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Published: 01 February 2015

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Is there, I ask, can there be, a more interesting study than that of alternating currents
Nicola Tesla
The selfcapacitance of a singlelayer air core solenoid
At high frequencies, the inductor would be correctly represented as a helical singlewire transmission line. As is known, such a structure is defined as the small inductances and capacitances distributed along the line length. When the length of this line is equal to λ / 4 (λ  wavelength of the electromagnetic wave in the coil), the line is in resonance mode and the frequency corresponding to this wavelength is called the selfresonant frequency of the coil. Coils rarely operate at frequencies close to the selfresonance or above. However, if this happens, such as in RF choke, the calculation of the coil is preferable to carry out in an RF electromagnetic simulators.
Typically, the design of the coil is selected such that it operates at a frequency much lower selfresonant. In such case, the coil with sufficient accuracy can be represented as an equivalent circuit of the ideal lumped elements. In this representation, the coil is connected in parallel its own parasitic capacitance. With the selfinductance of the coil, it forms parallel LC circuit, the resonant frequency of which is the selfresonant frequency of the coil. This capacitance depends on the material of the coilformer, shape of winding, availability of screen and core. To simplify the problem, we define to a first approximation, how a form of winding influences on a value of the parasitic capacitance.
In the evolution of the radio repeatedly attempts were made to pick up the empirical formula for the calculation of selfcapacitance.
 The first time the question of selfcapacitance of the coil inductance was raised by J.C. Hubbard in 1917. S.Butterworth, all known innovator and designer of frequency filtering circuits (remember Butterworth filters) in 1926 proposed a formula for the calculation, but it had serious limitations and could not count short coil. (What is meant by the term "short coil", read the article dedicated to the qfactor)
 There was invented a method of determining the selfcapacitance of the coil by indirect measurement. The resonant frequencies of the LC circuit with two different external capacitors are measured and then from the measurement results is calculated the selfcapacitance and selfinductance of the coil.However, it requires measurement equipment. Moreover, by this method, we can determine the selfcapacitance of the finishing made inductor but we desired to estimate this parameter at the stage of design.
The formula proposed by the engineer AJ Palermo in 1934 in the paper named "Distributed Capacity of SingleLayer Coils":
 С_{s}  selfcapacitance (pF);
 D  diameter of winding (cm);
 p  winding pitch (cm);
 d  wire diameter (cm);
This formula quite common can be found in a variety of references, but as shown by subsequent experience  the formula is not right! Palermo represents the coil as an electrostatic structure in which the selfcapacitance is obtained as the sum of series capacitance between adjacent turns. Logical conclusion  by increasing gap between adjacent turns we decrease this total capacitance.
Despite the apparent obviousness and clarity, this approach is wrong. As I noted at the outset, the RF coil is the distributed structure around which there is an electromagnetic field  the single entity in which the vectors of the electric and magnetic fields are always perpendicular and the electrical and magnetic components of the field can not exist independently of each other.
Since the magnetic field lines are mainly located along the axis of the coil, the electric vector of the wave is located perpendicular to the axis, and the components of the electric field parallel to the axis (between turns) are almost negligible.
Formula G. Grandi, published in article "Stray Capacitances of SingleLayer Solenoid AirCore Inductors  Grandi, Kazimierczuk, Massarini and Reggiani 1999":
 N  number of turns;
 ε_{0}  dielectric constant;
 D  coil diameter (cm);
 p  winding pitch (cm);
 d  wire diameter (cm).
It was the attempt of modern reanimation electrostatic model of the coil, that as shown in the case of formula Palermo, far from the reality. It is a vivid example, when "paper theory" is not rightly verified by experiment and nevertheless is accepted as the truth. From the formula we conclude, if a coil has the more turns, then the less a parasitic capacitance, what is absolute conflicts with practice.
In 1947 radio engineer R. G. Medhurst  an employee of research laboratory of the "General Electric Co.Ltd." has published a number of works related to experimental research of inductors. A careful measurement of the parasitic capacity of a large number of coils, Medhurst came to the conclusion that the selfcapacitance of the coil is only weakly dependent on the pitch of winding and model Palermo does not work. Also, Medhurst found a dependence of the selfcapacitance from the form factor of a coil that is the ratio of the length of the winding to its diameter l/D). On the basis of these measurements Medhurst derived his semiempirical formula:
 D coil diameter (cm),
 H  coefficient characterizing the form factor of the coil that was presented in the form of a table:
l/D  H  l/D  H  l/D  H 
50 
5.8 
5 
0.81 
0.7 
0.47 
40 
4.6 
4.5 
0.77 
0.6 
0.48 
30 
3.4 
4 
0.72 
0.5 
0.50 
25 
2.9 
3.5 
0.67 
0.45 
0.52 
20 
2.36 
3 
0.61 
0.4 
0.54 
15 
1.86 
2.5 
0.56 
0.35 
0.57 
10 
1.32 
2 
0.50 
0.3 
0.60 
9 
1.22 
1.5 
0.47 
0.25 
0.64 
8 
1.12 
1 
0.46 
0.2 
0.70 
7 
1.01 
0.9 
0.46 
0.15 
0.79 
6 
0.92 
0.8 
0.46 
0.1 
0.96 
However, the more famous the most simple formula for the estimation of the selfcapacitance of the onelayer air core coil, which was proposed by Medhurst:
 C_{s}  selfcapacitance [pF];
 D  coil diameter [cm];
The selfcapacitance of a singlelayer air core coil in pF is numerically approximately equal to the radius of the winding, but we need to remember that this is true when 0.5 < l/D < 2, otherwise we have to take the factor from the table.
A complete modern work that dedicated selfcapacitance of the coil is the article "The selfresonance and selfcapacitance of solenoid coils" Dr. David W Knight 2013" (G3YNH) [ref. 1 ⬇]
A few quotes from this work:

There is even a school of thought that says that the selfcapacitance is due to the capacitance between adjacent turns; and although that is partly true for multilayer coils and flat spirals, the hypothesis turns out to be a hopeless predictor of the reactance of singlelayer solenoids.

The solution, of course, lies in recognizing that the coil is a transmissionline; except that the line in question turns out to be a rather complicated one.

A trivial investigation involving a GridDip Oscillator and a set of engineer's callipers will confirm that the various resonances exhibited by a disconnected coil are associated primarily with the total conductor length. It is therefore extraordinary that the selfcapacitance of singlelayer coils is still routinely attributed to the static capacitance that is presumed exist between adjacent turns.
The calculation of the selfcapacitance of singlelayer coils in the program Coil32 is based on the Dr. David W Knight essay without considering the influence of the coilformer. It is understood that the inductor operates at a frequency much lower than the selfresonance frequency and it is an air core coil. Approximately can be considered that the presence of the coilformer increases the selfcapacitance compared with an estimated up to 15..30%, former with grooves under the wire increases the capacitance up to 40%, the impregnation and enveloping coils with varnish or compound increases the selfcapacitance of up to 50% and above.
Summing up, you can come to the following conclusions:
The concept of the selfcapacitance is directly related to the selfresonant frequency of the coil. Hence, when considering the physical processes in a singlelayer coil at frequencies close to the frequency of its selfresonance it is necessary to abandon the model of lumped elements as baseless and to consider the inductor as a transmission line. In the "straight transmission line", such parameters as a capacitance and inductance per length depends only on its geometry. In the spiral line, due to the influence of turns on each other, these parameters are also a function of frequency. This fact leads to important consequences:
 The resonance frequencies of this line though severely are depending on the total length of the conductor which is wound the coil, but not multiples of each other.
 The selfcapacitance of the coil depends on the frequency at which it is determined. Ignore the effect of frequency and talk about the calculation of the capacitance of the coil makes sense only when the coil operates at frequencies not exceeding 6070% from the frequency of its selfresonance. (In fairness it should be noted that this applies to the calculation of inductance!)
 At frequencies much lower than the frequency of the first resonance there is possible for simply using the lumped elements model for the coil. However, it should not be forgotten that their selfcapacitance, in this case, is the summary capacitance per length along the line and depends on the length of the conductor.
How the geometry of the coil affects its selfcapacitance?
 The selfcapacitance of a singlelayer air core coil is directly proportional to its diameter.
 The optimal coil in terms of parasitic capacitance (capacitance  minimum), is a coil with l/D ≈ 1. The same coil is optimal in terms of the quality factor. This is understandable because the coil with such a geometry winding has a maximum inductance at a minimum wire length.
 As shown above, increasing gap between the turns has almost no influence on the value of the selfcapacitance. Modifying the pitch of winding, first of all, we change the inductance of the coil, and not its selfcapacitance. This comes from the fact that adjacent turns are interacting with each other mainly through the magnetic field, and not through an electric.
References:
 The selfresonance and selfcapacitance of solenoid coils  © Dr. David W Knight 2013
 RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes  K.L. Corum and J.F. Corum 2001
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The value of SRF is always displaying in results field for singlelayer coils.
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