A high-voltage transmission line with a resistance of 0.30 /km carries a current of 1040 A. The line is at a potential of 680 kV at the power station and carries the current to a city located 159 km from the station.
(a) What is the power loss due to resistance in the line?
b) What fraction of the transmitted power does this loss represent?
Lets assume it is a three phase transmission line, three wires running 159000 m
Total power loss: 3*.30*159*1040^2=about 15megawatts.
Total power transmitted: 680k*3*1040= about 2 G watts The 3 is because there are three phases.
fraction=about 15M/2000M=less than 1 percent.
check my thinking
To find the power loss due to resistance in the transmission line, you can use the formula P = I^2 * R, where P is the power loss, I is the current, and R is the resistance.
Given:
Resistance (R) = 0.30 Ω/km (per kilometer)
Current (I) = 1040 A
Distance (d) = 159 km
(a) Power loss due to resistance in the line:
To calculate the total resistance (RT) in the transmission line, you'll need to multiply the resistance per kilometer (R) by the distance (d): RT = R * d.
RT = 0.30 Ω/km * 159 km = 47.7 Ω.
Now, plug the values of RT and I into the power loss formula: P = I^2 * RT.
P = (1040 A)^2 * 47.7 Ω = 54,873,600 W.
Therefore, the power loss due to resistance in the line is 54,873,600 watts or 54.87 MW.
(b) Fraction of the transmitted power represented by this loss:
The total power (PT) transmitted through the line can be calculated using the formula PT = V * I, where V is the potential difference (voltage) and I is the current.
Given:
Potential difference (V) = 680 kV (kilovolts)
Current (I) = 1040 A
PT = (680 kV) * (1040 A) = 707,200 kW.
To find the fraction of power loss, divide the power loss (P) by the total power (PT):
Fraction of transmitted power lost = P / PT = 54.87 MW / 707,200 kW.
To convert MW to kW, multiply by 1000:
54.87 MW = 54.87 x 1000 kW = 54,870 kW.
Now, divide the power loss (54,870 kW) by the total power (707,200 kW):
Fraction of transmitted power lost = 54,870 kW / 707,200 kW.
Divide the numerator and denominator by their greatest common divisor (GCD) to simplify the fraction if possible.
So, the fraction of transmitted power that represents the power loss due to resistance in the line is 0.0777 or 7.77%.