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:

	Temperature		Conductor Material
	Initial °C	Final°C	Copper	Aluminium	Steel
Thermoplastic 70°C (PVC)	70	160/140	115/103	76/78	42/37
Thermoplastic 90°C (PVC)	90	160/140	100/86	66/57	36/31
Thermosetting, 90°C (XLPE, EDR)	90	250	143	94	52
Thermosetting, 60°C (rubber)	60	200	141	93	51
Thermosetting, 85°C (rubber)	85	220	134	89	48
Thermosetting, 185°C (silicone rubber)	180	350	132	87	47

*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.