Conductor current

The permissible current rating of a power cable can be derived from the expression for the temperature rise above ambient temperature.

Cables in air consider the temperature rise by solar radiation and the thermal resistance to ambient is for a continuous load.
Buried cables consider the temperature rise by other buried cables and heat sources/sinks as well as the effect of drying-out of soil by applying the ratio of the thermal resistivities of dry and moist soils. The thermal resistance to ambient may have transient load variation and correction factors due to backfill material.
For DC cables, the dielectric losses $W_d$ are zero and $\Delta\theta_d$ disappears and without induction the factors $\lambda_1$ and $\lambda_2$ are zero.
The current rating for a four-core low-voltage cable may be taken to be equal to the current rating of a three-core cable for the same voltage and conductor size having the same construction, provided that the cable is to be used in a three-phase system where the fourth conductor is either a neutral conductor or a protective conductor. When it is a neutral conductor, the current rating applies to a balanced load.
Where it is desired that moisture migration be avoided by limiting the temperature rise of the cable surface to not more than $\Delta\theta_x$, the corresponding rating shall be obtained from the equation below. However, depending on the value of $\Delta\theta_x$ this may result in a conductor temperature which exceeds the maximum permissible value. The current rating used shall be the lower of the two values obtained.
For transient calculations, $I_c$ is the constant steady-state current applied to cable prior to application of a step function, a cyclic load or prior to emergency loading.

*** These formulae are valid when the outer temperature $\theta_{omax}$ defines the load of a system.

Symbol

$I_{c}$

Unit

Formulas

$\sqrt{\frac{\theta_{c}-\theta_{a}-\Delta \theta_{d}-\Delta \theta_{sun}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} \left(T_{4i}+T_{4ii}+T_{4iii}\right)\right)+n_{cc} \lambda_{4} \left(\frac{T_{4ii}}{2}+T_{4iii}\right)\right)}}$	Cables in air, in riser IEC 60287
$\sqrt{\frac{\theta_{c}-\theta_{a}+\left(v_{4}-1\right) \Delta \theta_{x}-v_{4} \Delta \theta_{p}-\Delta \theta_{d}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} \left(T_{4i}+T_{4ii}+T_{4\mu} v_{4}\right)\right)+n_{cc} \lambda_{4} \left(\frac{T_{4ii}}{2}+T_{4\mu} v_{4}\right)\right)}}$	Cables buried
$\sqrt{\frac{\theta_{c}-\theta_{de}-\Delta \theta_{d}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} \left(T_{4i}+T_{4ii}\right)\right)\right)}}$	cables in tunnel
$\sqrt{\frac{\theta_{c}-\theta_{a}-\Delta \theta_{d}-\Delta \theta_{0t}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} \left(T_{4i}+T_{4ii}+T_{4t}\right)\right)\right)}}$	Cables in tunnel (IEC 60287-2-3)
$\sqrt{\frac{\theta_{c}-\theta_{t}-\Delta \theta_{d}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} \left(T_{4i}+T_{4ii}+T_{4iii}\right)\right)\right)}}$	Cables in channel (Heinhold)
$\sqrt{\frac{\theta_{c}-\theta_{at}-\Delta \theta_{d}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} \left(T_{4i}+T_{4ii}+T_{4iii}\right)\right)\right)}}$	Cables in trough (air-filled)
$\sqrt{\frac{\theta_{c}-\theta_{a}-\Delta \theta_{d}-\Delta \theta_{p}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} T_{4iii}\right)\right)}}$	Cables subsea
$\sqrt{\frac{\theta_{c}-\theta_{e}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) n_{ph} T_{3}\right)}}$	Cables in riser
$I_{c} F_{red}$	With reduction (derating) factor
$\sqrt{\frac{\theta_{omax}-\theta_{a}-W_{d} n_{cc} T_{4iii}-\Delta \theta_{sun}}{R_{c} n_{cc} T_{4iii} \left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}\right)}}$	*** Cables in air, in riser IEC 60287
$\sqrt{\frac{\theta_{omax}-\theta_{a}+\left(v_{4}-1\right) \Delta \theta_{x}-v_{4} \Delta \theta_{p}-W_{d} n_{cc} T_{4ss} v_{4}}{R_{c} n_{cc} T_{4\mu} v_{4} \left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}\right)}}$	*** Cables buried
$\sqrt{\frac{\theta_{omax}-\theta_{a}-W_{d} n_{cc} T_{4t}-\Delta \theta_{0t}}{R_{c} n_{cc} T_{4t} \left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right)}}$	*** Cables in tunnel (IEC 60287-2-3)
$\sqrt{\frac{\theta_{omax}-\theta_{t}-W_{d} n_{cc} T_{4iii}}{R_{c} n_{cc} T_{4iii} \left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right)}}$	*** Cables in channel (Heinhold)
$\sqrt{\frac{\theta_{omax}-\theta_{at}-W_{d} n_{cc} T_{4iii}}{R_{c} n_{cc} T_{4iii} \left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right)}}$	*** Cables in trough (air-filled)
$\sqrt{\frac{\theta_{omax}-\theta_{a}-\Delta \theta_{d}-\Delta \theta_{p}}{R_{c} n_{cc} \frac{1}{U_{OHTC} \pi D_{ext}} \left(1+\lambda_{1}+\lambda_{2}\right)}}$	*** Cables subsea
$\sqrt{\frac{\theta_{omax}-\theta_{a}-\Delta \theta_{d}-\Delta \theta_{sun}}{R_{c} n_{cc} T_{4iii} \left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}+\lambda_{4}\right)}}$	*** Cables in riser
$\sqrt{\frac{\theta_{c}-\theta_{a}+\left(v_{4}-1\right) \Delta \theta_{x}-v_{4} \Delta \theta_{p}-\Delta \theta_{d}}{R_{c} \left(T_{1}+n_{ph} \left(1+\lambda_{1}\right) T_{2}+\left(1+\lambda_{1}+\lambda_{2}+\lambda_{3}\right) \left(n_{ph} T_{3}+n_{cc} T_{4\mu} v_{4}\right)+n_{cc} \lambda_{4} \left(\frac{T_{4ii}}{2}+T_{4\mu} v_{4}\right)\right)}}$	cables in duct with bentonite filling and cyclic