Power in Electric Circuits — Question 5

Question 5

High transmission voltage reduces the loss for a fixed delivered power

A load receives constant power $P_L$ at voltage $V_L>0$. The two supply wires together have resistance $R_w$. Use $I=P_L/V_L$ and source voltage $V_s=V_L+IR_w$.

1. Find wire loss, required source power and delivery efficiency.

2. For $P_L=12.0$ kW and $R_w=2.00\,\Omega$, compare losses at $V_L=240$ V and 2400 V.

Question 5 - Solution

High transmission voltage reduces the loss for a fixed delivered power

(a) Method and calculation

$$P_w=I^2R_w=P_L^2R_w/V_L^2,\quad P_s=V_sI=P_L+P_w.$$$$\eta=P_L/P_s=\frac1{1+P_LR_w/V_L^2}.$$

(b) Interpretation and further calculation

$$I_{240}=50.0\,\mathrm A,\quad P_{w,240}=5000\,\mathrm W.$$$$I_{2400}=5.00\,\mathrm A,\quad P_{w,2400}=50.0\,\mathrm W,\quad P_{w,2400}/P_{w,240}=.0100.$$

Physical check

The comparison holds delivered power fixed; raising voltage across a fixed resistance would be a different constraint.