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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:

U – voltage;
E – vector of electric field;
D – vector of electric displacement;
J – vector of active (conductive) current density;
J_d – vector of reactive (displacement) current density;
J_t – vector of total (apparent) current density.

Name, ActiveField constant	Formula and Description
Electric Charge qfInt_KGrad_n_ds	\[ Q_s = \oint(\mathbf D ·\mathbf n)ds \] 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	\[ I_{Active} = \int(\mathbf J ·\mathbf n)ds \] Active (Ohmic) electric current through a particular surface.
Reactive current through a given surface qfInt_Jreactive	\[ I_{Reactive} = \int(\mathbf J_d ·\mathbf n)ds \] Reactive (displacement) electric current through a particular surface.
Apparent current through a given surface qfInt_Japparent	\[ I_{Apparent} = \int(\mathbf J_t ·\mathbf n)ds \] Apparent (total) electric current through a particular surface.
Active power produced in a volume qfInt_PowerActive	\[ P_{Active} = \int(\mathbf E ·\mathbf J)dv \] Joule heat power produced in a particular volume.
Electric field energy qfInt_ElectrostaticEnergy	\[ W = \frac{1}{2}\int(\mathbf E ·\mathbf D)dv \] Electric field energy in a particular volume.
Mechanical force qfInt_MaxwellForce	\[ \mathbf f = \frac{1}{2}\oint\left( \mathbf E(\mathbf n · \mathbf D) + \mathbf D(\mathbf n · \mathbf E) - \mathbf n(\mathbf E · \mathbf D) \right) ds \] 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	\[ \mathbf T = \frac{1}{2}\oint\left( (\mathbf r × \mathbf E)(\mathbf n · \mathbf D) + (\mathbf r × \mathbf D)(\mathbf n · \mathbf E) - (\mathbf r × \mathbf n)(\mathbf E · \mathbf D) \right) ds \] 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	\[ U = \frac{1}{S}\int U ds \]
Average volume potential qfInt_Potential_dv	\[ U = \frac{1}{V}\int U dv \]
Root mean square value of the surface strength	\[ E = \sqrt {\frac{1}{S}\int \|\mathbf E\|^2 ds } \]
Root mean square value of the volume strength	\[ E = \sqrt {\frac{1}{V}\int \|\mathbf E\|^2 dv } \]
Root mean square value of the surface displacement	\[ D = \sqrt {\frac{1}{S}\int \|\mathbf D\|^2 ds } \]
Root mean square value of the volume displacement	\[ D = \sqrt {\frac{1}{V}\int \|\mathbf D\|^2 dv } \]