# Finding current over controllable voltage source using nodal analysis

I have the following circuit:

Let me explain the diagram a bit first: I0 is current in that direction, V0 is voltage and 2 V0 is also voltage times two. V1, V2, and V3 are my nodes. The 'S' by the resistor values stand for Siemens or conductance (reciprocal of Ohm).

What I want to do is find the current I0 and the voltage V0. I have found the voltage V0 to be $$V_0=6.29V$$ and the other voltages to be $$V_1=18.86, \;\;\;V_2=6.3V, \;\;\;V_3=13V$$ Which also means: $$2*V_0=12.58V$$

Where I'm stuck is what will be the value for I0? I really do not know how to go about finding it. I'm using nodal analysis on this problem.

Thank you for any help, I'm very stuck and would like to learn how to go about this problem.

These two are obvious by inspection:

\begin{align*} V_3&= 13 \:\text{V}\\\\ V_0&= V_2 \end{align*}

These are from simply writing out the nodal equations:

\begin{align*} V_1\cdot 1\:\text{S} + I_0 + V_1\cdot 2\:\text{S}&=2\:\text{A}+V_3\cdot 2\:\text{S}\\\\ V_2\cdot 4\:\text{S}+V_2\cdot 8\:\text{S}&=I_0+V_3\cdot 8\:\text{S}\\\\ V_2+2\cdot V_0&=V_1 \end{align*}

There are three unknowns: $$\I_0\$$, $$\V_1\$$ and $$\V_2\$$. ($$\V_0\$$ is $$\V_2\$$.)

It's really no harder than that.

Are you able to solve those three equations for the unknowns?