Supernode containing the reference node

shespuzzling

Joined Aug 13, 2009
88
Hi,

My question is related to nodal analysis and using supernodes. I don't have a pdf of the circuit in question but can try to get it if it helps.

Basically, I am unable to solve this circuit using nodal analysis when I write an equation around a supernode that contains a nonreference node and the reference node. I can't figure out why this won't work. If I write a different equation I'm able to get the correct answer, but why can't I do it around a supernode that contains the reference node? I just don't get it!

hgmjr

Joined Jan 28, 2005
9,029
Hi,

My question is related to nodal analysis and using supernodes. I don't have a pdf of the circuit in question but can try to get it if it helps.

Basically, I am unable to solve this circuit using nodal analysis when I write an equation around a supernode that contains a nonreference node and the reference node. I can't figure out why this won't work. If I write a different equation I'm able to get the correct answer, but why can't I do it around a supernode that contains the reference node? I just don't get it!

The best thing for you to do is go ahead and post your circuit. Once the circuit is available it will be possible to make suggestion that you can use teo solve the problem.

It is a good idea for you to try to solve the problem yourself and then post your efforts. Then the members can better determine where you are weak in your understanding of the solution approach.

hgmjr

shespuzzling

Joined Aug 13, 2009
88
Hi,

I've attached a PDF of the circuit and my work.

My problem is this: I thought that If you have 4 unknown voltages then you need 4 equations to solve but I am unable to solve with the 4 equations I have. The book skips writing an equation about the V1-Ref node and instead writes an equation which relates .2Vy with the node voltages V3 and V4 and then with V4 and V1 and when this equation is used you get the write answer. But they're still using 4 equations for 4 unknown voltages, just like I did.

My problem is, why doesn't it work with an equation written about the supernode at V1-Ref.?

Thanks! Please let me know if you have any questions.

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hgmjr

Joined Jan 28, 2005
9,029
You should be able to form an additional independent equation relating V4 and V3.

hgmjr

shespuzzling

Joined Aug 13, 2009
88
Right, I understand that another equation could be written relating V4-V1 and V3-V4, but I don't see why it doesn't work the way I've done it with the supernode about V1-Ref.

hgmjr

Joined Jan 28, 2005
9,029
I believe the reason it fails with the equation written around the supernode involving V1 and 0V is that the equation so formed does not relate one of the variable with one or more of the remaining three variables. Only with an equation that relates one of the variables to one or more of the remaining variables do you have a chance to substitute and reduce the number of unknowns by one variable.

hgmjr

shespuzzling

Joined Aug 13, 2009
88
I thought it might have had somethign to do with which variables were in the equations, but if you look at the Matrix that I wrote on the PDF, you can see that on the third line (that is for the equation about the ref. node) the only nodal voltage that isn't referenced is V3.

.2Vy = V3-V4 = .2(V4-V1)
V3-V4-.2V4+.2V1=0

.2(V1) + 1(V3) - 1.2(V4) = 0 (the book used this equation instead of the one I wrote for the supernode about V1-Ref.)

But, this equation also only relates 3 of the 4 variables present so I'm not understanding how it differs from mine. I've gone over my work (as shown on the PDF) about a million times and can't for the life of me figure out what I'm doing wrong.

P.S. thanks for all of the fast responses!

hgmjr

Joined Jan 28, 2005
9,029
Can you post the equation for the other supernode that you think should work?

hgmjr

shespuzzling

Joined Aug 13, 2009
88
Sure, it's written as equation #3 on the PDF but I will write it out for you here:

V1-V2 + V1-V4 - V4 = -14 - .5(Vx) ; Vx = V2-V1
(.5).......(2.5)....(1)

simplified, this gives you: 1.9(V1) - 1.5(V2) - 1.4(V4) = -14

(the formatting was a little weird when i tried to write the equation above, let me know if it doesn't make sense)

Last edited:

hgmjr

Joined Jan 28, 2005
9,029
Here is the equation written using the LaTex feature available in the "Go Advanced" reply box.

$\frac{V_1-V_2}{0.5}+\frac{V_1-V_4}{2.5}-V_4=14-0.5V_X$

where: $V_X=V_2-V_1$
This is one of the critical independent equations without which the problem could not be solved.

I thought the 4th equation you were referring to was

$V_1=-12$

hgmjr

shespuzzling

Joined Aug 13, 2009
88
I'm going to write out both my way of doing it and the books so you can more easily compare.

My way:

nodal analysis at node 2, at supernode between node 3 and 4, using the fact that V1=-12, and at the supernode between node 1 and the reference.

-2(V1) + 2.5(V2) - .5(V3) = 14 (@ node 2)
.1(V1) - 1(V2) + .5(V3) + 1.4(V4) = 0 (@ super node V3-V4)
V1 = -12
1.9(V1) - 1.5(V2) - 1.4(V4) = -14 (@ super node V1-Ref)

4 equations, 4 unknowns. I used my calcular to solve this matrix and couldn't come up with the right answer.

Book's way:

nodal analysis at node 2, at supernode between node 3 and 4, V1=-12, and then relating V4-V1 with V3-V4.

-2(V1) + 2.5(V2) - .5(V3) = 14 (@ node 2 - same as mine)
.1(V1) - 1(V2) + .5(V3) + 1.4(V4) = 0 (@ super node V3-V4 - same as mine)
V1 = -12 (same as mine)
.2(V1) + 1(V3) - 1.2(V4) = 0 (Here is where the book differs from my approach)

4 equations & 4 unknowns, but solving this matrix with my calculator gives me the correct answer.

So, why didn't my way of doing things work? As far as I can tell, it should.

hgmjr

Joined Jan 28, 2005
9,029
Are you classifying the two nodes V1 and 0V (ref) as making up a supernode? If so, I don't believe that is true.

hgmjr

shespuzzling

Joined Aug 13, 2009
88
Yes, that was my reasoning, since they are connected by a voltage source. Why doesn't it work? It seems to be the same logic as that used at nodes 3 and 4.

hgmjr

Joined Jan 28, 2005
9,029
In all of the writeups that I have seen on supernode usage, I have never seen any that classified a voltage source referenced to ground as a supernode. My understanding of supernodes are that they involve a voltage source that joins two nodes neither of which is the reference node. I believe supernodes are a way of taking advantage of the fact that an ideal voltage source has zero source impedance.

hgmjr

The Electrician

Joined Oct 9, 2007
2,724
hgmjr is quite correct. The mistake you are making is to assume that the reference node can be part of a supernode. This is not true.

shespuzzling

Joined Aug 13, 2009
88
Wow, so it's as simple as that! Thanks for your help! I'll remember never to do a supernode about a reference node then. So, theoretically then, if I chose a different node to be the reference node, then I should be able to do a supernode between node 1 and the center node without any problems?

hgmjr

Joined Jan 28, 2005
9,029
While you are probably right about changing the reference node opening up the potential to create a supernode, I believe the net effect would be to complicate rather than simplify the analysis.

As an aside, I would recommend you look into Millman's Theorem and add that to your circuit analysis tool kit.

hgmjr

The Electrician

Joined Oct 9, 2007
2,724
When using the nodal method, the thing to do would be to choose your reference node so that there are no voltage sources between any two non-reference nodes. In other words, make sure that all voltage sources have at least one terminal connected to the reference node.

You'll notice that you can't do that for your given circuit.

shespuzzling

Joined Aug 13, 2009
88
Thanks for all your help. The book's tend to gloss over these minor but important facts. I'll look into Millman's theorem as well.

hgmjr

Joined Jan 28, 2005
9,029
Here is the link to the material on Millman's Theorem in the AAC ebook. The Theorem is much more powerful than the simple explanation and examples might lead you to believe.

If you are interested in seeing some more interesting applications of the Theorem, have a look at my Blog here at AAC.

hgmjr