# Loop Rule

1. Aug 19, 2005

### hidayah

Why does the Loop Rule arise as a consequence of conservation of energy?

2. Aug 19, 2005

### Tide

Kirchoff's loop rule simply states that if you traverse a loop and return to given point then the potential at that point remains the same, i.e. the electrical potential is single-valued!

3. Aug 19, 2005

### rbj

what's the deal with the terminology here?? are we talking about Kirchoff's Voltage Law (KVL)?

if there is a net changing magnetic field inside the loop (of any reasonable quantity), it won't be a single electrical potential. this is why 60 Hz AC hum gets induced into audio circuits. but it should be small.

if there is no net changing magnetic field, then taking a small test charge from point "A" around the loop and back to point "A", then the electrostatic field is "conservative" and the integral or sum of all of the work done to that test charge will be zero and that is why, assigning the polarities consistently going around the loop clockwise, the sum of all of the voltages is zero.

Kirchoff's Current Law (KCL) for every node (less the "ground" node), Kirchoff's Voltage Law (KVL) for every loop (there are also redundant loops that need no separate equation), plus the volt-amp characteristics of every device connect between the nodes (that are also in the loops) are exactly the information one needs to analyze an electrical or electronic circuit.

Last edited: Aug 19, 2005
4. Aug 19, 2005

### Tide

rbj,

I'm not sure what you mean when you say there's not a single electrical potential when an oscillating magnetic field is present. If the electrical potential is multivalued at any given point then it is unphysical. Perhaps you meant there are different frequency components?

5. Aug 19, 2005

Are you implying that 'loop' or 'mesh' analysis does not work for AC circuits? If you are implying this, you might want to rethink your statement.

6. Aug 19, 2005

### rbj

no, i am not. (and i am not sure what i said to be construed to mean that.)

i am not sure if this is the offending concept, but one of Maxwell's Equations, expressed in integral form (i think it's Faraday's Law) says that when there is a changing magnetic field, going around that changing magnetic field in a closed loop gives you an induced potential (voltage) that is proportional to the rate of change of the magnetic field. do you agree with that? if so, then there is a deviation from Kirchoff's Voltage Law, the voltages don't all add up to zero.

like the deviation of Kirchoff's Current Law (where the currents going into a node don't sum to exactly zero resulting in a charge buildup), this KVL deviation is not normal and is not what we do when we analyze circuits (using the so-called "node-voltage" or "loop-current" methods) because we model that non-zero current or voltage as an additive source (and it's some kind of noise or hum or error source).

are we on the same page? i know what i'm typing about here.

Last edited: Aug 19, 2005
7. Aug 19, 2005

I still don't see how this is not saying that KVL, KCL, Loop analysis and Nodal Analysis does not work for AC circuits.

"Hum and buzz (50Hz/60Hz and it's harmonics) occur in unbalanced systems when currents flow in the cable shield connections between different pieces of equipment. Hum and buzz can also occur balanced systems even though they are generally much more"

http://www.epanorama.net/documents/groundloop/

I don't think that what you're trying to tell me is coming across correctly.

Regards,

8. Aug 19, 2005

### Tide

rbj,

You're thinking about induction and, since it's caused by varying magnetic flux, it's not quite correct to talk about electrical potential (vector potential will do, however!). In any case, the potential is single valued so traversing a loop will get you back to the same potential when you return to your starting point. Kirchoff works!

Induced fields are taken into account as the inductance (mutual and self) of the circuit or its components.

9. Aug 19, 2005

### rbj

i dunno what to say to you guys. i dunno if it's a communication problem or whatever. maybe i'm just trying too hard to dot the i's and cross the t's, but this is a real thing in real circuit design on real printed circuit boards. if you have a bunch of components laid out in a loop that is big enough in area, and in an environment where there is some 50 Hz or 60 Hz electromagnetic radiation about, this causes this hum to get into the voltage signal of those components laid out in a loop. this is precisely because of Faraday's Law and that, in fact traversing this loop will get you back to a different potential when you return to your starting point. but, what we do so we can use our normal node-voltage or loop-current analysis methods, is we model that difference in potential (that should be zero if we were more fortunate) as a "virtual" component: an additive independent voltage source. it might be a couple of microvolts.

this is not controversial among practicing electrical engineers. but it is something we don't put into the first sophmore physics and EE texts, because we don't want to confuse the hell out of students and we are able to model this added voltage as a lumped voltage source in the loop and then say all of the voltages add up to zero.

are we still in disagreement?

10. Aug 19, 2005

### Tide

rbj,

The hum is real but your explanation is not.

Please explain how the electrical potential at a point can have two different values?

11. Aug 19, 2005

### rbj

one more thing:

forget about circuits for the time being...

given just Coulomb's Law and a static electric field, then you have a conservative potential field and doing a line integral of work moving a test charge around any closed loop will get you zero. always.

this is not the same thing as a having a static electric field along with a changing magnetic field.

12. Aug 19, 2005

### rbj

this is not directly related to the issue at hand, but, in fact, the KVL, KCL, Loop and Node analyses we do for circuits at low frequencies does not work for extremely high frequency (like microwaves and above) AC circuits. you get to go to grad school in electrical engineering and take a sh1tload of really hard classes to learn how to do that stuff.

think of an antenna. there is current at one part of the radiating element where it is driven, but no current at the ends. that sure as hell does not satisfy KCL.

edit: just to be clear KVL, KCL, Loop and Node analyses work well when the wavelength of the signal in the circuit is much longer than the dimensions of the circuit and components. when that is the case, you gotta do physics rather than just circuit analysis.

hum and buzz can occur inside of a single box where the power supply components are not adequately shielded from the very low voltage analog signal processing components. you get some kind of nasty large varying E and M fields and the changing M fields induce spurious voltages in loops of components strung together. when the signal going into the op-amp is only a few microvolts (say it's coming out of an un-preamped microphone), that induced 60 Hz voltage can add up to something that competes well with the desired signal. and it is precisely because the voltages are not adding to exactly zero.

what happens when you have a single circular closed loop of wire in the presence of a non-zero and changing magnetic field? what happens when you apply

$$\oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt}$$ ?

that first integral, on the left hand side, is a voltage. do you get zero?

Last edited: Aug 20, 2005
13. Aug 20, 2005

### rbj

i'm sorry you don't like it.

the purely electrical potential at a point does not have two different values at the same instance of time.

but KVL isn't just that. KVL says that, assigning a consistent sense of polarity going around a loop of electrical components, that the voltages of all of the components of the loop add to zero. but if there is a changing magnetic flux that this loop of components is strung around, those voltages will not add to zero.

KVL works. but sometimes to make it work (when analyzing these noisy or hummy situations) we lump that non-zero induced voltage into a single virtual voltage source in that loop. sometimes we model it as several little voltage sources in series with each component in that loop (that might be more accurate, but is often not necessary).

14. Aug 20, 2005

### Tide

The loop rules apply to an instant of time.

15. Aug 20, 2005

### rbj

i never implied anything different. the left hand side of

$$\oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt}$$

is a line integral in space applied at a instant of time and the right hand side is an instantaneous rate of change at the same instant of time.

yet, if the right hand side is non-zero (that is my premise for the case of induced hum), then the left side is non-zero.

also, back to terminology: by "loop rules" i presume you mean Kirchoff's Voltage Law. if this is not the case, then all bets are off.

Last edited: Aug 20, 2005
16. Aug 20, 2005

### Antiphon

Rbj is right again. I have taken all those hard classes and then some.

When you analyze a circuit, it's an idealization where the
inductance, capacitance and resistance are concentrated in the components.

In reality they are distributed everywhere. So KVL works for an idealized
circuit because the idealization takes into account all the inductance. You can't treat
a real wire as if it were an ideal wire. Then KVL can't be used and you
must use the full induction relation for a physical wire.

The confusion with AC vs DC is because in a DC circuit the time dependent terms are
gone so you are only left with resistances. These are distributed too in
a real circuit but you can still apply KVL by integrating the differential VI
drop around the actual circuit- because different parts of the circuit are
decoupled at DC.

But you cannot do this in an AC circuit because the equivalent schematic will change.
Parts of the circuit will couple to other parts and in different ways depending
on the frequency.

A simple example is the power line in your house. At 500 MHz, the schematics
won't look anything like they do at 60 Hz if they are to accurately model the
circuit. And at 500 MHz, the component values of the equivalent circuit
will change depending on where I sit in my house. This is not the case at 60 Hz.

In sum,
-KVL, KCL work only for schematics, not physical circuits.
-KVL, KCL can still work for a DC but not AC physical circuit by integrating the differential VI drops around the circuit

Edit: Oh yes, and as for the OP, it has nothing to do with conservation of energy.

Last edited: Aug 20, 2005
17. Aug 20, 2005

### Tide

rbj,

Regarding my comment: "The loop rules apply to an instant of time." you said

Well, yes, you did. You said

followed by

If the electrical potential at a point has two different values and if those two different values cannot be at the same instant then you must have returned to the starting point at a different time than when you started.

In any case, I think we should put the burden back onto the original poster whose question is ambiguous and unclear.

18. Aug 20, 2005

### rbj

here i made the mistake of using the word "when". i do not offhand know how else to concisely word it, but "when" did not mean at a later time than starting. you can still, conceptually traverse the loop with a test charge in an instant of time (getting back to your starting point at the same instant of time) and you will have a line integral of the total amount of work done on that test charge. if there is no time-varying magnetic field, then the line integral done at that instant of time is zero because an electrostatic field is conservative. but if there is a non-zero

$$\frac {d\Phi_{\mathbf{B}}} {dt}$$

at that instant of time, then

$$\oint_{s} \mathbf{E} \cdot d\mathbf{s}$$

is also non-zero and the sum of all of the work done to the test charge is not zero meaning that the instantaneous sum of all of the voltages of the components in the loop do not add to zero and that this is a practical deviation from KVL.

i think you're stretching it a little to imply that by using "when" i meant that it had to be two different instances of time for starting and ending the closed loop. in fact, i think i proved to you that i was correct all the time and you're trying to save face. that's okay.

but i never once said that KVL was not meant to be applied at an instant of time. what i always said is, that under some kind of adverse circumstances, KVL is not always precisely valid. sometimes the voltages (at an instant of time) do not always add to zero, going around the closed loop and that circumstance can only be during an instant of time that there is a net changing magnetic field flowing through the hole of the loop. i also said that the practice often is to say KVL is exactly true anyway, but then introduce a phantom voltage source in the loop that would be equal to the line integral above.

i said "purely electrical" when i could have been more clear by saying "purely electrostatic". the point i was making was that at an instant of time there is one unambiguous electrostatic potential for that point because classical electrostatic fields are conservative fields (i think the OP was sorta referring to that when he says "Conservation of energy". in that case KVL is completely accurate. but a more general electromagnetic field is not necessarily conservative and in that case KVL is not perfectly valid.

if the OP meant "KVL" when he said "Loop rules", i think it was reasonably clear. it was the other thread (Convervation of charge) that i didn't get right away.

anyway, perhaps i shouldn't have offered these arcane caveats to KCL and KVL (we basically always take them to be exactly true when we do circuit analysis), but i just didn't want some other EE taking issue with what i said if i didn't bring up these caveats. i was trying to cover my butt and got it spanked (unjustly, i think) anyway.

19. Aug 20, 2005

### Tide

rbj,

I am stretching nothing and I understood exactly what you meant by the word "when." Reread what I wrote in my previous post.

Your first statement says (a) the potential at a given point has two different values - at the same time - while your second says (b) the different values at a point can only occur at different times. Which is it?

If (a) then the potential is multivalued and nonphysical which makes it moot.

If (b) then you were in fact implying some temporal element in the loop theorem.

We've already spent more time on the semantics than is warranted and, again, I suggest we return the burden to the original poster who posed an ill-stated and ambiguous question.

20. Aug 20, 2005

### rbj

i'm glad to read that. then there is no misunderstanding. nowhere did i mention anything about KVL at different times. (by "changing magnetic field" i mean the instantaneous rate of change at some instant of time, what the calculus prof. calls "the derivative".)

i've been very careful (except for saying "when" in a context of an instant of time and "purely electrical", meaning no magnetic component, when i should have said "purely electrostatic"). you said that i implied that this deviation from KVL had something to do with different times or that my explanation had something to do with different times and i never said that. it is you, Tide, that needs to carefully read what the other is saying and not inject something unsaid into it.

only if there is a changing magnetic field and then comparing the potential if there was no movement of the test charge to the resulting potential if the test charge was moved around a closed curve that encircled some of that changing magnetic field. same point in space, same instance of time, but different potentials if you include the magnetic effects. (of course they can't be different if you are only considering electrostatic effects.)

to be more clear, scaler "potential" doesn't have a well-defined meaning in cases other than purely electrostatic. with pure electrostatic situations, there is this unambiguous property of each point in space called its electrical potential and then we can call that electrostatic field a "conservative" field. and if you move a test charge around in that field and move it back to the starting point, no net work is done to it.

but it is not well-defined as a scaler field for situations with changing magnetic fields. you can take a test charge, move it around, return it to the original point (all in an instant of time) and been forced to do a net non-zero amount of work doing it. however if you just left that charge alone at that point (fixed it so it can't move), you would have done no work on it. same situation, identical starting and ending point, but different paths and different amounts of work done. that is not a "conservative field" and KVL will not be valid in that situation.

i never said anything about "different times" (until you brought it up and only to say that i ain't saying anything about "different times"). you are the first (and only) person to bring this up. i am saying something about different circumstances (for comparison purposes). like "compare scenario A at some point in space and some instance of time to scenario B at the same point in space and the same instance of time." that is what is different. i never said anything about different times.

ah! now you said something meaningful that we can talk about. i respectfully disagree with what you say here (it's not "nonphysical" at all, perhaps a nonconservative field, but certainly not nonphysical). and i have stated my alternative above. this might be something tangible to argue about. maybe not.

no, only you are.

fine by me. (it wasn't just semantics. you made a factual statement, i said i never implied otherwise, then you made another factual statement saying that i did imply otherwise which was not true.)

Last edited: Aug 20, 2005