When selecting a cable, the performance of the cable under fault conditions is an important consideration. It is important that calculations be carried out to ensure that any cable is able to withstand the effects of any potential fault or short circuit.  This note looks at how to do this.

The primary concern with cables under a fault condition is the heat generated, and any potential negative effect this may have on the cable insulation. 

Calculation of fault rating is based on the principle that the protective device will isolate the fault in a time limit such that the permitted temperature rise within the cable will not be exceeded.

In addition to the direct heating effect of fault currents, other considerations include:

• electro-mechanical stress and fault levels large enough to cause cable failure
• performance of joint and terminations under fault conditions

When calculating the fault ratings of a cable, it is generally assumed that the duration is short enough that no heat is dissipated by the cable to the surrounding.  Adopting this approach simplifies the calculation and errs on the safe side.

The normally used equation is the so-called adiabatic equation.  For a given fault of I, which lasts for time t, the minimum required cable cross-sectional area is given by:

 $A=\frac{\sqrt{I^{2}t}}{k}$

where: A - the nominal cross-section area, mm²
I - the fault current in, A
t -  duration of fault current, s
k - a factor dependant on cable type (see below)

Alternatively, given the cable cross-section and fault current, the maximum time allowable for the protective device can be found from:

 $t=\frac{k^2{A}^2}{I^2}$

The factor k is dependant on the cable insulation, allowable temperature rise under fault conditions, conductor resistivity and heat capacity.  Typical values of k are:

*where two values; lower value applied to conductor CSA > 300 mm²
* these values are suitable for durations up to 5 seconds, source: BS 7671, IEC 60364-5-54

Derivation  - Adiabatic Equation and k

The term adiabatic applies to a process where there is no heat transfer.  For cable faults, we are assuming that all the heat generated during the fault is contained within the cable (and not transmitted away).  Obviously, this is not fully true, but it is on the safe side.

From physics, the heat Q, required to rise a material ΔT is given by:

 $Q=cm\Delta T$

where Q - heat added, J
c - specific heat constant of the material, J.g^-1.K^-1
m - mass of the material, g
ΔT - temperature rise, K

The energy into the cable during a fault is given by:

 $Q=I^{2}Rt$

where  R - the resistance of the cable, Ω

From the physical cable properties, we can calculate m and R as:

 $m={\rho }_{c}Al$   and     $R=\frac{{\rho }_{r}l}{A}$

where ρ_c - material density in g.mm^-3
ρ_r - resistivity of the conductor, Ω.mm
l - length of the cable, mm

Combining and substituting we have:

 $I^{2}Rt=cm\Delta T$

 $I^{2}t\frac{{\rho }_{r}l}{A}=c{\rho }_{c}Al\Delta T$

and rearranging for A gives:

 $S=\frac{\sqrt{I^{2}t}}{k}$   by letting    $k=\sqrt{\frac{c{\rho }_c\Delta T}{{\rho }_r}}$

Note: ΔT is the maximum allowable temperature rise for the cable:

 $\Delta T={\theta }_f-{\theta }_i$

where θ_f - final (maximum) cable insulation temperature, °C
θ_i - initial (operating) cable insulation temperature, °C

Units:  are expressed in g (grams) and mm², as opposed to kg and m.  This is widely adopted by cable specifiers.  The equations can easily be redone in kg and m if required.

IEC 60364-5-54 gives the following formula for the calculation of the factor k:

$k=\sqrt{\frac{Q_c(\beta +20)}{{\rho }_{20}}\ln \left(\frac{\beta +{\theta }_f}{\beta +{\theta }_i}\right)}$

where Q_c = volumetric heat capacity of conductor at 20°C, J.K^-1.mm^-3
ß = reciprocal of temperature coefficient of resistivity at 0 °C 
ρ₂₀ =  electrical resistivity of conductor material at 20 °C, Ω.mm
ϴ_i = initial temperature of the conductor, °C
ϴ_f = final temperature of the conductor, °C

Note: strictly speaking ß is specified at 0 °C.  Within myCableEngineering calculations, ß is determined by taking the reciprocal of the 20 °C temperature coefficient.  This introduces a small, but negligible error. 

Typical Application

For manual calculations, the following will give realistic results:

Substituting the above values and rearranging the IEC equation slightly, gives:

$k=226\sqrt{\ln \left(1+\frac{{\theta }_f-{\theta }_i}{234.5+{\theta }_i}\right)}$- copper conductors

$k=148\sqrt{\ln \left(1+\frac{{\theta }_f-{\theta }_i}{228+{\theta }_i}\right)}$- aluminium conductors

$k=78\sqrt{\ln \left(1+\frac{{\theta }_f-{\theta }_i}{202+{\theta }_i}\right)}$- steel

For details on thermal withstand and I2t, see I2t Thermal Withstand. Energy limiting devices reduce the I2t let-through in the event of a fault.  The values are typically specified by the manufacturer:

Typical MCB Energy Limiting Characteristic

Using typical data given above, for a 10A, 10kA fault rated MCB would have an Ip/A of 1000 (10³), give an energy let-through of approximates 2,000 A²s.

The thermal withstand of a cable can be determined by comparing the maximum energy at the fault, with the maximum energy the cable can absorb.  The equation for this is:

$I^{2}t\le k^2{A}^2$

I²t is proportional to the energy let-through of the protective device
       I - fault current, A
       t - fault duration, S

k²A² is proportional to the energy withstand of the cable
        k - adiabatic constant
        A - cable cross-sectional area, mm²

To better understand the theory behind the above, please see The adiabatic equation.

As mentioned, the adiabatic equation assumes no heat is dissipated from the cable during a fault.  While putting the calculation on the safe side, in some situations, particularly for longer fault duration there is the potential to be able to get away with a smaller cross section.  In these instances, it is possible to do a more accurate calculation.

Considering non-adiabatic effects is more complex.  Unless there is some driver, using the adiabatic equations is just easier.  Software is available to consider non adiabatic effects, however, there is a cost, time and complexity associated with this.

The IEC also publish a standard which deals with non-adiabatic equations:

• IEC 60949 "Calculation of thermally permissible short-circuit current, taking into account non-adiabatic heating effects".

The method adopted by IEC 60949 is to use the adiabatic equation and apply a factor to cater for the non-adiabatic effects:

 $I=\epsilon I_{AD}$

where I - permissible short circuit current, A (or kA)
I_AD - adiabatic calculated permissible short circuit current, A (or kA)
ε - factor to allow for heat dissipation from cable

The bulk of the IEC 60949 standard is concerned with the calculation of ε.