Question

A generator supplies an average power of \(12 \mathrm{MW}\) through a transmission line that has a resistance of \(10.0 \Omega .\) What is the power loss in the transmission line if the rms line voltage \(8_{\text {rms }}\) is (a) \(15 \mathrm{kV}\) and (b) \(110 \mathrm{kV} ?\) What percentage of the total power supplied by the generator is lost in the transmission line in each case? (IMAGE CAN'T COPY)

Short answer

Question: Calculate the percentage of the total power lost in the transmission line for each case when the rms line voltage is 15kV and 110kV, considering an average power provided by the generator of 12MW and the resistance of the transmission line as 10 ohms.

Step-by-step solution

Find the current for each voltage value

First, we need to find the current when the rms line voltage is 15kV and 110kV. We can use the formula for power: \(P= V \times I\). Rearranging for current, we have \(I = \frac{P}{V}\). a) For \(15kV\), \(I_1 = \frac{12 \times 10^6}{15 \times 10^3}\) b) For \(110kV\), \(I_2 = \frac{12 \times 10^6}{110 \times 10^3}\) Calculate the values of \(I_1\) and \(I_2\).

Find the power loss for each current value

Now that we have the current values, we can use the formula for power loss in a transmission line: \(P_{loss} = I^2 \times R\). a) For \(I_1\), \(P_{loss1} = I_1^2 \times 10 \Omega\) b) For \(I_2\), \(P_{loss2} = I_2^2 \times 10 \Omega\) Calculate the values of \(P_{loss1}\) and \(P_{loss2}\).

Find the percentage of the total power lost for each case

Finally, we will find the percentage of the total power lost in the transmission line for each case. To do this, we will use the formula: \(\text{Percentage of power lost} = \frac{\text{Power loss}}{\text{Total power supplied}} \times 100\) a) Percentage power loss for \(15kV\): \(\frac{P_{loss1}}{12 \times 10^6} \times 100\) b) Percentage power loss for \(110kV\): \(\frac{P_{loss2}}{12 \times 10^6} \times 100\) Calculate the values of the percentage power loss for each case.