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Old 20th Oct 2017, 6:27 am   #341
Argus25
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

Just so that everyone reading the thread knows what we are talking about with the equations that describe the voltage across the capacitor Vc(t), with time, after the switch closes at t=0, in a circuit with R,C and L after a fixed voltage V from a battery say is applied, I have attached an image with the equations. (hopefully I haven't stuffed these up with an error)

As noted before if R is greater than twice the square root of L/C it is over damped, but say if R is less than twice the square root of L/C it is oscillatory.

This condition has to be tested before the correct equation for the voltage across the capacitor with time Vc(t) can be selected.

Otherwise, it can be seen that the value of the angular frequency w, or the value of B inside the square root sign will be negative and the imaginary value that results is of no help determining the voltage on the capacitor.

And back to my original point, perhaps the fact that a single equation solution for the voltage across the capacitor with time, cannot handle all values or proportions R, C and L, agrees with the remarks by Kurt Godel about mathematics.
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Old 20th Oct 2017, 7:22 am   #342
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

This fascinating thread would do well to be called “The placebo effect in audio engineering”
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Old 20th Oct 2017, 7:39 am   #343
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

Many a true word...
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Old 20th Oct 2017, 9:09 am   #344
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

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... And back to my original point, perhaps the fact that a single equation solution for the voltage across the capacitor with time, cannot handle all values or proportions R, C and L, agrees with the remarks by Kurt Godel about mathematics.
No, this really is fundamentally wrong.

The equation can handle all parameter values, it's just that those of us who are unfamiliar with its outputs struggle to interpret them. They are not incorrect though. And people like David (see the post above yours) can interpret them - he finds the reduction from complex outputs to 'recognisable solutions' to be 'fairly easy'.

When you say that the voltage across the capacitor is not imaginary you're missing the point that voltage as a function of time cannot be described by a single parameter but needs two - amplitude and phase angle (if oscillatory) or amplitude and growth/decay time (if exponential). One way of handling those two parameters is to combine them into a complex number. From an applications point of view this is perhaps the real utility of complex numbers - they can handle two parameters at the same time. My memory is that when we reach the end of a complex variable calculation we quite often take the real part of the result as representing physical reality.

Godel's theorem is something quite different. It is not about whether a particular formula can or can't deliver a result. It is about whether some results, despite being true, can be proved at all. Godel said that in any mathematical system built on axioms there will be some true results which can't be proved at all (I've simplified that somewhat - if there are any proper mathematicians reading this then please go outside and be sick there).

Cheers,

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Old 20th Oct 2017, 9:16 am   #345
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

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No, this really is fundamentally wrong.
Hi,

That is good, at least I got something wrong, I only threw in the Kurt Godel remark, which is why I said perhaps, for a bit of fun

But I agree with the points you made too.
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Old 20th Oct 2017, 10:11 am   #346
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

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When you say that the voltage across the capacitor is not imaginary you're missing the point that voltage as a function of time cannot be described by a single parameter but needs two - amplitude and phase angle (if oscillatory) or amplitude and growth/decay time (if exponential). One way of handling those two parameters is to combine them into a complex number. From an applications point of view this is perhaps the real utility of complex numbers - they can handle two parameters at the same time. My memory is that when we reach the end of a complex variable calculation we quite often take the real part of the result as representing physical reality.
Oh - maybe I can add something to this.

My undergrad students (not the brightest, at a not-very-high-ranking institution - so we were well suited; I mean that sincerely) used to get freaked out when we were using (a+jb) e^(jwt) as a solution to a lumped RLC system - electrical, mechanical or acoustic (or all three - loudspeakers are a nice context to teach this kind of transfer function analysis and the s-plane stuff which Radio Wrangler uses).

They were OK about the magnitude being complex, and visualising what it meant on a simple argand diagram. They were OK about the absolute phase of the signal changing with time, as 't' progressed, depending on 'w'. But they did object when the thought occurred 'what about the voltage (say) at t=.125ms; if we measure it, it has a (real) value, but this suggests it is a complex number'.

Telling them to 'take the real part in the (rare) case when you need to know the instantaneous value' was a bit of a nightmare, as I had spent a year telling them 'I know the imaginary part is imaginary, but if you just throw it away you'll get it all wrong; it's half the answer!'.

To this end it was worth spending a bit of time on the idea that "really, the solution to our equation ought to be written (a+jb) e^(jwt) + (a + jb) e^(-jwt). The negative frequency half gets left out to save ink and page space. But when you work it all through at time=t - you get a real value. And - it chimes the negative frequency spectrum which accompanies the positive one you want to see, every time you do a Fourier transform."

Well, I used to convince myself each time we worked through it, and a few of them too perhaps.
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Old 20th Oct 2017, 10:44 am   #347
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When you say that the voltage across the capacitor is not imaginary you're missing the point that voltage as a function of time cannot be described by a single parameter but needs two - amplitude and phase angle (if oscillatory) or amplitude and growth/decay time (if exponential).
Yes, but in the oscillatory case the phase, if you need it, is given in the solution.

But the question:

"what is the voltage at any specific time after the switch closes ? "

...only has a single answer (not two parameter) for either the over damped or oscillatory case. Because the time and therefore phase is specified by the inquiry.

There is no imaginary plane requirement here, nor can the imaginary component (when the proportions of L,C & R do not match the equation form) have any contribution to the desired data.
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Old 20th Oct 2017, 11:31 am   #348
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

Yes. It only has a single answer. But, once you've specified the time, that single answer will depend on two parameters -the amplitude of the voltage and the phase shift of the voltage. If you were to change L, say, then both the amplitude and the phase shift would change and the voltage would be different because they'd both changed. You could change C instead, to achieve the same change in amplitude as when you changed L. But you'd get a different change in phase shift. So the voltage at the time you've specified would be different. You need to keep track of both amplitude and phase shift and you can do this using a single variable as long as it's a complex variable.

The proportions of L, C and R always match the equation form, it's just that we're more familiar with, say, the sine (in the oscillatory case) of a real number than we are with the sine of a complex or even imaginary number. But those sines do exist and are perfectly valid. If we try to use the oscillatory formula in the case where the circuit is actually overdamped we will find ourselves trying to take the sine and/or cosine of complex numbers. But if we just grit our teeth and plough on, using the expressions under Trigonometric Functions here http://www.milefoot.com/math/complex/functionsofi.htm - the answer will come out right, to just the same final voltage as if we'd used the overdamped formulae and put real numbers into them. If the answer was 1.237V after 60ms using one set of formulae it will still be 1.237V after 60ms using the other set. It'll just have taken a different form of the calculations to get there.

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Old 20th Oct 2017, 12:10 pm   #349
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YYou need to keep track of both amplitude and phase shift and you can do this using a single variable as long as it's a complex variable.
I'm a bit lost here regarding this suggestion as it applies to the problem to be solved with the L,C,R resonant circuit with a step voltage applied.

The only requirement for the output of the equation is a magnitude for the voltage across the capacitor, at some specific time t after the switch closes.

I cannot see how it is of any advantage for this particular question to "keep track of amplitude and phase shift" as its looked after by the equations I attached, provided the correct form of the equation is selected to suit the damping.

If you have a single formula (to replace the two I attached in the image on my post above) that can give the voltage across the capacitor, at any time t after the switch closes (and has some advantage by keeping track of amplitude and phase shift) and on account of that, will give a solution providing the magnitude of the voltage at a specific time t, without caring about the relative proportions of L, C and R (and the damping), can you please attach this formula, so I can figure out how that works.

Thanks,
Hugo.
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Old 20th Oct 2017, 1:36 pm   #350
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

Considering a voltage as having a real plus imaginary term is a cartesian view, and just as valid is the equivalent polar view of resolving the same thing as a magnitude/angle pair.

We have three formats and we can flip between them to use whichever format makes any operation or consideration easiest. This can really confuse beginners but they have to get comfortable with hopping between cartesian, polar and complex-exponential forms.

The polar one is good for making one thing clear, voltage doesn't exist at a single node, you have to have a reference node. Voltage is a difference between two places. Ground is so often the default, people forget this. Phase also needs a reference, obviously. Cast signals into complex form as cartesian or complex-exponential and the phase thing doesn't go away, it becomes less obvious, but a reference is still needed.

Complex frequency is interesting to teach. A lot of fresh graduates get through university without a grip on it. I take the vandalism approach. Her's an equation for the transfer function of a simple network. I'm going to substitute in frequency as a complex number, light the blue touchpaper and stand well back, now I plot the result over a plane of real and imaginary values of frequency. Look what happens! A passive lowpass R-C section gives infinite gain! but it can only do so at an imaginary frequency. If not a proof, this is a solid indication that you can't make imaginary frequencies by normal means.

A next step is into I/Q resolution and processing of signals leading to the phasing method of SSB creation/demod. And then image cancelling mixers and recievers and the things in their pockets with NZIF receivers in them. It all becomes so much more real (pun alert) when they realise (another pun) they've already bought one!

Done right, you can turn it into fun.

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Old 20th Oct 2017, 1:41 pm   #351
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

Looking at the arguments about the solution to the damped oscillatory case, I do tend to agree with Argus.

The differential equation is of just one form. But the solution to it, could be of one form or 3, depending on whether your calculator can handle complex numbers or not.

But think of reality. In an underdamped case, you get an oscillatory response. As damping increases, the frequency drops (by 'frequency' I mean reciprocal of time between zero crossings). As the damping approaches a critical value, the frequency, defined in this way, drops smoothly to zero... and stays there. It doesn't head to zero asymptotically; it's a true zero. More than critical damping, and you never get a zero crossing so frequency stays at zero.

There's also the anomaly for the critical case. The formula for voltage with respect to time is, V = U exp(bt) + Vt exp (bt), which is distinct in form from the formula for the underdamped and overdamped cases, whether you allow complex numbers or not!
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Old 20th Oct 2017, 2:03 pm   #352
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I was so paranoid that my trusty old HP15C would give up the ghost that when the limited edition reissue came out with an ARM processor I squeezed my wallet and bought one.
The very one I have, super bit of kit, and being RPN it only gets borrowed once! The HP35s is also very good and Swiss Micros https://www.swissmicros.com/ make some very dinky copies of older HPs, I keep one of their weeny 15's in my jacket pocket.

You never know when you need to do some calculation.

Quote:
it is that the voltage with time, across the capacitor, is not imaginary, it is real
indeed it is but on the imaginary (in my opinion a silly name) axis for mathematical use. Take a capacitor across the mains, it takes current (real, measurable current) and does not dissipate power. So you have to use complex (yet another name that could be better) arithmetic to work it out, it is only 2D maths after all using ohms law.
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Old 20th Oct 2017, 11:10 pm   #353
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Looking at the arguments about the solution to the damped oscillatory case, I do tend to agree with Argus.

There's also the anomaly for the critical case. The formula for voltage with respect to time is, V = U exp(bt) + Vt exp (bt), which is distinct in form from the formula for the underdamped and overdamped cases, whether you allow complex numbers or not!
Thank you!

As I tried to explain, the argument about the utility of imaginary and complex numbers for AC theory calculations and solutions is not in dispute here. They only came up because it was suggested that you could find the square root of a negative number, or at least express it as a complex number, so the thread to some extent (already off the audiophoolery topic) got a bit sidetracked on the note that I might not understand complex numbers.

I did not include the case for the critical damping on my attachment, but as you say it is unique too.

As you point out also, as the boundary from a periodic function is crossed to a critically damped one and then over damping and the equation solution for the periodic or oscillatory case for frequency goes imaginary, the complex numbers generated keep increasing, but they have no meaning at all because the angular frequency, in reality, stays at zero.

It makes me wonder if it didn't we could have had an imaginary RF spectrum bandwidth and had heaps of radio stations on that. But of course this idea is nonsensical due to the fact (and students sometimes forget this) the imaginary component of a current can do no work, only the real part can do work. So of course we would not get any power out of our imaginary frequency radio transmitter, what a shame.

Why I brought this equation set up in the first place is that it always struck me as interesting that the reality was you could have just a simple R,C,L network in front of you on the bench and not be able to model it successfully with a single equation and succeed with your calculator. It sort of suggested a limit in the way mathematics could model not only circuits, but mechanical reality as well.

In a better system of mathematical analysis it might be possible to model the RLC network with a single equation that would work on a calculator, but I certainly don't know what that equation would look like, but I'm still searching for it.

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Old 21st Oct 2017, 12:11 am   #354
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There's also the anomaly for the critical case. The formula for voltage with respect to time is, V = U exp(bt) + Vt exp (bt), which is distinct in form from the formula for the underdamped and overdamped cases, whether you allow complex numbers or not!
Deep apologies everyone, I used the symbol 'V' twice... I should have used different letters, P and Q, or whatever, on the right-hand-side!

But... How did we get from audiophoolery to discussing damped oscillations? I looked back, Craig Sawyers was pointing out that the degree of knowledge about any aspect of science (which could include audio) could itself be unknowable. He then said this is certainly true for mathematics, citing Gödel's incompleteness theorem. Argus followed up by an illustration of damped oscillations, which although the basic equation can have its coefficients change smoothly and continuously, yields a solution which falls into three distinct sets.

Gödel proved we can't know every mathematical theorem. So mathematical knowledge must always be incomplete.

Is it possible that physical knowledge also must always be incomplete? Or can we prove that it is possible to decide that cable directionality is bunkum? I think the latter.

Suppose that I have a length of speaker cable, and I give it to an audiophool asking him to assess its optimum direction. Assuming it has one, he carefully marks the cable to show signal flow.

I thank him kindly, take it home, separate the two wires, and reverse ONE of them. By symmetry, whichever way I connect the 'new' cable between amp and speaker, I always have one conductor in the 'good' direction and one in the 'bad'. So it doesn't matter which way round I connect it. I've created a directionless cable!
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Old 21st Oct 2017, 12:44 am   #355
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Suppose that I have a length of speaker cable, and I give it to an audiophool asking him to assess its optimum direction. Assuming it has one, he carefully marks the cable to show signal flow.

I thank him kindly, take it home, separate the two wires, and reverse ONE of them. By symmetry, whichever way I connect the 'new' cable between amp and speaker, I always have one conductor in the 'good' direction and one in the 'bad'. So it doesn't matter which way round I connect it. I've created a directionless cable!
Ha Ha, I like this.

To finish this experiment you would hand the cable back to the audiophool take off its original markings and get him to repeat his experiment.

In theory he should re-mark the cable again, being confident that it was better one way.

The only fly in the ointment would be if he said; "something is really weird about this cable, it now seems to sound exactly the same each way around"......thank God we know that this would be a very very unlikely outcome.
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Old 21st Oct 2017, 12:48 am   #356
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Gödel proved we can't know every mathematical theorem.
Sorry, but he proved no such thing. He said that in any mathematical system based on axioms there will be some true results which can't be proved at all.

It's not theorems, it's results.

And he didn't prove that we can't know them. Indeed his whole point was that we would know them (or at least very strongly suspect them). But we wouldn't be able to prove them.

And the suggestion that the truth concerning mathematical systems tells us anything about the truth concerning physical reality, in particualr the completeness properties of the two of them, looks like a pretty big leap. Is there any hard academic support for it ? I believe there's quite a bit of debate about whether (and if so, why) all of physical reality can even be described mathematically. The two things (reality and mathematics) really are apples and pears.

Cheers,

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Old 21st Oct 2017, 1:40 am   #357
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Sorry, but he proved no such thing. He said that in any mathematical system based on axioms there will be some true results which can't be proved... And he didn't prove that we can't know them. Indeed his whole point was that we would know them (or at least very strongly suspect them). But we wouldn't be able to prove them.
A much appreciated correction and clarification, GJ!

Any observations on cable directionality? Assume every 2-core cable has an optimum direction, I then reverse one conductor and create a cable which is truly symmetric, thus violating the original assumption. So not every cable has an optimum direction. If we can find an audiophool who claims that every cable has an optimum direction, we are in a position to shoot him down in flames.
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Old 21st Oct 2017, 1:52 am   #358
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I'm grateful that Argus25 [post 353*] tries to explain what is going on, I really am Although it's often hard to understand the hard, or in fact, any detail of an advanced theoretical concept it might not be essential if someone [anyone] can translate some of it. In a way, Kalee [post 354*] states the obvious, which has brought me back in to this thread. I've met Marc In Manc and he doesn't hold back but seems, like every else in this thread, to really know his stuff! Basically, even science can boil down to belief [I'm not at all religious in the accepted sense but I have faith-now there's a concept]. Edison thought DC was the way to go until Tesla got in the way. As with Shakespeare's plays, you don't need advanced knowledge of the language but the story helps. That's my excuse anyway!

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Old 21st Oct 2017, 7:26 am   #359
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

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Sorry, but he proved no such thing. He said that in any mathematical system based on axioms there will be some true results which can't be proved at all.
David Hilbert in 1928, posed a question called der Entscheidungsproblem (the stopping problem), of whether an algorithm could be constructed that takes a general logical statement, and always answers whether or not it is true - ie returns YES or NO.

Godel answered that when he proved that an axiomatic mathematical system is either complete but has inconsistencies, or incomplete and consistent. In the latter case you can always construct a new axiom - and prove it. But since it is still incomplete you can still construct a new axiom - and prove it.

It has nothing to do with provability.

Alan Turing also proved it was impossible. The basis of his proof was to say: suppose we have an algorithm and a statement. How do we know if the algorithm will ever stop or just run on for ever? Well OK - let's write another algorithm which looks at the first one and tells you if it will ever stop. But how do we know that this second algorithm will ever stop? OK write a third algorithm....

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Old 21st Oct 2017, 7:39 am   #360
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It has nothing to do with provability.
It seems the word the mathematicians use is 'undecidable'. I know you're not going to spend the weekend correcting this bit of the Wikipedia article https://en.wikipedia.org/wiki/G%C3%B...ble_statements but are you saying someone needs to ?

I'm not going to be working on this either I'm afraid. Tomorrow I'm at Crackle's Basildon bash and today I'm visiting the Windsor Hi-Fi Show. I'll keep an out for wacky cables !

Cheers,

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