ElectricCurrentPDEComponent
ElectricCurrentPDEComponent[vars,pars]
yields an electric current PDE term with variables vars and parameters pars.
Details
![](Files/ElectricCurrentPDEComponent.en/details_1.png)
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- 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
.
![](Files/ElectricCurrentPDEComponent.en/Image_1.gif)
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Examples
open allclose allBasic Examples (4)
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:
Applications (5)
2D Stationary Analysis (1)
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
:
Specify ground potential at the lower boundary:
Specify an inward current flow at the upper boundary:
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:
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:
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:
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:
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:
Solve the harmonic PDE for [
]:
Transform the voltage at the upper boundary to the time domain:
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:
![](Files/ElectricCurrentPDEComponent.en/109.gif)
For frequency domain models, material parameters are not available when "Material" is specified:
![](Files/ElectricCurrentPDEComponent.en/110.gif)
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