Charge coefficient matrix with ground wires

The charge coefficient (Maxwell capacitance) matrix relates the line charges of the phase conductors to their voltages: $[Q] = [C][V]$. With ground wires on the tower, the ground wires also carry charges, but their potential is zero. So they are eliminated to give a reduced $P \times P$ matrix for the phase conductors only.

Method of images

The earth is treated as a perfect conductor at zero potential. Each conductor $i$ (radius $r_i$, height $H_i$, charge $q_i$ C/m) has an image of charge $-q_i$ at depth $H_i$ below ground.

   P1 o       o P2       (phase)   ^
         o g             (gnd wire) | H
 //////////////////////////////// ground
         o -g
   -P1 o       o -P2     (images)
   A_ij = conductor i to conductor j
   I_ij = conductor i to image of j

Maxwell's potential coefficients

The potential of conductor $i$ due to all charges is

$$V_i = \sum_{j} P_{ij}\, q_j$$

where

$$\begin{aligned} P_{ii} &= \frac{1}{2\pi\varepsilon_0}\ln\frac{2H_i}{r_i} \\ P_{ij} &= \frac{1}{2\pi\varepsilon_0}\ln\frac{I_{ij}}{A_{ij}} \end{aligned}$$

with $1/(2\pi\varepsilon_0) = 18 \times 10^9$ m/F. For bundles, $r_i$ is replaced by the equivalent bundle radius.

Step 1: Full matrix for P phases and g ground wires

With $n = P + g$ conductors, partition the matrix (subscript $p$ = phases, $g$ = ground wires):

$$\begin{bmatrix} [V_p] \\ [V_g] \end{bmatrix} = \begin{bmatrix} [P_{pp}] & [P_{pg}] \\ [P_{gp}] & [P_{gg}] \end{bmatrix} \begin{bmatrix} [Q_p] \\ [Q_g] \end{bmatrix}$$

Sizes: $P_{pp}$ is $P \times P$, $P_{pg}$ is $P \times g$, $P_{gg}$ is $g \times g$.

Step 2: Use the ground-wire condition

Ground wires are at (or near) ground potential, so $[V_g] = 0$:

$$\begin{aligned} 0 &= [P_{gp}][Q_p] + [P_{gg}][Q_g] \\ [Q_g] &= -[P_{gg}]^{-1}[P_{gp}][Q_p] \end{aligned}$$

The ground wires carry induced charges of opposite sign.

Step 3: Reduced potential coefficient matrix

Substitute into the first row:

$$\begin{aligned} [V_p] &= [P_{pp}][Q_p] + [P_{pg}][Q_g] \\ &= \big([P_{pp}] - [P_{pg}][P_{gg}]^{-1}[P_{gp}]\big)[Q_p] \\ &= [P_r][Q_p] \end{aligned}$$

Step 4: Charge coefficient matrix

$$[Q_p] = [P_r]^{-1}[V_p] = [C][V_p], \qquad [C] = [P_r]^{-1}$$

$[C]$ is a $P \times P$ symmetric matrix. Its diagonal elements are positive and its off-diagonal elements are negative.

One ground wire (g = 1)

$[P_{gg}]$ is a scalar, so each element is simply

$$P'_{ij} = P_{ij} - \frac{P_{ig}\,P_{gj}}{P_{gg}}, \quad i, j = 1 \ldots P$$

Two ground wires (g, h)

$$[P_{gg}]^{-1} = \frac{1}{P_{gg}P_{hh} - P_{gh}^2}\begin{bmatrix} P_{hh} & -P_{gh} \\ -P_{gh} & P_{gg} \end{bmatrix}$$

so

$$P'_{ij} = P_{ij} - \frac{P_{ig}\left(P_{hh}P_{gj} - P_{gh}P_{hj}\right) + P_{ih}\left(P_{gg}P_{hj} - P_{gh}P_{gj}\right)}{P_{gg}P_{hh} - P_{gh}^2}$$

Effect of ground wires

• The subtracted term makes $P'_{ii} < P_{ii}$, so the self capacitance of the phase conductors increases slightly.
• For the same voltage, the phase charge (and so the conductor surface gradient) rises a little.
• The ground wires also shield the phases from direct lightning strokes.