# 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,,xn],{x1,,xn}}.
• Frequency-dependent variables vars are vars={V[x1,,xn],ω,{x1,,xn}}.
• The current continuity equation is with volume charge density [], time variable [] and current density vector [].
• 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 [] is an externally generated current density vector and [] a current source.
• ElectricCurrentPDEComponent provides a frequency domain model:
• with vacuum permittivity [], polarization vector [], 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 [].
• The following parameters pars can be given:
•  parameter default symbol "CrossSectionalArea" 1 , cross-sectional area in [] "CurrentSource" 0 , current source in [] "ElectricalConductivity" 1 , electrical conductivity in [] "ExternalCurrent" {0,…} , external current density vector in [] "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 [] "RelativePermittivity" 1 , unitless relative permittivity "RemanentPolarization" {0,…} , remanent polarization vector in [] "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 ,keypi,pivi,], the parameters are replaced with .

# Examples

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## Basic Examples(4)

Define an stationary current PDE model:

Define a symbolic stationary current PDE:

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(6)

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 []:

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:

## 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:

Wolfram Research (2024), ElectricCurrentPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectricCurrentPDEComponent.html.

#### Text

Wolfram Research (2024), ElectricCurrentPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectricCurrentPDEComponent.html.

#### CMS

Wolfram Language. 2024. "ElectricCurrentPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ElectricCurrentPDEComponent.html.

#### APA

Wolfram Language. (2024). ElectricCurrentPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ElectricCurrentPDEComponent.html

#### BibTeX

@misc{reference.wolfram_2024_electriccurrentpdecomponent, author="Wolfram Research", title="{ElectricCurrentPDEComponent}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/ElectricCurrentPDEComponent.html}", note=[Accessed: 15-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_electriccurrentpdecomponent, organization={Wolfram Research}, title={ElectricCurrentPDEComponent}, year={2024}, url={https://reference.wolfram.com/language/ref/ElectricCurrentPDEComponent.html}, note=[Accessed: 15-August-2024 ]}