Charge matrix of 400 kV line

The charge matrix is $[Q] = [C][V] = 2\pi\varepsilon_0 [M]^{-1}[V]$. Here $[M]$ is Maxwell's coefficient matrix, built from the average height of the sagging conductor and the equivalent bundle radius.

Data

• 400 kV, flat horizontal, $S = 11.3$ m
• Bundle $2 \times 3.17$ cm diameter, so $r = 1.585$ cm; $B = 45.72$ cm
• Heights 11.875 m (tower) and 8.85 m (midspan)
• $2\pi\varepsilon_0 = 55.56$ pF/m

Step 1: Average height

$$H_{av} = 8.85 + \frac{11.875 - 8.85}{3} = 9.858\ \text{m}, \quad 2H = 19.717\ \text{m}$$

Step 2: Equivalent radius

$$r_{eq} = \sqrt{rB} = \sqrt{1.585 \times 45.72} = 8.513\ \text{cm}$$

Step 3: Maxwell coefficient matrix

$$\begin{aligned} M_{ii} &= \ln\frac{19.717}{0.08513} = 5.4451 \\ M_{12} = M_{23} &= \ln\frac{\sqrt{11.3^2 + 19.717^2}}{11.3} = 0.6987 \\ M_{13} &= \ln\frac{\sqrt{22.6^2 + 19.717^2}}{22.6} = 0.2830 \end{aligned}$$ $$[M] = \begin{bmatrix} 5.4451 & 0.6987 & 0.2830 \\ 0.6987 & 5.4451 & 0.6987 \\ 0.2830 & 0.6987 & 5.4451 \end{bmatrix}, \quad \det = 155.96$$

Step 4: Capacitance (charge-coefficient) matrix

$$[M]^{-1} = \begin{bmatrix} 0.18697 & -0.02312 & -0.00675 \\ -0.02312 & 0.18959 & -0.02312 \\ -0.00675 & -0.02312 & 0.18697 \end{bmatrix}$$ $$[C] = 55.56\,[M]^{-1} = \begin{bmatrix} 10.387 & -1.285 & -0.375 \\ -1.285 & 10.533 & -1.285 \\ -0.375 & -1.285 & 10.387 \end{bmatrix}\ \text{pF/m}$$

Step 5: Charge matrix

Balanced phase voltages: $V = 400/\sqrt{3} = 230.94$ kV rms at $0°, -120°, +120°$.

$$\begin{aligned} q_1 &= 230.94\,(10.387 - 1.285\angle{-120°} - 0.375\angle{120°}) \times 10^{-3} \\ &= 2.597\angle 4.02°\ \mu\text{C/m} \end{aligned}$$

Similarly:

$$[Q] = \begin{bmatrix} 2.597\angle 4.02° \\ 2.729\angle -120° \\ 2.597\angle 115.98° \end{bmatrix}\ \mu\text{C/m (rms)}$$

Peak values: 3.67, 3.86 and 3.67 μC/m.

The centre phase carries about 5% more charge than the outer phases, so its conductor surface gradient is the highest.

Answer: $[C]$ as above (diagonal 10.39/10.53 pF/m); charges ≈ 2.60, 2.73, 2.60 μC/m rms (3.67, 3.86, 3.67 μC/m peak).