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MagneticPDEComponent

MagneticPDEComponent[vars,pars]

yields a magnetic PDE term with variables vars and pars.

Details

MagneticPDEComponent generates an equation to model magnetostatics and low-frequency electromagnetics with model variables vars and model parameters pars.
MagneticPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:

MagneticPDEComponent models static magnetic fields produced by permanent magnets or magnetic and electric fields that are generated by low-frequency electric currents flowing in conductive materials.
MagneticPDEComponent is typically used to model electric motors, inductors and electromagnets.
MagneticPDEComponent creates PDE components for stationary, time, frequency and parametric analysis.

MagneticPDEComponent models magnetostatics and low-frequency electromagnetic phenomena with the dependent magnetic vector potential in units of [] and independent variables in units of [].
MagneticPDEComponent can model external currents also in the static case.
In the static case, when no currents are present, MagnetostaticPDEComponent should be used.
The vector-valued dependent variable is specified as a three-vector ={A_x₁,A_x₂,A_x₃}.
Stationary variables vars are vars={[x₁,…,x_n],{x₁,…,x_n}}.
Frequency-dependent variables vars are vars={[x₁,…,x_n],ω,{,…,x_n}}.
Time-dependent variables vars are vars={[t,x₁,…,x_n],t,{x₁,…,x_n}}.
MagneticPDEComponent provides a stationary magnetic model:

is the vacuum permeability in units of [], the magnetization vector in units of [] and the external current density vector in units of [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ].
The magnetization vector specifies the magnetic dipole moment per unit volume within a material, indicating the strength and direction of its magnetic properties.
MagneticPDEComponent provides a frequency domain model:

[] is the vacuum permittivity, relative permittivity [-], electrical conductivity [], angular frequency [] and the imaginary unit .
MagneticPDEComponent provides a time domain model:

An alternative model to the magnetization vector , is the remanent magnetic flux density vector in units of [ $TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ].
The stationary MagneticPDEComponent equation is given as:

is the unitless recoil permeability.
For linear materials, the stationary equation MagneticPDEComponent simplifies to:

is the unitless relative permeability.
can be isotropic, orthotropic or anisotropic.
can be a function of the magnetic field and describe nonlinear materials.
The units of the magnetic model terms are in [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ].
The following parameters pars can be given:

parameter	default	symbol
"ExternalCurrentSource"	{0,…}	, external current density vector in [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
"Magnetization"	{0,…}	, magnetization vector in []
"MagneticModelForm"	None
"RegionSymmetry"	None
"RelativePermeability"	1	, unitless relative permeability
"RemanentMagneticFluxDensity"	{0,…}	, remanent magnetic flux density in [ $TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
"Thickness"	1	, thickness in []
"VacuumPermeability"		, vacuum permeability in []

Additional parameters can be specified for the frequency and time domain models:
parameter default symbol

"ElectricalConductivity" 1 , electrical conductivity in []
The number of independent variables determines the dimensions of , and and the length of vectors , and .
The models are available in a 2D, a 2D axisymmetric and a 3D form.
For 3D stationary models with relative permeability , "MagneticModelForm" can be set to "FreeSpace", and with the Coulomb gauge condition, , the 3D operator simplifies to:

In 2D, with an out-of-plane direction, the magnetic vector potential has only a component. In the stationary linear case, the equation is given by:

[] is a variable denoting a "Thickness" in the direction and the dependent variable is specified as ={0,0,A_z}.
A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
"Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
dimension reduction e.g. linear stationary equation

2D
To solve this equation, the covariant formulation is made use of. The covariant formulation is a method in which a change of variable is applied to axisymmetric equation, given by :
The input specification for the parameters is exactly the same as for their corresponding operator terms.
If no parameters are specified, the default magnetic PDE is:

If the MagneticPDEComponent depends on parameters that are specified in the association pars as …,keyp_i…,p_iv_i,…, the parameters are replaced with .

Examples

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Basic Examples (3)

Define a symbolic 3D magnetic PDE model with vacuum permeability and relative permeability :

Wolfram Language code:

MagneticPDEComponent[{{Ax[x, y, z], Ay[x, y, z], Az[x, y, z]}, {x, y, z}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r]|>]

Set up the default magnetic PDE model with vacuum permeability

and relative permeability

Wolfram Language code: MagneticPDEComponent[{{Ax[x, y, z], Ay[x, y, z], Az[x, y, z]}, {x, y, z}}, <||>]

Define a symbolic time-dependent magnetic PDE model:

Wolfram Language code:

MagneticPDEComponent[{{Ax[t, x, y, z], Ay[t, x, y, z], Az[t, x, y, z]}, t, {x, y, z}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r], "ElectricalConductivity" -> σ|>]//MatrixForm

Scope (7)

Define a magnetic PDE model with a relative permeability set to 5:

Wolfram Language code: MagneticPDEComponent[{{Ax[x, y, z], Ay[x, y, z], Az[x, y, z]}, {x, y, z}}, <|"RelativePermeability" -> 5|>]

Activate a magnetic PDE model:

Wolfram Language code: MagneticPDEComponent[{{Ax[x, y, z], Ay[x, y, z], Az[x, y, z]}, {x, y, z}}, <|"RelativePermeability" -> 5|>]//Activate

Define a symbolic 2D out-of-plane magnetic PDE model with vacuum permeability , relative permeability and an external current in the direction:

Wolfram Language code:

MagneticPDEComponent[{{0, 0, Az[x, y]}, {x, y}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r], "ExternalCurrentSource" -> {0, 0, Jz[x, y]}|>]

Note that the and direction currents are not considered:

Wolfram Language code:

MagneticPDEComponent[{{0, 0, Az[x, y]}, {x, y}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r], "ExternalCurrentSource" -> {Jx[x, y], Jy[x, y], Jz[x, y]}|>]

Define a symbolic 2D axisymmetric magnetic PDE model:

Wolfram Language code:

MagneticPDEComponent[{{0, Aθ[r, z], 0}, {r, z}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r], "ExternalCurrentSource" -> {0, Jθ[r, z], 0}, "RegionSymmetry" -> "Axisymmetric"|>]

Note that the and direction currents are not considered:

Wolfram Language code:

MagneticPDEComponent[{{0, Aθ[r, z], 0}, {r, z}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r], "ExternalCurrentSource" -> {Jr[r, z], Jθ[r, z], Jz[r, z]}, "RegionSymmetry" -> "Axisymmetric"|>]

Define a 3D free-space magnetic PDE model with relative permeability of 1:

Wolfram Language code:

MagneticPDEComponent[{{Ax[x, y, z], Ay[x, y, z], Az[x, y, z]}, {x, y, z}}, <|"VacuumPermeability" -> Subscript[μ, 0], "ExternalCurrentSource" -> {Jx[x, y, z], Jy[x, y, z], Jz[x, y, z]}, "MagneticModelForm" -> "FreeSpace"|>]

Define a symbolic 2D-frequency magnetic PDE model:

Wolfram Language code:

MagneticPDEComponent[{{0, 0, Az[x, y]}, ω, {x, y}}, <|"VacuumPermeability" -> Subscript[μ, 0], "RelativePermeability" -> Subscript[μ, r], "ExternalCurrentSource" -> {0, 0, Jz[x, y]}, "ElectricalConductivity" -> σ|>]

Set up the default magnetic PDE model with vacuum permeability and relative permeability and maintain Quantity objects:

Wolfram Language code: MagneticPDEComponent[{{Ax[x, y, z], Ay[x, y, z], Az[x, y, z]}, {x, y, z}}, <|"Quantities" -> True|>]

Applications (2)

2D Stationary Analysis (1)

To model a long wire of circular cross section, define the mesh to use:

Wolfram Language code: mesh = [image];

Define a wire region:

Wolfram Language code: wireRegion = Disk[{0, 0}, 0.2];

Solve the magnetic PDE model with a uniform current density in the direction in the wire:

Wolfram Language code:

AzFun = NDSolveValue[{MagneticPDEComponent[{{0, 0, Az[x, y]}, {x, y}}, <|"ExternalCurrentSource" -> {0, 0, Piecewise[{{1, RegionMember[wireRegion][{x, y}]}}, 0]}|>] == 0, MagneticPotentialCondition[x ^ 2 + y ^ 2 >= 1, {{0, 0, Az[x, y]}, {x, y}}, <||>]}, Az, {x, y}∈mesh]

Visualize the magnetic field:

Wolfram Language code:

Show[StreamPlot[Evaluate[Curl[{0, 0, AzFun[x, y]}, {x, y, z}][[1 ;; 2]]], {x, y}∈mesh], Graphics[{Darker[Orange], wireRegion}]]

2D Frequency Analysis (1)

Define the mesh to model a long copper wire of circular cross section:

Wolfram Language code: mesh = \!$\*GraphicsBox[«3»]$;

Define a wire region:

Wolfram Language code: wireRegion = Disk[{0, 0}, 0.01];

Define the parameters of the model with:

Wolfram Language code:

pars = <|"ElectricalConductivity" -> Piecewise[{{5.9*^7, RegionMember[wireRegion][{x, y}]}}, 0], "RelativePermeability" -> 1|>;

Define the variables:

Wolfram Language code: vars = {{0, 0, Az[x, y]}, ω, {x, y}};

Define the uniform external current density in the direction:

Wolfram Language code:

Je0 = 100000;
pars["ExternalCurrentSource"] = {0, 0, Piecewise[{{Je0, RegionMember[wireRegion][{x, y}]}}]};

Define the magnetic equation:

Wolfram Language code: op = MagneticPDEComponent[vars, pars]

Define a zero magnetic potential condition at the exterior boundary:

Wolfram Language code: Subscript[Γ, zero] = MagneticPotentialCondition[x ^ 2 + y ^ 2 >= 0.1 ^ 2, vars, pars]

Define the PDE:

Wolfram Language code: pde = {op == 0, Subscript[Γ, zero]};

Set up an angular frequency of 800 Hz:

Wolfram Language code: ω0 = 2 * Pi * 800;

Replace the angular frequency and solve the PDE:

Wolfram Language code: AzFun = NDSolveValue[pde /. ω -> ω0, Az, {x, y}∈mesh]

Compute the electric field:

Wolfram Language code: Efield = {0, 0, -I * ω0 * AzFun[x, y]};

Compute the conduction current:

Wolfram Language code: Jc = pars["ElectricalConductivity"] * Efield;

Extract the external current:

Wolfram Language code: Je = pars["ExternalCurrentSource"];

Compute the total current:

Wolfram Language code: Jfield = Jc + Je;

Visualize the total current magnitude:

Wolfram Language code: DensityPlot[Sqrt[Total[Abs[Jfield] ^ 2]], {x, y}∈mesh, ...]

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MagneticPDEComponent

Details

Examples

Basic Examples (3)

Scope (7)

Applications (2)

2D Stationary Analysis (1)

2D Frequency Analysis (1)

Text

CMS

APA

BibTeX

BibLaTeX

	dimension	reduction	e.g. linear stationary equation
	2D

MagneticPDEComponent

Details

Examples

Basic Examples (3)

Scope (7)

Applications (2)

2D Stationary Analysis (1)

2D Frequency Analysis (1)

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Tech Notes

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Text

CMS

APA

BibTeX

BibLaTeX