2 Series Impedance 2.1 Carson’s Equations 2.3 Series Impedance Computations

2.2 Impedance of an N Conductor Transmission Line

The two conductor problem of Section 2.1 can be generalized to a group of n conductors with a common ground return. If currents i₁, i₂, …, i_n are flowing through the conductors, the voltage drop along conductor i is

\mathbf{V_{i}}=i_{1}Z_{i1-g}+\cdots+i_{i}Z_{ii-g}+\cdots+i_{n}Z_{in}

(16)

Similar equations can be constructed for all conductors in the group. Expressing the complete set of n voltage drop equations in matrix notation yields

\mathbf{V={Z}_{series}I}

(17)

where

V is the voltage vector.

I is the current vector.

Z_series is the series impedance matrix.

The elements of the impedance matrix Z_series are computed using Carson’s equations:

\mathbf{z_{ij}}=\begin{cases}R_{ii-g}+jX_{ii-g}&\operatorname{if}i=j\\ R_{ij-g}+jX_{ij-g}&\operatorname{if}i\neq j\end{cases}

(18)

where R_ii–g, R_ij–g, X_ii–g, and X_ij–g are defined by Equation 5 through Equation 8.

The series admittance of the n conductor configuration can be determined by inverting its impedance matrix, i.e.

\mathbf{Y_{series}=Z_{series}^{-1}}

(19)