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# Maximum Power Transfer Theorem

By BYJU'S Exam Prep

Updated on: September 25th, 2023

The **maximum power transfer theorem** is one of the important Network theorems. The theorem we use for solving the given electrical network/circuit is known as Network Theorem/Circuit theorem. The maximum power transfer theorem formula will help find the maximum power across the load resistance.

We can use the Maximum power transfer theorem whenever the load resistance is variable. In this article, get an overview of this theorem. Here, first will state the maximum power transfer theorem, then the derivation and important formulae.

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Table of content

## What is the Maximum Power Transfer Theorem?

Using the Maximum power transfer theorem, we will know in what condition we can get the maximum power across the variable load resistance. This theorem is just a continuation of Thevenin’s theorem. Thevenin’s equivalent circuit consists of a Thevenin’s voltage source, V_{Th,} in series with the Thevenin’s resistance, R_{Th}.

### Statement of Maximum Power Transfer Theorem

The statement of maximum power transfer theorem is that the value of load resistance, R_{L,} must and should be equal to the value of Thevenin’s resistance, R_{Th,} to get the maximum power across the load resistance, R_{L}. This is the statement of the maximum power transfer theorem. We can use this theorem for linear and Bilateral networks/circuits.

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## Maximum Power Transfer Theorem Formula

We have now discussed the derivation of the maximum power transfer theorem and the efficiency by using this theorem. All the important formulae we used and/or derived regarding this theorem are listed below.

- The current, I
_{L}flowing through the load resistor, R_{L}is

I_{L}=V_{Th}/(R_{Th}+ R_{L})

- The power dissipated by the load resistance, R
_{L}is

P_{L}=I_{L}^{2}R_{L}=>P_{L}=V_{Th}^{2}[RL/(R_{Th}+R_{L})^{2}]

- The condition for maximum power across the load resistance is RL=RTh.
- The maximum power across the load resistance is

(P_{L})_{max}=V_{Th}^{2}/4R_{Th}

- The total power supplied by the voltage source in the given circuit is

P_{S}=I_{L}^{2}(R_{Th}+ R_{L})

=>P_{S}=V_{Th}^{2}/2R_{Th}

- The maximum efficiency we can achieve is 50 % by using the maximum power transfer theorem.

We can find Maximum power transfer theorem applications in the receivers of communication systems. In all the real-time applications, we will use the condition of this theorem to get the maximum power across the variable load. In this article, we discussed the statement of the maximum power transfer theorem, its derivation, and the important formula of this theorem. Here, we applied this theorem to DC circuits. Similarly, we can apply this theorem to AC circuits as well.

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## Derivation of Maximum Power Transfer Theorem

From the derivation of this theorem, we will get to know not only the condition at which we can get the maximum power across the load resistance, R_{L} but also its value. Now, let’s see the maximum power transfer theorem derivation.

- In series, we can add the resistances. Since R
_{Th}and R_{L}are in series, the equivalent resistance will be R_{Th}+R_{L}. By using Ohm’s law, we will get the current I_{L}flowing through the circuit as

I_{L}=V_{Th}/(R_{Th}+R_{L}) - The power dissipated by load resistance, RL is

P_{L}=I_{L}^{2}R_{L}

=>P_{L}=[V_{Th}/(R_{Th}+R_{L})]^{2}R_{L}

=>P_{L}=V_{Th}^{2}[R_{L}/(R_{Th}+R_{L})^{2}] - In the given circuit, V
_{Th}and R_{Th}are constant, whereas R_{L}is variable. So, we must differentiate P_{L}concerning R_{L}and equate it to zero.dP

_{L}/dR_{L}=V_{Th}^{2}[{(R_{Th}+R_{L})^{2}(1)-2R_{L}(R_{Th}+R_{L})}/{R_{Th}+R_{L}}^{4}]=0=>(R

_{Th}+R_{L})^{2}-2R_{L}(R_{Th}+R_{L})=0=>(R

_{Th}+R_{L})(R_{Th}+R_{L}-2R_{L})=0=>(R

_{Th}-R_{L})=0=>R

_{L}=R_{Th} - So, the condition for maximum power across the load resistance is R
_{L}=R_{Th}. - We will get the maximum power dissipated by load resistance, R
_{L}by substituting R_{L}=R_{Th}in the P_{L}formula.(P

_{L})_{max}=V_{Th}^{2}/(R_{Th}+R_{Th})^{2}=>(P

_{L})_{max}=V_{Th}^{2}[R_{Th}/4R_{T}_{h}^{2}]=>(P

_{L})_{max}=V_{Th}^{2}[4R_{Th}^{2}]

## Efficiency of Maximum Power Transfer Theorem

Efficiency is a critical parameter in any circuit/network/system. It specifies how efficiently has produced the output for a given input. Here, we are dealing with both the output and input in terms of power. So, because of this, we can call it Power efficiency or efficiency of power.

- The total power supplied by the voltage source in the given circuit is
P

_{S}=I_{L}^{2}(R_{Th}+RL)=>P

_{S}=I_{L}^{2}(R_{Th}+R_{Th})=>P

_{S}=2I_{L}^{2}R_{Th}=>P

_{S}=2(V_{Th}/R_{Th})^{2}R_{Th}=>P

_{S}=V_{Th}^{2}/2R_{Th} - We will represent Power efficiency mathematically as
η=(P

_{L})_{max}/P_{S}=>η=(V

_{Th}^{2}/4R_{Th})/(V_{Th}^{2}/2R_{Th})=>η=0.5

- If we multiply the Power efficiency by 100, it is known as the percentage of power efficiency.
=> η=0.5×100 %

=> η=50 %

- The maximum efficiency we can achieve is 50 % by using the maximum power transfer theorem.

## Maximum Power Transfer Theorem Problems

**Question 1:** How many methods can we calculate the value of the load resistance, RL, to get the maximum power across it for the circuit, which has multiple independent DC sources and resistances?

**Answer:** 2

**Question 2:** How many methods can we calculate the value of the load resistance, RL, to get the maximum power across it for the circuit, which has multiple independent & dependent DC sources and resistances?

**Answer:** 2