Why Use |Z(Th)| Instead of R(Th) for Maximum Power Transfer in AC Circuits?

Hello, the maximum power transfer theorem states that maximum power is transferred from source to load when the load impedance Z(load)=R(load) + jX(load) is equal to the conjugate of the Thevenin impedance of the circuit, i.e, when R(load)=R(Th) and X(load) = -X(Th). Here it is asked to find a value of the load resistance R(load) that will absorb a maximum power for the circuit attached herewith. The problem is that, on analyzing it we find the Z(Th.) to be 4+j8 ohm, and since we know that the real or average power is absorbed by the resistance in an impedance, R(load) should be 4 ohm. However, the solution states that R(load) is 8.94 ohm which is simply the magnitude of Z(Th.) obtained. And the real power absorbed is also more if we take 8.94 ohm as the load resistance than the 4 ohm case, which means the solution is correct. Why are we talking about the magnitude of the load for we know that it is simply the resistor which absorbs the average power? w(omega)=1rad/s for this ckt. Help will be greatly appreciated.

Hi,

From a quick definition of Maximum Power Transfer in AC Circuits,

"Although these reactances do not absorb any average power, they will limit the circuit current unless the load reactance cancels the reactance of the Thévenin impedance."

That is maybe the reason we still need to bother about the reactances. And also the same reason why the answer to the question is the magnitude of Z(th).

Hello Kevin,

now the fact turned out to be crystal clear. If I find the Z(Th.) to be R(Th.)+jX(Th.), then I'm not going to help the load absorb a maximum power by putting in only R(Th.) as the load as the X(Th.) is not cancelled to increase the current.

Thanks a lot.

Hello Rajdeep!

Ah, I think you took the symbolism of R(load) too literally.
It doesn't mean that R(load) has to be a resistor.
It is just common sometimes to name load that way.

I think it would be better if you remembered and symbolized it as below:
Z(load)=|Z(th.)| = true "overall" impedance

R(load)=real(Z(load))

X(load)=im(Z(load))

For the example image you showed, the thevenin is 20+10j.
So the complex conjugate is 20-10j.
This means for maximum power transfer, you must connect "10j valued" capacitor
and a 20 ohm resistor.

Hope this helps.

Warm Regards and a Happy New Year!
Justin

Hi Justin,

Thank you. That really helped.

Happy New Year to you too.