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My directions for this problem are to find the power delivered when RL=3 and then find the maximum power.

schematic

simulate this circuit – Schematic created using CircuitLab

I found Rth = R1//R2 = 1 Ohm. Then I used Vth=InRth to find that Vth = 6V.

After doing this, I used P=RL*(Vth/(Rth+RL))^2 to get P=27W when RL=3.

From here I set RL=Rth to find the maximum power, then used Pmax=Rth*(Vth/2Rth)^2 to find that Pmax= 9W.

I know this can't be right since the maximum power is less than the power I found before, but I'm not sure where I'm going wrong.

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    \$\begingroup\$ Hint: when you wrote, "Rth = R1//R2 = 1 Ohm" you already got two things wrong. \$\endgroup\$
    – The Photon
    Mar 17 '19 at 22:06
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Your Vth and Rth value are incorrect.

One way to find Vth is to remove the load from your circuit. Vth will be the voltage between where the load used to be. In your case you need to remove RL and find the voltage between A and B.

enter image description here

The voltage between A and B is Vth.

Vth = I1 * R1 = 6A * 0.5 Ohm = 3V

Now to find Rth you can place a wire between where the load used to be and find the current flowing through that wire. Rth will be Vth divided by the current flowing through the wire. In your case you need to place a wire between A and B and find the current flowing through that wire.

enter image description here

The current flowing through R3 is Iwire.

Iwire = (R1 / (R1 + R3)) * I1 = (0.5 Ohm / (0.5 Ohm + 1 Ohm)) * 6A = 2A

Vth divided by Iwire is Rth.

Rth = Vth / Iwire = 3V / 2A = 1.5 Ohm

The correct value for Vth and Rth are:

Vth = 3V

Rth = 1.5 Ohm

Your P and Pmax formula are correct.

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