The purpose of Tesla Coil theory is to calculate the best possible coil with the transformer that you have. This will result in better performance and will take more punishment. First, find your transformers amp rating and impedance. The formula is:
VA/A=Z
V = Volts
A = Ampers
I= Impedance
E/I=Z
E = Volts
I = Ampers
Z = Transformers Impedance
To find the capacitive rating in microfarads that works best for your coil use the following formula:
C=1,000,000/2pFZ
C = Capacitance in microfarads
F = The AC frequency which is 60 Hz in the US
Z = Transformer Impedance
Now you must choose the type of primary coil you wish to use. For small and medium coils I would recommend the Helix and saucer coil. For large coils I would recommend the flat primary because there is less a chance of a arc striking it. If you are building a magnifier coil then I would recommend the helix design because there is lass of a chance of a arc reaching down to it.
Helix Coil
L=(NR)2 /9R+10H
L = inductance of the primary in microhenrys
N = Number of turns
R = Radius of Coil in inches
H = Height of coil in inches
Saucer Coil
L1=(NR)2 /9R+10H
L2=(NR)2 /8R+11W
L=(sqrt.)[L2 sin(x)]2 +[L2 cos(x)]2
L = Inductance in microhenrys
L1 = Helix factor
L2 = Spiral factor
N = Number of turns
R = Adverage radius in inches
H = Height of coil in inches
W = Width of coil in inches
X = degree of coil
Flat Primary
L=(NR)2 /9R+10H
L = Inductance of coil in microhenrys
N = Number of turns
R = Radius of coil in inches
H = Height of coil in inches
To find the resonate frequency of a resonate circuit such as a Tesal Coil, you must find the point where the inductive reactance equals the capacitive reactance.
Inductive Reactance
XL=2pFL
XL = Inductive reactance in ohms
F = The frequency in Hertz
L = The inductance in henrys
Capacitive Reactance
XC=1/2pFC
XC = Capacitive reactance in ohms
F = Frequency in ohms
C = Capacitance in farads
Tesla Coil Resonate Frequency
F=1/2p[(sqrt.)LC]
F = Resonate frequency in hertz
L = Inductance in hertz
C = Capacitance in farads
Secondary Coil
L=(NR)2/(9R+10H)
L = Inductance of coil in microhenrys
N = Number of turns
R = Radius of coil in inches
H = Height of coil in inches
Torrid Calculations
These equations will be used to find the best torrid for your coil. At the very high frequencies that a Tesla Coil operates everything becomes a virtual capacitor including the secondary coil itself! The formula below is used to find the self capacitance of the secondary coil.
C=.29L+.41R+1.94[(sqrt.)R3/L]
C = Self capacitance in picofarads
R = Radius of coil in inches
L = Length of coil in inches
Now to find the virtual capacitance of the Tesla Coil you add the self capacitance to the discharge capacitance.
Discharge Capacitance of a Torrid
C=1.4(1.2781-D2/D1)[(sqrt.)pD2(D1-D2)]
C = Capacitance in picofarads
D1 = Outside diameter of torrid in inches
D2 = Diameter of cross section of torrid in inches
Discharge Capacitance of a Sphere
C=25.4R/9
C = Capacitance in picofarads
R = Radius in inches
For smaller coils use a Sphere but for medium to large coils use a Torrid because its higher virtual capacitace will give longer arcs.
I must credit Matt Behrend for this information to go to his website click here.