■
Ground plane
Dielectric substrate
Fig. 3.40: A classical open microstrip antenna (left), ground plane and resonant patch have
the same dimensions (right).
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75
3.5.2
Calculation of Input Impedance and Near Fields
The microstrip antenna is realized using a lossless dielectric material having a
relative permittivity of Cp =
10.2 sandwiched between the
radiating patch and the
ground plane. The dielectric substrate and the ground plane are truncated to the
size of the radiating patch as shown in Fig. 3.41. The microstrip antenna dimensions
are L=30 mm, W=30 mm and H=5 mm. A coaxial feed probe is used to feed the
antenna which is located closely to the shorted edge for matching purposes. Moving
the feed point away from the shorted edge results in higher input resistance at the
resonance frequency.
Radiating patch
Dielectric substrate
Shorted edge
L
\
Coaxial feed
Fig. 3.41: A shorted and truncated microstrip antenna.
In FDTD simulations, spatial cell resolutions are selected as Aar = 2.5 mm, A y =
2.5 mm and Az = 1.25 mm. FDTD space contains 28 x 28 x 36 cells excluding the
region allocated for PML absorbing boundaries. The source of excitation is a Rayleigh
pulse with /3 = 350. Frequency spectrum of the source pulse exhibits bandpass filter
characteristics in which the passband exists at about 850 MHz. The feed model
described in section 3.3.2 is used to model the physical coaxial cable connection with
r = 0.25 mm.
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76
200
Micrnatrip antenna alone
Ground plane 50x50 (mm)
Ground plane oo
150
O.
50
0.80
0.86
0.84
0.82
0.88
0.90
Frequency (GHz)
Fig. 3.42: Input resistance of the shorted and truncated microstrip antenna for different
ground plane sizes. L=30 mm, W = 3 0 mm, H =5 mm, er = 10.2.
200
150
S ioo
50
0.8
0.9
1.0
1.1
1.3
1.4
1.5
1.6
1.7
1.8
Frequency (GHz)
Fig. 3.43: Effect o f substrate perm ittivity over the resonance o f the shorted and truncated
microstrip antenna. L=30 mm, W = 3 0 mm, H =5 mm.
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77
300
250
§ 150
■0
O
)
flu
J3 too
50
0.80
0.82
0 .8 4
0.86
0.88
0.90
Frequency (GHz)
Fig. 3.44: Effect o f substrate thickness over the resonance o f the shorted and truncated
microstrip antenna. L=30 mm, W = 3 0 mm, er = 10.2.
Fig. 3.42 shows the input resistances of the shorted and truncated microstrip
antenna for different sizes of ground planes. Input resistance is about 140 ohms
at about 845 MHz when the size of the ground plane is as same as the radiating
patch. If the ground plane is 50x50 (mm) large, the resonance frequency shifts to
860 MHz and the input resistance rises to 175 ohms. For an infinitely long ground
plane (ground plane is extended into PML layers) resonance occurs at about 860
MHz. The bandwidth is very narrow for all these cases. The truncated ground
plane yielded lower resonance frequency than the others. The effect of substrate
permittivity is investigated for er = 10.2, er = 5.6 and er = 2.2 for a 5 mm thick
substrate as shown in Fig. 3.43. Lower permittivity values yielded higher resonance
frequencies. This is due to the fact th a t the wavelength at resonance is determined
by the substrate permittivity. On the other hand, larger bandwidths are observed
by using lower perm ittivity substrates. For an ^ = 10.2 substrate, increasing the
substrate thickness from 5 mm to 10 m m will increase the bandwidth at the expense
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78
of a slight shift in the resonance frequency as shown in Fig. 3.44. The bandwidth of
the shorted and truncated microstrip antenna can be increased either by lowering the
dielectric permittivity or by increasing the th ickness of the substrate. Alternatively,
stacked antenna configurations can be used to increase the bandwidth.
45
T---------- 1---------- 1---------- 1---------- 1---------- 1---------- 1---------- 1
r
40
35
c 30
0
1
=5 25
>
c
20
®
O
15
10
5
j ____________ i____________ i____________i____________ i____________ i_____________ i____________ I____________ i _
5
10
15
20
25
30
Cells in X direction
35
40
45
Fig. 3.45: Near electric field distribution of the shorted and truncated microstrip antenna
in x-y plane.
3.6
Summary and Conclusions
In this chapter, various shielded antenna configurations were investigated to pro­
tect humans from excessive EM radiation. The first antenna considered was the
magnetically shielded antenna. It has been shown th a t using a magnetic coating
material inside the metallic shield can eliminate all the disadvantages of using the
metallic shield alone. These are the very low off-resonance input resistance and de-
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79
40
5
10
15
30
20
25
Cells in Y direction
35
40
45
Fig. 3.46: Near electric field distribution of the shorted and truncated microstri p antenna
in y-z plane.
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80
graded far fields. However, magnetic materials are expensive and heavy for cellular
applications. Even though the concept of magnetic shielding is found to be succesful,
a low cost antenna is desired for commercial applications.
The second antenna investigated was the dual-resonance shielded antenna. This
antenna blocks excessive EM radiation towards user by over 90% compared to con­
ventional monopole. Matching to 50 ohms transmission lines is not a problem because
the new antenna is inherently matched to 50 ohms transmission lines. This simplifies
the design and eliminates additional matching circuitry. In the absence of user, the
new antenna has 1.5 dB more gain than the conventional monopole antenna in far
zone. In the existence of user, far zone gain does not drop significantly and th e radia­
tion pattern seems to be almost unaltered. In the absence of user, E-plane (elevation
plane) far field gain of the new antenna is increased in the upper hemisphere and less
radiated energy is directed into ground. An input resistance of almost 50 ohms was
obtained within a bandwidth of 824-849 MHz. This band is known as the transm it
baud, and a minimum voltage standing wave ratio (VSWR) is required due to power
limitations of the transm itter unit. The 869-894 MHz is the receive band and VSWR
is better than 2.0. The input reactance is almost zero ohms in the vicinity of 870
MHz. Input impedance measurements show th at the shielded antenna is wideband
and an almost 50 ohms flat input resistance can be obtained in the 824-894 MHz
range. The measured input impedance and far fields of the dual-resonance shielded
antenna were compared with FDTD calculations and excellent agreement has been
achieved.
Finally, a shorted and truncated microstrip antenna was presented for wireless
systems demanding small, low profile and conformed antennas. By shorting the mi­
crostrip antenna, the resonance frequency of an open microstrip antenna was obtained
using half the resonant dimensions. The effect of dielectric permittivity and substrate
thickness were investigated. The bandwidth of the shorted and truncated microstrip
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81
antenna can be increased either by lowering the dielectric permittivity or by increas­
ing the thickness of th e substrate. Alternatively, stacked antenna configurations can
be used to increase the bandwidth.
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CHAPTER 4
TIME DOMAIN ANALYSIS OF MICROWAVE FERRITE DEVICES
4.1
Introduction
The well-known magnetic properties of ferrites are due to the existence of magnetic
dipole moments which are mainly caused by electron spin. Since the contribution to
overall magnetic moment from the orbited movement of an electron is very small,
this effect is often not considered [43]. A sufficient magnetic dc bias field can cause
magnetic dipoles to be aligned in the direction of the bias field and precess around
it. T he angular frequency of such precession is linearly related to dc bias field. This
phenomenon leads to magnetic anisotropy which is widely used to construct various
microwave devices such as isolators, circulators and phase shifters. However, ferrite
materials were introduced into microwave applications due to their high resistivity
at microwave frequencies. This property of ferrites reduces high eddy current losses
which are inversely proportional to their resistivity [44]. Increasing amounts of the
dc bias field will cause more magnetic dipoles to be aligned and precess freely around
the bias field, until all magnetic dipoles are aligned. The amount of magnetization in
this case is equal to saturation magnetization M a, a physical parameter for ferrites.
Currently, major full-wave techniques such as the Method of Moments (MoM) and
the Finite Elements (FEM) are frequency domain based techniques. Both MoM and
FEM codes need to be run at each frequency point within the desired bandwidth. A
time domain approach is computationally efficient in the analysis of wideband devices
such as circulators and isolators, since it is possible to obtain wide band frequency
domain results by frequency transforming the time domain data. Luebbers [45] ex­
tended FDTD method to isotropic and electrically dispersive media using a convo­
lution process to evaluate frequency dependent constitutive parameters. Hunsberger
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83
[46] extended FDTD for magnetized plasmas and used frequency domain permittiv­
ity tensor and recursive convolution. Pereda [47] used a time domain approach to
include magnetized ferrites and applied his technique to the calculation of propaga­
tion constants and dispersion characteristics of transversely magnetized ferrite loaded
waveguides [48] in 2-D. The tim e domain approach presented in this Dissertation is
similar to the one applied in [47]. However, full-wave analysis of isolators and circu­
lators using the FDTD method have not been reported previously.
The ferrite medium is modeled in time domain by solving Maxwell’s curl equations
and the equation of motion of the magnetization vector in consistency. New FDTD
updating equations are derived to include magnetized ferrite material within the reg­
ular FDTD time stepping algorithm. Magnetic losses are included in the formulation
by means of a phenomenological damping factor a . This technique is applied to a
thin film isolator [49] and a stripline disc junction circulator [50] in 3-D for validation
purposes. Both the isolator and the circulator have finite size of the magnetic mate­
rials as well as conductor discontinuities. The effect of non-uniform magnetization is
also included in the analysis of the stripline circulator.
4.2
Theory of Magnetized Ferrite
4.2.1
Maxwell's Equations in Anisotropic Media
Maxwell's equations in differential form for a homogeneous, linear, anisotropic,
source free, and electrically lossy medium Eire
V x l= -|
(4 .1 )
an
V x f f = <rE +
at
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(4.2)
84
V -D = 0
(4.3)
V •B = 0
(4.4)
•4
^
^
•#
The constitutive relations between B -H and D -B vector pairs in. a ferrite medium
are
B
H
(4.5)
D = eE
(4.6)
where J* is the permeability tensor. Since ferrites are electrically isotropic materials,
E is scaled by a scalar e only.
4.2.2
The Equation of Motion of the Magnetization Vector
Magnetic properties of ferrites are due to the existence of magnetic dipoles which
are mainly generated by the electron spin.
In this subsection, ferrite medium is
assumed to be infinite in extent. Assuming th a t a dc magnetic field of H0 is applied
to a ferrite medium, a torque T will be excerted on the magnetic dipole moment m
[51] given as
T = fioih x H q
(4.7)
The magnetic dipole moment rh and the spin angular momentum J of an individ­
ual electron share the same spin axis and are oppositely directed. The ratio of the
magnetic moment to the angular momentum is a constant known as the gyromagnetic
ratio
7 (1.759 x 1011C/Kg)
and is given as
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85
7=
—~J
(4-8)
T he presence of the torque causes a change in the spin angular momentum, since
the torque is equal to the time rate of change of the spin angular momentum
f
= w
(4-9)
Since the spin angular momentum and the magnetic dipole moment share the same
axis, the result is the precession of m around the axis of the applied dc magnetic field.
From (4.7), (4.8) and (4.9) the equation of motion of a single magnetic dipole [43],
[51] is
= - m ™ x H0
(4.10)
In a unit volume of ferrite there are N magnetic dipoles and the toted magnetic
moment of all aligned spins is simply M = N m . From (4.10), the equation of motion
takes the form of
^
=
(4.11)
where M is the toted magnetization and H is the toted magnetic field intensity vector
interned to ferrite. H can include the dc bias field Ho emd any rf signed superimposed
on it. (4.11) is known as the equation of motion of the magnetization vector in the
literature.
Magnetic loss mechemism in ferrites is still not well understood and formulated,
while dielectric losses cem easily be represented by a loss tangent of tern S. Magnetic
loss is defined in terms of the ferromagnetic resonance (FMR) linewidth A H , an
experimentally found parameter of ferrite. It is measured by either changing the
frequency of the applied rf field or varying the bias field Ho- Thus, it is possible
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86
to catch the ferromagnetic resonance condition of u> = u/0 in which the rf energy is
absorbed maximally by the ferrite material. The physical magnetic loss mechanism is
generally connected to the excitation of spin waves [52]. A phenomenological loss term
is introduced by Landau and Lifshitz [52] to account for magnetic losses. Including
the Landau-Lifshitz loss term, (4.11) can be w ritten as
^
= -w rM x H - ^ 2
{Mx{Mx
h ))
(4.12)
The phenomenological loss term scaled by the damping factor a is basically a
vector which acts like an opposing torque to the torque which causes precession and
has a duty of pulling M inward by decreasing the precession angle. The following
equation is mathematically simpler than (4.12) and known as the equation of motion
with Gilbert’s damping term [53].
dM
a ( 8M\
— ^ - ^ M x H + j ^ l M x - ^ j
4.2.3
,
.
(4.13)
Small Signal Analysis of Ferrite Biased in X Direction
Assuming that a sufficiently large dc magnetic field H q in x direction associated
with a small ac magnetic field H perpendicular to a x (perpendicular pumping) is
applied to an infinite ferrite medium. The total magnetic field intensity vector is
given as
Ht = axH 0 + H
(4.14)
The dc magnetic field causes ferrite material to magnetically saturate such that
the magnetization M is considered at saturation and referred as M„ (saturation magnetization). From (4.11), it can be seen th at M3 x H q = 0, and therefore M s |[ H q, in
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87
the absence of any ac magnetic field. In the presence of an ac magnetic field, an ac
magnetization is also generated. The total magnetization vector is then
M r = axM a + M
(4-15)
Using (4.14) and (4.15), the equation of m otion of the magnetization vector (4.13)
can be written as
^ - - ™
* x* +iB fi(* xT r)
( 4 -1 6 )
Following approximations are made
Ho » \H\
Ms » \M\
\Mt \ « M a
(4.17)
Obviously, these assumptions will linearize the equation of motion of the magnetiza­
tion vector since higher-order M H products are neglected. However, most microwave
ferrite devices can be characterized using the linearized equation of motion.
Substituting (4.14) and (4.15) in (4.16) and applying (4.17), the following scalar
differential equations are obtained.
I F
=
0
tj
./
dMg
- u mHz - U0M Z - ot—7—
dt
at
dM z
dM y
0^ = —uimHy + ujqMv + a —
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(4’18)
(4.19)
.
.
(4-20)
88
where u/0 = f*oyH0 is the free precession frequency (Larmor frequency) and uim =
to lM s.
The magnetic flux density B is linked to M as follows
B = to{H + M )
(4.21)
Thus the existence of M cause an increase in B. In cartesian coordinates (4.21) is
Bx = (i0 {Mx + Hx)
(4.22)
By = to {My +
Hy)
(4.23)
B s = to (M z +
H z)
(4.24)
Taking the time derivatives of (4.22)-(4.24) and from (4.18)
dBx
“5
eftT =
.
.
(4-25)
dHx
eft
dB,
dt
dB
dt
Multiplying (4.23) by u/q, and subtracting from (4.27), we can write
dBz
dt
=
+
(4-28)
If (4.20) is substituted in (4.28),
dBz
dt
UoBy = t o ( ^ - u o H y -u>„H„ + a ? ^ j
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(4.29)
89
(4.29) is still coupled to My. Using (4.26) and after some arrangement, (4.29) is
further simplified,
dB z
—
Leaving
n
U !()B y
=
dH z
,
p0
.
x rr .
(^0 +
(dB y
H y +
CX.
I fi
dH y\
o ^ I
/4 0 n ,
(4.30)
term on the left and passing all other term s to the right, (4.30) becomes
dH z
1 d B z wdD , ,
,
v„
a (d B y
- w r = -----^ r - - — By + (uo+u,m)H y ------- - V-o~zr
dt
no dt
Ho
Ho \ d t
dH y\
dt J
(4-31)
Following a similar procedure, another differential equation for Hy which is uncoupled
from M z and suitable for discretization can be derived. Multiplying (4.24) by u/o and
adding with (4.26),
dBy
dM ,
dt
_
( dHy
4- wqB z = Mo f
+
vqH z 4—
dM y
\
^ — h uiqM zJ
,
.
(4-32)
and M z can be uncoupled from (4.32) using (4.19) and (4.27),
dBy
n
dHy .
+ <
jJqB z = Ho ^
,
.
wr
(dB z
(wo + u/m) Hz — a. I
J
d H z\
Ho ^
fA ooX
(4.33)
Leaving - 1- term on the left, and passing all other term s to the right,
dHy
1 dBy H a/o B_ z — (u/o
,
x rr
=
+ u/m) Hz
dt
Ho dt
ho
ot f d B z
I ~ni
H o \ dt
d H z\
f*o p,. I
dt )
.
.
(4-34)
In addition, there is one more equation
Hx = — Bx
(4.35)
f*o
Equations (4.31) and (4.34) are uncoupled from M and will be discretized using central
finite-differences to obtain FDTD updating equations in a ferrite medium biased in x
direction.
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90
4.2.4
Small Signed Analysis of Ferrite Biased in Z Direction
The procedure of uncoupling M from (4.16) and (4.21) for z directed bias case is
quite similar to the procedure described in section 4.2.3. Total magnetic field intensity
and magnetization vectors for a ferrite medium biased in z direction are
H f =
clz H q
+
H
Mt = azM s + M
(4.36)
From (4.16) and (4.36), scalar components of M are
dM x
d M y
\r ,
tj
dMy
= ~ WoMy + WmHy — a ~Q^~
fA ov}
(4-37)
,r
rr ,
= UI0MX
- u mHx
+ a d—M *
l a oo\
(4.
38)
i t
= 0
(439)
By taking the time derivatives of (4.22)-(4.24) and using (4.39) we obtain
d B x
a t
=
^
dB y
dBz
,(
d M x
[
d t
~ d T
|(
d M y
dH y
{
d t
.
+
d H x
(4.40)
(4.41)
_ a T
dHz
s r = to -a r
,
.
(4A2)
After some arrangement, following equations which are uncoupled from M are ob­
tained
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91
dH x
1 dBx
u/0
,
\
tt
, a
I dBv
dH v \
SatT = -----*7"
-Z T)
hq at + fi~0 B v - (^o + wm) fly + —
ho \I ~at^ r ~ V o at
dHy
1 dBy ujq
.
a
( dB x
- 3r = —
- — b x + (ujq + um) h x —
dt
Ho at Vo
Ho \ at
dH x \
at J
H._ = — B z
Vo
4.3
t a a ->\
(4-43)
.
.
(4-44)
(4.45)
A Finite-Difference Time-Domain Algorithm for Ferrite
4.3.1 Introduction
The differential equations which are uncoupled from the magnetization M and
suitable for discretization are derived for x and z bias directions in section 4.2. How­
ever, the magnetic field components perpendicular to the direction of the bias field are
still coupled to each other. These field components will be uncoupled in section 4.3.2
during the discretization process. The numerical scheme developed for the solution
of EM fields in a ferrite medium is similar to the regular FDTD algorithm except
some small differences. The regular FDTD method is a two-step algorithm, i.e. H
is calculated from previously computed values of E, and E is found from the just
computed values of H , and so on. However, in the presence of ferrite, H is no longer
related to B with a scalar permeability of h - This will bring an extra computational
step to the regular FDTD algorithm as shown in Fig. 4.1. In the presence of ferrite,
B is calculated from the previous values of E. H is calculated from B and E is found
from H. In the ferrite algorithm, the magnetic field components which are perpen­
dicular to the direction of the bias field need to be updated using special updating
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92
equations. The FDTD time-stepping loop should last until all the transient response
is dissipated so th a t the frequency domain response can be obtained smoothly.
Actual B and H fields should exist at the same space point. Due to the loca­
tion of the fields in Yee’s cell, there is a half-cell distance between some of the field
components. A spatial averaging can be used to relocate these fields at the desired
spatial location as suggested in [47], or a finer mesh can be implemented instead. The
spatial averaging of some field components can cause reduction of accuracy [47]. This
is not a desirable feature specially in the analysis of slow wave devices which usually
require many time steps. Furthermore, B and H should be calculated at the same
time steps.
The finite-difference time domain technique developed for the analysis of mi­
crowave ferrite devices has the following features
• This algorithm is directly derived from the equation of motion of the magneti­
zation vector and eliminates lengthy convolution process as described in [46].
• It can be applied for finite size ferromagnetic materials having both conductor
and dielectric discontinuities.
• Non-uniform magnetization due to finite size ferrite sample demagnetization ef­
fects can be included in FDTD calculations easily by simply assigning a different
magnetization value to each cell of interest.
• This technique can handle nonlinearity arises when the rf magnetic field exceeds
a threshold value by simply modifying the damping factor a.
• W ith a single FDTD run, wide band frequency domain results can be obtained
by simply frequency transforming the time domain data.
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93
Initial values o f
E and H fields
Initial values of
E, H and B fields
Calculate B from E
Calculate H from E
Calculate H from B
Calculate E from H
Calculate E from H
n ^ n m ax
n>nmax
i
V
■
V
(a)
(b)
Fig. 4.1: (a) The regular F D T D algorithm and, (b) th e m odified FD TD algorithm in the
presence of ferrite.
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94
4.3.2
Derivation of Updating Equations for Ferrite Media
In sections 4.2.3 and 4.2.4, an infinite ferrite medium biased in x and z directions is
analyzed using the linearized equation of motion of the magnetization vector including
Gilbert’s phenomenological damping term to account for magnetic losses. Finally,
differential equations which are uncoupled from M and relate B and H only are
derived. These equations are discretized by applying a central finite-difference scheme
in time. Bias in x direction will be considered first in the following discussion.
Central finite difference scheme is applied to (4.34) and (4.31) for the x directed
bias case to obtain updating equations for Hy and H z field components to be used in
the regular FDTD algorithm. After some arrangement, the discrete form Hy and Hz
field components can be written as
+I/2-
( i w
Br
1/ 2) + 1
+ e r i/2) -
(uio+ M m )A t^ j y n + 1 / 2 _j_ j j n - 1 / 2 ^ _|_
^ oi(h \+1/2 -
( i(B
?+1/2-
H?+xn =
1 /2 _
\
£ n -l/2 ^ _
^
j
(4.46)
s r l/2) - i f w +1/ 2+ s r 1/2)+
\
+ h ;~ 112) - ^ ( b ; +1/2 - b ;~ 1/ 2)-iV
-
a ; - 1' 2 )
"
/
(4.47)
From (4.46), let A* represents
A* = / r - v * +
y
a y
+
(
I
_
3 r l/2 )+ I f ( B T l/2+ f r 1/2)+ \
^ n -l/2 j _
J
.
,
v •
1
Similarly, from (4.47)
A* -
+ ( * W+‘/2 “ S : ' ,/2) “
\ » W +1/2 - Br l/2>+ (
2
+
)
- “ ) H y ~ >n
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(4 49)
95
Let C denote
A (^o + u/m)A t
C=
j
+ <*
(4.50)
Then, from (4.46), (4.48) and (4.50) the following equation can be written
ffn + 1 /2 +
^ ffn + l/2 =
Ax
(4.51)
H ”+l'2- C#?+1/2 = A:
(4.52)
From (4.47), (4.49) and (4.50) similarly
As can be seen, (4.51) and (4.52) can be solved simultaneosly to obtain Hy *1?2
and i7”+1/2. Finally, updating equations for Hy and H z field components are
fjn+l/2 = Ay~CA:
ffn + 1 /2 _
(4.53)
CA„
s y + A*
(4.54)
i + c2
The second step is to derive updating equations for a z biased ferrite medium.
Similar to the x directed bias case, central finite difference scheme is applied to (4.43)
and (4.44) to obtain discrete Hx and Hy field components to be used in the regular
FDTD time stepping algorithm. (4.43) and (4.44) take the following form
n + i/2 =
ffn -1 /2 +
(u>o+wm )A f ^ n + i / 2 +
f f n - 1 /2 ) +
^ -(£ £ + 1 /2 -
J ^ " 1/ 2 ) -
^ a (H y+lf 2- H ^ 2)
(4.55)
H *+1/2 = H n - i /2 +
i (B?+l/2- -BJ'1_j_
'2) - ^(BT+I/2+Br1/2)+
(o<o+uim )At ^ f f n + 1 / 2
f f n —1 / 2 J
Q ^f f n + 1 / 2
f f n —l / 2 ^ _ ^
(4.56)
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96
The final updating equations for Hx and Hy field components for a ferrite medium
biased in z direction are
H i * 1'* =
(4-57)
H? i n =
(4 '5 8 )
where Q is given by (4.50). A* and
sue given as
* .. a - * + 1 i ; ^ +m -
+TfiHV/’+f r V2)+ \
n (B S*V2 - B ? "l/2) - (JisiaslS* - a ) ffJ-V 2 y
a. =
V
- f l T 172) - 1 ^ ( ^ +l/2 + ^ - l/2) y
'
Physicalmagnetic
I
a _ ^ g n + 1 /2 _
g n —1 / 2 ^ _|_ ^ (u o -W m )A t _
ffn -l/2
M59)
1’
'
(4.60)
loss mechanism which exists in an actual ferrite m aterial is
taken into account by the phenomenological damping factor a. This factor can be
related to the ferromagnetic resonance linewidth A H [51] as follows
a =
2u)
(4.61)
When the signal frequency is equal to the ferromagnetic resonance frequency (u> =
ujq),
a becomes
“ = m
(4-62>
(4.61) best represents physical magnetic losses around the ferromagnetic resonance
frequency. Below this frequency or for small values of the dc bias field, a. does not
necessarily yield accurate results [43].
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97
4.4
A Slow Wave Thin Film Isolator
4.4.1
Introduction
An ideal isolator is a unidirectional, two-port device which allows EM energy
transmission only in one direction. Isolators me widely used in microwave systems in
which a reasonable amount of power isolation from some part of the system is required.
This requirement can be due to enforce matching or isolation of high transmitted
power from the sensitive receiver modules as in radar systems. Ferrite isolators can be
constructed using classical metallic waveguides or in the form of planar transmission
lines on ferrite substrates.
Planar isolators me very desirable to be used in microwave integrated circuits
(MICs) due to their low cost and integrability with other planar microwave compo­
nents. A very well-known planar isolator is the edge-guided mode isolator made from
a microstrip line on a ferrite substrate [54]. The edge-guided modes develop when
the bias is perpendicular to the plane of the ferrite and travel on the edges of the mi­
crostrip conductor. These modes switch to other edge when the signal propagation is
reversed (or the dc bias field is reversed). Non-reciprocity can be obtained by putting
a resistive film on one of the edges as proposed in [54]. This will attenuate the modes
which cling to th at edge. T he edge-guided mode isolators were studied extensively
by many authors [54]-[55].
The isolator presented in this Dissertation is made from planar transmission lines
deposited on a thin ferrite film over a low loss dielectric substrate. T he device is very
small (in the order of ten microns) and can be integrated within MICs easily. The
ferrite film is biased in x direction to enable the magnetostatic surface wave mode
nonreciprocity which is the key operating mechanism of the device.
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98
4.4.2 Theory of Isolation Using Magnetostatic Surface Waves
Fig. 4.2 and Fig. 4.3 shows the cross-sectional and top views of the thin film isola­
tor. The operating mechanism of the isolator depends on the nonreciprocal behaviour
of the magnetostatic surface wave propagation. Basically, within a finite size ferrite
sample, for certain rf frequencies the wave number is much greater than the wave
number in free space \k\
|fc0[ and Maxwell's equations under this condition are
referred as the equations of magnetostatics [56].
Magnetostatic waves can be grouped in three categories; magnetostatic forward
volume waves (MSFVW), magnetostatic backward volume waves (MSBVW), and
magnetostatic surface waves (MSSW). MSFVW can support several modes having
the same cut-off frequency within the appropriate frequency limits. Similar is valid
for MSBVW, except the phase and the group velocities of the MSBVW modes are
oppositely directed. MSSW differs than the volume waves in the sense that there is
only one MSSW mode and propagation occurs on the surface of the ferrite sample.
MSSW mode can be switched from one surface of the ferrite substrate to the other
surface by changing the direction of the bias field [57]. Two different MSSW modes
will be enabled in thin film isolator based on the direction of the bias field. These
are the ground and ferrite-dielectric guided MSSW modes. The bandwidth of the
ground-guided MSSW mode is given in [57] as
(4.63)
while the ferrite-dielectric guided one’s is
(4.64)
A careful evaluation of (4.63) and (4.64) indicates that above
u /q
+
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ferrite-
99
dielectric guided MSSW will be cut-off. Since the two MSSW modes travel on different
surfaces of the ferrite film as function of the bias field direction, the frequencies above
u/0+
and below u/o + u/m will not be seen at the output port and isolation occurs.
W hen the bias direction is switched (in opposite direction) low loss signal transmission
to the output is made possible. It should be mentioned that, when the rf magnetic
field’s frequency is close to the MSSW upper frequency limit u/o + u/m) inverse group
velocity (delay/length) of the MSSW approaches to infinity. Such situation will result
in extremely slow propagating waves and numerical computation will usually require
many tim e steps.
4.4.3
Isolator Measurements and FDTD Analysis
T he thin film isolator measurements were provided by Dr. El-Badawy El-Sharawy
and Westinghouse Co. The cross-section and top views of the isolator are shown in
Fig. 4.2 and Fig. 4.3, respectively. A thin ferrite film stays on top of a dielectric
substrate backed by a ground plane.
The strip conductors are made from gold.
However, the electrical conductivity of the strip conductors are expected to be lower
than th a t of the gold. This is mainly due to the diffusion between the gold layer and
a very thin titanium holding layer which is used to stick the gold conductor to the
ferrite film [57].
T he physical dimensions of the isolator are hp = 47.0, ho = 400.0,
u>
g
— 250.0, u>t — 40.0, u/s = 30.0,
uje
= 500.0,
= 100.0, d = 40.0, and L = 2000.0 (fim)-
The relative permittivities of the dielectric substrate and the thin ferrite film are
€d = 10.0 and e/ = 15.0, respectively. The dc bias field is 1600 Oe and the saturation
magnetization is about 1780.0 G. The catalog value of the FMR linewidth is usually
45 Oe (for YIG). The strip conductors me about 2.0 fim thick. The device is biased
in x direction to excite surface magnetostatic wave mode. Insertion loss and isolation
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100
Dielectric substrate
Metal shielding
Fig. 4.2: Cross section o f the thin film isolator.
Port 2
Ferrite substrate
Conductor
Port I
wT ws
Fig. 4.3: Top view of the thin film isolator.
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101
measurements are shown in Fig. 4.6. Insertion loss is about
-6 dB
in the range of
7.2-8.2 GHz. Off-band rejection is about -15 dB below 6.3 GHz and between -15 and
-20 dB above 9.3 GHz. Isolation is about -30 dB at 7.6 GHz. The passband in which
the ground-guided MSSW enabled is similar to the characteristic curve of a bandpass
filter and covers a frequency range of approximately 6.3-9.3 GHz.
1
0
0
c*
Isolator dimensions and material
parameters used in FDTD simulations
1600 Oe
Ho
m3
1780 G
10eo
Cd
15e0
e/
hij
400 fim
hp
48 fim
hji
500 fim
100 fim
wp
250 fim
Wg
30 fim
ws
40 fim
Wt
d
40 fim
a
0.014
4.1 - 107S/m
&strip
tan Sf
tan 84
2.0 • 10"4
In FDTD simulations, magnetic loss of the thin YIG film is introduced in the
form of phenomenological damping factor a given by (4.62). A non-uniform mesh
in z direction is implemented to accurately model YIG-conductor and YIG-dielectric
interfaces. In the vicinity of the YIG-conductor interface, cell size is 2.0 /xm and
expands in size away from the interface. The mathematical relation between the two
consecutive cells is given by (2.59). Spatial resolutions in x-y plane are uniform in
size, and selected as Ax = 40 fim and A y = 10 fim. FDTD simulation parameters
are summarized in the above table.
The isolator is excited by a Gaussian pulse source voltage at the input (Port 1)
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102
using E y field components between the transducer and the ground conductor. Mur
1st order ABC is used to truncate both front and back planes.
1000
900
800
s 700
E
o
200
100
100
200
300
400
500
600
Distance in y direction (micrometer)
Fig. 4.4: A non-uniform, mesh is applied in z direction to accurately model ferrite sub­
strate.
Two frequency dependent parameters are needed to characterize the behaviour of
the isolator. These sure insertion loss (IL) and isolation (IS), both in dB scale. These
parameters are calculated simply by taking the ratio of the frequency domain output
and input voltages. An FFT algorithm is used to obtain frequency domain quantities
from the time domain data. For IL calculations, the dc bias field H
q
and the saturation
magnetization M a should be both negative quantities. For IS calculations, both
H
q
and M 3 should be positive. Fig. 4.6 shows the comparison of isolator measurements
and FDTD calculations of insertion loss and isolation. A good agreement has been
achieved except there is a slight dicrepancy between the calculated and measured
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103
isolation. This can be due to impedance mismatch during the measurements since
the source is m atched to isolator at the center frequency only. In addition, edge effects
can be considered since the bandwidth can be altered significantly [58].
Cells in y direction
0
0
Cells in x direction
Fig. 4.5: Various material levels of the thin film, isolator. Higher level is the gold conductor
level, lower level is the YIG substrate.
A parametric study is performed to investigate insertion loss and isolation over
various geometrical features and material parameters. Measurements show th at the
lowest insertion loss is about
-6 dB
in the passband. This value of IL is very high
for specific applications. IL should be decreased to reasonable values (<
1dB), while
a decent isolation should be maintained. In this analysis, effect of the ground strip
width, edge truncation, removed, of the shorts, dc bias field, and FMR linewidth are
investigated.
Effect of the ground strip width
w q
is investigated by varying
vjq
to be 250, 120
and 40 y.m as shown in Fig. 4.7 and Fig. 4.8. It is observed th a t the 3 dB bandwidth
of the insertion loss increases with decreasing wq- If vjq is decreased from 250 to
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120
104
-10
\
-30
-35
Frequency (GHz)
Fig. 4.6: Comparison of FDTD calculations and measurements of insertion loss and isola­
tion. .ffo=1600 Oe, Ms=1780 G, hp = 400 fim , hp = 48 fim ,
= 500 fim, w q = 250
fim , w p = 30 fim, w s = 40 fim, w p = 100 fim .
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105
f im ,
the 3 dB bandwidth increases by about 30%. Further decrease to Wq = 40 f i m
will almost double the bandwidth with respect to wq = 250 f i m case. On the other
hand, isolation gets worse and decreases almost 5 dB from
wq
= 250 to
vjq
= 40 f i m
case.
Effect of the dc bias field strength over th e isolator performance is investigated
as shown in Fig. 4.9 and Fig. 4.10. It is observed th at IL and IS curves are shifted
to a higher center frequency when the bias field strength is increased (with 400 Oe
intervals). While the characteristics of IL and IS curves did not change significantly.
The calculated shift of the center frequency is found to be about 1.2 GHz per 400 Oe.
This property implies that the thin film isolator is tunable simply by varying the dc
bias field strength.
It is shown in Fig. 4.3 th at the transducers at each side of the center ground plane
are shorted at opposite sides. These shorts are removed to investigate the performance
of open vs shorted structure as shown in Fig. 4.11. Insertion loss increased by about 4
dB and isolation decreased almost 9 dB. It can be concluded that a better separation
of MSSW modes traveling at opposite interfaces of the YIG layer can be obtained
using the shorted structure.
Insertion loss and isolation sire also calculated when the thin YIG film is truncated
to the edges of the transducers. Insertion loss decreased in the frequency range of
7.2-7.8 GHz considerably. This improvement is about 3 dB at 7.5 GHz. Isolation is
about -30 dB at 7.5 GHz and about 2 dB better above 9 GHz than the reference case.
Finally, the effect of the damping factor a is investigated. When a is reduced from
0.01400 to 0.00156, both IL and IS improved by about 2-3 dB.
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106
-10
<s -15
-20
-25
-30
-35
Frequency (GHz)
Fig. 4.7: Effect of w q over insertion loss. ifo=1600 Oe, Afa=1780 G, h o = 400 fim,
hp = 48 p m , fiA = 500 f im , wp = 30 p m , ws = 40 p m , we — 100 f im .
-10
•20
-25
-30
-35
Frequency (GHz)
Fig. 4.8: Effect of w q over isolation. #0= 1600 Oe, Ma=1780 G,
pm ,
= 500 p m , w p — 30 p m , w s = 40 f im , w p = 100 fim .
ho
= 400
fim , h p
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= 48
107
3 -IS
-zs
-30
Ho=1200 O e
Ho=1600 Oe
Hr=20000»
Frequency (GHz)
Fig. 4.9: Effect o f the dc bias field strength over insertion loss. A f,=1780 G, h p = 400
fim , h p = 48 fim , h& = 500 fim , w q = 250 fim , w t = 30 fim , w s = 40 fim , w p = 100
fim .
12000®
Ho=1600 o®
H„=2000 Oe
<0 -IS
1os S -zs
-30
-35
Frequency (GHz)
Fig. 4.10: Effect o f the dc bias field strength over isolation. M3=1780 G, h p = 400 fim ,
h p = 48 fim , Ha. = 500 fim , w q = 250 fim , w t = 30 fim , w s = 40 fim , w e = 100 fim .
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108
... E._
-10
-20
-25
-30
-35
Frequency (GHz)
Fig. 4.11: Comparison of insertion loss and isolation for open and shorted isolator struc­
tures. fTo=1600 Oe, Afj=1780 G, h g = 400 y m , h p = 48 yam, h a = 500 yam, w q = 250
y m , w t = 30 y m , w s = 40 y m , w g = 100 y m .
„
-10
CD
-15
-20
-25
-30
-35
F requency (GHz)
Fig. 4.12: Comparison of insertion loss and isolation for truncated YIG substrate.
£70=1600 Oe, M „ = 1780 G, h o = 400 y m , h p = 48 y m , h a = 500 y m , w q = 250
y m , w t = 30 y m , w $ = 40 y m , w g = 100 y m .
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109
o .15
1eo -30
-35
Fig. 4.13: Comparison, of insertion loss and isolation for different a values. Hq=1600 Oe,
M 3= 1780 G, h o = 400 /im , h p = 48 fim ,
= 500 fim , w q = 250 fim , w t = 30 fim ,
w s = 40 fim , w e = 100 fim .
4.5
A Stripline Disc Junction Circulator
4.5.1
Introduction
Circulators are three-port and wide band devices due to their dual resonant modes
generated by the magnetic dc bias field [51]. The primary application of a circulator
is the isolation of power. A 3-port circulator can be used as a two-port isolator simply
by terminating one of the ports with a matched load. Circulators are widely used
to prevent load pulling in microwave oscillators, to enforce impedance matching in
various microwave circuits, and as duplexing elements in receiver/transm itter modules
of radar systems [59].
Isolators which are used commercially are built using circulators term inated with a
50 ohms load at one port. These devices can not be integrated with other microwave
components due to their large dimensions. Recently, thin film YIG substrates are
introduced whose thicknesses are in the order of microns. While the ferrite disks used
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110
in traditional circulators are thick in the order of millimeter. Obviously, thin film
YIG substrates have potential advantages in building future microwave integrated
circuits containing non-reciprocal devices.
Early stripline circulator work goes back to 1960s [60], [61], [62]. Bosma [60] de­
rived frequency dependent circulator characteristics using a Green’s function approach
for a disk shaped stripline circulator. Simon [62] investigated experimentally better
matching techniques by adjusting the m aterial properties of the circulator. Miyoshi
analyzed arbitrarily shaped triplate ferrite planar circulators using a contour integral
approach [63] and further investigated disk shaped and triangular stripline circula­
tors for wideband operation [64]. Schloemann [65] found that for a broader band
operation of stripline circulators, low field losses due to nonuniform magnetization
inside the ferrite disks should be eliminated. He suggested using spherical domes to
provide uniform magnetization within the ferrite. Neidert [66] investigated intrinsic
losses of the stripline and microstrip ferrite circulators and found th a t magnetic losses
are the largest contributor among other intrinsic losses (dielectric and conductor). In
addition, wideband circulator losses are dominated by the matching circuitry losses.
A full-wave analysis of a circulator using Finite Element Method (FEM) was
presented by Lyon [67]. However, a FEM code needs to be nm a t each frequency
point within the desired bandwidth. In general, circulators are wide band devices
and with a single run of the FDTD code frequency domain results over a wide band
can be obtained by simply frequency transforming the time domain data.
4.5.2
Frequency Dependent Parameters of the Circulator
Typical frequency dependent parameters of a circulator are insertion loss I L ( f) ,
return loss R L ( f) and isolation I S ( f ) . These parameters are needed to evaluate the
frequency domain behaviour of any type of circulator. In the absence of m y discon-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
Port 2
Fig. 4.14: A stripline disc junction circulator (a) top view (b) cross section.
tinuities, an EM power associated with propagating electric and magnetic fields on
a transm ission line is considered as incident. In the presence of a discontinuity, the
power associated with EM waves will be the total due to reflections from the discon­
tinuity. The stripline circulator ports can generate reflections due to structural and
material discontinuities unless they are well matched. Since the circulator parameters
require the knowledge of the incident power at the input port, an accurate calculation
of the incident power is needed. Direct FDTD analysis of the circulator will not yield
the correct value of the incident power at the input port since the EM fields are total
(incident + reflected) quantities. In order to find the incident power at the input
port, a stripline structure alone (in the absence of ferrite disks) is analyzed first. This
reference structure will be used to calculate incident voltage and current at a specific
location. In addition, characteristic impedance of the stripline will be calculated. The
reference stripline structure has the same dimensions as well as spatial resolutions as
the stripline of the circulator.
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112
Let s denote any spatial point inside the FDTD com putational domain and s =
Si j k = s(xi,yj, Zk), then, following power quantities can be defined
Pinc(sin, f ) = Incident power
Pref(sin, f ) = Reflected power
Ptot(sin,f) = Total power
Pout{sovt, f ) — Output power (at matched port)
Frequency dependent circulator parameters can be written as
/ £ ( / ) = 10log (
(
4
.
6
I S ( f ) = 10 log (
\ ±inc\Sini J ) )
5
)
(4.66)
R L U ) = 10log ( % , / ,(* * ’ (,) )
(4.67)
\4
J)J
where n and m denote different ports separated by 120 degrees. I L ( f ) and I S ( f )
expressions are identical except the location at which the output power is recorded.
The power quantities in term s of voltages and currents are [68]
Pinc(sin, f ) = R e { V inc(sin, f )
P o u t[ s m itt
f') —
X / ’^ ( S i n , / ) }
/) ^
Ptot(sin, f ) = Re{Vtot(sin, f )
Pref(Sin, f ) = Re{Vref(sin, f )
X
X
Itot{ Sin,f)}
£ c / ( 5fn, / ) }
(4.68)
where * is the complex conjugate operator.
The time domain reflected voltage or current at s = stn is obtained by subtracting
the incident field from th e total one. Let F denotes either the voltage or th e current
at this location, the reflected field quantity can be written as
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113
Fref{sin, t) = Ftot(sin, t) - F ine( « « , t)
(4.69)
The time domain voltage and current at a specific s location are calculated using
the following expressions,
V (s,t) = - r E • dl
Jza
(4.70)
I(s,t)= f ^ H - d l
(4.71)
Here, stripline signed and ground conductors are located at za and
25, respectively.
(4.71) is simply the application of Ampere’s Law to find the current from the circu­
lating magnetic fields around the stripline conductor.
Characteristic impedance of the stripline can be calculated simply by dividing the
frequency domain voltage by current on the stripline. However, it is reported by Fang
[69] that the conventional impedance formula can yield inaccurate results as the fre­
quency of operation increases. This is due to the fact that time and space differences
between the voltage and current are not considered in conventional formulation. The
use of the following formula is suggested to obtain very accurate results.
U f ) =
(472)
f ) ■/ ( s i - i j . t , / )
Here stJifc and st-_ij,* are the two parallel planes perpendicular to x direction. The
denominator is the geometric average of the current sampled at these planes. This
form of (4.72) is specific to x directed transmission lines and for other directions it
can easily be modified by changing the subscripts of s.
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114
4.5.3
Non-Uniform Magnetization
Joseph and Schloemann [70] investigated demagnetizing fields in non-ellipsoidal
bodies back in 1960’s and found that demagnetizing fields are not necessarily uniform.
In fact, depending on the geometrical features of the structure of interest, demagne­
tizing fields can exhibit a strong non-uniformity. Schloemann [65] found that the low
field losses in stripline circulators are due to the non-uniform magnetization within
the ferrite disks. The strength of the dc bias field can be much higher at the perimeter
of the circulator them its center due to the non-uniform magnetization effect. In a
stripline circulator, the bias field is applied perpendicular to the plane of the ferrite
disks (in z direction). From the continuity of B at the boundary of the ferrite disks,
the internal dc bias field will differ from the external one by an amount equals to
N M [71],
H 0= H ext- N zM a
(4.73)
Here N z = 1.0 and is the demagnetizing factor as described in [51] under the assump­
tion of uniform magnetization throughout the ferrite sample and H ^ t is the externally
applied dc bias field.
A ferrite disk whose height and diam eter are L and 2a, respectively, is analyzed by
Joseph [70]. Joseph’s first order approximation to non-uniform demagnetizing factor
is
v m f z\
z ’
t
zM f(fci)
47r(ar)2
AoVkM )
4
(L - z ) k 2K (k 2)
4tt(ar)?
A p jfa h )
4
where
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,4
115
k\ =
k2
4ar
2~ (L-z
sin(fli) =
sin (/?2) =
Aar
z 2+ (a + r)2
)2+ (a + r )2
i j z 2+ (a —r )2
(£ - *)_______
----------------------\/(L — z )2+ (a — r )2
(4.75)
In (4.74), K (k ) is the complete elliptic integral of the first kind.
A0(/5, k) is
Heuman’s Lambda function and related to both complete and incomplete elliptic
integrals [72],[73]. /? is in radians. The solution of (4.74) can be used to find the
internal non-uniform dc bias field inside the finite size ferrite disk as follows
(4.76)
Ha{r,z) = H ^ - N i l\ r , z ) M ,
LO
0.9
0.8
M
Z
0.6
0.5
0.4
0.0
0.1
0.2
0J
0.4
0.5
r/a
0.6
0.7
0.8
0.9
LO
Fig. 4.15: The spatial variation of the demagnetizing factor
at z—0 as function of the
radial distance from the center of the disk for different values of q = L/a.
In Fig. 4.15 the spatial variation of the demagnetizing factor
at z=0 as a
function of the radial distance from the center of the disk is plotted using (4.74)
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116
LO
03
0.8
0.7
-S 0.6
0.5
Z 0 .4
0.3
03
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
LO
z/L
Fig. 4.16: The spatial variation o f the demagnetizing factor A^1' at r= 0 along the central
axis of the disk for different values o f q = L / a .
for different values of q = L /a. Similarly, Fig. 4.16 shows the spatial variation of
at r=0 along the central axis of the disk for different values of 9. It can be
seen from Fig. 4.15 th at smaller q values end up in stronger non-uniformity in the
demagnetizing factor. For <7= 4.0 and <7= 20.0,
that
is almost uniform. Fig. 4.16 shows
decreases toward the end faces of the ferrite disk for higher q values, at r = 0.
Meanwhile, it is almost uniform along the disk axis for 9= 0.1 and 9= 0.2.
4.5.4
FDTD Analysis and Validation
A traditional stripline circulator is shown in Fig. 4.14. The striplines which form
the three ports of the circulator merge a metal disk sandwiched by the two ferrite
disks on top and bottom . The magnetic dc bias field is applied perpendicular to the
plane of the ferrite disks. Such a structure is investigated by Gaukel and El-Sharawy
[71] using a subport eigenvalue m atrix analysis including some experimented work.
Circulator dimensions and physical param eters of the ferrite disks are summarized in
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117
the following table.
Disc junction stripline circulator
dimensions and m aterial parameters
2550.0 Oe
Hext
1780.0 G
4irM„
14.5e0
e/
22.45 mm
Df
11.43 mm
Dm
3.05 mm
yj,
b
2.16 mm
0.13 mm
t
b /2
L
a
Df / 2
A finite-difference time-domain analysis of the circulator is performed in 3-D in­
cluding the effect of the non-uniform magnetization described in section 4.5.3. The
direction of circulation will be in clockwise (CW) direction, from port 1 to port 2, 2
to 3, and 3 to 1 for -z bias and counter-clockwise direction for + z bias. A 3-D mesh
view of the circulator based on its m aterial content is shown in Fig. 4.17. The longer
stripline is going through the input port 1. Port 2 is the output and the next port
in CW direction, while port 3 is the isolated one. All ports are extended to PML
absorbing boundaries to numerically realize the matched load termination. Matching
of all the ports is compulsory to satisfy the ideal circulation condition.
The ferrite disks (YIG) used in [71] have a height to disk radius ratio of q =
L /a = 0.096 w 0.1. The spatial variation of
z) can be seen in Fig. 4.15 and
Fig. 4.16 for this value of q. As can be seen, the demagnetizing factor has a maximum
value of 0.95 and is uniform at the center of the disk along the disk sods. FDTD
analysis includes both uniform and non-uniform magnetization cases. For uniform
magnetization, N ^ — 0.95 is used. For the non-uniform case, spatisil distribution of
the dc bias field given by (4.76) is mapped onto the ferrite disks in 3-D.
The spatisil cell sizes sire chosen as Ax = Ay — 0.369 and Az = 0.135 (mm) based
on the geometrical features of the circulator. FDTD sinalysis is split into two psirts. In
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118
CaHs in y direction
Fig. 4.17: A three dimensional mesh view o f the stripline junction disc circulator.
the first part, an infinitely long stripline whose dimensions and spatial resolutions is as
same as the one used with ferrite disks is analyzed to obtain the incident voltage and
current quantities. A Gaussian pulse is launched and incident electric and magnetic
fields are collected at a reference location. This reference location is the junction
where the ferrite disks and stripline merge. From these fields, the incident voltage
and current are calculated as explained in section 4.5.2. In the second part, the
same Gaussian pulse is launched in the presence of the magnetized ferrite disks. The
voltage and current are calculated at the same reference location (port 1) and, at
port 2 and port 3. Following the procedure outlined in section 4.5.2, insertion loss,
isolation, return loss, and the characteristic impedance of the stripline are calculated
in frequency domain. Fig. 4.18 shows the characteristic impedance of the stripline
versus frequency. The calculated impedance is 49.47 ohms and almost a constant
in 1.0-2.0 GHz range, while the measured one is 48.06 ohms [71]. The circulator
measurements were provided by K. Gaukel [71].
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119
50.0
49.8
'X 4 9 .6
a 4 9 .4
49.0
1.0
1.4
1.6
Frequency (GHz)
1.8
2.0
Fig. 4.18: Characteristic impedance Zc of the stripline. The experimental value is 48.06
ohms.
The effect of the FMR linewidth AH over the frequency-dependent circulator
parameters is investigated for A H = 5,15,45 Oe. In Fig. 4.19, calculated insertion
loss is compared with the experimental curve. The agreement is good in magnitude
and frequency, except the calculated IL bandw idth increases and the minimum loss
decreases as the FMR linewidth decreases. The calculated bandwidth is closest to the
measurement when A H = 45 Oe. Calculated isolation is compared with the measure­
ment as shown in Fig. 4.20 for different values of A H . The two resonance frequencies
at 1.45 GHz and 1.75 GHz can be distinguished from the isolation measurement. As
the FMR linewidth is reduced from 45 Oe to 5 Oe, isolation at the second resonance
frequency increases by about 10 dB, while the first resonance seems to be affected
insignificantly. The agreement between the calculated isolation and measurement is
found to be very good in magnitude and frequency. Calculated return losses are
compared with the return loss measurement as shown in Fig. 4.21 for different A H
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120
values. Calculated RL curves reveal th at the input impedance of the ferrite-stripline
junction is closest to the characteristic impedance of the stripline when AH = 45 Oe
at 1.4 GHz.
o.o
-0.5
-3.0
DH=5 Oe
OH-15 Oe
DH=45 Oe
measured EL
-3.5
-4.0
1.1
12
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.19: Insertion loss of the circulator for different values o f A H . Hext — 2550 Oe,
1V2=0.95.
The effect of the externally applied dc bias field is investigated for fixed values of
N z = 0.95 and A H = 45 Oe. Insertion loss, isolation and return loss of the circulator
shift in frequency as the strength of the applied dc bias field varies. IL bandwidth
increases as Hcxt increases, while the minimal loss decreases as shown in Fig. 4.22.
When the bias field strength is increased by 50 Oe, IL curves shift by about 35 MHz
up at the lower end. Comparison of calculated and measured isolation is shown in
Fig. 4.23. When Hext = 2500 Oe, the first resonance peak is about -35 dB at 1.40
GHz and deeper than the second one. Increasing the external bias field from 2500 Oe
to 2550 Oe and, from 2550 Oe to 2600 Oe will result in a frequency shift of about 40
MHz in the isolation response, while the isolation magnitude drops by approximately
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121
BH=5 Oe
DH=15 Oe
DH=45 Oe
measured
-10
-20
-35
-40
-45
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.20: Isolation of the circulator for different values of A H. Hext = 2550 Oe, i\Ts=0.95.
-10
-15
3 -2 5
-30
-35
-40
1.1
1.3
1.4
1 .5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.21: Return loss o f the circulator for different values of A H .
iVz= 0.95.
H cxt — 2550 Oe,
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122
5-6 dB per 50 Oe increase. In a similar way, the second resonance peak shifts up
by about 80 MHz per 50 Oe increase in the bias field strength, while the isolation
increases from -30 dB for H at = 2500 Oe to -40 dB for H a t = 2600 Oe. Return losses
of the circulator sure shown in Fig. 4.24. Lower H „ t values yield improved response
at lower frequencies. The best return loss obtained was -35 dB for H at = 2500 Oe at
1.35 GHz.
o.o
-0.5
-
1.0
S -1-5
8 - 2.0
o
*"1 -2.5
G
Q
<o
’■£ -3.0
c -3.5
i-n
-4 .0
H«=250O Oe
H «=2550 Oe
Hat=2600 Oe
measured IL
-4 .5
-5.0
l.l
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.22: Insertion loss of the circulator for different values of the externally applied dc
field H a t - AiT = 45 Oe, N Z=Q.95.
The effect of the metallic disc diameter is also investigated as shown in Fig. 4.25,
Fig. 4.26 and Fig. 4.27. The metal disk diameter Dm is first increased by 25% and
then decreased by the same amount from its original value (11.81 mm). It has been
observed that all circulator parameters deteriorate significantly such th a t the circu­
lator loses its functionality.
Finally, the uniform and the non-uniform magnetization effects are studied. For
the uniform case N z = 0.95, Hext = 2550 Oe and A H = 45 Oe. For the non-uniform
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123
H^=2500 0 e
H.B=2550Oe
Hat=26000e
measured
-10
-15
TJ
-20
►3 -30
-35
-40
-45
1.1
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.23: Isolation of the circulator for different values of the externally applied dc field
A H = 45 Oe, N z= 0.95.
H ^M O O Oe
Hca=2550 Oe
H «=2600 0e
measured
-10
-30
-35
-40
1.1
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.24: Return loss of the circulator for different values o f the externally applied dc field
H ext. A H = 45 Oe, 1\TS=0.95.
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124
0.0
-0.5
-
1.0
-L 5
t
-
2.0
-
2.5
-3.0
Dm=8.86 mm
D„=I1.81 mm
D„=14.76 mm
measured IL
-3.5
-4.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.25: Insertion loss of the circulator for different values o f the metal disk diameter.
H ext = 2550 Oe, A H = 45 Oe, ATr= 0.95.
-10
£9-15
.2 -20
m -25
-30
-35
D„=8.86mm
D „=ll.81 mm
Dm=I4.76 mm
measured
-40
l.l
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.26: Isolation o f the circulator for different values of the m etal disk diameter. H ext —
2550 Oe, A H = 45 Oe, JV*=0.95.
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125
D„=8.86 mm
Dm=11.81 mm
D„= 14.76 mm
measured
-10
-15
J -20
-30
-35
-40
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.27: Return losses o f the circulator for different values of the m etal disk diameter.
H ext = 2550 Oe, A H = 45 Oe, N z= 0.95.
o.o
-0.5
-
1.0
-1.5
-
2.0
I s -2.5
-3.0
Uniform
Non-uniform
measured IL
-3.5
-4.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.28: Insertion losses o f the circulator for uniform and non-uniform magnetization
cases. H ext = 2550 Oe, A H = 45 Oe.
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126
Uniform
Non-uniform
measured
-10
-15
• O -20
o -25
-30
CD
-35
-40
-45
-50
1.1
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4.29: Isolation of the circulator for uniform and non-uniform magnetization cases.
H ext = 2550 Oe, A H = 45 Oe.
Uniform
Non-uniform
measured
-10
-15
CD
CD
-30
-35
-40
1.1
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Frequency (GHz)
Fig. 4 .3 0 : Return losses o f the circulator for u n if o r m and non-uniform magnetizatiion cases.
H txt = 2550 Oe, A H = 45 Oe.
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127
case, q = 0.1 and the maximum value of the demagnetizing factor is 0.95. Fig. 4.28
shows the comparison of calculated insertion loss versus measurement. Both the
uniform and non-uniform cases yield very close results for insertion loss. Isolation
calculations show th a t the amount of mode-splitting is higher in the non-uniform
case, and hence, the isolation bandwidth is wider as shown in Fig. 4.29. On the other
hand, the non-uniform return loss seems to indicate a better matching at the first
resonance peak than the uniform magnetization case as shown in Fig. 4.30.
4.6 The Non-Linear Phenomena in Ferrite
4.6.1
Introduction
The non-linear behaviour of ferrites were first observed experimentally in the
early 1950s. However, this unusual behaviour could not be explained immediately.
In 1957, Suhl [74] theoretically explained the physical processes occuring inside the
ferrite and established the theory of ferromagnetic resonance at high signal powers.
Ferrite materials exhibit non-linearity when the applied rf signal exceeds a threshold
value. During this state of the ferrite, two im portant phenomena occur. First, the
main resonance peak (FMR) is saturated, weakened in magnitude and broadened.
Second, a subsidiary resonance peak appears usually below the main resonance. Suhl
showed th at these interesting phenomena arise due to the transfer of energy from
the uniform precession to the spin wave modes. Basically, when the rf signal power
exceeds a critical value, certain spin wave modes are coupled to the uniform mode
through demagnetizing and exchange fields, and, hence excessive signal energy is
transferred into crystal lattice via spin waves in the form of heat.
Spin waves which are coupled through the first order amplitude of the uniform
mode causes the subsidiary resonance. This process is referred as the first-order non­
linear process. In general, these spin wave modes propagate at 9 degrees with respect
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128
to the dc bias field direction. For a maximal coupling, 0 « 45 degrees, and the spin
wave frequency is half th e signal frequency. On the other hand, spin waves which
propagate parallel to th e dc bias field (0 = 0) are responsible for the saturation of
the main resonance peak. For a maximum coupling, the spin wave frequency is equal
to the signal frequency for this case. This phenomenon is referred as the secondorder non-linear process since the spin waves are coupled through the second-order
amplitude of the uniform mode [74].
The non-linear properties of the ferrite are used to construct microwave signed,
processing devices such as frequency-selective limiters and signal-to-noise enhancers
[75], [76]. A frequency-selective limiter attenuates high-power signals and exhibits
low insertion loss to low-power signals. A signal-to-noise enhancer does the opposite
job by attenuating low-power signals more than the high-power ones. Both devices
are widely used in microwave receiver systems.
The equation of m otion of the magnetization vector (4.13) inherently includes th e
non-linear behaviour of ferrite. However, as discussed earlier, most ferrite devices op­
erate in the linear region and (4.13) is simplified to first-order to obtain the simplified
updating equations to be used in numerical simulations. In the following subsec­
tion, a simple and easy way of including the non-linearity into FDTD calculations
will be presented. T he aim of the study presented in consequent subsections is to
demonstrate that the non-linearity can be modeled using the same FDTD updating
equations derived for th e ferrite operating in linear region by modifying the dam p­
ing factor in a suitable way. Further development of this technique and a full-wave
analysis of various non-linear microwave ferrite devices are planned as future work.
4.6.2
Modified Damping Factor and FDTD Algorithm
W hen the signed am plitude exceeds a critical value, spin wave amplitudes grow
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129
exponentially as indicated in Suhl’s theory [74]. This reveals that the excessive energy
associated with an above-threshold signal is transferred into spin wave modes and
consequently, the uniform precession angle is pulled further toward the precession
axis. By analogy to Gilbert’s phenomenological damping factor, a secondary damping
factor ajt, which will be active only for above-threshold signals and zero for belowthreshold signals, can be introduced. In this and following subsections, Gilbert’s
damping factor a will be replaced by ao bi notation. The overall damping factor,
which includes both linear and non-linear factors, will be denoted as a and defined
in the following m anner
a = a 0 + ock
(4-77)
Here, a 0 is related to FMR linewidth A H by (4.62).
For the signals above-threshold level, it is shown in [77] that the increase in spin
wave amplitude is proportioned to an excess power parameter <j>. Using the original
notation of [77], <f>is given as
/ h2
\ 1/2
' - ( aT 1)
(478)
where h and hc are respectively the signal and critical (threshold) field amplitudes.
A suitable form of a/t should be in the form of (f>. Since the energy transfer to
spin waves occurs via (f>, experimentally observed increasing damping above threshold
should also be proportional to tf>. a* is defined as follows
(h2
\ 1/2
a k = K<t> = K { - - l j
(4.79)
where K is a coefficient and assumed to be K = 1.0 for the time being.
Both K and hc me related to very complex physicalphenomena inside the ferrite.
These include spin wave propagation angle with respect to the dc bias fielddirection,
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130
spin wave linewidth AH k, interned dc bias field and magnetization, and direction of
the input magnetic field. Since the value of a*, depends on the input signal amplitude,
comparison of transient magnetic field intensity with the critical field amplitude at
each cell inside the FDTD computational domain is required. If the critical signal
level is exceeded, a* is activated for th at FDTD cell. A relatively simple expression
of the critical field is given in [52] as
(4.80)
4.6.3
FDTD Analysis of a Ferrite-Filled Stripline
A ferrite-filled stripline structure, as shown in Fig. 4.31, is analyzed using the
FDTD method to observe the non-linear behaviour of the ferrite. The interned dc
bias field is 100 Oe in strength and applied in the plane of ferrite perpendicular to the
stripline. Ferrite m aterial is basically a YIG whose spin-wave linewidth A Hk is usually
in the order of 0.25 Oe [76]. Stripline dim ensions and ferrite material parameters are
given in the following table.
The spatial resolutions are selected as A x = 0.3125 mm, A y = 0.6250 mm and
A z = 0.3125 mm. FDTD computational domain has 49 x 68 x 33 cells. Mur 1st
order ABC is used at both front and back planes to truncate the FDTD space.
Magnetostatic waves exist approximately below 2.8 GHz. Application of a Gaussian
pulse will result in excitation of these very slow waves and in turn, transient energy
inside the FDTD space will need many time steps to dissipate. In order to suppress
the magnetostatic band, a Gaussian pulse modulated by a sine-wave (5.0 GHz) is
used as the source of excitation.
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131
Ferrite-filled stripline
dimensions and m aterial parameters
100.0 Oe
Hq
1780.0 G
4.1tM 3
15.0eo
e/
45.0 Oe
AH
0.25 Oe
A Hk
a
15.00 mm
6
10.00 mm
2.50 mm
Wj
I
31.85 mm
From (4.80) and the d ata outlined in the table above, the critical field magnitude
is about 0.266 Oe. Insertion loss of the stripline is calculated for different amplitudes
of the source pulse for a* = <fA as shown in Fig. 4.32 (referenced to input source
pulse amplitudes). T he applied sine modulated Gaussian pulse amplitude is A q. For
A q = 1.0, it is found th a t a* = 0.0 (h < hc) and the overall damping factor is equal to
ctQ. For A q = 50.0 and A0 = 100.0, the critical field is exceeded and a* is activated.
As can be seen from Fig. 4.32, insertion loss increases when the signal amplitude
increases beyond threshold. The critical field is reduced by half and the insertion
loss is calculated for the cases above as shown in Fig. 4.33. Since the threshold is
reduced, the non-linear damping occurs earlier, resulting in higher attenuation of the
input signal.
Metal enclosure
a
Fig. 4.31: A ferrite-filled stripline cross section. Stripline is 31.85 mm long.
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132
-5.0
-
6.0
4* -8-0
-9.0
-
10.0
3.0
3.5
4.0I
4.5
5.0
I
5.5
6.0
Frequency (GHz)
Fig. 4.32: Insertion loss o f a ferrite-filled stripline for different source pulse amplitudes.
hc = 0.266 Oe.
-5.0
-
6.0
8 -7 .0
-9.0
-
10.0
3.0
3.5
4.0I
4.5
5.0
!
5.5
6.0
Frequency (GHz)
Fig. 4.33: Insertion loss o f a ferrite-filled stripline for different source pulse amplitudes.
hc = 0.133 Oe.
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133
4.7 Summary and Conclusions
FDTD technique is extended to calculate EM fields in microwave devices contain­
ing magnetized ferrites. The ferrite medium is modeled in time domain by solving
Maxwell’s curl equations and the equation of motion of the magnetization vector in
consistency. New FDTD updating equations for the magnetic field components per­
pendicular to the direction of bias are derived. This technique was applied to a slow
wave thin film isolator and a stripline disc junction circulator. The magnetic loss of
ferrite associated with FMR linewidth is also included in FDTD calculations in the
form of phenomenological damping term a .
The tim e domain ferrite algorithm formulated in section 4.3 was applied to a slow
wave thin film isolator. The device is modeled in 3-D using a non-uniform expanding
grid architecture in z direction. Frequency-dependent insertion loss and isolation were
calculated for a number of material param eters and center conductor width. A good
agreement has been achieved in insertion loss calculation and measurement. Isolation
is in reasonably good agreement with the measurement in frequency and magnitude.
Except, a slight discrepancy was observed below 7.5 GHz. This can be due to source
mismatch in measurements and edge effects [58].
A stripline disc junction circulator is analyzed using FDTD technique in 3-D with
the application of linear ferrite algorithm presented in section 4.3. A parametric study
is conducted to study the effects of the external dc bias field, FMR linewidth, cen­
ter disk diam ater and non-uniform magnetization. Calculated frequency-dependent
circulator param eters were compared with the experimental data. A good agreement
has been achieved in insertion loss and isolation calculations. The slight discrepancy
visible in return loss calculations can be due to numerical errors associated w ith the
staircased approximation of the ferrite disks and the center conductor.
Finally, the non-linear behaviour of the ferrite is analyzed at introductory level. By
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134
analogy to Gilbert’s phenomenological damping factor, a secondary damping factor
otk, which will be active only for above-threshold signals and zero for below-threshold
signals, is introduced. However, both a*. and the critical field strength are related to
very complex physical phenomena inside the ferrite including spin wave propagation
angle, spin wave linewidth Aifjt, internal dc bias field and magnetization, operating
frequency and direction of the input magnetic field relative to the dc bias field. Further
development of this technique and full-wave analysis of various non-linear microwave
ferrite devices are planned as future work.
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CHA PTER 5
CONCLUSION
The research work presented in this dissertation includes analysis and design of
shielded mobile antenna systems and planar microwave ferrite devices using finitedifference time-domain method. In C hapter 1, a brief introduction is given about the
initiation, aims and motivation, and organization of this dissertation. T he theoretical
basics of the finite-difference time-domain method are given in C hapter 2. These
include the derivation of electric and magnetic field updating equations, obtaining
field solutions in a complex media, stability of the FDTD method, source of excitation
and numerical dispersion. Implementation of Mur and PML absorbing boundary
conditions, and an expanding non-unifo r m grid architecture to handle fine geometrical
features of the structure are also presented in Chapter 2.
The analysis and design of various shielded mobile antennas are presented in
Chapter 3. A near-to-far zone transformation, a necessary numerical tool required to
obtain far field radiation patterns of a general antenna structure from its transient
near fields, is described in section 3.2. A magnetically shielded, a dual-resonance
shielded, and, a shorted and truncated microstrip antenna are investigated. Input
impedance of the magnetically shielded antenna is calculated and compared to an un­
shielded monopole in section 3.3. A param etric study is performed to investigate the
effect of m aterial loss and thickness of the coating medium on the input impedance.
The dual-resonance shielded cefiular antenna is presented in section 3.4 including a
brief theory of operation. The effect of various geometrical features over the input
impedance of an initial design are parametrically investigated in section 3.4.2. A
proto-type of the dual-resonance shielded antenna has been built and tested at Ari­
zona State University’s Anechoic Chamber. Measured antenna parameters, such as
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136
the input impedance and far fields are compared to FDTD calculations. An excellent
agreement has been obtained between measurements and theory. In section 3.5, a
shorted and truncated microstrip antenna is introduced as an interned alternative of
the dual resonance shielded antenna. The bandwidth of this antenna is found to be
narrow for cellular phone applications. However, further investigation of this antenna
is planned as a future work to study stacked antenna configurations to increase the
bandwidth and enhance the overall performance.
In Chapter 4, an introductory level theory of magnetized ferrites, including the
equation of motion of the magnetization vector, magnetic loss mechanism in ferrite
and small signal analysis, are given. An algorithm is developed to solve electric and
magnetic fields inside the ferrite medium and its implementation is discussed in sec­
tion 4.3. In addition, special updating equations are derived for the magnetic field
components perpendicular to the direction of the dc bias field. Since the ferrite algo­
rithm is derived from the linearized equation of motion of the magnetization vector,
the analysis is limited to linear ferrite devices only. A slow wave thin film isolator,
whose operating mechanism depends on magnetostatic surface waves, and a stripline
disc junction circulator including the effect of non-uniform magnetization have been
analyzed in consequent sections. Non-linear behaviour of the ferrite is briefly men­
tioned in section 4.6. Extension of the regular ferrite algorithm to include non-linear
behaviour is discussed by introducing an alternative damping factor which will in­
crease with the increase in above-threshold signals. A simple ferrite-filled stripline
structure is analyzed for different input signal amplitudes and initial results are pre­
sented.
Plans for future work can be outlined as follows. Stacked microstrip antenna
configurations suitable for wireless communications will be investigated. The analysis
of non-linear phenomena in ferrite will be extended by investigating various non-linear
ferrite devices useful for microwave signal processing applications. These devices
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137
include frequency selective limiters and signal-to-noise enhancers.
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