Induced shield voltage phase a

The voltage gradient induced in a cable shield may be considered as a special case in which the parallel conductor is a shield at a spacing from the conductor that it embraces equal to the mean radius of the shield. When no other current-carrying conductor is in the vicinity.

Equations for the different cases:

Normal case and the three-phase symmetrical fault:

eq 1: General case of any cable formation.
eq 2: Trefoil formation, where $S {ab} = S {bc} = S {ac}$, the general equation simplifies.
eq 3: Flat formation, in which the axial spacing of adjacent cables is equal and $= S_{m}$.
eq 4: Trefoil or flat with regular transposition for which the $GMD$ can be used.

Phase-to-phase fault:

eq 5: In the general case of any cable formation, assuming a fault between phases a and b with no ground current flowing, when $I_{ka}$ is the fault current, the shield/sheath voltage gradients are shown.

Single-phase ground fault (solidly grounded neutral):

eq 6: Precise calculation of shield/sheath overvoltages for underground-fault conditions requires a knowledge of the proportion of the return current that flows in the ground itself and the proportion that returns by way of the parallel ecc. This depends on a number of factors, which are not usually accurately known.Fortunately, however, the overvoltages of practical interest are those between shields/sheaths and the parallel ecc, and these can be simply calculated by the assumption that this conductor carries the whole of the return current. This assumption is normally accurate and leads to shield/sheath overvoltages that are slightly higher than those observed in practice.
For a ground fault in phase a, and the general case of any cable formation when $I_{ka}$ is the fault current, the shield/sheath-to-ground conductor voltages are shown.

Magnitude of voltages:

eq 7: Typical maximum values of shield/sheath voltages calculated from these equations are given in Figure E.1 for a circuit in flat formation, for a current of 1000 A having a transposed ground conductor. For a three-phase symmetrical fault, the maximum voltage is reached in the outer cables and is the same as in Figure 1 but increased for higher current. For the phase-to-phase fault, the highest shield/sheath voltage results when the fault is between the outer cables so that $S {ac} = 2 \cdot S$. For a ground fault assuming the ground conductor to be laid as shown in Figure 2 of this guide, see Equation (E.8) and Equation (E.9).
The highest of the three-sheath voltages for a fault in phase a is $E_{a}$, and since the effect of $R_{c}$ can generally be neglected, the preceding equation for $E_{a}$ can be expressed as shown in Equation (E.10).
Figure E.1 shows the effect of varying $d/r_{g}$ over a typical range of values. It is clear that the overvoltages per meter due to the single-phase fault is much greater than for the other types of fault, for systems having solidly grounded neutral. For systems having impedance or resonant grounding of the neutral, the phase-to-phase fault is the most important.

Symbol

$E_a$

Unit

V/m

Formulas

$+j \omega I_{ka} 2{\cdot}{10}^{-7} \left(\frac{-1}{2} \ln\left(\frac{2{S_{ab}}^2}{d S_{ac}}\right)+j \frac{\sqrt{3}}{2} \ln\left(\frac{2S_{ac}}{d}\right)\right)$	three-phase symmetrical fault, general case, without transposition
$+j \omega I_{ka} 2{\cdot}{10}^{-7} \left(\frac{-1}{2}+j \frac{\sqrt{3}}{2}\right) \ln\left(\frac{2S_m}{d}\right)$	three-phase symmetrical fault, trefoil, without transposition
$+j \omega I_{ka} 2{\cdot}{10}^{-7} \left(\frac{-1}{2} \ln\left(\frac{S_m}{d}\right)+j \frac{\sqrt{3}}{2} \ln\left(\frac{4S_m}{d}\right)\right)$	three-phase symmetrical fault, flat, without transposition
$+j \omega I_{ka} 2{\cdot}{10}^{-7} \ln\left(\frac{GMD}{d}\right)$	three-phase symmetrical fault, with regular transposition
$+j \omega I_{ka} 2{\cdot}{10}^{-7} \ln\left(\frac{2S_{ab}}{d}\right)$	phase-to-phase fault between phases a + b
$I_{ka} j \omega 2{\cdot}{10}^{-7} \ln\left(\frac{2S_m}{d}\right)$	single-phase ground fault in phase a, both-side bonded
$I_{ka} \left(R_{ct}+j \omega 2{\cdot}{10}^{-7} \ln\left(\frac{2{S_{ap}}^2}{d r_g}\right)\right)$	single-phase ground fault in phase a, single-side bonded (solidly grounded neutral)
$I_{ka} \left(R_E+j \omega 2{\cdot}{10}^{-7} \ln\left(\frac{D_E}{d}\right)\right)$	single-phase ground fault in phase a, single-side bonded (without ecc)
$\frac{I_{kx}}{3} \left(Z_{ss}+2Z_{sg}\right)-I_{ka} \left(Z_{ss}-R_s\right)$	single-phase ground fault in phase a minor section 1, cross-bonded, trefoil
$\frac{I_{kx}}{3} \left(Z_{ss}+Z_{oog}+Z_{oig}\right)-I_{ka} \left(Z_{ss}-R_s\right)$	single-phase ground fault in phase a minor section 1, cross-bonded, flat