# Maximum Power Transfer Theorem

## What is the Maximum Power Transfer Theorem Statement?

In a DC circuit, the source will supply the power to the resistive  load and load receives the power and gets dissipated in the load. We will always try to increase the power transferred to the load by the source. The maximum power transfer theorem deals with the condition when the power supplied by the source is transferred to the resistive load at a maximum rate. We will discuss the maximum power transfer theorem in detail and also solve problems using maximum power transfer theorem.

### Maximum Power Transfer Theorem

Maximum power transfer theorem states that the voltage source in a DC circuit  will deliver maximum power to the resistive load connected to the voltage source when the load resistance is equal to the source resistance. We can take a resistive load in a DC circuit which is equal to the resistance of the source and a maximum power will be transferred to the load. By maximum power transfer theorem application, we are able to increase the signal strength in communication systems.

### Maximum Power Transfer Theorem Proof

Consider the circuit in which a DC source network is connected to the load resistance as shown in figure A below. We have to find the thevenin voltage and thevenin source of the source and the circuit is transformed to another circuit as shown in figure B.

To prove the maximum power transfer theorem, we use the circuit shown in figure B. First, we have to find the current passing through the circuit given by,

I = $\frac{V_{TH}}{R_{TH}+R_{L}}$…(1)
Where,

I- Current passing through the circuit

V$_{TH}$- Thevenin voltage of the source

R$_{TH}$ - Thevenin resistance of the source

R$_{L}$ - Resistance of the load
We can calculate the power delivered to the load resistance by the given equation,

P$_{L}$ = I$^{2}$R$_{L}$… (2)
Where,

P$_{L}$ - Power delivered to the load

Now, substitute the value of I from equation (1) in equation (2) to obtain the power delivered to the load resistance in terms of thevenin voltage.

P$_{L}$ = I$^{2}$R$_{L}$
P$_{L}$ =$\left ( \frac{V_{TH}}{R_{TH}+R_{L}} \right )^{2} R_{L}$…(3)

Differentiate the equation with respect to RL and equate to zero to obtain the condition for maximum power transfer

⇒$\frac{dP_{L}}{dR_{L}}$=0

$\frac{dP_{L}}{dR_{L}}$=$\frac{d}{dR_{L}}$$\left[\left(\frac{V_{TH}}{R_{TH}+R_{L}} \right )^{2} R_{L} \right ]$=0

$\frac{V^{2}_{TH}\left ( R_{TH}-R_{L} \right )}{\left ( R_{TH}+R_{L} \right )^{2}}$=0

(R$_{TH}$-R$_{L}$)=0

R$_{TH}$ =R$_{L}$

Therefore, the power delivered to the load resistance is maximum when the thevenin resistance is equal to the load resistance. This is in accordance with the maximum power transfer theorem. Hence, the maximum power transfer theorem is proved.

### Maximum Power Delivered to Load Resistance

We have seen the maximum power transfer theorem proof. Now, let's calculate the maximum power delivered to the load resistance when thevenin resistance is equal to the load resistance.

For the maximum power transfer to occur,

R$_{TH}$=R$_{L}$
Substitute R$_{TH}$ in the place of R$_{L}$ in the equation for power delivered to the load resistance to obtain the maximum power delivered.

Then maximum power delivered to the load resistance is given by,

P$_{max}$ =$\left ( \frac{V_{TH}}{R_{TH}+R_{TH}} \right )^{2} R_{TH}$

P$_{max}$ = $\left ( \frac{V_{TH}}{2R_{TH}} \right )^{2} R_{TH}$

P$_{max}$ = $\frac{V^{2}_{TH}}{4R_{TH}}$

The above equation is the maximum power transfer theorem formula to calculate the maximum power delivered to the load. Therefore, we can calculate the maximum power delivered to load by knowing the thevenin voltage and thevenin resistance.

### Steps to Solve Network using Maximum Power Transfer Theorem

Let us see the steps to calculate the maximum power transferred to the load for problems  using the maximum transfer theorem which are given below

Step 1: Identify the variable load resistance in the circuit and remove the load resistance.

Step 2: Replace the independent source voltage by a short circuiting the terminals and the independent current source by an open circuit.

Step 3: Find the thevenin resistance of the circuit by calculating the equivalent resistance between the terminals of the open circuited load resistance.

Step 4: Find the thevenin voltage by calculating the voltage across the terminals of the open circuited load resistance and find the maximum power delivered using the maximum power transfer theorem formula.

### Conclusion

Maximum power transfer theorem states that maximum power will be delivered by a DC source to the load resistance when the load resistance and source resistance are equal. So, if the load resistance is a variable resistance, we can vary the resistance of the load until it becomes equal to the source resistance and maximum power will be transferred to the load. For a given circuit, we can find the thevenin resistance of the circuit and the load resistance has to be equal to the thevenin circuit for a maximum power transfer. During maximum power transfer, the efficiency becomes 50%.

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1. State the maximum power transfer theorem for AC circuits.

Ans: According to the maximum power transfer theorem for the AC circuits, an AC voltage source will deliver maximum power to the complex load when the impedance of the load is equal to the complex conjugate of the source impedance. Unlike DC circuits, inductive reactance and capacitive reactive also come into the picture in AC circuits. If the source impedance is represented by,

ZS =  a + bj

Then the load impedance must be equal to the complex conjugate of the source impedance given by,

ZL =  a - bj

2. Give some examples for  maximum power transfer theorem applications.

Ans: In a communication system, the signal strength received by the antenna is low. Therefore, to receive the maximum signal from the antenna to the receiver, the impedance of the antenna and receiver are made complex conjugate.

A loudspeaker is used to produce sound waves that can reach to maximum. This is made possible by keeping the amplifier resistance and the resistance of the speaker equal.

To turn on the engine of a car using a car engine starting system, the resistance of the starter motor is made equal to the internal resistance of the battery to ensure maximum power transfer so as to turn on the engine of the car.

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