# 3.3 Shunt Admittance and Reactance Matrices

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:

1. Compute the configuration’s potential matrix P using Equation 33.

2. Compute its capacitance matrix C by inverting P.

3. Multiply the capacitance matrix C by the scalar j$\omega$ to obtain the shunt admittance matrix Yshunt.

4. Invert the the shunt admittance matrix Yshunt to determine the capacitive reactance Xshunt.