[Circuit theory] Help me solve this mathematically

studiot

Joined Nov 9, 2007
4,998
Bearing in mind this was posted in the homework section, I think the OP's course had introduced mesh analysis and was leading up to formal matrix methods. My post #7 furthers this aim.
I further think that the OP got these ideas confused with Kirchoffs laws, as often happens if Kirchoff is introduced at an earlier stage.
 

WBahn

Joined Mar 31, 2012
29,976
Mesh analysis is not a form of KVL and should not be attributed to him.
I'm not attributing anything to anyone. Saying that Method A is a formalized application of Method B is making no claim regarding who devised either method or even which method was devised first. There are tons of examples in which something developed first is later properly described as a special case of something that was developed much later. The methods and their relationships are what they are and depend on the physics, not who first recognized them and wrote them down.

Mesh analysis is most certainly a formalized application of KVL. In turn, KVL is a direct application of Faraday's Law in the special case of constant magnetic flux. Furthermore, it is entirely appropriate to say that KVL is a direct consequence of Maxwell's equations despite the fact that Kirchhoff formulated KVL (as a generalization of Ohm's work) something like 15 years before Maxwell published his work and nearly four decades before the equations were actually brought together and described as a set of equations my Heaviside and Hertz.

Consider the circuit fragment in the attached diagram. Ignore all of the loop currents for now and apply good old KVL:

EQN#1

\(
V_0-I_1R_1-I_2R_2-I_3R_3 = 0
\)

Now adopt the following formalisms:

Consider a circuit that can be drawn as a planar mesh (i.e., drawn on a piece of paper with no wires crossing). Define a mesh current for each mesh element with each mesh current flowing in the same sense (i.e., either clockwise or counterclockwise). For each mesh, apply KVL in the same direction as the mesh current, expression each branch current as the superposition of all mesh currents sharing that branch. Note: The "all" in the previous sentence actually extends this from the mesh current analysis applicable to planar circuits to the more general loop current analysis applicable to non-planar circuits as well.

Using this formalism and applying KVL, we then have:

EQN#2

\(
V_0-(I_{M5}-I_{M2})R_1-(I_{M5}-I_{M6})R_2-(I_{M5}-I_{M8})R_3 = 0
\)

This can then be rewritten as:

EQN#3

\(
-I_{M2}R_1+I_{M5}(R_1+R_2+R_3)-I_{M6}R_2-I_{M8}R_3 = V_0
\)

From which we start to see the structure that permits us to write the mesh equations by inspection; namely set the coefficient of the present mesh current equal to the positive sum of the resistances around the mesh and the coefficient of any adjacent mesh currents to the negative sum of any resistances shared and then set the sum of all of these to the sum of the voltage gains due to the voltage sources around the mesh.

Extending this to deal with current sources and/or dependent sources merely requires taking a step back and seeing how they affect the structure of the equation resulting from applying KVL under the formalism rules adopted.

It should be noted that there is nothing in the establishing formalism that fundamentally requires the circuit be linear. By this, I merely mean that we can still use the formalism to write equations of the form of EQN#2 above. However, if the constitutive relations for the components along the branches are not linear, then we will pick up cross terms and nonlinear terms when we write the equivalent of EQN#3. This may or may not result in a set of nonlinear equations that are easier to work with than those produced by other methods. As soon as you start talking non-linear systems, all bets are off as to what the best approach is since it is highly system dependent. In most cases, however, it isn't worth the trouble to even go down this path for a nonlinear system.
 

Attachments

studiot

Joined Nov 9, 2007
4,998
I'm not attributing anything to anyone. Saying that Method A is a formalized application of Method B is making no claim regarding who devised either method or even which method was devised first. There are tons of examples in which something developed first is later properly described as a special case of something that was developed much later. The methods and their relationships are what they are and depend on the physics, not who first recognized them and wrote them down.

Mesh analysis is most certainly a formalized application of KVL. In turn, KVL is a direct application of Faraday's Law in the special case of constant magnetic flux. Furthermore, it is entirely appropriate to say that KVL is a direct consequence of Maxwell's equations despite the fact that Kirchhoff formulated KVL (as a generalization of Ohm's work) something like 15 years before Maxwell published his work and nearly four decades before the equations were actually brought together and described as a set of equations my Heaviside and Hertz.
We have debated the exact form of KVL in this forum before, including going back to the original papers or at least the English translation of same.

I am sorry to see that you prefer the americanisation of KVL which arrived sometime in the latter half of the last century as it can be variance with Faraday's law as Professor Lewin of MIT ably showed. Kirchoff's original statement concurs with Faraday's Law and does not lead to Lewin's paradox. We have also discussed his video at great length here.

Yes it is true that you can obtain a set of mesh currents from KVL. However there is (mathematically) something more fundamental going on. This is a bit like the principal of virtual work in mechanics, which derives from the same underlying property of a system of linear simultaneous equations.
I understand that it was this mathematical process that led Maxwell to his mesh method. He had already acknowledged the work of Kirchoff to the extent of translating into English and commending it to others.

I am also aware that all these methods are engineering idealisations for the express purpose of calculating circuit properties. It is possible to draw a theoretical circuit that cannot be 'solved' using KVL.

You clearly understand the mesh method and the bald fact remains that it is the only method that proposes a single 'loop current' in place of an actual current and here the OP has equally clearly been told of and attempted to use such a loop current.
 

WBahn

Joined Mar 31, 2012
29,976
I may or may not be correct on the history. It doesn't matter, because I am ONLY talking about the methods and how they relate to one another irrespective of how or when either of them came about. The techniques and the relationships between them are what they are and have been since shortly after the Big Bang (or whatever you choose to mark the beginning of the present laws of physics).

My main point in even mentioning it is that I have seen lots of people that have been shown how to turn the crank on mesh current and node voltage analysis but who have absolutely no concept of what it is or what it means. They tend to therefore view it as some mysterious black box recipe that simply has to be applied to solve their problem. The same, of course, can be said of any number of techniques in any number of fields of study. My hope (and I have generally found it to work, but not always) is that by showing someone how various techniques relate to one another, then people are able to take their understanding, even if only partial, of each of the techniques and meld them into a much stronger understanding of all of them. This is especially true if they have a strong understanding of just any one of them. I have also found that it tends to reduce the likelihood of incorrectly mixing and matching parts of different techniques together when someone understands how they relate to one another.
 

studiot

Joined Nov 9, 2007
4,998
Hello WBahn, I think I agree with pretty much everything you say and have even drawn (positive) attention to it earlier in the thread.

However this is a teaching site and this particular section is dedicated to issues raised by students about their homework or coursework.

In particular this is the beginning of the original post.

Hi, the problem is attached and i'm trying to solve it and despite me doing circuit theory for the past weeks, this problem has me stumped.

I want to calculate the current I over R1 and R2.

Regarding R2 using the loop current method.
Nowhere in the original post is Kirchoff mentioned but reference is made to the 'loop current method'.

Judging from the complexity of the question I would hazard a guess that Zerotorrent is studying circuit theory as part of some engineering course with a large electrical/electronic component, rather than physics.

Many courses require a particular form of analysis in particular questions for academic reasons.
The student will receive no marks for solution by a different method.


When I entered this thread, perhaps a little late, several alternative methods had been offered but no attempt to lead Zerotorrent to a correct 'loop current solution' had been attempted.

So this was the done in my first post in this thread.

This in no way is meant to detract from the kind efforts of others to help. Maybe Zerotorrent can also learn something additional to outside his course from these.

Incidentally, I find it useful to talk about branch currents for methods that assign different currents to different points in the same loop, such as KVL. This avoids the confusion arising from the mesh method, that many beginners suffer.
 
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WBahn

Joined Mar 31, 2012
29,976
When I entered this thread, perhaps a little late, several alternative methods had been offered but no attempt to lead Zerotorrent to a correct 'loop current solution' had been attempted.
Huh? In the very paragraph that you objected to I pointed out on the basic problem he was having in applying the loop current method: "As such, you almost always need ALL of the equations to solve ANY of them, otherwise, you will have more unknowns than you have equations." He presented a single loop equation and was trying to work with it independently of the others.
 

studiot

Joined Nov 9, 2007
4,998
Originally Posted by studiot
When I entered this thread, perhaps a little late, several alternative methods had been offered but no attempt to lead Zerotorrent to a correct 'loop current solution' had been attempted. ?

Huh?
Yes exactly.

The situation was exactly as described when I made my first post - post#7.

This was about three hours before your post #8.
 

WBahn

Joined Mar 31, 2012
29,976
Okay, you are correct. I didn't look further back than my first post because I thought you were explaining why you were objecting to my first post. Now I'm a bit confused on what you were explaining. This, by the way, is not a dig; it is hard to keep track what people are talking about -- and I know my posts suffer the same malady -- because each person tends to come at the thread from the view point of a running conversation at a party and, like that conversation at a party, side conversations frequently arise that people meader through. While those can get just as confused, it is generally easier to keep the various conversations in better context than it is when you've just got a bunch of written replies. Which is somewhat counter-intuitive at first glance, because everyone is seeing the exact same information. But I think this is actually what makes it harder, because we too easily lose sight of the fact that everyone is interpretting and emphasizing that information differently.
 

studiot

Joined Nov 9, 2007
4,998
@WBahn

I'm sure we both have the best interests of helping zerotorrent at heart and can cooperate if he ever returns.

I am about to go on shift so will have to leave things now, but I have opened the door to help with post 7 for the eventuality that he actually means mesh analysis. Any (knowledgeable) one should be able to help him on from there.

On comment in your earlier post suggests you are a lecturer in the subject?

I agree it is often confusing for students and teachers alike as this thread shows.

http://forum.allaboutcircuits.com/showthread.php?t=68522

go well
 

WBahn

Joined Mar 31, 2012
29,976
On comment in your earlier post suggests you are a lecturer in the subject?
At times. For the last two decades I have been, mostly, a full-custom mixed-signal ASIC designer but I love teaching and have usually taught at the local universities on the side.

I have taught quite a few college courses and this has included courses in linear systems. I've also taught a senior level transistor design course where the students that learned via the "recipe book" approach really come into problems. I've learned to start that course with a one-class whirlwind tour of circuit analysis that starts with the defining relations for a resistor, capacitor, inductor, diode, and transistor and then show the application of KVL and KCL. From there, if I recall, I focus on the linear elements and introduce complex representations and show how phasors are merely shorthand notations for Fourier and/or Laplace transforms. I then move to superposition and, from there, to Mesh and Nodal analysis. I then step back to the time domain and bring in the non-linear elements and show the full application of KVL/KCL to get a differential equation but then show the expresion for an incremental change about an operating point and segregate the equation into the part involving the incremental voltages and currents and the parts that don't so that I can show them where the "large signal" and "small signal" circuit equivalents come from. At that point, I am ready to start with the exponential models for the diodes and transistors and develop the small signal models that are the focus of the course.

I do all of this in one class (which is actually about 75 min where I was teaching it) and it is amazing to watch the eyes open and the positive comments that come back. Some of it is because I really do think that I have given enough planning to the lecture and the presentation so that it is very coherent and transitions nicely from one topic to the next, but most of it, I think, is just that they have been introduced to all of these things largely in isolation and, since they were just learning each, they were focused on the little picture of how to use it to solve the problems on their homework right then. By seeing it all reviewed in a rapid fire continuous arc, I think many of them for the first time are seeing how it all ties together and fits into a larger whole. I know for sure that the confusion between what the large signal, small signal, and total solutions are and how they work together goes way down compared to before I started doing that.
 

vishnu.mec

Joined Aug 6, 2010
23
Hi

I dont understand why you would want to use mesh analysis here..
If you want to find the current (or voltage) when you have a lot of sources(like in here), go for the superpostion principle.

Considering u1 only(ie, replacing the other sources by their internal impedances), we can instantly see that it can provide current only to r1. that current is 250mA(from bottom to top).

Then considering u3 alone, we can see it cant drive a current through either r1 or r2 since both will be short by u2.

Again, considering u2 alone, we can see its connected across r1||r2||(r3+r4). thus the current through both r1 and r2 will be -500mA(the negative sign because its from top to bottom).

this is the total current through r2. but the current through r1 is
250mA+(-500mA)= -250mA.

That's it..! :)
 

DerStrom8

Joined Feb 20, 2011
2,390
Hi

I dont understand why you would want to use mesh analysis here..
If you want to find the current (or voltage) when you have a lot of sources(like in here), go for the superpostion principle.

Considering u1 only(ie, replacing the other sources by their internal impedances), we can instantly see that it can provide current only to r1. that current is 250mA(from bottom to top).

Then considering u3 alone, we can see it cant drive a current through either r1 or r2 since both will be short by u2.

Again, considering u2 alone, we can see its connected across r1||r2||(r3+r4). thus the current through both r1 and r2 will be -500mA(the negative sign because its from top to bottom).

this is the total current through r2. but the current through r1 is
250mA+(-500mA)= -250mA.

That's it..! :)
As studiot mentioned when I suggested using superposition, a mesh analysis is much easier in this case. There is a lot less work that would need to go into it.
 

WBahn

Joined Mar 31, 2012
29,976
I dont understand why you would want to use mesh analysis here..
Almost certainly, the reason is because that is the method that is being studied and/or evaluated. In other words, the OP has no choice regarding which method to use.

I would say, on this particular problem, the easiest method is very ad-hoc. Recognize that the voltage source U2 splits it into three separate problems that don't interact and solve each independently. All of the currents and voltages can be trivially solved in your head and written down directly.

Hopefully, the OP (and anyone else) gets into the habit of looking for alternate ways to solve any problem with an eye to pursuing the alternate path at least far enough to check if their original solution is correct or at least makes sense.
 
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