When sizing cables, the heat generated by losses within any sheath or armour need to be evaluated. When significant, it becomes a factor to be considered in the sizing of cables. To understand how sheath and armour losses affect the sizing of cables, you can review the post: IEC 60287 Current Capacity of Cables - Rated Current.
This note looks at how to obtain the necessary loss factors for use within the IEC 60287 calculation. In addition, the loss factors quantify the ratio of losses in the sheath to total losses in all conductors and have application outside the IEC 60287 standard.
Any cable sheath (or screen) the loss λ1, consists of two components:
$${\lambda}_{1}={\lambda}_{1}{}^{\prime \text{}\prime}+{\lambda}_{1}{}^{\prime \text{}\prime}$$
The loss in armour is considered as only one component, λ_{2}.
Sheath and armour losses are only applicable to alternating current (a.c.) cables. The actual formula for calculation of sheath and armour loss depend on the installation and arrangement of cables. The tables below presents some of the common installation situations and are based on equations given in IEC 60287:
For installations bonded only at one point, circulating currents are not possible and the loss is zero. Except in the case of large segmental type conductors (see Some Special Cases below), eddy current loss λ1'', for single core cables can be ignored.
$${\lambda}_{1}{}^{\prime}\text{=}\frac{{\text{R}}_{\text{S}}}{\text{R}}\frac{\text{1}}{\text{1+}{\left(\frac{{\text{R}}_{\text{s}}}{\text{X}}\right)}^{\text{2}}}$$
λ_{11}′ - loss factor for the outer cable with the greater losses $${\lambda}_{11}{}^{\prime}=\frac{{R}_{s}}{R}\left[\frac{0.75{P}^{2}}{{R}_{s}{}^{2}+{P}^{2}}+\frac{0.25{Q}^{2}}{{R}_{s}{}^{2}+{Q}^{2}}+\frac{2{R}_{s}PQ{X}_{m}}{\sqrt{3}\left({R}_{s}{}^{2}+{P}^{2}\right)\left({R}_{s}{}^{2}+{P}^{2}\right)}\right]$$ λ_{12}′ - loss factor for the outer cable with the least losses $${\lambda}_{12}{}^{\prime}=\frac{{R}_{s}}{R}\left[\frac{0.75{P}^{2}}{{R}_{s}{}^{2}+{P}^{2}}+\frac{0.25{Q}^{2}}{{R}_{s}{}^{2}+{Q}^{2}}-\frac{2{R}_{s}PQ{X}_{m}}{\sqrt{3}\left({R}_{s}{}^{2}+{P}^{2}\right)\left({R}_{s}{}^{2}+{P}^{2}\right)}\right]$$ λ_{1m}′ - loss factor for the middle cable $${\lambda}_{1m}{}^{\prime}=\frac{{R}_{s}}{R}\frac{{Q}^{2}}{{R}_{s}{}^{2}+{Q}^{2}}$$ where: $$P=X+{X}_{m}$$ $$Q=X+\frac{{X}_{m}}{3}$$
Due to any sheath or screen surrounding all cores, the possibility of circulating current does not exist, and the λ1' loss can be ignored. Eddy current loss, λ1'' does need to be considered.
- round or oval conductors, R_{s} ≤ 100 µΩ.m^{-1} $${\lambda}_{1}{}^{\prime \text{}\prime}=\frac{3{R}_{s}}{R}\left[{\left(\frac{2c}{d}\right)}^{2}\frac{1}{1+{\left(\frac{{R}_{s}}{\omega}{10}^{7}\right)}^{2}}+{\left(\frac{2c}{d}\right)}^{4}\frac{1}{1+4{\left(\frac{{R}_{s}}{\omega}{10}^{7}\right)}^{2}}\right]$$
- round or oval conductors, R_{s} >100 µΩ.m^{-1} $${\lambda}_{1}{}^{\prime \text{}\prime}=\frac{3.2{\omega}^{2}}{R{R}_{s}}{\left(\frac{2c}{d}\right)}^{2}{10}^{-14}$$ - for sector shaped conductors (any R_{s}) $${\lambda}_{1}{}^{\prime \text{}\prime}=0.94\frac{{R}_{s}}{R}{\left(\frac{2{r}_{1}+t}{d}\right)}^{2}\frac{1}{1+{\left(\frac{{R}_{s}}{\omega}{10}^{7}\right)}^{2}}$$
Multiple the unarmoured cable factor by: $${\left[1+{\left(\frac{d}{{d}_{A}}\right)}^{2}\frac{1}{1+\frac{{d}_{A}}{\mu \delta}}\right]}^{2}$$
For armoured cables, the losses are estimated as shown.
$${\lambda}_{2}=\frac{0.62{\omega}^{2}{10}^{-14}}{R\text{}{R}_{A}}+\frac{3.82\text{}A\text{}\omega \text{}{10}^{-5}}{R}{\left[\frac{1.48{r}_{1}+t}{{d}_{A}{}^{2}+95.7A}\right]}^{2}$$
Sheath (Rs) or armour (RA) resistance - values used above are calculated at their operating temperature. The operating temperature (in °C) and resistance can be determined from:
$${\theta}_{sc}=\theta -\left({I}^{2}R+0.5{W}_{d}\right)\times {T}_{1}$$ - for any sheath $${\theta}_{ar}=\theta -\left\{\left({I}^{2}R+0.5{W}_{d}\right)\times {T}_{1}+\left[{I}^{2}R\left(1+{\lambda}_{1}\right)+{W}_{d}\right]\times n{T}_{2}\right\}$$ - for any armour $${R}_{s}={R}_{s20}\left[1+{\alpha}_{20}\left({\theta}_{sc}-20\right)\right]$$ - for the cable sheath $${R}_{A}={R}_{A20}\left[1+{\alpha}_{20}\left({\theta}_{ar}-20\right)\right]$$ - for the cable armour
Note: for calculation of the dielectric loss W_{d}, refer to our Dielectric loss in cables note.
Cable Reactance - for single core cables, where there is significant spacing between conductors, it is necessary to use the reactance in the calculating of circulating current loss. Accurate values for reactance can be obtained from cable manufacturers or by using software. Alternatively, the following equations can be used to estimate the reactance (Ω.m-1):
Single core cable reactance estimates (assume bonded at both ends)
$$X=2\omega {10}^{-7}\mathrm{ln}\left(\frac{2s}{d}\right)$$ - trefoil or flat without transposition $$X=2\omega {10}^{-7}\mathrm{ln}\left({2}^{3}\sqrt{2}\frac{s}{d}\right)$$ - flat with transposition $${X}_{m}=2\omega {10}^{-7}\mathrm{ln}\left(2\right)$$- mutual reactance of flat formation cables
Steel tape armour resistance - depending on how steel tape is wound, the resistance can be estimated as follows:
Transposing of cables (see image) is a technique to reduce the circulating currents within cable sheaths and consequently increase the rating of the cable.
By cross bonding the sheath the induced currents are in opposite directions, cancelling each other out and significantly improving the current rating of the cable. Transposing the cables ensures that the reactance balance out and aids in implementation.
At intermediate transposition points, over voltage devices are installed to protect the cable and personnel in the event of voltage build up during faults.
In practice, three minor sections (part between the cross bond) would from a major section (three full transpositions). It makes sense to do these at each joint point - at each cable drum length.
Transposition and cross bonding are normally carried out in link boxes.
Eddy current losses λ1'', are normally small relative to other losses and can be ignored for single core cables. This changes for large conductors, which are of a segmented construction. Under these conditions, the eddy current loss should be considered.
For this condition, the value of λ1'' is derived from the circulating current loss factor λ1' by:
$${\lambda}_{1}{}^{\prime \text{}\prime}={\lambda}_{1}{}^{\prime}\times \frac{4{M}^{2}{N}^{2}+{\left(M+N\right)}^{2}}{4\left({M}^{2}+1\right)\left({N}^{2}+1\right)}$$
where: $$M=N=\frac{{R}_{s}}{X}$$ - for cables in trefoil $$M=\frac{{R}_{s}}{X+{X}_{n}}\text{and}N=\frac{{R}_{s}}{X-\frac{{X}_{m}}{3}}$$ - for cable in flat formation
If the spacing if not maintained the same for the full cable route than the reactance will vary along the route. In instances such as these, an equivalent overall reactance can be calculated from:
$$X=\frac{{l}_{a}{X}_{a}+{l}_{b}{X}_{b}+\dots +{l}_{n}{X}_{n}}{{l}_{a}+{l}_{b}+\dots +{l}_{n}}$$
- where l_{a}, l_{b}, ... are the section lengths and X_{a}, X_{b}, ... are the reactance of each section