Audible noise of 750 kV line using two formulas

The AN of each phase is computed from the empirical formula and the phase-to-microphone distance, then added on an energy basis. Both given formulas are evaluated. Formula (i), with the $26.4\log N$ term, is meant for bundles with $N \ge 3$. Formula (ii) is meant for $N < 3$, so (ii) is the appropriate one here ($N = 2$).

Data

• $H = 20$ m, $S = 15$ m, $N = 2$
• Sub-conductor diameter 0.30 m as printed, so $d = 30$ cm
• Gradient $E_m$ not given; assume 20 kV/cm on all phases (the value used in the companion question)
• Microphone at ground, 20 m horizontally beyond the outer phase

Step 1: Distances (horizontal 20, 35, 50 m)

$$\begin{aligned} D_1 &= \sqrt{20^2 + 20^2} = 28.28\ \text{m} \\ D_2 &= \sqrt{35^2 + 20^2} = 40.31\ \text{m} \\ D_3 &= \sqrt{50^2 + 20^2} = 53.85\ \text{m} \end{aligned}$$

Step 2: Common terms

$$\begin{aligned} 120\log 20 &= 156.12 \\ 55\log 30 &= 81.24 \\ 26.4\log 2 &= 7.95 \\ 11.4\log D &= 16.55,\ 18.30,\ 19.74 \end{aligned}$$

Formula (i)

$$\begin{aligned} AN_i &= 156.12 + 81.24 - 11.4\log D_i + 7.95 - 128.4 \\ &= 116.91 - 11.4\log D_i \end{aligned}$$

This gives 100.37, 98.61 and 97.18 dB(A).

$$\begin{aligned} AN &= 10\log\left(10^{10.037} + 10^{9.861} + 10^{9.718}\right) \\ &= 103.68\ \text{dB(A)} \end{aligned}$$

Formula (ii)

$$AN_i = 156.12 + 81.24 - 11.4\log D_i - 115.4 = 121.96 - 11.4\log D_i$$

This gives 105.42, 103.66 and 102.23 dB(A).

$$\begin{aligned} AN &= 10\log\left(10^{10.542} + 10^{10.366} + 10^{10.223}\right) \\ &= 108.74\ \text{dB(A)} \end{aligned}$$

The two formulas differ by a constant $26.4\log 2 - 128.4 + 115.4 = -5.05$ dB per phase.

Note on conductor size

A 30 cm sub-conductor is unrealistic; 0.030 m (3 cm) is probably intended. With $d = 3$ cm every value falls by exactly 55 dB, giving 48.68 dB(A) by (i) and 53.74 dB(A) by (ii). These are typical levels for a 750 kV line.

Answer: with d = 30 cm, AN = 103.7 dB(A) by (i) and 108.7 dB(A) by (ii); formula (ii) applies for N = 2. With d = 3 cm, 48.7 and 53.7 dB(A).