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 jω\omega yields

jω𝐐=jω𝐂𝐕j\omega\mathbf{Q}=j\omega\mathbf{CV} (38)

Recalling that the current phasor associated with a sinusoidal variation in charge is expressed as

𝐈=jω𝐐\mathbf{I}=j\omega\mathbf{Q} (39)

It is apparent that

𝐈=jω𝐂𝐕\mathbf{I}=j\omega\mathbf{CV} (40)

An alternate expression for the charging current is

𝐈=𝐘𝐬𝐡𝐮𝐧𝐭𝐕\mathbf{I=Y_{shunt}V} (41)

Therefore, the charging admittance (which is pure susceptance) must be

𝐘𝐬𝐡𝐮𝐧𝐭=jω𝐂\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 jω\omega to obtain the shunt admittance matrix Yshunt.

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