Cable Sheath Circulating Current Formulas
Sheath circulating current formulas for single-core cables bonded at both ends: the mutual reactance in trefoil, the sheath current and the sheath loss.
The formulas
| To find | Formula |
|---|---|
| Mutual reactance, trefoil | Xm = 2 × ω × 10⁻⁷ × ln 2 × 1000 Ω/km, with ω = 2π × f |
| Sheath current, both ends bonded | Is = I × Xm / √(Rs² + Xm²) |
| Sheath loss per cable | Ws = Is² × Rs, in W/km |
| Sheath current, single-point bonding | Is ≈ 0 (no closed loop) |
I is the conductor current, Rs the resistance of the sheath or screen in Ω/km at its working temperature and Xm the mutual reactance between a conductor and its sheath. The formula is for three single-core cables in trefoil with equal currents, where the spacing equals the sheath diameter, so the term ln(2s / d) becomes ln 2. For cables in flat formation the sheath currents differ from cable to cable, and the standard IEC 60287 gives the full method. The sheath current heats the cable, so it lowers the current rating, which is why long runs use single-point or cross bonding. The resistance and the current here are examples.
Worked example: 400 A in a trefoil group at 50 Hz, sheath resistance 0.3 Ω/km
- ω = 2π × 50 = 314.2 rad/s
- Xm = 2 × 314.2 × 10⁻⁷ × ln 2 × 1000 = 0.04355 Ω/km
- Is / I = 0.04355 / √(0.3² + 0.04355²) = 0.1437
- Is = 400 × 0.1437 = 57.47 A
- Ws = 57.47² × 0.3 = 990.7 W/km per cable
Sheath current share by sheath resistance (Xm as above)
| Sheath resistance | Is / I |
|---|---|
| 0.1 Ω/km | 39.93% |
| 0.3 Ω/km | 14.37% |
| 0.6 Ω/km | 7.24% |
| 1 Ω/km | 4.351% |
Frequently asked questions
Why does a current flow in the sheath at all?
The conductor current makes a magnetic field that links the sheath loop. If the sheaths are joined at both ends, the field drives a current round that loop, like the secondary of a transformer.
How is it avoided?
By bonding the sheaths at one point only, or by cross bonding them in sections so that the three induced voltages cancel. The choice depends on the length and the sheath voltage limit.
Why is a low sheath resistance worse here?
With a small Rs the sheath current rises toward the conductor current. A sheath of high resistance carries less current, but it loses more power per ampere.
Does the loss matter for the rating?
Yes. The sheath loss is heat that the cable must shed, so it lowers the current that the cable can carry. The rating method of the standard includes it.
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