Considerations Regarding
The Kandoian & Sichak Formula

by Andrťs Esteban de la Plaza


The formula given by Kandoian & Sichak to calculate the quarter-wave resonant frequency f of a coil in kilohertz is:

where ar is the aspect ratio (H/D), N the number of turns and D the diameter if the coil in inches. According to K&S, agreement is within 10% of experimental values.

It may be shown that this formula may well be obtained considering only the distributed capacitance of the coil.


In his paper "TESLA COIL MYTHS" (Tesla Coils Builder Association -TCBA News, Volume 6, #3. 1987), Mr. William M. Colb deals with The Quarter-Wave Coil and explains the difference between electrical and physical length, and introduces de Kandoian & Sichak well known formula.
I will show in this paper that the K&S formula may be obtained considering only the distributed capacity of the coil (Medhurst's formula) combined with the usual tank formulas...



This distributed capacitance, due to the electrostatic effects of the coilís winding may be obtained from the Medhurstís work. Thus, according to Medhurst (Medhurst, R. G. "High Frequency resistance and self-capacitance of single-layer solenoids") the self-capacitance value is given by C[pF] = K D, where the k parameter depends only on the length/diameter (aspect ratio) relationship, and D is the coil diameter in centimeters, thus C [pF] = k D

Then, for common aspect ratios found in Tesla coils ř 1 < ar < 7 , we have the following table of k:

aratio

k

7.0

1.01

6.0

0.92

5.0

0.81

4.5

0.77

4.0

0.72

3.5

0.67

3.0

0.61

2.5

0.56

2.0

0.50

1.5

0.47

1.0

0.46

A regression analysis of this table will give the following function for k as a function of ar:

k = 0.0965 ar + 0.3408

but a better approximation is given by:

k = 0.1ar + 0.32.



On the other hand if we want a power form for the regression curve, instead of the linear one, then after some "number crunching" we will obtain:

 

 

 

 

Thus the Medhurst formula may be written as:

C [pF] = k D[in]2.54

C [pF] = 0.4277 ar 0.4277 D[in]2.54

C [pF] = 1.08636 ar 0.4277 D[in]

Now letís continue with the Wheelerís formula for the inductance of the coil:


Now letís calculate the resonant frequency using Wheeler & Medhurst:

 

Now letís see if we can obtain a regression formula for W in the power form y = . Thus, we plot W for different ar values, 0.5 < ar < 10 , and obtain the following table:

ar

W

0.5

7.14936

1.0

7.61577

1.5

8.09823

2.0

8.53567

2.5

8.92980

3.0

9.28768

3.5

9.61567

4.0

9.91881

4.5

10.20101

5.0

10.46534

5.5

10.71425

6.0

10.94970

7.0

11.38633

8.0

11.78508

9.0

12.15293

10.0

12.49505

The best regression is a power curve given by

W = 7.65464 ar0.19815

Now we replace this formula of W in the f [kHz] one, thus:

f [Hz] = 0.1527 7.65464 109 ar0.19815 / DN

f [Hz] = 1.16886 109 ar0.19815 / DN

or




Now letís compare this formula with the Kandoian & Sichak one:

řthereís only 0.523% difference...

řthereís only 0.925% difference...

Should we conclude that both formulas are the same? I believe the answer is YES!

 

The approach taken has some interesting points we should consider:

  1. We used the tank formula (f = 1/ ), the Wheelerís formula for the inductance and the approximation of the capacitance of the coil as a distributed parameter...



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