Solving for v_1, v_2, and v_3 with Nodal Analysis

[VinnyCee]

Find $v_1,\,v_2,\,and\,v_3$ in the circuit below using nodal analysis:

My work so far:

$$I\,=\,\frac{v_1}{2\Omega},\,\,I_1\,=\,\frac{v_2}{4\Omega},\,\,I_2\,=\,\frac{v_3}{3\Omega},\,\,I_3\,=\,\frac{v_1\,-\,v_3}{6\Omega}$$

KVL @ loop1 => $$-I\,(2\Omega)\,+\,10\,V\,+\,I_1(4\Omega)\,=\,0$$

Which equals:
$$-\left(\frac{v_1}{2\Omega}\right)(2\Omega)\,+\,10\,V\,+\,\left(\frac{v_2}{4\Omega}\right)(4\Omega)\,=\,0$$

Which equals:
$$-v_1\,+\,10\,V\,+\,v_2\,=\,0$$

KVL @ loop2 => $$-v_2\,-\,5\,I\,+\,v_3\,=\,0$$

KVL @ loop3 => $$-10\,V\,+\,v_1\,-\,v_3\,+\,5\,I\,=\,0$$

KCL @ v1 => $$I\,+\,I_3\,+\,I_4\,=\,0$$

KCL @ v2 => $$I_4\,=\,I_1\,+\,I_5$$

KCL @ v3 => $$I_2\,=\,I_5\,+\,I_3$$

KCL @ Super Node 1 => $$I_4\,+\,I_3\,=\,I_1\,+\,I_2$$

When I combine these equations to get 4 equations with 4 variables, I get the following matrix:

$$\left[\begin{array}{cccc|c} -1 & 1 & 0 & 0 & -10 \\ 0 & -1 & 1 & -5 & 0 \\ 1 & 0 & -1 & 5 & 10 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{3} & 0 & 0 \end{array}\right]$$

The columns go like this: v1, v2, v3, I, constant

But this matrix has infinite solutions! How do I solve?

 

[VinnyCee]

I figured it out!

The last variable column of the matrix can be eliminated because $$I\,=\,\frac{v_1}{2\Omega}$$

This gives the matrix:

$$\left[\begin{array}{ccc|c}-1 & 1 & 0 & -10 \\-\frac{5}{2} & -1 & 1 & 0 \\ \frac{7}{2} & 0 & -1 & 10 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{3} & 0 \end{array}\right]$$

This RREF's out to:

$$v_1\,=\,\frac{70}{23}\,=\,3.043\,V$$

$$v_2\,=\,-\frac{160}{23}\,=\,-6.956\,V$$

$$v_3\,=\,\frac{15}{23}\,=\,0.6522\,V\,V$$

NOTE: The third row of the matrix is not required!

 

[kyle8921]

As a scientist, it is important to note that the matrix obtained from nodal analysis does not always have a unique solution. This is due to the fact that the circuit may have multiple paths for current to flow, leading to multiple possible combinations of voltage values at different nodes. However, we can still obtain a set of solutions by using algebraic manipulation or by setting certain variables to a known value. Additionally, we can use simulation software or physical measurements to verify the accuracy of our solutions. It is also important to keep in mind that nodal analysis is just one method of solving circuits and there may be other techniques that can provide a unique solution.