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I am trying to see if diode1 and diode2 is off or on.

I assumed off/ on . I tried to do node analysis in Vo to figure out Vo.

But I don't know the current flowing in blue circled branch in the picture.

My professor just assumed that the current flowing in that branch is 0.

Why is the current flowing through the blue branch zero?

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\$ V_O \$ is the open-circuit output voltage. It is all you are asked for in this case so you don't need to worry about current in that branch. It makes the solving the problem much simpler as now you only have to calculate the voltages.

Many real-world circuits could be considered like that. For example, if that circuit was feeding an op-amp input with an input impedance of 10 MΩ then negligible current would flow so you could consider it as an open circuit.

How to proceed:

  • Finish your calculation for \$ V_O \$ as shown in your second photo.
  • Then think what happens when the diodes are replace (a) if \$ V_O > 0 \$ in the previous step and (b) if \$ V_O < 0 \$.
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The blue branch doesn't conduct any current. The node voltage Vo indicates the voltage with respect to ground. What you need to do is apply KVL to obtain Vo and conclude whether your initial guess were correct.

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