Chapter 25 Problem 35

Problem

An electric power plant can produce electricity at a fixed power $P$, but the plant operator is free to choose the voltage $V$ at which it is produced. This electricity is carried as an electric current $I$ through a transmission line (resistance $R$) from the plant to the user, where it provides the user with electric power $P^{\prime }$

(a) Show that the reduction in power $\Delta P=P-P^{\prime }$ due to transmission losses is given by

$\Delta P=P^{2}R{\big /}V^{2}$

(b) In order to reduce power losses during transmission, should the operator choose $V$ to be as large or as small as possible?

Solution

(a)

The current through the transmission line is $I=P/V$. The resistance of the wire will result in a transmission loss $I^{2}R$

 $\Delta P=I^{2}R=P^{2}R{\big /}V^{2}$

Equivalently, the voltage drop due to resistance of the transmission line is $V^{\prime }=IR$. The end-user sees $V-V^{\prime }$, and thus, the power they receive is:

$P^{\prime }=(V-V^{\prime })I=VI-V^{\prime }I=VI-I^{2}R=P-I^{2}R$

The power loss is

$\Delta P=P-P^{\prime }=P-(P-I^{2}R)=I^{2}R$

 $=P^{2}R{\big /}V^{2}$

(b)

$\Delta P\propto {\frac {1}{V^{2}}}$

thus $V$ should be as large as possible