The scalar potential formulations are restricted to static field analysis with partial orthotropic nonlinear permeability. The degrees of freedom (DOFs), element matrices, and load vectors are presented here in the following form (Zienkiewicz([75]), Chari([73]), and Gyimesi([141])):

5.2.1.2. Coefficient Matrix

(5–85)

(5–86)

(5–87)

5.2.1.3. Applied Loads

(5–88)

where:

{N} = element shape functions (φ = {N}T{φe})

vol = volume of the element

{Hg} = preliminary or “guess” magnetic field (see Electromagnetic Field Fundamentals)

{Hc} = coercive force vector (input as MGXX, MGYY, MGZZ on MP command))

[μ] = permeability matrix (derived from input material property MURX, MURY, and MURZ (MP command) and/or material curve B versus H (accessed with TB,BH))(see Equation 5–7, Equation 5–8, and Equation 5–9)

= derivative of permeability with respect to magnitude of the magnetic field intensity (derived from the input material property curve B versus H (accessed with TB,BH))

The material property curve is input in the form of B values versus H values and is then converted to a spline fit curve of μ versus H from which the permeability terms μ_h and are evaluated.

The coercive force vector is related to the remanent intrinsic magnetization vector as:

(5–89)

where:

μo = permeability of free space (input as MUZRO on EMUNIT command)

The Newton-Raphson solution technique (Option on the NROPT command) is necessary for nonlinear analyses. Adaptive descent is also recommended (Adaptky on the NROPT command). When adaptive descent is used Equation 5–85 becomes:

(5–90)

where:

ξ = descent parameter (see Newton-Raphson Procedure)

The vector potential formulation is applicable to both static and dynamic fields with partial orthotropic nonlinear permeability. The basic equation to be solved is of the form:

(5–91)

The terms of this equation are defined below (Biro([120])).

5.2.2.1. Degrees of Freedom

(5–92)

where:

= magnetic vector potentials (input/output as AZ)

= time integrated electric scalar potential (input/output as VOLT)

The VOLT degree of freedom is a time integrated electric potential to allow for symmetric matrices.

5.2.2.2. Coefficient Matrices

(5–93)

(5–94)

(5–95)

(5–96)

(5–97)

(5–98)

(5–99)

(5–100)

(5–101)

5.2.2.3. Applied Loads

(5–102)

(5–103)

(5–104)

(5–105)

(5–106)

where:

[NA] = matrix of element shape functions for {A}

[N] = vector of element shape functions for {V} (V = {N}T{Ve})

{Js} = source current density vector (input as JS on BFE command)

{Jt} = total current density vector (input as JS on BFE command)

vol = volume of the element

{Hc} = coercive force vector (input as MGXX, MGYY, MGZZ on MP command)

νo = reluctivity of free space (derived from value using MUZRO on EMUNIT command)

[ν] = partially orthotropic reluctivity matrix (inverse of [μ], derived from input material property curve B versus H (input using TB,BH command))

= derivative of reluctivity with respect to the magnitude of magnetic flux squared (derived from input material property curve B versus H (input using TB,BH command))

[σ] = orthotropic conductivity (input as RSVX, RSVY, RSVZ on MP command (inverse)) (see Equation 5–12).

{v} = velocity vector

The coercive force vector is related to the remanent intrinsic magnetization vector as:

(5–107)

The material property curve is input in the form of B values versus H values and is then converted to a spline fit curve of ν versus |B|² from which the isotropic reluctivity terms ν_h and are evaluated.

The above element matrices and load vectors are presented for the most general case of a vector potential analysis. Many simplifications can be made depending on the conditions of the specific problem. Restricting the formulation to 2-D, there is only one component of the vector potential (AZ).

Combining some of the above equations, the variational equilibrium equations may be written as:

(5–108)

(5–109)

Here T denotes transposition.

Static analyses require only the magnetic vector potential degrees of freedom (KEYOPT controlled) and the K coefficient matrices. If the material behavior is nonlinear then the Newton-Raphson solution procedure is required (Option on the NROPT command (see Newton-Raphson Procedure)).

For 2-D dynamic analyses, a current density load of either source ({J_s}) or total {J_t} current density is valid. J_t input represents the impressed current expressed in terms of a uniformly applied current density. This loading is only valid in a skin-effect analysis with proper coupling of the VOLT degrees of freedom. The electric scalar potential must be constrained properly in order to satisfy the fundamentals of electromagnetic field theory. This can be achieved by direct specification of the potential value (using the D command) as well as with coupling and constraining (using the CP and CE commands).

The general transient analysis (ANTYPE,TRANS accepts nonlinear material behavior and permanent magnets (MGXX, MGYY, MGZZ). Harmonic transient analyses (ANTYPE,HARMIC (see Harmonic Analysis)) is a linear analyses with sinusoidal loads; therefore, it is restricted to linear material behavior without permanent magnets.

The following section describes the derivation of the electromagnetic finite element equations used by SOLID236 and SOLID237 elements.

In an edge-based electromagnetic analysis, the magnetic vector potential {A} is approximated using the edge-based shape functions:

(5–110)

where:

[W] = matrix of element vector (edge-based) shape functions.

{A_e} = edge-flux at the element mid-side nodes (input/output as AZ). Edge-flux is defined as the line integral = of the magnetic vector potential along the element edge L.

The electric scalar potential V is approximated using scalar (node-based) element shape functions:

(5–111)

where:

{N} = vector of element scalar (node-based) shape functions,

{V_e} = electric scalar potential at the element nodes (input/output as VOLT).

Applying the variational principle to the governing electromagnetic equations (see Equation 5–48 - Equation 5–50), we obtain the system of finite element equations:

(5–112)

where:

= element magnetic reluctivity matrix,

= element linear magnetic reluctivity matrix,

= element nonlinear magnetic reluctivity matrix,

= element electric conductivity matrix,

= element magneto-electric coupling matrix,

= element electromagnetic coupling matrix,

= element eddy current damping matrix,

= element displacement current damping matrix,

= element magneto-dielectric coupling matrix,

= element displacement current mass matrix,

= element source current density vector,

= element remnant magnetization load vector,

vol = element volume,

[ν] = reluctivity matrix (inverse of the magnetic permeability matrix input as MURX, MURY, MURZ on MP command or derived from the B-H curve input on TB command),

= derivative of reluctivity with respect to the magnitude of magnetic flux squared (derived from the B-H curve (input via TB,BH command))

[σ] = electrical conductivity matrix (inverse of the electrical resistivity matrix input as RSVX, RSVY, RSVZ on MP command),

[ε]= dielectric permittivity (input as PERX, PERY, PERZ on MP command) (applicable to a harmonic electromagnetic analysis (KEYOPT(1) = 1) only),

{v} = velocity vector (input as VELO on BF command) (applicable to electromagnetic analysis option (KEYOPT(1) = 1) only),

{J_s} = source current density vector (input as JS on BFE command) (applicable to the stranded conductor analysis option (KEYOPT(1) = 0) only),

{H_c} = coercive force vector (input as MGXX, MGYY, MGZZ on MP command),

{I_e} = nodal current vector (input/output as AMPS).

Equation 5–112 describes the strong coupling between the magnetic edge-flux and the electric potential degrees of freedom is nonsymmetric. It can be made symmetric by either using the weak coupling option (KEYOPT(2) = 1) in static or transient analyses or using the time-integrated electric potential (KEYOPT(2) = 2) in transient or harmonic analyses. In the latter case, the VOLT degree of freedom has the meaning of the time-integrated electric scalar potential , and Equation 5–112 becomes:

(5–113)

The electric scalar potential V is approximated over the element as follows:

(5–114)

where:

5.2.4.1. Quasistatic Electric Analysis

The application of the variational principle and finite element discretization to the differential Equation 5–80 produces the matrix equation of the form:

(5–115)

where:

vol = element volume

[σeff] = "effective" conductivity matrix (defined by Equation 5–83)

{Ie} = nodal current vector (input/output as AMPS)

Equation 5–115 is used in the finite element formulation of PLANE230, SOLID231, and SOLID232. These elements model both static (steady-state electric conduction) and dynamic (time-transient and time-harmonic) electric fields. In the former case, matrix [C^v] is ignored.

A time-harmonic electric analysis can also be performed using elements PLANE121, SOLID122, and SOLID123. In this case, the variational principle and finite element discretization are applied to the differential Equation 5–82 to produce:

(5–116)

where:

5.2.4.2. Electrostatic Analysis

The matrix equation for an electrostatic analysis using elements PLANE121, SOLID122, and SOLID123 is derived from Equation 5–84:

(5–117)

{ρ} = charge density vector (input as CHRGD on BF command)

{ρs} = surface charge density vector (input as CHRGS on SF command)