by alireza vashghani farahani a thesis submitted in ......5.2.4 fluxmetric and magnetometric...

Numerical Modeling of Wave Propagation in Strip Lineswith Gyrotropic Magnetic Substrate and Magnetostaic

Waves

by

Alireza Vashghani Farahani

A thesis submitted in conformity with the requirements

for the degree of Doctor of PhilosophyGraduate Department of Electrical and Computer Engineering

University of Toronto

Copyright c© 2011 by Alireza Vashghani Farahani

Abstract

Numerical Modeling of Wave Propagation in Strip Lines with Gyrotropic Magnetic

Substrate and Magnetostaic Waves

Alireza Vashghani Farahani

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2011

Simulating wave propagation in microstrip lines with Gyrotropic magnetic substrate is

considered in this thesis. Since the static internal field distribution has an important

effect on the device behavior, accurate determination of the internal fields are considered

as well. To avoid the losses at microwave frequencies it is assumed that the magnetic

substrate is saturated in the direction of local internal field. An iterative method to

obtain the magnetization distribution has been developed. It is applied to a variety of

nonlinear nonuniform magnetic material configurations that one may encounter in the

design stage, subject to a nonuniform applied field.

One of the main characteristics of the proposed iterative method to obtain the static

internal field is that the results are supported by a uniqueness theorem in magnetostatics.

The series of solutions Mn,Hn, where n is the iteration number, minimize the free Gibbs

energy G(M) in sequence. They also satisfy the constitutive equationM = χH at the end

of each iteration better than the previous one. Therefore based on the given uniqueness

theorem, the unique stable equilibrium state M is determined.

To simulate wave propagation in the Gyrotropic magnetic media a new FDTD for-

mulation is proposed. The proposed formulation considers the static part of the electro-

magnetic field, obtained by using the iterative approach, as parameters and updates the

dynamic parts in time. It solves the Landau-Lifshitz-Gilbert equation in consistency with

Maxwell’s equations in time domain. The stability of the initial static field distribution

ii

ensures that the superposition of the time varying parts due to the propagating wave will

not destabilize the code.

Resonances in a cavity filled with YIG are obtained. Wave propagation through a

microstrip line with YIG substrate is simulated. Magnetization oscillations around local

internal field are visualized. It is proved that the excitation of magnetization precession

which is accompanied by the excitation of magnetostatic waves is responsible for the

gap in the scattering parameter S12. Key characteristics of the wide microstrip lines are

verified in a full wave FDTD simulation. These characteristics are utilized in a variety

of nonreciprocal devices like edgemode isolators and phase shifters.

iii

To my parents

iv

Acknowledgements

I would like to express special gratitude to my supervisors Professor J. D. Lavers

and the late Professor A. Konrad. The continuous support, encouragement and helpful

advice provided by Professor A. Konrad both in the M.A. Sc and in the first part of

the PhD programs are always remembered. I would like to thank Professor J. D. Lavers

in particular for his unmatched encouragement and invaluable insight. Professor Lavers

kindly accepted the responsibility at the later stage of my PhD program. Although I

didn’t have the opportunity to work under his supervision for a longer period of time,

the time I spent with him greatly influenced me.

The comments made by the committee members, Professor M. R. Iravani, Professor

A. Prodic, Professor M. Pugh and Professor M. Mojahedi have helped me to clarify the

thesis presentation. I greatly appreciate their help and valuable comments.

I would like to extend my appreciation to Professor Isaac Mayergoyz, the external ex-

aminer, whose deep understanding of the research field, insightful comments and nice

remarks added to the quality and value of the work.

Last, but not the least, I would like to express my sincere appreciation to my wife,

Marzieh. Without her understanding, patience and continuous support this work could

not be completed.

v

Contents

1 Introduction 1

1.1 Loss Mechanisms in Microwave Ferrite Devices . . . . . . . . . . . . . . . 4

1.2 Literature Review: Numerical Methods and Proposed Designs . . . . . . 6

1.2.1 Nonuniform Internal Field Distributions Hi and M . . . . . . . . 7

1.2.2 FDTD Formulations . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 Designs to Improve Frequency Bandwidth . . . . . . . . . . . . . 11

1.3 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Internal Distributions M and H 23

2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Distant Magnetic Field, Magnetic Dipole Moment and Magnetization . . 25

2.4 Magnetic Scalar Potential ϕ . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Series Solution for the Magnetic Scalar Potential . . . . . . . . . . . . . . 28

2.6 Fluxmetric and Magnetometric Demagnetizing Factors . . . . . . . . . . 32

2.7 Other Existing Methods to Obtain Internal Field Distributions . . . . . . 34

2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Micromagnetics 37

3.1 Micromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Helmholtz Potential and Free Gibbs Energy . . . . . . . . . . . . . . . . 39

3.3 Contributions to the Helmholtz Potential A . . . . . . . . . . . . . . . . 41

3.3.1 Magnetostatic Contribution to the Helmholtz Potential . . . . . . 41

3.3.2 Magneto-crystalline Anisotropy Energy Contribution . . . . . . . 42

vi

3.3.3 Contribution to A by the Exchange Interaction . . . . . . . . . . 43

3.4 Variational Approach, the Equilibrium Condition . . . . . . . . . . . . . 45

3.5 Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Stability of the Distribution M = (Ms/|H|)H . . . . . . . . . . . . . . . 503.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Field Distributions M and H in Ferrite Substrates (1) 54

4.1 The Iterative Method to Obtain Internal Fields M and H . . . . . . . . 55

4.1.1 Geometrical Factors, Rectangular Geometry . . . . . . . . . . . . 61

4.1.2 Field Distributions in Ferrite Substrates . . . . . . . . . . . . . . 62

4.2 Fluxmetric and Magnetometric Demagnetizing Factors in Nonlinear Mag-

netic Media: Rectangular Geometry . . . . . . . . . . . . . . . . . . . . . 66

4.3 Finite Element Method Based on a Minimization Theorem to Obtain M

and H, Rectangular Geometry . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Reduced Set of Equations to Obtain Unknown Nodal Components

of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Field Distributions M and H in Ferrite Substrates (2) 80

5.1 Demagnetizing Field Distribution, Cylindrical Geometry . . . . . . . . . 80

5.2 Iterative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.1 H and M in Cylindrical Magnetic Media, Ha = Hak . . . . . . . 88

5.2.2 H and M in Cylindrical Magnetic Media, Ha = Haρ(ρ, z)aρ +

Haz(ρ, z)k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.3 H and M in Cylindrical Magnetic Media, Radial Step Change in

Ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2.4 Fluxmetric and Magnetometric Demagnetizing Factors in Nonlin-

ear Nonuniform Magnetic Media: Cylindrical Geometry . . . . . . 115

5.3 Finite Element Method Based on a Minimization Theorem to Obtain M

and H, Cylindrical Geometry . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3.1 Results, Discussion and New Perspectives . . . . . . . . . . . . . 121

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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6 Cavity Resonant Frequencies 124

6.1 FDTD Formulation to Simulate Wave Propagation in Gyrotropic Magnetic

Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Resonance in Rectangular Cavities Filled with Ferrimagnetic Materials . 129

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7 Strip Lines with Magnetic Substrate, Magnetization Precession 140

7.1 Plane Wave Propagation in Magnetic Medium . . . . . . . . . . . . . . . 141

7.1.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.1.2 Polder’s Susceptibility and Permeability Tensors . . . . . . . . . . 143

7.2 The Magnetostatic Approximation . . . . . . . . . . . . . . . . . . . . . 145

7.2.1 Magnetostatic Waves . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.3 Absorbing Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . 148

7.3.1 PML ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.4 Microstrip Line with Magnetic Substrate, Magnetization Precession . . . 154

7.4.1 FDTD Implementation . . . . . . . . . . . . . . . . . . . . . . . . 155

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8 Edgemode Isolator 169

8.1 Dominant Mode Pattern, Propagation Constant . . . . . . . . . . . . . . 171

8.2 FDTD Simulation, Edgemode Isolator . . . . . . . . . . . . . . . . . . . 175

8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

9 Conclusion 188

A Mathematical Formulaes 197

B Mathematical Integrals 198

C Series Expansion of the Unit Vector α = H/|H| 201

D Symmetry Considerations 203

E Mur’s ABC 210

F FDTD Implementation of the UPML 214

References 220

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List of Figures

1.1 Y-junction circulator subject to applied field in axial direction. The ferrite

disks are backed by the metallic plates at both ends. The middle plate is

connected to the strip lines that are separated by 120◦. . . . . . . . . . 2

1.2 Edgemode isolator. While propagating along the line, the electromagnetic

field shows transverse displacement normal to the direction of propagation. 3

1.3 Strip line edgemode isolator as proposed by Schloemann to increase the

operating frequency bandwidth. The central ferrite layer has a saturation

magnetization which is two times larger than the saturation magnetiza-

tion of the input/output ferrite substrates. In the strip line geometry the

central strip conductor is positioned between two ferrite platelets and two

metallic ground planes at the top and the bottom. This is in contrast with

the microstrip line geometry in which only one metallic ground plane is

present, Figure 1.2. The ferrite layers in contact with the high permeabil-

ity poles and the ground metallic plates ensure uniformity of the internal

field inside the guiding structure between metallic plates. The high perme-

ability poles and the permanent bar magnet constitute a magnetic circuit

that provides the applied field in the z direction normal to the ground

planes and ferrite substrates. The electromagnetic signal propagating in

the y direction, passes through ferrite layers. The central layer has a higher

saturation magnetization. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Rectangular magnetic material subject to a uniform applied field Ha = Haẑ. 31

2.2 Cylindrical magnetic material subject to a uniform applied field Ha = Haẑ. 32

ix

4.1 Brick source and field elements. Source point position vector rj scans

the surface of the jth element when one is integrating the contributions

of surface magnetic pole densities to the demagnetizing field at the field

point ri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Flowchart showing each step of the iterative approach to obtain M and H. 58

4.3 Cosine of angle θ between M and H along corner line parallel to the y

axis. The ferrite slab is divided into 40×40×4 cuboids. Iterations resultsfor which n ≥ 3, are zoomed in the inset. Ha = Hax̂. . . . . . . . . . . . 63

4.4 Internal field Hx along the corner line parallel to the y axis. The horizontal

axis shows the the values of y normalized by the slab length in the y

direction. The applied field is in the x direction. The inset shows the

decrease in WM as n increases. . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Cosine of the angle θ between M and H along the corner line parallel to

the strip conductor. The angle θ changes from 33◦ for n = 1 to 0◦ for n = 7. 65

4.6 Hz along the positive x axis, starting from the center of substrate parallel

to the strip conductor is shown. The solid line indicated by ”A first order”,

is the first order analytic result obtained by Schloemann. Higher iteration

solutions converge as n increases. To clarify the picture, results for which

n is even are not shown. The inset shows the decrease in WM as n increases. 65

4.7 Nf and Nm along the z axis for a bar of dimensions 2a×2a×2c, γ = c/a =0.1 and χ = 1×10−6. The square bar is thin and thus Nf ≈ Nm. Ha = Haẑand µ0Ha = 0.35T . χ ≈ 0 means a uniform M and H. Therefore theconstitutive equation M = χH is satisfied quickly. As a result Nf1 → Nf2and Nm1 → Nm2 after just the first iteration. WM corresponding to eachiteration is shown in the figure for different n. The initial guess for the

magnetization, M = χHa/(1 + χ), is obtained by applying the continuity

of normal component of B across the air/YIG interface. Given the initial

guess for the magnetization M, the surface method yields Nf,m = 1. The

results for Nf,m in the case of surface method are shown by index 2. Pardo

et. al have reported Nf = 0.79331 and Nm = 0.80508 for this case [26].

The iterative results are shown in the figure: Nf = 79591, Nm = 0.80508. 67

x

4.8 Nf,m for a square bar with aspect ratio γ = c/a = 0.1. As iteration number

n increases, the results by the surface and volume methods indicated by

”s” and ”v” approach each other. The inset shows WM decreases with n.

In this example χ = 1.5. The iterative method yields Nf = 0.777 and

Nm = 0.789. These results are very close to the reported results in the

literature [26], Figure 4.13. . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.9 Nf and Nm for a square bar of dimensions a× a× c, γ = c/a = 0.1. Theinset shows the assumed magnetization curve of the material. The square

bar is thin and thus Nf ≈ Nm. Ha = Hak and µ0Ha = 0.35T . WMcorresponding to different iterations is shown in Figure 4.10. . . . . . . . 68

4.10 WM as a function of n for the configuration of Figure 4.9. The inset shows

in a closer look the decrease in WM for n ≥ 3. . . . . . . . . . . . . . . . 69

4.11 Rectangular magnetic material centered at the origin. The first octant Ω1

is shown by dotted edges. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.12 cos(θ) as a function of grid points for a rectangular slab. Unique distribu-

tions M and H satisfy the constitutive equation M = χH. cos(θ), where

θ is the angle between H and M, along the edges for two different χ are

shown. Pardo et. al. have reported Nf = 0.79331 and Nm = 0.80508 [26].

Ha = Haẑ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.13 cos(θ) as a function of grid points for a rectangular slab. Unique distribu-

tions M and H satisfy the constitutive equation M = χH. cos(θ), where

θ is the angle between H and M, along the edges for two different χ are

shown. Pardo et. al. have reported Nf = 0.77297 and Nm = 0.78819 [26].

Ha = Haẑ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.14 cos(θ) and χ as a function of grid points for a rectangular slab. The

configuration of applied field and magnetic material show no symmetry.

Ha is uniform but is not parallel with any of the edges. Figure shows

that numerical solutions M and H are parallel and satisfy the constitutive

equation M = χH. Therefore unique distributions M andH are obtained.

cos(θ) along an edge is shown. Inset shows |M|/|H| = χ for the numericalsolutions M and H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

xi

5.1 Cylindrical magnetic material subject to an applied field Ha in axial di-

rection. In the more general case the applied field may have azimuthal

symmetry, Ha = Haz(ρ, z)k +Haρ(ρ, z)âρ. . . . . . . . . . . . . . . . . . 82

5.2 Source and field rings in case of azimuthal symmetry. Position vector r′

scans both the surface and volume of the source ring to integrate contribu-

tions of surface and volume magnetic pole densities to the demagnetizing

field at point r. In each field element, only one point is needed to obtain

the internal fields for the whole element. This point is chosen to be at the

center of element’s cross section with the xz plane. . . . . . . . . . . . . 82

5.3 cos(θ), where θ is the angle between H and M, as a function of ρ on the

top cylinder base at z ≈ L. Cylinder radius is a = L/2 in which L is itsheight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 cos(θ), where θ is the angle between H and M, as a function of ρ on the

midplane cross section. z ≈ L/2 and a = L/2. . . . . . . . . . . . . . . . 90

5.5 cos(θ), where θ is the angle between H and M, as a function of z at ρ ≈ a.a = L/2. The inset magnifies the top right corner to show the convergence

between directions of H and M as n increases. . . . . . . . . . . . . . . . 91

5.6 Hz as a function of ρ on the midplane cross section at z = L/2. . . . . . 91

5.7 Hz as a function of ρ on the lower cylinder base at z ≈ 0. Symmetry andthe numerical results show that Hz on the top is the same as Hz at the

bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.8 Hz as a function of z at ρ ≈ 0. . . . . . . . . . . . . . . . . . . . . . . . . 92

5.9 Hz as a function of z at ρ ≈ a. . . . . . . . . . . . . . . . . . . . . . . . . 93

5.10 Hρ as a function of ρ on the bottom surface. Hρ is an odd function of z

if the coordinate system is positioned at the center of cylinder. Hρ,L =

−Hρ,0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.11 Hρ as a function of z at ρ ≈ a. . . . . . . . . . . . . . . . . . . . . . . . . 94

5.12 The applied field in the z direction, Haz as a function of ρ at different

heights inside the magnetic disk. Coordinate z is measured from the sur-

face of the north pole below the disk. . . . . . . . . . . . . . . . . . . . . 96

5.13 The applied field in the z direction, Haz as a function of z at different radii

close to axes and the disk surface. . . . . . . . . . . . . . . . . . . . . . . 96

xii

5.14 The applied field in radial direction, Haρ as a function of ρ at different

heights inside the magnetic disk. Coordinate z is measured from the sur-

face of the north pole below the disk. . . . . . . . . . . . . . . . . . . . . 97

5.15 The applied field in radial direction, Haρ as a function of z at different

radii close to axes and the disk surface. . . . . . . . . . . . . . . . . . . . 97

5.16 cos(θ) as a function of ρ close to the disk surface on the top, z ≈ z1 + L.z1 is the distance between the lower disk surface (base) and the surface of

the north pole. The inset magnifies the top right corner. . . . . . . . . . 98

5.17 cos(θ) as a function of ρ on the midplane cross section of magnetic disk. . 99

5.18 cos(θ) as a function of ρ close to the disk surface at the bottom, z ≈ z1. . 995.19 cos(θ) as a function of z close to axis. Inset magnifies the top right corner. 100

5.20 cos(θ) as a function of z close to disk surface at ρ ≈ a. . . . . . . . . . . 1005.21 Hz as a function of ρ at the midplane disk cross section. . . . . . . . . . 101

5.22 Hz as a function of ρ on the lower disk surface. . . . . . . . . . . . . . . 101

5.23 Hz as a function of ρ on the top disk surface. . . . . . . . . . . . . . . . . 102

5.24 Hz as a function of z along disk axis. . . . . . . . . . . . . . . . . . . . . 102

5.25 Hz as a function of z close to disk surface at ρ ≈ a. . . . . . . . . . . . . 1035.26 Hρ as a function of ρ at the midplane disk cross section. . . . . . . . . . 103

5.27 Hρ as a function of ρ on the lower disk surface. . . . . . . . . . . . . . . 104

5.28 Hρ as a function of ρ on the top disk surface. . . . . . . . . . . . . . . . . 104

5.29 Hρ as a function of z along disk axis. . . . . . . . . . . . . . . . . . . . . 105

5.30 Hρ as a function of z close to disk surface at ρ ≈ a. . . . . . . . . . . . . 1055.31 cos(θ) as a function of ρ on the top surface. The inset is to magnify

different iteration results close to the discontinuity in Ms. . . . . . . . . . 107

5.32 cos(θ) as a function of ρ on the midplane cross section. . . . . . . . . . . 107

5.33 cos(θ) as a function of z at ρ ≈ 0. . . . . . . . . . . . . . . . . . . . . . . 1085.34 cos(θ) as a function of z at ρ ≈ a. . . . . . . . . . . . . . . . . . . . . . . 1085.35 Hz as a function of ρ on the midplane cross section. . . . . . . . . . . . . 109

5.36 Hz as a function of ρ on the base at z ≈ 0. . . . . . . . . . . . . . . . . . 1095.37 Hz as a function of z at ρ ≈ 0. . . . . . . . . . . . . . . . . . . . . . . . . 1105.38 Hz as a function of z at ρ ≈ a. . . . . . . . . . . . . . . . . . . . . . . . . 1105.39 Hρ as a function of ρ on the midplane cross section. . . . . . . . . . . . . 111

5.40 Hρ as a function of ρ on the base at z ≈ 0. . . . . . . . . . . . . . . . . . 1115.41 Hρ as a function of ρ on the top surface at z ≈ L. . . . . . . . . . . . . . 112

xiii

5.42 Hρ as a function of z at ρ ≈ 0. . . . . . . . . . . . . . . . . . . . . . . . . 1125.43 Hρ as a function of z at ρ ≈ a. . . . . . . . . . . . . . . . . . . . . . . . . 1135.44 Bρ as a function of ρ on the midplane cross section. . . . . . . . . . . . . 113

5.45 Bρ as a function of ρ on the base at z ≈ 0. . . . . . . . . . . . . . . . . . 1145.46 Nf and Nm for a cylindrical magnetic material subject to uniform axial

applied field. χ = 1. As n increases, the results obtained by surface

and volume methods for fluxmetric Nf , and magnetometric Nm, converge

to the results shown in the picture. In the literature Nf = 0.2315 and

Nm = 0.2942 [24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.47 Nf and Nm for a cylindrical magnetic material with radial step change in

χ. Uniform applied field is in the axial direction, Ha = Hak. . . . . . . . 117

5.48 Nf andNm for a cylindrical nonlinear magnetic material subject toHa(ρ, z).

See Figure 4.9 for the material magnetization characteristics. . . . . . . . 118

5.49 Internal field Hz(z, ρ = a) for the configuration considered in Figure 5.48. 118

5.50 cos(θ), where θ is the angle between H and M, along the length at ρ ≈ aand along ρ at z ≈ L is shown. Also |M|/|H| in axial direction at ρ ≈ a isshown. For χ = 0.0001 and γ = 1, Chen et. al. have obtained Nf = 0.2322

and Nm = 0.3116 [24]. The same reference for χ = 1 and γ = 1 gives

Nf = 0.2315 and Nm = 0.2942. . . . . . . . . . . . . . . . . . . . . . . . 122

6.1 Standard Yee cell supplemented by the sampling points for components of

b,m,B and M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2 Component of magnetization Mx on the midplane cross section parallel to

the xz plane inside the ferrite cavity resonator at a given instant of time. 132

6.3 Component of electric field Ey on the midplane cross section parallel to

the xz plane inside the ferrite cavity resonator at a given instant of time.

The direction of propagation is along the z axis. . . . . . . . . . . . . . . 132

6.4 Resonance frequencies of the ferrite filled resonator as given by the pro-

posed FDTD formulation. The results are based on uniform internal field

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.5 Dispersion curve and resonance frequencies. The dispersion curve is plot-

ted by using (6.20). The circles show the resonances that are predicted

by the FDTD method shown in Figure 6.4. They satisfy the condition

βlz/π = n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xiv

6.6 Cosine of φ, the angle between M and H, along the corner line inside

the rectangular magnetic prism for different iteration numbers n. Inset:

iterations 4− 7 are zoomed in. . . . . . . . . . . . . . . . . . . . . . . . . 1366.7 Cosine of φ, the angle between M and H, along the center line in the rect-

angular magnetic prism for different iteration numbers n. Inset: iterations

5− 7 are zoomed in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.8 Distribution ofHz as a function of normalized coordinate η for a 4c×4c×2c

sample and different iteration numbers n (A: analytical, N: numerical).

The analytical solution is obtained from the first-order solution in [14].

Inset: continuity of Bz (normal component of B) across cavity boundaries

( |η| > 1 is air, |η| < 1 is YIG). . . . . . . . . . . . . . . . . . . . . . . . 1376.9 Component of electric field along the direction of applied field inside the

rectangular cavity on the midplane cross section. . . . . . . . . . . . . . . 137

6.10 FFT ofEz(t). Resonant frequencies in the cases of uniform and nonuniform

internal field distributions of H are shown for comparison. . . . . . . . . 138

6.11 Dispersion curve and resonance frequencies. The dispersion curve is plot-

ted by using (6.20). The circles show the resonances that are predicted

by the FDTD method shown in Figure 6.10. They satisfy the condition

βlz/π = n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.1 Berenger’s proposed PML for the two-dimensional problems. Outer bound-

aries are perfect electric conductors. In the corners the layers overlap. . . 150

7.2 3D region surrounded by the PML layers. The outer surfaces are per-

fect electric conductors. Only four PML out of the six layers are shown.

The interface of these layers with the main simulation domain are at the

surfaces y = y1, y = y2, x = x1, and z = z2. In overlapping regions the

zero conductivities are replaced by the relevant conductivities of the layers

making the overlapped regions. . . . . . . . . . . . . . . . . . . . . . . . 151

7.3 Incident wave interacting with the PML region. . . . . . . . . . . . . . . 152

7.4 Microstrip line with YIG and GGG substrate. . . . . . . . . . . . . . . . 155

7.5 Magnetic field H right below the microstrip line inside the YIG substrate.

µ0Ms = 0.178T and µ0Ha = 0.5T . . . . . . . . . . . . . . . . . . . . . . . 156

7.6 Cosine of the angle between M and H, θ in the YIG substrate on a plane

just below the strip conductor. . . . . . . . . . . . . . . . . . . . . . . . . 156

xv

7.7 Magnetization magnitude as a function of time at a given point within the

YIG substrate. The tip of vector M is always on a sphere with radius Ms. 157

7.8 Input Gaussian pulse. The pulse width is less than 0.1 nanosecond. . . . 159

7.9 Fourier spectrum of the input pulse. . . . . . . . . . . . . . . . . . . . . . 159

7.10 Scattering parameter S12 as a function of frequency. Two different cases of

applied field and YIG width along the direction of propagation are shown.

The solid lines show the results for the nonuniform internal field distribution.161

7.11 Component of dynamic magnetization in the x direction, mx(t). . . . . . 162

7.12 mx(t) is zoomed in between 3.5 to 5 nanoseconds. . . . . . . . . . . . . . 163

7.13 Component of dynamic magnetization in the z direction, mz(t). . . . . . 163

7.14 mz(t) is zoomed in between 3.5 to 5 nanoseconds. . . . . . . . . . . . . . 164

7.15 Component of dynamic magnetization in the y direction, my(t). . . . . . 164

7.16 my(t) is zoomed in between 3.5 to 5 nanoseconds. . . . . . . . . . . . . . 165

7.17 Magnetization precession about the direction of internal field H. Static

internal field H and M are with a very good approximation in the y

direction. Dynamic contribution m is mainly in the xz plane. M + m

rotates about the y axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.18 Fourier amplitude of mx for different frequencies. . . . . . . . . . . . . . 166

7.19 Fourier amplitude of mz for different frequencies. . . . . . . . . . . . . . 166

7.20 Fourier amplitude of my for different frequencies. Compared to the fourier

amplitude for mx and mz, the magnetization excitation in the y direction

is negligible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.1 Edgemode isolator. While propagating along the line, the electromagnetic

field shows transverse displacement normal to the direction of propagation. 171

8.2 Input Gaussian pulse propagates through the input strip line with the

nonmagnetic dielectric substrate. The input line resembles the 50Ω coaxial

line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.3 Input Gaussian pulse propagates through the input strip line. A differentt

view shows the entrance to the wider strip line with magnetic substrate. 180

8.4 The Gaussian pulse propagates through the edgemode isolator. Trans-

verse displacement is clearly achieved. The applied external field is in the

positive y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xvi

8.5 The Gaussian pulse is transfering to the output 50Ω line with dielectric

substrate. The transverse field displacement is being removed and the

uniform Gaussian shape is gradually retrieved. . . . . . . . . . . . . . . . 182

8.6 Propagation of the Gaussian pulse through the edgemode isolator. The

External applied field Ha = −Haj is in the negative y direction. Thechange in the direction of applied field changes the direction of field dis-

placement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.7 The Gaussian pulse propagating in the edgemode isolator subject to an

applied field in negative y direction. This is the same pulse as in Figure 8.6

from a different view to show total mirror geometry compared to Figure 8.4.184

8.8 Propagation of the Gaussian pulse in the reverse direction compared to

Figure 8.4. The applied field like in Figure 8.4 is in the positive y direc-

tion. Changing the direction of propagation changes the direction of field

displacement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.9 Gaussian pulse propagating in the edgemode isolator in the negative z

direction. Like in Figure 8.8, the applied field is in the +y direction. An

observer that has changed his position from the y axis in Figure 8.4 to the

opposite mirror axis in the diagonal direction in this figure, should not see

any difference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.10 Field decay in the transverse direction. Dashed line is based on the ap-

proximate analytic results given by Hines [8]. The solid line is the result of

FDTD simulation. In the analytic approach, magnetic wall boundary con-

dition is applied at the edges of the strip conductor normal to the metallic

plate at x = 8mm and x = 12mm. Analytic solution outside this interval

is not given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

xvii

Nomenclature

α, β, γ Direction cosines in the x, y and z directions respectively

χ̄ Susceptibility tensor

µ̄ Permeability tensor

χ Susceptibility

γ Gyromagnetic ratio

h̄ Reduced Planck’s constant, h/2π

λ, η Constants showing the extent of damping

∏

Effective field

B Magnetic induction

D Electric displacement

F Mechanical force

H Magnetic field intensity

h′i Lorentz local field

Ha Applied magnetic field intensity

Hi Internal magnetic field intensity

JM Magnetic charge current density

J Current density

L Orbital angular momentum

xviii

M Magnetization

N Mechanical torque

P Variable introduced in equation (3.31)

p Momentum

Ps Variable introduced in equation (3.31)

S Spin angular momentum

v Unit vector in the direction of magnetization M

µ0 Permeability of free space

ω Angular frequency

ω0 Angular frequency defined in equation (7.21)

ωM Angular frequency defined in equation (7.20)

ρ Volume electric charge density

ρs Surface electric charge density

ρ∗s Surface magnetic pole density

ρ∗v Volume magnetic pole density

σ Electric conductivity

σ∗ Magnetic conductivity

A Helmholtz (free energy) function

Am Magnetostatic contribution to the Helmholtz function

fM Observed lower limit of the frequency bandwidth

fmax Upper limit of the frequency bandwidth

fmin Lower limit of the frequency bandwidth

G Free Gibbs energy

xix

g Lande factor

h Planck’s constant

J Constant showing the extent of overlap of electronic wave functions

K1, K2 Constants in the anisotropy energy

Ms Saturation magnetization

Nf Fluxmetric demagnetizing factor

Neff Effective demagnetizing factor

Nm Magnetometric demagnetizing factor

Nzz Demagnetizing factor in the z direction

Q Heat exchanged between a thermodynamic system and its environment

S Entropy

S12 Scattering parameter between ports 1 and 2

T Absolute Kelvin temperature

U Internal energy of a thermodynamic system

Um Internal energy contributed by the magnetostatic interaction

W Work on a thermodynamic system

wa Anisotropy energy density

we The density of excess exchange energy

ABC Absorbing boundary condition

FDTD Finite difference time domain

FEM Finite element method

FETD Finite element time domain

GGG Gadolinium Gallium Garnet

xx

LLG Landau-Lifchitz-Gilbert equation

MSW Magneto static wave

PML Perfectly matched layer

TEM Transverse electromagnetic

YIG Yttrium Iron Garnet

xxi

Chapter 1

Introduction

Microwave ferrite devices permit the control of microwave propagation by a static mag-

netic field [1]. Unlike magnetic metal, ferrite is a magnetic dielectric that allows propaga-

tion of electromagnetic waves. The Landau-Lifchitz-Gilbert (LLG) equation governs the

dynamics of the magnetization vector M inside the ferrite and determines its response

to the existing time varying magnetic field intensity H [2][3]. In the linear regime, this

response is given by the susceptibility tensor χ̄, M = χ̄H, which in turn determines the

Polder’s permeability tensor µ̄, B = µ̄H [4], where B is the magnetic induction field. It

turns out that the permeability tensor is not diagonal and its elements depend on fre-

quency. Moreover, the diagonal elements are not equal. While the frequency dependence

causes a dispersive behavior, the difference between the diagonal elements makes the ma-

terial anisotropic. On the other hand, the off diagonal elements add another peculiarity

to the ferrite behavior. In this case, the magnetic field component in, for example, the

y direction, Hy, contributes to the magnetization in the x direction, Mx. These complex

behaviors provide a wide range of applications for the ferrite materials while at the same

time make their analysis more difficult.

Ferrites are used in two classes of devices [5]. The first class includes signal con-

trol devices like circulators and isolators in which the nonreciprocal behavior of wave

propagation is essential. The second class includes signal control devices such as phase

shifters, tunable filters, delay lines, switches and variable attenuators. In the second

class, nonreciprocity does not play an important role in the device performance. An

alternative semiconductor based device that satisfies similar requirements exists for most

of the devices in the second class. However, this is not the case for the first class in

which nonreciprocity plays an important role. It therefore appears much more certain

1

Chapter 1. Introduction 2

Figure 1.1: Y-junction circulator subject to applied field in axial direction. The ferrite

disks are backed by the metallic plates at both ends. The middle plate is connected to

the strip lines that are separated by 120◦.

that these devices will continue to be used extensively in microwave technology.

Since its introduction in 1964, the Y-junction Circulator has been used as a three

port device to rout electromagnetic signals in microwave circuitry [6][7]. It is made of

two ferrite discs of the same diameter that are separated by a metal disk of negligible

thickness, Figure 1.1. Backed by similar metal discs at both ends, they form a cylindrical

resonant cavity. The middle metal disc is connected to three conductor strips separated

by 120◦. A DC magnetic field normal to the ground plane in the axial direction is used

to rout the signal. If the magnitude of the DC field in conjunction with the disc radius is

chosen appropriately, the signal entering from the first port exits the second port while

the output from the third port is zero [4]. The planar geometry of the microstrip line

circulators has the convenience that they can be integrated easily in the planar microwave

circuits.

Like circulators, isolators are used to guide the electromagnetic wave propagation.

They permit wave propagation in one direction and block its propagation in the reverse

direction. Not all isolators utilize a planar geometry. In contrast with resonance isola-

tors or field displacement isolators that are using waveguide geometries, the edge mode

isolator benefits from planar geometry and therefore can be integrated into microwave


Figure 1.2: Edgemode isolator. While propagating along the line, the electromagnetic

field shows transverse displacement normal to the direction of propagation.

planar circuits. Arguably, edge mode isolators are the most promising isolator type for

broadband applications [1].

Hines introduced an edge or peripheral mode isolator in 1974, Figure 1.2. At the same

time, he developed a theoretical description of its behavior, based on Maxwell’s equations

and certain physical assumptions [8]. This description predicts sideways displacement

of the electromagnetic field in the ferrite substrate, normal to the direction of wave

propagation. Specifically, the zeroth order mode resembles a Transverse Electromagnetic,

or TEM, wave that propagates along the line while its amplitude decays exponentially

transverse to the strip conductor. The direction of sideways displacement changes when

the direction of the applied magnetic field, which is normal to the ground plane, is

reversed. The direction of field displacement also changes for a given direction of applied

field when the wave propagates in the opposite direction. One side of the strip conductor

is shunted to the ground plane by a thin layer of finite resistance. In the case of a wide

strip conductor, wave propagation shows a clear nonreciprocal behavior. The insertion

loss when the wave propagates in the forward direction and interacts weekly with the

resistive sheet is lower than when it is propagating in the reverse direction.

The developed theory predicts a wide operating frequency band [8]. The high fre-

quency limit fmax is imposed by the propagation of the first higher order mode. Although


the theory given by Hines does not predict a lower frequency limit, his experiments show

that there is actually a lower limit fmin = fM for the device operation. Hines attributed

the observed lower frequency limit to the low field low frequency loss mechanism. This is

a well known mechanism that occurs when the magnetic material is partially saturated

[9][10]. In microwave devices with ferrite components, to avoid this loss mechanism, the

applied field is adjusted to produce the net internal field necessary to saturate the mate-

rial. However, this approach doesn’t eliminate the low frequency limit present in the edge

mode isolators. This shows that there is another physical process that is responsible for

the observed low frequency limit fM in the edgemode isolators. The quantitative descrip-

tion of the underlying physical phenomena has been the subject for further investigation

[5].

In order to comply with the current trends towards miniaturization and higher fre-

quency bandwidths, it is necessary to have a better understanding of microstrip lines with

ferrite substrate [1]. There are some proposed mechanisms to account for the observed

low frequency limit that are introduced briefly in the next section. It is the purpose of this

thesis to verify their validity. It turns out that the existing tools to analyze and design

these devices need to improve accordingly. As will become clear, the complexity of the

problem makes it impossible to use analytical approaches and therefore the use of numer-

ical methods is inevitable. Because the approach used in this thesis is mostly theoretical,

it is necessary to use sound assumptions and powerful numerical methods. The Finite

Difference Time Domain method, FDTD, has developed as an efficient method in the

time domain during the past 40 years [33]. With the available memory and speed in per-

sonal computers, allowing using finer grids in the order of 1% of the smallest wavelength

propagating in the structure, the borders between analytical and numerical methods are

disappearing.

1.1 Loss Mechanisms in Microwave Ferrite Devices

Different mechanisms have been proposed to describe the onset of losses responsible for

the previously mentioned lower frequency limit. Here a summary of these mechanisms is

given.

• Non-uniform bias field

The static bias field internal to a ferrite sample is in general different from the


external applied field, Ha. The difference is caused by the volume magnetic pole

density and/or the surface magnetic pole density induced on the boundary inter-

faces. The internal magnetic field Hi is affected by the shape of the ferrite sample

and its orientation with respect to the applied magnetic field. Even in the case of

uniform applied field, for non ellipsoidal bodies like ferrite platelet substrates in

rectangular or cylindrical geometries, the internal field is nonuniform. For smaller

samples, the nonuniformity becomes more prominent. A nonuniform internal field

distribution has been proposed to be one of the sources of the observed losses at

the low frequency limit [1][5]. It should be possible to minimize this type of losses

by careful device design. The methods developed in this thesis will enable one to

examine the quantitative impact of such modifications.

• Magnetostatic waves

When the spin waves have a wavelength comparable to the size of the ferrite speci-

men, they are called Magneto Static Waves, MSWs [11]. Magnetostatic waves that

are excited at the interfaces between the ferrite and the conducting plates or other

adjacent dielectric substrates are proposed to contribute to the ferrite losses in the

microwave region. However, to dates there is no quantitative analysis in the case

of edgemode isolators to support this proposition [1][5]. An objective of this thesis

is to develop a numerical modeling approach that will allow such an analysis to be

undertaken.

• Low field low frequency loss

One important process occurs when magnetic domains of opposite polarity are

present in the material, i.e. when the material is unmagnetized or partially magne-

tized. Under these conditions, an unusual type of ferromagnetic resonance occurs

where the gyroscopic motion of the domain magnetization generates magnetic poles

at the domain boundaries. This in turn causes magnetic losses below a character-

istic frequency fM [12]. This mechanism was considered by Hines to justify the

observed loss at the lower frequency limit of the edgemode isolators. However,

usually the internal field is adjusted to saturate the ferrite substrate and therefore

it seems that low field loss mechanism cannot be a source for the observed low fre-

quency limit. This motivated the research by Schloemann and others to investigate

other possible mechanisms causing the observed lower frequency limit [1].


• Intrinsic low field loss

Intrinsic low field loss is another mechanism that has been proposed theoretically

but has not been verified experimentally [13]. As the name suggests, this is an

intrinsic mechanism which is not avoidable. However, theoretical analysis shows

that the effect of this mechanism is quite small and can be ignored [13].

1.2 Literature Review: Numerical Methods and Pro-

posed Designs

In this section, a review of the previous work that is related to this research is given. It

includes these different but related areas:

• Methods that are developed to obtain nonuniform internal field distributions Hiand M.

• FDTD formulations to simulate wave propagation in microwave devices that utilizeferrimagnetic materials.

• Attempts to give quantitative descriptions of different physical mechanisms thatmight be responsible for losses in magnetic substrates at low frequencies. Emphasis

is on the planar geometries like microstrip lines, edgemode isolators and Y-junction

circulators.

• Ideas to design microwave ferrite devices with improved frequency bandwidth. Theedgemode isolator and Y-junction circulator are considered in particular.

The first two parts are related since the FDTD code uses the static nonuniform internal

field distribution as the initial input for the fields. The components of the electromagnetic

wave are dealt with as time varying superpositions on the static fields. Therefore, the

method to obtain the nonuniform internal field distribution and the FDTD formulation

are considered together and should be compatible. They are used to investigate the effect

of different proposed loss mechanisms. The methods developed will help in designing

microwave ferrite devices with appropriate characteristics. Specifically, improving the

frequency bandwidth of edgemode isolators is of particular interest.


1.2.1 Nonuniform Internal Field Distributions Hi and M

The magnetostatic boundary value problem to be solved for the internal field distributions

is presented in chapter 2. It is assumed that the magnetic material is saturated in the

direction of the local internal field. Therefore the appropriate constitutive equation is

M =MsHi/|Hi| in whichMs is the saturation magnetization. This is a valid assumptionas long as the material is not very small in the range of one micron, 1µm, and the

hysteresis effects can be ignored. In the former case, one may ignore the exchange

interactions and the latter case is valid if the applied magnetic field is much larger than

the coercive force, which is the case for microwave ferrite devices with thin hysteresis

loops and Ha > Ms [14].

Joseph et. al. [14] have used a series expansion for the magnetic scalar potential

in ascending powers of Ms/Ha to obtain the internal magnetic field distribution Hi and

magnetizationM. They have found a mathematical expression for the first order solution.

Second order solutions are given for the special cases of infinitely long prisms or cylinders.

It was shown that for nonellipsoidal bodies like cylindrical or rectangular geometries, the

demagnetizing factors are a function of position. Moreover, it was shown that the first

order solution satisfies the sum rule for the demagnetizing tensor [15]. According to this

rule, the sum of diagonal elements in the demagnetizing tensor is 1. Although it was

assumed that the material is saturated in the direction of local internal field, the first

order solution for the H and M do not satisfy this assumption. The first order solution

corresponds to uniform magnetization in the direction of applied field.

In another approach, the magnetized material was divided into small cubical elements

[16]. Each element was considered as a source of magnetic field. Because the elements

were very small, it is reasonable to assume that the magnetization is uniform in each

element and therefore only the magnetic surface pole densities contribute to the magnetic

field. Adding the contributions from all elements at a given point inside the material gives

the demagnetizing field at that point. By assuming uniform magnetization, it is possible

to define the demagnetizing tensor and find its elements as a function of position. It

can be shown that this is a first order approach and the results agree with the first

order analytic solution reported in [14]. Although for rectangular geometries the cubical

cells are appropriate, in the case of cylindrical geometries curved boundaries have been

modeled with an error that depends on the size of elements.

There have been approaches to obtain internal field distributions which are based on


a finite element method, (FEM). The FEM is used to solve the integral equation for

magnetization M [17]. In the case of non-linear material properties, the resulting system

of equations are considered to be quasi linear and the Newton-Raphson method is used.

Alternatively, FEM is used to solve the Poisson’s equation for total scalar magnetic po-

tential, ϕ, subject to the appropriate boundary conditions [18][19]. In another approach,

the magnetostatic energy is formulated in terms of magnetic scalar potential and is mini-

mized to obtain ϕ [20]. In both of these methods, one needs to take numerical derivative

of ϕ to obtain the internal fields. This numerical differentiation introduces error in the

internal field distribution calculations.

In magnetic measurements, instead of precise distributions of internal magnetic field

Hi and magnetization M, one is interested in the average response of the magnetic

material to the applied field Ha. In these cases, fluxmetric Nf , and magnetometric Nm,

demagnetizing factors, corresponding to different approaches in takeing the field averages,

are defined [21]-[24]. In a series of papers, results for Nf and Nm in graphical and tabular

formats are reported [24]-[28]. Both cylindrical and rectangular geometries with a wide

range of aspect ratios and susceptibilities are considered. It is assumed that the magnetic

susceptibility χ is constant. Therefore ∇ ·M = 0 and the main problem is reduced tothat of obtaining the surface magnetic pole density ρ∗s. In these papers, the main focus

is on the average response of the magnetic material and the field distributions are not

obtained. However, given the magnetic surface pole density, it is possible to obtain the

demagnetizing field and the internal magnetic field distribution. It can be shown that in

this approach, the magnetization distribution and the magnetic field do not satisfy the

constitutive equation M = χH.

In [29], the effective demagnetizing factorNeff for non-ellipsoidal bodies is defined and

measured experimentally. The experiment is based on the measurement of the in plane

and out of plane major hysteresis loops using a vibrating sample magnetometer. Neff in

the z and x directions for the rectangular geometries and in the axial and radial directions

for the cylindrical geometries are obtained. These measured values are compared with

the average demagnetizing factors in rectangular and cylindrical geometries [14][22][24].

Good agreement for the case of rectangular materials is shown. For cylindrical geometries

an error up to 20% is reported.

In a different approach, micromagnetic methods are used to obtain the magnetiza-

tion distribution M near saturation [30]. The Landau-Lifchitz-Gilbert equation, LLG,

is solved iteratively to obtain the ground state of the magnetization near saturation.


Exchange and anisotropy effects are ignored and it is assumed that the change in mag-

netization direction is relatively slow from one cell to the next. It is also assumed that

the magnetization is parallel with the local internal field. To relate the demagnetizing

field and hence the internal field with the magnetization, the first order demagnetizing

tensor obtained in [16] and [30] is used.

1.2.2 FDTD Formulations

The Finite Difference Time Domain Method, FDTD, is an efficient numerical method

that has expanded significantly since its introduction in 1966 [31][32][33]. It is used in a

variety of scientific and engineering disciplines [33]. As a time domain method, it gives a

real time record of the physical process being studied. Once the program is run for enough

time steps and the steady state has been achieved, the frequency domain response can be

obtained by applying the discrete Fourier Transform. Constructing the main loop that is

used to update the field components in time needs considerable attention in the case of a

complex material response such as is observed in ferrites. There are also other important

issues related to the simulation of a physical process in the time domain. These include

satisfying interface boundary conditions, choosing an appropriate Absorbing Boundary

Condition ABC, source excitation and managing numerical dispersion and stability.

Farahani and Konrad have shown that the FDTD method is a special case of the

Finite Element method in the time domain, FETD, with rectangular edge elements [34].

Therefore, it is easier to apply interface boundary conditions compared to other FETD

methods based on nodal elements. An ABC is needed because it is not possible to

include all space in the simulation of open boundary problems. This boundary surrounds

completely the simulation domain and should ideally have zero reflection and absorb

completely the outgoing waves. In some cases, like microstrip lines, it also simulates the

role of matched loads connected to the output ports. In this thesis, both first order Mur’s

ABC and Perfectly Matched Layer, PML, are used.

Anisotropic materials where one of their constitutive parameters depends strongly on

frequency are called Gyrotropic. In the case of Gyrotropic magnetic materials, elements

of the susceptibility tensor χ̄ or the permeability tensor µ̄ show strong dependence on

frequency. Plasmas and ferrites in the presence of an applied static magnetic field are

examples of gyrotropic media. In simulating wave propagation in gyrotropic media like

plasmas and ferrites, the presence of off diagonal elements adds more complexity com-


pared to other dispersive and anisotropic media. In this case, one needs to add more

sampling points to the Yee cell and spatial averaging is inevitable. Kunz and Luebbers

were the first to simulate wave propagation in gyrotropic media [35]. They used Polder’s

permeability tensor to characterize the medium’s response in the frequency domain. By

using convolution and appropriate time domain representations for susceptibility compo-

nents, they could formulate a method to solve problems involving ferrites and plasmas.

Although it is possible in principle to extend their method to three dimensions, they

only used the method to find the reflection and transmission coefficient for a one dimen-

sional problem. They obtained these coefficients for waves with right and left circular

polarizations incident on an infinite air/ferrite interface.

Polder’s permeability tensor is obtained by assuming a steady state solution of the

form A(r)ejωt and linearizing the LLG equation [4]. This tensor gives the response of

the magnetic medium in the frequency domain. Instead of going from the time domain

represented by the LLG equation to the frequency domain represented by the permeabil-

ity tensor, and then returning back to the time domain by means of convolution, it is

possible to consider the LLG equation in consistency with Maxwell’s equations both in

the time domain. This method was used to obtain phase and attenuation constants of

the TE10 mode of a rectangular waveguide loaded with ferrite [36][37]. In this method, to

consider the LLG equation together with the Maxwell’s equations in 3D, it is necessary

to modify the original Yee’s cell by adding more sampling points for the magnetization

M and the magnetic field B components. This formulation was used later to simulate

wave propagation in a microstrip isolator and stripline junction circulator [38][39]. In the

most recent approach, an equivalent circuit was obtained for the unit Yee cell and the

results for the geometries considered in [36] are reproduced [40].

All these approaches consider the case where, in the absence of a propagating electro-

magnetic wave, the applied fieldHa, the magnetization M and the internal magnetic field

Hi are parallel and uniform inside the magnetic material. It is assumed that the applied

field is along one of the coordinate axes and that the magnetic medium is saturated.

Boundary conditions on B and H at the ferrite dielectric interfaces are used to obtain

a uniform internal field Hi. Based on these simplifications, it is clear that these FDTD

formulations cannot be used to simulate wave propagation in magnetic media where all

the relevant static field vectors are nonuniform.

By using the magnetic field induction B instead of H in the LLG equation, it is pos-

sible to consider cases in which the applied field is not parallel with the coordinate axes.


Resonant frequencies of a rectangular cavity filled with ferrite and scattering parame-

ters of a microstrip line with YIG/GGG substrate were obtained by using this approach

[41][42]. In both of these cases, Ha was not restricted to any specific direction. However,

the internal field Hi was obtained by using the interface boundary conditions and was

assumed uniform inside the magnetic material. The ferrite material was considered to

be saturated in the direction of local internal field. In this method, the update equations

in the main FDTD loop were given in terms of total fields. This means that at each

time step, the total magnetic field components that are composed of static and dynamic

parts, are updated. The dynamic parts are the components of the superposed propa-

gating electromagnetic wave. This formulation has a problem when the internal static

magnetic field is nonuniform. In this case, even in the absence of an incident wave, the

nonuniformity of the static internal field causes the displacement current density D to

change with time. Regardless of the accuracy, to obtain the nonuniform static internal

field distribution, the discretization results in a nonzero integral of H around any closed

loop. This in turn causes a change in the displacement vector D and thus the electric

field in time.

In [43], an iterative method based on the magnetic scalar potential is used to obtain

the nonuniform internal field distribution. In this method, more sampling points are

introduced into the YEE cell. By obtaining the magnetic scalar potential at these new

points, the static magnetic field distribution is obtained by differentiation. Scattering

parameters of a microstrip line with ferrite substrate where the nonuniformity of the

internal field distributions are considered are obtained.

1.2.3 Designs to Improve Frequency Bandwidth

In section 1.1 the proposed loss mechanisms were introduced briefly. It seems that the

nonuniformity of the internal field distribution and the excitation of magnetostatic waves

are the two mechanisms that are responsible for the losses at the low frequency limit

[45]. In a series of graphs, Schloemann has shown the effect of the internal bias field

and its nonuniformity on the insertion loss of the edgemode isolator [5]. It is shown

that when the magnitude of the internal bias field increases, the insertion loss increases

too. This is why the microwave ferrite devices work at a low bias internal field that

is high enough to keep the material saturated. When the applied field is normal to

the broad face of the ferrite platelet, Figure 1.2, the internal field is almost uniform deep


inside. The nonuniformity is appreciable near the boundaries. Therefore the propagating

wave interacts with the nonuniformity along the direction of wave propagation when it

reaches the output port. In this port, there is an interface between the ferrite platelet

and an adjacent dielectric like air. Based on a first order theory [14], Schloemann has

calculated the nonuniform internal field near this interface. It is shown that the insertion

loss for lower frequency components is higher and increases as the wave approaches the

interface where it encounters more field nonuniformity [5]. These results are the basis for

the proposed magnet designs that are supposed to yield week uniform internal fields to

achieve higher frequency bandwidths.

Unlike the nonuniformity of the internal field distribution, there is no quantitative

analysis to show the effect of magnetostatic waves on the low field low frequency loss in the

case of the edgemode isolator. When the applied field is parallel with the strip conductor

and the direction of wave propagation, it is shown that Magneto Static Surface Waves,

MSSWs, are excited and their frequency is given [44]. In this geometry, the internal bias

field is parallel with the interface between the ferrite substrate and the ground planes,

strip conductor and the air or other possible dielectrics above the strip conductor. Based

on this result, Schloemann has anticipated that in the case of an edgemode isolator where,

at the output port, the applied field is parallel with the interface between the ferrite and

the adjacent dielectric, Figure 1.2, the same physical situation as in [44] is realized and

the interface magnetostatic waves are excited [1].

In the more general case where the magnetic substrate is composed of two different

ferrite platelets with different saturation magnetization, Figure 1.3, the analysis in a

recent patent, [45], based on the results given in [44], shows that the frequency of the

excited magnetostatic waves at the ferrite/ferrite interface is given by:

f = γµ0Hi +1

2(fM1 − fM2) (1.1)

in which Hi is the internal field at the interface and fM1,2 is the characteristic frequency

corresponding with the saturation magnetization Ms1,2 of each medium:

fM = γµ0Ms (1.2)

Here, γ is the Gyromagnetic ratio and µ0 is the permeability of the free space. Equations

(1.1)-(1.2) can be used to obtain the frequency for magnetostatic waves that are excited

at the interface between a ferrite layer and a dielectric with Ms = 0. In this case the


frequency of the excited magnetostatic wave at the ferrite dielectric interface would be:

f = γµ0Hi +1

2fM (1.3)

The first order demagnetizing factor of a ferrite substrate in the direction normal to its

broad face varies from 1 at the center to around 0.5 close to the perimeter. Therefore

to achieve low internal field at the center of the ferrite platelet, the uniform applied field

should be close in magnitude to Ms. The internal field is given by:

Hi = Ha −NzzMs (1.4)

where Nzz is the demagnetizing factor in the z direction and the broad face of the

ferrite platelet is parallel with the xy plane. The applied field is in the z direction. As

stated above, Nzz ≈ 1 implies that the internal field is close to zero at the center of theferrite platelet if the applied field is close to the saturation magnetization. However, for

this arrangement, the internal field near the perimeter will be close to 0.5Ms. Equations

(1.2)-(1.4) predict the frequency of the excited magnetostatic waves to be fM which is the

observed experimental low frequency limit. These considerations have led Schloemann to

propose a new design for the edgemode isolator to increase its frequency bandwidth [45].

In the proposed design, instead of using a single ferrimagnetic material in the substrate,

a series of parallel slabs normal to the direction of wave propagation is used. In one of

these configurations the substrate is composed of a ferrite platelet with higher saturation

magnetization at the center and two similar ferrites of lower saturation magnetization

at the input and output ports, Figure 1.3. The ratio of saturation magnetizations is

Ms1/Ms2 = 2. Equations (1.1)-(1.3) predict that in both of the ferrite/ferrite and the

ferrite/dielectric interfaces the frequency of the excited magnetostatic surface waves is

given by:

f = γµ0Hi +1

4fM1 (1.5)

where fM1 = γµ0Ms1 and Ms1 is the saturation magnetization of the central layer. By

using appropriate magnetic poles to reduce the internal field in each magnetic layer

compared to the lowest saturation magnetization in the layered structure, in each layer

γµ0Hi ≈ 0 and (1.5) suggests that a lower operating frequency of 14fM1 compared tofM = fM1 can be obtained.

These recent ideas to decrease the lower frequency limit of the edgemode isolator

were not verified experimentally. The theoretical background is an extension of the in


Figure 1.3: Strip line edgemode isolator as proposed by Schloemann to increase the

operating frequency bandwidth. The central ferrite layer has a saturation magnetization

which is two times larger than the saturation magnetization of the input/output ferrite

substrates. In the strip line geometry the central strip conductor is positioned between

two ferrite platelets and two metallic ground planes at the top and the bottom. This is

in contrast with the microstrip line geometry in which only one metallic ground plane is

present, Figure 1.2. The ferrite layers in contact with the high permeability poles and the

ground metallic plates ensure uniformity of the internal field inside the guiding structure

between metallic plates. The high permeability poles and the permanent bar magnet

constitute a magnetic circuit that provides the applied field in the z direction normal

to the ground planes and ferrite substrates. The electromagnetic signal propagating in

the y direction, passes through ferrite layers. The central layer has a higher saturation

magnetization.


plane magnetization where the applied field is parallel with the strip line conductor [44].

Although they seem reasonable, there are some points that one needs to consider. In the

proposed design, there is no reference to the actual dimensions of the ferrite layers and

the geometry, as a whole. Physical properties such as the saturation magnetization or the

applied field distribution are not given. In other words, the design is quite qualitative.

An appropriate numerical method that is able to obtain the internal field distributions in

the cases where the applied field is not uniform and the substrate is composed of a layered

structure of nonlinear materials had not been developed. Therefore it is not clear how

much uniformity of the internal field is achievable given a proposed magnet pole design

or the device layered structure. Also, using different ferrite layers might cause reflections

when the electromagnetic signal passes from one layer to the other. These reflections can

deteriorate the device behavior and are not taken into account in the proposed design.

These are all questions that remain open.

The proposed design follows earlier works to improve frequency bandwidth of strip

line circulators. Two hemispherical caps of the same magnetic material that is used in

the central magnetic discs of a Y-junction microstrip circulator are used to produce a

uniform internal bias field [46]. By placing these caps over the two faces of the central

thin magnetic discs a magnetic sphere is formed. As an ellipsoidal body, the field inside

the magnetic sphere, including the ferrite discs, is uniform. By adjusting the applied

magnetic field one can reduce the magnitude of the uniform internal field compared to the

saturation magnetization. This approach has resulted in a broader frequency band width

for the proposed circulator which has been verified experimentally. In another approach

that is used again for a Y-junction circulator, the central ferrite disc is surrounded by a

ferrite ring with lower saturation magnetization [47]. Again higher frequency bandwidths

are obtained.

The review given in this section shows that while it is possible to use different distri-

butions of the applied field or a combination of different magnetic materials to improve

the behavior of microwave ferrite devices, the appropriate numerical methods to obtain

the internal field distributions in these cases have not been developed appropriately. The

FDTD formulations also need to improve. They should be able to integrate with the

method that is used to find the nonuniform internal field. There are also some con-

siderations about stability and the correct choice of ABC. In the context of the FDTD

formulation, when one talks about the stability condition, usually it refers to numerical

stability. It is shown that there is another type of stability that might have a physical


origin that relates to the stability of the initial static internal field distribution. Although

it is claimed that there are some PML methods that have been developed to terminate

simulation domains involving arbitrarily anisotropic and dispersive media, they have not

been applied to terminate open boundary problems involving gyrotropic media. The

nonuniformity of the internal field adds more problems because this is equivalent to a

permeability tensor with elements that vary with position. Similarly, the attempts to

give a quantitative description of different loss mechanisms in the case of edgemode iso-

lator are not yet complete. The quantitative description of the magnetostatic wave’s

excitation is given for in plane internal field. Its continuation to normal bias field needs

verification.

1.3 Thesis Objectives

To verify the effect of nonuniform internal field on the magnetic losses and the behavior

of the microwave devices with magnetic elements in general, it is necessary to have first of

all an accurate solution to the internal field distributions. At the design stage, one might

need to consider a variety of nonuniform applied field distributions or a combination of

different magnetic materials. The methods to obtain internal field distributions should

be able to consider cases involving nonlinear media as well. However, the existing FDTD

formulations do not handle appropriately the cases in which the internal fields are not

uniform. Therefore, it is necessary to have an improved FDTD formulation that inte-

grates appropriately with the methods that are proposed to obtain nonuniform internal

field distributions. Extending the PML as the most recent and efficient ABC to termi-

nate simulation domains involving gyrotropic media with local variation of permeability

tensor elements is another problem to be solved. Given the appropriate numerical tools,

one can examine the effect of the proposed loss mechanisms and visualize the presence

of magnetostatic waves. It would also be possible to examine the proposed designs to

achieve higher frequency bandwidth or alternatively come up with a new design.

According to these considerations, the thesis objectives are as follows:

• To obtain nonuniform internal field distributions Hi and M in the ferrite substratessubject to a nonuniform applied field. The substrate might be composed of differ-

ent layers of ferrimagnetic materials with different saturation magnetizations Ms.

The proposed method could be applied to the nonlinear case where the magnetic


material is saturated in the direction of local internal field. Both cylindrical and

rectangular geometries are considered.

• To improve the existing FDTD formulations. The improved FDTD formulationintegrates appropriately with the method that is used to obtain the internal field

distributions. It is able to simulate wave propagation in ferrite media with nonuni-

form internal field distributions.

• To validate the FDTD formulation; wave propagation in microstrip lines with mag-netic substrate and resonant cavities filled with magnetic material are simulated.

The simulation results can be used to verify both the FDTD formulation and also

indirectly the solution to the internal nonuniform field distributions.

• To simulate wave propagation in the edgemode isolator to verify the effect of differ-ent proposed loss mechanisms. Given the appropriate method to find the internal

fields in the most general case of nonuniform nonlinear media and with an efficient

FDTD formulation, one can verify the performance of various edgemode isolator

designs that are proposed to increase their frequency bandwidth.

1.4 Contributions

• An iterative method has been used to obtain nonuniform internal field distribu-tions inside nonuniform nonlinear magnetic media subject to a nonuniform applied

field [48][49]. The scope of cases that are considered is general. The nonunifor-

mity of χ can be caused by a layered structure of magnetic materials with different

constant susceptibilities or saturation magnetization. It can be caused by a non-

linear magnetization curve as well [49]. The nonlinearity where the ferrite media

are saturated in the direction of local internal field is of particular interest. Due

to their wide range of applications in microwave circuits, magnetic materials with

both cylindrical and rectangular geometries are considered.

When the applied field is close to the saturation magnetization, the iterative method

may have convergence difficulties. One can compare this with the Schloemann’s ap-

proach where the magnetic scalar potential is expanded in a power series in Ms/Ha

[14]. When Ha ≈ Ms, the series solution converges much more slowly. Although


even the first few iterations yield a solution that is much more accurate than the

first iteration results, one can use successive relaxation to improve convergence [67].

The distinctive feature of the iterative approach is that the results are supported

by a uniqueness theorem in magnetostatics. It is shown that the series of solutions

Mn,Hn, where n is the loop iteration number, minimize consecutively the given

functional WM and better satisfies the constitutive equation M = χH after each

iteration. Therefore, according to the given theorem in magnetostatics, one can

claim that the unique magnetization distribution is obtained.

This is specially important since in microwave magnetic devices it is common to

use an applied magnetic field which is close to saturation magnetization. In cases

like this, higher order solutions as given by Schloemann’s series solutions, are not

negligible. While the first iteration yields the first order analytic solution by Schloe-

mann, the given theorem validates the higher order iteration corrections. These

higher order corrections are usually ignored in the literature [5].

• The iterative approach has been used to obtain the average demagnetizing factorsboth for linear and nonlinear nonuniform magnetic materials subject to a nonuni-

form applied field [49]. This is a new approach that unlike the existing methods is

not restricted to linear and uniform materials.

• A new approach to obtain the internal field distributionsM andH is introduced. Inthis new approach, the minimization theorem that is noted previously in this section

has been used in a finite element formulation [51]. In principle the method is not

restricted to a given class of magnetic materials. However in the thesis it is applied

to the cases where the magnetic susceptibility is constant. It is shown that the

results satisfy the constitutive equation and minimize the given energy functional.

Therefore, one is confident that the unique internal distribution is obtained.

It is possible to extend the finite element method to the cases where the magnetic

material is saturated in the direction of the local internal field. For this important

class of applications the new finite element method has the advantage that one

can ignore the nonlinear term in WM that includes susceptibility χ. This is in the

expense of adding a constraint term toWM to keep the magnitude of magnetization

|M| =Ms constant. This new term involves only unknown nodal components ofM.The necessary formulation is given where one can use the constrained minimization


methods to obtain the final field distributions. As an example, a simple gradient

method can be used. This new FE approach also opens a new and more powerful

method to calculate the demagnetizing factors. As a special case, the fluxmetric and

magnetometric demagnetizing factors for χ = 0, 1 are obtained and compared with

the published results. Using a slightly different version of the given magnetization

theorem, one can consider negative susceptibilities χ as well [50].

• An improved FDTD formulation that integrates with the methods which are usedto obtain nonuniform static internal field distributions is developed. In this for-

mulation the static and dynamic parts of the electromagnetic fields are separated.

The static parts enter the update equations as parameters. The main FDTD loop

is iterated to update the dynamic parts in time. The formulation is used to obtain

resonant frequencies of rectangular cavities filled with ferrite material [55]. The

method is also applied to obtain scattering parameters of a microstrip line with

YIG/GGG substrate [56]. This problem has previously been solved by ignoring

the nonuniformity of the internal field distribution. The uniform field distribution

was found by applying boundary conditions at the substrate interfaces. Compared

to the published results, the new results that are now based on the real nonuni-

form internal field distributions show better agreement with both experiment and

the theoretical expectations. The results validate both the nonuniform distribution

that is obtained by the iterative method and the improved FDTD formulation. The

distinctive feature of the FDTD formulation that is presented here is the fact that

it introduces the concept of physical stability. Since the initial static field distri-

bution is proved to be physically stable, the components of the electromagnetic

wave considered as small deviations superposed on the static fields will not make

the code unstable. Applying the PML ABC in the above example made it possible

to validate nicely the excitation of magnetization oscillations [56]. This is the first

time that these excitations have been visualized. They are excited in the same fre-

quency bandwidths where one observes a gap in the transmitted signal frequency

spectrum. Therefore, it is postulated that the excitation of magnetization oscilla-

tions is the source of the observed notch in the scattering parameter S12. These

magnetization oscillations are further damped because of the effects represented by

the damping coefficient α in the LLG equation.

• The FDTD method is used to simulate wave propagation in the edgemode isolator.


The key features of the device that are predicted by Hines using some simplify-

ing assumptions are verified successfully. Specifically the transverse displacement

of the propagating wave is observed. For a given direction of applied field, the

direction of field displacement reverses when the wave propagates in the opposite

direction. It also reverses when the applied magnetic field is switched in the op-

posite direction. The exponential decrease of the wave amplitude confirms the

theoretical expectation that was given by Hines [8]. This is the first time that a full

wave FDTD simulation of the edgemode isolator has been done and the key fea-

tures of the device are visualized. The results suggest that one can use the FDTD

method, in conjunction with the methods to obtain unique stable solution to the

internal field distributions, as a strong numerical approach in both analysis and

design stage. The approach can be used to analyze the device behavior, verify the

source of losses and improve the device performance by developing better designs

of the edgemode isolators.

1.5 Thesis Outline

Chapter 1 has introduced the research topic, its objectives and the contributions. Rele-

vant magnetic properties of ferrites are introduced briefly. These properties are utilized

in a wide range of planar geometries to route the electromagnetic signal. Y-junction

circulators and edgemode isolators are mentioned in particular. The proposed loss mech-

anisms at the low frequency limit were reviewed. Relevant previous works in developing

the appropriate numerical methods or increasing the frequency bandwidth of microwave

ferrite devices were described.

In Chapter 2, the fundamental magnetostatic problem to be solved for the nonuniform

static field distribution inside magnetic substrates subject to an external applied field is

defined. A more detailed review of the existing methods to solve for the internal field

distribution is given. This review helps to realize their shortcomings and the advantages

of the new approaches that are proposed in the thesis. The review also forms a basis to

compare and validate the results that are obtained by different proposed approaches.

Chapter 3 introduces the Landau-Lifchitz-Gilbert equation, LLG, that governs the

magnetization dynamics. The presentation in this chapter mainely follows the pioneer-

ing works of Brown, [50][52]. A variational approach is used to obtain the general form

of the equilibrium and stability conditions for the time evolution of M. This approach is


based on a thermodynamic concept that all natural processes are following the direction

in which the free Gibbs energy of the system decreases. To follow this approach, terms of

the free Gibbs energy of a system that is comprised of magnetic material and an external

applied field are obtained. A semiclassical approach is adopted. It turns out that the

equilibrium condition is the same as the constitutive equation M = χH. The important

contribution in this chapter is that by using a minimization theorem in magnetostatics,

it is proved that in the case χ = Ms/|H|, the constitutive equation is also the condi-tion for the stability. Therefore one has the full magnetization dynamics problem that

includes the LLG and Maxwell’s equations together. Propagation of the electromagnetic

waves in the magnetic media includes the interaction between the magnetization and the

electromagnetic field and thus both sets of equations should be considered at the same

time.

by alireza vashghani farahani a thesis submitted in ......5.2.4 fluxmetric and magnetometric...

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