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V E R S I O N 3 . 2

E l e c t r oma g n e t i c s M o d u l eM o d e l L i b r a r y

™COMSOL

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Electromagnetics Module Model Library© COPYRIGHT 1994–2005 by COMSOL AB. All rights reserved

Patent pending

The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from COMSOL AB.

COMSOL, COMSOL Multiphysics, and COMSOL Script are trademarks of COMSOL AB.

Other product or brand names are trademarks or registered trademarks of their respective holders.

Version: September 2005 COMSOL 3.2

C O N T E N T S

C h a p t e r 1 : E l e c t r o m a g n e t i c s M o d u l e M o d e l L i b r a r y

Introduction 2

Model Library Guide . . . . . . . . . . . . . . . . . . . . . 3

Typographical Conventions . . . . . . . . . . . . . . . . . . . 7

C h a p t e r 2 : T u t o r i a l M o d e l s

Electromagnetic Forces on Parallel Current-Carrying Wires 10

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 10

Model Definition . . . . . . . . . . . . . . . . . . . . . . . 10

Results and Discussion. . . . . . . . . . . . . . . . . . . . . 11

Modeling Using the Graphical User Interface . . . . . . . . . . . . 11

Inductive Heating of a Copper Cylinder 17

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 17




Modeling Using the Programming Language . . . . . . . . . . . . . 22

Permanent Magnet 24

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 24




C h a p t e r 3 : M o t o r s a n d D r i v e s M o d e l s

Linear Electric Motor of the Moving Coil Type 32

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 32

C O N T E N T S | i

ii | C O N T E N T S




Modeling Using the Programming Language . . . . . . . . . . . . . 38

Generator 41

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 41

Modeling in COMSOL Multiphysics . . . . . . . . . . . . . . . . 41



Generator with Mechanical Dynamics and Symmetry 55

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 55




Generator in 3D 66

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 66




C h a p t e r 4 : R F a n d M i c r o w a v e M o d e l s

Three-port Ferrite Circulator 74

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 74



References . . . . . . . . . . . . . . . . . . . . . . . . . 79


Monoconical RF Antenna 85

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 85




Antenna Impedance and Radiation Pattern . . . . . . . . . . . . . 93

Far-Field Computation . . . . . . . . . . . . . . . . . . . . . 95

Magnetic Dipole Antenna 96

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 96




Modeling Using the Programming Language . . . . . . . . . . . . 100

Thermal Drift in a Microwave Filter 102

Introduction . . . . . . . . . . . . . . . . . . . . . . . 102

Model Definition . . . . . . . . . . . . . . . . . . . . . . 102

Results and Discussion. . . . . . . . . . . . . . . . . . . . 103

Modeling Using the Graphical User Interface . . . . . . . . . . . 105

Waveguide H-Bend with S-parameters 116

Introduction . . . . . . . . . . . . . . . . . . . . . . . 116




Waveguide Adapter 121

Introduction . . . . . . . . . . . . . . . . . . . . . . . 121


Results. . . . . . . . . . . . . . . . . . . . . . . . . . 123

Modeling in COMSOL Multiphysics . . . . . . . . . . . . . . . 125


Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . 126

S-Parameter Analysis . . . . . . . . . . . . . . . . . . . . 128

Creating the Geometry in MATLAB. . . . . . . . . . . . . . . 131

Microwave Cancer Therapy 132



References . . . . . . . . . . . . . . . . . . . . . . . . 138


C O N T E N T S | iii

iv | C O N T E N T S


Balanced Patch Antenna for 6 GHz 144

Introduction . . . . . . . . . . . . . . . . . . . . . . . 144


Results and Discussions . . . . . . . . . . . . . . . . . . . 147


References . . . . . . . . . . . . . . . . . . . . . . . . 151

Modeling using the Graphical User Interface . . . . . . . . . . . 151

C h a p t e r 5 : O p t i c s a n d P h o t o n i c s M o d e l s

Photonic Microprism 166

Introduction . . . . . . . . . . . . . . . . . . . . . . . 166



References . . . . . . . . . . . . . . . . . . . . . . . . 168


Photonic Crystal 173

Introduction . . . . . . . . . . . . . . . . . . . . . . . 173



References . . . . . . . . . . . . . . . . . . . . . . . . 175



Step-Index Fiber 182

Introduction . . . . . . . . . . . . . . . . . . . . . . . 182



References . . . . . . . . . . . . . . . . . . . . . . . . 184



Stress-Optical Effects in a Silica-on-Silicon Waveguide 189

Introduction . . . . . . . . . . . . . . . . . . . . . . . 189

The Stress-Optical Effect and Plane Strain . . . . . . . . . . . . 189

Perpendicular Hybrid-Mode Waves . . . . . . . . . . . . . . . 190

Plane Strain Analysis . . . . . . . . . . . . . . . . . . . . 191

Optical Mode Analysis . . . . . . . . . . . . . . . . . . . . 199

Convergence Analysis . . . . . . . . . . . . . . . . . . . . 201

Stress-Optical Effects with Generalized Plane Strain 206

Introduction . . . . . . . . . . . . . . . . . . . . . . . 206

Generalized Plane Strain . . . . . . . . . . . . . . . . . . . 206

Plane Strain Analysis . . . . . . . . . . . . . . . . . . . . 212

Optical Mode Analysis . . . . . . . . . . . . . . . . . . . . 220

Second Harmonic Generation of a Gaussian Beam 223

Introduction . . . . . . . . . . . . . . . . . . . . . . . 223



References . . . . . . . . . . . . . . . . . . . . . . . . 227


Transient Simulation for FFT Analysis . . . . . . . . . . . . . . 233

Propagation of a 3D Gaussian Beam Laser Pulse 236

Introduction . . . . . . . . . . . . . . . . . . . . . . . 236




C h a p t e r 6 : E l e c t r i c a l C o m p o n e n t M o d e l s

A Tunable MEMS Capacitor 246

Introduction . . . . . . . . . . . . . . . . . . . . . . . 246





C O N T E N T S | v

vi | C O N T E N T S

Induction Currents from Circular Coils 256

Introduction . . . . . . . . . . . . . . . . . . . . . . . 256



Harmonic Analysis in the Graphical User Interface . . . . . . . . . 259

Transient Analysis . . . . . . . . . . . . . . . . . . . . . 263


Magnetic Field of a Helmholtz Coil 267

Introduction . . . . . . . . . . . . . . . . . . . . . . . 267




Inductance in a Coil 275

Introduction . . . . . . . . . . . . . . . . . . . . . . . 275




Integrated Square-Shaped Spiral Inductor 284


Results. . . . . . . . . . . . . . . . . . . . . . . . . . 287



Bonding Wires to a Chip 296

Introduction . . . . . . . . . . . . . . . . . . . . . . . 296





C h a p t e r 7 : G e n e r a l I n d u s t r i a l A p p l i c a t i o n M o d e l s

Eddy Currents in 3D 306

Introduction . . . . . . . . . . . . . . . . . . . . . . . 306




Change to Aluminum and Stainless Steel . . . . . . . . . . . . . 311

Change to Magnetic Iron . . . . . . . . . . . . . . . . . . . 311

Cold Crucible 314

Introduction . . . . . . . . . . . . . . . . . . . . . . . 314




Electric Impedance Sensor 326

Introduction . . . . . . . . . . . . . . . . . . . . . . . 326




One-sided Magnet and Plate 334

Introduction . . . . . . . . . . . . . . . . . . . . . . . 334



References . . . . . . . . . . . . . . . . . . . . . . . . 336


Modeling using a nonlinear magnetic material in the plate . . . . . . 341

Magnetic Signature of a Submarine 345

Introduction . . . . . . . . . . . . . . . . . . . . . . . 345




C O N T E N T S | vii

viii | C O N T E N T S

3D Quadrupole Lens 355

Introduction . . . . . . . . . . . . . . . . . . . . . . . 355


Results. . . . . . . . . . . . . . . . . . . . . . . . . . 356


Superconducting Wire 362

Introduction . . . . . . . . . . . . . . . . . . . . . . . 362



Reference . . . . . . . . . . . . . . . . . . . . . . . . 365


INDEX 371

1

E l e c t r o m a g n e t i c s M o d u l e M o d e l L i b r a r y

1

2 | C H A P T E R

I n t r o du c t i o n

The Electromagnetics Module Model Library consists of a set of models from various areas of electromagnetic field simulation. Their purpose is to assist you in learning, by example, how to model sophisticated electromagnetic components, systems and effects. Through them, you can tap the expertise of the top researchers in the field, examining how they approach some of the most difficult modeling problems you might encounter. You can thus get a feel for the power that COMSOL Multiphysics offers as a modeling tool. In addition to serving as a reference, the models can also give you a big head start if you are developing a model of a similar nature.

We have divided these models into six groups: electrical components, general industrial applications, motor and drives, optics and photonics, RF and microwave engineering, and finally tutorial models. The models also illustrate the use of the various electromagnetics-specific application modes from which we built them. These specialized modes are unavailable in the base COMSOL Multiphysics package, and they come with their own graphical-user interfaces that make it quick and easy to access their power. You can even modify them for custom requirements. COMSOL Multiphysics itself is very powerful and, with sufficient expertise in a given field, you certainly could develop these modes by yourself—but why spend the hundreds or thousands of hours that would be necessary when our team of experts has already done the work for you?

Note that the model descriptions in this book do not contain every detail on how to carry out every step in the modeling process. Before tackling these in-depth models, we urge you to first read the second book in the Electromagnetics Module documentation set. Titled the Electromagnetics Module User’s Guide, it introduces you to the basic functionality in the module, reviews new features in the version 3.2 release, covers basic modeling techniques and includes reference material of interest to those working in electromagnetics. For more information on how to work with the COMSOL Multiphysics graphical user interface, please refer to the COMSOL Multiphysics User’s Guide or the COMSOL Multiphysics Quick Start manual. An explanation on how to model with a programming language is available in yet another book, the COMSOL Multiphysics Scripting Guide.

The book in your hands, the Electromagnetics Module Model Library, provides details about a large number of ready-to-run models that illustrate real-world uses of the module. Each entry comes with theoretical background as well as instructions that illustrate how to set it up. They were written by our staff engineers who have years of

1 : E L E C T R O M A G N E T I C S M O D U L E M O D E L L I B R A R Y

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experience in electromagnetics; they are your peers, using the language and terminology needed to get across the sophisticated concepts in these advanced topics.

Finally note that we supply these models as COMSOL Multiphysics Model MPH-files so you can import them into COMSOL Multiphysics for immediate execution, allowing you to follow along with these examples every step along the way.

Model Library Guide

The table below summarizes key information about the entries in the Electromagnetics Module Model Library. A series of columns states the application mode (such as Perpendicular Currents) used to solve the corresponding model. The solution time is the elapsed time measured on a machine running Windows XP with a 2.6 GHz Pentium 4 processor and 2 GB of RAM. For models with a sequential solution strategy, the Solution Time column shows the elapsed time for the longest solution step. Additional columns point out the solution properties a given example highlights. The choices here include the type of analysis (such as time dependent) and whether multiphysics or parametric studies are included.

TABLE 1-1: ELECTROMAGNETICS MODULE MODEL LIBRARY

MODEL PAGE APPLICATION MODE

SOLUTION TIME

ST

AT

ION

AR

Y

TIM

E-H

AR

MO

NIC

TIM

E D

EP

EN

DE

NT

EIG

EN

FR

EQ

UE

NC

Y/E

IGE

NM

OD

E

NO

NL

INE

AR

MU

LT

IPH

YS

ICS

TUTORIAL MODELS

Eddy Currents 24* Azimuthal Induction Currents

2 s √

Floating Potentials and Electric Shielding

51* Conductive Media DC

17 s √

Microstrip 100* Electrostatics, In-Plane Electric and Induction Currents, Potentials

√ √

I N T R O D U C T I O N | 3

4 | C H A P T E R

PA

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Periodic Boundary Conditions

57* 3D Electromagnetic Waves

7 s √

Far-Field Postprocessing 64* Axisymmetric TE Waves

1 s √

Dielectric scattering PML 72* In-Plane TE Waves

1 s √

Electromagnetic Forces on Parallel Current-Carrying Wires

10 Perpendicular Currents

1 s √

Inductive Heating of a Copper Cylinder

17 Azimuthal Currents, Heat Transfer

11 s √ √ √ √

Permanent Magnet 24 Magnetostatics, No Currents

15 s √

MOTORS AND DRIVES MODELS

Linear Electric Motor of the Moving Coil Type


1 s √

Generator 41 Perpendicular Currents

6 min √ √ √

Generator with Mechanical Dynamics and Symmetry


3 min √ √ √

Generator in 3D 66 Magnetostatics, No Currents

4 min √ √

RF AND MICROWAVE MODELS

Waveguide H-Bend 37* 3D Electromagnetic Waves, In-Plane TE Waves

4 s √



SOLUTION TIME

ST

AT

ION

AR

Y

TIM

E-H

AR

MO

NIC

TIM

E D

EP

EN

DE

NT

EIG

EN

FR

EQ

UE

NC

Y/E

IGE

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IPH

YS

ICS


√

√

√

√

√

PA

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TU

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Three-port Ferrite Circulator

74 In-Plane TE Waves

1 s √

Monoconical RF Antenna 85 Axisymmetric TM Waves

2 min √

Magnetic Dipole Antenna 96 Axisymmetric TE Waves

2 s √

Thermal Drift in a Microwave Filter

102 3D Electromagnetic Waves, Moving Mesh, Shell

2 min √ √ √

Waveguide H-Bend with S-parameters

116 In-Plane TE Waves

38 s √

Waveguide Adapter 121 Boundary Mode Analysis, 3D Electromagnetic Waves

15 min √ √ √

Microwave Cancer Therapy

132 Axisymmetric TM Waves, Bioheat Equation

45 s √ √

Balanced Patch Antenna for 6 GHz

144 3D Electromagnetic Waves

¤ √

OPTICS AND PHOTONICS MODELS

Photonic Microprism 166 In-Plane TE Waves

3 s √

Photonic Crystal 173 In-Plane TE Waves

2 s √

Step-Index Fiber 182 Perpendicular Hybrid-Mode Waves

5 s √



SOLUTION TIME

ST

AT

ION

AR

Y

TIM

E-H

AR

MO

NIC

TIM

E D

EP

EN

DE

NT

EIG

EN

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6 | C H A P T E R

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Stress-Optical Effects in a Silica-on-Silicon Waveguide

189 Plane Strain, Perpendicular Hybrid-Mode Waves

4 s √ √ √

Stress-Optical Effects with Generalized Plane Strain

206 Plane Strain, Perpendicular Hybrid-Mode Waves, Weak Form, Point

18 s √ √ √

Second Harmonic Generation of a Gaussian Beam

223 In-Plane TE Waves, ODE

9 min √ √

Propagation of a 3D Gaussian Beam Laser Pulse

236 3D Electromagnetic Waves

79 min √

ELECTRICAL COMPONENT MODELS

A Tunable MEMS Capacitor

246 Electrostatics 15 s √

Induction Currents from Circular Coils

256 Azimuthal Currents

16 s √ √

Magnetic Field of a Helmholtz Coil

267 3D Quasi-Statics 7 min √

Inductance in a Coil 275 Azimuthal Currents

17 s √

Integrated Square-Shaped Spiral Inductor

284 3D Magnetostatics

14 s √

Bonding Wires to a Chip 296 3D Quasi-Statics ¤ √GENERAL INDUSTRIAL APPLICATION MODELS

Eddy Currents in 3D 306 3D Quasi-Statics 3 min √



SOLUTION TIME

ST

AT

ION

AR

Y

TIM

E-H

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TIM

E D

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* this page number refers to the Electromagnetic Module User’s Guide. ¤ this model requires a 64-bit platform.

Typographical Conventions

All COMSOL manuals use a set of consistent typographical conventions that should make it easy for you to follow the discussion, realize what you can expect to see on the screen, and know which data you must enter into various data-entry fields. In particular, you should be aware of these particular conventions:

• A boldface font of the shown size and style indicates that the given word(s) appear exactly that way on the COMSOL graphical user interface (for toolbar buttons in the corresponding tooltip). For instance, we often refer to the Model Navigator, which is the window that appears when you start a new modeling session in COMSOL; the corresponding window on the screen has the title Model Navigator. As another example, the instructions might say to click the Multiphysics button, and the boldface font indicates that you can expect to see a button with that exact label on the COMSOL user interface.

Cold Crucible 314 3D Quasi-Statics 16 s √

Electric Impedance Sensor 326 Small In-Plane Currents

2 min √

One-sided Magnet and Plate

334 Magnetostatics, No Currents

37 s √

Magnetic Signature of a Submarine

345 Magnetostatics, No Currents

11 s √

3D Quadrupole Lens 355 Magnetostatics, No Currents

28 s √

Superconducting Wire 362 PDE, General Form

42 s √ √



SOLUTION TIME

ST

AT

ION

AR

Y

TIM

E-H

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TIM

E D

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8 | C H A P T E R

• The names of other items on the graphical user interface that do not have direct labels contain a leading uppercase letter. For instance, we often refer to the Draw toolbar; this vertical bar containing many icons appears on the left side of the user interface during geometry modeling. However, nowhere on the screen will you see the term “Draw” referring to this toolbar (if it were on the screen, we would print it in this manual as the Draw menu).

• The symbol > indicates a menu item or an item in a folder in the Model Navigator. For example, Physics>Equation System>Subdomain Settings is equivalent to: On the Physics menu, point to Equations System and then click Subdomain Settings. COMSOL>Heat Transfer>Conduction means: Open the COMSOL folder, open the Heat

Transfer folder, and select Conduction.

• A Code (monospace) font indicates keyboard entries in the user interface. You might see an instruction such as “Type 1.25 in the Current density edit field.” The monospace font also indicates COMSOL Script codes.

• An italic font indicates the introduction of important terminology. Expect to find an explanation in the same paragraph or in the Glossary. The names of books in the COMSOL documentation set also appear using an italic font.


2

T u t o r i a l M o d e l s

This chapter contains a selection of models that demonstrate important features and modeling techniques. Therefore, this chapter contains models from many different application areas. For pedagogical reasons, these models are kept as simple as possible with the main focus on the feature that is demonstrated.

9

10 | C H A P T E R

E l e c t r omagn e t i c F o r c e s on Pa r a l l e l C u r r e n t - C a r r y i n g W i r e s

Introduction

One ampere is defined as the constant current in two straight parallel conductors of infinite length and negligible circular cross section, placed one meter apart in vacuum, that produces a force of 2⋅10-7 newton per meter of length. This model shows a setup of two parallel wires in the spirit of this definition, but with the difference that the wires do not have a negligible cross section. However, by successively shrinking the radius of the wires, the force would approach 2⋅10-7 newton per meter of length.

The force between the wires is computed using two different methods: by integrating the volume force density over the wire cross section and by integrating the stress tensor on the boundary. The results are in good agreement and are close but not equal to the number 2⋅10-7 for the 1 ampere definition, as expected.

Model Definition

The model is built using the 2D Perpendicular Currents application mode. The modeling plane is a cross section of the two wires and the surrounding air.

D O M A I N E Q U A T I O N S

In the 2D Perpendicular Currents application mode, it is assumed that the only non-zero component of the magnetic vector potential is Az. This corresponds to all currents being perpendicular to the modeling plane. The following equation is solved:

where µ is the permeability of the medium and the externally applied current. is set so that the applied current in the wires equals 1 A, but with different signs.

B O U N D A R Y C O N D I T I O N S

At the exterior boundary of the air domain, Az is fixed to zero. The interior boundaries between the wires and the air only assume continuity, corresponding to a homogenous Neumann condition.

µ Az∇×( )∇× Jze

=

Jze Jz

e

2 : TU T O R I A L M O D E L S

Results and Discussion

The expression for the surface stress reads

where n1 is the boundary normal pointing out from the conductor wire and T2 the stress tensor of air. The closed line integral of this expression around the circumference of either wire evaluates to -1.92e-7 N. The minus sign indicates that the force between the wires is repulsive.

The volume force density is given by

The surface integral of the x component of the volume force on the cross section of a wire gives the result -1.92e-7 N. The accuracy in the calculations is ultimately limited by the number of mesh elements and the finite size of the geometry.

Model Library path: Electromagnetics_Module/Tutorial_Models/parallel_wires

Modeling Using the Graphical User Interface

Select the 2D>Electromagnetics Module>Statics>Magnetostatics>Perpendicular Induction

Currents, Vector Potential mode in the Model Navigator.

O P T I O N S A N D S E T T I N G S

1 In the Constants dialog box, enter the following names and expressions.

The radius r of each wire is 0.2 m, and the area is A = πr2. The total current is I0 = 1 ampere and the current density is J0 = I0/A.

NAME EXPRESSION DESCRIPTION

r 0.2 Wire radius (m)

A pi*r^2 Wire area (m2)

I0 1 Wire current (A)

J0 I0/A Current density (A/m2)

n1T212--- H B⋅( )n1– n1 H⋅( )BT

+=

F J B× Jze

– By ,⋅ Jze Bx 0,⋅= =

E L E C T R O M A G N E T I C F O R C E S O N P A R A L L E L C U R R E N T - C A R R Y I N G W I R E S | 11

12 | C H A P T E R

2 Give axis and grid settings according to the following table.

G E O M E T R Y M O D E L I N G

1 Draw a circle C1 with radius 0.2 centered at (-0.5, 0).

2 Draw a circle C2 with radius 0.2 centered at (0.5, 0).

3 Draw a circle C3 with radius 5 centered at (0, 0).

Figure 2-1: Geometry.

P H Y S I C S S E T T I N G S

Boundary ConditionsThe default boundary conditions are used.

AXIS GRID

x min -6 x spacing 0.5

x max 6 Extra x -0.3 0.7

y min -6 y spacing 0.5

y max 6 Extra y 0.2


Subdomain SettingsApply an External current density and define Electromagnetic force variables according to the following table.

Leave the other parameters at their default values.

M E S H G E N E R A T I O N

1 Initialize the mesh.

SUBDOMAIN 1 2 3

0 J0 -J0

Electromagnetic force variables wire1 wire2

Jze


14 | C H A P T E R

2 Refine the mesh once.

Figure 2-2: Mesh.

C O M P U T I N G T H E S O L U T I O N

Click the Solve button.

P O S T P R O C E S S I N G A N D V I S U A L I Z A T I O N

1 The default plot shows the magnetic flux density. To visualize the magnitude of the volume force density, use a Surface plot with the expression sqrt(Jez_emqa^2*(Bx_emqa^2+By_emqa^2)). Add the predefined quantity Magnetic field, norm as a Contour plot. Figure 2-3 shows the result. The volume force density is unevenly distributed within each wire due to the slight inhomogeneous magnetic field from the other wire. If you decrease the radii of the wires, the force density becomes evenly distributed. In the definition of 1 A it is assumed that the


radii are negligible. The field becomes homogeneous as the radii approaches zero. The effects of the self-induced field of a wire then cancel out.

Figure 2-3: Force density and magnetic field.

2 To visualize the air stress tensor on the boundary, clear the Contour check box and select the Arrow check box. Click the Arrow tab and select Boundaries from the Plot

arrows on list. Click the Boundary tab in the Arrow data area and select Maxwell

surface stress tensor (wire1) from the Predefined quantities list. Use a Scale factor of 0.1.


16 | C H A P T E R

Figure 2-4: Stress tensor and magnetic field.

Compute the resulting force on the wires.

1 To compute the resulting force in the x direction on the left wire using the stress tensor method, open the Data Display dialog box and enter the expression wire1_forcex_emqa. Since this is a scalar variable you can evaluate it in any coordinate within the geometry. You should get a result near -1.92e-7.

2 To alternatively compute the resulting force by integrating , open the Subdomain Integration dialog box and integrate the expression -Jez_emqa*By_emqa on subdomain 2. The result is again approximately -1.92e-7. The force in the y direction should be negligible. Confirm this by integrating Jez_emqa*Bx_emqa on subdomain 2. The result is approximately -4.5e-15.

J B×


I n du c t i v e Hea t i n g o f a Coppe r C y l i n d e r

Introduction

The induced currents in a copper cylinder produce heat, and when the temperature rises, the electrical conductivity of the copper changes. Solving the heat transfer simultaneously with the field propagation is therefore crucial for an accurate description of this process.

The heating caused by the induced currents is called inductive heating. Generally heating due to currents is also called resistive heating or ohmic heating.

Model Definition

The system to be solved is given by

where ρ is the density, Cp is the specific heat capacity, k is the thermal conductivity, and Q is the inductive heating.

The electrical conductivity of copper, σ, is given by the expression

where ρ0 is the resistivity at temperature T0, and α is the temperature coefficient of the resistivity. T0 is the temperature 293 K, and T is the actual temperature in the domain.

The time average of the inductive heating over one period, is given by

jωσ T( )A ∇ µ 1– ∇ A×( )×+ 0=

ρCp t∂∂T ∇ k T∇⋅– Q T A,( )=

σ 1ρ0 1 α T T0–( )+( )[ ]

---------------------------------------------------=

Q 12---σ E 2

=

I N D U C T I V E H E A T I N G O F A C O P P E R C Y L I N D E R | 17

18 | C H A P T E R


The temperature after 1200 seconds is shown in Figure 2-5 below. The average temperature of the copper cylinder has been raised from 293 K up to 300 K during this time. The current in the coil has an amplitude of 1 kA.

Figure 2-5: Temperature distribution of the copper cylinder and its surroundings after 1200 second.

Model Library path: Electromagnetics_Module/Tutorial_Models/inductive_heating


1 Select the Axial symmetry (2D) in the Space dimension list.

2 Click the Multiphysics button in the Model Navigator.

3 Select the Electromagnetics Module>Quasi-Statics>Magenetic>Azimuthal Induction

Currents>Vector Potential> time-harmonic analysis application mode, and add it to the model.

4 Select the Heat Transfer>Conduction>Transient analysis application mode and click Add to add it to the model.


5 Set the Ruling application mode to Heat Transfer by Conduction. This makes the heat transfer application mode choose the solver.


1 From the Options menu, choose Axes/Grid Settings.

2 Set axis and grid settings according to the following table.

3 From the Options menu, choose Constants.

4 In the Constants dialog box, enter the following variable names and expressions.


1 Draw a circle C1 with radius 0.01 centered at (0.05,0).

2 Draw a rectangle R1 with opposite corners at (0,-0.25) and (0.2,0.25).

3 Draw a rectangle R2 with opposite corners at (0,-0.05) and (0.03,0.05).

AXIS GRID

r min -0.05 r spacing 0.05

r max 0.5 Extra r 0.03

z min -0.3 z spacing 0.05

z max 0.3 Extra z -0.01 0.01


I0 1e3 Current (A)

diam 0.02 Diameter (m)

circ pi*diam Circumference (m)

Js0 I0/circ Surface current density (A/m)

r0 1.754e-8 Resistivity at T = T0 (Ωm)

T0 293 Reference temperature (K)

alpha 0.0039 Temperature coefficient (K−1)

rho1 1.293 Density of air (kg/m3)

Cp1 1.01e3 Heat capacity of air (J kg−1 K−1)

k1 0.026 Thermal conductivity of air (J s−1 m−1 K−1)

rho2 8930 Density of copper (kg/m3)

Cp2 340 Heat capacity of copper (J kg−1 K−1)

k2 384 Thermal conductivity of copper (J s−1 m−1 K−1)


20 | C H A P T E R


Application Scalar VariablesIn the Application Scalar Variables dialog box set the frequency nu_qa to 500 Hz.

Boundary Conditions1 Select the Azimuthal Currents application mode from the Multiphysics menu.

2 Open the Boundary Settings dialog box, and select the Interior boundaries check box.

3 Enter boundary coefficients according to the following table.

4 Select the Heat Transfer application mode from the Multiphysics menu.

5 Enter boundary coefficients according to the following table.

Subdomain Settings1 Select the Azimuthal Currents application mode from the Multiphysics menu.

2 Enter the electrical conductivity according to the following table.

3 Select the Heat Transfer application mode from the Multiphysics menu.

4 Enter the expressions according to the following table.

There is a variable Qav_qa defined by the Azimuthal Currents application mode, that could have been used as the heat source in the heat transfer equation. However,

BOUNDARY 1,3,5 2,7,9 10-13

Boundary condition Axial symmetry Magnetic insulation Surface current

Jsϕ Js0

BOUNDARY 1,3,5 2,7,9

Boundary condition Axial symmetry Temperature

T0 T0

SUBDOMAIN 1 2 3

σ 0 1/(r0*(1+ alpha*(T-T0)))

0

SUBDOMAIN 1 2 3

k k1 k2 k2

ρ rho1 rho2 rho2

Cp Cp1 Cp2 Cp2

Q 0 0.5*sigma_qa*normE_qa^2 0


Qav_qa is defined using the complex-conjugation function conj, which has no derivative defined. When using the nonlinear solver, COMSOL Multiphysics needs to calculate the derivatives of all coefficients to find the Jacobian of the problem. To avoid this problem, a more explicit expression for the heat source is used.

Initial ConditionsOn the Init tab select all domains in the Domain selection list and set the initial conditions to T0.


Initialize the mesh.


1 In the Solver Parameters dialog box, set the Times to linspace(0,1200,21).

2 Select the Use complex functions with real input check box on the Advanced tab.

3 Solve the problem.

PO S T P R O C E S S I N G A N D V I S U A L I Z A T I O N

Plot the temperature as a surface plot. To see the temperature in the domain as a function of time, select the Animation tab and click Start Animation.


22 | C H A P T E R

Modeling Using the Programming Language

% Start by removing any old FEM and application structures with % the same names that will be used in this model.clear fem a1 a2

% Define the space coordinates of the model.fem.sdim='r' 'z';

% Define workspace variables to be used as parameters in the model % setup.I0=1e3;rad=0.01;

% Define the FEM structure constants to be used in the modeling % process.fem.const=... 'Js0',I0/(2*pi*rad),... 'r0',1.754e-8,... 'T0',293,... 'alpha',0.0039,... 'rho1',1.293,... 'Cp1',1.01e3,... 'k1',0.026,... 'rho2',8930,... 'Cp2',340,... 'k2',384;

% Create the geometry.c1=circ2(0.05,0,rad);r1=rect2(0,0.2,-0.25,0.25);r2=rect2(0,0.03,-0.05,0.05);fem.geom=geomcomp(c1 r1 r2);

% Create the axisymmetric electromagnetics mode.a1.mode.class='AzimuthalCurrents';a1.mode.type='axi';a1.prop.analysis='harmonic';

a1.var.nu=500;

a1.bnd.type='ax' 'A0' 'Js';a1.bnd.Js0phi=[] [] 'Js0';a1.bnd.ind=[1 2 1 0 1 0 2 0 2 3 3 3 3];a1.border='on';

a1.equ.sigma='0' '1/(r0*(1+alpha*(T-T0)))' '0';

% Create the axisymmetric heat transfer mode.a2.mode.class='HeatTransfer';


a2.mode.type='axi';a2.assign.T0='T0_ht';

a2.bnd.type='ax' 'T';a2.bnd.T0=[] 'T0';a2.bnd.ind=[1 2 1 0 1 0 2 0 2 0 0 0 0];

a2.equ.k='k1' 'k2' 'k2';a2.equ.rho='rho1' 'rho2' 'rho2';a2.equ.C='Cp1' 'Cp2' 'Cp2';a2.equ.Q=0 '0.5*sigma*normE^2' 0;a2.equ.init='T0';

% Add both modes to the FEM structure, and generate the coupled % system.fem.appl=a1 a2;fem=multiphysics(fem);

% Generate the mesh.fem.mesh=meshinit(fem);fem.xmesh=meshextend(fem);

% Solve the time-dependent problem.fem.sol=femtime(fem,'tlist',linspace(0,1200,21),... 'complex','on','report','on');

% Finally, visualize the heating process by generating a movie.postmovie(fem,'tridata','T','tribar','on')


24 | C H A P T E R

Pe rmanen t Magne t

Introduction

In magnetostatic problems, where no currents are present, the problem can be solved using a scalar magnetic potential. This model illustrates this technique for a horse shoe-shaped permanent magnet placed near an iron rod.

Figure 2-6: A full 3D view of the geometry. Left-right and top-down symmetry is used to minimize the problem size.

Model Definition

In a current free region, where

it is possible to define the scalar magnetic potential, Vm, from the relation

This is analogous to the definition of the electric potential for static electric fields.

Using the constitutive relation between the magnetic flux density and magnetic field

∇ H× 0=

H ∇Vm–=

B µ0 H M+( )=


together with the equation

you can derive an equation for Vm,


The force on the rod is calculated by integrating the surface stress tensor over all boundaries of the rod. The expression for the stress tensor reads,

where n1 is the boundary normal pointing out from the rod and T2 the stress tensor of air. The integration gives 148 N, which corresponds to one quarter of the rod. The actual force on the rod is therefore four times this value, 592 N.

Model Library path: Electromagnetics_Module/Tutorial_Models/permanent_magnet_and_rod


Select the 3D>Electromagnetics Module>Statics>Magnetostatics, No Currents mode in the Model Navigator.


In the Constants dialog box enter the following names and expressions.


You can use symmetries to reduce the size of the model. Only one arm of the magnet needs to be included in the model, and a horizontal cut can also be made. Thus, only one quarter of the magnet and rod will be modeled.


murFe 5000 Rel. permeability of iron

Mpre 750000 Magnetization of magnet (A/m)

∇ B⋅ 0=

∇ µ0∇Vm µ0M0–( )⋅– 0=

n1T212--- H B⋅( )n1– n1 H⋅( )BT

+=

P E R M A N E N T M A G N E T | 25

26 | C H A P T E R

1 Create a work plane by opening the Work Plane Settings dialog box and clicking OK to obtain the default work plane in the x-y plane.

2 In the Axes/Grid Settings dialog box, clear the Auto check box on the Grid tab and set both the grid x spacing and y spacing to 0.2.

3 Draw a rectangle R1 with corners at (0, 0.2) and (0.4, 0.4).

4 Draw another rectangle R2 with corners at (0.4, 0.2) and (0.6, 0.4).

5 Select the Ellipse/Circle (Centered) tool, and draw a circle C1 with center at (0.6, 0) and radius 0.2. Use the right mouse button to make sure a circle is obtained rather than an ellipse.

6 Draw another circle C2 also with center at (0.6, 0) but with radius 0.4.

7 Select both circles and click the Difference button to create the object CO1.

8 Draw a rectangle R3 with corners at (0.2, 0) and (0.6, 0.4), and a rectangle R4 with corners at (0.2, -0.4) and (1, 0).

9 Select CO1, R3, and R4 and click the Difference button to create the object CO2.

10 Select CO2 and R2 and click the Union button. Then click the Delete Interior

Boundaries button.

11 Open the Extrude dialog box and select both R1 and CO2. Enter the Distance 0.1 and click OK to extrude the magnet.

12 Create the rod by combining a cylinder and a sphere. Click the Cylinder button to open the Cylinder dialog box, and enter the Radius 0.1 and Height 0.4. Set the Axis


base point to (-0.3, 0, 0) and the Axis direction vector to (0, 1, 0). Click OK to create the cylinder.

13 Click the Sphere button and enter the Radius 0.1. Set the Axis base point to (-0.3, 0.4, 0) and click OK to create the sphere.

14 Half of the rod should now be removed. This can be done by subtracting a block from the cylinder and sphere. Click the Block button and enter the Length coordinates (0.4, 0.6, 0.2). Set the Axis base point to (-0.5, -0.1, -0.2) and click OK to create a block.

15 Open the Create Composite Object dialog box, and type the formula CYL1+SPH1-BLK1 in the Set formula edit field. Clear the Keep interior boundaries check box and then click OK.

16 Finally add a block surrounding the magnets and rod. In the Block dialog box use the Length coordinates (5, 2, 2) and the Axis base point (-2, 0, 0).


Boundary ConditionsAlong the boundaries far away from the magnet, the magnetic field should be tangential to the boundary as the flow lines should form closed loops around the magnet. The natural boundary condition is

n µ0∇Vm µ0M0–( )⋅ n B⋅ 0= =


28 | C H A P T E R

Thus the magnetic field is made tangential to the boundary by a Neumann condition on the potential.

Along the symmetry boundary below the magnet, the magnetic field should also be tangential, and therefore you can apply the same Neumann condition there.

On the other hand, the magnetic field should be normal to the other symmetry boundary at the front in the figure above in order for the magnetic flow lines to form closed loops around the magnet. This means that the magnetic field is symmetric with respect to the boundary. This can be achieved by setting the potential to zero along the boundary, and thus making the potential antisymmetric with respect to the boundary.

Enter these boundary conditions according to the following table.

Subdomain SettingsIn the Subdomain Settings dialog box, enter the coefficients according to the table below.


1 Open the Mesh Parameters dialog box, and select Fine mesh. In order to make the mesh finer in the magnet and in the rod, set Maximum element size for subdomains 2, 3, and 4 to 0.025.





The default plot shows a slice plot of the magnetic potential.

BOUNDARY 2, 8, 24 ALL OTHER

Boundary condition Zero potential Magnetic insulation

SUBDOMAIN 1 2 3 4

Constitutive relation

µr 1 murFe murFe

M Mpre 0 0

Force variables rod

B µ0µrH= B µ0µrH= B µ0H µ0M+= B µ0µrH=


1 In the Plot Parameters dialog box, select Slice and Arrow plot.

2 On the Slice tab set the x levels to 0, y levels to 0 in the Slice positioning area. For the z levels select Vector with coordinates and enter 0.05. Select bone from the Colormap list.

3 Click the Arrow tab, select the Magnetic flux density as Arrow data. Set the x points to 200 and the y points to 100 in the Arrow positioning area. For the z points select Vector with coordinates and enter 0.051. Select Arrow from the Arrow type list.

4 Click OK to close the dialog and make the plot.

5 Click the button Goto XY View and use the zoom tool to zoom in on the magnet and rod.

Force CalculationTo calculate the force on the rod the surface stress tensor is used, which is derived in “Electromagnetic Forces” on page 118,

where n1 is the boundary normal pointing out from the rod and T2 the stress tensor of air. The electromagnetic force variable rod, which was defined on the rod in the Subdomain Settings dialog box, generates the variables rod_forcex_emnc, rod_forcey_emnc, and rod_forcez_emnc which are the forces on the rod in the x, y, and z directions.

n1T212--- H B⋅( )n1– n1 H⋅( )BT

+=


30 | C H A P T E R

To evaluate the force in the x direction, open the Data Display dialog box and enter the expression rod_forcex_emnc. Because this is a scalar you can evaluate it in any point. The result is about 148 N. As only one quarter of the rod has been modeled, the actual force is four times this value, 592 N.


3

M o t o r s a n d D r i v e s M o d e l s

This chapter contains examples of models in the area of electric motors and drives. You will find static models, with force computations as well as transient models with motion.

31

32 | C H A P T E R

L i n e a r E l e c t r i c Mo t o r o f t h e Mov i n g Co i l T y p e 1

Introduction

Linear electric motors (LEMs) are electromechanical devices that produce motion in a straight line, without the use of any mechanism to convert rotary motion to linear motion. The benefits over conventional rotary-to-linear counterparts are the absence of mechanical gears and transmission systems, which results in a higher dynamic performance and improved reliability. The acceleration available from these motors is remarkable compared to traditional motor drives that convert rotary to linear motion.

A LEM is similar to a rotary motor whose stator and rotor have been cut along a radial plane and unrolled so that it provides linear thrust. The same electromagnetic force that produces torque in a rotary motor produces a direct linear force in a linear motor.

The force that develops in the motor depends on the magnetic field and the current in the coils and of course on the shape of the different parts. A parameter of interest is the maximum force in standstill operation, that is, when the magnetic field is constant and the force depends only on the current density distribution. The current density in the coils is limited by the maximum temperature allowed in the coils. For a certain motor topology you can estimate the maximum electromagnetic force in standstill by first performing a heat transfer simulation to get the maximum current density, and then using this value in the magnetostatic problem where the forces are computed.

Model Definition

A LEM of moving coil type consists of a stationary primary and a moving secondary. The stationary primary is a permanent magnet assembly consisting of a double magnet array with altering polarities bounded inside a U-shaped iron former. The moving secondary is centered within the U-shaped magnet assembly, and consists of 3-phase core coils mounted on a T-shaped aluminum armature. The cross-sectional view of a moving coil is shown in the following figure.

1. Model provided by Hans Johansson and Mikael Hansson, The Royal Institute of Technology, Stockholm.

3 : M O T O R S A N D D R I V E S M O D E L S

The permanent magnets in the primary have a residual induction Br of 1.23 T, giving the magnetization

The current in the coil is obtained via a heat transfer simulation. If a temperature of 65° C is allowed in the coil and the surrounding temperature is 25° C, the simulation in the heat transfer mode gives a maximum heat source Q of 230 000 W/m3.

The heat source is equivalent to the resistive losses in the copper, which are given by the expression

where ρCu is the resistivity of copper, which is 1.72·10-8 Ωm. This gives the maximum current density Jz of 3.66 A/mm2.

E Q U A T I O N

COMSOL Multiphysics solves the problem using a magnetostatic equation for the magnetic potential A. This is a 2D model, so it is possible to formulate an equation for the z component of the potential,

where µ0 is the permeability of vacuum, µr the relative permeability, Br the remanent flux density, and is the current density.


The force on the coils can be calculated either using the method of virtual displacement making a small perturbation and calculating the change in magnetic energy

M Br µ0⁄ 0.98 106A/m⋅= =

Q J E⋅ 1ρCu---------- Jz

2⋅= =

∇ µ01– µr

1– Az∇µ0

1– µr1– Br– y

µ01– µr

1– Brx

–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

⋅– Jze

=

Jze

L I N E A R E L E C T R I C M O T O R O F T H E M O V I N G C O I L TY P E | 33

34 | C H A P T E R

or using the Lorentz force expression

The resulting force in the x direction is Fx = 276 N/m. The air gap shear stress τ is the force developed per unit air gap area and gives an indication of the specific force capability. F/l·A is the force per moving volume. This can be compared with the force per moving mass, which indicates the acceleration capabilities. The air gap length is 40 mm and the air gap area is 190 mm2.

Model Library path: Electromagnetics_Module/Motors_and_Drives/coil_LEM


Select the 2D, Electromagnetics Module>Statics>Magnetostatics>Perpendicular Induction

Currents, Vector Potential mode in the Model Navigator. From the Element list, select Lagrange - Linear as element type.


1 Set axis and grid settings according to the following table:

QUANTITY RESULT

F/l 276 N/m

τ 0.69 N/cm2

F/l·A 1.45 N/cm3

AXIS GRID

x min -0.06 x spacing 0.002

x max 0.06 Extra x

y min -0.01 y spacing 0.01

y max 0.03 Extra y 0.0025 0.0035 0.0075 0.0105

WmB H⋅

2--------------=

F J B×=


2 In the Constants dialog box, enter the following variable names and expressions:


1 Draw a rectangle R1 with opposite corners at (-0.05,0) and (0.05,0.0105).

2 Draw a rectangle R2 with opposite corners at (-0.05,0.0075) and (0.05,0.0105).

3 Draw a rectangle R3 with opposite corners at (-0.058,0.0035) and (-0.042,0.0075).

4 Select R3 and click the Array button. Set the Displacement in the x direction to 0.02 and the Array size in the x direction to 6.

5 Draw a rectangle R9 with opposite corners at (-0.018,0) and (0.022,0.0025).

6 Copy R9 into two new rectangles, pasting them at the same location.

7 Select R11 and click the Scale button on the draw toolbar. Set the Scale factor in the x direction to 2/3 and the Scale factor in the y direction to 1 and change the Scale base point to [0.002 0].


sigCu 5.99e7 Conductivity of copper (S/m)

sigFe 1.03e7 Conductivity of iron (S/m)

muFe 2e3 Relative permeability of iron

Br 1.23 Remanent flux density of the magnets (T)

Jin 3.66e6 Current density (A/m2)


36 | C H A P T E R

Br+

8 Select R10 and click the Scale button on the Draw toolbar. Set the Scale factor in the x direction to 1/3 and the Scale factor in the y direction to 1 and change the Scale base point to [0.002 0].

9 Press Ctrl+A to select all objects and then open the Create Composite Object dialog box. Enter the expression (R1+...R11)*R1 in the Set formula text field.

10 Draw a line L1 from (0.002,0) to (0.002,0.0025).


Boundary ConditionsSpecify boundary conditions according to the following table.

Subdomain SettingsSpecify the subdomain properties according to the following table.

BOUNDARY 2,16,22,25,30,36,39,44 ALL OTHER

Boundary condition Electric insulation Magnetic insulation

SUBDOMAIN 1 3 2,6,13 4,10,14


σ 0 sigFe sigFe sigFe

µr 1 muFe 1 1

B µ0µrH= B µ0µrH= B µ0µrH Br+= B µ0µrH=



Initialize the mesh.


1 Select the Adaptive mesh refinement check box in the Solver Parameters dialog box.

2 Click the Solve button to start the simulation.


1 In the Plot Parameters dialog box, select Contour plot and select the magnetic potential as Contour data. Set the Colormap to cool on the Contour tab.

Br 0 0 0 0 0 Br 0 -Br

Je 0 0 0 0

SUBDOMAIN 1 3 2,6,13 4,10,14

SUBDOMAIN 5,7 9,11 8,12


σ sigCu sigCu sigCu

µr 1 1 1

Br 0 0 0 0 0 0

Je Jin -Jin 0

B µ0µrH= B µ0µrH= B µ0µrH=


38 | C H A P T E R

From the results obtained with the simulation, you can compute the force on the coils with cemforce from the command line. This requires that you run COMSOL Multiphysics together with COMSOL Script or MATLAB.

1 Select Export>FEM Structure as 'fem' from the File menu.

2 Give the following command at the prompt.

F=cemforce(fem,'Wm_emqa','dl',[5 7 8 9 11 12])

which gives the result

F = -137.1164 3753.5013

Note that only the force in the x direction makes sense in this case. The reason is that when moving the coils in the y direction the total area of the simulation structure changes, which causes the energy balance to break down.

3 In this case you can also obtain the forces using the Lorentz expression

Compute the total integrated force in the x direction with the command

Fx=postint(fem,'-Jez_emqa*By_emqa','dl',[5 7 8 9 11 12])

This gives the result

Fx =

-137.6029

Since only half of the linear motor is modeled, you need to multiply the results by 2 to obtain the total force density.


% Clear the FEM and application structures.clear fem appl

% Define the FEM structure constants.fem.const=... 'sigCu',5.99e7,... 'sigFe',1.03e7,... 'muFe',2e3,... 'Br0',1.23,... 'Jin',3.66e6;

F J B× Jin– By ,⋅ Jin Bx 0,⋅ .= =


% Create the geometry, by first setting all solid object in a cell % array and then calling geomcsg.ss=cell(6,1);ss1=rect2(-0.05,0.05,0,0.0105);ss2=rect2(-0.05,0.05,0.0075,0.0105);dx=0;for i1=3:8 ssi1=rect2(-0.058+dx,-0.042+dx,0.0035,0.0075); dx=dx+0.02;endss9=rect2(-0.018,0.022,0,0.0025);ss10=scale(ss9,2/3,1,0.002,0);ss11=scale(ss9,1/3,1,0.002,0);cc1=curve2([0.002 0.002],[0 0.0025]);g1=solid2(geomcsg(ss,cc));fem.geom=solid2(g1*ss1);fem.sdim='x' 'y';

% Create the application structure.appl.mode='PerpendicularCurrents';appl.bnd.type='A0' 'tH0';appl.bnd.ind=ones(1,55);appl.bnd.ind([2 16 22 25 30 36 39 44])=2;appl.equ.sigma='0' 'sigFe' 'sigFe' 'sigFe'... 'sigCu' 'sigCu' 'sigCu';appl.equ.mur='1' 'muFe' '1' '1'... '1' '1' '1';appl.equ.Br='0' '0' '0' '0' '0' 'Br0'... '0' '-Br0' '0' '0' '0' '0' '0' '0';appl.equ.magconstrel='mur' 'mur' 'Br' 'Br' 'mur' 'mur' 'mur';appl.equ.Jez='0' '0' '0' '0'... 'Jin' '-Jin' '0';appl.equ.ind=[1 3 2 4 5 3 5 7 6 4 6 7 3 4];appl.prop.elemdefault='lag1';

% Generate the PDE coefficients and boundary conditions for the % FEM structure.fem.appl=appl;fem=multiphysics(fem);

% Generate the mesh.fem.mesh=meshinit(fem);fem.xmesh=meshextend(fem);

% Solve the problem using the adaptive solver with two % refinements.fem=adaption(fem,'ngen',3,'report','on');

% Visualize the solution.postplot(fem,'contdata','Az','contmap','cool','geom','on');


40 | C H A P T E R

% You can compute the forces in the same way as for the GUI-based % analysis.F=cemforce(fem,'Wm','dl',[5 7 8 9 11 12]);Fx=postint(fem,'-Jez*By','dl',[5 7 8 9 11 12]);


Gene r a t o r

Introduction

This example shows how the circular motion of a rotor with permanent magnets generates an induced EMF in a stator winding. The generated voltage is calculated as a function of time during the rotation. The model also shows the influence on the voltage from material parameters, rotation velocity, and number of turns in the winding.

The center of the rotor consists of annealed medium carbon steel, which is a material with a high relative permeability. The center is surrounded with several blocks of a permanent magnet made of Samarium Cobalt, creating a strong magnetic field. The stator is made of the same permeable material as the center of the rotor, confining the field in closed loops through the winding. The winding is wound around the stator poles. In Figure 3-1 the generator is shown with part of the stator sliced, in order to show the winding and the rotor.

Figure 3-1: A schematic drawing of the generator showing how the rotor, stator, and stator winding are constructed. The winding is also connected between the loops, interacting to give the highest possible voltage.

Modeling in COMSOL Multiphysics

The COMSOL Multiphysics model of the generator is a time-dependent 2D problem on a cross section through the generator. This is a true time-dependent model where

G E N E R A T O R | 41

42 | C H A P T E R

the motion of the magnetic sources in the rotor is accounted for in the boundary condition between the stator and rotor geometries. Thus there is no Lorentz term in the equation, resulting in the PDE

where the magnetic vector potential only has a z component.

Achieve rotation through coupling variables, where the center part of the geometry, containing the rotor and part of the air-gap, is coupled with a rotation transform to the coordinate system of the stator. Extrusion coupling variables are used for this, and the vector potential is coupled to a boundary between the stator and rotor with the transform

A boundary condition is then used to force the vector potential along the boundary in the air-gap to the rotated potential. The same transform is also used for some postprocessing variables, so that the fields from the rotor can be viewed together with the fields from the stator. Another way to describe the modeling approach is that the rotor problem is solved in a rotating coordinate system where the rotor is fixed (rotor frame), whereas the stator problem is solved in a coordinate system that is fixed with respect to the stator (stator frame). The two frames are then connected via time dependent boundary conditions and suitable postprocessing variables are transformed from the rotor frame to the stator frame.

The material in the stator and the center part of the rotor has a nonlinear relation between the magnetic flux, B and the magnetic field, H, the so called B-H curve. This is introduced by using a relative permeability, µr which is made a function of the norm of the magnetic flux, |B|.

Note: Here µ is defined as the ratio of |B| to |H|, that is not as the local slope of the B-H curve.

It is important that the argument for the permeability function is the norm of the magnetic flux, |B| rather than the norm of the magnetic field, |H|. In this problem B is calculated from the dependent variable A according to

σ A∂t∂

------- ∇ 1µ---∇ A×⎝ ⎠

⎛ ⎞×+ 0=

wt–( )cos wt–( )sin–

wt–( )sin wt–( )cos

xrot

yrot

⋅xstat

ystat

=


H is then calculated from B using the relation

resulting in an implicit or circular definition of µr had |H| been used as the argument for the permeability function.

In COMSOL Multiphysics, the B-H curve is introduced as an interpolation function, see Figure 3-2.

Figure 3-2: The relative permeability versus the norm of the magnetic flux, |B|, for the rotor and stator materials.

The generated voltage is computed as the line integral of the electric field, E, along the winding. Since the winding sections are not connected in the 2D geometry, a proper line integral cannot be carried out. A simple approximation is to neglect the voltage contributions from the ends of the rotor, where the winding sections connect. The voltage is then obtained by taking the average z component of the E field for each winding cross section, multiplying it by the axial length of the rotor, and taking the sum over all winding cross sections.

B ∇ A×=

HB Br–

µ0µr-----------------=

Vi NN LA---- Ez Ad∫

windings∑=


44 | C H A P T E R

where L is the length of the generator in the 3rd dimension, NN is the number of turns in the winding, and A is the total area of the winding cross-section.


The generated voltage in the rotor winding is a sine-shaped signal. At a rotation speed of 60 rpm the voltage will have an amplitude around 2.3 V for a single turn winding, see Figure 3-3.

Figure 3-3: The generated voltage over one quarter of a revolution. This simulation used a single turn winding.


The norm of the magnetic flux, |B|, and the field lines of the B field are shown below in Figure 3-4 at time 0.20 s.

Figure 3-4: The norm and the field lines of the magnetic flux after 0.2 s of rotation. Note the brighter regions, which indicate the position of the permanent magnets in the rotor.

Model Library path: Electromagnetics_Module/Motors_and_Drives/generator


1 In the Model Navigator, click the Multiphysics button.

2 In the Electromagnetics Module folder, select Quasi-Statics, Magnetic>Perpendicular

Induction Currents, Vector Potential>Transient analysis.

3 Click Add twice to add two copies of this application mode.

4 Click Application Mode Properties to open the Application Mode Properties dialog box for the second application mode.

5 In the Application Mode Properties dialog box, select Weak constraints: ideal.

6 Click OK.

7 Click OK to close the Model Navigator.




46 | C H A P T E R

2 In the Constants dialog box, define the following constants with names and expressions. The description field is optional.


Please follow the steps in the geometry modeling carefully. In the following sections, it is assumed that the boundary numbering is that resulting from these steps. After completing the geometry, there should be a total of 148 boundaries.

1 Go to Draw>Specify Objects>Circle to create circles with the following properties:

2 Go to Draw>Specify Objects>Rectangle to create rectangles with the following properties:

3 In the Create Composite Object dialog box, clear the Keep interior boundaries check box.

4 Type C2+C1*(R1+R2+R3+R4) in the Set formula edit field

5 Click OK.


t 0 Time for stationary solution (s)

w 2*pi Angular frequency of rotation (radians/s)

A pi*0.02^2 Area of wires in the stator (m^2)

L 0.4 Length of the generator (m)

NN 1 Number of turns in stator winding

BrSmCo 0.84 Remanent flux in permanent magnets (T)

murSmCo 1 Relative permeability in permanent magnets

NAME RADIUS BASE X Y ROTATION ANGLE

C1 0.3 Center 0 0 22.5

C2 0.235 Center 0 0 0

NAME WIDTH HEIGHT BASE X Y ROTATION ANGLE

R1 0.1 1 Center 0 0 0

R2 0.1 1 Center 0 0 45

R3 0.1 1 Center 0 0 90

R4 0.1 1 Center 0 0 135


6 Create a circle with the following properties:

7 Draw four rectangles with the following properties:

8 In the Create Composite Object dialog box, type C1*(R1+R2+R3+R4) in the Set formula edit field.

9 Click OK.


11 In the Create Composite Object dialog box, make sure that the Keep interior

boundaries check box is not selected.

12 Enter the formula C1+CO2.

13 Click OK.

14 Draw circles according to the table below.

15 Click the Zoom Extents button to get a better view of the geometry.

16 Select the circles C4 and C5.

17 Press Ctrl+C to copy the circles.


C1 0.215 Center 0 0 0


R1 0.1 1 Center 0 0 22.5

R2 0.1 1 Center 0 0 -22.5

R3 0.1 1 Center 0 0 67.5

R4 0.1 1 Center 0 0 -67.5


C1 0.15 Center 0 0 22.5


C1 0.15 Center 0 0 22.5

C2 0.225 Center 0 0 0

C3 0.4 Center 0 0 0

C4 0.015 Center 0.025 0.275 0

C5 0.015 Center -0.025 0.275 0


48 | C H A P T E R

18 Press Ctrl+V to paste the circles. Make sure that the displacements are zero and click OK.

19 Go to Draw>Modify>Rotate and enter 45 in the Rotation angle text field before clicking OK.

20 Repeat the last two steps, but with rotation angles 90, 135, 180, -45, -90, and -135.


1 From the Options menu, point to Expressions and click Subdomain Expressions.

2 In the Subdomain Expressions dialog box, define the following expressions:

3 Click OK.

4 From the Options menu, point to Extrusion Coupling Variables and click Subdomain

Variables.

5 In the Subdomain Extrusion Variables dialog box, select subdomain 3 and define a new variable named, A0z, with the expression Az.

6 Click the General transformation radio button.

NAME EXPRESSION IN SUBDOMAINS 3-11,18 EXPRESSION IN ALL OTHER SUBDOMAINS

Az_all AZ Az2

Ez_all EZ Ez_emqa2

normB_all NB normB_emqa2

R sqrt(x^2+y^2) sqrt(x^2+y^2)

u -y*w 0

v x*w 0


7 Click the Destination tab, select Boundary from the Level list. Select boundaries 48, 49, 94, 95, and check the box Use selected boundaries as destination.

8 Enter the following expressions in the Destination transformation area.

9 Click the Source tab, select the subdomains 3–11 and 18, and define the variables from the table below. To enable the edit fields for the destination transformation, click the General transformation radio button. Leave all other settings at their default values.

10 Click the Destination tab, and make sure that Subdomain is selected in the Level list. Select subdomains 3–11 and 18, select the Use selected subdomains as destination check box, and enter transformation from the table above.

11 Click OK.

12 From the Options menu, point to Integration Coupling Variables and click Subdomain

Variables.

X Y

x*cos(-w*t)-y*sin(-w*t) x*sin(-w*t)+y*cos(-w*t)

NAME EXPRESSION DESTINATION TRANSFORMATION X

DESTINATION TRANSFORMATION Y

AZ Az x*cos(-w*t)-y*sin(-w*t) x*sin(-w*t)+y*cos(-w*t)

NB normB_emqa x*cos(-w*t)-y*sin(-w*t) x*sin(-w*t)+y*cos(-w*t)

EZ Ez_emqa

-u*By_emqa+

v*Bx_emqa

x*cos(-w*t)-y*sin(-w*t) x*sin(-w*t)+y*cos(-w*t)


50 | C H A P T E R

13 In the Subdomain Integration Variables dialog box, define a variable with integration order 4 and global destination according to the table below.

14 Click OK.

15 Go to Options>Functions.

16 In the Functions dialog box, define a new Interpolation function MUR with the interpolation method Linear, the extrapolation method Constant and data from Table.

17 Enter interpolation points according to the list below.

NAME EXPRESSION IN SUBDOMAINS 14-17,23-26

EXPRESSION IN SUBDOMAINS 12-13, 19-22,27-28

Vi -L*Ez_emqa2/A L*Ez_emqa2/A

X F(X)

1 1200

1.1 820

1.2 560

1.3 420

1.4 290

1.5 220

1.6 160

1.7 110

1.8 70


18 Click OK.

Subdomain Settings1 From the Multiphysics menu, select Perpendicular Induction Currents (emqa).

2 In the Subdomain Settings dialog box, select subdomains 1, 2, 12–17, and 19–28 from the Subdomain selection list and clear the Active in this domain check box.

3 Enter subdomain settings for the active subdomains according to the table below. In order to set a remanent flux density, you need to click the radio button for

first.

4 On the Init tab, type R in the Magnetic potential, z component edit field for all active subdomains.

5 Click OK.

6 From the Multiphysics menu, select Perpendicular Induction Currents (emqa2).

7 In the Subdomain Settings dialog box, select subdomains 3-11,18 from the Subdomain selection list and clear the Active in this domain check box.

8 Select subdomain 2 and type MUR(normB_emqa2) in the Relative permeability edit field.

9 On the Init tab, enter R in the Magnetic potential, z component edit field for all active subdomains.

10 Click OK.

Boundary Conditions1 From the Multiphysics menu, select Perpendicular Induction Currents (emqa).

1.9 47

2.0 26

2.1 15

2.2 10

2.3 7

2.4 6

SUBDOMAIN 3 4, 7, 8, 11 5, 6, 9, 10 18

Rel. permeability 1 murSmCo murSmCo MUR(normB_emqa)

Rem. flux density x -BrSmCo*x/R BrSmCo*x/R

Rem. flux density y -BrSmCo*y/R BrSmCo*y/R

X F(X)

B µ0µrH Br+=


52 | C H A P T E R

2 In the Boundary Settings dialog box, enter the following settings:

3 Click OK.

4 From the Multiphysics menu, select Perpendicular Induction Currents (emqa2).


6 Select boundaries 33-34,93,96. On the Weak Constr. tab, clear the Use weak

constraints check box.

7 Click OK.


1 In the Mesh Parameters dialog box, select Finer from the Predefined mesh sizes list and enter 2 in the Resolution of narrow regions edit field.

2 Click Remesh and then OK.


1 In the Solver Parameters dialog box, select Stationary nonlinear from the Solver list.

2 Click OK.

3 Click the Solve button.

4 Open the Solver Parameters dialog box again, and select Time dependent from the Solver list.

BOUNDARY 48-49,94-95

Boundary condition Electric insulation

BOUNDARY 48-49,94-95 33-34,93,96

Boundary condition Magnetic potential Magnetic insulation

A0z A0z


5 Enter linspace(0,0.25,26) in the Times edit field, and enter Az 1e-3 Az2 1e-3 lm2 5e3 in the Absolute tolerance edit field.

6 Click OK.

7 Click the Restart button.


1 Open the Plot Parameters dialog box. Select the solution for time 0.20 s from the Solution at time list.

2 On the Surface tab, type normB_all in the Expression edit field.

3 On the Contour tab, type Az_all in the Expression edit field and 15 in the Levels text field. Click to select the Contour plot check box.

4 In the Contour color frame, select the Uniform color radio button. Click the Color button and select a black color. Clear the Color scale check box.


54 | C H A P T E R

5 Click OK to close the Plot Parameters dialog box.

To view the induced voltage as a function of time, proceed with the steps below.

6 From the Postprocessing menu, select Domain Plot Parameters.

7 On the Point tab, select an arbitrary point in the Point selection list, and type Vi*NN in the Expression edit field.

8 Click OK.

To create an animation of the rotating rotor, do the following steps.

1 Open the Plot Parameters dialog box, click the Animate tab, and make sure that all time steps are selected.

2 Click the Start Animation button. When the movie has been generated, the COMSOL Movie Player will open it.


Gene r a t o r w i t h Me chan i c a l D ynam i c s and S ymme t r y

Introduction

This is an extension of the generator model including an ordinary differential equation for the mechanical dynamics and computation of the torque resulting from the magnetic forces. In addition it exploits symmetry to reduce the model size to 1/8 of the original size. The model also involves a lumped winding resistance.


The reader is supposed to be familiar with the generator model in the preceding section. Here the focus is on concepts which have not already been covered in that model. The smallest possible model of the generator is a sector obtained by cutting radially through two adjacent rotor poles, see Figure 3-5.

Figure 3-5: The smallest possible sector model.

As adjacent rotor poles are magnetized radially but in opposite directions, such a sector model will exhibit antisymmetry. This is readily accounted for in COMSOL Multiphysics by the support for periodic boundary conditions. In addition, the sliding

G E N E R A T O R W I T H M E C H A N I C A L D Y N A M I C S A N D S Y M M E T R Y | 55

56 | C H A P T E R

mesh coupling must be modified in order to keep the coupling coordinates within the modeled geometry sector. This is done by modifying the rotational transform to

where

The mod and atan2 functions are standard numerical functions available in COMSOL Multiphysics and COMSOL Script. The α variable is the rotation angle, which is defined by an ordinary differential equation (ODE) for the mechanical dynamics. As α increases, the sliding mesh coupling reaches the edge of the modeled sector where the mod function causes it to jump back to the beginning of the sector. As the sliding mesh coupling also must be antisymmetric, the coupled quantities are multiplied by a factor G

that will be either +1 or -1. The ODE for the mechanical dynamics

involves the torque Tz+M and the moment of inertia I0+Irotor. M is the externally applied driving torque and Tz is the electromagnetic torque. The model computes the latter by boundary integration of the cross product between the radius vector (x,y) and the normal Maxwell stress components (nTx,nTy) over the rotor boundary times the model depth L. The factor 8 is introduced to account for the entire device.

The moment of inertia is composed of an external moment of inertia I0 (flywheel and turbine) and the moment of inertia of the rotor itself Irotor. The model computes the latter by subdomain integration of the material density ρ times radius vector squared times the model depth L. Again, the factor 8 is introduced to account for the entire generator.

θ( )cos θ( )sin–

θ( )sin θ( )cos

xrot

yrot

⋅xstat

ystat

=

θ mod ϕ α– π 4⁄( , ) ϕ–=

ϕ 2 y x( , )atan π 8⁄–=

G 2 4 ϕ α+( )( ) 0>( )sin( ) 1–=

t2

2

d

d α Tz M+

I0 Irotor+------------------------------=

Tz 8L nTyx nTxy+( ) ldC∫=


The model accounts for the lumped winding resistance by setting the winding current to the induced voltage divided by the winding resistance. This corresponds to a generator operating with the output terminals in short circuit. The model computes the induced voltage using a subdomain integration coupling variable.


The startup of the generator was simulated for 2s which is sufficient for reaching steady state. The time evolution of the rotation angle is shown in Figure 3-6.

Figure 3-6: The time evolution of the rotation angle is shown. After about 1 s, the rotation angle increases linearly with time corresponding to constant rotation speed.

Model Library path: Electromagnetics_Module/Motors_and_Drives/generator_sector_dynamic

Irotor 8ρL x2 y2+( ) Ad

A∫∫=


58 | C H A P T E R


Start by loading the generator model from the model library: Electromagnetics_Module/Motors_and_Drives/generator



2 In the Constants dialog box, delete w and t and add M, Rc and I0 to make the table look as below.


1 Go to Draw>Specify Objects>Line and select Closed polyline (solid) in the Style list. Enter the Coordinates x: 0 0.4*cos(67.5*pi/180) 0.4 0.4*cos(22.5*pi/180) 0 and Coordinates y: 0 0.4*sin(67.5*pi/180) 0.4 0.4*sin(22.5*pi/180) 0.

2 Click OK and a composite object CO2 is created.

3 Draw a circle C20 with the radius 0.4 and the center at the origin (0,0).

4 Select both objects (CO2 and C20), then press the Intersection button to create CO4.

5 Open the Create Composite Object dialog by pressing the corresponding toolbar button in the Draw toolbar.

6 Select all objects in the list except CO4. Select Keep interior boundaries and click Union followed by OK.

7 Select both objects (CO2 and CO4) by pressing Ctrl+A, then click the Intersection button to create CO1

8 Click the Zoom Extents button.


A pi*0.02^2 Area of wires in the stator

L 0.4 Length of the generator

NN 1 Number of turns in stator winding

BrSmCo 0.84 Remanent flux in permanent magnets


M 1400 Externally applied torque (Nm)

Rc 1e-4 Resistance of winding (ohms)

I0 100 External moment of inertia (kg*m^2)



1 From the Options menu, point to Expressions and click Scalar Expressions.

2 In the Scalar Expressions dialog box, define the following expressions:

3 From the Options menu, point to Expressions and click Subdomain Expressions.

4 In the Subdomain Expressions dialog box, you need to edit u and v

5 Add another variable rho for the material density and set it to 7870 in subdomain 1 and 8400 in subdomains 2 and 6. Click OK.

6 From the Options menu, point to Integration Coupling Variables and click Subdomain

Variables.

7 In the Subdomain Integration Variables dialog box, delete the variable Vi and enter the variables according to the table below with default global destination and default integration order 4. The variable TVI, is the time integral of the induced voltage in the winding. It may seem more intuitive to try integrating the time derivative Az2t to get this voltage directly. However, models using time derivatives directly in multiphysics couplings will not be properly assembled so an extra ODE variable must be used to define the time derivative of TVI.


phi atan2(y,x)-pi/8 angular offset from right edge

theta mod(phi-alpha,pi/4)-phi angular source position for sliding mesh coupling

G 2*(sin(4*(phi+alpha))>0)-1 antisymmetric factor in sliding mesh coupling

NAME EXPRESSION IN SUBDOMAINS 1-3, 6 EXPRESSION IN ALL OTHER SUBDOMAINS

Az_all AZ Az2

Ez_all EZ Ez_emqa2

normB_all NB normB_emqa2

R sqrt(x^2+y^2) sqrt(x^2+y^2)

u -y*alphat 0

v x*alphat 0

NAME EXPRESSION IN SUBDOMAINS 1, 2, 6 EXPRESSION IN SUBDOMAINS 7, 8

TVI -8*L*NN*Az2/A

I 8*L*rho*(x^2+y^2)


60 | C H A P T E R

8 From the Options menu, point to Integration Coupling Variables and click Boundary

Variables.

9 In the Boundary Integration Variables dialog box, enter the following variable with default global destination and default integration order 4. This is where the electromagnetic torque is computed by boundary integration of the cross product between the radius vector (x,y) and the normal Maxwell stress components (F_nTx,F_nTy) over the rotor boundary.

10 From the Options menu, point to Extrusion Coupling Variables and click Subdomain

Variables.

11 In the Subdomain Extrusion Variables dialog box, select subdomain 3 and the variable named, A0z, with the expression Az.

12 Change the expression to Az*G.

13 Click the Destination tab and select the destination boundary 17.

14 Edit the expressions in the Destination transformation area to.

15 Click the Source tab, select the subdomains 1–3 and 6, and change the remaining variables according to the table below.

16 For each of these variables (not for A0z), click the Destination tab, and make sure that Subdomain is selected in the Level list. Select subdomains 1-3 and 6 and enter new x and y transformations:

17 Click OK.

NAME EXPRESSION ON BOUNDARIES 7, 8, 16, 19, 26

Tz (x*F_nTy_emqa-y*F_nTx_emqa)*8*L

X Y

x*cos(theta)-y*sin(theta) x*sin(theta)+y*cos(theta)

NAME EXPRESSION

AZ Az*G

NB normB_emqa

EZ G*(Ez_emqa-u*By_emqa+v*Bx_emqa)

X Y

x*cos(theta)-y*sin(theta) x*sin(theta)+y*cos(theta)


Subdomain Settings1 From the Multiphysics menu, select Perpendicular Induction Currents, Vector Potential

(emqa2).

2 Open the Subdomain Settings dialog box, select subdomains 7, 8 from the Subdomain

selection list and set the External current density to NN*vi/(Rc*A).

3 Click OK.

4 From the Multiphysics menu, select Perpendicular Induction Currents, Vector Potential

(emqa).

5 Open the Subdomain Settings dialog box. On the Forces tab, select subdomains 1, 2

and 6 from the Subdomain selection list and set the Name field to F.

6 Click OK.

Periodic Boundary ConditionsYou only need to set the new antisymmetric boundaries.


(emqa).


3 Click OK.


(emqa2).


6 Click OK.

7 Open the Periodic Boundary Conditions dialog from the Physics menu.

8 Select boundaries 1,3 and 4 from the Boundary selection list on the Source tab.

9 Enter the Expression Az.

10 Click in the Constraint name field and the name pconstr1 appears automatically.

11 Click the Destination tab, and select boundaries 2,10 and 12.

BOUNDARY 1-4, 10, 12


BOUNDARY 5, 6, 13, 14



62 | C H A P T E R

12 Select the Use selected boundaries as destination check box and enter the expression -Az.

13 Click the Source Vertices tab, and add source vertices 1 and 4 in that order.

14 Click the Destination Vertices tab, and add destination vertices 1 and 22 in that order.

15 Go back to the Source tab.

16 Select boundaries 5 and 6 from the Boundary selection list on the Source tab.

17 Enter the Expression Az2 on a new line (not the pconstr1 line).

18 Click in the Constraint name field and a name pconstr2 will appear automatically.

19 Click the Destination tab, and select boundaries 13 and 14.

20 Select the Use selected boundaries as destination check box and enter the expression -Az2.

21 Click the Source Vertices tab, and add source vertices 4 and 11 in that order.

22 Click the Destination Vertices tab, and add destination vertices 22 and 27 in that order.

23 Click OK.

Periodic Point ConditionsYou must also make the Lagrange multiplier variable lm2, associated with the weak boundary constraint, antisymmetric.

1 Open the Periodic Point Conditions dialog from the Physics menu.

2 Select point 4 from the Point selection list on the Source tab.

3 Enter the Expression lm2.

4 Click in the Constraint name field and a name pconstr1 appears automatically.

5 Click the Destination tab, and select point 22.

6 Select the Use selected points as destination check box and enter the expression -lm2.

7 Click the Source Vertices tab, and add source vertex 4.

8 Click the Destination Vertices tab, and add destination vertex 22.

9 Click OK.

ODE Settings1 From the Physics menu, open the ODE Settings dialog box and enter entities

according to the following table. The description field is optional. Note that the


time derivative of TVI cannot be addressed unless the algebraic state variable tvi is

introduced.

2 Click OK.


1 In the Mesh Parameters dialog box, select Finer from the Predefined mesh sizes list and enter 1 in the Resolution of narrow regions edit field.

2 On the Boundary tab, select boundaries 2, 7, 8, 10, 12-14, 16, 17, 19 and 26 from the Boundary selection list and enter a Maximum element size of 0.005.




2 Click OK.

NAME EQUATION Init(u) Init(ut) DESCRIPTION

tvi tvi-TVI 0 0 time integral of induced voltage in winding

vi vi-tvit 0 0 induced voltage in winding

alpha alphatt-(M+Tz)/(I+I0) 0 0 rotation angle


64 | C H A P T E R

3 Open the Solver Manager, click the Initial value expression evaluated using current solution button in the Initial value and click the Use setting from Initial value frame button in the Values of variables not solved for and

linearization point.

4 Click the Solve for tab and deselect the variables vi and alpha under ODE.

5 Click OK.


7 Open the Solver Parameters dialog box again, and select Time dependent from the Solver list.

8 Enter linspace(0,2,101) in the Times edit field, and enter Az 1e-3 Az2 1e-3 lm2 5e3 alpha 0.015 tvi inf vi inf in the Absolute tolerance edit field.

9 Click OK.

10 Open the Solver Manager, click on the Solve for tab and select all variables.

11 Click OK.



1 Open the Plot Parameters dialog box. Select the solution for time 0.20 s from the Solution at time list.

2 On the Surface tab, type normB_all in the Expression edit field.

3 On the Contour tab, type Az_all in the Expression edit field and 15 in the Levels text field. Click to select the Contour plot check box.

4 In the Contour color frame, click the Uniform color radio button. Click the Color button and select a black color. Clear the Color scale check box.



To view the rotation angle as a function of time, proceed with the steps below.

6 From the Postprocessing menu, select Domain Plot Parameters.

7 On the Point tab, select an arbitrary point in the Point selection list, and type alpha in the Expression edit field.

8 Click OK.

9 Repeat steps 6 through 8 for alphat, Tz,(M+Tz)/(I+I0) and vi/Rc to plot angular velocity, electromagnetic torque, acceleration, and winding current, respectively.

To create an animation of the rotating rotor, do the following steps:

1 Open the Plot Parameters dialog box, click the Animate tab, and make sure that all time steps are selected.

2 Click the Start Animation button. When the movie has been generated, the COMSOL Movie Player will open it.


66 | C H A P T E R

Gene r a t o r i n 3D

Introduction

This model is a 3D version of the 2D generator model on page 41, so most of the details about the model are explained there. The main difference is that this is a static example, calculating the magnetic fields around and inside the generator.


The model has some differences compared to the 2D generator model. The PDE is simplified to solve for the magnetic scalar potential, Vm, instead of the vector potential, A. It is based on the assumption that currents can be neglected, which holds true when the generator terminals are open. The equation for Vm becomes

The stator and center of the rotor are made of annealed medium-carbon steel, which is a nonlinear magnetic material. This is implemented in COMSOL Multiphysics as an interpolation function for the relative permeability, µr. The difference compared to the 2D version is that the argument to this function needs to be changed to the norm of the magnetic field, |H|, in this model. The reason is that in this problem, H is calculated from the dependent variable Vm according to

B is then calculated from H using the relation

resulting in an implicit or circular dependence of µr if B were used as the argument for the interpolation function.


In Figure 3-7, the norm of the magnetic flux is plotted for a slice through a centered cross-section of the generator. The streamlines of the magnetic flux are also plotted. The starting points of the streamlines have been carefully selected to show the closed

∇– µ Vm∇ Br–( )⋅ 0=

H Vm∇–=

B µ0µrH Br+=


loops between neighboring stator and rotor poles. A few streamlines are also plotted at the edge of the generator, to illustrate the field there.

Figure 3-7: A combined slice and streamline plot of the magnetic flux density.

Model Library path: Electromagnetics_Module/Motors_and_Drives/generator3d



1 In the Model Navigator, select 3D from the Space dimensions list.

2 In the Electromagnetics Module folder, select Statics>Magnetostatics, No Currents.



1 Go to Draw>Work Plane Settings.

2 In the dialog box that appears, click OK.

3 Go to Draw>Specify Object>Circle to create circles with the following properties:


C1 0.3 Center 0 0 22.5

C1 0.235 Center 0 0 0

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4 Go to Draw>Specify Objects>Rectangle to create rectangles with the following properties:

5 In the Create Composite Object dialog box, clear the Keep interior boundaries check box.

6 Enter the formula C2+C1*(R1+R2+R3+R4).

7 Click OK.


9 Draw four rectangles with the following properties:

10 In the Create Composite Object dialog box, enter the formula C1*(R1+R2+R3+R4).

11 Click OK.


13 In the Create Composite Object dialog box, make sure that the Keep interior

boundaries check box is not selected.

14 Enter the formula C1+CO2.

15 Click OK.


R1 0.1 1 Center 0 0 0

R2 0.1 1 Center 0 0 45

R3 0.1 1 Center 0 0 90

R4 0.1 1 Center 0 0 135


C1 0.215 Center 0 0 0


R1 0.1 1 Center 0 0 22.5

R2 0.1 1 Center 0 0 -22.5

R3 0.1 1 Center 0 0 67.5

R4 0.1 1 Center 0 0 -67.5


C1 0.15 Center 0 0 22.5


16 Draw a circle with parameters according to the table below.


18 Go to Draw>Extrude.

19 In the dialog box that appears, select all objects and type 0.4 in the Distance edit field.

20 Click OK.

21 Use the Cylinder tool to create cylinders with parameters according to the table below.



1 From the Options menu, select Constants.


C1 0.15 Center 0 0 22.5

NAME AXIS BASE POINT Z

RADIUS HEIGHT

CYL1 -0.2 0.4 0.2

CYL2 0 0.4 0.4

CYL3 0.4 0.4 0.2


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2 In the Constants dialog box, define the following constants with names and expressions:

3 Click OK.

4 Go to Options>Functions.

5 In the Functions dialog box, define a new Interpolation function MUR with the interpolation method Linear, the extrapolation method Constant and data from Table.

6 Input interpolation points according to the list below.


BrSmCo 0.84 Remanent flux in permanent magnets (T)


X F(X)

4000 281

8000 158

16000 87.0

24000 60.9

32000 47.1

40000 38.5

48000 32.6

64000 24.9

80000 20.3

96000 17.2

112000 14.9

128000 13.2

144000 11.9

176000 9.93

208000 8.56

240000 7.56

272000 6.78

304000 6.18

336000 5.68

396000 4.96


7 Click OK.

Subdomain Settings1 Enter subdomain settings for the active subdomains according to the table below.

In order to set a remanent flux density, you need to click the radio button for first.

2 Click the Init tab, select all subdomains, and enter R in the Vm(t0) edit field.

3 Click OK.

Boundary Conditions1 In the Boundary Settings dialog box, apply the following boundary condition:

2 Click OK.


1 In the Mesh Parameters dialog box, select Finer from the Predefined mesh sizes list.



1 In the Solver Parameters dialog box, select Stationary linear from the Solver list.

2 Click OK.



5 Click OK.



1 Open the Plot Parameters dialog box.

2 Click the check boxes for Slice plot and Streamline plot.

SUBDOMAIN 1, 3, 4 5, 9, 10, 13 6, 8, 11, 12 2, 7

Rel. permeability 1 murSmCo murSmCo MUR(normH_emnc)

Rem. flux density x -BrSmCo*x/R BrSmCo*x/R

Rem. flux density y -BrSmCo*y/R BrSmCo*y/R

BOUNDARY ALL

Boundary condition Magnetic insulation

B µ0µrH Br+=


72 | C H A P T E R

3 Click the Slice tab. From the Predefined quantities list, select Magnetic flux density,

norm.

4 Type 0 in the edit field for x levels, and 1 in the edit field for z levels.

5 Click the Streamline tab, and select Magnetic flux density from the Predefined

quantities list.

6 Click the Specify start point coordinates radio button, and enter the values according to the table below.

7 Select tube from the Line type list, and click OK.

FIELD EXPRESSION

x 0 0 0 0 0 0 0

y 0.35 0.35 0.35 0.35 0.35 0.3 0.2

z linspace(0.1,0.59,7)


4

R F a n d M i c r o w a v e M o d e l s

In this chapter, you will find models within RF and Microwave Engineering. From the mathematical viewpoint, the same physics formulations are used in the chapter on Optics and Photonics. The terminology and the way numerical results are presented differs slightly but anyone interested in RF and Microwave Engineering may benefit from also reading that chapter.

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Th r e e - p o r t F e r r i t e C i r c u l a t o r

Introduction

A microwave circulator is a multiport device that has the property that a wave incident in port 1 is coupled into port 2 only, a wave incident in port 2 is coupled into port 3 only and so on. Circulators are used to isolate microwave components e.g. to couple a transmitter and a receiver to a common antenna. They typically rely on the use of anisotropic materials, most commonly ferrites. In this example, a three-port circulator is constructed from three rectangular waveguide sections joining at 120º where a ferrite post is inserted at the center of the joint. The geometry is shown in the figure below.

To match the junction, identical dielectric tuning elements are inserted into each branch (not shown above). The ferrite post is magnetized by a static H0 bias field along the axis. The bias field is usually supplied by external permanent magnets. Here, the focus is on the modeling of the ferrite and how to minimize reflections at the inport by matching the junction by the proper choice of tuning elements. For a general introduction to the modeling of rectangular waveguide structures, see the waveguide H-bend model found on page 37 in the Electromagnetics Module User’s Guide and the corresponding 2D model found on page 116 in this document. Matching the

4 : R F A N D M I C R O W A V E M O D E L S

circulator junction involves calculating how well a TE10 wave propagates between ports in the circulator for different materials in the tuning element. This is done by calculating the scattering parameters, or S-parameters, of the structure as a function of the permittivity of the tuning elements for the fundamental TE10 mode. The S-parameters are a measure of the transmittance and reflectance of the circulator. For a theoretical background on S-parameters, see the section “S-Parameters” on page 86 in the Electromagnetics Module User’s Guide.

This model only includes the TE10 mode of the waveguide. Thus the model can be made in 2D as the fields of the TE10 mode have no variation in the transverse direction. The figure below shows the 2D geometry including the dielectric tuning elements.

Model Definition

The dependent variable in this application mode is the z component of the electric field E. It obeys the following relation:

where µr denotes the relative permeability, ω the angular frequency, σ the conductivity, ε0 the permittivity of vacuum, εr the relative permittivity, and k0 is the free space wave number. Losses are neglected so the conductivity is zero everywhere.

∇ µr1– ∇ Ez×( )× εr

jσωε0---------–⎝ ⎠

⎛ ⎞ k02Ez– 0=

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The magnetic permeability is of key importance in this model as it is the anisotropy of this parameter that is responsible for the nonreciprocal behavior of the circulator. The theory of the magnetic properties of ferrites is described in references [1] and [2]. It is assumed that the static H0 magnetic bias field is much stronger than the alternating magnetic field of the microwaves so the quoted results are a linearization for a small signal analysis around this operating point. Further it is assumed that the applied magnetic bias field is strong enough for the ferrite to be in magnetic saturation. Under those assumptions and neglecting losses, the anisotropic permeability of a ferrite magnetized in the positive z-direction is given by:

where

and

Here µ0 denotes the permeability of free space, ω is the angular frequency of the microwave field, ω0 is the precession frequency or Larmor frequency of a spinning electron in the applied H0 magnetic bias field, ωm is the electron Larmor frequency at the saturation magnetization Ms of the ferrite, and finally γ is the gyromagnetic ratio of the electron. For a lossless ferrite, the permeability clearly becomes unbounded at ω=ω0. In a real ferrite, this resonance becomes finite and is broadened due to losses. For complete expressions including losses, the interested reader is referred to references [1] and [2]. In this analysis the operating frequency is chosen sufficiently off from the Larmor frequency to avoid the singularity. The material data, Ms=2.39e5 A/m and εr=12.9, were taken for Magnesium ferrite from reference [2]. The applied bias field was set to H0=2.72e5 A/m, that is well above saturation. The electron

µ[ ]µ jκ 0jκ– µ 00 0 µ0

=

κ µ0ωωm

ω02 ω2–

----------------------

⎝ ⎠⎜ ⎟⎛ ⎞

=

µ µ0 1ω0ωm

ω02 ω2–

----------------------+⎝ ⎠⎜ ⎟⎛ ⎞

=

ω0 µ0γH0=

ωm µ0γMs=


gyromagnetic ratio was set to 1.759e11 C/kg. Finally an operating frequency of 10 GHz was selected to be well above the cutoff for the TE10 mode which for a waveguide cross-section of 2 cm by 1 cm is at about 7.5 GHz. At the ports, matched port boundary conditions were used to make the boundaries transparent to the wave.


The S11 parameter as a function of the relative permittivity of the matching elements, eps_r is shown in the figure below. The S11 parameter corresponds to the reflection coefficient at port 1. Thus matching the junction is equivalent to minimizing the magnitude of S11.

By choosing eps_r to about 1.28, a reflection coefficient of about -35 dB is obtained which is a good value for a circulator design. Judging from the absence of standing wave patterns in the magnitude plot of the electric field and by looking at the direction


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of the microwave energy flow in the figure below it is clear that the circulator behaves as desired.

By feeding the circulator at a different port, the nonreciprocal behavior of the device becomes quite apparent.


References

[1] Robert E. Collin: Foundations for Microwave Engineering, Second Edition, IEEE Press/Wiley-Interscience

[2] David M. Pozar: Microwave Engineering, Third Edition, John Wiley & Sons Inc.

Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/circulator


In the Model Navigator, select 2D in the Space dimension list and then select the Electromagnetics Module>In-Plane Waves>TE Waves>Harmonic propagation application mode.


Define the following constants in the Constants dialog box. The description field is optional and can be omitted.


Use the toolbar buttons and the Draw and Edit menus.


gamma 1.759e11 Gyromagnetic ratio (C/kg)

mu0 4e-7*pi Permeability of free space (Vs/Am)

freq 3e8/0.03 Frequency (Hz)

w 2*pi*freq Angular frequency (radians/s)

H0 w/(gamma*mu0+1e4) Applied magnetic bias field (A/m)

w0 mu0*gamma*H0 Larmor frequency (radians/s)

Ms 0.3/mu0 Saturation magnetization (A/m)

wm mu0*gamma*Ms Larmor frequency at saturation limit (radians/s)

mur 1+w0*wm/(w0^2-w^2) Relative permeability tensor element

kr w*wm/(w0^2-w^2) Relative permeability tensor element

eps_r 1 Relative permittivity in matching layer


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1 Draw a Rectangle with the lower left corner in (-0.06,-0.01) and the width 0.06 and height 0.02.

2 Copy the rectangle and paste it without any displacement.

3 Rotate the pasted rectangle 120º around (0,0).

4 Paste a new copy of the original rectangle without any displacement.

5 Rotate the pasted rectangle -120º around (0,0).

6 Select all objects and click the Union button in the toolbar.

7 Click the Delete Interior Boundaries button in the toolbar.

8 Draw another Rectangle with the lower left corner in (-0.02,-0.01) and the width 0.0142264973081037 and height 0.02. The width must be entered to all decimal places, or the two right corners of the rectangle will not connect to the corners in the previously created object. An alternative way to define this rectangle is: Go to the Draw menu and select Draw Objects>Rectangle/Square. Point the mouse to (-0.02,-0.01). Click the left mouse button and keep pressing it down while you move the pointer diagonally up and to the right. When it eventually snaps to the wanted corner in the existing object, you release the mouse button.

9 Repeat steps 2-5 above for the latest rectangle.

10 Draw a Circle with a radius of 0.0023 centered at the origin.

S C A L A R V A R I A B L E S

In the Application Scalar Variables dialog box available from the Physics menu, set the Frequency to freq.


The default boundary condition is perfect electric conductor which is fine for all external boundaries except at the ports. Interior boundaries are by default not active meaning that continuity is imposed.

At boundaries number 1, 20 and 21 specify the Port boundary condition. On the Port tab set the values according to the table below:

BOUNDARY 1 20 21

Port number 1 3 2

Wave excitation at this port Selected Cleared Cleared

Mode specification Analytic Analytic Analytic

Mode number 1 1 1


S U B D O M A I N S E T T I N G S

Click the Groups tab in the Subdomain Settings dialog box and create three groups called air, dielectric and ferrite respectively. Keep the default subdomain settings for the air group. For the dielectric group, change the Relative permittivity to eps_r. For the ferrite group, set the Relative permittivity to 12.9, select the Relative permeability to be anisotropic and enter: mur -i*kr i*kr mur. Then click the Subdomains tab and assign subdomains 1,3 6 and 7 to the air group. Assign subdomains 2,4 and 5 to the dielectric group and subdomain 8 to the ferrite group.


Open the Mesh Parameters dialog box and click on the Subdomain tab and set the Maximum element size to 0.003 for subdomains 1 to 7 and to 0.0005 for subdomain 8. This will make sure that the waves are properly resolved everywhere. Especially in the ferrite, the wavelength will be much shorter than in free space, calling for a much denser mesh here. Click OK and click the Initialize Mesh button in the toolbar.


Use the parametric solver in order to study the S-parameters as a function of the relative permittivity in the matching elements.

1 In the Solver Parameters dialog box set Solver to Parametric linear.

2 Enter eps_r as Name of parameter and linspace(1,1.5,51) as List of parameter

values. The solver will calculate the solution using 51 equidistant permittivity values in the range 1 to 1.5.

3 Solve by clicking the Solve button.

The solution time is about 30 s on a PC with a 2.2 GHz P4 processor running Windows XP.


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The default plot of the instantaneous electric field is shown above. It is more instructive to look at the magnitude of the electric field and to set the color range manually as the plot otherwise will be dominated by the strong electric field in the ferrite post. Open the Plot Parameters dialog, click on the Surface tab and select Electric field, norm from the list of Predefined quantities. Click the Range button and set a range from 0 to 350 before clicking OK. Then click the Arrow tab, check Arrow plot and select Power flow,


time average from the list of Predefined quantities. Set the arrow Color to black before clicking OK.

The presence of standing waves at the input is clearly visible for this value (1.5) of the relative permittivity in the matching elements. The next step is to plot the reflection coefficient S11 as a function of eps_r. You can do this using a domain plot.

1 In the Domain Plot Parameters dialog box select all solutions in the Solutions to plot list.

2 Select the Point tab and select S_parameter dB(S11) from the list of Predefined

quantities.


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3 S11 is defined everywhere, but you only need to evaluate it at a single point to make a plot as a function of eps_r. Select point 1 and click Apply to plot S11.

We see that S11 is minimized by setting eps_r to 1.28. Make this change in the Constants dialog, open the Solver Parameters dialog and select the Stationary linear solver and solve again. Any standing wave pattern should barely be visible now. Change the inport from port 1 to port 2 by editing the boundary settings for these boundaries and solve again to see the nonreciprocal behavior of the device.


Mono c on i c a l R F An t e nna

Introduction

Conical antennas are useful for many applications due to their broadband characteristics and relative simplicity. This example includes an analysis of the antenna impedance and the radiation pattern as functions of the frequency for a monoconical antenna with a finite ground plane and a 50 Ω coaxial feed. The rotational symmetry makes it possible to model this in 2D using one of the axisymmetric electromagnetic wave propagation application modes. When modeling in 2D, you can use a dense mesh, giving an excellent accuracy for a wide range of frequencies.

Model Definition

The antenna geometry consists of a 0.2 m tall metallic cone with a top angle of 90 degrees on a finite ground plane of a 0.282 m radius. The coaxial feed has a central conductor of 1.5 mm radius and an outer conductor (screen) of 4.916 mm radius separated by a teflon dielectric of relative permittivity of 2.07. The central conductor of the coaxial cable is connected to the cone, and the screen is connected to the ground plane.

Figure 4-1: The geometry of the antenna. The central conductor of the coaxial cable is connected to the metallic cone, and the cable screen is connected to the finite ground plane.

The COMSOL Multiphysics model takes advantage of the rotational symmetry of the problem, which allows modeling in 2D, using cylindrical coordinates. When modeling in 2D you can use a very fine mesh, to achieve an excellent accuracy.

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An electromagnetic wave propagating in a coaxial cable is characterized by transverse electromagnetic fields (TEM). Assuming time-harmonic fields with complex amplitudes containing the phase information, you have:

where z is the direction of propagation and r, ϕ, and z are cylindrical coordinates centered on axis of the coaxial cable. Z is the wave impedance in the dielectric of the cable, and C is an arbitrary constant. The angular frequency is denoted by ω. The propagation constant, k, relates to the wavelength in the medium λ as

In the air, the electric field also has a finite axial component whereas the magnetic field is purely azimuthal. Thus it is possible to model the antenna using an axisymmetric transverse magnetic (TM) formulation, and the wave equation becomes scalar in Hϕ:


The boundary conditions for the metallic surfaces are:

At the feed point, a matched coaxial port boundary condition is used to make the boundary transparent to the wave. The antenna is radiating into free space, but you can only discretize a finite region. Therefore, truncate the geometry some distance from the antenna using a scattering boundary condition allowing for outgoing spherical waves to pass without being reflected. Finally, apply a symmetry boundary condition for boundaries at r = 0.


Figure 4-2 shows the antenna impedance as a function of frequency. Ideally, the antenna impedance should be matched to the characteristic impedance of the feed,

E erCr----ej ωt kz–( )=

H eϕCrZ-------ej ωt kz–( )=

k 2πλ------=

1ε--- Hϕ∇×⎝ ⎠

⎛ ⎞ µω2Hϕ–∇× 0=

n E× 0=


50 Ω, to obtain maximum transmission into free space. This is quite well fulfilled in the high frequency range.

Figure 4-2: The antenna impedance as a function of frequency from 200 MHz to 1.5 GHz. The solid line shows the radiation resistance, whereas the dashed line represents the reactance.

Figure 4-3 shows the antenna radiation pattern in the near-field for three different frequencies. The effect of the finite diameter of the ground plane is to lift the main lobe from the horizontal plane. For an infinite ground plane or in the high frequency limit, the radiation pattern is symmetric around zero elevation. This is easy to understand, as an infinite ground plane can be replaced by a mirror image of the monocone below the plane. Such a biconical antenna is symmetric around zero elevation and has its main


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lobe in the horizontal direction. The decreased lobe lifting at higher frequencies is just about visible in Figure 4-3.

Figure 4-3: The antenna radiation pattern in the near-field as a function of the elevation angle for 200 MHz (solid line), 863 MHz (dotted line) and 1.5 GHz (dashed line).


Figure 4-4 shows the antenna radiation pattern in the far-field for the same frequencies as the radiation pattern at the boundary in Figure 4-3.

Figure 4-4: The antenna radiation pattern for the far-field as function of elevation angle for 200 MHz (solid line), 863 MHz (dotted line) and 1.5 GHz (dashed line).

Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/conical_antenna


Select the Axial symmetry (2D), Electromagnetics Module, Electromagnetic Waves, TM

Waves, Harmonic propagation mode in the Model Navigator.


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1 In the Constants dialog box, enter the following variable name and expression.

2 In the Axis/Grid Settings dialog box, give grid settings according to the following table.


To create the model geometry, draw the right half of the cross section of the antenna and the coaxial feed in a truncated half space.

1 Open the Rectangle dialog box by shift-clicking the Rectangle/Square button. Set Width to 0.282, Height to 0.301, set Base to Corner and set r to 0, and z to -0.101.

2 Using the same approach, draw a second rectangle and set the Width to 0.003416, Height to 0.1, r to 0.0015, and z to -0.1.

3 Draw a third rectangle and set the Width to 0.276, Height to 0.091, r to 0.006, and z to -0.101.


NAME EXPRESSION

frequency 5e8

GRID

r spacing 0.05

Extra r 0.0015 0.004916

z spacing 0.05

Extra z


5 Click the Line button, and draw a polygon with corners at (0.3, 0), (0.3, 0.2), (0.2, 0.2) and (0.0015, 0). Finally right-click to create a solid object CO1.

6 Open the Create Composite Object dialog box, and enter the formula R1-(R2+R3+CO1). Click OK to create the composite object.

7 Zoom out and draw a circle with radius 0.6 centered at (0, 0).

8 Draw a rectangle with opposite corners at (-0.6, -0.6) and (0, 0.6).

9 Select the three objects R1, C1, and CO1 and click the Difference button.

10 Zoom in around (0, 0) and draw a line from (0.0015, 0) to (0.004916, 0).


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Scalar Variables1 Set the frequency to frequency in the Scalar Variables dialog box to obtain a 500

MHz wave.

Boundary Conditions1 Enter boundary conditions according to the following table.

2 In the Boundary Settings dialog box, select boundary 6 and click the Port tab.

3 Check the Wave excitation at this port box.

4 Set Mode specification to Coaxial.

5 Click OK.

Subdomain SettingsThe inner of the coaxial line is made of teflon. The subdomain parameters need to be specified accordingly.

1 Enter the subdomain settings according to the following table.


1 Open the Mesh Parameters dialog box and set Maximum element size to 2.5e-2.

2 To make the mesh finer inside the coaxial line, where the wavelength is shorter due to higher permittivity, click on the Subdomain tab, select subdomain 2, and set Maximum element size to 5e-4.

3 To make the mesh finer near the antenna, click on the Boundary tab, select boundaries 4 and 8, and set Maximum element size to 2.5e-3.

4 Click the Remesh button.

BOUNDARY 1, 3 6 14, 15 ALL OTHER

Boundary condition

Axial symmetry

Port Scattering boundary condition

Perfect electric conductor

Wave type Spherical wave

SUBDOMAIN 1 2

µr 1 1

εr 1 2.07





The default plot shows the azimuthal magnetic field component of the transmitted wave. Due to the strong field in the coaxial line, you need to manually adjust the plot parameters.

1 Click on Range on the Surface tab in the Plot Parameters dialog box, clear the Auto check box and set Min and Max to -0.5 and 0.5 respectively.

2 Go to the Contour tab, check Contour plot and choose Magnetic field, phi component from the Predefined quantities list. Then in the Contour color frame, select Uniform

color and set the color to white. Finally, under Contour levels, select Vector with

isolevels, and type -20:4:20. Click OK.

Antenna Impedance and Radiation Pattern

A frequency sweep for the radiation pattern and the antenna impedance can be made from the graphical user interface or using a script file. The S11 scattering parameter is automatically computed when using the port boundary condition at the feed boundary. From S11, the antenna impedance is deduced using the relation

where Ztl = 50 Ω is the characteristic impedance of the coaxial line.

Z Ztl1 S11+

1 S11–-------------------=


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1 In the Constants dialog box, enter the following variable name and expression.

2 In the Scalar Expressions dialog box, define the following variables.

S O L V I N G T H E M O D E L

1 Open the Solver Parameters dialog box and select the Parametric linear solver.

2 Set Name of parameter to frequency and List of parameter values to linspace(200e6,1.5e9,50).

3 Click OK and solve the problem.


To plot the antenna impedance as a function of frequency

1 Open the Domain Plot Parameters dialog box.

2 Set Plot type to Point plot and select all frequencies in the Solutions to plot list.

3 Click on the Title/Axis button. Enter Antenna impedance (ohm) as Title and click OK.

4 Go to the Point tab and select point 1. Then enter real(Z) as Expression to plot the radiation resistance.

5 Click on Line settings and select blue for Line color and Solid line for Line style.

6 Click Apply to make the plot.

7 In order to make a second plot in the same figure window, go to the General tab and check Keep current plot.

8 Go to the Point tab and enter imag(Z) as Expression to plot the reactance.

9 Click on Line settings and select green for Line color and Dashed line for Line style.

10 Click Apply to make the plot in Figure 4-2.

To plot the radiation pattern as a function of frequency

1 On the General tab, set Plot type to Line/Extrusion and select a few frequencies in the Solutions to plot list, for example 2e8, 8.632653e8 and 1.5e9.

NAME EXPRESSION

Z_tl 50

VARIABLE NAME EXPRESSION

Z Z_tl*(1+S11_emwh)/(1-S11_emwh)


2 Clear Keep current plot.

3 Click on the Title/Axis button and enter Radiation pattern (dB) as Title and click OK.

4 Go to the Line/Extrusion tab and select Boundaries 14 and 15. Then enter 10*log10(nPoav_emwh) as Expression and clear Smooth.

5 Click on Expression for x-axis data and enter atan2(z,r)*180/pi as Expression and click OK.

6 Click on Line settings and select cycle for Line color. Change the Line style to cycle.

7 Click OK to make the plot in Figure 4-3.

Far-Field Computation

You can easily add far-field calculation to the model, and then plot the radiation pattern in the far-field as shown in Figure 4-4.


1 Select Far-Field from the Physics menu to open the Far-Field Variables dialog box.

2 Select boundaries 14 and 15 in the Boundary selection.

3 Enter far-field variable with name Efar and select the Source boundaries check box.

4 Click the Destination tab.

5 Change the Level to Boundary and select boundaries 14 and 15.

6 Click OK.


To evaluate the far-field variables, select Update Model from the Solver menu.


To plot the radiation pattern of the far-field

1 Open the Domain Plot Parameters dialog box from the Postprocessing menu.

2 Select a few frequencies in the Solution to use list, for example 2e8, 8.632653e8 and 1.5e9.

3 Click the Line/Extrusion tab and select boundaries 14 and 15.

4 Enter 10*log10(abs(Efarr)^2+abs(Efarz)^2+1e-2) as Expression.

5 Click on Line Settings and make sure that Line color and Line style are cycle.

6 Click OK to make the plot in Figure 4-4.


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Magne t i c D i p o l e An t e nna

Introduction

A magnetic dipole consists in its simplest form of a current carrying coil. Three such coils coupled together in an AC power circuit constitute a radiating system. Changing the phases of the coils alters the transmission angle of the waves.

Model Definition

An array of three magnetic dipoles with a radius of 5 cm will be modeled. The dipoles are placed 0.1 m apart. The cylindrical symmetry of the problem allows us to set up a 2D axisymmetric model. The dipoles then appear as points in the r-z modeling plane.

The emitted waves from the dipoles are axisymmetric TE waves. The governing equation for the waves is

where µr is the relative permeability, εr the relative permittivity, and k0 the free space wave number. In the axisymmetric case this equation can be reduced to a scalar equation for the azimuthal component of the electric field Eϕ.

∇ µr1– ∇ E×( )× k0

2εrE– 0=



The figure below shows the radiation pattern from an array of three coils, each carrying a current of 1 A. The relative phases of the currents are ϕ1 = 0, ϕ2 = 0.7 rad, ϕ3 = 1.4 rad.

Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/dipole_array


Select the Axial symmetry (2D), Electromagnetics Module, Electromagnetic Waves, TE

waves, harmonic propagation mode in the Model Navigator.



NAME EXPRESSION

I0 1

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2 Use the default frequency of 1 GHz, given in the Application Scalar Variables dialog box.


1 Draw a circle with the radius 1 centered at (0,0).

2 Draw a rectangle with opposite corners at (0,-1) and (1,1).

3 Select both the circle and rectangle and click the Intersection button.

4 Draw three points at the coordinates (0.05, -0.1), (0.05, 0), and (0.05, 0.1).


Point SettingsDefine the three dipoles by entering a current at the points according to the following table.

th1 -0.7

th2 2*th1

POINT 3 4 5

I0 I0 I0*exp(j*th1) I0*exp(j*th2)

NAME EXPRESSION


Boundary ConditionsEnter boundary conditions according to the following table.

Subdomain SettingsUse the default values applicable for air.


1 To obtain a uniform mesh, set Maximum element size to 0.03 in the Mesh Parameters dialog box.

2 Generate the mesh.




The default plot shows the electric field component of the transmitted wave.

To show the radiation pattern from the antenna array, make a plot along the half-circle boundary.

BOUNDARY 1 2, 3

Boundary condition Axial symmetry Scattering boundary condition


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1 In the Domain Plot Parameters dialog box select Line plot. Set the y-axis data to 10*log10(nPoav_emwe+1e-3) and select boundaries 2 and 3. The term 1e-3 is to avoid singularities at the end points.


% When programming the same model you start by defining the % constants. Use a constant named I1 rather than I0 to avoid% conflict with the point variable I0 generated by the % application mode.clear femfem.const=... 'I1',1,... 'th1',-0.7,... 'th2',-1.4;

% You can now define the coordinates and the geometry.fem.sdim='r' 'z';clear drawdraw.s.objs=circ2(0,0,1)*rect2(0,1,-1,1);draw.s.name='C01';draw.p.objs=point2(0.05,-0.1),point2(0.05,0),point2(0.05,0.1);draw.p.name='PT1','PT2','PT3';fem.draw=draw;fem.geom=geomcsg(fem);

R 4 : R F A N D M I C R O W A V E M O D E L S

% Create the application structure and compute the corresponding % FEM coefficients.clear applappl.mode.class='AxisymmetricWaves';appl.mode.type='axi';appl.bnd.type='ax' 'SC';appl.bnd.ind=[1 2 2];appl.pnt.I0='0' 'I1' 'I1*exp(j*th1)' 'I1*exp(j*th2)';appl.pnt.ind=[1 1 2 3 4 1];fem.appl=appl;fem=multiphysics(fem);

% You can now mesh the problem using the proper parameters, and % then solve it.fem.mesh=meshinit(fem,'hmax',... 0.03 [] [] []);fem.xmesh=meshextend(fem);fem.sol=femlin(fem);

% Make a visualization of the field by the following command.postplot(fem,'tridata','Ephi')

% The radiation pattern is visualized using a cross-section plot.postcrossplot(fem,1,[2 3],'lindata','10*log10(nPoav)');


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Th e rma l D r i f t i n a M i c r owa v e F i l t e r

Introduction

Microwave filters serve to suppress unwanted frequencies in the output of microwave transmitters. Amplifiers are in general nonlinear and produce harmonics that must be suppressed using one or several narrow passband filters on the output. High frequency stability of such filters can be hard to achieve as microwave systems may be subject to thermal drift caused by high power loads or harsh environmental conditions like exposure to direct sunlight in the desert. Thus, system engineers need to estimate the drift of the passband frequency that arises due to thermal expansion of a filter.

Model Definition

Figure 4-5shows the filter geometry. It consists of a box with a cylindrical post centered on one face. It is typically made of brass covered with a thin layer of silver to minimize losses. The silver layer is sufficiently thin to have a negligible influence on the device’s thermal and mechanical properties.

Figure 4-5: A microwave filter consists of a box-shaped thin metallic shell, typically made of brass, that contains a cylindrical post. This configuration forms a closed air-filled electromagnetic cavity between the box walls and the post.

The thermal expansion and the associated drift in eigenfrequency is caused by a uniform increase of the temperature of the cavity walls. The thermal expansion is


readily computed using the Shell application mode from the Structural Mechanics Module. The electromagnetic cavity mode analysis is easily performed using the 3D Electromagnetic Waves application mode in the Electromagnetics Module. In principle, it is possible to determine the thermal drift in the passband frequency by repeated calculation of the thermal expansion followed by an electromagnetic mode analysis of the deformed geometry. A complication arises, however, because it is not wise to regenerate the mesh—doing so would introduce too much numerical noise.

A better approach is to use a special set of equations to smoothly deform the original mesh according to the thermal expansion. Thus, the Arbitrary Lagrangian-Eulerian (ALE) application for deformed geometry in COMSOL Multiphysics is used to map the thermal expansion to the geometry used for the electromagnetic analysis.


M O D E L I N G T H E T H E R M A L E X P A N S I O N

The filter’s temperature can rise due to power dissipation in the filter itself, in the surrounding electronics, or due to external heating. Figure 4-6 shows the thermal expansion results for a filter entirely made of brass.

Figure 4-6: Thermal expansion at 100 °C above the reference temperature.

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E L E C T R O M A G N E T I C M O D E A N A L Y S I S

An actual filter usually consists of multiple cavities cascaded, but this discussion limits the analysis to one cell. Figure 4-7 shows the filter’s lowest eigenmode. The typical quarter-wave resonance of the cylindrical post is clearly visible. A strong capacitive coupling between the top of the post and the nearby face of the box is also obvious.

Figure 4-7: Results from the electromagnetic mode analysis. The plot shows the electric and surface current patterns of the fundamental mode.


E I G E N F R E Q U E N C Y V E R S U S T E M P E R A T U R E

By repeating the structural and electromagnetic analysis for a number of operating temperatures, an eigenfrequency versus temperature curve is obtained. The results for two different designs appear in Figure 4-8.

Figure 4-8: Eigenfrequency versus temperature for two different designs.

For the first design, the entire filter is made of brass whereas for the second design the post is made of steel. It is obvious that the combination of steel and brass is superior to a design using brass alone. The reason is the reduced capacitive coupling between the top of the post and the nearby face of the box, which results from the different coefficients of thermal expansion for the two materials. This coupling has a strong influence on the resonant frequency and, when reduced, it counteracts the effect of an overall increase in cavity size.

Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/mw_filter


1 In the Model Navigator, click the Multiphysics button.

2 Set the Space dimension to 3D.

3 In the Structural Mechanics Module folder, select Shell>Static analysis

4 Click Add to add a copy of this application mode.


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5 In the COMSOL Multiphysics folder, select Deformed Mesh>Moving Mesh (ALE)>Static

analysis. Change Element to Lagrange -Linear.

6 Click Add.

7 In the Electromagnetics Module folder, select Electromagnetic Waves>Eigenfrequency

analysis.

8 Click Add.

9 The Electromagnetic Waves>Eigenfrequency analysis application mode will now appear under Frame (ale) in the displayed tree structure.

10 Change Ruling application mode to Electromagnetic Waves (emw).




2 In the Constants dialog box, define the following constant. The description field is optional.


1 Go to the Draw menu and click the block symbol and define a block with Length X:

0.02, Length Y: 0.02, Length Z: 0.07. Leave the other parameters at their defaults.

2 Click OK.


4 Select Work Plane Settings from the Draw menu.

5 Click OK to create a default xy work plane at z=0.


7 In the 2D work plane (Geom 2), go to Draw>Specify Objects>Circle to create a circle C1 with radius 0.004 and center at (0.01,0.01).

8 Go to Draw>Extrude and extrude C1 a Distance of 0.067.

9 Press Ctrl+A to select all objects in the 3D geometry.

10 Click the Difference button in the draw toolbar.


T0 273.15 Reference temperature (K)



B O U N D A R Y C O N D I T I O N S ( S H E L L )

1 From the Multiphysics menu, select Geom1: Shell (smsh).

2 In the Boundary settings dialog box, click the Groups tab.

3 Select the unnamed1 group and change the Name to Brass.

4 Change the Young’s modulus to 1.05e11 (Pa).

5 Change the Poisson’s ratio to 0.35.

6 Change the Density to 8400 (kg/m3).

7 Change the Thermal expansion coeff. to 2.1e-5 (K-1).

8 Change the Thickness to 0.002 (m).

9 Click the Load tab and click Include thermal expansion.

10 Set Strain temperature to T1 and Strain ref. temperature to T0. T1 is the “operating temperature” and will be defined later in the solver settings.

11 Click New to create a new group that will inherit the settings for “Brass” and name it “Steel”.

12 Click the Material tab.

13 Change the Young’s modulus to 2e11 (Pa).

14 Change the Density to 7800 (kg/m3).

15 Change the Thermal expansion coeff. to 1.15e-5 (K-1).


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16 Click the Boundaries tab. By now, all boundaries, should belong to the “Brass” group and “Steel” is only there for future use.

17 Click OK.

P O I N T S E T T I N G S

Appropriate point settings must be applied to eliminate any translation or rotation of the structure.

1 In the Point settings dialog box, select point 1.

2 Constrain Rx, Ry and Rz to 0.

3 Select point 3 and constrain Rx and Rz to 0.

4 Select point 13 and constrain Ry and Rz to 0.

5 Click OK.

B O U N D A R Y C O N D I T I O N S ( A L E )

The default subdomain settings in ALE is “free displacement” which is right for this model so you only need to set the ALE boundary displacements to the displacements from the shell application mode.

1 From the Multiphysics menu, select Geom1: Moving Mesh ALE (ale).

2 In the Boundary settings dialog box, select all boundaries.

3 Set dx, dy and dz to u, v and w, respectively.

4 Click OK.

B O U N D A R Y C O N D I T I O N S ( E M W A V E S )

The default boundary and subdomain settings in the Electromagnetic Wave application mode are “perfect electric conductor” and “air” which is fine for this analysis.


Open the Mesh Parameters dialog and select Finer from the list of Predefined mesh sizes

and close the dialog by clicking OK. Click the Initialize Mesh button in the toolbar to generate the mesh. In order to see the mesh and later the solution better, go to the


Options menu and select Suppress>Suppress Boundaries and select boundaries 1, 2 and 4. Click OK and go to mesh mode by clicking the Mesh Mode toolbar button.

C O M P U T I N G T H E S O L U T I O N ( S H E L L )

Use the parametric solver in order to study thermal expansion as a function of temperature. The solver settings are adapted for the ruling application mode which is the electromagnetic problem. Thus, you need to change these to fit also the structural problem. You also need to use the option to record the solver calls in a log that can be replayed later.

1 In the Solver Parameters dialog box select the Parametric linear solver.

2 Enter T1 as Name of parameter and linspace(273.15,373.15,11) as List of

parameter values. The solver will calculate the solution using 11 equidistant temperature values in the range 273.15 K to 373.15 K.

3 Change the Linear system solver from Direct (SPOOLES) to Direct (UMFPACK).

4 Change Matrix symmetry to Nonsymmetric.

5 Click OK.

6 Open the Solver Manager.

7 Change Initial value to Initial value expression.

8 Click the Solve For tab and select only the Shell (smsh) variables.

9 Click the Script tab and select Automatically add commands when solving.

10 Click OK.



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1 Open the Plot Parameters dialog box and deselect Slice under Plot type on the General tab. Instead, select the Boundary and Deformed shape plots.

2 Click the Boundary tab and select Total displacement (smsh) from the list of Predefined

quantities.

3 Click the Deform tab and deselect Subdomain from Domain types to deform.

4 Click OK.

C O M P U T I N G T H E S O L U T I O N ( A L E )

In order to map the shell deformations smoothly to the volume mesh, the ALE problem must be solved. The parametric solver will again be used, now starting from the previous solutions.


2 Click the Initial value tab.

3 Change the Value of variables not solved for and linearization point to Stored solution.

4 Click Store solution and click OK in the dialog that opens (store all solutions).

5 Change Parameter value to All.

6 Click the Solve For tab and select only the Moving Mesh ALE (ale) variables.

7 Click OK to close the solver manager.




1 Open the Plot Parameters dialog box and select also Slice under Plot type on the General tab.

2 Click the Slice tab and change Number of levels to 1 X level and type sqrt(dx^2+dy^2+dz^2) in the slice Expression field.

3 Click OK.

E X P O R T I N G D A T A T O C O M S O L S C R I P T

1 Go to the File menu and select Export>Fem Structure.

2 Type fem_brass in the dialog that appears.

3 Click OK and close the COMSOL Script window that appears.

B O U N D A R Y C O N D I T I O N S ( S H E L L )

Change the material in the post from Brass to steel.

1 From the Multiphysics menu, select Geom1: Shell (smsh).

2 In the Boundary settings dialog box, change Group from Brass to Steel for boundaries 6-10.

3 Click OK.

C O M P U T I N G T H E S O L U T I O N U S I N G A S O L V E R S C R I P T

Now, you must repeat the solution steps above. To automate this, use the solver script that has been recorded.


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2 Click the Script tab.

3 Select the Solve using a script check box.

4 Click OK.


E X P O R T I N G D A T A T O C O M S O L S C R I P T

1 Go to the File menu and select Export>Fem Structure.

2 Type fem_steel in the dialog that appears.

3 Click OK and close the COMSOL Script window that appears.

C O M P U T I N G T H E S O L U T I O N ( E M WAV E S )

From the graphical user interface, the electromagnetic problem can only be solved for one temperature at a time. Perform this for one temperature. Then go to the COMSOL Script command window for the rest of the analysis.


2 Click the Script tab.

3 Deselect Solve using a script.

4 Click the Solve For tab and select only the Electromagnetic Waves (emw) variables.

5 Click the Initial value tab.

6 Change Parameter value to 373.15.


7 Click the Store solution button and click OK in the dialog that opens (store all solutions).

8 Click OK to close the solver manager.

9 Open the Solver Parameters dialog.

10 Change Solver to Eigenfrequency.

11 Set Desired number of eigenfrequencies to 1 and Search for eigenfrequencies around to 1e9.

12 Click OK to close the solver parameters dialog.



1 Open the Plot Parameters dialog box and deselect Deformed shape under Plot type on the General tab. Keep the Boundary and Slice plots.

2 Click the Boundary tab and select Surface current density, norm (emw) from the list of Predefined quantities.

3 Click the Slice tab and select Electric field, norm (emw) from the list of Predefined

quantities.

4 Click OK. Note that the results may differ slightly from below as the mesh on your computer will be different and the plotted quantities are rather mesh sensitive.


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C O M P U T I N G T H E S O L U T I O N I N C O M S O L S C R I P T

Open the COMSOL Script command window from the File menu. The following commands are now issued to solve the electromagnetic problem for all temperature values for the two filter designs. The solver calls can essentially be copied from the Script tab in the Solver Manager. As COMSOL Script can open and run M-files, the most convenient procedure is to save the commands in a text file that is saved with the file extension .m, for example, eigloop.m.

fem = fem_brass;fem1 = fem;T = fem.sol.plist;freq = zeros(size(T));for k=1:length(T) u = asseminit(fem1,'init',fem.sol,'solnum',k); fem1.sol=femeig(fem1,'u',u, ... 'solcomp','tExEyEz', ... 'outcomp','z','thz','w','lm9','lm8', ... 'v','tExEyEz','thx','lm7','u','y', ... 'thy','x','neigs',1, ... 'shift',439.256636); freq(k) = postinterp(fem1,'nu_emw',zeros(0,1),'dom',1,'solnum',1);end;plot(T,freq,'+-');grid on;hold on;

fem = fem_steel;fem1 = fem;T = fem.sol.plist;freq = zeros(size(T));for k=1:length(T) u = asseminit(fem1,'init',fem.sol,'solnum',k); fem1.sol=femeig(fem1,'u',u, ... 'solcomp','tExEyEz', ... 'outcomp','z','thz','w','lm9','lm8', ... 'v','tExEyEz','thx','lm7','u','y', ... 'thy','x','neigs',1, ... 'shift',439.256636); freq(k) = postinterp(fem1,'nu_emw',zeros(0,1),'dom',1,'solnum',1);end;plot(T,freq,'o-');grid on;hold on;legend('Brass','Steel');


.


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Wavegu i d e H -B end w i t h S - p a r ame t e r s

Introduction

This example is an extension of the waveguide H-bend model found on page 37 in the Electromagnetics Module User’s Guide. For a general introduction to the model, see the Electromagnetics Module User’s Guide.

The purpose of the model is to calculate how well a TE10 wave propagates through a bend in a waveguide filled with a dielectric medium. This is done by calculating the scattering parameters, or S-parameters, of the waveguide bend as a function of the frequency, for the fundamental TE10 mode. The S-parameters are a measure of the transmittance and reflectance of the waveguide bend. For a theoretical background on S-parameters, see the section “S-Parameters” on page 86 in the Electromagnetics Module User’s Guide.

This model only includes the TE10 mode of the waveguide, and you can therefore make the model in 2D, because the fields of the TE10 mode have no variation in the transverse direction. The figure below shows the modeling geometry.

The bend of the waveguide is made of a dielectric medium with the refractive index n = 1.45.


Model Definition

The governing equation for TE waves with the transversal component Ez is given by

where n is the refractive index and k0 is the free space wave number.

The entrance and exit ports do not represent physical boundaries in the geometry. To avoid reflection at these boundaries, matched boundary conditions make the boundaries transparent to the wave.


The S-parameters as a function of frequency are shown in the figures below. The S11 parameter corresponding to the reflectance is given by the curve with plus sign markers. The S21 parameter corresponding to the transmittance is given by the curve with circle markers.

The resonances at 4.6 GHz, 5.3 GHz, and 6.0 GHz are the cavity resonances of the dielectric region in the bend.

∇– ∇Ez n2k02Ez–⋅ 0=

W A V E G U I D E H - B E N D W I T H S - P A R A M E T E R S | 117

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Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/hbend_S_parameters


Begin by following the steps in the section “2D Modeling Using the Graphical User Interface” on page 48 in the Electromagnetics Module User’s Guide. Then continue as follows.


1 Click the Line button, and click at (0,0.0174) and at (0,-0.0174) with the left mouse button. Then click with the right mouse button to add the line to the geometry.

2 Draw another line from (0.0526,0.075) to (0.0526,0.0924).

S C A L A R V A R I A B L E S

In the Application Scalar Variables dialog box, set the frequency nu_emwe to fq1.


The default boundary condition is perfect electric conductor which is fine for all external boundaries except at the ports. Interior boundaries are by default not active meaning that continuity is imposed.

At boundaries number 1 and 7, specify the Port boundary condition. On the Port tab set the values according to the table below:


Set the refractive index to 1.45 in subdomain 2 in the Subdomain Settings dialog box.


Initialize a mesh and refine it twice.

BOUNDARY 1 7

Port number 1 2

Wave excitation at this port Selected Cleared

Mode specification Analytic Analytic

Mode number 1 1



Use the parametric solver in order to study the S-parameters as functions of the frequency.

1 In the Solver Parameters dialog box select the Parametric linear solver.

2 Enter fq1 as Name of parameter and linspace(4.31e9,6e9,50) as List of

parameter values. The solver will calculate the solution using 50 equidistant frequency values in the range 4.31 GHz to 6 GHz. The frequency 4.31 GHz is just above the cut-off frequency.



You can plot the S-parameters using a domain plot.

1 In the Domain Plot Parameters dialog box select all solutions in the Solutions to plot list.


quantities.

3 S11 is defined everywhere, but you only need to evaluate it at a single point to make a plot as a function of frequency. Select point 1 and click Apply to plot S11.

4 On the General tab select Keep current plot.

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quantities and click Apply.


Wavegu i d e Adap t e r

Introduction

This is a model of an adapter for microwave propagation in the transition between a rectangular and an elliptical waveguide. Such waveguide adapters are designed to keep energy losses due to reflections at a minimum for the operating frequencies. To investigate the characteristics of the adapter, the simulation includes a wave traveling from a rectangular waveguide through the adapter and into an elliptical waveguide. The S-parameters are calculated as functions of the frequency. The involved frequencies are all in the single-mode range of the waveguide, that is, the frequency range where only one mode is propagating in the waveguide.

Model Definition

The waveguide adapter consists of a rectangular part smoothly transcending into an elliptical part, as seen in Figure 4-9.

Figure 4-9: The geometry of the waveguide adapter.

The walls of manufactured waveguides are typically plated with a good conductor, such as silver. The COMSOL Multiphysics model approximates the walls by perfect conductors. This is represented by the boundary condition .n E× 0=

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The rectangular port is excited by a transversal electric (TE) wave, which is a wave that has no electric field component in the direction of propagation. This is what an incoming wave would look like after traveling through a straight rectangular waveguide with the same cross section as the rectangular part of the adapter. The excitation frequencies are selected so that the TE10 mode is the only propagating mode through the rectangular waveguide. The cut-off frequencies for the different modes are given analytically from the relation

where m and n are the mode numbers, and c is the speed of light. For the TE10 mode, m = 1 and n = 0. With the dimensions of the rectangular cross section (a = 2.286 cm and b = 1.016 cm), the TE10 mode is the only propagating mode for frequencies between 6.6 and 14.7 GHz.

At the elliptical port of the waveguide, the solution of an eigenmode problem on the port boundary gives the propagating mode. The elliptical port is passive, but the eigenmode is still used in the boundary condition to the 3D propagating wave simulation. Using the Electromagnetics Module’s Boundary Mode Analysis application mode, the eigenmode equation is

Here Hn is the component of the magnetic field perpendicular to the boundary, n the refractive index, β the propagation constant in the direction perpendicular to the boundary, and k0 the free space wave number. The eigenvalues are . The dimensions of the elliptical end of the waveguide are such that the frequency range for the lowest propagating mode overlaps that of the rectangular port.

With the stipulated excitation at the rectangular port and the numerically established mode shape at the elliptical port as boundary conditions, the following equation is solved for the electric field vector E inside the waveguide adapter:

where µr denotes the relative permeability, j the imaginary unit, σ the conductivity, ω the angular frequency, εr the relative permittivity, and ε0 the permittivity of free space. The model uses material parameters for free space: σ = 0 and µr = εr = 1.

νc( )mnc2---

ma-----⎝ ⎠

⎛ ⎞ 2 nb---⎝ ⎠

⎛ ⎞ 2+=

n 2– Hn∇×( )∇× n 2– β2 k02

–( )Hn+ 0=

λ β2–=

µr1– E∇×( )∇× k0

2 εrjσ

ωε0---------–⎝ ⎠

⎛ ⎞ E– 0=


Results

The first eigenmode of the elliptical port is displayed in Figure 4-10. The effective mode index is β/k0 = 0.535.

Figure 4-10: The x component of the electric field for the first eigenmode of the elliptical port.

Figure 4-11 shows a single mode wave propagating through the waveguide.

Figure 4-11: The x component of the propagating wave inside the waveguide adapter at the frequency 10 GHz.


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Naming the rectangular port 1 and the elliptical port 2, the S-parameters describing the reflection and transmission of the wave are defined as follows:

Here Ec is the calculated total field. E1 is the analytical field for the port excitation, and E2 is the eigenmode calculated from the boundary mode analysis and normalized with respect to the outgoing power flow. Figure 4-12 and Figure 4-13 show the S11 and S21 parameters as functions of the frequency.

Figure 4-12: The S11 parameter (in dB) as a function of the frequency. This parameter describes the reflections when the waveguide adapter is excited at the rectangular port.

S11

Ec E1–( ) E1*⋅( ) A1d

port 1∫

E1 E1∗⋅( ) A1dport 1∫

-----------------------------------------------------------------=

S21

Ec E2*⋅( ) A2d

port 2∫

E2 E2*⋅( ) A2d

port 2∫

-----------------------------------------------=


Figure 4-13: The S21 parameter (in dB) as a function of the frequency. This parameter is a measure of the part of the wave that is transmitted through the elliptical port when the waveguide adapter is excited at the rectangular port.


This model uses the Electromagnetics Module’s Port boundary condition for the wave propagation problem. With this boundary condition, the S-parameters are calculated automatically. The mode of the incoming wave is specified. The mode of the outgoing wave is given from an eigenmode analysis on the cross section of the elliptical part of the waveguide adapter.

The geometry of the waveguide adapter was created using the loft command on the MATLAB command line. This makes a smooth connection between the box and the ellipse possible. The resulting geometry is provided in a model file, so that you can use it without MATLAB access. The script used for creating the geometry is provided in the end of the model description.

Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/waveguide_adapter


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The waveguide adapter model is set up in two steps: first the boundary mode analysis is performed on the elliptical port of the adapter, then the wave propagation problem is solved using the mode shape as a boundary condition.

Mode Analysis

Open the file waveguide_adapter_geometry.mph in the Model Library. The geometry in the file is a waveguide transition with one elliptical end and one rectangular end. Use the following steps to calculate the S-parameters for this geometry when the wave enters from the rectangular end and exits at the elliptical end.

1 Open the Model Navigator from the Multiphysics menu.

2 In the Model Navigator select the Electromagnetics Module> Boundary Mode

Analysis> TE Waves application mode and add it to the model.

3 Then select the Electromagnetics Module> Electromagnetic Waves> Harmonic

propagation application mode and add it to the model.


First use the Boundary Mode Analysis application mode to compute the eigenmode of the elliptical end. This eigenmode is then part of the boundary condition for the electromagnetic wave propagation simulation.


First set up the boundary mode analysis problem. Select Boundary Mode Analysis from the Multiphysics menu.

Scalar VariablesIn the Scalar Variables dialog box, set the frequency nu_emweb to 7e9. This is a frequency above the cut-off frequency, which ensures that you find a propagating wave mode when doing the mode analysis.

Boundary SettingsIn the Boundary Settings dialog box, clear the Active in this domain check box for all boundaries except boundary 6. This boundary is the elliptical end of the waveguide, where the mode analysis will be done.


Edge SettingsIn the Edge Settings dialog box select the Perfect electric conductor boundary condition at edges 5, 6, 10, 14, 29, 39, 52 and 55. These are the exterior edges of the ellipse.


1 In the Mesh Parameters dialog box, enter 0.003 in the the Maximum element size edit field.



1 Open the Solver Parameters dialog box.

2 On the General tab, select the Eigenvalue solver and enter 1 in the Search for effective

mode indices around edit field.

3 Click OK to close the dialog box.

4 Open the Solver Manager dialog box and select only the Boundary Mode Analysis application mode on the Solve For tab.



Visualize the eigenmodes as a boundary plot.

1 In the Plot Parameters dialog box, clear the Slice check box and select the Boundary check box.

2 Select the Effective mode index 0.535.

3 On the Boundary tab, select the Tangential electric field, x component. There are two field variables one for each of the two application modes. Pick the tEx_emweb variable.

4 Click OK to plot the field and click the Go to XY view button to better see the field.


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5 Make the same type of plot of the y component of the electric field, tEy_emweb.

Figure 4-14: The y component of the electric field for the first eigenmode of the elliptical port.

6 To see the normal component of the magnetic field, start by selecting Normal

magnetic field (wev) in the Predefined quantities list.

7 On the General tab, enter 90 in the Solution at angle (phase) edit field. This compensates for the phase shift of 90 degrees between the electric and the magnetic field.

8 Click OK to see the plot.

S-Parameter Analysis

Now continue to set up the 3D problem to do the S-parameter analysis. Start by clicking the Go to Default 3D View button in order to see the full geometry.

B O U N D A R Y S E T T I N G S

1 Select Electromagnetic Waves in the Multiphysics menu.

2 Open the Boundary Settings dialog box.


3 At the rectangular end, boundary 13, select the Port boundary condition. On the Port tab set the values according to the table below:

4 At the elliptical end, boundary 6, also select the Port boundary condition. On the Port tab, set the values according to the table below:

5 Use the default Perfect electric conductor boundary condition at all other boundaries.


1 Open the Solver Manager dialog box. On the Initial Value tab, click Store Solution. In the dialog box that appears, select the solution with the effective mode index 0.535 and click OK.

2 Set the Initial Value to Stored solution.

3 On the Solve For tab select only the Electromagnetic Waves application mode.

4 Click OK to close the Solver Manager.

5 In the Solver Parameters dialog box, select the Parametric linear solver.

6 Set Name of parameter to freq and List of parameter values to linspace(6.6e9,1e10,50).

7 In the Linear system solver list, select Direct (SPOOLES).

8 Select the Symmetric in the Matrix symmetry list in order to save memory.

9 On the Advanced tab, clear the Use Hermitian transpose check box.

DESCRIPTION VALUE

Port number 1

Wave excitation at this port selected

Mode specification Rectangular

Mode type Transverse electric (TE)

Mode number 10

DESCRIPTION VALUE

Port number 2

Wave excitation at this port cleared

Mode specification Numeric

Mode type Automatic

Use numeric data from Boundary Mode Analysis, TE Waves (wev)


130 | C H A P T E

10 Click OK to close the Solver Parameters dialog box.

11 Open the Application Scalar Variables dialog box from the Physics menu and clear the Synchronize equivalent variables check box. This allows you to set different values for the two frequency variables nu_emwev and nu_emw. nu_emweb is the frequency variable for the Boundary Mode Analysis application mode. It is important that it retains its value of 7e9 in order for the data from the mode analysis solution to be evaluated correctly. Set the frequency variable nu_emw to freq. This is the frequency variable of the 3D Electromagnetic Waves application mode.

12 The eigenvalue variable lambda from the mode analysis solution is not available in postprocessing after solving the second problem. Therefore you need to define a constant lambda with the value -6169. This lambda value corresponds to the effective mode index 0.535. Define lambda in the Constants dialog box. To find the value of lambda , you could also use the Data Display dialog box after doing the mode analysis but before solving the second problem.



1 To visualize the x component of the electric field in the waveguide, start by opening the Plot Parameters dialog box.

2 On the General tab, make sure that the Solution at angle (phase) text field reads 0.

3 Clear the Boundary check box, and select the Slice check box.

4 Click the Slice tab and select Electric field, x component (weh) in the Predefined

quantities list.

5 Under Slice Positioning, set the number of x levels, y levels, and z levels to 1, 1, and 0 respectively.

6 Click OK to see the plot.

The port boundary conditions generates two S-parameter variables, S11_emw and S21_emw as well as two variables S11dB_emw and S21dB_emw, which are the S-parameters on a dB scale. The names of the variables are derived from the port numbers entered in the Boundary Settings dialog box. Make a domain plot in a point to plot the S-parameters as functions of frequency.

1 On the Point tab in Domain Plot Parameters dialog box select point 1 and select S-parameter dB (S11) (emw). Click Apply to plot S11 on a dB scale.

2 When plotting S21 first deselect the two first data points on the General tab to better see the variation. Select the variable S-parameter dB (S21) (emw) and click OK to make the plot.


Creating the Geometry in MATLAB

To create the geometry, start COMSOL Multiphysics with MATLAB, and create a new 3D model in the graphical user interface. Then run the script below from the command line. After running the script, go to File>Import>Geometry Objects. In the dialog box that appears, enter the geometry object g7. The loft function is not available in COMSOL Script.

lx = 1.827/2; ly = 3.015/2; e1 = ellip2(lx,ly);y0 = 1.142999999817202; x0 = sqrt((1-y0^2/ly^2)*lx^2);p1 = point2(x0, y0); p2 = scale(p1,1,-1);p3 = rotate(p1,pi); p4 = scale(p3,1,-1);g1 = solid2(geomcsg(e1,,p1,p2,p3,p4));lx2 = 1.016; ly2 = 2.286;r1 = solid2(geomcsg(rect2(lx2,ly2,'Base','center'),...point2(0,-2.286/2),point2(1.016/2,0),...point2(0,2.286/2),point2(-1.016/2,0)));z0 = 3.0; z1 = 11.0;wrk0 = geomgetwrkpln('quick','xy',z0);wrk1 = geomgetwrkpln('quick','xy',z1);t = cell(1,8);t1 = 3; t2 = 2; t3 = 4; t4 = 5; t5 = 6; t6 = 7; t7 = 8; t8 = 1;g2 = loft(r1 g1,'LoftEdge',t,...'wrkpln',wrk0 wrk1,'loftmethod','linear');g3 = extrude(r1,'distance',3);g4 = extrude (g1,'distance',3,'wrkpln',wrk1);g5 = solid3(geomcsg(g2,g3,g4));g6 = geomdel(g5);g7=scale(g5,0.01,0.01,0.01);geomplot(g7);


132 | C H A P T E

M i c r owa v e Can c e r Th e r ap y

Electromagnetic heating appears in a wide range of engineering problems and is ideally suited for modeling in COMSOL Multiphysics, because of its multiphysics capability. This specific model is an example from the area of hyperthermic oncology, where the electromagnetics field is coupled to the bioheat equation. The modeling issues and techniques are generally applicable to any problem involving electromagnetic heating.

In hyperthermic oncology, cancer is treated by applying localized heating to the tumor tissue, often in combination with chemotherapy or radiotherapy. Some of the challenges associated with the selective heating of deep-seated tumors without damaging surrounding tissue are:

• Control of heating power and spatial distribution

• Design and placement of temperature sensors

Among possible heating techniques, RF and microwave heating have attracted much attention from clinical researchers. Microwave coagulation therapy is one such technique where a thin microwave antenna is inserted into the tumor. The microwaves heat up the tumor, producing a coagulated region where the cancer cells are killed.

The purpose of this model is to compute the temperature field, radiation field and the specific absorption rate (SAR) in the liver tissue, when using a thin coaxial slot antenna used in microwave coagulation therapy. It closely follows the analysis found in reference [1]. The temperature distribution in the tissue is computed using the bioheat equation.

Model Definition

The antenna geometry is shown in Figure 1-1. It consists of a thin coaxial cable with a 1 mm wide ring shaped slot cut on the outer conductor 5 mm from the short-circuited tip. For hygienic purposes, the antenna is enclosed in a sleeve (catheter) made of PTFE (polytetrafluoroethylene). Material data and geometrical dimensions are given in the table below. The antenna is operated at 2.45 GHz, a frequency widely used in microwave coagulation therapy.

TABLE 4-1: DIMENSIONS AND MATERIAL PROPERTIES OF THE COAXIAL SLOT ANTENNA.

ENTITY DIMENSION [MM]

Diameter of the central conductor 0.29


Figure 4-15: The antenna geometry. The coaxial cable with a ring shaped slot cut on the outer conductor is short-circuited at the tip. A plastic catheter is surrounding the antenna.

The model takes advantage of the rotational symmetry of the problem, which allows modeling in 2D, using cylindrical coordinates as indicated in Figure 4-16. When modeling in 2D, a fine mesh is used, giving excellent accuracy. A frequency-domain

Inner diameter of the outer conductor

0.94

Outer diameter of the outer conductor

1.19

Diameter of catheter 1.79

ENTITY RELATIVE PERMITTIVITY

Inner dielectric of the coaxial cable 2.03

Catheter 2.60

Liver tissue 43.03

ENTITY ELECTRICAL CONDUCTIVITY [S/M]

Liver tissue 1.69

TABLE 4-1: DIMENSIONS AND MATERIAL PROPERTIES OF THE COAXIAL SLOT ANTENNA.

M I C R O W A V E C A N C E R T H E R A P Y | 133

134 | C H A P T E

problem formulation is used, with the complex-valued azimuthal component of the magnetic field as the unknown.

Figure 4-16: The computational domain is obtained as a rectangle in the r-z plane.

The radial and axial extent of the computational domain is in reality larger than indicated in Figure 4-16. The interior of the metallic conductors is not modeled. Metallic parts are modeled using boundary conditions setting the tangential component of the electric field to zero.

D O M A I N A N D B O U N D A R Y E Q U A T I O N S — E L E C T R O M A G N E T I C S

An electromagnetic wave propagating in a coaxial cable is characterized by transverse electromagnetic fields (TEM). Assuming time-harmonic fields with complex amplitudes containing the phase information you have:

where z is the direction of propagation and r, ϕ and z are cylindrical coordinates centered on the axis of the coaxial cable. Pav is the time average power flow in the

r

z

Computationaldomain

E erCr----ej ωt kz–( )=

H eϕCrZ-------ej ωt kz–( )=

Pav Re 12---E H*×⎝ ⎠

⎛ ⎞ 2πr rdrinner

router

∫= ezπC2

Z-------

routerrinner-------------⎝ ⎠

⎛ ⎞ln=


cable, Z is the wave impedance in the dielectric of the cable and rinner and router are the inner and outer radii respectively of this dielectric. The angular frequency is denoted by ω. The propagation constant k relates to the wavelength in the medium λ as

In the tissue, the electric field also has a finite axial component whereas the magnetic field is purely in the azimuthal direction. Thus you can model the antenna using an axisymmetric transverse magnetic (TM) formulation, and so the wave equation becomes scalar in Hϕ

The boundary conditions for the metallic surfaces are

The feed point is modeled using a first order low-reflecting boundary condition combined with an excitation field Hϕ0

where

for an input power of Pav W, deduced from the time-average power flow.

The antenna is radiating into the tissue where a damped wave propagates. Because it is only possible to discretize a finite region, truncate the geometry some distance from the antenna using a similar absorbing boundary condition without excitation. Use this boundary condition for all external boundaries. Finally, apply a symmetry boundary condition for boundaries at r = 0.

k 2πλ------=

εrjσ

ωε0---------–⎝ ⎠

⎛ ⎞ 1–Hϕ∇×⎝ ⎠

⎛ ⎞ µrk02Hϕ–∇× 0=

n E× 0=

n εE µHϕ–× 2 µHϕ0–=

Hϕ0

PavZ

πrrouterrinner-------------⎝ ⎠

⎛ ⎞ln

--------------------------------

r------------------------------------=


136 | C H A P T E

D O M A I N A N D B O U N D A R Y E Q U A T I O N S — H E A T T R A N S F E R

The stationary heat transfer problem is described by the bioheat equation:

where k is the liver thermal conductivity, ρb is the blood density, Cb is the blood specific heat, ωb is the blood perfusion rate. Qmet is the heat source from metabolism and Qext an external heat source.

This model neglects the heat source from metabolism. The external heat source is equal to the resistive heat generated by the electromagnetic field:

Assume that the blood perfusion rate is ωb = 0.0036 s-1 and that the blood enters the liver at the body temperature Tb = 37+273 K, and is heated to the temperature T. The blood heat capacity is Cb = 3639 J/kg/K.

For a more realistic model, you may consider letting ωb be a function of the temperature. At least for external organs like hands and feet, it is evident that a temperature increase does give an increasing blood flow.

The heat transfer problem is modeled only in the liver domain. Where this domain is truncated, insulation is used, that is:


Figure 4-17 shows the resulting steady-state temperature distribution in the liver tissue for an input microwave power of 10 W. The temperature is highest near the antenna. It then decreases with the distance from the antenna and reaches 310 K closer to the

Er 0=

r∂∂Ez 0=

∇ k T∇–( )⋅ ρbCbwb Tb T–( ) Qmet Qext+ +=

Qext12---Re σ jωε–( )E E*⋅[ ]=

n T∇⋅ 0=


outer boundaries of the computational domain. The perfusion of relatively cold blood seems to limit the extent of the area that is heated.

Figure 4-17: The temperature in the liver tissue.

Figure 4-18 shows the distribution of the microwave heat source. It is clear that the temperature field follows the heat source distribution quite well. That is, near the antenna the heat source is strong, which leads to high temperatures, while far from the antenna, the heat source is weaker and the blood manages to keep the tissue at normal body temperature.

Figure 4-18: The computed microwave heat source density takes on its highest values near the tip and the slot. The scale is cut off at 1e6 W/m3.


138 | C H A P T E

In Figure 4-19, the normalized specific absorption (SAR) value is computed along a line parallel to the antenna and 2.5 mm from the antenna axis. The results are in good agreement with those found in reference [1].

Figure 4-19: Normalized SAR value along a line parallel to the antenna and 2.5 mm from the antenna axis. The tip of the antenna is located at 70 mm and the slot at 65 mm.

References

[1] K. Saito, T. Taniguchi, H. Yoshimura, K. Ito, “Estimation of SAR Distribution of a Tip-Split Array Applicator for Microwave Coagulation Therapy Using the Finite Element Method”, IEICE Transactions on Electronics, VOL.E84-C, 7, pp. 948-954, July 2001.


The COMSOL Multiphysics implementation is straightforward. Drawing the geometry is best done entering rectangles and setting the dimensions directly from the Draw menu. The scale differences together with the strong radial dependence of the electromagnetic fields make some manual adjusting of the mesh parameters needed. 4th order elements for the electromagnetic problem and a dense mesh in the dielectric


result in well-resolved fields. The solutions for both the electromagnetic problem and the heat transfer problem are computed in parallel. This takes the coupling of the resistive heating from the electromagnetic solution into the bioheat equation into account. In principle, the two problems could be solved in sequence, as there is only a one-way coupling from the electromagnetic problem to the bioheat problem.

Model Library path: Electromagnetics_Module/RF_and_Microwave_Engineering/microwave_cancer_therapy

Note: This model requires both the Electromagnetics Module and the Heat Transfer Module to be installed.


M O D E L N A V I G A T O R

1 In the Model Navigator, select Axial Symmetry 2D in the Space dimension list.

2 Go to the Heat Transfer Module and select Bioheat Equation>Steady-state analysis.

3 Click the Multiphysics button and add the application mode to the model by clicking the Add button.

4 Similarly, go to the Electromagnetics Module and select Electromagnetic Waves>TM

Waves>Harmonic propagation.

5 Select Lagrange - Quartic in the Element list.

6 Click Add and then OK.


In the Constants dialog box, enter the following variable names and expressions.

NAME EXPRESSION

k_liver 0.56

rho_blood 1000

c_blood 3639

omega_blood 3.6e-3


140 | C H A P T E


1 Go to Draw>Specify Objects>Rectangle to specify two rectangles with the following parameters:

2 Open the Create Composite Object dialog box. Click to clear the Keep interior

boundaries check box. Create a union of the rectangles.

3 Specify two more rectangles, with the following properties.

4 Go to Draw>Specify Objects>Line. Specify a line with the r coordinates 0 8.95e-4 8.95e-4 and the z coordinates 9.5e-3 0.01 0.08.

5 Finally, specify another rectangle with the following parameters:

T_blood 310

epsilon0 8.8542e-12

mu0 4*pi*1e-7

eps_diel 2.03

eps_cat 2.6

eps_liver 43.03

sig_liver 1.69

Z0 sqrt(mu0/epsilon0)

Z Z0/sqrt(eps_diel)

r_inner 1.45e-4

r_outer 4.7e-4

Pin 10

C sqrt(Z*Pin/(pi*log(r_outer/r_inner)))

WIDTH HEIGHT BASE CORNER R BASE CORNER Z

0.595e-3 0.01 0 0

29.405e-3 0.08 0.595e-3 0


0.125e-3 1e-3 0.47e-3 0.0155

3.35e-4 0.0699 0.135e-3 0.0101


1.25e-4 1e-3 4.7e-4 0.0155

NAME EXPRESSION



Subdomain Settings—Bioheat Equation1 Select 1 Bioheat Equation (htbh) in the Multiphysics menu.

2 From the Physics menu, choose Subdomain Settings.

3 Select subdomains 2, 3, and 4, and then clear the Active in this domain check box.

4 Select subdomain 1 and enter subdomain settings according to the following table.

Boundary Conditions—Bioheat Equation1 Select Boundary Settings in the Physics menu.

2 Select all outer boundaries and set the boundary condition to Thermal insulation.

3 Click OK.

Note: As this is a steady-state model, the properties ρ and C are not present in the bioheat equation. They are both set to 1. The metabolic heat generation is neglected and therefore set to 0. The variable Qav_emwh is a subdomain expression for the resistive heating provided by the TM Waves application mode.

Scalar Variables—TM Waves1 Select 2 TM Waves (wh) in the Multiphysics menu.

2 Select Scalar Variables from the Physics>Application Scalar Variables menu.

3 Set the frequency to 2.45e9.

SUBDOMAIN 1

k k_liver

ρ 1

C 1

ρb rho_blood

Cb c_blood

ωb omega_blood

Tb T_blood

Qmet 0

Qext Qav_emwh


142 | C H A P T E

Boundary Conditions—TM Waves1 Enter boundary conditions according to the following table

2 Click OK.

Subdomain Settings—TM Waves1 Enter subdomain settings according to the following table.

2 Click OK.


1 Open the Mesh Parameters dialog box. Set the Global Maximum element size to 3e-3.

2 Go to the Subdomain page and set the Maximum element size on subdomain 3 to 1.5e-4.

3 Click Remesh and then click OK.


Click the Solve button to compute the solution.


The default plot shows the temperature field. The following steps describe how you can visualize the resistive heating of the tissue:

1 Open the Plot Parameters dialog box and go the Surface tab. Select Resistive heating,

time average in the Predefined quantities list and click Apply.

BOUNDARY 1,3 2,14,18,20-21 8 ALL OTHER

Boundary condition

Axial Symmetry Scattering boundary conditon

Matched boundary

Perfect electric conductor

0 C/Z/r

β k_emwh

Wave type

Spherical wave

SUBDOMAIN 1 2 3 4

(isotropic) eps_liver eps_cat eps_diel 1

(isotropic) sig_liver 0 0 0

1 1 1 1

H0ϕ

εr

σ

µr


2 The heating decreases rapidly in the liver tissue. To get a better feeling for the heating at a distance from the antenna, enter min(Qav_emwh,1e6) in the Expression

text field and click OK. This reproduces the plot in Figure 4-18.

3 To compute the total heating power deposited in the liver, open the Subdomain

Integration dialog box from the Postprocessing menu. Select the Compute the volume

integral check box and integrate the resistive heating over subdomain 1. You will get a value of around 9.94, indicating that almost all of the 10 W input effect is absorbed.

4 To reproduce the plot in Figure 4-19, go to the Cross-Section Plot Parameters dialog box and make a Line plot of Qav_emwh/3.2e6 from (r0, z0) = (2.5e-3, 0.08) to (r1, z1) = (2.5e-3, 0).


144 | C H A P T E

Ba l a n c e d Pa t c h An t e nna f o r 6 GHz

Introduction

Patch antennas are becoming more common in wireless equipment, like wireless LAN access points, cellular phones, and GPS handheld devices. The antennas are small in size and can be manufactured with simple and cost effective techniques. Due to the complicated relationship between the geometry of the antenna and the electromagnetic fields, it is very difficult to estimate the properties of a certain antenna shape. At this stage of the antenna design the engineer can benefit a lot from using computer simulations. The changes in the shape of the patch can be directly related to the changes in radiation pattern, antenna efficiency, and antenna impedance.

Balanced antennas are fed using two inputs, resulting in less disturbances on the total system through the ground. Balanced systems also provide a degree of freedom to alter antenna properties, by adjusting the phase and magnitude of the two input signals. In Figure 4-20 below you can see the antenna that was modeled using the Electromagnetics Module.

Figure 4-20: A picture of the real antenna that we extract the properties for.


Model Definition

The patch antenna is fabricated on a printed circuit board (PCB) with a relative dielectric constant of 5.23 [1]. The entire backside is covered with copper, and the front side has the pattern of the drawing shown in Figure 4-21 below.

Figure 4-21: The printed pattern of the patch antenna shown as a top view. The large square is 10x10 mm, the smaller rectangles are 5.2x3.8 mm, the thicker lines are 0.6 mm wide, and the thinner lines are 0.2 mm wide and 5.2 mm long. The entire pcb is 50x50 mm and 0.7 mm thick.

The coaxial cables have an outer conductor with an inner diameter of 4 mm and a center conductor with a diameter of 1 mm. The gap between the conductors is filled with a material with a dielectric constant of 2.07 (teflon). This gives an impedance of the cables close to 58 Ω. There are two coaxial cables feeding the patch antenna from two sides, resulting in a balanced feed. The shape of the field at the outer boundary of the cables are known TEM modes, which can be specified with analytical formulas expressed in the local cylindrical coordinate system of each cable. The following formulas are used for the incident fields at the input boundaries of both cables,

E0 erErn

r---------ejωt

=

H0 eϕHrn

r----------ejωt eϕ

Ernrη---------ejωt

= =

P012---E

0H0∗× ez

12---

Ern2

ηr2---------= =

B A L A N C E D P A T C H A N T E N N A F O R 6 G H Z | 145

146 | C H A P T E

where η is the wave impedance, and n denotes the input number, 1 or 2. The signals in the cables have the same magnitude but are shifted 180 in phase.

The entire antenna is modeled in 3D, using the electric field as solution variable. The time-harmonic nature of the signals makes it possible to solve the Vector-Helmholtz equation for the electric field everywhere in the geometry,

where k0 is the wave number for free space and is defined as,

All metallic object are defined as perfect electric conductors (PEC), and the input to the coaxial cables uses the low-reflective boundary condition. The latter is a first-order approximation of an absorbing boundary, which lets all scattered waves pass the boundary “without” reflections. In addition, you can supply an incident wave, which in this case is the TEM mode of a coaxial cable described earlier in the text. The low-reflective boundary condition usually works well for waves propagating in cables, but for radiating waves from an antenna they are not very accurate and create reflections. Perfectly Matched Layers (PMLs) are more suitable in such cases, and for this antenna we will use the spherical PML. The details of the PML are described in “Perfectly Matched Layers (PMLs)” on page 72 of the Electromagnetics Module User’s Guide, and we will only describe the damping of the PML here, since it is important when designing the length of the PML. A wave entering the PML will be damped without reflections as it travels through the layer. When the wave reaches the outer boundary, the entire wave can be reflected in a worst case scenario. During the propagation back to the simulation volume the wave will continue to get damped, so it always has to travel twice the distance of the PML layer. The damping is determined by the wave number, k0, the PML constant bPML, and the length of the PML layer, dPML, using the following formula,

For frequencies around 6 GHz the damping parameter α is 0.007, so less than 1% of the radiated signal is reflected back.

°

∇ µ 1– ∇ E×( )× k02εrE– 0=

k0 ω ε0µ0=

α e2bPMLdPMLk0–

=


Results and Discussions

Figure 4-22: The final geometry of the patch antenna plotted together with a slice plot of the Electric field in the surrounding air. The spherical PML is located outside the air domain and outside all cable regions.

Typical parameters important for an antenna are optimum radiation frequency, antenna impedance, and radiation pattern. Firstly, we use the parametric solver to find the optimum radiation frequency, sweeping the frequency. It is most efficient to do a coarse sweep first, and then successively more resolved sweep in order to find the peak in the antenna efficiency. The antenna efficiency is the radiated power divided by the


148 | C H A P T E

input power. In Figure 4-23 this efficiency is plotted against the frequency, and the optimum frequency was located at 6.16 GHz.

Figure 4-23: A resolved sweep in the range around 6 GHz, where the region between 6.0 to 6.2 Ghz was resolved in detail to locate the peak.


The antenna impedance is plotted against the frequency in Figure 4-24, and at the optimum frequency the impedance almost match the cable impedance.

Figure 4-24: The antenna impedance seen from one port plotted versus frequency. Note that this is the impedance seen from the cable. The pattern on the PCB takes care of the matching between the cable and the antenna plate.

The radiation pattern is usually plotted in a lobe pattern, which is a surface representing the directional strength in all directions of the antenna. This plot is interpreted as follows; the vector between the origin and a point on the surface has a magnitude that represents the radiated energy in the direction parallel to the vector. The energy is given in dB, and the 0 dB level is the maximum radiated energy (a point on the unit sphere). The lobe pattern can be created with a deformation plot, where 0 dB is no deformation and -40 dB is the maximum deformation to origin. The radiation pattern


150 | C H A P T E

for the modeled patch antenna is shown in Figure 4-25 for the optimum frequency, 6.16 GHz.

Figure 4-25: The radiation pattern plotted as a lobe pattern, using the deformed plot feature. It is clear that the wave mainly propagates in two lobes, and only a small radiation out of the backside of the PCB.

Model Library Electromagnetics_Module/RF_and_Microwave_Engineering/patch_antenna_balanced


Antenna simulations in 3D often get very large, so this model should be solved on a 64-bit platform. The final mesh results in about 360 000 complex degrees of freedom, and problems of such large size can seldom solve without iterative solvers. On a 64-bit platform it solves with a direct solver consuming between 3 and 4 Gb of memory.


References

[1] E. Recht and S. Shiran, "A Simple Model for Characteristic Impedance of Wide Microstrip Lines for Flexible PCB," Proceedings of IEEE EMC Symposium 2000, pp. 1010-1014

Modeling using the Graphical User Interface


1 In the Model Navigator, select 3D in the Space dimension list.

2 In the Electromagnetics Module folder, select Electromagnetic Waves and Harmonic

propagation. Click OK.



2 In the Constants dialog box, define the following constants with names and expressions:

3 Click OK.


The geometry modeling of this model is rather extensive, so it is possible to import the patch antenna geometry from a binary file. Select the section that you prefer.

NAME VALUE DESCRIPTION

a_pml 1 Parameter for PML

b_pml 1 Parameter for PML

a_coax 5e-4 Radius for the inner conductor of the coaxial cable (m)

b_coax 2e-3 Inner radius for the outer conductor (m)

epsilonr_coax 2.07 Rel. permittivity of the insulator in the cable

epsilonr_pcb 5.23 Rel. permittivity of the printed circuit board for the patch antenna

Er1 1 Magnitude and phase for the radial electric field of the first feeding cable (V/m)

Er2 -1 Magnitude and phase of the second cable (V/m)

freq 6e9 Default frequency (1/s)


152 | C H A P T E

Importing the Geometry from a Binary File1 From the File menu, select Import>CAD Data From File.

2 In the Import CAD Data From File dialog box, make sure that the COMSOL native format or All 3D CAD files is selected in the Files of type drop-down menu.

3 Locate the patch_antenna_balanced_geom.mphbin file, and click Import.

4 Skip section “Creating the Geometry from Scratch” and begin at “Add Geometry for Radiation Pattern” on page 156.

Creating the Geometry from ScratchThe geometry will be drawn using the unit mm in a separate geometry. As a final step the entire drawn geometry will be converted to meters by copying it to the first geometry followed by a scale operation. All the dialog boxes for specifying the primitive objects are accessed from the Draw menu and Specify Object. The software generates the Name column in the tables below, so you do not have to enter them. Just check that you get the correct name for the objects that you create.

We begin by creating a work plane for the patch antenna cross-section:

1 From the Draw menu, select Work Plane Settings.

2 In the Work Plane Settings dialog box, select the x-y plane at z equal to 0. Click OK.

3 Draw squares with the properties according to the table below.

4 Draw rectangles with the properties according to the following table.

5 Click the Zoom Extents toolbar button.

6 Select all the rectangles, R1, R2, and R3, and press the Union toolbar button.

7 Click the Delete Interior Boundaries toolbar button.

8 Copy the selected object by pressing Ctrl-C, and then paste it by pressing Ctrl-V. In the dialog box that appears, just click OK to create the copy in the same location as the original.

NAME WIDTH BASE (X,Y) DESCRIPTION

SQ1 50 Center (0,0) PCB

SQ2 10 Center (0,0) Patch

NAME WIDTH HEIGHT BASE CORNER DESCRIPTION

R1 0.4 5.2 Corner (-0.2,-5-5.2) Conductor

R2 5.2 3.8 Center (0,-5-5.2-3.8/2) Conductor

R3 0.8 25-5-5.2-3.8 Corner (-0.4,-25) Conductor


9 Click on the Rotate toolbar button. In the Rotate dialog box, enter -90 in the Rotation angle edit field.

10 Select the objects SQ2, CO1, and CO2. Press the Union toolbar button and then the Delete Interior Boundaries toolbar button.

Now it is time to draw the patch antenna in a new 3D geometry that we call patch_antenna.

11 Open the Model Navigator from the Multiphysics menu.

12 Click on the Add Geometry button. In the Add Geometry dialog box, enter patch_antenna in the Geometry Name edit field, and select 3D from the Space

dimension drop-down menu. Click OK.

13 Click OK to close the Model Navigator window.

14 Click on Geom2 tab in the main window.

15 Select the large square, SQ1, and select Extrude from the Draw menu.

16 In the Extrude dialog box, enter -0.8 in the Distance edit field, and select patch_antenna from the Extrude to geometry list. Click OK.

17 Go back to Geom2 to do an embedding.

18 Select the patch object C01, and select Embed from the Draw menu. Select patch_antenna in the Embed to geom list. Click OK.

The following steps will create the coaxial cables that feeds the antenna. The connectors are also drawn to resemble the real device as closely as possible:


20 In the Work Plane Settings dialog box, select the z-x plane and set y equal to -25. Click Apply, then Add to create a new work plane geometry. Click OK.

21 Draw circles with the properties according to the table below.

NAME RADIUS BASE (X,Y) DESCRIPTION

C1 0.5 Center (0,0) Inner conductor

C2 2 Center (0,0) Outer conductor inner border

C3 2.5 Center (0,0) Outer conductor outer border

C4 1.25 Center (12.5/2-1.75,12.5/2-1.75) One of the holes of the connector


154 | C H A P T E

22 Extrude the objects C1, C2, and C3, to the patch_antenna geometry. Enter the expression -45+25 in the Distance edit field.

23 Go back to the geometry Geom3 by clicking on the Geom3 tab.

24 Specify a rectangle with the following properties:

25 From the Draw menu, select Fillet/Chamfer. In the Fillet/Chamfer dialog box, select all the points of object R1, make sure that the Fillet radio button is selected, and enter 1.75 in the Radius edit field. Click OK.

26 Select circle C4 and press the Array toolbar button. In the Array dialog box, enter the expression -12.5+3.5 in both the Displacement x and y edit fields, and enter 2 in both the x and y Array size edit fields. Click OK.

27 Select circle C3 and copy it by pressing Ctrl-C. This will save the circle for later use.

28 Select the circles C3, C4, C5, C6, and C7. Press the Union toolbar button.

29 Press Ctrl-V to get the circle back. Leave all the displacements at zero and click OK.

30 Select the objects CO1 and CO2. Press the Difference toolbar button.

31 Extrude the result to the patch_antenna geometry, using a distance of -1.5.

32 Go back to geometry Geom3.

33 Open the Work Plane Settings from the Draw menu. Enter -45 in the y edit field. This will change the y-location of the work plane in the 3D geometry.

34 Extrude the circle C3 to the patch_antenna geometry, using a distance of -30.

35 Select the extruded objects, EXT2, EXT3, EXT4, EXT5, and EXT6. Copy them by pressing Ctrl-C.

36 Paste them with Ctrl-V and leave the displacements at zero.

37 Click the Rotate toolbar button. In the Rotate dialog box, enter -90 in the Rotation

angle edit field. Click OK.

The final step is to create the surrounding air volume, and the PML region. This is done with two spheres.

38 Specify two spheres with the following properties:

NAME WIDTH HEIGHT BASE CORNER DESCRIPTION

R1 12.5 12.5 Center (0,0) Conductor

NAME RADIUS BASE CORNER DESCRIPTION

SPH1 60 Center (0,0) PML+Air

SPH2 40 Center (0,0) Air


The entire geometry in now completed in the patch_antenna geometry. To help the software coerce the geometry to its minimal regions (subdomains) you can do step-by-step Union and Difference operations until we only have one solid object (and one face object). Do all the operations below in the specified order, otherwise the names of the geometry objects will not match.

1 Select all objects in the patch_antenna geometry by pressing Ctrl+A.

2 Press Ctrl+C to copy, then go the Geom1 by clicking on the Geom1 tab, and paste the objects with Ctrl+V.

3 Do all the geometry operations according to the table below, starting from the first row:

4 Select all objects with Ctrl+A.

5 Press the Scale toolbar button, and enter 1e-3 in all the Scale factor edit fields. Click OK.

STEP OBJECTS OPERATION

1 EXT1, EXT3, EXT8 Union

2 CO1, EXT4, EXT9 Union

3 CO2, EXT5, EXT10 Union

4 CO1, SPH2 Union

5 CO2, SPH1 Union

6 CO1, EXT6 Difference





156 | C H A P T E

6 Click the Zoom Extents toolbar button. Turn on the head light and increase the transparency slightly to get the following image of the final geometry.

Add Geometry for Radiation Pattern1 Open the Model Navigator from the Multiphysics menu.

2 Click on the Add Geometry button. In the Add Geometry dialog box, enter Far-field in the Geometry Name edit field, and select 3D from the Space dimension drop-down menu. Click OK.

3 Click OK to close the Model Navigator window.

4 Click the Sphere toolbar button. In the Sphere dialog box, select the Face radio button. Click OK.


1 Go to geometry Geom1 by clicking on the Geom1 tab.

1 From the Options menu, choose Expressions>Scalar Expressions.


2 In the Scalar Expressions dialog box, define the following variables with names and expressions:

3 Click OK.

4 From the Options menu, select Expressions>Subdomain Expressions.


r sqrt(x^2+y^2+z^2) Spherical coordinate r for the PML

theta acos(z/r) Spherical coord. θ for the PML

phi atan2(y,x) Spherical coord. ϕ for the PML

Lxx stheta*sphi/sr*sin(theta)^2 *cos(phi)^2+sr* (cos(theta)^2*cos(phi)^2+ sin(phi)^2)

xx element of the anisotropic PML matrix

Lxy (stheta*sphi/sr* sin(theta)^2+sr*( cos(theta)^2-1))*cos(phi)* sin(phi)

xy element

Lxz (stheta*sphi/sr-sr)* sin(theta)*cos(theta)* cos(phi)

xz element

Lyx Lxy yx element

Lyy stheta*sphi/sr*sin(theta)^2 *sin(phi)^2+sr*( cos(theta)^2*sin(phi)^2+ cos(phi)^2)

yy element

Lyz (stheta*sphi/sr-sr)* sin(theta)*cos(theta)* sin(phi)

yz element

Lzx Lxz zx element

Lzy Lyz zy element

Lzz stheta*sphi/sr*cos(theta)^2 +sr*sin(theta)^2

zz element

S11 (sqrt(P1in)-sqrt(P1))/sqrt(P1in)

S11 parameter for first input

Z_coax sqrt(mu0_emw/(epsilon0_emw* epsilonr_coax))/(2*pi)* log(b_coax/a_coax)

Impedance for the coaxial cables

Z0 Z_coax*(1+S11)/(1-S11) Input impedance for the system on the first cable


158 | C H A P T E

5 In the Subdomain Expressions dialog box, define the following variables:

6 Click OK.

7 From the Options menu, choose Expressions>Boundary Expressions.

8 In the Boundary Expressions dialog box, define the following variables with names and expressions:

9 Click OK.

10 From the Options menu, point to Integration Coupling Variable and then click Boundary Variables.

11 In the Boundary Integration Variables dialog box, define the following variables with names and expressions. Use Global destination and Integration order 4 for all variables.

12 From the Physics menu, select Scalar Variables.

13 In the Scalar Variables dialog box, enter freq in the expression column for variable nu_emw. Click OK.

SUBDOMAIN 1 6 ALL OTHER

sr a_pml-i*b_pml

stheta 1

sphi 1

normE_air normE_emw

BOUNDARY 12 130 ALL OTHER

r2 (y-yc)^2+(z-zc)^2 (x-xc)^2+(z-zc)^2

xc 0

yc 0

zc 0 0

BOUNDARY 12 130 ALL OTHER

P1 -nPoav_emw

P1in 0.5*sqrt(epsilonr_coax* epsilon0_emw/mu0_emw)* Er1*conj(Er1)/r2

P2 -nPoav_emw

P2in 0.5*sqrt(epsilonr_coax* epsilon0_emw/mu0_emw)* Er2*conj(Er2)/r2


Boundary Conditions1 From the Physics menu, open the Boundary Settings dialog box, select the Interior

boundaries check box, and enter the settings according the following two tables (leave all fields not specified at their default values):

2 Leave the Boundary Settings dialog box open. A special variable is needed later to plot all the metal objects in the geometry. Use the boundary settings to select all boundaries with the Perfect electric conductor condition. Mark boundary 91 and then select the Select by group check box. All metal boundaries should now be selected. Click OK.

3 Go to the Options menu and select Expressions>Boundary Expressions.

4 In the Boundary Expressions dialog box, add the variable boundary_plot and type 1 in the Expression column.

5 Select the boundaries, 76, 77, 79, 92, 95, 96, 185 and 186. Type 2 in the Expression column for the boundary_plot variable.

6 Select the boundaries, 1, 3, 5, 7, 114, 115, 144, and 146. Clear the Expression column for the boundary_plot variable.

7 Click OK.

BOUNDARY 12 130 2, 4, 6, 8, 145, 147, 168, 171

Boundary condition Scattering boundary

Scattering boundary

Scattering boundary

Wave type Plane wave Plane wave Spherical wave

E0,x 0 Er2*(x-xc)/r2

0

E0,y Er1*(y-yc)/r2

0 0

E0,z Er1*(z-zc)/r2

Er2*(z-zc)/r2

0

BOUNDARY 78, 91

Boundary condition Perfect electric conductor


160 | C H A P T E

Subdomain Settings1 From the Physics menu, open the Subdomain Settings dialog box and enter the

following settings:

2 Select the subdomains 2, 5, 7, 9, 10, and 11. Uncheck the check box Active in this

domain. This will deactivate all the metal objects in the simulation.

3 Click OK.

4 From the Physics menu, select Far-Field. In the Far-Field Variables dialog box, select the boundaries 40, 42, 44, 46, 163, 165, 169, and 170. Enter Efar at the first row of the Name column, and check the Source boundaries check box for this row.

5 Click the Destination tab. Select geometry Far-field from the Geometry drop-down menu, and select Boundary from the Level drop-down menu.

6 Select all boundaries and check the Use the selected subdomains as destination check box. Click OK.


1 Click the Geom1 tab.

2 From the Mesh menu, choose Mesh Parameters.

3 Enter 1.6 in the Maximum element size scaling factor edit field, and 1.5 in the Element growth rate edit field.

4 Click the Boundary tab.

5 Select boundaries 40, 42, 44, 46, 163, 165, 169, and 170, and type 6e-3 in the Maximum element size edit field.

6 Select boundaries 15, 16, 19, 20, 24, 26, 28, 30, 89, 90, 138–143, 150, 151,155, and 157, and type 0.002 in the Mesh curvature cut off edit field.

7 Click the Edge tab.

SUBDOMAIN 1 3, 4, 12, 13 8 ALL OTHER

εr(isotropic) epsilonr_coax epsilonr_pcb 1

εr(anisotropic) Lxx Lxy Lxz Lyx Lyy Lyz Lzx Lzy Lzz

µr(isotropic) 1 1 1

µr(anisotropic) Lxx Lxy Lxz Lyx Lyy Lyz Lzx Lzy Lzz


8 Select the edges 165, 173, 199, 200, 201, 202, 203, 204, 205, 206, 229, 230, 231, 232, 253, 254, 255, 295, 297, 354, 355, 356, 357, 359, 379, and 408, and type 5e-4 in the Maximum element size edit field. The edges listed above are all edges of the patch in the patch antenna, and there is a technique to select all of them with the mouse. Click on Go to XY View toolbar button. You now see a top view of the patch antenna. Then click on the Orbit/Pan/Zoom toolbar button. This is a toggle button that shifts between rotating the view (button pressed down) and selecting objects. The button must now be in the selecting objects state. With the mouse, select all edges belonging to the patch (mouse click and drag around these edges). Right click on one of the selected edges to lock the selection (colored blue). The figure below shows how the selection should look like. Now just select the remaining four edges closest to the connectors.

9 Select edges 204, 206, 297, and 354, and type 3e-4 in the Maximum element size edit field.

10 Click the Advanced tab.

11 Enter 1.2 in the z-direction scale factor edit field.

12 Click the Remesh button and then click OK.


1 From the Solve menu, choose Solver Parameters

2 Select the Parametric Linear solver from the Solver list.

3 In the Name of parameter edit field type freq, and in the List of parameter values edit field type 6.1e9:0.01e9:6.2e9.

4 Make sure that the Symmetric option is selected from the Matrix symmetry drop-down menu.


162 | C H A P T E

5 Click OK.

6 Press the Solve toolbar button.

The solve process takes some time, because this is a fairly large problem and we perform a parameter sweep of 11 steps.


1 Select Plot Parameters from the Postprocessing menu.

2 Make sure that the Boundary, Slice, and Geometry edges check boxes are checked under the General tab.

3 Click on the Slice tab, and enter normE_air in the Expression edit field.

4 Click on the Range button, deselect the Auto check box, and type 500 in the Max edit field and 0 in the Min edit field. Click OK.

5 Click on the Boundary tab, and enter boundary_plot in the Expression edit field. Select the Hot color map from the Colormap list.

6 Click on the Range button, deselect the Auto check box, and type 2 in the Max edit field and 0 in the Min edit field. Click OK.

7 Click OK. You should now see the plot in Figure 4-22 on page 147.

The steps below plots the extracted parameters from the frequency sweep, antenna efficiency and input impedance.

1 To plot the antenna efficiency open the Domain Plot Parameters dialog box from the Postprocessing menu.

2 Choose the Point plot type and check that all parameters are selected in the Parameter value list.

3 On the Point tab, type (P1in+P2in)/(P1+P2) in the Expression edit field.

4 Select vertex 1 and click OK. In a separate window you should see the plot in Figure 4-23 on page 148.

5 To plot the antenna impedance open the Domain Plot Parameters dialog box from the Postprocessing menu.

6 Choose the Point plot type and check that all parameters are selected in the Parameter value list.

7 On the Point tab, type Z0 in the Expression edit field.

8 Select vertex 1 and click OK. In a separate window you should see the plot in Figure 4-24 on page 149.


The final step is to plot the radiation pattern shown in Figure 4-25 on page 150. This is done in the geometry labeled Far-Field. This plot includes a far-field calculation of all the solutions and can take 10-20 minutes to perform. Do the following steps to get the plot:

1 Click on the Far-field geometry tab in the main window.

2 Open the Scalar Expressions dialog box from the Options>Expressions menu.

3 In this dialog box enter the expressions according to the table below:

4 Open the Constants dialog box from the Options menu.

5 In this dialog box, add the constants specified in the table below:


7 Make sure that the Boundary, Edge, Deformed shape, and Geometry edges check boxes are checked under the General tab.

8 Click on the Boundary tab, and enter normE2 in the Expression edit field.

9 Click on the Edge tab, and enter 1 in the Expression edit field. Select the Uniform color radio button, press the Color button, and select the white color from the palette. Click OK.

10 Click on the Deform tab. Deselect the Subdomain check box. In the Deformation data frame, click on the Boundary Data tab, and enter deform*x, deform*y, and deform*z in the x component, y component, and z component edit fields. Click on the Edge Data tab, and enter the same expressions in the fields there.

11 Click OK. Please note that it may take up to 20 minutes to see the plot. When done you should see the plot in Figure 4-25 on page 150.


normE2 Efarx*conj(Efarx)+Efary* conj(Efary)+Efarz*conj(Efarz)

Magnitude of the Far-field.

normE2_dB 10*log10(normE2) The magnitude in dB

deform (normE2_dB-maxE2_dB)/ (maxE2_dB-minE2_dB)

The deform variable for the radiation pattern


maxE2_dB 10.361 Maximum value for normE2 in dB

minE2_dB -21 Minimum value for normE2 in dB


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5

O p t i c s a n d P h o t o n i c s M o d e l s

In this chapter, you will find models within Optics and Photonics. From the mathematical viewpoint, these models use the same physics formulations as in the previous chapter on RF and Microwave Engineering. The terminology and the way numerical results are presented differs slightly though. Anyone interested in RF and Microwave Engineering may benefit from also reading this chapter.

165

166 | C H A P T E

Pho t on i c M i c r o p r i sm

Introduction

A microprism is used for reducing radiative losses in photonic waveguide bends. If you place a microprism between two waveguides forming a sharp bend, light will be guided between the waveguides through the prism.

For a certain refractive index of the prism, the light propagating through the prism will couple to the respective mode under just the appropriate resonance angle (see Reference [1]). If the initial field distribution does not diffract while propagating through the prism, the coupling from the prism to the guide is the inverse to the light transfer from the guide to the prism. Therefore, the efficiency of the process is very high.

The interface between the guide and the prism must be sufficiently long to allow almost all the power to exit from the guide into the prism and vice versa. On the other hand, to avoid diffraction, the size of the prism should be kept as small as possible.

This model also makes use of so called perfectly matched layers, PMLs. These are domains with artificial absorption that reduce nonphysical reflections. For more information on PMLs, see Reference [2] and the section “Perfectly Matched Layers (PMLs)” on page 72 in the Electromagnetics Module User’s Guide.

Microprism

R 5 : O P T I C S A N D P H O T O N I C S M O D E L S

Model Definition

The model is built using the 2D In-Plane TE Waves application mode. The modeling takes place in the x-y plane.


The dependent variable in this application mode is the z component of the electric field E. It obeys the following relation:

where µr denotes the relative permeability, ω the angular frequency, σ the conductivity, ε0 the permittivity of vacuum, εr the relative permittivity, and k0 the wave number. Different refractive indices are used for the prism and the guides. The wave is dampened by PMLs where the wave enters and exits the setup. The whole geometry is surrounded by another PML, which decreases reflections from the non-physical exterior boundary. The solution is calculated for IR light with a wavelength in vacuum of 870 nm.


The exterior boundaries in this model use a scattering boundary condition to terminate the PML. Inside the geometry, continuity is applied everywhere except for at the boundary where the wave is entering the structure. This boundary is excited with a cosine function fitted to match the width of the waveguide.

∇ µr1– ∇ Ez×( )× εr

jσωε0---------–⎝ ⎠

⎛ ⎞ k02Ez– 0=

P H O T O N I C M I C R O P R I S M | 167

168 | C H A P T E


Figure 5-1 shows the geometry and the solution of the model. The wave enters the horizontal guide from the right, and exits at the top of the vertical guide. The circles surrounding the entry and the exit are PMLs.

Figure 5-1: The z component of the electric field.

Several variables govern the transmission of the wave through the bend, for example the relation between the refractive indices of the guide and the prism, the size of the prism, and the gap between the guides and the prism. The optimal values of the parameters also depend on the angle of the bend.

References

[1] CLEO 2001, Conference on Lasers and Electro-Optics, OSA-Optical Society of America, p. 129-130.

[2] Jianming Jin: The Finite Element Method in Electromagnetics, Second Edition, Wiley-Interscience

Model Library path: Electromagnetics_Module/Optics_and_Photonics/photonic_micro_prism



Select 2D, Electromagnetic Module, In-Plane Waves, TE Waves, Harmonic propagation in the Model Navigator.


1 In the Axes/Grid Settings dialog box, set axis and grid settings according to the following table.



Start by defining the solid objects for the two guides and the prism.

1 Select Draw Line and click at (0.3,-0.1), (4.5,-0.1), (4.5,0.1), and (0.1,0.1). Create a solid object CO1 by clicking the right mouse button.

2 Select Draw Line and click at (0.1,0.1), (0.1,4.5), (-0.1,4.5), and (-0.1,0.3). Create a solid object CO2 by clicking the right mouse button.

3 Select Draw Line and click at (0.18,3.18), (0.18,0.18), and (3.18,0.18). Create a solid object CO3 by clicking the right mouse button.

Then, define circular domains for the absorbing layers.

1 Draw a circle C1 with radius 5 and a circle C2 with radius 5.5, both centered at (0,0).

2 Draw a circle C3 centered at (4.5,0) and a circle C4 centered at (0,4.5), both with radius 0.5.

3 Unite all objects into one solid object by selecting all objects and clicking the Union toolbar button.

The geometry of the model is now ready, except for the size. The dimensions should be given in µm.

AXIS GRID

x min -6 x spacing 0.5

x max 6 Extra x -0.1 0.1 0.18 0.3 3.18

y min -6 y spacing 0.5

y max 6 Extra y -0.1 0.1 0.18 0.3 3.18

NAME EXPRESSION

n_Guide 1.5

n_Prism 2.5


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1 Click the Scale toolbar button, and give the value 1e-6 for both scaling factors.

2 Click the Zoom Extents button to be able to see the resulting object.

Figure 5-2: Model geometry.


Scalar VariablesSet the frequency to 3e8/870e-9 in the Scalar Variables dialog box.

Boundary ConditionsEnter the boundary conditions according to the following table. To apply boundary conditions to interior boundaries, it is necessary to select the Interior boundaries check box.

Subdomain Settings1 Subdomains 1 and 5 represent air or vacuum. This is the default setting.

2 Subdomain 7 is the prism. Choose to represent the material properties in terms of the refractive index, and enter n_Prism in the text field.

BOUNDARY 15 16,17,23,28 ALL OTHER

Boundary condition Electric field Scattering boundary condition Continuity

E0z cos(y*pi/0.4e-6) 0

Wave type Cylindrical wave


3 Subdomains 2, 4, and 6 are the free part of the waveguide. Use n_Guide as the refractive index here.

4 Subdomains 3, 8, and 9 are used for damping the wave. Choose to represent the material properties in terms of εr, µr,and σ. Enter n_Guide^2*Izz for the Relative

permittivity and Ixx 0 0 Iyy for the Relative permeability (anisotropic).

5 Subdomain 10 also damps the wave, in order to reduce the effect of the exterior boundary. Enter Izz for the Relative permittivity and Ixx 0 0 Iyy for the Relative

permeability (anisotropic).

Expression Variables1 Open the Subdomain Expressions dialog box from the Options menu. Mark all

subdomains and specify the expressions for diagonal indices of the PML tensor.

2 For best results, the wave is damped along the expected direction of propagation, except for in the beginning of the waveguide, where it is damped in the perpendicular direction. This is achieved by setting the following expressions.


1 Change the default mesh parameters to get an applicable mesh. Open the Mesh

Parameters dialog box. On the Global tab, set Max element size to 250e-9. On the Subdomain tab, select the subdomains 2, 4, 6, 7, and 8, and set Maximum element

size to 120e-9.

NAME EXPRESSION

Ixx sy*sz/sx

Iyy

Izz

sx*sz/sy

sx*sy/sz

SUBDOMAIN 3,4,8 9,10 ALL OTHER

sx 1 1-i 1

sy 1-i 1-i 1

sz 1 1 1


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Figure 5-3: Mesh


1 Click the Solve button to compute the solution.


By default, the z component of the electric field is visualized. It shows clearly that the wave propagates between the two guides, through the prism; see Figure 5-1. Another interesting entity for visualization is the electric energy density. Select Electric energy

density, time average on the Surface tab of the Plot Parameters dialog box. The resulting plot (Figure 5-4) shows that the energy density is mainly localized to the waveguide and the prism, and that there are signs of resonances in the prism.

Figure 5-4: Electric energy density.


Pho t on i c C r y s t a l

Photonic crystal devices are periodic structures of alternating layers of materials with different refractive indices. Waveguides that are confined inside of a photonic crystal can have very sharp low-loss bends, which may enable an increase in integration density of several orders of magnitude.

Introduction

This model describes the wave propagation in a photonic crystal that consists of GaAs pillars placed equidistant from each other. The distance between the pillars prevents light of certain wavelengths to propagate into the crystal structure. Depending on the distance between the pillars, waves within a specific frequency range are reflected instead of propagating through the crystal. This frequency range is called the photonic band gap [1]. By removing some of the GaAs pillars in the crystal structure you can create a guide for the frequencies within the band gap. Light can then propagate along the outlined guide geometry.

Pillar (GaAs)

Air

Incoming wave

Outgoing wave

P H O T O N I C C R Y S T A L | 173

174 | C H A P T E

Model Definition

The geometry is a square of air with an array of circular pillars of GaAs as described above. Some pillars are removed to make a waveguide with a 90o bend.

The objective of the model is to study TE waves propagating through the crystal. To model these, use a scalar equation for the transverse electric field component Ez,

where n is the refractive index and k0 is the free space wave number.

Because there are no physical boundaries, you can use the scattering boundary condition at all boundaries. Set the amplitude Ez to 1 on the boundary of the incoming wave.


Figure 5-5 contains a plot of the z component of the electric field. It clearly shows the propagation of the wave through the guide.

Figure 5-5: The z component of the electric field showing how the wave propagates along the path defined by the pillars.

∇ ∇Ez n2k02Ez–⋅– 0=


If the angular frequency of the incoming wave is less than the cut-off frequency of the waveguide, the wave does not propagate through the outlined guide geometry. In Figure 5-6 the wavelength has been increased by a factor of 1.17.

Figure 5-6: A longer wavelength will not propagate through the guide.

References

[1] Joannopoulus J. D., Meade R. D., Winn J. N., Photonic Crystals (Modeling the Flow of Light), Princeton university press, 1995.

[2] Chuang Shun Lien, Physics of Optoelectronic Devices, Wiley series in pure and applied optics, 1995.

Model Library path: Electromagnetics_Module/Optics_and_Photonics/photonic_crystal



1 Select 2D in the Space dimension list.


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2 In the list of application modes, select Electromagnetics Module>In-Plane Waves> TE

Waves>Harmonic propagation.

3 Click OK.

A P P L I C A T I O N M O D E P R O P E R T I E S

For convenience, specify that the wavelength rather than the frequency should be used as input. In the Application Mode Properties dialog box set the property Specify wave

using to Free space wavelength.



2 Set axis and grid settings according to the following table (clear the Auto check box to enter the grid settings).

3 Click OK.


The easiest way to create the crystal geometry is using a geometry array.

1 Start by drawing a circle with the radius 7e-8 and the center at x = 0, y = 0.

2 Select the circle and click the Array button. In the Displacement edit fields, type 3.75e-7 for the displacements in both directions and in the Array size edit fields, type 10 in both the x and y directions

3 Create the guide as a 90º bend by removing some pillars from the array. Remove the circles C3, C13, C23, C33, C43, C53, C63, C64, and C74 to C80.

4 Use the Rectangle dialog box to create a rectangle intersecting with all pillars. Set both the Width and Height to 9*3.75e-7.

5 Select all objects and open the Create Composite Object dialog box. Type R1*(C1+C2+...+C99+C100+R1) in the Set formula edit field and click OK to create an object with the regions outside the rectangle removed. The sum within

AXIS GRID

x min 0 x spacing 5e-7

x max 5e-6 Extra x 7e-8

y min -1e-6 y spacing 5e-7

y max 4e-6 Extra y


parentheses in the set formula creates a union of all objects. The * operator then takes the intersection of this union and the rectangle.


Scalar VariablesIn the Application Scalar Variables dialog box, set the wavelength to 1e-6.

Boundary ConditionsUse low-reflecting boundary conditions on all exterior boundaries. Apply a source at the input port. These settings are summarized in the following table.

Subdomain SettingsEnter subdomain settings according to the following table.

Since the refractive index of GaAs is frequency dependent, the refractive indices used in the model are defined among the expression variables. The frequency dependency

BOUNDARY ALL OTHER 5

Boundary condition Scattering boundary condition Scattering boundary condition

E0z 0 1

Wave type Plane wave Plane wave

SUBDOMAIN 1, 3-86 2

n n_GaAs n_Air


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of the refractive index of GaAs is linearized between the refractive index values corresponding to the wavelengths 1.0332 µm and 1.2339 µm according to Ref. [2]. In the Scalar Expressions dialog box in the Options menu, define the variables n_Air and n_GaAs


1 Change the default mesh parameters to get an applicable mesh. Open the Mesh

Parameters dialog box and set Element growth rate to 1.55 and Mesh curvature factor to 0.65.



Solve the problem with the default solver.


By default, the z component of the electric field is visualized. This clearly shows the propagation of the wave through the guide.

VARIABLE NAME EXPRESSION

n_Air 1

n_GaAs -3.3285e5*lambda0_emwe+3.5031


If the angular frequency of the incoming wave is less than the cut-off frequency of the waveguide, the wave will not propagate through the outlined guide geometry.

1 Open the Application Scalar Variables dialog box and multiply the wavelength by 1.17.


Use a cross-section line plot to visualize the evanescent electric field. Position the plot along the outlined guide between the inlet and the bend.

1 Open the Cross-Section Plot Parameters dialog box and select Line plot.


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2 As Cross-section line data set x0 to 0, x1 to 2.5e-6, y0 to 0.75e-6, and y1 to 0.75e-6. Set the x-axis data to x.

The resulting plot shows that the z component of the electric field declines exponentially along the plot line.


% Start by defining the variables.clear fem applfem.expr.n_GaAs = '-3.3285e5*lambda0+3.5031';fem.expr.n_Air = '1';

% Define the geometry.latt_const = 3.75e-7; oPillar = circ2(0,0,7e-8);N = 10; cPillars = geomarrayr(oPillar,latt_const,latt_const,N,N);aRm_Pillars = [3 13 23 33 43 53 63 64 74:80];bKeep_Pillars = logical(ones(1,N^2));bKeep_Pillars(aRm_Pillars) = logical(0);oPillars = geomcomp(cPillars(bKeep_Pillars));oBox = rect2(0,(N-1)*latt_const,0,(N-1)*latt_const);fem.geom = (oBox+oPillars)*oBox;


% Create the application structure and compute the corresponding % FEM coefficients.appl.mode = 'InPlaneWaves';appl.prop.inputvar = 'lambda';appl.var.lambda0 = 1e-6;appl.bnd.type = 'cont' 'SC' 'SC';appl.bnd.ind = ones(1,340);appl.bnd.ind(1:72) = 2;appl.bnd.ind(5) = 3;appl.bnd.E0 = '0' '0' '0' '0' '0' '0' '0' '0' '1';appl.equ.matparams = 'n';appl.equ.n = 'n_GaAs','n_Air';appl.equ.ind = ones(1,86);appl.equ.ind(2) = 2;fem.appl = appl;fem = multiphysics(fem);

% Mesh and solve the problem using the proper parameters.fem.mesh = meshinit(fem,'hgrad',1.55,'hcurve',0.65);fem.xmesh = meshextend(fem);fem.sol = femlin(fem);

% Visualize the resulting electric field.postplot(fem,'tridata','Ez','geom','on','tribar','on');

% Modify the angular frequency and resolve the problem.fem.appl.var.lambda0 = 1e-6*1.17;fem = multiphysics(fem);fem.xmesh = meshextend(fem);fem.sol = femlin(fem);

% Visualize the new solution.postplot(fem,'tridata','Ez','geom','on','tribar','on');

% Make the cross section line plot.postcrossplot(fem,1,[0 2.5e-6;... 0.75e-6 0.75e-6],'lindata','Ez','title','',... 'axislabel','width [m]' 'electric field [V/m]' '');


182 | C H A P T E

S t e p - I n d e x F i b e r

Introduction

The transmission speed of optical waveguides is superior to microwave waveguides, because optical devices have a much higher operating frequency than microwaves, enabling a far higher bandwidth.

Today the silica glass (SiO2) fiber is forming the backbone of modern communication systems. Before 1970, optical fibers suffered from large transmission losses, making optical communication technology merely an academic issue. In 1970, researchers showed, for the first time, that low-loss optical fibers really could be manufactured. Earlier losses of 2000 dB/km now went down to 20 dB/km. Today’s fibers have losses near the theoretical limit of 0.16 dB/km at 1.55 µm (infrared light).

One of the winning devices has been the single-mode fiber, having a step-index profile with a higher refractive index in the centre core, and a lower index in the outer cladding. Numerical software is playing an important role in the design of single mode waveguides and fibers. For a fiber cross-section, even the most simple shape is difficult and cumbersome to deal with analytically. A circular step-index waveguide is a basic shape where benchmark results are available, see reference [1].

Here, a single step-index waveguide made of silica glass is studied. The inner core is made of pure silica glass with refractive index n1 = 1.4457 and the cladding is doped, with a refractive index of n2 = 1.4378. These values are valid for free-space wavelengths of 1.55 µm. The radius of the cladding is chosen to be large enough so that the field of confined modes is zero at the exterior boundaries.

For a confined mode there is no energy flow in the radial direction, thus the wave must be evanescent in the radial direction in the cladding. This is true only if

On the other hand, the wave cannot be radially evanescent in the core region. Thus

The waves are more confined when neff is close to the upper limit in this interval.

neff n2>

n2 neff n1< <


Model Definition

The mode analysis is made on a cross-section in the x-y plane of the fiber. The wave propagates in the z direction and has the form

where ω is the angular frequency and β the propagation constant. An eigenvalue equation for the magnetic field H is derived from Helmholtz equation

which is solved for the eigenvalue λ = −β2.

As boundary condition along the outside of the cladding the magnetic field is set to zero. Because the amplitude of the field decays rapidly as a function of the radius of the cladding this is a valid boundary condition.


When studying the characteristics of optical waveguides, the effective mode index of a confined mode,

as a function of the frequency is an important characteristic. A common notion is the normalized frequency for a fiber. This is defined as

H x y z t, , ,( ) H x y,( )ej ωt βz–( )=

∇ n 2– ∇ H×( )× k02H– 0=

neffβk0------=

V 2πaλ0

---------- n12 n2

2– k0a n1

2 n22

–= =

S T E P - I N D E X F I B E R | 183

184 | C H A P T E

where a is the radius of the core of the fiber. For this simulation, the effective mode index for the fundamental mode, 1.4444 corresponds to a normalized frequency of 4.895. The electric and magnetic fields for this mode is shown in Figure 5-7 below.

Figure 5-7: The color scale is the z component of the electric field and the contour lines is the z component of the magnetic field. This plot is for the effective mode index 1.4444.

References

[1] Amnon Yariv, Optical Electronics in Modern Communications, 5th edition, Oxford University Press, 1997

Model Library path: Electromagnetics_Module/Optics_and_Photonics/step_index_fiber




2 Select the Electromagnetics Module>Perpendicular Waves>Hybrid-Mode Waves>Mode

analysis application mode .


3 Click OK.


1 In the Application Mode Properties dialog box, set Field components to In-plane

components. This selects the two-components equation formulation for hybrid-mode waves. The default three-component equation formulation can also be used for this problem. See “Selecting Equation Formulation” on page 286 in the Electromagnetics Module User’s Guide for general information about the different options.

2 Set the Default element type to Lagrange - Quadratic.

3 For convenience set the property Specify wave using to Free space wavelength. This makes the wavelength available in the Application Scalar Variables dialog box instead of the frequency.


1 In the Constants dialog box, enter the following names and expressions for the refractive indices.


1 Draw a circle C1 centered at (0,0) with radius 8e-6.

2 Draw a second circle C2 centered at (0,0). This time with the radius 40e-6.

3 Click the Zoom Extents button for a full visualization of the two circles.


Scalar VariablesIn the Application Scalar Variables dialog box, set the free space wavelength lambda0_emwv to 1.55e-6.

Boundary ConditionsUse the default boundary conditions on all exterior boundaries.

NAME EXPRESSION

nSilicaGlass 1.4457

nSilicaGlassDoped 1.4378


186 | C H A P T E

Subdomain SettingsIn the Subdomain Settings dialog box enter the refractive index in the two domains according to the following table.



2 Refine the mesh once.


The modes of interest have an effective mode index somewhere between the refractive indices of the two materials, that is,

1 In the Solver Parameters dialog box set the parameter Search for effective mode

indices around to 1.446. This guarantees that the solver will find the fundamental mode, which has the largest effective mode index.



The default plot shows the power flow in the z direction for the fundamental mode. This is the HE11 mode, which is verified by visualizing the transversal components of both the magnetic and the electric field.

1 In the Plot Parameters dialog box, select surface and contour plots.

2 Set Solution at angle (phase) to 45.

3 On the Surface tab, set the Surface data to Electric field, z component.

SUBDOMAIN 1 2

n nSilicaGlassDoped nSilicaGlass

1.4378 neff 1.4457< <


4 On the Contour tab, give Magnetic field, z component as Contour data.


% The model can also be simulated by programming. First set up the % structure with the geometry and the refractive indices as % constants.clear femfem.sdim='x' 'y';fem.geom=circ2(0,0,8e-6)+circ2(0,0,40e-6);fem.const=... 'nSilicaGlassDoped',1.4378,... 'nSilicaGlass',1.4457;

% Then define the application mode structure and generate the FEM % structure from it.clear applappl.mode='PerpendicularWaves';appl.prop.comps='2';appl.prop.inputvar='lambda';appl.var.lambda0=1.55e-6;appl.equ.n='nSilicaGlassDoped' 'nSilicaGlass';fem.appl=appl;fem=multiphysics(fem);

% Now generate the mesh, and the extended mesh structure.fem.mesh=meshinit(fem);fem.mesh=meshrefine(fem);


188 | C H A P T E

fem.xmesh=meshextend(fem);

% Solve the problem and visualize the solution.% Note that the 'eigpar' parameter to the solver% must be expressed as the eigenvalue lambda = -beta^2.fem.sol=femeig(fem,'eigpar',6 -3.436e13);postplot(fem,'tridata','Ez','contdata','Hz','phase',45);


S t r e s s -Op t i c a l E f f e c t s i n a S i l i c a - o n - S i l i c o n Wavegu i d e

Introduction

Planar photonic waveguides in silica (SiO2) have a great potential for use in wavelength routing applications. The major problem with this kind of waveguides is birefringence. Anisotropic refractive indices result in splitting of the fundamental mode and pulse broadening. The goal is to minimize birefringence effects by adapting materials and manufacturing processes. One source of birefringence is the use of a silicon (Si) wafer on which the waveguide structure is deposited. After annealing at high temperature (approximately 1000 °C), mismatch in thermal expansivity between the silica and silicon layers results in thermally induced stresses in the structure at the operating temperature (typically room temperature, 20 °C).

The Stress-Optical Effect and Plane Strain

The general linear stress-optical relation can be written, using tensor notation, as

where ∆nij = nij − n0Iij, nij is the refractive index tensor, n0 is the refractive index for a stress-free material, Iij is the identity tensor, Bijkl is the stress-optical tensor, and σkl is the stress tensor. Due to symmetry, it is possible to reduce the number of independent parameters in the stress-optical tensor that characterizes this constitutive relation. Since nij and σkl are both symmetric, Bijkl = Bjikl and Bijkl = Bijlk. In many

Air

Cladding (Silica)

Buffer (Silica)

Silicon Wafer Core

Optical Computation Domain Boundary

∆nij B– ijklσkl=

S T R E S S - O P T I C A L E F F E C T S I N A S I L I C A - O N - S I L I C O N W A V E G U I D E | 189

190 | C H A P T E

cases you can further reduce the number of independent parameters, and this model includes only two independent parameters, B1 and B2. The stress-optical relation simplifies to

where nx = n11, ny = n22, nz = n33, σx = σ11, σy = σ22, and σz = σ33.

This translates to

Using the two parameters B1 and B2, it is assumed that the non-diagonal parts of nij and σkl are negligible. This means that the shear stress corresponding to σ12 = τxy is neglected. In addition, the shear stresses corresponding to σ13 = τxz and σ23 = τyz are neglected by using the plane strain approximation. The plane strain approximation holds in a situation where the structure is free in the x and y direction but where the z strain is assumed to be zero. Note that this deformation state is not correct if the structure is free also in the z direction. In such a case a modified deformation state equation is needed where the x and z directions are equivalently handled.

The first part of this model utilizes the Plane Strain application mode from the Structural Mechanics Module. The resulting birefringent refractive index is computed using expression variables and can be considered a postprocessing step of the plane strain model. The refractive index tensor is used as material data for the second part of the model, the mode analysis.

The model “Stress-Optical Effects with Generalized Plane Strain” on page 206 demonstrates a computation where the structure is free also in the z direction, using a formulation called generalized plane strain.

Perpendicular Hybrid-Mode Waves

For a given frequency ν, or equivalently, free-space wavelength λ0 = c0/ν, the Perpendicular Hybrid-Mode Waves application mode of the Electromagnetics Module can be used for the mode analysis. In this model the free-space wavelength is chosen

∆nx

∆ny

∆nz

B2 B1 B1

B1 B2 B1

B1 B1 B2

σx

σy

σz

–=

nx n0 B2σx– B1 σy σz+( )–=

ny n0 B2σy– B1 σz σx+( )–=

nz n0 B2σz– B1 σx σy+( )–=


to be 1.55 µm. The simulation is set up with the normalized magnetic field components H=(Hx, Hy, Hz) as dependent variables. The effective mode index neff = β/k0 is obtained from the eigenvalues.

Using this application mode, the wave is assumed to have the form

The computations show a shift in effective mode index due to the stress-induced change in refractive index. The birefringence causes the otherwise two-fold degenerate fundamental mode to split.

Model Library path: Electromagnetics_Module/Optics_and_Photonics/stress_optical

Note: This model requires both the Electromagnetics Module and the Structural Mechanics Module to be installed.

Plane Strain Analysis

1 Select the Structural Mechanics Module, Plane Strain, static analysis application mode.

2 Click Multiphysics then Add to add this application mode to the model.

3 Next select the Electromagnetics Module, Perpendicular Waves, Hybrid-Mode Waves,

mode analysis application mode. Add this mode to the model.

4 Open the Application Mode Properties dialog box and set Field components to In-plane



H H x y,( )ej ωt βz–( ) Hx x y,( ) Hy x y,( ) Hz x y,( ), ,( )ej ωt βz–( )= =


192 | C H A P T E

6 Set the Ruling application mode to the Perpendicular Hybrid-Mode Waves application mode. This makes this mode specify the interpretation of the parameters to the eigenvalue solver given in the Solver Parameters dialog box.


1 In the Constants dialog box enter the following names and expressions.

Notice that the temperatures are given in degrees Celsius. This works well since this is a linear model. (Nonlinear models need to have temperatures in Kelvin to get the correct thermodynamics.)


nSi 3.5 Refractive index, silicon (Si)

nBuf 1.445 Refractive index, silica (SiO2)

nClad 1.445 Refractive index, silica

deltan 0.0075 Relative index difference:

nCore nClad/sqrt(1-2*deltan) Refractive index, core

alphaSi 2.5e-6 Thermal expansion coefficient, silicon (K−1)

alphaSiO2 0.35e-6 Thermal expansion coefficient, silica (K−1)

ESi 110e9 Young’s modulus, silicon (N/m2)

ESiO2 78e9 Young’s modulus, silica (N/m2)

nuSi 0.19 Poisson’s ratio, silicon

nuSiO2 0.42 Poisson’s ratio, silica

B1 4.2e-12 First stress optical coefficient (m2/N)

B2 0.65e-12 Second stress optical coefficient (m2/N)

T1 20 Operating temperature (°C)

T0 1000 Reference temperature (°C)

∆ncore

2 ncladding2

–( )

2ncore2

--------------------------------------------=


2 Give axis and grid settings according to the following table.


Draw rectangles corresponding to the different regions.

AXIS GRID

x min -0.2e-3 x spacing 1e-5

x max 0.2e-3 Extra x

y min -0.1e-3 y spacing 1e-5

y max 0.03e-3 Extra y

REGION LOWER LEFT CORNER UPPER RIGHT CORNER

Silicon Wafer (-16e-5,-1e-4) (16e-5,-1.7e-5)

Buffer (-16e-5,-1.7e-5) (16e-5,-3e-6)

Cladding (-16e-5,-3e-6) (16e-5,1.3e-5)

Core (-3e-6,-3e-6) (3e-6,3e-6)


194 | C H A P T E


Point Settings and Boundary ConditionsAll regions have free boundaries, which also is the default boundary condition. However, these conditions will not suffice in creating a unique solution since the computational domain is allowed to move and rotate freely; the problem is ill-posed. The problem becomes well-posed by adding constraints at points to keep the domain fixed.

1 Select the Plane Strain application mode from the Multiphysics menu.

2 Open the Point Settings dialog box, and check the constraints Rx and Ry at point 1, the lower left corner, for both the x and y direction. This will keep the domain from moving (translational move).

3 Check the constraint Ry at point 9, the lower right corner, for the y direction only, keeping the domain from rotating but allowing it to slide in the x direction.

Subdomain SettingsThe air domain need not be part of the plane strain model.

1 Specify the subdomain settings for the Plane Strain application mode according to the following table.

The other material parameters, ρ, αdM, and βdK, need not be specified because they do not enter the equation for static problems.

SUBDOMAIN 1 1 2,3,4 2,3,4

Tab Material Loads Material Loads

E ESi E ESiO2

ν nuSi Temp T1 ν nuSiO2 Temp T1

α alphaSi Tempref T0 α alphaSiO2 Tempref T0


E X P R E S S I O N V A R I A B L E S

In the Subdomain Expressions dialog box in the Options menu, define the following variables.


Initialize a mesh and refine it twice.


1 In the Solver Parameters dialog box select the Stationary linear solver.

2 On the Solve for tab in the Solver Manager dialog box, select only the Plane Strain application mode. This will make sure that only the Plane Strain application mode equations are solved in the first run.


VARIABLE NAME SUBDOMAIN EXPRESSION

N 1

2

3

4

nSi

nBuf

nClad

nCore

Nx 1

2,3,4

N

N-B1*sx_pn-B2*(sy_pn+sz_pn)

Ny 1

2,3,4

N

N-B1*sy_pn-B2*(sx_pn+sz_pn)

Nz 1

2,3,4

N

N-B1*sz_pn-B2*(sx_pn+sy_pn)


196 | C H A P T E


The default plot shows the von Mises effective stress as a colored surface plot.

1 To view the von Mises stress along a horizontal line through the entire structure, use a cross-section plot. In the Cross-Section Plot Parameters dialog box, select von


Mises stress as y-axis data on the Line/Extrusion tab. As Cross-section line data set x0 to -16e-5, y0 to 0, x1 to 16e-5, and y1 to 0. Use x as x-axis data.

Notice that the effective stress is varying slowly on the horizontal cut. This means that the significant influence on the stress induced changes in refractive index comes from the stress variations in the vertical direction. This is expected since the extension of the domains in the x direction is chosen to minimize effects of the edges.


198 | C H A P T E

2 To view the stress-induced birefringence nx − ny, type Nx-Ny as Surface data in the Plot Parameters dialog box.

3 Now create a cross-section plot of the birefringence nx − ny on the vertical line from (0,-3e-6) to (0,3e-6). Do this by entering Nx-Ny as y-axis data, and setting x0 to 0, y0 to -3e-6, x1 to 0, and y1 to 3e-6 in the Cross-Section Plot Parameters dialog box. Set the x-axis data to Arc-length. The birefringence varies linearly from about 2.9⋅10-4 at the bottom of the core to 2.4⋅10-4 at the top.

4 Now create another cross-section plot of the birefringence on the horizontal line from (-3e-6,0) to (3e-6,0). On the General tab check Keep current plot to make the


plot in the same window as the previous one. On the Line/Extrusion tab set x0 to -3e-6, y0 to 0, x1 to 3e-6, and y1 to 0.

The birefringence is constant on the horizontal line, thus the influence of the edges is indeed reduced to a minimum.

Optical Mode Analysis

Select the Perpendicular Hybrid-Mode Waves application mode from the Multiphysics menu.


The computational domain can be reduced significantly for the optical mode analysis due to the fact that the energy of the fundamental modes will be concentrated in the core region and the energy density is rapidly decaying in the cladding and buffer regions.

1 Draw a rectangle with corners at (-1e-5,-1e-5) and (1e-5,1e-5).

The rectangular region containing the active domains can be enlarged later on for validating the results. The rectangular region should be chosen large enough so that the computed propagation constants do not change significantly if the region is extended.


200 | C H A P T E


Occasionally all variables and settings are not updated exactly as expected when modifying the geometry. In this case the expression for N has got the wrong value in domain 6. In the Subdomain Expressions dialog box change the value of N in subdomain 6 to nCore.


Scalar VariablesIn the Application Scalar Variables dialog box, set the free space wavelength to 1.55e-6.

Subdomain Settings1 Deactivate the Perpendicular Hybrid-Mode Waves application mode in all

subdomains except 4, 5, and 6. This will make sure the problem is only solved in the newly drawn rectangle.

2 For all active domains, select n (anisotropic) and set Nx, Ny, and Nz, respectively, as anisotropic refractive indices by changing the entries on the diagonal in the refractive index matrix.

Boundary ConditionsThe only available perfect magnetic conductor boundary condition will suffice. Both the E and the H fields are assumed to be of negligible size at the boundaries, hence any one of them can be set equal to zero at the boundary.


1 On the Subdomain tab in the Mesh Parameters dialog box, set the Maximum element

size to 2e-6 for subdomains 4 and 5, and to 1e-6 for subdomain 6. This will make the mesh dense in the activated regions and coarse in the deactivated regions. The mesh in the deactivated regions will only affect the interpolated values of the expression variables and not the mode computation results.



1 In the Solver Parameters dialog box, select the Eigenvalue solver.

2 Enter 1.46 in the text field Search for effective mode indices around. The default Desired number of effective mode indices is set to 6, which is fine in this case. These settings will make the eigenmode solver search for the 6 eigenmodes with effective mode indices closest to the value 1.46. This value is an estimate of the effective mode index for the fundamental mode. For propagating modes it must hold that


3 In the Solver Manager dialog box, select only the Perpendicular Hybrid-Mode Waves application mode on the Solve For tab. This will ensure that the Plane Strain application mode will not be part of the eigenmode computation.

4 On the Initial Value tab, set the initial value to Current solution. This will take the plane strain solution and use it to evaluate the refractive indices for the mode analysis.



1 On the Surface tab in the Plot Parameters dialog box, set the Surface data to Power

flow, time average, z component. This will visualize the power flow, also called optical intensity or the Poynting vector, in the z direction (out of plane direction).

2 The default eigenmode is the one with the highest effective mode index corresponding to one of the fundamental modes.

Convergence Analysis

To check the accuracy of the eigenvalues, you need to examine the sensitivity of these values to changes in the mesh density.

neff ncore< 1.456=


202 | C H A P T E

1 Lower the value of the maximum element size for the core subdomain, number 6, to 5e-7 and solve again.

2 Repeat this for the maximum element sizes 2.5e-7 and 1.25e-7.

The effective mode indices found are

It is evident from the table that the fundamental modes, corresponding to the two lowest eigenmodes with the highest effective mode indices, are nearly degenerate. It is also clear that the split of the degenerate fundamental modes that is expected due to the stress-induced anisotropic refractive index has been properly resolved with these

ELEMENT SIZE, CORE SUBDOMAIN

1E-6 5E-7 2.5E-7 1.25E-7

EFFECTIVE MODE INDEX

neff1 1.451148 1.45114 1.451136 1.451135

neff2 1.450882 1.450874 1.450872 1.450871

neff3 1.445409 1.445407 1.445405 1.445405

neff4 1.445143 1.44514 1.445139 1.445139

neff5 1.445138 1.445134 1.445132 1.445131

neff6 1.444872 1.444867 1.444865 1.444865


mesh sizes. The convergence of the two fundamental modes is shown in the figure below.

Now, run the model without the stress-induced birefringence.

1 Change the refractive index of subdomains 4, 5, and 6 to an isotropic index by selecting the n (isotropic) radio button in the Subdomain Settings dialog box, and typing N as refractive index.

2 Solve the problem for a maximum element size of 2.5e-7 for subdomain 6.

Compare these ideal values of the propagation constants with the stress-optical case.

ELEMENT SIZE, CORE SUBDOMAIN:

2.5E-7 (STRESS) 2.5E-7 (NO STRESS) DIFFERENCE


*1e-4

neff1 1.451136 1.450821 3.15

neff2 1.450872 1.450821 .51

neff3 1.445405 1.445093 57.79

neff4 1.445139 1.445083 .56

neff5 1.445132 1.444832 3.00

neff6 1.444865 1.444799 .66

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 71.4509

1.4509

1.451

1.451

1.4511

1.4511

1.4511

1.4512

1.4512Computed value of effective mode index vs. mesh size h in core subdomain

−log10(h)

effe

ctiv

e m

ode

inde

x n ef

f


204 | C H A P T E

The difference is significant; thus it can be concluded that the shift in the effective mode indices due to the stress-optical effect is indeed resolved.

Visual inspection of the four higher eigenmodes indicates that these are probably degenerate in the ideal case of no thermally induced stresses. Moreover, the higher eigenmodes have a larger portion of energy leaking into the cladding and buffer, and are thus more affected than the fundamental modes of the distance to the air and silicon layers. Because of this leakage, the boundary condition will affect the higher eigenmodes more than the fundamental mode. Thus a refined analysis of the higher modes would make it necessary to enlarge the computation domain.

The figure below shows the z component of the Poynting vector for the fourth eigenmode.

M O D E L I N G E R R O R S A N D S E N S I T I V I T Y A N A L Y S I S

Although the fundamental modes have converged to 5 decimal places the known modeling errors makes the exactness of the numbers uncertain. One major modeling error is due to the fact that the model uses plane strain in a case where the real-world model does not necessarily conform to this deformation state. This modeling error is reduced in the refined model “Stress-Optical Effects with Generalized Plane Strain” on page 206 presented below. Moreover, the material properties are only known to a few decimal places and the computed magnitudes of the effective mode indices are correspondingly uncertain. A standard way of examining the effects of uncertainty in material parameters is to perform a sensitivity analysis. That is, one of the material


parameters is slightly perturbed and the resulting perturbation in the computed parameter is examined. Another source of uncertainty is whether the stress from the thermal expansion is large with respect to the other sources of stress in the material originating from the manufacturing process.


206 | C H A P T E

S t r e s s -Op t i c a l E f f e c t s w i t h Gen e r a l i z e d P l a n e S t r a i n

Introduction

The assumptions made for plane strain in the previous analysis of the waveguide structure, in the model “Stress-Optical Effects with Generalized Plane Strain” on page 206, do not hold in a situation where the silicon-silica laminate is free to expand in the z direction. Instead a generalized plane strain model need to be used that allows for free expansion in the z direction. The boundary conditions in the x-y plane already allow the structure to expand freely in all directions in the plane. When the different materials in a laminate expands with different expansion coefficients, the laminate bends. In this model, the silica-silicon laminate bends in both the x and z directions. The bending in the z direction is not covered by the Plane Strain application mode but modifications to the plane strain equations are needed.

Note: This model requires both the Electromagnetics Module and the Structural Mechanics Module to be installed.

Generalized Plane Strain

One possible extension of the plane strain formulation is to assume that the strain in the z direction has the form

That is, the strain is linearly varying throughout the cross section. This approximation is expected to be good when the bending curvature is small with respect to the extents of the structure in the x-y plane and corresponds to a small rotation that is representative of each cross section of the structure along the z axis. (A more general model would include second degree terms in x and y.)

One way of validating the result of this simulation is to plot the εx and εz strains on a cross-section on the symmetry-line at x = 0. Because you can expect an identical bending in the x and z directions, these strains should be identical and vary linearly.

εz e0 e1x e2y+ +=


E X T E N S I O N O F T H E P L A N E S T R A I N E Q U A T I O N S

The objective is to extend the current Plane Strain application mode with additional PDE terms that represent the generalized plane strain deformation state. In COMSOL Multiphysics, the coefficients e0, e1, and e2 in the expression for the εz strain is modeled with coupling variables associated with a point geometry object (zero-dimensional object). This is a convenient way to include degrees of freedom that are constant throughout a subdomain. Because the intentions of this model is to expand the equations for the ordinary Plane Strain application mode, you need to add the new εz strain contributions n as weak terms to the ordinary plane strain equations. The reason for this is that for COMSOL Multiphysics to be able to solve the equations as a linear problem, the assembler requires that you enter all contributions from coupling variables as weak terms.

Start from the 3D stress-strain relation for linear isotropic conditions including thermal effects,

where σx, σy, σz, τxy, τyz, and τxz are the components of the stress tensor, εx, εy, εz, γxy, γyz, and γxz are the components of the strain tensor, α is the thermal expansion coefficient, T is the operating temperature, and Tref is the reference temperature corresponding to the manufacturing temperature. The 6-by-6 matrix D (tensor) has entries that contain expressions in the Young’s modulus E and the Poisson’s ratio ν. The matrix D is presented in the User’s Guide of the Structural Mechanics Module.

For generalized plane strain (and ordinary plane strain), assume that γyz = γxz = 0 and two of the equations vanish

σx

σy

σz

τxy

τyz

τxz

D

εx

εy

εz

γxy

γyz

γxz

α

T Tref–

T Tref–

T Tref–

000

–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

σx

σy

σz

τxy

D

εx

εy

εz

γxy

α

T Tref–

T Tref–

T Tref–

0

–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

S T R E S S - O P T I C A L E F F E C T S W I T H G E N E R A L I Z E D P L A N E S T R A I N | 207

208 | C H A P T E

The matrix D shrinks accordingly.

The PDE solved is Navier’s equation

where

in the case of generalized plane strain

and u = (u, v, w) are the unknown displacement variables solved for.

The coefficient c above is a tensor containing the elements of D in a pattern that is listed in the User’s Guide of the Structural Mechanics Module and

contains the contributions from the thermal strains.

In the case of ordinary plane strain the upper left 2-by-2 submatrix of σ, γ, and D is used.

It is assumed here that the variables T and Tref are constants. For information on how the temperature variables enter Navier’s equation as unknowns in a heat transfer problem, see the Reference Guide of the Structural Mechanics Module.

To expand the plane strain equation to a generalized plane strain formulation one needs to consider the weak form of Naiver’s equation. The weak form of a system of equations is a scalar equation involving integrals. See “The Weak Form” on page 253 in the COMSOL Multiphysics Modeling Guide for more information on the appearance of the weak form for systems of equations.

To get the weak form of Navier’s equation, multiply by a test function v = [utest, vtest, wtest] and integrate

∇– σ⋅ K=

σ

σx τxy 0

τyx σy 0

0 0 σz

c u∇ γ–= =

γ E1 ν+( ) 1 2ν–( )

---------------------------------------

1 ν+( )α T Tref–( ) 0 0

0 1 ν+( )α T Tref–( ) 0

0 0 1 ν+( )α T Tref–( )

=


Integration by parts gives

or in more detail, with K = 0

where uy,test = ∂utest/∂y and similarly for the other test function derivatives. The dot product in the domain integral is interpreted as a dot product operating on the rows of the matrices so that

where

v ∇– σ⋅( ) ΩdΩ∫ v K Ωd⋅

Ω∫=

0 vσn Γ v∇ σ⋅ Ω v K Ωd⋅Ω∫+d

Ω∫–d

Ω∂∫=

0 utest vtest wtest

σx τxy 0

τyx σy 0

0 0 σz

nx

ny

nz

ΓdΩ∂∫

ux test, uy test, uz test,

vx test, vy test, vz test,

wx test, wy test, wz test,

σx τxy 0

τyx σy 0

0 0 σz

⋅

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

ΩdΩ∫–

=

ux test, uy test, uz test,

vx test, vy test, vz test,

wx test, wy test, wz test,

σx τxy 0

τyx σy 0

0 0 σz

⋅

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

ΩdΩ∫

ux test, uy test, uz test, σx τxy 0

vx test, vy test, vz test, τyx σy 0 wx test, wy test, wz test, 0 0 σz⋅+⋅

+⋅⎝

⎠

⎛

⎞ Ωd

Ω∫=


210 | C H A P T E

and

Note that the displacement variable w in the z direction is not available in the Plane Strain application mode. However, it is not needed since the generalized formulation assumes a certain fix shape of the deformation in the z direction.

The same relations hold for the test functions,

In COMSOL Multiphysics, the test functions utest and vtest are denoted u_test and v_test, respectively. Their derivatives ux,test, uy,test, vx,test, and vy,test are denoted ux_test, uy_test, vx_test, and vy_test, respectively. Note that the z components of the test functions are not available in the Plane Strain application mode.

Using this notation, the weak form of Navier’s equation becomes (with K = 0)

σxE

1 ν+( ) 1 2ν–( )--------------------------------------- 1 ν–( )εx νεy νεz 1 ν+( )α T Tref–( )–+ +( )=

σyE

1 ν+( ) 1 2ν–( )--------------------------------------- νεx 1 ν–( )εy νεz 1 ν+( )α T Tref–( )–+ +( )=

σzE

1 ν+( ) 1 2ν–( )--------------------------------------- νεx νεy 1 ν–( )εz 1 ν+( )α T Tref–( )–+ +( )=

τxyE

1 ν+( ) 1 2ν–( )---------------------------------------

1 2ν–2

----------------γxy⎝ ⎠⎛ ⎞=

εx x∂∂u=

εy y∂∂v=

εz z∂∂w=

γxy y∂∂u

x∂∂v+=

εx test, ux test,=

εy test, vy test,=

εz test, wz test,=

γxy test, uy test, vx test,+=

0 vσn ΓdΩ∂∫ εx test, σx εy test, σy εz test, σz γxy test, τxy+ + +( ) Ωd

Ω∫–=


This expression for the weak equation shows which terms should be added as weak terms to generalize the plane strain formulation.

To see the difference compared to the ordinary plane strain formulation, the equation for σz can be separated out according to

where D has been reduced to

The ordinary plane strain equation is obtained if εz = 0.

To expand the Plane Strain application mode, weak contributions from εz need to be added to the stress components σx and σy according to

In addition, σz needs to be included in the weak equation from being merely a post processing quantity in the original Plane Strain application mode,

where

and σz was presented earlier.

In COMSOL Multiphysics, the coefficients e0, e1, and e2 in the expression for the εz strain are modeled with coupling variables as weak contributions to the ordinary plane strain equations. These coefficients represent three additional unknown variables that prevail throughout the modeled domain. In practice, these variables need to be

σx

σy

τxy

D

εx

εy

γxy

1 ν+( )α T Tref–( )

1 ν+( )α T Tref–( )

0

–

νεz

νεz

0

+

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

=

σzE

1 ν+( ) 1 2ν–( )--------------------------------------- νεx νεy 1 ν–( )εz 1 ν+( )α T Tref–( )–+ +( )=

D E1 ν+( ) 1 2ν–( )

---------------------------------------

1 ν– ν 0ν 1 ν– 0

0 0 1 2ν–2

----------------

=

E1 ν+( ) 1 2ν–( )

--------------------------------------- εx test, νεz εy test, νεz+( )–

εz test, σz–

εz test, e0 test, e1 test, x e2 test, y+ +=


212 | C H A P T E

associated with an arbitrary point of the geometry. When using coupling variables in this way one specifies a map from this point onto the entire domain on which the generalized plane strain equations hold. The test functions corresponding to e0, e1, and e2 are automatically created.

Model Library path: Electromagnetics_Module/Optics_and_Photonics/stress_optical_generalized

Plane Strain Analysis

Many of the modeling steps will be similar to the model “Stress-Optical Effects in a Silica-on-Silicon Waveguide” on page 189, but some steps are made in a different order, and there are obviously differences when generalizing the plane strain problem.

M U L T I P H Y S I C S S E T T I N G S

1 Select the Structural Mechanics Module, Plane Strain, static analysis application mode.

2 Click Multiphysics then Add to add this application mode to the model.

3 Next select the Electromagnetics Module, Perpendicular Waves, Hybrid-Mode Waves,

mode analysis application mode. Add this mode to the model.

4 Open the Application Mode Properties dialog box and set Field components to In-plane



The unknown parameters e0, e1, and e2 should be constant within the whole modeling domain. This can be achieved by adding a new geometry with only a single point to the model, and define e0, e1, and e2 as the dependent variables at this point. Later on coupling variables will be defined which map these dependent variables from the point to the whole model.

1 Click the Add Geometry button to add a 2D geometry (2D is the default space dimension).


2 Select the PDE Modes, Weak Form, Point application mode from the application mode tree.

3 Type e0 e1 e2 in the Dependent variables text field.

4 Click Add to add the weak point application mode to the second geometry. This application mode now contains the three degrees of freedom for the parameters e0, e1, and e2 needed for the z strain.

5 Set the Ruling application mode to the Perpendicular Hybrid-Mode Waves application mode. This makes this mode specify the interpretation of the parameters to the eigenvalue solver given in the Solver Parameters dialog box.

6 Click the OK button to close the Model Navigator.

The Point application mode will be the active mode when the Model Navigator has been closed.


The three degrees of freedom e0, e1, and e2 need to be associated with a point. The location of this point is insignificant. This means you could equally well have chosen to add a 1D or 3D geometry. Using the programming language, there is also an option of using a 0D geometry.

1 Add a point at any position, say at the origin (0,0).


In the Constants dialog box enter the following names and expressions. They are identical to the constants defined on page 192.


nSi 3.5 Refractive index, silicon (Si)

nBuf 1.445 Refractive index, silica (SiO2)

nClad 1.445 Refractive index, silica

deltan 0.0075 Relative index difference:

nCore nClad/sqrt(1-2*deltan) Refractive index, core

alphaSi 2.5e-6 Thermal expansion coefficient, silicon (K−1)

∆ncore

2 ncladding2

–( )

2ncore2

--------------------------------------------=


214 | C H A P T E


1 Switch to Geom1 by clicking its tab.

2 Draw rectangles corresponding to the different regions.


Point Settings and Boundary ConditionsAll regions have free boundaries, which also is the default boundary condition. However, these conditions will not suffice in creating a unique solution since the computational domain is allowed to move and rotate freely; the problem is ill-posed. The problem becomes well-posed by adding constraints at points to keep the domain fixed.

1 Select the Plane Strain application mode from the Multiphysics menu.

2 Open the Point Settings dialog box, and check the constraints Rx and Ry at point 1, the lower left corner, for both the x and y direction. This will keep the domain from moving (translational move).

alphaSiO2 0.35e-6 Thermal expansion coefficient, silica (K−1)

ESi 110e9 Young’s modulus, silicon (N/m2)

ESiO2 78e9 Young’s modulus, silica (N/m2)

nuSi 0.19 Poisson’s ratio, silicon

nuSiO2 0.42 Poisson’s ratio, silica

B1 4.2e-12 First stress optical coefficient (m2/N)

B2 0.65e-12 Second stress optical coefficient (m2/N)

T1 20 Operating temperature (°C)

T0 1000 Reference temperature (°C)

REGION LOWER LEFT CORNER UPPER RIGHT CORNER

Silicon Wafer (-16e-5,-1e-4) (16e-5,-1.7e-5)

Buffer (-16e-5,-1.7e-5) (16e-5,-3e-6)

Cladding (-16e-5,-3e-6) (16e-5,1.3e-5)

Core (-3e-6,-3e-6) (3e-6,3e-6)



3 Check the constraint Ry at point 9, the lower right corner, for the y direction only, keeping the domain from rotating but allowing it to slide in the x direction.

Subdomain SettingsThe air domain need not be part of the plane strain model.

1 Specify the subdomain settings for the Plane Strain application mode according to the following table.

The other material parameters, ρ, αdM, and βdK, need not be specified because they do not enter the equation for static problems.

SUBDOMAIN 1 1 2,3,4 2,3,4

Tab Material Loads Material Loads

E ESi E ESiO2

ν nuSi Temp T1 ν nuSiO2 Temp T1

α alphaSi Tempref T0 α alphaSiO2 Tempref T0


216 | C H A P T E


In the Subdomain Expressions dialog box in the Options menu, define the following variables.

The variables ex, ey, and ez are the strains εx, εy, and εz, and the variables sx, sy, and sz are the stresses σx, σy, and σz. The refractive index variables Nx, Ny, and Nz are defined slightly differently compared to the previous model. In this case they are expressed in the generalized stress variables sx, sy, and sz.

VARIABLE NAME SUBDOMAIN EXPRESSION

E 1

2,3,4

ESi

ESiO2

nu 1

2,3,4

nuSi

nuSiO2

alpha 1

2,3,4

alphaSi

alphaSiO2

ex 1,2,3,4 ux

ey 1,2,3,4 vy

ez 1,2,3,4 e00+e11*x+e22*y

sx 1,2,3,4 sx_pn+E/((1+nu)*(1-2*nu))* nu*ez

sy 1,2,3,4 sy_pn+E/((1+nu)*(1-2*nu))* nu*ez

sz 1,2,3,4 E/((1+nu)*(1-2*nu))* (nu*ex+nu*ey+(1-nu)*ez- (1+nu)*alpha*(T1-T0))

N 1

2

3

4

nSi

nBuf

nClad

nCore

Nx 1

2,3,4

N

N-B1*sx-B2*(sy+sz)

Ny 1

2,3,4

N

N-B1*sy-B2*(sx+sz)

Nz 1

2,3,4

N

N-B1*sz-B2*(sx+sy)


C O U P L I N G V A R I A B L E S

The unknown parameters e0, e1, and e2 should be constant within the whole modeling domain. This can be achieved by defining coupling variables which map these dependent variables from the point to the entire model.


2 Open the Point Integration Variables dialog box by choosing Scalar Coupling Variables>Point Variables from the Options menu.

3 Enter the following names and expressions.

The variables e00, e11, and e22 will take the value of e0, e1, and e2 in the entire model.


Finally the additional terms in the generalized plane strain problem have to be added.


2 Select Equation System>Subdomain Settings from the Physics menu.

3 On the Weak tab in the Subdomain Settings dialog box, add

-E/((1+nu)*(1-2*nu))*(ex_test*nu*ez+ey_test*nu*ez)-ez_test*sz

in the second row of the weak term for all subdomains. The two first rows correspond to the plane strain problem. The expression can actually be entered in any of these two fields since they are added together by COMSOL Multiphysics while assembling the problem.



2 Refine it twice.


4 Initialize the mesh in this geometry too.


1 In the Solver Parameters dialog box select the Stationary linear solver.

NAME EXPRESSION

e00 e0

e11 e1

e22 e2


218 | C H A P T E

2 On the Solve for tab in the Solver Manager dialog box, select only the Plane Strain and the Weak Form, Point application modes. This will exclude the Hybrid-Mode Waves application mode from the solving.



1 On the Surface tab in the Plot Parameters dialog box, change the Surface expression to abs((ex-ez)/ez)<0.05. Clear the Smooth check box. Click OK to visualize the area where the relative difference between the x and z strain is within 5%. The model is most accurate in the regions close to the core, far from the boundaries on the far left and right.

2 For symmetry reasons the strains εx and εz should be equal. To see how well this model has achieved this, make a cross-section plot of them. Open the Cross-Section

Plot Parameters dialog box. Enter the Title ex and ez. On the Line tab, set x0 to 0, x1 to 0, y0 to -1e-4, and y1 to 1.3e-5 to make a vertical line across the structure. Enter the y-axis data expression ex, and set the Line resolution to 25. Set the x-axis data to y. Click the Line Settings button and select circle as Line marker. Click Apply to plot ex.

3 On the General tab check Keep current plot to be able to plot ez in the same figure window.


4 Then change the Line expression to ez. Use plus sign as Line marker for this plot. Click OK to make the plot.

5 To view the stress-induced birefringence nx − ny together with the deformation, open the Plot Parameters dialog box and select Surface and Deformation plot. On the Surface tab enter the Surface expression Nx-Ny and check Smooth again. As


220 | C H A P T E

Deformation data use Displacement (pn) which is the default selected on the Deform tab. Click OK.

Note: The plane strain PDE system has been extended in this model. This means that some of the postprocessing variables that are provided in the Plane Strain application mode are no longer valid and will give wrong results if used. It is recommended that only the variables entered in the Subdomain Expressions dialog box are used for postprocessing.

Optical Mode Analysis

Now continue with the mode analysis, very much in the same way as was done in the previous model on page 199.


The computational domain can be reduced significantly for the optical mode analysis due to the fact that the energy of the fundamental modes will be concentrated in the core region and the energy density is exponentially decaying in the cladding and buffer regions.

1 Draw a rectangle with corners at (-1e-5,-1e-5) and (1e-5,1e-5).



Next, set up the parameters for the hybrid-mode wave problem.

1 Select the Perpendicular Hybrid-Mode Waves application mode in the Multiphysics menu.

Scalar VariablesIn the Application Scalar Variables dialog box, set the free space wavelength to 1.55e-6.

Subdomain Settings1 Deactivate the Perpendicular Hybrid-Mode Waves application mode in all

subdomains except 4, 5, and 6. This will make sure the problem is only solved in the newly drawn rectangle.

2 For all active domains, select n (anisotropic) and set Nx, Ny, and Nz, respectively, as anisotropic refractive indices by changing the entries on the diagonal in the refractive index matrix.


Occasionally all variables and settings are not updated exactly as expected when modifying the geometry. In this case the expression for N has got the wrong value in domain 6. In the Subdomain Expressions dialog box change the value of N in subdomain 6 to nCore.


1 On the Subdomain tab in the Mesh Parameters dialog box, set the Maximum element

size to 2e-6 for subdomains 4 and 5, and to 2.5e-7 for subdomain 6.



1 In the Solver Parameters dialog box, select the Eigenvalue solver.

2 Enter 1.46 in the text field Search for effective mode indices around. The default Desired number of effective mode indices is set to 6, which is fine in this case. These settings will make the eigenmode solver search for the 6 eigenmodes with effective mode indices closest to the value 1.46. This value is an estimate of the effective mode index for the fundamental mode. For propagating modes it must hold that

neff ncore< 1.456=


222 | C H A P T E

3 In the Solver Manager dialog box, select only the Perpendicular Hybrid-Mode Waves application mode on the Solve For tab. This will ensure that only this application mode will not be part of the eigenmode computation.

4 On the Initial Value tab, set the initial value to Current solution. This will take the plane strain solution and use it to evaluate the refractive indices for the mode analysis.


The effective mode indices which are found are listed in the table below together with the effective mode indices obtained from the analysis in the previous model without the generalization of the plane strain equations.

There is a systematic shift in the propagation constants when the strain in the z direction is taken into account.

GENERALIZED PLANE STRAIN

PLANE STRAIN DIFFERENCE


*1e-3

neff1 1.451216 1.451136 -.08

neff2 1.450928 1.450872 -.06

neff3 1.445487 1.445405 -.08

neff4 1.445213 1.445139 -.07

neff5 1.445195 1.445132 -.06

neff6 1.444923 1.444865 -.06


S e c ond Ha rmon i c Gen e r a t i o n o f a Gau s s i a n Beam

Introduction

Laser systems are an important application area in modern electronics. There are several ways to generate a laser beam, but they all have one thing in common: the wave length is determined by the stimulated emission, which depends on material parameters. It is especially difficult to find lasers that generate short wave lengths (e.g. ultraviolet light). With nonlinear materials it is possible to generate harmonics that are an even multiple of the frequency of the laser light. With such materials it is straightforward to generate a laser beam with half the wave length of the original beam. This model shows how a second harmonic generation can be set up as a transient wave simulation, using nonlinear material properties. A YAG (λ=1.06 µm) laser beam is focused on a nonlinear crystal, so that the waist of the beam is inside the crystal.

Model Definition

When a laser beam propagates it travels as an approximate plane wave, with a Gaussian shape of the cross-section intensity. The waist of the beam is defined as the characteristic length of the Gaussian function. A focused laser beam has a minimum waist, w0, at a certain point along the direction of propagation. The solution of the time-dependent Maxwell’s equations gives the following Electric field (x-component):

,

where the following expressions have been defined,

.

E x y z, ,( ) E0 x,w0

w z( )------------e

r2

w z( )2--------------–

ωt η z( )– r2 k2R z( )----------------–⎝ ⎠

⎛ ⎞ excos=

w z( ) w0 1 zz0-----⎝ ⎠

⎛ ⎞ 2+=

η z( ) zz0-----⎝ ⎠

⎛ ⎞atan=

R z( ) z 1z0z-----⎝ ⎠

⎛ ⎞2

+⎝ ⎠⎛ ⎞=

S E C O N D H A R M O N I C G E N E R A T I O N O F A G A U S S I A N B E A M | 223

224 | C H A P T E

In these expressions, w0 is the minimum waist, ω is the angular frequency, r is the radial cylindrical coordinate, and k is the wave number. The wave fronts of the beam is not exactly planar, it propagates like a spherical wave with radius R(z). Close to the point of the minimum waist the wave is almost plane. These expressions are used to excite a plane wave at one end of a cylinder.

In this example the problem is solved as a 2D cross section that resembles the axial symmetry of the problem. However, due to the x-polarization of the E-field, true axial symmetry cannot be used (available as an application mode in COMSOL Multiphysics). Here the the ϕ-direction is approximates with a standard 2D cross section, neglecting the revolution around the z-axis. So this actually models an infinitely long Gaussian beam plane. The second harmonics generation is probably not affected significantly by this approximation. In the model “Propagation of a 3D Gaussian Beam Laser Pulse” on page 236, propagation of a true 3D Gaussian beam is calculated without any nonlinear effects.

The nonlinear properties for 2nd harmonic generation in a material can be defined with the following matrix,

,

where P is the polarization. The model only uses the d11 parameter for simplicity. To keep the problem size small the nonlinear parameter is magnified by some orders of magnitude. The crystal here has a value of 1e-17 F/V is used, when the values for most

w0z

P

d11 d12 d13 d14 d15 d16

d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

Ex2

Ey2

Ez2

2EzEy

2EzEx

2ExEy

⋅=


materials usually are in the 1e-22 F/V range. Without this magnification, a detectable 2nd harmonics requires the z-direction of the geometry to be much longer, resulting in a large problem.

The Gaussian beam is also excited as a pulse in time, using a Gaussian envelope function. This produces a wave package with a Gaussian frequency spectrum. The frequency spectrum of a series of time samples can be calculated using the Fast Fourier Transform (FFT), which can be done at the COMSOL Script command line using the built in function fft.


The main purpose of this simulation is to calculate the second harmonic generation when the pulse travels along the 20 µm geometry. So you have to solve for the time it takes for the pulse to enter, pass, and disappear from the volume. The pulse has a characteristic time of 10 fs, and below you can see the pulse after it has traveled 61 fs.

Figure 5-8: The pulse after 61 fs.


226 | C H A P T E

After 90 fs the pulse has reached the output boundary, see Figure 5-9. The simulation have to continue for another 30 fs so the pulse completely disappears.

Figure 5-9: The pulse after 90 fs. It has now reached the output boundary.

The simulation stores the times between 60 fs and 120 fs, which is when the pulse passes the output boundary. The electric field at this boundary has a second harmonic component that is extracted using FFT. The extraction and transformation is performed at the COMSOL Script command line. The result is shown in Figure 5-10 below. There are two groups of peaks in the figure, where the peaks on the right are


mirrors of the ones to the left. This mirror effect is due to the sampling of the harmonic signals, which always create a mirror spectrum around the sampling frequency.

Figure 5-10: The result from the FFT on the beam at the output boundary. The two small peaks between the large peaks are the second harmonic generation.

References

[1] A. Yariv, Quantum Electronics, 3rd Edition, John Wiley & Sons, 1988

Model Library Electromagnetics_Module/Optics_and_Photonics/second_harmonic_generation




2 Click the Multiphysics button, and then Click the Add Geometry button.

3 In the Add Geometry dialog box, type z r phi in the Independent variables edit field. Click OK.


228 | C H A P T E

4 Select the Electromagnetics Module>In-Plane TE Waves>Transient propagation application mode.

5 Type Aphi in the Dependent variables edit field, and click Add.

6 Click OK.



2 In the Constants dialog box, define the following constants with names, expressions, and descriptions (optional):

3 Click OK.


All the dialog boxes for specifying the primitive objects are accessed from the Draw menu and Specify Objects. The first column in the table below contains the labels of the geometric objects. These are automatically generated by COMSOL Multiphysics, and you do not have to enter them. Just check that you get the correct label for the objects that you create.

1 Draw a rectangle with the properties according to the table below.





w0 2e-6 Minimum waist of laser beam (m)

lambda0 1.06e-6 Wave length of laser beam (m)

E0 3e4 Peak electric field (V/m)

z0 pi*w0^2/lambda0 (m)

k0 2*pi/lambda0 Wave number (1/m)

t0 2.5e-14 Time delay of pulse (s)

dt 1e-14 Characteristic time of pulse (s)

d11 1e-17 Matrix element for 2nd generation (F/V)

LABEL WIDTH HEIGHT BASE CORNER

R1 20e-6 6e-6 Corner (-1e-5,-6e-6)



3 Click OK.



6 Click OK.

7 From the Options menu, point to Integration Coupling Variable and then click Point

Variables.

8 In the Point Integration Variables dialog box, define the following variables with names and expressions. Use Global destination for all variables.


w w0*sqrt(1+(z/z0)^2) Analytical waist function along z (m)

eta atan(z/z0) Analytical angle along z (m)

R z*(1+(z0/z)^2) Analytical radius along z (m)

c0 1/sqrt(epsilon0_emwe*mu0_emwe) Speed of light (m/s)

omega0 2*pi*c0/lambda0 Angular frequency (rad/s)

BOUNDARY 1 2 3

E_bnd w0/w*exp(-r^2/w_bnd^2)* cos(omega0*t-k0*z+eta_bnd- r^2*k0/(2*R_bnd))

E_pulse exp(-(t-t0)^2/dt^2)

E_lim E0*1.2 -E0*1.2

POINT 2 ALL OTHER

w_bnd w

eta_bnd eta

R_bnd R


230 | C H A P T E

Boundary Conditions1 From the Physics menu, open the Boundary Settings dialog box and enter the settings

according the following two tables (leave all fields not specified at their default values):

2 Click OK.

Subdomain Settings1 From the Physics menu, open the Subdomain Settings dialog box, select subdomain

1, and click on the D=ε0εrE+Dr radio button for the constitutive relation.

2 Define the subdomain settings according to the following table:

BOUNDARY 1 4

Boundary condition Scattering boundary Scattering boundary

E0,ϕ E0*E_pulse*E_bnd 0

BOUNDARY 2 3

Boundary condition Perfect electric conductor Perfect magnetic conductor

SUBDOMAIN 1

εr 1

Dr d11*Ephi_emwe^2


3 Click OK.


1 From the Mesh menu, choose Map mesh.

2 Click on the Boundary tab, and select boundary 1.

3 Select the Contrained edge element distribution check box.


232 | C H A P T E

4 Click the Edge vertex distribution radio button. In the edit field below, type sin(linspace(0,pi/2,21)). This creates a denser mesh closer to the upper boundary.

5 Select boundary 4 and do the same operations there.

6 Select boundary 2, click the Edge vertex distribution radio button, and type 0:5e-8:20e-6 in the edit field.





2 In the Times edit field type 0 6.1e-14 9e-14.

3 Click on the Time Stepping tab, and select the Manual tuning of step size check box. Enter 1e-16 in the Initial time step and Maximum time step edit fields.

4 Click OK.

5 From the Solve menu, choose Solver Manager. Click the Output tab and select the Include time derivatives check box. This ensures that the time derivatives of the A-field, which is equal to the negative E-field, is calculated more accurately.

6 Click OK.

7 Click the Solve toolbar button.



2 Make sure that the Surface, Boundary, and Geometry edges check boxes are selected under the General tab.

3 Select time 6.1e-14 form the Solution at time list menu.

4 Click the Surface tab, and enter Ephi_emwe in the Expression edit field.

5 Click the Height Data tab, select the Hight data check box, and enter Ephi_emwe in the Expression edit field.

6 Click the Boundary tab, type E_lim in the Expression edit field.

7 Click the Height Data tab, select the Hight data check box, and enter E_lim in the Expression edit field.

8 Click the Uniform Color radio button.

9 Click the Color button and select white from the palette in the dialog box that appears. Click OK to close the dialog box.

10 Click Apply to get the plot in Figure 5-8.

11 Click on the General tab, and select time 9e-14. Click OK. You should now see the plot in Figure 5-9.

Transient Simulation for FFT Analysis

The amount of time steps that has to be saved for the FFT analysis is very large, and to store all solutions of the A-field would result in a huge model file size. For the FFT, it is only interesting to look at the field at the output boundary, so you only need to store


234 | C H A P T E

these values. In COMSOL Multiphysics you can use the ODE Settings to store a field value at a certain point.



2 In the Scalar Expressions dialog box, add the following variable:

3 Click OK.

4 From the Options menu, point to Integration Coupling Variable and then click Point

Variables.

5 In the Point Integration Variables dialog box, add the following variable, and use Global destination.

6 Click OK.


1 From the Physics menu, choose ODE Settings.

2 In the ODE Settings dialog box, define the following variable and equation:


Ephi_out Aphi_outt Output electric field for FFT (V/m)

POINT 4 ALL OTHER

Aphi_pnt Aphi

NAME (U) EQUATION DESCRIPTION

Aphi_out Aphi_out-Aphi_pnt Output A-field for FFT


3 Click OK.



2 In the Times edit field type 0:1e-16:1.2e-13.

3 Click OK.

4 From the Solve menu, choose Solver Manager. Click the Output tab and select the Aphi_out variable in the Output for variables area.

5 Click OK.


PO S T P R O C E S S I N G U S I N G C O M S O L S C R I P T

To get the Fourier transform of the output we need to execute some commands in COMSOL Script.

1 Export the FEM structure to the command line by pressing Ctrl-F. The COMSOL Script window opens automatically.

2 In the COMSOL Script window, enter the following commands:

T = 6e-14:1e-16:1.2e-13;Eout = postint(fem,'Ephi_out','edim',0,'dl',4,'T',T);freq = ((1:length(Eout))-1)/(length(Eout)*1e-16);xEout = fft(Eout);plot(freq,abs(xEout));

This produces the plot in Figure 5-10 on page 227.


236 | C H A P T E

P r opaga t i o n o f a 3D Gau s s i a n Beam La s e r Pu l s e

Introduction

This is a 3D version of the model “Second Harmonic Generation of a Gaussian Beam” on page 223. The differences are that the nonlinear material parameters have been removed, the geometry is smaller, and the laser pulse is shorter. This is necessary to reduce the execution time and the model file size shipped with the product.

Model Definition

All the details about Gaussian laser beams are covered in the model “Second Harmonic Generation of a Gaussian Beam” on page 223. This model uses the same YAG laser pulse, but with a characteristic time of 40 fs. Because the Gaussian pulse has spherical wave fronts when it approaches and leaves the minimum waist, the input and output boundaries are made spherical.

The model also use the symmetry of the cylindrical shape so that it is only necessary to simulate one quarter of the full volume. The perfect electric and perfect magnetic boundary conditions are used as symmetry conditions one these boundaries. It is not straightforward to use axial symmetry, because the polarization of the beam is in the x-direction.



The spherical fronts can be noted in the figures below. These figures shows the pulse that propagates at two different times, at the entry and just after passing the minimum waist.

Figure 5-11: The pulse entering the domain at t=10 fs. The spot size is shown on the input boundary to the left.

P R O P A G A T I O N O F A 3 D G A U S S I A N B E A M L A S E R P U L S E | 237

238 | C H A P T E

Figure 5-12: After 20 fs the pulse has passed the minimum waist and soon reaches the output boundary.

Model Library Electromagnetics_Module/Optics_and_Photonics/gaussian_beam_3d




2 Select the Electromagnetics Module>Electromagnetic Waves>Transient propagation application mode. Click OK.




2 In the Constants dialog box, define the following constants with names, expressions, and description (optional):

3 Click OK.


All the dialog boxes for specifying the primitive objects are accessed from the Draw menu and Specify Objects. The first column in the tables below contains the labels of the geometric objects. These are automatically generated by COMSOL Multiphysics, and you do not have to enter them. Just check that you get the correct label for the objects that you create.

Begin by drawing the cross section of the simulation domain followed by an extrusion to 3D. The input and output boundaries are then adjusted so they get a slight spherical shape, consistent with the spherical wave fronts of the Gaussian pulse.


2 In the Work Plane Settings dialog box, select the x-y plane and enter -2e-6 in the z edit field. Click OK.

Draw a circle with the properties according to the table below.

3 Draw rectangles with the properties according to the following table.



w0 2e-6 Minimum waist of laser beam (m)

lambda0 1.06e-6 Wave length of laser beam (m)

E0 3e4 Peak electric field (V/m)

z0 pi*w0^2/lambda0 (m)

k0 2*pi/lambda0 Wave number (1/m)

t0 1e-14 Time delay of pulse (s)

dt 4e-14 Characteristic time of pulse (s)

NAME RADIUS BASE (X,Y)

C1 6e-6 Center (0,0)

NAME WIDTH HEIGHT BASE CORNER

R1 6e-6 1.2e-5 Corner (-6e-6,-6e-6)

R2 6e-6 6e-6 Corner (0,-6e-6)


240 | C H A P T E

5 Select all the rectangles, R1 and R2, and click the Union toolbar button.

6 Click the Delete Interior Boundaries toolbar button.

7 Select all objects by pressing Ctrl-A, and click the Create Composite Object toolbar button.

8 In the dialog box that appears, enter C1-CO1 in the Set formula edit field. Click OK.

9 Select Extrude from the Draw menu.

10 In the Extrude dialog box, enter 4e-6 in the Distance edit field, and click OK.

11 It is desired to have the spherical wave fronts of the beam coincide with the input and output boundaries. Do this with two spheres with a radius from the value of the R expression evaluated at one of the boundaries. Specify the two spheres using the following properties:

12 Select all objects by pressing Ctrl-A, and click the Intersection toolbar button.

13 Click the Headlight toolbar button. You should now see the geometry shown in the figure below.


1 Go to geometry Geom1 by clicking the Geom1 tab.

NAME RADIUS BASE CORNER

SPH1 7.22713e-5 Center (7.22713e-5-2e-6,0)

SPH2 7.22713e-5 Center (-7.22713e-5+2e-6,0)




4 Click OK.



7 Click OK.

8 From the Options menu, select Integration Coupling Variable>Point Variables.

9 In the Point Integration Variables dialog box, define the following variables with names and expressions. Use Global destination for all variables.


w w0*sqrt(1+(z/z0)^2) Analytical waist function along z

eta atan(z/z0) Analytical angle along z

R z*(1+(z0/z)^2) Analytical radius along z

r sqrt(x^2+y^2) Radial coordinate

c0 1/sqrt(epsilon0_emw*mu0_emw) Speed of light

omega0 2*pi*c0/lambda0 Angular frequency

BOUNDARY 3 ALL OTHER

E_bnd w0/w*exp(-r^2/w_bnd^2)* cos(omega0*t-k0*z+eta_bnd- r^2*k0/(2*R_bnd))

E_pulse exp(-(t-t0)^2/dt^2)

POINT 1 ALL OTHER

w_bnd w

eta_bnd eta

R_bnd R


242 | C H A P T E


according the following two tables (leave all fields not specified at their default values):

2 Click OK.

Subdomain SettingsUse the default values for the subdomain settings.



2 Enter 5e-8 in the Maximum element size edit field.

3 Click the Advanced tab.

4 Enter 0.12 in the x-direction scale factor and y-direction scale factor edit fields. This keeps the small mesh size in the z-direction, but stretch the elements in the x- and y-directions.



1 From the Solve menu, choose Solver Parameters.

2 In the Times edit field type 0:1e-14:3e-14.

3 Select the GMRES from the Linear system solver list.

4 Select Algebraic multigrid from the Preconditioner list.

5 Make sure that the Symmetric option is selected from the Matrix symmetry list.

6 Click on the Time Stepping tab, and select the Manual tuning of step size check box. Enter 1e-16 in the Initial time step and Maximum time step edit fields.

7 Click OK.

BOUNDARY 3 4,

Boundary condition Scattering boundary Scattering boundary

E0,x E0*E_pulse*E_bnd 0

E0,y 0 0

E0,z 0 0

BOUNDARY 2 1, 5

Boundary condition Perfect magnetic conductor Perfect electric conductor


8 From the Solve menu, choose Solver Manager. Click the Output tab and select the Include time derivatives check box. This ensures that the time derivatives of the A-field, which is the E-field, is calculated more accurately.

9 Click OK.


The solve process takes some time, because this is a large problem that takes about 300 time steps.


Plot the x-component of the E-field on the boundaries. Some boundaries are suppressed so it is possible to get a view into the simulation volume.


2 Make sure that the Boundary and Geometry edges check boxes are selected under the General tab.

3 Click the Boundary tab and enter Ex in the Expression edit field.

4 Click OK.


244 | C H A P T E

5 From the Options menu, select Suppress>Suppress Boundaries. In the Suppress

Boundaries dialog box, select boundaries 4 and 5. Click OK.

6 Compare the results at times 1e-14 and 2e-14 to see how the pulse propagates. These results are plotted in Figure 5-11 on page 237 and in Figure 5-12 on page 238.


6

E l e c t r i c a l C o m p o n e n t M o d e l s

This chapter contains models of electrical components where in most cases the static and quasi-static application modes in the Electromagnetics Module have been used.

245

246 | C H A P T E

A Tunab l e MEMS Capa c i t o r

Introduction

In an electrostatically tunable parallel plate capacitor, you can modify the distance between the two plates when the applied voltage changes. For tuning of the distance between the plates the capacitor includes a spring that attaches to one of the plates. For a given voltage between the plates, it is possible to compute the distance of the two plates, if you know the characteristics of the spring. This model includes an electrostatic simulation for a given distance. A postprocessing step then computes the capacitance.

The capacitor in this model is a typical component in various microelectromechanical systems (MEMS) for electromagnetic fields in the radio frequency range 300 MHz to 300 GHz.

Figure 6-1: The tunable MEMS capacitor consists of two metal plates. The distance between the plates is tuned via a spring that is connected to one of the plates.

Model Definition

To solve the problem, use the 3D Electrostatics application mode in the Electromagnetic Module. The capacitance is available directly as a variable for postprocessing.


The electric scalar potential V satisfies Poisson’s equation,

R 6 : E L E C T R I C A L C O M P O N E N T M O D E L S

where ε0 is the permittivity of free space, εr is the relative permittivity and ρ is the space charge density. The electric field and displacement are obtained from the gradient of V.

Figure 6-2: The computational domain. The interior of the capacitor plates and connecting bars is not included.


Potential boundary conditions are applied to the capacitor plates and bars. For the upper plate and connecting bars a port condition is setting the potential to 1V whereas the lower plate is kept at ground potential. For the surface of the surrounding box, apply conditions corresponding to zero surface charge at the boundary,


Figure 6-3 shows the computed electric field and potential distribution in the capacitor. The potential on each capacitor plate is constant, as dictated by the boundary condition. Right between the plates, the electric field is almost uniform, but

∇ ε0εr V∇( )⋅– ρ=

E V∇–=

D ε0εrE=

n D⋅ 0=

A TU N A B L E M E M S C A P A C I T O R | 247

248 | C H A P T E

close to the edges of the plates, you can see fringing fields that change magnitude and direction rapidly in space.

Figure 6-3: The electric potential is shown as a surface plot while the cones indicate the strength and orientation of the electric field.

The capacitance, C, from the simulation is approximately 0.09 pF.

Model Library path: Electromagnetics_Module/Electrical_Components/capacitor


Select 3D, Electromagnetics Module, Statics, Electrostatics mode in the Model Navigator.


1 In the Axes/Grid Settings dialog box, clear the Axis auto check box and give the axis settings according to the following table.

AXIS MIN MAX

x -50 300



The geometry of this model can be created either by drawing in a work plane and then extruding the 2D geometry, or by using 3D primitives. The advantage with the latter approach is that the coordinates of the blocks can be given explicitly.

1 Draw solid blocks according to the following table.

y -50 300

z -25 75

NAME LENGTH AXIS BASE POINT

X Y Z X Y Z

BLK1 22 60 8 0 240 46

BLK2 40 22 8 22 259 46

BLK3 176 262 8 62 19 46

BLK4 40 22 8 238 259 46

BKL5 22 60 8 278 240 46

BLK6 40 22 8 238 19 46

BLK7 22 60 8 278 0 46

BLK8 40 22 8 22 19 46

BLK9 22 229 8 0 41 0

BLK10 40 22 8 -40 139 0

AXIS MIN MAX


250 | C H A P T E

2 Draw a solid cylinder with radius 5.5 and height 38, with the axis parallel to the z axis, and the base point of the axis located at (11, 250, 8).

3 Select all geometry objects using the Select All command from the Edit menu. Then open the Create Composite Object dialog box. Be sure to clear the Keep interior

boundaries check box, and verify that all objects are selected. Then click OK to create a composite object as the union of all objects. The advantage with removing the interior boundaries in this geometry is that it makes it easier to create a high quality mesh later on.

4 Now, draw two more solid block objects according to the following table.

5 Using the Shift key, select both block objects and then click the Union button to form another composite object. The interior boundaries of this object are not kept, because the settings in the Create Composite Object dialog box are applied for all set operations until modified again.


X Y Z X Y Z

BLK1 176 262 8 62 19 8

BLK2 181 22 8 139 139 0


6 Now draw a block surrounding the two objects corresponding to the two plates in the capacitor. Create a block of size 360-by-340-by-94, with the base point of the axis at (-40, -20, -20).

7 Once again open the Create Composite Object dialog box, and now enter the formula BLK1-(CO1+CO2) in the Set formula edit field.

8 The geometry of the model is now ready, except for the size. The capacitor dimensions you have entered should be given in µm. This is easily corrected by clicking the Scale button, and giving the value 1e-6 for all three scaling factors. Then click the Zoom Extents button to be able to see the object.


Boundary ConditionsThe lower plate is grounded, and a positive potential is applied to the upper plate. The surrounding boundaries are kept electrically insulated. The following table shows the corresponding boundary settings.

BOUNDARY ALL OTHER 37-40 45 47-51 57 1-4 9 76

Boundary condition

Port Ground Zero charge/Symmetry

Port tab Use as inport


252 | C H A P T E

Subdomain SettingsBecause there is silica-glass between the plates, you need to modify the permittivity. Enter the following values for the subdomain settings:


1 Open the Mesh Parameters dialog box, and select Coarse from the predefined mesh sizes list on the General tab. Click the Advanced tab, and set the Resolution of narrow

regions parameter to 1.5 to help the mesh generator resolve the thin regions in the geometry.





The electric field distribution is used together with a plot of the potential to visualize the result of the simulation. You have access to the capacitance as a variable C11_emes that you can plot or display directly.

1 Start by suppressing some boundaries not to be used for visualization. Select Suppress Boundaries from the Options menu, and select boundary 1, 2, and 4. Click OK.

SUBDOMAIN 1

εr 4.2

ρ 0


2 In the Plot Parameters dialog box clear Slice, then select Boundary and Arrow as Plot

type. Use the default settings for both plot types.

3 Open the Data Display dialog box and enter C11_emes in the Expression edit field. This gives the capacitance.


% Start the modeling by clearing the fem variable in % the COMSOL Script or MATLAB workspace.clear fem

% Begin by specifying space dimensions and defining a geometry.fem.sdim = 'x','y','z';

blks1 = block3(22,60,8,'corner',[0 240 46]);blks2 = block3(40,22,8,'corner',[22 259 46]);blks3 = block3(176,262,8,'corner',[62 19 46]);blks4 = block3(40,22,8,'corner',[238 259 46]);blks5 = block3(22,60,8,'corner',[278 240 46]);blks6 = block3(40,22,8,'corner',[238 19 46]);blks7 = block3(22,60,8,'corner',[278 0 46]);blks8 = block3(40,22,8,'corner',[22 19 46]);blks9 = block3(22,218,8,'corner',[0 41 0]);blks10 = block3(40,22,8,'corner',[-40 139 0]);cyl1 = cylinder3(5.5,38,[11 250 8]);co1 = geomcomp(blks: cyl1,'face','all','edge','all');


254 | C H A P T E

blk1 = block3(176,262,8,'corner',[62 19 8]);blk2 = block3(181,22,8,'corner',[139 139 0]);co2 = blk1 + blk2;

blk3 = block3(360,340,94,'corner',[-40 -20 -20]);co3 = geomcomp(co1 co2 blk3,'ns','CO1','CO2','BLK1',... 'sf','BLK1-(CO1 + CO2)','face','all','edge','all');

fem.geom = scale(co3,1e-6,1e-6,1e-6,0,0,0);

% The application mode structure contains all the physical% properties of the model. Create this with the following % commands:clear appl

appl.mode = 'EmElectrostatics';

appl.equ.rho0 = '0';appl.equ.epsilonr = '4.2';

appl.bnd.inport = 0,1,0;appl.bnd.type = 'nD0','port','V0';

appl.bnd.ind = ones(1,76)*2;appl.bnd.ind(1,[1:4 9 76]) = 1;appl.bnd.ind(1,[37:40 45 47:51 57]) = 3;

% Generate the coefficients in the FEM structure.fem.appl = appl;fem = multiphysics(fem);

% Create a mesh for the geometry with the following commands. The % extended mesh structure contains all the necessary data % defining the different elements.fem.mesh = meshinit(fem,... 'Hcurve', 0.5,... 'Hcutoff',0.02,... 'Hgrad', 1.5,... 'Hmaxfact',1.6,... 'Hnarrow',1.5);

fem.xmesh = meshextend(fem);

% Solve the problem and visualize the result.fem.sol = femiter(fem,... 'report', 'on',... 'prefun', 'amg');

bdlList = setdiff(1:76,[1 2 4]);postplot(fem,...


'bdl', bdlList,... 'tridata','V','cont','on',... 'tribar', 'on',... 'arrowdata','Ex','Ey','Ez',... 'axisequal','on',... 'axisvisible','off')

% The capacitance value:I1 = postinterp(fem,'C11',[0;0;0]);


256 | C H A P T E

I n du c t i o n Cu r r e n t s f r om C i r c u l a r C o i l s

Introduction

A time-varying current induces a varying magnetic field. This field induces currents in neighboring conductors. The induced currents are called eddy currents. The following model illustrates this phenomenon by a time-harmonic field simulation as well as a transient analysis, which provides a study of the eddy currents resulting from switching on the source.

Two current-carrying coils are placed above a copper plate. They are surrounded by air, and there is a small air gap between the coils and the metal plate. A potential difference provides the external source. To obtain the total current density in the coils you must take the induced currents into account. The time-harmonic case shows the skin effect, that is, that the current density is high close to the surface and decreases rapidly inside the conductor.

Model Definition

E Q U A T I O N

To solve the problem in the Electromagnetics Module, use a quasi-static equation for the magnetic potential A.

σ A∂t∂

------- ∇ µ01– µr

1– ∇ A×( )×+ σVloop2πr-------------=


where µ0 is the permeability of vacuum, µr the relative permeability, σ the electrical conductivity, and Vloop is the voltage over one turn in the coil. In the time-harmonic case the equation reduces to

F O R C E S

It is also possible to compute the forces on the plate caused by the eddy currents. The following expression gives the Lorentz force on the plate:

When dealing with time-harmonic fields, you must take the complex nature of the field representation into special consideration. In the time domain, the force in the axial direction in this axisymmetric model is

where and are the current and magnetic flux in the frequency domain, obtained from the harmonic equation.

jωσA ∇ µ01– µr

1– ∇ A×( )×+ σVloop2πr-------------=

F J B×=

Fz JϕBr– Re J′ϕejωt( ) Re B′rejωt( )⋅–= =

J′ϕ B′r

I N D U C T I O N C U R R E N T S F R O M C I R C U L A R C O I L S | 257

258 | C H A P T E


The applied voltage is constant across each wire, but the induced current makes the current density higher toward the end facing the copper plate. Figure 6-4 below shows this and also displays the field lines from the magnetic flux.

Figure 6-4: The ϕ component of the current density plotted together with the field lines of the magnetic flux.


The eddy currents in the wires and in the copper plate give rise to a force between the parts. Figure 6-5 below shows a plot of the force on the plate over one period.

Figure 6-5: The force on the copper plate over one period. Although the solution only gives the amplitude and phase, these can easily be converted to a real signal in the time domain.

Model Library path: Electromagnetics_Module/Electrical_Components/coil_above_plate

Harmonic Analysis in the Graphical User Interface


1 From the Space dimension list, select Axial symmetry (2D).

2 Select the Electromagnetics Module>Quasi-Statics, Magnetic>Azimuthal Induction

Currents, Vector Potential>Time-harmonic analysis application mode.



0 1 2 3 4 5 6 7−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−6

angle (wt)

axia

l for

ce (

Fz)


260 | C H A P T E

2 Set axis and grid settings according to the following table.


1 Draw the surrounding rectangle R1 with opposite corners at (0,-0.05) and (0.1,0.05).

2 Draw the rectangle representing the metal plate R2 with the corners at (0,-0.02) and (0.08,0). Now change the position of R2 by pressing the Move button on the Draw toolbar or selecting it in the Draw menu under Modify. Set the z displacement to -0.001.

3 Draw a circle C1 representing the first current-carrying coil, centered at (0.0125,0.0025) with radius 0.0025.

4 Select C1 and click the Array button. Set the Displacement in the r direction to 0.006 and the Array size in the r direction to 2.

AXIS GRID

r min -0.05 r spacing 0.01

r max 0.15 Extra r 0.0125

z min -0.08 z spacing 0.01

z max 0.08 Extra z 0.0025 0.005



Scalar VariablesIn the Application Scalar Variables dialog box, set the frequency variable nu_emqa to 50.

Boundary ConditionsSet boundary conditions according to the following table.

Subdomain SettingsIn this model, use the COMSOL Multiphysics material library to specify material parameters in the subdomains. Only the conductivity, the permeability, and the permittivity are given by the selected material in the library.

1 Open the Material Library dialog box from the Options menu.

2 Select Copper from the material list in Library 1. Click Copy. Select Geom1 and click Paste to add copper to the model materials.

3 Open the Subdomain Settings dialog box and enter subdomain settings according to the following table.



2 Refine it once.




The magnetic potential is the default visualization quantity. In this case it is of interest to plot the eddy currents together with the magnetic flux density. These fields are used to examine the magnetic forces that arise in current carrying conductors.

1 In the Plot Parameters dialog box select Surface plot and Streamline plot.

2 Select Induced current density, phi component as Surface data.

BOUNDARY 1,3,5 2,7,9

Boundary condition Axial symmetry Magnetic insulation

SUBDOMAIN 1 2 3,4

material Copper Copper

Vloop 0 0 1e-4


262 | C H A P T E

3 Select Magnetic flux density as Streamline data.

4 For a better view of the current intensive region, use the Zoom Window button on the main toolbar. The current density is highest near the edge of the plate.

If you are running COMSOL Multiphysics with COMSOL Script or MATLAB, you can visualize the force over one period.

1 Select Export>FEM structure from the File menu.

2 Use the following commands to show the magnetic force on the plate over one period.

wt=linspace(0,2*pi,40);for i=1:length(wt) Fv(i)=postint(fem,'-2*pi*r*real(Jiphi_emqa)*real(Br_emqa)',... 'dl',2,'phase',wt(i));endfigureplot(wt,Fv)


xlabel('angle (wt)'),ylabel('axial force (Fz)')

Transient Analysis

The previous model was an example of harmonic field variations. To study a transient case where the current increases abruptly from zero to a constant value, you only need to make a few alterations. To adapt the application mode to a transient analysis, you must first change the analysis type.


1 Open the Solver Parameters dialog box and set the Analysis to Transient. Enter the vector 0 logspace(-4,-1,10) in the Times edit field.

2 Set the Absolute tolerance to 1e-8.



In a way similar to the harmonic analysis, you can visualize the magnetic flux density, the induced currents, and the magnetic forces.

1 Select Surface and Arrow plots.

0 1 2 3 4 5 6 7−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−6

angle (wt)

axia

l for

ce (

Fz)


264 | C H A P T E

2 Set the Arrow data to the magnetic flux density. Choose Normalized as Arrow length, set the Arrow positioning to 25 in both directions, and enter 0.5 as Scale factor.

3 Use the zoom features to see the eddy current distribution.

4 Change the evaluation time under the General tab in the Plot Parameters dialog box, or click Start Animation under the Animate tab, to see how the current distribution changes in time.


% Start modeling by defining the names of the cylindrical% coordinates and by defining the geometry of the structure.clear femfem.sdim='r' 'z';fem.geom=rect2(0,0.1,-0.05,0.05)+... rect2(0,0.08,-0.019,-0.001)+... circ2(0.0125,0.0025,0.0025)+... circ2(0.0185,0.0025,0.0025);

% Create the application structure.clear applappl.mode.class='AzimuthalCurrents';appl.mode.type='axi';appl.prop.analysis='harmonic';appl.var.nu=50;appl.bnd.type='ax' 'A0';appl.bnd.ind=[1 2 1 0 1 0 2 0 2 0 0 0 0 0 0 0 0];


appl.equ.sigma='0' '5.99e7' '5.99e7';appl.equ.Vloop='0' '0' '1e-4';appl.equ.ind=[1 2 3 3];fem.appl=appl;fem=multiphysics(fem);

% Create the mesh.fem.mesh=meshinit(fem);fem.mesh=meshrefine(fem);fem.xmesh = meshextend(fem);

% Solve and visualize the solution.fem.sol=femlin(fem);postplot(fem,'tridata','Jiphi',... 'arrowdata','Br' 'Bz','arrowscale',0.3,... 'arrowstyle','normal')

% You can handle the postprocessing in exactly the same way as % above, when solving the problem in the graphical user interface.wt=linspace(0,2*pi,40);for i=1:length(wt) Fv(i)=postint(fem,'-2*pi*r*real(Jiphi)*real(Br)','dl',2,... 'phase',wt(i));endfigureplot(wt,Fv)xlabel('angle (wt)'),ylabel('axial force (Fz)')

% To simulate the transient case, the analysis type has% to be modified.fem.appl.prop.analysis='transient';

% Generate the PDE coefficients for the transient analysis.fem=multiphysics(fem);

% Regenerate the extended mesh.fem.xmesh = meshextend(fem);

% To solve the problem, the solver femtime is invoked. You can % visualize the result with the same command as in the time % harmonic case. To study the result at a specific output time, % set the parameter solnum.fem.sol=femtime(fem,'tlist',[0 logspace(-4,-1,10)],'atol',1e-8);postplot(fem,'tridata','Jiphi',... 'arrowdata','Br' 'Bz','arrowscale',0.3,... 'arrowstyle','normal','solnum',6)

% The axial force is visualized with the following commands.F=postint(fem,'abs(2*pi*r*Jiphi*Br)',... 'dl',2,'solnum',1:length(fem.sol.tlist));


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loglog(fem.sol.tlist(:),F(:),'k');xlabel('time (t)');ylabel('axial force (F_z)');


Magne t i c F i e l d o f a He lmho l t z C o i l

Introduction

A Helmholtz coil is a parallel pair of identical circular coils spaced one radius apart and wound so that the current flows through both coils in the same direction. This winding results in a uniform magnetic field between the coils with the primary component parallel to the axis of the two coils. The uniform field is the result of the sum of the two field components parallel to the axis of the coils and the difference between the components perpendicular to the same axis.

The purpose of the devices is to allow scientists and engineers to perform experiments and tests that require a known ambient magnetic field. Helmholtz field generation can be static, time-varying DC or AC, depending on the applications.

Applications range from cancelling the earth’s magnetic field for certain experiments, via generating magnetic fields for applications such as determining magnetic shielding effectiveness and determining the susceptibility of electronic equipment to magnetic fields, to calibration of magnetometers and navigational equipment, and bio-magnetic studies.

Figure 6-6: The Helmholtz coil consists of two coaxial circular coils, one radius apart along the axial direction. The coils carry parallel currents of equal magnitude.

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Model Definition

The model is built using the 3D Quasi-statics time-harmonic application mode. The model geometry is shown in Figure 6-7.

Figure 6-7: The model geometry.


Assuming time-harmonic (AC) currents and fields, the magnetic vector potential A must satisfy the following equation:

where ω is the angular frequency, σ is the conductivity, and µ is the permeability. Je denotes the externally applied current density.

The relations between fields and potentials are given by

This model uses the following parameter values:

jωσA ∇ µ 1– ∇ A×( )×+ Je=

B ∇ A×=

H µ 1– B=

ν 50= (Hz)

µ 4π10 7–= (Vs/Am)


In order to avoid numerical instability, the conductivity is given a small value, which in this case is 1 S/m everywhere. The external current density is zero except in the circular coils, where an AC current density of 1 A/m2 (peak value) is specified. This corresponds to a coil current of 2.5 mA (peak value). The currents are specified to be parallel and in phase for the two coils.


The only boundary conditions that you need to specify is for the exterior boundary, that is, the spherical surface, where you apply conditions corresponding to zero magnetic flux:


Figure 6-8 shows the magnetic flux density between the coils. You can see that the flux is fairly uniform between the coils, except for the region close to the edges of the coil.

Figure 6-8: The surface color plot shows the magnetic flux density. The arrows indicate the magnetic field (H) strength and direction.

The main property of the Helmholtz coil is that the magnetic flux becomes uniform in a reasonably large region with a rather simple coil system. This is illustrated in Figure 6-9 (a), which shows a radial flux density profile for an axial position right

n A× 0=


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between the coils. Figure 6-9 (b) shows an axial magnetic flux density profile. The model clearly demonstrates the highly uniform magnetic field obtained in a Helmholtz coil.

In the absence of any test object, this model is fully axisymmetric and could be implemented as a 2D axisymmetric model, which would be much less computationally demanding. However, this full 3D model has the advantage that a non axisymmetric test object could be included in the analysis as a slight modification of the model.

Figure 6-9: The magnetic flux density profile. In the graph at the left, the profile is taken along a radial cross sectional line through the axis right between the coils. In the graph on the right, the profile is taken along the axis. The high degree of uniformity is clearly shown.

Model Library path: Electromagnetics_Module/Electrical_Components/helmholtz_coil


Select the 3D, Electromagnetics Module, Quasi-Statics mode in the Model Navigator.


Open the Application Mode Properties dialog box from the Physics menu. Set the Potentials property to Magnetic, the Default element type to Vector.

(a) (b)



In the Constants dialog box, enter the following names and expressions.


The coils will be made from squares in a 2D work plane, which are revolved into the 3D space.

1 Open the Work Plane Settings dialog box. Click OK to obtain the default work plane in the x-y plane.

2 In the 2D geometry set axis and grid settings according to the following table.

3 Draw a square with corners at (-0.425, 0.175) and (-0.375, 0.225).

4 Draw another square with corners at (-0.425, -0.225) and (-0.375, -0.175).

5 Select both squares and use the Revolve dialog box to revolve into the 3D geometry.

6 Finally add a sphere with radius 1 and center at the origin to the 3D geometry.


Boundary ConditionsUse the default boundary condition.

Subdomain SettingsApply the current in the coils as a given source current. Use a constant value of 1 S/m for the electrical conductivity both in air and the coils. This will neglect the induced current but guarantee numerical stability. A too small a value of the conductivity makes the problem numerically unstable.


J0 1 Current density in coil (A/m2)

AXIS GRID


x max -0.2 Extra x


y max 0.4 Extra y -0.225 0.225


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In the Subdomain Settings dialog box enter the electrical conductivity and source current according to the following table.


1 Open the Mesh Parameters dialog box and select Fine from the predefined mesh sizes list on the General tab. Click the Subdomain tab, select the subdomains 2 and 3 and set the Maximum element size to 0.05.



Solve using the default solver.


1 Suppress the surface of the sphere, that is, boundaries 1, 2, 3, 4, 21, 22, 31, and 32 using the Suppress Boundaries dialog box in the Options menu.

2 In the Plot Parameters dialog box select Slice, Boundary, and Arrow plot and clear Geometry edges.

3 On the Slice tab use the default Slice data which is magnetic flux density, norm. Set the number of x levels to 0 and the number of z levels to 1.

4 On the Boundary tab set the Boundary data to 1 and select a Uniform color of your choice.

5 On the Arrow tab select Magnetic field as Arrow data, set the x points to 24, the y points to 10, and the z points to 1. Enter 0.5 as Scale factor.

6 To add some lighting to the plot open the Visualization/Selection Settings dialog box and select Scenelight. Disable lights 1 and 3.

SUBDOMAIN 1 2, 3

σ 1 1

Je 0 0 0 -J0*z/sqrt(x^2+z^2) 0 J0*x/sqrt(x^2+z^2)


You can visualize the magnetic flux density profile using cross-section plots.

1 Open the Cross-Section Plot Parameters dialog box. Click the Title/Axis button and set the title to Magnetic flux density.

2 On the Line tab use the default y-axis data which is Magnetic flux density, norm. Set the x-axis data to x and set x0 to -0.8 and x1 to 0.8. Click Apply to make a plot of the magnetic flux density along a line through the axis right between the coils.


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To make a plot in the other direction change the x-axis data to y and set x0 to 0, x1 to 0, y0 to -0.8 and y1 to 0.8.


I n du c t a n c e i n a Co i l

Introduction

This is a model of a simple AC coil: a single-turn thick copper wire. A parametric study is performed of the current distribution in the coil at different frequencies. The skin effect becomes increasingly important at the higher frequencies.

Model Definition

The model is built using the Axisymmetric Quasi-statics Azimuthal Currents application mode. A time harmonic formulation is used. The modeling takes place in the r-z plane. Only the part where r > 0 is drawn, in a manner such that the full 3D geometry is retrieved by revolving this view around the z axis.


The dependent variable in this application mode is the azimuthal component of the magnetic vector potential A, which obeys the following relation:

where ω denotes the angular frequency, σ the conductivity, µ the permeability, ε the permittivity, and Vloop the voltage applied to the coil. Outside the coil, σ is set to zero.


The model includes boundary conditions for the exterior boundary and the symmetry axis. For the exterior boundary, use magnetic insulation. This implies that the magnetic vector potential is zero at the boundary, corresponding to a zero magnetic flux. For the symmetry boundary use a symmetry condition. The magnetic potential is continuous across the interior boundary between the copper wire and the surrounding air.

jωσ ω2ε–( )Aϕ ∇ µ 1– ∇ Aϕ×( )×+σVloop

2πr-----------------=

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Figure 6-10 and Figure 6-11 show the total current density in the coil, at frequencies of 10 Hz and 2 kHz respectively.

Figure 6-10: The current density at 10 Hz.

Figure 6-11: The current density at 2 kHz.


At 10 Hz, the current density is largely inversely proportional to the radial distance. At 2 kHz, the current is concentrated to the surface of the coil, due to the skin effect.

The inductance versus frequency is plotted against frequency in Figure 6-12. Two approaches have been used to calculate the inductance, the virtual work and Ohm’s law. The difference between the two approaches is less than 0.01 percent.

Figure 6-12: The inductance as a function of the frequency calculated using the virtual work method.

Model Library path: Electromagnetics_Module/Electrical_Components/inductance



3 On the New tab of the Model Navigator, select Axial symmetry 2D as Space dimension. Click on Electromagnetics Module, then select Quasi-Statics, Azimuthal Currents and finally Time-harmonic analysis.



Start your modeling session by adjusting the GUI main axes area to the size of the geometry to draw. You can also make your simulation easier by defining variables that you can use later when defining your problem.


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1 Select Axes/Grid Settings from the Options menu to open the Axes/Grid Settings dialog box.

2 On the Axis tab set r min to -0.25 and r max to 0.25. Set the same interval limits for the z axis.

3 Now select the Grid tab and then clear the Auto check box, to manually define new grid settings.

4 Set r spacing to 0.05 and give the value 0.03 in the Extra r text field. Set z spacing to 0.05, and Extra z to 0.01.

5 Click Apply to see how the new settings are invoked. Note that the r axis settings are adjusted to keep the correct aspect ratio.

6 To define global constants, select Constants from the Options menu. This will open the Constants dialog box. This model uses only one constant parameter, namely the voltage applied to the coil.

7 Give Name and Expression according to the table below, by first entering the constant name and then the expression in the edit fields.

8 The numerical value of the voltage that will be used in the model is now visible in the dialog box.



You can now define the geometry of the structure, using the CAD tools that are available in COMSOL Multiphysics.

1 Start by drawing the half circle representing the air domain. A half circle is conveniently created as the union of a circle and a rectangle. First draw the circle. This is done by selecting Draw Objects and then Ellipse/Circle (Centered) from the Draw menu or by pressing the Draw Centered Ellipse button on the draw toolbar to the left of the main axes area. With the right mouse button, click and hold at (0,0). Move the cursor to (0.2,0) and release the button. The circle is now created.

2 Click the Draw Rectangle button on the draw toolbar or select the corresponding entry in the Draw menu. Draw a rectangle with opposite corners at (0,-0.2) and (0.2,0.2), this time using the left mouse button.

3 Click the Create Composite Object button or select Create Composite Object in the Draw menu. In the dialog box that appears, mark C1 and R1 and click the Intersection


V0 1 Voltage applied to coil (V)


button, or type C1*R1 in the Set formula text field. Click OK to apply the changes and close the dialog box.

4 Continue by drawing the circle representing the coil. Select Draw Centered Ellipse again. With the right mouse button, click at (0.03,0) and move the cursor to (0.03,0.01), where you release the button.

5 Double-click GRID at the bottom of the window to hide the grid lines.

Figure 6-13: Geometry.


Scalar Variables1 Open the Application Scalar Variables dialog box by selecting Scalar Variables from the

Physics menu.

2 The solution should be calculated for a range of frequencies. Type freq in the Expression field for nu_emqa. Leave the permittivity and the permeability of vacuum at their default values.


Boundary Conditions1 Open the Boundary Settings dialog box by selecting Boundary Settings from the

Physics menu

2 Give the boundary conditions according to the following table.

BOUNDARY 1 2-3

Boundary condition Axial symmetry Magnetic insulation


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Boundary 1 is the vertical boundary along the z axis. The axial symmetry boundary condition makes sure that the solution is symmetric around this axis. The magnetic insulation condition sets the magnetic potential Aϕ to zero along the boundaries 2 and 3.

Subdomain Settings1 Select Subdomain Settings from the Physics menu.

2 The domain properties for this model are given in the following table. Use the default values for the properties that are not given here for subdomain 2, and for all properties for subdomain 1.


Coupling VariablesScalar coupling variables are used for integrating the current density and the magnetic energy, in order to calculate the inductance and resistance of the coil.

1 Select Integration Coupling Variables>Subdomain Variables from the Options menu to open the Subdomain Integration Variables dialog box.

2 Specify the variable names and expressions according to the following tables. Use Integration order 4 and Destination global.


Expression VariablesExpression variables are used for completing the inductance and resistance calculations.

1 Select Expressions> Scalar Expressions from the Options menu to open the Scalar Expressions dialog box.

SUBDOMAIN 2

σ 5.998e7

Vloop V0

SUBDOMAIN 1, 2

Wm 2*pi*r*Wmav_emqa

SUBDOMAIN 2

Itot Jphi_emqa


2 Specify the variable names and expressions according to the following table.



In this model, as in many others dealing with electromagnetic phenomena, the effects on the fields near interfaces between materials are of special interest. To get accurate results make sure you generate a mesh that is very fine in these areas. To achieve this, refine the mesh once.

1 Select Initialize Mesh in the Mesh menu or use the corresponding button on the main toolbar above the main axes area to generate a mesh.

2 Select Refine Mesh in the Mesh menu or use the corresponding button on the main toolbar.

3 To see the mesh in the interesting region better, select Zoom Window from the Options menu. You can now draw a rectangular window around the coil and the cylinder to get a better view of the mesh.

Figure 6-14: Mesh.

NAME EXPRESSION

L_w 4*Wm/abs(Itot)^2

L_e imag(V0/Itot)/(2*pi*freq)

Resist real(V0/Itot)


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1 Open the Solver Parameters dialog box by selecting Solver Parameters in the Solve

menu, or by pressing the Solver Parameters button. In the Solver list, select Parametric linear. Type freq in the Name of parameter text field, and linspace(10,2000,20) in the List of parameter values text field. This means that the solution is computed for 20 equally spaced values of freq, running from 10 through 2000.


3 Select Solve in the Solve menu, or click the Solve button, to calculate the solution.


After the solution is calculated, a surface plot is automatically shown for the magnetic flux density. The buttons on the plot toolbar are shortcuts to get plots of other types. Full control over the plot generation is provided in the Plot Parameters dialog box.

1 Select Plot Parameters from the Post menu to open the Plot Parameters dialog box.

2 Select Surface and Geometry edges as Plot type on the General tab.

3 On the Surface tab, set the Surface Expression to Total current density, phi component

(Jphi_emqa).

4 Click Apply to generate the plot.

5 By default, the plot is rendered for the last entry in the frequency list. To view another solution, select the desired frequency from the Parameter value list on the General tab. Click OK to generate the plot and close the dialog box.

The resistance and the inductance are calculated for all frequencies and can be visualized using point plots.

1 Open the Cross-Section Plot Parameters dialog box from the Postprocessing menu. On the General tab, click Point plot and mark all the solutions.

2 On the Point tab, type Resist in the Expression text field. The coordinates can be left at their default values. The value of the resistance is evaluated in the specified coordinates, but since Resist is a scalar variable, it does not depend on which


coordinates are being used, as long as they are inside the geometry. Click Apply to create a plot of the resistance as a function of the frequency.

Figure 6-15: Resistance (Ω) as a function of the frequency (Hz).

3 The calculated values of the inductance can be visualized the same way as the resistance. The relative difference between the results from the two calculation methods gives an indication of the errors introduced from the discretization and the finite geometry. Type abs((L_w-L_e)/L_w) in the Expression text field on the Point tab and click Apply to visualize this entity.


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I n t e g r a t e d S qua r e - S ha p ed S p i r a l I n du c t o r

This example presents a model of a micro-scale square inductor, used for LC bandpass filters in microelectromechanical systems (MEMS).

The modeling is carried out in two steps. Because this is a static simulation, it is possible to compute the electric and magnetic fields separately. The first step computes the currents in the inductor. You then introduce these currents as sources for the magnetic flux computations in a second step.

The purpose of the model is to calculate the self-inductance of the micro-inductor. Given the magnetic field, you can compute the self-inductance, L, from the relation

where Wm is the magnetic energy, and I is the current. This model includes a boundary condition that sets the current to 1. The self-inductance L is the L11 component of the inductance matrix and is available as a predefined variable (L11_emqa) for postprocessing.

Model Definition

The model geometry consists of the spiral-shaped inductor and the air surrounding it. The subdomain corresponding to air can be done arbitrary large depending on the position of the device. Figure 6-16 below shows the inductor and air subdomains used in the model. The outer dimensions of the model geometry are around 0.3 mm.

L2Wm

I2-------------=


Figure 6-16: Inductor geometry and the surrounding subdomain.

The model equations are the following:

In the equations above, σ denotes the electric conductivity, A the magnetic vector potential, V the electric scalar potential, Je the externally generated current density vector, µ0 the permittivity in vacuum, and µr the relative permeability.

The electric conductivity in the coil is set to 106 S/m and 1 S/m in air. The conductivity of air is arbitrarily set to a small value in order to avoid singularities in the solution, but the error becomes small as long as the value of the conductivity is small.

The constitutive relation is specified with the following expression:

where H denotes the magnetic field.

∇– σ∇V Je–( )⋅ 0=

∇ µ01– µr

1– ∇ A×( )× σ V∇+ Je=

B µ0µrH=

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(c)

The boundary conditions are of three different types corresponding to the three different boundary groups that you see in Figure 6-17 (a), (b), and (c) below.

Figure 6-17: Boundaries with the same type of boundary conditions.

The boundary conditions for the boundaries highlighted in Figure 6-17 (a) are magnetic insulating boundaries with a port boundary condition. For the boundaries in Figure 6-17 (b), both magnetic and electric insulation prevail. The last set of boundary conditions, Figure 6-17 (c), are magnetically insulating but set to a constant potential of 0 V (ground).

(a) (b)


Results

Figure 6-18 shows the electric potential in the inductor and the magnetic flux lines. The thickness of the flow lines represents the magnitude of the magnetic flux. As expected this flux is largest in the middle of the inductor.

Figure 6-18: Electric potential in the device and magnetic flux lines around the device.

The model gives a self-inductance of 7.21e-10 nH.

Model Library path: Electromagnetics_Module/Electrical_Components/inductor


1 Start COMSOL Multiphysics.

2 In the Model Navigator click the Multiphysics button.

3 Select the 3D, Electromagnetics Module, Statics, Magnetostatics mode.

4 Click the Add button.

5 Open the Application mode properties dialog box by clicking the button with this name.


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6 Set the property Potentials to Electric and magnetic.

7 Set the property Default element type to A-vector, V-Linear.

8 Click OK


Give the axis settings according to the following table.


1 Start the modeling of this geometry by creating a number of solid block objects that form the base of the inductor. Use the data for the blocks according to the table below.

AXIS MIN MAX

x -50 300

y -50 300

z -25 25


X Y Z X Y Z

BLK1 51 22 20 -20 99 0

BLK2 22 98 20 9 121 0

BLK3 232 22 20 9 219 0

BLK4 22 188 20 219 31 0

BKL5 192 22 20 49 9 0

BLK6 22 148 20 49 31 0

BLK7 152 22 20 49 179 0

BLK8 22 108 20 179 71 0

BLK9 112 22 20 89 49 0

BLK10 22 68 20 89 71 0

BLK11 62 22 20 89 139 0

BLK12 51 22 20 249 139 0


2 Open the Work Plane Settings dialog box and choose BLK12 face number 3 on the Face Parallel tab.

3 Give the axis and grid setting as in the following table.

AXIS GRID

x min -30 x spacing 5

x max 10 Extra x -27

y min -130 y spacing 5

y max 30 Extra y -98 22


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4 Click the Line button on the Draw toolbar, and click at (-27,-120), (-27,22), (0,22), (0,0), (-5,0), (-5,-98), (0,-98), and (0,-120). Finally right-click to create a composite solid object.

5 Open the Extrude dialog box and set the Distance to 22. Click OK to create the extruded object.

6 Select all geometry objects, by using the Select all option from the Edit menu. Then open the Create Composite Object dialog box. Be sure to clear the Keep interior

boundaries check box, and verify that all objects are selected. Then click OK to create the composite object as the union of all objects.

7 Now, draw a block that surrounds the inductor. Set the length of the sides to 320, 300, and 200, for the x, y, and z coordinates respectively. Set the axis base point to (-20, -25, -100).

8 Select both the block object and the composite object CO1 by using the Shift key.

9 Open the Create Composite Object dialog box. Select the Keep interior boundaries check box and click OK.

10 All distances in this model are given in µm, so the geometry should now be scaled with a factor of 10-6, in order to obtain the correct size. Do this by clicking the Scale


button, and then giving the value 1e-6 for all three scaling factors. After that, click the Zoom Extents button to be able to see the object.


Boundary ConditionsGive the boundary settings according to the following table.

The boundaries 5 and 76 are the two ends of the inductor.

Subdomain SettingsEnter the following values in the Subdomain Settings dialog box.

Setting the conductivity to zero in air would lead to a numerically singular problem. Therefore the conductivity is arbitrarily set to 1 S/m. Although not physically correct,

BOUNDARY 5 75,76 1, 2, 3, 4, 10

Electric Parameters Port Ground Electric insulation

Magnetic Parameters Magnetic insulation

Magnetic insulation

Magnetic insulation

Port tab Use port as inport

SUBDOMAIN 1 2

σ 1 1e6


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this value is much smaller than the electrical conductivity in the inductor, 106 S/m. The electric and magnetic fields are only marginally affected by the value of the conductivity in air as long as it is small.


1 Open the Mesh Parameters dialog box form the Mesh menu.

2 Change the Predefined mesh size to Coarser and click the Remesh button.

3 On the Subdomain tab, select subdomain 2 and type 1.8e-5 in the Maximum element

size field.


1 Open the Solver Manager dialog box and select the electric potential V as the only dependent variable on the Solve For tab. By doing this, you ensure that the first analysis only solves the Poisson’s equation for the electrostatic problem.

2 Then click the Solve button in the Main toolbar.

3 Open the Solver Manager dialog box. Because the next simulation computes the magnetic properties, change the variables to solve for to tAxAyAz and psi on the Solve For tab.

4 On the Initial Value tab set the Initial value to Current solution. This ensures that the solver uses the current solution to calculate the current source in the magnetostatic problem.

5 Now click either the Restart button or the Solve button to start the solution process. If you clicked the former button, you could have omitted the step to set the initial value in the Solver Parameters dialog box, because the initial solution is then automatically updated with the current solution.


Now add a visualization of the magnetic flux density lines to illustrate the results from the simulation.

1 Open the Suppress Boundaries dialog box from the Options menu. Select boundaries 1-4, 10, and 75.

2 Open the Plot Parameters dialog box and clear the Slice and Geometry edges check boxes, and select Streamline and Boundary as plot types.

3 On the Boundary tab set the Boundary expression to V.

4 Click the Streamline tab and choose the Magnetic flux density as Streamline data and set the Number of start points to 31.


5 Then change the Line type to Tube and click the Tube Radius button. Select the Radius

data check box and select Magnetic flux density, norm as Radius data. Set the Radius

scale factor to 0.7. Click OK to close the Tube Radius Parameters dialog box.

6 Click the Use expression to color streamlines radio button.

7 Click the Color Expression button and select Electric potential as Streamline color data. Clear the Color scale check box. Click OK to close the Streamline Color Expression dialog box.

8 Click the Advanced button and set the Maximum number of integration steps to 100. Click OK to close the Advanced Streamline Parameters dialog box.


10 For improved visualization click the Scenelight button.

To extract the self-inductance, select Display Data from the Postprocessing menu and then type L11_emqa in the Expression edit field, or:

1 Open the Domain Plot Parameters dialog box from the Postprocessing menu.

2 Click the Point tab and plot Inductance matrix, element 11.


% The inductor model can just as easily be created from COMSOL % Script or MATLAB, using the COMSOL Multiphysics functions.% Start by defining the geometry.


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clear femg = ... block3(51,22,20,'corner',[-20 99 0]),... block3(22,98,20,'corner',[9 121 0]),... block3(232,22,20,'corner',[9 219 0]),... block3(22,188,20,'corner',[219 31 0]),... block3(192,22,20,'corner',[49 9 0]),... block3(22,148,20,'corner',[49 31 0]),... block3(152,22,20,'corner',[49 179 0]),... block3(22,108,20,'corner',[179 71 0]),... block3(112,22,20,'corner',[89 49 0]),... block3(22,68,20,'corner',[89 71 0]),... block3(62,22,20,'corner',[89 139 0]),... block3(51,22,20,'corner',[249 139 0]),... ;

l2 = line2([-25 -25 0 0 -5 -5 0 0],[-120 22 22 0 0 -98 -98 -120]);

gend+1 = extrude(l2,'Distance',22,... 'Wrkpln',[249 249 600;139 139 139;0 351 0]);

co1 = geomcomp(g,'face','all','edge','all');

b1 = block3(320,300,200,'corner',[-20 -25 -100]);

co2 = geomcomp(co1 b1);

fem.geom = scale(co2,1e-6,1e-6,1e-6);

% The physical properties of the model is defined in the % application mode structure. Continue the modeling by creating % this structure. The commands to use are given below.clear appl

appl.mode='QuasiStatics';appl.prop.analysis='static';

appl.equ.sigma='1','1e6';

appl.bnd.inport='1',’0’,’0’;appl.bnd.eltype='port','V0','nJ0';appl.bnd.magtype='A0';appl.bnd.ind=zeros(1,76);appl.bnd.ind([5])=1;appl.bnd.ind([75 76])=2;appl.bnd.ind([1 2 3 4 10])=3;

fem.appl=appl;

fem=multiphysics(fem);


% Generate the mesh, by issuing the following command.fem.mesh=meshinit(fem, ... 'hmaxfact',1.9, ... 'hcutoff',0.03, ... 'hgrad',1.6, ... 'hcurve',0.6, ... 'hmaxsub',[2,1.8e-5]);

% As always, when working at the command prompt, you need to % generate the extended mesh structure, containing the element % data.fem.xmesh=meshextend(fem);

% You can now solve the electrostatic equation, to compute the % currents.fem.sol=femlin(fem,'solcomp','V');

% Follow this by solving the magnetostatic equations % together with the gauge fixing equation to obtain the % magnetic flux density around the coils.fem.sol=femiter(fem,... 'init',fem.sol.u,... 'solcomp','tAxAyAz','psi');

% Visualize the result of the simulation, with the following % properties to the plot function.nBdl = setdiff(1:76,[1:4 10 75]);postplot(fem,... 'bdl',nBdl,... 'cont', 'internal',... 'tridata','V',... 'tribar', 'on',... 'flowtubescale',0.3,... 'flowcolor','k',... 'flowdata','Bx','By','Bz',... 'flowlines',50,... 'flowback','off',... 'flowsteps',150,... 'scenelight','on',... 'axisequal','on',... 'axisvisible','off');

% The inductance can be calculated with the commandinductance=postinterp(fem,'L11',[0;0;0]);


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Bond i n g W i r e s t o a Ch i p

Electronics manufacturers use bonding wires to connect pads on an integrated circuit (IC) to the pins of the package that houses it. These bonding wires introduce relatively large inductances that can introduce crosstalk and signal distortion. This model calculates the inductance and the mutual inductance between several bonding wires.

Introduction

In today’s electrical systems, high-speed communications place high demands on the packages that allow the connection of an IC to a circuit board. The most common method is to use bonding wires, which connect the pads on a semiconductor chip to the pins of a package. These bonding wires introduce relatively large inductances that can result in crosstalk and signal distortion. This model investigates the inductance of single wires and the mutual inductance between several bonding wires. The model extracts these parameters and presents them in the form of a lumped-parameter matrix, from which you can determine an impedance value. In addition, the harmonic nature of the signals results in a complex-valued matrix. Although the model extracts admittance or impedance parameters, you can also convert them to an S-parameter matrix.

Model Definition

B O N D I N G - W I R E M O D E L

Each bonding wire carries a current that results from an applied current. One end of the wire connects to the package, and the other end connects to a small pad on a chip;


these pads cause a capacitive load (not included in this model). This example studies four wires at the corner of a package (Figure 6-19).

Figure 6-19: The model geometry, which studies the corner of an IC where four bonding wires connect a chip to a package.

The model tackles the problem using the Quasi-static application mode, where the solution variables are the electrostatic potential, V, and the magnetic vector potential, A. Each wire has a finite conductivity, which the model handles as a separate subdomain. Because this simulation investigates effects due to the wire, it excludes any effects from vias (vertical interconnections between layers in a multilayer circuit board) and pins in the package. As a result, the model applies the potential directly to the boundary on a side of the package close to the wire, whose other side ties to the chip. The chip is modeled as a grounded plate.

L U M P E D - P A R A M E T E R C A L C U L A T I O N

This model uses the port boundary condition with the fixed current density mode to model the terminals that connects each bonding wire. The fixed current density mode sets a fixed current density, so the total current gets equal to 1 A. The voltage is taken as the average over each port boundary.

B O N D I N G W I R E S T O A C H I P | 297

298 | C H A P T E

The extracted voltages represent a column in the impedance matrix, Z, which determines the relationship between the applied currents and the corresponding voltages with the formula

.

Thus, when I1 = 1 and all the other currents are zero, the V vector equals the first column of Z. You get the full matrix by setting each of the currents to one. With this matrix, you can calculate other types of parameter matrices. For more information on lumped parameter calculations in the Electromagnetics module, see “Lumped Parameters” on page 96 in the Electromagnetics Module User’s Guide.

To calculate the entire matrix, you must export the model’s entire FEM structure to the COMSOL Script command line. A script called cemlumped.m performs the calculation, which solves all the necessary port conditions to extract the parameter matrix.


The inductive effects clearly dominate in this model. At a frequency of 10 MHz, the inductance matrix is

(in nH)

where the model calculates L from the imaginary part of the impedance matrix, Z, using the relationship . The L matrix is symmetric, which is a property most lumped parameter matrices have.

It is always important to check the extracted Z matrix before analyzing the other parameters, especially the ratio between the real and complex parts, and the condition

V1

V2

V4

V4

Z11 Z12 Z13 Z14

Z21 Z22 Z23 Z24

Z31 Z32 Z33 Z34

Z41 Z42 Z43 Z44

I1

I2

I3

I4

⋅=

L

2.6 0.59 0.0057 0.00520.59 3.5 0.029 0.018

0.0056 0,029 3,6 0,940,0052 0,018 0,94 3,5

=

Z( )imag ωL=


number of the matrix. Furthermore, the boundary conditions create mirror wires outside the simulation domain, and they influence the extracted parameters.

Figure 6-20: This plot shows a slice plot, an arrow plot of the magnetic flux density, and a boundary plot of the current density. For this solution, the third bonding wire is used as input, giving the third column in the parameter matrix.

Model Library path: Electromagnetics_Module/Electrical_Components/bond_wires_to_chip




2 In the Electromagnetics Module>Quasi-Statics, Electromagnetic folder, select the Electric and Induction Currents application mode.

3 Click OK.


300 | C H A P T E



2 In the Constants dialog box, define the following constants with names, expressions, and descriptions (optional):

3 Click OK.


1 From the File menu, select Import>CAD Data From File.

2 In the Import CAD Data From File dialog box, make sure that the COMSOL native

format or All 3D CAD files is selected in the Files of type list.

3 Locate the bond_wires_to_chip_geom.mphbin file, and click Import.



1 From the Physics menu, choose Scalar Variables.

2 In the Scalar Variables dialog box, change the value of the frequency, nu_emqav, to 10e6.

3 Click OK.

Boundary Conditions1 From the Physics menu, open the Boundary Settings dialog box.

2 Select all boundaries, click the Electrical Parameters tab, and select the Electric

insulation boundary condition from the Boundary Condition list.

3 Select the Interior boundaries check box to allow settings on interior boundaries.

4 Enter the other boundary settings according the following two tables (leave all fields not specified at their default values):


epsilonr_pcb 2.5 Relative permittivity of package material

sigma_pcb 10 Conductivity of package material (S/m)

BOUNDARY 30 42

Electric boundary condition Port Port

Port number 1 2


5 Click OK.

Subdomain Settings1 From the Physics menu, open the Subdomain Settings dialog box.

2 Define the subdomain settings according to the following table:

Use port as input cleared cleared

Input property Fixed current density

Fixed current density

BOUNDARY 81 80 9, 33, 45, 50, 52

Electric boundary condition Port Port Ground

Port number 3 4

Use port as input selected cleared


Fixed current density

SUBDOMAIN 1, 4, 5 2 6-13

Material Silicon Aluminum

εr epsilonr_pcb

σ sigma_pcb

BOUNDARY 30 42


302 | C H A P T E

3 Click OK.



2 In the Element growth rate edit field, type 1.4.


4 Select Suppress>Suppress Boundaries from the Options menu. Select boundaries 7, 8, 10, 24, and 79. Click OK. Click the Mesh mode button to see the following figure.



2 Select the UMFPACK solver from the Linear solver list.

3 Click OK.

4 Press the Solve toolbar button.



2 Make sure that the Slice, Boundary, Arrow, and Geometry edges check boxes are selected under the General tab.

3 Click on the Slice tab, and select Magnetic flux density, norm from the Predefined

quantities list.


4 Type 0 in the x levels edit field. Click the Vector with coordinates radio button for the y levels. Enter 4e-3 in the corresponding edit field.

5 Click on the Boundary tab, and type nJ_emqav*(nJ_emqav<5e6)+

5e6*(nJ_emqav>=5e6) in the Expression edit field.

6 Click the Range button. Deselect the Auto check box and enter 0 and 5e6 in the Min and Max edit fields. Click OK.

7 Click the Arrow tab, and select Magnetic flux density from the Predefined quantities list.

8 Click the Vector with coordinates radio button for the y points. Type the coordinate 3.9e-3 in the edit field next to that radio button.

9 In the x points and z points edit fields for the Number of points, type 40.

10 Select the 3D arrow in the Arrow type list.

11 Press the Color button, and select the white color from the palette. Click OK.

12 Click OK in the Plot Parameters dialog box to get the plot in Figure 6-20 on page 299.


P A R A M E T E R - M A T R I X C A L C U L A T I O N

The cemlumped script goes through all the port boundary conditions, setting the Use port as input property on one port at a time and solves the problem. To execute this script, you first need to export the FEM structure to the COMSOL Script command line. Do the following steps to get the full Z lumped parameter matrix. Note that this requires a 64-bit platform and that the calculation of the matrix takes about 2 hours.

1 Export the FEM structure by pressing Ctrl-F. The COMSOL Script window opens automatically.

2 Enter the following at the command prompt:

Z = cemlumped(fem,’Z’);freq = postint(fem,’nu_emqav’,’edim’,0,’dl’,1);L = imag(Z)/(2*pi*freq)*1e9

L =

2.636013 0.586400 0.005667 0.005164 0.586400 3.467757 0.028766 0.017782 0.005667 0.028766 3.640835 0.942190 0.005164 0.017782 0.942190 3.476300


304 | C H A P T E

This is the inductance matrix given in nH presented in “Results and Discussion” on page 298.


7

G e n e r a l I n d u s t r i a l A p p l i c a t i o n M o d e l s

In this chapter, you will find models of general industrial applications which do not fit into any of the other chapters. Typically, these models deal with eddy currents, electric impedance and other static or low to intermediate frequency applications.

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Eddy Cu r r e n t s i n 3D

Introduction

Induced eddy currents and associated thermal loads is of interest in many high power AC applications. This example is of general nature and illustrates some of the involved physics as well as suitable modeling techniques in the Electromagnetics Module.

Model Definition

A metallic plate is placed near a 50 Hz AC conductor. The resulting eddy current distribution in the plate depends on the conductivity and permeability of the plate. Four different materials will be considered: copper, aluminum, stainless steel, and magnetic iron.The model consists of a single wire and a plate with dimensions as shown below.

1 0 0 m m

6 0 m m

2 4 k A , 5 0 H z

1 0 0 m m

1 0 m m

R 7 : G E N E R A L I N D U S T R I A L A P P L I C A T I O N M O D E L S

As you cannot afford meshing an infinite volume, it is necessary to specify a finite volume to mesh and solve for. In this case, it makes sense to enclose the wire and the plate in a cylinder with the wire on the axis of this cylinder.

The conductor is modeled as a line current with 0 degrees phase and an effective value of 24 kA.

In the subdomains the magnetic vector potential is calculated from,

where σ is the conductivity, ε the permittivity, and µ the permeability.

An important parameter in eddy current modeling is the skin depth, δ.

The table below lists the skin depth for the different materials at a frequency of 50 Hz.

MATERIAL REL. PERMEABILITY CONDUCTIVITY SKIN DEPTH

copper 1 5.998e7 9 mm

aluminum 1 3.774e7 12 mm

stainless steel

1 1.137e6 67 mm

iron 4000 1.12e7 0.34 mm

jωσ ω2ε–( )A ∇ 1µ---∇ A×⎝ ⎠

⎛ ⎞×+ 0=

δ 2ωµσ-----------=

E D D Y C U R R E N T S I N 3 D | 307

308 | C H A P T E

The combination of a high permeability and a high conductivity can make it impossible to resolve the skin depth when modeling. When the skin depth is small in comparison to the size of conducting objects, the interior of those may be excluded from the model and replaced with an impedance boundary condition accounting for induced surface currents. This condition uses the following relation between the magnetic and electric field at the boundary:

.

The distribution of the dissipated power, Pd, has the unit W/m2, and can be calculated from

where JS is the induced surface current density, and the asterisk denotes the complex conjugate.


The induced eddy current distribution for a plate made of copper is shown as streamlines, whereas the distribution of the ohmic losses is shown as a slice plot.

n H× ε jσ ω⁄–µ

----------------------n E n×( )×+ 0=

Pd12--- Js E*⋅( )=


A total dissipated power of 6 W was obtained from integration through the plate. By repeating the simulation for different materials, the model shows that lowering the conductivity decreases the dissipated power. However, for high permeability materials like soft iron, the dissipated power is higher than in a copper despite a much lower conductivity.

Model Library path: Electromagnetics_Module/General_Industrial_Applications/eddy_currents_3D


Create a new 3D model using the application mode, Electromagnetics Module>

Quasi-Statics, Magnetic>Induction Currents.


The entire geometry is drawn in mm for simplicity. The final step in the geometry creation, is a conversion of all dimensions to meters, since COMSOL’s physical constants are given in SI units. Use the toolbar buttons and the Draw menu.

2D Cross SectionThe geometry is created from a 2D cross section, that lies in the yz-plane of the final geometry.

1 Begin by specifying a yz work plane at x=0 from the Draw menu.

2 In the 2D work plane, draw a circle with radius=250 mm.

3 Add a point at the center of the circle.

4 Draw a 60 mm tall and 100 mm wide rectangle that is centered at x=0, y=100 mm.

Extruding to 3DCreate the 3D geometry from the cross-section.

1 Select both the circle and the point and Extrude (from the Draw menu) these objects a distance of 300 mm.

2 Go back to the 2D view and Extrude the rectangle by 10 mm.

3 In the 3D view, select the resulting box and Move it by 145 mm in the x direction.

The next step is to convert the length scale to meters. Select all objects and Scale them with the factor 1E-3 in all directions.


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Open the Application Scalar Variables dialog from the Physics menu, and make sure that the Frequency is set to 50 Hz.

Subdomain SettingsOpen the Subdomain Settings dialog box and select the plate. Click the Electric

Parameters tab and Load the Library material copper. For the air domain, enter a small conductivity of 100 S/m to improve convergence for the iterative solver we are going to use. Any parameters not specified can be left at their default values.

Boundary SettingsThe Boundary Settings can be left at the default value Magnetic insulation.

Edge SettingsOpen the Edge Settings dialog box and specify a line current of 24e3*sqrt(2) A for the conductor (edge 6).


The default settings for the mesh are not adequate to resolve the solution properly, so you have to change some parameters in the Mesh Parameters dialog box. Click the Global tab in that dialog box and set the Predefined mesh sizes list to Extremely coarse. Enter 0.05 in the Maximum element size edit field. Then click the Subdomain tab and select the plate subdomain. Enter 2.5e-3 in the Maximum element size edit field.

These settings force a fine mesh in the plate but allow for a quick growth of the element size outside of the plate. In order not to get a too coarse mesh in the air, the settings limit the element size to 5 cm. Click the Remesh button.


1 Open the Solver Parameters dialog box and set the Linear system solver to GMRES and the Preconditioner to SSOR vector.

2 Click the Settings button and increase the Number of Iterations before restart to 150.

3 Change the Matrix symmetry setting to Nonsymmetric.

4 Click the Solve button to solve the model. This takes a few minutes.


The default plot is not what you want so open the Plot Parameters dialog box. Create a slice plot of the Resistive heating (Qav_emqa) at x=0.1451 only, and do a streamline


plot of the Total current density. Change the Number of start points to 50. In order to plot the streamlines only in the plate, Suppress the air domain from the Options menu.

The final step is to calculate the total power dissipated in the plate by the induced eddy currents. This is done by going to the Postprocessing menu and select Subdomain

integration. In the dialog that appears, select the plate subdomain and from the list of Predefined quantities select Resistive heating (Qav_emqa) before clicking OK. The result should be close to 6 W.

Change to Aluminum and Stainless Steel

Repeating the analysis for aluminum and stainless steel is simple — just change the material properties for the plate subdomain and solve again. Aluminum is available from the material library whereas you have to enter stainless steel by typing the corresponding conductivity 1.137e6 (S/m) value in the dialog box.

Change to Magnetic Iron

For magnetic iron, the skin depth is too small to be resolved with reasonable computational effort. The solution is to use the impedance boundary condition.


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Subdomain SettingsOpen the Subdomain Settings dialog and select the plate. Clear the Active in this domain check box. This tells the solver not to include this domain in the solution process and also makes the impedance boundary condition available on the boundaries of this domain.

Boundary SettingsSelect the Impedance boundary condition on the boundaries of the plate. Click the Material Properties tab and click the Load button to load iron from the material library.


For this analysis, you also need to define a postprocessing variable for the resistive losses on the boundaries with the impedance boundary condition. Go to the Options menu and select Expressions>Boundary Expressions and define an expression Qav_bnd according to the table below. Only enter the expression for the boundaries given in the table..


Use the same settings as before. Click the Solve button to solve the model. This will take a few minutes.

NAME BOUNDARY EXPRESSION DESCRIPTION

Qav_bnd plate boundaries

0.5*real(Jsx_emqa

*conj(Ex_emqa)+ Jsy_emqa

*conj(Ey_emqa)+ Jsz_emqa*

conj(Ez_emqa))

Surface power density of the plate



The default plot may now give an error message as the plotted quantity no longer is defined in the plate domain. Open the Plot Parameters dialog box. Create a boundary plot of the resistive heating (Qav_bnd).

The final step is to calculate the total power dissipated in the plate. To do this, perform a Boundary integration of the resistive heating (Qav_bnd) from the Postprocessing

menu. The result should be close to 25 W.


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Co l d C r u c i b l e 1

Introduction

A cold crucible is used for the manufacturing of alloys that require a high degree of purity, like titanium alloys for the aeronautic industry. The cold crucible in this model is made of several water-cooled copper sectors forming the container in which the alloy is manufactured.

Model Definition

The crucible is sketched in the figure below. The harmonic current flowing in the coils induces a current flowing in the opposite direction along the edge of the crucible. The induced currents produced in each of the sectors make the cold crucible act as field concentrator.

To save computation time, exploit the symmetries of the model and set up only one quarter of the crucible. The flow lines of the magnetic field are normal to the symmetry plane inserted in the figure below. To understand this, note that the geometry is symmetric with respect to the plane, and that the current is tangential to the plane.

1. Model made in cooperation with Roland Ernst, Centre National de la Recherche Scientifique, Grenoble, France.


The first symmetry can be achieved by letting this plane be a boundary of the model. To this boundary, apply a boundary condition setting the tangential component of the magnetic field to zero.

The figure below indicates a second symmetry plane. As the flow lines of the magnetic field make closed loops around the coils, the field must be tangential to the symmetry plane.

To model this symmetry, use a boundary condition that sets the tangential component of the magnetic potential to zero.

The crucible operates at 10 kHz. At this frequency the currents flow very close to the surface of the conducting regions. The skin depth

C O L D C R U C I B L E | 315

316 | C H A P T E

is less than one millimeter, and the inner radius of the crucible is 5 cm. You can therefore model the currents as surface currents.

Because the electric potential does not need to be specified to generate currents, it is possible to work with the potential

and thereby reduce the set of four equations for the A and V potentials to three equations for the Ã potential (see “3D and 2D Quasi-Statics Application Modes” on page 166 in the Electromagnetics Module User’s Guide).


To compute the surface currents flowing into and out of the end of the crucible, you can use integrals at that boundary. From Kirchhoff ’s current law the total current flowing into the boundary should be equal to the currents flowing out of it. The figure below shows the integrations along the x side and z side, where the start and end points of each curve correspond to the flow in and out of the boundary, respectively.

The total inflow to the boundary is 1.38 A and the total outflow is 1.40 A, which agrees with Kirchhoff’s power law within 2 percent.

δ 2ωµσ-----------=

A˜ A jω---- V∇–=


Model Library path: Electromagnetics_Module/General_Industrial_Applications/crucible



1 In the Model Navigator, select 3D from the Space dimension list.

2 Then click on Electromagnetics Module>Quasi-Statics, Magnetic>Induction Currents to select the Quasi-Statics application mode.

3 Click OK.


1 In the Constants dialog box in the Options menu, enter the following names and expressions.

The electrical conductivity σ = 1 S/m for air is obviously not the physically correct value, but too large a contrast between the conductivities makes it difficult for the numerical algorithms to find a solution. The error you make by using an incorrect value for the conductivity in air is negligible for the currents on the surfaces.

2 In the Application Scalar Variables dialog box in the Physics menu, set the frequency variable nu_emqav to 10000.


The crucible and inductor will first be drawn in a 2D work plane and then extruded into 3D.

1 Create a work plane by opening the Work Plane Settings dialog box in the Draw menu and clicking OK to obtain the default work plane in the x-y plane.

2 Open the Axes/Grid Settings dialog box from the Options menu. Enter the axis settings in the table below on the Axis tab.

NAME EXPRESSION

DESCRIPTION

sigMetal 5e7 Conductivity of metal (S/m)

sigAir 1 Conductivity of air (S/m)

Js0 100 Coil surface current (A/m)


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3 Then click on the Grid tab and clear the Auto check box and enter the grid settings according to the table below.

4 Draw a circle C1 with radius 0.17 and a circle C2 with radius 0.16, both centered at the origin.

5 Select C1 and C2 and press the Difference button to create the solid object CO1.

6 Draw a circle C1 with radius 0.15 and a circle C2 with radius 0.05, both centered at the origin.

7 Select C1 and C2 and press the Difference button to create the solid object CO2.

8 Draw a rectangle R1 with the lower left corner at (-0.2, -0.2) and width 0.4 and height 0.2.

9 Open the Create Composite Object dialog box and enter the set formula CO1-R1 to create the composite object CO3.

10 Draw a new rectangle R1 with the lower left corner at (-0.2, -0.2) and width 0.4, and height 0.2.


12 Draw a rectangle R1 with the lower left corner at (0, 0) and width 0.155, and height 0.01.

AXIS GRID


x max 0.3 Extra x -0.17 -0.16 0.155


y max 0.2 Extra y 0.01



14 Select Extrude from the Draw menu to open the Extrude dialog box. Select the object CO3 and set the Distance to 0.05. Click OK to close the dialog box and create the extruded object EXT1.

15 Click on the Geom2 tab to return to the work plane.

16 Open the Extrude dialog box again, and select the object CO2 and set the Distance to 0.1 to create the extruded object EXT2.

17 Now create a region of air surrounding the crucible. Press the Block button to open the Block dialog box and set the Length coordinates to (1, 0.5, 0.5). Click More and set the coordinates of the Axis base point to (-0.5, 0, 0). Click OK to close the dialog box and create the block BLK1.

18 Press the Cylinder button to open the Cylinder dialog box and set the Radius to 0.5 and the Height to 0.5. Click OK to close the dialog box and create the cylinder CYL1.

19 Open the Create Composite Object dialog box and enter the set formula BLK1*CYL1. Click the Apply button to create the object CO1.

20 Enter the set formula CO1-EXT1-EXT2 and click OK.


320 | C H A P T E

y


Boundary ConditionsEnter boundary conditions according to the following table.

Subdomain SettingsEnter PDE coefficients according to the following table.


After solving the problem, the postprocessing includes a study of how the current flows on the rectangular end of the crucible (boundary 17). To prepare for that, define two coupling variables constituting the integrated current over a cross section of this boundary. Jsx_emqav is the x component of the surface current, and Isx is the integral of Jsx_emqav in the vertical direction across boundary 17,

Jsz_emqav is the z component of the surface current, and Isz is the integral of Jsz_emqav in the horizontal direction across boundary 17,

BOUNDARY 1, 2, 4, 16 3, 8 5, 6, 7, 14, 15 9-13, 17

Boundary condition

Magnetic insulation Electric insulation Surface current Impedance boundarcondition

Js -y*Js0 x*Js0 0

σ sigMetal

µr 1

εr 1

SUBDOMAIN 1

σ sigAir

Isx x( ) Jsx x z,( ) zd∫=

Isz z( ) Jsz x z,( ) xd∫=


The table below summarizes the variable definitions.

1 In the Boundary Projection Variables dialog box, select boundary 17, enter the Name Isx, set the Expression to Jsx_emqav, and use the Integration order 2. Select the General transformation and fill in the x and y fields with x and z, respectively. The integration will be performed over the second coordinate, z, and Isx will therefore be a function of x.

2 Under the Destination tab, set the Level to Edge, and select edge 38, which is one of the edges of boundary 17, and set the Destination transformation to x.

3 Go back to the Source tab and add a second projection variable, Isz, select boundary 17 and set the Integrand to Jsz_emqav, the Integration order to 2. Select the General

transformation, and fill in z and x in the fields x and y. In this case the integration will be performed over x.

4 On the Destination tab, select edge 40, and set the Destination transformation to z.


To get a good quality mesh, set the maximum mesh element size manually on the boundaries adjacent to the thin regions in the model.

1 Open the Mesh Parameters dialog box, select Fine in the Predefined mesh sizes list, and set the Resolution of narrow regions parameter to 1.

2 Go to the Boundary page, select the boundaries 5, 7, 9, 13, 14, and 15 and set the Maximum element size to 0.02.



1 Press the Solve button. (use the default solver)

VARIABLE

TYPE SOURCE DESTINATION

Isx Projection Boundary: 17 Integrand: Jsx_emqav Integration order: 2 Source transformation, x: x Source transformation, y: z

Edge: 38 Destination transformation: x

Isz Projection Boundary: 17 Integrand: Jsz_emqav Integration order: 2 Source transformation, x: z Source transformation, y: x

Edge: 40 Destination transformation: z


322 | C H A P T E


1 To see the currents on the crucible, open the Suppress Boundaries dialog box from the Options menu and suppress the boundaries 1-4, 8 and 16.

2 Open the Plot Parameters dialog box.

3 The current on the surface of the crucible can be visualized with a surface plot and an arrow plot. Clear Slice plot and select Boundary plot and Arrow plot in the Plot

Parameters dialog box. On the Boundary tab select Surface current density, norm as Boundary data. On the Arrow tab select to plot arrows on boundaries. Set Type to arrow, clear the Auto check box, and set the Scale factor to 3.

To verify the current conservation on the end boundary of the crucible, boundary 17, make use of the coupling variables Isx and Isz:

1 In the Domain Plot Parameters dialog box, select Line plot.

2 Click on the Line tab and select edge 38. Then enter the Line expression abs(Isx). Click Apply to make the plot.

3 The left side of the plot corresponds to x = 0.15 and the right side to x = 0.05. The value at x = 0.15 is the total current flowing in through the right edge of the


boundary, and the value at x = 0.05 is the total current flowing out through the left edge of the boundary.


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4 Select edge 40 and change the Line expression to abs(Isz). The left side of the plot corresponds to z = 0 and the right side to z = 0.1. The value at z = 0.1 is the total current flowing out through the top edge of the boundary.

5 By studying the plots, you can see that the current flowing in equals the current flowing out. To more accurately verify this, export the FEM structure to the MATLAB workspace by selecting Export FEM structure as 'fem' from the File menu. (This requires that you run COMSOL Multiphysics together with COMSOL Script or MATLAB.) Then enter the commands

Iinx=postinterp(fem,'abs(Isx)',0.1,'dom',38);Iinz=postinterp(fem,'abs(Isz)',0.0,'dom',40);Ioutx=postinterp(fem,'abs(Isx)',0.0,'dom',38);Ioutz=postinterp(fem,'abs(Isz)',0.1,'dom',40);(Ioutx+Ioutz)/(Iinx+Iinz) ans = 1.0132


The evaluation points along the edges are given in the local coordinate of the edge. As the length of these edges is 0.1, the coordinate lie in the range 0 to 0.1. The current flowing in and the current flowing out differ by less than two percent.


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E l e c t r i c Impedan c e S e n s o r

Introduction

Electric impedance measurements are used for imaging and detection. The applications range from nondestructive testing and geophysical imaging to medical imaging. One example is to monitor the lung function of babies in neonatal intensive care. The frequency ranges from less than one Hz to about one GHz depending on the application.

This model simulates a single electrode placed on a conducting block with an air filled cavity inside. The block is connected to ground on the lower face and on the sides. The analysis shows how the lateral position of the cavity affects the measured impedance, which you compute in a postprocessing step.

Figure 7-1: The electrode is placed on a conducting block with an air filled cylindrical cavity inside.

Model Definition

In the Electromagnetics Module you solve the problem using the 2D Small In-Plane Currents application mode. This application mode is useful for the modeling of AC problems when inductive effects are negligible. A sufficient requirement for this is that the skin depth is large compared to the size of the geometry. The skin depth δ, is given by


where ω is the angular frequency, µ is the permeability, and σ is the conductivity. This model uses nonmagnetic materials with a frequency of 1 MHz and a typical electrical conductivity of 1 mSm-1, so the skin depth is about 15 m. The size of the geometry is about 1 m.


When induction is neglected, the electric field is curl free and can be assigned a scalar potential V. The equation of continuity for the conduction and displacement currents then becomes

where ε0 is the permittivity of free space, and εr is the relative permittivity. The electric field E and displacement D are obtained from the gradient of V


Ground potential boundary conditions are applied on the lower and vertical edges of the domain. The upper edge is set to insulation except at the electrode, where a uniformly distributed current source of 1 A is applied.

δ 2ωµσ-----------=

∇ σ jωεrε0+( ) V∇( )⋅– 0=

E V∇–=

D ε0εrE=

n J⋅ Jn=

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Figure 7-2 shows the calculated impedance and its phase angle as functions of air cavity coordinate. When the cavity passes under the electrode, a sharp peak can be seen in the impedance value.

Figure 7-2: The absolute value and phase for the impedance as function of air cavity coordinate. The peak indicates when the cavity is beneath the electrode.

Model Library path: Electromagnetics_Module/General_Industrial_Applications/electric_impedance


Select the 2D, Electromagnetics Module, Quasi-Statics, Electric, In-Plane Electric Currents

mode in the Model Navigator.


1 Select Axes/Grid Settings from the Options menu to open the Axes/Grid Settings dialog box.

2 On the Axis tab set x min to -0.55 and x max to 0.55. Set the interval limits on the y axis to -0.55 and 0.

3 Click the Grid tab and then clear the Auto check box to manually define new grid settings.


4 Set x spacing to 0.1 and give the values -0.01 0.01 in the Extra x text field. Set y spacing to 0.1.

5 To define some global constants to be used in your model, select Constants from the Options menu.



1 Click the Rectangle/Square button on the Draw toolbar or select the corresponding entry in the Draw menu, then draw a rectangle with opposite corners at (-0.5,-0.5) and (0.5,0), using the left mouse button.

2 Click the Point button on the Draw toolbar and draw a point at (-0.01,0) and another point at (0.01,0).


sig_bulk 1e-3 bulk conductivity (S/m)

eps_r_bulk 5 relative permittivity in bulk

x0 0 x position of cavity center (m)

y0 -0.1 y position of cavity center (m)

r0 0.09 radius of cavity (m)


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Scalar VariablesOpen the dialog box by selecting Scalar Variables from the Physics menu and set the frequency nu_emqvw to 1e6.

Boundary ConditionsGround potential boundary conditions are applied on the lower and vertical edges of the domain. The upper edge is set to insulation except at the electrode, where a port is defined using the fixed current density property. This applies a uniformly distributed current source of 1 A to the electrode. The following table shows the corresponding boundary settings.

Subdomain SettingsThe cavity is modeled using functions and logical expressions for the permittivity and conductivity distributions. This has the advantage that you can change the position of the cavity without modifying the geometry. Use the default constitutive relation,

and enter the material parameters according to the table below.


1 Open the Mesh Parameters dialog box.

2 Select Normal from the predefined mesh sizes list on the Global tab. Specify a Maximum element size of 0.01.


BOUNDARY 4 1, 2, 6 3, 5

Boundary condition Port Ground Electric insulation

Port number 1

Use port as input selected


SUBDOMAIN 1

σ sig_bulk*(((x-x0)^2+(y-y0)^2)>r0^2)

εr 1+(eps_r_bulk-1)*(((x-x0)^2+(y-y0)^2)>r0^2)

D ε0εrE=



1 In the Solver Parameters dialog box, select Parametric linear in the Solver list and enter x0 as the Name of parameter and linspace(-0.5,0.5,21) for Parameter list.



After solving the problem, a surface plot is automatically shown for the electric potential. The buttons on the plot toolbar can be used to get plots of other types. To visualize the current distribution on a dB scale, open the Plot Parameters dialog box.

1 Select Plot Parameters from the Postprocessing menu to open the Plot Parameters dialog box.

2 Select Surface as Plot type and set Parameter value to 0 on the General tab.

3 On the Surface tab, set the Surface Expression to 20*log10(normJ_emqvw) and Colormap to hot.

4 Click Apply to generate a plot.

5 To better see the variations the color range can be changed. Click Range to open the Color Range dialog box.

6 Clear the Auto check box to manually define new range settings.

7 Keep the value for Max but change the Min value to -35. Click OK.


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The impedance is defined as the ratio of voltage to total current at the electrode. You can visualize it as a function of the cavity position using the cross-section plot tool available from the Postprocessing menu. The impedance is available as a postprocessing variable, Z11_emqvw.

1 Open the Domain Plot Parameters dialog.

2 Select all solutions from the Solution to use list by pressing Ctrl+A.

3 Now select the Point tab and set Expression to abs(Z11_emqvw) and select point number 1. Click Apply to get the figure below.


4 Repeat the last step but set the Expression to 180*angle(Z11_emqvw)/pi to get the phase angle of the impedance.


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One - s i d e d Magne t and P l a t e

Introduction

Permanent magnets with a one-sided flux are used to attach posters and notes to refrigerators and notice boards but can also be found in advanced physics applications like particle accelerators. The one-sided flux behavior is obtained by giving the magnet a magnetization that varies in the lateral direction, reference Ref. 1. As no currents are present, a permanent magnet can be modeled using a scalar magnetic potential formulation. We have used this technique to model a cylindrical one-sided permanent magnet. A special technique to model thin sheets of high permeability material was used to model a thin µ-metal plate next to the magnet. This circumvents the difficulty of volumetric meshing of thin extended structures in 3D.

Figure 7-3: A cylindrical magnet above a µ-metal plate is modeled.

Model Definition

In a current free region, where

we can define the scalar magnetic potential, Vm, from the relation

This is analogous to the definition of the electric potential for static electric fields.

∇ H× 0=

H ∇Vm–=


Using the constitutive relation between the magnetic flux density and magnetic field


we can derive an equation for Vm,

It can be shown that applying a laterally periodic magnetization of

results in a magnetic flux that only emerges on one side of the magnet.


Figure 7-4 shows the calculated magnetic flux density and direction for the one-sided magnet. The resulting force on the plate is considerably higher for the case with the one-sided magnetization compared to the case with uniform magnetization of the same amplitude.

Figure 7-4: The magnetic flux density and direction is plotted in a cross-section of the geometry. The one-sided behavior is clearly seen as the flux does not emerge on the top of the magnet.

B µ0 H M+( )=

∇ B⋅ 0=

∇ µ0∇Vm µ0M–( )⋅– 0=

M Mpre kx( )sin 0 Mpre kx( )cos, ,( )=

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References

1. Shute, Mallinson, Wilton and Mapps, “One-Sided Fluxes in Planar, Cylindrical and Spherical Magnetized Structures”, IEEE Transactions on Magnetics, Vol 36, No. 2, March 2000, pp. 440-451,

Model Library path: Electromagnetics_Module/General Industrial Applications/one_sided_magnet


Select the 3D, Electromagnetics Module, Statics, Magnetostatics, No Currents mode in the Model Navigator. Click OK.


In the Constants dialog box enter the following names and expressions. The description field is optional and can be omitted.


1 Use the Cylinder tool to create a cylinder with the radius 0.01 and the height 0.005.

2 Use the Sphere tool to create a sphere with the radius 0.02.

3 Open the Work Plane Settings dialog from the Draw menu and use the default work plane in the x-y plane. Set z to -0.005 and click OK.

4 First click the Projection of All 3D Geometries toolbar button and then click the Zoom

Extents toolbar button to adapt the axis settings of the drawing board.

5 Draw a circle C1 with radius 0.01 centered at (0,0).

NAME EXPRESSION

DESCRIPTION

k pi/0.01 Wave number in x-direction (1/m)

Mpre 5e5 Magnetization amplitude in magnet (A/m)

mur 4e4 Relative permeability in plate

th 5e-4 Plate thickness (m)


6 Select C1 and then select Embed from the Draw menu and click OK.


Subdomain SettingsIn the Subdomain Settings dialog box, change the Constitutive relation in subdomain 2 to allow for a premagnetization vector M and enter the values according to the table below.

This kind of premagnetization will result in a magnetic flux that only emerges on the lower side of the magnet. That is, on the side where the µ-metal plate is. Click OK.

Boundary ConditionsAlong the external boundaries, the magnetic field should be tangential to the boundary as the flow lines should form closed loops around the magnet. The natural boundary condition from the equation is

Thus the magnetic field is made tangential to the boundary by a Neumann condition on the potential. On the internal boundary representing the µ-metal plate, we will apply a special boundary condition for thin sheets of highly permeable material. Such plates are often used for the purpose of magnetic shielding.

SUBDOMAIN 2

M Mpre*sin(k*x) 0 Mpre*cos(k*x)

n µ0∇Vm µ0M–( )⋅ n B⋅ 0= =


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Enter these boundary conditions according to the following table. In order to access interior boundaries, check Interior boundaries.

Point SettingsSo far the magnetic potential is not constrained anywhere and the problem will have an infinite number of solutions. Thus, we must supply a condition fixing it to a specific value somewhere. In the Point Settings dialog box, constrain the Magnetic potential to zero in point 1.


1 Open the Mesh Parameters dialog box, and select Normal mesh.



There is no need to change the solver settings. The default solver settings are the Conjugate gradients solver and the Algebraic multigrid Preconditioner. This is normally the most efficient solver setting for Poisson’s equation, which we are solving in this model. This problem will generate Symmetric matrices so this option is checked as default. Click the Solve Problem button. The computation time is about 6 seconds on a 3 GHz P4 machine.


The default plot shows a slice plot of the magnetic potential.

1 In the Plot Parameters dialog box, select Slice and Arrow plots. On the Slice tab set the Expression to Magnetic flux density, norm, normB_emnc and the Slice positioning x levels to 0, y levels to 1, z levels to 0. Then on the Arrow tab, select the magnetic flux density from Predefined quantities under Arrow data. Set the Arrow positioning x points to 50, y points to 1, the z points to 50, and select cone as Arrow type. Click OK to close the dialog and make the plot.

BOUNDARY 1-4, 10,11,13,14 5 6-9, 12, 15

Type Magnetic insulation Magnetic shielding Continuity

Relative permeability mur

Thickness th


2 Use the zoom tool to zoom in on the magnet and plate.

Magnetic field in the plateThe magnetic field in the plate is not available as predefined postprocessing variables. In order to make this quantity more readily available, we will define extra expressions that can be used for postprocessing.

From the Options menu, select Expressions -> Boundary Expressions.

1 Specify the variable names and expressions on boundary 5 (plate) according to the following table.


3 From the Solve menu, select Update Model in order to compute values for the newly defined variables using the existing solution.

The magnetic flux density in the plate can now be displayed using the available postprocessing functionality.


2 Select Boundary and Geometry edges as Plot type on the General tab.

3 On the Boundary tab, set the Expression to NORMB.

NAME EXPRESSION

NORMH sqrt(VmTx*VmTx+VmTy*VmTy+VmTz*VmTz)

NORMB mu0_emnc*mur*NORMH


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4 Click OK to generate the plot.

Force CalculationTo calculate the force on the plate we use the surface stress tensor,

where n1 is the boundary normal pointing out from the plate and T2 the stress tensor of air.

In this model the H and B fields are discontinuous across the plate and we must evaluate the fields on both sides of the plate.

All surfaces have an up and a down side. In order to find out which side is up and which is down, one can make an arrow plot of the components of the up normal (unx, uny, unz) on the boundaries. The up normal is the outward normal from the up side. In this case we will evaluate the fields on both sides and there is no need to know the direction of the up normal. The application mode defines variables for the surface stress tensor on the up and down side of the boundaries, for example, unTz_emnc and dnTz_emnc for the z component of the surface stress tensor.

Now, the x and y components of the force vanish for symmetry reasons. To find the force in the z direction, integrate the surface stress tensor over the plate. In the Boundary Integration dialog box, select boundary 5 and enter the Expression unTz_emnc+dnTz_emnc. Click OK to evaluate the integral, which gives 1.2 N. This result is displayed in the log field at the very bottom of the COMSOL window.

n1T212--- H B⋅( )n1– n1 H⋅( )BT

+=


Force Calculation with Uniform MagnetizationIn the Constants dialog box change the constant k to 0 in order to obtain a uniform magnetization of (0,0,Mpre). Solve and calculate the force on the plate again. What is the conclusion?

Modeling using a nonlinear magnetic material in the plate

Magnetic saturation effects are important in many applications. This step of the modeling process will show how to include a nonlinear magnetic material with saturation in the plate. We will enter a relative permeability µr that is a function of the magnitude, |H| of the magnetic field as described by the following equation.

Bsat is the magnetic saturation flux, which for µ-metal is about 1.2 (T).

Before we start modifying the model, it is recommended to save it under a new name. This is done from the File menu by selecting Save as...


1 In the Constants dialog box enter the following names and expressions.

NAME EXPRESSION

DESCRIPTION

k pi/0.01 Wave number in x-direction (1/m)

Mpre 5e5 Magnetization amplitude in magnet (A/m)

µr 1 µrmax 1–( ) 1– µ0 HBsat--------------⎝ ⎠

⎛ ⎞

1n---

+

1–

+=


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Expression variables are used for the nonlinear permeability.

From the Options menu, select Expressions -> Boundary Expressions.

1 Specify the variable names and expressions on boundary 5 (plate) according to the following table.

2 Click OK to close the dialog box


Initial ConditionsThe nonlinear solver works by performing iterations, linearizing and solving the model for a sequence of operating points that converges to the final solution. In order for this to work properly in the first iteration we need to start with an initial guess with a nonzero gradient. On the Init page of the Subdomain settings dialog, select all domains in the Domain selection list and set the initial conditions to sqrt(x^2+y^2+z^2). Click OK.


1 On the General page in the Solver Parameters dialog box, change to the Stationary nonlinear solver. Keep all other settings as before.

2 Click OK.

3 Click the Solve Problem button.

The computation time is about 35 seconds on a 3 GHz P4 machine.

murmax 4e4 Maximum relative permeability in plate

th 5e-4 Plate thickness (m)

Bsat 1.2 Saturation flux density (T)

N 1 Model order parameter

NAME EXPRESSION


NORMB mu0_emnc*mur*NORMH

mur 1-1/(1/(1-murmax)-(mu0_emnc*NORMH/Bsat)^(1/N))

NAME EXPRESSION

DESCRIPTION



The relative permeability in the plate can now be displayed using the available postprocessing functionality.

1 In the Plot Parameters dialog box, select Boundary and Arrow plots. On the Boundary

tab set the Expression to mur. Then on the Arrow tab, select the magnetic flux density from Predefined quantities under Arrow data. Set the Arrow positioning x

points to 50, y points to 1, the z points to 50, and select cone as Arrow type. Click OK to close the dialog and make the plot.

The plate has begun to saturate as the relative permeability is well below 4e4 in the blue areas. Try increasing the value of the premagnetization Mpre to 1e6 in the Constants dialog box and solve again to see a more pronounced saturation effect.

The computation time is about 100 seconds on a 3 GHz P4 machine.


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Force CalculationIn the Boundary Integration dialog box, select boundary 5 and enter the Expression unTz_emnc+dnTz_emnc. Click OK to evaluate the integral, which gives 3.3 (N). This result is displayed at the very bottom of the COMSOL window.


Magne t i c S i g na t u r e o f a S ubma r i n e

Introduction

A vessel traveling on the surface or under water gives rise to detectable local disturbances in the Earth’s magnetic field. These disturbances can be used to trigger weapon systems. The magnetic signature of a ship can be reduced by generating a counteracting magnetic field of suitable strength and direction based on prior knowledge of the magnetic properties of the vessel. An important step in the design of a naval ship is therefore to predict its magnetic signature. Another application where magnetic signatures are of great importance is in urban traffic control. Magnetic sensors, buried in our streets, are used to sense vehicles and control traffic lights. Ships and cars are both to a large extent made of sheet metal. This makes them hard to simulate using standard finite element analysis because volume meshes of thin extended structures are difficult to generate and tend to become very large. This model demonstrates a powerful technique that circumvents the problem by modeling the sheet metal as 2D faces embedded in a 3D geometry. Thus only comparatively inexpensive 2D face meshes need to be created in addition to the 3D volume mesh used for the surrounding medium. A tangential projection of the 3D equation is then solved on the 2D face mesh.

Figure 7-5: Submarine HMAS Collins, courtesy of Kockums AB.

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Model Definition

In magnetostatic problems, where no currents are present, the problem can be solved using a scalar magnetic potential. A special technique to model thin sheets of high permeability materials is also demonstrated. The model geometry is shown in Figure 7-6. It consists of face objects representing the submarine enclosed in a 3D box representing the surrounding water.

Figure 7-6: The model geometry.


In a current-free region, where

it is possible to define the scalar magnetic potential, Vm, from the relation

This is analogous to the definition of the electric potential for static electric fields. Using the constitutive relation between the magnetic flux density and magnetic field


you can derive an equation for Vm,

∇ H× 0=

H ∇Vm–=

B µ0 H M+( )=

∇ B⋅ 0=

∇ µ0∇Vm µ0M–( )⋅– 0=



Along the vertical external boundaries of the surrounding box, the magnetic field is assumed to be tangential to the boundaries. The natural boundary condition from the equation is

Thus the magnetic field is made tangential to the boundary by a Neumann condition on the potential. In order to emulate the Earth’s magnetic field a potential condition is applied on the top boundary

whereas the bottom boundary is set to a magnetic potential of zero.

This corresponds to a vertical magnetic flux density of about 0.5 G—a typical value for the Earth’s magnetic field. On the face objects representing the hull of the submarine, you apply a 2D tangential projection of the 3D domain equation where the thickness and permeability of the hull are introduced as parameters. This is readily available in the used formulation as a shielding boundary condition, which is useful for modeling of highly permeable thin sheets. Corresponding boundary conditions are available in the Conductive Media DC and Electrostatics application modes for modeling of thin sheets with high conductance and high permittivity respectively.


Figure 7-7 shows the magnetic flux density in a horizontal slice plot 7.5 m below the keel of the submarine. A distinct field perturbation due to the presence of the vessel

n µ0∇Vm µ0M–( )⋅ n B⋅ 0= =

Vm 1989 A( )=

Vm 0=


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can be seen. The magnitude and direction of the tangential magnetic field in the hull of the vessel is shown using color and arrows.

Figure 7-7: The slice color plot shows the magnetic flux density. The arrows and boundary color plot show the direction and strength of the tangential magnetic field in the hull.

Model Library path: Electromagnetics_Module/General_Industrial_Applications/submarine


Select the 3D>Electromagnetics Module>Statics>Magnetostatics, No Currents application mode in the Model Navigator. Click OK.

Open the Work Plane Settings dialog box from the Draw menu, use the default x-y work plane and click OK.



1 On the Grid tab in the Axes/Grid Settings dialog box, clear the Auto check box and enter the grid settings according to the following table.


1 Use the Rectangle tool to create a rectangle with its lower left corner at (-12.5,-2.5) with the width 22.5 and height 5.

2 Use the Ellipse/Circle (Centered) tool to create a circle with the radius 2.5 and its center at (-12.5,0).

3 Select the 3rd Degree Bezier Curve tool from the and click in sequence at the following points (10.0, 2.5), (13.0, 2.5), (13.0, 1.0) and (17.0, 1.0). Then, select the Line tool from the toolbar and click at (17.0, -1.0). Finally, go back to the 3rd

Degree Bezier Curve tool and click at (13.0, -1.0), (13.0, -2.5) and (10.0, -2.5). Close the curve by clicking the right mouse button.

4 Press Ctrl+A to select all objects and then click the Union button on the toolbar.

5 Click the Delete Interior Boundaries button on the toolbar. Then click the Zoom

Extents button.

6 Use the Rectangle tool to create a rectangle with its lower left corner at (-15,-2.5), with the width 32 and height 2.5.

GRID

x spacing 0.5

y spacing 0.5


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7 Press Ctrl+A to select all objects and then click the Difference button on the toolbar.


9 Select the composite object CO2.

10 Open the Revolve dialog box from the Draw menu. Set the coordinates for the Second

point to (1,0) and click OK to revolve into the 3D geometry.

11 Go back to the work plane Geom2.

12 Select Ellipse/Circle (Centered) from the Draw menu and create an ellipse with the A-semiaxis 2.5 and the B-semiaxis 0.625 and its center at (0,0).

13 With the ellipse selected, open the Extrude dialog box from the Draw menu and extrude it a Distance of 2 m.

14 In the 3D view, use the Move tool to move the extruded object a distance of 2 m in the z direction.

15 Press Ctrl+A to select all objects and then click the Union button on the toolbar.

16 Click the Delete Interior Boundaries button on the toolbar.


17 Click the Coerce to Face button on the toolbar. Then click the Zoom Extents button

18 Click the Block button on the toolbar and set the Base to Center, the Axis base point to (0,0,0), Length to (100,50,50) and click OK.


Subdomain SettingsKeep the default subdomain settings.

Boundary ConditionsAlong the vertical boundaries of the surrounding box, the magnetic field should be tangential to the boundaries. In order to emulate the Earth’s magnetic field, apply a potential condition of 1989 A on the top boundary. At the bottom boundary, set a magnetic potential of zero. On the internal boundaries representing the hull of the submarine, apply a special boundary condition denoted shielding. This makes use of a 2D tangential projection of the 3D domain equation where the thickness and permeability of the hull are introduced as parameters.

Enter the boundary conditions according to the following table. In order to access the interior boundaries, select the Interior boundaries check box.

BOUNDARY 1, 2, 5, 27 6–26 4 3

Boundary condition

Magnetic insulation

Magnetic shielding

Magnetic potential

Zero potential

µr 700

d 0.05

Vm0 1989


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1 Open the Mesh Parameters dialog box, and type 0.5 in the field Maximum element

size scaling factor.



1 On the General tab in the Solver Parameters dialog box, use the default solver Conjugate gradients and the default preconditioner, Algebraic multigrid. This is normally the most efficient solver setting for Poisson’s equation, which you are solving in this model.

2 This problem generates Symmetric matrices so this option is selected in the Matrix

symmetry list by default.

3 Click OK.


The computation time is about 1 minute on a 3 GHz P4 machine.


The default plot shows a slice plot of the magnetic potential with 5 slices perpendicular to the x axis. A more interesting plot can easily be obtained.

1 Select Plot Parameters from the Postprocessing menu to open the Plot Parameters dialog box and click the Slice tab.

2 Select Magnetic flux density, norm as Expression.

3 Set the numbers of x-levels and y-levels to 0.

4 For z, select Vector with coordinates and set the value to -10. Click OK to close the dialog and make the plot.

Tangential Magnetic Field in the HullThe magnetic field in the hull is not available as predefined postprocessing variables so you need to define extra expressions involving the tangential derivatives of the magnetic potential that you can use for postprocessing.

From the Options menu, select Expressions>Boundary Expressions.


1 Specify the variable names and expressions on the hull boundaries 6-26, according to the following table.


3 From the Solve menu, select Update Model in order to compute values for the newly defined variables using the existing solution.

The magnetic field in the hull can now be displayed using the available postprocessing functionality in COMSOL Multiphysics.


2 Select the Slice, Boundary, and Arrow check boxes in the Plot type area.

3 On the Slice tab, keep the previous settings.

4 On the Boundary tab, type NORMH in the Expression edit field.

5 On the Arrow tab, select Boundaries from the Plot arrows on list. For Arrow data, click the Boundary tab and enter HX, HY and HZ as the x, y and z components.

6 Enter a Scale factor of 0.2.

NAME EXPRESSION


HX -VmTx

HY -VmTy

HZ -VmTz


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7 Click OK to generate the plot.


3D Quad r upo l e L en s

Introduction

This is a full 3D version of the Quadrupole Lens model in the COMSOL Multiphysics Model Library. For a general overview of the setup and the physics, see ???????. In this version of the model, we also take fringing fields into account, and the forces on the ions are calculated using all components of their velocities.

Model Definition

The geometry of the quadrupole lens is the same as outlined in the documentation to the 2D model: three quadrupoles in a row, followed by 1 m of empty space, where the ions are left to drift. The Electromagnetics module features an application mode for magnetostatics without currents. This reduces the memory usage considerably compared to the formulation including currents.


The magnetic field is described using the Magnetostatics equation, solving for the magnetic scalar potential Vm [Wb/m]:

where µ0 = 4πe-7 H/m denotes the permeability of vacuum and M is the magnetization [A/m]. In the iron subdomain, the following formulation is used:

,

where µr = 4000 is the relative permeability. The magnetic scalar potential is everywhere defined so that .


The magnetic insulation boundary condition, reading , is used all around the iron cylinder, and at the lateral surfaces of the air domain that encloses the drift length. At the base surface of the same air domain, we fix the magnetic potential to zero.

µ0 Vm µ0M–∇( )∇•– 0=

µ0µr Vm∇∇•– 0=

H Vm∇–=

n B• 0=

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Results

The magnetic field density and arrows showing its direction are depicted in the figure below.

Figure 7-8: Slice plot and arrows of the magnetic field density in the quadrupole lens.

Each ion passing through the assembly will experience Maxwell forces equal to , where v [m/s] is the velocity of the ion. To find the transverse position

as a function of time, we are solving Newton’s second law for each ion: . F qv B×=

qv B× ma=


where t [s] is the time of flight. Figure 7-9 here below shows the traces of the ions as the fly through the quadrupole lens.

Figure 7-9: Particle tracing plot of the ions. The color of the lines show the local force acting on each ion. The force grows larger (red) far away from the center of the beam line and smaller (blue) where two oppositely polarized quadrupoles join.

Model Library path: Electromagnetics_Module/General_Industrial_Applications/quadrupole3d



1 In the Model Navigator, select 3D in the Space dimension drop list.

2 Select Electromagnetics Module > Statics > Magnetostatics, No Currents from the list of application modes.



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1 Enter the following constant names and expressions in the Constants dialog box. The descriptions are optional.



1 Go to Draw > Work Plane Settings and click OK to get the default xy-plane.

1 Go to Draw > Specify Objects and specify a rectangle with the following properties:

2 Go to Draw > Modify > Rotate and rotate your rectangle by 45 degrees around the origin.

3 Specify a circle with the following properties:

4 Select both the circle and the rectangle, and click the Intersection button.


M 11 Ion mass number

Z 5 Ion charge number

vz 0.01*3e8 Ion velocity (m/s)

mp 1.672e-27 Proton mass (kg)

Ze 1.602e-19 Proton charge (C)

m M*mp Ion mass (kg)

q Z*Ze Ion charge (C)

MQ 5.8e3 Quadrupole magnetization (A/m)

PROPERTY EXPRESSION

Width 0.177

Height 0.07

Position: Base Corner

Position: x 0

Position: y -0.035

PROPERTY EXPRESSION

Radius 0.2

Position: Base Center

Position: x 0.2

Position: y 0.2


5 Make a copy of your composed object (CO1) and paste it at the same location, that is with zero displacements.

6 Rotate the copied object (CO2) by 90 degrees around the origin.

7 Paste two more copies of CO1 at the same location, and rotate them by 180 and 270 degrees respectively.

8 Create a circle centered in the origin with a radius of 0.2 m.

9 Create another circle centered in the origin, with a radius of 0.12 m.

10 Click the Create Composite Object button and create an object using the formula C1+C2-(CO1+CO2+CO3+CO4).

11 Once more, draw a circle centered in the origin with a radius of 0.2 m.

The geometry should now look like the figure below.

Figure 7-10: Workplane view of the geometry.

12 Select all objects and go to Draw > Extrude. Click OK to extrude the geometry to a height of 1 m.

13 In the 3D view, copy the object you just extruded, and paste it to the same location. You can find the copy and paste commands in the Edit menu.

14 Go to Draw > Modify > Scale and scale the pasted object by a factor of 2 in the z direction.


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15 Go to Draw > Modify > Move and move the scaled object 1 m in the z direction.

16 Paste another copy of the first object into the geometry, but now with a displacement of 3 m in the z direction.

17 Create a cylinder with the following specifications:


Boundary Conditions1 In the Boundary Settings dialog box, set the boundary conditions according to the

table below. The easiest way to do this is to first select all the exterior boundaries, by selecting the Select by group check box and select one of them. Apply magnetic insulation to all the boundaries, and change the condition on boundary 13 back to zero potential.

Subdomain Settings1 In the Subdomain Settings dialog box, select subdomains 1-3 and load Iron from the

list of materials.

2 Leave the default settings in subdomain 4 and 11-13.

3 In all the other subdomains, set the constitutive relation to and set the magnetization according to the table below.


1 In the Mesh Parameters dialog box, select Finer among the Predefined mesh sizes.

2 On the Subdomain tab, select subdomains 4 and 11-13, and enter 0.03 for the maximum element size (m).

3 On the Advanced tab, set the z-direction scale factor to 0.2.

PROPERTY EXPRESSION

Radius 0.2

Height 1

Axis base point z 4

BOUNDARY 13 ALL OTHER

Zero potential Magnetic insulation

SUBDOMAIN 5,7,18 8,10,15 9,14,16 6,17,19

M (x component) MQ/sqrt(2) -MQ/sqrt(2) MQ/sqrt(2) -MQ/sqrt(2)

M (y component) MQ/sqrt(2) MQ/sqrt(2) -MQ/sqrt(2) -MQ/sqrt(2)

B µ0H µ0M+=


4 Click the Remesh button.





The default plot shows the magnetic potential. Follow the instructions to get an overview of the magnetic field.

1 In the Plot Parameters dialog box, go to the Slice tab and select Magnetic field, norm

among the Predefined quantities. Set the Number of levels to (1,1,0).

2 On the Arrow tab, select the Arrow plot check box. Set the Number of points to (2,2,20) and the Scale factor to 0.2.

3 Click Apply to see the magnetic field.

To see how the ions travel through the system of quadrupoles, do the following:

4 On the General tab, clear the Slice check box and the Arrow check box, and select the Particle tracing check box.

2. On the Particle tracing tab, select Electromagnetic force in the Predefined forces list and apply the following settings:

5 Click the Color Expression button. Enter the expression q*vz*normB_emnc. This is an approximate value of the local force acting on each ion.

6 Click the Advanced button and set the Relative tolerance to 1e-6.

7 Click OK to see the particle tracing plot.

PROPERTY EXPRESSION

m m

Start points x 0.03*cos(linspace(0,2*pi,41))

Start points y 0.03*sin(linspace(0,2*pi,41))

Initial velocity z vz

Pathline color Use expression


362 | C H A P T E

S up e r c ondu c t i n g W i r e

Current can flow in a superconducting wire with practically zero resistance, although factors including temperature, current density, and magnetic field can limit this phenomenon. This model solves a time-dependent problem of a current building up in a superconducting wire close to the critical current density. This model is based on a suggestion by Dr. Roberto Brambilla, CESI - Superconductivity Dept., Milano, Italy.

Introduction

The Dutch physicist Heike Kamerlingh Onnes discovered superconductivity in 1911. He cooled mercury to the temperature of liquid helium (4 K) and observed that its resistivity suddenly disappeared. Research in superconductivity reached a peak during the 1980s in terms of activity and discoveries, especially when scientists uncovered the superconductivity of ceramics. In particular, it was during this decade that researchers discovered YBCO—a well-known ceramic superconductor composed of yttrium, barium, copper, and oxygen that has a critical temperature above the temperature of liquid nitrogen. However, researchers have not yet created a room-temperature superconductor, so much work remains for the broad commercialization of this area.

This model illustrates how current builds up in a cross section of a superconducting wire; it also shows where critical currents produce a swelling in the non-superconducting region.

Model Definition

The dependence of resistivity on the amount of current makes it difficult to solve the problem using the standard application modes in the Electromagnetics Module. The reason is this: A circular dependency arises because the current-density calculation contains the resistivity, leading to a resistivity that is dependent on itself.

An alternate approach uses the magnetic field as the dependent variable, and you can then calculate the current as

.

The electric field is a function of the current, and Faraday’s law determines the complete system as in

J ∇ H×=


where E(J) is the current-dependent electric field. The model calculates this field with the empirical formula

where E0 and α are constants determining the nonlinear behavior of the transition to zero resistivity, and JC is the critical current density, which decreases as temperature increases.

For the superconductor YBCO, this model uses these parameter values (Ref. 1):

Systems with two curl operators are best dealt with using vector elements (edge elements). This is the default element for the application modes in the Electromagnetics Module that solve similar equations. This particular formulation for the superconducting system is not available in the Electromagnetics Module, so you must define it using the PDE General Form application mode. In addition, the model uses higher order vector elements, called curl elements in COMSOL Multiphysics. The resulting system becomes

.

The current density has only a z-component for symmetry reasons.

The model controls current through the wire with its outer boundary condition. Because Ampere’s law must hold around the wire, a line integral around it must add

PARAMETER VALUE

E0 0.0836168 V/m

α 1.449621256

JC 17 MA

TC 92 K

∇ E J( )× µ H∂t∂

--------–=

E J( )0 J JC<

E0J JC–

JC--------------------⎝ ⎠

⎛ ⎞α J

J------ J JC≥

=

DaHd

dt-------- ∇+ Γ⋅ F

µ0 0

0 µ0

Hxdtd

-----------

Hydtd

-----------

⋅ ∇+0 Ez Jz( )

Ez Jz( )– 0⋅⇒ 0

0= =

S U P E R C O N D U C T I N G W I R E | 363

364 | C H A P T E

up to the current through the wire. Cylindrical symmetry results in a known magnetic field at the outer boundary

.

For vector elements, the Dirichlet boundary conditions puts a constraint on the tangential component of the vector field. If the field components are called Hx and Hy, the tangential counterparts are tHx and tHy.


The model applies a simple transient exponential function as the current through the wire, reaching a final value of 1 MA. This extremely large current is necessary if the superconducting wire is to reach its critical current density. Figure 7-11 plots the current density at two times after the current starts.

Figure 7-11: The current density at t = 0.02, 0.05, and 0.1. The swelling of the ring comes from the transition out of the superconducting state at current densities exceeding JC.

H ld⋅∫° 2πrHφ Iwire Hφ⇒Iwire2πr-------------= = =


Reference

1. R. Pecher, M.D. McCulloch, S.J. Chapman, L. Prigozhin and C.M. Elliotth, “3D-modelling of bulk type-II superconductors using unconstrained H-formulation,” 6th European Conf. Applied Superconductivity (EUCAS 2003).

Model Library path: Electromagnetics_Module/General_Industrial_Applications/superconductive_wire




2 Select the COMSOL Multiphysics>PDE Modes>PDE, General Form>Time-dependent

analysis application mode.

3 In the Dependent variables edit field, type Hx Hy. Click OK.



2 In the Constants dialog box, define the following constants with names, expressions, and description (optional):

3 Click OK.


mu0 4*pi*1e-7 Permeability of vacuum (H/m)

alpha 1.449621256 Parameter for resistivity model

Jc 1.7e7 Critical current density (A/m2)

I0 1e6 Applied current (A)

rho_air 1e6 Resistivity of air (Ω m)

tau 2e-2 Time constant of applied current (s)

Tc 92 Critical temperature (K)

dT 4 Parameter for resistivity model (K)

dJ Jc/1e4 Parameter for resistivity model (A/m2)

E0 0.0836168 Parameter for resistivity model (V/m)


366 | C H A P T E


All the dialog boxes for specifying the primitive objects are accessed from the Draw menu and Specify Objects. The first column in the table below contains the labels of the geometric objects. These are automatically generated by COMSOL Multiphysics, and you do not have to enter them. Just check that you get the correct label for the objects that you create.

1 Draw two circles with the properties according to the table below.





3 Click OK. The expression HyxHxy is a short name for the z-component of the curl of the H-field, which is equal to:

.



6 Click OK.

7 From the Options menu, choose Expressions>Subdomain Expressions.

NAME RADIUS BASE (X,Y)

C1 0.1 Center (0,0)

C2 1 Center (0,0)


Jz_sc HyxHxy Current density (A/m2)

normH_sc sqrt(Hx^2+Hy^2) Norm of the H-field (A/m)

I1 I0*(1-exp(-t/tau)) Applied current (A)

Q_sc Ez_sc*Jz_sc Generated heat in superconductor

BOUNDARY 1, 2, 5, 8 ALL OTHER

H0phi I1/(2*pi*sqrt(x^2+y^2))

x∂∂Hy

y∂∂Hx–


8 In the Subdomain Expressions dialog box, define the following variables with names and expressions:


according the following table:

2 Click OK.

Subdomain Settings1 From the Physics menu, open the Subdomain Settings dialog box and define the

subdomain settings according to the following table:

SUBDOMAIN 1 2

Ez_sc rho_air*Jz_sc E0*((Jz_sc-Jc)* flc2hs(Jz_sc-Jc-dJ,dJ)/Jc)^alpha

BOUNDARY 2, 8 1, 5

Boundary condition Dirichlet Dirichlet

R1 H0phi*t1x+tHz H0phi*t1x-tHx

R2 H0phi*t1y+tHy H0phi*t1y-tHy

SUBDOMAIN 1, 2

Γ11 0

Γ12 Ez_sc

Γ21 -Ez_sc

Γ22 0

F1 0

F2 0

Da,11 mu0

Da,12 0

Da,21 0

Da,22 mu0


368 | C H A P T E

2 Click on the Element tab, make sure that both subdomains are selected, and enter shcurl(2,‘Hx’,’Hy’) in the shape edit field. Second order vector elements are now used to solve the model.

3 Click OK.



2 Select the Coarse mesh size from the Predefined mesh sizes list.

3 Click the Subdomain tab, select subdomain 2, and type 2e-2 in the Maximum element

size edit field.




2 In the Times edit field type linspace(0,0.1,21).

3 Click the Time Stepping tab, and select the Manual tuning of step size check box. Enter 1e-9 in the Initial time step, and enter 1e-3 in the Maximum time step edit fields.

4 Click OK.





2 Make sure that the Surface, and Geometry edges check boxes are checked under the General tab.

3 Click the Surface tab, and check that Hx is in the Expression edit field.

4 Click OK. You should now see the plot in the figure below.

5 Select Domain Plot Parameters from the Postprocessing menu.

6 Under the General tab, select the solutions at times 0.02, 0.05, and 0.1.

7 Click on the Surface tab. Type Jz_sc in the Expression edit field and select subdomain 2.

8 Click OK. You should now see the plot shown in Figure 7-11 on page 364.


370 | C H A P T E

I N D E X

3D electromagnetic waves model 121

3D electrostatics model 246

3D magnetostatics model 66, 284, 345

3D quasi-statics model 267, 317

3D waveguide 121

A absorbing layers 166

adaption 37

adaptive solver 37

algebraic multigrid preconditioner 338,

352

antenna

conical 85

model 85, 96

RF 85

antenna impedance 86, 93

application mode

Axisymmetric TE waves 97

Axisymmetric TM waves 89, 139

Azimuthal currents 18, 259, 277

Bioheat equation 139

Boundary mode analysis 126

Electromagnetic waves 126

Electrostatics (3D) 248

Heat transfer 18

In-plane TE waves 169, 176

Magnetostatics 25, 67, 287, 348

Perpendicular currents 11, 34, 45

Perpendicular hybrid-mode waves 184,

191, 212

Plane strain 191, 212

Quasi-statics (3D) 270, 317

Small in-plane currents 328

Weak form 213

Application Scalar Variables dialog box

279

arrow plot 263

axisymmetric TE waves model 96

axisymmetric TM waves model 85, 132

azimuthal currents model 17, 256, 275

B bioheat equation model 132

birefringence 189

stress-induced 198

boundary integration 340, 344

boundary mode analysis model 121

boundary plot 322

C capacitance

computing 246

capacitor

for MEMS applications 246

cemforce 38

characteristic impedance 93

cladding 182

confined mode 182

contour plot 14

convergence analysis 201

coupling variable

extrusion 42

projection 321

scalar 94

cross-section plot 162

crucible 314

cut-off frequency 122, 175, 179

D dipole

magnetic 96

E eddy currents

forces 257

model 256

effective mode index 183

electromagnetic force 32

electromagnetic wave model 3D 121

electrostatics model 246

I N D E X | 371

372 | I N D E X

errors in modeling 204

expansion coefficient 206

F far-field

computation 95

fiber

optical 182

silica glass 182

single mode 182

force computation 16, 29, 38, 340, 341,

344

frequency

normalized 183

G generalized plane strain 206

generator 41, 55, 66

H H bend 74, 116

Helmholtz coil 267

I impedance

of antenna 86, 93

inductance 284

inductive heating 17

in-plane TE waves model 74, 116, 166,

173

intensity

optical 201

interpolation function 50, 66, 70

L linear electric motor 10

Lorentz force 38, 257

M magnetic dipole 96

magnetic material

nonlinear 42

magnetic potential

scalar 24, 334, 346

magnetic shielding 347

magnetostatics model 10, 24, 32, 66, 284,

345

material library 261

Maxwell stress tensor 11, 29, 340

MEMS 246

inductor 284

microprism 166

microwave waveguide 121

mode analysis 199

modeling errors 204

motor

electric 32

moving geometry 41, 55

N Navier’s equation 208

weak form 208

nonlinear permeability 42, 66

normalized frequency 183

O ohmic heating 17

optical fiber 182

optical intensity 201

optical waveguide 166, 182

optics 182

optoelectronics 182

P parametric solver 94

particle tracing 357

perfectly matched layers 166

permanent magnet 24

permeability

nonlinear 42

perpendicular currents model 10, 32, 41,

55

perpendicular hybrid-mode waves model

182, 189, 206

perturbation

of material parameters 205

photonic band gap 173

photonic microprism 166

photonic waveguide 189

plane strain 206

generalized 206

plot

arrow 263

boundary 322

contour 14

streamlines 282, 339

surface 282, 331, 339

PMLs 166

power flow 201

Poynting vector 201

preconditioner

algebraic multigrid 338, 352

Q quasi-statics model 267, 317

small currents 326

R radiation pattern 93

reactance 94

refractive index 166, 182

residual induction 33

resistance 94

resistive heating 17

resistive loss

computing 33

RF applications 246

S scalar magnetic potential 24, 334, 346

scattering parameter 75, 93, 116

sensitivity analysis 204

shielding boundary condition 347

silica glass fiber 182

single mode fiber 182

skin depth 315, 326

solver

adaptive 37

parametric 94

S-parameter 75, 93, 116, 121

Specific absorption rate (SAR) 132

streamline plot 282, 339

stress tensor

Maxwell 11, 29, 340

stress-induced birefringence 198

stress-optical effect 189

stress-optical tensor 189

surface plot 282, 331, 339

surface stress 11, 29, 340

T typographical conventions 7

W waveguide

3D 121

H bend 74, 116

microwave 121

optical 166, 182

photonic 189

weak form 208

I N D E X | 373

374 | I N D E X

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