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Integral Quantities in Transient Electric Analysis

For the transient electric analysis, the most interesting integral values are: active and reactive current through particular surface, Joule heat, mechanical force and torque, field energy.

The following notations are used in formulas:

E – vector of electric field;
D – vector of electric displacement,
j_A – vector of active (conductive) current density,
j_RE – vector of reactive (displacement) current density,
U – voltage.

Name,
ActiveField constant

Formula and Description

Electric Charge

qfInt_KGrad_n_ds

Q_s = _s∮ (D·n)ds

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

Active current through a given surface

qfInt_Jactive

I_A = _s∫ (j_A·n)ds

Active (Ohmic) electric current through a particular surface.

Reactive current through a given surface

qfInt_Jreactive

I_RE = _s∫ (j_RE·n)ds

Reactive (displacement) electric current through a particular surface.

Active power produced in a volume

qfInt_PowerActive

P_A = _v∫ (E·j_A)dv

Joule heat power produced in a particular volume.

Reactive power produced in a volume

qfInt_PowerReactive

P_RE = _v∫ (E·j_RE)dv

Reactive power produced in a particular volume.

Mechanical force

qfInt_MaxwellForce

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

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

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

Electric field energy in a particular volume.

Surface energy

qfInt_GradKGrad_n_ds

W_S = 1/2·_s∫ (E·D)ds

Potential difference

qfInt_Grad_t_dl

ΔU = _L∫ (E·t)dl

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

Average volume potential

qfInt_Potential_dv

U_V = 1/V·_v∫ U·dv

Average volume strength

qfInt_Grad_dv

E_a = 1/V·_v∫ E·dv

Average electric field strength vector in a particular volume.

Average volume displacement

qfInt_KGrad_dv

D_a = 1/V·_v∫ D·dv

Average electric displacement vector in a particular volume.

Mean square strength

qfInt_Grad2_dv

E_a² = 1/V·_v∫ E²·dv

Mean square dispacement

qfInt_KGrad2_dv

D_a² = 1/V·_v∫ D²·dv

Line integral of displacement

qfInt_KGrad_t_dl

x = _L∫ (D·t)dl