# TESLA MATH FORMULARY

• To: tesla-at-grendel.objinc-dot-com
• Subject: TESLA MATH FORMULARY
• From: richard.quick-at-slug-dot-org (Richard Quick)
• Date: Thu, 10 Aug 1995 03:14:00 GMT
• >Received: from ns-1.csn-dot-net (root-at-ns-1.csn-dot-net [199.117.27.21]) by uucp-1.csn-dot-net (8.6.12/8.6.12) with SMTP id VAA01145 for <tesla-at-grendel.objinc-dot-com>; Wed, 9 Aug 1995 21:30:24 -0600

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EXPLANATION OF EQUATIONS:

x = Multiply by
/ = divide by ( may also use line seperating terms above and below
line as in standard mathematics).
( ) = Terms in parentheses should be calculated first as in standard
algebraic equations.
pi = 3.141592654. The circumferance of a circle divided by it's
diameter will always give you this constant.

n
Z   = means the term "Z" multiplied by itself n times.

Note: In some cases I do not use the symbol "x" but instead simply
put the terms close together, example: "LC" instead of "L x C".
This is standard for algebraic equations and means "multiply by".

=======================================================================
MATH FOR TESLA COILS

1. Determine your neon sign transformer (or other transformer's)
Impedence:

E
Z  =  ---
I

Z = Impedence
E = volts
I = current in Amps
Note: divide milliamps by 1000 to get Amps. 30 milliamps = .030 Amps.

The Impedence of the primary capacitor should match the Impedence of the
transformer at 60 Hz (60 Hz is the AC cycle rate of common household
wall sockets, at least in America).

2. To match Impedence and determine capacitor value:

1
C =   -------------------
2 x pi x Z x .00006

C = capacitance in microfarads needed for primary capacitor.
Z = Impedence from equation one (Transformer Impedence)
pi = 3.141592654
Note: The .00006 is the 60 Hz AC, if you live outside the US then

Next you need to find the Reactance of the primary capacitor at the
frequency you have choosen. Many times the frequency is decided by the
length of wire used on the secondary coil. See below for equations that
determine frequency by length of wire used on secondary.
When we find the Reactance , we can then find your needed Inductance
for the Primary coil.

3. To determine Reactance of capacitor:

1
X(C) = ---------------
2 x pi x C x F

X(C) = capacitor Reactance
C = Capacitor value in microfarads, from equation 2)
F = Frequency in Mhz (megahertz)
pi = 3.141592654
Note: To convert kilohertz to megahertz simply divide by 1000.
190 Khz = .190 Mhz

4. To determine the Inductance needed for Primary coil:

Set X(L) = X(C)

X(L)
L = ------------
2 x pi x F

L = Inductance in microhenries needed for Primary Coil.
To get millihenries divide the answer by 1000.
X(L) = Reactance from equation 3, same as X(C).
F = frequency in Megahertz.  Divide Khz by 1000 to get Mhz.
pi = 3.141592654

Now you know the values for your capacitor and primary coil. These
values will give you the best ringing for your circuit (ie. more bang
for your buck)! Use the equations below to finnish the project.

Note: Many people don't go to the trouble to work these equations out.
They simply slap the parts together and then try to tune. If you work
the equations out first you will save lots of time in tuning, you will
at least be in the right ball park! Also, just because you worked it
out on paper that doesn't mean it will work the first time you plug it
in. Trial and error is a large part of the Tesla Coil hobby!

=======================================================================
CAPACITORS IN PARALLEL:
|      |
|__C1__|
|      |
|__C2__|
|      |
|__C3__|
|      |
Capacitance = C1 + C2 + C3, etc...

Maximum voltage rating will be equal to the voltage rating of the
lowest voltage capacitor of the group.
=======================================================================
CAPACITORS IN SERIES:
|              |
|__C1__C2__C3__|
|              |
|              |

1     1     1
Capacitance = 1 / --- + --- + ---, etc...
C1    C2    C3

The total capacitance of several capacitors in series will always be
LESS than that of the smallest capacitor.

Total voltage rating increases with number of capacitors in series.
Simply add the voltage ratings together.
When capacitors are placed in series to increase voltage rating they
should have the same capacitance and voltage rating else voltages will
divide unevenly, most likely causing failure.
=======================================================================
EQUATION 1:  PLATE TYPE CAPACITORS

Capacitance (in microfarads) = (0.224 KA / d) (N-1)

0.224 x Dielectric Strength x Area of plate
C = -----------------------------------------  x (Number of plates - 1)
distance between plates in inches

NOTE: For two plate rolled capacitors the value derived from this
formula must be multiplied by two, or, the value must be computed
with the number of plates (N) equal to three (3), so (N-1 = 2)
=======================================================================
EQUATION 2: LEYDEN JAR or SALT WATER TYPE CAPACITORS (jar/bottle type)

2
C = .0884 k ( pi  r   + 2  pi  r  l )
------------------------------
1,000,000 t

k = dielectric strength
r = jar radius in centimeters
l = height of the jar portion used (in centimeters)
t = thickness of the jar wall in centimeters
pi = 3.141592654

2
r   = r x r  (radius squared)

=======================================================================
EQUATION 3: FREQUENCY OF A CIRCUIT

1
f =  --------------------
__________
2 pi   / L C

f = frequency in cycles per second
L = circuit inductance in henries
C = circuit capacitance in farads
pi = 3.141592654

_________
The symbol " /          " means the square root
For a result "f" in Khz: enter "C" in microfarads, "L" in microhenries
and multiply result by 1000.
=======================================================================
EQUATION 4: INDUCTANCE OF A FLAT PANCAKE COIL

Picture a 1 inch flat ribbon that is about 30 feet long. Now, roll
that ribbon into a spiral that has all its sides about 1/2 inch apart.
Most common material is Aluminum Roof Flashing. Use plastic bolts to
hold sections of strips together if you have short pieces of ribbon.
This makes a good mechanical connection (you can't solder aluminum).

center axis
|
| | | | |    |    | | | | |       <---cross section
|                        of flat spiral
|---A----|    |---W---|           coil.
|

2         2
a     x   n
L = ---------------
8 a   +   11w

L = inductance in microhenries.
a = average radius in inches as measured from the central axis to
the middle of the winding.
n = number of turns in the winding.
w = width of the coil in inches.
Note: Make sure you measure "a" from center axis - the very middle

=======================================================================
EQUATION 5: NUMBER OF TURNS FOR A HELICAL PRIMARY
_________________________
/
N =     /  L  [( 9 x R) + (10 x H)]
/  --------------------------
/            2
\/            R

N = Number of turns needed.
L = inductance in microhenries desired.
H = height (inches).

_________
The symbol " /          " means the square root, in this case of whole
equation.

=======================================================================
EQUATION 5: LENGTH OF WIRE NEEDED FOR DESIRED FREQUENCY OF COIL

300,000
L =     -------  / 4 x (3 / .9144)
f

f = frequency, in Khz, that is desired for coil.
L = length of wire needed, in feet, for desired frequency.
/ = divided by.

Note - 300,000 is the speed of light in Kilometers per second. the
term "3/.9144" is a conversion factor to turn meters to feet.
You don't have to understand this. Just thought I would tell
those who were wondering.

=======================================================================
EQUATION 6: FRQUENCY OF COIL

300,000
f =  ------------------------------
T x W x pi x (.9144 / 36) x 4

f = frequency of coil in Khz
T = number of turns on coil
W = width of the coil in inches
pi = 3.141592654

=======================================================================
EQUATION 7: CAPACITANCE OF A SPHERE IN SPACE

R
C =   -------
9
9 x 10

9
9x10  = 9,000,000,000
6
Note: To convert Farads to microfarads simply multiply by 10   or in
other words by 1,000,000.

=======================================================================
EQUATION 8: CAPACITANCE OF A SPHERE SUSPENDED IN A DIELECTRIC

K x R
C = ---------
9
9 x 10

K = dielectric constant
6
Note: To convert Farads to microfarads simply multiply by 10   or in
other words by 1,000,000.

=======================================================================
EQUATION 9: CAPACITANCE OF A TOROID
___________________
/    2
C =(1+ (0.2781 - d2/d1)) x  2.8  x      /  2 pi  (d1-d2)(d2/2)
/   -------------------
\/      4 pi

C = capacitance in picofarads (+- 5% )
d1 = outside diameter of toroid in inches
d2 = diameter of cross section (cord) of toroid in inches

Equation courtesy of Bert Pool
=======================================================================

EQUATION 10: POWER FACTOR CORRECTION FOR NEON SIGN TRANSFORMERS

Neons typically have an efficiency of about 50%, in that they
draw twice as much power as they put out. This problem can be
resolved with the use of power factor correction (pfc) capaci-
tance across the line. The pfc capacitors used are the same as
for capacitor starting motors. The voltage rating should be at
least twice the line current used, and I like a 4x voltage margin
for long life. The formula used to determine ballpark pfc is as
follows:
9
10^
C = Corrected kVA  ------ 2
2(pi)fe^

This should read C = Corrected kVA times (10 to the ninth power)
over, (2 pi times f times e squared)

C = required capacitance in microfarads
f = frequency of applied voltage
e = applied voltage

CORRECTED KVA is determined by dividing the corrected power factor
output of the neon sign xfrmr (below) by 1000

Corrected Power Factor

Secondary Rating         Volt-Amps
kV,mA

15,120                 900
15,60                  450
15,30                  250
12,120                 775
12,60                  400
12,30                  200
9,120                 600
9,60                  300
9,30                  150
7.5,120                 500
7.5,60                  250
7.5,30                  125
6,60                  200
6,30                  100

Equation courtesy of Richard Quick

======================= END OF FILE ===================================

This was originally from Keelynet I think. I cut it down quite a bit
to produce a file that covers only the math. RQ
___ Blue Wave/QWK v2.12

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