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, …, in are flowing through the conductors, the voltage drop along conductor i is

𝐕𝐢=i1Zi1-g++iiZii-g++inZin\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

𝐕=𝐙𝐬𝐞𝐫𝐢𝐞𝐬𝐈\mathbf{V={Z}_{series}I} (17)


V is the voltage vector.

I is the current vector.

Zseries is the series impedance matrix.

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

𝐳𝐢𝐣={Rii-g+jXii-gifi=jRij-g+jXij-gifij\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 Rii–g, Rij–g, Xii–g, and Xij–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. 

𝐘𝐬𝐞𝐫𝐢𝐞𝐬=𝐙𝐬𝐞𝐫𝐢𝐞𝐬-𝟏\mathbf{Y_{series}=Z_{series}^{-1}} (19)