Critical system length

The series inductance of an overhead line is about 2-3 times larger compared to an underground circuit but the shunt capacitance of an underground line is about 10-20 times larger. These factors depend on the geometrical configuration of the cable system and material properties of the cable. It is possible to define the concept of critical length considering only the capacitance of the cable.

To feed a purely resistive load in a radial network with a given current $I_{L}$ trough an underground line, it is necessary to inject a higher current $I_{Z}$ at the source to compensate for the cable capacitance. The difference being the capacitive current $I_{C}$ generated in the line, in quadrature with the current in load $I_{Z}^2=I_{L}^2+I_{C}^2$.

With increasing length, the capacitive charging current will reach the value of the maximum allowable current of the cable, so the charging current accounts for all the available heat losses in the cable. This length is called the critical length and occurs when the thermal rated current for the line is equal to the capacitive current for the cable $I_{Z}=I_{C}$. The equation can be re-arranged to find the length at which the charging current is equal to the thermal rating $I_{c}$ of the cable.

It can be concluded that the critical length is determined by the system voltage and frequency and by cable rating which is determined by the conductor size, environmental and installation conditions and cable capacitance.

$10\frac{\sqrt{3}}{\omega C_b U_o} I_c{\cdot}{10}^6$