homework and exercises - The sign of the current flowing in a circuit

I was doing the following problem:

And I was asked to find Iy.

I found Iy to be 2.64 using KCL. However, the right answer was negative 2.64.

Is it negative only because there is a dependant voltage source with "+ -" ? And why must it be negative? Does "-" in the final answer play a significant role? If I put positive 2.64 instead of negative, would that be wrong?

Also, here is another question

I found V1 to be 11.9. However, the right answer was - 11.9.

Can someone clarify for me when do you put negative sign?

Answer

To solve circuit problems all you need to do is choose a convention and then apply KVL and KCL.

When labelling the circuit the choice of current directions and voltages is arbitrary and if you find that after all the algebra a current comes out to be negative than all that means is that the initial current direction chosen when labelling was wrong and the current actually flows in the opposite direction.

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KCL

Label the nodes making one of them the zero volt node.

KCL is derived from the law of conservation of charge and requires the algebraic sum of the currents entering (or leaving) a node to be zero.

Choosing to sum the currents leaving node $nV_x$ you get

$\dfrac {nv_x - V_1}{R_1} + \dfrac {nv_x - 0}{R_2} + i_y = 0$

Notice that since the currents leaving the junction are being found the voltage across any resistor is the voltage at the node minus the voltage at the other end of the resistor ($V_{\text{node}} - V_{\text {other end}}$).

For currents entering the node the voltage across any resistor would be the voltage remote from the node minus the node voltage. ($V_{\text{other end}} - V_{\text {node}}$)

So the equation for currents entering the node would be $\dfrac {V_1 - nv_x}{R_1} + \dfrac {0- nv_x}{R_2} - i_y = 0$

The same equation as before.

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KVL

A convention which is often used to label circuit diagram prior to using KVL is called the associated variables convention.

Label a current direction for every circuit component and then the voltage label with a plus sign where the current enters the component.
Instead the voltage can be labelled and the current then labelled going in to the circuit element at the plus sign end.

KVL is derived from the law of conservation of energy.

When going round a loop assign a sign to the voltage which is the sign at the end where you are entering the circuit element and the sum of these voltages should be zero.

Starting at the bottom left hand corner and going clockwise gives

$-V_1 =i_xR_1+nv_x = 0$

You now have two equations and two unknowns $v_x$ and $i_y$ which give $v_x = 13.2$ V and $i_y = -2.64$ A.

The current direction was guessed incorrectly by the setter of the problem.

So the analysis of the circuit has yielded this: