VERSION 4.3b

User s Guide

AC/DC Module

C o n t a c t I n f o r m a t i o n

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Part No. CM020101

A C / D C M o d u l e U s e r s G u i d e 19982013 COMSOL

Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending.

This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agreement.

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Version: May 2013 COMSOL 4.3b

N T E N T S | i

C o n t e n t s

C h a p t e r 1 : I n t r o d u c t i o n

About the AC/DC Module 2

What Can the AC/DC Module Do? . . . . . . . . . . . . . . . . 2

AC/DC Module Physics Guide . . . . . . . . . . . . . . . . . . 3

Where Do I Access the Documentation and Model Library? . . . . . . 9

Overview of the Users Guide 12

C h aC O

p t e r 2 : R e v i e w o f E l e c t r o m a g n e t i c s

Fundamentals of Electromagnetics 16

Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . 16

Constitutive Relations . . . . . . . . . . . . . . . . . . . . . 17

Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . 19

Reduced Potential PDE Formulations . . . . . . . . . . . . . . . 19

Electromagnetic Energy . . . . . . . . . . . . . . . . . . . . 20

The Quasi-Static Approximation and the Lorentz Term . . . . . . . . 21

Material Properties . . . . . . . . . . . . . . . . . . . . . . 22

About the Boundary and User Interface Conditions . . . . . . . . . 23

Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Electromagnetic Forces 26

Overview of Forces in Continuum Mechanics . . . . . . . . . . . . 26

Forces on an Elastic Solid Surrounded by Vacuum or Air . . . . . . . . 28

Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Forces in Stationary Fields . . . . . . . . . . . . . . . . . . . 29

Forces in a Moving Body . . . . . . . . . . . . . . . . . . . . 33

Electromagnetic Energy and Virtual Work . . . . . . . . . . . . . 35

ii | C O N T E N T S

Electromagnetic Quantities 37

References for the AC/DC User Interfaces 39

C h a p t e r 3 : M o d e l i n g w i t h t h e A C / D C M o d u l e

Preparing for Modeling 42

What Problems Can You Solve? . . . . . . . . . . . . . . . . . 43

Selecting the Space Dimension for the Model Geometry . . . . . . . . 44

Simplifying the Geometry Using Boundary Conditions . . . . . . . . . 46

Applying Electromagnetic Sources . . . . . . . . . . . . . . . . 47Selecting a Study Type . . . . . . . . . . . . . . . . . . . . . 48

2D Field Variables . . . . . . . . . . . . . . . . . . . . . . 49

About the AC/DC Material Database . . . . . . . . . . . . . . . 49

Meshing and Solving . . . . . . . . . . . . . . . . . . . . . . 49

Force and Torque Computations 51

Calculating Electromagnetic Forces and Torques . . . . . . . . . . . 51

Model ExamplesElectromagnetic Forces . . . . . . . . . . . . . 52

Coil Domains 54

About the Single-Turn Coil and the Multi-Turn Coil Features . . . . . . 54

About the Coil Name . . . . . . . . . . . . . . . . . . . . . 56

Coil Excitation . . . . . . . . . . . . . . . . . . . . . . . . 56

Coil Groups. . . . . . . . . . . . . . . . . . . . . . . . . 59

Lumped Parameter Calculations . . . . . . . . . . . . . . . . . 60

Using Coils in 3D Models . . . . . . . . . . . . . . . . . . . 62

Computing Coil Currents . . . . . . . . . . . . . . . . . . . 64

Lumped Parameters 66

Calculating Lumped Parameters with Ohms Law . . . . . . . . . . . 66

Calculating Lumped Parameters Using the Energy Method . . . . . . . 68

Studying Lumped Parameters . . . . . . . . . . . . . . . . . . 69

Lumped Parameter Conversion . . . . . . . . . . . . . . . . . 70

N T E N T S | iii

Lumped Ports with Voltage Input 71

About Lumped Ports . . . . . . . . . . . . . . . . . . . . . 71

Lumped Port Parameters . . . . . . . . . . . . . . . . . . . . 72

S-Parameters and Ports 74

S-Parameters in Terms of Electric Field . . . . . . . . . . . . . . 74

S-Parameter Calculations: Lumped Ports . . . . . . . . . . . . . . 75

S-Parameter Variables . . . . . . . . . . . . . . . . . . . . . 75

Connecting to Electrical Circuits 76

About Connecting Electrical Circuits to Physics User Interfaces . . . . . 76

Connecting Electrical Circuits Using Predefined Couplings . . . . . . . 77

Connecting Electrical Circuits by User-Defined Couplings . . . . . . . 77

C h aC O

p t e r 4 : T h e E l e c t r i c F i e l d U s e r I n t e r f a c e s

The Electrostatics User Interface 80

Domain, Boundary, Edge, Point, and Pair Nodes for the Electrostatics User

Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Charge Conservation . . . . . . . . . . . . . . . . . . . . . 84

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . 86

Space Charge Density . . . . . . . . . . . . . . . . . . . . . 86

Force Calculation. . . . . . . . . . . . . . . . . . . . . . . 86

Zero Charge . . . . . . . . . . . . . . . . . . . . . . . . 88

Ground . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Electric Potential . . . . . . . . . . . . . . . . . . . . . . . 89

Surface Charge Density . . . . . . . . . . . . . . . . . . . . 90

External Surface Charge Accumulation . . . . . . . . . . . . . . 90

Electric Displacement Field . . . . . . . . . . . . . . . . . . . 91

Periodic Condition . . . . . . . . . . . . . . . . . . . . . . 92

Thin Low Permittivity Gap . . . . . . . . . . . . . . . . . . . 93

Dielectric Shielding . . . . . . . . . . . . . . . . . . . . . . 94

Terminal . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Floating Potential . . . . . . . . . . . . . . . . . . . . . . . 97

Distributed Capacitance . . . . . . . . . . . . . . . . . . . . 98

Line Charge . . . . . . . . . . . . . . . . . . . . . . . . . 98

iv | C O N T E N T S

Line Charge (on Axis) . . . . . . . . . . . . . . . . . . . . . 99

Line Charge (Out-of-Plane) . . . . . . . . . . . . . . . . . . 100

Point Charge . . . . . . . . . . . . . . . . . . . . . . . 101

Point Charge (on Axis) . . . . . . . . . . . . . . . . . . . 102

Change Cross-Section . . . . . . . . . . . . . . . . . . . . 102

Change Thickness (Out-of-Plane). . . . . . . . . . . . . . . . 103

Electrostatic Point Dipole . . . . . . . . . . . . . . . . . . 104

Archies Law . . . . . . . . . . . . . . . . . . . . . . . 105

Porous Media . . . . . . . . . . . . . . . . . . . . . . . 106

The Electric Currents User Interface 108

Domain, Boundary, Edge, Point, and Pair Nodes for the Electric Currents User

Interface . . . . . . . . . . . . . . . . . . . . . . . . . 110Current Conservation . . . . . . . . . . . . . . . . . . . . 113

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 115

External Current Density. . . . . . . . . . . . . . . . . . . 115

Current Source . . . . . . . . . . . . . . . . . . . . . . 116

Electric Insulation . . . . . . . . . . . . . . . . . . . . . 116

Floating Potential . . . . . . . . . . . . . . . . . . . . . . 117

Boundary Current Source . . . . . . . . . . . . . . . . . . 118

Normal Current Density . . . . . . . . . . . . . . . . . . . 119

Distributed Impedance. . . . . . . . . . . . . . . . . . . . 120

Terminal . . . . . . . . . . . . . . . . . . . . . . . . . 121

Electric Shielding . . . . . . . . . . . . . . . . . . . . . . 122

Contact Impedance . . . . . . . . . . . . . . . . . . . . . 123

Electrical Contact . . . . . . . . . . . . . . . . . . . . . 126

Sector Symmetry . . . . . . . . . . . . . . . . . . . . . . 127

Line Current Source . . . . . . . . . . . . . . . . . . . . 128

Line Current Source (on Axis). . . . . . . . . . . . . . . . . 129

Point Current Source . . . . . . . . . . . . . . . . . . . . 129

Point Current Source (on Axis) . . . . . . . . . . . . . . . . 130

Electric Point Dipole . . . . . . . . . . . . . . . . . . . . 131

Electric Point Dipole (on Axis). . . . . . . . . . . . . . . . . 132

The Electric Currents, Shell User Interface 133

Boundary, Edge, Point, and Pair Nodes for the Electric Currents, Shell User

Interface . . . . . . . . . . . . . . . . . . . . . . . . . 135

Current Conservation . . . . . . . . . . . . . . . . . . . . 136

N T E N T S | v

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 138

Current Source . . . . . . . . . . . . . . . . . . . . . . 139

Change Shell Thickness . . . . . . . . . . . . . . . . . . . 139

Normal Current Density . . . . . . . . . . . . . . . . . . . 139

Electric Shielding . . . . . . . . . . . . . . . . . . . . . . 140

Theory of Electric Fields 142

Charge Relaxation Theory . . . . . . . . . . . . . . . . . . 142

Theory for the Electrostatics User Interface 146

Electrostatics Equations . . . . . . . . . . . . . . . . . . . 146

The Electrostatics User Interface in Time-Dependent or Frequency-Domain

Studies. . . . . . . . . . . . . . . . . . . . . . . . . . 147

C h aC O

Effective Material Properties in Porous Media and Mixtures . . . . . . 148

Effective Conductivity in Porous Media and Mixtures . . . . . . . . 148

Effective Relative Permittivity in Porous Media and Mixtures . . . . . 149

Effective Relative Permeability in Porous Media and Mixtures . . . . . 150

Archies Law Theory . . . . . . . . . . . . . . . . . . . . 151

Reference for Archies Law . . . . . . . . . . . . . . . . . . 152

Theory for the Electric Currents User Interface 153

Electric Currents Equations in Steady State . . . . . . . . . . . . 153

Dynamic Electric Currents Equations . . . . . . . . . . . . . . 154

Theory for the Electrical Contact Feature . . . . . . . . . . . . 155

Theory for the Electric Currents, Shell User Interface 158

Electric Currents, Shell Equations in Steady State. . . . . . . . . . 158

Dynamic Electric Currents Equations . . . . . . . . . . . . . . 158

p t e r 5 : T h e M a g n e t i c F i e l d U s e r I n t e r f a c e s

The Magnetic Fields User Interface 160

Domain, Boundary, Point, and Pair Nodes for the Magnetic Fields User Interface

163

Ampres Law . . . . . . . . . . . . . . . . . . . . . . . 166

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 168

vi | C O N T E N T S

External Current Density. . . . . . . . . . . . . . . . . . . 169

Velocity (Lorentz Term) . . . . . . . . . . . . . . . . . . . 170

Magnetic Insulation . . . . . . . . . . . . . . . . . . . . . 171

Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . 172

Surface Current . . . . . . . . . . . . . . . . . . . . . . 173

Magnetic Potential . . . . . . . . . . . . . . . . . . . . . 174

Perfect Magnetic Conductor . . . . . . . . . . . . . . . . . 174

Line Current (Out-of-Plane) . . . . . . . . . . . . . . . . . 175

Electric Point Dipole . . . . . . . . . . . . . . . . . . . . 176

Gauge Fixing for A-Field . . . . . . . . . . . . . . . . . . . 177

Multi-Turn Coil . . . . . . . . . . . . . . . . . . . . . . 178

Reference Edge . . . . . . . . . . . . . . . . . . . . . . 182

Automatic Current Calculation . . . . . . . . . . . . . . . . 183Electric Insulation . . . . . . . . . . . . . . . . . . . . . 184

Input . . . . . . . . . . . . . . . . . . . . . . . . . . 184

Output . . . . . . . . . . . . . . . . . . . . . . . . . 185

Harmonic Perturbation . . . . . . . . . . . . . . . . . . . 185

Single-Turn Coil . . . . . . . . . . . . . . . . . . . . . . 186

Gap Feed . . . . . . . . . . . . . . . . . . . . . . . . . 188

Boundary Feed . . . . . . . . . . . . . . . . . . . . . . 189

Ground . . . . . . . . . . . . . . . . . . . . . . . . . 190

Domain Group . . . . . . . . . . . . . . . . . . . . . . 190

Reversed Current Direction . . . . . . . . . . . . . . . . . 191

Lumped Port . . . . . . . . . . . . . . . . . . . . . . . 191

Lumped Element . . . . . . . . . . . . . . . . . . . . . . 194

Edge Current . . . . . . . . . . . . . . . . . . . . . . . 195

External Magnetic Vector Potential . . . . . . . . . . . . . . . 195

Impedance Boundary Condition . . . . . . . . . . . . . . . . 196

Transition Boundary Condition . . . . . . . . . . . . . . . . 198

Thin Low Permeability Gap . . . . . . . . . . . . . . . . . . 199

Magnetic Point Dipole . . . . . . . . . . . . . . . . . . . . 200

Magnetic Point Dipole (on Axis) . . . . . . . . . . . . . . . . 201

Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . 201

The Magnetic Field Formulation User Interface 203

Domain, Boundary, Point, and Pair Nodes for the Magnetic Field Formulation

User Interface . . . . . . . . . . . . . . . . . . . . . . . 205

Faradays Law . . . . . . . . . . . . . . . . . . . . . . . 207

T E N T S | vii

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 207

Magnetic Gauss Law . . . . . . . . . . . . . . . . . . . . 207

Magnetic Insulation . . . . . . . . . . . . . . . . . . . . . 208

Electric Field . . . . . . . . . . . . . . . . . . . . . . . 208

Surface Magnetic Current Density . . . . . . . . . . . . . . . 209

The Magnetic Fields, No Currents User Interface 210

Domain, Boundary, Point, and Pair Nodes for the Magnetic Fields, No Currents

User Interface . . . . . . . . . . . . . . . . . . . . . . . 212

Magnetic Flux Conservation. . . . . . . . . . . . . . . . . . 214

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 215

Magnetic Insulation . . . . . . . . . . . . . . . . . . . . . 215

Magnetic Scalar Potential . . . . . . . . . . . . . . . . . . . 216C O N

Magnetic Flux Density . . . . . . . . . . . . . . . . . . . . 216

Zero Magnetic Scalar Potential. . . . . . . . . . . . . . . . . 217

External Magnetic Flux Density . . . . . . . . . . . . . . . . 218

Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . 218

Thin Low Permeability Gap . . . . . . . . . . . . . . . . . . 219

The Rotating Machinery, Magnetic User Interface 221

Domain, Boundary, Edge, Point, and Pair Nodes for the Rotating Machinery,

Magnetic User Interface . . . . . . . . . . . . . . . . . . . 223

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 224

Electric Field Transformation . . . . . . . . . . . . . . . . . 225

Prescribed Rotation . . . . . . . . . . . . . . . . . . . . . 225

Prescribed Rotational Velocity . . . . . . . . . . . . . . . . . 225

Mixed Formulation Boundary . . . . . . . . . . . . . . . . . 226

The Magnetic and Electric Fields User Interface 227

About the Magnetic and Electric Field Interface Boundary Conditions . . 230

Domain, Boundary, Edge, Point, and Pair Nodes for the Magnetic and Electric

Fields User Interface . . . . . . . . . . . . . . . . . . . . 233

Ampres Law and Current Conservation . . . . . . . . . . . . 235

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 236

Magnetic Insulation . . . . . . . . . . . . . . . . . . . . . 236

Surface Current . . . . . . . . . . . . . . . . . . . . . . 237

Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . 237

Magnetic Continuity. . . . . . . . . . . . . . . . . . . . . 238

viii | C O N T E N T S

RLC Coil Group . . . . . . . . . . . . . . . . . . . . . . 239

Theory of Magnetic and Electric Fields 243

Maxwells Equations . . . . . . . . . . . . . . . . . . . . . 243

Magnetic and Electric Potentials . . . . . . . . . . . . . . . . 243

Gauge Transformations . . . . . . . . . . . . . . . . . . . 244

Selecting a Particular Gauge. . . . . . . . . . . . . . . . . . 244

The Gauge and the Equation of Continuity for Dynamic Fields. . . . . 245

Explicit Gauge Fixing/Divergence Constraint . . . . . . . . . . . 245

Ungauged Formulations and Current Conservation . . . . . . . . . 246

Time-Harmonic Magnetic Fields . . . . . . . . . . . . . . . . 246

Theory for the Magnetic Fields User Interface 248

C h aMagnetostatics Equation . . . . . . . . . . . . . . . . . . . 248

Frequency Domain Equation . . . . . . . . . . . . . . . . . 249

Transient Equation . . . . . . . . . . . . . . . . . . . . . 249

Theory for the Magnetic Field Formulation User Interface 251

Equation System . . . . . . . . . . . . . . . . . . . . . . 251

Theory for the Magnetic Fields, No Currents User Interface 254

Theory for the Magnetic and Electric Fields User Interface 255

Magnetostatics Equations . . . . . . . . . . . . . . . . . . . 255

Frequency Domain Equations . . . . . . . . . . . . . . . . . 256

p t e r 6 : T h e E l e c t r i c a l C i r c u i t U s e r I n t e r f a c e

The Electrical Circuit User Interface 258

Ground Node . . . . . . . . . . . . . . . . . . . . . . . 259

Resistor . . . . . . . . . . . . . . . . . . . . . . . . . 259

Capacitor. . . . . . . . . . . . . . . . . . . . . . . . . 260

Inductor . . . . . . . . . . . . . . . . . . . . . . . . . 260

Voltage Source. . . . . . . . . . . . . . . . . . . . . . . 260

Current Source . . . . . . . . . . . . . . . . . . . . . . 261

Voltage-Controlled Voltage Source . . . . . . . . . . . . . . . 262

N T E N T S | ix

Voltage-Controlled Current Source . . . . . . . . . . . . . . . 262

Current-Controlled Voltage Source . . . . . . . . . . . . . . . 263

Current-Controlled Current Source . . . . . . . . . . . . . . 263

Subcircuit Definition . . . . . . . . . . . . . . . . . . . . 264

Subcircuit Instance . . . . . . . . . . . . . . . . . . . . . 264

NPN BJT . . . . . . . . . . . . . . . . . . . . . . . . . 264

n-Channel MOSFET . . . . . . . . . . . . . . . . . . . . . 265

Diode . . . . . . . . . . . . . . . . . . . . . . . . . . 266

External I vs. U . . . . . . . . . . . . . . . . . . . . . . 266

External U vs. I . . . . . . . . . . . . . . . . . . . . . . 267

External I-Terminal . . . . . . . . . . . . . . . . . . . . . 268

SPICE Circuit Import . . . . . . . . . . . . . . . . . . . . 269

C h a

C h aC O

Theory for the Electrical Circuit User Interface 271

Electric Circuit Modeling and the Semiconductor Device Models. . . . 271

NPN Bipolar Transistor . . . . . . . . . . . . . . . . . . . 272

n-Channel MOS Transistor . . . . . . . . . . . . . . . . . . 274

Diode . . . . . . . . . . . . . . . . . . . . . . . . . . 277

SPICE Import . . . . . . . . . . . . . . . . . . . . . . . 280

References for the Electrical Circuit User Interface . . . . . . . . . 280

p t e r 7 : T h e H e a t T r a n s f e r B r a n c h

The Induction Heating User Interface 282

Domain, Boundary, Edge, Point, and Pair Nodes for the Induction Heating User

Interface . . . . . . . . . . . . . . . . . . . . . . . . . 285

Induction Heating Model . . . . . . . . . . . . . . . . . . . 287

Electromagnetic Heat Source . . . . . . . . . . . . . . . . . 288

Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 289

Coil Group Domain. . . . . . . . . . . . . . . . . . . . . 289

Reversed Current Direction . . . . . . . . . . . . . . . . . 290

p t e r 8 : G l o s s a r y

Glossary of Terms 294

x | C O N T E N T S

1

1

About the AC/DC Module

Overview of the Users GuideI n t r o d u c t i o n

This guide describes the AC/DC Module, an optional add-on package for COMSOL Multiphysics designed to assist you to solve and model low-frequency electromagnetics.

This chapter introduces you to the capabilities of the module including an introduction to the modeling stages and some realistic and illustrative models. A summary of the physics interfaces and where you can find documentation and model examples is also included. The last section is a brief overview with links to each chapter in this guide.

In this chapter:

2 | C H A P T E R 1 : I N T R

Abou t t h e AC /DC Modu l e

In this section:

What Can the AC/DC Module Do?

AC/DC Module Physics Guide

Where Do I Access the Documentation and Model Library?

W

Telcoqin

Tsi

Mgpvip

Lqda

TothqO D U C T I O N

hat Can the AC/DC Module Do?

he AC/DC Module provides a unique environment for simulation of AC/DC ectromagnetics in 2D and 3D. The module is a powerful tool for detailed analysis of ils, capacitors, and electrical machinery. With this module you can run static,

uasi-static, transient, and time-harmonic simulations in an easy-to-use graphical user terface.

he available physics interfaces cover the following types of electromagnetics field mulations:

Electrostatics

Electric currents in conductive media

Magnetostatics

Low-frequency electromagnetics

aterial properties include inhomogeneous and fully anisotropic materials, media with ains or losses, and complex-valued material properties. Infinite elements makes it ossible to model unbounded domains. In addition to the standard results and sualization functionality, the module supports direct computation of lumped arameters such as capacitances and inductances as well as electromagnetic forces.

ike all COMSOL modules, there is a library of ready-to-run models that make it uicker and easier to analyze discipline-specific problems. In addition, any model you evelop is described in terms of the underlying partial differential equations, offering unique way to see the underlying physical laws of a simulation.

he AC/DC physics interfaces are fully multiphysics enabledcouple them to any ther interface in COMSOL Multiphysics or the other modules. For example, to find e heat distribution in a motor, first find the current in the coils using one of the

uasi-static interfaces in this module and then couple it to a heat equation in the main

L E | 3

COMSOL Multiphysics package or the Heat Transfer Module. This forms a powerful multiphysics model that solves all the equations simultaneously.

The AC/DC Module also provides interfaces for modeling electrical circuits.

A

Tfodefieelpo

Esideth

Tcrmin

Insy

Building a COMSOL Model in the COMSOL Multiphysics Reference Manual

AC/DC Module Physics GuideA B O U T T H E A C / D C M O D U

C/DC Module Physics Guide

he physics interfaces in the AC/DC Module form a complete set of simulation tools r electromagnetic field simulations. To select the right physics interface for scribing the real-life physics, the geometric properties and the time variations of the lds need to be considered. The interfaces solve for these physical quantitiesthe

ectric scalar potential V, the magnetic vector potential A, and the magnetic scalar tential Vm.

ach interface has a tag which is of special importance when performing multiphysics mulations. This tag helps distinguish between physics interfaces and the variables fined by the interface have an underscore plus the physics interface tag appended to eir names.

he Model Wizard is an easy way to select the physics interface and study type when eating a model for the first time, and physics interfaces can be added to an existing odel at any time. Full instructions for selecting interfaces and setting up a model are the COMSOL Multiphysics Reference Manual.

2D, in-plane and out-of-plane variants are available for problems with a planar mmetry as well as axisymmetric interfaces for problems with a cylindrical symmetry.

Where Do I Access the Documentation and Model Library?

When using an axisymmetric interface it is important that the horizontal axis represents the r direction and the vertical axis the z direction, and that the geometry in the right half-plane (that is, for positive r only) must be created.

4 | C H A P T E R 1 : I N T R

See What Problems Can You Solve? and Table 1-1 for information about the available study types and variables. See also Overview of the Users Guide for links to the chapters in this guide.

In the COMSOL Multiphysics Reference Manual:

Studies and the Study Nodes

The Physics User Interfaces

PHYSI

Elect

Elect

Elect

Elect

Magn

MagnForm

MagnCurr

MagnFieldO D U C T I O N

For a list of all the interfaces included with the COMSOL Multiphysics

basic license, see Physics Guide.

CS USER INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

AC/DC

ric Currents* ec all dimensions stationary; frequency domain; time dependent; small signal analysis, frequency domain

ric Currents - Shell ecs 3D, 2D, 2D axisymmetric

stationary; frequency domain; time dependent; small signal analysis, frequency domain

rical Circuit cir Not space dependent

stationary; frequency domain; time dependent

rostatics* es all dimensions stationary; time dependent; eigenfrequency; frequency domain; small signal analysis, frequency domain

etic Fields* mf 3D, 2D, 2D axisymmetric

stationary; frequency domain; time dependent; small signal analysis, frequency domain; coil current calculation (3D only)

etic Field ulation

mfh 3D, 2D, 2D axisymmetric

stationary; frequency domain; time dependent; small signal analysis, frequency domain

etic Fields, No ents

mfnc 3D, 2D, 2D axisymmetric

stationary; time dependent

etic and Electric s

mef 3D, 2D, 2D axisymmetric

stationary; frequency domain

L E | 5

A

Rotating Machinery, Magnetic

rmm 3D, 2D stationary; time dependent, coil current calculation (3D only)

Heat Transfer

Electromagnetic Heating

In

* Thiadde

PHYSICS USER INTERFACE ICON TAG SPACE DIMENSION

AVAILABLE PRESET STUDY TYPE

TABLE

PHYSI

Elect

Elect

ElectShell

MagnA B O U T T H E A C / D C M O D U

C / D C M O D U L E S T U D Y A V A I L A B I L I T Y

duction Heating ih 3D, 2D, 2D axisymmetric

stationary; time dependent; frequency-stationary; frequency-transient

s is an enhanced interface, which is included with the base COMSOL package but has d functionality for this module.

1-1: AC/DC MODULE DEPENDENT VARIABLES, FIELD COMPONENTS, AND PRESET STUDY AVAILABILITY

CS INTERFACE TAG DEPENDENT VARIABLES

FIELD COMPONENTS

* PRESET STUDIES**

MA

GN

ET

IC F

IEL

D

EL

EC

TR

IC F

IEL

D

MA

GN

ET

IC P

OT

EN

TIA

L

CU

RR

EN

T D

EN

SIT

Y

ST

AT

ION

AR

Y

TIM

E D

EP

EN

DE

NT

FR

EQ

UE

NC

Y D

OM

AIN

SM

AL

L-S

IGN

AL

AN

AL

YS

IS,

FR

EQ

UE

NC

Y D

OM

AIN

FR

EQ

UE

NC

Y-S

TA

TIO

NA

RY

FR

EQ

UE

NC

Y-T

RA

NS

IEN

T

rostatics es V x y z

ric Currents ec V x y z

x y z

ric Currents, ecs V x y z

x y z

etic Fields mf A x y z

x y z

x y z

x y z

6 | C H A P T E R 1 : I N T R

MagnForm

MagnElect

MagnCurr

RotaMagn

Elect

Induc

*Thefor c**CuEigen

TABLE 1-1: AC/DC MODULE DEPENDENT VARIABLES, FIELD COMPONENTS, AND PRESET STUDY AVAILABILITY

PHYSICS INTERFACE TAG DEPENDENT VARIABLES

FIELD COMPONENTS

* PRESET STUDIES**

TIC

FIE

LD

IC F

IEL

D

TIC

PO

TE

NT

IAL

T D

EN

SIT

Y

NA

RY

PE

ND

EN

T

NC

Y D

OM

AIN

IGN

AL

AN

AL

YS

IS,

NC

Y D

OM

AIN

NC

Y-S

TA

TIO

NA

RY

NC

Y-T

RA

NS

IEN

TO D U C T I O N

etic Field ulation

mfh H x y z

etic and ric Fields

mef V, A x y z

x y z

x y z

x y z

etic Fields, No ents

mfnc Vm x y z

ting Machinery, etic

rmm A, Vm x y z

x y z

x y z

x y z

rical Circuit cir not applicable

tion Heating ih A,T, J x y z

x y z

x y z

x y z

se are the nonzero field components. For Cartesian coordinates, these are indexed by x, y, and z; ylindrical coordinates, r, , and z are used.stom studies are also available based on the interface, for example, Eigenfrequency and value.

Studies and Solvers in the COMSOL Multiphysics Reference Manual

What Can the AC/DC Module Do?

AC/DC Module Physics Guide

Where Do I Access the Documentation and Model Library?

MA

GN

E

EL

EC

TR

MA

GN

E

CU

RR

EN

ST

AT

IO

TIM

E D

E

FR

EQ

UE

SM

AL

L-S

FR

EQ

UE

FR

EQ

UE

FRE

QU

E

L E | 7

S H O W M O R E P H Y S I C S O P T I O N S

There are several general options available for the physics user interfaces and for individual nodes. This section is a short overview of these options, and includes links to additional information when available.

Ttrop

Awmdian

Yosodi

FavEq

Ath

The links to the features described in the COMSOL Multiphysics Reference Manual (or any external guide) do not work in the PDF, only from within the online help. A B O U T T H E A C / D C M O D U

o display additional options for the physics interfaces and other parts of the model ee, click the Show button ( ) on the Model Builder and then select the applicable tion.

fter clicking the Show button ( ), additional sections get displayed on the settings indow when a node is clicked and additional nodes are available from the context enu when a node is right-clicked. For each, the additional sections that can be splayed include Equation, Advanced Settings, Discretization, Consistent Stabilization, d Inconsistent Stabilization.

u can also click the Expand Sections button ( ) in the Model Builder to always show me sections or click the Show button ( ) and select Reset to Default to reset to splay only the Equation and Override and Contribution sections.

or most nodes, both the Equation and Override and Contribution sections are always ailable. Click the Show button ( ) and then select Equation View to display the uation View node under all nodes in the Model Builder.

vailability of each node, and whether it is described for a particular node, is based on e individual selected. For example, the Discretization, Advanced Settings, Consistent

To locate and search all the documentation for this information, in COMSOL Multiphysics, select Help>Documentation from the main menu and either enter a search term or look under a specific module in the documentation tree.

8 | C H A P T E R 1 : I N T R

Stabilization, and Inconsistent Stabilization sections are often described individually throughout the documentation as there are unique settings.

O

AoD

AoSar

SECTION CROSS REFERENCE

Show More Options and Expand Sections

Advanced Physics Sections

The Model Wizard and Model Builder

Discretization Show Discretization

Discretization (Node)

Dc

CIn

C

O

S

CS

Da

In

MO D U C T I O N

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

t the main level, some of the common settings found (in addition to the Show ptions) are the Interface Identifier, Domain, Boundary, or Edge Selection, and ependent Variables.

t the nodes level, some of the common settings found (in addition to the Show ptions) are Domain, Boundary, Edge, or Point Selection, Material Type, Coordinate ystem Selection, and Model Inputs. Other sections are common based on application ea and are not included here.

iscretizationSplitting of omplex variables

Compile Equations

onsistent and consistent Stabilization

Show Stabilization

Numerical Stabilization

onstraint Settings Weak Constraints and Constraint Settings

verride and Contribution Physics Exclusive and Contributing Node Types

ECTION CROSS REFERENCE

oordinate System election

Coordinate Systems

omain, Boundary, Edge, nd Point Selection

About Geometric Entities

About Selecting Geometric Entities

terface Identifier Predefined Physics Variables

Variable Naming Convention and Scope

Viewing Node Names, Identifiers, Types, and Tags

aterial Type Materials

L E | 9

W

Alicth

T

Tfuhael

T

Model Inputs About Materials and Material Properties

Selecting Physics

Adding Multiphysics Couplings

Pair Selection Identity and Contact Pairs

Continuity on Interior Boundaries

SECTION CROSS REFERENCEA B O U T T H E A C / D C M O D U

here Do I Access the Documentation and Model Library?

number of Internet resources provide more information about COMSOL, including ensing and technical information. The electronic documentation, context help, and e Model Library are all accessed through the COMSOL Desktop.

H E D O C U M E N T A T I O N

he COMSOL Multiphysics Reference Manual describes all user interfaces and nctionality included with the basic COMSOL Multiphysics license. This book also s instructions about how to use COMSOL and how to access the documentation

ectronically through the COMSOL Help Desk.

o locate and search all the documentation, in COMSOL Multiphysics:

Press F1 or select Help>Help ( ) from the main menu for context help.

If you are reading the documentation as a PDF file on your computer, the blue links do not work to open a model or content referenced in a different guide. However, if you are using the online help in COMSOL Multiphysics, these links work to other modules, model examples, and documentation sets.

10 | C H A P T E R 1 : I N T

Press Ctrl+F1 or select Help>Documentation ( ) from the main menu for opening the main documentation window with access to all COMSOL documentation.

Click the corresponding buttons ( or ) on the main toolbar.

and then either enter a search term or look under a specific module in the documentation tree.

T

Estasinap

Ind

Tthandbb

Tmu

Ifth

If you have added a node to a model you are working on, click the Help R O D U C T I O N

H E M O D E L L I B R A R Y

ach model comes with documentation that includes a theoretical background and ep-by-step instructions to create the model. The models are available in COMSOL MPH-files that you can open for further investigation. You can use the step-by-step structions and the actual models as a template for your own modeling and plications.

most models, SI units are used to describe the relevant properties, parameters, and imensions in most examples, but other unit systems are available.

o open the Model Library, select View>Model Library ( ) from the main menu, and en search by model name or browse under a module folder name. Click to highlight y model of interest, and select Open Model and PDF to open both the model and the

ocumentation explaining how to build the model. Alternatively, click the Help utton ( ) or select Help>Documentation in COMSOL to search by name or browse y module.

he model libraries are updated on a regular basis by COMSOL in order to add new odels and to improve existing models. Choose View>Model Library Update ( ) to

pdate your model library to include the latest versions of the model examples.

you have any feedback or suggestions for additional models for the library (including ose developed by you), feel free to contact us at info@comsol.com.

button ( ) in the nodes settings window or press F1 to learn more about it. Under More results in the Help window there is a link with a search string for the nodes name. Click the link to find all occurrences of the nodes name in the documentation, including model documentation and the external COMSOL website. This can help you find more information about the use of the nodes functionality as well as model examples where the node is used.

E | 11

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

For general product information, contact COMSOL at info@comsol.com.

To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to support@comsol.com. An automatic notification and case number is sent to you by email.

C O M S O L WE B S I T E S

C

C

S

D

S

P

CA B O U T T H E A C / D C M O D U L

OMSOL website www.comsol.com

ontact COMSOL www.comsol.com/contact

upport Center www.comsol.com/support

ownload COMSOL www.comsol.com/support/download

upport Knowledge Base www.comsol.com/support/knowledgebase

roduct Updates www.comsol.com/support/updates

OMSOL Community www.comsol.com/community

12 | C H A P T E R 1 : I N T

Ove r v i ew o f t h e U s e r s Gu i d e

The AC/DC Module Users Guide gets you started with modeling using COMSOL Multiphysics. The information in this guide is specific to this module. Instructions how to use COMSOL in general are included with the COMSOL Multiphysics Reference Manual.

T

T

T

Inthino

M

InthME

E

Tu

R O D U C T I O N

A B L E O F C O N T E N T S , G L O S S A R Y , A N D I N D E X

o help you navigate through this guide, see the Contents, Glossary, and Index.

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

the Review of Electromagnetics chapter contains an overview of the theory behind e AC/DC Module. It is intended for readers that wish to understand what goes on the background when using the physics interfaces and discusses the Fundamentals f Electromagnetics, Electromagnetic Forces, and Electromagnetic Quantities.

O D E L I N G W I T H T H E A C / D C M O D U L E

the Modeling with the AC/DC Module chapter, the goal is to familiarize you with e modeling procedure using this particular module. Topics include Preparing for odeling, Force and Torque Computations, Lumped Parameters, Connecting to

lectrical Circuits.

L E C T R I C F I E L D S

he Electric Field User Interfaces chapter describes these interfaces and includes the nderlying theory for each interface at the end of the chapter:

The Electrostatics User Interface, which simulates electric fields in dielectric materials with a fixed or slowly-varying charge present. Preset stationary, time dependent, frequency domain, and small-signal analysis study types are available.

As detailed in the section Where Do I Access the Documentation and Model Library? this information can also be searched from the COMSOL Multiphysics software Help menu.

E | 13

The Electric Currents User Interface, which simulates the current in a conductive and capacitive material under the influence of an electric field. All three study types (stationary, frequency domain, and time dependent) are available.

The Electric Currents, Shell User Interface, which simulates the current in a conductive and capacitive shell under the influence of an electric field. All three study types (stationary, frequency domain and time dependent) are available.

M A G N E T I C F I E L D S

Tun

E

TeqO V E R V I E W O F T H E U S E R S G U I D

he Magnetic Field User Interfaces chapter describes these interfaces and includes the derlying theory for each interface at the end of the chapter:

The Magnetic Fields User Interface, which handles problems for magnetic fields with prescribed currents, solving for the magnetic vector potential. All three study types (stationary, frequency domain, and time dependent) are available. This is the recommended primary choice for modeling of magnetic fields involving source currents.

The Magnetic Field Formulation User Interface has the equations, boundary conditions, and currents for modeling magnetic fields, solving for the magnetic field. It is especially suitable for modeling involving nonlinear conductivity effects, for example in superconductors. All three study types (stationary, frequency domain, and time dependent) are available.

The Magnetic Fields, No Currents User Interface, which handles magnetic fields without currents. When no currents are present, the problem is easier to solve using the magnetic scalar potential. The stationary and time dependent study types are available.

The Rotating Machinery, Magnetic User Interface combines a Magnetic Fields formulation (magnetic vector potential) and Magnetic Fields, No Currents formulation (magnetic scalar potential) with a selection of predefined frames for prescribed rotation or rotational velocity - most of its features are taken either from the Magnetic Fields or the Magnetic Fields, No Currents interfaces.

The Magnetic and Electric Fields User Interface handles problems for magnetic and electric fields. It is based on the magnetic vector potential and the electric scalar potential. The stationary and frequency domain study types are available.

L E C T R I C A L C I R C U I T

he Electrical Circuit User Interface chapter describes the interface, which has the uations for modeling electrical circuits with or without connections to a distributed

14 | C H A P T E R 1 : I N T

fields model, solving for the voltages, currents, and charges associated with the circuit elements. The underlying theory for the interface is included at the end of the chapter.

H E A T TR A N S F E R

The Heat Transfer Branch chapter describes the interface, which combines all physics features from the Magnetic Fields interface in the time harmonic formulation with the Heat Transfer interface for modeling of induction and eddy current heating.

Heat transfer through conduction and convection in solids and free media (fluids) is suR O D U C T I O N

pported by physics interfaces shipped with the basic COMSOL Multiphysics license.

The Heat Transfer User Interface, The Joule Heating User Interface, and Heat Transfer Theory in the COMSOL Multiphysics Reference Manual.

15

2R e v i e w o f E l e c t r o m a g n e t i c s

This chapter contains an overview of the theory behind the AC/DC Module. It is intended for readers that wish to understand what goes on in the background when using the physics interfaces.

In this chapter:

Fundamentals of Electromagnetics

Electromagnetic Forces

Electromagnetic Quantities

References for the AC/DC User Interfaces

16 | C H A P T E R 2 : R E V

Fundamen t a l s o f E l e c t r omagne t i c s

In this section:

Maxwells Equations

Constitutive Relations

Potentials

M

TMa b

TfomwI E W O F E L E C T R O M A G N E T I C S

Reduced Potential PDE Formulations

Electromagnetic Energy

The Quasi-Static Approximation and the Lorentz Term

Material Properties

About the Boundary and User Interface Conditions

Phasors

axwells Equations

he problem of electromagnetic analysis on a macroscopic level is that of solving axwells equations subject to certain boundary conditions. Maxwells equations are set of equations, written in differential or integral form, stating the relationships etween the fundamental electromagnetic quantities. These quantities are:

Electric field intensity E

Electric displacement or electric flux density D

Magnetic field intensity H

Magnetic flux density B

Current density J

Electric charge density he equations can be formulated in differential form or integral form. The differential rm is presented here because it leads to differential equations that the finite element ethod can handle. For general time-varying fields, Maxwells equations can be ritten as:

S | 17

The first two equations are also referred to as Maxwell-Ampres law and Faradays lam

A

Ocofo

C

Tth

wel4th

Tas

Tel

H J Dt-------+=

E Bt-------=

D = B 0=F U N D A M E N T A L S O F E L E C T R O M A G N E T I C

w, respectively. Equation three and four are two forms of Gauss law: the electric and agnetic form, respectively.

nother fundamental equation is the equation of continuity

ut of the five equations mentioned, only three are independent. The first two mbined with either the electric form of Gauss law or the equation of continuity rm such an independent system.

onstitutive Relations

o obtain a closed system, the equations include constitutive relations that describe e macroscopic properties of the medium. They are given as

(2-1)

here 0 is the permittivity of vacuum, 0 is the permeability of vacuum, and the ectrical conductivity. In the SI system, the permeability of vacuum is chosen to be 107 H/m. The velocity of an electromagnetic wave in a vacuum is given as c0 and e permittivity of a vacuum is derived from the relation:

he electromagnetic constants 0, 0, and c0 are available in COMSOL Multiphysics predefined physical constants.

he electric polarization vector P describes how the material is polarized when an ectric field E is present. It can be interpreted as the volume density of electric dipole

J t------=

D 0E P+=B 0 H M+ =

J E=

0 1c0

20---------- 8.854 10 12 F/m 1

36--------- 109 F/m= =

18 | C H A P T E R 2 : R E V

moments. P is generally a function of E. Some materials can have a nonzero P also when there is no electric field present.

The magnetization vector M similarly describes how the material is magnetized when a magnetic field H is present. It can be interpreted as the volume density of magnetic dipole moments. M is generally a function of H. Permanent magnets, for instance, have a nonzero M also when there is no magnetic field present.

For linear materials, the polarization is directly proportional to the electric field, Pmth

Tmtear

G

FTd

S

ww

FI E W O F E L E C T R O M A G N E T I C S

0 e E , where e is the electric susceptibility. Similarly in linear materials, the agnetization is directly proportional to the magnetic field, Mm H , where m is e magnetic susceptibility. For such materials, the constitutive relations are:

he parameter r is the relative permittivity and r is the relative permeability of the aterial. Usually these are scalar properties but can, in the general case, be 3-by-3 nsors when the material is anisotropic. The properties and (without subscripts) e the permittivity and permeability of the material, respectively.

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

or nonlinear materials, a generalized form of the constitutive relationships is useful. he relationship used for electric fields is D orE + Dr where Dr is the remanent isplacement, which is the displacement when no electric field is present.

imilarly, a generalized form of the constitutive relation for the magnetic field is

here Br is the remanent magnetic flux density, which is the magnetic flux density hen no magnetic field is present.

or some materials, there is a nonlinear relationship between B and H such that

D 0 1 e+ E 0rE E= = =B 0 1 m+ H 0rH H= = =

The Charge Conservation node describes the macroscopic properties of the medium (relating the electric displacement D with the electric field E) and the applicable material properties.

B 0rH Br+=

B f H =

S | 19

The relation defining the current density is generalized by introducing an externally generated current Je. The resulting constitutive relation is J E + Je.

Potentials

Under certain circumstances it can be helpful to formulate the problems in terms of the electric scalar potential V and the magnetic vector potential A. They are given by the equalities:

Tmmrepo

R

TexexantoreAM

D

TFpoF U N D A M E N T A L S O F E L E C T R O M A G N E T I C

he defining equation for the magnetic vector potential is a direct consequence of the agnetic Gauss law. The electric potential results from Faradays law. In the agnetostatic case where there are no currents present, Maxwell-Ampres law duces to H0. When this holds, it is also possible to define a magnetic scalar tential by the relation H Vm.

educed Potential PDE Formulations

he reduced potential option is useful for models involving a uniform or known ternal background field, usually originating from distant sources that may be pensive or inconvenient to include in the model geometry. A typical example is when alyzing induced magnetization in ferromagnetic objects such as ships or vehicles due the Earths magnetic field. The strategy is then to solve only for the induced fields presented by the reduced vector potential Ared, introducing the substitution AredAext, where Aext represents the known background field, into axwell-Ampres law:

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

ime-Harmonicor time-harmonic quasi-static systems solving for an A formulation, the reduced tential formulation results in the following PDE:

B A=E V A

t-------=

1 A Jtd

dD+=

j 2 Aext Ared+ 1 Aext Ared+ + Je=

20 | C H A P T E R 2 : R E V

Here it is possible to interpret the term Aext as an additional remanent magnetic flux density and the term (j2Aext as an additional external current source.TransientSimilarly to the time-harmonic formulation, in the transient formulation, the above substitution results in the reduced equation

SIn

Inm

E

T

T

Tth

t Aext Ared+ 1 Aext Ared+ + Je=I E W O F E L E C T R O M A G N E T I C S

tatic static formulations, the induced current is zero. Maxwell-Ampres law reduces to:

this case it is also possible to express the external field through a known external agnetic flux density, Bext. The domain equation in reduced form then reads:

lectromagnetic Energy

he electric and magnetic energies are defined as:

he time derivatives of these expressions are the electric and magnetic power:

hese quantities are related to the resistive and radiative energy, or energy loss, rough Poyntings theorem (Ref. 1)

1 Aext Ared+ Je=

1 Ared Bext+ Je=

We E Dd0

D VdV E D t------- td0T VdV= =

Wm H Bd0

B VdV H B t------- td0T VdV= =

Pe EDt------- VdV=

Pm HBt------- VdV=

E Dt------- H

Bt-------+

VdV J E VdV E H n dSS+=

S | 21

where V is the computation domain and S is the closed boundary of V.

The first term on the right-hand side represents the resistive losses,

which result in heat dissipation in the material. (The current density J in this expression is the one appearing in Maxwell-Ampres law.)

Tlo

T

U

Bthre

T

T

Aarfiespca

Ph J E VdV=F U N D A M E N T A L S O F E L E C T R O M A G N E T I C

he second term on the right-hand side of Poyntings theorem represents the radiative sses,

he quantity SE H is called the Poynting vector.nder the assumption the material is linear and isotropic, it holds that:

y interchanging the order of differentiation and integration (justified by the fact that e volume is constant and the assumption that the fields are continuous in time), the sult is:

he integrand of the left-hand side is the total electromagnetic energy density:

he Quasi-Static Approximation and the Lorentz Term

consequence of Maxwells equations is that changes in time of currents and charges e not synchronized with changes of the electromagnetic fields. The changes of the lds are always delayed relative to the changes of the sources, reflecting the finite eed of propagation of electromagnetic waves. Under the assumption that this effect n be ignored, it is possible to obtain the electromagnetic fields by considering

Pr E H n dSS=

E Dt------- E

Et------- t

12---E E = =

H Bt-------

1---B

Bt------- t

12-------B B

= =

t 1

2---E E 1

2-------B B+ Vd

V J E VdV E H n dSS+=

w we wm+=12---E E 1

2-------B B+=

22 | C H A P T E R 2 : R E V

stationary currents at every instant. This is called the quasi-static approximation. The approximation is valid provided that the variations in time are small and that the studied geometries are considerably smaller than the wavelength (Ref. 5).

The quasi-static approximation implies that the equation of continuity can be written as Jand that the time derivative of the electric displacement Dt can be disregarded in Maxwell-Ampres law.

There are also effects of the motion of the geometries. Consider a geometry moving wg

Tpm

w

M

w

M

UTmb

AI E W O F E L E C T R O M A G N E T I C S

ith velocity v relative to the reference system. The force per unit charge, Fq, is then iven by the Lorentz force equation:

his means that to an observer traveling with the geometry, the force on a charged article can be interpreted as caused by an electric field E'EvB. In a conductive edium, the observer accordingly sees the current density

here Je is an externally generated current density.

axwell-Ampres law for quasi-static systems is consequently extended to

hereas Faradays law remains unchanged.

aterial Properties

ntil now, there has only been a formal introduction of the constitutive relations. hese seemingly simple relations can be quite complicated at times. There are four ain groups of materials where they require some consideration. A given material can

elong to one or more of these groups. The groups are:

Inhomogeneous Materials

Anisotropic Materials

Nonlinear Materials

Dispersive Materials

material can belong to one or more of these groups.

Fq---- E v B+=

J E v B+ Je+=

H E v B+ Je+=

S | 23

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

Inhomogeneous materials are the least complicated. An inhomogeneous medium is one in which the constitutive parameters vary with the space coordinates so that different field properties prevail at different parts of the material structure.

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

For anisotropic materials the field relationships at any point differ for different directions of propagation. This means that a 3-by-3 tensor is necessary to properly derediubiexso

N

Nofonal

D

Ddo

A

TbemF U N D A M E N T A L S O F E L E C T R O M A G N E T I C

fine the constitutive relationships. If this tensor is symmetric, the material is often ferred to as reciprocal. In such cases, rotate the coordinate system such that a agonal matrix results. If two of the diagonal entries are equal, the material is niaxially anisotropic. If none of the elements has the same value, the material is axially anisotropic (Ref. 2). Anisotropic parameters are needed, for example, to amine permittivity in crystals (Ref. 2) and when working with conductivity in lenoids.

O N L I N E A R M A T E R I A L S

onlinearity is the effect of variations in permittivity or permeability with the intensity the electromagnetic field. Nonlinearity also includes hysteresis effects, where not ly the current field intensities influence the physical properties of the material, but

so the history of the field distribution.

I S P E R S I V E M A T E R I A L S

ispersion describes changes in a waves velocity with wavelength. In the frequency main dispersion is expressed with a frequency dependence of the constitutive laws.

bout the Boundary and User Interface Conditions

o get a full description of an electromagnetics problem, boundary conditions must specified at material interfaces and physical boundaries. At interfaces between two edia, the boundary conditions can be expressed mathematically as

n2 E1 E2 0=n2 D1 D2 s=n2 H1 H2 Js=n2 B1 B2 0=

24 | C H A P T E R 2 : R E V

where s and Js denote surface charge density and surface current density, respectively, and n2 is the outward normal from medium two. Of these four conditions, only two are independent. This is an overdetermined system of equations, so it needs to be reduced. First select either equation one or equation four. Then select either equation two or equation three. Together these selections form a set of two independent conditions.

From these relationships, the interface condition is derived for the current density,

I N

AfifucosureHseca

P

W

InuI E W O F E L E C T R O M A G N E T I C S

T E R F A C E B E T W E E N A D I E L E C T R I C A N D A P E R F E C T C O N D U C T O R

perfect conductor has infinite electrical conductivity and thus no internal electric eld. Otherwise, it would produce an infinite current density according to the third ndamental constitutive relation. At an interface between a dielectric and a perfect nductor, the boundary conditions for the E and D fields are simplified. Assume that bscript 1 corresponds to a perfect conductor; then D10 and E10 in the lationships just given. If, in addition, it is a time-varying case, then B10 and 10, as well, as a consequence of Maxwells equations. The result is the following t of boundary conditions for the fields in the dielectric medium for the time-varying se:

hasors

henever a problem is time-harmonic the fields can be written in the form:

stead of using a cosine function for the time dependence, it is more convenient to se an exponential function, by writing the field as:

n2 J1 J2 st--------=

n 2 E2 0=n 2 H2 Js=n 2 D2 s=n 2 B2 0=

E r t E r t + cos=

E r t E r t + cos Re E r ejejt Re E r ejt = = =

S | 25

The field is a phasor, which contains amplitude and phase information of the field but is independent of t. One thing that makes the use of phasors suitable is that a time derivative corresponds to a multiplication by j,

This means that an equation for the phasor can be derived from a time-dependent equation by replacing the time derivatives by a factor j. All time-harmonic equations indr

E r

Et------- Re jE r e

jt =F U N D A M E N T A L S O F E L E C T R O M A G N E T I C

the AC/DC Module are expressed as equations for the phasors. (The tilde is opped from the variable denoting the phasor.)

When analyzing the solution of a time-harmonic equation, it is important to remember that the field that has been calculated is a phasor and not a physical field.

For example, all plot functions visualize by default, which is E at time t0. To obtain the solution at a given time, specify a phase factor in all results pages and in the corresponding functions.

Re E r

26 | C H A P T E R 2 : R E V

E l e c t r omagne t i c F o r c e s

There are several ways to compute electromagnetic forces in COMSOL Multiphysics. In the most general case, the calculation of electromagnetic forces involves the computation of volume forces acting on a body, and of surface forces originating from jumps in the electromagnetic fields on the boundaries. The volume and surface forces are derived from a general stress tensor that includes electromagnetic terms.

Ttheltefo

Ain

In

O

C

wteeqliI E W O F E L E C T R O M A G N E T I C S

he derivation of the expressions for the electromagnetic stress tensor utilizes ermodynamic potential (energy) principles (Ref. 1 and Ref. 3). The distribution of ectromagnetic forces in a system depends on the material. Accordingly, the chniques and expressions used when calculating electromagnetic forces are different r different types of materials.

nother technique for calculating forces using the method of virtual work is described the section Electromagnetic Energy and Virtual Work.

this section:

Overview of Forces in Continuum Mechanics

Forces on an Elastic Solid Surrounded by Vacuum or Air

Torque

Forces in Stationary Fields

Forces in a Moving Body

Electromagnetic Energy and Virtual Work

verview of Forces in Continuum Mechanics

auchys equation of continuum mechanics reads

here is the density, r denotes the coordinates of a material point, Tis the stress nsor, and fext is an external volume force such as gravity (fextg). This is the uation solved in the structural mechanics physics interfaces for the special case of a

near elastic material, neglecting the electromagnetic contributions.

t2

2

dd r T fext+=

S | 27

In the stationary case there is no acceleration, and the equation representing the force balance is

The stress tensor must be continuous across a stationary boundary between two materials. This corresponds to the equation

wisri

Inel

Ftoteif

Itis

T

0 T fext+=

n1 T2 T1 0=E L E C T R O M A G N E T I C F O R C E

here T1 and T2 represent the stress tensor in Materials 1 and 2, respectively, and n1 the normal pointing out from the domain containing Material 1. This relation gives se to a surface force acting on the boundary between Material 1 and 2.

certain cases, the stress tensor T can be divided into one part that depends on the ectromagnetic field quantities and one part that is the mechanical stress tensor,

or the special case of an elastic body, the mechanical stress tensor is proportional only the strain and the temperature gradient. The exact nature of this split of the stress nsor into an electromagnetic and a mechanical part depends on the material model, it can be made at all.

is sometimes convenient to use a volume force instead of the stress tensor. This force obtained from the relation

his changes the force balance equation to

Material 1

Material 2

n1

T TEM M+=

For more information on the mechanical stress tensor for elastic materials, see the documentation for the interfaces. For example, Structural Mechanics in the COMSOL Multiphysics Reference Manual.

fem TEM=

0 M fem fext+ +=

28 | C H A P T E R 2 : R E V

or, as stated in the structural mechanics physics interfaces,

Forces on an Elastic Solid Surrounded by Vacuum or Air

Consider a solid (Material 1) surrounded by vacuum (Material 2). It is natural to associate the surface force on the boundary between the materials with the solid. In m

Inbtoooinm

O

Tth

T

Fth

A

M f where f fem fext+==I E W O F E L E C T R O M A G N E T I C S

any applications air can be approximated by vacuum.

practice, the equation for the force balance also needs to include an external oundary force gext. It is nonzero on those parts of the boundary where it is necessary compensate for the contributions to the stress tensor that you are not interested in

r do not have enough information on. These contributions come from the influence f the adjacent domains. By approximating the surroundings by vacuum or air, the fluence of these boundaries and their adjacent domains (that are not part of our odel) on the electromagnetic fields are neglected.

n the boundary, the following equations apply:

he external boundary force gext can represent the reaction force from another body at the solid is attached to.

he equations for the balance of forces on the solid now become

or calculating the total force F on the solid these equations need to be integrated over e entire solid and the solid/vacuum boundary

ccording to Gauss theorem:

n1 T2 T1 0=n1T2 n1T2 gext+=

T1 fext+ 0=n1 T2 T1 gext+ 0=

T1 fext+ Vd1 n1 T2 T1 gext+ Sd

1+ 0=

T1 Vd1 n1T1 Sd

1 0=

S | 29

this means that the external force

is needed to balance the term for the boundary integral of the stress tensor in the surrounding vacuum

torean

w

Ttefrw

T

Tgi

wth

F

T

Fext fext Vd1 gext Sd

1+=E L E C T R O M A G N E T I C F O R C E

keep the solid stationary. That is Fext F 0. If the external forces are suddenly moved, the solid is no longer stationary, but F causes the solid to begin to move with initial acceleration according to

here m is the total mass and a is the acceleration of the solid.

o summarize, the total force, F, is computed as a boundary integral of the stress nsor in vacuum on the outside of the solid. To obtain this result, the contribution om the air pressure gradient has been neglected. This is equivalent of assuming that T20. A more detailed treatment shows that the pressure gradient contributes ith a lifting (buoyancy) force on the solid.

orque

he torque in the case of Forces on an Elastic Solid Surrounded by Vacuum or Airis ven by

here rO is a point on the axis of rotation. This follows from a derivation similar to e one made for forces.

orces in Stationary Fields

he electromagnetic fields are stationary if

F n1T2 Sd1=

ma d2r

dt2--------- Vd

1 F= =

MO r rO n1T2 Sd1=

30 | C H A P T E R 2 : R E V

that is, if the fields vary so slowly that the contributions from induced currents and displacement currents can be neglected.

Also assume that the objects modeled are not moving v 0 so that there is no co

T

Ta elmterebm

AF

w3nTte

Uca

tB 0=

tD 0=I E W O F E L E C T R O M A G N E T I C S

ntributions from Lorentz forces. These are treated later on.

H E E L E C T R O M A G N E T I C S T R E S S TE N S O R

he expressions for the stress tensor in a general electromagnetic context stems from fusion of material theory, thermodynamics, continuum mechanics, and ectromagnetic field theory. With the introduction of thermodynamic potentials for echanical, thermal, and electromagnetic effects, explicit expressions for the stress nsor can be derived in a convenient way by forming the formal derivatives with spect to the different physical fields (Ref. 1 and Ref. 3). Alternative derivations can e made for a vacuum (Ref. 4) but these cannot easily be generalized to polarized and agnetized materials.

ir and Vacuumor air, the stress tensor is

here p is the air pressure, I is the identity 3-by-3 tensor (or matrix), and E and B are -by-1 vectors. In this expression of the stress tensor, air is considered to be onpolarizable and nonmagnetizable. When air is approximated by vacuum, p = 0. his expression, with p = 0, of the stress tensor is also known as the Maxwell stress nsor.

sing the fact that, for air, D = 0E and B =0H the expression for the stress tensor n be written as

To apply the stress tensor in air to calculate the total force and torque on a magnetizable rod close to a permanent magnet, see Permanent Magnet: Model Library path ACDC_Module/Magnetostatics/permanent_magnet.

T2 pI02-----E E 1

20---------B B+ I 0EE

T 10------BBT+ +=

S | 31

The equation for the balance of forces becomes

Maxwells equations in free space give that the contribution of the electromagnetic pa

Tdeap

W

iseq

Cpr

wexca

E

T2 pI12---E D 1

2---H B+ I ED

T HBT+ +=

0 pI 12---E D 1

2---H B+ I ED

T HBT+ + fext+=E L E C T R O M A G N E T I C F O R C E

rt of the stress tensor is zero, and the resulting expression is

hus, using the same terminology as earlier, fem0 for air, with MpI. In the rivation of the total force on an elastic solid surrounded by vacuum or air, the proximation p0 has been used.hen operating with the divergence operator on the stress tensor, the relation

useful (and similarly for B). From the right-hand side it is clear (using Maxwells uations) that this is zero for stationary fields in free space.

onsider again the case of a solid surrounded by air. To compute the total force, the ojection of the stress tensor on the outside of the solid surface is needed,

here n1 is the surface normal, a 1-by-3 vector, pointing out from the solid. This pression can be used directly in the boundary integral of the stress tensor for lculating the total force F on the solid.

lastic Pure Conductor

0 p fext+=

EET 12---E EI E E E E =

n1T2 pn112---E D 1

2---H B+ n1 n1 E D

T n1 H BT+ +=

For an example of how to compute the total force on two parallel wires either by integrating the volume force or by integrating the stress tensor on the surrounding surface, see Electromagnetic Forces on Parallel Current-Carrying Wires: Model Library path ACDC_Module/Verification_Models/parallel_wires.

32 | C H A P T E R 2 : R E V

A material that is nonpolarizable and nonmagnetizable (P0 and M0) is called a pure conductor. This is not necessarily equivalent to a perfect conductor, for which E0, but merely a restriction on the dielectric and magnetic properties of the material. The stress tensor becomes identical to the one for air, except for pI being replaced by the purely mechanical stress tensor M:

w

Tvo

an

T

an

GF(n

wngty

T1 M 12---E D12---H B+ I ED

T HBT+ +=I E W O F E L E C T R O M A G N E T I C S

here D0E and B0H.he situation is slightly different from the case of air because there can be currents and lume charges in the conductor. The current density is

d the volume charge density

he equation for the balance of forces now becomes

d this means that

eneral Elastic Materialor an elastic solid, in the general case of a material that is both dielectric and magnetic onzero P and M), the stress tensor is given by the expression

here in (E, B) the dependence of E and B has not been separated out. Thus is ot a purely mechanical stress tensor in this general case. Different material models ive different appearances of (E, B). The electromagnetic contributions to (E, B) pically represent pyroelectric, pyromagnetic, piezoelectric, piezomagnetic, dielectric,

J H 10------ B= =

D 0 E= =

0 M E J B fext+++=

fem E J B+=

T1 E B 02-----E E 1

20---------B B M B+ I=

0EET 10------BBT EPT MBT+ + +

S | 33

and magnetization effects. The expression for the stress tensor in vacuum, air, and pure conductors can be derived from this general expression by setting MP0.T1 must be symmetric. The terms EP

T and MBT are symmetric in the case of a linear dielectric and magnetic material because

HexcotoMTinca

F

Cotcotrelin

F

Aeqveth

Intr

T

P 0eE=M BB=E L E C T R O M A G N E T I C F O R C E

ere, the magnetic susceptibility B differs slightly from the classical m. The other plicit terms are all symmetric, as is (E, B). In the general case this imposes nstraints on the properties of (E, B). For a nonlinear material (E, B) might need include terms such as EPT or +MBT to compensate for asymmetric EPT or

BT.

o instantiate the stress tensor for the general elastic case, an explicit material model cluding the magnetization and polarization effects is needed. Such material models n easily be found for piezoelectric materials (Ref. 3).

orces in a Moving Body

alculating forces in moving objects is important, especially for electric motors and her moving electromagnetic devices. When performing the computations in a ordinate system that moves with the object, the electromagnetic fields are

ansformed. The most well-known relation for moving objects is the one for the ectric field. The transformed quantity of the electric field is called the electromotive tensity.

I E L D TR A N S F O R M A T I O N S A N D G A L I L E I I N V A R I A N T S

ssume that the object modeled is moving with a constant velocity, v = v0. The uations now take on a slightly different form that includes the Galilei invariant rsions of the electromagnetic fields. The term Galilei invariant is used due to the fact at they remain unchanged after a coordinate transformation of the type

continuum mechanics, this transformation is commonly referred to as a Galilei ansformation.

he Galilei invariant fields of interest are

r' r v0t+=

34 | C H A P T E R 2 : R E V

TmnTthar

AT

T

ET

w

TtocoC

Tre

E

E v B (Electromotive intensity)+=J

J v (Free conduction current density)=P P

t------- v P v P (Polarization flux derivative)+=

M

M v P (Lorentz magnetization)+=H B

0------ 0v E M

(Magnetomotive intensity)=I E W O F E L E C T R O M A G N E T I C S

he electromotive intensity is the most important of these invariants. The Lorentz agnetization is significant only in materials for which neither the magnetization M

or the polarization P is negligible. Such materials are rare in practical applications. he same holds for the magnetization term of the magnetomotive intensity. Notice at the term 0v E is very small compared to B/0 except for cases when v and E e both very large. Thus in many practical cases this term can be neglected.

ir and Vacuumhe stress tensor in the surrounding air or vacuum on the outside of a moving object is

here is an additional term in this expression compared to the stationary case.

lastic Pure Conductorhe stress tensor in a moving elastic pure conductor is

here D0E and B0H.o get the equation for the balance of forces the divergence of this expression needs be computed. This requires an introduction of an extra term in Cauchys equation rresponding to an additional electromagnetic contribution to the linear momentum. auchys equation with this extra term is

he extra term is canceled out by the additional term in the stress tensor, and the final sult is

T2 pI12---E D 1

2---H B+ I ED

T HBT D B vT+ + +=

T1 M 12---E D12---H B+ I ED

T HBT D B vT+ + +=

t2

2

dd r D B+ T fext+=

S | 35

For the case of no acceleration, with the explicit appearance of the transformed quantities,

T

w

GT

T

Tin

E

Asykn

Tse

M

Tco

t2

2

dd r M E J B fext+++=

0 M E v B+ J v B fext+++=E L E C T R O M A G N E T I C F O R C E

he terms containing v B cancel out, which yields the following equation:

hich is the same expression as for the stationary case.

eneral Elastic Materialhe stress tensor for a moving general elastic material is

he magnetization M and the polarization P occur explicitly in this expression.

o instantiate the stress tensor for the general elastic case a material model explicitly cluding the magnetization and polarization effects is needed.

lectromagnetic Energy and Virtual Work

nother technique to calculate forces is to derive the electromagnetic energy of the stem and calculate the force by studying the effect of a small displacement. This is own as the method of virtual work or the principle of virtual displacement.

he method of virtual work is used for the electric energy and magnetic energy parately for calculating the total electric or magnetic force as follows.

A G N E T I C F O R C E A N D TO R Q U E

he method of virtual work utilizes the fact that under constant magnetic flux nditions (Ref. 5), the total magnetic force on a system is computed as

0 M E J B fext+++=

T1 E

B 02-----E E 1

20---------B B M

B+ I +=

0EET 10------BBT E

PT M

BT 0 E B vT+ + + +

F Wm=

36 | C H A P T E R 2 : R E V

If the system is constrained to rotate about an axis the torque is computed as

where is the rotational angle about the axis.Under the condition of constant currents, the total force and torque are computed in the same way but with opposite signs,

E

Usy

Usy

T Wm

=

I E W O F E L E C T R O M A G N E T I C S

L E C T R I C F O R C E A N D TO R Q U E

nder the condition of constant charges, the total electric force and torque on a stem are computed as

nder the condition of constant potentials, the total electric force and torque on a stem are computed as

FI Wm=

TI Wm

=

FQ We=

TQ We

=

FV We=

TV We

=

The method of virtual work can be employed by using the features for deformed mesh and sensitivity analysis in COMSOL Multiphysics.

See Deformed Geometry and Moving Mesh and Sensitivity Analysis in the COMSOL Multiphysics Reference Manual.