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 [TemplateBox[{InterpretationBox[, 1], "V", volts, "Volts"}, QuantityTF]] as dependent variable and independent variables [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]].
  • 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 [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 3}}, coulombs per meter cubed, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]], time variable [TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]] 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 [TemplateBox[{InterpretationBox[, 1], {"S", , "/", , "m"}, siemens per meter, {{(, "Siemens", )}, /, {(, "Meters", )}}}, QuantityTF]] is the electrical conductivity and [TemplateBox[{InterpretationBox[, 1], {"V", , "/", , "m"}, volts per meter, {{(, "Volts", )}, /, {(, "Meters", )}}}, QuantityTF]] 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 frequency domain model:
  • with vacuum permittivity [TemplateBox[{InterpretationBox[, 1], {"F", , "/", , "m"}, farads per meter, {{(, "Farads", )}, /, {(, "Meters", )}}}, QuantityTF]], polarization vector [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]], angular frequency [TemplateBox[{InterpretationBox[, 1], {"rad", , "/", , "s"}, radians per second, {{(, "Radians", )}, /, {(, "Seconds", )}}}, QuantityTF]] 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:
  • parameterdefaultsymbol
    "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 [TemplateBox[{InterpretationBox[, 1], {"S", , "/", , "m"}, siemens per meter, {{(, "Siemens", )}, /, {(, "Meters", )}}}, QuantityTF]]
    • "ExternalCurrent"{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 [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]]
  • 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:
  • parameterdefaultsymbol
    "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 [TemplateBox[{InterpretationBox[, 1], {"F", , "/", , "m"}, farads per meter, {{(, "Farads", )}, /, {(, "Meters", )}}}, QuantityTF]]
  • 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:
  • dimensionreductione.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 [TemplateBox[{InterpretationBox[, 1], {"S", , "/", , "m"}, siemens per meter, {{(, "Siemens", )}, /, {(, "Meters", )}}}, QuantityTF]] 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:

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 [TemplateBox[{InterpretationBox[, 1], "A", amperes, "Amperes"}, QuantityTF]] 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 vars and pars:

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 [TemplateBox[{InterpretationBox[, 1], "V", volts, "Volts"}, QuantityTF]]. Set up the stationary current PDE model variables vars and pars:

Set up the stationary current PDE:

The radius of the tungsten wire is [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]] 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 [TemplateBox[{InterpretationBox[, 1], "V", volts, "Volts"}, QuantityTF]] 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 vars and pars:

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 [TemplateBox[{InterpretationBox[, 1], "Hz", hertz, "Hertz"}, QuantityTF]], 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 vars:

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 [TemplateBox[{InterpretationBox[, 1], "Hz", hertz, "Hertz"}, QuantityTF]]:

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: 24-November-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: 24-November-2024 ]}