ample: -phase, \( 220 \mathrm{kV}, 50 \mathrm{~Hz} \) transmission line consists of 1.5 cm radius ductor spaced 2 meters apart in equilateral triangular formation. temperature is \( 40^{\circ} \mathrm{C} \) and atmospheric pressure is 76 cm , calculate corona loss per km of the line. Take mo \( =0.85 \). ution: e corona loss is given.by:

To calculate the corona loss per kilometer of a transmission line, we can use the following formula: \[ W_c = 0.005 \cdot \frac{(V^2)}{d^2} \cdot \left( \frac{1}{\sqrt{T}} \right) \cdot \left( \frac{1}{\sqrt{P}} \right) \] Where: - \( W_c \) = corona loss in watts per kilometer (W/km) - \( V \) = phase voltage in kilovolts (kV) - \( d \) = geometric mean distance (GMD) in meters (m) - \( T \) = temperature in degrees Celsius (°C) - \( P \) = atmospheric pressure in cm of mercury (cmHg) ### Given Data: - Phase voltage, \( V = 220 \, \text{kV} \) - Temperature, \( T = 40 \, \text{°C} \) - Atmospheric pressure, \( P = 76 \, \text{cmHg} \) - Radius of the conductor, \( r = 1.5 \, \text{cm} = 0.015 \, \text{m} \) - Distance between conductors, \( d = 2 \, \text{m} \) ### Step 1: Calculate the Geometric Mean Distance (GMD) For an equilateral triangular formation, the GMD can be calculated using the formula: \[ d_{g} = \frac{d}{\sqrt{3}} \cdot \left( \frac{1}{\sqrt{3}} \right) \cdot \left( \frac{1}{\sqrt{3}} \right) = \frac{d}{\sqrt{3}} \approx 1.1547 \, \text{m} \] ### Step 2: Substitute Values into the Formula Now we can substitute the values into the corona loss formula: 1. Convert the temperature to Kelvin for the calculation: \[ T = 40 + 273.15 = 313.15 \, \text{K} \] 2. Substitute the values into the formula: \[ W_c = 0.005 \cdot \frac{(220)^2}{(2)^2} \cdot \left( \frac{1}{\sqrt{313.15}} \right) \cdot \left( \frac{1}{\sqrt{76}} \right) \] ### Step 3: Calculate Each Component 1. Calculate \( \frac{(220)^2}{(2)^2} \): \[ \frac{(220)^2}{(2)^2} = \frac{48400}{4} = 12100 \] 2. Calculate \( \sqrt{313.15} \): \[ \sqrt{313.15} \approx 17.7 \] 3. Calculate \( \sqrt{76} \): \[ \sqrt{76} \approx 8.72 \] ### Step 4: Final Calculation Now substitute these values back into the formula: \[ W_c = 0.005 \cdot 12100 \cdot \left( \frac{1}{17.7} \right) \cdot \left( \frac{1}{8.72} \right) \] Calculating this step-by-step: 1. Calculate \( \frac{1}{17.7} \approx 0.0565 \) 2. Calculate \( \frac{1}{8.72} \approx 0.114 \) Now substitute these values: \[ W_c = 0.005 \cdot 12100 \cdot 0.0565 \cdot 0.114 \] Calculating: \[ W_c \approx 0.005 \cdot 12100 \cdot 0.00644 \approx 0.389 \, \text{W/km} \] ### Conclusion The corona loss per kilometer of the transmission line is approximately **0.389 W/km**.