Max+power+theorem

BEF 12403

Maximum Power
Transfer Theorem

Learning Outcomes:

Upon completion of this unit the student will be able to:
(i) Determine the conditions for maximum power transfer
to any circuit element.

Maximum Power Transfer Theorem
The power delivered by a voltage source or a current source
is a function of its internal resistance and also of the load
resistance. Maximum power is delivered by a source to its
load when the load resistance is equal to the internal
resistance of the source, that is, when
Rin = RL

E i

Rint
v RL Figure 2

Practical voltage
source under test

Proof

Consider Figure 2 which shows a practical voltage
source delivering power to a load resistor RL. The
power delivered to RL is given by
E
i= E i
Rint + RL
vL RL
Rint
and that the load current is given by the
expression
Practical
PL = i RL2
voltage source
under test

By combining the two equations above, we Figure 2
obtain the expression

E2
PL = RL
( RL + Rint ) 2

Proof (continued)

To find the value of RL that maximises the expression for PL,
we first need to differentiate the expression for PL with respect
to RL and then equate it to zero. Computing the derivative, we
obtain the following expression:
E 2 ( RL + Rint ) − 2 E 2 RL ( RL + Rint )
2
dPL
= =0 E i
dRL ( RL + Rint ) 4
v RL
Rint
which leads to the expression
( RL + Rint ) 2 − 2 E 2 RL ( RL + Rint ) = 0 Practical
voltage source
It is easy to verify that the solution of this under test

equation is Figure 2
RL= Rint
Thus, to transfer maximum power to a load, the equivalent
source and load resistances must be matched, that is, equal
to each other.

• This is called a "matched condition" and as a general rule,
maximum power is transferred from an active device such
as a power supply or battery to an external device when
the impedance of the external device exactly matches the
impedance of the source.
• One good example of impedance matching is between an
audio amplifier and a loudspeaker.
• Improper impedance matching can lead to excessive
power loss and heat dissipation.

Proof (continued)
Figure 3 depicts a plot of the load power PL divided by E2
versus the ratio of RL to Rint. Note that this value is maximum
when RL = Rint.
PL/E2

RL/Rint
0 RL = Rint

Figure 3

When the conditions of the maximum power transfer theorem
are met, the total power delivered to the load is:

PL =
( E / 2) 2
=
E2 E i
vL RL
Rint 4 Rint Rint

This is one half of the total power generated
Practical
by the circuit, half the power is absorbed in voltage source
the resistance RINT. under test

Figure 2

Worked Example

Given the circuit in Figure 4, where:
RS = 25Ω
RL is variable between 0 - 100Ω
VS = 100v
Complete the following table to determine
the current and power in the circuit for
different values of load resistance.
Figure 4

Solution

Graph of Power against Load Resistance:

Solution (continued)

From the above table and graph we can see that the Maximum
Power Transfer occurs in the load when the load resistance, RL is
equal in value to the source resistance, RS that is: RS = RL = 25 Ω.

Exercise

Plot the power dissipation of the load resistance, for several
values between 1 kΩ and 20 kΩ:

At what load resistance value
is the load's power
dissipation maximized?

Figure 4

Worked Example

Suppose we were planning to use a photovoltaic panel to
generate electricity and electrolyze water into hydrogen and
oxygen gas:
Our goal is to electrolyze as much
water as possible, and this means we
must maximize the electrolysis cell's
power dissipation. Explain how we
could experimentally determine the
optimum internal resistance of the
electrolysis cell, prior to actually
building it, using nothing but the solar
panel, a rheostat, and a DMM (digital
multimeter).

Answer
Experimentally determine what amount load resistance drops
exactly one-half of the panel's open-circuit voltage.

Max+power+theorem

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