Integral quantities in AC magnetics

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For the AC magnetic field analysis, the most interesting integral values are: the total, eddy and external current, the mechanical force and torque, the magnetic flux and flux linkage, the magnetomotive force, the field energy.
The following notations are used in formulas:

A – z-component of complex magnetic vector potential;
B – complex vector magnetic flux density;
H – complex vector magnetic field intensity (field strength);
J_total, J_eddy, J_external – complex values of total, eddy and external current density.

Since AC magnetic problems' formulations use complex values that represent the real world quantities sinusoidally changing with time, the integral values might appear in the following different ways:

As a complex value, with amplitude and phase (e.g. current, flux linkage, magnetomotive force).
As a complex vector, with the endpoint sweeping an ellipse in any complete time period (e.g. induction, magnetic field strength). The characteristics of complex vectors are: the amplitude (per coordinate), the phase, and the polarization coefficient.
As a oscillated value (e.g. ohmic loss power, field energy, etc.) pulsing around its mean value with double frequency. The characteristics of oscillated values are: the mean value, the phase, and the pulsation amplitude.
As a oscillated vector (e.g. mechanical forces) with magnitude and direction varying around its mean value with double frequency. The characteristics of oscillated vectors are: the mean value (length, slant and coordinates), and the variation amplitude (used, for example, to estimate the mechanical force limit for a period).

Name, ActiveField constant	Formula and Description
Total current qfInt_Jtotal	\[ I = \int_{S_c} J_{total} ds, \] Complex value. 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`_total - is a total current density in z-direction.
External current qfInt_Jextern	\[ I_{external} = \int_{S_c} J_{external} ds, \] Complex value. External current through a particular surface.
Eddy current qfInt_Jeddies	\[ I_{eddy} = \int_{S_c} J_{eddy} ds, \] Complex value. Eddy current through a particular surface.
Magnetomotive force qfInt_KGrad_t_dl	\[ F = \int_L (\mathbf H · \mathbf t) dl, \] Complex value. According Ampere's law the magnetomotive force around a closed line is equal to total current through the contour.
Flux linkage per one turn qfInt_FluxLinkage	For planar case \[ Ψ = \frac{1}{S_c} \oint_{S_c} Ads \] For axisymmetric case \[ Ψ = \frac{1}{S_c} 2π\oint_{S_c} (rA)ds \] Complex value. The integral has to be evaluated over a cross section of the coil, and S_c is the area of the cross section.
Joule heat qfInt_Power	\[ P = \int \frac{J_{total}^2}{σ} dv, \] Oscillated value. Joule heat power in a particular volume. σ – electric conductivity of the media.
Core loss qfInt_Steinmetz	\[ P_{core} = \int \left( k_hfB^2 + k_cf^2B^2 + k_e(f·B)^{1.5} \right ) dv \] Average power of core loss, which is the sum of the hysteresis losses, eddy current losses and excess magnetic losses (not covered by first two loss types). `B` - magnitude of the module of the flux density vector per period, `f` - problem frequency, `k`_h, `k`_c, `k`_e - volume power loss coefficients for specific magnetic material.
Magnetic field energy qfInt_MagneticEnergy	\[ W = \frac{1}{2} \int \left (\mathbf H · \mathbf B \right )dv \] Oscillated value. This formula is used for both linear and nonlinear cases.
Power flow qfInt_EnergyFlow	\[ P_S = \int \left (\mathbf S · \mathbf n \right )ds \] Oscillated value. Power flow through the given surface (Poynting vector flow). Here S = [E×H] is a Pointing vector.
Maxwell force qfInt_MaxwellForce	\[ \mathbf f = \frac{1}{2}\oint\left( \mathbf H(\mathbf n · \mathbf B) + \mathbf B(\mathbf n · \mathbf H) - \mathbf n(\mathbf H · \mathbf B) \right) ds \] Oscillated vector. Maxwell force acting on bodies contained in a particular volume. The integral is evaluated over the boundary of the volume, and n denotes the vector of the outward unit normal.
Maxwell torque qfInt_MaxwellTorque	\[ \mathbf T = \frac{1}{2}\oint\left( (\mathbf r × \mathbf H)(\mathbf n · \mathbf B) + (\mathbf r × \mathbf B)(\mathbf n · \mathbf H) - (\mathbf r × \mathbf n)(\mathbf H · \mathbf B) \right) ds \] Oscillated value. Maxwell force torque 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.
Lorentz force qfInt_LorentzForce	\[ \mathbf f = \int [\mathbf J_{total} × \mathbf B] dv\] Oscillated vector. The Lorentz force acting on conductors contained in a particular volume.
Lorentz torque qfInt_LorentzTorque	\[ \mathbf T = \int [\mathbf r × [\mathbf J_{total} × \mathbf B]] dv\] Oscillated value. The Lorentz force torque acting on bodies contained in a particular volume. The torque is considered relative to the origin of the coordinate system.

Note. The Maxwell force incorporates both the force acting on ferromagnetic bodies and Lorentz force, which acts only on conductors.
If the first component is negligible or is not considered, we recommend calculating the electromagnetic force as Lorentz force. Its precision is less sensitive to the contour path, and you can simply select conductors via block selection to calculate the force.
With Maxwell force, this method leads to very rough results, and you are recommended to avoid coinciding of your contour parts and material boundaries as described in Calculating integrals.

Name, ActiveField constant	Formula and Description
Average surface potential qfInt_Potential_ds	\[ A = \frac{1}{S}\int A ds \] Complex value.
Average volume potential qfInt_Potential_dv	\[ A = \frac{1}{V}\int A dv \] Complex value.
Root mean square value of the surface flux density	\[ B = \sqrt {\frac{1}{S}\int \|\mathbf B\|^2 ds } \] Complex value.
Root mean square value of the volume flux density	\[ B = \sqrt {\frac{1}{V}\int \|\mathbf B\|^2 dv } \] Complex value.
Root mean square value of the surface strength	\[ H = \sqrt {\frac{1}{S}\int \|\mathbf H\|^2 ds } \] Complex value.
Root mean square value of the volume strength	\[ H = \sqrt {\frac{1}{V}\int \|\mathbf H\|^2 dv } \] Complex value.