Question

The series-sequence impedance matrix of a completely transposed three-phase line is ___________ with its nondiagonal elements equal to ___________.

Short answer

The series-sequence impedance matrix of a completely transposed three-phase line is diagonal with its nondiagonal elements equal to zero.

Step-by-step solution

Determine the sequence matrix for the series impedance.

The sequence impedance matrix for the series impedance is given as follows.

${\mathit{Z}}_{\mathbf{s}}\mathbf{=}\left[\begin{array}{ccc}{Z}_{aa}+2Z_{ab}& 0& 0\\ 0& Z_{aa}-Z_{ab}& 0\\ 0& 0& Z_{aa}-Z_{ab}\end{array}\right]$ …… (1)

Here is the series impedance and same for all phases and is the mutual impedance between the phases.

Determine the diagonal and non-diagonal elements of the sequence impedance matrix.

Determine the sequence impedance.

The zero-sequence impedance from equation (1).

${\mathit{Z}}_{\mathbf{0}}\mathbf{=}{\mathit{Z}}_{\mathbf{a}\mathbf{a}}\mathbf{-}\mathbf{2}{\mathit{Z}}_{\mathbf{a}\mathbf{b}}$

The positive and negative sequence impedance from equation (1).

$$\begin{gathered} {\mathit{Z}}_{\mathbf{1}}\mathbf{=}{\mathit{Z}}_{\mathbf{2}} \\ {\mathit{Z}}_{\mathbf{1}}\mathbf{=}{\mathit{Z}}_{\mathbf{a}\mathbf{a}}\mathbf{-}{\mathit{Z}}_{\mathbf{a}\mathbf{b}} \end{gathered}$$

Since the impedances are only present at the diagonal places of the matrix. Therefore, sequence matrix is the diagonal matrix and non- diagonal elements are zero.