The first derived result is the magnetic field intensity which is divided into two parts (see Electromagnetic Field Fundamentals); a generalized field {H_g} and the gradient of the generalized potential - ϕg. This gradient (referred to here as {H_ϕ) is evaluated at the integration points using the element shape function as:

(5–118)

where:

{N} = shape functions

{ωg} = nodal generalized potential vector

The magnetic field intensity is then:

(5–119)

where:

Then the magnetic flux density is computed from the field intensity:

(5–120)

where:

Nodal values of field intensity and flux density are computed from the integration points values as described in Nodal and Centroidal Data Evaluation.

Magnetic forces are also available and are discussed below.

The magnetic flux density is the first derived result. It is defined as the curl of the magnetic vector potential. This evaluation is performed at the integration points using the element shape functions:

(5–121)

where:

{B} = magnetic flux density (output as B)

x = curl operator

[NA] = shape functions

{Ae} = nodal magnetic vector potential

Then the magnetic field intensity is computed from the flux density:

(5–122)

where:

Nodal values of field intensity and flux density are computed from the integration point value as described in Nodal and Centroidal Data Evaluation.

Magnetic forces are also available and are discussed below.

For a vector potential transient analysis current densities are also calculated.

(5–123)

where:

(5–124)

where:

and

(5–125)

where:

{Js} = current density component due to V

= divergence operator

{Ve} = electric scalar potential

{N} = element shape functions for V evaluated at the integration points

and

(5–126)

where:

The following section describes the results derived from an edge-based electromagnetic analysis using SOLID236 and SOLID237 elements.

The electromagnetic fields and fluxes are evaluated at the integration points as follows:

(5–127)

(5–128)

(5–129)

(5–130)

(5–131)

where:

{B} = magnetic flux density (output as B at the element nodes),

{H} = magnetic field intensity (output as H at the element nodes),

{E} = electric field intensity (output as EF at the element nodes),

{J_c} = conduction current density (output as JC at the element nodes and as JT at the element centroid),

{J_s} = total (conduction + displacement) current density (output as JS at the element centroid; same as JT in a static or transient analysis),

{A_e}= edge-flux at the element mid-side nodes (input/output as AZ),

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

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

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

[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),

[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).

{ν} = velocity vector (input as VELO on BF command) (applicable to electromagnetic analyses (KEYOPT(1) = 1) only)

Nodal values of the above quantities are computed from the integration point values as described in Nodal and Centroidal Data Evaluation.

Magnetic forces are computed by elements using the vector potential method (PLANE13, PLANE233, SOLID236 and SOLID237) and the scalar potential method (SOLID5, SOLID96, and SOLID98). Three different techniques are used to calculate magnetic forces at the element level.

5.3.4.1. Lorentz forces

Magnetic forces in current carrying conductors (element output quantity FJB) are numerically integrated using:

(5–132)

where:

{N} = vector of shape functions

For a 2-D analysis, the corresponding electromagnetic torque about +Z is given by:

(5–133)

where:

{Z} = unit vector along +Z axis

{r} = position vector in the global Cartesian coordinate system

In a time-harmonic analysis, the time-averaged Lorentz force and torque are computed by:

(5–134)

and

(5–135)

respectively.

where:

{J}* = complex conjugate of {J}

5.3.4.2. Maxwell Forces

The Maxwell stress tensor is used to determine forces on ferromagnetic regions. Depending on whether the magnetic forces are derived from the Maxwell stress tensor using surface or volumetric integration, one distinguishes between the surface and the volumetric integral methods.

5.3.4.2.1. Surface Integral Method

This method is used by PLANE13, SOLID5, SOLID96, and SOLID98 elements.

The force calculation is performed on surfaces of air material elements which have a nonzero face loading specified (MXWF on SF commands) (Moon([77])). For the 2-D application, this method uses extrapolated field values and results in the following numerically integrated surface integral:

(5–136)

where:

{Fmx} = Maxwell force (output as FMX)

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

T12 = Bx By

T21 = Bx By

3-D applications are an extension of the 2-D case.

For a 2-D analysis, the corresponding electromagnetic torque about +Z axis is given by:

(5–137)

where:

= unit surface normal in the global Cartesian coordinate system

In a time-harmonic analysis, the time-averaged Maxwell stress tensor force and torque are computed by:

(5–138)

and

(5–139)

respectively.

where:

{B}* = complex conjugate of {B}

Re{ } = denotes real part of a complex quantity

5.3.4.2.2. Volumetric Integral Method

This method is used by PLANE233, SOLID236, and SOLID237 elements with KEYOPT(8) = 0.

The Maxwell forces are calculated by the following volumetric integral:

(5–140)

where:

= element magnetic Maxwell forces (output as FMAG at all the element nodes with KEYOPT(7) = 0 or at the element corner nodes only with KEYOPT(7) = 1),

[B] = strain-displacement matrix

{T^mx} = Maxwell stress vector = {T₁₁ T₂₂ T₃₃ T₁₂ T₂₃ T₁₃}^T

The EMFT macro can be used with this method to sum up Maxwell forces and torques.

5.3.4.3. Virtual Work Forces

Electromagnetic nodal forces (including electrostatic forces) are calculated using the virtual work principle. The two formulations currently used for force calculations are the element shape method (magnetic forces) and nodal perturbations method (electromagnetic forces).

5.3.4.3.1. Element Shape Method

Magnetic forces calculated using the virtual work method (element output quantity FVW) are obtained as the derivative of the energy versus the displacement (MVDI on BF commands) of the movable part. This calculation is valid for a layer of air elements surrounding a movable part (Coulomb([76])). To determine the total force acting on the body, the forces in the air layer surrounding it can be summed. The basic equation for force of an air material element in the s direction is:

(5–141)

where:

Fs = force in element in the s direction

s = virtual displacement of the nodal coordinates taken alternately to be in the X, Y, Z global directions

vol = volume of the element

For a 2-D analysis, the corresponding electromagnetic torque about +Z axis is given by:

(5–142)

In a time-harmonic analysis, the time-averaged virtual work force and torque are computed by:

(5–143)

and

(5–144)

respectively.

5.3.4.3.2. Nodal Perturbation Method

This method is used by PLANE121, SOLID122 and SOLID123 elements.

Electromagnetic (both electric and magnetic) forces are calculated as the derivatives of the total element coenergy (sum of electrostatic and magnetic coenergies) with respect to the element nodal coordinates (Gyimesi et al.([347])):

(5–145)

where:

Fxi = x-component (y- or z-) of electromagnetic force calculated in node i

xi = nodal coordinate (x-, y-, or z-coordinate of node i)

vol = volume of the element

Nodal electromagnetic forces are calculated for each node in each element. In an assembled model the nodal forces are added up from all adjacent to the node elements. The nodal perturbation method provides consistent and accurate electric and magnetic forces (using the EMFT command macro).

Joule heat is computed by elements using the vector potential method (PLANE13, SOLID236, and SOLID237) if the element has a nonzero resistivity (material property RSVX) and a nonzero current density (either applied J_s or resultant J_t). It is available as the output power loss (output as JHEAT) or as the coupled field heat generation load (LDREAD,HGEN).

Joule heat per element is computed as:

The first derived result in this analysis is the electric field. By definition (Equation 5–73), it is calculated as the negative gradient of the electric scalar potential. This evaluation is performed at the integration points using the element shape functions:

(5–148)

Nodal values of electric field (output as EF) are computed from the integration points values as described in Nodal and Centroidal Data Evaluation. The derivation of other output quantities depends on the analysis types described below.

5.3.6.1. Quasistatic Electric Analysis

The conduction current and electric flux densities are computed from the electric field (see Equation 5–74 and Equation 5–75):

(5–149)

(5–150)

where:

Both the conduction current {J} and electric flux {D} densities are evaluated at the integration point locations; however, whether these values are then moved to nodal or centroidal locations depends on the element type used to do a quasistatic electric analysis:

• In a current-based electric analysis using elements PLANE230, SOLID231, and SOLID232, the conduction current density is stored at both the nodal (output as JC) and centoidal (output as JT) locations. The electric flux density vector components are stored at the element centroidal location and output as nonsummable miscellaneous items;

• In a charge-based analysis using elements PLANE121, SOLID122, and SOLID123 (harmonic analysis), the conduction current density is stored at the element centroidal location (output as JT), while the electric flux density is moved to the nodal locations (output as D).

The total electric current {J_tot} density is calculated as a sum of conduction {J} and displacement current densities:

(5–151)

The total electric current density is stored at the element centroidal location (output as JS). It can be used as a source current density in a subsequent magnetic analysis (LDREAD,JS).

The Joule heat is computed from the centroidal values of electric field and conduction current density. In a steady-state or transient electric analysis, the Joule heat is calculated as:

(5–152)

where:

Q = Joule heat generation rate per unit volume (output as JHEAT)

In a harmonic electric analysis, the Joule heat generation value per unit volume is time-averaged over a one period and calculated as:

(5–153)

where:

Re = real component

{E}* = complex conjugate of {E}

The value of Joule heat can be used as heat generation load in a subsequent thermal analysis (LDREAD,HGEN).

In a transient electric analysis, the element stored electric energy is calculated as:

(5–154)

where:

W = stored electric energy (output as SENE)

In a harmonic electric analysis, the time-averaged electric energy is calculated as:

(5–155)

Electrostatic forces are determined using the nodal perturbation method (recommended) described in Nodal Perturbation Method or the Maxwell stress tensor described here. This force calculation is performed on surfaces of elements which have a nonzero face loading specified (MXWF on SF commands). For the 2-D application, this method uses extrapolated field values and results in the following numerically integrated surface integral:

(5–156)

where:

εo = free space permittivity (input as PERX on MP command)

T12 = Ex Ey

T21 = Ey Ex

n1 = component of unit normal in x-direction

n2 = component of unit normal in y-direction

s = surface area of the element face

3-D applications are an extension of the 2-D case.