Transformers — Question 9

Question 9

Stepping up transmission voltage reduces line loss quadratically

Ideal transformers deliver 50.0 kW at unity power factor through a line of total resistance 20.0 ohms. Compare receiving-end line voltages 5.00 kV and 25.0 kV. Neglect transformer losses.

1. Find line current and resistive loss in each case.

2. Find total transmission efficiency and the ratio of line losses.

Question 9 - Solution

Stepping up transmission voltage reduces line loss quadratically

(a) Method and calculation

$$I_1=P/V_1=10.0\,A,\quad I_2=P/V_2=2.00\,A.$$$$\boxed{P_{{\mathrm{loss}},1}=2000\,W,\quad P_{{\mathrm{loss}},2}=80.0\,W}.$$

(b) Interpretation and further calculation

$$\eta=P/(P+P_{\mathrm{loss}})\Rightarrow\eta_1=50000/52000,\quad\eta_2=50000/50080.$$$$\boxed{\eta_1=.961538,\quad\eta_2=.998403,\quad P_{{\mathrm{loss}},2}/P_{{\mathrm{loss}},1}=1/25}.$$

Physical check

Receiving-end voltage is specified so current follows directly from delivered power; source power must also cover line heat.