understanding maximum power transfer theorem

[PG1995]

Hi

I'm trying to understand 'background' working of the maximum power transfer which says that power transfer takes place across R_L when it is equal to R_th (R_L is load resistance and R_th is Thevenin resistance). It is easily proved by doing some math.

P=VI, P=I^2.R, P=V^2/R, V=IR

Let's focus on P=V^2/R_L. To have maximum power transfer to R_L, V should be as large as power (V is volts dropped across R_L). V across R_L could be found using voltage divider rule: (R_L x E)/(R_L + R_th). Further R_L should be as little as possible because it is denominator. But also note that making R_L smaller would reduce the volts dropped across R_L.

But now focus on P=I^2.R_L. Compare it with the previous analysis of P=V^2/R_L. In (I^2.R_L), R_L should be as large as possible which is in contrast with the previous analysis which required R_L to be minimum. So some compromise is needed. But there is another point to note. R_L and R_th are in series so if R_L is made too big then the value of I would drop.

I hope you could see where I'm coming from (or, rather trying to come from! ). Could you please help me to really understand the working of maximum power transfer theorem? Many thanks.

 

[carbonzit]

Power is one thing, and you've correctly identified the formulas for calculating it.

But determining maximum power is a calculus problem (finding maxima/minima: remember that?).

 

[RCinFLA]

The calculus is the incremental change as you look for the highest power to the load resistor.

You can do the solution by incrementally varying the load resistance. Given a fixed idea voltage source in series with a fixed source resistance feeding a load resistance, vary the load resistance from zero to infinity and calculate V^2/R for each incremental step. Do some manual coarse steps first then home in on the maximum power point.

In calculus, it is where the rate of change of the rate of change goes to zero, called second derivative = zero. If the rate of change of the rate of change confuses you think of travel. There is distance, speed, and acceleration. The rate of change of distance (like miles per hour) is the first derivative, the rate of change of speed is the acceleration which is the first derivative of speed and second derivative of distance vs. time. The math is differential equations which is a cousin of calculus.

Most computer algorythms use a scheme of incremental discrete calculations that zero's in on the answer, similar process to your multiple manual trials.

 

[KeepItSimpleStupid]

Even if there wasn't Calculus, think of a 50 ohm output transmitter operating into a load. If Z(load) isn't equal to the transmitter's Zout, there will be reflections etc. This is, in essence a transmission line.

 

[mbarazeen]

you have soem what explaiend the answer, you are right about you calculations too, say we always prefer R_th to be as lower as possible to get more power out by decreasing R_L, since R_th is there it makes voltage drop so that at a point your drawn power will start to decrease with lowering R_L.

you work with some practical values, say your source is 5V, R_th= 2, now you connect a 1 ohm to the source and calculate the power, then 3 ohms, keep on approaching until 2, the maximum value would be at 2. as i remember this method is called successive approximation...