DerAlte, first of all, my apologies for mis-understanding your statement earlier and blatantly opposing it- "More turns, more R, more Pdiss (R*I^2)".
I read it by mistake as "
For a given number of turns, more R means more Pdis" (and a proof is that I did not highlight "more turns" part when I quoted it last time
). But then none of my posts/equations are going against that fact. If you increase R by adding more turns to the VC (or if you increase the number of rings in the other case), then damping has to be more, no two ways on this.
Anyways, rest of the discussion is interesting, so here it goes....
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but the scene is not that bad that we have to discretize the parameters involved and make independent (and most likely incorrect) assumptions of 2nd and 3rd orders
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I don't think there was any mistake in my assumptions. I did not really "discretize" any parameter, unlike what you are thinking. All I did was to analyze only an instantaneous condition, so as to isolate time. In fact, isn't that how differential equations like the one mentioned by you are derived? The differential equation explains what a sequence of ALL such instantaneous conditions sum upto, and how everything changes with time. I, for simplicity, considered only one such time instance, rather than all of them- which are infinite in number. I am absolutely not loosing on anything by doing that, as compared to what your differential equation would have explained (as far as the purpose of our argument is concerned- amount of damping- , of course!). Also, I agree with 4 points made by you, and I haven't ever gone against them.
"Discrete" analysis is fundamentally different in the sense delta-x or delta-t are not infinitesimally small as in my "instantaneous" case, they are FINITE, measurable and greater than zero. For example, in digital audio the time between two samples is discretization per se. BTW, I believe discrete system analysis is more complicated than analog domain, definitely not lesser.
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I have a problem in reconciling to a set of time-dependent entities being into a lumped constants K1, K1' and K2.
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These K's symbolize fixed parameter of the system like- length of the wire, magnetic field strength, velocity (being instantaneous) etc. - I am not brute forcing time dependent parameters into constants.
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Gut feel, but by your logic all motors and generators could have used 1 turn poles instead of multi-turn, and the alu former could have been the VC itself! Help, OEO!
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If that's what you concluded from my posts, then probably you did not read them as seriously as I would have expected you to. If that was the conclusion, then I would have doubted myself and checked my postings 100 times to find out mistakes. All I tried to prove was that a shorted VC having N turns with resistance R per turn should provide the same amount of damping as N separate turns (rings) with resistance R each.
If you want to match the same level of damping with just one turn, then you have to make it much thicker and reduce its resistance to R/N.
Regarding speakers and motors with 1 turn, yes one can make such transducers and they will work too. But the difference would be that they will need much more current and less voltage to operate at the expected levels. It will be something like converting home appliance (high voltage, low current) to operate with car electricals (low voltage, high current), and you would probably need 0-gauge wires to drive them too! In short, practically there are more turns due to similar reasons for which transformers are required in electric world and gearboxes in mechanical world.
Once upon a time, I had made such a funny, practically useless DIY. I mean a "driver", not "enclosure" as it is almost implied in audio DIY community. When I read about the basic principle of a loudspeaker's working for the first time, I felt like it wasn't so complicated and that I could easily make one such device. So, I took a small plastic casing, attached a small steel/iron cap (bottom plate of a dry cell) on one side, made an electromagnet with an iron nail and "N" turns of a fine wire, mounted it as close to the steel plate as possible but without touching it (this part was quite tough). Connected the coil to headphone jack of a walkman, expecting it to reproduce the song. And it did indeed produce sound to make me happy, albeit at a very very low level. I could clearly recognize lyrics when holding it next to my ear... ahaa... I wasn't so familiar with a term called SQ then! Much later I realized that it could have been called (probably) a form of planar electromagnetic speakers. One turn speaker should be something like this one, I guess.