Passive Convention Specifics With Regards to Resistors (Problem linked)

circuit analysis

I was wondering how exactly passive convention works with respect to unlabeled resistors. In the problem attached, I realized you only get the answer if you label the resistor terminals one way, regardless of the direction you choose for the current. I was wondering, then, how you know which way to label these resistors. The initial method I used was wrong, which was to go clockwise and label the unlabelled resistor terminals positive to negative in that direction, but that turned out to be incorrect. Is there a general rule on how to label them, and if so what is it? I know it works the other way, (setting them negative to positive in the clockwise direction starting from the top left corner), but I don't understand why, or what dictates that. I am not concerned about the answer since I know the method to get it after trial and error, just the reasoning behind it, as this was a question of mine throughout other problems as well. I truly appreciate any help!

Problem:

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Best Answer

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I annotated your circuit a bit. In red the loop current \$I\$, while in green currents and voltage references for each resistor. Please note that I followed no particular rule to do the latter.

First of all, when solving a circuit you need to choose a reference for the voltage. I did so on bottom left corner, just under the \$50V\$ generator.

Now since this circuit has only one loop you would normally need only one equation, plus one additional equation for each controlled source: that makes two for this problem.

Starting from the reference: $$ 50 - 2I + 2V_x -3I-10I-10-5I=0 $$ While the additional equation is: $$ V_x = -10I $$ I am sure you can solve it from here.

What about all the green currents then? Whell I've drawn them to prove that how you choose them does not matter at all. For each resistor you can write its voltage and current with respect to the green reference: $$ I_2=-I\qquad V_2=-2I\\ I_3=I\qquad V_3=-3I\\ I_{10}=-I\qquad V_{10}=-10I\\ I_5=-I\qquad V_5 = 5I $$

easy as pie. Please note that usually (as in please do so) for passive components you use a reference for which a positive IV product means dissipated power, i.e. currents enters at the plus terminal, while for generators and such you do the opposite. That's because that's what usually happens: generators source current from the positive terminal, passives drain currents in the positive terminal... But that's just a convention: look at your 10V generator. If we choose a reference with the same direction as I then it has the same references a passive would have... It is actually dissipating power as a matter of fact.

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