Shunt admittances play a crucial role in the analysis of transmission lines, particularly for three-phase systems with neutral conductors. When a uniformly charged conductor is positioned above the Earth, it induces an equal but opposite charge on its surface. This interaction creates electric field lines between the conductor and the Earth.

To model this effect, the method of images is employed. This method involves replacing the Earth with an image conductor that mirrors the original conductor's properties but is symmetrically beneath it. The image conductor maintains the same radius and charge magnitude as the original, ensuring that the electric field and voltage distribution above the Earth remain consistent with the actual scenario.

In three-phase lines with neutral conductors, separate image conductors are utilized for each phase. The voltage between a conductor and its corresponding image conductor is determined by their distance. Due to symmetry, the voltage between each conductor and the Earth is half of this value. Neutral conductors, which are grounded, carry no charge and simplify the calculation process.

Matrix equations express the relationships between phase-to-neutral voltages and conductor charges. These equations are further partitioned and rewritten to establish the relationships between phase-conductor charges and phase-to-neutral voltages. The equations reveal that a positive line-to-neutral voltage on one phase induces positive and negative charges on different phases.

For transposed lines, the elements of the capacitance matrix are averaged, allowing the derivation of the shunt phase admittance matrix. Similar methods yield the equivalent three-by-three shunt admittance matrix in the case of double-circuit lines with parallel, non-transposed configurations. These principles can be extended to multiple parallel circuits, facilitating comprehensive analysis and ensuring the efficient operation of complex power transmission systems.