Maximum power transfer theorem

Maximum power theorem is a theorem or technique used in Electrical Network Analysis and Electrical circuit designing.  It was invented by a German engineer Moritz von Jacobi in 1840. He invented the theorem in the process of finding a way to maximize the output of the battery to a motored boat which he designed to travel in the river Neva ; Thus the theorem is also sometimes refereed to as Jacobi’s Law.

Maximum power transfer theorem deals with the power transferred to the load on a circuit with a network of various sources or components on it. The maximum power transfer theorem defines the condition under which the maximum power is transferred to the load in a circuit.

Note: Here we are talking about maximum power transferred to the load only, not about the maximum power transferred to the load and internal components or resistance of the source combined, Under the condition of Maximum power transfer we only deal with the power transferred to the load and does not consider the power dissipated in internal circuits or resistance of the source so we are not talking about the maximum efficiency of power transfer but instead maximum possible power transfer from a source to a load.
The Maximum Power Transfer Theorem states that:

The power transferred from a source or circuit to a load is maximum when the resistance of the load is made equal or matched to the internal resistance of the source or circuit providing the power to the load.

For Example:

In the following Circuit:

maximum power transfer theorem example
maximum power transfer theorem example

According to maximum power transfer theorem the maximum power will be yielded to the load RL when RL is equal to the internal resistance of the circuit or R1+R2.

The Maximum power transfer theorem holds true in any kind of circuit may it be linear, non-linear, active, DC or AC. In the case of DC circuits the load resistance is matched with internal resistance of the source by making both resistance equal and in case of AC the Load impedance is matched with the internal impedance of the circuit or source by making the load impedance the complex conjugate of the source impedance. For eg: load impedance will be R_1 - jX if the internal impedance of the source is  R_1 + jX

Proof of Maximum Power Transfer Theorem:

Maximum power transfer theorem can be proved in DC networks or resistive circuits as following:

Let,
V = EMF supplied to the load.
R_L = Load resistance.
R_i = Internal resistance of the source.

I = Current flowing through the load, internal resistance and the source of the circuit.
P_L = Power transferred to the load.
P_i = Power dissipated at internal resistances.

Then,

Power transferred to the load = P_LI^2 R_L
or,
P_L\left( \dfrac{V}{R_i + R_L}\right)^2 \times R_L\dfrac{V^2}{\frac{R_i^2}{R_L}+2R_i +R_L}

Now using the theorems of Differential calculus , If we keep the RL variable and want to calculate the maximum value of PL then we need to differentiate the PL with respect to RL and equate it with zero.
Thus,
Under Maximum power transfer to load condition:
\dfrac{d}{dR_L}P_L = \dfrac{d}{dR_l}\dfrac{V^2}{ \frac{R_i^2}{R_L}+2R_i +R_L}= 0
or,
-\dfrac{R_i^2}{R_L^2} +1= 0

or,

R_i = R_L

And in AC networks using same mathematical technique we can prove:

if,
Z_i = R_i + X_i = Internal impedance of reactive circuits.
Z_L = R_L + X_L = Load impedance.
Then, Under the condition of maximum power transfer to load:
Z_i = Z_lR_i = R_L and X_i = - X_L

Power Transfer Efficiency:

Power transfer efficiency is the efficiency of any source or circuit in transferring it’s power to the load. Or it is the ratio of power transferred to the load over total power transferred by the source.
It is denoted by the Greek letter \eta

And Mathematically:  \eta = \dfrac{P_L}{P_T} , Where P_L is the power transferred to the load and P_T is the total power transferred by the source.
We can expand this expression as:
P_L = I^2 R_L and P_T = P_L + P_I = I^2R_L + I^2 R_I , where, I is the total current flowing through the circuit  or Thevenin equivalent of the circuit, P_I is the power dissipated in internal circuits of the source.
Thus,
\eta=\dfrac{P_L}{P_T}=\dfrac{I^2R_L}{I^2R_L+I^2R_I}=\dfrac{R_L}{R_L+R_I}=\dfrac{1}{1+\frac{R_I}{R_L}}

As we know that under the condition of maximum power transfer R_LR_I , we can derive from above formula that the efficiency under the condition of maximum power transfer is only 0.5.

The overall efficiency decreases if the R_L is kept very low and it increases up to  one when the R_L is increased to infinity as shown on the graph below:

Graph: Power transfer efficiency Vs Load resistance
Graph: Power transfer efficiency Vs Load resistance

And the power transferred to load becomes minimum when R_L is kept very low and very high. The maximum power transfer to load is obtained when R_L = R_I. As shown on the following graphs:

Graph: Power transfer to load Vs Load Resistance.
Graph: Power transfer to load Vs Load Resistance.

 

Applications of Maximum Power Transfer Theorem:

The Maximum Power Transfer Theorem has a wide range of usage on real life situation. The theorem is used to maximize the power output to a load from any circuit. So they can be used to design circuits where the maximum output performance is desired for example to match an Amplifier with a Loudspeaker to yield maximum power to the speaker and thus produce maximum sound.
In some situations Transformer Coupling are also used to yield maximum power to the load when the matching of Load and Source impedance is not possible for example is the amplifier is of 1000 Ohms and the speaker if of 10 ohms.

The application of Maximum Power Theorem is done only under the conditions when the maximum performance is desired over the overall efficiency of the circuit because as we discussed above the efficiency of a circuit under maximum power transfer condition is only 0.5. So, Maximum power transfer theorem is applied in radio electronics; for example: In Antenna Signal amplifier for radio and TV receivers; and various other fields where maximum performance is required but the maximum efficiency is not desired.

Elaboration of Maximum power transfer theorem with example:
let an amplifier circuit provides 20 voltage with 5 ohms internal resistance.

If you connect a 1 ohm speaker to the circuit:

Total current flowing through the system( internal resistance and speaker) = 20 V / (5+1) ohm = 3.333 amps.
Total power usage by the whole system = I^2 * Total resistance = 1^2 * 6 = 66.6666 watts
Power usage by speaker = I^2 * speaker resistance = 11.11 watts.
Power usage by internal resistance = I^2 * internal resistance = 55.55 watts
That means 16% of power is transferred to speaker.

Now, If you connect a 5 ohm speaker to the circuit:

Total current flowing through the system ( internal resistance and speaker) = 20 V / (5+5) ohm = 2 amps.
Total power usage by the whole system = I^2 * Total resistance = 2^2 * 10 = 40 watts.
Power usage by speaker = I^2 * Speaker resistance = 20 watt.
Power usage by internal resistance = I^2 * Internal resistance = 20 watts again. 
That means 50% power is transferred to speaker.

Again, If you connect a 15 ohm speaker:

Total current flowing through the system( internal resistance and speaker) = 20 V / (5+15) ohm = 1 amps.
Total power usage by the whole system = I^2 * Total resistance = 1^2 * 20 = 20 watts.
Power usage by speaker = I^2 * speaker resistance = 15 watts.
Power usage by internal resistance = I^2 * internal resistance = 5 watts
That means 66% of power is transferred to speaker.

In above condition when the one ohm resistance load is connected only 16% power is transferred to the speaker which is 11.11 watts! in second condition when the speaker resistance is matched with internal resistance 50% power is transferred to the speaker which is 20 watts. This is the condition of maximum power transfer. And again in third condition with 15 ohm speaker 15 watt power is transferred to the speaker but it is 66% of the total power transferred.