Simple Circuits: Understanding Node Voltage Analysis

[galaxy_twirl]

Homework Statement

Homework Equations

##v₁ + v₂ + v₃ = 0## (KCL)

The Attempt at a Solution

I don't understand why my solution is wrong. I was taught that if I were to use Node Voltage analysis, I should put nodes with voltage sources into supernodes which was what I did, however, my teacher showed a rough guide in class and he did not use supernode.

Hence, I am a bit confused now -- when do we use supernodes and if we use supernodes, how big should they be? For the question above, if I were to use supernode, assuming I made some calculation errors above, will I get the same answer as my teacher? Thanks!

My teacher's partial solution:

 

[gneill]

A Supernode in this case would include the reference node and the source 2v. It does not include R4. But you don't need to write an equation for the reference node, just the two other nodes, v and v2.

 

[galaxy_twirl]

A Supernode in this case would include the reference node and the source 2v. It does not include R4.

What do you mean by the reference node? Does it mean that I must not cover the resistor R₄?

[/QUOTE]But you don't need to write an equation for the reference node, just the two other nodes, v and v2.[/QUOTE]

I see. Hmm.. Then when we do know when not to use supernode? This is because, I usually would put voltage sources in supernodes whenever I see them in a circuit.. May I know if this is bad practice/a misconception? :(

Thanks! :D

 

[gneill]

First you select a reference node. In this circuit the bottom rail is the natural choice.

Any sources directly connected to the reference node are effectively part of a supernode. But you don't need to write a node equation for the reference node since by definition it has the known potential 0V.

In other locations a supernode comprises a set of nodes whose potentials with respect to each other are fixed by voltage sources and thus independent of any current flowing between them. A resistor in series with a voltage source doesn't work because a change in current changes the potential across the resistor, thus altering the end-to-end potential difference of the combination.

 

[galaxy_twirl]

I see. OH YA! Sorry about the issue on reference node. I had always called it the "ground -ve 0" node because I always fix it, usually on the negative end of independent voltage sources, and where the end lands on the other nodes with nothing in between the nodes so I can put 0V across left to right and vice versa, so the term reference node slipped my mind. ><

Any sources directly connected to the reference node are effectively part of a supernode. But you don't need to write a node equation for the reference node since by definition it has the known potential 0V.

Ah yes. My teacher said that supernodes can be of any size, since a supernode is a collection of nodes.

In other locations a supernode comprises a set of nodes whose potentials with respect to each other are fixed by voltage sources and thus independent of any current flowing between them. A resistor in series with a voltage source doesn't work because a change in current changes the potential across the resistor, thus altering the end-to-end potential difference of the combination.

I see. So that means I cannot put a supernode at the dependent voltage source, because I will not be able to loop it after and around R₄, which makes it look funny, and hence I should not do it. Am I right to say this? I attempted to draw a supernode below, according to what you have said above, to be sure I did not interpret your message wrongly. :)

 

[gneill]

The reference node and the directly attached sources can be considered a supernode.

The potential at the top of v1 is, well, v1. And the potential at the top of the 2v source is 2v. These are fixed potentials with respect to the reference node as set by those sources.

 

[galaxy_twirl]

The reference node and the directly attached sources can be considered a supernode.

View attachment 75824

The potential at the top of v1 is, well, v1. And the potential at the top of the 2v source is 2v. These are fixed potentials with respect to the reference node as set by those sources.

I see. Thank you for the illustration. :) I shall try solving the question again.

Just wondering, if I were to not use supernodes at all, is it generally okay for Node Voltage Analysis? This is just in case where I get confused in the exam and use supernodes inappropriately. :X (But I will still keep in mind on your advice on how to set the boundaries of supernodes. :D)

 

[galaxy_twirl]

Hmm.. I still got 3/16 as my answer. :(

 

[gneill]

I see. Thank you for the illustration. :) I shall try solving the question again.

Just wondering, if I were to not use supernodes at all, is it generally okay for Node Voltage Analysis? This is just in case where I get confused in the exam and use supernodes inappropriately. :X (But I will still keep in mind on your advice on how to set the boundaries of supernodes. :D)

You'll find that you can't avoid using supernodes if there are voltage sources connecting two nodes (unless you choose an analysis method other than nodal analysis). When two nodes are connected by a voltage source there's no way to write an expression for the current through it in terms of the node potentials and a resistance because there's no resistance value to use.

 

[gneill]

Hmm.. I still got 3/16 as my answer. :(

Type out your work so we can have a look.

 

[galaxy_twirl]

Here is my working: (I was typing out when you posted this. Haha.)

Thanks! :D

 

[galaxy_twirl]

You'll find that you can't avoid using supernodes if there are voltage sources connecting two nodes (unless you choose an analysis method other than nodal analysis). When two nodes are connected by a voltage source there's no way to write an expression for the current through it in terms of the node potentials and a resistance because there's no resistance value to use.

I see. Thank you for your advice. :D I will keep a lookout for those cases and apply supernodes appropriately.

 

[gneill]

Your "supernode" equation is not correct; it has no term for the path through v1 and R1.

But your supernode includes the reference node, so you do not have to write an equation for it. The second node to use is where R3, R4, and R5 come together (that is, v2).

 

[galaxy_twirl]

Your "supernode" equation is not correct; it has no term for the path through v1 and R1.

Oh gosh! @.@ Sorry. I made a careless mistake there. :( I have to add (v-v₁)/1 in front of the first equation under SN.

But your supernode includes the reference node, so you do not have to write an equation for it. The second node to use is where R3, R4, and R5 come together (that is, v2).

And hmm.. I shouldn't have written an equation for v cos it overlaps with my equation for the supernode..

I finally got the answer~ Thank you very much! ^^