Question

77 kW is to arrive at a town over two 0.140 ? lines. Estimate how much power is saved if the voltage is stepped up from 240 V to 2400 V and then down again, rather than simply transmitting at 240 V . Assume the transformers are each 99% efficient.

Answer 1

Without the transformers, we can find the delivered current, which is the current in the transmission lines, from the delivered power:
                        P_out= IV_out

77*10^3 W = I*(240V), which gives I = 320.833A.

The power loss in the lines is

                        P_L0= I^2*R_lines= ((320.833)^2)*2*(0.140) = 28821.46 = 28.821 kW

With the transformers, to deliver the same power at 240V, the delivered current from the step-down transformer is still 320.833A.

If the step-down transformer is 99% efficient, we have

                        (0.99)I_p2V_p2 = I_s2V_s2;

                        (0.99)I_p2(2400V) = (320.833 A)(240V), which gives I_p2 = 32.40 A.

Because this is the current in the lines, the power loss in the lines is

                        P_L2 = I_p2²R_lines = (32.40 A)^2*(2)(0.140W) = 293.93 W = 0.293 kW

For the 1% losses in the transformers, we approximate the power in each transformer.

                        P_Lt = (0.01)(77 kW + 77.14 kW) = 1.54kW.

The total power loss using the transformers is

                         P_L = 0.293kW + 1.54kW = 1.833 kW.

The power saved by using the transformers is

                          P_saved = P_L0 - P_L = 28.821 kW - 1.833kW = 26.988 kW.