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Re: Formulas wanted!(toroid, helical)

Subject:  Re: Formulas wanted!(toroid, helical)
  Date:  Sat, 10 May 1997 14:18:34 -0700 (PDT)
  From:  "Edward V. Phillips" <ed-at-alumni.caltech.edu>
    To:  tesla-at-pupman-dot-com


Matt:
        Accurate formulae are VERY complicated.  The following
data out of TCBA news should be useful, and the definition of
diameters is unambiguous:

   CAPACITANCE OF ISOLATED TERMINALS, DIMENSIONS IN INCHES

(3" RING = TOROID OF GIVEN OD,MADE OF 3" DIAMETER TUBING, ETC.)

                TERMINAL C (uufd), OUTSIDE DIAMETER EQUALS 

              6"      12"      18"      24"      30"      36"

DISK         5.3     10.7     16.0     21.3     26.7     32.0

3" RING      6.6     12.9     18.8     24.5     30.1     35.7

4" RING      ----    13.2     19.5     25.1     30.7     36.4

6" RING      ----    13.2     19.8     25.9     31.7     37.2

8" RING      ----    ----     19.9     26.4     32.5     38.4

10" RING     ----    ----     ----     26.5     33.0     39.2

12" RING     ----    ----     ----     ----     33.1     39.6

SPHERE       8.4     16.8     25.1     33.5     41.9     50.3

C, uufd, OD = 6"      12"      18"      24"      30"      36"

    (--- INDICATES O.D. LESS THAN TWICE RING DIAMETER.)

>From Michael Schoessow, TCBA NEWS Volume 6, #2, 1987, pp 12-15

NOTE:

     The effective capacitance (the value to add to the
distributed capacitance of the coil to determine the resultant
resonant frequency) of a terminal on top of a coil will be less
than the capacitance given here.  These capacitances are the
values which would result if all of the electric field lines
from the terminal terminated in ground infinitely far removed
from the terminal.  In the case of the coil, some of the field
lines will terminate along the coil, so that some of the capacitance
will be across only part of the coil, reducing its effect on the
resonant frequency.  This is kind of a lame explanation, but isn't
too far from being correct.

            Notice that the capacitance of the toroids falls between
that of the thin disk (smallest) and that of the sphere (highest),
and that it is surprisingly independent of the diameter of the
ring from which the toroid is made.  This suggests that the main
benefit of larger diameter rings is in increasing the "breakout
voltage" of the terminal, not in increasing its capacitance.  Notice
further that the capacitance of the toroid is pretty close to its O.D.
in inches, probably close enough for a working estimate of its
effect.

     I have measured the resonant frequency of a small coil here
with several different terminals.  The measurements were made in my
basement, with the coil rather closer to surrounding objects than
is desirable for accurate results.

COIL WOUND 11/12/95:
FORM: CARDBOARD TUBE, HEAVILY SHELLAC COATED.
COIL HEAVILY COATED WITH URETHANE VARNISH
DIAMETER: 3.07"
LENGTH: 16.16"
WIRE SIZE: #28 B&S

N calculated: 1131 TURNS (from geometry, 1 kc inductance)
L 1kc: 17.37 mH
Q 1kc: 1.82

UNLOADED:
        fr: about 489.6
        Ff 3dB (unloaded): 3 kHz
        Q: 163 (very approximate)
        Cd: 6.08 uufd (calculated from fr)
        Ccalc: 6.16 uufd (Medhurst)

LOADED 9" TOROID OF 3" DUCT:
        fr : 330.9 kc
        Ff 3dB: 1.6 kHz or less (approximate, hard to measure)
        Q: 208 (very approximate)
        Cd: 13.32 uufd (calculated from fr)
        Ctoroid: 7.24 uufd

LOADED 12" TOROID OF 3" DUCT:
        fr : 300.66 kc
        Ff 3dB: less than above, hard to measure
        Q: ?
        Cd: 16.13 uufd (calculated from fr)
        Ctoroid: 10.05 uufd

LOADED 9"+12" TOROID:
        fr : 285.6 kc
        Cd: 17.88 uufd (calculated from fr)
        Ctotal: 11.8 uufd

LOADED 12"+9" TOROID:
        fr : 284.0 kc
        Cd: 18.08 uufd (calculated from fr)
        Ctotal: 12.0 uufd

     From this it appears that the 9" toroid adds a bit over
7 uufd and the 12" toroid adds about 10 uufd.  When the two
toroids are stacked together the added capacitance is about
12 uufd.  Taking this data, which is not necessarily representative
of any other configuration, I come up with this rule of thumb
which may be wildly in error, but which I suspect will give useful
results:

     "The effective capacitance of a toroidal upper terminalis 
approximately equal to the O.D. of the toroid, less about half of
the diameter of the coil."  

This is for measurement at low voltage where there is no discharge
with resultant ion cloud with its added capacitance, 

     Does anyone have any measurements to confirm or modify
this?   

SOME USEFUL EQUATIONS:

WHEELER'S FORMULA FOR INDUCTANCE OF A SOLENOID:

     L = d N^2 / (18 + 40 (l/d))

     or

     L = d^2 N^2 / (18 d + 40 l)

     Where:
     L = inductance in microhenries, N = number of turns,
     d = diameter in inches, and l = length in inches.

Stated to be valid to within 1% for all lengths greater than
1/3 of diameter.  This is the "low frequency current sheet
inductance", which is not exactly good at any frequency but
which is useful enough as a working formula.

MEDHURST'S FORMULA FOR DISTRIBUTED CAPACITANCE OF A SOLENOID:

Cd = 5.08R(.0563L/R+.08+.38SQR(R/L)) uufd. R,L IN INCHES
Cd =    2R(.0563L/R+.08+.38SQR(R/L)) uufd. R,L IN CENTIMETERS

     Where:
     R = Radius of solenoid, L = length of solenoid,
     and SQR(XX) means square root of xx.

     I have found that the calculated resonant frequency using
these equations matches measured values within a few percent,
which is the experimental error.

        Well, that had more than toroids in it.  Hope it helps.
Ed