Shunt Admittances

Consider a uniformly charged conductor above Earth. It induces an equal negative charge on Earth, creating electric field lines.

Earth's effect is modeled using an image conductor, identical but beneath the original, keeping the electric field and voltage constant.

For three-phase lines with neutral conductors, separate image conductors are used. The voltage between a conductor and its image depends on their distance. By symmetry, the voltage between this conductor and Earth is half this value.  Grounded neutral conductors carry no charge.

Matrix equations express these relationships, enabling calculations of phase-to-neutral voltages and conductor charges.

Further partitioning yield equations relating phase-conductor charges to phase-to-neutral voltages.

Equations reveal that a positive line-to-neutral voltage on one phase induces positive and negative charges on different phases.

For transposed lines, capacitance matrix elements are averaged, deriving the shunt phase admittance matrix.

For a double-circuit line with parallel, non-transposed lines, similar methods yield the equivalent three-by-three shunt admittance matrix. These concepts extend to multiple parallel circuits.