Capacitance and charge matrices (9.81 m clearance)

The capacitance matrix is $[C] = 2\pi\varepsilon_0[M]^{-1}$, where $[M]$ is Maxwell's coefficient matrix built with the image method and the equivalent bundle radius. The charge matrix is $[Q] = [C][V]$.

Data

• 400 kV, flat horizontal, $S = 11.3$ m, $H = 9.81$ m (average)
• Sub-conductor radius $r = 3.17$ cm; $B = 45.72$ cm; $N = 2$
• $2\pi\varepsilon_0 = 1/(18 \times 10^9) = 55.56$ pF/m

Step 1: Equivalent radius

$$r_{eq} = \sqrt{rB} = \sqrt{3.17 \times 45.72} = 12.04\ \text{cm} = 0.1204\ \text{m}$$

Step 2: Maxwell coefficients ($2H = 19.62$ m)

$$\begin{aligned} M_{ii} &= \ln\frac{19.62}{0.1204} = 5.0936 \\ M_{12} = M_{23} &= \ln\frac{\sqrt{11.3^2 + 19.62^2}}{11.3} = \ln\frac{22.641}{11.3} = 0.6950 \\ M_{13} &= \ln\frac{\sqrt{22.6^2 + 19.62^2}}{22.6} = \ln\frac{29.928}{22.6} = 0.2809 \end{aligned}$$ $$[M] = \begin{bmatrix} 5.0936 & 0.6950 & 0.2809 \\ 0.6950 & 5.0936 & 0.6950 \\ 0.2809 & 0.6950 & 5.0936 \end{bmatrix}, \quad \det = 127.10$$

Step 3: Inverse and capacitance matrix

$$[M]^{-1} = \begin{bmatrix} 0.20033 & -0.02632 & -0.00746 \\ -0.02632 & 0.20351 & -0.02632 \\ -0.00746 & -0.02632 & 0.20033 \end{bmatrix}$$ $$[C] = 55.56\,[M]^{-1} = \begin{bmatrix} 11.129 & -1.462 & -0.414 \\ -1.462 & 11.306 & -1.462 \\ -0.414 & -1.462 & 11.129 \end{bmatrix}\ \text{pF/m (nF/km)}$$

Step 4: Charge matrix

Balanced phase voltages $V = 400/\sqrt{3} = 230.94$ kV rms at $0°, -120°, 120°$; $[Q] = [C][V]$:

$$\begin{aligned} q_1 &= 230.94\,\big(11.129 - 1.462\angle{-120°} - 0.414\angle{120°}\big) \times 10^{-3} \\ &= 2.795\angle 4.30°\ \mu\text{C/m} \end{aligned}$$ $$[Q] = \begin{bmatrix} 2.795\angle 4.30° \\ 2.949\angle -120° \\ 2.795\angle 115.70° \end{bmatrix}\ \mu\text{C/m (rms)}$$

Peak values: 3.95, 4.17 and 3.95 μC/m.

If the line is transposed (for reference)

$$M_s = 5.0936,\quad M_m = \frac{2(0.6950) + 0.2809}{3} = 0.5569$$ $$C_1 = \frac{55.56}{5.0936 - 0.5569} = 12.25\ \text{nF/km}, \quad q = 12.25 \times 230.94 \times 10^{-3} = 2.83\ \mu\text{C/m (rms)}$$

Answer: $[C]$ as above; charges ≈ 2.79∠4.3°, 2.95∠−120°, 2.79∠115.7° μC/m rms.