A three phase fully transposed transmission line ha
s solid conductors of 2 cm diameter,
spaced at distances between centres of 3.65 m, 5.5
m and 8.2 m. Determine the inductance of
the complete line per km.
To determine the inductance of the complete line per km, we need to calculate the self-inductance and mutual inductance of the transmission line.
The self-inductance is the inductance of each phase conductor due to its own current. The mutual inductance is the inductance between the phase conductors due to the interaction of their magnetic fields.
To calculate the self-inductance, we can use the formula:
L_self = (μ₀ / 2π) * ln(D_out / D_in)
Where:
- L_self is the self-inductance per meter of a single phase conductor
- μ₀ is the permeability of free space (4π * 10^-7 H/m)
- D_out is the outer diameter of the conductor
- D_in is the inner diameter of the conductor
In this case, the conductor has a solid shape, so the inner diameter D_in is 0 cm. The outer diameter D_out is given as 2 cm. Plugging these values into the formula, we can calculate the self-inductance per meter.
L_self = (4π * 10^-7 H/m) / (2π) * ln(2 cm / 0 cm)
L_self = (2 * 10^-7 H/m) * ln(2)
To calculate the mutual inductance, we can use the formula:
L_mutual = (μ₀ / π) * ln(D_s / D_r)
Where:
- L_mutual is the mutual inductance per meter
- D_s is the average spacing between the phase conductors
- D_r is the radius of the phase conductor
The average spacing D_s is given as 3.65 m, 5.5 m, and 8.2 m. To calculate the radius D_r, we divide the outer diameter by 2, since the conductor is assumed to have a circular cross-section.
D_r = D_out / 2 = 2 cm / 2 = 1 cm
Now, we can plug in the values into the formula and calculate the mutual inductance per meter for each spacing.
L_mutual_1 = (4π * 10^-7 H/m) / π * ln(3.65 m / 1 cm)
L_mutual_2 = (4π * 10^-7 H/m) / π * ln(5.5 m / 1 cm)
L_mutual_3 = (4π * 10^-7 H/m) / π * ln(8.2 m / 1 cm)
Finally, we can calculate the total inductance per meter by summing the self-inductance and the mutual inductances for each spacing:
L_total = L_self + L_mutual_1 + L_mutual_2 + L_mutual_3
To convert the inductance per meter to inductance per kilometer, we multiply the result by 1000:
L_total_per_km = L_total * 1000