# Re: 1/4 WAVE SECONDARIES

• To: tesla-at-grendel.objinc-dot-com
• Subject: Re: 1/4 WAVE SECONDARIES
• From: EDHARRIS-at-MPS.OHIO-STATE.EDU
• Date: Fri, 20 Oct 1995 17:50:20 -0400 (EDT)
• >Received: from phyas1.mps.ohio-state.edu (phyas1.mps.ohio-state.edu [128.146.37.10]) by uucp-1.csn-dot-net (8.6.12/8.6.12) with ESMTP id PAA05438 for <tesla-at-grendel.objinc.COM>; Fri, 20 Oct 1995 15:50:29 -0600

```Malcolm writes:

SECONDARY ANALYSIS 2
********************
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.

EH> I just want to make sure I understand: Are you saying that for the
EH>given length of wire you then compute what frequency corresponds
EH>to that wavelength (using free-space  speed of light?)

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

EH> By self capacitance, do you mean the interturn winding capacitance?
EH> There is also the capacitance from each winding to ground which depends
EH> specifiaclly on the environment near the coil...

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

EH> If you use one of the standard formulas like Nakaoga's (sp?) it will
EH> probably be in error since the reduction of inductance due to the skin and
EH> proximity effects are almost never taken into accout

(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)

EH> There is also the proximity effect, could you elaborate?

(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).

EH> I have a couple of papers, one theoretical, and one experimental which
EH>make a crack at it. Do you really think radiation is a major loss? I
EH>haven't seriously esimated it, but since tesla coils dimensions are all a
EH>small fraction of the working wavelength, they sould be very inefficient

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

NOTES :
- For a chosen h/d ratio, the capacitance required to force ANY
coil

EH> Is it true though, that you have based this condition on measurements
EH> done on coils with a fixed form diameter? For given H/D, but different
EH> D, I'd have expected a change in the interwinding capacitance.

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.

```