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Conductor current

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

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

Symbol
$I_c$
Unit
A
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
$\sqrt{\frac{\theta_c-\theta_a-\Delta \theta_d-\Delta \theta_{0t}}{R_c \left(T_1+n_{ph} \left(1+\lambda_1\right) T_2+n_{ph} \left(1+\lambda_1+\lambda_2+\lambda_3\right) \left(T_3+T_{4t}\right)\right)}}$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
$\sqrt{\frac{\theta_c-\theta_a-\Delta \theta_d-\Delta \theta_p}{R_c \left(T_1+n_{ph} \left(1+\lambda_1\right) T_2+\left(1+\lambda_1+\lambda_2+\lambda_3\right) \left(n_{ph} T_3+n_{cc} T_{4iii}\right)\right)}}$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
$\sqrt{\frac{\theta_{omax}-\theta_a-W_d n_{cc} T_{4t}-\Delta \theta_{0t}}{R_c n_{ph} T_{4t} \left(1+\lambda_1+\lambda_2+\lambda_3\right)}}$*** 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
$\sqrt{\frac{\theta_{omax}-\theta_a-\Delta \theta_d-\Delta \theta_p}{R_c n_{cc} \frac{1}{U_{OHTC} \pi D_{ext}} \left(1+\lambda_1+\lambda_2\right)}}$*** 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 buried in ducts with bentonite filling and transient loading
Related
$D_{ext}$
Diameter external [m]
$\Delta \theta_{0t}$
Air temperature increase [K]
$\Delta \theta_d$
Temperature rise dielectric losses [K]
$\Delta \theta_p$
Temperature rise by other buried objects [K]
$\Delta \theta_{sun}$
Temperature difference conductor—enclosure [K]
$\Delta \theta_x$
Critical soil temperature rise [K]
$F_{red}$
Derating factor
$\lambda_1$
Loss factor shield (screen/sheath)
$\lambda_2$
Loss factor armour
$\lambda_3$
Loss factor steel pipe pipe-type cable
$\lambda_4$
Loss factor magnetic duct
$n_{cc}$
Number of conductors combined
$n_{ph}$
Number of phases in a cable
$\pi$
Archimedes' constant $\pi$
$R_c$
Electrical resistance conductor [$\Omega$/m]
$T_1$
Thermal resistance conductor—sheath [K.m/W]
$T_2$
Thermal resistance armour bedding [K.m/W]
$T_3$
Thermal resistance jacket [K.m/W]
$T_{4i}$
Thermal resistance medium in the duct [K.m/W]
$T_{4ii}$
Thermal resistance duct wall [K.m/W]
$T_{4iii}$
Thermal resistance ambient [K.m/W]
$T_{4\mu}$
Thermal resistance ambient [K.m/W]
$T_{4ss}$
Thermal resistance steady-state [K.m/W]
$T_{4t}$
Equivalent thermal resistance for tunnel [K.m/W]
$\theta_a$
Ambient temperature [°C]
$\theta_{at}$
Air temperature with load [°C]
$\theta_c$
Temperature conductor [°C]
$\theta_{de}$
Temperature duct outer surface [°C]
$\theta_e$
External temperature object [°C]
$\theta_{omax}$
Max. temperature outer surface [°C]
$\theta_t$
Temperature wall (inner) [°C]
$U_{OHTC}$
Overall heat transfer coefficient [W/(K.m$^2$)]
$v_4$
Ratio thermal resistivity dry/moist soil
$W_d$
Dielectric losses (phase) [W/m]
Used in
$\Delta W$
Incremental heat generated [W]
$I_{ar}$
Induced circulating current armour [A]
$I_{c,LF}$
Conductor root mean square current [A]
$I_{c,peak}$
Permissible peak cyclic load current [A]
$I_{c,sum}$
Total current for all parallel systems [A]
$I_{EMF}$
Phase current for EMF calculation [A]
$I_{ka}$
Complex conductor current a [A]
$I_{kb}$
Complex conductor current a [A]
$I_{kc}$
Complex conductor current c [A]
$I_s$
Induced circulating current shield [A]
$I_{sp}$
Induced circulating current pipe [A]
$L_{crit}$
Critical length [km]
$S_G$
Apparent power generator-side [kVA]
$W_c$
Conductor losses (phase) [W/m]
$W_{encl}$
Ohmic losses enclosure [W/m]
$W_h$
Heat generated by external object [W/m]
$W_{sar}$
Total loss in shield and magnetic armour (phase) [W/m]