Nodal Voltage Analysis: Correcting an Error in Circuit Solution

[alexmath]

I tried solving the following circuit using nodal analysis, but the answer I've got is wrong. Can you tell me please which one of these 3 equation is wrong? Thank you!

 

[gneill]

Please make use of the posting template when you begin a new thread.

Looks like only your equation (2) is a correct node equation.

 

[alexmath]

Ok, sorry for that. Can you also tell me how the equation 1 and 3 should look like, in 3 days a have a test and I'm really really struggling understanding this.

 

[gneill]

I can't show you directly, but I can give you hints.

In your equation (1), note that node 1 has only three connections. So there should be just three terms in total, where you have four, and one of them is a duplicate of another while yet another involves nodes 2 and 3 alone for some reason...

In your equation (3), as above it has just three connections so there should be just three terms in total. The last term on the right cancels the term on the left hand side; you only want to account for a given current once. The middle term on the right side seems to involve a branch not even connected to node 3.

A simple way to write node equations is to simply assume that all currents are flowing out of the given node and write their sum. Let's say you want to write the equation for node 2 (for which you have a correct equation already). Then you could write it as:
$$\frac{V2 - 2 - V1}{2} + \frac{V2 - 4}{2} + \frac{V2 - V2}{2} = 0$$
The first term accounts for the current on the branch between nodes 2 and 1, the second term accounts for the current between node 2 and ground, while the third term accounts for the current between node 2 and node 3. That's all three connections accounted for.

The math will automatically take care of the current directions if you do it this way. No need to guess current directions in any branches. Simply assume that all currents are flowing outwards from the chosen node and that the sum of them will be zero.