# Thermal resistance between one conductor and sheath

All three-core cables require fillers to fill the space between insulated cores and the belt insulation or a sheath. The fillers of extruded cables usually have higher thermal resistivities than the insulation, unlike paper-insulated cables In a paper by G. Anders from 1998, a formula was developed to take into account the different thermal resistivities. This formula has been included in the new editions of the IEC standards.

For three-core cables with a touching metallic screen made of copper tapes each core the thermal resistance of the insulation is obtained in two steps:

1. First, the cables are considered as belted cables for which $t_1/t$ = 0.5.
2. Second, the resulting $T_1$ is multiplied by a factor $K$, called the screening factor.
In a paper by G. Anders from 1999, a more precise formula was developed. This formula has not been included in the IEC, but is used by Cableizer.

Type of screen/sheath of multi-core cables

• SS: with separate screen and separate sheaths
• SC: with separate screen and common sheath
• CC: with common screen and common sheath

For calculation of the cyclic and emergency current rating of cables acc. IEC 60853, the multi-core cable is replaced by an equivalent single-core construction dissipating the same total conductor losses. The space between the equivalent single-core conductor and the sheath is considered to be completely occuied by insulation (for oil-filled cables, this space is filled partly by the total volume of oil in the ducts and the remainder is oil-impregnated paper).

Symbol
$T_1$
Unit
K.m/W
Formulae
 $\frac{\rho_i}{2\pi} \ln\left(1+\frac{2t_1}{d_c}\right)$ Single-core cables $\frac{\rho_i}{2\pi} G_1$ Multi-core cables with sector-shaped conductors / multi-core cables type SS with sheath $\frac{K \rho_i}{2\pi} G_1$ screened three-core cables $0.89T_1+K \left(\frac{\sqrt{3} \rho_f}{\rho_i}-0.12d_c+2.25\right) e^{0.13+\frac{t_1}{d_c}-7K}$ more accurate formula for screened cables with round conductors $\frac{\rho_i}{2\pi} G_1+0.031\left(\rho_f-\rho_i\right) e^{\frac{0.67t_1}{d_c}}$ unscreened multi-core cables with round conductors $1.07T_1$ part-metallic sheathed single-core cables in trefoil formation, up to 35 kV $1.16T_1$ part-metallic sheathed single-core cables in trefoil formation, from 35 to 150 kV $\frac{\theta_c-\theta_{encl}}{W_{conv,ce}+W_{rad,ce}}$ PAC/GIL $0.385\rho_i \frac{2t_i}{d_c+2t_i}$ three-core oil-filled cables with circular conductors and metallized paper core screens and circular oil ducts between the cores $0.35\rho_i \left(0.923-\frac{2d_c}{d_c+2t_i}\right)$ three-core oil-filled cables with circular conductors and metal tape core screens and circular oil ducts between the cores $\frac{475}{{D_{sc}}^{1.74}} \left(\frac{0.5\left(\frac{D_{shb}+D_{sh}-2t_{sh}}{2}-2.16D_{sc}\right)}{D_{sc}}\right)^{0.62}+\frac{\rho_i}{2\pi} \ln\left(\frac{d_c-2t_{sc}}{d_c}\right)$ three-core oil-filled cables with circular conductors, metal tape core screens, without fillers and oil ducts, having a copper woven fabric tape binding the cores together and a corrugated aluminium sheath $\frac{\rho_i}{2\pi} \ln\left(\frac{D_{ins}}{d_c}\right)$ 4-loop
Related
$d_c$
$D_{ins}$
$D_{sc}$
$D_{sh}$
$D_{shb}$
$\rho_f$
$\rho_i$
$t_1$
$t_i$
$t_{sc}$
$t_{sh}$
$\theta_c$
$\theta_{encl}$
$W_{conv,ce}$
$W_{rad,ce}$
Used in
$I_c$
$Q_{B,i}$
$Q_{B,s}$
$T_A$
$T_B$
$T_d$
$T_{eq}$
$T_{int}$
$T_{tot}$
$\theta_{ar}$
$\theta_{c,z}$
$\theta_e$
$\theta_s$
$\theta_{sc}$
$\theta_{sh}$
$\theta_{sp}$