This article describes legacy postprocessor

Integral quantities in DC and Transient magnetics

Generally the integral quantities of interest in magnetic analysis are: mechanical force and torque, magnetic flux and flux linkage, magnetomotive force (MMF), magnetic field energy.
The following notations are used in formulas:

B - magnetic flux density vector;
H - vector of magnetic field strength;
A - z-component of magnetic vector potential;
B(H) - magnetization curve of a ferromagnetic, that assumed to be isotropic.

Name,
ActiveField constant

Formula and Description

Mechanical force

qfInt_MaxwellForce

F = 1/2·_s∮(H·(n·B) + B·(n·H) - n·(H·B))ds

Total magnetic force acting on bodies contained in a particular volume, where integral is evaluated over the boundary of the volume.

Mechanical torque

qfInt_MaxwellTorque

T = 1/2·_s∮([r×H]·(n·B) + [r×B]·(n·H) - [r×n]·(H·B))ds

Total torque of magnetic forces acting on bodies contained in a particular volume. where r is a radius vector of the point of integration.
The torque vector is parallel to z-axis in the planar case, and is identically equal to zero in the axisymmetric one. The torque is considered relative to the origin of the coordinate system. The torque relative to any other arbitrary point can be obtained by adding extra term of [F×r₀], where F is the total force and r₀ is the radius vector of the point.

Flux linkage per one turn

qfInt_FluxLinkage

Ψ = 1/S_c· _{s_c}∫A·ds	for planar case
Ψ = 2π/S_c· _{s_c}∫rA·ds	for axisymmetric case
The integral has to be evaluated over a cross section of the coil, and S_c is the area of the cross section.

Magnetomotive force

qfInt_KGrad_t_dl

F = _L∫ (H·t)dl

Magnetomotive force is a line integral around the contour of magnetic field strength.
According Ampere's law the magnetomotive force around a closed line is equal to total current through the contour.

Magnetic flux

qfInt_Grad_n_ds

Φ = _s∫ (B·n)ds

Magnetic flux over the surface defined by the contour.

Magnetic field energy

qfInt_MagneticEnergy

W = 1/2·_v∫H·Bdv	linear case
W = _v∫ ( ^{^B}∫₀H(B' )dB' ) dv	nonlinear case

Magnetic field co-energy

qfInt_MagneticCoenergy

W = _v∫ ( ^{^H}∫₀B(H' )dH' ) dv	nonlinear case
For linear problem the co-energy is equal to the magnetic energy.

Linearized field energy

qfInt_ElectrostaticEnergy

W_lin = 1/2·_v∫H·Bdv	nonlinear case
For linear case the linearized energy is equal to the ordinary magnetic energy.

Surface energy

qfInt_GradKGrad_n_ds

W_s = 1/2·_s∫ (B·H)ds

The (B·H) product is integrated is over a surface defined by the contour.

Average surface potential

qfInt_Potential_ds

A_s = 1/S·_s∫ A·ds

Average volume potential

qfInt_Potential_dv

A_v = 1/V·_v∫ A·dv

Average volume flux density

qfInt_Grad_dv

B_a = 1/V·_v∫ B·dv

Average volume strength

qfInt_KGrad_dv

H_a = 1/V·_v∫ H·dv

Mean square flux

qfInt_Grad2_dv

B_a² = 1/V·_v∫ B²·dv

Mean square strength

qfInt_KGrad2_dv

H_a² = 1/V·_v∫ H²·dv

Line integral of flux density

qfInt_KGrad_t_dl

x = _L∫ (B·t)dl

Line integral over the contour of magnetic flux densisy.

Surface integral of strength

qfInt_Grad_n_ds

x = _s∫ (H·n)ds

With transient problems only:

Total current

qfInt_Jtotal

I = _{s_c}∫ j·ds

Electric current through a particular surface. The integral is evaluated over a cross section of the coil, and S_C is the area of the cross section. j - is a total current density in z-direction.

External current

qfInt_Jextern

I_ext = _{s_c}∫ j_ext·ds

External current through a particular surface. j_ext - density of external current.

Eddy current

qfInt_Jeddies

I_eddy = _{s_c}∫ j_eddy·ds

Eddy current through a particular surface. j_eddy - density of eddy current.

Joule heat

qfInt_Power

P = _v∫ j²/g·dv

Joule heat power in a volume. g - electric conductivity, j - total current density.