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In many practical situations, a circuit is designed to provide power to a load. There are applications in areas such as communications where it is desirable to maximize the power delivered to a load. This method is called the maximum power transfer theorem.

**What is the Maximum Power Transfer Theorem?**

We now address the problem of delivering the maximum power to a load when given a system with known internal losses. It should be noted that this will result in significant internal losses greater than or equal to the power delivered to the load.

Maximum power transfer theorem is one of the popular electric circuit analysis. This theorem will help us to simplify a complex electrical circuit into a maximum external power generated with an internal resistance.

That is the basic idea of it.

**Maximum Power Transfer Theorem Statement**

The maximum power transfer theorem states that:

In order to generate the maximum external power with a finite internal resistance, this finite internal resistance has to be equal to the resistance of the available sources in the same circuit.

This statement shows that the said resistance is equal to the Thevenin equivalent resistance.

## Maximum Power Transfer Theorem Formula

The Thevenin equivalent is useful in finding the maximum power a linear circuit can deliver to a load. We assume that we can adjust the load resistance R_{L}.

If the entire circuit is replaced by its Thevenin equivalent except for the load, as shown above, the power delivered to the load is

For a given circuit, V_{Th} and R_{Th} are fixed. By varying the load resistance R_{L}, the power delivered to the load varies as sketched below. We notice that the power is small for small or large values of R_{L }but maximum for some value of R_{L }between 0 and ∞.

Below is the maximum power transfer theorem graph based on our discussion above.

Let me show that this maximum power occurs when R_{L }is equal to R_{Th}. This is known as the maximum power theorem.

Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load (R_{L }= R_{Th})

To prove the maximum power transfer theorem, we differentiate p in Equation.(1) with the respect to R_{L }and set the result equal to zero. We obtain

which yields

**Maximum Power Transfer Theorem Formula**

Just as stated above, the maximum power transfer theorem formula will focus on an electrical circuit with a source and a variable load resistance R_{L }acts as a finite internal resistance.

Observe the circuit below.

The maximum power transferred is obtained by substituting Equations.(3) to (1), for

Equation.(4) applies only when R_{L }= R_{Th}. When R_{L }≠ R_{Th}, we compute the power delivered to the load using Equation.(1).

**Maximum Power Transfer Theorem Solved Problems**

For better understanding, let us review the example below :

1. Find the value of R_{L }for maximum power transfer in the circuit below

Answer:

We need to find the Thevenin resistance R_{Th }and the Thevenin voltage V_{Th }across the terminals a-b.

To get R_{Th}, we use the circuit in above and obtain

To get V_{Th}, we consider the circuit below

Applying mesh analysis gives

Solving for i_{1}, we get i_{1} = -2/3. Applying KVL around the outer loop to get V_{Th }across terminals a-b, we obtain

For maximum power transfer,

and the maximum power is