How can I solve this circuit with voltage and current sources using the node voltage method?

Question

Here is the circuit.

^{simulate this circuit – Schematic created using CircuitLab}

What do I have to do when there are two different sources?

Find all the voltages in the resistors.

Supposing every current going out of the nodes, here are the equations:

NODE1) \$V1=-15\$

NODE2) \$V2/R2+(V2-V3)/R3-I1=0\$

NODE3) \$(V3-V1)/R1+(V3-V2)/R3-I2=0\$

• I don't know if the first equation is right, or if it has to be \$-15+(V1-V3)/R1+I1=0\$
• I don't know what am I supposed to do with the \$V1\$ in the NODE3 equation, using -15 instead of \$V1\$ or simply don't put it. (?)

Considering only the NODE2 and NODE3 equations: \$G*V=I\$ I have to find \$G^-1\$:

\$G^-1=1/detG*(G^a)^T\$

\$G^a\$ = matrix composed by algebraic complements, \$G^T\$ = transposed.

By doing that I find the wrong results.

I know there are many ways to resolve this problem, but I have to practice the node voltage method.

It may be that I made some algebra or numerical errors, I'm trying again in the meantime.

Answer 1

I've added a ground reference to your schematic on the left and then redrew it, on the right:

^{simulate this circuit – Schematic created using CircuitLab}

You already know \$V_1=+15\:\text{V}\$. That's "free." At this point, you need only quickly write out the other two equations:

$$\begin{align*} \frac{V_2}{R_2}+\frac{V_2}{R_3}&=I_1+\frac{V_3}{R_3}\\\\ \frac{V_3}{R_1}+\frac{V_3}{R_3}&=I_2+\frac{V_1}{R_1}+\frac{V_2}{R_3} \end{align*}$$

I divide up my equations (shown above) using a "mental model" I acquired from reading Spice source code: I place out-flowing currents on the left and in-flowing currents on the right. A current source is either an out-flowing current or else it is an in-flowing current. So I place that on whichever side it belongs, depending on which way it points.

You can re-arrange the above equations per the standard format for matrix solution, if you want.

$$\left[{\begin{array}{cc}\frac{1}{R_2}+\frac{1}{R_3}&\frac{-1}{R_3}\\\frac{-1}{R_3}&\frac{1}{R_2}+\frac{1}{R_3}\end{array}}\right]\left[{\begin{array}{cc}V_2\vphantom{\frac{V_1}{R_3}}\\V_3\vphantom{\frac{1}{R_3}}\end{array}}\right]=\left[{\begin{array}{cc}I_1\vphantom{\frac{V_1}{R_3}}\\I_2+\frac{V_1}{R_1}\vphantom{\frac{1}{R_3}}\end{array}}\right]$$

Now use Cramer's rule, if by hand. Or use Sage/sympy or some other software product, if you prefer. Regardless, the simultaneous solution of the above will produce \$V_2\$ and \$V_3\$ for you.

Answer 2

When you have a voltage source with one end at ground and you are using the node voltage method you immediately know the voltage at one node by inspection. You do need to be careful with the sign. Look in your textbook or course notes for an example.

When writing node voltage equations you are actually writing a KCL equation. It looks like you are adding all of the current leaving each node and setting the sum equal to zero, which is fine.

We won't solve your homework problems for you so I've just given some hints. You almost have the equations written correctly.