Hi,

Here is a network where we have to calculate the value of the three resistors, R1, R2, and R3.
There is one main source and two internal sources. V2 is a regular constant voltage source, but it is to be changed manually. For example, if it started out as 1v then we want to go down to maybe 0.1v and up to maybe 10v.
The key point is, no matter what we change V2 voltage to, the output Vout stays constant. If we chose say resistors 1, 2, and 3 for the respective values and Vout somehow came out to 1v (it wont I dont think this is just an off the wall example) then when we change V2 the output voltage Vout must remain at 1v.
V3 is a dependent source, dependent on V2 and it's polarity. The 'gain' is just 2, so when V2 goes from say 1v to 1.1v then V3 goes from 2v to 2.2v because it is just double.

I don't think this is too hard but it's a little tricky. If you feel like trying it, see what you can come up with. There 'could' be more than one solution too but one single solution is good enough.

 

Most obvious solution:

Spoiler

Set R1=R3=0,

Now the voltage across Vout is forced to be V2.

But wait, you say, isn't Vout also V1-V2, which is only V2 if V1=2*V2?

Well, yes. The issue is that two ideal voltage sources, V1 and V3, are now in parallel. So who wins?

This is easily resolved by letting R1=R3=an infinitesimal resistance non-zero resistance.

But, since the system is linear, we can choose any "infinitesimal" resistance, even if it might not be so "infinitesimal", such as 1 Ω or 1 MΩ.

So, really, we just have the requirement that R1=R3.

If that generalization is hard to grasp (and it's NOT the most obvious thing in the world), then just analyze the circuit and solve for Vout. Then, if it isn't obvious (which it should be at that point) how to make the dependence on V2 go away, set the derivative of Vo with respect to V2 to zero and see what constraint that yields. It should yield R1=R3.

If you didn't want to do that, you could walk the effects through.

If the goal has been met, then changing V2 doesn't change Vout at all, which means the current in R2 doesn't change, which means the current in V2 doesn't change.

In order for the current in V2 to remain constant, whatever amount the current in the branch through V1 changes by, the current in the bottom branch must change by the same amount (or an equal and opposite amount, depending on your reference direction assignments).

If V2 goes up by ΔV, the current up through R1 goes up by ΔV/R1.

If V2 goes up by ΔV, the voltage across R3 goes down by ΔV and so the current down through R3 goes down by ΔV/R3.

For these to cancel out, they need to be the same, which happens when R1 = R3.