# Calculation of k Factor

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

$k=\sqrt{\frac{{Q}_{c}\left(\beta +20\right)}{{\rho }_{20}}\mathrm{ln}\left(\frac{\beta +{\theta }_{f}}{\beta +{\theta }_{i}}\right)}$

where Qc = 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
ϴi = initial temperature of the conductor, °C
ϴf = 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:

β [°C]    Qc [J.K-1.mm-3 ρ20 [Ω.mm]
Copper 234.5 3.45 x 10-3 17.241 x 10-6
Aluminium 228 2.5 x 10-3 28.267 x 10-6
Steel 202 3.8 x 10-3 138 x 10-6

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

$k=226 ln( 1+ θ f − θ i 234.5+ θ i )$- copper conductors

$k=148 ln( 1+ θ f − θ i 228+ θ i )$- aluminium conductors

$k=78 ln( 1+ θ f − θ i 202+ θ i )$- steel