I was reading a copy of Art of Electronics that i had borrowed from the IIWC librar when i saw a very interesting problem of which i had never thought of :
Prove that maximum power is transferred to the load only when source impedance is equal to load impedance.
A.K.A. the maximum power transfer theorem.
Instead of blindly googling the proof , i decided to find the proof myself. Thankfully , AOE gave a hint that the proof involved calculus. Otherwise , it is doubtful if my math challenged mind would have been able to work this out.
Not finding any ‘noob – understandable’ proofs for the MPTT on the internet , i have posted the proof here.
The problem involves finding when maximum power is delivered to load L.
It is pretty clear that power delivered to the load will be zero when both Rl is 0 and infinity. So , the maximum power should be transferred somewhere in between.
At the point where the power transfer peaks , the tangent to that point will be horizontal. Therefore , the derivative at that point will be zero.
So , we can find the point of maximum power transfer by finding the maxima of the derivative of the power function.
First step : Find the Power Function
The power transferred to the load will be equal to the voltage across it times the current in it.
Rl and Rs form a voltage divider , hence the voltage across Rl will be
The current through Rl will be
So , the power dissipated by Rl will be the product of the above two equations bcoz P=IV.
Second Step : Differentiate the Power Function.
if we try to differentiate this entire equation , it will result in a reasonably complex derivative for my noob brain due to the presence of the variables V and Rs.
Instead , to simply the process , i have taken V and Rs as two constants and then have found the derivative.
I have taken V = 5V and Rs = 7Ω.
So , we get the Power as :
When this is differentiated , you get the derivative as :
Third Step : Find the Maxima
Setting the derivative to zero to find the maxima :
The solution to this equation is 7 !! Which is precisely the same as the value of Rs !!!
If you find the solution to the equation by taking Rl and V differently , you will still find that Rl = Rs each and everytime!!
So , it is indeed true that maximum power is transferred when source resistance = load resistance.
