Nodal Analysis - Difficult

Thread Starter

Godsninja

Joined Apr 30, 2016
16
I posted a thread last night with some nodal analysis problems. I was going to edit the thread today but based on the feedback I got I'll make a new thread altogether. I'll paste a single photo of a single problem, however I will not include any work (in the photo itself, besides Va, Vb, and Vc, marked in red). I can upload a second photo after some suggestions, which will include the developments, if it comes to it.

To the problem:

20170708_nodal1.jpg

So, looking at the circuit it's obvious there are 4 nodes, so will expect 3 equations for 3 unknowns. I will consider the bottom node as reference. I tried other circuit configurations with different reference but it didn't help.

But one of the nodes, Va, is known to be 10V. I'm not exactly sure but I believe this means that since now it's only 2 unknowns, that we only need 2 equations.

Node a:
-iΔ + (Va-Vb)/10 + (Va-Vc)/30 = 0

Node b:
(Vb-Va)/10 + Vb/40 + (Vb-Vc)/20 = 0

Node c:
(Vc-Va)/30 + (Vc-Vb)/20 +/- .... = 0

I made 3 equations because I believe there should be a supernode around Vc and reference, leading to 2 eqn's, but I've never seen a problem before with a supernode that connects a regular essential node to reference, or if that even makes sense. In the first equation, I'm not sure if -iΔ is correct, but I'm much more confident in that than the last one, where I didn't even write anything for the CCVS..

This is my absolute best. I know I am suppost to make a substituation for iΔ, but I don't know what iΔ since there is not resistance, and that doesn't help for replacing 20iΔ.

I'm looking for feedback based on what I did wrong. Then I'll take that and try to solve.

MOD EDIT: Inserted image into post body.
 

WBahn

Joined Mar 31, 2012
25,760
You not only know that the voltage on Va is 10 V, you know that the voltage on Vc is -(20 V/A) iΔ.

Because you have a controlled source, you have an additional unknown, namely the control signal. So you still have two unknowns, Vb and iΔ.

You can write the normal node equation for Node B (which, other than ignoring units, you did correctly).

You can use your first node equation for iΔ.
 

Thread Starter

Godsninja

Joined Apr 30, 2016
16
Ok got it. Thankyou. I was completely oblivious to Vc being -20V/AiΔ....
I see how using units helped solve for iΔ.
For node 2, would you have written the units on everything? i.e. (Vb-Va)/10Ω + Vb/40Ω + (Vb-Vc)/20Ω = 0A 20170708_nodal2.jpg
 

WBahn

Joined Mar 31, 2012
25,760
Ok got it. Thankyou. I was completely oblivious to Vc being -20V/AiΔ....
I see how using units helped solve for iΔ.
For node 2, would you have written the units on everything? i.e. (Vb-Va)/10Ω + Vb/40Ω + (Vb-Vc)/20Ω = 0A
Absolutely!

Although units on an explicit zero are optional and frequently left out (even by a Units Nazi like myself) because zero is zero (as long as it is an absolute measure).

If you find tracking units of physical numbers gets to be a pain, take that has an indication that it is usually better to work with symbolic values such as R1 and R2 throughout the bulk of the work and only substitute in physical values at the end. This not only makes the work more generally applicable, it let's you see under the hood of how different components affect the final result. It also makes correcting errors a lot easier.

Tracking units has so many advantages it's hard to overstate them.

It is perhaps the single most effective error detection tool available to the engineer. If the units don't work out, then you know the answer is wrong. Most mistakes that you make (not all, but most) will mess up the units. If you are tracking and checking the units as you go, you will catch most of your mistakes almost immediately. Then, you can examine when and where the units stopped working out and that is almost always the point at which the error was made. If nothing else, when you get down to the final answer and the units have not worked out to what you know they need to be, you at least know that there is something wrong with your answer. If, instead, you just tack on the units that you want the answer to have then you are foregoing this ability to check your work and that is, in my opinion, an inexcusable degree of negligence for an engineer. People can and do die as a consequence.

After a while you will get so used to having coherent units that it will be hard for you to work through anyone's work that abuses them and you won't have to explicitly look for the discrepancies, they will simply jump out at you.

So why, you might reasonably ask, do instructors and textbooks not make a big deal out of it? Even texts that claim to emphasize the proper treatment of units almost always just mean tacking the desired units onto the final answer and their own examples fail to actually track the units properly. I believe the reason is that most instructors and most textbook authors have little, if any, actual work experience in the real world designing real things that have the potential to cause real death and destruction; the bulk of their entire professional life has been in academia where a missed mistake in their work only means that they have to regrade a problem set or add yet another entry to the errata for the text.

Where you tend to see professors and texts that properly emphasize their units are in physics and chemistry, probably because most of those people, even if they never worked in industry, have spent a significant amount of time with physical lab exercises and research, many of which have the potential for causing real damage if done improperly.
 

Thread Starter

Godsninja

Joined Apr 30, 2016
16
I appreciate the reply. I find myself wondering why I continually stop and start using units. I can understand why it's hard to overstate it when so many problems can be solved, and little mistakes avoided, by just substituting the numerical values in at the end. I solved all the problems after considering this and the voltage thing you mentioned.

THANKS!!!

Absolutely!

Although units on an explicit zero are optional and frequently left out (even by a Units Nazi like myself) because zero is zero (as long as it is an absolute measure).

If you find tracking units of physical numbers gets to be a pain, take that has an indication that it is usually better to work with symbolic values such as R1 and R2 throughout the bulk of the work and only substitute in physical values at the end. This not only makes the work more generally applicable, it let's you see under the hood of how different components affect the final result. It also makes correcting errors a lot easier.

Tracking units has so many advantages it's hard to overstate them.

It is perhaps the single most effective error detection tool available to the engineer. If the units don't work out, then you know the answer is wrong. Most mistakes that you make (not all, but most) will mess up the units. If you are tracking and checking the units as you go, you will catch most of your mistakes almost immediately. Then, you can examine when and where the units stopped working out and that is almost always the point at which the error was made. If nothing else, when you get down to the final answer and the units have not worked out to what you know they need to be, you at least know that there is something wrong with your answer. If, instead, you just tack on the units that you want the answer to have then you are foregoing this ability to check your work and that is, in my opinion, an inexcusable degree of negligence for an engineer. People can and do die as a consequence.

After a while you will get so used to having coherent units that it will be hard for you to work through anyone's work that abuses them and you won't have to explicitly look for the discrepancies, they will simply jump out at you.

So why, you might reasonably ask, do instructors and textbooks not make a big deal out of it? Even texts that claim to emphasize the proper treatment of units almost always just mean tacking the desired units onto the final answer and their own examples fail to actually track the units properly. I believe the reason is that most instructors and most textbook authors have little, if any, actual work experience in the real world designing real things that have the potential to cause real death and destruction; the bulk of their entire professional life has been in academia where a missed mistake in their work only means that they have to regrade a problem set or add yet another entry to the errata for the text.

Where you tend to see professors and texts that properly emphasize their units are in physics and chemistry, probably because most of those people, even if they never worked in industry, have spent a significant amount of time with physical lab exercises and research, many of which have the potential for causing real damage if done improperly.
 

WBahn

Joined Mar 31, 2012
25,760
I appreciate the reply. I find myself wondering why I continually stop and start using units.
It's one of those things that usually doesn't really scream out its value until we get royally bit by it. Sure, we can understand on an intellectual level that it has value, but we don't really believe it. I was lucky (in more ways than one) because I was almost killed because someone couldn't be bothered to track his units (and the someone WAS killed). That gave me a very personal perspective on the value of doing so, but I also saw my grades when I went back to college go from being solid to being almost straight A's. I also can't count the number of mistakes, both mine and others, that I've been able to catch because of attention to units.
 

MrAl

Joined Jun 17, 2014
7,593
I appreciate the reply. I find myself wondering why I continually stop and start using units. I can understand why it's hard to overstate it when so many problems can be solved, and little mistakes avoided, by just substituting the numerical values in at the end. I solved all the problems after considering this and the voltage thing you mentioned.

THANKS!!!

Hi,

You can also check your work by doing the problem using a different technique. For this circuit you might consider using superposition to find the result. The topology looks horizontally symmetrical so you can probably write one equation, then use that equation to generate the second equation, then solve for Vo.
If you get the same result as with Nodal, then you can be more certain you got it right.

Because this circuit has a dependent source, it is a little more challenging :)
 
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