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schematic

simulate this circuit – Schematic created using CircuitLab

There is this circuit where I am tasked to get the maximum power transfer of Ro. Based on my understanding, I have to get its resistance and the voltage across it.

I calculated its resistance to be 1.674,(((2||3)+1 || 7)), ohms by using the concept of Thevenin resistance. My problem is that I don't know how to get the voltage across Ro to get its maximum power transfer.

I think using nodal analysis would help but I am confused as to what nodes I should be using to find Ro since essential nodes does not go to Ro.

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  • \$\begingroup\$ Use Superposition theorem for voltage across Ro \$\endgroup\$
    – user215805
    Mar 31, 2021 at 10:40
  • \$\begingroup\$ can you show me for one of them? im really confused on what node to look at \$\endgroup\$ Mar 31, 2021 at 10:42
  • \$\begingroup\$ This is not a homework solution service, we won't do your homework for you. You need to demonstrate that you have made a significant effort to solve the problem yourself and show all of your work. In this case, I don't think you need to find the voltage across Ro at all. \$\endgroup\$ Mar 31, 2021 at 11:51
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    \$\begingroup\$ @ElliotAlderson why dont i need the voltage? isnt the power formula v^2/r? I already have its resistance so shouldnt i solve for voltage to get its power? \$\endgroup\$ Mar 31, 2021 at 11:54
  • \$\begingroup\$ You need to research "maximum power transfer" and Thevenin equivalent circuits. \$\endgroup\$ Mar 31, 2021 at 12:19

2 Answers 2

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Re-drawn, your schematic looks about like this:

schematic

simulate this circuit – Schematic created using CircuitLab

I made a choice about ground. You get to pick exactly one node and call it ground. Since this is a maximum-power question, I wanted to pick one side of \$R_O\$ and call that ground, so that I could just focus on the output impedance as seen from the other side. Single-ended analysis is easier.

I've also left \$R_O\$ disconnected, but pointing to where it will be connected, to highlight that this is a maximum-power question. You probably already know that the maximum power will be obtained when \$R_O\$ is the same resistance as the output resistance of the circuit without \$R_O\$ attached across \$R_2\$.

In laying out the schematic I've kind of assumed, without knowing for sure, that \$V_1\$ and \$V_2\$ are probably negative values. And since I want positive at top and negative at bottom, I've arranged things according to my guess.

Finally, there's no need at all to show where that voltage-dependent current source sinks. It sinks to a known voltage reference and it's entirely irrelevant to the circuit. Besides, current sources/sinks have infinite impedance (open circuit, basically.) So I just left one end hanging. It's simpler to consider and doesn't alter a single thing.

Once you are at this point, you just have two very simple nodal equations to develop. (For \$V_1\$ and for \$V_2\$.) You will want to analyze the above circuit to get the node voltage for \$V_2\$. Then you will want to analyze the circuit a second time, but looking like this:

schematic

simulate this circuit

By adding a one amp current into the \$V_2\$ node, you will see that its resulting voltage changes, as a result of it. The change in voltage, the difference between the two solutions for \$V_2\$, is then divided by the current change (which is obvious), and that tells you the Thevenin source impedance for the entire circuit -- absent \$R_O\$, of course.

That source impedance will then also be the impedance you want for \$R_O\$ to obtain the maximum power in \$R_O\$ for this circuit.

(That value, I can assure you, is not \$\approx 1.674\:\Omega\$.)

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R0 must be equal to the resistance that R0 itself has in parallel. Let' call it Req.

Let's calculate Req:

1) Turn off V1 and V2. Short them.

2) V3/2 is tricky. You can't turn it off.

3) Add a source current I = 1 A, in parallel to R0. 

4) Disconnect R0 from the circuit. Just cut its 2 wires.

5) Use LKV and LKC to calculate the voltage across R2. Let's call it VR2. 

6) Be careful with your calculations becuase V3/2 sinks current proportional to VR3.

Req = VR2 / I = VR2 / 1 A

R0 must be equal to Req for maximum power transfer.

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  • \$\begingroup\$ By LKV and LKC do you meen kirchoffs current law and kirchoffs voltage law? \$\endgroup\$ Apr 1, 2021 at 7:16
  • \$\begingroup\$ Yes, I do. Don't be scared of Req is a negative number. It may happen with voltage controlled current sources like V3/2. \$\endgroup\$ Apr 1, 2021 at 7:40
  • \$\begingroup\$ may i ask how is negative resistance possible? \$\endgroup\$ Apr 1, 2021 at 11:48
  • \$\begingroup\$ It's not possible if your circuit has R,L,C components only. It's possible if your circuit has transistors which are voltage controlled current sources. Transistors can also transform capacitors in inductors. \$\endgroup\$ Apr 1, 2021 at 11:55
  • \$\begingroup\$ may i ask if VR2 is also used to calculate the maximum power transfer? Im pretty sure it is im just clarifying, thanks! \$\endgroup\$ Apr 1, 2021 at 13:36

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