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Re: Dr. Antonio's Papers - Questions regarding mode of operation in Tesla Coils



Original poster: Rob Maas <robm-at-nikhef.nl> 

 >It happens also that complete energy transfer is only possible when
 >b and a are integers with odd difference, as 1:2, 2:3, 1:4, etc.
 >This causes the existence of a set of values of the coupling coefficient
 >that result in maximum energy transfer:
 >Mode 1:2, energy transfer in  1.0 cycles: k= 0.6000000000
 >Mode 2:3, energy transfer in  1.5 cycles: k= 0.3846153846
 >Mode 3:4, energy transfer in  2.0 cycles: k= 0.2800000000
 >Mode 4:5, energy transfer in  2.5 cycles: k= 0.2195121951
 >Mode 5:6, energy transfer in  3.0 cycles: k= 0.1803278689
 >Mode 6:7, energy transfer in  3.5 cycles: k= 0.1529411765
 >Mode 7:8, energy transfer in  4.0 cycles: k= 0.1327433628
 >Mode 8:9, energy transfer in  4.5 cycles: k= 0.1172413793
 >Mode 9:10, energy transfer in  5.0 cycles: k= 0.1049723757
 >Mode 10:11, energy transfer in  5.5 cycles: k= 0.0950226244
 >Mode 11:12, energy transfer in  6.0 cycles: k= 0.0867924528
 >Mode 12:13, energy transfer in  6.5 cycles: k= 0.0798722045
 >Mode 13:14, energy transfer in  7.0 cycles: k= 0.0739726027
 >Mode 14:15, energy transfer in  7.5 cycles: k= 0.0688836105
 >Mode 15:16, energy transfer in  8.0 cycles: k= 0.0644490644
 >The use of these exact values is only important when a and b are low,
 >as 1:2 or 2:3, maybe 3:4. Above this, there is little difference.
 >Most Tesla coils designed to produce sparks operate with a:b above
 >7:8, and so exact coupling coefficients are not important, but it
 >doesn't cost to design aiming at a certain optimal mode.
 >To chose a particular a:b mode, consider that the energy transfer
 >takes b/2 full cycles of the oscillation at the primary.
 >Use the lowest possible value, to minimize the number of oscillations,
 >since with long energy transfer times most of the energy is
 >dissipated in the primary circuit before being transferred.
 >The value of a is always b-1 in practical systems. If a=b-3, there is
 >an incomplete first notch before the true first notch of the primary
 >voltage, and anyway it's difficult to force operation on these modes
 >(k12 becomes too high). With a=b-5 there are two incomplete notches,
 >and so on.
 >If a=b-2, or the ratio reduces to a ratio of odd integers, with
 >a "premature complete first notch" at b/4 cycles, or the result is a
 >mode where energy transfer is never completed.
 >It's difficult to build a coil operating in a mode below, maybe, 5:6.
 >The coupling coefficient becomes too high, forcing small distance
 >between primary and secondary, what causes all sorts of insulation
 >problems.
 >To get a better feeling about what happens, try my programs mrn4
 >and teslasim, that you can download from:
 >http://www.coe.ufrj.br/~acmq/programs
 >
 >Antonio Carlos M. de Queiroz
 >
 >
Hello Antonio,

Some aspects regarding these 'modes' is puzzling me. I first summarize
a particular case according to your method, as outlined in your ICES 2001
paper:
take a=7, and b=8, then k_12=(b^2-a^2)/(b^2+a^2)=0.1327
then the two frequencies are given by
w1 = a.w0, and w2 = b.w0, so the ratio w1/w2 = a/b = 7/8 = 0.875

The frequnecy w0 is defined as w0 = (1/a.b).SQRT((a^2+b^2)/(2.L_2.C_2))
If I now define w_res = 1/SQRT(L.C), then we have

w0 = (1/a.b).SQRT((a^2+b^2)/2).w_res = (1/56).SQRT(113/2).w_res = 0.1342.w_res

So far so good. Now I go to the paper of Kenneth D. Skeldon et al. "A high
potential Tesla coil impulse generator for lecture demonstrations and
science exhibitions", Am. J. Phys. 65 (8),744-754, 1997.

In this paper there are also two frequencies (here called w_l and w_u, l and u
refer to 'lower' and 'upper' respectively), which depend on the coupling k
according to

w_l = w_res / SQRT(1+k^2), and w_u = w_res / SQRT(1-k^2), where the coupling
constant k is being defined in the usual way via the mutual inductance M
and the two inductances L_1 and L_2 via

M = k.SQRT(L_1.L_2)

If I now try to make a connection between these two methods (i.e. yours and
Skeldon's) via the assumption

k = k_12 = 0.1327, I find

w_l = 0.9913.w_res and w_u = 1.0089.w_res

from the very definition of the two omega's:

w_l/w_u = SQRT((1-k^2)/(1+k^2) = 0.9826


Now the questions:

1) do w_l and w_u have the same meaning as 'your' w1 and w2 ?

2) is Skeldon's coupling coeff. 'k' identical to your k_12

3) what is the PHYSICAL meaning of 'your' w0

In Skeldon's approach the picture is rather straighforward:
the Tesla system comprises two circuits, both tuned to the same
frequency w_res = 1/SQRT(L1.C1) = 1/SQRT(L2.C2)

When the system becomes excited, there is a shift from the original
frequncy w_res into two frequncies w_l and w_u which are VERY CLOSE
(in the given example w_l and w_u are less than 2% apart), and the
beat between these two frequncies determines the energy transfer
between primary and secundary circuit.

According to 'your' definition the w1 and w2 are 13% apart. It seems
the two methods do describe two different systems.
What am I missing here?

I like to hear your comment on this.


thanks in advance,    Rob Maas