# Maximum power theory in electronics

#### logearav

Joined Aug 19, 2011
243
Please see my attachment. From the circuit
Energy transferred to the load E = Ir + IR, where r is the internal resistance and R is Load Resistance. so
I = E/(r+R)
The Power W is W = I^2R = E^2R/(r+R)^2
Upto this i understand. I can't understand the following steps
As R changes, so does W. W will be maximum when dW/dR is zero
that is d/dR(E^2R/(r+R)^2) = 0
My question is why W is maximum when dW/dR = 0. Why dW/dR concept is introduced here

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#### t_n_k

Joined Mar 6, 2009
5,455
The derivative (slope) of a function with respect to some independent variable indicates that either a maximum or a minimum occurs when the slope is zero. In this case the derivative of power in relation to the load resistance necessitates that a maximum occurs when the derivative is equated to zero. This occurs when R=r.

#### steveb

Joined Jul 3, 2008
2,436
t_n_k is correct (as usual ), but keep in mind that, in general, zero slope can be a local maximum, a local minimum or even an inflection point that is not a local maximum nor a local minimum. There is also the concept of a global maximum which is the point with the highest value, and this point could even be at plus or minus infinity, or any point where the function goes to infinity. See the following for more details.

http://en.wikibooks.org/wiki/Calculus/Extrema_and_Points_of_Inflection

So, there are two levels to your question. The first level is what t_n_k is addressing, which is just the basic understanding of what derivative means and why a maximum may occur at a point where the slope is zero.

But, your question is very good a one. The answer of "the maximum occurs at dW/dR=0" is not complete. The full answer is that the maximum may occur at a point where the slope is zero, but you have to do more tests using the second derivative, taking formal limits, and/or by plotting the function, to characterize the function and identify for sure that the point you think is the maximum, really is the maximum.

So, the question is really to ask where the global maximum is, but in your case, the global maximum is also a local maximum, and there is only one local maximum. With experience, you would not need any detailed analysis to see that an inverted parabolic-like shape (which is what you have over a practical range of R) is like a simple hill with the highest point at the top of the hill. With experience you will also notice when the function is very complex and a more careful analysis is needed to not get fooled (as often happens to me when I'm not careful).

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#### hgmjr

Joined Jan 28, 2005
9,029