Loop Calculations, Part 2

Continuing the Quest for an Accurate Coil Inductance
             -and The Mysteries of Distributed Capacitance

In Part 1 of this series we began our search for an accurate inductance formula. Accuracy is imperative in order to correctly predict our passive loop's inductance value and thus its tuning range. Passive loops for mediumwave are a special breed of coil animal. They could be termed "very large, very short coils", and they don't fit the common formulas found for calculating coil inductance.

Coils can basically be divided into two groups, long and short. The dividing line seems to be that 0.5 ratio of length/diameter. By coil length we mean the length of the winding as opposed to the diameter of the winding. If a coil's length is greater than half its diameter, it can be considered a long coil. If its length is shorter than half its diameter, it can be considered a short coil. The long coil inductance formulas put forth in the 1920s really didn't work well for short coils, those having a length/diameter ratio less than about 0.4. And the short coil formulas of the day only worked down to a length/diameter ratio of about 0.2, greatly losing accuracy below this. Our passive loop lies well south of 0.2, usually in the range 0.05 to 0.1. And it is usually square - not to our advantage as we swim in a sea of mostly circular coil formulas.

Wheeler's and other long coil formulas of the late 1920s were simple and accurate within the range of coil geometries for which they were intended. Many inductance calculators you find on the web use these formulas, despite the fact that there are far more accurate formulas for short coils. The problem with these calculators is there is no indication of coil geometry limitation, leaving the uninitiated to believe the result as accurate. Even Wheeler's short coil formula of this era, thought to be accurate to within 2%, was in later years revised upwards nearly 10% for some coil geometries!

At 79 years old and fourteen years before his death at age 93, Wheeler published in 1982 his continuous inductance formula, a single encompassing formula which calculates the coil inductance of all cylindrical coils, long and short. Using Grover, Wheeler and others work, and refined by Rosa's corrections and unmentioned scientists Lundin, Nagaoka, Knight and more, we have an accurate formula for predicting the inductance of coils with length/diameter ratios approaching that of a passive loop.

One would think we were done now - simply figure the tuning range of the passive loop using the calculated inductance. We have one more complication, however, and that is the loop's unknown self-capacitance. It must be accounted for in the second formula which calculates the tuning range of the loop.

Now we open that can of worms, that of self-capacitance, or commonly but erroneously known as distributed capacitance. Every coil also has self-capacitance as well as inductance. Self-capacitance is the inherant internal capacitance of the wire-formed coil itself. This spurious capacitance adds extra capacitance to the tuned circuit, changing (in fact further lowering) the resonant frequency. Additionally, this self-capacitance reduces the overall tuning range of the coil because it alters the range of the tuning capacitor. For example, if our common 10-365pF tuning capacitor is used in parallel with a proper loop happening to have 30pF of self-capacitance, the capacitive tuning range in reality is 40-395pF. What may have been a perfect, calculated tuning range of 530-1700 KHz is now altered downward to 500-1450 KHz. So, it is advantageous to design a loop with as little self capacitance as possible, or at least account for it in our design.

As stated, few formulas are commonly found which accurately calculate the inductance of a very large, very short coil like our passive loop antenna, whether it be circular, square, or other polygonal shaped. Virtually none of these account for coil self-capacitance, which, as we have seen, greatly changes the tuned circuit's operating parameters. An accurate self-capacitance figure is necessary to predict an accurate tuning range for our loop. Now we must wrestle with attempting to calculate or measure the loop's self-capacitance, which conceivably may change our tuning range by perhaps an additional 5-10 percent.

Much fact and fiction has been written concerning the cause and calculation of self-capacitance of coils. Once again, a little history is in order. Bear with me as I think you will find it interesting, especially if you like a bit of intrigue.

Little work if any had been done in this area of radio science much before 1920 - that of predicting coil self-capacitance. In 1917, J. C. Hubbard publishes a little-known paper in Physical Review entitled, "On the Effect of Distributed Capacity in Single-layer Solenoids". He measures coils of varying geometry down to a 0.2 length/diameter ratio, some with as few as 35 turns. He ponders that, "There is no evidence that the variation of ratio of pitch to diameter of wire [turns spacing] has a measurable effect on the distributed capacity in the region studied, though some effect is to be expected for coils of a smaller number of turns than those studied here."

S. Butterworth, famous innovator and designer of frequency filtering circuitry, tackles the problem again from 1922 - 1926. He publishes self-capacitance formulas which purport to cover single layer solenoids wound with round wire, the stipulations being that the number of turns have to be large (our long coil again) and well spaced.

In 1934, engineer A. J. Palermo enters with his text, "Distributed Capacity of Single-layer Coils", published in the proceedings of the Institute of Radio Engineers. In that text he presents a formula for calculating the self-capacitance of coils, based on his theory of accumulated inner-winding capacitance of a coil's adjacent turns. The paper is widely accepted by the engineering community of the day in both the scientific and popular press. The premise derived from his theory is that spacing of coil turns plays a major part in self capacitance, in that wider turn spacing lowers self capacitance, and tighter turn spacing increases it. Palermo claims to have found the solution. Has he?

Experimentation in the field continues. General Electric engineer R. J. Medhurst suddenly springs onto the scene in 1947 with his two-part article, "H.F. Resistance and Self-Capacitance of Single-Layer Solenoids" in Wireless Engineer. After exhaustive testing of self-capacitance on some 40 coils, Medhurst summarizes:

"These measurements show a very considerable divergence from the formula of A. J. Palermo though they are in quite good agreement with other previous experimental work [inferring Butterworth]. Self-capacitance of coils of this type is shown to be substantially independent of the spacing of the turns. It is given by an expression of this form:

Cd(picofarads) = H * D

where D(cm) is the mean coil diameter and H depends on the length/diameter. A table of H is given, based on these measurements."

"....to better than 5%, the measured values [of self-capacitance] fit the expression:

Cd(picofarads) = 0.46 * D

"....being independent of the spacing ratio."

Medhurst presents a table of values for H, covering coil geometries down to 0.1 length/diameter ratio. Accuracy below about 0.2 is still in question. It also seems that the most efficient length/diameter ratio causing the lowest self-capacitance in a coil is near 1.0, that is, a coil having a length equal to its diameter.

Medhurst explains further:

"Measurements of self-capacitance of a wide range of single-layer coils were consequently carried out. The results failed to confirm Palermo's claim that the self-capacitance varies steeply with the spacing of turns. They are, instead, in quite good agreement with previous work which had shown the self-capacitance to be very nearly independent of d/s (spacing ratio of diameter/turns)."

"Thus, the capacitance between adjacent turns will be less than that predicted by Palermo. The fact that self-capacitance is substantially independent of spacing of turns suggests that the part of the self-capacitance considered by Palermo is actually negligible."

At this, J. C. Hubbard remarked, "....we apparently have two quite independent factors [determining the self-capacitance of coils], one predominating greatly in very short coils, the other, in very long coils."

Medhursts's findings are approximate of course, though a rudimentary start to the overall understanding of the self-capacitance of coils. Experimentation continues through the next decades, each overwhelmingly refuting Palermo's theory. However, the myth of accumulated inter-turn capacitance has been cast and continues through the years, propagated through popular writing and idle commentary.

On his exceptional web site Dr. David Knight (G3YNH) delves into great detail concerning loop inductance and self-capacitance, both explaining and expanding upon current knowledge in the field. His articles on the current state of inductance calculation and self-capacitance of coils are well worth the read if you have a mathematical curiosity about such things.

The general thinking in past years has pointed to transmission line theory as being the root driving mechanism of coil self-capacitance. So interesting I find the huge amount of evidence against Palermo's supposition of turns spacing driving coil self-capacitance, I'll reproduce some of David Knight's eloquent analysis here. It makes for very interesting reading.

From David Knight's article, "The Self-Resonance and Self-Capacitance of Solenoid Coils", Version 0.01 (provisional), 9th May 2010, and "From Transmitter to Antenna, Inductors and Transformers: Solenoids, Part 1":

"Attributing self-capacitance to the static turn-to-turn electric field is a fallacy akin to taking the coil apart and trying to find the capacitor....The solution, of course, lies in recognising that the coil is a transmission line; except that the line in question turns out to be a rather complicated one."

"Unfortunately, the electrical literature abounds with articles which claim that the self capacitance of a coil is due to the capacitance between adjacent turns. This hypothesis is easily refuted, because it makes the wholly incorrect prediction that coils which have closely-spaced turns will have much greater self-capacitance than those which do not. The static component of self capacitance is small in single-layer coils, because a wave travelling along the wire does so with its electric vector nearly perpendicular to the coil axis, i.e., the electric field component parallel to the axis is almost negligible in comparison to the radial component. Nevertheless, the static capacitance idea appears to be so intellectually compelling, that there are at least two examples, in the peer-reviewed literature, where researchers have been motivated to fabricate or selectively report experimental evidence in order to support it."

"There is even a school of thought which says that the self-capacitance is due to the capacitance between adjacent turns; and although this is partly true for multi-layer coils, the hypothesis turns out to be a hopeless predictor of the reactance of single-layer coils. Experimentally, it transpires that self-capacitance increases as the spacing between turns increases...."

"Palermo gives a formula based on the hypothesis that the self-capacitance can be deduced by considering the capacitance between adjacent turns. Medhurst, being a meticulous experimenter, soon ran into difficulties with that approach; and so was forced to 'find out whether Palermo's formula did in fact agree with experiment'". He concluded that the data supporting Palermo's theory were suspect; and fell only a little short of accusing Palermo of scientific fraud.....Medhurst was aware that self-capacitance is substantially independent of turn-spacing provided that the coil has plenty of turns. He therefore chose to keep the number of turns per unit length high to eliminate pitch effects."

"A trivial investigation involving a Grid-Dip oscillator and a set of engineer's callipers will confirm that the various resonances exhibited by a disconnected coil are associated with the total conductor length. It is therefore extraordinary that the self-capacitance of single-layer coils is still routinely attributed to the static capacitance which is presumed exist between adjacent turns."

"Palermo reported a total of 19 self-capacitance measurements, 12 of which he carried out himself, and 7 of which were communicated to him by F W Grover of the National Bureau of Standards. It was in the group of measurements performed by Palermo himself that Medhurst found some of the numbers to be unreproducibly large....Medhurst was right to cry foul; but, in fact, the extent of the tampering was even greater than Medhurst had suspected."

"His [Palermo] formula often produces values which are much too large. In such cases, he appears to have adopted the habit of adjusting the calculated value downwards and the measured value upwards in order to obtain plausible agreement. Since he acknowledges the help of F W Grover however, he was evidently not in a position to tamper with the NBS data; and so in that case he confined himself to writing down false calculation results. In the worst instance, his formula gives 27pF, but he reports 12.9pF to confer with an NBS measurement of 12.8pF. There are other sleights of hand for those who wish to pursue the issue, but overall the paper is a travesty."

"That then is the insalubrious basis on which the inter-turn capacitance hypothesis became part of electromagnetic folklore. What Palermo hoped to gain by promoting his defective theory is difficult to guess; but he may have been motivated by inability to accept failure after an early success. His formula was subsequently turned into tables and abacs to 'assist' the radio engineer; and his dogma diffused naturally into the textbooks to lie in wait for the unwary. The 'capacitance between adjacent turns' hypothesis re-emerged in a new guise in 1999, in a paper by Grandi, Kazimierczuk, Massarini and Reggiani. These authors cite Medhurst, only to dismiss his work for being empirical; and make no mention of Palermo despite the strong parallel and Medhurst's barbed discussion."

"In summary, it is fair to say that theories which attempt to attribute the self-capacitance of single-layer solenoids to the inter-turn capacitance are wrong. In Palermo's case, the problem lies firstly in the assumption that a single wire can behave like two wires lying parallel, and secondly that the resulting capacitance should be divided by N [number of coil turns]. Logically, his theory is no better than a guess; which happens to work roughly for some coils, but has no actual predictive power."

How interesting and so anti the commonplace conception we have about self-capacitance so often found in the idle writing of today! We see that number of coil turns and spacing of coil turns matters very little in single-layer solenoid coils of many turns, so-called long coils. It is only when we get to the very short coil that matters begin to get sticky. Passive loop territory, again.

So where does this leave us? Even today, commonly-found formula predictors of self-capacitance still fall short of the mark of accuracy when dealing with in very short coils, coils having less than about 0.2 length/diameter ratio. Their result is often high by a factor of two and sometimes more. Over the years, Medhurst produced not one but several formulas for calculating self-capacitance for different coil geometries. Medhurst's original simple formula of Cd(picofarads) = 0.46 * D seems to give a nearly workable result in the region for very short coils. Two other Medhurst formulas purportedly come even closer in accuracy.

We have one additional way to predict, actually calculate, the self-capacitance of a coil. It is fail-safe in its accuracy. We first wind a test loop, then wire a tuning capacitor with known range to the loop. Using a receiver, we record the lower and upper frequency of the capacitor's tuning range by listening for signal or background noise peak at either extent of its range. Plugging the capacitor's known low and high values and the low and high tuned frequency extent into a simple (okay, maybe not so simple) formula, we can directly calculate the coil's exact inductance and self-capacitance. We then have a basis upon which to refine our calculation and thus the number of loop turns to satisfy our goal of tuning the mediumwave band completely.

Our last problem now is predicting the inductance of a very short, very large diameter polygonal coil to a high degree of accuracy. We will take our best shot with Wheeler's Continuous inductance formula and Nagaoka's formula for circular coils, using Rosa's corrections for both, and then interpolate for a polygonal loop. This, because we know that inductance varies very nearly proportionally with the area of the loop. Two other formulas we will use for comparison are Grover's old formulas for polygonal loops.

Next up in Part 3: The Inductance Calculator Program - Loop Calculator One

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Bibliography and of Interest:

Frederick W. Grover "Additions to Inductance Formulas", Scientific Paper #320, Bulletin of the Bureau of Standards 14, pp. 555-570 (1918).

Frederick W. Grover "Tables for the Calculation of the Inductance of Circular Coils of Rectangular Cross Section", Scientific Paper #455, Scientific Papers of the Bureau of Standards 18, pp. 451-487 (1922).

Frederick W. Grover "Formulas and Tables for the Calculation of the Inductance of Coils of Polygonal Form", Scientific Paper #468, Scientific Papers of the Bureau of Standards 18, pp. 737-762 (1922).

Frederick W. Grover "The Calculation of the Inductance of Single-Layer Coils and Spirals Wound with Wire of Large Cross Section", Proceedings of the Institute of Radio Engineers (1929).

Frederick W. Grover "Inductance Calculations: Working Formulas and Tables", (Van Nostrand, 1946 and Dover, 1962 and 2004).

Harold A. Wheeler "Simple Inductance Formulas for Radio Coils", Proceedings of the Institute of Radio Engineers, Vol. 16, No. 10, October 1928.

A. J. Palermo. "Distributed Capacity of Single-layer Coils", Proceedings of the Institute of Radio Engineers, Vol. 22, pp. 897 (1934).

H. Nagaoka, "The Inductance Coefficients of Solenoids", Tokyo, Vol. 27, No. 6, (1909).

E. B. Rosa, "The Self and Mutual Inductances of Linear Conductors", BBS Vol. 4, No. 2, (1908).

E. B. Rosa, Bulletin of the Bureau of Standards, Vol. 2, pp. 161-187 (1906).

E. B. Rosa and F. W. Grover, "Formulas and Tables for the Calculation of Mutual and Self Induction", [Revised], Bulletin of the Bureau of Standards, Vol. 8, No. 1, p. 122 (1911).

R. G. Medhurst, "H.F. Resistance and Self-Capacitance of Single-Layer Solenoids", Wireless Engineer, Feb. 1947, pp. 35-43 & Mar. 1947 pp. 80-92.

J. C. Hubbard, "On the Effect of Distributed Capacity in Single-layer Solenoids", Physical Review, 1917, Vol. 9, p. 529.

R. Lundin, "A Handbook Formula for the Inductance of a Single-Layer Circular Coil", Proceedings IEEE, Vol. 73, No. 9, pp. 1428-1429 (1985).

David W. Knight, G3YNH, "From Transmitter to Antenna, Inductors and Transformers: Solenoids, Part 1"

David W. Knight, G3YNH, "The Self-Resonance and Self-Capacitance of Solenoid Coils", Version 0.01 (provisional), 9th May 2010.