Maximum Power Transfer Theorem

The concept of maximum power transfer theorem was proposed by “Moritz Von Jacobi” in the mid 19th century. The other name given to this theorem is Jacobi’s law. The main scenario is to transfer the maximum power and not maximum efficiency. 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”. It can be used in the applications of both AC and DC circuits.

Maximum Power Transfer Theorem Circuit

In the basic circuit diagram it consists of DC voltage source, a series resistance and a load resistance.

In this theorem, the load resistance ‘RL’ will be equal to the internal resistance of the circuit or the addition of R1 and R2 (R1+R2). This theorem is applicable for

• Linear and non-linear circuits
• Active circuits
• AC and DC Circuits

Considering the DC circuits then the load resistance should be equal to the internal resistance of the source by making both the resistance equal.

For AC circuits the load impedance is equal to the internal impedance of source by making the load impedance the complex conjugate of the source.

The load impedance of the circuit is given by R1 – jX
The internal impedance of the source ids R1 + jX

In a maximum power transfer theorem when the circuit is very complex then to solve it will take much time. So, to overcome it, we basically follow to the Thevenin’s theorem, i.e., we replace the complex system into the Thevenin’s equivalent circuit as shown in the below fig.

Let RL be the load resistance and variable resistance

I=VTH/(RL+RTH) VTH/(RL+RTH)

PL = I2RL=(Vth/(RL+RTH))2.RL

V2/(R2/RL +2RI+RL)

Now RL will be varied by using the theorem of differential calculus. To calculate the PL then it has to be differentiated

d/dRL PL=d/dRl V^2/(〖Ri〗^2/RL+2Ri+RL)

Maximum Power Transfer Theorem for DC Circuits

Let’s solve a numerical problem by which its function can be easily understood

RL dissipates more power when RL = RTH = RN

RL = 240Ω

RT = RTH + RL = 480Ω

VL = 6V

Maximum Power

VL = ETH/2

TL = IN/2

IL = VL/RL = 6/240 = 25mA

VL = IL * RL = 25mA * 240

= 6V

PL = VL*IL = 6*25 = 150mW

IL = 33.3 mA

Maximum Power Transfer Theorem for AC Circuits

The maximum power transfer theorem gives an impedance in AC circuit load. The active AC network will have a source of internal impedance ZS which is will be connected to a load ZL. In this theorem, maximum power will transfer from source to load only when the load impedance is equal to the complex conjugate of source impedance ZS.

I=VTH/(ZTH+ZL)

By substituting above given impedance, we get

I=VTH/(RTH+jXTH+RL+jXL)

I=VTH/((RTH+ RL)+j( XTH+ +XL))

Now power delivered to the load is

PL=I2RL

PL=(V2THxRL)/((RTH+ RL)2+( ZTH+ +ZL)2)

Power delivered to load is to be differentiated with respect to XL and equals to Zero to get maximized power.

XL+XTH=0

XL=- XTH

By putting XL in the power equation then we get it as

PL =( V2THxRL)/(RTH+ RL)2

Derivation and equating to zero

RL + RTH= 2RL

RL = RTH

If XL = – XTH RL = RTH then from source to load maximum power transfer takes place. From this we can also conclude that the impedance of the load is complex conjugate of the source impedance that is ZL = Z*TH

Example Problem

ZAB =( (2j)/(2+2j))-1j

= ((4j)/2+2j))-1j

= (4j-2j+2)/2+2j

= 2+2j/2+2j

= 1Ω

ZL=Zth=2Ω

VTH = VAB = 20/2(1+j) X 2

= 14.15∟45

Yet to complete the problem, Pmax = VTH2/4RTH
=196/4

= 49W

Impedance matching of reflection less in the radio, transmission lines, some of the electronic there will be the necessity of matching the source impedance just like transmitter to the load impedance to avoid reflection in the transmission line.

In the reactive circuits this maximum power transfer theorem applies to the source or load are not totally resistive. In any reactive components of source and load should be of equal magnitude but opposite phase. This implies that impedance of both load and source should be complex conjugates of each other

For resistive circuits these two concepts are similar. If in case of source is totally inductive and load to be total capacitive, if resistive losses are absent, then it will receive 100% of the energy from the source but send it back after a quarter cycle.

Power Transfer  Efficiency

Basically, in the maximum power transfer theorem results in maximum power, but not maximum efficiency. If the source resistance is greater than load resistance, power dissipated in the load is decreased while most of the power is dissipated at the source. Then the efficiency gets reduced, in fact becomes lower.

Efficiency = output/input *100

= I2LRL/(2I2LRL)*100
= 50 %

Hence it can be concluded that the efficiency of the maximum power transfer system is 50%.

The graph Power transfer to load Vs Load resistance

Applications of Power Transfer Theorem

In the real time application let’s say in Loud Speaker, we use maximum power transfer theorem. The design of the circuit is made in such a way that amplifies the loudspeaker to get maximum power to the speaker and thus produce the maximum sound. This will really useful in the public meeting.

In many of the Transformers coupling this maximum power transfer theorem is applied to sending the maximum power to the load when the matching of the load and the source impedance is not possible.
Generally in electronics equipments like Radio and Television receivers will be Antenna which amplifies the signal of TV and Radio receiver.

We know it usually in a car engine, the power delivered to the start button of the car will generally depend upon the resistance of the motor and the internal resistance of the battery. Now it will check the condition that if two resistances are equal, maximum power will be transferred to the motor to turn on the engine

Thus, this is all about maximum power transfer theorem. We believe that you have got a better understanding of this concept. Furthermore, any queries regarding this concept or electrical and electronics projects, please give your feedback by commenting in the comment section below. Here is a question for you, what is the main principle of the maximum power transfer theorem?