The characteristic or fictitious diameter $D_x$ is the location at which the effect of the loss factor commences. Inside the circle of diameter $D_x$, the temperature changes according to the peak value of the losses. Outside this circle, it changes with the average losses. The characteristic diameter is a function of the diffusivity of the medium and the length of the loss cycle. In the majority of cases, the soil diffusivity $\delta_{soil}$ will not be known. In these cases, a value of 0.5 of 0.5*10-6 m²/s can be used. This value is based on a soil thermal resistivity of 1.0 K.m/W and a moisture content of about 7% of dry weight.

The calculation of the characteristic (or fictitious) diameter for sinusoidal load is based on the IEEE paper 'Ampacity calculation for deeply installed cables' by E. Dorison et al, dated 2010. Three different methods can be choses:

1. Neher McGrath: Neher evaluated constants epirically to best fit the temperature rises calculated over a range of cable sizes. Using measured data, Neher obtained the value 61200 when $\tau$ is expressed in hours and $\delta_{soil}$ is expressed in m²/s. Based on the evaluations by Neher, the characteristic diameter becomes 212 mm.
2. Dorison: The calculation uses the modified Bessel function K of order 0: k0() and the modified Bessel function of the first kind of order 1: k1(). The IEC standard 60853 uses an approximation with an exponential integral valid for an infinite thin wire whereas the formulas presented in the paper the cable diameter influences the diameter of the area affected by load variations. This is particularily important for cables in tunnels, where the tunnel diameter replaces the diameter of the cable.
3. Heinhold: Alternative expressions for $D_x$ are given in the book 'Kabel und Leitungen für Starkstrom' by L. Heinhold, 5th edition 1999. Heinhold defined three equations to calculate the characteristic diameter $D_x$ depending on the type of load curve (sinusoidal, rectilinear, and other load curve)
4. Symbol
$D_{\mathrm{x}}$
Unit
mm
Formulae
 $1020 \sqrt{\delta _{\mathrm{soil}} \tau}$ Method by Neher McGrath $\frac{\sqrt{6} k_{\mathrm{H}}}{720 \rho_{\mathrm{4}}^{0.4} \sqrt{\frac{1}{\tau}}}$ Method by Heinhold $D_{\mathrm{e}} e^{\frac{2000}{D_{\mathrm{e}} q_{\mathrm{x}}} \operatorname{abs}{\left (\frac{\operatorname{k_{0}}{\left (\frac{D_{\mathrm{e}} q_{\mathrm{x}}}{2000} \right )}}{\operatorname{k_{1}}{\left (\frac{D_{\mathrm{e}} q_{\mathrm{x}}}{2000} \right )}} \right )}}$ Method by Dorison
Related
$D_{\mathrm{e}}$
$\delta _{\mathrm{soil}}$
$\rho_{\mathrm{4}}$
$\tau$
$T_{\mathrm{4d}}$