From: "Terrell W. Fritz"
Date: Mon, 16 Oct 2000 19:23:38 -0600
Subject: Re: [TSSP] Surprising secondary voltage profiles
Hi Paul, At 07:04 AM 10/16/2000 +0100, you wrote: >Terry, > >Yes, the measured at predicted curves are stunningly different, >something we really must resolve. > >Terry wrote: >> I used a tiny antenna placed very near the coil and measured >> the RMS voltage... > >How was the RMS extracted from the RF probe signal? Computerized digital scope... I pushed the little button :-)) > >> I retuned the coil for >> maximum signal to sort of compensate for detuning by the antenna. > >The closeness of the probe, and the retuning, might upset the >measurements, if the probe presents a capacitance >approaching that of the topload + the self cap of the part of the coil >above the probe, you may have been making the probe position the >effective electrical 'end' of the coil, ie the part of the coil >above the probe is starved of current and there would be little or >no V rise above the probe. > A very good point. I was desperate for the data at the time and it's the best I could come up with. >eg > > V | > | . . . . . > | . > | . > | . > | . > |. > ------------------------- X > ^ probe > >Did you have to retune more than a few percent? Argh! I have the raw data but it does not include the frequencies... I don't remember tuning much but a few percent is pushing it. http://63.225.104.49/TeslaCoils/Misc/voltagedist.xls > >You could check for this with a simultaneous V probe on the toroid. >The toroid volts should remain constant-ish as the moving probe works >its way up the coil. I now have one of those 40kV Tektronix freon filled probes. High voltage but low contact capacitance. Although, the probe itself is a big one and probably like setting the secondary next to a cow! > >Referring to Malcolm's Ruler, I'm not at all convinced that >the mechanical analogy can be applied. The differential equations >which apply to the beam displacement are very different to those >which apply to the electrical potential. It's a long time >since I did mechanics theory, but I managed to look up some >formulae describing the resonating beam. I have probed around coils a bit and the concave curve seems closer to me than the convex. But that is just a "feeling". The Malcolm's ruler thing seemed like a convenient fit. > >Apparently, the displacement V(x,t) of position x, time t, is >governed by > > EI d^4 V(x,t) / dx^4 = - lambda d^2 V(x,t)/dt^2 > >where the d are partial differentials. E is the uniform >elastic modulus, I is the uniform inertia, and lamda is >the uniform mass density per unit length. > >The x and t dependency of displacement V(x,t) can be separated, >to get, for the x dependency, ie the displacement profile, > > d^4 V(x)/dx^4 = k^4 V(x) > >where k is a constant for a particular mode of vibration. > >The solution to this 4th order equation, with the boundary conditions >as for Malcolm's Ruler, is an evil looking thing, > >V(x) = C * ( cos(kx) - cosh(kx) + > (sin(kx) - sinh(kx)((-cos(kL) - cos(kL)))/(sin(kL)-sinh(kL))) > >and the plotted solution looks exactly like the curve of the ruler. > >An approximation to V(x) is that which occurs when the beam is >bent by application of a force at the top: > > V(x) = 0.5 Vmax ( 3(x/L)^2 - (x/L)^3) for 0 <= x <= length L > >None of this is very reminiscent of the much simpler differential >equations governing a transmission line in the absence of >longitudinal coupling. The sine profiles of the uniform transmission >line are distored in the x direction - stretched and squeezed when >the line made is non-uniform, but there is no point of inflection >allowed in the absence of longitudinal coupling. (I think I can >prove that.) Interesting! > >The only hope of rescuing Malcolm's Ruler is, as you say, the >possibility that the longitudinal coupling, ie internal C and >mutual inductance, comes to the rescue and enables the V profile >to go concave. > >Let's look at what it would take to do this. In my last email >I said that > > dV(x) = w L I(x) dx > >and so for V to be concave, I(x) needs to be increasing. > >The mutual inductance modifies this so that the I(x) is replaced >by a weighted average of the I in the region around x - the >weighting being the mutual inductance profile. If that region, >ie the 'span' of the mutual coupling were to extend from the >top half to the bottom half of the coil, the large currents lower >down might induce a significant extra dV/dx in regions higher up. >The problem is however, if you now slide that coupling region a >little way up the coil in order to examine the dV/dx at a point >higher up, all the contributing currents going into the weighting >region are reduced a little from what they were lower down. >Thus, even if a substantial portion of dV(x) is due to mutual >inductance from higher current regions lower down, the dV(x) >will still reduce with height, ie the V slope will be convex. > >Thus, on the face of it, a uniform mutual inductance profile >cannot make V concave. On the contrary, mutual inductance >I think will act to flatten the V profile towards linear, by >making dV/dx depend on a local weighted average of the currents >around x, rather than just the spot current at x - a kind of >smoothing process. > >So that leaves the hope that the displacement currents of the >internal capacitance can come to the rescue. The problem here is >that the direction of the displacements currents is the wrong way >around to do this, it acts to remove current just when you >want it to insert current. Consider any two points A and B on >the coil, A higher than B, and therefore at a higher voltage. >Consider a quarter cycle during which the voltage is rising >from zero towards a peak. Current is leaving both A and B to >charge up the external capacitance. But since A is at a >higher voltage than B, current will also leave A towards B as >the internal capacitance between A and B is charged up, thus >taking more current from the high point A than would be the >case without internal capacitance. >The two points could have been chosen anywhere, so the general >flow of internal capacitance displacement current is from high >to low as the voltage is rising. >Thus internal capacitance acts to divert current away from the >coil higher up, and insert it lower down, thus reducing series >current with height, and tending to make the V profile more >convex. > >I believe this 'descending' internal C displacement current is >the cause of the current peak which occurs just above the bottom >of the coil, which *is* associated with a small concavity in the V >profile near the bottom, ie the effect you wanted is there, but >at the wrong end of the coil. > >My understanding of this is summarised in pn1310 page 4, eq 5.5 >which gives differential equations applying to the V and I profiles >of a tesla coil. > >The results from tsim are based on a model which takes into account >the mutual inductance and internal C. According to this model, they >have an effect on dispersion, but so far all the coils looked at >the V profiles have been an almost linear rise over say the lower >60% of the coil, with a gentle levelling off above that. The current >is always a very distorted cosine, with no tendency to rise in the top >half of the coil - it falls quite rapidly in this region. > >The possibility remains that I've dropped a minus sign in the program >(or in my head!) which might reverse the effects described above, so >I'll try to prove this one way or another without reference to tsim. > >> One thing that does trouble me is that E-Tesla5 uses the curves I >> measured and gives very good accuracy for loaded and unloaded coils. >> I tried your graphs back then as secondary voltage profiles and the >> program showed significant errors. Not just simple scaling but it >> appeared the voltage profiles were not working... > >Concerning E-Tesla5, I have a gut feeling that you should be >weighting with V^2 rather than V but I'm unable to explain myself >at the moment, but its something to do with weighting to get >an equivalent stored energy, rather than a 'mean' capacitance. >I'll ponder this further. The V^2 idea is interesting because it would suggest I made two mistakes. The first one, and another that corrected the first... That is possible but the profile really should be a plain "V" boundary condition. Thanks for the wonderful insights. I am still thinking of a solid way to measure the profile... Don't get hung up on the convex/concave thing. If you think it is convex, just "go for it" and continue onward! Cheers, Terry > >Cheers, > >-- >Paul Nicholson, >Manchester, UK. >--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.