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.
contact_resistance
Contact resistance between two connected materials. Requires electrical_contact_resistance.
electrical_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.
electrical_contact_resistance (real) [=1.0e-6]
electrical_contact resistance between two connected materials. Used with contact_resistance type. Currently supported only for electric_potential = battery_joule_heating.

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)
∇ · j = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaey4bIeTaeS4JPFMaaeOAaiabg2da9iaaicdaaaa@3CAF@
Where j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOAaaaa@36F9@ 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 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdmhaaa@37CF@ ) of the material. The current density can then be expressed as:(2)
j = − σ ∇ Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOAaiabg2da9iabgkHiTiabeo8aZjabgEGirlaabA6aaaa@3D61@

where Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaeOPdaaa@3738@ (units: V) is the electric potential field. The inverse of conductivity σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaeq4Wdmhaaa@37CF@ is the electrical resistivity ρ E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKaaGcbaaaaaaa aapeGaeqyWdiNcpaWaaSbaaKqaGfaapeGaamyraaWcpaqabaaaaa@3A0D@ 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 = − ∇ ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaaCyraiabg2da9iabgkHiTiabgEGirlabew9aMbaa@3C1B@ ). For example:(3)
S = j ⋅ E = σ | ∇ ϕ | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaae4uaiabg2da9iaabQgacqGHflY1caWHfbGaeyypa0Jaae4Wdmaa emaapaqaa8qacqGHhis0cqaHvpGzaiaawEa7caGLiWoapaWaaWbaaS qabeaapeGaaGOmaaaaaaa@45D3@
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 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamyraaWdaeqaaOWdbmaabmaapaqa a8qacaWGubaacaGLOaGaayzkaaGaeyypa0JaeqyWdi3cpaWaaSbaae aapeGaamyraaWdaeqaaOWaaSbaaeaapeGaaGimaaWcpaqabaGcpeWa aeWaa8aabaWdbiaaigdacqGHRaWkcqaHXoqycaGGOaGaamivaiabgk HiTiaadsfapaWaaSbaaSqaa8qacaWGsbGaamyraiaadAeaa8aabeaa aOWdbiaawIcacaGLPaaaaaa@4BAD@

Where ρ E 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqyWdi3damaaBaaaleaapeGaamyraaWdaeqaaOWaaSbaaSqaa8qa caaIWaaapaqabaaaaa@39FF@ is the reference temperature resistivity, α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape GaeqySdegaaa@37AB@ the temperature coefficient, T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamivaaaa@36E5@ the local temperature and T R E F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaabaaaaaaaaape Gaamiva8aadaWgaaWcbaWdbiaadkfacaWGfbGaamOraaWdaeqaaaaa @39AA@ 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.

Electrical Contact Resistance

Electrical contact resistance occurs at the interface between connected electrical components, for example, welded joints. Contact resistance creates ohmic losses that lead to thermal hotspots and other undesirable thermal properties.

Contact resistance is enabled through the material model by setting the type = contact_resistance in the ELECTRICAL_RESISTIVITY_MODEL and linking this material to a THERMAL_SHELL. The electrical resistance is then given in Ω ⋅ m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyQdCLaey yXICTaamyBamaaCaaaleqabaGaaGOmaaaaaaa@3BAB@ . Currently, this is only supported by electric_potential = battery_joule_heating.

An example contact resistance input:
MATERIAL_MODEL("weld") {
	type = solid
	...
	electrical_resistivity_model = "weld"
}
ELECTRICAL_RESISTIVITY_MODEL("weld") {
	type = contact_resistance
	electrical_contact_resistance = 1e2
}