Maximum power transfer in DC circuit happens when the load resistance is equal to the source resistance. How can you prove this?

To prove this you need to consider a very simple DC circuit with a DC signal source, source resistor and load resistor. This is shown below.

How to proof maximum power transfer in DC circuit
In the above figure Vs is DC voltage source, say of 1V. RS and RL are source and load resistors. Now assume some resistance value for RS, say 1Ohm.

Then maximum power transfer theorem for DC circuit says that RL is also 1Ohm for maximum power transfer. We can prove it by two different method actually. One is by simply graphical method and the other is using derivative. Here we show via graphical method.

To prove via graphical method, find the power at the load(PL). Plot this power at the load against load resistor value and you will find that power is maximum when the load resistance is also 1Ohm.

You will get something like this-

How to proof maximum power transfer in DC circuit

 You get the equation PL vs RL as follows.

\[P_{L}=\frac{V_{L}^{2}}{R_{L}}\]
using,
\[V_{L}=\frac{R_{L}}{1+R_{L}}\]
We get,
\[P_{L}=\frac{R_{L}}{(1+R_{L})^2}\]
Now if you plot this equation you will get the above graph.

You could have taken any value for source voltage and source resistor but no matter what, you will always get maximum value of power at the load when the load resistance is equal to source resistance. Hence in this way you can prove maximum power transfer theorem in DC circuit. In AC circuit we will have similar arguments for maximum power transfer.

Now, for those who are interested in using derivative to find the condition for maximum power transfer, you just have to take the derivative of PL(the last equation), then you equate it to 0 for maximum power. And then you find the value of RL which will also give you the same 1Ohm.

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