Paradoxical circuit behavior

studiot

Joined Nov 9, 2007
4,998
the Electrician
What you describe is ohmic losses, which were postulated to be zero.
You did not address my comments in post#9 concerning this situation.

Further what happens if you take the limit as R tends to zero in my post #15 analysis?

I agree with Professor Shrive that there is mathematically no difference between resonance in mechanical and electrical (or any other) systems.

Resonance requires a periodic forcing function that can supply drive at precisely the right timing to be accepted by the driven system.
Forcing functions with a period close to this will also excite the driven system but the response will be at a maximum at the resonant frequency.

The drive and the response may be measured in different units depending upon the situation.
 
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nsaspook

Joined Aug 27, 2009
13,081
What you describe is ohmic losses, which were postulated to be zero. Current thinking is that radiation accounts for the loss of energy; here are the calculations:

http://kirchhoff.weebly.com/uploads/1/6/3/0/1630371/t.choy_ajp72-662.pdf
I can't access the first paper but this is the conclusion of the second:
VIII. CONCLUSION
We have extended the discussion of the radiation from the
transient switching of charges between two capacitors. We
have shown that the capacitors themselves can radiate, using
a point electric dipole model. We found this radiation to be
small but not insignificant,
..
The calculation of the radiation
would be a good exercise for an undergraduate student using
the methods developed here. Our results show that although
the details of the capacitor radiators are unimportant for the
recovery of the missing energy, they are important for the
study of the transient response and electromagnetic compat-
ibility.
...
The analysis shows again that
the two-capacitor problem with radiation still remains elu-
sive
If we step the potential causing the electrons in the cross-section of the conductor to move so fast as a group that there aren't collisions (reducing PE by conversion to thermal energy) with other electrons in the conductor (low electrical resistance) this means they will hit the capacitor plate first and the potential energy of total electron flow can become a significant factor of total energy like we see in a high energy beam in vacuum. This PE will be converted to some type of KE when it hits the plates.
 

Thread Starter

The Electrician

Joined Oct 9, 2007
2,971

Thread Starter

The Electrician

Joined Oct 9, 2007
2,971
Unless the potential is so great that we see accelerations close to a large fraction of light speed the losses from EM radiation will be small.
This appears not to be so. The papers I referenced show that EM radiation can account for the loss. Here's another reference that doesn't include the full paper:

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=231509

but the abstract says:

"The circuit then behaves as a loop antenna and radiates energy in the form of electromagnetic radiation (EMR). All loss of energy in the system can be accounted for through EMR considerations."
 

nsaspook

Joined Aug 27, 2009
13,081
I don't doubt it's EM radiation in the end because thermal radiation (energy from atomic level vibrations as energy is increased ) is usually involved but it's not the kind normally associated with an antenna loop circuit IMO as this is in a completely ideal circuit without electrical resistivity or inductance. Inductance here is like elasticity, removing it creates a inelastic collision between two objects where there is conservation of momentum but kinetic energy is lost.
http://arxiv.org/pdf/1309.5034.pdf

What this shows is there are limits to that circuit theory can explain just as there are limits to what current physics can explain at the extremes of the very big (the creation of the universe) and very small (why quantum processes behave as they do).
 
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studiot

Joined Nov 9, 2007
4,998
Hello, Electrician,

My comment in post=9 concerned overidealisation.

I am basically suggesting that just as trying to solve a beam, built in at both ends by using the three equations of static equilibrium (Resolving vertically, horizontally and taking moments) will fail,

So will trying to solve the two capacitor problem by excessive idealisation of circuit elements and the use of KVL/KCL or equivalent circuit theorems.

In the case of the beam the real world elastic properties of the beam need to be invoked, in the case of the two capacitor circuit, the real world 'imperfections' of capacitors and wires are needed.
 

wayneh

Joined Sep 9, 2010
17,496
Further what happens if you take the limit as R tends to zero in my post #15 analysis?
I think that's the key here. The "paradox" only exists because of an apparent failure of the mathematics. Of course the math doesn't really fail, and using the right math will resolve the paradox.
 

Thread Starter

The Electrician

Joined Oct 9, 2007
2,971
Singal: http://arxiv.org/pdf/1309.5034.pdf
says that the "missing" energy is converted to heat at the capacitor plate where the collisions occur. I don't see whether he has published the paper in a refereed journal, and whether any of the proponents of the radiation theory have published criticisms. What if the capacitor plates are also superconductors?

Pankovic and Kapor: http://arxiv.org/pdf/0912.0650.pdf
have a theory which specifically rejects the need to convert the missing energy into heat, or to invoke EM radiation either. Their paper also shows no evidence of having been peer reviewed.

A google search finds many papers on the topic, and even now no consensus among the community of experts exists.

If the capacitors are made of superconductor material, as are the connecting wires of finite length, when the connection is made an oscillating current will exist forever. Surely EM radiation (not thermal radiation) will eventually carry away the energy.
 
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studiot

Joined Nov 9, 2007
4,998
If the capacitors are made of superconductor material, as are the connecting wires of finite length, when the connection is made an oscillating current will exist forever. Surely EM radiation (not thermal radiation) will eventually carry away the energy.
All this is like saying

If my beam is too narrow, like the Tay bridge, it will twist and topple over,
If it is to flexible, like the Tacoma Narrows bridge, it will wobble into failure,
and so on

That is the lost energy will be dissipated by whatever means available to it in the real world.
 

Thread Starter

The Electrician

Joined Oct 9, 2007
2,971
Hello, Electrician,

My comment in post=9 concerned overidealisation.

I am basically suggesting that just as trying to solve a beam, built in at both ends by using the three equations of static equilibrium (Resolving vertically, horizontally and taking moments) will fail,

So will trying to solve the two capacitor problem by excessive idealisation of circuit elements and the use of KVL/KCL or equivalent circuit theorems.

In the case of the beam the real world elastic properties of the beam need to be invoked, in the case of the two capacitor circuit, the real world 'imperfections' of capacitors and wires are needed.
I explained how idealization leads to the "paradox" in the first 4 paragraphs of post #1, and how real world imperfections allow the problem to be solved. You've reiterated and given further examples, but it didn't seem to call for a response from me.

All this is like saying

If my beam is too narrow, like the Tay bridge, it will twist and topple over,
If it is to flexible, like the Tacoma Narrows bridge, it will wobble into failure,
and so on

That is the lost energy will be dissipated by whatever means available to it in the real world.
Of course. I can't help but agree--if the ordinary resistance of the wires and material of the capacitors are made zero, as with superconductors, some other means will operate. It seems to me that it goes without saying. Some people find it interesting to explore what those means might be. A google search finds many papers on the topic, and there doesn't seem to be a consensus yet. When the connecting wires are of finite extent (with finite inductance and non-zero radiation resistance), I find the radiation hypothesis appealing.
 

Thread Starter

The Electrician

Joined Oct 9, 2007
2,971
Sometimes there may be a "reasonable" solution to one of these "over-idealized" circuits. Consider the example I gave in post #1:
ParCir5.png
Suppose the 6 and 12 volt batteries are composed of a series connection of some kind of cells--lead acid, or perhaps alkaline primary cells. Real cells will have an internal resistance R. The 6 volt battery will be a series connection of n cells, and the 12 volt battery will be a series connection of 2*n cells.

If R is not zero, then we have an easy solution; the voltage at the top of the 1 ohm resistor will be 24/(3+2*n*R). This expression has a well behaved, easy to calculate limit as R approaches zero--8 volts.

The absolutely ideal case has no solution, but if my life depended on coming up with a number, I'd go with 8 volts, and explain my reasoning as given.
 

wayneh

Joined Sep 9, 2010
17,496
Some people find it interesting to explore what those means might be.
The interesting or paradoxical part of it is that the loss is predicted with great precision by a formula, and the result is exactly the same no matter what the mechanism is. It's hard to imagine that aggressively eliminating losses by, for instance, using short, superconducting conductors, is all predicted to have exactly zero impact.
 

studiot

Joined Nov 9, 2007
4,998
Of course. I can't help but agree--if the ordinary resistance of the wires and material of the capacitors are made zero, as with superconductors, some other means will operate. It seems to me that it goes without saying. Some people find it interesting to explore what those means might be. A google search finds many papers on the topic, and there doesn't seem to be a consensus yet. When the connecting wires are of finite extent (with finite inductance and non-zero radiation resistance), I find the radiation hypothesis appealing.
I'm adding to, not contradicting, anything you are saying here.

As soon as you have connecting wires and ground or the physical body of a real capacitor etc etc you have transmission lines, further capacitors, inductors etc etc.

So these may well be available to launch EM radiation.

Again in a parallel with structural engineering there is a theorem known as 'the shakedown theorem' which basically says that a structure acts in the strongest mode available to it.
ie the stress paths are attracted to the strongest elements.

A similar mechanism operates in electrical engineering with the disposition of currents (real or complex) between available paths, although I don't know of a name for this.
 

nsaspook

Joined Aug 27, 2009
13,081
A similar mechanism operates in electrical engineering with the disposition of currents (real or complex) between available paths, although I don't know of a name for this.
I think it's called the second law of thermodynamics. :)

The principle of minimum dissipated power/energy might be what you mean.
 
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