If the charge density along the transmission line is sinusoidal rather than linear, Equation 35 is a phasor equation. Multiplying Equation 35 by j$\omega$ yields
$j\omega\mathbf{Q}=j\omega\mathbf{CV}$ | (38) |
Recalling that the current phasor associated with a sinusoidal variation in charge is expressed as
$\mathbf{I}=j\omega\mathbf{Q}$ | (39) |
It is apparent that
$\mathbf{I}=j\omega\mathbf{CV}$ | (40) |
An alternate expression for the charging current is
$\mathbf{I=Y_{shunt}V}$ | (41) |
Therefore, the charging admittance (which is pure susceptance) must be
$\mathbf{Y_{shunt}}=j\omega\mathbf{C}$ | (42) |
The preceding discussion suggests a computational procedure for determining the capacitive parameters of a conductor configuration:
Compute the configuration’s potential matrix P using Equation 33.
Compute its capacitance matrix C by inverting P.
Multiply the capacitance matrix C by the scalar j$\omega$ to obtain the shunt admittance matrix Y_{shunt}.
Invert the the shunt admittance matrix Y_{shunt} to determine the capacitive reactance X_{shunt}.