# Maximum Power Transfer Theorem

#### fine1988

Joined Feb 10, 2017
5
Hi. I know that maximum power transfer theorem states that the power transferred to the load will be maximum if source and load resistance is equal. I also know that this is not the same as efficiency which will increase if load resistance is greater than source resistance. However, I do not understand WHY this theorem is true. My common sense tells me that the power on the load should increase if its resistance increases.

#### Papabravo

Joined Feb 24, 2006
19,617
Hi. I know that maximum power transfer theorem states that the power transferred to the load will be maximum if source and load resistance is equal. I also know that this is not the same as efficiency which will increase if load resistance is greater than source resistance. However, I do not understand WHY this theorem is true. My common sense tells me that the power on the load should increase if its resistance increases.
You can approach the problem in two ways. You can use calculus to compute the maximum power transfer as a function of the ratio of source and load resistance. You can also make a table and convince your self that the maximum occurs when the resistances are equal.

#### crutschow

Joined Mar 14, 2008
31,151
.....However, I do not understand WHY this theorem is true. My common sense tells me that the power on the load should increase if its resistance increases.
Common sense doesn't always work where electricity is concerned.

The Theorem assumes a fixed supply voltage and source resistance.
For a constant supply voltage the load power will not increase with an increase in resistance, it will actually drop as the resistance increases (V²/R).
Going in the other direction, the source resistance will start to reduce the load voltage as the load resistance drops and the current increases, limiting the increase in load power at some point.
If you do the math to determine the maximum load power you will find that it occurs when the two resistances are equal.

Below is the LT spice simulation of the load resistance power (yellow trace) versus load resistance.
As you can see, it peaks when the load resistance equals the source resistance of 10 ohms.

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#### BobTPH

Joined Jun 5, 2013
6,095
If you can change the source resistance, the maximum power will always be when the source resistance is lowest. The theorem applies with the source resistance is fixed. Then, the maximum power will be transferred when the load resistance is the same as the source resistance.

A good example is audio amplifiers. Suppose you want to drive an 8 Ohm load with 10V. Let's see what the power is with different source resistances.

Source resistance 0. Voltage across speaker 10V. Power to speaker 10 ^ 2 / 8 = 12.5W.

Source resistance 8 Ohms. Voltage across speaker 5V. Power to speaker 5 * 2 / 8 = 3.1W

Source resistance 16 Ohms. Voltage across speaker 3.33V Power to speaker 3.3 ^ 2 / 8 = 1.4W

Now lets take a fixed source resistance of 4 Ohms and try different speakers.

Speaker impedance 2 Ohms. Voltage across speaker 3.33V. Power to speaker = 3.3 ^ 2 / 2 = 5.5W

Speaker impedance 4 Ohms. Voltage across speaker 5V. Power to speaker 5 ^ 2 / 4 = 6.3W

Speaker impedance 8 Ohms. Voltage across speaker 6.66. Power to speaker 6.66 ^ 2 / 8 = 5.5W

The common misinterpretation of theorem is that you can get the most power into a fixed load by making the source resistance the same as the load resistance. As my examples above show, this is not the case. To get the most power to a given load, the best source will have 0 resistance.

Bob

#### crutschow

Joined Mar 14, 2008
31,151
The theorem applies with the source resistance is fixed.
The theorem is true whether the source resistance is fixed on not.
But it's usually applied when the source resistance is fixed.

#### MrAl

Joined Jun 17, 2014
9,639
The theorem is true whether the source resistance is fixed on not.
But it's usually applied when the source resistance is fixed.

Hi,

Yes i agree fully.

To the OP:

The main reason for having a theorem sometimes is so that we can understand a given phenomenon from the preexisting set of assumptions. This can not be overstated i dont think.

In any problem you have to know what the assumptions are beforehand so that you can come up with reasonable solutions. In the case of the MPT Theorem, it was made so that we can understand how to get the maximum transfer of energy from some source to some load, when the source is fixed. So the assumption here is that we have NO CONTROL over the source, only over the load. Once we understand this then we can easily understand the reason for this theorem.

Note that to ask the question about the efficiency when the source resistance changes is counter to following the set of assumptions for the MPT Theorem. Thus the question of efficiency becomes a separate issue. The main idea is that given a source that we can not change, find a load that allows for the best possible extraction of energy from that source, and the best usually means to get the most out of the source.

If this still isnt clear, consider the case of a source of 10v with a series resistance of 9 ohms.
Now the MPTT tells us that the best load is 9 ohms, but gee that leads to a pretty nasty efficiency of only about 50 percent. Who wants that?
Now if we are allowed to change the SOURCE series resistance, then why not go down to 1 ohm, and then the same resistance of 9 ohms gives us more voltage and that's got to allow more power to flow to the output. In fact the power before was 2.77 watts and now it is 9 watts, much better right?
But now let's decrease the output resistance to 1 ohm. Now the power out is 25 watts.
But then why not go down to 0.1 ohms for the source resistance, or 0.01 ohms or 0.001 ohms. We almost never have that option otherwise we'd get an infinite power output with 0 ohms source resistance. The power output with 0.001 ohms is 25000 watts with a matching load resistance. This helps ti illustrate the problem we encounter. Most power stages have a power rating associated with them with a certain load impedance, and that's usually the matching impedance. If we did change teh source resistance and then adjusted the load accordingly, we'd blow the stage.

So in each case once we match the source impedance we get the maximum power transferred to the output. It's almost always true though that we can not change the source resistance so we have to just calculate the best load.

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#### BobTPH

Joined Jun 5, 2013
6,095
The theorem is true whether the source resistance is fixed on not.
But it's usually applied when the source resistance is fixed.
It's a matter of semantics. For any given source resistance, the max power is when the load resistance is the same as the source resistance.

But if you hold the load resistance constant and vary the source resistance, the max power is when you make the source resistance zero. This is the misunderstanding I have seen many beginners struggle with.

Bob