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-

21, pp.496-499, Jul. 1973.

[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

thickness containing anisotropic media," IEEE Trans. Microwave Theory Tech., vol.

MTT-39, pp. 1426-1433, Aug. 1991.

[4] J. Wang, A. L. Geiler, V. G. Harris, and C. Vittoria, “Numerical simulation of wave

propagation in Y- and Z-type hexaferrites for high frequency applications,” J. Appl.

Phys., vol. 107, 09A515, 2010.

[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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129

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

131

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.

132

References

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disk resonators,” IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 800-806,

May, 1982.

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barium ferrite quasi-single-crystals for microwave application,” J. Appl. Phys., vol

101, 09M501, 2007.