Transverse Magnetic Potential Model Reference

This model solves for the transverse magnetic potential according to Eqn. (4319). The magnetic flux density $\mathbf{\text{B}}$ and the magnetic field $\mathbf{\text{H}}$ are calculated on demand from the magnetic potential-Z.

Material Properties

Magnetic Permeability

Specifies the magnetic permeability $\mu$ of the material (see Eqn. (4220)).

Method	Associated Value Node
Constant, Field Function Suitable for linear isotropic materials (see Eqn. (4220)). Available for fluids and solids.	Magnetic Permeability > Constant, Field Function Specify $\mu$ as a scalar profile.
Table (B,H) Suitable for nonlinear isotropic materials (see Eqn. (4223)). Available for solids.	Magnetic Permeability > Table (B,H) > Tabular Data Specifies a nonlinear $B$ - $H$ curve as a table of $B,H$ values, from which Simcenter STAR-CCM+ determines the profile of $\mu$ . See Using Table(B,H) Method for Permeability. Table: Magnetic Flux density—specifies the table column that contains the values of $B$. Table: Magnetic Field—specifies the table column that contains the values of $H$. Input Table—allows you to select the table with the $B,H$ data. You can select an imported file table or a $B,H,\mu$ table from the material database.
Tabular Suitable for nonlinear isotropic materials (see Eqn. (4223)). Available for solids.	Tabular > Table Allows you to provide a $B,H$ curve using a table, from which Simcenter STAR-CCM+ determines the profile of $\mu$. See Using Tabular Data. If the provided $B,H$ curve is monotonous and convex, that is: $$\{\begin{array}{l}\frac{dB}{dH}>0\\ \frac{d^{2}B}{d{H}^2}<0\end{array}$$ set $\mathbf{\text{H}}$ as the interpolating variable in the method. If the curve is concave, set the magnetic flux density $\mathbf{\text{B}}$ as the interpolating variable.

For guidelines on setting the magnetic permeability, see Defining Electromagnetic Material Properties.

Susceptibility Temperature Factor

In thermal analyses, specifies the susceptibility temperature factor $S\left(T\right)$ , as defined in Eqn. (4225).

This property is only available for solid materials, when you both:

• activate either an energy model or the Specified Temperature model in the solid physics continuum
• define the magnetic permeability using the Table (B,H) method

The available methods are:

Constant, Field Function Allow you to specify $S\left(T\right)$ using a constant value or a field function (typically, a function of temperature).

Susceptibility Temperature Factor > Constant, Field Function Specifies $S\left(T\right)$ as a scalar profile.

Table (T) Allows you to specify $S\left(T\right)$ using a table of $S,T$ values. For more information and instructions, see Defining Temperature-Dependent Properties.

Susceptibility Temperature Factor > Table (T) > Table (T) Specifies the $S,T$ table through the following properties: Table: Data—specifies the table column that contains the values of $S$ . Table: Temperature—specifies the table column that contains the values of $T$ . Input Table—allows you to select the table with the $S,T$ data. You can select an imported file table or a table from the material database.

Electrical Conductivity

Specifies the electrical conductivity $\sigma$ (see Electrical Conductivity: Generalized Ohm's Law) of the material, in transient simulations.

The available methods for defining the electrical conductivity depend on the physics models that you activate in the physics continuum.

For heat transfer analysis, Simcenter STAR-CCM+ provides specific methods for defining $\sigma$ as a function of temperature.

Method	Corresponding Physics Value Nodes
Constant, Field Function Available for fluids and solids.	Electrical Conductivity > Constant, Field Function Specify $\sigma$ as a scalar profile.
Polynomial in T Available for fluids and solids, when you activate an energy model in the physics continuum. This method can produce non-positive values for electrical conductivity.	Electrical Conductivity > Polynomial in T Specifies $\sigma$ as a polynomial function of temperature, for heat transfer analysis. See Using Polynomial in T.
Resistivity Polynomial(T) Available for fluids and solids, when you activate an energy model in the physics continuum. Use this method for materials where the resistivity $\rho$ (see Eqn. (4229)) has a polynomial dependency with temperature. This method can produce non-positive values for resistivity.	Electrical Conductivity > Resistivity Polynomial in T Specifies $\rho$ as a polynomial function of temperature.
Table (T) Available for fluids and solids, when you activate an energy model in the physics continuum. This method does not extrapolate outside of the bounds you define in the table. If the table contains non-positive values for the electrical conductivity a warning message is displayed and the simulation does not proceed until all non-positive conductivities are fixed.	Electrical Conductivity > Table (T) Allows you to define $\sigma$ as a function of temperature by providing a table of $\sigma ,T$ values, from which Simcenter STAR-CCM+ determines the profile of $\sigma \left(T\right)$. See Using Table(T). The temperature range in the table must be consistent with the Minimum/Maximum Allowable Temperature settings that are specified under the Reference Values node for the physics continuum. This requirement is specific to electrical conductivity. Electrical Conductivity > Resistivity Interpolation Option When On, calculates the electrical conductivity of the material by interpolation of the resistivity values. The tabular conductivity values are converted to resistivity values to create an internal resistivity vs temperature table. Tabular resistivity values are first interpolated and then converted back to conductivity values.
Table (T,P) Available for compressible gases, when you activate an energy model in the physics continuum.	Electrical Conductivity > Table (T,P) Allows you to define $\sigma$ as a function of temperature and pressure by providing a table of $\sigma ,T,p$ values, from which Simcenter STAR-CCM+ determines the profile of $\sigma (T,p)$. See Using Table (T,P).

Initial Conditions

Magnetic Vector Potential Z

Allows you to initialize the transverse magnetic vector potential, that is, the component of the magnetic vector potential normal to the 2D domain, to a specified scalar profile.

Boundary Settings

Magnetic Vector Potential Specification

In transverse magnetic mode, the magnetic vector potential and electric current sheet are normal to the 2D domain. As only the normal component is non-zero, you specify the magnetic vector potential and the current sheet at a boundary using scalar profiles (see Boundary and Interface Conditions).

Method	Corresponding Physics Value Nodes
Electric Current Sheet Z Neumann b. c. that sets the electric current sheet at the boundary to: $$J_S={\overline{J}}_{S}n$$ where $n$ is the unit vector normal to the boundary and ${\overline{J}}_S$ is a specified scalar profile.	Electric Current Sheet Allows you to specify the normal electric current sheet as a scalar profile, ${\overline{J}}_S$.
Magnetic Vector Potential Z Dirichlet b. c. that specifies the magnetic vector potential at the boundary as: $$A=\overline{A}n$$ where $n$ is the unit vector normal to the boundary and $\overline{A}$ is a specified scalar profile.	Magnetic Vector Potential Allows you to specify the transverse magnetic vector potential as a scalar profile, $\overline{A}$.

Region Settings

Applies to fluid, porous, and solid regions.

Current Conservation Option

When activated, Simcenter STAR-CCM+ calculates the transverse magnetic potential while enforcing electric current conservation within the region. Deactivate this option when you explicitly specify the transverse magnetic potential at the outer boundaries of conducting regions (that is, when you use the Magnetic Vector Potential Z boundary condition).

To facilitate rapid convergence for conservation of eddy currents in two-dimensional, transverse geometries, set AMG Linear Solver properties as follows when the current conservation option is used:

Property	Value
Max Cycles	200
Convergence Tolerance	1.0E-5
Acceleration Method	Bi Conjugate Gradient Stablized

This also corrects for ohmic heating losses. Only available for transient simulations, when eddy currents are not suppressed.

Electric Current Density Source Option

Allows you to specify the electric current density for the region. This option is not available when you use the model in combination with the Excitation Coil model. In this case, the electric current density is supplied by the Excitation Coil model.

In axisymmetric simulations, you can calculate the volume integral of the electrical current density using a Sum report (see Sum under Reports).

Method	Corresponding Physics Value Nodes
None No electric current density source is defined.	None
Transverse Electric Current Density Specifies a user-defined electric current density source ( $J_u$ in Eqn. (4311)). As in Transverse Magnetic mode the current density is normal to the 2D region, you only specify the normal component of the electric current density.	Electric Current Density Source Allows you to specify the normal component of the electric current density as a scalar profile. Electric Current Density Source Potential Derivative When the current source is a nonlinear function of magnetic vector potential, specifying the derivative of the electric current density with respect to the magnetic vector potential improves the convergence.
Total Current Specifies the total current passing through the 2D region.	Total Electric Current Source Allows you to specify the total electric current passing through the 2D region, using a scalar profile.

Interface Settings

Magnetic Vector Potential Z Periodicity

In 2D domains, allows you to turn periodic and repeating interfaces into anti-periodic interfaces (see Boundary and Interface Conditions).

For example, in electrical machine simulations, the cross-sectional field analysis can be reduced to either an odd number or an even number of poles by employing anti-periodic or periodic interface conditions, respectively.

Method	Corresponding Physics Value Nodes
Periodic The magnetic potential has the same sign at each side of the interface (see Eqn. (4323)).	None
Anti Periodic The magnetic potential has the opposite sign at each side of the interface (see Eqn. (4322)).	None

Transverse Magnetic Potential Solver

Solves for the transverse magnetic potential according to Eqn. (4319).

Under-Relaxation Factor

During the iterative process, the Transverse Magnetic Potential solver under-relaxes the changes in the solution, as:

$${\mathbf{A}}^{n+1}={\mathbf{A}}^n+\omega {\mathbf{A}}^{\prime }$$

where $\omega$ is the under-relaxation factor. At each iteration, the under-relaxation factor governs the extent to which the new solution replaces the old one. The default value is 1.0.

If residuals show solution divergence and do not decrease, reduce the under-relaxation factor.

Solver Frozen

When On, the solver does not update any quantity during an iteration. It is Off by default. This is a debugging option that can result in non-recoverable errors and wrong solutions due to missing storage. See Finite Volume Solvers Reference for details.

Reconstruction Freeze

Specifies whether reconstruction gradients are updated with each iteration. When deactivated, the solver updates the reconstruction gradients with each iteration. When you activate both this property and Temporary Storage Retained, the solver uses the reconstruction gradients from the last iteration in which they were updated.

Reconstruction Zeroed

Allows you to set reconstruction gradients to zero. When activated, the solver sets reconstruction gradients to zero at the next iteration. Gradients remain zeroed.

When deactivated, the solver does not reset reconstruction gradients. The gradients are recomputed at the next iteration if they have been zeroed.

Temporary Storage Retained

When activated, the solver retains the following data, computed at each iteration, and makes them available as field functions:

• Magnetic Vector Potential Correction-Z Ap Coefficient
• Magnetic Vector Potential Correction-Z Residual
• Magnetic Vector Potential-Z Gradient
• Magnetic Vector Potential-Z Recon

Permeability Solver

Controls the solution of the magnetic permeability in all the continua that include either the Finite Volume Magnetic Vector Potential model or the Transverse Magnetic Potential model.

Reports

Magnetic Force

Calculates the total electromagnetic force acting on one or more parts or regions, or a combination of them (see Eqn. (4350)), along a specified Direction. If the specified direction is [0, 0, 0], the report returns the force magnitude.

The input parts or regions must be surrounded by a force-free medium. A medium is considered force-free when:

• It does not have user-defined electric current density sources
• In unsteady simulations, you either activate the Eddy Current Suppression model in the region physics continuum, or you suppress eddy currents at the region level (see Eddy Current Suppression Model Reference)
• The associated physics continuum does not include the Excitation Coil model or the Permanent Magnet model

Magnetic Torque

Calculates the magnitude of the total electromagnetic torque acting on a region, or group of regions (see Eqn. (4352)), that is surrounded by a force-free medium. For this report, you specify the axis with respect to which torque is calculated ($r$ in Eqn. (4352)). Set the Axis Origin and Axis properties with respect to the appropriate coordinate system. If the specified axis is [0, 0, 0], the report returns the torque magnitude.

Sum

In axisymmetric simulations, you can calculate the volume integral of the electric current density using a Sum report. Specifically, use the Sum report with the field function ${ElectricCurrentDensity_Z}*$Volume/($${Centroid}[1]) where $${Centroid}[1] is the distance of the cell centroid away from the axis of symmetry.

Field Functions

Electric Current Density-Z

Vector field function that represents the component of the electric current density $J$ normal to the 2D domain.

Electromagnetic Force Density

Electromagnetic force density at an interface between two materials ($f_{EM}$ in Eqn. (4349)).

Electromechanical Stress Tensor

Electromechanical stress tensor ${\sigma }_{EM}$, as defined in Eqn. (4347) (for linear materials) and Eqn. (4348) (for nonlinear materials).

Magnetic Field

Vector field function that represents the magnetic field $H$, which is related to the magnetic flux density, $B$, through Eqn. (4220) or Eqn. (4223).

Magnetic Flux Density

Vector field function that represents the magnetic flux density, $B$, which is related to the magnetic vector potential, $A$, through Eqn. (4233).

Magnetic Vector Potential-Z

Vector field function that represents the transverse magnetic vector potential, that is, the component of $A$ normal to the 2D domain. See $A_z$ in Eqn. (4319).

Permeability

Represents the scalar magnetic permeability $\mu$ of isotropic materials (see Eqn. (4220) or Eqn. (4223)).

For information on the Transverse Magnetic Potential solver field functions, see Temporary Storage Retained.