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i just realize i solve this circuit: enter image description here

The problem asks to find Va, Vb and Vc, so i used the concept of 'SuperNode' in the V1 branch and i it got solved right (Vb=-12V, Va=14V and Vc=4V) , but according to Sadikus book:

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it

This was wrong, because the supernode V1 that i considered cant actually be a supernode because one of its nodes is a reference node (GND node)

So i guess this was solved correctly because a casuality? or im skipping something important and V1 can actually be a supernode? If this was just a casuality, how can i solve this without the supernode concept?

Also, what does it mean the bolded part in sadikus book statement?, i can get it

Thanks in advance!

EDIT I add my solution considering V1 source as a supernode: enter image description here

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  • \$\begingroup\$ Somewhere else in the explanation of MNA in your book it should tell you how to deal with voltage sources connected to the reference node. In Nagel (p. 62) he talks about dealing with grounded voltage sources with "reduced equations" (which you can read more about at the linked document). \$\endgroup\$
    – The Photon
    Commented Sep 22 at 5:34
  • \$\begingroup\$ Cblue X, I'm just a little confused by your writing: (1) "i used the concept of 'SuperNode' in the V1 branch and i it got solved right"; and, (2) "the supernode V1 that i considered cant actually be a supernode because one of its nodes is a reference node (GND node)". Could you show us your work here? Show us how you "used the concept of 'SuperNode' in the V1 branch", please. (Especially when you seem confused by the bolded part of the quote and need an explanation for it, since it is unclear to me what you did apply when you also claim you don't understand that part.) \$\endgroup\$ Commented Sep 22 at 6:41
  • \$\begingroup\$ @ThePhoton I may be wrong, but the concept of supernode is part and parcel with the idea of the "node voltage method" (nodal analysis), and explicitly is not part of modified nodal analysis. MNA obviates the need for the supernode concept. \$\endgroup\$ Commented Sep 22 at 7:48
  • \$\begingroup\$ @periblepsis the way I was taught, "nodal analysis" does not allow for voltage sources at all, and "modified nodal analysis" introduces supernodes to allow analyzing circuits that contain voltage sources. Nagel (see above link) says (p. 62) "The most serious disadvantage of Nodal Analysis is the inability to handle voltage-defined branches" which agrees with my memory. However he also describes MNA without ever using the term "supernode" so that term may be a later introduction to the pedagogy. \$\endgroup\$
    – The Photon
    Commented Sep 22 at 14:22
  • \$\begingroup\$ @ThePhoton MNA intentionally adds current unknowns to the matrix. At least as it is taught today. I provide a link showing that. In such cases there is no need for a supernode. And it isn't easy to write code to identify such cases, anyway. That said, if you write up an answer here where you show me what you call MNA for this case, I'd read it and either admit I'm wrong or show you why I think it isn't MNA and extend my answer to show you what I think the MNA matrix would look like. \$\endgroup\$ Commented Sep 22 at 14:40

2 Answers 2

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ground

For nodal analysis, pick a reference node and call it ground or \$0\:\text{V}\$. This way, the remaining node voltages have an implied reference point. (All voltages are really voltage-differences.)

But keep in mind you can move that reference to any node that is convenient. Don't always accept the given reference, if a different reference point would make things simpler to solve by reducing the number of equations.

That said, the reference you have is fine.

re-draw the schematic

Next, get in the practice of re-drawing schematics. Even if doing so means you have a schematic that the teacher will not accept as a valid answer. The skills you develop in re-drawing schematics will serve you well.

The process isn't formulaic and there isn't only one way to go with it. This is another reason to practice a lot. Doing so develops your own unique style that communicates to your mind better and, because of that, may allow you to communicate ideas to others more readily, as well.

Let's take your schematic and look at one possible result:

schematic

simulate this circuit – Schematic created using CircuitLab

Current sources can be broken into two parts, as shown in the 3rd rendition above. And there's no need to analyze the ground node. So that entire lower section in the 3rd rendition can be chucked out the window, as shown in the 4th rendition above.

super-node

Now, this does have a super-node, as shown in the green oval below:

enter image description here

The only reason for a supernode here would be if you don't care about knowing \$I_{V_2}\$ and just want to develop a single KCL statement for the entire supernode -- using either one of \$V_a\$ or else \$V_c\$ as the reference. This would be a good way to go if solving this without a computer and just using paper and pen, instead. (You could then later work out the current from that result.)

$$\begin{align*} \underbrace{\frac{V_c+V_2}{R_1}+\frac{V_c}{R_2}+I_2}\quad&=\quad\underbrace{\frac{V_b}{R_1}+\frac{V_b}{R_2}+I_1}\\ \text{leaving super-node}\quad&\quad\:\:\text{entering super-node} \end{align*}$$

This solves out as \$V_c=V_b-V_2\cdot\frac{R_2}{R_1+R_2}-\left(I_2-I_1\right)\cdot\left(R_1\vert\vert R_2\right)\$. Which does work out to \$V_c=+4\:\text{V}\$. And from there you can get the rest. (\$V_b=-12\:\text{V}\$.)

without using a super-node

On the other hand, if you want to include the current through \$V_2\$ then a supernode isn't required and you can instead just assign a new variable, \$I_{V_2}\$, to the current (pick a direction, of course) in \$V_2\$. Here, you then leave both \$V_a\$ and \$V_c\$ as unknowns and add a third unknown, \$I_{V_2}\$. This will mean solving for three equations and three unknowns, but you will know in advance that \$V_a=V_c+V_2\$ so this gives you one of the equations needed. The KCL rule will let you develop the other two equations.

$$\begin{align*} \frac{V_a}{R_1}&=\frac{V_b}{R_1}+I_1+I_{V_2}\\\\ \frac{V_c}{R_2}+I_2+I_{V_2}&=\frac{V_b}{R_2}\\\\ V_a&=V_c+V_2 \end{align*}$$

That solves out to \$V_a=+14\:\text{V}\$, \$V_c=+4\:\text{V}\$, and \$I_{V_2}=-2.4\:\text{A}\$ (meaning I got the current arrow direction wrong.) But it required three equations and three unknowns. (\$V_b=-12\:\text{V}\$.)

modified nodal analysis

This isn't either of the above, exactly. And I believe this page is pretty nice for discussing MNA. MNA is really more targeted at computerized (rote algorithmic development) analysis, though it works just fine by hand.

That page shows you how the unknowns are laid out into matrix form for developing solutions. MNA would definitely consider \$I_{V_2}\$ to be an unknown to solve and, because it takes a rote algorithmic approach that is easy to code as software and solves all of the unknowns, isn't interested in the use of super-nodes.

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    \$\begingroup\$ This doesn't address OP's key question of why and how voltage sources connected to the reference node are treated differently than ones that are not; and why, according to their textbook, a supernode can't encompass the reference node. \$\endgroup\$
    – The Photon
    Commented Sep 22 at 14:54
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    \$\begingroup\$ @ThePhoton The author hasn't written out their process. So all I can do is hope it may. I already admitted my confusion in comments. \$\endgroup\$ Commented Sep 22 at 14:57
  • \$\begingroup\$ I have added how i solved! \$\endgroup\$
    – Cblue X
    Commented Sep 22 at 21:51
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    \$\begingroup\$ @periblepsis thank you so much dude! i think this is a really great explanation, i have posted my first solution and i think the reason i got it right was because i casually ended with the same equation \$\endgroup\$
    – Cblue X
    Commented Sep 22 at 22:17
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I can answer one part of your question,

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it

...

what does it mean the bolded part in sadikus book statement?

The currents in parallel with the voltage source do not leave the supernode. Therefore, they don't contribute any terms to the KCL equation for the supernode. So we can consider them "part of" the supernode in the sense that we don't need to consider their branch currents when writing the KCL equation for the supernode.

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  • \$\begingroup\$ Thanks man, i got it!! \$\endgroup\$
    – Cblue X
    Commented Sep 22 at 22:17

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