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I'm reviewing material from a lower div circuits class in preparation for the upper div class by redoing problems, and this is one of them.

Does it make sense for the branch with the dependent current source to have a voltage Va(t) across it? Wouldn't the voltage measured across it be zero assuming the current source is ideal and the wire has no resistance (since it's not modeled in the problem.)enter image description here

Can someone explain why these dependent sources make sense, please?

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A current source can certainly have a voltage across it. If the voltage across a current source is zero, then it is not delivering or absorbing any power. However, if the voltage across the source is not zero, then it is either sourcing or sinking power into the rest of the circuit.

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Think of an extremely simple circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

There is obviously a voltage of 1V across R1, so by Kirchhoff's voltage law, there also must be a voltage of 1V across I1.

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Yes. An ideal current source is a device that always produces the given current regardless of what voltage is applied across it.

A device that always has 0 V across it is called a 0 V voltage source, or, less formally, a short circuit.

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