This part of the exercise examines the effects of different windings
for a given coil former. Three different formers are considered, each
having the same diameter but a different height (and hence h/d ratio).
     The investigation revealed some remarkable facts about the effect of
forcing each coil to resonate at a frequency such that the wire in each
coil is 1/4 wavelength long at that frequency.
     The diameter of all coils is 12". The h/d ratios considered are 2,3
and 4. For a given h/d ratio and diameter, the self-capacitance of the
coils are pre-determined prior to adding a terminal. Changing the wire
diameter for a given former size changes the number of turns, inductance
and therefore resonant frequency of each coil in addition to wirelength.
The self-capacitance does not change under these conditions.

The procedure for generating the coils is as follows :-
(1) Choose h/d ratio and Dsec, hence Hsec and self-capacitance follow
(2) Choose a wire gauge
(3) From the table Richard Q. supplied, the number of turns for the
    chosen wire gauge is defined (this table takes wire insulation
    thickness into account)
(4) Calculate the coil inductance
(5) Calculate the length of wire used
(6) Calculate the 1/4 wavelength frequency based on wirelength
(7) Calculate the total capacitance (Ctot = Csec+Cterm) required to
    make the coil resonate at this frequency
(8) Calculate the thickness of wire in terms of skin depths for this
    frequency (this determines the suitability of the chosen wire gauge
    at this frequency)
(9) Calculate Lsec/Ctot ratio (mH/pF)

NOTE : Q is considered incalculable due to factors such as radiation
       resistance (i.e. I haven't figured out how to do it yet).

TABLE 1 :   h/d ratio = 2,  Hsec = 24",  Csec = 15.2pF
Dwire(mm) | Nsec | Lsec(mH) | Lwire(in) | f(kHz) | Ctot(pF) | Skin | L/C
   0.8    |  696 |   59.5   |   26239   | 112.5  |   33.6   | 4.1  | 1.77
   1.0    |  552 |   37.4   |   20810   | 141.9  |   33.6   | 5.7  | 1.11
   1.2    |  480 |   28.3   |   18096   | 163.2  |   33.6   | 7.3  | 0.84
TABLE 2 :   h/d ratio = 3,  Hsec = 36",  Csec = 18.6pF
Dwire(mm) | Nsec | Lsec(mH) | Lwire(in) | f(kHz) | Ctot(pF) | Skin | L/C
   0.8    | 1044 |   95.0   |   39358   |  75.0  |   47.4   | 3.3  | 2.00
   1.0    |  828 |   60.0   |   31215   |  94.6  |   47.4   | 4.7  | 1.27
   1.2    |  720 |   45.2   |   27143   | 108.8  |   47.4   | 6.0  | 0.95

TABLE 3 :   h/d ratio = 4,  Hsec = 48",  Csec = 22pF
Dwire(mm) | Nsec | Lsec(mH) | Lwire(in) | f(kHz) | Ctot(pF) | Skin | L/C
   0.8    | 1392 |  130.9   |   52477   |  56.3  |  61.0    | 2.85 | 2.15
   1.0    | 1104 |   82.3   |   41620   |  71.0  |  61.0    | 4.3  | 1.35
   1.2    |  960 |   62.3   |   36191   |  81.6  |  61.0    | 5.2  | 1.02

- For a chosen h/d ratio, the capacitance required to force ANY
  coil to resonate so as to make the wire 1/4 wavelength long is
  identical!! Since self-capacitance is constant for any coil with
  the same diameter and h/d ratio, the terminal capacitance is
  always the same as well! It would appear that this constancy has
  been the basis for the many attempts to base 1/4 wave length on
  h/d ratio.
- Based on the above finding, the total capacitance required for
  a coil to be 1/4 wavelength long for a particular former may
  be found by designing ANY coil for that former and going through
  the calculation steps as shown above to find the required cap-
  acitance value.
- There is a clear trade-off between skin resistance and L/C ratio
  for the coils in each group since an increase in wire size implies
  greater current carrying area as well as reduced inductance due
  to fewer turns.
- For some sacrifice in skin performance, the first coil in TABLE3
  has a much higher inductance than the first coil in TABLE2. The
  L/C ratio is also higher but not by much (Q is proportional to
  the square root of L/C and inversely proportional to total coil
  resistance which suggests that the first coil in TABLE2 will
  have the higher Q of the two - its radiating area is less than
  3/4 that of the larger coil if the terminal size is taken into
  account as well).
- As coil size increases (from table 1 to 3) the size of the
  terminal required to make the coil resonate at the 1/4 wavelength
  frequency increases also.
- My pick for highest power operation is the second coil in TABLE3.
  Since a higher power throughput means longer sparks, the longer
  coils will allow longer sparks before they show a preference for
  striking the ground or primary strike ring. This coil will have
  a better skin performance than the first coil in TABLE3.
- The large terminals not only allow a higher secondary voltage
  before "letting go" if they have the largest possible radius-of-
  curvature for their capacitance but also a high discharge
  current due to high capacitance (greater energy storage).
- The proximity of the terminal to the top of the coil is going
  to affect these figures somewhat.
- The additional capacitance due to corona has been ignored in
  this analysis. If an attempt is made to allow for additional 
  corona/ion cloud capacitance, I suggest a reduction in total
  coil capacitance should be based on the intended operating
  power level as this affects the amount of ionization. As power
  level increases, total coil capacitance should be decreased. 
- This last point conflicts with the normal practice of increasing
  terminal size to get the best results at higher power levels.
  Also as Richard has previously said, the coil compensates for
  a lack of terminal size by forming a larger ion cloud. So at a
  higher the power level, the larger the coil itself should be.
- Is it worth designing for 1/4 wavelength like this ?  Given
  that ionization can vary wildly it is debatable as the relation
  holds only for one degree of ionization. For example, varying
  spark length implies varying total secondary capacitance. The
  nett effect of increasing capacitance is to reduce wire length
  as the resonant frequency falls.

     I did this analysis with a view to building a coil that would allow
a number of key experiments to be performed, including using terminals
with different capacitances but the same radius-of-curvature to determine
whether resonating the coil at, above, or below the 1/4 wavelength frequency
was critical to performance (this changes the capacitance distribution in
the system. It can also be viewed as effecting a change in the wirelength).
For a set amount of primary energy, the results will be clouded by the 
increased primary L/C ratio needed to make the secondary resonate at lower
frequencies. This can be offset by reducing the coupling constant k as 
primary inductance increases to keep the mutual inductance M constant.