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Volumetric density of dielectric losses

Refernce is made to Zbigniew Nadolny 'Electric Field Distribution and Dielectric Losses in XLPE Insulation and Semiconductor Screens of High-Voltage Cables', 2022 ().

Symbol
$w_{d}$
Unit
W/m$^3$
Formulas
${E_{Fi}}^2 \omega \epsilon_{0} \epsilon_{i} \mathrm{tan} \delta$General equation (Nadolny2022)
${E_{Fi}}^2 \kappa_{i}$General equation (Ohm's law)
$\frac{{\delta_{i}}^2 {U_{o}}^2 \left(\frac{r}{r_{osc}}\right)^{2\left(\delta_{i}-1\right)}}{{r_{osc}}^2 \left(1-\left(\frac{r_{isc}}{r_{osc}}\right)^{\delta_{i}}\right)^2} \kappa_{i} e^{\alpha_{i} \theta_{i}+\gamma_{i} E_{Fi}}$rewritten using $\alpha$ and $\kappa$
Related
$\alpha_{i}$
Temperature coefficient of conductivity insulation material [1/K]
$\delta_{i}$
Electrical thickness of insulation material [m]
$E_{Fi}$
Electrical field strength [kV/mm]
$\epsilon_{0}$
Vacuum permittivity [F/m]
$\epsilon_{i}$
Relative permittivity insulation material
$\gamma_{i}$
Electrical field coefficient insulation material [mm/kV]
$\kappa_{i}$
Electrical conductivity insulation material [S/m]
$\omega$
Angular frequency [rad/s]
$r_{isc}$
Radius above the inner semi-conducting layer [mm]
$r_{osc}$
Radius over capacitive insulation layers [mm]
$\theta_{i}$
Temperature of insulation [°C]
$U_{o}$
Operating voltage [kV]