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Toward Self-biased Ferrite Microwave Devices
A Dissertation Presented
by
Jianwei Wang
To
The Department of Electrical and Computer Engineering
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the field of
Electrical Engineering
Northeastern University
Boston, Massachusetts
May, 2011
NORTHEASTERN UNIVERSITY
Graduate School of Engineering
Thesis Title: Toward Self-biased Ferrite Microwave Devices
Author: Jianwei Wang
Department: Electrical and Computer Engineering
Approved for Dissertation Requirement for the Doctor of Philosophy Degree
______________________________________________ ____________________
Dissertation Advisor: Carmine Vittoria Date
______________________________________________ ____________________
Thesis Reader: Vincent Harris Date
______________________________________________ ____________________
Thesis Reader: Anton Geiler Date
______________________________________________ ____________________
Department Chair: Ali Abur Date
Graduate School Notified of Acceptance:
______________________________________________ ____________________
Director of the Graduate School: Sara Wadia Fascetti Date
Acknowledgements
I want to thank my advisor, Professor Carmine Vittoria. He does not only impart me
with precious knowledge and give me guidance, but also be a model of a scientist to me.
He is always creative and energetic. He cares about his student sincerely. I feel really
lucky to be one of his students.
I want to thank my advisor, Professor Vince Harris. He provided me with many helpful
suggestions and technical guidance through out my graduate career.
I want to thank Professor Yajie Chen for his help and guidance in my research project.
I want to thank Doctor Anton Geiler. He always gave me help whenever I needed.
I want to thank Doctor Zhaohui Chen and Doctor Aria Yang for their cooperation in my
research work.
I want to thank Doctor Soack Yoon for his sincerely help.
I want to thank other collegues in our group, Andrew Daigle, Scott Gillette, Bolin Hu,
Khabat Ebnabbasi, etc..
I want to thank my wife for her continously care and support.
I want to thank my parents for their praying for me every day.
ABSTRACT
Circulators are important components in modern radar systems. Its non-reciprocal
chracteristics make it possible for the radar systems to receive and transmit the signals at
the same time and same frequency. However, bulky permanent magnets are required to
provide a large static magnetic field in order for the traditional circulator to function
properly. This thesis focuses on how to remove the permanent magnets so that size,
weight and cost of systems can be reduced.
In-plane circulator requires low-biasing field due to the shape anisotropy of the
magnetic substrate. This thesis presents a spectral domain method to assist the analysis of
magnetic microstrip line and magnetic coupled microstrip lines. The latter is the main
component of the in-plane circulator. The ferrite modeled by this method could be cubic ,
M-type , Y-type and Z-type ferrites. An in-plane circulator based on YIG(yttrium iron
garnet) operating at C band is designed with this method and simulated with Ansoft®
HFSS. The reflection and isolation is less than 15 dB from 6.3 GHz to 7.8 GHz with a
200 Oe biasing field.
A self biased junction circulator based on oriented M-type hexaferrite was designed,
fabricated and tested. A new topology structure was used to ease the fabrication process
and integration with other components. An isolation of 21 dB with corresponding
insertion loss of 1.52 dB was measured, which render itself to the first hexaferrite-based
self-biased circulator operating below 20 GHz.
Theoretical models were developed to design self-biased Y-junction circulators
operating at UHF frequencies. The proposed circulator design consisted of insulating
nanowires of YIG embedded in high permittivity BSTO(barium-strontium titanate)
substrates. The model represents the nanowires and the BSTO substrate by an equivalent
medium with effective properties inclusive of the average saturation magnetization,
dynamic demagnetizing fields, and permittivity. The effective medium approach was
validated against the exact calculations and good agreement was observed between the
two simulations in terms of calculated S-parameters. Using the proposed approach, a self-
biased junction circulator consisting of YIG nanowires embedded in a BSTO substrate
was designed and simulated. The center frequency insertion loss was calculated to be as
low as 0.16 dB with isolation of -42.3 dB at 1 GHz.
TABLE OF CONTENTS
Chapter 1. Introduction 1
1.1 Magnetic Materials 1
1.2 Magnetic Properties 2
1.2.1 Demagnetizing Field 2
1.2.2 Anisotropy Magnetic Field 5
1.2.3 Remanence magnetization 7
1.3 Microwave Properties 8
1.4 Introduction of Circulator 12
1.4.1 Junction Circulator 12
1.4.2 In-plane Circulator 19
1.5 Limitation of Ferrite Devices 26
References 28
Chapter 2. Full-wave EM Simulation 31
2.1 Introduction of Hexagonal Y-type Ferrite 31
2.2 Application of Spectral Domain Method to Devices on Y-type
Ferrites 38
2.3 Y-type Ferrite Phase Shifter: Design and Experiment 78
2.4 Conclusions 83
References 85
Chapter 3. In-plane Circulator 86
3.1 Application of Spectral Domain Method to In-plane Circulator 86
3.1.1 Revised Spectral Domain Method 87
3.1.2 Current and Voltage in Coupled Microstrip Lines 90
3.2 In-plane Circulator Design and HFSS simulation 96
3.3 Conclusions 103
References 104
Chapter 4 Hexaferrites-based Self-biased Y-Junction Circulator 105
4.1 New Microstrip Y-Junction Circulator Design 107
4.2 HFSS Simulation 109
4.3 Experiment 110
4.4 Results and Discussion 111
4.5 Conclusions 112
References 114
Chapter 5 Nanowire-based Y-Junction Circulator 116
5.1 Modeling of YIG-nanowires 118
5.2 Equivalent Modeling of the YIG Nanowire Substrate 120
5.3 Nanowire-based Y-Junction Circulator Design 124
5.4 Conclusions 125
References 127
Chapter 6 Conclusions 129
References 131
1
Chapter 1. Introduction
The first application of magnetic materials can be dated back to 1000 years ago, when
the first compass was invented by Chinese as a magnetic direction finder [1]-[3]. Since
that time, efforts to utilize magnetic materials and magnetism has never stopped to date.
In this paper, the application of magnetic materials was focused on RF magnetic devices.
Characteristics of magnetic materials were discussed in sections 1.1, 1.2 and 1.3 and RF
magnetic devices were introduced in sections 1.4 and 1.5.
1.1 Magnetic Materials
Ferrimagnetic materials, or ferrites, are the most popular magnetic materials in RF and
microwave application. There are three practical types of ferrites: spinels, garnets and
hexaferrites [4]. Spinels and garnets have cubic crystal structure whereas hexaferrites
have a hexagonal one.
Although spinel ferrites exhibit a large static initial permeability, in the range of
10<µr<1000, its permeability at high frequencies drops down to one at around 2 GHz.
Spinel ferrites are known as high relaxation loss materials, with typical ferrimagnetic loss
(∆H) in the order of 2-1000 Oe. Therefore, applications of spinel ferrites are usually
limited to low frequencies.
The garnet ferrites have many applications in RF and microwave devices in past 20
years. G. Menzer first studied the cubic crystal structure of garnet ferrites in 1928. The
most famous garnet ferrite, yttrium iron garnet (Y3Fe5O12, or YIG), was first prepared by
F. Bertaut and F. Forrat. YIG is a very low loss material at high frequencies. The FMR
2
linewidth, ∆H, of YIG was measured to be ~ 0.2 Oe at 3 GHz. Many commercial
magnetic microwave devices are made of YIG substrates.
Hexagonal ferrites or hexaferrites have a hexagonal crystal structure, which gives this
type of ferrite many interesting characteristics. The hexaferrites have a large saturation
magnetization (4πMs) and large magnetocrystalline uniaxial anisotropy field(HA). The
large HA can help to bias the ferrite at high frequencies so that the hexaferrites-based
devices usually can operate from Ka up to Ku bands. The hexaferrits can be
subcategorized into M-type(BaFe12O19), Y-type(Ba2Me2Fe12O22), Z-
type(Ba3Me2Fe24O41), etc.. M-type ferrites are usually used in junction circulators due to
the fact that the easy axis of magnetization is along the c-axis, whereas for Y-type and Z-
type ferrites the plane of easy magnetization is perpendicular to the c-axis( within the
basal plane).
1.2 Magnetic Properties
In this section, some important magnetic properties are introduced in order to provide
some background knowledge.
1.2.1 Demagnetizing Field
The demagnetizing field is a magnetic field due to the surface magnetic charges on the
interface between the magnetic material and non-magnetic material. It tends to reduce the
total magnetic moments inside the magnetic material and the internal magnetic field.
Let's investigate a thin magnetic plate shown in Fig. 1.1. The thickness of this plate, t, is
assumed to be infinitely small. No DC magnetic field is applied, so all the magnetic field
3
is generated by the surface magnetic charge. It is assumed that all the magnetic moment
was aligned along the z direction. Due to Gauss' theorem, B is continuous on the surface,
z=t.
Bo=Bi,
Bo= H0, (4πMair=0)
Bi=4πM−Hi,
So, H0=−Hi +4πM.
On the two sides of the surface magnetic charge, the magnetic field is opposite in
direction and equal in magnitude,
2Hi=4πM,
Hi=2πM.
Taking into account the contribution from the plane t=0, we can conclude Hi=4πM with
the direction along the z-axis. Therefore, the demagnetizing field of an infinite thin
magnetic plate is equal to 4πM, where M is the magnetization normal to the surface of
the plate. In practical situations, the magnetization lies in the film plane in order to
minimize the magnetostatic energy. Generation of surface charges on demagnetizing field
implies generation of energy which nature does not comply. Hence, this is unstable
magnetization configuration and, M, in this case would lie in the plane, representing a
lower energetic state.
4
The formula to calculate the internal field is
Hi=Ha−NM,
where Ha is the applied field, N is the magnetizing factor (in this case, N is equal to 4π),
M is the magnetization along the normal direction. If we are investigating a three
dimensional object, as shown in fig. 1.2, the calculation of demagnetizing factor would
be more complicated. A practical formula to calculate the demagnetizing factor of a
rectangular ferromagnetic prisms was given by Aharoni in [5]
( )abc3ab
c
abc3
c2ba
abc3
c2ba
c
abarctan2
a
aln
b2
c
b
bln
a2
c
b
bln
c2
a
a
aln
c2
b
b
bln
ac2a
aln
bc2D
3a,c
3c,b
3b,a
c,bc,a
222333
c,a
c,a
c,b
c,b
b,a
b,a
b,a
b,a2
c,a2
c,bz
∆+∆+∆−∆+∆+∆
−++
−++
∆+
+∆
−∆+
+∆
−∆+
−∆
+∆+
−∆
+∆+
+∆
−∆∆+
+∆
−∆∆=π
−−
where
222 cba ++=∆ ,
t M
z
Bo
Bi
Ho
Hi M
y x
Air
Air
Magnetic
Material
Fig. 1.1 Demagnetizing field of a magnetic plate
5
22
b,a ba +=∆ ,
22
c,b cb +=∆ ,
22
c,a ca +=∆ ,
22
c,a ca −=∆ − ,
22
c,b cb −=∆ − .
The other two demagnetizing factors can be easily derived by interchanging a, b, and c
accordingly. Notice, when calculating Dz, a and b are commutative.
A simpler derivation may be found in [6] by Vittoria.
1.2.2 Anisotropy magnetic Field
The magnetic anisotropy energy implies that the magnetic potential energy depends on
the direction of the magnetization. There are mainly three types: magnetocrystalline
anisotropy, shape anisotropy(demagnetizing) and stress anisotropy magnetic energies.
The first two will be introduced in this section.
Fig. 1.2 The rectangular ferromagnetic prisms under investigation. The field Happl is along the z
axis
6
Magnetocrystalline anisotropy energy
The uniaxial magnetic anisotropy energy can be expressed as [6]
Fu=Kusin2θsin
2φ.
If Ku>0, Fu is minimum for Mv
perpendicular to c-axis. If Ku<0, Fu is minimum for Mv
parallel to c-axis. The magnitude of the magnetic anisotropy field is derived as
Hk=2|Ku|/Ms,
where Ms is the saturation magnetization.
The cubic magnetic anisotropy energy can be expressed as
( )2
1
2
3
2
3
2
2
2
2
2
11A kF αα+αα+αα= ,
where φθ=α 222
1 cossin , φθ=α 222
2 sinsin , and θ=α 22
3 cos . The maximum magnetic
anisotropy field is given as
Hk=2K1/Ms, for K1>0, and
φ
θ
x
y, c-axis
z
Mv
Fig. 1.3 Magnetization with uniaxial crystal symmetry
7
Hk=4|K1|/(3Ms), for K1<0.
Shape anisotropy energy
The demagnetizing field can be expressed in general as
( )zzzyyyxxxD aMNaMNaMNHvvvv
++−= .
The free energy from demagnetizing field can be derived from
∫ ⋅−= MdHF DD
vv,
so that
( )2
zz
2
yy
2
xxD MNMNMN2
1F ++= .
The total free energy can be expressed as
( )2
zz
2
yy
2
xxD MNMNMN2
1HMF +++⋅−=vv
( )θ+ϕθ+ϕθ+ϕθ−= 2
z
22
y
22
x
2 cosNsinsinNcossinNM2
1cossinMH
Following the procedure in Chapter 5 of [6], the ferromagnetic resonance (FMR) can be
derived from
( )[ ]0
22
2
2
sinM
1FFF
θ−=
γ
ωθϕϕϕθθ ,
where 0θ is obtained from the equilibrium condition
0FF
=ϕ∂
∂=
θ∂
∂.
8
1.2.3 Remanence magnetization
The remanence magnetization, Mr, is the residue magnetization when the applied field is
reduced to zero. The position of Mr in a hysteresis loop is shown in Fig. 1.4. The
remanence magnetization is important to the self-biased junction circulator.
1.3 Microwave Properties
Microwave permeability
Let’s assume that there is a magnetic dipole immersed in a static magnetic field along
the z-axis. The equation of motion of the magnetic dipole moments can be derived
as(MKS units were used in this section) [7]
HM
dt
Md0
vvv
×γµ−= . (1.1)
Assume the total magnetic field and total magnetization can be expressed as
hzHH 0t
vv+= , (1.2)-a
Mr
Fig. 1.4 Hysteresis loops of magnetically oriented M-type strontium hexaferrite.
9
mzMM 0t
vv+= , (1.2)-b
where H0 is the applied bias field, M0 is the DC magnetization, hv
is the applied AC field,
and mv
is the AC magnetization caused by hv
. Substituting (1.2) into (1.1) gives the
following equations
( ) ( )hzHmzMdt
md000
vvv
+×+γµ−= ,
( ) ( )( )
( ) ( )( )
( )
−γµ−=
+−+γµ−=
+−+γµ−=
⇒
xyyx0z
0zxx0z0
y
y0z0zy0x
hmhmdt
dm
HhmhMmdt
dm
hMmHhmdt
dm
(1.3)
Assuming hz<<H0 and mz<<M0, and ignoring mxhy and myhx terms, (1.3) reduces into
=
γµ−γµ=
γµ+γµ−=
0dt
dm
hMmHdt
dm
hMmHdt
dm
z
x00x00
y
y00y00x
,
=
ω−ω=
ω+ω−=
⇒
0dt
dm
hmdt
dm
hmdt
dm
z
xmx0
y
ymy0x
(1.4)
where 000 Hγµ=ω , and 00m Mγµ=ω .
Taking the derivative over t on both sides of the first two equations in (1.4) gives the
following equations
10
ω−ω=
ω+ω−=
xmx02
y
2
ymy02
x
2
hdt
dm
dt
d
dt
md
hdt
dm
dt
d
dt
md
( )
( )
ω−ω+ω−ω=
ω+ω−ωω−=⇒
xmymy002
y
2
ymxmx002
x
2
hdt
dhm
dt
md
hdt
dhm
dt
md
ωω+ω−=ω+
ω+ωω=ω+⇒
ym0xmy
2
02
y
2
ymxm0x
2
02
x
2
hhdt
dm
dt
md
hdt
dhm
dt
md
. (1.5)
Assuming hv
and mv
are tje ω dependent, (1.5) reduces to
( )( )
ωω+ωω−=ω−ω
ωω+ωω=ω−ω
ym0xmy
22
0
ymxm0x
22
0
hhjm
hjhm,
( ) ( )
( ) ( )
ω−ω
ωω
ω−ω
ωω−ω−ω
ωω
ω−ω
ωω
=
⇒
z
y
x
22
0
m0
22
0
m
22
0
m
22
0
m0
z
y
x
h
h
h
000
0j
0j
m
m
m
,
h
000
0
0
m yyyx
xyxxvv
χχ
χχ
=⇒ ,
where ( )22
0
m0xx
ω−ω
ωω=χ , ( )22
0
mxy
j
ω−ω
ωω=χ , ( )22
0
myx
j
ω−ω
ωω−=χ and ( )22
0
m0yy
ω−ω
ωω=χ .
bv
and hv
are related by
( )hmb 0
vvv+µ=
[ ]( )hh0
vv+χµ=
11
[ ] [ ]( )hU0
v+χµ=
h
00
0j
0j
0
v
µ
µκ−
κµ
=
where ( )
ω−ω
ωω+µ=µ
22
0
m00 1 and ( )22
0
m0
ω−ω
ωωµ=κ .
Wave propagation along the bias field direction
Assume a plane wave propagates in an infinite magnetic medium along the bias
direction, z axis. The plane wave has no distribution along x axis and y axis. The
electromagnetic fields can be expressed as [7]
( ) zj
yx eEyExE β−+=v
(1.6)-a
( ) zj
yx eHyHxH β−+=v
(1.6)-b
Substituting (1.6) into Maxwell equations
[ ]
ωε=×∇
µω−=×∇
EjH
HjEvv
vv
( )
( ) ( )
+ωε=+×
∂
∂+
∂
∂+
∂
∂
µ
µκ−
κµ
ω−=+×
∂
∂+
∂
∂+
∂
∂
⇒
yxyx
y
x
0
yx
EyExjHyHxz
zy
yx
x
0
H
H
00
0j
0j
jEyExz
zy
yx
x
( ) ( ) ( )( )
( ) ( )
+ωε=+×
∂
∂+
∂
∂+
∂
∂
µ+κ−+κ+µω−=+×
∂
∂+
∂
∂+
∂
∂
⇒
yxyx
yxyxyx
EyExjHyHxz
zy
yx
x
HHjyHjHxjEyExz
zy
yx
x
( ) ( )
( )
+ωε=∂
∂+
∂
∂−
µ+κ−ω−κ+µω−=∂
∂+
∂
∂−
⇒
yxxy
yxyxxy
EyExjHz
yHz
x
yHHjjxHjHjEz
yEz
x
12
( )
( )
ωε=∂
∂
ωε=∂
∂−
µ+κ−ω−=∂
∂
κ+µω−=∂
∂−
⇒
yx
xy
yxx
yxy
EjHz
EjHz
HHjjEz
HjHjEz
( )( )
β
ωε−=
β
ωε=
µ+κ−ω−=β−
κ+µω−=β
⇒
yx
xy
yxx
yxy
EH
EH
HHjjEj
HjHjEj
β
ωεµ+
β
ωεκω−=β−
β
ωεκ+
β
ωεµ−ω−=β
⇒
xyx
xyy
EEjjEj
EjEjEj
( )( )
=κεω+β−µεω
=β−µεω+κεω⇒
0EjE
0EjE
y
2
x
22
y
22
x
2
For non-trivial solution, the determinant is set to zero. So
( )εκ±µω=β± ,
which means there are two basis modes for a plane electromagnetic wave propagating in
an infinite magnetic material. This knowledge is so important that it is the basis for many
magnetic devices.
1.4 Introduction of Circulator
1.4.1 Y-Junction Circulator
Y-junction circulator is a non-reciprocal device used in wireless communications
systems. The nonreciprocal property of ferrite materials makes it possible for the
13
transmission and reception of wireless signals occurring at the same time and frequency,
as shown in Fig. 1.5. The Y-junction circulator can also be used to isolate the reflection
from the transmission signal to protect the high frequency amplifier in a communication
system.
Fig. 1.5 Radar system
14
Fig. 1.6 shows the structure of a stripline Y-junction circulator. In the circular region, the
metallic signal trace is separated from the ground planes with two ferrite disks. These two
ferrite disks are biased with two permanent magnets to magnetically saturate the ferrite.
The whole structure has a vertical symmetry. Depending on the direction of the bias
magnetic field, the circulation can be either clockwise(Port I→ Port III→ Port II→ Port
I) or counterclockwise. For a clockwise circulation, the signal enters at Port I and leaves
at Port III, and Port II is the isolation port. The flow is reversed as the bias field is
reversed.
Bosma first gave a theoretical analysis to the stripline junction circulator [8]. As shown
in Fig. 1.7, it is assumed that the electric field has only z component in the disk area and
the striplines only support TEM wave propagation and the microwave magnetic field has
Fig. 1.6 Y-junction stripline circulator
15
no horizontal distribution with the azimuth angle. Therefore, the boundary condition for
Hφ is
( )
Ψ+π<ϕ<Ψ−π
Ψ+π<ϕ<Ψ−π
Ψ+π−<ϕ<Ψ−π−
=ϕϕ
.elsewhere,d
,c
,3/3/,b
,3/3/,a
,RH (1.7)
The relative permeability of ferrite has a tensor form:
[ ]
µκ
κ−µ
=µ
100
0i
0i
.
An effective scalar permeability ueff was introduced as
µ
κ−µ=µ
22
eff.
The intrinsic wave number k can be expressed as
Fig. 1.7 The configuration of the junction of the circulator
16
εµεµω= eff00
22k ,
where ε is the relative permittivity of ferrite.
( )ϕ,rEz satisfies the homogeneous Helmholtz equation in the disk area
( ) 0,rEkr
1
rr
1
rz
2
2
2
22
2
=ϕ
+
ϕ∂
∂+
∂
∂+
∂
∂. (1.8)
The tangential component of magnetic field in the disk area can be derived as
( )eff0
rr E
r
1i
r
E
i,rHµωµ
ϕ∂
∂
µ
κ+
∂
∂
=ϕϕ
( )ϕ,rEz can also be derived from ( )ϕϕ ,RH using
( ) ( ) ( ) ϕ′ϕ′ϕ′ϕ=ϕ ϕ
π
π−∫ d,RH,R:,rG,rEz .
For small Ψ, the z component of the electric intensity at the interface where the metal
disk and stripline meet can be derived as
( ) ( ) ( ) ( )[ ]c;3/Gb3/;3/Ga3/;3/G2A3/,REz ππ−+ππ−+π−π−Ψ==π−
( ) ( ) ( ) ( )[ ]c;3/Gb3/;3/Ga3/;3/G2B3/,REz ππ+ππ+π−πΨ==π
( ) ( ) ( ) ( )[ ]c;Gb3/;Ga3/;G2C,REz ππ+ππ+π−πΨ==π
After consideration of boundary condition at the edge of the disk, the green function can
be expressed as
( ) ( )( )
( ) ( ) ( ) ( )
( ) ( )( )∑
∞
=
µ
κ−′
ϕ′−ϕ′−ϕ′−ϕµ
κ
π
ξ+
′π
ξ−=ϕ′ϕ
1n
n2
n2
n
nn
eff
0
0eff krJ
x
xnJxJ
ncosxJinsinx
xnJ
xJ2
krJi,R;,rG .
In order for resonance to occur,
17
( ) ( )0
x
xnJxJ n
n =µ
κ−′ , (1.9)
where n is either positive or negative.
Fig 8 shows the solution to equation (1.9) in terms of κ/µ. The practical circulation is
located in the neighborhood of region A.
Fay and Comstock used a different method to study the junction circulator model that
Bosma used and came up with similar result [9]. Assume the solution to equation (1.8)
has the following form
( ) 0,rEkr
1
rr
1
rz
2
2
2
22
2
=ϕ
+
ϕ∂
∂+
∂
∂+
∂
∂. (1.8)
( )( )ϕ−
−ϕ
+ += jn
n
jn
nnzn eaeaxJE . (1.10)
where x=kr.
From Maxwell equations, the φ component of magnetic field can be derived as
Fig 1.8. x values of the resonant mode versus κ/µ.
18
( ) ( ) ( ) ( )
µ
κ−−+
µ
κ+−= −
ϕ−−−
ϕ+ϕ 1
x
xnJxJea1
x
xnJxJeajYH n
1n
jn
nn
1n
jn
neffn
where eff0
0effY
µµ
εε= .
The boundary condition is the same as Bosma's model and is rewritten as
( )
Ψ+π−<ϕ<Ψ−π−
Ψ+π<ϕ<Ψ−π
Ψ<ϕ<Ψ−
=ϕϕ
.elsewhere,0
,3/23/2,0
,3/23/2,H
,,H
,RH1
1
(1.11)
and
−=ϕ
=ϕ
=ϕ
=
.120,0
,120,E
,0,E
E 1
1
z
o
o (1.12)
Assume only n=1 mode is considered, after combination of (1.10) and (1.12) the two
coefficient a+n and a-n can be obtained,
( )
+=+
3
j1
kRJ2
Ea
1
1 ,
( )
−=−
3
j1
kRJ2
Ea
1
1 .
Therefore, Hφ1 can be expressed as
( )
( ) ( )
( ) ( )
µ
κ−−
−+
µ
κ+−
+
=ϕ−
ϕ
ϕ
j10
j10
1
1eff1
e1kR
kRJkRJ
3
j1
e1kR
kRJkRJ
3
j1
kRJ2
EjYH . (1.13)
Equation (1.13) can be expanded into Fourier series,
19
ϕ
π
Ψ+ϕ
π
Ψ+
π
Ψ= ∑
∞
=1n
1 nsinn
nsin3ncos
n
nsin2HH .
The n=1 mode can be written as
( ) ( )[ ]ϕ−ϕ
ϕ ++−π
Ψ= jj
11e3j1e3j1
2
sinHH . (1.14)
Comparing (1.13) and (1.14), it is necessary that
( ) ( ) ( ) ( )
µ
κ+−−=
µ
κ−− 1
kR
kRJkRJ1
kR
kRJkRJ 1
01
0,
( ) ( )0
kR
kRJkRJ 1
0=− ,
which is in agreement with Bosma's conclusion.
1.4.2 In-plane Circulator
Although Y-junction circulators are very popular, there are some limitations when used
at high frequencies. The drop-in technology used to fabricate the junction circulator
becomes very demanding and expensive due to the dependence of the circulator’s
diameter on the wavelength. Moreover, a strong bias field is required to overcome the
large demagnetization factor of the thin ferrite disk if the ferrite used is not self-biased,
which makes the applications of junction circulators very inconvenient. A possible
alternative is the distributed in-plane circulator configuration. The advantage of the in-
plane non-reciprocal device over the traditional junction device is that there is no need for
a strong applied field to overcome the large demagnetization field resulting from the
shape of the ferrite disc.
In-plane circulators and isolators, which include coupled slot-lines sections with a
longitudinally magnetized ferrite, were discovered by L. E. Davis and D. B. Sillars in
20
1986 [10]. P. Kwan et al. [11] demonstrated non-reciprocal properties in a ferrite device
consisting of two dielectric image lines coupled to each other via a ferrite slab. The
principle of operation of the in-plane circulator is explained by J. Mazur and M.
Mrozowski in 1989 [12]-[13] using the coupled mode model developed by D. Marcuse in
1973 [14] and I. Awai and T. Itoh in 1981 [15]. The coupled mode theory claims that the
solution to the investigated structure is represented by the coupling between the even and
odd modes supported by the basis structure (basis structure is the same as the investigated
structure except that its permeability is a scalar).
From the coupled mode theory, we can arrive at the following conclusions:
a) As the wave propagates, the energy of one mode converts to the other. Over the
distance 2π=Cz , we observe the total exchange of energy between the two modes.
b) If the structure is excited by the even mode (see Fig. 1.9-a), the field in guide 1 (the
left one) vanishes at 4π=Cz from the excitation plane and the field is concentrated in
guide 2 (the right one). The converse effect occurs if the biasing magnetic field ( 0H ) is
reversed.
c) The odd excitation (See Fig. 1.9-b) causes an effect similar to the change of
magnetization direction in case b. Over the distance 4π=Cz from the excitation plane
the field in guide 2 becomes zero and the field in guide 1 reaches maximum. Again the
change of magnetization direction results in the converse effect.
d) If the structure is excited at port 1(see Fig 1.9-c), over a distance of 4π=Cz , an
even mode signal will output at port 1’ and 3’. While the excitation at port 3 (see Fig 1.9-
21
d) will result in an odd mode signal at port 1’ and 3’. Again the change of magnetization
direction results in the converse effect.
The ferrite coupled lines can be used as a three port circulator when cascaded with a T
junction. The optimum length is 4π=Cz , and the operating theory can be illustrated by
Fig. 1.10.
+ =
Fig. 1.9 a) Even mode excitation, b) Odd mode excitation,
c) Excitation at port 1, d) Excitation at port 3.
d)
= +
c)
b) a)
22
If the structure is excited at port 1 (see Fig. 1.10-a), an even mode will be generated at
port 1’ and 3’. The even mode can pass through the T junction and reach port 2. So for the
circulator, signal couples from port 1 to port 2 with port 3 as the isolation port. If the
structure is excited at port 2 (see Fig. 1.10-b), an even mode signal will be formed at port
=
Fig. 1.10 a) Excitation at port 1, b) Excitation at port 2, c) Excitation at port
c)
= =
a) b)
23
1’ and 3’, and arrive at port 3 through ferrite coupled lines. So for the circulator, signal
couples from port 2 to port 3. The condition for excitation at port 3 is a little complicated
(see Fig. 1.10-c). First, the signal at port 3 will output as odd mode at port 1’ and 3’. But
for the T junction, odd mode will be totally reflected back instead of arriving at port 2.
Then the odd mode signal at port 1’ and 3’ will output at port 1. So for a circulator, signal
couples from port 3 to port 1.
C. S. Teoh and L. E. Daivs also presented a different method in 1995 [16] and [17] to
solve the problem using the superposition of two dominant normal modes. Just as the
dielectric coupled microstrip lines support even mode and odd mode, the microstrip
ferrite coupled lines (FCL) also support two normal modes: clockwise elliptical-polarized
mode and counterclockwise elliptical-polarized mode. These two modes are normal to
each other, which means they do not affect each other while propagating along the
structure. Assume the two modes supported by the structure are mode 1 and mode 2, and
the signals at line 1 and line 2 for mode 1 and mode 2 can be expressed as
Mode 1 Mode 2
Line 1 ( ) zj
11p111eVzV
β−= ( ) ( )θ+β−= zj
12p122eVzV
( ) zj
11
11p
111e
Z
VzI
β−= ( ) ( )θ+β−= zj
12
12p
122e
Z
VzI
Line 2 ( ) ( )11zj
21p21 eVzVϕ+β−= ( ) ( )22zj
22p22 eVzVϕ+θ+β−=
( ) ( )11zj
21
21p
21 eZ
VzI
ϕ+β−= ( ) ( )22zj
22
22p
22 eZ
VzI
ϕ+θ+β−=
24
The power on line 1 can be derived as
( ) ( ) ( )[ ]zIzVRe2
1zP
1line1line1
∗=
( )( ) ( )
++= θ+ββθ+β−β− zj
12
12pzj
11
11pzj
12p
zj
11p2121 e
Z
Ve
Z
VeVeVRe
2
1
( )[ ] ( )[ ]
+++= θ+β−β−θ+β−β zj
11
12p11pzj
12
12p11p
12
2
12p
11
2
11p1212 e
Z
VVe
Z
VV
Z
V
Z
VRe
2
1
( )( )θ+β−β
++
+= zcos
Z
1
Z
1VV
2
1
Z
V
Z
V
2
112
1112
12p11p
12
2
12p
11
2
11p (1.15)-a
The power on line 2 can be derived as
( ) ( ) ( )[ ]zIzVRe2
1zP
2line2line2
∗=
( ) ( )( ) ( ) ( )
++= ϕ+θ+βϕ+βϕ+θ+β−ϕ+β− 22112211 zj
22
22pzj
21
21pzj
22p
zj
21p eZ
Ve
Z
VeVeVRe
2
1
( )( ) ( )( )
+++= θ+ϕ−ϕ+β−β−θ+ϕ−ϕ+β−β 12121212 zj
21
22p21pzj
22
22p21p
22
2
22p
21
2
21pe
Z
VVe
Z
VV
Z
V
Z
VRe
2
1
( )( )θ+ϕ−ϕ+β−β
++
+= 1212
2221
22p21p
22
2
22p
21
2
21pzcos
Z
1
Z
1VV
2
1
Z
V
Z
V
2
1. (1.15)-b
The total power is the combination of the power on line 1 and line 2,
( ) ( ) ( )( )
( )( )θ+ϕ−ϕ+β−β
++
++
θ+β−β
++
+=+
1212
2221
22p21p
22
222p
21
221p
12
1112
12p11p
12
212p
11
211p
21
zcosZ
1
Z
1VV
2
1
Z
V
Z
V
2
1
zcosZ
1
Z
1VV
2
1
Z
V
Z
V
2
1zPzP
( ) ( )
( )( ) ( )( )θ+ϕ−ϕ+β−β
++θ+β−β
++
++
+=+
1212
2221
22p21p12
1112
12p11p
22
222p
21
221p
12
212p
11
211p
21
zcosZ
1
Z
1VV
2
1zcos
Z
1
Z
1VV
2
1
Z
V
Z
V
2
1
Z
V
Z
V
2
1zPzP
.
From the viewpoint of power conservation, the total power should be independent of z.
So the third and fourth term should cancel each other,
25
+=
+
2221
22p21p
1112
12p11pZ
1
Z
1VV
Z
1
Z
1VV
o18012 =ϕ−ϕ .
For a symmetrical FCL, 1pm21p11p VVV == , 2pm22p12p VVV == , 1m2111 ZZZ == , and
2m2212 ZZZ == . Also, we assume 1pm
2pm
V
Vk = , and substitute into (1.15), we can obtain
( ) ( )( )θ+β−β
++
+= zcos
Z
1
Z
1kV
2
1
Z
k
Z
1V
2
1zP 12
1m2m
2
1pm
2m
2
1m
2
1pm1 (1.16)-a
( ) ( )( )θ+β−β
+−
+= zcos
Z
1
Z
1kV
2
1
Z
k
Z
1V
2
1zP 12
2m1m
2
1pm
2m
2
1m
2
1pm2 . (1.16)-b
In order for the FCL to function correctly as shown in Fig. 1.9(c), the following
conditions must be imposed: if the signal enters at port 1, which indicates P2(0)=0 and
0z
)0(P2 =∂
∂, the voltage at z=L must be either even mode or odd mode, P1(L)=P2(L) or
P1(L)=−P2(L).
Applied P2(0)=0 and 0z
)0(P2 =∂
∂ to (1.16)-b, we can derive θ=0 and k=1 or
1m
2m
Z
Zk =
(ignored). Applied P1(L)=P2(L) to (1.16), we can derive
( )( ) 0Lcos 12 =β−β ,
L,2,1,0n,2
1nL12 =π
+=β−β .
Therefore, the total voltage on line 1 and line 2 at z=L can be expressed as
( ) ( )[ ]LjLj
1pm1121 e1eVLV
β−β−β− += ,
( ) ( ) ( )[ ]LjLj
1pm21211 e1eVLV
β−β−ϕ+β− −= .
The phase difference between line 1 and line 2 is
26
( ) ( )
=π
+ϕ
=π
−ϕ=∠−∠
L
L
,5,3,1n,2
,4,2,0n,2LVLV
1
1
21 .
If the signal is odd mode at z=L, we can conclude φ1=-90° and φ2=90
°. For even mode at
z=L, φ1=90° and φ2=-90
°.
1.5 Limitation of Ferrite Devices
The current commercialized circulator usually need a permanent magnet to provide DC
bias field. Fig. 1.11 shows a microstrip junction circulator biased with a permanent
magnet from DORADO, Inc.. The permanent magnet increased the size, weight and cost
to the system, which may be unfavorable when the trend in modern technologies is
toward miniature and efficient devices.
One solution is to use the self-biased magnetic material. Previously, self-biased junction
circulator designs were demonstrated at frequencies above 30 GHz at Ka and V band
Fig. 1.11 a) microstrip junction circulator b) microstrip junction circulator with permanent magnet
a) b)
27
utilizing magnetically oriented M-type hexaferrite compacts [18]-[22]. In this thesis, the
first hexaferrite-based self-biased circulator operating below 20 GHz will be addressed in
Chapter 4.
Another solution is to take the advantage of the shape anisotropy of the magnetic
nanowires to achieve self-biasing. Self-biased junction circulators based on metal
ferromagnetic nanowires have been fabricated and tested. Relatively high insertion loss
of up to 10dB was measured at X and Ku band [23]-[24]. In this thesis, self-biased
junction circulator based on YIG-nanowires operating at 2 GHz and below are presented
at Chapter 5. The porous BSTO membrane and embedded YIG-nanowires make it
possible to design circulators at such a low frequency band.
As a different technology, the in-plane circulator takes advantage of the low
demagnetizing field along the longitudinal direction and a small biasing field can bias
this type of device. However, it is not easy to obtain the parameters of the normal mode
directly from the commercial EM software packages, such as Ansys® HFSS. Those
parameters are very important for the design of these devices. Chapter 2 will discuss how
to use Galerkin's method in spectral domain, or spectral domain method, to analyze the
longitudinally biased magnetic devices. Chapter 3 will discuss how to use the simulation
result obtained from the method discussed in Chapter 2 to design in-plane circulator.
28
References
[1] Allan H. Morrish, The Physical Principles of Magnetism (John Wiley & Sons, Inc.,
New York, 1965).
[2] Soshin Chikazumi, Physics of Ferromagnetism Second Edition (Oxford University
Press Inc., New York, 1997).
[3] Raul Valenzuela: Magnetic Ceramics (Cambridge University Press, New York,
1994).
[4] B. Lax and K. J. Button, Microwave Ferrites and Ferrimagnetics (McGraw-Hill
Book Company, Inc., 1962).
[5] A. Aharoni, “Demagnetizing factors for rectangular ferromagnetic prisms,” J. Appl.
Phys., 83, pp. 3432-3434, Mar. 1998.
[6] Carmine Vittoria, Magnetics, Dielectrics, and Wave Propagation with MATLAB®
Codes (CRC Press, Taylor & Francis Group, Boca Raton, 2011).
[7] David M. Pozar, Microwave Engineering, Second Edition (John Wiley & Sons, Inc.,
New York, 1998).
[8] H. Bosma, “On stripline Y-Circulation at UHF,” IEEE Trans. Microwave Theory
Tech., vol. 12, pp. 61-72, Jan. 1964.
[9] C. E. Fay, and R. L. Comstock, “Operation of the ferrite junction circulator,” IEEE
Trans. Microwave Theory Tech., vol. 13, pp. 15-27, Jan. 1965.
[10] L. E. Davis and D. B. Sillars, “Millimetric nonreciprocal coupled-slot fin-line
components,” IEEE Trans, Microwave theory Tech., vol. MTT-34, pp.804-808, Jul.
29
1986.
[11] P. Kwan,H.How and C. Vittoria, “Non-reciprocal coupling structure of a ferrite
loaded dielectric image line”, IEEE Trans. Magn., vol. 28, pp. 3222-3224, Sep.
1992.
[12] J. Mazur and M. Mrozowski, “On the mode coupling in longitudinally magnetized
waveguide structures,” IEEE Trans. Microwave Theory Tech., vol. 37, pp. 159-164,
Jan. 1989.
[13] J. Mazur and M. Mrozowski, “Nonreciprocal operation of structures comprising a
section of coupled ferrite lines with longitudinal magnetization direction,” IEEE
Trans. Microwave Theory Tech., vol. 37, pp. 1012-1019, July 1989.
[14] D. Marcuse, “Coupled-mode theory for anisotropic optical guide,” Bell Syst. Tech. J.,
vol. 54, pp. 985-995, May 1973.
[15] I. Awai and T. Itoh, “Coupled-mode theory analysis of distributed nonreciprocal
structures,” IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 1077-1086, Oct.
1981.
[16] C. S. Teoh and L. E. Davis, “Normal-mode analysis of ferrite-coupled lines using
microstrips and slotlines,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 2991-
2998, Dec. 1995.
[17] C. S. Teoh and L. E. Davis, “Normal-mode analysis of ferrite-coupled lines using
microstrips and slotlines,” IEEE MTT-S Int. Microwave Symp. Dig., Orlando, FL,
pp.99-102, May 1995.
30
[18] J. A. Weiss, N. G. Watson, and G. F. Dionne, "New uniaxial-ferrite millimeter-wave
junction circulators," IEEE MTT-S Int. Microwave symp. Dig., pp.145-148, 1989.
[19] Y. Akaiwa, and T. Okazaki, "An application of a hexagonal ferrite to a millimeter-
wave Y circulator," IEEE Trans. Magn., vol. 10, pp. 374-378, Jun. 1974.
[20] N. Zeina, H. How, and C. Vittoria, " Self-biasing circulators operating at Ka-band
utilizing M-type hexagonal ferrites," IEEE Trans. Magn., vol. 28, pp. 3219-3221,
Jan. 1992.
[21] B.K. O’Neil, and J. L. Young, “Experimental investigation of a self-biased
microstrip circulator,” IEEE Trans. Microwave Theory Tech., vol. MTT-57, pp.
1669-1674, Jul. 2009.
[22] S. A. Oliver, P. Shi, W. Hu, H. How, S. W. McKnight, N. E. McGruer, P. M.
Zavracky, and C. Vittoria, "Integrated self-biased hexaferrite microstrip circulators
for milimeter-wavelength applications," IEEE Trans. Microwave Theory Tech., vol.
MTT-49, pp. 385-387, Feb. 2001.
[23] A. Saib. M. Darques, L. Piraux, D. Vanhoenacker-Janvier, and I. Huynen, "An
unbiased integrated microstrip circulator based on magnetic nanowired substrate,"
IEEE Trans. Microwave Theory Tech., vol. MTT-53, pp. 2043-2049, Jun. 2005.
[24] M. Darques, J. De la Torre Medina, L. Piraux, L. Cagnon and I. Huynen,
"Microwave circulator based on ferromagnetic nanowires in an alumina template,"
Nanotechnology 21, 145208, 2010.
31
Chapter 2 Full-wave EM Simulation
In this chapter, the spectral domain method is used to model the longitudinally biased
magnetic material and devices. The cubic ferrite and hexagonal M-type ferrite can be
modeled with similar equations as discussed in section 1.3. However, hexagonal Y-type
and Z-type ferrites need to be characterized with different set of equations. Y-type and Z-
type ferrites have an easy plane of magnetization perpendicular to the crystallographic c-
axis, which is more favorable for in-plane devices. Therefore, section 2.1 is an
introduction to the special permeability of hexagonal Y-type ferrites. In section 2.2,
spectral domain method is used to model hexagonal Y-type ferrite and devices, which is
also applicable to cubic and hexagonal M-type ferrites. Section 2.3 shows a comparison
between simulation results and experimental results in terms of a Y-type hexaferrite phase
shifter.
2.1 Introduction of hexagonal Y-type ferrite
Y-type hexaferrites have a negative uniaxial magnetocrystalline anisotropy constant,
which results in an easy magnetization plane in its basal plane, which is in contrast to an
easy magnetization axis of M-type hexaferrites due to a positive uniaxial
magnetocrystalline anisotropy constant.
Assuming the applied field and propagation direction are along the z axis, and the film
lies in x-z plane, as shown in Fig 2.1.
32
Fig 2.1 ( in this plot replace Ha with c-axis)
The permeability of Y-type material can be expressed as
[ ] [ ]χπµ 4+= Ir
+
=
000
0
0
4
100
010
001
yyyx
xyxx
χχ
χχ
π
+
+
=
100
0414
0441
yyyx
xyxx
πχπχ
πχπχ
=
100
0
0
yyyx
xyxx
µµ
µµ
(2.1)
where
( )
( )2
2
4
4
γ
ωπ
πχ
−++
++=
sA
sAsxx
MHHH
MHHM (2.1-a)
x
z, H, Ms
y, Ha
33
( )2
2
4γ
ωπ
χ
−++
=
sA
syy
MHHH
HM (2.1-b)
( )2
2
4γ
ωπ
γ
ω
χχ
−++
=−=
sA
s
xyxy
MHHH
jM
(2.1-c)
( )
( )2
2
4
441
γ
ωπ
ππµ
−++
+++=
sA
sAsxx
MHHH
MHHM (2.1-d)
( )2
2
4
41
γ
ωπ
πµ
−++
+=
sA
syy
MHHH
HM (2.1-e)
( )2
2
4
4
γ
ωπ
γ
ωπ
µµ
−++
=−=
sA
s
yxxy
MHHH
M
j (2.1-f)
Then ,we bring in damping by substituting γ
ω with
2
Hj
∆−
γ
ω
( )
( )2
24
441
∆−−++
+++=
HjMHHH
MHHM
sA
sAsxx
γ
ωπ
ππµ
( ) ( )
( )2
222
22
44
4444
1
∆+
∆+
−++
∆+
−++++
+=
HH
MHHH
HMHHHMHHM
sA
sAsAs
γ
ω
γ
ωπ
γ
ωπππ
( )
( )2
222
44
44
∆+
∆+
−++
∆++
−
HH
MHHH
HMHHM
j
sA
sAs
γ
ω
γ
ωπ
γ
ωππ
34
xxxx jµµ ′′−′= (2.2)
( ) ( )
( )2
222
22
44
4444
1
∆+
∆+
−++
∆+
−++++
+=′
HH
MHHH
HMHHHMHHM
sA
sAsAs
xx
γ
ω
γ
ωπ
γ
ωπππ
µ (2.2-a)
( )
( )2
222
44
44
∆+
∆+
−++
∆++
=′′
HH
MHHH
HMHHM
sA
sAs
xx
γ
ω
γ
ωπ
γ
ωππ
µ (2.2-b)
( )2
24
41
∆−−++
+=H
jMHHH
HM
sA
syy
γ
ωπ
πµ
( )
( )2
222
22
44
444
1
∆+
∆+
−++
∆+
−++
+=
HH
MHHH
HMHHHHM
sA
sAs
γ
ω
γ
ωπ
γ
ωππ
( )2
222
44
4
∆+
∆+
−++
∆
−
HH
MHHH
HHM
j
sA
s
γ
ω
γ
ωπ
γ
ωπ
yyyy jµµ ′′−′= (2 .3)
( )
( )2
222
sA
22
sAs
yy
H4
HM4HHH
4
HM4HHHHM4
1
∆
γ
ω+
∆+
γ
ω−π++
∆+
γ
ω−π++π
+=µ′ (2.3-a)
35
( )2
222
sA
s
yy
H4
HM4HHH
HHM4
∆
γ
ω+
∆+
γ
ω−π++
∆γ
ωπ
=µ′′ (2.3-b)
( )2
24
24
∆−−++
∆−
=−=H
jMHHH
HjjM
sA
s
yxxy
γ
ωπ
γ
ωπ
µµ
( )
( )
( )
( )2
222
222
22
22
22
44
2444
44
44
24
∆+
∆+
−++
∆−
∆+
−++
+
∆+
∆+
−++
∆+
+++
∆
=
HH
MHHH
HHMHHHM
j
HH
MHHH
HMHHH
HM
sA
sAs
sA
sAs
γ
ω
γ
ωπ
γ
ωπ
γ
ωπ
γ
ω
γ
ωπ
γ
ωππ
xyxy jµµ ′′−′= (2.4)
( )
( )2
222
22
44
44
24
∆+
∆+
−++
∆+
+++
∆
=′
HH
MHHH
HMHHH
HM
sA
sAs
xy
γ
ω
γ
ωπ
γ
ωππ
µ (2.4-a)
( )
( )2
222
222
44
2444
∆+
∆+
−++
∆−
∆+
−++
−=′′
HH
MHHH
HHMHHHM
sA
sAs
xy
γ
ω
γ
ωπ
γ
ωπ
γ
ωπ
µ (2.4-b)
In summary, for a Y-type ferrite
xxxxxx jµµµ ′′−′=
( ) ( )
( )2
222
22
44
4444
1
∆+
∆+
−++
∆+
−++++
+=′
HH
MHHH
HMHHHMHHM
sA
sAsAs
xx
γ
ω
γ
ωπ
γ
ωπππ
µ
36
( )
( )2
222
44
44
∆+
∆+
−++
∆++
=′′
HH
MHHH
HMHHM
sA
sAs
xx
γ
ω
γ
ωπ
γ
ωππ
µ
yyyyyy jµµµ ′′−′=
( )
( )2
222
22
44
444
1
∆+
∆+
−++
∆+
−++
+=′
HH
MHHH
HMHHHHM
sA
sAs
yy
γ
ω
γ
ωπ
γ
ωππ
µ
( )2
222
44
4
∆+
∆+
−++
∆
=′′
HH
MHHH
HHM
sA
s
yy
γ
ω
γ
ωπ
γ
ωπ
µ
xyxyxy jµµµ ′′−′=
( ) ( )
( )2
222
222
44
44
2
14
∆+
∆+
−++
∆+
−+++
∆
=′
HH
MHHH
HMHHHHM
sA
sAs
xy
γ
ω
γ
ωπ
γ
ωπ
γ
ωπ
µ
( )
( )2
222
22
44
444
∆+
∆+
−++
∆−
−++
−=′′
HH
MHHH
HMHHHM
sA
sAs
xy
γ
ω
γ
ωπ
γ
ωπ
γ
ωπ
µ
Assume OeH A 10000= , GM s 20004 =π , OeH 250=∆ , OeH 100= . The plots of each
component of permeability are shown below.
37
Fig 2.2
Fig 2.3
38
Fig 2.4
From these plots, we deduce that the FMR is at 3.08 GHz, and the operating frequency,
where both xxµ and yyµ are larger than zero, is 17 GHz. The permeability comonents are
322.0=′xxµ , 1=′
yyµ , 34.0=′′xyµ .
2.2 Application of Spectral Domain Method to the Devices on Y-type
Ferrites
In this section, the dispersion characteristics of microstrip lines on magnetically
anisotropic substrates are studied utilizing the Galerkin's method in the spectral domain.
The application of the proposed approach to a magnetically tunable hexagonal Y-type
ferrite(Zn2Y) phase shifter allowed the calculation of phase constants, differential phase
shifts, instantaneous bandwidths, and tuning factors as a function of applied magnetic
field. Numerical results are compared with experimental data as a function of frequency
and magnetic field. The proposed approach is effective in modeling magnetically
39
anisotropic materials which are becoming increasingly important for the engineering of
next generation microwave devices. Such a theoretical treatment of anisotropic magnetic
materials is presently unavailable in commercial numerical simulation tools.
Galerkin's method has been successfully applied in the analysis of microstrip lines on
scalar permittivity substrates by Itoh [1]. This method was later extended to the analysis
of microstrip and slotlines on anisotropic permittivity substrates by Geshiro [2]. At the
same time, isotropic ferrite materials was analyzed by Kitazawa and Itoh using the
spectral domain approach [3]. In this section, Galerkin's method is applied to microstrip
lines on anisotropic hexagonal Y-type ferrite substrates [4]. Resulting dispersion
characteristics are used in the analysis of a phase shifter device, which allows for the
evaluation of key design parameters, such as phase constant and differential phase shift.
For operation at microwave frequencies, low loss ferrites such as yttrium garnets and
lithium spinels, require strong magnetic bias fields that can be realized with a
combination of permanent magnets and current driven coils. The magnitude of the
magnetic bias field necessary to operate a ferrite device at high frequency can be greatly
reduced if an anisotropic material, such as hexagonal Y- or Z- type ferrite, is utilized. The
strong magnetocrystalline anisotropy field in these materials can be used to compensate
for the external magnetic bias field requirement. Recently, a KU band microstrip phase
shifter was demonstrated in which fields as low as 100 Oe were required [5]. Cubic
ferrites, like YIG, require 3000 to 4000 Oe to operate at the same frequency band. As
such, anisotropic magnetic materials are expected to play an important role in the design
40
and development of future microwave systems as they allow greatly reduced component
size, weight, cost, and dc power consumption.
The permeability in magnetic materials assumes a tensor form. For ferrites that possess
a cubic crystal structure, such as garnets or spinels, the precessional motion of the
magnetization vector under an applied magnetic bias field is circular, with the diagonal
components of the permeability tensor being equal. This is not the case in hexagonal
ferrites where the motion is elliptical due to strong magnetocrystalline anisotropy fields.
As such, the diagonal elements of the permeability tensor are no longer equal. The easy
plane is the x-z plane, normal to the c-axis which is along the y axis. The microwave
permeability tensor of a hexagonal Y-type ferrite magnetized in the direction
perpendicular to the crystallographic c-axis is given in the CGS system of units by
[ ]
µ
µµ−
µµ
=µ
ZZ
YYXY
XYXX
00
0
0
, (2.5)
where
( )
( )2
2
A
Asxx
HHH
HHM41
γ
ω−+
+π+=µ ,
( )2
2
A
syy
HHH
HM41
γ
ω−+
π+=µ ,
( )2
2
A
s
xy
HHH
M4
j
γ
ω−+
γ
ωπ
=µ ,
41
1zz =µ .
where 4πMs is the saturation magnetization, HA is the magnetocrystalline anisotropy field,
H is the internal field, ω is the radial frequency, and γ is the electron gyromagnetic ratio.
Equation (2.5) is applicable for a infinite medium of Y-type hexaferrites whereas (2.1) is
applicable for Y-type plate.
Fig. 2.5 shows the structure under consideration as a microstrip line on top of a
longitudinally biased Y-type ferrite substrate with the crystallographic c-axis aligned
perpendicular to the plane and a ground plane on the bottom of the substrate.
Y-type ferrite substrate region
The Ev
field and Hv
field are assumed tje ω dependent, and can be expressed as
zyx EzEyExE)))v
++= and
zyx HzHyHxH)))v
++= .
Fig. 2.5. Cross-section of the microstrip line phase shifter, where the microstrip line is on top of the hexagonal Y-type ferrite
substrate with anisotropic permeability tensor and scalar permittivity. The crystallographic c-axis is along y axis and the
biasing field is along the z axis.
42
From Maxwell equations
EjHvv
ωε=×∇ ,
HjEvtv
⋅µω−=×∇ ,
the following equations can be derived:
xyz EjH
zH
yωε=
∂
∂−
∂
∂, (2.6)-a
yzx EjH
xH
zωε=
∂
∂−
∂
∂, (2.6)-b
zxy EjH
yH
xωε=
∂
∂−
∂
∂, (2.6)-c
( )yxyxxx0yz HHjE
zE
yµ+µωµ−=
∂
∂−
∂
∂, (2.6)-d
( )yyyxxy0zx HHjEx
Ez
µ+µ−ωµ−=∂
∂−
∂
∂, (2.6)-e
zzz0xy HjE
yE
xµωµ−=
∂
∂−
∂
∂. (2.6)-f
Assume the wave is propagating along z direction, the derivative over z can be
substituted with -jβ term. Also, we define a Fourier transform between the spatial
coordinate x and the spectral domain parameter α [1],[2]
( ) ( )∫+∞
∞−
α=α dxey,xfy,f~ xj ,
where ( )y,xf represents any component of either the electric or the magnetic field. The
advantage of this transformation is that the spatial derivative with respect to x is reduced
to multiplication by a factor -jα in the spectral domain, which drastically simplifies the
problem. After the above operation, (2.6) can be derived as
43
xyz E
~jH
~jH
~
yωε=β+
∂
∂, (2.7)-a
yzx E~
jH~
jH~
j ωε=α+β− , (2.7)-b
zxy E
~jH
~
yH~
j ωε=∂
∂−α− , (2.7)-c
( )yxyxxx0yz H~
H~
jE~
jE~
yµ+µωµ−=β+
∂
∂, (2.7)-d
( )yyyxxy0zx H
~H~
jE~
jE~
j µ+µ−ωµ−=α+β− , (2.7)-e
zzz0xy H~
jE~
yE~
j µωµ−=∂
∂−α− . (2.7)-f
From (2.7)-d
( )yxyxxx0yz H~
H~
jE~
jE~
yµ+µωµ−=β+
∂
∂
xx
yxyy
0
z
0x
H~
E~
E~
y
j
H~
µ
µ−ωµ
β−
∂
∂
ωµ=⇒ . (2.8)-a
From equation (2.7)-a
ωε
β+∂
∂
=j
H~
jH~
yE~
yz
x . (2.8)-b
In summary,
xx
yxyy
0
z
0x
H~
E~
E~
y
j
H~
µ
µ−ωµ
β−
∂
∂
ωµ= , (2.8)-a
ωε
β+∂
∂
=j
H~
jH~
yE~
yz
x . (2.8)-b
Substitute equation (2.8) into the other equations in equation (2.7)
44
From (2.7)-b
yzx E~
jH~
jH~
j ωε=α+β−
yz
xx
yxyy
0
z
0 E~
jH~
j
H~
E~
E~
y
j
j ωε=α+µ
µ−ωµ
β−
∂
∂
ωµβ−⇒
0H~
jH~
jE~
jE~
yzxx0yxy0y
2
xxz =µαωµ+µβωµ+µ−∂
∂β⇒ . (2.9)-a
From (2.7)-c
zxy E~
jH~
yH~
j ωε=∂
∂−α−
zxx
2
0ry0xyyz2
2
yxx0 E~
kjH~
yE~
yE~
yjH
~j µε=
∂
∂ωµµ+
∂
∂β+
∂
∂−µαωµ−⇒ . (2.9)-b
From (2.7)-e
( )yyyxxy0zx H~
H~
jE~
jE~
j µ+µ−ωµ−=α+β−
µ+µ
µ−ωµ
β−
∂
∂
ωµµ−ωµ−=α+
ωε
β+∂
∂
β−⇒ yyy
xx
yxyy
0
z
0xy0z
yz
H~
H~
E~
E~
y
j
jE~
jj
H~
jH~
yj
( ) y
2
xx
22
0r0zxx0 H~
kjH~
yβµ−µεωµ+
∂
∂µβωµ−⇒
0E~
kjE~
yjk y
2
0rxyzxyxx
2
0r =εβµ+
∂
∂µ+αµε+ . (2.9)-c
From (2.7)-f
zzz0xy H~
jE~
yE~
j µωµ−=∂
∂−α−
45
0H~
yjH
~k
yE~
yzzz
2
0r2
2
y =∂
∂β−
µε+
∂
∂−αωε⇒ . (2.9)-d
In summary,
0H~
jH~
jE~
jE~
yzxx0yxy0y
2
xxz =µαωµ+µβωµ+µ−∂
∂β (2.10)-a
0E~
yE~
ky
jH~
jy
yzxx
2
0r2
2
yxxxy0 =∂
∂β+
µε+
∂
∂−
αµ−
∂
∂µωµ (2.10)-b
( ) 0E~
kjE~
yjkH
~kjH
~
yy
2
0rxyzxyxx
2
0ry
2
xx
22
0r0zxx0 =εβµ+
∂
∂µ+αµε+βµ−µεωµ+
∂
∂µβωµ−
(2.10)-c
0H~
yjH
~k
yE~
yzzz
2
0r2
2
y =∂
∂β−
µε+
∂
∂−αωε (2.10)-d
From (2.10)-a
0H~
jH~
jE~
jE~
yzxx0yxy0y
2
xxz =µαωµ+µβωµ+µ−∂
∂β
2
xx
zxx0yxy0z
yj
H~
jH~
jE~
yE~
µ
µαωµ+µβωµ+∂
∂β
=⇒ (2.11)-a
Substitute (2.11)-a into (2.10)-c
( ) 0E~
kjE~
yjkH
~kjH
~
yy
2
0rxyzxyxx
2
0ry
2
xx
22
0r0zxx0 =εβµ+
∂
∂µ+αµε+βµ−µεωµ+
∂
∂µβωµ−
( )
0
H~
jH~
jE~
yk
E~
yjkH
~kjH
~
y
2
xx
zxx0yxy0z
2
0rxy
zxyxx
2
0ry
2
xx
22
0r0zxx0
=µ
µαωµ+µβωµ+∂
∂β
εβµ+
∂
∂µ+αµε+βµ−µεωµ+
∂
∂µβωµ−⇒
46
( )
0H~
jkH~
jkE~
yk
E~
yjkH
~kjH
~
y
zxx0
2
0rxyyxy0
2
0rxyz
2
0rxy
2
zxyxx
2
xx
2
0ry
2
xx
22
0r
2
xx0z
2
xxxx0
=µαωµεβµ+µβωµεβµ+∂
∂εµβ+
∂
∂µ+αµµε+βµ−µεµωµ+
∂
∂µµβωµ−⇒
( )( )
( ) z
2
0rxy
22
xxxx
2
xx
2
0rzxx0
2
0rxyz
2
xxxx0
y
2
xy
2
0r
22
xx
22
0r
2
xx0
E~
ykkjH
~jkH
~
y
H~
kkj
∂
∂εµβ+µ+αµµε+µαωµεβµ+
∂
∂µµβωµ−=
µεβ+βµ−µεµωµ−⇒
( )
( ) z
2
0rxy
22
xxxx
2
xx
2
0rzxx0
2
xx
2
0rxy
y
2
yy
2
xx
2
xy
4
0
2
rxx0
E~
ykkjH
~
ykj
H~
kj
∂
∂εµβ+µ+µµαε+µβωµ
∂
∂µ−εαµ=
µµ+µεµωµ−⇒
( )
( )2
yy
2
xx
2
xy
4
0
2
rxx0
z
2
0rxy
22
xxxx
2
xx
2
0rzxx0
2
xx
2
0rxy
ykj
E~
ykkjH
~
ykj
H~
µµ+µεµωµ−
∂
∂εµβ+µ+µµαε+µβωµ
∂
∂µ−εαµ
=⇒
(2.11)-b
Substitute (2.11)-b into (2.11)-a
2
xx
zxx0yxy0z
yj
H~
jH~
jE~
yE~
µ
µαωµ+µβωµ+∂
∂β
=
( )
( )
( ) z0xxxy
22
xx
2
0rxy
2
yy
2
xx
2
xy
4
0
2
rxx
zxyxx
2
xx
2
0r
22
0r
2
xy
2
0r
2
xy
2
xx
2
yy
2
xxxxxx
2
xy
4
0
2
r
y
2
xx
2
yy
2
xx
2
xy
4
0
2
rxx
H~
ykjkj
E~
kjy
kkk
E~
kj
ωµµ
µβ
∂
∂µ−εαµ−µµ+µεαµ+
µµµαβε−
∂
∂ββεµ−εµµ−µµµ+µµε=
µµµ+µεµ⇒
( )
( ) z0xx
2
xxxy
22
xx
2
yyxx
2
0
2
xyr
zxyxx
2
xx
2
0rxx
2
yy
2
xx
y
2
xx
2
yy
2
xx
2
xy
4
0
2
rxx
H~
yjk
E~
kjy
E~
kj
ωµµ
∂
∂µµβ+µαµµ+µε+
µµµαβε−
∂
∂βµµµ=
µµµ+µεµ⇒
47
( )
( ) z0xx
2
xxxy
22
xx
2
xx
2
0r
2
zxyxx
2
xx
2
0rxx
2
yy
2
xx
y
2
xx
2
yy
2
xx
2
xy
4
0
2
rxx
H~
yjk
E~
kjy
E~
kj
ωµµ
∂
∂µµβ+µαβµ−εµ+
µµµαβε−
∂
∂βµµµ=
µµµ+µεµ⇒
(2.11)-c
In summary,
( )
( )2
yy
2
xx
2
xy
4
0
2
rxx0
z
2
0rxy
22
xxxx
2
xx
2
0rzxx0
2
xx
2
0rxy
ykj
E~
ykkjH
~
ykj
H~
µµ+µεµωµ−
∂
∂εµβ+µ+µµαε+µβωµ
∂
∂µ−εαµ
=
(2.11)-b
( )
( ) 2
xx
2
yy
2
xx
2
xy
4
0
2
rxx
z0xx
2
xxxy
22
xx
2
xx
2
0r
2
zxyxx
2
xx
2
0rxx
2
yy
2
xx
y
kj
H~
yjkE
~kj
y
E~
µµµ+µεµ
ωµµ
∂
∂µµβ+µαβµ−εµ+
µµµαβε−
∂
∂βµµµ
=
(2.11)-c
Substitute (2.11)-b and (2.11)-c into (2.10)-b and (2.10)-d. From (2.10)-b
0E~
yE~
ky
jH~
jy
yzxx
2
0r2
2
yxxxy0 =∂
∂β+
µε+
∂
∂−
αµ−
∂
∂µωµ
( )
( )
( )
( ) 0kj
H~
yjkE
~kj
y
y
E~
ky
j
kj
E~
ykkjH
~
ykj
jy
2
xx
2
yy
2
xx
2
xy
4
0
2
rxx
z0xx
2
xxxy
22
xx
2
xx
2
0r
2
zxyxx
2
xx
2
0rxx
2
yy
2
xx
zxx
2
0r2
2
2
yy
2
xx
2
xy
4
0
2
rxx0
z
2
0rxy
22
xxxx
2
xx
2
0rzxx0
2
xx
2
0rxy
xxxy0
=µµµ+µεµ
ωµµ
∂
∂µµβ+µαβµ−εµ+
µµµαβε−
∂
∂βµµµ
∂
∂β+
µε+
∂
∂−
µµ+µεµωµ−
∂
∂εµβ+µ+µµαε+µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µωµ⇒
( )
( )
( ) 0H~
yyjkjE
~
ykj
yj
E~
kky
j
E~
ykkjj
yjH
~
ykjj
yj
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxyxx
2
xx
2
0r2
2
xx
2
yy
2
xx0
z
2
xxxx0
2
yy
2
xx
2
xy
4
0
2
rxx
2
0r2
2
z
2
0rxy
22
xxxx
2
xx
2
0rxxxy
2
xx0zxx0
2
xx
2
0rxyxxxy
2
xx0
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−
∂
∂µµµαβε−
∂
∂βµµµβωµ−
µµωµµµ+µε
µε+
∂
∂−
∂
∂εµβ+µ+µµαε
αµ−
∂
∂µµωµ+µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ⇒
48
( ) ( )
( ) 0H~
yyjkjH
~
ykjj
yj
E~
ykj
yj
E~
kky
jE~
ykkjj
yj
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxx0
2
xx
2
0rxyxxxy
2
xx0
zxyxx
2
xx
2
0r2
2
xx
2
yy
2
xx0
z
2
xxxx0
2
yy
2
xx
2
xy
4
0
2
rxx
2
0r2
2
z
2
0rxy
22
xxxx
2
xx
2
0rxxxy
2
xx0
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ
∂
∂µµµαβε−
∂
∂βµµµβωµ−
µµωµµµ+µε
µε+
∂
∂−
∂
∂εµβ+µ+µµαε
αµ−
∂
∂µµωµ+⇒
( ) ( )
( )
( ) 0H~
yyjkjH
~
ykjj
yj
E~
j
ykj
y
kky
yk
ykj
ykjk
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxx0
2
xx
2
0rxyxxxy
2
xx0
z0
xyxx
2
xx
2
0r2
2
xx
2
yy
2
xx
2
xxxx
2
yy
2
xx
2
xy
4
0
2
rxx
2
0r2
2
2
22
0r
2
xy
22
xx
2
0rxy
22
xxxxxyxx
2
xx
2
0rxx
2
xx
2
0rxx
2
xx
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ
ωµ
∂
∂µµµαβε−
∂
∂βµµµβ−
µµµµ+µε
µε+
∂
∂−
∂
∂εµβ+µ+
∂
∂εµβ+µαµ−
∂
∂µµµαε+µµαεαµ+µ+
⇒
( )
( ) ( )
( ) 0H~
yyjkjH
~
ykjj
yj
E~
j
yk
yyk
ykj
ykj
kkk
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxx0
2
xx
2
0rxyxxxy
2
xx0
z0
2
22
xx
2
yy
2
xx
2
xxxx
2
yy
2
xx
2
xy
4
0
2
r2
2
2
22
0r
2
xx
2
xy
22
xx
xyxx
2
xx
2
0r
222
xxxyxx
2
0r
2
xxxx
2
yy
2
xx
2
xy
4
0
2
rxx
2
0rxx
2
xx
2
xx
2
0rxx
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ+
ωµ
∂
∂βµµµ−µµµµ+µε
∂
∂−
∂
∂εµµβ+µ+
∂
∂µµµεαβ+
∂
∂βµµµαε−
µµµµ+µεµε−µµµαεαµ+
⇒
( )
( ) ( )( )
( ) 0H~
yyjkjH
~
ykjj
yj
E~
j
ykk
kk
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxx0
2
xx
2
0rxyxxxy
2
xx0
z0
2
22
xx
2
yy
2
xx
2
xxxx
2
yy
2
xx
2
xy
4
0
2
r
2
0r
2
xx
2
xy
22
xx
2
xx
2
xx
2
0r
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ+
ωµ
∂
∂βµµµ−µµµµ+µε−εµµβ+µ+
µµεµµ−µε−µα+
⇒
( )
( ) ( )( )
( ) 0H~
yyjkjH
~
ykjj
yj
E~
j
ykkk
kk
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxx0
2
xx
2
0rxyxxxy
2
xx0
z0
2
222
xxxx
2
yy
2
0r
2
xy
2
xx
2
yy
2
xxxx
2
xyxx
4
0
2
r
2
0r
2
xx
2
xy
2
xx
2
xx
2
0r
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ+
ωµ
∂
∂βµµµ−εµ+µµµµ−µµε−εµµ+
µµεµµ−µε−µα+
⇒
( )
( ) 0H~
yyjkjH
~
ykjj
yj
E~
kjy
k
z0xx2
22
xxxy
22
xx
2
xx
2
0r
2
0zxx0
2
xx
2
0rxyxxxy
2
xx0
z0
2
xx
2
xx
2
0r2
22
yy
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=ωµµ
∂
∂µµβ+
∂
∂µαβµ−εµβωµ−µβωµ
∂
∂µ−εαµ
αµ−
∂
∂µµωµ+
ωµµµε
∂
∂µ−µµ−µε−µα+⇒
( )
( ) 0H~
jyy
jky
kjjy
E~
kjy
k
z0xx02
22
xxxy
22
xx
2
xx
2
0r
22
xx
2
0rxyxxxy
2
xx
z0
2
xx
2
xx
2
0r2
22
yy
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=ωµµβωµ
∂
∂µµβ−
∂
∂µαβµ−εµ−
∂
∂µ−εαµ
αµ−
∂
∂µµ+
ωµµµε
∂
∂µ−µµ−µε−µα+⇒
( )
( ) 0H~
yyjk
yjk
yykj
E~
yk
z2
22
xxxy
22
xx
2
xx
2
0r
22
xxxx
2
xx
2
0rxyxx
2
xx
2
2
2
xy
2
xx
2
xxxy
2
0r
2
xxxy
z
2
xxxx0r2
22
yy
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=β
∂
∂µµβ−
∂
∂µαβµ−εµ−
∂
∂µµµα+εµµµα+
∂
∂µµµ−
∂
∂µεµαµ+
µµωεε
∂
∂µ−µµ−µε−µα+⇒
49
( )
( ) 0H~
yyykjk
E~
yk
z2
22
xxxy
2
2
2
xy
4
xx
2
0r
2
xxxxyyxx
2
0rxyxx
2
xx
2
z
2
xxxx0r2
22
yy
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=β
∂
∂µµβ−
∂
∂µµ−
∂
∂εµαµµ−µ+εµµµα++
µµωεε
∂
∂µ−µµ−µε−µα+⇒
( )
( ) 0H~
yyj
E~
yk
z02
2
xyyyxxxy
2
z2
22
yy
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
=βωµ
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−µµ−µε−µα+⇒
. (2.12)-a
From (2.10)-d
( )
( )
( )
( ) 0kj
E~
ykkjH
~
ykj
yj
H~
ky
kj
H~
yjkE
~kj
y
2
yy
2
xx
2
xy
4
0
2
rxx0
z
2
0rxy
22
xxxx
2
xx
2
0rzxx0
2
xx
2
0rxy
zzz
2
0r2
2
2
xx
2
yy
2
xx
2
xy
4
0
2
rxx
z0xx
2
xxxy
22
xx
2
xx
2
0r
2
zxyxx
2
xx
2
0rxx
2
yy
2
xx
=µµ+µεµωµ−
∂
∂εµβ+µ+µµαε+µβωµ
∂
∂µ−εαµ
∂
∂β−
µε+
∂
∂−
µµµ+µεµ
ωµµ
∂
∂µµβ+µαβµ−εµ+
µµµαβε−
∂
∂βµµµ
αωε
( )
( )
( ) 0E~
yk
ykjH
~
yykj
H~
kky
H~
yjkkjE
~kj
ykj
z
2
xx2
22
0rxy
22
xxxx
2
xx
2
0rzxx0
2
xx2
22
xx
2
0rxy
z
2
xxxx0
2
yy
2
xx
2
xy
4
0
2
rzz
2
0r2
2
z
2
xxxy
22
xx
2
xx
2
0r
2
0xx
2
0rzxyxx
2
xx
2
0rxx
2
yy
2
xx
2
0r
=µβ
∂
∂εµβ+µ+
∂
∂µµαε+µωµµββ
∂
∂µ−
∂
∂εαµ+
µµωµµµ+µε
µε+
∂
∂−
∂
∂µµβ+µαβµ−εµωµµαε−
µµµαβε−
∂
∂βµµµαε−⇒
( )
( ) ( )
0H~
yykj
H~
kky
H~
yjkkj
E~
yk
ykjE
~kj
ykj
zxx0
2
xx2
22
xx
2
0rxy
z
2
xxxx0
2
yy
2
xx
2
xy
4
0
2
rzz
2
0r2
2
z
2
xxxy
22
xx
2
xx
2
0r
2
0xx
2
0r
z
2
xx2
22
0rxy
22
xxxx
2
xx
2
0rzxyxx
2
xx
2
0rxx
2
yy
2
xx
2
0r
=µωµµββ
∂
∂µ−
∂
∂εαµ+
µµωµµµ+µε
µε+
∂
∂−
∂
∂µµβ+µαβµ−εµωµµαε−
µβ
∂
∂εµβ+µ+
∂
∂µµαε+
µµµαβε−
∂
∂βµµµαε−⇒
( )
( ) ( )
0H~
yykj
H~
kky
H~
yjkkj
E~
kky
jy
k
zxx0
2
xx2
22
xx
2
0rxy
z
2
xxxx0
2
yy
2
xx
2
xy
4
0
2
rzz
2
0r2
2
z
2
xxxy
22
xx
2
xx
2
0r
2
0xx
2
0r
z
2
xxxx
2
0rxy
2
0r
22
yy
2
xx2
2
xy
2
0r
=µωµµββ
∂
∂µ−
∂
∂εαµ+
µµωµµµ+µε
µε+
∂
∂−
∂
∂µµβ+µαβµ−εµωµµαε−
βµµε
µεα−
∂
∂αµ−µ+
∂
∂µε⇒
50
( )
( ) ( )
0
H~
kkyy
jkkj
yykj
E~
kky
jy
k
zxx0
2
xx
2
yy
2
xx
2
xy
4
0
2
rzz
2
0r2
22
xxxy
22
xx
2
xx
2
0r
22
0r
2
xx
2
2
22
xx
2
0rxy
z
2
xxxx
2
0rxy
2
0r
22
yy
2
xx2
2
xy
2
0r
=
µωµ
µµµ+µε
µε+
∂
∂−
∂
∂µµβ+µαβµ−εµαε−
µβ
∂
∂µ−
∂
∂εαµ
+
βµµε
µεα−
∂
∂αµ−µ+
∂
∂µε⇒
( )
( ) ( )
( )
0
H~
yyk
ykj
ykj
kkjkjk
E~
kky
jy
k
zxx0
2
22
xx
22
xx2
22
xx
2
yy
2
xx
2
xy
4
0
2
r
2
0r
2
xxxy
22
xx
22
0rxy
zz
2
0r
2
xx
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
0r
2
xx
2
0r
2
z
2
xxxx
2
0rxy
2
0r
22
yy
2
xx2
2
xy
2
0r
=
µωµ
∂
∂µβµ−
∂
∂µµµ+µε−
∂
∂αεµµβ−
∂
∂µβεαµ+
µεµµµ+µε−µααεβµ−εµ−
+
βµµε
µεα−
∂
∂αµ−µ+
∂
∂µε⇒
( )
( ) ( )
( )
0
H~
ykk
kkjkjk
E~
kky
jy
k
zxx0
2
22
xx
2
0r
2
yy
2
0r
2
zz
2
0r
2
xx
2
yy
2
xx
2
xy
4
0
2
r
2
xx
2
0r
2
xx
2
0r
2
z
2
xxxx
2
0rxy
2
0r
22
yy
2
xx2
2
xy
2
0r
=
µωµ
∂
∂µεβµ−εµ−
µεµµµ+µε−µααεβµ−εµ−
+
βµµε
µεα−
∂
∂αµ−µ+
∂
∂µε⇒
( )
( ) ( )( ) ( )
0
H~
ykkk
E~
yj
y
z2
22
yy
2
0r
2
zz
2
yy
2
xx
2
xy
4
0
2
r
22
xx
2
0r
2
z0rxy
2
yyxx2
2
xy
=
∂
∂βµ−εµ−µµµ+µε−αβµ−εµ+
βωεε
µα−
∂
∂αµ−µ+
∂
∂µ⇒
, (2.12-b)
where 2
xx
2
xx
2
0rk µ=β−µε , 2
yy
2
yy
2
0rk µ=β−µε , yyxx
2
xy
2 µµ+µ=µ .
In summary,
( )
( ) 0H~
yyj
E~
y
z02
2
xyyyxxxy
2
z2
22
yy
22
xx
2
=βωµ
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−µ−µα+
(2.12)-a
51
( )
( )
0
H~
y
E~
yyj
z2
22
yyzz
222
xx
z0r2
2
xyyyxxxy
2
=
∂
∂µ−µµ−αµ+
βεωε
∂
∂µ+
∂
∂αµ−µ+µα−
(2.12)-b
where
22
0
2 βµεµ −= xxrxx k
22
0
2 βµεµ −= yyryy k
yyxxxy µµµµ +=22
22
0
22 βµεµµxxrxx
k −=
22
0
22 βµεµµ yyryy k −=
2224
0
22
yyxxxyr k µµµεµ +=
Equation (2.12) is the wave equations for longitudinal field components ( )y,E~
z α and
( )y,H~
z α . Next, the other field components can be expressed in terms of z
E~
and z
H~
.
Rewrite equation (2.11)-b and (2.11)-c
2
0
222
0
2 ~~
~
µ
ωµµβαµβµαεµ
j
Hy
jEkjy
E
zxyxxzxyryy
y
∂
∂++
−
∂
∂
= (2.13)-a
2
z0rrxy
2
0
2
xxz
2
xx
2
0rxy
yj
E~
ykjH
~
ykj
H~
µ−
εωε
∂
∂εµ+µα+β
∂
∂µ−εαµ
= (2.13)-b
Substitute (2.13)-a and and (2.13)-b into (2.8)-a
xx
yxyy
0
z
0x
H~
E~
E~
y
j
H~
µ
µ−ωµ
β−
∂
∂
ωµ=
52
yxyy
0
z
0
xxx H~
E~
E~
y
jH~
µ−ωµ
β−
∂
∂
ωµ=µ⇒
2
z0rrxy
2
0
2
xxz
2
xx
2
0rxy
xy
2
z0xy
22
xxzxy
2
0r
2
yy
0
z
0
xxx
j
E~
ykjH
~
ykj
j
H~
yjE
~kj
yE~
y
jH~
µ−
εωε
∂
∂εµ+µα+β
∂
∂µ−εαµ
µ−
µ
ωµ
∂
∂µβ+αµ+β
µαε−
∂
∂µ
ωµ
β−
∂
∂
ωµ=µ⇒
z0rxyrxy
2
0
2
xxzxy
2
xx
2
0rxy
z0
0
xy
22
xxz
0
xy
2
0r
2
yyz
0
2
xxx
2
E~
ykjH
~
ykj
H~
yjE
~kj
yE~
yH~
j
εεωµ
∂
∂εµ+µα+βµ
∂
∂µ−εαµ+
ωµωµ
β
∂
∂µβ+αµ−β
ωµ
β
µαε−
∂
∂µ−
∂
∂
ωµ
µ−=µµ⇒
z0
0
xy
22
xxzxy
2
xx
2
0rxy
z
0
22
yy
0
2
rxy
2
00rxy
2
xx0rxyxy
2
0r
0
2
xxx
2
H~
yjH
~
ykj
E~
ykjkjH
~j
ωµωµ
β
∂
∂µβ+αµ−βµ
∂
∂µ−εαµ+
∂
∂
ωµ
βµ−
ωµ
µ−εµεεωµ+
µαεεωµ+µαε
ωµ
β+=µµ⇒
( ) zxyxx
2
0r
2
xx
2
0r
2
xy
z
0
22
yy
0
2
rxy
2
00rxy
2
xx0rxyxy
2
0r
0
2
xxx
2
H~
ykjk
E~
ykjkjH
~j
β
∂
∂µµε−αµ−εµ+
∂
∂
ωµ
βµ−
ωµ
µ−εµεεωµ+
µαεεωµ+µαε
ωµ
β+=µµ⇒
( ) zxyxx
2
0r
2
xx
2
0r
2
xy
z
0
22
yy
0
2
rxy
2
00rxy
2
00
2
rxyxxxxx
2
H~
ykjk
E~
ykkjH
~j
β
∂
∂µµε−αµ−εµ+
∂
∂
ωµ
βµ−
ωµ
µ−εµεεωµ+εωεµαµ=µµ⇒
zxyxx
2
0rxx
2
yyz0rxx
2
yy
2
0rxyxxx
2H~
ykjE
~
ykjH
~j β
∂
∂µµε−µµα−+εεωµ
∂
∂µ−εαµ=µµ⇒
zxy
2
0r
2
yyz0r
2
yy
2
0rxyx
2H~
ykjE
~
ykjH
~j β
∂
∂µε+µα−εωε
∂
∂µ−εαµ=µ⇒
2
zxy
2
0r
2
yyz0r
2
yy
2
0rxy
xj
H~
ykjE
~
ykj
H~
µ
β
∂
∂µε+µα−εωε
∂
∂µ−εαµ
=⇒ . (2.13)-c
Substitute (2.13)-a and (2.13)-b into (2.8)-b
53
ωε
β+∂
∂
=j
H~
jH~
yE~
yz
x
2
z0rrxy
2
0
2
xxz
2
xx
2
0rxy
zxj
E~
ykjH
~
ykj
jH~
yE~
jµ−
εωε
∂
∂εµ+µα+β
∂
∂µ−εαµ
β+∂
∂=ωε⇒
z0rrxy
2
0
2
xxz
2
xx
2
0rxyz
2
x
2E~
jy
kjH~
jy
kjH~
yjE
~jj εβωε
∂
∂εµ+µα−ββ
∂
∂µ−εαµ−
∂
∂µ=ωεµ⇒
z0rrxy
2
0
2
xxz
2
0r
2
yy
2
0rxy
2
x
2 E~
jy
kjH~
yjkkE
~jj εβωε
∂
∂εµ+µα−
∂
∂εµ+εαµβ=ωεµ⇒
z0rrxy
2
0
2
xxz
2
0r
2
yyxy
2
x0r
2 E~
jy
kjH~
ky
jE~
jj εβωε
∂
∂εµ+µα−ε
∂
∂µ+αµβ=εωεµ⇒
zrxy
2
0
2
xxz0
2
yyxy
2
x
2 E~
jy
kjH~
yjE
~jj β
∂
∂εµ+µα−ωµ
∂
∂µ+αµβ=µ⇒
2
z0
2
yyxy
2
zrxy
2
0
2
xx
xj
H~
jy
jE~
ykj
E~
µ−
ωµ
∂
∂µ+αµβ+β
∂
∂εµ+µα+
=⇒ (2.13)-d
In summary,
( ) ( ) 0H~
yyjE
~
yz02
2
xyyyxxxy
2
z2
22
yy
22
xx
2 =βωµ
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−µ−µα+ , (2.11)-a
( ) ( ) 0H~
yE~
yyj z2
22
yyzz
222
xxz0r2
2
xyyyxxxy
2 =
∂
∂µ−µµ−αµ+βεωε
∂
∂µ+
∂
∂αµ−µ+µα− , (2.11)-b
2
z0xy
22
xxzxy
2
0r
2
yy
yj
H~
yjE
~kj
yE~
µ
ωµ
∂
∂µβ+αµ+β
µαε−
∂
∂µ
= , (2.12)-a
2
z0rrxy
2
0
2
xxz
2
xx
2
0rxy
yj
E~
ykjH
~
ykj
H~
µ−
εωε
∂
∂εµ+µα+β
∂
∂µ−εαµ
= , (2.12)-b
54
2
zxy
2
0r
2
yyz0r
2
yy
2
0rxy
xj
H~
ykjE
~
ykj
H~
µ
β
∂
∂µε+µα−εωε
∂
∂µ−εαµ
= , (2.12)-c
2
z0
2
yyxy
2
zrxy
2
0
2
xx
xj
H~
jy
jE~
ykj
E~
µ−
ωµ
∂
∂µ+αµβ+β
∂
∂εµ+µα+
= . (2.12)-d
Next, (2.11) should be sovled. Assume both ( )y,E~
z α and ( )y,H~
z α have an yeγ
dependence in the transverse direction, so that
zzEE
yγ=
∂
∂,
zz EEy
2
2
2
γ=∂
∂,
zz HHy
γ=∂
∂,
zz HHy
2
2
2
γ=∂
∂.
Substitute above equations into (2.11), we can obtain
( )( ) ( )( ) 0H~
jE~
z0
2
xyyyxxxy
2
z
22
yy
22
xx
2 =βωµγµ−αγµ−µ+µα+γµ−µ−µα+ (2.13)-a
( )( ) ( )( ) 0H~
E~
j z
22
yyzz
222
xxz0r
2
xyyyxxxy
2 =γµ−µµ−αµ+βεωεγµ+αγµ−µ+µα− (2.13)-b
The determinant of the above homogeneous equation is set to be zero in order to get a
non-trivial solution.
( )( ) ( )( )( )( ) ( )( ) 0
j
j22
yyzz
222
xx0r
2
xyyyxxxy
2
0
2
xyyyxxxy
222
yy
22
xx
2
=γµ−µµ−αµβεωεγµ+αγµ−µ+µα−
βωµγµ−αγµ−µ+µαγµ−µ−µα+
( )( ) ( )( )( )( ) ( )( ) 0jj 0r
2
xyyyxxxy
2
0
2
xyyyxxxy
2
22
yyzz
222
xx
22
yy
22
xx
2
=βεωεγµ+αγµ−µ+µα−βωµγµ−αγµ−µ+µα−
γµ−µµ−αµγµ−µ−µα+⇒
55
( ) ( )( ) ( )( )( )( )( )
( ) ( )( )( )( )
0k
j
jj
j
22
0r
2
xyyyxxxy
22
xy
2
xyyyxxxy
2
yyxx
2
xyyyxxxy
2
xy
2
22
yyzz
222
xx
22
yy
22
yyzz
222
xx
22
xx
2
=βε
γµ+αγµ−µ+µα−γµ−
γµ+αγµ−µ+µα−αγµ−µ+
γµ+αγµ−µ+µα−µα
−
γµ−µµ−αµγµ−γµ−µµ−αµµ−µα+⇒
( )( ) ( ) ( )( )
( ) ( )( )( )
( ) ( )0k
jj
jj
22
0r
42
xy
3
yyxxxy
3
yyxxxy
22
xy
222
yyxxyyxx
222
xy
3
yyxxxy
3
xyyyxx
2
xy
4
42
yy
2
yy
22
yyzz
222
xx
22
xx
22
yy
22
xx
2
zz
222
xx
=βε
γµ−
αγµ−µµ−αγµ−µµ+
γµα+γαµ−µµ−µ−γαµ+
γαµ−µµ−γαµµ−µ+
µα−
−
γµµ+γµµµ−αµ+µ−µαµ−µ−µαµµ−αµ⇒
( )( ) ( ) ( )( )( )( )( ) 0k2
22
0r
42
xy
222
yy
2
xx
22
xy
4
42
yy
2
yy
22
yyzz
222
xx
22
xx
22
yy
22
xx
2
zz
222
xx
=βεγµ−γαµ+µ−µ+µα−−
γµµ+γµµµ−αµ+µ−µαµ−µ−µαµµ−αµ⇒
( )( ) ( ) ( )( )( )( ) 0kk2k
42
xy
22
0r
2222
0r
2
yy
2
xx
22
xy
422
0r
42
yy
2
yy
22
yyzz
222
xx
22
xx
22
yy
22
xx
2
zz
222
xx
=γµβε+γαβεµ+µ−µ−µαβε+
γµµ+γµµµ−αµ+µ−µαµ−µ−µαµµ−αµ⇒
( )( )( ) ( )( ) ( )( )
0k
k2
k
42
xy
22
0r
42
yy
2
yy
2222
0r
2
yy
2
xx
222
yyzz
222
xx
22
xx
22
yy
2
xy
422
0r
22
xx
2
zz
222
xx
=γµβε+γµµ+
γβαεµ+µ−µ−γµµµ−αµ+µ−µαµ−
µαβε+µ−µαµµ−αµ⇒
( )( ) 2
xy
422
0r
22
xx
2
zz
222
xx k µαβε+µ−µαµµ−αµ⇒
( ) ( ) ( )( )( ) 2222
0r
2
yy
2
xx
22
yyzz
222
xx
2
yy
22
xx
2 k2 γβαεµ+µ−µ+µµµ−αµ+µµ−µα−
( ) 0k 42
xy
22
0r
2
yy
2
yy =γµβε+µµ+ . (2.14)
This equation is solvable, and we assume solutions are of the form 1γ , 1γ− , 2γ , 2γ− .
zE~
and zH~
can be expressed as
y
4
y
3
y
2
y
1z2211 eAeAeAeAE
~ γ−γγ−γ +++= , (2.15)-a
y
4
y
3
y
2
y
1z2211 eBeBeBeBH
~ γ−γγ−γ +++= . (2.15)-b
Substitute (2.15) into (2.11)-a
56
( ) ( ) 0H~
yyjE
~
yz02
2
xyyyxxxy
2
z2
22
yy
22
xx
2 =βωµ
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−µ−µα
( ) ( )
( ) ( ) 0eBeBeBeByy
j
eAeAeAeAy
y
4
y
3
y
2
y
102
2
xyyyxxxy
2
y
4
y
3
y
2
y
12
22
yy
22
xx
2
2211
2211
=+++βωµ
∂
∂µ−
∂
∂αµ−µ+µα+
+++
∂
∂µ−µ−µα⇒
γ−γγ−γ
γ−γγ−γ
( )
( )
( )
( )
( )
( )
( )
( )
0
eByy
j
eByy
j
eByy
j
eByy
j
eAy
eAy
eAy
eAy
y
42
2
xyyyxxxy
2
y
32
2
xyyyxxxy
2
y
22
2
xyyyxxxy
2
y
12
2
xyyyxxxy
2
0
y
42
22
yy
22
xx
2
y
32
22
yy
22
xx
2
y
22
22
yy
22
xx
2
y
12
22
yy
22
xx
2
2
2
1
1
2
2
1
1
=
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−
∂
∂αµ−µ+µα
βωµ+
∂
∂µ−µ−µα+
∂
∂µ−µ−µα+
∂
∂µ−µ−µα+
∂
∂µ−µ−µα
⇒
γ−
γ
γ−
γ
γ−
γ
γ−
γ
57
( )
( )
( )
( )
( )
( )
( )
( )
0
Bey
ey
je
Bey
ey
je
Bey
ey
je
Bey
ey
je
Aey
e
Aey
e
Aey
e
Aey
e
4
y
2
2
xy
y
yyxx
y
xy
2
3
y
2
2
xy
y
yyxx
y
xy
2
2
y
2
2
xy
y
yyxx
y
xy
2
1
y
2
2
xy
y
yyxx
y
xy
2
0
4
y
2
22
yy
y22
xx
2
3
y
2
22
yy
y22
xx
2
2
y
2
22
yy
y22
xx
2
1
y
2
22
yy
y22
xx
2
222
222
111
111
22
22
11
11
=
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−
∂
∂αµ−µ+µα+
∂
∂µ−
∂
∂αµ−µ+µα
βωµ+
∂
∂µ−µ−µα+
∂
∂µ−µ−µα+
∂
∂µ−µ−µα+
∂
∂µ−µ−µα
⇒
γ−γ−γ−
γγγ
γ−γ−γ−
γγγ
γ−γ−
γγ
γ−γ−
γγ
( )( )( )( )( )( )( )( )
( )( )( )( )( )( )( )( )
0
eBj
eBj
eBj
eBj
eA
eA
eA
eA
y
4
2
2xy2yyxxxy
2
y
3
2
2xy2yyxxxy
2
y
2
2
1xy1yyxxxy
2
y
1
2
1xy1yyxxxy
2
0
y
4
2
2
2
yy
22
xx
2
y
3
2
2
2
yy
22
xx
2
y
2
2
1
2
yy
22
xx
2
y
1
2
1
2
yy
22
xx
2
2
2
1
1
2
2
1
1
=
γµ−αγµ−µ−µα+
γµ−αγµ−µ+µα+
γµ−αγµ−µ−µα+
γµ−αγµ−µ+µα
βωµ+
γµ−µ−µα+
γµ−µ−µα+
γµ−µ−µα+
γµ−µ−µα
⇒
γ−
γ
γ−
γ
γ−
γ
γ−
γ
( )( )( )( )( )( )( )( )
( )( )( )( )( )( )( )( )
0
eBj
eBj
eBj
eBj
eA
eA
eA
eA
y
40
2
2xy2yyxxxy
2
y
30
2
2xy2yyxxxy
2
y
20
2
1xy1yyxxxy
2
y
10
2
1xy1yyxxxy
2
y
4
2
2
2
yy
22
xx
2
y
3
2
2
2
yy
22
xx
2
y
2
2
1
2
yy
22
xx
2
y
1
2
1
2
yy
22
xx
2
2
2
1
1
2
2
1
1
=
βωµγµ−αγµ−µ−µα+
βωµγµ−αγµ−µ+µα+
βωµγµ−αγµ−µ−µα+
βωµγµ−αγµ−µ+µα+
γµ−µ−µα+
γµ−µ−µα+
γµ−µ−µα+
γµ−µ−µα⇒
γ−
γ
γ−
γ
γ−
γ
γ−
γ
58
( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )0
eBjeA
eBjeA
eBjeA
eBjeA
y
40
2
2xy2yyxxxy
2y
4
2
2
2
yy
22
xx
2
y
30
2
2xy2yyxxxy
2y
3
2
2
2
yy
22
xx
2
y
20
2
1xy1yyxxxy
2y
2
2
1
2
yy
22
xx
2
y
10
2
1xy1yyxxxy
2y
1
2
1
2
yy
22
xx
2
22
22
11
11
=
βωµγµ−αγµ−µ−µα+γµ−µ−µα+
βωµγµ−αγµ−µ+µα+γµ−µ−µα+
βωµγµ−αγµ−µ−µα+γµ−µ−µα+
βωµγµ−αγµ−µ+µα+γµ−µ−µα⇒
γ−γ−
γγ
γ−γ−
γγ
⇒
( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( ) 0BjA
0BjA
0BjA
0BjA
40
2
2xy2yyxxxy
2
4
2
2
2
yy
22
xx
2
30
2
2xy2yyxxxy
2
3
2
2
2
yy
22
xx
2
20
2
1xy1yyxxxy
2
2
2
1
2
yy
22
xx
2
10
2
1xy1yyxxxy
2
1
2
1
2
yy
22
xx
2
=βωµγµ−αγµ−µ−µα+γµ−µ−µα
=βωµγµ−αγµ−µ+µα+γµ−µ−µα
=βωµγµ−αγµ−µ−µα+γµ−µ−µα
=βωµγµ−αγµ−µ+µα+γµ−µ−µα
⇒
( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( ) 42
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
4
32
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
3
2
0
2
1xy1yyxxxy
2
2
1
2
yy
22
xx
2
2
1
0
2
1xy1yyxxxy
2
2
1
2
yy
22
xx
2
1
Bj
A
Bj
A
Aj
B
Aj
B
γµ−µ−µα
βωµγµ−αγµ−µ−µα−=
γµ−µ−µα
βωµγµ−αγµ−µ+µα−=
βωµγµ−αγµ−µ−µα
γµ−µ−µα−=
βωµγµ−αγµ−µ+µα
γµ−µ−µα−=
. (2.16)-a
Substitute (2.15) into (2.11)-b
( ) ( ) 0H~
yE~
yyj z2
22
yyzz
222
xxz0r2
2
xyyyxxxy
2 =
∂
∂µ−µµ−αµ+βεωε
∂
∂µ+
∂
∂αµ−µ+µα−
( ) ( )
( ) ( ) 0eBeBeBeBy
eAeAeAeAyy
j
y
4
y
3
y
2
y
12
22
yyzz
222
xx
y
4
y
3
y
2
y
10r2
2
xyyyxxxy
2
2211
2211
=+++
∂
∂µ−µµ−αµ+
+++βεωε
∂
∂µ+
∂
∂αµ−µ+µα−⇒
γ−γγ−γ
γ−γγ−γ
59
( )
( )
( )
( )
( )
( )
( )
( )
0
eBy
eBy
eBy
eBy
eAyy
j
eAyy
j
eAyy
j
eAyy
j
y
42
22
yyzz
222
xx
y
32
22
yyzz
222
xx
y
22
22
yyzz
222
xx
y
12
22
yyzz
222
xx
y
42
2
xyyyxxxy
2
y
32
2
xyyyxxxy
2
y
22
2
xyyyxxxy
2
y
12
2
xyyyxxxy
2
0r
2
2
1
1
2
2
1
1
=
∂
∂µ−µµ−αµ+
∂
∂µ−µµ−αµ+
∂
∂µ−µµ−αµ+
∂
∂µ−µµ−αµ
+
∂
∂µ+
∂
∂αµ−µ+µα−+
∂
∂µ+
∂
∂αµ−µ+µα−+
∂
∂µ+
∂
∂αµ−µ+µα−+
∂
∂µ+
∂
∂αµ−µ+µα−
βεωε⇒
γ−
γ
γ−
γ
γ−
γ
γ−
γ
( )( )( )( )( )( )( )( )
( )( )( )( )( )( )( )( )
0
eB
eB
eB
eB
eAj
eAj
eAj
eAj
y
4
2
2
2
yyzz
222
xx
y
3
2
2
2
yyzz
222
xx
y
2
2
1
2
yyzz
222
xx
y
1
2
1
2
yyzz
222
xx
y
4
2
2xy2yyxxxy
2
y
3
2
2xy2yyxxxy
2
y
2
2
1xy1yyxxxy
2
y
1
2
1xy1yyxxxy
2
0r
2
2
1
1
2
2
1
1
=
γµ−µµ−αµ+
γµ−µµ−αµ+
γµ−µµ−αµ+
γµ−µµ−αµ
+
γµ+αγµ−µ−µα−+
γµ+αγµ−µ+µα−+
γµ+αγµ−µ−µα−+
γµ+αγµ−µ+µα−
βεωε⇒
γ−
γ
γ−
γ
γ−
γ
γ−
γ
( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )
0
eBeAj
eBeAj
eBeAj
eBeAj
y
4
2
2
2
yyzz
222
xx
y
40r
2
2xy2yyxxxy
2
y
3
2
2
2
yyzz
222
xx
y
30r
2
2xy2yyxxxy
2
y
2
2
1
2
yyzz
222
xx
y
20r
2
1xy1yyxxxy
2
y
1
2
1
2
yyzz
222
xx
y
10r
2
1xy1yyxxxy
2
22
22
11
11
=
γµ−µµ−αµ+βεωεγµ+αγµ−µ−µα−+
γµ−µµ−αµ+βεωεγµ+αγµ−µ+µα−+
γµ−µµ−αµ+βεωεγµ+αγµ−µ−µα−+
γµ−µµ−αµ+βεωεγµ+αγµ−µ+µα−⇒
γ−γ−
γγ
γ−γ−
γγ
⇒
( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( ) 0BAj
0BAj
0BAj
0BAj
4
2
2
2
yyzz
222
xx40r
2
2xy2yyxxxy
2
3
2
2
2
yyzz
222
xx30r
2
2xy2yyxxxy
2
2
2
1
2
yyzz
222
xx20r
2
1xy1yyxxxy
2
1
2
1
2
yyzz
222
xx10r
2
1xy1yyxxxy
2
=γµ−µµ−αµ+βεωεγµ+αγµ−µ−µα−
=γµ−µµ−αµ+βεωεγµ+αγµ−µ+µα−
=γµ−µµ−αµ+βεωεγµ+αγµ−µ−µα−
=γµ−µµ−αµ+βεωεγµ+αγµ−µ+µα−
60
⇒
( )( )( )( )
( )( )( )( )( )( )
( )( )( )( )
( )( ) 4
0r
2
2xy2yyxxxy
2
2
2
2
yyzz
222
xx
4
3
0r
2
2xy2yyxxxy
2
2
2
2
yyzz
222
xx
3
22
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
2
12
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
1
Bj
A
Bj
A
Aj
B
Aj
B
βεωεγµ+αγµ−µ−µα−
γµ−µµ−αµ−=
βεωεγµ+αγµ−µ+µα−
γµ−µµ−αµ−=
γµ−µµ−αµ
βεωεγµ+αγµ−µ−µα−−=
γµ−µµ−αµ
βεωεγµ+αγµ−µ+µα−−=
. (2.16)-b
In summary,
( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )( ) 42
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
4
32
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
3
2
0
2
1xy1yyxxxy
2
2
1
2
yy
22
xx
2
2
1
0
2
1xy1yyxxxy
2
2
1
2
yy
22
xx
2
1
Bj
A
Bj
A
Aj
B
Aj
B
γµ−µ−µα
βωµγµ−αγµ−µ−µα−=
γµ−µ−µα
βωµγµ−αγµ−µ+µα−=
βωµγµ−αγµ−µ−µα
γµ−µ−µα−=
βωµγµ−αγµ−µ+µα
γµ−µ−µα−=
(2.16)-a
( )( )( )( )
( )( )( )( )( )( )
( )( )( )( )
( )( ) 4
0r
2
2xy2yyxxxy
2
2
2
2
yyzz
222
xx
4
3
0r
2
2xy2yyxxxy
2
2
2
2
yyzz
222
xx
3
22
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
2
12
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
1
Bj
A
Bj
A
Aj
B
Aj
B
βεωεγµ+αγµ−µ−µα−
γµ−µµ−αµ−=
βεωεγµ+αγµ−µ+µα−
γµ−µµ−αµ−=
γµ−µµ−αµ
βεωεγµ+αγµ−µ−µα−−=
γµ−µµ−αµ
βεωεγµ+αγµ−µ+µα−−=
(2.16)-b
However, these two sets of equations are not independent, so we choose
61
( )( )( )( )( )( )( )( )
( )( )( )( )
( )( )( )( ) 22
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
2
12
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
1
42
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
4
32
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
3
Aj
B
Aj
B
Bj
A
Bj
A
γµ−µµ−αµ
βεωεγµ+αγµ−µ−µα−−=
γµ−µµ−αµ
βεωεγµ+αγµ−µ+µα−−=
γµ−µ−µα
βωµγµ−αγµ−µ−µα−=
γµ−µ−µα
βωµγµ−αγµ−µ+µα−=
to relate these coefficients. Substitute above equations into (2.15), we can obtain the
expression for zE~
and zH~
y
42
y
31
y
2
y
1z2211 eBZeBZeAeAE
~ γ−γγ−γ −−+= , (2.17)-a
y
4
y
3
y
22
y
11z2211 eBeBeAYeAYH
~ γ−γγ−γ ++−−= , (2.17)-b
where,
( )( )( )( )2
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
1
jZ
γµ−µ−µα
βωµγµ−αγµ−µ+µα= ,
( )( )( )( )2
2
2
yy
22
xx
2
0
2
2xy2yyxxxy
2
2
jZ
γµ−µ−µα
βωµγµ−αγµ−µ−µα= ,
( )( )( )( )2
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
1
jY
γµ−µµ−αµ
βεωεγµ+αγµ−µ+µα−= ,
( )( )( )( )2
1
2
yyzz
222
xx
0r
2
1xy1yyxxxy
2
2
jY
γµ−µµ−αµ
βεωεγµ+αγµ−µ−µα−= .
Substitute (2.17)-a and (2.17)-b into (2.12)-c
2
zxy
2
0r
2
yyz0r
2
yy
2
0rxy
xj
H~
ykjE
~
ykj
H~
µ
β
∂
∂µε+µα−εωε
∂
∂µ−εαµ
=
62
( )
( )2
y
4
y
3
y
22
y
11xy
2
0r
2
yy
y
42
y
31
y
2
y
10r
2
yy
2
0rxy
j
eBeBeAYeAYy
kj
eBZeBZeAeAy
kj
2211
2211
µ
++−−β
∂
∂µε+µα−
−−+εωε
∂
∂µ−εαµ
=
γ−γγ−γ
γ−γγ−γ
2
y
4xy
2
0r
2
yy
y
3xy
2
0r
2
yy
y
22xy
2
0r
2
yy
y
11xy
2
0r
2
yy
y
42
2
yy
2
0rxy
y
31
2
yy
2
0rxy
y
2
2
yy
2
0rxy
y
1
2
yy
2
0rxy
0r
j
eBy
kj
eBy
kj
eAYy
kj
eAYy
kj
eBZy
kj
eBZy
kj
eAy
kj
eAy
kj
2
2
1
1
2
2
1
1
µ
∂
∂µε+µα+
∂
∂µε+µα+
∂
∂µε+µα−
∂
∂µε+µα−
β−
∂
∂µ−εαµ−
∂
∂µ−εαµ−
∂
∂µ−εαµ+
∂
∂µ−εαµ
εωε
=
γ−
γ
γ−
γ
γ−
γ
γ−
γ
( ) ( )( )( ) ( )( )( ) ( )( )( ) ( )( )
2
y
42xy
2
0r
2
yy20r2
2
yy
2
0rxy
y
32xy
2
0r
2
yy10r2
2
yy
2
0rxy
y
221xy
2
0r
2
yy0r1
2
yy
2
0rxy
y
111xy
2
0r
2
yy0r1
2
yy
2
0rxy
j
eBkjZkj
eBkjZkj
eAYkjkj
eAYkjkj
2
2
1
1
µ
βγµε−µα+εωεγµ+εαµ−
βγµε+µα+εωεγµ−εαµ−
βγµε−µα+εωεγµ+εαµ+
βγµε+µα+εωεγµ−εαµ
=γ−
γ
γ−
γ
. (2.17)-c
Substitute (2.17)-a and (2.17)-b into (2.12)-d
2
z0
2
yyxy
2
zrxy
2
0
2
xx
xj
H~
jy
jE~
ykj
E~
µ−
ωµ
∂
∂µ+αµβ+β
∂
∂εµ+µα+
=
63
( )
( )2
y
4
y
3
y
22
y
110
2
yyxy
2
y
42
y
31
y
2
y
1rxy
2
0
2
xx
j
eBeBeAYeAYjy
j
eBZeBZeAeAy
kj
2211
2211
µ−
++−−ωµ
∂
∂µ+αµβ+
−−+β
∂
∂εµ+µα+
=
γ−γγ−γ
γ−γγ−γ
2
y
4
2
yyxy
2
y
3
2
yyxy
2
y
22
2
yyxy
2
y
11
2
yyxy
2
0
y
42rxy
2
0
2
xx
y
31rxy
2
0
2
xx
y
2rxy
2
0
2
xx
y
1rxy
2
0
2
xx
j
eBy
j
eBy
j
eAYy
j
eAYy
j
j
eBZy
kj
eBZy
kj
eAy
kj
eAy
kj
2
2
1
1
2
2
1
1
µ−
∂
∂µ+αµβ+
∂
∂µ+αµβ+
∂
∂µ+αµβ−
∂
∂µ+αµβ−
ωµ+
∂
∂εµ+µα−
∂
∂εµ+µα−
∂
∂εµ+µα+
∂
∂εµ+µα
β+
=
γ−
γ
γ−
γ
γ−
γ
γ−
γ
( ) ( )( )( ) ( )( )
( ) ( )( )( ) ( )( )
2
y
402
2
yyxy
2
22rxy
2
0
2
xx
y
302
2
yyxy
2
12rxy
2
0
2
xx
y
2201
2
yyxy
2
1rxy
2
0
2
xx
y
1101
2
yyxy
2
1rxy
2
0
2
xx
j
eBjjZkj
eBjjZkj
eAYjjkj
eAYjjkj
2
2
1
1
µ−
ωµγµ−αµβ+βγεµ−µα−+
ωµγµ+αµβ+βγεµ+µα−+
ωµγµ−αµβ−βγεµ−µα+
ωµγµ+αµβ−βγεµ+µα
=γ−
γ
γ−
γ
(2.17)-d
In summary,
y
42
y
31
y
2
y
1z2211 eBZeBZeAeAE
~ γ−γγ−γ −−+= , (2.17)-a
y
4
y
3
y
22
y
11z2211 eBeBeAYeAYH
~ γ−γγ−γ ++−−= , (2.17)-b
2
y
44
y
33
y
22
y
11
xj
eBYeBYeAYeAYH~ 2211
µ
−−+=
γ−γγ−γ
, (2.17)-c
64
2
y
44
y
33
y
22
y
11x
j
eBZeBZeAZeAZE~ 2211
µ−
+++=
γ−γγ−γ
. (2.17)-d
where,
( ) ( )( )11xy
2
0r
2
yy0r1
2
yy
2
0rxy1 YkjkjY βγµε+µα+εωεγµ−εαµ=
( ) ( )( )21xy
2
0r
2
yy0r1
2
yy
2
0rxy2 YkjkjY βγµε−µα+εωεγµ+εαµ=
( ) ( )( )βγµε+µα+εωεγµ−εαµ= 2xy
2
0r
2
yy10r2
2
yy
2
0rxy3 kjZkjY
( ) ( )( )βγµε−µα+εωεγµ+εαµ= 2xy
2
0r
2
yy20r2
2
yy
2
0rxy4 kjZkjY
( ) ( )( )101
2
yyxy
2
1rxy
2
0
2
xx1 YjjkjZ ωµγµ+αµβ−βγεµ+µα=
( ) ( )( )201
2
yyxy
2
1rxy
2
0
2
xx2 YjjkjZ ωµγµ−αµβ−βγεµ−µα=
( ) ( )( )02
2
yyxy
2
12rxy
2
0
2
xx3 jjZkjZ ωµγµ+αµβ+βγεµ+µα−=
( ) ( )( )02
2
yyxy
2
22rxy
2
0
2
xx4 jjZkjZ ωµγµ−αµβ+βγεµ−µα−=
2
xx
2
xx
2
0rk µ=β−µε , 2
yy
2
yy
2
0rk µ=β−µε , yyxx
2
xy
2 µµ+µ=µ
2
xx
2
0r
22
xx k βµ−εµ=µ , 2
yy
2
0r
22
yy k βµ−εµ=µ , 2
yy
2
xx
2
xy
4
0
2
r
2 k µµ+µε=µ .
Next, apply the boundary conditions at the interface between the Y-type ferrite slab and
the ground plane
0
0=
=yzsE , (2.18)-a
00 ==
−=∂
∂
y
xs
yy
xy
y
zs HjHy µ
µβ . (2.18)-b
From (2.18)-a
0E0yzs =
=,
0eBZeBZeAeA0y
y
42
y
31
y
2
y
12211 =−−+
=
γ−γγ−γ ,
0BZBZAA 423121 =−−+ . (2.19)-a
From (2.18-b)
65
0y
xs
yy
xy
0y
zs HjHy
==µ
µβ−=
∂
∂,
( )
0y
2
y
44
y
33
y
22
y
11
yy
xy
0y
y
4
y
3
y
22
y
11
j
eBYeBYeAYeAYj
eBeBeAYeAYy
2211
2211
=
γ−γγ−γ
=
γ−γγ−γ
µ
−−+
µ
µβ−=
++−−∂
∂
,
( )
µ
−−+
µ
µβ−=γ−γ+γ+γ−
2
44332211
yy
xy
4232221111j
BYBYAYAYjBBAYAY ,
( ) ( ) 0BYBYAYAYjBBAYAYj 44332211xy4232221111
2
yy =−−+βµ+γ−γ+γ+γ−µµ ,
0BYjBYjAYjAYj
BjBjAYjAYj
44xy33xy22xy11xy
42
2
yy32
2
yy221
2
yy111
2
yy
=βµ−βµ−βµ+βµ+
γµµ−γµµ+γµµ+γµµ−,
( ) ( )( ) ( ) 0BYjjBYjj
AYjYjAYjYj
44xy2
2
yy33xy2
2
yy
22xy21
2
yy11xy11
2
yy
=βµ+γµµ−βµ−γµµ+
βµ+γµµ+βµ+γµµ−. (2.19)-b
In summary,
0BZBZAA 423121 =−−+ , (2.19)-a
( ) ( )( ) ( ) 0BYjjBYjj
AYjYjAYjYj
44xy2
2
yy33xy2
2
yy
22xy21
2
yy11xy11
2
yy
=βµ+γµµ−βµ−γµµ+
βµ+γµµ+βµ+γµµ−. (2.19)-b
1A and 3B need to be expressed in terms of 2A and 4B . From (2.19)-a,
423121 BZBZAA ++−= .
Substitute into (2.19)-b,
( ) ( )( ) ( ) 0BYjjBYjj
AYjYjAYjYj
44xy2
2
yy33xy2
2
yy
22xy21
2
yy11xy11
2
yy
=βµ+γµµ−βµ−γµµ+
βµ+γµµ+βµ+γµµ−,
( ) ( )( )( ) ( )( )( ) ( )( ) 0BYjjYjYjZ
BYjjYjYjZ
AYjYjYjYj
44xy2
2
yy1xy11
2
yy2
33xy2
2
yy1xy11
2
yy1
22xy21
2
yy1xy11
2
yy
=βµ+γµµ−βµ+γµµ−+
βµ−γµµ+βµ+γµµ−+
βµ+γµµ+βµ+γµµ−−
,
66
( ) ( )( )( ) ( )( )
( ) ( )( ) 0BYYZjYZj
BYYZjYZj
AjYYjYY
4412xy2121
2
yy
3311xy1112
2
yy
2xy211
2
yy21
=−βµ+γ+γµµ−+
−βµ+γ−γµµ+
βµ−−γµµ+
,
( ) ( )( ) ( ) ( )( )( ) ( )( )
3Y
1Y
1Zxyj
1Y
1Z
122
yyj
4B
1Y
2Z
4Yxyj
21Y
2Z
12
yyj2
A1
2yyj
2Y
1Yxyj
2Y
1Y
3B
−βµ+γ−γµµ
−βµ+γ+γµµ+γµµ+−βµ−= .
Substitute back to (2.19)-a,
0BZBZAA 423121 =−−+
( ) ( ) ( )( )( ) ( )( )
( ) ( )( ) ( ) ( )( )( ) ( )( )
3Y1Y1Zxyj1Y1Z122
yyj
4B
1Y
2Z
4Yxyj
21Y
2Z
12
yyj2
A1
2yyj
2Y
1Yxyj
2Y
1Y
1Z
3Y
1Y
1Zxyj
1Y
1Z
122
yyj
3Y1Y1Zxyj1Y1Z122
yyj4B2Z2A
1A
−βµ+γ−γµµ
−βµ+γ+γµµ+γµµ+−βµ−+
−βµ+γ−γµµ
−βµ+γ−γµµ+−=
( ) ( )( ) ( ) ( )( )( ) ( )( )
( ) ( )( ) ( ) ( )( )( ) ( )( )3Y1Y1Zxyj1Y1Z12
2yyj
4B1Z1Y2Z4Yxyj21Y2Z12
yyj4B3Y1Y1Zxyj1Y1Z122
yyj2Z
3Y1Y1Zxyj1Y1Z122
yyj
2A1Z12
yyj2Y1Yxyj2Y1Y2A3Y1Y1Zxyj1Y1Z122
yyj
1A
−βµ+γ−γµµ
−βµ+γ+γµµ+−βµ+γ−γµµ+
−βµ+γ−γµµ
γµµ+−βµ−+−βµ+γ−γµµ−=
( ) ( )( ) ( ) ( )( )( ) ( )( )
( ) ( )( ) ( ) ( )( )( ) ( )( ) 4
311xy1112
2
yy
1124xy2121
2
yy311xy1112
2
yy2
2
311xy1112
2
yy
11
2
yy21xy21311xy1112
2
yy
1
BYYZjYZj
ZYZYjYZjYYZjYZjZ
AYYZjYZj
ZjYYjYYYYZjYZjA
−βµ+γ−γµµ
−βµ+γ+γµµ+−βµ+γ−γµµ+
−βµ+γ−γµµ
γµµ+−βµ−+−βµ+γ−γµµ−=
In summary,
( ) ( )( ) ( ) ( )( )( ) ( )( )
( ) ( )( ) ( ) ( )( )( ) ( )( ) 4
311xy1112
2
yy
1124xy2121
2
yy311xy1112
2
yy2
2
311xy1112
2
yy
11
2
yy21xy21311xy1112
2
yy
1
BYYZjYZj
ZYZYjYZjYYZjYZjZ
AYYZjYZj
ZjYYjYYYYZjYZjA
−βµ+γ−γµµ
−βµ+γ+γµµ+−βµ+γ−γµµ+
−βµ+γ−γµµ
γµµ+−βµ−+−βµ+γ−γµµ−=
,
( ) ( )( ) ( ) ( )( )( ) ( )( )3Y1Y1Zxyj1Y1Z12
2yyj
4B1Y2Z4Yxyj21Y2Z12
yyj2A12
yyj2Y1Yxyj2Y1Y
3B−βµ+γ−γµµ
−βµ+γ+γµµ+γµµ+−βµ−= ,
or
42211 BFAFA += , (2.20)-a
67
44233 BFAFB += , (2.20)-b
where,
( ) ( )( ) ( ) ( )( )( ) ( )( )
( ) ( )( ) ( )( )311xy1112
2
yy
213xy2112
2
yy
311xy1112
2
yy
11
2
yy21xy21311xy1112
2
yy
1
YYZjYZj
YZYjYZj
YYZjYZj
ZjYYjYYYYZjYZjF
−βµ+γ−γµµ
−βµ+γ−γ−µµ=
−βµ+γ−γµµ
γµµ+−βµ−+−βµ+γ−γµµ−=
,
( ) ( )( ) ( ) ( )( )( ) ( )( )
( ) ( )( ) ( )( )
3Y1Y1Zxyj1Y1Z122
yyj
4Y1Z3Y2Zxyj2Z1Z22
yyj
3Y1Y1Zxyj1Y1Z122
yyj
1Z1Y2Z4Yxyj21Y2Z12
yyj3Y1Y1Zxyj1Y1Z122
yyj2Z
2F
−βµ+γ−γµµ
+−βµ++γµµ=
−βµ+γ−γµµ
−βµ+γ+γµµ+−βµ+γ−γµµ=
,
( ) ( )( )( ) ( )( )311xy1112
2
yy
1
2
yy21xy21
3YYZjYZj
jYYjYYF
−βµ+γ−γµµ
γµµ+−βµ−= ,
( ) ( )( )( ) ( )( )311xy1112
2
yy
124xy2121
2
yy
4YYZjYZj
YZYjYZjF
−βµ+γ−γµµ
−βµ+γ+γµµ= .
Substitute (2.20) into (2.17), we can obtain
( ) ( ) 4
y
2
y
41
y
22
y
31
yy
1z BeZeFZeFAeFZeeFE~
221211 γ−γγγγ−γ −−+−+= ,
( ) ( ) 4
yy
4
y
212
y
3
y
2
y
11z BeeFeFYAeFeYeFYH~
221211 γ−γγγγ−γ ++−++−−= ,
( ) ( )42
y
4
y
43
y
21
22
y
33
y
2
y
11
xB
j
eYeFYeFYA
j
eFYeYeFYH~ 221211
µ
−−+
µ
−+=
γ−γγγγ−γ
,
( ) ( )42
y
4
y
43
y
2122
y
33
y
2
y
11x B
j
eZeFZeFZA
j
eFZeZeFZE~ 221211
µ−
+++
µ−
++=
γ−γγγγ−γ
.
In order to be consistent with previous notation, change A2 to A1 and B4 to C1,
1211z CTATE~
+= , (2.21)-a
1413z CTATH~
+= , (2.21)-b
1615x CTATH~
+= , (2.21)-c
1817x CTATE~
+= , (2.21)-d
where,
68
( )y
31
yy
11211 eFZeeFT γγ−γ −+= ,
( )y
2
y
41
y
22221 eZeFZeFT
γ−γγ −−= ,
( )y
3
y
2
y
113211 eFeYeFYT γγ−γ +−−= ,
( )yy
4
y
214221 eeFeFYT
γ−γγ ++−= ,
( )2
y
33
y
2
y
115
j
eFYeYeFYT
211
µ
−+=
γγ−γ
,
( )2
y
4
y
43
y
216
j
eYeFYeFYT
221
µ
−−=
γ−γγ
,
( )2
y
33
y
2
y
117
j
eFZeZeFZT
211
µ−
++=
γγ−γ
,
( )2
y
4
y
43
y
218
j
eZeFZeFZT
221
µ−
++=
γ−γγ
.
Air region
In air region, the problem can be simplified with the condition that εr=1 and µr=1. The
equations for longitudinal components are simplified to
0E~
yza
2
a2
2
=
γ−
∂
∂ , (2.22)-a
0H~
yza
2
a2
2
=
γ−
∂
∂. (2.22)-b
where
2
0
222ka −+= βαγ .
By solving the above equation and assuming the electric field and magnetic field vanish
at infinity, we can obtain zaE~
and zaH~
,
y
2zaaeAE
~ γ−= , (2.23)-a
y
2zaaeCH
~ γ−= . (2.23)-b
In the same way, the x components can be derived as
69
22
0
za0za
xak
H~
dy
djE
~
E~
β−
ωµ−αβ−
= ,
22
0
y
20
y
2
k
eCdy
djeA aa
β−
ωµ−αβ−
=
γ−γ−
,
22
0
y
2a0
y
2
k
eCjeA aa
β−
γωµ+αβ−=
γ−γ−
, (2.23)-c
22
0
z0z
xak
E~
dy
djH
~
H~
β−
ωε+αβ−
= ,
22
0
y
20
y
2
k
eAdy
djeC aa
β−
ωε+αβ−
=
γ−γ−
,
22
0
y
2a0
y
2
k
eAjeC aa
β−
γωε−αβ−=
γ−γ−
. (2.23)-d
In summary, in air region
y
2zaaeAE
~ γ−= , (2.23)-a
y
2zaaeCH
~ γ−= , (2.23)-b
22
0
y
2a0
y
2xa
k
eCjeAE~ aa
β−
γωµ+αβ−=
γ−γ−
, (2.23)-c
22
0
y
2a0
y
2xa
k
eAjeCH~ aa
β−
γωε−αβ−=
γ−γ−
, (2.23)-d
or
29za ATE~
= , (2.23)-a
29za CTH~
= , (2.23)-b
211210
~CTATExa += , (2.23)-c
70
213212xa CTATH~
+= , (2.23)-d
where
y
9aeT γ−= ,
22
0
y
10k
eT
a
β−
αβ−=
γ−
,
22
0
y
a011
k
ejT
a
β−
γωµ=
γ−
,
22
0
y
a012
k
ejT
a
β−
γωε−=
γ−
,
22
0
y
13k
eT
a
β−
αβ−=
γ−
.
Now we put the field components in air region and in substrate region together.
Substrate region:
1211zs CTATE~
+= , (2.21)-a
1413zs CTATH~
+= , (2.21)-b
1615xs CTATH~
+= , (2.21)-c
1817xs CTATE~
+= . (2.21)-d
Air region:
29za ATE~
= , (2.23)-a
29za CTH~
= , (2.23)-b
211210xa CTATE~
+= , (2.23)-c
213212xa CTATH~
+= . (2.23)-d
The boundary conditions on the interface between the substrate and the air( dy = ) are
dyza
dyzs E
~E~
=== , (2.24)-a
71
dy
xady
xs EE==
=~~
, (2.24)-b
x
dyzs
dyza JHH
~~~=−
==, (2.24)-c
z
dyxs
dyxa JHH
~~~=+−
==. (2.24)-d
Substitute (2.21) and (2.23) into (2.24),
0291211 =−+ ATCTAT , (2.25)-a
02112101817 =−−+ CTATCTAT , (2.25)-b
xJCTCTAT~
291413 =+−− , (2.25)-c
zJATCTCTAT~
2122131615 =−−++ . (2.25)-d
From (2.25)-a and (2.25) -b,
( )( )7281
21122981021
TTTT
CTTATTTTA
−
−+−= , (2.26)-a
( )( )7281
21112971011
TTTT
CTTATTTTC
−
+−= . (2.26)-b
Substitute (2.26) into (2.25)-c and (2.25)-d,
( ) ( )( )( )
( ) ( )( )( ) x2
7281
728191423112
7281
971014981023 J~
CTTTT
TTTTTTTTTTA
TTTT
TTTTTTTTTT=
−
−+−+
−
−−−,
(2.27)-a
( ) ( ) ( )( ) 2
7281
127281697101598102 ATTTT
TTTTTTTTTTTTTTT
−
−−−++−
( ) ( )( ) z2
7281
137281115261 J~
CTTTT
TTTTTTTTTT=
−
−−−+ . (2.27)-b
Rewrite (2.27) as
x212211 J~
CSAS =+ , (2.28)-a
z222221 J~
CSAS =+ , (2.28)-b
72
( ) ( )( )( )7281
97101498102311
TTTT
TTTTTTTTTTS
−
−−−= ,
( ) ( )( )( )7281
7281914231112
TTTT
TTTTTTTTTTS
−
−+−= ,
( ) ( ) ( )( )7281
12728169710159810221
TTTT
TTTTTTTTTTTTTTTS
−
−−−++−= ,
( ) ( )( )7281
13728111526122
TTTT
TTTTTTTTTTS
−
−−−= .
From (2.23)-a and (2.23)-c
9
za2
T
E~
A = , (2.29)-a
za
911
10xa
11
2 E~
TT
TE~
T
1C −= . (2.29)-b
Substitute (2.29) into (2.28). From (2.28)-a,
11222112
922912
911
1012
9
11
12
11
12
11
~~~~
SSSS
JTSJTS
TT
TS
T
S
S
TJ
S
TE xz
xxa−
−
−−= ,
zxxa J
SSSS
TSSTJ
SSSS
TSSTE
~~~
11222112
10121111
11222112
10222111
−
−−
−
−= . (2.30)-a
From (2.28)-b
11222112
922912
~~~
SSSS
JTSJTSE xz
za−
−= ,
zxza J
SSSS
TSJ
SSSS
TSE
~~~
11222112
912
11222112
922
−+
−
−= . (2.30)-b
In summary,
zxxa JGJGE~~~
1211 += ,
zxza JGJGE~~~
2221 += ,
11222112
1022211111
SSSS
TSSTG
−
−= ,
73
11222112
1111101212
SSSS
STTSG
−
−= ,
11222112
92221
SSSS
TSG
−
−= ,
11222112
91222
SSSS
TSG
−= ,
21122211
12222
~~
SSSS
JSJSA zx
−
−= ,
21122211
21112
~~
SSSS
JSJSC xz
−
−= ,
( )( )7281
21122981021
TTTT
CTTATTTTA
−
−+−= ,
( )( )7281
21112971011
TTTT
CTTATTTTC
−
+−= .
We have now completed the modeling of the magnetic microstrip line problem. Before
we continue to the next step, the important formulas and parameters are listed as follows:
zxxa JGJGE~~~
1211 += ,
zxza JGJGE~~~
2221 += ,
11222112
1022211111
SSSS
TSSTG
−
−= ,
11222112
1111101212
SSSS
STTSG
−
−= ,
11222112
92221
SSSS
TSG
−
−= ,
11222112
91222
SSSS
TSG
−= ,
74
( ) ( )( )( )7281
97101498102311
TTTT
TTTTTTTTTTS
−
−−−= ,
( ) ( )( )( )7281
7281914231112
TTTT
TTTTTTTTTTS
−
−+−= ,
( ) ( ) ( )( )7281
12728169710159810221
TTTT
TTTTTTTTTTTTTTTS
−
−−−++−= ,
( ) ( )( )7281
13728111526122
TTTT
TTTTTTTTTTS
−
−−−= ,
( )yyy eFZeeFT 211
3111
γγγ −+= −,
( )yyyeZeFZeFT 221
24122
γγγ −−−= ,
( )yyy eFeYeFYT 211
32113
γγγ +−−= −,
( )yyyeeFeFYT 221
4214
γγγ −++−= ,
( )2
332115
211
µ
γγγ
j
eFYeYeFYT
yyy −+=
−
,
( )2
443216
221
µ
γγγ
j
eYeFYeFYT
yyy −−−= ,
( )2
332117
211
µ
γγγ
j
eFZeZeFZT
yyy
−
++=
−
,
( )2
443218
221
µ
γγγ
j
eZeFZeFZT
yyy
−
++=
−
,
yaeT γ−=9 ,
22
0
10β
αβ γ
−
−=
−
k
eT
ya
,
22
0
011
β
γωµ γ
−=
−
k
ejT
y
aa
,
75
22
0
012
β
γωε γ
−
−=
−
k
ejT
y
aa
,
22
0
13β
αβ γ
−
−=
−
k
eT
ya
,
( ) ( )( ) ( ) ( )( )( ) ( )( )
3111112
2
11
2
21213111112
2
1YYZjYZj
ZjYYjYYYYZjYZjF
xyyy
yyxyxyyy
−+−
+−−+−+−−=
βµγγµµ
γµµβµβµγγµµ,
( ) ( )( ) ( ) ( )( )( ) ( )( )311xy1112
2yy
1124xy21212
yy311xy11122
yy2
2YYZjYZj
ZYZYjYZjYYZjYZjZF
−βµ+γ−γµµ
−βµ+γ+γµµ+−βµ+γ−γµµ= ,
( ) ( )( )( ) ( )( )3111112
2
1
2
2121
3YYZjYZj
jYYjYYF
xyyy
yyxy
−+−
+−−=
βµγγµµ
γµµβµ,
( ) ( )( )( ) ( )( )3111112
2
1242121
2
4YYZjYZj
YZYjYZjF
xyyy
xyyy
−+−
−++=
βµγγµµ
βµγγµµ,
( ) ( )( )11
2
0
2
01
22
01 YkjkjY xyryyryyrxy βγµεµαεωεγµεαµ ++−= ,
( ) ( )( )21
2
0
2
01
22
02 YkjkjY xyryyryyrxy βγµεµαεωεγµεαµ −++= ,
( ) ( )( )βγµεµαεωεγµεαµ 2
2
0
2
102
22
03 xyryyryyrxy kjZkjY ++−= ,
( ) ( )( )βγµεµαεωεγµεαµ 2
2
0
2
202
22
04 xyryyryyrxy kjZkjY −++= ,
( ) ( )( )101
22
1
2
0
2
1 YjjkjZ yyxyrxyxx ωµγµαµββγεµµα +−+= ,
( ) ( )( )201
22
1
2
0
2
2 YjjkjZ yyxyrxyxx ωµγµαµββγεµµα −−−= ,
( ) ( )( )02
22
12
2
0
2
3 ωµγµαµββγεµµα jjZkjZ yyxyrxyxx +++−= ,
( ) ( )( )02
22
22
2
0
2
4 ωµγµαµββγεµµα jjZkjZ yyxyrxyxx −+−−= ,
2
0
222ka −+= βαγ ,
( )( )( )( )2
2
2222
0
2
22
2
1γµµµα
βωµγµαγµµµα
yyxx
xyyyxxxy jZ
−−
−−+= ,
76
,
( )( )( )( )2
1
2222
0
2
11
2
1γµµµαµ
βεωεγµαγµµµα
yyzzxx
rxyyyxxxy jY
−−
+−+−= ,
( )( )( )( )2
1
2222
0
2
11
2
2γµµµαµ
βεωεγµαγµµµα
yyzzxx
rxyyyxxxy jY
−−
+−−−= ,
( )( )( ) ( ) ( )( )( )( ) 0
2
4222
0
22
2222
0
22222222222
2422
0
222222
=++
+−+−+−−
+−−⇒
γµβεµµ
γβαεµµµµµµαµµµµα
µαβεµµαµµαµ
xyryyyy
ryyxxyyzzxxyyxx
xyrxxzzxx
k
k
k
,
2
xx
2
0r
2
xx k β−µε=µ ,
2
yy
2
0r
2
yy k β−µε=µ ,
yyxx
2
xy
2 µµ+µ=µ ,
2
xx
2
0r
22
xx k βµ−εµ=µ ,
2
yy
2
0r
22
yy k βµ−εµ=µ ,
2
yy
2
xx
2
xy
4
0
2
r
2 k µµ+µε=µ .
Galerkin's method is used to obtain a set of linear equation group, from which, by
setting the determinant to be equal zero, the phase constant β is derived. First, xJ~
and zJ~
are expanded in terms of basis functions [2]
∑=
=N
n
nnx cJ1
~~η ,
∑=
=M
m
mmz dJ1
~~ξ .
The basis functions are chosen such that they are only non-zero on the top microstrip
conductor,
( )( )( )( )2
2
2222
0
2
22
2
2γµµµα
βωµγµαγµµµα
yyxx
xyyyxxxy jZ
−−
−−−=
77
( )
( )
( )
=−
−
=−
−
=
L
L
,6,4,2,2
1sin
,5,3,1,2
1cos
22
22
mxw
w
xm
mxw
w
xm
xm π
π
ξ
( )
( )
( )
=−
π−
=−
π+
=η
L
L
,6,4,2n,xw
w2
x1ncos
,5,3,1n,xw
w2
x1nsin
x
22
22
n
The Fourier transforms of the basis functions are
( )
( ) ( )
( ) ( )
=
−−−
−+−
=
−−+
−+
=
L
L
,6,4,2,2
1
2
1
2
1
,5,3,1,2
1
2
1
2
1
~
00
00
mm
wJm
wJj
mm
wJm
wJ
m πα
παπ
πα
παπ
αξ
( )
( ) ( )
( ) ( )
=
−−+
−+
=
+−−
++−
=
L
L
,6,4,2,2
1
2
1
2
1
,5,3,1,2
1
2
1
2
1
~
00
00
nn
wJn
wJ
nn
wJn
wJj
n πα
παπ
πα
παπ
αη
where ( )xJ0 is the Bessel function of the first kind of order zero. Next, we use the basis
functions to multiply xJ~
and zJ~
, and integrate in terms of α from -∞ to +∞ to obtain a
linear equation group
,0d~
G~dd~G~cM
1m
m12pm
N
1n
n11pn =αξη+αηη ∑ ∫∑ ∫=
+∞
∞−
∗
=
+∞
∞−
∗ (2.31)-a
N,,3,2,1p L=
,0d~
G~
dd~G~
cM
1m
m22qm
N
1n
n21qn =αξξ+αηξ ∑ ∫∑ ∫=
+∞
∞−
∗
=
+∞
∞−
∗ (2.31)-b
M,,3,2,1q L=
where cn and dm are the unknowns in this linear equation group. By setting the
determinant to be equal zero, we can solve for the phase constant β. Once β is known, cn
78
and dm can be calculated, and subsequently, the electric and magnetic fields can be
determined.
2.3 Y-type Ferrite Phase Shifter: Design and Experiment
In order to verify the validity of our proposed method, the problem was first simplified
with the condition HA=0 so that we could verify our results against previously published
data. A good agreement was observed between dispersion characteristics obtained by our
method and those obtained by the finite element method [6] and the infinite line method
[7] for microstrip lines on magnetically isotropic YIG substrates. For example, a phase
constant of 452.2 rad/m was calculated for applied field of 300 Oe by the proposed
method, compared to measured values of 445 rad/m in [6] and 452.2 rad/m in [7].
To verify the numerical results for a microstrip phase shifter on an anisotropic Y-type
ferrite substrate measurements were carried out using an Agilent E8364A PNA Series
Network Analyzer on a microstrip test fixture as a function of magnetic field generated
by an electromagnet [5], as shown in Fig. 2.6. A single crystal zinc substituted hexagonal
Y-type barium ferrite slab (Ba2Zn2Fe12O22, Zn2Y) with a 30 µm thick copper microstrip
deposited on top was placed in the fixture. TRL calibration was utilized to establish
reference planes at the connectors of the test fixture. The parameters of the device used in
numerical calculations were d = 0.635 mm, w = 0.35 mm, 4πMs = 2000 Oe, HA = 9000
Oe, ferromagnetic resonance linewidth ∆H = 25 Oe, and relative permittivity εr= 19. The
first two basis functions ( )αξ1
~, ( )αξ2
~, ( )αη1
~ and ( )αη2~ were used to expand xJ
~ and zJ
~,
respectively.
79
Phase constant β of the phase shifter for different values of the magnetic bias field is
shown in Fig. 2.7.
When performing the integration in (2.31), care must be taken. For example, at H = 300
Oe, there are singularities for frequencies below 14 GHz when integrating in the spectral
Fig. 2.7 Calculated phase constant β as a function of frequency for different values of magnetic bias field H.
Fig. 2.6 Photograph of the test fixture used in the S parameters measurement. The Y-type barium hexaferrite slab is
located in the center.
80
domain. We avoided the singularities by adding artificially a small imaginary component
to µxx or µyy. However, above 14 GHz there is no singularity which is the practical region
of interest. At high frequencies (f>15 GHz), the curves are nearly linear since the
permeability of Y-type ferrite at these frequencies asymptotically approaches unity and
the slope is determined predominantly by the dielectric constant. Phase constant
dispersions exhibit cut-off behavior at low frequencies corresponding to the
ferromagnetic antiresonance frequency of the ferrite. The frequency range between
ferromagnetic resonance and antiresonance is characterized by negative permeability
responsible for the cut-off behavior. Near the anti-resonance frequency the permeability
changes rapidly and, by tuning the magnetic bias field, differential phase shifts can be
realized. Fig. 2.8 shows the differential phase shift calculated numerically and measured
experimentally.
81
We define differential phase shifts of S21 as follows
( ) ( ) ( )200HH 2121 φ−φ=φ∆ ,
where 200 Oe ≤< H 500 Oe. Good agreement was observed between simulation and
experiment above cut-off frequency. For a 300 Oe magnetic bias field, the simulation and
experimental curves nearly overlap. The difference between calculated and measured
( )40021φ∆ is 5~11 deg/cm in the frequency range of 14.5 ~ 16 GHz while ( )50021φ∆ was
29 deg at 14.8 GHz and 11.4 deg at 16 GHz.
Fig. 2.8 Calculated and experimental differential phase shift per unit length as a function of frequency for different
values of magnetic bias field H.
82
Fig. 2.9 shows the differential phase shift as a function of bias field for select frequencies.
The difference between simulated and measured phase shifts was observed to increase
with increasing bias field, or equivalently, as the operating frequency approaches the cut-
off frequency. This difference is likely to be associated with the fact that the
demagnetizing fields in the finite ferrite slab used in this experiment are not fully and
accurately represented in the numerical simulation. There are two factors that give rise to
the discrepancy between theory and experiment:1. non-saturation and 2. uncertainty in
the demagnetizing factors in a non-ellipsoidally shaped sample as ours. Nevertheless, the
discrepancy is at most 10% on average which indeed is remarkable in view of the non-
uniformities in the sample. A more detailed model of the demagnetizing fields is expected
to improve the accuracy of this method at greater magnitudes of the magnetic bias field
( H > 500 Oe).
Fig. 2.9 Calculated and experimental differential phase shift per unit length as a function of magnetic bias
field H for different operating frequencies.
83
Instantaneous bandwidth is another important figure of merit for phase shifters. We
define it as the frequency range over which the deviation in the phase shift does not
exceed 10% divided by the center frequency. As evident from Fig. 2.10, both simulation
and experiment show that the instantaneous bandwidth increases with frequency for a
fixed value of bias field and decreases with bias field at a fixed frequency.
The reason for this is that at higher frequencies the phase constant is nearly linear, thus
the differential phase shift is correspondingly more uniform in frequency.
Fig. 2.10 Calculated and experimental instantaneous bandwidth as a function of frequency for different values of
magnetic bias field H.
84
Fig. 2.11 shows the tuning factor of the phase shifter. The tuning factor is defined as the
amount of phase shift incurred per unit length per unit bias field. In Fig. 2.11, the tuning
factor is higher at low frequency (near cut-off) and decreases monotonically with
increasing frequency. Therefore there is a trade-off between the tuning factor and
instantaneous bandwidth in practical use. The insertion loss of ~6.5 dB was measured in
the frequency range of 15 to 16 GHz corresponding to the bias field in the range of 200 to
500 Oe. High losses stem from impedance mismatches in the test fixture (Fig. 2.6).
Practical insertion losses can be achieved through proper impedance matching and
improved fabrication techniques.
2.4 Conclusion
A spectral domain method was developed to analyze the planar magnetic microstrip line
devices. Magnetic materials, such as cubic ferrites, M-type hexaferrites, and Y-type and
Fig. 2.11 Calculated and experimental tuning factor as a function of frequency for different values of magnetic bias
field H.
85
Z-type hexaferrites, can be modeled with this approach. A Y-type microstrip phase shifter
was fabricated and tested to compare with the proposed approach. Good agreement
between the experimental and simulated results was obtained.
86
References
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characteristics of microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTT-
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[2] M. Geshiro and S. Yagi, "Analysis of slotlines and microstrip lines on anisotropic
substrates ," IEEE Trans. Microwave Theory Tech., vol. MTT-39, pp. 64-69, Jan.
1991.
[3] T. Kitazawa and T. Itoh, "Asymmetrical coplanar waveguide with finite metallization
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[4] J. Wang, A. L. Geiler, V. G. Harris, and C. Vittoria, “Numerical simulation of wave
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[5] A. L. Geiler, J. Wang, J. Gao, et al., "Development of Low Magnetic Bias Field
Hexagonal Y-type Ferrite Phase Shifters at Ku Band," IEEE Trans. Magn., vol. 45,
pp. 4179-4182, Oct. 2009.
[6] C. S. Teoh and L. E. Davis, "A Comparison of the Phase Shift Characteristics of
Axially-magnetized Microstrip and Slotline on Ferrite," IEEE Trans. Magn., vol. 31,
No. 6, pp. 3464-3466, Nov. 1995.
[7] H. Yang, "Microstrip Open-End Discontinuity on a Nonreciprocal Ferrite Substrate,"
IEEE Trans. Microwave Theory Tech., vol. 42, No. 12, pp. 2423-2428, Dec. 1994.
87
Chapter 3. In-plane Circulator
This chapter will focus on the in-plane circulator design and simulation. First, in section
3.1 the spectral domain method was accommodated to coupled microstrip lines on
magnetic substrate. Second, in section 3.2, important parameters of normal modes
obtained from spectral domain method were used to design the in-plane circulator and the
design was verified with HFSS simulations.
3.1 Application of Spectral Domain Method to in-plane circulator
The spectral domain method was discussed in Chapter II regarding the modeling of a
single microstrip line on magnetic substrate. In this section, the spectral domain method
will be accommodated to coupled microstrip lines on magnetic substrate. All the eqations
in substrate region and air region do not need to be revised. The basis function used to
expand the current density need to be changed to include the coupled lines.
88
3.1.1 Revised Spectral Domain Method
Fig. 3.1 shows the cross-section of the magnetic coupled microstrip lines. The basis
functions for single line are (the width of the strip is 2w)
( )
( )
( )
=−
−
=−
−
=
L
L
,6,4,2,2
1sin
,5,3,1,2
1cos
22
22
mxw
w
xm
mxw
w
xm
xm π
π
ξ , (3.1)-a
( )
( )
( )
=−
−
=−
+
=
L
L
,6,4,2,2
1cos
,5,3,1,2
1sin
22
22
nxw
w
xn
nxw
w
xn
xn π
π
η . (3.1)-b
In the α domain,
z x
y, c-axis
h
2L
d W
S
0ε 0µ
ε µt
-W W
Fig. 3.1 Cross-section of the magnetic coupled microstrip lines, where the microstrip lines are on top of the hexagonal Y-
type ferrite substrate with anisotropic permeability tensor and scalar permittivity. The crystallographic c-axis is along y
axis and the biasing field is along the z axis.
89
( )
( ) ( )
( ) ( )
=
π−−α−
π−+απ−
=
π−−α+
π−+απ
=αξ
L
L
,6,4,2m,2
1mwJ
2
1mwJj
2
1
,5,3,1m,2
1mwJ
2
1mwJ
2
1
~
00
00
m , (3.2)-a
( )
( ) ( )
( ) ( )
=
π−−α+
π−+απ
=
π+−α−
π++απ−
=αη
L
L
,6,4,2n,2
1nwJ
2
1nwJ
2
1
,5,3,1n,2
1nwJ
2
1nwJj
2
1
~
00
00
n . (3.2)-b
For coupled lines, the basis functions can be expressed as
( ) ( ) 0x,Sxx mm1 <+ξ=ξ , (3.3)-a
( ) ( ) 0x,Sxx mm2 >−ξ=ξ , (3.3)-b
( ) ( ) 0x,Sxx nn1 <+η=η , (3.3)-c
( ) ( ) 0x,Sxx nn2 >−η=η . (3.3)-d
In the α domain,
( ) ( )αξ=αξ α−m
Sj
m1
~e
~, (3.4)-a
( ) ( )αξ=αξ αm
Sj
m2
~e , (3.4)-b
( ) ( )αη=αη α−n
Sj
n1
~e , (3.4)-c
( ) ( )αη=αη αn
Sj
n2
~e . (3.4)-d
Expand xJ and zJ with the basis functions
( ) ( )∑∑==
η+η=N
1n
n2n2
N
1n
n1n1x xcxcJ , (3.5)-a
( ) ( )∑∑==
ξ+ξ=M
1m
m2m2
M
1m
m1m1z xdxdJ . (3.5)-b
Performing the Fourier Transform on both sides
90
( ) ( )∑∑==
αη+αη=N
1n
n2n2
N
1n
n1n1x
~c~cJ~
, (3.6)-a
( ) ( )∑∑==
αξ+αξ=M
1m
m2m2
M
1m
m1m1z
~d
~dJ
~. (3.6)-b
Substitute (3.6) into the expressions for xaE~
and zaE~
,
( ) ( ) ( ) ( )
αξ+αξ+
αη+αη= ∑∑∑∑
====
M
1m
m2m2
M
1m
m1m112
N
1n
n2n2
N
1n
n1n111xa
~d
~dG~c~cGE
~, (3.7)-a
( ) ( ) ( ) ( )
αξ+αξ+
αη+αη= ∑∑∑∑
====
M
1m
m2m2
M
1m
m1m122
N
1n
n2n2
N
1n
n1n121za
~d
~dG~c~cGE
~. (3.7)-b
(3.7) can be rewritten as
( ) ( ) ( ) ( )∑∑∑∑====
αξ+αξ+αη+αη=M
1m
m212m2
M
1m
m112m1
N
1n
n211n2
N
1n
n111n1xa
~Gd
~Gd~Gc~GcE
~, (3.8)-a
( ) ( ) ( ) ( )∑∑∑∑====
αξ+αξ+αη+αη=M
1m
m222m2
M
1m
m122m1
N
1n
n221n2
N
1n
n121n1za
~Gd
~Gd~Gc~GcE
~. (3.8)-b
Next, using the conjugate of the basis function to muliply both sides of the equation,
( ) ( )∑∑==
αηη+αηη=ηN
1n
n211
*
p1n2
N
1n
n111
*
p1n1xa
*
p1~G~c~G~cE
~~
( ) ( )∑∑==
αξη+αξη+M
1m
m212
*
p1m2
M
1m
m112
*
p1m1
~G~d
~G~d , N,,3,2,1p L= (3.9)-a
( ) ( )∑∑==
αηη+αηη=ηN
1n
n211
*
p2n2
N
1n
n111
*
p2n1xa
*
p2~G~c~G~cE
~~
( ) ( )∑∑==
αξη+αξη+M
1m
m212
*
p2m2
M
1m
m112
*
p2m1
~G~d
~G~d , N,,3,2,1p L= (3.9)-b
( ) ( )∑∑==
αηξ+αηξ=ξN
1n
n221
*
q1n2
N
1n
n121
*
q1n1za
*
q1~G
~c~G
~cE
~~
( ) ( )∑∑==
αξξ+αξξ+M
1m
m222
*
q1m2
M
1m
m122
*
q1m1
~G
~d
~G
~d , M,,3,2,1q L= (3.9)-c
91
( ) ( )∑∑==
αηξ+αηξ=ξN
1n
n221
*
q2n2
N
1n
n121
*
q2n1za
*
q2~G
~c~G
~cE
~~
( ) ( )∑∑==
αξξ+αξξ+M
1m
m222
*
q2m2
M
1m
m122
*
q2m1
~G
~d
~G
~d . M,,3,2,1q L= (3.9)-d
Next integrating in terms of α from ∞− to ∞+
∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
αηη+αηηN
1n
n211
*
p1n2
N
1n
n111
*
p1n1 d~G~cd~G~c
0d~
G~dd~
G~dM
1m
m212
*
p1m2
M
1m
m112
*
p1m1 =αξη+αξη+ ∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
, N,,3,2,1p L= (3.10)-a
∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
αηη+αηηN
1n
n211
*
p2n2
N
1n
n111
*
p2n1 d~G~cd~G~c
0d~
G~dd~
G~dM
1m
m212
*
p2m2
M
1m
m112
*
p2m1 =αξη+αξη+ ∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
, Np ,,3,2,1 L= (3.10)-b
∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
αηξ+αηξN
1n
n221
*
q1n2
N
1n
n121
*
q1n1 d~G~
cd~G~
c
0d~
G~
dd~
G~
dM
1m
m222
*
q1m2
M
1m
m122
*
q1m1 =αξξ+αξξ+ ∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
, Mq ,,3,2,1 L= (3.10)-c
∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
αηξ+αηξN
1n
n221
*
q2n2
N
1n
n121
*
q2n1 d~G~
cd~G~
c
0d~
G~
dd~
G~
dM
1m
m222
*
q2m2
M
1m
m122
*
q2m1 =αξξ+αξξ+ ∑ ∫∑ ∫=
−∞
∞−=
−∞
∞−
. Mq ,,3,2,1 L= (3.10)-d
By setting the determinant to be zero, we can solve for β .
3.1.2 Current and Voltage of the Coupled Microstrip Lines
The current density on the interface between air and substrate can be expressed as
( ) ( )∑∑==
αη+αη=N
1n
n2n2
N
1n
n1n1x
~c~cJ~
, (3.11)-a
92
( ) ( )∑∑==
αξ+αξ=M
1m
m2m2
M
1m
m1m1z
~d
~dJ
~. (3.11)-b
The longitudinal current density components of the coupled lines are
( )∑=
αξ=M
1m
m1m11z
~dJ
~, (3.12)-a
( )∑=
αξ=M
1m
m2m22z
~dJ
~. (3.12)-b
The other coefficients are
21122211
z12x222
SSSS
J~
SJ~
SA
−
−= (3.13)-a
21122211
x21z112
SSSS
J~
SJ~
SC
−
−= (3.13)-b
( )( )7281
2112298102
1TTTT
CTTATTTTA
−
−+−= (3.14)-c
( )( )7281
2111297101
1TTTT
CTTATTTTC
−
+−= (3.14)-d
The field components can be expressed as
In substrate region
1211zs CTATE~
+= (3.15)-a
1413zs CTATH~
+= (3.15)-b
1615xs CTATH~
+= (3.15)-c
1817xs CTATE~
+= (3.15)-d
ωε
α+β−= zsxs
ys
H~
H~
E~
(3.15)-e
93
yy0
xsxy0zsxs
ys
H~
E~
E~
H~
µωµ
µωµ+α−β= (3.15)-f
In air region
29za ATE~
= (3.16)-a
29za CTH~
= (3.16)-b
211210xa CTATE~
+= (3.16)-c
213212xa CTATH~
+= (3.16)-d
ωε
α+β−= zaxa
ya
H~
H~
E~
(3.16)-e
0
zsxsys
E~
E~
H~
ωµ
α−β= (3.16)-f
The voltage of line 2 can be derived as
( )∫−=d
ys dyySEV0
2 , ,
where ( ) ( )∫+∞
∞−
−= ααπ
αdeyEyxE
xj
ysys ,2
1, ,
( )∫ ∫+∞
∞−
−−=d
Sj
ys dydeyE0
,2
1αα
πα ,
where ωε
α+β−= zsxs
ys
H~
H~
E~
, 1413zs CTATH~
+= ,1615xs CTATH
~+= ,
∫ ∫+∞
∞−
−+−−=
d
Sjzsxs dydeHH
0
~~
2
1α
ωε
αβ
πα
( ) ( )∫ ∫
+∞
∞−
−+++−−=
d
Sj dydeCTATCTAT
0
14131615
2
1α
ωε
αβ
πα
( ) ( )∫ ∫
+∞
∞−
−−+−−=
d
Sj dydeCTTATT
0
164153
2
1α
ωε
βαβα
πα
94
( ) ( )∫ ∫
+∞
∞−
−−+−−=
d
Sj dydeCTTATT
0
164153
2
1α
ωε
βαβα
πα
( ) ( )∫ ∫
+∞
∞−
− −+−−=
d
Sj dydCTTATT
e0
164153
2
1α
ωε
βαβα
πα
( )( )
( )( )
∫ ∫∞+
∞−
γ−γγ
γ−γγ
γγ−γ
γγ−γ
α− αωε
µ
−−β−
++−α
+
µ
−+β−
+−−α
π−=
d
0
1
2
y4
y43
y21
yy4
y21
1
2
y33
y2
y11
y3
y2
y11
Sj dyd
C
j
eYeFYeFY
eeFeFY
A
j
eFYeYeFY
eFeYeFY
e2
1
221
221
211
211
αωε
µ
−−
β−
++−α
+
µ
−+
β−
+−−α
π−=
γ−γγ
γ−γγ
γγ−γ
γγ−γ
∞+
∞−
α−
∫∫∫
∫∫∫
∫∫∫
∫∫∫
∫ d
C
j
edyYedyFYedyFY
edyedyFedyFY
A
j
edyFYedyYedyFY
edyFedyYedyFY
e2
1
1
2
y
d
0
4y
d
0
43y
d
0
21
y
d
0
y
d
0
4y
d
0
21
1
2
y
d
0
33y
d
0
2y
d
0
11
y
d
0
3y
d
0
2y
d
0
11
Sj
221
221
211
211
,
where
1
dd
0y1
yy
d
0
1eeedy
11
1
γ
−=
γ=
γ
=
γγ
∫ ,
1
dd
0y1
yy
d
0
1eeedy
11
1
γ−
−=
γ−=
γ−
=
γ−γ−
∫ ,
2
dd
0y2
yy
d
0
1eeedy
22
2
γ
−=
γ=
γ
=
γγ
∫ ,
2
dd
0y2
yy
d
0
1eeedy
22
2
γ−
−=
γ−=
γ−
=
γ−γ−
∫ ,
95
αωε
µ
γ−
−−
γ
−−
γ
−
β−
γ−
−+
γ
−+
γ
−−α
+
µ
γ
−−
γ−
−+
γ
−
β−
γ
−+
γ−
−−
γ
−−α
π−=
γ−γγ
γ−γγ
γγ−γ
γγ−γ
∞+
∞−
α−
∫ d
C
j
1eY
1eFY
1eFY
1e1eF
1eFY
A
j
1eFY
1eY
1eFY
1eF
1eY
1eFY
e2
1
1
2
2
d
4
2
d
43
1
d
21
2
d
2
d
4
1
d
21
1
2
2
d
33
1
d
2
1
d
11
2
d
3
1
d
2
1
d
11
Sj
221
221
211
211
αωε
µ
γ−
−−
γ
−−
γ
−
β−
γ−
−+
γ
−+
γ
−−α
+
µ
γ
−−
γ−
−+
γ
−
β−
γ
−+
γ−
−−
γ
−−α
π−=
γ−γγ
γ−γγ
γγ−γ
γγ−γ
∞+
∞−
α−
∫ d
C
j
1eY
1eFY
1eFY
1e1eF
1eFY
A
j
1eFY
1eY
1eFY
1eF
1eY
1eFY
e2
1
1
2
2
d
4
2
d
43
1
d
21
2
d
2
d
4
1
d
21
1
2
2
d
33
1
d
2
1
d
11
2
d
3
1
d
2
1
d
11
Sj
221
221
211
211
.
In the same way,
( )∫ −−=d
ys dyySEV0
1 ,
( )∫ ∫+∞
∞−
−=d
Sj
ys dydeyE0
,2
1αα
πα
αωε
µ
γ−
−−
γ
−−
γ
−
β−
γ−
−+
γ
−+
γ
−−α
+
µ
γ
−−
γ−
−+
γ
−
β−
γ
−+
γ−
−−
γ
−−α
π−=
γ−γγ
γ−γγ
γγ−γ
γγ−γ
∞+
∞−
α
∫ d
C
j
1eY
1eFY
1eFY
1e1eF
1eFY
A
j
1eFY
1eY
1eFY
1eF
1eY
1eFY
e2
1
1
2
2
d
4
2
d
43
1
d
21
2
d
2
d
4
1
d
21
1
2
2
d
33
1
d
2
1
d
11
2
d
3
1
d
2
1
d
11
Sj
221
221
211
211
.
96
The current in line 2 can be derived as
( )∫+
−
=wS
wS
zz dxxJI 12
( )∫ ∫+
−
+∞
∞−
−=wS
wS
xj
z dxdeJ ααπ
α1
2
1
( ) ααπ
αdJdxe z
xj
12
1∫ ∫
+∞
∞−
+∞
∞−
−
= ,
where ( )απδα2=∫
+∞
∞−
−dxe
xj
( ) ( ) αααπδπ
dJ z122
1∫
+∞
∞−
=
( )02 == αz
J .
In the same way
( )011 == αzz
JI .
In summary,
α
ωε
µ
γ−
−−
γ
−−
γ
−
β−
γ−
−+
γ
−+
γ
−−α
+
µ
γ
−−
γ−
−+
γ
−
β−
γ
−+
γ−
−−
γ
−−α
π−=
γ−γγ
γ−γγ
γγ−γ
γγ−γ
∞+
∞−
α
∫ d
C
j
1eY
1eFY
1eFY
1e1eF
1eFY
A
j
1eFY
1eY
1eFY
1eF
1eY
1eFY
e2
1V
1
2
2
d
4
2
d
43
1
d
21
2
d
2
d
4
1
d
21
1
2
2
d
33
1
d
2
1
d
11
2
d
3
1
d
2
1
d
11
Sj
1
221
221
211
211
. (3.17)-a
97
α
ωε
µ
γ−
−−
γ
−−
γ
−
β−
γ−
−+
γ
−+
γ
−−α
+
µ
γ
−−
γ−
−+
γ
−
β−
γ
−+
γ−
−−
γ
−−α
π−=
γ−γγ
γ−γγ
γγ−γ
γγ−γ
∞+
∞−
α−
∫ d
C
j
1eY
1eFY
1eFY
1e1eF
1eFY
A
j
1eFY
1eY
1eFY
1eF
1eY
1eFY
e2
1V
1
2
2
d
4
2
d
43
1
d
21
2
d
2
d
4
1
d
21
1
2
2
d
33
1
d
2
1
d
11
2
d
3
1
d
2
1
d
11
Sj
2
221
221
211
211
. (3.17)-b
( )0JI 1z1z =α= . (3.17)-c
( )0JI 2z2z =α= . (3.17)-d
3.2 In-plane Circulator Design and HFSS simulation
According to [1], the procedure to design in-plane circulator is
1) Plot the phase difference between the two line for two basis modes. Adjust the physical
parameters of the FCL, such as the width of the copper or gap between the two lines and
the thickness of the substrate, until the frequency of the crossover point is close to 90°.
2) Plot the characteristic impedance of the two basis modes. The crossover point should
be close to 50 ohms.
3) Calculate the optimum length of the FCL from
−+ β−β
π=
2/L .
In this section, an in-plane circulator operating at 7 GHz was designed and simulated. A
biasing field of 200 G is applied longitudinally to the YIG substrate. The width of the
copper lines was 0.15 mm. The gap between these two lines were 0.5 mm. The thickness
of the substrate was 0.5 mm. Fig. 3.2 shows the phase difference between the two lines
98
for the two basis modes.
The two curves cross at 7.29 GHz with 96.5°. Fig. 3.3 shows the characteristic impedance
of the two basis modes. These two curves cross at 6 GHz with 46.8 Ω. From Figs. 3.2 and
3.3, the operating frequency is approximately at 7 GHz.
Fig. 3.2 Phase difference between the two lines in FCL section for the two basis modes. RHCP
means right-handed circular polarization, and LHCP means left-handed circular polarization.
Only the absolute values are shown.
99
Fig. 3.4 shows the phase constant for the two basis modes. At 7 GHz, abs(βRHCP-
βLHCP)=38 rad/m. Therefore, the length of the FCL section is
mm3.412/
LLHCPRHCP
=β−β
π= .
Fig. 3.4 Phase constant of the two basis modes. RHCP means right-handed circular polarization,
and LHCP means left-handed circular polarization.
Fig. 3.3 Characteristic impedance of the two basis modes. RHCP means right-handed circular
polarization, and LHCP means left-handed circular polarization.
100
The FCL section is connected with a T junction to consist of an in-plane circulator. The
biasing field, Ha, is 200 G. The T junction is used to combine the even mode signal and
reject the odd mode signal. However, the characteristic impedance at port 1 is 25 Ω
instead of 50 Ω. A impedance matching network is required to match port 1 to 50 Ω.
YIG Alumina Alumina
T junction FCL section
Port2
Port3
Port1
Fig. 3.5 FCL section connected with a T junction. The applied biasing field, Ha is along the
longitudinal direction. The circulation is port1-port3-port2-port1.
Hext
101
Fig. 3.7 S parameters for port 2 with FCL section's length of 41.3mm.
Fig. 3.6 S parameters for port 1 with FCL section's length of 41.3mm.
102
Fig. 3.6 to Fig. 3.8 shows the S parameters of this circulator design with FCL section's
length of 41.3mm. In order to reduce the reflection and isolation below 15 dB, the FCL
section's length was extended to 45mm.
Fig. 3.9 S parameters for port 1 with FCL section's length of 45mm.
Fig. 3.8 S parameters for port 3 with FCL section's length of 41.3mm.
103
Fig. 9 to Fig 11 shows the S parameters for the circulator with FCL section's length of 45
mm. The reflection and isolation is less than 15 dB from 6.3 GHz to 7.8 GHz with a
bandwidth of 1.5 GHz. The insertion loss is less than 1.12 dB for excitation at port 1 and
port 2, and is less than 2.2 dB for excitation at port 3. The reason for that is the wave
Fig. 3.11 S parameters for port 3 with FCL section's length of 45mm.
Fig. 3.10 S parameters for port 2 with FCL section's length of 45mm.
104
excited at port 3 would travel twice as long as excited at ports 1 and 2.
3.3 Conclusion
An in-plane FCL circulator was designed on YIG substrate according to the normal
mode theory. The important parameters of the basis modes were calculated with a matlab
code based on spectral domain method. The reflection and isolation is less than 15 dB
from 6.3 GHz to 7.8 GHz with a bandwidth of 1.5 GHz. The insertion loss is less than
1.12 dB for excitation at ports 1 and 2, and is less than 2.2 dB for excitation at port 3.The
problem with the in-plane circulator design is that the length of the microstrip lines are
too great to overcome practical constraints on insertion loss. Some in-plane circulator
designs with reduced length have been reported recently [2]-[3]. But the short length of
the device is at the cost of narrow bandwidth. New structures are still need to investigate
to both have short length and wide bandwidth characteristics.
105
References
[1] C. S. Teoh, L. E. Davis, "Design and measurement of microstrip ferrite coupled line
circulators," International Journal of RF and Microwave Computer-Aided
Engineering, vol. 11, No. 3, pp. 121-130, May. 2001.
[2] M. Cao, and R. Pietig, “Ferrite coupld-line circulator with reduced length,” IEEE
Trans.Microwave theory Tech., vol. 53, pp. 2572-2579, Aug. 2005.
[3] S. D. Yoon, Jianwei Wang, Nian Sun, C. Vittoria and V. G. Harris, "Ferrite-coupled
line circulator simulations for application at X-band frequency," IEEE Trans.
Magn., vol 43, pp. 2639-2641, Jun. 2007.
106
Chapter 4 Hexaferrites-based Self-biased Y-Junction Circulator
Ferrite based junction circulators are indispensable components in modern radar
systems[1]-[2].One advantage of this configuration is that they offer shorter microstrip
lines and,therefore, lower losses compared to the in plane design. The nonreciprocal
properties of ferrite materials make it possible for the transmission and reception of
wireless signals occuring at the same time and frequency. Traditional junction circulators
need strong bias fields provided by permanent magnets. As a result, size, weight and cost
is increased, which may be unfavorable when the trend in modern technologies is toward
miniature and efficient devices. The objective of self-biasing ferrite materials is to be able
to remove permanent magnets in future designs of circulator devices. Previously, self-
biased junction circulator designs were demonstrated at frequencies above 30 GHz at Ka
and V band utilizing magnetically oriented M-type hexaferrite compacts [3]-[7]. In this
chapter, we present a self-biased circulator at 13.6 GHz, which represents the first
hexaferrite-based self-biased circulator operating below 20 GHz [8]. The push toward the
operation of self-biased circulators to lower frequencies is important because the L, S, C
and X bands are popular radar, satellite and wireless communications bands for military
and commercial applications. Other types of self-biased circulators based on
ferromagnetic nanowires were demonstrated recently [9]-[10]. These designs are
typically characterized by relatively high insertion loss due to the high conductivity of
metallic ferromagnetic materials. In this work, a hexaferrite material is utilized in the
design of a self-biased junction circulator because of its low eddy current loss compared
107
with conducting nanowires. While self-biasing of ferromagnetic nanowires is made
possible by the shape anisotropy, in hexaferrites self-biasing is achieved by taking
advantage of the large uniaxial magnetic anisotropy field intrinsic to magnetically
oriented polycrystalline M-type hexaferrites. Due to the lack of need for permanent
biasing magnets the overall circulator size and weight can be substantially reduced.
In this chapter, a self-biased microstrip Y-junction circulator operating at Ku band was
designed and fabricated for the first time utilizing the magnetically oriented M-type
strontium hexaferrite. We adopted a novel composite design consisting of a dielectric
substrate resting upon a hexaferrite slab. Previous self-biased circulator designs consisted
of only one ferrite slab. The advantage of our design is that it lends itself to simpler
fabrication procedure. The microstrip circuit can be fabricated on a copper clad dielectric
substrate, which is then positioned on top of a hexaferrite slab. The self-biased strontium
M-type hexaferrite employed in the present circulator was prepared by the conventional
ceramic sintering technique. The perpendicularly c-axis oriented M-type hexaferrite
compact possessed high density, high saturation magnetization and especially high
remanence, which became the cornerstone for the self-biased devices fabricated by us.
The permeability tensor of the strontium M-type ferrite disk oriented perpendicular to
the disk plane has the following form [6]:
[ ]
µ
µκ
κ−µ
=µ
000
0j
0j
.
108
ω−ω
ωω+µ=µ
22
0
m00 1 ,
220
m0
ω−ω
ωωµ=κ ,
( )rA00 MH −γµ=ω ,
and r0m Mγµ=ω ,
where Mr is the remnant magnetization, HA is the uniaxial magnetocrystalline anisotropy
field, ω is the radial frequency, and γ is the electron gyromagnetic ratio.
With the properties of the strontium hexaferrite, our junction circulator needed to
operate above FMR frequency in terms of field sweeps, which corresponds to a small κ/µ
value. As a result, the coupling angle would also be small if the surrounding medium and
the substrate under the junction are the same material [11]-[12] limiting the bandwidth of
the device. Impedance matching was needed in order to increase the bandwidth.
4.1 New Microstrip Y-Junction Circulator Design
The microstrip lines in the Y junction circulator circuit were patterned on a copper clad
Duroid® dielectric substrate, which was placed upon a polished disk of strontium barium
Fig. 4.1 Cross-section view of the junction resonator and quarter-wave
microstrip line.
109
ferrite, as shown in Fig. 4.1. In traditional Y-junction microstrip circulator designs, the
microstrip circuit is usually deposited directly on the ferrite surface, which has numerous
constrains imposed by lithographic processing. First, the minumum feature size is only
0.06~0.08 mm which reduces the tolorance during the fabrication; second, to deposite
copper on dielectric material is more convenient and simpler than to deposite gold on
ferrite material. In addition, a conventional ferrite microstrip circulator requires a hole to
be machined in the center of the dielectric substrate. The ferrite disc or puck needs to be
tightly inserted and fitted into the hole at the point where the microstrip lines need to be
coupled across the gap. To ensure structural and signal transmission continuity, the gap
between the ferrite and dielectric needs to be filled with dielectric paste. Clearly, this
presents extra fabrication costs. The composite design we employed removes the need for
complex lithography fabrication on ferrite substrates and removes the need for the costly
embedding of ferrites in dielectric substrates. We believe that this fabrication process is
compatible with integrated circuit (IC) fabrication processes.
In this junction circulator design, the quarter-wave microstrip line is used to match the
input impedance only at the center frequency. Since κ/µ is very small over the frequency
of interest, which is between 0.1 and 0.5, the coupling angle is insensitive to the
circulation condition. Therefore, the quarter-wave microstrip line is directly connected
with the junction resonator. The design procedure was simulated in HFSS® based on
Bosma's theory [13]. First, the radius of the junction resonator at the center of the circuit
was estimated according to the first circulation equation, kR=1.84 [13] at operation
110
frequency of 13.65 GHz, where k is the wavenumber in the ferrite medium. Second, the
quarter-wave microstrip line was included in the simulation. The characteristic
impedance of the quarter-wave microstrip line was estimated from 0int ZZZ = , where
Zin is the input impedance of the junction resonator at operation frequency and Z0 is equal
to 50 Ohms. Third, the width of the quarter-wave microstrip line was tuned using HFSS®
until the isolation was maximized.
4.2 HFSS Simulation
Static and microwave measurements for the oriented hexaferrite compacts were
performed using vibrating sample magnetometer (VSM) and a shorted waveguide
ferromagnetic resonance technique. The hysteresis loop is shown in Fig. 4.2. The
Fig. 4.2 Hysteresis loops of magnetically oriented M-type strontium hexaferrite.
"Perp" refers to the measurement performed with the external field perpendicular to
the sample surface, and "par" to the measurement with the field applied within the
sample surface.
111
following magnetic properties were measured: Mr=302.4±15.9 kA/m, HA=1.4-1.6 MA/m,
Mr/Ms>92%, FMR linewidth ∆H=47.7-119.4 kA/m at U-band (42-55 GHz). The
strontium M-type hexaferrite was polished down to a thin disk with a thickness of 0.353
mm. The circulator circuit was designed and fabricated on top of Duroid® dielectric
substrate. The center operation frequency was designed at f=13.6 GHz, which is above
FMR resonance operation in terms of field sweeps. The corresponding radius of the
junction resonator based on Bosma theory was 2.7mm. The thickness of the Duroid® was
0.254 mm. The width and the length of the quarter-wave microstrip line were 0.16mm
and 3.3mm, respectively. The size of the 50 Ω microstrip line were optimized by HFSS
resulting in 1.4 mm width and 1.03 mm length, respectively. The HFSS model is shown
in Fig. 4.3.
Ferrite
Duroid
Copper
Junction resonator
Quarter-wave
microstrip line
50 Ω magnetic microstrip
line
Fig. 4.3 Perspective view of copper-duroid-ferrite structure in HFSS.
112
4.3 Experiment
The junction circulator was prepared with standard photolithography processing and
was tested using the Agilent E8364A PNA Series Network Analyzer. Fig. 4.4 shows the
copper test fixture used in the S parameter measurement. The test fixture also served as
the device ground. A 50 Ohms load was connected to one of the ports while the other two
ports were connected to the Network Analyzer.
Fig. 4.4 Photograph of device and test fixture used in S parameter
measurements
113
4.4 Results and Discussion
Fig. 4.5 shows the comparison between simulated and measured S parameters. An
isolation of 21 dB with corresponding insertion loss of 1.52 dB was obtained with center
frequency of operation of 13.65GHz. The experimental results matched simulation results
well except the 15 dB isolation bandwidth was measured to be 220 MHz compared to
420 MHz from simulation. This discrepancy may be attributed to uncertainty of the
material parameters. In this experiment, the Duroid® and the ferrite are only loosely
stacked without applying paste.
4.5 Conclusion
A self-biased microstrip Y-junction circulator operating at Ku band was designed and
fabricated on a composite consisting of Duroid® and ferrite substrates for the first time.
The advantage of this structure is that it provides ease of fabrication and integration with
Fig. 4.5 Simulated and experimental
114
other passive and active circuits, such as filters and amplifiers. Future work will focus on
lowering the frequency of operation of self-biased circulators below 10 GHz.
115
References
[1] B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics. New York:
McGraw-Hill, 1962, ch. 12.
[2] V. G. Harris, A. Geiler, Y. Chen, S. D. Yoon, M. Wu, A. Yang, etc.,"Recent advances
in processing and applications of microwave ferrites," J. Mag. Mag. Mat., vol. 321,
pp. 2035-2047, 2009.
[3] J. A. Weiss, N. G. Watson, and G. F. Dionne, "New uniaxial-ferrite millimeter-wave
junction circulators," IEEE MTT-S Int. Microwave symp. Dig., pp.145-148, 1989.
[4] Y. Akaiwa, and T. Okazaki, "An application of a hexagonal ferrite to a millimeter-
wave Y circulator," IEEE Trans. Magn., vol. 10, pp.374-378, Jun. 1974.
[5] N. Zeina, H. How, and C. Vittoria, " Self-biasing circulators operating at Ka-band
utilizing M-type hexagonal ferrites," IEEE Trans. Magn., vol. 28, pp. 3219-3221,
Jan. 1992.
[6] B.K. O’Neil, and J. L. Young, “Experimental investigation of a self-biased
microstrip circulator,” IEEE Trans. Microwave Theory Tech., vol. MTT-57, pp.
1669-1674, Jul. 2009.
[7] S. A. Oliver, P. Shi, W. Hu, H. How, S. W. McKnight, N. E. McGruer, P. M.
Zavracky, and C. Vittoria, "Integrated self-biased hexaferrite microstrip circulators
for milimeter-wavelength applications," IEEE Trans. Microwave Theory Tech., vol.
MTT-49, pp. 385-387, Feb. 2001.
[8] J. Wang, A. Yang, Y. Chen, Z. Chen, A. Geiler, S. M. Gillette, etc., "Self biased Y-
116
junction circulator at Ku band," IEEE Microw. Wireless Compon. Lett., to be
published.
[9] A. Saib. M. Darques, L. Piraux, D. Vanhoenacker-Janvier, and I. Huynen, "An
unbiased integrated microstrip circulator based on magnetic nanowired substrate,"
IEEE Trans. Microwave Theory Tech., vol. MTT-53, pp. 2043-2049, Jun. 2005.
[10] M. Darques, J. De la Torre Medina, L. Piraux, L. Cagnon and I. Huynen,
"Microwave circulator based on ferromagnetic nanowires in an alumina template,"
Nanotechnology 21, pp. 145208, 2010.
[11] Y. S. Wu, and F. J. Rosenbaum, "Wide-band operation of microstrip circulators,"
IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 849-856, Oct. 1974.
[12] S. Ayter, and Y. Ayasli, "The frequency behavior of stripline circulator junctions,"
IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 197-202, Mar. 1978.
[13] H. Bosma, "On strip line Y-circulation at UHF," IEEE Trans. Microwave Theory
Tech., vol. MTT-12, pp. 61-72, Jan. 1964.
117
Chapter 5 Nanowire-based Y-Junction Circulator
Ferrite based junction circulators and isolators are critical components in modern
wireless communication systems [1]-[2]. However, the need for permanent magnets
increases the size, weight and cost of these systems, especially in platforms where
thousands of circulators are required. The phased array antenna system is one example.
Vast efforts have been made to remove or minimize the use of permanent magnets that
are required to bias the ferrite-based circulator. Magnetically oriented M-type hexaferrite
compacts have been utilized in the self-biased junction circulator design from Ka to V
band [3]-[7]. Recently, an innovative layered hexaferrite-dielectric self-biased junction
circulator operating at Ka band has also been demonstrated [8]. Other types of self-biased
junction circulators based on metal ferromagnetic nanowires have been fabricated and
tested. However, this type of circulator is characterized by relatively high insertion loss
due to the high conductivity of the metallic ferromagnetic material. Relatively high
insertion loss of up to 10 dB was measured at X and Ku band [9]-[10] compared to < 2
dB for some of the hexaferrite-based designs [3]-[7]. The self-biasing field in the
ferromagnetic nanowires-based designs was realized by the shape anisotropy of the
nanowires. Clearly, the insertion loss is of concern in the use of metal nanowires. To date,
no self-biased junction circulators have been realized for frequencies below X-band.
There is, however, a strong need for high performance, compact, lightweight, and cost-
effective circulator devices to meet the demands of rapidly growing wireless
communication markets with frequencies of operation typically in the L and S bands.
118
In order to utilize hexaferrites for self-biased junction circulators at frequencies below
X-band, the uniaxial magnetocrystalline anisotropy field, HA, needs to be reduced
considerably. However, low HA usually implies non-collinear magnetic ordering [11], and
consequently higher insertion loss and lower Néel temperature, compared to
ferrimagnetic hexaferrites). For the self-biased junction circulators utilizing metal
ferromagnetic nanowires [9]-[10], the saturation magnetization of the magnetic
nanowires is usually very high, on the order of 10 kG, which limits the application to X-
band or higher.
In this chapter, a self-biased junction circulator operating at 2 GHz and below (L band)
was designed on a composite substrate consisting of insulating yttrium iron garnet (YIG)
ferrite nanowires embedded in a microporous barium-strontium titanate (BSTO)
membrane [12]. The proposed composite substrate is depicted schematically in Fig. 5.1.
By taking advantage of shape anisotropy of the YIG nanowires to achieve self-bias, the L
band junction circulator can be designed to operate above ferromagnetic resonance
(FMR) frequency in terms of magnetic field. This allowed the use of a material with
higher saturation magnetization and Curie temperature, thus resulting in a more
temperature stable design. Operation above FMR also yields smaller junction size due to
higher effective permeability of the ferrite nanowires. The porous membrane further
reduces the size of the Y-junction resonator due to the high permittivity of the BSTO. In
order to achieve a relatively low circulation frequency (i.e., L band) and reduce insertion
losses YIG was selected as the rod material. YIG is characterized by relatively low FMR
119
frequency, narrow FMR linewidth, ∆H < 10 Oe, and low dielectric loss tangent, tanδd <
0.001. The relatively low FMR frequency is the result of the moderate saturation
magnetization of YIG, 4πMS = 1750 G, and the nanowire form factor. The FMR
frequency of YIG nanowires can be approximated by ωFMR = γ×2πM.
5.1 Modeling of YIG-nanowires
YIG nanowires provide a shape anisotropy field that helps alleviate the need of a strong
magnetic bias field. It is well known that the demagnetizing field along the axis of a
needle-shaped ferrite wire is very small. The external magnetic field required to
magnetically saturate the ferrite sample is on the order of the magnitude of the
demagnetizing field
+π=
n21
1M4H Sd (5.1)
where n is the ratio of the length to the diameter of the nanowire [13]. For YIG
nanowires, Hd is on the order of 5 Oe assuming n = 100. In order to saturate the needle-
shaped sample of YIG, the externally applied magnetic field has to overcome the
demagnetizing and the coercive fields. With the coercive field of polycrystalline YIG
(a) (b) (c)
Fig. 5.1 Proposed approach to the formation of YIG/BSTO composite. (a) Seed layer growth
(Pulsed laser deposition) (b) Template positioning (microporous BSTO membrane) (c) Embedded
pillar growth (Liquid phase epitaxy).
120
being on the order of 1.5 Oe, the external magnetic field on the order of 5-7 Oe would be
needed in this case.
Unfortunately, the interaction between nanowires lowers the magnetization, since dipole
fields from each wire oppose the magnetization direction in adjacent wires. This means
that for a fixed applied field, the magnetization of an isolated nanowire is higher than that
of a nanowire within a composite. This effect is estimated in the following. We define P
as the volume loading factor, or the combined volume of the nanowires divided by the
total volume of the composite substrate. The effect of nanowire interactions on the
average saturation magnetization of the composite was calculated by performing static
magnetic simulations using Ansys® Maxwell 3D. This approach allowed the calculation
of the magnetization of the center nanowire positioned within a 5 by 5 array. Table I
shows the magnetization, M, of the nanowires for different values of volume loading
factor, P, assuming 20 Oe magnetic bias field. It is evident from Table I that with
increasing volume loading factor, P, the magnetization of a single wire within the array is
decreased due to dipole-dipole interactions.
121
TABLE I. Magnetization (M) versus volume loading factor (P). Last row includes a
simple estimate of equivalent magnetization (Meq). Meq=M·P.
5.2 Equivalent Modeling of the YIG Nanowire Substrate
It is computationally intensive and time consuming to model and simulate junction
circulators based on YIG nanowires using commercially available numerical simulation
packages, such as Ansys®
HFSS. This is because of extremely fine meshes required to
calculate, for example, 1 million nanowires with a radius of 5 nm embedded within a
composite forming a circulator junction with a radius of 5 mm.
To address this limitation, we propose a simple model to predict the performance of
junction circulators involving as many as 105 wires or more. The model reduces the
calculation time by a factor of 700. This is due to the fact that HFSS has to break up the
nanowires into a large number of tetrahedra, i.e., 1,066,570 in this case, whereas it only
takes 4,998 tetrahedra for the single slab.
According to our simple model we can represent the composite of a multitude of wires
as a single slab of ferrite material characterized by the average saturation magnetization
of the composite, see third row of Table I. The other advantage of our model is the
P (%) 8.73 30.68 54.54 64.9
4πM (G) 1622 1486 1455 1454
4πMeq (G) 142 456 794 944
122
inclusion of the ‘dynamic’ demagnetizing field due to the precessional motion of the
magnetization in a nanowire. In the conventional simulation, where Maxwell boundary
conditions are applied simultaneously to all the wires in the composite, the dynamic
demagnetizing field is calculated automatically by the HFSS software. That is not the
case for the model adopted here. In order to correctly model the YIG nanowires, one must
explicitly incorporate the dynamic demagnetizing field in the magnetic bias source in
HFSS. The dynamic demagnetizing field comes from the expression for the magnetic
susceptibility,
[ ]( )
γ
ωγ
ω−
γ
ω−
=χ
rf
rf
2
22
rfHj
jH
H
M (5.2)
where Hrf = 2πM is the dynamic demagnetizing field perpendicular to the wire axis and
M is the magnetization of the nanowire.
The other approximations include the following equivalent permittivity and average
saturation magnetization of the substrate used in the model
( ),P1P
,PMM
dielectric,rYIG,req,r
eq
−⋅ε+⋅ε=ε
⋅= (5.3)
where
Meq is the average saturation magnetization of the composite substrate (see Table I),
εr,eq is the equivalent relative permittivity of the composite substrate,
M is the magnetization of the YIG nanowires as calculated in Table I (taking into
account the interaction between wires),
123
εr,YIG is the relative permittivity of the YIG nanowires,
and εr,dielectric is the relative permittivity of the porous membrane.
Equation (3) allows us to take over all of the formulations developed in the 1960s and
70s as well as by special software like as HFSS, for the design of two dimensional Y-
junction circulators. Instead of applying Maxwell boundary conditions for each nanowire
they are now applied for the composite as a whole. Effectively, we have reduced the
multi-dimensional boundary value problem to a standard two dimensional boundary
value problem.
We verified our equivalent model by calculating the electromagnetic scattering S-
parameters for both calculational approaches. In approach (a) no approximations as
described above were included. Approximately 1,000 YIG wires with radius of 0.095 mm
and height of 1 mm were included in the Y-junction resonator in HFSS simulation. The
volume loading factor, P, was 30.68%. Duroid was used as the low permittivity dielectric
material in this simulation to save the simulation time. Additionally, the following
parameters were used: εr,YIG=14.7, εr,dielectric=2.2, M=1485.7 G, ∆H=5 Oe. In calculational
approach (b) we used the internal field derived from susceptibility tensor in (2) as well as
other approximations described above. The equivalent magnetization and equivalent
dielectric constant were calculated according to (3) to be 398.5 G and 5.5 respectively.
The internal dynamic magnetic field of 650 Oe corresponding to the dynamic
demagnetizing field of the YIG wires was utilized. In both calculations the following
parameters were assumed: the thickness of the substrate was 1 mm and the radius of the
124
copper disk was 5 mm. The method described in [14]-[15] was used to design this
junction circulator. In Fig. 2 we plot S21 and S31 as a function of frequency for both
approaches (a) and (b). The calculation of approach (a) involved only one computer with
execution time of 22 minutes per frequency point, and memory of 50 GB, however, as the
number of wires increase for tenfold, the memory increase by about 7 times and the
execution time increased by about 6 times. As a comparison, it only take 2 seconds to
calculate one frequency point for approach (b). Clearly, approach (a) is prohibitive, and
approach (b) is more effective and, therefore, more preferred.
It is important to note that in Fig. 2 the circulation frequency calculated by the
approximate method (b) is about 4% higher than the one from approach (a). The
Fig. 5.2 Simulation results of the original model and equivalent model. S21 is the
insertion loss, and S31 is the isolation.
125
difference may be explained solely in terms of our choice of Hd in the susceptibility
tensor(see above definition). The magnetizing field within the single wires due to the
interaction with other wires is very non-uniform. The assumption of uniform
demagnetizing field may result in a certain amount of error.
Also, from Fig. 2 approach (a) overestimates the insertion loss by 1.08 dB in
comparison to approach (b). This is mainly attributed to the fact that the number of YIG
wires in approach (a) is not enough to represent the model with many wires at the input
port of the junction circulator disk. Increasing the number of YIG wires in approach (a)
would finally optimize the matching condition and reduce the insertion loss.
The datasets of these two approaches reasonably agree with each other. We can conclude
that the simple equivalent model (b) can effectively represent the exact model calculation
(a).
5.3 Nanowire-based Y-Junction Circulator Design
Whereas Fig. 2 applies to a circulator design operating near 2 GHz , we now calculate,
using approach (b) only, S-parameters for a circulator design operating near 1 GHz. The
following parameters were used in the simulation: P=30.68%, Meq=398.5 Gs, εr,eq=77.1,
∆H=5 Oe and Hi=743 Oe ( Hi is the internal field or magnetic bias source used in HFSS.
In this case, it is equal to the Hrf in (2)). The thickness of the substrate was 1 mm, and the
radius of the copper disk was 8.25 mm. Fig. 3 shows the simulation results of this
junction circulator. This circulator operates at 1.006GHz, and the -20 dB bandwidth is 50
MHz. The insertion loss was calculated to be 0.16 dB.
126
5.4 Conclusion
With the inclusion of shape anisotropy of magnetic insulating nanowires embedded in
high dielectric constant composites, it is indeed feasible to design self-biased circulators
operating at UHF frequencies. Designing and modeling such devices requires proper
form of the susceptibility tensor and accurate expressions of internal field introduced in
this manuscript. The radii of the circulators are 5 and 8.25 mm at frequencies of 2 and 1
GHz, respectively. The insertion losses are very low, thus lending themselves to
practicality. An approximate model (b) is proposed to calculate S-parameters for a given
design. The equivalent model was compared with the exact model (a) and the results
Fig. 3 Simulation results of the low-bias junction circulator. S21 is the insertion loss,
and S31 is the isolation.
127
agree reasonably well with each other. Future work will focus on the fabrication of these
types of junction circulators.
128
References
[1] B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics. New York:
McGraw-Hill, 1962, ch. 12.
[2] V. G. Harris, A. Geiler, Y. Chen, S. D. Yoon, M. Wu, A. Yang, etc.,"Recent advances
in processing and applications of microwave ferrites," J. Mag. Mag. Mat., vol. 321,
pp. 2035-2047, 2009.
[3] J. A. Weiss, N. G. Watson, and G. F. Dionne, "New uniaxial-ferrite millimeter-wave
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junction circulator at Ku band," IEEE Microw. Wireless Compon. Lett.,accepted.
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Chapter 6 Conclusions
This thesis presented mainly three types of circulators: in-plane FCL circulator on YIG
substrate, self-biased junction circulator on M-type strontium hexaferrite substrate, and
self-biased junction circulator on YIG nanowires substrate. Although promising
characteristics, such as self-biasing or low-biasing, were found out of these three types of
circulators, there are still rooms for them to improve in order to compete with the YIG
junction circulator which dominates the commercial market.
The in-plane circulator takes advantage of the low demagnetizing field in the substrate’s
plane to realize a low-biasing wide-band circulator. However, this device is characterized
by large dimensions due to the small coupling between the even and odd mode(coupled
mode theory) or the small difference of the phase constant between the two basis
modes(normal mode theory). A new structure is strongly needed to take the full
advantage of the Faraday rotation to reduce the device’s length and reduce the insertion
loss at the same time.
The self-biased M-type hexaferrite junction circulator is a self-biased circulator. Since
the biasing magnet is removed, the weight, volume, and therefore the cost of the system
can be drastically reduced. However, compared to the YIG junction circulator, the
hexaferrite junction circulator has a large insertion loss ~2 dB and narrow bandwidth. The
bandwidth can be increased with a better circuit design. The junction can be absorbed
into the matching network to achieve wideband characteristics[1]. The insertion loss has
to be decreased from the breakthrough of the ferrite material. A quasi-single-crystal
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Barium hexaferrites with ∆H≈300 Oe has been reported [2].
The junction circulator on YIG nanowires substrate is also a self-biased circulator. Due
to the low magnetocrystalline anisotropy field(HA) and magnetization(M) of YIG, this
device can work at lower frequency band, such as L band. Although the simulation results
has been presented in this thesis, a prototype need to be demonstrated to prove the low
insertion loss characteristics. The successful experimental demonstration would render
this type of circulator to be a perfect candidate for the commercial YIG junction
circulator.
Hexagonal Y- and Z- type ferrite is suitable for phase shifter design at above X band due
to the fact that the magnitude of the magnetic bias field necessary to operate a ferrite
device at high frequency can be greatly compensated by the magnetocrystalline
anisotropy field in these materials. This thesis presented the method to model Y- and Z-
type ferrite in microstrip devices. The important parameters for basis modes, such as
phase constant, characterisitc impedance, can be obtained by this method. In the future,
efforts should be focused on 3 dimensional modeling of these material in the devices to
assist more complicated design of microwave magnetic devices, such as meaderline
phase shifter.
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References
[1] J. Helszajn, “Quarter-wave coupled junction circulators using weakly magnetized
disk resonators,” IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 800-806,
May, 1982.
[2] Y. Chen, A.L. Geiler, T. Chen, T. Sakai, C. Vittoria, and V.G. Harris, “Low-loss
barium ferrite quasi-single-crystals for microwave application,” J. Appl. Phys., vol
101, 09M501, 2007.
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