# Re: Coupling Questions & coil measurements

```Ross wrote:

> In my reading, I have found several references that discuss "Magic K"
> values.  Brent's  book explanation says that Magic Ks relate to even
> integers (multiples) of "RF cycles". That makes no sense to me because
> "RF cycles" are different for each coil and it's unique Fsec.

The transfer of energy from the primary capacitance to the secondary
top capacitance occurs in a series of oscillations, or "RF cycles",
where the amplitude of the primary cycles decays cosinusoidally and
the amplitude of the peaks at the secondary increases sinusoidally.
Eventually, after some cycles, all the energy is at the secondary
capacitance. If the primary gap doesn't "quench" at this instant
("first notch"), the process continues, and after the same number
of cycles the energy is again concentrated in the primary capacitance.
Note that while these transfers occur, somee energy is dissipated
in resistive losses, so less and less energy is transferred at each
full cycle.

> The paper "Tesla Transformers" by Von Werner Heise also describes
> "values of K to strive for".  One of the magic K values is 1.33.  I have
> helped measure K on 3 well performing coils in the last week and all had
> a K value of almost exactly 1.33!?!  The coils varied in size from 3 ft
> long secondaries to secondaries as tall as me.  I may be
> misunderstanding Mr Heise, but it seems that his argument is based on
> first notch quenching.

Something wrong. k is always smaller than 1.

> Can someone explain why some of these K values are supposedly better.
> Has anyone seen evidence of superior performance after moving from a sub
> optimal K to a "magic" K?  In other words, is it an artifact of the math
> or does it really matter?

The "magic" Ks are the ones that cause some the peaks of the secondary
"RF cycles" coincide with the peaks of the sinusoidal envelope of the
"beating" waveform that is the secondary voltage. In this condition,
all the energy in the system is concentrated in the secondary
capacitance.
This is important for high Ks, as 0.6 (the largest), and progressively
less important for lower values, as there are more "RF cycle" peaks
close to the peaks of the beating secondary waveform.

> On a similar topic, I'm not sure that I understand the "double hump"
> response.  Can someone please describe how the resonant freq of the
> primary/secondary system is different from the resonant freq of either
> circuit measured independently and how this relates to coupling?

so they become tuned to different frequencies. The optimum values of
the coupling coefficient cause the two new resonance frequencies to
be at ratios of successive integers, as 1:2 (K=0.6), 2:3 (K=0.385),
3:4 (K=0.28), etc.
Actually, any relation a:b where b-a is odd is allowed, resulting
in -other sets- of "magical ks", not mentioned in the usual literature.
Considering this, the general formula for the ideal coupling coefficient
is:
k=(b^2-a^2)/(b^2+a^2)
Where b-a is odd and both are integers. The energy is transferred
in b semicycles of the secondary voltage.

> Once I get more of this figured out, I hope to devote a web page to
> proper Tesla Coil measurement techniques and interpretation.  I intend
> to discuss it in layman's terms and then spend a little time describing
> the differential equations and showing how a few things are derived (for
> us engineer/math types).  If anyone already has a page like this, please
> let me know.  Also, any references to this type of information is
> appreciated.

I have set up a page with some expressions for the magnifier case,
considering triple resonance instead of double resonance. But the
same formulation works for double resonance, if the third mode is
considered at infinite frequency (This is how I obtained the
expressions above):
http://www.coe.ufrj.br/~acmq/tesla/magnifier.html