Introduction to Electric Circuits

3.8 The maximum power transfer theorem 55

Ro

A

IL

RE

Figure 3.28

load resistor R L is given by PL -- /L2RL 9 But I L -- Eo/(Ro + RL) SO that

PL = Eo2RL/(Ro + RL) 2 (3.1.4)

As RE varies, with E0 and R0 being constant, this will be a maximum (or a

minimum) when dPc/dRc = 0. Using the technique for differentiating a

quotient we get

dPL/dRL = {(Ro + RL)2Eo 2- Eo2RL[2(Ro + RL)]J/(Ro + RL) 4

This will be zero when the numerator is zero, i.e. when

(R0 + RL)ZEo 2: 2Eo2RL(Ro + RE)

Ro + RL = 2RL

R0 = RL (3.15)

This can be confirmed as a maximum, rather than a minimum, by showing that

d2pL/dRL 2 is negative.

The power delivered to the load is therefore a maximum when the resistance

of the load is equal to the internal resistance of the source or network, and this

is called the maximum power transferred theorem. The actual value of the

maximum power transferred is obtained by putting RE = R0 into the equation

for Pc. This gives

emax-- Eo2RL//(RL + RE) 2= Eo2RL//(2RL) 2

emax-- Eo2/4RL 9 (3.16)

Example 3.8

For the circuit of Fig. 3.18 (Example 3.6) determine:

(1) the value of the load resistor, r, which would give the maximum power

transfer; and

(2) the maximum power transferred to the load.

Solution

1 Using the Thevenin equivalent circuit of Fig. 3.19, the maximum power