When metallic cable sheaths or armour of three-phase single core cables are unbonded or bonded at only one end, a voltage is induced at the unbonded ends. Should both ends be bonded, a circulating current flows in the sheath or armour as a result of the induced voltage.

This note presents a method to calculate the induced voltage and circulating currents. While presented for cable sheaths, the method is equally applicable to armouring.

## INDUCED VOLTAGE

The inductive voltage induced per unit length in the sheath of a single core cable is given by:

with and

For a three phase set of cables this is expanded to[2]:

, ,

with and

and

## CIRCULATING CURRENTS

Circulating current is made up of two components:

- Capacitive - the conductor and sheath, coupled with the insulation (dielectric), act as a capacitor. Capacitive current flows from the conductor into the sheath and to the ground.
- Inductive - sheaths bonded at both ends, transformer coupling between the sheath and conductor results in a sheath current flow.

### CAPACITIVE CURRENT

For a single core cable, the capacitance is given by:

Leakage current in the insulation can be ignored and the capacitive current for each phase per unit length is given by:

, ,

For sheaths bonded to ground at one end only, the total capacitive current is given by multiplying the above by the total cable length.

For sheaths bonded to ground at both ends the, the capacitive current can flow in two directions towards the ground. For simplicity, it can be assumed that the current divides equally (Is1/2 for example). This can be added (or subtracted) to the inductive current to give an estimate of maximum sheath circulating current.

#### INDUCTIVE CURRENT

For sheaths bonded at only one end, no inductive current can flow. The inductive circulating current for sheaths bonded at both ends is given by dividing the induced sheath voltage by the impedance:

The sheath resistance can be estimated from:

with

The sheath reactance Xs, for cables bonded at both ends, depends on the configuration and can be approximated by[3]:

trefoil -

flat, no transposition -

flat, regular transposition -

Note: the calculation of sheath voltage and currents is complex and is affected by the conductor current, the physical construction of the cable, installation arrangements and deliberate or accidental parallel current paths. Given, this complexity the results obtained by the calculation method presented, should be considered as indicative of magnitude rather than a measurable value.

## CROSS BONDING AND TRANSPOSITION

To reduce sheath induced voltages and circulating current, cables are often cross bonded and transposed. Figure 1 illustrates the cross bonding and transposition of cables.

By cross bonding as shown, over three sections the induced voltage in each section is 120° phase shifted. Summation of the phase shifted voltages reduces the overall induced voltage and circulating currents.

Figure 1. Cross Bonding and Transposition of cables

For balanced cables in a trefoil, the induced sheath currents are symmetrical, and cross bonding only of the sheaths is required. For flat formation, the induced voltages vary across phases and to balance out the induced voltages it is necessary to transpose (rearrange) the cables.

The calculation of induced voltages and sheath currents described can be extended to cover differing arrangements of cross bonding and transposing of cables.

## REFERENCES AND SYMBOLS

### REFERENCES

[1] Moore G. Electric cables handbook/BICCCables. Oxford: BSCI; 2000.

[2] Chen, Wu, Cheng, Yan. Sheath circulating current calculations and measurements of underground power cables. Xi'an Jiaotong University;

[3] IEC 60287-1. Electric cables - calculation of the current rating, part 1-1: current rating equations (100% load factor) and calculation of losses - general, IEC; 2006

### SYMBOLS

*As* - cross-sectional area of sheath, m^{2}

*C* - capacitance, F.m^{-1}

*d _{c}* - diameter of the conductor, m

*d*- inside diameter of the sheath, m

_{s}*f*- frequency, Hz

*I*- cable conductor current, A

*I*,

_{1}*I*,

_{2}*I*- conductor phase current of L1, L2 and L3, A

_{3}*I*,

_{s1}*I*,

_{s2}*I*- sheath phase current of L1, L2 and L3, A

_{s3}*L*- inductance of sheath, H.m

_{s}^{-1}

*R*- resistance of the sheath, Ω.m

_{s}^{-1}

*S*- distance between cable centres, m

*S*- distance of cable centres between L1 and L2, m

_{12}*S*- distance of cable centres between L2 and L3, m

_{23}*S*- distance of cable centres between L3 and L1, m

_{31}*t*- the thickness of sheath, m

_{s}*X*,

_{1}*X*,

_{3}*X*,

_{a}*X*- reactance formulas; sheath induced voltage, Ω.m

_{b}^{-1}

*X*- mutual reactance between conductor and sheath, Ω/unit length

_{m}*X*- sheath reactance, Ω.m

_{s}^{-1}

*U*- sheath voltage, V

_{s}*U*,

_{1}*U*,

_{2}*U*- phase voltage of L1, L2 and L3, V

_{3}*U*,

_{s1}*U*,

_{s2}*U*- sheath inductive phase voltage of L1, L2 and L3, V

_{s3}*ɑ*- temperature coefficient of resistivity, per °C

_{s}*η*- sheath/conductor temperature ratio (typically 0.7-0.8)

*ϵ*- permittivity of freespace = 8.854187819x10

_{o}^{-12}F/m

*ϵ*- relative permittivity of a dielectric

_{r}*θ*- service temperature of conductor, °C

*ρ*- resistivity of sheath, Ω.m

_{s}*ω*- angular frequency = 2πf, s

^{-1}