# Supernodes in Series

I have this problem 3.32 from Fundamentals of Electric Circuits 6th edition by Alenxander and Sadiku. I have drawn the circuit in LTspice. My goal is to solve this problem using nodal analysis and supernodes. I am not sure how to handle have two supernodes in series, can someone give me a suggestion?

This problem is for an intro to circuits class. Its asking me to identify the voltages at N1, N2, and N3. In this section, we were dealing with super nodes. From the supernode definition I know that a supernode is when a voltage source is connected between two nonreference nodes. Notice, and the part that is confusing me is that between N1 and N2 there is a supernode, and there also is one between N2 and N3. This problem is very easy as I know the solution doesn't even yield a KCL equation. I don't understand how the supernodes are working in this problem. I am really thankful for all the quick responses, hopefully that clarified what's going on in my head.

• I don't see what the problem is - can you explain why you are having some form of problem. Just by visual inspection, the current through R1 is 2 mA, I mean, it's that easy BUT I am not inside your head so I don't see how you might see the circuit. May 3 '20 at 14:59
• devdev: just start by calling the bottom wire 0 V. Then follow upward through the 12 V source to the shared node above it. What's the voltage at that node? (It's easy.) Then follow through the voltage source on the left to the node on the left side of R1. What must be the voltage there? Then, instead, follow through the voltage source on the right to the node on the right side of R1. What must be the voltage there? If you know the voltages on both sides (and you do, if you think this through) of R1, you have the current in R1 and R2 and the voltages at all the nodes. Must you use nodal?
– jonk
May 3 '20 at 15:20
• @jonk Thank you so much for this wonderful intuitive explanation. that was really insightful for building up my intuition. I asked about supernodes because that was the problem this section belongs to and there is probably some important insight about supernodes buried in this problem, but still I am happy you walked me through an intuitive solution May 3 '20 at 17:08
• It's a trick question but you can find all node voltages just by inspection (no calculations are needed here) and notice that we know that N2 = 12V. Thus, N1 and N3 are? Do you see it?
– G36
May 3 '20 at 19:09
• Let me quote OP: "This problem is very easy as I know the solution doesn't even yield a KCL equation." ... *"My goal is to solve this problem using nodal analysis and supernodes." May 3 '20 at 19:33

When performing nodal analysis, you must create a supernode if there is a voltage source which is not directly connected to ground or connected to ground through other voltage sources. I will leave you with that hint.

But, as always with nodal analysis, the very first thing you should do is select a reference (ground) node. LTspice will also expect you to designate one of the nodes as ground.

• I added the ground node, but notice how there are two super nodes connected in series. Between N1 and N2 and between N2 and N3 May 3 '20 at 17:03
• You are never forced to use supernodes. You can always use standard nodal analysis without involving that concept. In fact, I consider it more prone to error because it is easier to get it wrong using the idea, than not. That said, if a class requires the usage, then I suppose one must do so.
– jonk
May 3 '20 at 17:12
• @Jonk If you don't use supernodes "when needed", you must use a variable current, which makes the equations a big harder to solve. May 3 '20 at 17:42
• @Huisman Yes, that's true. But one is less likely to make mistakes, my experience, as it just flows out from well-worn principles without adding any new thinking steps. And robust matrix solution tools abound. So the only justification for reducing the equations by adding something you might not get right (notice how many have trouble with supernodes?), is if you are using paper and pencil and Cramer's rule. Then I'd agree.
– jonk
May 3 '20 at 17:47

OP wrote:

From the supernode definition I know that a supernode is when a voltage source is connected between two nonreference nodes

Semi-quoted from the wiki, the wiki definition is better to my opinion.

A supernode has two nodes, one non-reference node and another node that may be a second non-reference node or the reference node. Supernodes containing the reference node have one node voltage variable. For nodal analysis, the supernode construct is only required between two non-reference nodes.

So, not required, but still solvable!

The circuit of OP has 3 supernodes:

• V1: has a referenced node and an non-reference node node
• V2: has 2 non-reference nodes
• V3: has 2 non-reference nodes

The supernodes yield the following relations(1)

• $$\V_{N2} = V3 = 12 \text{ V}\$$
• $$\V_{N1} = V_{N2} - V1 = 2 \text{ V}\$$
• $$\V_{N3} = V_{N2} - V2 = -8 \text{ V}\$$

Normally these additional equations help solving the KCL, but this example makes things worse to understand.