Re: Magnifer vs. Tesla Coil

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

Tesla list wrote:

 > Original poster: Paul Nicholson <paul-at-abelian.demon.co.uk>

 > With the heavily toploaded maggy, the topload itself takes up most of
 > 90 degrees, at all resonant modes. Thus the quarter wave mode shows
 > almost uniform current along the secondary-tertiary and the 3/4 wave
 > mode looks more like a half-wave, with the voltage peak appearing
 > around half way along the secondary-tertiary electrical length.

For an optimized design this would be at the middle of the secondary
coil (a very small maximum, ideally null), with a zero at the
transmission line, and the voltage rising along the third coil and
reaching the maximum at the topload. This is the condition reached
when the energy transfer is complete.

 > Just how that electrical length divides up between the physical
 > lengths of the two coils depends on h/d ratios, heights above ground,
 > and turn counts, that sort of thing.

It's possible to model even a two coils system as a magnifier,
by considering the resonator coil as composed of two coils with
a grounded capacitor at the center and another at the top. The
magnetic coupling adds some complications, but it's probably
possible to find an equivalent without magnetic coupling to the
top part (I didn't try).

 >  > I would say the first zero (of voltage) of the 3/4 wave mode at
 >  > the transmission line.
 >
 > No, the voltage zero of this mode is not far below the topload,
 > and the voltage peak is about half way.

This would not then correspond to the ideal lumped design.

 > Ideally, the voltage peak of
 > the 3/4 wave adds to the 1/4 wave mode giving an instantaneous zero
 > volts on the transmission line, coincident with peak topvolts.

Ok. You consider that there are two "wave modes" that add up.
There are three, actually. A conventional Tesla coil can be operated
in 3/4 wave mode too, with just 2 oscillations, although the 4th-
order lumped model doesn't show this. The magnifier adds
a third oscillation. A lumped model must be of 6th-order, at least,
to show the 3/4 wave mode.

 > In this 'perfect tuning' scenario, both 1/4 wave and 3/4 wave modes
 > peak together in the time domain.  On the transmission line they sum
 > to zero, but on the topload they sum to a peak (because the 3/4 wave
 > mode has an extra half-wave of electrical length in between, so is
 > inverted at the top wrt to the transmission line).
 >
 > Hope that makes sense!  I think this distributed description is
 > equivalent to the lumped model tuned for complete energy transfer to
 > the topload, in terms of the relative phases of the three normal modes
 > involved:
 >    a) All three peak simultaneously, and at that instant...
 >    b) sum to zero on the t-line,
 >    c) zero on Cpri,
 >    d) no current anywhere,
 >    e) and a max volts on the topload.

Correct.

 > Perhaps I should prepare some animations of the voltage/current
 > distributions along the maggy.  That would make things a lot clearer,
 > I think, for many of the list members interested in maggy operation.
 > A picture says a thousand words, or something.  It's really a lot
 > simpler than it all sounds!!

This would be interesting. Maybe something made with transmission lines
can be exactly designed too. With the theory that I have now I can
design a magnifier with the third coil split in many LC sections,
but at the end of the energy transfer all the energy will be at the
last capacitor, and not distributed along the coil with a notch at
the transmission line. I also can't model a secondary that stores
charge at other point than at its end, the "transmission line".

 > The top voltage of the maggy, in which the 3/4 wave mode is supported
 > on the pedestal of the 1/4 wave beat waveform, might turn out to
 > be beneficial to streamer formation - due to the extra heating
 > effect of the higher frequency current.  Could it be that the large
 > 1/4 wave voltage provides the main impetus for streamer extension,
 > and the 3/4 wave ripple into the streamer capacitance (at quite a low
 > impedance from the topload [*]) serves then to get the channels nicely
 > heated up?

Interesting possibility. The magnifiers that appear to work well
have really a third mode at high frequency (modes a:a+1:a+n>2).

 > Some basic questions about maggies that need to be answered:
 >
 > 1/ Is there any point to tuning so that the 3/4 wave voltage peak is
 >     on the transmission line.

The lumped model shows three oscillations, not two. They add in
complicated ways along the three coils as the time proceeds. The
primary voltage decays and the top voltage rises much as in
a conventional Tesla coil. The high-frequency oscillations of
the third mode are clearly observable only at the transmission line.

 > 2/ Should the maggy be designed to enhance relative amplitude of the
 >     3/4 wave mode, ie is this mode beneficial, or is it just something
 >     to be contained efficiently?
 > 3/ Does the relative phase of 1/4 and 3/4 wave modes actually matter,
 >     ie do they really need to come together to peak at the same time?
 >     Perhaps only the relative 3/4 mode *amplitude* is significant, if
 >     at all?

A possibility would be to design the system to maximize the amplitude
of the high-frequency mode. This is possible with modes as a:a+n:a+n+1,
n odd >1, still keeping the complete energy transfer condition.
But the networks result quite strange.
Let's see:
My 3:4:5 magnifier:
C1=    5.0800000000 nF
L1=   62.0177952756 µH
C2=   79.6444444444 pF (requires a lumped capacitor)
L2=    3.9480000000 mH
C3=    9.8000000000 pF
L3=   28.2000000000 mH
k12=     0.3504383220
Voltage gain: 22.7676820722
Energy transfer time: 6.6061 µs

Changing to 3:6:7:
C1=    5.0800000000 nF
L1=   76.0511811024 µH (still problematic)
C2=   72.3692307692 pF
L2=   11.2224489796 mH (too big)
C3=    9.8000000000 pF
L3=   28.2000000000 mH
k12=     0.5335461862 (difficult)
Voltage gain:   22.7676820722
Energy transfer time:     9.9092 µs

 > [*] The severely shortened top 1/4 wave of the 3/4 wave mode makes
 >      the topload quite a low impedance 'source' at that mode.

Consistent with the idea of the high-frequency oscillation feeding
the streamers. But why then not simply design the entire system
to oscillate at high frequencies? Considerations of size, probably.

 > [+] Fredholm integral operators transform voltage distribution
 >      'vectors' into current distributions, and vice versa.  The
 >      eigenvectors of the 'round-trip' operator are the normal modes of
 >      the resonator.  These operators are parameterised by a complex
 >      frequency, and the complex frequencies at which the eigenvectors
 >      occur give us the resonant frequencies and decay rates of each
 >      mode. A least squares fit allocates mode amplitudes and phases
 >      to match the initial 'firing' conditions of the resonator. The
 >      subsequent time domain evolution is then trivial to compute.

Seems complicated...

Antonio Carlos M. de Queiroz