Easy access to optimal resonance frequency but the benefit is quite small

[VNTC]

After finished winding the secondary coil we can immediate determine the optimal resonance frequency of a tesla coil.
Used this formula:
Fo=3000000/(L*4)  Fo= optimize resonance frequency
L= length of wire in m
For example :
A coil 1050turns; 9cm diameter
L=1050*9*π=296.8m
L*4=1187.2m
Fo=300000/1187.2=252.7kHz
I have done the test with this coil ruing at a deviation 28% of this frequency and this is the result:
Toroi d   Frequency       Input power   Burst duration   Spark length
20cm   Fr=250kHz   240V/2A        100us               100%
40cm   Fr=180kHz   240V/1.6A         100us                 90%

Fr =RF radiation frequency measure by 5cm antenna with a distance from discharge wire.
Although the difference is not large, but it is a reality exists it reflects exactly the 1/4λ principal in the illustrated picture: with the difference of 180/250 = 72 %( compared to the original reference)
1/4λ=90 degree
¼λ*.72=64.8 degree
Sin 90=1 amplitude =1
Sin 64.8=.9 amplitude =.9
Power is proportional with V*V
Spark length proportional with square root of power output so it is proportional with amplitude V. (last column)

With deviation 28% of frequency will drop 10% of spark length, which is a weak affect so we can say: λ/4 rule is not important role in Tesla coil, hhowever  it is still better to apply it because it is not harmful

[dexter]

I've seen a paper about calculating the self capacitance of a solenoid and in one of the methods they used the self resonance frequency, measured inductance and the length of wire to calculate the self capacitance.
but in their calculation they used l=λ/2 (l= length of wire)

i'm confused... which formula λ/2 or λ/4 describe the self resonance of a solenoid?

[VNTC]

Hi Dexter,
Thank you for your reply please read the qote following you will see something.
  Quote:
   Tesla was not the first to invent this circuit.[21][15] Henry Rowland built a spark-excited resonant transformer circuit (above) in 1889[2] and Elihu Thomson had experimented with similar circuits in 1890, including one which could produce 64 inch (1.6 m) sparks,[9][22][23] [1] and other sources confirm     Tesla was not the first.[14][24][15] However he was the first to see practical applications for it and patent it. He even realized that the secondary coil functioned as a quarter-wave resonator; he specified the length of the wire in the secondary coil must be a quarter wavelength at the resonant frequency.[25][8]
A coil with the balance shape will working ok on a wide range of frequency my testing coil can run with toroid 40cm to 15cm with the different 10% only at power out.(180kHz-275kHz)