Mode Splitting Discussion
This thread morphed into the "SSTC Modes and Soft Switching" Discussion, link at the bottom of page
Date : Thu, 19 Aug 2004 14:05:07 -0600. Subject : Mode Splitting
Original poster: "Bob (R. A.) Jones"
<a1accounting@bellsouth.net>
Hi all,
I wrote this description of mode (some times called frequency) splitting while
exploring how an SSTC can be driven with soft switching while maintaining high
voltage gain (no break out). I thought it may be of interest to some readers
A naked (zero k to primary) Tesla coil secondary has oscillation modes usually
described as 1/4 wave, 3/4 wave etc. all truncated if its top loaded.
As the k to the primary is increased from zero the 1/4 wave mode splits in two
modes. One mode is higher and one mode is lower in frequency than the original
mode. They also have opposite polarity at the top of the secondary for a given
polarity at the primary. That probably needs expansion. The reflected impedance
of the primary is either inductive or capacitive hence the wave of one mode is
shortened and the other is lengthened. The shortened one will have the same
polarity at both end while the lengthened one will have the opposite polarity at
its ends with one null near the primary end. Of cause the real effect is
distributed along the coil with the distributed inductive coupling from the
primary. Incidentally I don't think the higher order modes of the secondary
split because at those higher frequencies the reflected impedance of primary is
always inductive so they are just shifted. In the case of a top load coil all
modes are truncated at the top.
Initially the polarity of all modes including the two fundamental modes of a
standard coil and the three modes of a magi are the same at the primary cap and
sum every where else to zero. A number of half cycles later the two
fundamental (or three in the magi) modes now sum to a maximum at the top load
and sum to zero at the primary cap. Initially all the modes sum to zero every
where else but its only the first two (or three modes) that have the correct
phase to peak at the top load and sum to zero at the primary C. Fortunately the
amplitudes of the higher modes are small which is why they can neglected in most
models and analysis. Bob.
Date : Thu, 19 Aug 2004 18:28:07 -0600. Subject : Re: Mode Splitting
Original poster: "Antonio Carlos M. de Queiroz"
<acmdq@uol.com.br>
Tesla list wrote:
>
> Original poster: "Bob (R.A.) Jones"
<a1accounting@bellsouth.net>
Some notes:
> I wrote this description of mode (some times called frequency) splitting
while exploring how an SSTC can be driven >with soft switching while maintaining
high voltage gain (no break out). I thought it may be of interest to some
readers
>
> A naked(zero k to primary) Tesla coil secondary has oscillation modes
> usually described as 1/4 wave, 3/4 wave etc. all truncated if its top loaded.
I think about these modes as voltage profiles along the secondary coil when the
output voltage is maximum. They are not (directly) related to the oscillation at
multiple frequencies that causes the energy transfer from the primary to the
secondary circuits.
> As the k to the primary is increased from zero the 1/4 wave mode splits in two
modes. One mode is higher and one >mode is lower in frequency than the original
mode.
Better to say: With k=0 the primary and the secondary systems resonate
(oscillate) at the same frequency. When the coils become coupled, this single
frequency splits in two, one above and the other below the original frequency,
and both systems oscillate at both frequencies simultaneously.
> They also have opposite polarity at the top of the secondary for a given
polarity at the primary.
They add constructively at the primary initially, while adding destructively at
the secondary. When the energy transfer is complete, they add constructively at
the secondary while adding destructively at the primary. The actual polarities
depend on the directions of the windings of the coils, and how you measure the
primary voltage.
> That probably needs expansion. The
> reflected impedance of the primary is either inductive or capacitive hence
> the wave of one mode is shortened and the other is lengthened. The
> shortened one will have the same polarity at both end while the lengthened
> one will have the opposite polarity at its ends with one null near the
> primary end. Of cause the real effect is distributed along the coil with
> the distributed inductive coupling from the primary. Incedently I don't
> think the higher order modes of the secondary split because at those higher
> frequencies the reflected impedance of primary is always inductive so they
> are just shifted. In the case of a top load coil all modes are truncated at
> the top.
I don't see much use in considering steady state impedances in this case, where
there are two frequencies involved and the waveforms are all transient.
> Initially the polarity of all modes including the two fundamental modes of
> a standard coil and the three modes of a magi are the same at the primary
> cap and sum every where else to zero. A number of half cycles later the two
> fundamental (or three in the magi) modes now sum to a maximum at the top
> load and sum to zero at the primary cap.
Ok. Antonio Carlos M. de Queiroz.
Subject: Re: Mode Splitting. Date: Fri, 20 Aug 2004 13:01:48 -0600
Original poster: "Bob (R.A.) Jones" <a1accounting-at-bellsouth-dot-net>
----- Original Message -----
From: "Tesla list" <tesla-at-pupman-dot-com>
To: <tesla-at-pupman-dot-com>
Sent: Thursday, August 19, 2004 5:28 PM
Subject: Re: Mode Splitting
> Original poster: "Antonio Carlos M. de Queiroz" <acmdq-at-uol-dot-com.br>
>
> Better to say: With k=0 the primary and the secondary systems resonate
> (oscillate) at the same frequency. When the coils become coupled, this
> single frequency splits in two, one above and the other below the
original
> frequency, and both systems oscillate at both frequencies
simultaneously.
Perhaps very clumsily I was trying state that the two modes are independent.
Yes its true that in the usual impulsive system they are excited simultaneously
but in a master oscillator SSTC or using a signal generator either mode can be
driven independently to the extent of their Q and separation. It has been
incorrectly stated that mode splitting does not occur in an SSTC as if some how
its a property of the drive signal as opposed to a property of the system.
> > That probably needs expansion. The
> > reflected impedance of the primary is either inductive or
capacitive hence
> > the wave of one mode is shortened and the other is lengthened.
The
> > shortened one will have the same polarity at both end while the
lengthened
> > one will have the opposite polarity at its ends with one null
near the
> > primary end. Of cause the real effect is distributed along the
coil with
> > the distributed inductive coupling from the primary.
Incidentally I don't
> > think the higher order modes of the secondary split because at
those higher
> > frequencies the reflected impedance of primary is always
inductive so they
> > are just shifted. In the case of a top load coil all modes are
truncated at
> > the top.
>
> I don't see much use in considering steady state impedances in this
case,
> where there are two frequencies involved and the waveforms are all
transient.
I think we would agree that the system is linear (assuming a closed spark
gap) just a collection of Ls Cs and Rs so it can be completely characterized by
it complex impedances which I assume you refer to as steady state impedances.
Hence in the usual complex circuit analysis whether we get a transient or not is
a function of the excitation signal.
>From the perspective of the secondary we can replace coupled primary
circuit
with a parallel tuned circuit in series with primary end of the secondary
using the relationship ((Lm.s)^2)/Zp were Lm is the mutual
inductance and Zp is the series impedance of the primary. Similar to the way
you eliminate the coupling in several of your papers.
The referred impedance is equivalent to a parallel tuned circuit
which is high impedance (assuming both are at the same frequency) at the
frequency of the 1/4 wave mode of the secondary.
But that mode requires a low impedance so even though the
impedance is real (something that's bugged me for a long time) the mode can not
be supported.
Either side of the 1/4 wave frequency the impedance is low and either
inductive or capacitive hence it can support a 1/4 plus a bit or a
truncated 1/4 wave mode.
Using this description its also easy to visualize what happens if you vary
the primary and secondary frequencies. As the separation frequency
increase
one mode move down the resonance curve of the referred primary and one moves
up to the peak. At the peak the impedance is too high to support that mode
and it disappears leaving only the other mode and the uncoupled resonance
of the primary that has very little feed thru to the secondary.
Well easy for me to visualize.
Apart from finding the roots of the transfer function. I have not read any
explanation of the mode splitting that even come close to holding water.
One surprise, for me anyway, was that one of the split modes has null.
But
in fact that's the only way the two orthogonal modes can sum to a maximum at
the primary while summing to zero at the top end and be almost zero along the
secondary.
Then (if the frequencies and phase are right) after a time interval sum to zero
at the
primary and a max at the top of the coil. There was minor problem
because
two orthogonal modes or for that matter three or any finite number can not
initially sum to zero over the length of secondary. To do that you need you
need a contribution from all the higher order modes.
An other intersting point is that the closer the two spilt modes are in
frequency the better they cancel along the secondary so less contribution is
required from the higher order modes.
Putting this an other way. In an impulsive system the tighter the coupling the
more energy is wasted in the higher order modes.
This ignores the distributed effects of the coupling so it may only be partially
true.
In any case I do not mean to suggest that the wasted energy is necessarily
significant relative to other losses only that its inevitable. Bob.
Subject: Re: Mode Splitting. Date: Sat, 21 Aug 2004 07:56:14 -0600
Original poster: "Antonio Carlos M. de Queiroz" <acmdq-at-uol-dot-com.br>
Tesla list wrote:
>
> Original poster: "Bob (R.A.) Jones" <a1accounting-at-bellsouth-dot-net>
> Perhaps very clumsily I was trying state that the two modes are
independent.
> Yes its true that in the usual impulsive system they are excited
> simultaneously but in a master oscillator SSTC or using a signal
generator
> either mode can be driven
> independently to the extent of their Q and separation.
> It has been incorrectly stated that mode splitting does not occur in an
SSTC
> as if
> some how its a property of the drive signal as opposed to a property of
the
> system.
The splitting always occurs (I am assuming a system with two capacitors
and
two inductors). In an SSTC where the rise time is fast and the losses
small, the output voltage rising transient has three frequencies on it.
The two natural oscillations of the system and the driving frequency.
If the driving frequency coincides with one of the natural frequencies,
the rising transient is a sum of the two natural oscillations with an
oscillatory term that rises linearly with the time. But this combination
may not be the one that produces the fastest rise until breakout occurs.
> I think we would agree that the system is linear (assuming a
closed spark
> gap) just a collection of Ls Cs and Rs so it can be completely
characterized
> by it complex impedances which I
> assume you refer to as steady state impedances. Hence in the usual
complex
> circuit analysis whether we get a transient
> or not is a function of the excitation signal.
Ok for impedances in Laplace transform ("s"). Steady-state sinusoidal
analysis ("jw") can't be directly used in these systems.
> >From the perspective of the secondary we can replace coupled
primary
circuit
> with a parallel tuned circuit in series with primary end of the
secondary
> using the relationship ((Lm.s)^2)/Zp were Lm is the mutual
> inductance and Zp is the series impedance of the primary. Similar to the
way
> you eliminate the coupling in several of your papers.
Yes. The transformer can be eliminated to simplify the analysis, or
included back after a simplified synthesis.
> The referred impedance is equivalent to a parallel tuned circuit
> which is high impedance (assuming both are at the same frequency) at the
> frequency of the 1/4 wave mode of the secondary.
> But that mode requires a low impedance so even though the
> impedance is real (something that's bugged me for a long time) the mode
can
> not be supported.
> Either side of the 1/4 wave frequency the impedance is low and either
> inductive or capacitive hence it can support a 1/4 plus a bit or a
> truncated 1/4 wave mode.
What you call 1/4 wave mode I assume that is the natural oscillation
frequency of the secondary system alone.
> Using this description its also easy to visualize what happens if you
vary
> the primary and secondary frequencies. As the separation frequency
increase
> one mode move down the resonance curve of the referred primary and one
moves
> up to the peak. At the peak the impedance is too high to support that
mode
> and it disappears leaving only the other mode and the uncoupled
resonance
> of the primary that has very little feed thru to the secondary.
> Well easy for me to visualize.
> Apart from finding the roots of the transfer function. I have not
read any
> explanation of the mode splitting that even come close to holding water.
I don't see how one of the modes can disappear. Any combination of two
capacitors and two inductors (assuming that they form a 4th-order
system)
must resonate at two different frequencies. The frequencies can't be
equal,
because this would generate waveforms that grow linearly with the time,
without limit, violating energy conservation. This explains why mode
splitting must occur. Antonio Carlos M. de Queiroz.
Subject: Re: Mode Splitting. Date: Sat, 21 Aug 2004 07:57:34 -0600
Original poster: "Malcolm Watts"
Hi Bob,
On 20 Aug 2004, at 13:01, Tesla list wrote:
> Original poster: "Bob (R.A.) Jones"
>
> ----- Original Message -----
> From: "Tesla list" <tesla-at-pupman-dot-com>
> To: <tesla-at-pupman-dot-com>
> Sent: Thursday, August 19, 2004 5:28 PM
> Subject: Re: Mode Splitting
>
>
> > Original poster: "Antonio Carlos M. de Queiroz"
>
> >
> > Better to say: With k=0 the primary and the secondary systems
> resonate > (oscillate) at the same frequency. When the coils
become
> coupled, this > single frequency splits in two, one above and the
> other below the > original > frequency, and both systems oscillate
at
> both frequencies > simultaneously.
>
> Perhaps very clumsily I was trying state that the two modes are
> independent. Yes its true that in the usual impulsive system they are
> excited simultaneously but in a master oscillator SSTC or using a
> signal generator either mode can be driven independently to the extent
> of their Q and separation. It has been incorrectly stated that mode
> splitting does not occur in an SSTC as if some how its a property of
> the drive signal as opposed to a property of the system.
Agreed. It is easy to show mathematically that for any waveform other than
constant amplitude sinewave, other frequencies are present.
> > > That probably needs expansion. The
> > > reflected impedance of the primary is either inductive
or
> capacitive hence
> > > the wave of one mode is shortened and the other is
lengthened.
> The > > shortened one will have the same polarity at both
end while the lengthened
> > > one will have the opposite polarity at its ends with one
null
> near the > > primary end. Of cause the real effect is
distributed
> along the coil with
> > > the distributed inductive coupling from the primary.
Incedently
> I
> don't
> > > think the higher order modes of the secondary split
because at
> those
> higher
> > > frequencies the reflected impedance of primary is always
> inductive so
> they
> > > are just shifted. In the case of a top load coil all
modes are
> truncated at
> > > the top.
> >
> > I don't see much use in considering steady state impedances in
this
> > case, > where there are two frequencies involved and the
waveforms
> are all > transient.
>
> I think we would agree that the system is linear (assuming a
closed
> spark gap) just a collection of Ls Cs and Rs so it can be completely
> characterized by it complex impedances which I assume you refer to as
> steady state impedances. Hence in the usual complex circuit analysis
> whether we get a transient or not is a function of the excitation
> signal.
>
> >From the perspective of the secondary we can replace coupled
primary
> circuit
> with a parallel tuned circuit in series with primary end of the
> secondary using the relationship ((Lm.s)^2)/Zp were Lm is the mutual
> inductance and Zp is the series impedance of the primary. Similar to
> the way you eliminate the coupling in several of your papers. The
> referred impedance is equivalent to a parallel tuned circuit which is
> high impedance (assuming both are at the same frequency) at the
> frequency of the 1/4 wave mode of the secondary. But that mode
> requires a low impedance so even though the impedance is real
> (something that's bugged me for a long time) the mode can not be
> supported. Either side of the 1/4 wave frequency the impedance is low
> and either inductive or capacitive hence it can support a 1/4 plus a
> bit or a truncated 1/4 wave mode.
>
> Using this description its also easy to visualize what happens if you
> vary the primary and secondary frequencies. As the separation
> frequency increase one mode move down the resonance curve of the
> referred primary and one moves up to the peak. At the peak the
> impedance is too high to support that mode and it disappears leaving
> only the other mode and the uncoupled resonance of the primary
that
> has very little feed thru to the secondary. Well easy for me to
> visualize. Apart from finding the roots of the transfer function.
I
> have not read any explanation of the mode splitting that even come
> close to holding water.
My take on it is that the incrementing/decrementing waveforms *must* produce
sidebands because the change in amplitude is altering the slope of the
sinusoidal waveform continuously. Malcolm.
This is continued in the "SSTC Modes and Soft Switching" discussion