Calculating series impedances for a three-phase overhead line involves evaluating resistances and inductive reactances in a network with three-phase and multiple neutral conductors grounded at regular intervals.

Using Kirchhoff's laws, an integro-differential equation for the network is derived. This equation accounts for unbalanced phase currents, which may induce return currents through neutral wires and the earth, seeking the least impedance path. Earth return conductors can replace the actual earth in calculations, mirroring the negative current of the overhead conductors and having identical geometric mean radius (GMR) and resistance.

Overhead conductors are renumbered, starting with phase conductors, followed by neutral conductors. The total current across all conductors in a transmission line is zero, a condition that defines the flux linkage for each overhead conductor. The vector of voltage drops across the conductors can be calculated by representing a one-meter section of the circuit.

The equations derived from Kirchhoff's laws are partitioned and solved to yield voltage drop vectors across phase conductors and the corresponding phase currents. These relationships allow for the formulation of a three-by-three series-phase impedance matrix. This matrix encapsulates the line's electrical behavior, providing critical insights into the interactions between phase conductors and their impedance characteristics.

For fully transposed lines, where conductor positions are regularly rotated to balance inductance and capacitance, the impedance matrix can be further simplified. By averaging the diagonal and off-diagonal elements, a generalized view of the line's behavior is obtained. This approach helps in predicting the performance of the transmission line under various operating conditions, ensuring efficient and reliable power transmission.

Understanding series impedances in a three-phase line through detailed modeling and solving complex equations is crucial for designing and operating efficient power transmission systems.