How do I get the voltage of Ro to calculate the maximum power transfer?

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.

• Use Superposition theorem for voltage across Ro Mar 31, 2021 at 10:40
• can you show me for one of them? im really confused on what node to look at Mar 31, 2021 at 10:42
• 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. Mar 31, 2021 at 11:51
• @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? Mar 31, 2021 at 11:54
• You need to research "maximum power transfer" and Thevenin equivalent circuits. Mar 31, 2021 at 12:19

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:

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\$$.)

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.

• By LKV and LKC do you meen kirchoffs current law and kirchoffs voltage law? Apr 1, 2021 at 7:16
• Yes, I do. Don't be scared of Req is a negative number. It may happen with voltage controlled current sources like V3/2. Apr 1, 2021 at 7:40
• may i ask how is negative resistance possible? Apr 1, 2021 at 11:48
• 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. Apr 1, 2021 at 11:55
• may i ask if VR2 is also used to calculate the maximum power transfer? Im pretty sure it is im just clarifying, thanks! Apr 1, 2021 at 13:36