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ElectricCurrentPDEComponent

ElectricCurrentPDEComponent[vars,pars]

yields an electric current PDE term with variables vars and parameters pars.

Details

ElectricCurrentPDEComponent is typically used to generate an electric current continuity equation with model variables vars and model parameters pars.
ElectricCurrentPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:

ElectricCurrentPDEComponent creates PDE components for stationary, frequency and parametric analysis.
ElectricCurrentPDEComponent models electric fields produced by direct or alternating currents in conductive materials when magnetic and inductive effects are negligible.
The results of ElectricCurrentPDEComponent can be use to compute current density magnitude values. »

ElectricCurrentPDEComponent models stationary or harmonic electric fields with the electric scalar potential [] as dependent variable and independent variables [].
Stationary variables vars are vars={V[x₁,…,x_n],{x₁,…,x_n}}.
Time-dependent variables vars are vars={V[t,x₁,…,x_n],t,{x₁,…,x_n}}.
Frequency-dependent variables vars are vars={V[x₁,…,x_n],ω,{x₁,…,x_n}}.
The current continuity equation is with volume charge density [ $TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 3}}, coulombs per meter cubed, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ], time variable [] and current density vector [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ].
The constitutional material model equation, known as Ohm's law, is where [] is the electrical conductivity and [] the electric field with .
ElectricCurrentPDEComponent provides a stationary electric current model where [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ] is an externally generated current density vector and [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 3}}, amperes per meter cubed, {{(, "Amperes", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ] a current source:

ElectricCurrentPDEComponent provides a time domain model with time variable [], vacuum permittivity [] and relative permittivity []:

ElectricCurrentPDEComponent provides a frequency domain model with vacuum permittivity [], polarization vector [ $TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ], angular frequency [] and the imaginary unit :

For linear materials, the frequency domain model simplifies to:

is the unitless relative permittivity.
can be isotropic, orthotropic or anisotropic.
The implicit default boundary condition for the electric current model is a 0 ElectricCurrentDensityValue.
The units of the electric current model terms are in [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 3}}, amperes per meter cubed, {{(, "Amperes", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ].
The following parameters pars can be given:

parameter	default	symbol
"CrossSectionalArea"	1	, cross-sectional area in [ $TemplateBox[{InterpretationBox[, 1], {{"m", ^, 2}}, meters squared, {"Meters", ^, 2}}, QuantityTF]$ ]
"CurrentSource"	0	, current source in [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 3}}, amperes per meter cubed, {{(, "Amperes", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ]
"ElectricalConductivity"	1	, electrical conductivity in []
"ExternalCurrentSource"	{0,…}	, external current density vector in [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
"Material"	-	none
"RegionSymmetry"	None
"Thickness"	1	, thickness in []

If a "Material" is specified, material constants are extracted from the material data; otherwise, relevant material parameters need to be specified.
Additional parameters can be specified for the frequency domain models:

parameter	default	symbol
"Polarization"	{0,…}	, polarization vector in [ $TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
"RelativePermittivity"	1	, unitless relative permittivity
"RemanentPolarization"	{0,…}	, remanent polarization vector in [ $TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]
"VacuumPermittivity"		, vacuum permittivity in []

All parameters may depend on the spacial variable and dependent variable .
The number of independent variables determines the dimensions of , and , and the length of vectors , and .
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. stationary equation

1D

2D
In 1D, when a "CrossSectionalArea" is specified, the ElectricCurrentPDEComponent equation is given as:

In 2D, when a "Thickness" is specified, the ElectricCurrentPDEComponent equation is given as:

In a 1D axisymmetric case, when a "Thickness" is specified, the ElectricCurrentPDEComponent equation is given as:

The input specification for the parameters is exactly the same as for their corresponding operator terms.
If no parameters are specified, the default electric current PDE is:

If the ElectricCurrentPDEComponent 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 (5)

Define an stationary current PDE model:

Define a symbolic stationary current PDE:

Define a symbolic time-dependent current PDE model:

Define a symbolic frequency current PDE model:

Solve for the electric scalar potential in a constricted rectangular plate with an electrical conductivity of :

Compute the current density vector:

Visualize the current density vector:

Scope (7)

Define a stationary current PDE model for a specific material:

Specify an stationary current PDE with an electrical conductivity of in units of [] and an external current density of in units of [ $TemplateBox[{InterpretationBox[, 1], {"A", , "/", , {"m", ^, 2}}, amperes per meter squared, {{(, "Amperes", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]$ ]:

Activate a stationary current PDE model for a specific material:

Define a symbolic stationary current PDE with electrical conductivity an external current, a current source and a thickness:

Define a symbolic 2D axisymmetric stationary current PDE:

Define a 3D frequency current PDE model:

Define a 3D time-dependent current PDE model:

Applications (5)

2D Stationary Analysis (1)

Solve for the electric scalar potential in a 3-bar electric switch with an electrical conductivity of :

Compute the current density vector:

Visualize the current density vector:

3D Stationary Analysis (3)

Model a copper wire that is excited with a direct current (DC) of [] with a current density boundary condition at the upper boundary and with a zero electric potential condition at the lower boundary.

Set up the stationary current PDE model variables and :

Set up the equation:

Specify ground potential at the lower boundary:

Specify an inward current flow at the upper boundary:

Define the cylinder:

Solve the PDE:

Visualize the electric potential:

Model a tungsten wire with a potential difference of []. Set up the stationary current PDE model variables and :

Set up the stationary current PDE:

The radius of the tungsten wire is [] and the geometric shape of the wire is s-shaped. Specify the parameters of the geometry:

The simulation domain:

An electric potential boundary condition of [] is applied at the left end boundary and a zero electric potential condition is applied at the right end boundary. A tolerance 0f is applied at both ends to account for numerical errors in the discretized domain.

Set the electric potential boundary conditions at both ends of the wire:

Solve the PDE:

Compute the current density vector:

Visualize the current density magnitude:

Model a copper spiral inductor that is excited with a current density normal to the left boundary and has a zero electric potential boundary condition at the right boundary.

Define the spiral inductor geometry:

Set up the stationary current PDE model variables and :

Specify an inward current flow on the left boundary:

Specify a ground potential:

Solve the PDE:

Compute the current density vector:

Visualize the current density magnitude:

Frequency Analysis (1)

Model a dielectric material of a cylindrical capacitor that is excited with an alternating current (AC) of [], with a current density boundary condition at the upper electrode, and with a zero electric potential boundary condition at the lower boundary.

Set up the frequency current PDE model variables :

Define the frequency and the period:

Set up a region :

Specify an electrical conductivity and a relative permittivity :

Specify the ground potential at the lower boundary:

Specify an inward current flow at the upper boundary:

Set up the equation:

Solve the harmonic PDE for []:

Transform the voltage at the upper boundary to the time domain:

Visualize the voltage at the upper plate of the capacitor:

Possible Issues (2)

For a symbolic computation, the "ElectricalConductivity", "VacuumPermittivity" or "RelativePermittivity" parameters should be given as a matrix:

For numeric values, the "ElectricalConductivity", "VacuumPermittivity" or "RelativePermittivity" parameter is automatically converted to a matrix of proper dimensions:

This automatic conversion is not possible for symbolic input:

Not providing the properly dimensioned matrix will result in an error:

For frequency domain models, material parameters are not available when "Material" is specified:

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ElectricCurrentPDEComponent

Details

Examples

Basic Examples (5)

Scope (7)