Series Impedances: Three-Phase Line

To calculate series impedances, including resistances and inductive reactances, consider a network of three-phase and N-neutral conductors grounded regularly for a three-phase overhead line.

Using Kirchhoff's laws, an integro-differential equation for the network is derived.

Unbalanced phase currents may induce return currents through neutral wires and the earth, following the path of least impedance.

Earth return conductors, mirroring the overhead conductor's negative current, can replace the earth.

They have the same geometric mean radius and resistance as their overhead counterparts.

Overhead conductors are renumbered, starting with phase conductors, followed by neutral ones.

In a transmission line, the total current across conductors is zero, defining the flux linking each overhead conductor.

Using a one-meter section's circuit representation, the vector of voltage drops across the conductors is calculated.

Equations are partitioned and solved, yielding voltage drop vectors across phase conductors and phase currents and their relationship.

This gives the three-by-three series-phase impedance matrix to comprehend the line's behavior.

For fully transposed lines, averaging diagonal and off-diagonal elements offers a generalized line behavior view.