COMSOL Multiphysics AC/DC MODULE REFERENCE GUIDE V ERSION 3.5

Transcription

1 COMSOL Multiphysics AC/DC MODULE REFERENCE GUIDE V ERSION 3.5

2 How to contact COMSOL: Benelux COMSOL BV Röntgenlaan DX Zoetermeer The Netherlands Phone: +3 (0) Fax: +3 (0) info@comsol.nl Denmark COMSOL A/S Diplomvej Kgs. Lyngby Phone: Fax: info@comsol.dk Finland COMSOL OY Arabianranta 6 FIN Helsinki Phone: Fax: info@comsol.fi France COMSOL France WTC, 5 pl. Robert Schuman F Grenoble Phone: +33 (0) Fax: +33 (0) info@comsol.fr Germany COMSOL Multiphysics GmbH Berliner Str. 4 D Göttingen Phone: Fax: info@comsol.de Italy COMSOL S.r.l. Via Vittorio Emanuele II, Brescia Phone: Fax: info.it@comsol.com Norway COMSOL AS Søndre gate 7 NO-7485 Trondheim Phone: Fax: info@comsol.no Sweden COMSOL AB Tegnérgatan 23 SE- 40 Stockholm Phone: Fax: info@comsol.se Switzerland FEMLAB GmbH Technoparkstrasse CH-8005 Zürich Phone: +4 (0) Fax: +4 (0) info@femlab.ch United Kingdom COMSOL Ltd. UH Innovation Centre College Lane Hatfield Hertfordshire AL0 9AB Phone:+44-(0) Fax: +44-(0) info.uk@comsol.com United States COMSOL, Inc. New England Executive Park Suite 350 Burlington, MA 0803 Phone: Fax: COMSOL, Inc Wilshire Boulevard Suite 800 Los Angeles, CA Phone: Fax: COMSOL, Inc. 744 Cowper Street Palo Alto, CA 9430 Phone: Fax: info@comsol.com For a complete list of international representatives, visit Company home page COMSOL user forums AC/DC Module Reference Guide COPYRIGHT 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, COMSOL Script, COMSOL Reaction Engineering Lab, and FEMLAB are registered trademarks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their respective holders. Version: September 2008 COMSOL 3.5 Part number: CM02003

3 CONTENTS Chapter : Introduction Typographical Conventions Chapter 2: Application Mode Reference Application Mode Variables 4 Common Variables 6 Electrostatic Fields 7 Conductive Media DC Shell, Conductive Media DC Electrostatics Magnetostatic and Quasi-Static Fields 2 3D and 2D Quasi-Statics Application Modes Electric Currents (3D) and In-Plane Electric Currents (2D) Perpendicular Induction Currents, Vector Potential Azimuthal Induction Currents, Vector Potential In-Plane Induction Currents, Magnetic Field Meridional Induction Currents, Magnetic Field Magnetostatics, No Currents Chapter 3: Programming Reference The Programming Language 62 The Application Structure Eddy Currents in the Programming Language Application Mode Programming Reference 79 Conductive Media DC CONTENTS i

4 Shell, Conductive Media DC Electrostatics D and 2D Quasi-Statics Perpendicular and Azimuthal Currents In-Plane and Meridional Currents, Magnetic Field Magnetostatics, No Currents Chapter 4: Function Reference Summary of Commands 2 Commands Grouped by Function 3 Force Computation Lumped Parameter Computation File Exchange cemforce cemtorque rlcmatrix spiceimport INDEX 9 ii CONTENTS

5 Introduction The AC/DC Module 3.5 is an optional package that extends the COMSOL Multiphysics modeling environment with customized user interfaces and functionality for the analysis of electromagnetic effects, s, and systems. This particular module solves problems in the general areas of electrostatic fields, magnetostatic fields, and quasi-static fields. The application modes included here are fully multiphysics enabled, making it possible to couple them to any other physics application mode in COMSOL Multiphysics or the other modules. For example, to find the heat distribution in a motor you would first find the current in the coils using one of the quasi-static application modes in this module, and then couple it to a heat equation in the main COMSOL Multiphysics package or the Heat Transfer Module. The underlying equations for electromagnetics are automatically available in all of the application modes a feature unique to COMSOL Multiphysics. This also makes nonstandard modeling easily accessible. The documentation set for the AC/DC Module consists of two printed books, the AC/DC Module User s Guide and the AC/DC Module Model Library, and this AC/DC Module Reference Guide. All three books are available in PDF and HTML versions from the COMSOL Help Desk. This book contains reference

6 information such as application mode implementation details, information about command-line programming, and details about the command-line functions that are specific to the AC/DC Module (for example, functions for electromagnetic force computations and import of SPICE netlists). 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 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. 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 Equation System and then click Subdomain Settings. COMSOL Multiphysics>Heat Transfer>Conduction means: Open the COMSOL Multiphysics 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.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 CHAPTER : INTRODUCTION

7 2 Application Mode Reference This chapter provides listings of the application mode variables that you have access to in the AC/DC Module s application modes. 3

8 Application Mode Variables The application modes in the AC/DC Module define a large set of variables. The purpose of this reference chapter is to list all the variables that each application mode define. Other information, like the theoretical background for the application mode, can be found in the chapter The Application Modes on page 49 of the AC/DC Module User s Guide. The application mode variables listed in the following sections are all available in postprocessing and when formulating the equations. You can use any function of these variables when postprocessing the result of the analysis. It is also possible to use these variables in the for the physical properties in the equations. The application mode variable tables are organized as follows: The Name column lists the names of the variables that you can use in the in the equations or for postprocessing. The indices i and j (using an italic font) in the variable names can mean any of the spatial coordinates. For example, Ei means either Ex, Ey, Ez in 3D when the spatial coordinates are x, y, and z. In 2D axisymmetry Ei stands for either Er or Ez. The variable names of vector and tensor s are constructed using the names of the spatial coordinates. For example, if you use x, y, and z as the spatial coordinate names, the variables for the vector s of the electric field are Ex, Ey, and Ez. In a COMSOL Multiphysics model, the variable names get an underscore plus the application mode name appended to the names listed in the tables. For example, the default name of the Electrostatics application mode is emes. With this name the variable for the x of the electric field is Ex_emes. The Type column indicates if the variable is defined on subdomains (S), boundaries (B), edges (E), or points (P). The column indicates the top level where the variables is defined. Many variables that are available on subdomains also exist on boundaries, edges, and points, but take the average value of the values in the subdomains around the boundary, edge, or point in question. The Analysis column specifies for which type of analysis the variable is defined. The available analysis types are, for example, static, transient, harmonic, and eigenfrequency. The available analysis types are application-mode dependent. Some variables are defined differently depending on the analysis type, and others are only available for some analysis types. 4 CHAPTER 2: APPLICATION MODE REFERENCE

9 The Constitutive Relation column indicates the constitutive relation for which the variable definition applies. The abbreviations used are defined in the table below. ABBREVIATION epsr P Dr mur CONSTITUTIVE RELATION D = ε 0 ε r E D = ε 0 E + P D = ε 0 ε r E + D r B = µ 0 µ r H M B = µ 0 (H + M) Br B = µ 0 µ r H + B r The Material Parameters column appears in the tables for the electromagnetic waves application modes. It indicates if the refractive index n or the relative permeability ε r, the conductivity σ, and the relative permeability µ r are used as material parameters in the expression for the variable. The Description column gives a description of the variables. The Expression column gives the expression of the variables in terms of other physical quantities. In these, the subscripts i and j of vector and tensor s stand for one of the spatial coordinates. For example, E i is either E x, E y, or E z. When two equal subscripts appear in an expression this implies a summation. For example σ ij E j = σ ix E x + σ iy E y + σ iz E z. APPLICATION MODE VARIABLES 5

10 Common Variables There are a couple of variables that the application modes share. These variables are listed in the tables below. In some cases, the variables only exist when a certain boundary condition or application mode property has been selected. See page 4 for a description of the notation used in the tables. APPLICATION SCALAR VARIABLES There are no common scalar variables, see the corresponding section for application modes. APPLICATION SUBDOMAIN VARIABLES The common subdomain variables are given the table below. TABLE 2-: COMMON APPLICATION MODE SUBDOMAIN VARIABLES NAME DIMENSION DESCRIPTION EXPRESSION d 2D thickness d dr_guess R0_guess default guess for width in radial direction of infinite-element domain default guess for inner radius of infinite-element domain r R 0 Si infinite-element x i coordinate s i S0i_guess default guess for inner x i coordinate of infinite-element domain S 0i Sdi_guess default guess for width along x i i coordinate of infinite-element domain dvol Volume integration contribution dv APPLICATION BOUNDARY VARIABLES The common boundary variables are given the table below. TABLE 2-2: COMMON APPLICATION MODE BOUNDARY VARIABLES NAME DESCRIPTION EXPRESSION dbnd boundary thickness d bnd dvolbnd Area integration contribution da 6 CHAPTER 2: APPLICATION MODE REFERENCE

11 Electrostatic Fields A number of variables and physical quantities are available for postprocessing and for use in equations and boundary conditions. They are all given in the following tables. See page 4 for a description of the notation used in these tables. The up and down subscripts indicate that the variable should be evaluated on the geometrical up or down side of the boundary. Conductive Media DC The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for Conductive Media DC are given the table below. TABLE 2-3: APPLICATION MODE SUBDOMAIN VARIABLES, CONDUCTIVE MEDIA DC NAME DESCRIPTION EXPRESSION V electric potential V sigma electric conductivity σ sigmaij electric conductivity, x i x j σ ij Qj current source Q j d thickness d Jei normje Jii normji Ji normj Ei external current density, x i external current density, norm potential current density, x i potential current density, norm total current density, x i total current density, norm electric field, x i J i e J e σ ij E j J i J e J i J i e + J i i J J V x i ELECTROSTATIC FIELDS 7

12 TABLE 2-3: APPLICATION MODE SUBDOMAIN VARIABLES, CONDUCTIVE MEDIA DC NAME DESCRIPTION EXPRESSION norme electric field, norm E E Q resistive heating J E APPLICATION BOUNDARY VARIABLES The boundary variables for Conductive Media DC are given in the table below. TABLE 2-4: APPLICATION MODE BOUNDARY VARIABLES, CONDUCTIVE MEDIA DC NAME DESCRIPTION EXPRESSION tei tangential electric field, x i t i t V normte tangential electric field, norm E t E t nj current density outflow n J njs source current density n up ( J down J up ) Jsi surface current density, x i dσ E ti normjs surface current density, norm J s J s sigmabnd electric conductivity on boundary σ bnd Qs surface resistive heating J s E t Qjl line current source Q jl Qj0 point current source Q j0 APPLICATION POINT VARIABLES The point variable for the Conductive Media DC application mode appears in the following table. TABLE 2-5: APPLICATION MODE POINT VARIABLES, CONDUCTIVE MEDIA DC NAME TYPE DESCRIPTION EXPRESSION Qj0 P Point current source Q j0 Shell, Conductive Media DC For application mode variables see the section Conductive Media DC on page 7. 8 CHAPTER 2: APPLICATION MODE REFERENCE

13 Electrostatics The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. APPLICATION MODE SCALAR VARIABLES The application-specific scalar variable in this mode is given in the following table. TABLE 2-6: APPLICATION MODE SCALAR VARIABLES, ELECTROSTATICS, NAME DESCRIPTION EXPRESSION epsilon0 permittivity of vacuum ε 0 APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for Electrostatics are given the table below. TABLE 2-7: APPLICATION MODE SUBDOMAIN VARIABLES, ELECTROSTATICS, NAME CONSTITUTIVE RELATION DESCRIPTION V electric potential V epsilonr epsr, Dr relative permittivity ε r epsilonr P relative permittivity epsilonrij epsr, Dr relative permittivity, x i x j ε rij epsilonrij P relative permittivity, x i x j epsilon permittivity ε 0 ε r EXPRESSION epsilonij permittivity, x i x j ε 0 ε rij Pi P electric polarization, x i P i Pi epsr, Dr electric polarization, x i D i ε 0 E i normp electric polarization, norm P P Dri epsr remanent displacement, x i Dri P remanent displacement, x i Dri Dr remanent displacement, x i 0 P i D ri normdr remanent displacement, norm D r D r rho space charge density ρ ELECTROSTATIC FIELDS 9

14 TABLE 2-7: APPLICATION MODE SUBDOMAIN VARIABLES, ELECTROSTATICS, NAME Ei CONSTITUTIVE RELATION DESCRIPTION electric field, x i EXPRESSION V x i norme electric field, norm E E Di epsr electric displacement, x i ε 0 ε rij E j Di P electric displacement, x i ε 0 E i + P i Di Dr electric displacement, x i ε 0 ε rij E j + D ri normd We APPLICATION BOUNDARY VARIABLES electric displacement, norm electric energy density The boundary variables for Electrostatics are given in the table below. TABLE 2-8: APPLICATION MODE BOUNDARY VARIABLES, ELECTROSTATICS, NAME DESCRIPTION EXPRESSION nd surface charge density n up ( D down D up ) epsilonbnd relative permittivity on boundary ε bnd unti Maxwell surface stress tensor, x i, up side of boundary D D E D 2 -- ( E 2 up D up )n idown + ( n down D up )E iup dnti untei dntei Maxwell surface stress tensor, x i, down side of boundary electric Maxwell surface stress tensor, x i, up side of boundary electric Maxwell surface stress tensor, x i, down side of boundary -- ( E 2 down D down )n iup + ( n up D down )E idown -- ( E 2 up D up )n idown + ( n down D up )E iup -- ( E 2 down D down )n iup + ( n up D down )E idown 0 CHAPTER 2: APPLICATION MODE REFERENCE

15 APPLICATION EDGE VARIABLES The edge variable for Electrostatics appears in the following table: TABLE 2-9: APPLICATION MODE EDGE VARIABLES, ELECTROSTATICS, NAME DESCRIPTION EXPRESSION Ql line charge density Q l APPLICATION POINT VARIABLES The point variable for Electrostatics appears in the following table: TABLE 2-0: APPLICATION MODE POINT VARIABLES, ELECTROSTATICS, NAME DESCRIPTION EXPRESSION Q0 charge Q 0 ELECTROSTATIC FIELDS

16 Magnetostatic and Quasi-Static Fields 3D and 2D Quasi-Statics Application Modes APPLICATION MODE VARIABLES The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. In the containing the electric potential, set this potential to zero if it is not a dependent variable. See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-: APPLICATION MODE SCALAR VARIABLES, 2D AND 3D QUASI-STATICS NAME ANALYSIS DESCRIPTION EXPRESSION nu harmonic frequency ν omega harmonic angular frequency 2πν epsilon0 permittivity of vacuum ε 0 mu0 permeability of vacuum µ 0 psi0 gauge fixing on scaling for gauge fixing variable ψ 0 extb0i reduced potential, background field given by B External magnetic flux density, x i B 0i,ext exta0i reduced potential, background field given by A External magnetic vector potential, x i A 0i,ext 2 CHAPTER 2: APPLICATION MODE REFERENCE

17 APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for 2D and 3D Quasi-Statics are given the table below. TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION V 2D, 3D electric potential V Ai 2D, 3D magnetic potential, x i A i Ai 2D, 3D reduced potential, background field given through A total magnetic potential, x i A i,red +A i,ext extai 2D, 3D reduced potential, background field given through A external magnetic potential, x i in non- infinite element subdomains curlai 3D, 2D curl of magnetic potential, x i mur 2D, 3D mur, Br relative permeability mur 2D, 3D M relative permeability murij 3D mur, Br relative permeability, x i x j murij 3D M relative permeability, x i x j A 0i,ext ( A) i µ r µ rij mu 2D, 3D permeability µ 0 µ r muij 3D permeability, x i x j epsilonr 2D, 3D harmonic epsr, Dr relative permittivity epsilonr 2D, 3D harmonic P relative permittivity µ 0 µ rij ε r MAGNETOSTATIC AND QUASI-STATIC FIELDS 3

18 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION epsilonrij 2D, 3D harmonic epsr, Dr relative ε rij permittivity, x i x j epsilonrij 2D, 3D harmonic P relative permittivity, x i x j epsilon 2D, 3D harmonic permittivity ε 0 ε r epsilonij 2D, 3D harmonic permittivity, x i x j sigma 2D, 3D transient, harmonic, static, electric potential sigmaij 2D, 3D transient, harmonic, static, electric potential electric conductivity electric conductivity, x i x j delta 2D, 3D harmonic skin depth Pi 2D, 3D harmonic P electric polarization, x i Pi 2D, 3D harmonic epsr, Dr electric polarization, x i normp 2D, 3D harmonic polarization, norm Dri 2D, 3D harmonic epsr remanent displacement, x i Dri 2D, 3D harmonic P remanent displacement, x i Dri 2D, 3D harmonic Dr remanent displacement, x i ε 0 ε rij σ σ ij Re( jωµ ( σ + jωε) ) P i e j phase D i ε 0 E i 0 P i P P * D ri e jphase 4 CHAPTER 2: APPLICATION MODE REFERENCE

19 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION normdr 2D, 3D harmonic remanent displacement, norm Mi 3D static, transient M magnetization, x i Mi 3D harmonic M magnetization, x i M i * D r D r M i e j phase Mi 3D mur, Br magnetization, x i normm 3D magnetization, norm Mi 2D static, transient M magnetization, x i out of plane Mi 2D harmonic M magnetization, x i out of plane Mi 2D mur, Br magnetization, x i out of plane normm 2D magnetization, norm Bri 3D mur remanent flux density, x i Bri 3D M remanent flux density, x i Bri 3D static, transient Br remanent flux density, x i Bri 3D harmonic Br remanent flux density, x i normbr 3D remanent flux density, norm B i /µ 0 H i M i M i e j phase B i /µ 0 H i M i 0 µ 0 M i B ri M M * B ri e j phase * B r B r MAGNETOSTATIC AND QUASI-STATIC FIELDS 5

20 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Bri 3D M remanent flux density, x i out of plane Bri 3D static, transient Br remanent flux density, x i out of plane Bri 3D harmonic Br remanent flux density, x i out of plane normbr 2D remanent flux density, norm Jei 3D, 2D static, transient external current density, x i Jei 3D, 2D harmonic external current density, x i normje 3D, 2D external current density, norm vi 3D, 2D electric potential normv 3D, 2D electric potential velocity, x i velocity, norm Ei 3D, 2D harmonic electric field, x i µ 0 M i B ri B ri e j phase B r J i e J i e e jphase v i J e J e* v v jωa i V x i Ei 3D, 2D transient electric field, x i Ei 3D, 2D static, electric potential electric field, x i norme 3D, 2D electric field, norm Di 3D, 2D harmonic epsr electric displacement, x i A i t V x i E E * ε 0 ε rij E j 6 CHAPTER 2: APPLICATION MODE REFERENCE

21 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Di 3D, 2D harmonic P electric displacement, x i Di 3D, 2D harmonic Dr electric displacement, x i normd 3D, 2D harmonic electric displacement, norm Bi 3D magnetic flux density, x i normb 3D magnetic flux density, norm Bi 2D magnetic flux density, x i out of plane normb 2D magnetic flux density, norm Bi 3D, 2D reduced potential extbi extbi reduced potential background field given by B reduced potential, background field given by A total magnetic flux density, x i external magnetic flux density, x i in non-infinite element subdomains external magnetic flux density, x i Hi 3D mur magnetic field, x i Hi 3D M magnetic field, x i ε 0 E i + P i ε 0 ε rij E j + D ri D D * ( A) i B B * ( A) i B i ( A red ) i + ( B ext ) i B 0i,ext ( A ext ) i µ rijbj µ 0 B i /µ 0 M i MAGNETOSTATIC AND QUASI-STATIC FIELDS 7

22 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Hi 3D Br magnetic field, x i normh 3D magnetic field, norm Hi 2D mur magnetic field, x i out of plane Hi 2D M magnetic field, x i out of plane Hi 2D Br magnetic field, x i out of plane normh 2D magnetic field, norm, x i out of plane Jii 3D, 2D harmonic induced current density, x i normji 3D, 2D harmonic induced current density, norm Jii 3D, 2D transient induced current density, x i normji 3D, 2D transient induced current density, norm Jdi 3D, 2D harmonic displacement current density, x i normjd 3D, 2D harmonic displacement current density, norm Jpi 3D, 2D electric potential normjp 3D, 2D electric potential potential current density, x i potential current density, norm µ rij ( B j B rj ) µ 0 H H * µ r Bi µ 0 B i /µ 0 M i µ r H i jωσ ij A j σ ij E j jωd i ( B i B ri ) µ 0 J i J i* J i J i J d J d* σ ij V x j J p J p* 8 CHAPTER 2: APPLICATION MODE REFERENCE

23 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Jvi 3D, 2D electric potential normjv 3D, 2D electric potential Ji 3D, 2D static, electric potential velocity current density, x i velocity current density, norm total current density, x i Ji 3D, 2D static total current density, x i Ji 3D, 2D transient total current density, x i Ji 3D, 2D harmonic, electric potential total current density, x i Ji 3D, 2D harmonic total current density, x i normj 3D, 2D total current density, norm Evi 3D, 2D electric potential normev 3D, 2D electric potential Gfi 3D, 2D static, harmonic Lorentz electric field, x i Lorentz electric field, norm gauge fixed field, x i Gfi 3D, 2D transient gauge fixed field, x i Weav 3D, 2D harmonic time average electric energy density Wm 3D, 2D static magnetic energy density σ ij ( v B ) j ( v B ) i A σa J v J v* e v p J i + J i +J i J i e J i e + J i i e v p i d J i + J i + J i + J i +J i e i d J i + J i +J i J J * * E v E v Re D E* 4 H B 2 MAGNETOSTATIC AND QUASI-STATIC FIELDS 9

24 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Wmav 3D, 2D harmonic time average magnetic energy density Wav 3D, 2D harmonic time average total energy density Q 3D, 2D static, electric potential --Re( H B * ) 4 W e av + W m av resistive heating J ( E + v B + σ J e ) Q 3D, 2D transient resistive heating J ( E + σ J e ) Qav 3D, 2D harmonic, electric potential time average resistive heating Qav 3D, 2D harmonic time average resistive heating Qmav 3D, 2D harmonic time average magnetic hysteresis losses Poi 3D, 2D static, electric potential normpo 3D, 2D static, electric potential power flow, x i power flow, norm Poiav 3D, 2D harmonic time average power flow, x i normpoav 3D, 2D harmonic time average power flow, norm FLtzi 3D, 2D Lorentz force, x i (Note: for harmonic analysis with angular frequency ω, the Lorentz force s angular frequency is 2ω) FLtzavi 3D, 2D harmonic cycle average Lorentz force, x i --Re( J ( E * + v B * + σ J e* )) 2 --Re( J ( E * + σ J e* )) 2 -- Re( jωh B * ) 2 ( E H ) i S S * --Re( E H * ) i 2 S av (J B) i S av* -- ( J B * ) i 2 20 CHAPTER 2: APPLICATION MODE REFERENCE

25 APPLICATION MODE BOUNDARY VARIABLES The boundary variables for 2D and 3D Quasi-Statics are given the table below. TABLE 2-3: APPLICATION MODE BOUNDARY VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS DESCRIPTION EXPRESSION Jsi 3D, 2D surface current density, x i normjs 3D, 2D surface current density, norm nj 3D, 2D electric potential njs 3D, 2D electric potential sigmabnd 3D, 2D static, harmonic epsilonrbn d normal current density source current density conductivity on boundaries 3D, 2D harmonic permittivity on boundaries murbnd 3D, 2D harmonic permeability on boundaries tei 3D, 2D electric potential normte 3D, 2D electric potential tdi 3D, 2D electric potential normtd 3D, 2D electric potential tangential electric field, x i tangential electric field, norm tangential electric displacement, x i tangential electric displacement, norm Qs 3D, 2D transient surface resistive heating Qsav 3D, 2D harmonic time average surface resistive heating npo 3D, 2D static, electric potential npoav 3D, 2D harmonic time average power outflow n up ( H down H up ) n J n up ( J down J up ) σ bnd ε rbnd µ rbnd D ti J s E power outflow n S * J s J s t i n S av t V * E t E t * D t D t --Re( J 2 s E * ) MAGNETOSTATIC AND QUASI-STATIC FIELDS 2

26 TABLE 2-3: APPLICATION MODE BOUNDARY VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS DESCRIPTION EXPRESSION deltabnd 3D, 2D harmonic, impedance cond. unti 3D static, transient dnti 3D static, transient skin depth Maxwell surface stress tensor, x i, up side of boundary Maxwell surface stress tensor, x i, down side of boundary untiav 3D harmonic Maxwell surface stress tensor, x i, up side of boundary dntiav 3D harmonic Maxwell surface stress tensor, x i, down side of boundary unti 2D static, transient dnti 2D static, transient Maxwell surface stress tensor, x i out of plane, up side of boundary Maxwell surface stress tensor, x i out of plane, down side of boundary untiav 2D harmonic Maxwell surface stress tensor, x i out of plane, up side of boundary dntiav 2D harmonic Maxwell surface stress tensor, x i out-of-plane, down side of boundary Re( jωµ ( σ + jωε) ) -- ( H 2 up B up )n idown + ( n down H up )B iup -- ( H 2 down B down )n iup + ( n up H down )B idown * --Re( H 4 up B up )n idown * + --Re(( n 2 down H up )B iup ) * --Re( H 4 down B down )n iup * + --Re(( n 2 up H down )B idown ) --H 2 iup B iup n idown --H 2 idown B idown n iup --Re( H 4 iup B iup )n idown --Re( H 4 idown B idown )n iup 22 CHAPTER 2: APPLICATION MODE REFERENCE

27 TABLE 2-3: APPLICATION MODE BOUNDARY VARIABLES, 2D AND 3D QUASI-STATICS NAME DIMENSION ANALYSIS DESCRIPTION EXPRESSION extbni 3D, 2D reduced potential External magnetic flux density, normal x i FsLtzi 3D, 2D Lorentz surface force, x i (Note: for harmonic analysis with angular frequency ω, the Lorentz force s angular frequency is 2ω) FsLtzavi 3D, 2D harmonic cycle average Lorentz surface force, x i (n B ext ) i J s B -- ( J 2 s B * ) i Electric Currents (3D) and In-Plane Electric Currents (2D) The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-4: APPLICATION MODE SCALAR VARIABLES, ELECTRIC CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION nu harmonic Frequency ν omega harmonic Angular frequency 2πν epsilon0 Permittivity of vacuum ε 0 mu0 Permeability of vacuum µ 0 MAGNETOSTATIC AND QUASI-STATIC FIELDS 23

28 APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for Electric Currents are given the table below. TABLE 2-5: APPLICATION MODE SUBDOMAIN VARIABLES, ELECTRIC CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION V Electric potential V epsilonr epsr, Dr relative permittivity epsilonr P relative permittivity epsilonrij epsr, Dr relative permittivity, x i x j epsilonrij P relative permittivity, x i x j ε r ε rij epsilon permittivity ε 0 ε r epsilonij permittivity, x i x j sigma electric conductivity sigmaij electric conductivity, x i x j Pi harmonic P electric polarization, x i Pi transient P electric polarization, x i Pi epsr, Dr electric polarization, x i normp polarization, norm Dri epsr remanent displacement, x i ε 0 ε rij σ σ ij P i e jphase P i D i ε 0 E i 0 P P * 24 CHAPTER 2: APPLICATION MODE REFERENCE

29 TABLE 2-5: APPLICATION MODE SUBDOMAIN VARIABLES, ELECTRIC CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Dri P remanent P i displacement, x i Dri harmonic Dr remanent D ri e jphase displacement, x i Dri transient Dr remanent displacement, x i D ri normdr remanent displacement, norm Jei harmonic external current density, x i Jei transient external current density, x i normje external current density, norm Ei electric field * D r D r J i e e jphase J i e J e J e* V x i norme electric field, norm Di epsr electric displacement, x i Di P electric displacement, x i Di Dr electric displacement, x i normd electric displacement, norm E E * ε 0 ε rij E j ε 0 E i + P i ε 0 ε rij E j + D ri D D * MAGNETOSTATIC AND QUASI-STATIC FIELDS 25

30 TABLE 2-5: APPLICATION MODE SUBDOMAIN VARIABLES, ELECTRIC CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Jdi harmonic displacement current density, x i Jdi transient displacement current density, x i normjd displacement current density, norm Jpi potential current density, x i normjp potential current density, norm Ji total current density, x i normj total current density, norm Weav harmonic time average electric energy density We transient electric energy density Qav harmonic time average resistive heating Q transient resistive heating APPLICATION MODE BOUNDARY VARIABLES jω D i The boundary variables for Electric Currents are given the table below. TABLE 2-6: APPLICATION MODE BOUNDARY VARIABLES, ELECTRIC CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION tei t i t V tangential electric field, x i normte harmonic tangential electric field, norm D i t J d J d* σ ij V x j J p J p* e d p J i + J i +J i J J * --Re( D E * ) 4 -- ( D E * ) 2 --Re( J ( E * + σ J e* )) 2 * E t E t J ( E * + σ J e* ) 26 CHAPTER 2: APPLICATION MODE REFERENCE

31 TABLE 2-6: APPLICATION MODE BOUNDARY VARIABLES, ELECTRIC CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION normte transient tangential electric field, norm tdi tangential electric displacement, x i normtd harmonic tangential electric displacement, norm normtd transient tangential electric displacement, norm sigmabnd electric conductivity on boundary epsilonrbnd relative permittivity on boundary Jsi Jsi Qsav Qs nj njs harmonic, shielding cond. transient, shielding cond. harmonic, shielding cond. transient, shielding cond. surface current density, x i surface current density, x i time average surface resistive heating surface resistive heating normal current density Source current density ε E ti σ bnd ε rbnd d( σ E ti + jωd ti ) J s E Perpendicular Induction Currents, Vector Potential The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. n J n up ( J down J up ) At the edges of the modeled structure, the magnetic potential, electric field and tangential magnetic field can be used for postprocessing. The energy flow in the normal direction is also available. E t E t * D t D t D t D t d D ti σe ti + t --Re( J 2 s E * ) MAGNETOSTATIC AND QUASI-STATIC FIELDS 27

32 See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-7: APPLICATION MODE SCALAR VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION nu harmonic frequency ν omega harmonic angular frequency 2πν epsilon0 permittivity of vacuum ε 0 mu0 permeability of vacuum µ 0 exta0z reduced potential, background field given through A external magnetic vector potential, out-of-plane A 0z,ext extb0i reduced potential, background field given through B external magnetic flux density, in-plane x i B 0i,ext APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for Perpendicular Currents are given the table below. TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. Az Az extaz reduced potential, background field given through A reduced potential, background field given through A DESCRIPTION magnetic potential, z total magnetic potential, z external magnetic potential, z in non-infinite element subdomains EXPRESSION A z A z,red + A z,ext A 0z,ext 28 CHAPTER 2: APPLICATION MODE REFERENCE

33 TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION curlai curl of magnetic potential, x i mur mur, Br relative permeability mur M relative permeability murij mur, Br relative permeability, x i x j murij M relative permeability, x i x j EXPRESSION ( A) i µ r µ rij mu permeability µ 0 µ r muij permeability, x i x j µ 0 µ rij epsilonr harmonic epsr, Dr relative ε r permittivity epsilonr harmonic P relative permittivity epsilon harmonic permittivity ε 0 ε r sigma electric conductivity σ delta harmonic skin depth Re( jωµ ( σ + jωε) ) deltav static, transient potential difference deltav harmonic potential difference V V e j phase L length L Pz harmonic P electric P z e j phase polarization, z Pz harmonic epsr, Dr electric polarization, z D z ε 0 E z MAGNETOSTATIC AND QUASI-STATIC FIELDS 29

34 TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. Drz harmonic epsr remanent displacement, z Drz harmonic P remanent displacement, z Drz harmonic Dr remanent displacement, z Mi static, transient M magnetization, x i Mi harmonic M magnetization, x i Mi mur, Br magnetization, x i normm magnetization, norm Bri mur remanent flux density, x i Bri M remanent flux density, x i Bri static, transient Br remanent flux density, x i Bri harmonic Br remanent flux density, x i normbr Jez static, transient DESCRIPTION remanent flux density, norm external current density, z Jez harmonic external current density, z EXPRESSION 0 P z D r z e j phase M i M i e j phase B i /µ 0 H i 0 µ 0 M i B ri B ri e j phase J z e M M * * B r B r J z e e jphase 30 CHAPTER 2: APPLICATION MODE REFERENCE

35 TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. vi normv velocity, x i velocity, norm Ez transient electric field, z Ez harmonic electric field, z norme transient, harmonic DESCRIPTION electric field, norm Dz harmonic epsr electric displacement, z Dz harmonic P electric displacement, z Dz harmonic Dr electric displacement, z normd harmonic electric displacement, norm Bx magnetic flux density, x By magnetic flux density, y Bx reduced potential total magnetic flux density, x By reduced potential total magnetic flux density, y normb magnetic flux density, norm EXPRESSION v i v v A z t jωa z E z ε 0 ε r E z ε 0 E z + P z ε 0 ε r E z + D rz D z A z y A z x A z + B y ext,x A z B x ext, y B B * MAGNETOSTATIC AND QUASI-STATIC FIELDS 3

36 TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. extbi extbi reduced potential, background field specified through B reduced potential, background field specified through A external magnetic flux density, x i in non-infinite element subdomains external magnetic flux density, x i in non-infinite element subdomains Hi mur magnetic field, x i Hi M magnetic field, x i Hi Br magnetic field, x i normh magnetic field, norm Jpz potential current, z Jiz transient, harmonic DESCRIPTION induced current density, z Jdz harmonic displacement current density, z Jvz velocity current density, z Jz static total current density, z Jz transient total current density, z EXPRESSION B 0ext,i ( A ext ) i µ rijbj µ 0 B i /µ 0 M i µ rij ( B j B rj ) µ 0 H H * σ V/L σ E z jω D z σ ( v x B y v y B x ) e v p J z + J z +J z e v p i J z + J z + J z +J z 32 CHAPTER 2: APPLICATION MODE REFERENCE

37 TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. Jz harmonic total current density, z normj total current density, norm Evz harmonic Lorentz electric field, z normev Wm transient, harmonic static, transient DESCRIPTION Lorentz electric field, norm magnetic energy density Weav harmonic time average electric energy density Wmav harmonic time average magnetic energy density Wav harmonic time average total energy density Q static resistive heating Q transient resistive heating Qav harmonic time average resistive heating Qmav harmonic time average magnetic hysteresis losses Pox transient power flow, x Poy transient power flow, y normpo transient power flow, norm EXPRESSION e v p i d J z + J z + J z + J z +J z J z ( v x B y v y B x ) E vz H B 2 * --Re( E 4 z D z ) --Re( H B * ) 4 W e av + W m av J V z v x B y v y B x σ L E z H y E z H x J z e J V z E z + v x B y v y B x σ L --Re J * 2 z E z -- Re( jωh B * ) 2 S S * J z e * * V * + v x B y v y B x σ L J z e* MAGNETOSTATIC AND QUASI-STATIC FIELDS 33

38 TABLE 2-8: APPLICATION MODE SUBDOMAIN VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION Poxav harmonic time average power flow, x Poyav harmonic time average power flow, y normpoav harmonic time average power flow, norm FLtzi Lorentz force, x i (Note: for harmonic analysis with angular frequency ω, the Lorentz force s angular frequency is 2ω) FLtzavi harmonic cycle average Lorentz force, x i EXPRESSION * --Re( E 2 z H y ) * --Re( E 2 z H x ) S av (J B) i S av* -- ( J B * ) i 2 APPLICATION MODE BOUNDARY VARIABLES The boundary variables for Perpendicular Currents are given the table below. TABLE 2-9: APPLICATION MODE BOUNDARY VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION Jsz surface current density, z murbnd relative permeability on boundary sigmabnd harmonic conductivity on boundary extbni reduced potential External magnetic flux density, normal x i n x up ( H y down H y up ) n y up ( H x down H x up ) µ rbnd ε rbnd (n B ext ) i 34 CHAPTER 2: APPLICATION MODE REFERENCE

39 TABLE 2-9: APPLICATION MODE BOUNDARY VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION harmonic σ bnd epsilonr bnd relative permittivity on boundary Qs transient surface resistive heating J s E Qsav harmonic time average surface resistive --Re( J 2 s E * ) heating npo transient power outflow n S npoav harmonic time average power outflow n S av unti dnti static, transient static, transient Maxwell surface stress tensor, x i, up side of boundary Maxwell surface stress tensor, x i, down side of boundary untiav harmonic Maxwell surface stress tensor, x i, up side of boundary dntiav harmonic Maxwell surface stress tensor, x i, down side of boundary -- ( H 2 up B up )n idown + ( n down H up )B iup -- ( H 2 down B down )n iup + ( n up H down )B idown * --Re( H 4 up B up )n idown * + --Re(( n 2 down H up )B iup ) * --Re( H 4 down B down )n iup * + --Re(( n 2 up H down )B idown ) MAGNETOSTATIC AND QUASI-STATIC FIELDS 35

40 TABLE 2-9: APPLICATION MODE BOUNDARY VARIABLES, PERPENDICULAR CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION FsLtzi Lorentz surface force, x i (Note: for harmonic analysis with angular frequency ω, the Lorentz force s angular frequency is 2ω) (J s B) i FsLtzavi harmonic cycle average -- Lorentz surface ( J 2 s B * ) i force, x i Azimuthal Induction Currents, Vector Potential The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. At the edges of the modeled structure, the magnetic potential, electric field and tangential magnetic field can be used for postprocessing. The energy flow in the normal direction is also available. See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-20: APPLICATION MODE SCALAR VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION nu harmonic frequency ν omega harmonic angular frequency 2πν epsilon0 permittivity of vacuum ε 0 mu0 permeability of vacuum µ 0 36 CHAPTER 2: APPLICATION MODE REFERENCE

41 TABLE 2-20: APPLICATION MODE SCALAR VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION exta0phi A 0ϕ,ext extb0i reduced potential, background field given through A reduced potential, background field given through B external magnetic vector potential, out-of-plane external magnetic flux density, in-plane x i APPLICATION MODE SUBDOMAIN VARIABLES B 0i,ext The subdomain variables for Azimuthal Currents are given the table below. TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. Aphi Aphi extaphi Aphir Aphiz reduced potential, background field given through A reduced potential, background field given through A DESCRIPTION magnetic potential, ϕ total magnetic potential, ϕ external magnetic potential, ϕ in non-infinite element subdomains magnetic potential, r derivative of ϕ magnetic potential, z derivative of ϕ EXPRESSION A ϕ A ϕ,red + A 0ϕ,ext A ϕ r A ϕ z A ϕ,ext MAGNETOSTATIC AND QUASI-STATIC FIELDS 37

42 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION Aphit transient magnetic potential, time derivative of ϕ curlai curl of magnetic potential, x i mur mur, Br relative permeability mur M relative permeability murij mur, Br relative permeability, x i x j murij M relative permeability, x i x j EXPRESSION A ϕ t ( A) i µ r µ rij mu permeability µ 0 µ r muij permeability, x i x j µ 0 µ rij epsilonr harmonic epsr, relative ε r Dr permittivity epsilonr harmonic P relative permittivity epsilon harmonic permittivity ε 0 ε r sigma electric conductivity σ delta harmonic skin depth Re( jωµ ( σ + jωε) ) Vloop static, loop potential V loop transient Vloop harmonic loop potential V loop e j phase Pphi harmonic P electric polarization, ϕ P ϕ e jphase 38 CHAPTER 2: APPLICATION MODE REFERENCE

43 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. Pphi harmonic epsr, Dr electric polarization, ϕ Drphi harmonic epsr remanent displacement, ϕ Drphi harmonic P remanent displacement, ϕ Drphi harmonic Dr remanent displacement, ϕ Mi static, M magnetization, x i transient Mi harmonic M magnetization, x i Mi mur, Br magnetization, x i normm magnetization, norm Bri mur remanent flux density, x i Bri M remanent flux density, x i Bri static, transient Br remanent flux density, x i Bri harmonic Br remanent flux density, x i normbr Jephi static, transient DESCRIPTION remanent flux density, norm external current density, ϕ EXPRESSION D ϕ 0 P ϕ M i M i e j phase B i / µ 0 H i 0 µ 0 M i B ri ε 0 E ϕ D rϕ e jphase M M * B ri e j phase J ϕ e * B r B r MAGNETOSTATIC AND QUASI-STATIC FIELDS 39

44 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. Jephi harmonic external current density, ϕ vi velocity, x i normv velocity, norm Ephi transient electric field, ϕ Ephi harmonic electric field, ϕ norme transient, harmonic electric field, norm Dphi harmonic epsr electric displacement, ϕ Dphi harmonic P electric displacement, ϕ Dphi harmonic Dr electric displacement, ϕ normd harmonic electric displacement, norm Br magnetic flux density, r Bz magnetic flux density, z Br Bz reduced potential reduced potential DESCRIPTION total magnetic flux density, r total magnetic flux density, z EXPRESSION J ϕ e e jphase v i v v A ϕ t jωa ϕ E ϕ ε 0 ε r E ϕ ε 0 E ϕ + P ϕ ε 0 ε r E ϕ + D ϕ A ϕ z A ϕ r A ϕ r D rϕ A ϕ B z ext,r A ϕ A ϕ B r r ext,y 40 CHAPTER 2: APPLICATION MODE REFERENCE

45 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. extbi extbi normb reduced potential, background field specified through B reduced potential, background field specified through A DESCRIPTION external magnetic flux density, x i in non-infinite element subdomains external magnetic flux density, x i in non-infinite element subdomains magnetic flux density, norm Hi mur magnetic field, x i Hi M magnetic field, x i Hi Br magnetic field, x i normh magnetic field, norm Jpphi loop current, ϕ Jiphi transient, harmonic induced current density, ϕ Jdphi harmonic displacement current density, ϕ Jvphi velocity current density, ϕ Jphi static total current density, ϕ EXPRESSION B 0ext,i ( A ext ) i B B * µ rijbj µ 0 B i /µ 0 M i µ rij ( B j B rj ) µ 0 H H * σv loop /2πr σe ϕ jωd ϕ σ ( v z B r v r B z ) J ϕ e v p + J ϕ +J ϕ MAGNETOSTATIC AND QUASI-STATIC FIELDS 4

46 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. Jphi transient total current density, ϕ Jphi harmonic total current density, ϕ normj total current density, norm Wm static, transient DESCRIPTION magnetic energy density Weav harmonic time average electric energy density Wmav harmonic time average magnetic energy density Wav harmonic time average total energy density Q static resistive heating Q transient resistive heating Qav harmonic time average resistive heating Qmav harmonic time average magnetic hysteresis losses Por transient power flow, r Poz transient power flow, z normpo transient power flow, norm EXPRESSION e v p i J ϕ + J ϕ + J ϕ +J ϕ e v p i d J ϕ + J ϕ + J ϕ + J ϕ +J ϕ J ϕ H B 2 * --Re( E 4 ϕ D ϕ ) --Re( H B * ) 4 W e av + W m av V loop J ϕ v r B z v z B r σ 2πr V loop J ϕ e J ϕ E ϕ + v r B z v z B r σ 2πr * --Re J 2 ϕ E ϕ -- Re( jωh B * ) 2 E ϕ H z E ϕ H r S S * * V loop J ϕ e * * + v r B z v z B r σ 2πr J ϕ e* 42 CHAPTER 2: APPLICATION MODE REFERENCE

47 TABLE 2-2: APPLICATION MODE SUBDOMAIN VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION Porav harmonic time average power flow, r Pozav harmonic time average power flow, z normpoav harmonic time average power flow, norm FLtzi Lorentz force, x i (Note: for harmonic analysis with angular frequency ω, the Lorentz force s angular frequency is 2ω) FLtzavi harmonic cycle average Lorentz force, x i EXPRESSION * --Re( E 2 ϕ H z ) * --Re( E 2 ϕ H z ) S av (J B) i S av* -- ( J B * ) i 2 APPLICATION MODE BOUNDARY VARIABLES The boundary variables for Azimuthal Currents are given the table below. TABLE 2-22: APPLICATION MODE BOUNDARY VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION Jsphi surface current density, n z up ( H r down H r up ) n r up ( H z down H z up ) ϕ murbnd relative µ rbnd permeability on boundary sigmabnd harmonic conductivity on ε rbnd boundary epsilonr bnd harmonic relative permittivity on boundary σ bnd MAGNETOSTATIC AND QUASI-STATIC FIELDS 43

48 TABLE 2-22: APPLICATION MODE BOUNDARY VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION Qs transient surface resistive heating J s E Qsav harmonic time average surface resistive --Re( J 2 s E * ) heating npo transient power outflow n S npoav harmonic time average power outflow n S av unti dnti static, transient static, transient Maxwell surface stress tensor, x i, up side of boundary Maxwell surface stress tensor, x i, down side of boundary untiav harmonic Maxwell surface stress tensor, x i, up side of boundary dntiav harmonic Maxwell surface stress tensor, x i, down side of boundary extbni reduced potential External magnetic flux density, normal x i -- ( H 2 up B up )n idown + ( n down H up )B iup -- ( H 2 down B down )n iup + ( n up H down )B idown * --Re( H 4 up B up )n idown * + --Re(( n 2 down H up )B iup ) * --Re( H 4 down B down )n iup * + --Re(( n 2 up H down )B idown ) (n B ext ) i 44 CHAPTER 2: APPLICATION MODE REFERENCE

49 TABLE 2-22: APPLICATION MODE BOUNDARY VARIABLES, AZIMUTHAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION FsLtzi Lorentz surface force, x i (Note: for harmonic analysis with angular freqency ω, the Lorentz force s angular freqency is 2ω) J s B FsLtzavi harmonic cycle average -- Lorentz surface ( J 2 s B * ) i force, x i In-Plane Induction Currents, Magnetic Field All the nonzero s of the fundamental electromagnetic field quantities can be used in visualization and postprocessing of the results and when defining the equations and boundary conditions. Both magnetic and electric energy and resistive heating can be computed, as well as the two s of the Poynting vector. At the boundaries, the magnetic field, the transversal electric field and the normal energy are available. See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-23: APPLICATION MODE SCALAR VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION nu harmonic frequency ν omega harmonic angular frequency 2πν epsilon0 permittivity of vacuum ε 0 mu0 permeability of vacuum µ 0 MAGNETOSTATIC AND QUASI-STATIC FIELDS 45

50 APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for In-Plane Currents are given the table below. TABLE 2-24: APPLICATION MODE SUBDOMAIN VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS CONST REL. DESCRIPTION EXPRESSION Hz H z magnetic field, z normh magnetic field, H z norm nu harmonic frequency ν omega harmonic angular frequency 2πν d thickness d epsilon0 permittivity of vacuum mu0 permeability of vacuum mur mur, Br relative permeability mur M relative permeability ε 0 µ 0 µ r mu permeability µ 0 µ r epsilonr harmonic epsr, Dr relative permittivity epsilonr harmonic P relative permittivity epsilonrij harmonic epsr, Dr relative permittivity, x i x j epsilonrij harmonic P relative permittivity, x i x j ε r ε rij epsilon harmonic permittivity ε 0 ε r epsilonij harmonic permittivity, x i x j ε 0 ε rij sigma sigmaij electric conductivity electric conductivity, x i x j σ σ ij 46 CHAPTER 2: APPLICATION MODE REFERENCE

51 TABLE 2-24: APPLICATION MODE SUBDOMAIN VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS CONST REL. DESCRIPTION EXPRESSION Pi harmonic P external polarization, x i Pi harmonic epsr, Dr electric polarization, x i normp harmonic electric polarization, norm Dri harmonic epsr remanent displacement, x i Dri harmonic P remanent displacement, x i Dri harmonic Dr remanent displacement, x i normdr harmonic remanent displacement, norm Mz static, transient M magnetization, z Mz harmonic M magnetization, z Mz mur, Br magnetization, z Brz mur remanent flux density, z Brz M remanent flux density, z Brz static, transient Br remanent flux density, z P i e j phase D i ε 0 E i 0 P i D ri e j phase M z M z e j phase B z / µ 0 H z 0 µ 0 M z B rz P P * * D r D r MAGNETOSTATIC AND QUASI-STATIC FIELDS 47

52 TABLE 2-24: APPLICATION MODE SUBDOMAIN VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS CONST REL. DESCRIPTION EXPRESSION Brz harmonic Br remanent flux density, z Jei static, transient external current density, x i Jei harmonic external current density, x i normje external current density, norm vi velocity, x i normv velocity, norm Bz mur magnetic flux density, z Bz M magnetic flux density, z Bz Br magnetic flux density, z normb Ex static, transient magnetic flux density, norm electric field, x Ex harmonic electric field, x Ey static, transient electric field, y Ey harmonic electric field, y norme electric field, norm B rz e j phase J i e J i e e jphase v i e e* J i Ji v v µ 0 µ r H z µ 0 ( H z + M z ) µ 0 µ r H z + B rz B z σ xi σ cxi σ yi σ cyi e ( J i J i ) v y B z ( J i jωd ri + ( σ iy v x σ ix v y )B z ) J i e e ( J i J i ) + v x B z ( J i jωd ri + ( σ iy v x σ ix v y )B z ) E E * J i e 48 CHAPTER 2: APPLICATION MODE REFERENCE

53 TABLE 2-24: APPLICATION MODE SUBDOMAIN VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS CONST REL. DESCRIPTION EXPRESSION Di harmonic epsr electric displacement, x i Di harmonic P electric displacement, x i Di harmonic Dr electric displacement, x i normd harmonic electric displacement, norm Jii induced current density, x i normji induced current density, norm Jvx velocity current density, x Jvy velocity current density, y normjv velocity current density, norm Jdi harmonic displacement current density, x i normjd harmonic displacement current density, norm Jx total current density, x Jy total current density, y ε 0 ε rij E j ε 0 E i + P i ε 0 ε rij E j + D ri D D * σ ij E j J i J i* σ xx v y B z σ xy v x B z σ yx v y B z σ yy v x B z J v J v* jωd i J d J d* H z y H z y MAGNETOSTATIC AND QUASI-STATIC FIELDS 49

54 TABLE 2-24: APPLICATION MODE SUBDOMAIN VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS CONST REL. DESCRIPTION EXPRESSION normj Wm static, transient total current density, norm magnetic energy density Wmav harmonic time average magnetic energy density Weav harmonic time average electric energy density Wav harmonic time average total energy density Q static, transient Qav harmonic time average resistive heating Pox Poy normpo static, transient static, transient static, transient resistive heating J ( E + σ J e ) + J x v y B z J y v x B z power flow, x power flow, y power flow, norm Poxav harmonic time average power flow, x Poyav harmonic time average power flow, y normpoav harmonic time average power flow, norm J J * H z B z 2 * --Re( H 4 z B z ) --Re( E D * ) 4 W e av + W m av --Re J E * σ J e* * * ( ( + ) + J 2 x v y B z J y v x B z ) E y H z E x H z S S * * --Re( E 2 y H z ) * -- Re( E 2 x H z ) S av S av* 50 CHAPTER 2: APPLICATION MODE REFERENCE

55 APPLICATION MODE BOUNDARY VARIABLES The boundary variables for In-Plane Currents are given the table below. TABLE 2-25: APPLICATION MODE BOUNDARY VARIABLES, IN-PLANE CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION npo static, transient power outflow n S npoav harmonic time average power outflow n S av Meridional Induction Currents, Magnetic Field The magnetic field is available for postprocessing and for use in equations and boundary conditions at both subdomains and boundaries. At boundaries, you can also work with the tangential electric field and the normal of the Poynting vector. Energy and resistive heating are available in subdomains beside the standard electromagnetic field densities. See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-26: APPLICATION MODE SCALAR VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION nu harmonic frequency ν omega harmonic angular frequency 2πν epsilon0 permittivity of vacuum ε 0 mu0 permeability of vacuum µ 0 APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for Meridional Currents are given the table below. TABLE 2-27: APPLICATION MODE SUBDOMAIN VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Hphi magnetic field, H ϕ normh Hphir ϕ magnetic field, norm magnetic field, r derivative of ϕ H ϕ H ϕ r MAGNETOSTATIC AND QUASI-STATIC FIELDS 5

56 TABLE 2-27: APPLICATION MODE SUBDOMAIN VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Hphiz magnetic field, z derivative of ϕ Hphit transient magnetic field, time derivative of ϕ mur mur, Br relative permeability mur M relative permeability H ϕ z H ϕ t mu permeability µ 0 µ r epsilonr harmonic epsr, Dr relative ε r permittivity epsilonr harmonic P relative permittivity epsilonrij harmonic epsr, Dr relative ε rij permittivity, x i x j epsilonrij harmonic P relative permittivity, x i x j epsilon harmonic permittivity ε 0 ε r epsilonij harmonic permittivity, ε 0 ε rij x i x j sigma electric σ conductivity sigmaij electric σ ij conductivity, x i x j Pi harmonic P external P i e j phase polarization, x i Pi harmonic epsr, Dr electric polarization, x i D i ε 0 E i µ r 52 CHAPTER 2: APPLICATION MODE REFERENCE

57 TABLE 2-27: APPLICATION MODE SUBDOMAIN VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION normp harmonic electric polarization, norm Dri harmonic epsr remanent displacement, x i Dri harmonic P remanent displacement, x i Dri harmonic Dr remanent displacement, x i normdr harmonic remanent displacement, norm Mphi static, transient M magnetization, ϕ Mphi harmonic M magnetization, ϕ Mphi mur, Br magnetization, ϕ Brphi mur remanent flux density, ϕ Brphi M remanent flux density, ϕ Brphi static, Br remanent flux transient density, ϕ Brphi harmonic Br remanent flux density, ϕ Jei static, transient external current density, x i 0 P i D ri e j phase 0 P P * * D r D r M ϕ M ϕ e jphase B ϕ µ 0 H ϕ µ 0 M ϕ B rϕ B rϕ e jphase J i e MAGNETOSTATIC AND QUASI-STATIC FIELDS 53

58 TABLE 2-27: APPLICATION MODE SUBDOMAIN VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Jei harmonic external current density, x i normje external current density, norm vi velocity, x i normv velocity, norm Bphi mur magnetic flux density, ϕ Bphi M magnetic flux density, ϕ Bphi Br magnetic flux density, ϕ normb magnetic flux density, norm Bi extbi extbphi Er reduced potential reduced potential background field given by B reduced potential, background field given by A static, transient total magnetic flux density, x i external magnetic flux density, x i in non-infinite element subdomains external magnetic flux density, ϕ electric field, r Er harmonic electric field, r J i e e jphase v i e e* J i Ji v v µ 0 µ r H ϕ µ 0 ( H ϕ + M ϕ ) µ 0 µ r H ϕ + B rϕ B ϕ σ ri σ cri e ( J i J i ) + v z B ϕ ( J i jωd ri + ( σ ir v z σ iz v r )B ϕ ) J i e 54 CHAPTER 2: APPLICATION MODE REFERENCE

59 TABLE 2-27: APPLICATION MODE SUBDOMAIN VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Ez static, transient electric field, z Ez harmonic electric field, z norme electric field, norm Di harmonic epsr electric displacement, x i Di harmonic P electric displacement, x i Di harmonic Dr electric displacement, x i normd harmonic electric displacement, norm Jii induced current density, x i normji induced current density, norm Jvr velocity current density, r Jvz velocity current density, z normjv velocity current density, norm Jdi harmonic displacement current density, x i normjd harmonic displacement current density, norm σ zi σ czi ε 0 ε rij E j ε 0 E i + P i ε 0 ε rij E j + D ri σ ij E j jωd i e ( J i J i ) v r B ϕ ( J i jωd ri + ( σ ir v z σ iz v z )B ϕ ) E E * D D * J i J i* J i e σ rz v r B ϕ σ rr v z B ϕ σ zz v r B ϕ σ zr v z B ϕ J v J v* J d J d* MAGNETOSTATIC AND QUASI-STATIC FIELDS 55

60 TABLE 2-27: APPLICATION MODE SUBDOMAIN VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Jr Jz normj Wm static, transient total current density, r total current density, z total current density, norm magnetic energy density Wmav harmonic time average magnetic energy density Weav harmonic time average electric energy density Wav harmonic time average total energy density Q static, resistive heating transient Qav harmonic time average resistive heating Por Poz normpo static, transient static, transient static, transient power flow, r power flow, z power flow, norm Porav harmonic time average power flow, r Pozav harmonic time average power flow, z normpoav harmonic time average power flow, norm H ϕ z H ϕ r H ϕ r J J * H ϕ B ϕ 2 * --Re( H 4 ϕ B ϕ ) --Re( E D * ) 4 W e av + W m av J ( E+ σ J e ) + J z v r B ϕ J r v z B ϕ --Re J E * σ J e* * * ( ( + ) + J 2 z v r B ϕ J r v z B ϕ ) E z H ϕ E r H ϕ S S * * -- Re( E 2 z H ϕ ) * --Re( E 2 r H ϕ ) S av S av* 56 CHAPTER 2: APPLICATION MODE REFERENCE

61 APPLICATION MODE BOUNDARY VARIABLES The boundary variables for Meridional Currents are given the table below. TABLE 2-28: APPLICATION MODE BOUNDARY VARIABLES, MERIDIONAL CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION npo static, transient power outflow n S npoav harmonic time average power outflow n S av Magnetostatics, No Currents The fundamental fields that can be derived from the electric potential are available for postprocessing and for use in equations and boundary conditions. See page 4 for a description of the notation used in this table. APPLICATION MODE SCALAR VARIABLES The application-specific variables in this mode are given in the following table. TABLE 2-29: APPLICATION MODE SCALAR VARIABLES, MAGNETOSTATICS, NO CURRENTS NAME DESCRIPTION EXPRESSION mu0 permeability of vacuum µ 0 exth0i External magnetic field (only present in reduced potential formulation), x i H 0ext,i APPLICATION MODE SUBDOMAIN VARIABLES The subdomain variables for Magnetostatics, No Currents are given the table below. TABLE 2-30: APPLICATION MODE SUBDOMAIN VARIABLES, MAGNETOSTATICS, NO CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Vm magnetic potential V m mur mur, Br relative permeability µ r mur M relative permeability murij mur, Br relative permeability, x i x j µ rij murij M relative permeability mu permeability µ 0 µ r muij permeability, x i x j µ 0 µ rij Mi M magnetization, x i M i MAGNETOSTATIC AND QUASI-STATIC FIELDS 57

62 TABLE 2-30: APPLICATION MODE SUBDOMAIN VARIABLES, MAGNETOSTATICS, NO CURRENTS NAME ANALYSIS CONST. REL. DESCRIPTION EXPRESSION Mi mur, Br magnetization, x i normm magnetization, norm Bri mur remanent flux density, x i Bri M remanent flux density, x i Bri Br remanent flux density, x i normbr remanent flux density, norm Hi total potential magnetic field, x i B i /µ 0 H i 0 µ 0 M i B ri M M * * B r B r V m x i redhi reduced potential reduced magnetic field, x i V m x i Hi reduced potential total magnetic field, x i V m H x ext,i i exthi reduced potential external magnetic field, x i in non-infinite element subdomain H 0ext,i normh magnetic field, norm H H * Bi mur magnetic flux density, x i Bi M magnetic flux density, x i Bi Br magnetic flux density, x i normb Wm magnetic flux density, norm magnetic energy density µ 0 µ rij H j µ 0 ( H i + M i ) µ 0 µ rij H j + B ri B B * H B 2 58 CHAPTER 2: APPLICATION MODE REFERENCE

63 APPLICATION MODE BOUNDARY VARIABLES The boundary variables for Magnetostatics, No Currents are given the table below. TABLE 2-3: APPLICATION MODE BOUNDARY VARIABLES, MAGNETOSTATICS, NO CURRENTS NAME ANALYSIS DESCRIPTION EXPRESSION thi total potential tangential magnetic field, x i extthi redthi thi normth reduced potential reduced potential reduced potential external magnetic field, tangential x i reduced magnetic field, tangential x i total magnetic field, tangential x i tangential magnetic field, norm t i t i t i t V m H ext, i t V m H ext,t +H red,t * H t H t nb normal magnetic flux density n B unti Maxwell surface stress tensor, x i, up side of boundary -- ( H 2 up B up )n idown + ( n down H up )B iup dnti Maxwell surface stress tensor, x i, down side of boundary -- ( H 2 down B down )n iup + ( n up H down )B idown MAGNETOSTATIC AND QUASI-STATIC FIELDS 59

64 60 CHAPTER 2: APPLICATION MODE REFERENCE

65 3 Programming Reference This chapter, which is a complement to the COMSOL Multiphysics Scripting Guide, describes and exemplifies how to use the application modes provided by the AC/DC Module in command-line modeling. 6

66 The Programming Language In other parts of this documentation set, the examples use the COMSOL Multiphysics graphical user interface for solving problems with the AC/DC Module. Although this user interface provides a convenient environment for modeling many problems, it can sometimes be useful to work with a programming tool. A summary of the application structure, and how application objects are used for a convenient transformation of application mode data to PDE and boundary coefficients, is presented in the following section. Thereafter the same problem as solved in An Example Eddy Currents on page 2 of the AC/DC Module User s Guide is built using the programming language. For details on specific functions, see the chapter Function Reference on page in this manual and the Command Reference chapter of the COMSOL Multiphysics Reference Guide. The Application Structure The process of performing a simulation using the application modes available through the AC/DC Module includes the correct setup of the application structure. The application structure contains the necessary information for the model setup in several fields. This section describes the application structure in the context of the AC/DC Module. See also the section Application Structures on page 54 in the COMSOL Multiphysics Scripting Guide. Most fields have corresponding entries in the FEM structure, described in the section Specifying a Model on page 3 in the COMSOL Multiphysics Scripting Guide. Table 3- gives an overview of the fields in the application structure. TABLE 3-: APPLICATION STRUCTURE FIELDS FIELD appl.mode appl.dim appl.sdim appl.border appl.name DESCRIPTION Application mode class Cell array of dependent variable names Cell array of spatial coordinates Assembly on interior boundaries; turn on/off assembly on interior boundaries Application mode name 62 CHAPTER 3: PROGRAMMING REFERENCE

67 TABLE 3-: APPLICATION STRUCTURE FIELDS FIELD appl.var appl.assign appl.assignsuffix appl.equ appl.bnd appl.edg DESCRIPTION Cell array or structure with application-specific scalar variables. Assigned variable names Suffix to append to all application mode variable names Structure containing domain properties Structure containing boundary conditions Structure containing edge conditions Most of these fields have default values and need not be specified when solving a problem using the programming language. The function multiphysics transforms the application structure data to the FEM structure to generate the complete set of equations; see the entry for multiphysics on page 350 of the COMSOL Multiphysics Reference Guide for details. The application mode specific names of the fields in the structures in the table above can be found in the chapter Application Mode Programming Reference on page 79. APPLICATION MODE CLASS The application modes are specified via a corresponding class name. To specify the class, write the name of the class as a string, for example, appl.mode.class='perpendicularcurrents'; which specifies that the Perpendicular Currents application mode will be used. Some application modes, for example the Conductive Media DC and Electrostatics application modes, can be used both for Cartesian and axisymmetric problems. The default is to use Cartesian coordinates. To specify an axisymmetric problem, use appl.mode.type = 'axi'; DEPENDENT VARIABLES The dim field in the application structure states the names of the dependent variables, and hence gives the dimension of the corresponding PDE system. If the dim field is missing, the corresponding standard names of the electromagnetic field quantities solved for are used. Note that in some axisymmetric modes, there is a variable transformation in the equations solved. In those modes, the default name of the dependent variable name has dr added to the name of the electromagnetic quantity, since the dependent variable is divided by r. THE PROGRAMMING LANGUAGE 63

68 The default name of the dependent variable in the Perpendicular currents quasi-statics application mode is 'Az'. You can set a new name of the dependent variable by typing appl.dim = {'A'}; SPATIAL COORDINATES The names of the spatial coordinates are given in fem.sdim. However, fem.sdim only gives the names of the spatial coordinates of the geometry: one variable in D, two variables in 2D, and three variables in 3D. To specify all three spatial coordinates, use appl.sdim. The additional spatial coordinates are used when giving names to vector variables. For example, in 2D Cartesian coordinates fem.sdim = {'x' 'y'}; appl.sdim = {'x' 'y' 'z'}; defines the spatial coordinates x, y, z. In cylindrical coordinates fem.sdim = {'r 'z }; appl.sdim = {'r' 'phi' 'z'}; defines the spatial coordinates r, ϕ, z. The coordinates defined both in fem.sdim and appl.sdim should match. APPLICATION NAME This field is used for giving the application a name. If no name is specified, a default name is used. appl.name = 'magnet'; APPLICATION MODE PROPERTIES Some application modes define properties to, for example, specify which type of analysis to perform or which dependent variables to use. These are specified in the appl.prop structure. For example, the Perpendicular Currents application mode defines the field appl.prop.analysis, which specifies if a static, transient, or time-harmonic analysis should be performed. For instance, appl.prop.analysis = 'harmonic'; indicates time-harmonic analysis. 64 CHAPTER 3: PROGRAMMING REFERENCE

69 The default element type used for domains and boundaries can be specified in the appl.prop.elemdefault field. For example, you can write appl.prop.elemdefault='lag'; to obtain linear Lagrange elements. APPLICATION SCALAR VARIABLES For scalar variables that are valid in the whole model, such as permittivity and permeability of vacuum and angular frequency, the values can be specified in the appl.var field. Note that only predefined variable names can be used in this field. For the Perpendicular Currents application mode, the variables can be set as appl.var.epsilon0 = ; appl.var.mu0 = ; appl.var.nu = 00; The appl.var field can also be specified as a cell array: appl.var = {'epsilon0' 'mu0', 'nu' 00}; The default values in the Perpendicular Currents application mode in the harmonic formulation are appl.var.epsilon0 = e-2; appl.var.mu0 = 4*pi*e-7; appl.var.nu = 50; ASSIGNED VARIABLE NAMES To avoid duplicate variable names, you can use the appl.assign field to state the relation between the assigned name of a variable available in postprocessing and in the PDE and boundary coefficients on one hand, and the default name that is used in the COMSOL Multiphysics and AC/DC Module algorithms on the other. This field is thus only necessary when you solve a multiphysics problem where variable name conflicts may arise. If you want to use the variable name w for the intrinsic variable omega in the Perpendicular currents quasi-statics mode, all you have to enter is appl.assign.omega = 'w'; An alternative syntax uses a cell array where the intrinsic and new variable names are listed pairwise: appl.assign = {'omega' 'w'}; THE PROGRAMMING LANGUAGE 65

70 The odd-index entries in the assign field state the default application scalar variable names or the postprocessing variable names; these are listed for all application modes in the chapter Application Mode Reference on page 3 of this manual. The even-index entries in appl.assign are the variable names that you want to use for the physical entities when modeling. There is also a field appl.assignsuffix, which can be used to add a suffix to the name of all application mode variables. Variables appearing in appl.assign will take the given assigned name, while the others will get the suffix added to their names. DOMAIN PROPERTIES The application structure field appl.equ is used for specifying electromagnetic properties that will be transformed into PDE coefficients. For example the Perpendicular Currents application mode defines the fields appl.equ.sigma appl.equ.mur appl.equ.epsilonr appl.equ.v appl.equ.jez appl.equ.m appl.equ.br appl.equ.pz appl.equ.drz appl.equ.deltav appl.equ.l appl.equ.magconstrel appl.equ.elconstrel Now consider an example with two materials. To use other values than the default for the physical properties, enter them as follows. Scalars such as sigma are given as a cell array with the values, appl.equ.sigma={'5.9e7' '0'}; Vectors such as M are given as a nested cell array with the vector s as elements in the inner cell array, appl.equ.m={{'0' '0'} {'0' '500'}}; In the 3D case, the vector has three s, appl.equ.m={{'0' '0' '0'} {'0' '500' '0'}}; Tensors or matrix variables are given as cell arrays with the tensor s, appl.equ.mur={{'' '2'; '3' '4'} {'5' '6' '7' '8'}}; 66 CHAPTER 3: PROGRAMMING REFERENCE

71 This example shows two ways of writing a 2-by-2 tensor. The first tensor is written row by row with a semicolon separating the rows, 2 34 while the second tensor is written column by column, There are a number of short-hand ways of writing a tensor. In 2D {'' '2'} is a diagonal tensor {'' '2' '3'} is a symmetric tensor 0 02 and {'' '2' '3' '4'} is the full tensor Similarly in 3D {'' '2' '3'} is a diagonal tensor {'' '2' '3' '4' '5' '6'} is a symmetric tensor and {'' '2' '3' '4' '5' '6' '7' '8' '9'} is the full tensor THE PROGRAMMING LANGUAGE 67

72 Constitutive Relations In the static and quasi-static application modes three different constitutive relations between both the D and E fields and the B and H fields can be used. To specify which ones to use, use the fields appl.equ.elconstrel and appl.equ.magconstrel. Their values are listed in the table below. CONSTITUTIVE RELATION VARIABLE VALUE D = ε 0 ε r E elconstrel epsr D = ε 0 E + P elconstrel P D = ε 0 ε r E + D r elconstrel Dr B = µ 0 µ r H magconstrel mur B = µ 0 ( H + M) magconstrel M B = µ 0 µ r H + B r magconstrel Br H = f( B ) e B magconstrel fh H = f(b) magconstrel aniso_fh Refractive Index and Relative Permittivity In the wave application modes, the material data can be given either by the refractive index or the three properties relative permittivity, relative permeability and conductivity. To specify which of the two to use, use the field appl.equ.matparams. It takes the value n for refractive index and the value epsr for the other three properties. For example, appl.equ.matparams = {'n' 'epsr'}; appl.equ.n = {'.2' ''}; appl.equ.epsilonr = {'' '2'}; appl.equ.sigma = {'0' 'e4 }; appl.equ.mur = {'' '.'}; specifies the refractive index n =.2 for the first domain and ε r = 2, σ = 0 4 S/m, and µ r =. for the second one. BOUNDARY CONDITIONS The boundary conditions for the simulation is given in the appl.bnd field. First you have to select which type of boundary condition you want at each boundary. This is done using the field appl.bnd.type. Then some boundary conditions requires 68 CHAPTER 3: PROGRAMMING REFERENCE

73 certain boundary variables to be set, unless the default value is sufficient. Note that not all types of boundary conditions require a boundary variable. For example, the boundary variables defined by the Perpendicular Currents application mode are appl.bnd.j0 appl.bnd.jn appl.bnd.h0 appl.bnd.js0z appl.bnd.a0z appl.bnd.eta appl.bnd.esz appl.bnd.sigmabnd appl.bnd.murbnd appl.bnd.murtensorbnd appl.bnd.mutype appl.bnd.epsilonrbnd appl.bnd.type The variables are specified using the same form for scalars, vectors, and tensors as above for the subdomain variables. DOMAIN GROUPS Often you have a model with several subdomains having the same physical properties and many boundaries where you want to apply the same boundary condition. To simplify the notation, the index vectors in the fields equ.ind and bnd.ind are used. Consider this example: appl.equ.sigma = {'5.9e7' '0'}; appl.equ.ind = [ 2 ]; Here you have five subdomains. Subdomain 2 is nonconducting, whereas the other four subdomains have the electric conductivity S/m. Here is another example where three different boundary conditions are applied to six boundaries: appl.bnd.type = {'th0' 'H' 'A'}; appl.bnd.a0z = {[] [] '00'}; appl.bnd.h0 = {[] {'0' '5'} []}; appl.bnd.ind = [ ]; This specifies the magnetic field at Boundaries 2, 3, and 4; the magnetic potential at Boundaries 5 and 6; and electric insulation at Boundary. Note that the electric insulation condition does not need any boundary variable to be defined and that we have set the variables to an empty vector where the value is ignored. THE PROGRAMMING LANGUAGE 69

74 There is an alternative syntax for the index vector using a cell array of numeric vectors instead of a single numeric vector as in the examples above. In this case the numeric vectors in the cell array list the domains having the same settings. For example, appl.equ.ind = {[ 3 4 5] 2}; is equivalent to and appl.equ.ind = [ 2 ]; appl.bnd.ind = { [2 3 4] [5 6]}; is equivalent to appl.bnd.ind = [ ]; Eddy Currents in the Programming Language Next, reproduce the model describing eddy currents in a cylinder in the programming language by creating an application structure and an FEM structure. The application structure contains the physical properties stated above, and the FEM structure contains the geometry, the mesh, and finally the solution vector. This section extends the example by doing some postprocessing that is not included in the modeling procedure using the graphical user interface. For this purpose, add lines around the cylinder and a circle around the coil as shown in the figure below. These curves do not represent physical boundaries, but you can use them for evaluating a line integral of the magnetic field, in order to calculate the current flowing inside the curve from the integral form of Ampère s law, C H dl = I 70 CHAPTER 3: PROGRAMMING REFERENCE

75 PREPROCESSING Start by clearing the FEM and application structures to avoid accidental use of old data. clear fem appl Give the space dimensions of the problem. fem.sdim={'r' 'z'}; Then define the variables to be used in the solution process. When programming it is easy to parameterize a model using workspace variables. If several physical quantities in the model depend on one variable, you can define this variable as a workspace variable. In this model the current flowing through the coil is I0, and the radius of the coil is rad. To define these variables and give them a value, just enter I0=e3; rad=0.0; You can now define the FEM structure constants that you want to use when defining the domain properties and boundary conditions. fem.const.sigcoil=3.7e7; fem.const.sigcyl=3.7e7; fem.const.j0=i0/(pi*rad^2); fem.const.js0=i0/(2*pi*rad); THE PROGRAMMING LANGUAGE 7

76 The geometry data is stored in the geom field of the FEM structure. For more details on how to create geometries, see Geometry Modeling and CAD Tools on page 23 in the COMSOL Multiphysics User s Guide. c=circ2(0.05,0,rad); r=rect2(0,0.2,-0.25,0.25); r2=rect2(0,0.03,-0.05,0.05); r3=rect2(0,0.035,-0.06,0.06); c5=circ2(0.05,0,0.02); fem.geom=geomcomp({c r r2 c2 c3 c4 c5}); To verify that the geometry is correct, and to find the domain numbers of the subdomains, enter the commands geomplot(fem,'sublabels','on'); axis([ ]); Similarly you can find the domain numbers of the boundaries by entering the command geomplot(fem,'edgelabels','on'); SKIN EFFECT MODELING It is now time to create the application structure. This is the part where you put the physics into the code. First, define the application mode to be used. Use the Azimuthal currents application mode, just as when modeling in the graphical user interface. The name of that class is AzimuthalCurrents. appl.mode.class='azimuthalcurrents'; 72 CHAPTER 3: PROGRAMMING REFERENCE

77 To indicate that the problem is axisymmetric, use the type field of the application mode. appl.mode.type='axi'; The Azimuthal Currents application can handle static, transient, and time-harmonic problems. To specify that a time-harmonic analysis should be performed, set the analysis property. appl.prop.analysis='harmonic'; Static analysis is the default that COMSOL Multiphysics would use if you do not specify the analysis property. The frequency of the source in this model is 00 Hz. Enter the following to use a correct value of the application scalar variable nu, the frequency. appl.var.nu=00; Set the symmetry condition at the symmetry axis, and magnetic insulation on all the other boundaries. At the interior boundaries no boundary condition is invoked. Note that the field appl.bnd.ind specifies the condition to be used at each boundary. The following means that the first boundary condition applies to Boundaries, 3, 5, 7, and 9, whereas the second condition is employed at Boundaries 2,, and 4. The command zeros(,22) creates a vector with 22 zeros, and the next two lines modify some of the elements in the vector. appl.bnd.type={'ax' 'A0'}; appl.bnd.ind=zeros(,22); appl.bnd.ind([ ])=; appl.bnd.ind([2 4])=2; In the field appl.bnd.type, ax stands for the axial symmetry boundary condition, and A0 for the electric insulation boundary condition, which fixes A ϕ to zero at the boundary. See page 0 for a complete list of the available boundary conditions. Apply the current in the coil by using the variable J0. Specify the electric conductivity in the coil and cylinder too, but use the default values for the dielectric and magnetic properties. The field appl.equ.ind specifies that the first value applies to Subdomains, 2, and 4; the second value to Subdomain 3; and the last value to Subdomain 5. appl.equ.sigma={'0' 'sigcyl' 'sigcoil'}; appl.equ.jephi={'0' '0' 'J0'}; appl.equ.ind=[ 2 3]; Generate the corresponding fields in the FEM structure by calling the function multiphysics. THE PROGRAMMING LANGUAGE 73

78 fem.appl=appl; fem=multiphysics(fem); Initialize the mesh by invoking the function meshinit and then create the extended mesh by invoking meshextend. For more details on creating meshes see the entries of the functions meshinit, meshrefine, and meshextend in the Command Reference chapter of the COMSOL Multiphysics Reference Guide. fem.mesh=meshinit(fem); fem.xmesh=meshextend(fem); Use the adaptive solver for solving this problem. Set the number of adaptive refinements to 2. fem=adaption(fem,... 'ngen',2,'report','on'); Visualize the result as a surface plot. postplot(fem,'tridata','jphi','tribar','on') axis([ ]); SURFACE CURRENTS MODELING This problem can also be modeled using surface currents, as described on page 2 of the AC/DC Module User s Guide. You simply set the electric conductivity in the coil to zero, and specify boundary conditions on the interior boundaries surrounding the coil. The conductivity in the coil is modified by entering fem.const.sigcoil=0; 74 CHAPTER 3: PROGRAMMING REFERENCE

79 The new boundary conditions are invoked by entering the following. To enable the conditions at interior boundaries, the appl.border field is used. appl.bnd.type={'ax' 'A0' 'Js'}; appl.bnd.js0phi={[] [] 'Js0'}; appl.bnd.ind([ ])=3; appl.border='on'; You must also remove the source on the subdomain representing the coil appl.equ.jephi={'0' '0' '0'}; The new PDE system can be generated when the fem.appl field is updated. fem.appl=appl; fem=multiphysics(fem); fem.xmesh=meshextend(fem); The adaptively refined mesh can be used for solving this problem too. Simply call the linear solver. fem.sol=femlin(fem); A similar plot as before, but without any current density in the coil domain can be obtained. To see the surface currents, use a line plot. postplot(fem,'tridata','jphi','tribar','on',... 'lindata','jsphi','linmap','hot') axis([ ]); THE PROGRAMMING LANGUAGE 75