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All physics textbooks wrong in the setup and derivation of the RLC series circuit equation?

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Ash Small

Sat Jun 20 2015, 01:32PM

Registered Member #3414 Joined: Sun Nov 14 2010, 05:05PM
Location: UK
Posts: 4245

All mathematical models are wrong, but they can be useful approximations, sometimes, if you use the correct model.....Who was it who said 'In theory, theory is the same as practice, but in practice, it isn't'?

Hazmatt_(The Underdog)

Sat Jun 20 2015, 05:30PM

Registered Member #135 Joined: Sat Feb 11 2006, 12:06AM
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Posts: 1735

This is the justification from my Physics Book, Halliday/Resnick/Walker 6th Edition

Signification

Sat Jun 20 2015, 06:59PM

Registered Member #54278 Joined: Sat Jan 17 2015, 04:42AM
Location: Amite, La.
Posts: 367

Hazmatt_(The Underdog) wrote ...

This is the justification from my Physics Book, Halliday/Resnick/Walker 6th Edition

Great!! Starting there...

We know that:
1) Emf = closed path integral of Eâ€¢dl
and since, in general, the flux (Î¦) in a circuit is proportional to the current (i), where the proportionality constant is the self inductance (L), we have:
2) Î¦ = -L di/dt (with Lenz'); So from your physics book:
3) Emf = -dÎ¦/dt; now, equating 1) and 2) yields:

***** Emf = closed path integral of Eâ€¢dl = - L di/dt (NOT 0) ***** THIS IS THE GENERAL LAW OF FARADAY:

Why not just use this whenever an inductor is (or is not) in the loop and settle for the right answer obtained every time and in a valid way. AND, unlike several applications in other fields, where such an argument leads to non-classical techniques to get the "exact / precise / perfect" answer is impractical, here it is no more difficult.

...

Sat Jun 20 2015, 10:51PM

Registered Member #56 Joined: Thu Feb 09 2006, 05:02AM
Location: Southern Califorina, USA
Posts: 2445

Signification wrote ...

***** Emf = closed path integral of Eâ€¢dl = - L di/dt (NOT 0) ***** THIS IS THE GENERAL LAW OF FARADAY:

Why not just use this whenever an inductor is (or is not) in the loop and settle for the right answer obtained every time and in a valid way. AND, unlike several applications in other fields, where such an argument leads to non-classical techniques to get the "exact / precise / perfect" answer is impractical, here it is no more difficult.

You can, it is certainly not wrong on a technical level to solve the circuits presented in this thread that way. Of course if your professor tells you to correctly apply Kirchoff's laws and you solve it using Faraday's law you would still be marked down

The point of Kirchhoff laws are to make circuit analyses simpler. Sure, for this exceedingly simple case of a few components in a single loop you do not save a whole lot of time by using the simplified method proposed by Kirchhoff, but I would love to see you use Faraday's law to solve something as simple as a differential amplifier, without going through roughly the same steps Kirchhoff did to derive his laws, and then solving it as he proposes.

I do feel that a point which has been made several times (by Klugesmith, myself, Uspring, and others) needs further attention, which is that you need to be consistent with your method. If you apply Kirchhoff's voltage law just like you would Faraday's law (as has been done several times in this thread) you will get the wrong answer, because that is not the correct way to apply Kirchhoff's laws (as has been pointed out by the aforementioned people). If you are applying Kirchhoff's voltage law on a circuit which is under the influence of an external magnetic field, you need to put in a model for the coupling between the magnetic field and the wires in your circuit. If you do not do this, you are breaking the assumptions that are used to derive Kirchhoff's laws, and you will get the wrong answer, as you would expect.

Personally, I do not like to see Kirchhoff's law written as a path integral, because I find it misleading. Some physics books have an unhealthy obsession with calculus and try to re-write the usual Kirchhoff's formulation (...The directed sum of the electrical potential differences...) as a path integral, but as you have pointed out this can be confusing. I have yet to see a book written by an electrical engineer that does it this way, since the whole benefit to Kirchhoff's laws is that you can beak the circuit down into a number of abstract 'components' (resistances, voltage controlled current sources, small signal models for nonlinear components, etc) and easily manipulate the resulting system of equations to quickly extract the circuits behavior.

Antonio

Sun Jun 21 2015, 02:15AM

Registered Member #834 Joined: Tue Jun 12 2007, 10:57PM
Location: Brazil
Posts: 644

I like to think that Kirchhoff's laws only work perfectly with small or slow purely resistive circuits. I the circuit happens to have a pair of conductors with large area one close to the other, electrical charges accumulate on them when the voltage between them varies, violating the current law. We then add a capacitor there and the law holds again. A long conductor, possibly coiled, presents a voltage over it when the current varies, violating the voltage law. We add then an inductor to the model and the law continues to hold. If this same conductor produces a voltage when the current in a nearby conductor varies, we add a transformer to the circuit to account for the effect.

DerAlbi

Sun Jun 21 2015, 04:31AM

Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727

Kirchhoff works with every circuit element there is. There are no restrictions as long as you have a V-I-relationship that can be expressed as equation.
If such relation happens to be nonlinear, interpolated tables or a differential or even an integral so be it.
Best example of applied kirchhoffs law is a circuit simulator. And magically you even can simulate inductors!

klugesmith

Mon Jun 22 2015, 12:16AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716

Ash Small wrote ...
All mathematical models are wrong, but they can be useful approximations, sometimes, if you use the correct model.....Who was it who said 'In theory, theory is the same as practice, but in practice, it isn't'?

An Internet search just found it attributed to one Chuck Reid: In theory, there is no difference between theory and practice; In practice, there is.

The aphorism about models belongs to statistician George E. P. Box, FRS. His writings include several variants, today well covered by Wikipedia. Referring to books from 1987 and 2005,
p.74: "Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful." p.424: "Essentially, all models are wrong, but some are useful". p.440: "The most that can be expected from any model is that it can supply a useful approximation to reality: All models are wrong; some models are useful".

Signification

Mon Jun 22 2015, 01:58PM

Registered Member #54278 Joined: Sat Jan 17 2015, 04:42AM
Location: Amite, La.
Posts: 367

I would like to make a brief summary of my original intentions, of a circuit involving an (L), to be looked upon:
There are four possibilities:
----------------------------------- ---------------------------------------
BTW, I like to use the following shorthand integral notation:
"INTEG" = any integral
"PATH" = line integral
"C.L." = closed loop (line integral)
"O.L." = open loop (line integral)
"C.S." = closed surface (surface integral)
"O.S." = open surface (surface integral)
---------------------------------------- ------------------------------------
"L"'s contribution to:

1) Emf of C.L. PATH of Eâ€¢dl = -L di/dt.........L's contribution in transversing the loop= 0
2) Emf of C.L. PATH of Eâ€¢dl = 0................L's contribution in transversing the loop = -L di/dt
3) Emf of C.L. Path of Eâ€¢dl = -L di/dt.........L's contribution in transversing the loop = -L di/dt
4) Emf of C.L. PATH of Eâ€¢dl = 0.................L's contribution in transversing the loop = 0

1) Is "correct"
2) Incorrect, but yields the proper answer ( identical to 1) )
3) Incorrect: would employ the "L" twice! (I have never seen this used--possibly because of the absurd answer)!
4) Incorrect: would "NEVER" consider "L" (resulting in just an RC behavior) "L" would be missing from the differential equation, and, from the obviously wrong, solution!

So, whether 1) or 2) is used, we get the same, correct, answer. 3) and 4) yield nonsense.

...

Mon Jun 22 2015, 06:02PM

Registered Member #56 Joined: Thu Feb 09 2006, 05:02AM
Location: Southern Califorina, USA
Posts: 2445

Your expressions without further justification of your assumptions used to derive them are insufficient to conclude 'correct' or 'incorrect'.

1) is correct under the usual assumptions of Faraday's law (infinitely thin conductor, not moving at relativistic speeds, no external fields acting on the circuit, yatta yatta, maybe we can all agree to use Halliday/Resnick/Walker's derivation?)
1) is incorrect under several of the usual assumptions of Kirchoff's law, mainly the assumption made in the derivation of Kirchoff's laws that the 'wires' (which refer to the abstract concept of an electrical connection between 2 components, not the physical wires in your circuit) have exactly the same voltage at all points (on a given wire) for a given time. Clearly in this circuit if we model the inductor as a coiled up 'wire' we have a contradiction.

2) is incorrect under the usual assumptions of Faraday's law, because in order to set E*dl = 0 there needs to be no flux in the circuit (neither external or internal).
2) is correct under the usual assumptions made for Kirchoff's law, where one has modeled the inductance of the physical inductor as a lumped element with inductance L, and additional 'wires' (which recall, are not representative of the physical conductors in the circuit, but rather the abstract concept of the electrical connections between the lumped components) which do not drop any voltage.

3) and 4) are wrong under either set of assumptions, but I would argue that with some creative assumptions they could be correct. In any case, they aren't particularly useful to us since they won't be modeling an inductor in this universe

BigBad

Mon Jun 22 2015, 08:06PM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
Location:
Posts: 600

Is it just me, or have you guys not thought about why complex numbers are used to analyse circuits?

Kirchoff's law work just fine with inductors and capacitors in the circuit if you use complex impedances; provided also that there's no significant unmodelled stray inductances or capacitances etc.

In that case, the voltages do indeed zero out around loops.

The ideas that Kirchoff falls apart if there's inductance, or that Faraday's law is not already part of Kirchoff are total bullshit. You just have to use complex impedances and currents and voltages.

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