Integral quantities in DC conduction
Generally the integral quantities of interest in DC conduction analysis are: electric current, Joule heat.
The following notations are used in formulas:
E – electric field strength;
D – vector of electric flux density (electric displacement);
j – vector of current density;
ρ – electric resistivity;
U – scalar electric potential.
Name, |
Formula and Description |
Current through a surface qfInt_KGrad_n_ds |
I = s∫ (j·n)ds Electric current through a particular surface. |
Joule heat in a volume qfInt_GradKGrad_dv |
W = v∫ (E·j)dv Power losses in a particular volume |
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. |
Surface Joule heat qfInt_GradKGrad_n_ds |
WS = s∫ (E·j)ds |
Average surface potential qfInt_Potential_ds |
US = 1/S·s∫ U·ds |
Average volume potential qfInt_Potential_dv |
Ua = 1/V·v∫ U·dv |
Average volume strength qfInt_Grad_dv |
Ea = 1/V·v∫ E·dv |
Average volume current density qfInt_KGrad_dv |
ja = 1/V·v∫ j·dv |
Mean square strength qfInt_Grad2_dv |
Ea2 = 1/V·v∫ E2·dv |
Mean square current density qfInt_KGrad2_dv |
ja2 = 1/V·v∫ j2·dv |
Surface integral of strength qfInt_Grad_n_ds |
x = s∫ (E·n)ds |
Line integral of current density qfInt_KGrad_t_dl |
x = L∫ (j·t)dl |