Halbach3D 24 FF075PLosONE
Date Mar 1, 2016 12:30:31 PM
Contents
1. Global
1.1. Definitions
2. Component 1
2.1. Definitions
2.2. Geometry 1
2.3. Materials
2.4. Magnetic Fields, No Currents
2.5. Mesh 1
3. Component 2
3.1. Definitions
3.2. Geometry 2
4. Study 1
4.1. Stationary
4.2. Solver Configurations
5. Results
5.1. Data Sets
5.2. Tables
5.3. Plot Groups

1. Global

Date Mar 1, 2016 9:51:04 AM
Global settings
Name Halbach3D_24_FF075PLosONE.mph
Path R:\projects\MVogel_ULF_Halbach\COMSOL Models Halbach\3D\Halbach3D_24_FF075PLosONE.mph
Program COMSOL 5.0 (Build: 244)
Unit system SI
Used products
COMSOL Multiphysics
AC/DC Module

1.1. Definitions

1.1.1. Parameters 1

Parameters
Name Expression Value Description
BpRem 1[T] 1.0000 T
BpState 1 1.0000 1 = ON, -1 = OFF
halbach 1 1.0000 1 = Yes 0 = NO
argBp BpState*(1 + halbach)*pi/12 0.52360
tickn 0 0.0000 1 = Tangential, 0 = Radial
tick tickn*pi/2 0.0000 pi/2 = tang, 0= radial
dm 0.0216[m] 0.021600 m
length 0.7[m] 0.70000 m
RA 0.11[m] 0.11000 m
FOV 0.05[m] 0.050000 m

2. Component 1

Date Mar 1, 2016 9:51:04 AM
Component settings
Unit system SI
Geometry shape order automatic

2.1. Definitions

2.1.1. Coordinate Systems

Boundary System 1
Coordinate system type Boundary system
Tag sys1
Settings
First (t1) Second (t2) Third (n)
t1 t2 n
Settings
Name Value
Create first tangent direction from Global Cartesian

2.2. Geometry 1

geom_geom1.png

Geometry 1

Units
Length unit m
Angular unit deg
Geometry statistics
Property Value
Space dimension 3
Number of domains 26
Number of boundaries 158
Number of edges 312
Number of vertices 206

2.2.1. Cylinder 1 (Cyl1)

Position
Name Value
Position {RA, 0, -length/2}
Axis
Name Value
Axis type z - axis
Size and shape
Name Value
Radius dm/2
Height length

2.2.2. Rotate 1 (Rot1)

Selections of resulting entities
Name Value
Keep input objects On
Rotation {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345}
Point on axis of rotation {0, 0, 0}
Axis type z - axis

2.2.3. Cylinder 2 (Cyl2)

Position
Name Value
Position {0, 0, -0.75}
Axis
Name Value
Axis type z - axis
Size and shape
Name Value
Radius 0.4
Height 1.5

2.2.4. Sphere 1 (Sph1)

Position
Name Value
Position {0, 0, 0}
Axis
Name Value
Axis type z - axis
Size
Name Value
Radius FOV/2

2.3. Materials

2.3.1. Air

material_mat1.png

Air

Selection
Geometric entity level Domain
Selection Domains 1, 13
Basic Settings
Description Value
Heat capacity at constant pressure Cp(T[1/K])[J/(kg*K)]
Speed of sound cs(T[1/K])[m/s]
Dynamic viscosity eta(T[1/K])[Pa*s]
Ratio of specific heats 1.4
Thermal conductivity {{k(T[1/K])[W/(m*K)], 0, 0}, {0, k(T[1/K])[W/(m*K)], 0}, {0, 0, k(T[1/K])[W/(m*K)]}}
nu0 nu0(T[1/K])[m^2/s]
Density rho(pA[1/Pa], T[1/K])[kg/m^3]
Electrical conductivity {{0[S/m], 0, 0}, {0, 0[S/m], 0}, {0, 0, 0[S/m]}}
Functions
Function name Type
Cp Piecewise
cs Analytic
eta Piecewise
k Piecewise
nu0 Piecewise
rho Analytic

material_mat1_def_Cp.png

Cp

material_mat1_def_cs.png

cs

material_mat1_def_eta.png

eta

material_mat1_def_k.png

k

material_mat1_def_nu0.png

nu0

2.3.2. Soft Iron (Without Losses)

material_mat2.png

Soft Iron (without losses)

Selection
Geometric entity level Domain
Selection Domains 2–12, 14–26
Material parameters
Name Value Unit
Relative permeability 1.05 1
Basic Settings
Description Value
Electrical conductivity {{0.666e6[S/m], 0, 0}, {0, 0.666e6[S/m], 0}, {0, 0, 0.666e6[S/m]}}
Relative permittivity {{1.05, 0, 0}, {0, 1.05, 0}, {0, 0, 1.05}}
Relative permeability {{1.05, 0, 0}, {0, 1.05, 0}, {0, 0, 1.05}}
BH curve Settings
Description Value
Magnetic flux density norm BH(normH[m/A])[T]
normH sqrt(H1^2 + H2^2 + H3^2)
Functions
Function name Type
BH Interpolation

material_mat2_BHCurve_BH.png

BH

HB curve Settings
Description Value
Magnetic field norm HB(normB[1/T])[A/m]
normB sqrt(B1^2 + B2^2 + B3^2)
Functions
Function name Type
HB Interpolation

material_mat2_HBCurve_HB.png

HB

2.4. Magnetic Fields, No Currents

physics_emnc.png

Magnetic Fields, No Currents

Selection
Geometric entity level Domain
Selection Domains 1–26
Equations

equ_emnc_1.png

equ_emnc_2.png

Settings
Description Value
Magnetic scalar potential Quadratic
Value type when using splitting of complex variables Complex
Used products
COMSOL Multiphysics
AC/DC Module
Variables
Name Expression Unit Description Selection
emnc.Bbx 0 T Background magnetic flux density, x component Domains 1–26
emnc.Bby 0 T Background magnetic flux density, y component Domains 1–26
emnc.Bbz 0 T Background magnetic flux density, z component Domains 1–26
emnc.Hbx 0 A/m Background magnetic field, x component Domains 1–26
emnc.Hby 0 A/m Background magnetic field, y component Domains 1–26
emnc.Hbz 0 A/m Background magnetic field, z component Domains 1–26
emnc.nx nx Normal vector, x component Boundaries 5–82, 84–91, 93–158
emnc.ny ny Normal vector, y component Boundaries 5–82, 84–91, 93–158
emnc.nz nz Normal vector, z component Boundaries 5–82, 84–91, 93–158
emnc.nx dnx Normal vector, x component Boundaries 1–4, 83, 92
emnc.ny dny Normal vector, y component Boundaries 1–4, 83, 92
emnc.nz dnz Normal vector, z component Boundaries 1–4, 83, 92
emnc.nmeshx nxmesh Mesh normal vector, x component Boundaries 5–82, 84–91, 93–158
emnc.nmeshy nymesh Mesh normal vector, y component Boundaries 5–82, 84–91, 93–158
emnc.nmeshz nzmesh Mesh normal vector, z component Boundaries 5–82, 84–91, 93–158
emnc.nmeshx dnxmesh Mesh normal vector, x component Boundaries 1–4, 83, 92
emnc.nmeshy dnymesh Mesh normal vector, y component Boundaries 1–4, 83, 92
emnc.nmeshz dnzmesh Mesh normal vector, z component Boundaries 1–4, 83, 92
emnc.unmeshx unxmesh Mesh normal vector, upside, x component Boundaries 5–82, 84–91, 93–158
emnc.unmeshy unymesh Mesh normal vector, upside, y component Boundaries 5–82, 84–91, 93–158
emnc.unmeshz unzmesh Mesh normal vector, upside, z component Boundaries 5–82, 84–91, 93–158
emnc.dnmeshx dnxmesh Mesh normal vector, downside, x component Boundaries 5–82, 84–91, 93–158
emnc.dnmeshy dnymesh Mesh normal vector, downside, y component Boundaries 5–82, 84–91, 93–158
emnc.dnmeshz dnzmesh Mesh normal vector, downside, z component Boundaries 5–82, 84–91, 93–158
emnc.intWm emnc.int_Wm(emnc.d*emnc.dWm) J Total magnetic energy Global
emnc.d 1 1 Contribution Domains 1–26
emnc.unTmx 0 Pa Maxwell upward magnetic surface stress tensor, x component Boundaries 1–158
emnc.unTmy 0 Pa Maxwell upward magnetic surface stress tensor, y component Boundaries 1–158
emnc.unTmz 0 Pa Maxwell upward magnetic surface stress tensor, z component Boundaries 1–158
emnc.dnTmx 0 Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 1–158
emnc.dnTmy 0 Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 1–158
emnc.dnTmz 0 Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 1–158

2.4.1. Magnetic Flux Conservation 1

physics_emnc_mfc1.png

Magnetic Flux Conservation 1

Selection
Geometric entity level Domain
Selection Domains 1, 13
Equations

equ_emnc_mfc1_1.png

equ_emnc_mfc1_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component 0
Remanent flux density, y component 0
Remanent flux density, z component 0
Relative permeability User defined
Relative permeability {{1.1, 0, 0}, {0, 1.1, 0}, {0, 0, 1.1}}
Magnetic flux density norm User defined
Variables
Name Expression Unit Description Selection
emnc.unTmx -0.5*emnc.dnx*(real(up(emnc.Bx))*real(up(emnc.Hx))+real(up(emnc.By))*real(up(emnc.Hy))+real(up(emnc.Bz))*real(up(emnc.Hz)))+real(up(emnc.Bx))*(real(up(emnc.Hx))*emnc.dnx+real(up(emnc.Hy))*emnc.dny+real(up(emnc.Hz))*emnc.dnz) Pa Maxwell upward magnetic surface stress tensor, x component Boundaries 71–74, 86–89
emnc.unTmy -0.5*emnc.dny*(real(up(emnc.Bx))*real(up(emnc.Hx))+real(up(emnc.By))*real(up(emnc.Hy))+real(up(emnc.Bz))*real(up(emnc.Hz)))+real(up(emnc.By))*(real(up(emnc.Hx))*emnc.dnx+real(up(emnc.Hy))*emnc.dny+real(up(emnc.Hz))*emnc.dnz) Pa Maxwell upward magnetic surface stress tensor, y component Boundaries 71–74, 86–89
emnc.unTmz -0.5*emnc.dnz*(real(up(emnc.Bx))*real(up(emnc.Hx))+real(up(emnc.By))*real(up(emnc.Hy))+real(up(emnc.Bz))*real(up(emnc.Hz)))+real(up(emnc.Bz))*(real(up(emnc.Hx))*emnc.dnx+real(up(emnc.Hy))*emnc.dny+real(up(emnc.Hz))*emnc.dnz) Pa Maxwell upward magnetic surface stress tensor, z component Boundaries 71–74, 86–89
emnc.unTmx -0.5*emnc.dnx*(real(up(emnc.Bx))*real(up(emnc.Hx))+real(up(emnc.By))*real(up(emnc.Hy))+real(up(emnc.Bz))*real(up(emnc.Hz)))+real(up(emnc.Bx))*(real(up(emnc.Hx))*emnc.dnx+real(up(emnc.Hy))*emnc.dny+real(up(emnc.Hz))*emnc.dnz) Pa Maxwell upward magnetic surface stress tensor, x component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.unTmy -0.5*emnc.dny*(real(up(emnc.Bx))*real(up(emnc.Hx))+real(up(emnc.By))*real(up(emnc.Hy))+real(up(emnc.Bz))*real(up(emnc.Hz)))+real(up(emnc.By))*(real(up(emnc.Hx))*emnc.dnx+real(up(emnc.Hy))*emnc.dny+real(up(emnc.Hz))*emnc.dnz) Pa Maxwell upward magnetic surface stress tensor, y component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.unTmz -0.5*emnc.dnz*(real(up(emnc.Bx))*real(up(emnc.Hx))+real(up(emnc.By))*real(up(emnc.Hy))+real(up(emnc.Bz))*real(up(emnc.Hz)))+real(up(emnc.Bz))*(real(up(emnc.Hx))*emnc.dnx+real(up(emnc.Hy))*emnc.dny+real(up(emnc.Hz))*emnc.dnz) Pa Maxwell upward magnetic surface stress tensor, z component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 71–74, 86–89
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 71–74, 86–89
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 71–74, 86–89
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 1–4, 83, 92
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 1–4, 83, 92
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 1–4, 83, 92
emnc.Hx -Vmx A/m Magnetic field, x component Domains 1, 13
emnc.Hy -Vmy A/m Magnetic field, y component Domains 1, 13
emnc.Hz -Vmz A/m Magnetic field, z component Domains 1, 13
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 1–158
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 1–158
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 1–158
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domains 1, 13
emnc.murxx 1.1 1 Relative permeability, xx component Domains 1, 13
emnc.muryx 0 1 Relative permeability, yx component Domains 1, 13
emnc.murzx 0 1 Relative permeability, zx component Domains 1, 13
emnc.murxy 0 1 Relative permeability, xy component Domains 1, 13
emnc.muryy 1.1 1 Relative permeability, yy component Domains 1, 13
emnc.murzy 0 1 Relative permeability, zy component Domains 1, 13
emnc.murxz 0 1 Relative permeability, xz component Domains 1, 13
emnc.muryz 0 1 Relative permeability, yz component Domains 1, 13
emnc.murzz 1.1 1 Relative permeability, zz component Domains 1, 13
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domains 1, 13
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domains 1, 13
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domains 1, 13
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domains 1, 13
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domains 1, 13
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domains 1, 13
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domains 1, 13
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domains 1, 13
emnc.Ixx 1 1 Spatial identity matrix, xx component Domains 1, 13
emnc.Iyx 0 1 Spatial identity matrix, yx component Domains 1, 13
emnc.Izx 0 1 Spatial identity matrix, zx component Domains 1, 13
emnc.Ixy 0 1 Spatial identity matrix, xy component Domains 1, 13
emnc.Iyy 1 1 Spatial identity matrix, yy component Domains 1, 13
emnc.Izy 0 1 Spatial identity matrix, zy component Domains 1, 13
emnc.Ixz 0 1 Spatial identity matrix, xz component Domains 1, 13
emnc.Iyz 0 1 Spatial identity matrix, yz component Domains 1, 13
emnc.Izz 1 1 Spatial identity matrix, zz component Domains 1, 13
emnc.Brx 0 T Remanent flux density, x component Domains 1, 13
emnc.Bry 0 T Remanent flux density, y component Domains 1, 13
emnc.Brz 0 T Remanent flux density, z component Domains 1, 13
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domains 1, 13
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domains 1, 13
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domains 1, 13
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domains 1, 13
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domains 1, 13
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domains 1, 13
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domains 1, 13
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domains 1, 13
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domains 1, 13
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domains 1, 13
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 1–4, 83, 92
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 1–4, 83, 92
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 1–4, 83, 92
emnc.unTx emnc.unTmx Pa Maxwell upward surface stress tensor, x component Boundaries 71–74, 86–89
emnc.unTy emnc.unTmy Pa Maxwell upward surface stress tensor, y component Boundaries 71–74, 86–89
emnc.unTz emnc.unTmz Pa Maxwell upward surface stress tensor, z component Boundaries 71–74, 86–89
emnc.unTx emnc.unTmx Pa Maxwell upward surface stress tensor, x component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.unTy emnc.unTmy Pa Maxwell upward surface stress tensor, y component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.unTz emnc.unTmz Pa Maxwell upward surface stress tensor, z component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 1–4, 83, 92
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 1–4, 83, 92
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 1–4, 83, 92
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 71–74, 86–89
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 71–74, 86–89
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 71–74, 86–89
emnc.dnTx 0 Pa Maxwell downward surface stress tensor, x component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.dnTy 0 Pa Maxwell downward surface stress tensor, y component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.dnTz 0 Pa Maxwell downward surface stress tensor, z component Boundaries 5–70, 75–82, 84–85, 90–91, 93–158
emnc.unx unx Normal vector up direction, x component Boundaries 1–158
emnc.uny uny Normal vector up direction, y component Boundaries 1–158
emnc.unz unz Normal vector up direction, z component Boundaries 1–158
emnc.dnx dnx Normal vector down direction, x component Boundaries 1–158
emnc.dny dny Normal vector down direction, y component Boundaries 1–158
emnc.dnz dnz Normal vector down direction, z component Boundaries 1–158
emnc.W emnc.Wm J/m^3 Energy density Domains 1, 13
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domains 1, 13
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domains 1, 13
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domains 1, 13
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domains 1, 13

2.4.2. Magnetic Insulation 1

physics_emnc_mi1.png

Magnetic Insulation 1

Selection
Geometric entity level Boundary
Selection No boundaries
Equations

equ_emnc_mi1.png

Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Boundaries 5–82, 84–91, 93–158

2.4.3. Initial Values 1

physics_emnc_init1.png

Initial Values 1

Selection
Geometric entity level Domain
Selection Domains 1–26
Settings
Description Value
Magnetic scalar potential 0

2.4.4. Magnetic Flux Conservation 2

physics_emnc_mfc2.png

Magnetic Flux Conservation 2

Selection
Geometric entity level Domain
Selection Domain 26
Equations

equ_emnc_mfc2_1.png

equ_emnc_mfc2_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 1) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 1) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 147–150, 157–158
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 147–150, 157–158
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 147–150, 157–158
emnc.Hx -Vmx A/m Magnetic field, x component Domain 26
emnc.Hy -Vmy A/m Magnetic field, y component Domain 26
emnc.Hz -Vmz A/m Magnetic field, z component Domain 26
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 147–150, 157–158
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 147–150, 157–158
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 147–150, 157–158
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 26
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 26
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 26
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 26
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 26
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 26
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 26
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 26
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 26
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 26
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 26
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 26
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 26
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 26
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 26
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 26
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 26
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 26
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 26
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 26
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 26
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 26
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 26
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 26
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 26
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 26
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 26
emnc.Brx BpRem*cos(tick) T Remanent flux density, x component Domain 26
emnc.Bry BpRem*sin(tick) T Remanent flux density, y component Domain 26
emnc.Brz 0 T Remanent flux density, z component Domain 26
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 26
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 26
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 26
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 26
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 26
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 26
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 26
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 26
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 26
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 26
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 147–150, 157–158
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 147–150, 157–158
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 147–150, 157–158
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 147–150, 157–158
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 147–150, 157–158
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 147–150, 157–158
emnc.unx unx Normal vector up direction, x component Boundaries 147–150, 157–158
emnc.uny uny Normal vector up direction, y component Boundaries 147–150, 157–158
emnc.unz unz Normal vector up direction, z component Boundaries 147–150, 157–158
emnc.dnx dnx Normal vector down direction, x component Boundaries 147–150, 157–158
emnc.dny dny Normal vector down direction, y component Boundaries 147–150, 157–158
emnc.dnz dnz Normal vector down direction, z component Boundaries 147–150, 157–158
emnc.W emnc.Wm J/m^3 Energy density Domain 26
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 26
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 26
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 26
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 26

2.4.5. Magnetic Flux Conservation 3

physics_emnc_mfc3.png

Magnetic Flux Conservation 3

Selection
Geometric entity level Domain
Selection Domain 25
Equations

equ_emnc_mfc3_1.png

equ_emnc_mfc3_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 2) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 2) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 143–146, 154, 156
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 143–146, 154, 156
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 143–146, 154, 156
emnc.Hx -Vmx A/m Magnetic field, x component Domain 25
emnc.Hy -Vmy A/m Magnetic field, y component Domain 25
emnc.Hz -Vmz A/m Magnetic field, z component Domain 25
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 143–146, 154, 156
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 143–146, 154, 156
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 143–146, 154, 156
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 25
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 25
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 25
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 25
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 25
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 25
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 25
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 25
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 25
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 25
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 25
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 25
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 25
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 25
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 25
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 25
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 25
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 25
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 25
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 25
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 25
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 25
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 25
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 25
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 25
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 25
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 25
emnc.Brx BpRem*cos(argBp+tick) T Remanent flux density, x component Domain 25
emnc.Bry BpRem*sin(argBp+tick) T Remanent flux density, y component Domain 25
emnc.Brz 0 T Remanent flux density, z component Domain 25
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 25
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 25
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 25
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 25
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 25
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 25
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 25
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 25
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 25
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 25
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 143–146, 154, 156
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 143–146, 154, 156
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 143–146, 154, 156
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 143–146, 154, 156
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 143–146, 154, 156
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 143–146, 154, 156
emnc.unx unx Normal vector up direction, x component Boundaries 143–146, 154, 156
emnc.uny uny Normal vector up direction, y component Boundaries 143–146, 154, 156
emnc.unz unz Normal vector up direction, z component Boundaries 143–146, 154, 156
emnc.dnx dnx Normal vector down direction, x component Boundaries 143–146, 154, 156
emnc.dny dny Normal vector down direction, y component Boundaries 143–146, 154, 156
emnc.dnz dnz Normal vector down direction, z component Boundaries 143–146, 154, 156
emnc.W emnc.Wm J/m^3 Energy density Domain 25
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 25
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 25
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 25
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 25

2.4.6. Magnetic Flux Conservation 4

physics_emnc_mfc4.png

Magnetic Flux Conservation 4

Selection
Geometric entity level Domain
Selection Domain 23
Equations

equ_emnc_mfc4_1.png

equ_emnc_mfc4_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 3) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 3) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 133–136, 138, 152
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 133–136, 138, 152
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 133–136, 138, 152
emnc.Hx -Vmx A/m Magnetic field, x component Domain 23
emnc.Hy -Vmy A/m Magnetic field, y component Domain 23
emnc.Hz -Vmz A/m Magnetic field, z component Domain 23
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 133–136, 138, 152
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 133–136, 138, 152
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 133–136, 138, 152
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 23
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 23
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 23
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 23
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 23
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 23
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 23
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 23
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 23
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 23
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 23
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 23
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 23
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 23
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 23
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 23
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 23
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 23
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 23
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 23
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 23
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 23
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 23
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 23
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 23
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 23
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 23
emnc.Brx BpRem*cos(2*argBp+tick) T Remanent flux density, x component Domain 23
emnc.Bry BpRem*sin(2*argBp+tick) T Remanent flux density, y component Domain 23
emnc.Brz 0 T Remanent flux density, z component Domain 23
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 23
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 23
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 23
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 23
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 23
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 23
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 23
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 23
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 23
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 23
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 133–136, 138, 152
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 133–136, 138, 152
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 133–136, 138, 152
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 133–136, 138, 152
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 133–136, 138, 152
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 133–136, 138, 152
emnc.unx unx Normal vector up direction, x component Boundaries 133–136, 138, 152
emnc.uny uny Normal vector up direction, y component Boundaries 133–136, 138, 152
emnc.unz unz Normal vector up direction, z component Boundaries 133–136, 138, 152
emnc.dnx dnx Normal vector down direction, x component Boundaries 133–136, 138, 152
emnc.dny dny Normal vector down direction, y component Boundaries 133–136, 138, 152
emnc.dnz dnz Normal vector down direction, z component Boundaries 133–136, 138, 152
emnc.W emnc.Wm J/m^3 Energy density Domain 23
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 23
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 23
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 23
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 23

2.4.7. Magnetic Flux Conservation 5

physics_emnc_mfc5.png

Magnetic Flux Conservation 5

Selection
Geometric entity level Domain
Selection Domain 21
Equations

equ_emnc_mfc5_1.png

equ_emnc_mfc5_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 4) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 4) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 122–126, 128
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 122–126, 128
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 122–126, 128
emnc.Hx -Vmx A/m Magnetic field, x component Domain 21
emnc.Hy -Vmy A/m Magnetic field, y component Domain 21
emnc.Hz -Vmz A/m Magnetic field, z component Domain 21
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 122–126, 128
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 122–126, 128
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 122–126, 128
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 21
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 21
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 21
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 21
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 21
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 21
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 21
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 21
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 21
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 21
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 21
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 21
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 21
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 21
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 21
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 21
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 21
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 21
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 21
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 21
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 21
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 21
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 21
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 21
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 21
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 21
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 21
emnc.Brx BpRem*cos(3*argBp+tick) T Remanent flux density, x component Domain 21
emnc.Bry BpRem*sin(3*argBp+tick) T Remanent flux density, y component Domain 21
emnc.Brz 0 T Remanent flux density, z component Domain 21
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 21
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 21
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 21
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 21
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 21
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 21
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 21
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 21
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 21
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 21
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 122–126, 128
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 122–126, 128
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 122–126, 128
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 122–126, 128
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 122–126, 128
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 122–126, 128
emnc.unx unx Normal vector up direction, x component Boundaries 122–126, 128
emnc.uny uny Normal vector up direction, y component Boundaries 122–126, 128
emnc.unz unz Normal vector up direction, z component Boundaries 122–126, 128
emnc.dnx dnx Normal vector down direction, x component Boundaries 122–126, 128
emnc.dny dny Normal vector down direction, y component Boundaries 122–126, 128
emnc.dnz dnz Normal vector down direction, z component Boundaries 122–126, 128
emnc.W emnc.Wm J/m^3 Energy density Domain 21
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 21
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 21
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 21
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 21

2.4.8. Magnetic Flux Conservation 6

physics_emnc_mfc6.png

Magnetic Flux Conservation 6

Selection
Geometric entity level Domain
Selection Domain 19
Equations

equ_emnc_mfc6_1.png

equ_emnc_mfc6_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 5) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 5) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 109–112, 114, 116
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 109–112, 114, 116
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 109–112, 114, 116
emnc.Hx -Vmx A/m Magnetic field, x component Domain 19
emnc.Hy -Vmy A/m Magnetic field, y component Domain 19
emnc.Hz -Vmz A/m Magnetic field, z component Domain 19
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 109–112, 114, 116
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 109–112, 114, 116
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 109–112, 114, 116
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 19
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 19
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 19
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 19
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 19
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 19
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 19
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 19
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 19
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 19
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 19
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 19
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 19
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 19
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 19
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 19
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 19
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 19
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 19
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 19
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 19
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 19
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 19
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 19
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 19
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 19
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 19
emnc.Brx BpRem*cos(4*argBp+tick) T Remanent flux density, x component Domain 19
emnc.Bry BpRem*sin(4*argBp+tick) T Remanent flux density, y component Domain 19
emnc.Brz 0 T Remanent flux density, z component Domain 19
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 19
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 19
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 19
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 19
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 19
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 19
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 19
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 19
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 19
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 19
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 109–112, 114, 116
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 109–112, 114, 116
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 109–112, 114, 116
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 109–112, 114, 116
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 109–112, 114, 116
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 109–112, 114, 116
emnc.unx unx Normal vector up direction, x component Boundaries 109–112, 114, 116
emnc.uny uny Normal vector up direction, y component Boundaries 109–112, 114, 116
emnc.unz unz Normal vector up direction, z component Boundaries 109–112, 114, 116
emnc.dnx dnx Normal vector down direction, x component Boundaries 109–112, 114, 116
emnc.dny dny Normal vector down direction, y component Boundaries 109–112, 114, 116
emnc.dnz dnz Normal vector down direction, z component Boundaries 109–112, 114, 116
emnc.W emnc.Wm J/m^3 Energy density Domain 19
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 19
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 19
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 19
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 19

2.4.9. Magnetic Flux Conservation 7

physics_emnc_mfc7.png

Magnetic Flux Conservation 7

Selection
Geometric entity level Domain
Selection Domain 17
Equations

equ_emnc_mfc7_1.png

equ_emnc_mfc7_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 6) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 6) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 97–100, 102, 104
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 97–100, 102, 104
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 97–100, 102, 104
emnc.Hx -Vmx A/m Magnetic field, x component Domain 17
emnc.Hy -Vmy A/m Magnetic field, y component Domain 17
emnc.Hz -Vmz A/m Magnetic field, z component Domain 17
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 97–100, 102, 104
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 97–100, 102, 104
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 97–100, 102, 104
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 17
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 17
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 17
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 17
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 17
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 17
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 17
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 17
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 17
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 17
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 17
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 17
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 17
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 17
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 17
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 17
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 17
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 17
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 17
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 17
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 17
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 17
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 17
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 17
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 17
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 17
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 17
emnc.Brx BpRem*cos(5*argBp+tick) T Remanent flux density, x component Domain 17
emnc.Bry BpRem*sin(5*argBp+tick) T Remanent flux density, y component Domain 17
emnc.Brz 0 T Remanent flux density, z component Domain 17
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 17
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 17
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 17
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 17
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 17
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 17
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 17
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 17
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 17
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 17
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 97–100, 102, 104
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 97–100, 102, 104
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 97–100, 102, 104
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 97–100, 102, 104
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 97–100, 102, 104
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 97–100, 102, 104
emnc.unx unx Normal vector up direction, x component Boundaries 97–100, 102, 104
emnc.uny uny Normal vector up direction, y component Boundaries 97–100, 102, 104
emnc.unz unz Normal vector up direction, z component Boundaries 97–100, 102, 104
emnc.dnx dnx Normal vector down direction, x component Boundaries 97–100, 102, 104
emnc.dny dny Normal vector down direction, y component Boundaries 97–100, 102, 104
emnc.dnz dnz Normal vector down direction, z component Boundaries 97–100, 102, 104
emnc.W emnc.Wm J/m^3 Energy density Domain 17
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 17
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 17
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 17
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 17

2.4.10. Magnetic Flux Conservation 8

physics_emnc_mfc8.png

Magnetic Flux Conservation 8

Selection
Geometric entity level Domain
Selection Domain 15
Equations

equ_emnc_mfc8_1.png

equ_emnc_mfc8_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 7) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 7) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 79–82, 90–91
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 79–82, 90–91
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 79–82, 90–91
emnc.Hx -Vmx A/m Magnetic field, x component Domain 15
emnc.Hy -Vmy A/m Magnetic field, y component Domain 15
emnc.Hz -Vmz A/m Magnetic field, z component Domain 15
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 79–82, 90–91
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 79–82, 90–91
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 79–82, 90–91
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 15
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 15
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 15
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 15
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 15
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 15
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 15
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 15
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 15
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 15
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 15
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 15
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 15
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 15
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 15
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 15
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 15
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 15
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 15
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 15
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 15
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 15
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 15
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 15
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 15
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 15
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 15
emnc.Brx BpRem*cos(6*argBp+tick) T Remanent flux density, x component Domain 15
emnc.Bry BpRem*sin(6*argBp+tick) T Remanent flux density, y component Domain 15
emnc.Brz 0 T Remanent flux density, z component Domain 15
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 15
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 15
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 15
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 15
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 15
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 15
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 15
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 15
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 15
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 15
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 79–82, 90–91
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 79–82, 90–91
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 79–82, 90–91
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 79–82, 90–91
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 79–82, 90–91
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 79–82, 90–91
emnc.unx unx Normal vector up direction, x component Boundaries 79–82, 90–91
emnc.uny uny Normal vector up direction, y component Boundaries 79–82, 90–91
emnc.unz unz Normal vector up direction, z component Boundaries 79–82, 90–91
emnc.dnx dnx Normal vector down direction, x component Boundaries 79–82, 90–91
emnc.dny dny Normal vector down direction, y component Boundaries 79–82, 90–91
emnc.dnz dnz Normal vector down direction, z component Boundaries 79–82, 90–91
emnc.W emnc.Wm J/m^3 Energy density Domain 15
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 15
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 15
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 15
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 15

2.4.11. Magnetic Flux Conservation 9

physics_emnc_mfc9.png

Magnetic Flux Conservation 9

Selection
Geometric entity level Domain
Selection Domain 12
Equations

equ_emnc_mfc9_1.png

equ_emnc_mfc9_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 8) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 8) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 63–66, 68, 70
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 63–66, 68, 70
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 63–66, 68, 70
emnc.Hx -Vmx A/m Magnetic field, x component Domain 12
emnc.Hy -Vmy A/m Magnetic field, y component Domain 12
emnc.Hz -Vmz A/m Magnetic field, z component Domain 12
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 63–66, 68, 70
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 63–66, 68, 70
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 63–66, 68, 70
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 12
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 12
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 12
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 12
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 12
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 12
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 12
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 12
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 12
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 12
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 12
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 12
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 12
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 12
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 12
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 12
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 12
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 12
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 12
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 12
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 12
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 12
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 12
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 12
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 12
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 12
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 12
emnc.Brx BpRem*cos(7*argBp+tick) T Remanent flux density, x component Domain 12
emnc.Bry BpRem*sin(7*argBp+tick) T Remanent flux density, y component Domain 12
emnc.Brz 0 T Remanent flux density, z component Domain 12
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 12
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 12
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 12
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 12
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 12
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 12
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 12
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 12
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 12
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 12
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 63–66, 68, 70
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 63–66, 68, 70
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 63–66, 68, 70
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 63–66, 68, 70
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 63–66, 68, 70
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 63–66, 68, 70
emnc.unx unx Normal vector up direction, x component Boundaries 63–66, 68, 70
emnc.uny uny Normal vector up direction, y component Boundaries 63–66, 68, 70
emnc.unz unz Normal vector up direction, z component Boundaries 63–66, 68, 70
emnc.dnx dnx Normal vector down direction, x component Boundaries 63–66, 68, 70
emnc.dny dny Normal vector down direction, y component Boundaries 63–66, 68, 70
emnc.dnz dnz Normal vector down direction, z component Boundaries 63–66, 68, 70
emnc.W emnc.Wm J/m^3 Energy density Domain 12
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 12
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 12
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 12
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 12

2.4.12. Magnetic Flux Conservation 10

physics_emnc_mfc10.png

Magnetic Flux Conservation 10

Selection
Geometric entity level Domain
Selection Domain 10
Equations

equ_emnc_mfc10_1.png

equ_emnc_mfc10_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 9) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 9) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 51–54, 56, 58
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 51–54, 56, 58
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 51–54, 56, 58
emnc.Hx -Vmx A/m Magnetic field, x component Domain 10
emnc.Hy -Vmy A/m Magnetic field, y component Domain 10
emnc.Hz -Vmz A/m Magnetic field, z component Domain 10
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 51–54, 56, 58
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 51–54, 56, 58
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 51–54, 56, 58
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 10
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 10
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 10
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 10
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 10
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 10
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 10
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 10
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 10
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 10
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 10
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 10
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 10
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 10
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 10
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 10
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 10
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 10
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 10
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 10
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 10
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 10
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 10
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 10
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 10
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 10
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 10
emnc.Brx BpRem*cos(8*argBp+tick) T Remanent flux density, x component Domain 10
emnc.Bry BpRem*sin(8*argBp+tick) T Remanent flux density, y component Domain 10
emnc.Brz 0 T Remanent flux density, z component Domain 10
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 10
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 10
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 10
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 10
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 10
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 10
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 10
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 10
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 10
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 10
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 51–54, 56, 58
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 51–54, 56, 58
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 51–54, 56, 58
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 51–54, 56, 58
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 51–54, 56, 58
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 51–54, 56, 58
emnc.unx unx Normal vector up direction, x component Boundaries 51–54, 56, 58
emnc.uny uny Normal vector up direction, y component Boundaries 51–54, 56, 58
emnc.unz unz Normal vector up direction, z component Boundaries 51–54, 56, 58
emnc.dnx dnx Normal vector down direction, x component Boundaries 51–54, 56, 58
emnc.dny dny Normal vector down direction, y component Boundaries 51–54, 56, 58
emnc.dnz dnz Normal vector down direction, z component Boundaries 51–54, 56, 58
emnc.W emnc.Wm J/m^3 Energy density Domain 10
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 10
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 10
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 10
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 10

2.4.13. Magnetic Flux Conservation 11

physics_emnc_mfc11.png

Magnetic Flux Conservation 11

Selection
Geometric entity level Domain
Selection Domain 8
Equations

equ_emnc_mfc11_1.png

equ_emnc_mfc11_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 10) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 10) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 40–44, 46
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 40–44, 46
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 40–44, 46
emnc.Hx -Vmx A/m Magnetic field, x component Domain 8
emnc.Hy -Vmy A/m Magnetic field, y component Domain 8
emnc.Hz -Vmz A/m Magnetic field, z component Domain 8
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 40–44, 46
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 40–44, 46
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 40–44, 46
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 8
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 8
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 8
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 8
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 8
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 8
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 8
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 8
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 8
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 8
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 8
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 8
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 8
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 8
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 8
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 8
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 8
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 8
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 8
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 8
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 8
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 8
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 8
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 8
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 8
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 8
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 8
emnc.Brx BpRem*cos(9*argBp+tick) T Remanent flux density, x component Domain 8
emnc.Bry BpRem*sin(9*argBp+tick) T Remanent flux density, y component Domain 8
emnc.Brz 0 T Remanent flux density, z component Domain 8
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 8
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 8
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 8
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 8
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 8
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 8
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 8
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 8
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 8
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 8
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 40–44, 46
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 40–44, 46
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 40–44, 46
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 40–44, 46
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 40–44, 46
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 40–44, 46
emnc.unx unx Normal vector up direction, x component Boundaries 40–44, 46
emnc.uny uny Normal vector up direction, y component Boundaries 40–44, 46
emnc.unz unz Normal vector up direction, z component Boundaries 40–44, 46
emnc.dnx dnx Normal vector down direction, x component Boundaries 40–44, 46
emnc.dny dny Normal vector down direction, y component Boundaries 40–44, 46
emnc.dnz dnz Normal vector down direction, z component Boundaries 40–44, 46
emnc.W emnc.Wm J/m^3 Energy density Domain 8
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 8
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 8
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 8
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 8

2.4.14. Magnetic Flux Conservation 12

physics_emnc_mfc12.png

Magnetic Flux Conservation 12

Selection
Geometric entity level Domain
Selection Domain 6
Equations

equ_emnc_mfc12_1.png

equ_emnc_mfc12_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 11) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 11) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 25–28, 32, 34
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 25–28, 32, 34
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 25–28, 32, 34
emnc.Hx -Vmx A/m Magnetic field, x component Domain 6
emnc.Hy -Vmy A/m Magnetic field, y component Domain 6
emnc.Hz -Vmz A/m Magnetic field, z component Domain 6
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 25–28, 32, 34
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 25–28, 32, 34
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 25–28, 32, 34
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 6
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 6
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 6
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 6
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 6
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 6
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 6
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 6
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 6
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 6
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 6
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 6
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 6
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 6
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 6
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 6
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 6
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 6
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 6
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 6
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 6
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 6
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 6
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 6
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 6
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 6
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 6
emnc.Brx BpRem*cos(10*argBp+tick) T Remanent flux density, x component Domain 6
emnc.Bry BpRem*sin(10*argBp+tick) T Remanent flux density, y component Domain 6
emnc.Brz 0 T Remanent flux density, z component Domain 6
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 6
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 6
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 6
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 6
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 6
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 6
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 6
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 6
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 6
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 6
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 25–28, 32, 34
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 25–28, 32, 34
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 25–28, 32, 34
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 25–28, 32, 34
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 25–28, 32, 34
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 25–28, 32, 34
emnc.unx unx Normal vector up direction, x component Boundaries 25–28, 32, 34
emnc.uny uny Normal vector up direction, y component Boundaries 25–28, 32, 34
emnc.unz unz Normal vector up direction, z component Boundaries 25–28, 32, 34
emnc.dnx dnx Normal vector down direction, x component Boundaries 25–28, 32, 34
emnc.dny dny Normal vector down direction, y component Boundaries 25–28, 32, 34
emnc.dnz dnz Normal vector down direction, z component Boundaries 25–28, 32, 34
emnc.W emnc.Wm J/m^3 Energy density Domain 6
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 6
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 6
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 6
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 6

2.4.15. Magnetic Flux Conservation 13

physics_emnc_mfc13.png

Magnetic Flux Conservation 13

Selection
Geometric entity level Domain
Selection Domain 4
Equations

equ_emnc_mfc13_1.png

equ_emnc_mfc13_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 12) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 12) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 13–16, 20, 30
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 13–16, 20, 30
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 13–16, 20, 30
emnc.Hx -Vmx A/m Magnetic field, x component Domain 4
emnc.Hy -Vmy A/m Magnetic field, y component Domain 4
emnc.Hz -Vmz A/m Magnetic field, z component Domain 4
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 13–16, 20, 30
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 13–16, 20, 30
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 13–16, 20, 30
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 4
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 4
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 4
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 4
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 4
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 4
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 4
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 4
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 4
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 4
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 4
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 4
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 4
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 4
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 4
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 4
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 4
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 4
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 4
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 4
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 4
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 4
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 4
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 4
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 4
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 4
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 4
emnc.Brx BpRem*cos(11*argBp+tick) T Remanent flux density, x component Domain 4
emnc.Bry BpRem*sin(11*argBp+tick) T Remanent flux density, y component Domain 4
emnc.Brz 0 T Remanent flux density, z component Domain 4
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 4
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 4
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 4
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 4
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 4
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 4
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 4
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 4
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 4
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 4
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 13–16, 20, 30
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 13–16, 20, 30
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 13–16, 20, 30
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 13–16, 20, 30
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 13–16, 20, 30
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 13–16, 20, 30
emnc.unx unx Normal vector up direction, x component Boundaries 13–16, 20, 30
emnc.uny uny Normal vector up direction, y component Boundaries 13–16, 20, 30
emnc.unz unz Normal vector up direction, z component Boundaries 13–16, 20, 30
emnc.dnx dnx Normal vector down direction, x component Boundaries 13–16, 20, 30
emnc.dny dny Normal vector down direction, y component Boundaries 13–16, 20, 30
emnc.dnz dnz Normal vector down direction, z component Boundaries 13–16, 20, 30
emnc.W emnc.Wm J/m^3 Energy density Domain 4
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 4
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 4
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 4
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 4

2.4.16. Magnetic Flux Conservation 14

physics_emnc_mfc14.png

Magnetic Flux Conservation 14

Selection
Geometric entity level Domain
Selection Domain 2
Equations

equ_emnc_mfc14_1.png

equ_emnc_mfc14_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 13) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 13) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 5–8, 17–18
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 5–8, 17–18
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 5–8, 17–18
emnc.Hx -Vmx A/m Magnetic field, x component Domain 2
emnc.Hy -Vmy A/m Magnetic field, y component Domain 2
emnc.Hz -Vmz A/m Magnetic field, z component Domain 2
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 5–8, 17–18
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 5–8, 17–18
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 5–8, 17–18
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 2
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 2
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 2
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 2
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 2
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 2
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 2
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 2
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 2
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 2
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 2
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 2
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 2
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 2
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 2
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 2
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 2
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 2
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 2
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 2
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 2
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 2
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 2
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 2
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 2
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 2
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 2
emnc.Brx BpRem*cos(12*argBp+tick) T Remanent flux density, x component Domain 2
emnc.Bry BpRem*sin(12*argBp+tick) T Remanent flux density, y component Domain 2
emnc.Brz 0 T Remanent flux density, z component Domain 2
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 2
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 2
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 2
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 2
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 2
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 2
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 2
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 2
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 2
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 2
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 5–8, 17–18
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 5–8, 17–18
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 5–8, 17–18
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 5–8, 17–18
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 5–8, 17–18
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 5–8, 17–18
emnc.unx unx Normal vector up direction, x component Boundaries 5–8, 17–18
emnc.uny uny Normal vector up direction, y component Boundaries 5–8, 17–18
emnc.unz unz Normal vector up direction, z component Boundaries 5–8, 17–18
emnc.dnx dnx Normal vector down direction, x component Boundaries 5–8, 17–18
emnc.dny dny Normal vector down direction, y component Boundaries 5–8, 17–18
emnc.dnz dnz Normal vector down direction, z component Boundaries 5–8, 17–18
emnc.W emnc.Wm J/m^3 Energy density Domain 2
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 2
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 2
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 2
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 2

2.4.17. Magnetic Flux Conservation 15

physics_emnc_mfc15.png

Magnetic Flux Conservation 15

Selection
Geometric entity level Domain
Selection Domain 3
Equations

equ_emnc_mfc15_1.png

equ_emnc_mfc15_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 14) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 14) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 9–12, 19, 29
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 9–12, 19, 29
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 9–12, 19, 29
emnc.Hx -Vmx A/m Magnetic field, x component Domain 3
emnc.Hy -Vmy A/m Magnetic field, y component Domain 3
emnc.Hz -Vmz A/m Magnetic field, z component Domain 3
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 9–12, 19, 29
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 9–12, 19, 29
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 9–12, 19, 29
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 3
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 3
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 3
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 3
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 3
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 3
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 3
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 3
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 3
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 3
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 3
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 3
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 3
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 3
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 3
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 3
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 3
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 3
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 3
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 3
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 3
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 3
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 3
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 3
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 3
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 3
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 3
emnc.Brx BpRem*cos(13*argBp+tick) T Remanent flux density, x component Domain 3
emnc.Bry BpRem*sin(13*argBp+tick) T Remanent flux density, y component Domain 3
emnc.Brz 0 T Remanent flux density, z component Domain 3
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 3
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 3
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 3
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 3
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 3
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 3
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 3
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 3
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 3
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 3
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 9–12, 19, 29
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 9–12, 19, 29
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 9–12, 19, 29
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 9–12, 19, 29
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 9–12, 19, 29
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 9–12, 19, 29
emnc.unx unx Normal vector up direction, x component Boundaries 9–12, 19, 29
emnc.uny uny Normal vector up direction, y component Boundaries 9–12, 19, 29
emnc.unz unz Normal vector up direction, z component Boundaries 9–12, 19, 29
emnc.dnx dnx Normal vector down direction, x component Boundaries 9–12, 19, 29
emnc.dny dny Normal vector down direction, y component Boundaries 9–12, 19, 29
emnc.dnz dnz Normal vector down direction, z component Boundaries 9–12, 19, 29
emnc.W emnc.Wm J/m^3 Energy density Domain 3
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 3
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 3
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 3
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 3

2.4.18. Magnetic Flux Conservation 16

physics_emnc_mfc16.png

Magnetic Flux Conservation 16

Selection
Geometric entity level Domain
Selection Domain 5
Equations

equ_emnc_mfc16_1.png

equ_emnc_mfc16_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 15) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 15) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 21–24, 31, 33
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 21–24, 31, 33
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 21–24, 31, 33
emnc.Hx -Vmx A/m Magnetic field, x component Domain 5
emnc.Hy -Vmy A/m Magnetic field, y component Domain 5
emnc.Hz -Vmz A/m Magnetic field, z component Domain 5
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 21–24, 31, 33
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 21–24, 31, 33
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 21–24, 31, 33
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 5
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 5
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 5
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 5
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 5
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 5
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 5
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 5
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 5
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 5
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 5
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 5
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 5
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 5
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 5
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 5
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 5
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 5
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 5
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 5
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 5
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 5
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 5
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 5
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 5
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 5
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 5
emnc.Brx BpRem*cos(14*argBp+tick) T Remanent flux density, x component Domain 5
emnc.Bry BpRem*sin(14*argBp+tick) T Remanent flux density, y component Domain 5
emnc.Brz 0 T Remanent flux density, z component Domain 5
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 5
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 5
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 5
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 5
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 5
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 5
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 5
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 5
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 5
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 5
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 21–24, 31, 33
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 21–24, 31, 33
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 21–24, 31, 33
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 21–24, 31, 33
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 21–24, 31, 33
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 21–24, 31, 33
emnc.unx unx Normal vector up direction, x component Boundaries 21–24, 31, 33
emnc.uny uny Normal vector up direction, y component Boundaries 21–24, 31, 33
emnc.unz unz Normal vector up direction, z component Boundaries 21–24, 31, 33
emnc.dnx dnx Normal vector down direction, x component Boundaries 21–24, 31, 33
emnc.dny dny Normal vector down direction, y component Boundaries 21–24, 31, 33
emnc.dnz dnz Normal vector down direction, z component Boundaries 21–24, 31, 33
emnc.W emnc.Wm J/m^3 Energy density Domain 5
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 5
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 5
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 5
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 5

2.4.19. Magnetic Flux Conservation 17

physics_emnc_mfc17.png

Magnetic Flux Conservation 17

Selection
Geometric entity level Domain
Selection Domain 7
Equations

equ_emnc_mfc17_1.png

equ_emnc_mfc17_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 16) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 16) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 35–39, 45
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 35–39, 45
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 35–39, 45
emnc.Hx -Vmx A/m Magnetic field, x component Domain 7
emnc.Hy -Vmy A/m Magnetic field, y component Domain 7
emnc.Hz -Vmz A/m Magnetic field, z component Domain 7
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 35–39, 45
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 35–39, 45
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 35–39, 45
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 7
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 7
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 7
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 7
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 7
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 7
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 7
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 7
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 7
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 7
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 7
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 7
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 7
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 7
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 7
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 7
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 7
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 7
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 7
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 7
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 7
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 7
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 7
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 7
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 7
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 7
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 7
emnc.Brx BpRem*cos(15*argBp+tick) T Remanent flux density, x component Domain 7
emnc.Bry BpRem*sin(15*argBp+tick) T Remanent flux density, y component Domain 7
emnc.Brz 0 T Remanent flux density, z component Domain 7
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 7
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 7
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 7
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 7
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 7
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 7
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 7
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 7
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 7
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 7
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 35–39, 45
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 35–39, 45
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 35–39, 45
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 35–39, 45
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 35–39, 45
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 35–39, 45
emnc.unx unx Normal vector up direction, x component Boundaries 35–39, 45
emnc.uny uny Normal vector up direction, y component Boundaries 35–39, 45
emnc.unz unz Normal vector up direction, z component Boundaries 35–39, 45
emnc.dnx dnx Normal vector down direction, x component Boundaries 35–39, 45
emnc.dny dny Normal vector down direction, y component Boundaries 35–39, 45
emnc.dnz dnz Normal vector down direction, z component Boundaries 35–39, 45
emnc.W emnc.Wm J/m^3 Energy density Domain 7
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 7
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 7
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 7
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 7

2.4.20. Magnetic Flux Conservation 18

physics_emnc_mfc18.png

Magnetic Flux Conservation 18

Selection
Geometric entity level Domain
Selection Domain 9
Equations

equ_emnc_mfc18_1.png

equ_emnc_mfc18_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 17) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 17) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 47–50, 55, 57
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 47–50, 55, 57
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 47–50, 55, 57
emnc.Hx -Vmx A/m Magnetic field, x component Domain 9
emnc.Hy -Vmy A/m Magnetic field, y component Domain 9
emnc.Hz -Vmz A/m Magnetic field, z component Domain 9
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 47–50, 55, 57
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 47–50, 55, 57
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 47–50, 55, 57
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 9
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 9
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 9
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 9
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 9
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 9
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 9
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 9
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 9
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 9
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 9
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 9
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 9
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 9
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 9
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 9
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 9
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 9
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 9
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 9
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 9
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 9
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 9
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 9
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 9
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 9
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 9
emnc.Brx BpRem*cos(16*argBp+tick) T Remanent flux density, x component Domain 9
emnc.Bry BpRem*sin(16*argBp+tick) T Remanent flux density, y component Domain 9
emnc.Brz 0 T Remanent flux density, z component Domain 9
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 9
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 9
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 9
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 9
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 9
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 9
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 9
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 9
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 9
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 9
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 47–50, 55, 57
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 47–50, 55, 57
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 47–50, 55, 57
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 47–50, 55, 57
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 47–50, 55, 57
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 47–50, 55, 57
emnc.unx unx Normal vector up direction, x component Boundaries 47–50, 55, 57
emnc.uny uny Normal vector up direction, y component Boundaries 47–50, 55, 57
emnc.unz unz Normal vector up direction, z component Boundaries 47–50, 55, 57
emnc.dnx dnx Normal vector down direction, x component Boundaries 47–50, 55, 57
emnc.dny dny Normal vector down direction, y component Boundaries 47–50, 55, 57
emnc.dnz dnz Normal vector down direction, z component Boundaries 47–50, 55, 57
emnc.W emnc.Wm J/m^3 Energy density Domain 9
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 9
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 9
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 9
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 9

2.4.21. Magnetic Flux Conservation 19

physics_emnc_mfc19.png

Magnetic Flux Conservation 19

Selection
Geometric entity level Domain
Selection Domain 11
Equations

equ_emnc_mfc19_1.png

equ_emnc_mfc19_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 18) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 18) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 59–62, 67, 69
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 59–62, 67, 69
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 59–62, 67, 69
emnc.Hx -Vmx A/m Magnetic field, x component Domain 11
emnc.Hy -Vmy A/m Magnetic field, y component Domain 11
emnc.Hz -Vmz A/m Magnetic field, z component Domain 11
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 59–62, 67, 69
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 59–62, 67, 69
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 59–62, 67, 69
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 11
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 11
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 11
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 11
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 11
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 11
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 11
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 11
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 11
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 11
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 11
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 11
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 11
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 11
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 11
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 11
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 11
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 11
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 11
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 11
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 11
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 11
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 11
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 11
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 11
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 11
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 11
emnc.Brx BpRem*cos(17*argBp+tick) T Remanent flux density, x component Domain 11
emnc.Bry BpRem*sin(17*argBp+tick) T Remanent flux density, y component Domain 11
emnc.Brz 0 T Remanent flux density, z component Domain 11
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 11
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 11
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 11
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 11
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 11
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 11
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 11
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 11
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 11
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 11
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 59–62, 67, 69
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 59–62, 67, 69
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 59–62, 67, 69
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 59–62, 67, 69
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 59–62, 67, 69
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 59–62, 67, 69
emnc.unx unx Normal vector up direction, x component Boundaries 59–62, 67, 69
emnc.uny uny Normal vector up direction, y component Boundaries 59–62, 67, 69
emnc.unz unz Normal vector up direction, z component Boundaries 59–62, 67, 69
emnc.dnx dnx Normal vector down direction, x component Boundaries 59–62, 67, 69
emnc.dny dny Normal vector down direction, y component Boundaries 59–62, 67, 69
emnc.dnz dnz Normal vector down direction, z component Boundaries 59–62, 67, 69
emnc.W emnc.Wm J/m^3 Energy density Domain 11
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 11
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 11
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 11
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 11

2.4.22. Magnetic Flux Conservation 20

physics_emnc_mfc20.png

Magnetic Flux Conservation 20

Selection
Geometric entity level Domain
Selection Domain 14
Equations

equ_emnc_mfc20_1.png

equ_emnc_mfc20_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 19) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 19) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 75–78, 84–85
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 75–78, 84–85
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 75–78, 84–85
emnc.Hx -Vmx A/m Magnetic field, x component Domain 14
emnc.Hy -Vmy A/m Magnetic field, y component Domain 14
emnc.Hz -Vmz A/m Magnetic field, z component Domain 14
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 75–78, 84–85
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 75–78, 84–85
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 75–78, 84–85
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 14
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 14
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 14
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 14
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 14
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 14
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 14
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 14
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 14
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 14
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 14
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 14
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 14
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 14
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 14
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 14
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 14
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 14
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 14
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 14
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 14
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 14
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 14
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 14
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 14
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 14
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 14
emnc.Brx BpRem*cos(18*argBp+tick) T Remanent flux density, x component Domain 14
emnc.Bry BpRem*sin(18*argBp+tick) T Remanent flux density, y component Domain 14
emnc.Brz 0 T Remanent flux density, z component Domain 14
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 14
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 14
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 14
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 14
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 14
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 14
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 14
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 14
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 14
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 14
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 75–78, 84–85
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 75–78, 84–85
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 75–78, 84–85
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 75–78, 84–85
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 75–78, 84–85
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 75–78, 84–85
emnc.unx unx Normal vector up direction, x component Boundaries 75–78, 84–85
emnc.uny uny Normal vector up direction, y component Boundaries 75–78, 84–85
emnc.unz unz Normal vector up direction, z component Boundaries 75–78, 84–85
emnc.dnx dnx Normal vector down direction, x component Boundaries 75–78, 84–85
emnc.dny dny Normal vector down direction, y component Boundaries 75–78, 84–85
emnc.dnz dnz Normal vector down direction, z component Boundaries 75–78, 84–85
emnc.W emnc.Wm J/m^3 Energy density Domain 14
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 14
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 14
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 14
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 14

2.4.23. Magnetic Flux Conservation 21

physics_emnc_mfc21.png

Magnetic Flux Conservation 21

Selection
Geometric entity level Domain
Selection Domain 16
Equations

equ_emnc_mfc21_1.png

equ_emnc_mfc21_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 20) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 20) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 93–96, 101, 103
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 93–96, 101, 103
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 93–96, 101, 103
emnc.Hx -Vmx A/m Magnetic field, x component Domain 16
emnc.Hy -Vmy A/m Magnetic field, y component Domain 16
emnc.Hz -Vmz A/m Magnetic field, z component Domain 16
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 93–96, 101, 103
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 93–96, 101, 103
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 93–96, 101, 103
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 16
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 16
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 16
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 16
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 16
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 16
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 16
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 16
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 16
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 16
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 16
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 16
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 16
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 16
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 16
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 16
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 16
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 16
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 16
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 16
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 16
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 16
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 16
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 16
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 16
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 16
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 16
emnc.Brx BpRem*cos(19*argBp+tick) T Remanent flux density, x component Domain 16
emnc.Bry BpRem*sin(19*argBp+tick) T Remanent flux density, y component Domain 16
emnc.Brz 0 T Remanent flux density, z component Domain 16
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 16
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 16
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 16
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 16
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 16
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 16
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 16
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 16
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 16
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 16
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 93–96, 101, 103
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 93–96, 101, 103
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 93–96, 101, 103
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 93–96, 101, 103
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 93–96, 101, 103
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 93–96, 101, 103
emnc.unx unx Normal vector up direction, x component Boundaries 93–96, 101, 103
emnc.uny uny Normal vector up direction, y component Boundaries 93–96, 101, 103
emnc.unz unz Normal vector up direction, z component Boundaries 93–96, 101, 103
emnc.dnx dnx Normal vector down direction, x component Boundaries 93–96, 101, 103
emnc.dny dny Normal vector down direction, y component Boundaries 93–96, 101, 103
emnc.dnz dnz Normal vector down direction, z component Boundaries 93–96, 101, 103
emnc.W emnc.Wm J/m^3 Energy density Domain 16
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 16
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 16
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 16
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 16

2.4.24. Magnetic Flux Conservation 22

physics_emnc_mfc22.png

Magnetic Flux Conservation 22

Selection
Geometric entity level Domain
Selection Domain 18
Equations

equ_emnc_mfc22_1.png

equ_emnc_mfc22_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 21) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 21) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 105–108, 113, 115
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 105–108, 113, 115
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 105–108, 113, 115
emnc.Hx -Vmx A/m Magnetic field, x component Domain 18
emnc.Hy -Vmy A/m Magnetic field, y component Domain 18
emnc.Hz -Vmz A/m Magnetic field, z component Domain 18
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 105–108, 113, 115
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 105–108, 113, 115
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 105–108, 113, 115
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 18
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 18
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 18
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 18
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 18
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 18
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 18
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 18
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 18
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 18
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 18
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 18
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 18
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 18
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 18
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 18
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 18
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 18
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 18
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 18
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 18
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 18
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 18
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 18
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 18
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 18
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 18
emnc.Brx BpRem*cos(20*argBp+tick) T Remanent flux density, x component Domain 18
emnc.Bry BpRem*sin(20*argBp+tick) T Remanent flux density, y component Domain 18
emnc.Brz 0 T Remanent flux density, z component Domain 18
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 18
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 18
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 18
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 18
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 18
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 18
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 18
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 18
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 18
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 18
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 105–108, 113, 115
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 105–108, 113, 115
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 105–108, 113, 115
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 105–108, 113, 115
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 105–108, 113, 115
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 105–108, 113, 115
emnc.unx unx Normal vector up direction, x component Boundaries 105–108, 113, 115
emnc.uny uny Normal vector up direction, y component Boundaries 105–108, 113, 115
emnc.unz unz Normal vector up direction, z component Boundaries 105–108, 113, 115
emnc.dnx dnx Normal vector down direction, x component Boundaries 105–108, 113, 115
emnc.dny dny Normal vector down direction, y component Boundaries 105–108, 113, 115
emnc.dnz dnz Normal vector down direction, z component Boundaries 105–108, 113, 115
emnc.W emnc.Wm J/m^3 Energy density Domain 18
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 18
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 18
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 18
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 18

2.4.25. Magnetic Flux Conservation 23

physics_emnc_mfc23.png

Magnetic Flux Conservation 23

Selection
Geometric entity level Domain
Selection Domain 20
Equations

equ_emnc_mfc23_1.png

equ_emnc_mfc23_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 22) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 22) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 117–121, 127
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 117–121, 127
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 117–121, 127
emnc.Hx -Vmx A/m Magnetic field, x component Domain 20
emnc.Hy -Vmy A/m Magnetic field, y component Domain 20
emnc.Hz -Vmz A/m Magnetic field, z component Domain 20
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 117–121, 127
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 117–121, 127
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 117–121, 127
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 20
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 20
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 20
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 20
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 20
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 20
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 20
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 20
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 20
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 20
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 20
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 20
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 20
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 20
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 20
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 20
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 20
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 20
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 20
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 20
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 20
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 20
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 20
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 20
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 20
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 20
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 20
emnc.Brx BpRem*cos(21*argBp+tick) T Remanent flux density, x component Domain 20
emnc.Bry BpRem*sin(21*argBp+tick) T Remanent flux density, y component Domain 20
emnc.Brz 0 T Remanent flux density, z component Domain 20
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 20
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 20
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 20
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 20
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 20
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 20
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 20
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 20
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 20
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 20
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 117–121, 127
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 117–121, 127
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 117–121, 127
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 117–121, 127
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 117–121, 127
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 117–121, 127
emnc.unx unx Normal vector up direction, x component Boundaries 117–121, 127
emnc.uny uny Normal vector up direction, y component Boundaries 117–121, 127
emnc.unz unz Normal vector up direction, z component Boundaries 117–121, 127
emnc.dnx dnx Normal vector down direction, x component Boundaries 117–121, 127
emnc.dny dny Normal vector down direction, y component Boundaries 117–121, 127
emnc.dnz dnz Normal vector down direction, z component Boundaries 117–121, 127
emnc.W emnc.Wm J/m^3 Energy density Domain 20
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 20
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 20
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 20
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 20

2.4.26. Magnetic Flux Conservation 24

physics_emnc_mfc24.png

Magnetic Flux Conservation 24

Selection
Geometric entity level Domain
Selection Domain 22
Equations

equ_emnc_mfc24_1.png

equ_emnc_mfc24_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 23) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 23) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 129–132, 137, 151
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 129–132, 137, 151
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 129–132, 137, 151
emnc.Hx -Vmx A/m Magnetic field, x component Domain 22
emnc.Hy -Vmy A/m Magnetic field, y component Domain 22
emnc.Hz -Vmz A/m Magnetic field, z component Domain 22
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 129–132, 137, 151
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 129–132, 137, 151
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 129–132, 137, 151
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 22
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 22
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 22
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 22
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 22
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 22
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 22
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 22
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 22
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 22
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 22
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 22
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 22
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 22
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 22
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 22
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 22
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 22
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 22
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 22
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 22
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 22
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 22
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 22
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 22
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 22
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 22
emnc.Brx BpRem*cos(22*argBp+tick) T Remanent flux density, x component Domain 22
emnc.Bry BpRem*sin(22*argBp+tick) T Remanent flux density, y component Domain 22
emnc.Brz 0 T Remanent flux density, z component Domain 22
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 22
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 22
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 22
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 22
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 22
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 22
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 22
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 22
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 22
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 22
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 129–132, 137, 151
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 129–132, 137, 151
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 129–132, 137, 151
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 129–132, 137, 151
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 129–132, 137, 151
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 129–132, 137, 151
emnc.unx unx Normal vector up direction, x component Boundaries 129–132, 137, 151
emnc.uny uny Normal vector up direction, y component Boundaries 129–132, 137, 151
emnc.unz unz Normal vector up direction, z component Boundaries 129–132, 137, 151
emnc.dnx dnx Normal vector down direction, x component Boundaries 129–132, 137, 151
emnc.dny dny Normal vector down direction, y component Boundaries 129–132, 137, 151
emnc.dnz dnz Normal vector down direction, z component Boundaries 129–132, 137, 151
emnc.W emnc.Wm J/m^3 Energy density Domain 22
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 22
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 22
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 22
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 22

2.4.27. Magnetic Flux Conservation 25

physics_emnc_mfc25.png

Magnetic Flux Conservation 25

Selection
Geometric entity level Domain
Selection Domain 24
Equations

equ_emnc_mfc25_1.png

equ_emnc_mfc25_2.png

Settings
Description Value
Constitutive relation Remanent flux density
Remanent flux density, x component BpRem*cos(argBp*(-1 + 24) + tick)
Remanent flux density, y component BpRem*sin(argBp*(-1 + 24) + tick)
Remanent flux density, z component 0
Relative permeability From material
Relative permeability {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}
Properties from material
Property Material Property group
Relative permeability Soft Iron (without losses) Basic
Variables
Name Expression Unit Description Selection
emnc.dnTmx -0.5*emnc.unx*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bx))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, x component Boundaries 139–142, 153, 155
emnc.dnTmy -0.5*emnc.uny*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.By))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, y component Boundaries 139–142, 153, 155
emnc.dnTmz -0.5*emnc.unz*(real(down(emnc.Bx))*real(down(emnc.Hx))+real(down(emnc.By))*real(down(emnc.Hy))+real(down(emnc.Bz))*real(down(emnc.Hz)))+real(down(emnc.Bz))*(real(down(emnc.Hx))*emnc.unx+real(down(emnc.Hy))*emnc.uny+real(down(emnc.Hz))*emnc.unz) Pa Maxwell downward magnetic surface stress tensor, z component Boundaries 139–142, 153, 155
emnc.Hx -Vmx A/m Magnetic field, x component Domain 24
emnc.Hy -Vmy A/m Magnetic field, y component Domain 24
emnc.Hz -Vmz A/m Magnetic field, z component Domain 24
emnc.tHx -VmTx A/m Tangential magnetic field, x component Boundaries 139–142, 153, 155
emnc.tHy -VmTy A/m Tangential magnetic field, y component Boundaries 139–142, 153, 155
emnc.tHz -VmTz A/m Tangential magnetic field, z component Boundaries 139–142, 153, 155
emnc.normH sqrt(realdot(emnc.Hx,emnc.Hx)+realdot(emnc.Hy,emnc.Hy)+realdot(emnc.Hz,emnc.Hz)) A/m Magnetic field norm Domain 24
emnc.murxx model.input.mur11 1 Relative permeability, xx component Domain 24
emnc.muryx model.input.mur21 1 Relative permeability, yx component Domain 24
emnc.murzx model.input.mur31 1 Relative permeability, zx component Domain 24
emnc.murxy model.input.mur12 1 Relative permeability, xy component Domain 24
emnc.muryy model.input.mur22 1 Relative permeability, yy component Domain 24
emnc.murzy model.input.mur32 1 Relative permeability, zy component Domain 24
emnc.murxz model.input.mur13 1 Relative permeability, xz component Domain 24
emnc.muryz model.input.mur23 1 Relative permeability, yz component Domain 24
emnc.murzz model.input.mur33 1 Relative permeability, zz component Domain 24
emnc.Bx mu0_const*emnc.Ixx*emnc.Hx+mu0_const*emnc.Ixy*emnc.Hy+mu0_const*emnc.Ixz*emnc.Hz+mu0_const*emnc.chimxx*emnc.Hx+mu0_const*emnc.chimxy*emnc.Hy+mu0_const*emnc.chimxz*emnc.Hz+emnc.Brx T Magnetic flux density, x component Domain 24
emnc.By mu0_const*emnc.Iyx*emnc.Hx+mu0_const*emnc.Iyy*emnc.Hy+mu0_const*emnc.Iyz*emnc.Hz+mu0_const*emnc.chimyx*emnc.Hx+mu0_const*emnc.chimyy*emnc.Hy+mu0_const*emnc.chimyz*emnc.Hz+emnc.Bry T Magnetic flux density, y component Domain 24
emnc.Bz mu0_const*emnc.Izx*emnc.Hx+mu0_const*emnc.Izy*emnc.Hy+mu0_const*emnc.Izz*emnc.Hz+mu0_const*emnc.chimzx*emnc.Hx+mu0_const*emnc.chimzy*emnc.Hy+mu0_const*emnc.chimzz*emnc.Hz+emnc.Brz T Magnetic flux density, z component Domain 24
emnc.normB sqrt(realdot(emnc.Bx,emnc.Bx)+realdot(emnc.By,emnc.By)+realdot(emnc.Bz,emnc.Bz)) T Magnetic flux density norm Domain 24
emnc.Mx emnc.Bx/mu0_const-emnc.Ixx*emnc.Hx-emnc.Ixy*emnc.Hy-emnc.Ixz*emnc.Hz A/m Magnetization, x component Domain 24
emnc.My emnc.By/mu0_const-emnc.Iyx*emnc.Hx-emnc.Iyy*emnc.Hy-emnc.Iyz*emnc.Hz A/m Magnetization, y component Domain 24
emnc.Mz emnc.Bz/mu0_const-emnc.Izx*emnc.Hx-emnc.Izy*emnc.Hy-emnc.Izz*emnc.Hz A/m Magnetization, z component Domain 24
emnc.normM sqrt(realdot(emnc.Mx,emnc.Mx)+realdot(emnc.My,emnc.My)+realdot(emnc.Mz,emnc.Mz)) A/m Magnetization norm Domain 24
emnc.Ixx 1 1 Spatial identity matrix, xx component Domain 24
emnc.Iyx 0 1 Spatial identity matrix, yx component Domain 24
emnc.Izx 0 1 Spatial identity matrix, zx component Domain 24
emnc.Ixy 0 1 Spatial identity matrix, xy component Domain 24
emnc.Iyy 1 1 Spatial identity matrix, yy component Domain 24
emnc.Izy 0 1 Spatial identity matrix, zy component Domain 24
emnc.Ixz 0 1 Spatial identity matrix, xz component Domain 24
emnc.Iyz 0 1 Spatial identity matrix, yz component Domain 24
emnc.Izz 1 1 Spatial identity matrix, zz component Domain 24
emnc.Brx BpRem*cos(23*argBp+tick) T Remanent flux density, x component Domain 24
emnc.Bry BpRem*sin(23*argBp+tick) T Remanent flux density, y component Domain 24
emnc.Brz 0 T Remanent flux density, z component Domain 24
emnc.normBr sqrt(realdot(emnc.Brx,emnc.Brx)+realdot(emnc.Bry,emnc.Bry)+realdot(emnc.Brz,emnc.Brz)) T Remanent flux density norm Domain 24
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Domain 24
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Domain 24
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Domain 24
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Domain 24
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Domain 24
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Domain 24
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Domain 24
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Domain 24
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Domain 24
emnc.unTx 0 Pa Maxwell upward surface stress tensor, x component Boundaries 139–142, 153, 155
emnc.unTy 0 Pa Maxwell upward surface stress tensor, y component Boundaries 139–142, 153, 155
emnc.unTz 0 Pa Maxwell upward surface stress tensor, z component Boundaries 139–142, 153, 155
emnc.dnTx emnc.dnTmx Pa Maxwell downward surface stress tensor, x component Boundaries 139–142, 153, 155
emnc.dnTy emnc.dnTmy Pa Maxwell downward surface stress tensor, y component Boundaries 139–142, 153, 155
emnc.dnTz emnc.dnTmz Pa Maxwell downward surface stress tensor, z component Boundaries 139–142, 153, 155
emnc.unx unx Normal vector up direction, x component Boundaries 139–142, 153, 155
emnc.uny uny Normal vector up direction, y component Boundaries 139–142, 153, 155
emnc.unz unz Normal vector up direction, z component Boundaries 139–142, 153, 155
emnc.dnx dnx Normal vector down direction, x component Boundaries 139–142, 153, 155
emnc.dny dny Normal vector down direction, y component Boundaries 139–142, 153, 155
emnc.dnz dnz Normal vector down direction, z component Boundaries 139–142, 153, 155
emnc.W emnc.Wm J/m^3 Energy density Domain 24
emnc.dWm emnc.Wm J/m^3 Integrand for total magnetic energy Domain 24
emnc.Wm 0.5*mu0_const*((emnc.murxx*emnc.Hx+emnc.murxy*emnc.Hy+emnc.murxz*emnc.Hz)*emnc.Hx+(emnc.muryx*emnc.Hx+emnc.muryy*emnc.Hy+emnc.muryz*emnc.Hz)*emnc.Hy+(emnc.murzx*emnc.Hx+emnc.murzy*emnc.Hy+emnc.murzz*emnc.Hz)*emnc.Hz) J/m^3 Magnetic energy density Domain 24
Shape Functions
Name Shape function Unit Description Shape frame Selection
Vm Lagrange (Quadratic) A Magnetic scalar potential Material Domain 24
Weak Expressions
Weak expression Integration frame Selection
emnc.d*(-emnc.Bx*test(Vmx)-emnc.By*test(Vmy)-emnc.Bz*test(Vmz)) Material Domain 24

2.4.28. Magnetic Shielding 1

physics_emnc_ms1.png

Magnetic Shielding 1

Selection
Geometric entity level Boundary
Selection Boundaries 1–4, 83, 92
Equations

equ_emnc_ms1.png

Settings
Description Value
Surface thickness 0.12
Relative permeability User defined
Relative permeability {{5500, 0, 0}, {0, 5500, 0}, {0, 0, 5500}}
Variables
Name Expression Unit Description Selection
emnc.murxx 5500 1 Relative permeability, xx component Boundaries 1–4, 83, 92
emnc.muryx 0 1 Relative permeability, yx component Boundaries 1–4, 83, 92
emnc.murzx 0 1 Relative permeability, zx component Boundaries 1–4, 83, 92
emnc.murxy 0 1 Relative permeability, xy component Boundaries 1–4, 83, 92
emnc.muryy 5500 1 Relative permeability, yy component Boundaries 1–4, 83, 92
emnc.murzy 0 1 Relative permeability, zy component Boundaries 1–4, 83, 92
emnc.murxz 0 1 Relative permeability, xz component Boundaries 1–4, 83, 92
emnc.muryz 0 1 Relative permeability, yz component Boundaries 1–4, 83, 92
emnc.murzz 5500 1 Relative permeability, zz component Boundaries 1–4, 83, 92
emnc.Ixx 1 1 Spatial identity matrix, xx component Boundaries 1–4, 83, 92
emnc.Iyx 0 1 Spatial identity matrix, yx component Boundaries 1–4, 83, 92
emnc.Izx 0 1 Spatial identity matrix, zx component Boundaries 1–4, 83, 92
emnc.Ixy 0 1 Spatial identity matrix, xy component Boundaries 1–4, 83, 92
emnc.Iyy 1 1 Spatial identity matrix, yy component Boundaries 1–4, 83, 92
emnc.Izy 0 1 Spatial identity matrix, zy component Boundaries 1–4, 83, 92
emnc.Ixz 0 1 Spatial identity matrix, xz component Boundaries 1–4, 83, 92
emnc.Iyz 0 1 Spatial identity matrix, yz component Boundaries 1–4, 83, 92
emnc.Izz 1 1 Spatial identity matrix, zz component Boundaries 1–4, 83, 92
emnc.chimxx -1+emnc.murxx 1 Magnetic susceptibility, xx component Boundaries 1–4, 83, 92
emnc.chimyx emnc.muryx 1 Magnetic susceptibility, yx component Boundaries 1–4, 83, 92
emnc.chimzx emnc.murzx 1 Magnetic susceptibility, zx component Boundaries 1–4, 83, 92
emnc.chimxy emnc.murxy 1 Magnetic susceptibility, xy component Boundaries 1–4, 83, 92
emnc.chimyy -1+emnc.muryy 1 Magnetic susceptibility, yy component Boundaries 1–4, 83, 92
emnc.chimzy emnc.murzy 1 Magnetic susceptibility, zy component Boundaries 1–4, 83, 92
emnc.chimxz emnc.murxz 1 Magnetic susceptibility, xz component Boundaries 1–4, 83, 92
emnc.chimyz emnc.muryz 1 Magnetic susceptibility, yz component Boundaries 1–4, 83, 92
emnc.chimzz -1+emnc.murzz 1 Magnetic susceptibility, zz component Boundaries 1–4, 83, 92
emnc.ds 0.12 m Surface thickness Boundaries 1–4, 83, 92
emnc.tBx mu0_const*(emnc.Ixx*emnc.tHx+emnc.Ixy*emnc.tHy+emnc.Ixz*emnc.tHz+emnc.chimxx*emnc.tHx+emnc.chimxy*emnc.tHy+emnc.chimxz*emnc.tHz) T Tangential magnetic flux density, x component Boundaries 1–4, 83, 92
emnc.tBy mu0_const*(emnc.Iyx*emnc.tHx+emnc.Iyy*emnc.tHy+emnc.Iyz*emnc.tHz+emnc.chimyx*emnc.tHx+emnc.chimyy*emnc.tHy+emnc.chimyz*emnc.tHz) T Tangential magnetic flux density, y component Boundaries 1–4, 83, 92
emnc.tBz mu0_const*(emnc.Izx*emnc.tHx+emnc.Izy*emnc.tHy+emnc.Izz*emnc.tHz+emnc.chimzx*emnc.tHx+emnc.chimzy*emnc.tHy+emnc.chimzz*emnc.tHz) T Tangential magnetic flux density, z component Boundaries 1–4, 83, 92
emnc.normtB sqrt(realdot(emnc.tBx,emnc.tBx)+realdot(emnc.tBy,emnc.tBy)+realdot(emnc.tBz,emnc.tBz)) T Tangential magnetic flux density norm Boundaries 1–4, 83, 92
Weak Expressions
Weak expression Integration frame Selection
(-emnc.tBx*test(VmTx)-emnc.tBy*test(VmTy)-emnc.tBz*test(VmTz))*emnc.d*emnc.ds Material Boundaries 1–4, 83, 92

2.5. Mesh 1

Mesh statistics
Property Value
Minimum element quality 0.06942
Average element quality 0.739
Tetrahedral elements 3644457
Triangular elements 260242
Edge elements 22220
Vertex elements 206

mesh_mesh1.png

Mesh 1

2.5.1. Size (Size)

Settings
Name Value
Maximum element size 0.0825
Minimum element size 0.006
Curvature factor 0.4
Resolution of narrow regions 0.7
Maximum element growth rate 1.4
Predefined size Finer

2.5.2. Free Tetrahedral 1 (Ftet1)

Selection
Geometric entity level Domain
Selection Domain 13

mesh_mesh1_ftet1.png

Free Tetrahedral 1

Size 1 (Size1)
Selection
Geometric entity level Domain
Selection Domain 13

mesh_mesh1_ftet1_size1.png

Size 1

Settings
Name Value
Maximum element size 0.03
Minimum element size 3.0E-4
Curvature factor 0.2
Maximum element growth rate 1.3
Predefined size Extremely fine

2.5.3. Free Tetrahedral 2 (Ftet2)

Selection
Geometric entity level Remaining
Size 1 (Size1)
Selection
Geometric entity level Domain
Selection Domains 2–12, 14–26

mesh_mesh1_ftet2_size1.png

Size 1

Settings
Name Value
Maximum element size 0.0525
Minimum element size 0.00225
Curvature factor 0.3
Resolution of narrow regions 0.85
Maximum element growth rate 1.35
Predefined size Extra fine

2.5.4. Free Tetrahedral 3 (Ftet3)

Selection
Geometric entity level Remaining
Size 1 (Size1)
Selection
Geometric entity level Domain
Selection Geometry geom1

mesh_mesh1_ftet3_size1.png

Size 1

Settings
Name Value
Maximum element size 0.0525
Minimum element size 0.00225
Curvature factor 0.3
Resolution of narrow regions 0.85
Maximum element growth rate 1.35
Predefined size Extra fine

3. Component 2

Date Mar 1, 2016 9:51:04 AM
Component settings
Unit system SI
Geometry shape order automatic

3.1. Definitions

3.1.1. Coordinate Systems

Boundary System 2
Coordinate system type Boundary system
Tag sys2
Settings
First (t1) Second (n) Third (to)
t1 n to
Settings
Name Value
Create first tangent direction from Global Cartesian

3.2. Geometry 2

geom_geom2.png

Geometry 2

Units
Length unit m
Angular unit deg
Geometry statistics
Property Value
Space dimension 2
Number of domains 1
Number of boundaries 6
Number of vertices 6

3.2.1. Composite Object 2 (CO2)

Selections of resulting entities
Name Value
Filename $FILENAME$_geom2_CO2.mphbin

4. Study 1

4.1. Stationary

Study settings
Property Value
Include geometric nonlinearity Off
Physics and variables selection
Physics interface Discretization
Magnetic Fields, No Currents (emnc) physics
Mesh selection
Geometry Mesh
Geometry 1 (geom1) mesh1
Geometry 2 (geom2) nomesh

4.2. Solver Configurations

4.2.1. Solution 1

Compile Equations: Stationary (St1)
Study and step
Name Value
Use study Study 1
Use study step Stationary
Dependent Variables 1 (V1)
General
Name Value
Defined by study step Stationary
Initial values of variables solved for
Name Value
Solution COMSOL 3.5a Solution
Values of variables not solved for
Name Value
Solution COMSOL 3.5a Solution
Magnetic Scalar Potential (Mod1.Vm) (Mod1_Vm)
General
Name Value
Field components mod1.Vm
Stationary Solver 1 (S1)
General
Name Value
Defined by study step Stationary
Relative tolerance 1.0E-6
Log
Stationary Solver 1 in Solution 1 started at 1-Mar-2016 11:43:26.
Linear solver
Number of degrees of freedom solved for: 4869978.
Symmetric matrices found.
Scales for dependent variables:
Magnetic scalar potential (mod1.Vm): 1
Orthonormal null-space function used.
Iter      SolEst     Damping    Stepsize #Res #Jac #Sol LinIt   LinErr   LinRes
   1        0.94   1.0000000        0.94    1    1    1    37 4.7e-007 3.6e-009
Stationary Solver 1 in Solution 1: Solution time: 156 s (2 minutes, 36 seconds)
                                   Physical memory: 10.15 GB
                                   Virtual memory: 10.35 GB
Fully Coupled 1 (Fc1)
General
Name Value
Linear solver Iterative 1
Method and termination
Name Value
Initial damping factor 1.0
Minimum damping factor 1.0e-4
Restriction for step-size update 10.0
Termination criterion Solution
Iterative 1 (I1)
General
Name Value
Solver Conjugate gradients
Multigrid 1 (Mg1)
General
Name Value
Solver Algebraic multigrid
Presmoother (Pr)
SOR 1 (So1)
General
Name Value
Relaxation factor 1.0
Postsmoother (Po)
SOR 1 (So1)
General
Name Value
Relaxation factor 1.0
Coarse Solver (Cs)
Direct 1 (D1)
Error
Name Value
Check error estimate No

4.2.2. COMSOL 3.5a Solution

5. Results

5.1. Data Sets

5.1.1. Study 1/Solution 1

Solution
Name Value
Solution Solution 1
Component Save Point Geometry 1

dataset_dset1.png

Data set: Study 1/Solution 1

5.1.2. Cut Plane 1

Data
Name Value
Data set Study 1/Solution 1
Plane data
Name Value
Plane type Quick
Plane xy - planes
z-coordinate 0
Advanced
Name Value
Space variables {cpl1x, cpl1y}

dataset_cpl1.png

Data set: Cut Plane 1

5.1.3. Cut Line 3D 1

Data
Name Value
Data set Study 1/Solution 1
Line data
Name Value
Line entry method Two points
Points {{-0.1, 0, 0}, {0.1, 0, 0}}
Advanced
Name Value
Space variable cln1x

dataset_cln1.png

Data set: Cut Line 3D 1

5.2. Tables

5.2.1. Pg3/Mmv1

Max/min volume

pg3/mmv1
x y z Magnetic flux density norm (T)
0.0000 0.0000 0.025000 0.11812
-0.025000 0.0000 0.0000 0.11815

5.3. Plot Groups

5.3.1. 2D Plot Group 1

plotgroup_pg1.png

Surface: Magnetic flux density norm (T) Arrow Line: Remanent flux density

5.3.2. 1D Plot Group 2

plotgroup_pg2.png

Line Graph: (emnc.normB) (T)

5.3.3. 3D Plot Group 3

plotgroup_pg3.png

Isosurface: (emnc.normB) (T) Max/Min Volume: Magnetic flux density norm (T)