From: Paul
Date: Wed, 18 Oct 2000 11:25:30 +0100
Subject: Re: [TSSP] Re: Some resonator theory notes
I've revised pn1310.ps once again to fix a few more errors. Current version is now 0.2e. Is anyone relying on the gif file copies? Mark wrote: > You may want to model this to have a look at Cdis. It's instructive to examine these results in the framework of the theory notes pn1310, as the resulting conclusions differ from Mark's. One thing to note first is that the Cdis referred to is really an equivalent parallel capacitance, since it is obtained from Fres and Ldc. The relation between this equivalent capacitance and the physical capacitance distribution depends on the V/I amplitude profiles prevailing at the time of the Fres measurement. Reordering the components of the staged inductor leaves Ldc unchanged, and since the shape of all three coils is the same, the capacitance distribution of the staged inductor also stays the same between the two experiments. What changes between experiments 1 and 2 is the equivalent series inductance. Assuming a cosine current profile and using the approximation for stored energy in pn1310 equ 8.1, we can estimate the change in the equivalent series inductance that occurs between experiments 1 and 2. Split the integral into three parts, each of pi/6 radians length. The three integrals of cos^2 are integral( 0 to pi/6 of cos^2) = 0.479 (lower 3rd) integral( pi/6 to pi/3 of cos^2) = 0.262 (middle 3rd) integral( pi/3 to pi/2 of cos^2) = 0.044 (top 3rd) We can now form the sum for total effective series inductance (ie Les as defined in equ 6.2) for each experiment L1 and L2, from the component DC inductances La, Lb, Lc, each weighted with the appropriate integral, L1 = ( La * 0.044 + Lb * 0.262 + Lc * 0.479) * 6/pi L2 = ( La * 0.479 + Lb * 0.262 + Lc * 0.044) * 6/pi and we get the effective inductances L1 = 7.26 mH L2 = 2.94 mH With the equivalent parallel cap remaining the same (unknown) value, the frequency goes up in the ratio sqrt(L1/L2) = 1.57 The actual frequency ratio was measured as 578/408 = 1.42, a prediction error of 11%, which is reasonable considering we ignored a few things. Thus the frequency change comes about due to the redistribution of non-uniform inductance and not a change in self capacitance. The capacitance Cdis is of little use in itself as an equivalent capacitance, since not all of the Ldc inductance is available for it to resonate with. It does not correctly represent the stored energy for a given topvolts, so cannot be used reliably for topload frequency change predictions. Nor does it represent a meaningful distributed capacitance either. Its use in this instance is an attempt to carry the lumped approximation a little too far. > It appears that one can reduce Cdis substantially using a > tapered winding approach. Only in appearance. The coiler wants to minimise the stored energy for a given voltage, as given by equ 6.1, and that remains unchanged in this experiment. If there is a small change in the internal capacitance due to interturn distances changing, it is substantialy masked by the effective series inductance change. Generally the energy stored in the interturn cap is very small, say of order 1% so it's completely ignored in the theory notes pn1310. The tsim simulator has an option to include it but it makes only around 1% difference in Fres predictions. So, out of interest, what is the equivalent parallel capacitance of say, coil B, if it isn't 5.68pF. Well it would have to be the value calculated by equ 6.1, and if we assume the 'usual' linear V profile, then the equivalent cap becomes the Medhurst cap. If at the same time we assume a cosine current profile, then the approximation 8.4 can be used, and we get 1/Cmed = (2 PI F sqrt(8)/PI)^2 * Ldc so Cmed for coil B is 7pF. Note that this is quite a bit higher than the Cmed extracted from Medhurst tables, the extra in this case is presumably due to external capacitance presented to the coil in the measurement setup. What use is this equivalent (energy) capacitance? Well since its defined by refering all the capacitance to the hot end, you can use it approximately to estimate the frequency shift when a small top load capacitance is added, simply by paralleling the equivalent with the lumped topload cap. In doing so however a uniform component is added to the current profile and eventualy the resonant frequency is better described by equ 7.7 instead of 8.4. The equivalent parallel capacitance also becomes useful when calculating output voltages, either by transimpedance, or by ratio of primary:secondary capacitance. The quantity Cdis is not suitable for either of these. Referring now to the the Oct 97 non-uniform coil measurements, Terry wrote : > Mark took some very nice measurements and made addition comments > on these coils at: > http://users.better.org/tfritz/site/papers/nonlinearcoils/markphd.txt If we just look at one set of measurements on a non-uniform coil from markphd.txt: > Coil #2 upright: Fres=1844 kHz Cdis=3.71 pF CHowe=4.65 pF > Coil #2 inverted: Fres=1420 kHz Cdis=6.27 pF CHowe=7.73 pF Again, the change in frequency here should be interpreted in terms of a change in value of the integral in 8.1, not a change in capacitance. Mark wrote (3 years ago): > I conclude that the effect is real and appears to be primarily > capacitive in nature. If pn1310 is accepted then probably not. One can see the potential for confusion here. If in doubt, look to the integrals 6.1 and 6.2 and decide which contributions are altered by the change in configuration. In this case inverting the non-uniform coil changes M(). As a result V() and I() will also change to form a new solution of equ 5.5, so clearly any calculations like those above which assume the V and I profiles stay the same are only going to be rough approximations. BTW, one last thing, the reason I have confidence in 6.1, 6.2 and 5.5 as descriptions of a tesla coil, and thus my interpretation of Marks results is that: a) They are physically justifiable in a straightforward way. b) When these are applied with sufficient precision so that all the approximations above are removed, the predictions of Fres come out to within a percent or two without needing any fiddle factors. I'm hoping that pn1310 can be made to provide a firm enough basis for consistent interpretation of future measurements and, as we've just seen, a reinterpretation of existing results. Regards, -- Paul Nicholson, Manchester, UK. --
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.