This article describes legacy postprocessor

Integral quantities in AC conduction

For the AC conduction analysis, the most interesting integral values are: active, reactive and apparent current through particular surface, Joule heat, mechanical force and torque, field energy.
The following notations are used in formulas:

E – complex vector of electric field strength;
D – complex vector of electric displacement;
j_A – complex vector of active current density;
j_RE – complex vector of reactive current density;
j_APP – complex vector of reactive current density;
U – complex value of electric potential;

Since AC conduction 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, potential).
As a complex vector, with the endpoint sweeping an ellipse in any complete time period (e.g. current density, 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

Active current through a given surface

qfInt_Jactive

I_A = _s∫ (j_A·n)ds

Complex value. Active electric current through a particular surface.

Reactive current through a given surface

qfInt_Jreactive

I_RE = _s∫ (j_RE·n)ds

Complex value. Reactive electric current through a particular surface.

Apparent current through a given surface

qfInt_Japparent

I_APP = _s∫ (j_APP·n)ds

Complex value. Apparent electric current through a particular surface.

Active power produced in a volume

qfInt_PowerActive

P_A = _v∫ (E·j_A)dv

Oscillated value. Joule heat power produced in a particular volume.

Reactive power produced in a volume

qfInt_PowerReactive

P_RE = _v∫ (E·j_RE)dv

Oscillated value. Reactive power produced in a particular volume.

Apparent power produced in a volume

qfInt_PowerApparent

P_APP = _v∫ (E·j_APP)dv

Oscillated value. Apparent power produced in a particular volume.

Mechanical force

qfInt_MaxwellForce

F = 1/2·_s∮(E·(n·D) + D·(n·E) - n·(E·D))ds

Oscillated vector. Electric force acting on bodies contained in a particular volume. Evaluated by calculating of Maxwell stress tensor over volume�s bounding surface.

Mechanical torque

qfInt_MaxwellTorque

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

Oscillated value. Electric 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.

Electric field energy

qfInt_ElectrostaticEnergy

W = 1/2·_v∫ (E·D)dv

Oscillated value. Electric field energy in a particular volume.

Surface energy

qfInt_GradKGrad_n_ds

W_S = 1/2·_s∫ (E·D)ds

Oscillated value.

Potential difference

qfInt_Grad_t_dl

ΔU = _L∫ (E·t)dl

Complex value. The potential difference between the ending and started points of a contour can be calculated as a line integral over the contour of electric field strength.

Average surface potential

qfInt_Potential_ds

U_s = 1/S·_s∫ U·ds

Complex value.

Average volume potential

qfInt_Potential_dv

U_v = 1/V·_v∫ U·dv

Complex value.

Average volume strength

qfInt_Grad_dv

E_a = 1/V·_v∫ E·dv

Complex vector. Average electric field strength vector in a particular volume.

Average volume displacement

qfInt_KGrad_dv

D_a = 1/V·_v∫ D·dv

Complex vector. Average electric displacement vector in a particular volume.

Mean square strength

qfInt_Grad2_dv

E_a² = 1/V·_v∫ E²·dv

Oscillated value.

Mean square displacement

qfInt_KGrad2_dv

D_a² = 1/V·_v∫ D²·dv

Oscillated value.

Electric charge

qfInt_Grad_n_ds

Q_s = _s∫ (E·n)ds

Complex value. The total electric charge in a particular volume can be calculated as a flux of electric displacement over the volume�s closed boundary.

Line integral of displacement

qfInt_KGrad_t_dl

x = _L∫ (D·t)dl

Complex value.