Integral quantities in AC conduction

Help contents

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:

U – complex value of electric potential;
E – complex vector of electric field strength;
D – complex vector of electric displacement;
J – complex vector of active current density;
J_d – complex vector of reactive current density;
J_t – complex vector of reactive current density.

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
Electric charge qfInt_Grad_n_ds	\[ Q_s = \oint(\mathbf D ·\mathbf n)ds \] 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.
Active current through a given surface qfInt_Jactive	\[ I_{Active} = \int(\mathbf J ·\mathbf n)ds \] Complex value. Active electric current through a particular surface.
Reactive current through a given surface qfInt_Jreactive	\[ I_{Reactive} = \int(\mathbf J_d ·\mathbf n)ds \] Complex value. Reactive electric current through a particular surface.
Apparent current through a given surface qfInt_Japparent	\[ I_{Apparent} = \int(\mathbf J_t ·\mathbf n)ds \] Complex value. Apparent electric current through a particular surface.
Active power produced in a volume qfInt_PowerActive	\[ P_{Active} = \int(\mathbf E ·\mathbf J)dv \] Oscillated value. Joule heat power produced in a particular volume.
Electric field energy qfInt_ElectrostaticEnergy	\[ W = \frac{1}{2}\int(\mathbf E ·\mathbf D)dv \] Oscillated value. Electric field energy in a particular volume.
Mechanical force qfInt_MaxwellForce	\[ \mathbf f = \frac{1}{2}\oint\left( \mathbf E(\mathbf n · \mathbf D) + \mathbf D(\mathbf n · \mathbf E) - \mathbf n(\mathbf E · \mathbf D) \right) ds \] 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	\[ \mathbf T = \frac{1}{2}\oint\left( (\mathbf r × \mathbf E)(\mathbf n · \mathbf D) + (\mathbf r × \mathbf D)(\mathbf n · \mathbf E) - (\mathbf r × \mathbf n)(\mathbf E · \mathbf D) \right) ds \] 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.

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