ELECTRICAL_RESISTIVITY_MODEL

Specifies an electrical resistivity model for the charge conservation equation. The electrical resistivity represents the ability of a material to resist the flow of current. Electrical resistivity model applies to solid materials only.

Type

AcuSolve Command

Syntax

ELECTRICAL_RESISTVITY_MODEL("name"){parameters...}

Qualifier

User-given name.

Parameters

type (enumerated) [=none]

Type of the electrical resistivity model.

constant or const: Constant electrical resistivity. Requires electrical_resistivity.
linear_temperature: Linear temperature dependent resistivity.
piecewise_linear or linear: Piecewise linear curve fit. Requires curve_fit_values and curve_fit_variable.
cubic_spline or spline: Cubic spline curve fit. Requires curve_fit_values and curve_fit_variable.
user_function or user: User-defined function. Requires user_function, user_values and user_strings.

electric_resistivity (real) [=1.72e-8]

Constant value of electrical resistivity. When type is constant.

linear_temperature_reference_temperature (real) [=293.15]

Reference temperature for linear_temperature resistivity type.

linear_temperature_reference_temperature_resistivity (real) [=1.72e-8]

Resistivity of the material at the reference temperature for linear_temperature resistivity type.

linear_temperature_temperature_coefficient (real) [=0.00393]

Relative change of resistivity for a given change in temperature for linear_temperature resistivity type.

curve_fit_values or curve_values (array) [={0,0}]

A two-column array of independent-variable/isotropic electrical resistivity data values. Used with piecewise_linear and cubic_spline types.

curve_fit_variable or curve_var (enumerated) [=temperature]

Independent variable of the curve fit for isotropic electrical resistivity. Used with piecewise_linear and cubic_spline types.

user_function or user (string) [no default]

Name of the user-defined function. Used with user_function type.

user_values (array) [={}]

Array of values to be passed to the user-defined function. Used with user_function type.

user_strings (list) [={}]

Array of strings to be passed to the user-defined function. Used with user_function type.

Description

The resistivity model is used for both the charge conservation equation and for the coupling of charge conservation to the energy equation. A brief description of both models in the context of electrical resistivity is given below.

Conservation of charge is a fundamental physical principle and is given by:(1)

\nabla \cdot j = 0

Where

j

is the current density vector (units: A/m²). Typically, it is assumed that the current density is proportional to the electric field, where the proportionality constant is the electrical conductivity (

σ

) of the material. The current density can then be expressed as:(2)

j = - σ \nabla Φ

where $Φ$ (units: V) is the electric potential field. The inverse of conductivity $σ$ is the electrical resistivity $ρ_{E}$ of the material.

To couple the electrical model into the thermal simulation in AcuSolve, a source term (S) is introduced into the energy equation. The source term is based on Joule’s first law which says that the heat generated per unit volume is equal to the product of current density and the electric field (

E = - \nabla ϕ

). For example:(3)

S = j \cdot E = σ {| \nabla ϕ |}^{2}

The available resistivity model is set in the ELECTRICAL_RESISTIVITY_MODEL command and is referenced by MATERIAL_MODEL commands. For example:

ELECTRICAL_RESISTIVITY_MODEL( "my resistivity model" ) { 
    type = constant
    electrical_resistivity = 2.9e-8
}
MATERIAL_MODEL( "my material model" ) { 
    electrical_resistivity_model = "my resistivity model"
}

The simplest case is type = constant. This applies a constant resistivity in both the conservation of charge equation and the source term in the energy equation. An example of a constant electrical resistivity value is shown above.

The second type is a linear temperature dependent resistivity (type=linear_temperature). For this model the resistivity is a linear function of temperature given by:(4)

ρ_{E} (T) = ρ_{E}_{0} (1 + α (T - T_{R E F})

Where $ρ_{E}_{0}$ is the reference temperature resistivity, $α$ the temperature coefficient, $T$ the local temperature and $T_{R E F}$ the reference temperature. The temperature coefficient describes the relative change of resistivity for a given change in temperature.

In the input file a linear temperature dependent resistivity would be defined as follows:

ELECTRICAL_RESISTIVITY_MODEL( "my resistivity model" ) { 
    type = linear_temperature
    linear_temperature_reference_temperature = 293.15
    linear_temperature_reference_temperature_resistivity = 1.754e-8
    linear_temperature_temperature_coefficient = 0.0039
}

Electrical resistivity models of types piecewise_linear and cubic_spline may be used to define electrical resistivity as a function of a single independent variable. These types of electrical resistivity models are consistent with other material models. Currently the only curve_fit_variable supported is temperature.

An electrical resistivity model of type user_function may be used to model more complex behaviors; see the AcuSolve User-Defined Functions Manual for a detailed description of user-defined functions.