



Since writing the article and publishing it on ResearchGate,
a number of points have emerged from rereading and correspondence.
I also have some new ideas for measurements and testpiece fabrication.
Should another article be forthcoming, some of those issues might
be addressed; but until that happens, the following comments
might be of interest Too long for a physics paper Several people have pointed out that there needs to be a drastically cut version containing only the new results. This matter is under consideration but will not happen immediately. If I do produce such an article, I would like to do some more experiments and attend to the issues outlined below: Free coil SRF calculation: Weighted average of internal and external permittivities In section 11 of the article, in the subsection on coil former dielectric; a weighting function is used to give the effective average permittivity (equation 11.18). This uses Nagaoka's coefficient, unrigorously, for want of any better idea at the time. It might be possible to devise a better weighting function by noting the difference between the results obtained for the characteristic resistance of the free helix and a helix carrying uniform current. Compare equations (11.8) and (11.8a). On going from the uniform current to the free helix case, Nagaoka's coefficient is replaced by a combination of modified Bessel functions. The factor of 2 inside the squareroot bracket is not correct; but the following function, using the argument x as defined in the article, has the correct form and limits: k_{w} = √[ K_{1}(x) I_{1}(x) K_{0}(x) I_{0}(x) ] Free coil SRF calculation: Long coil correction for coils on formers Given the limited amount of data, it is not possible to be sure, but it appears that the long coil correction might need to be increased as the coil former dielectric constant is increased. The required adjustment applies only to the region in which the Gompertz function is having an effect, and could be implemented by increasing the k_{1} coefficient. Does the Gompertz function have a physical basis? It causes the curve to veer in a manner characteristic of combinations of Bessel functions. The Ollendorff model is obviously a simplification and a more complicated function is indicated. I have not yet tried to calculate the velocity factor using the Sensiper tape helix model. (The characteristic resistance and axial field functions have little effect on the behaviour of long thin coils and so are not responsible for this particular kink in the curve). Dielectric constant of ebonite On page 93, I used the dielectric constant of ebonite as 2.8, based on information from Kaye & Laby. When I was writing that part however, I was pretty sure that Drude had given it somewhere in his Tesla Transformatoren paper; but I couldn't find it at the time. Actually, it is on p41 of the translation, footnote 5913. Drude's measurement was 2.79. Nonuniform current distribution Bart McGuyer's paper on nonreciprocal mutual inductance in systems having nonuniform current distribution^{[1]} was not not known to me prior to writing the article. It is supportive of the conclusions reached here, and contains interesting quantitative theory and additional discussion. Failure to cite it was an unfortunate omission [1] Paul Drude's Prediction of Nonreciprocal Mutual Inductance for Tesla Transformers, Bart McGuyer, 2014, PLoS ONE. 9(12):e115397. doi: 10.1371/journal.pone.0115397 Transmission Line models A simple transmission line approach to coil resonance was used by J M Miller and first reported in the Bulletin of the Bureau of Standards ^{[2][3]}. [2] Electrical oscillations in antennas and inductance coils, J M Miller, BBS 14, 1918, p677696. [3] Electrical oscillations in antennas and inductance coils, J M Miller, Proc. IRE 7(3), 1919, p299326. Modern coil measurements The main point of the article was to take all of the solenoid coil resonance and selfcapacitance measurements made over the preceding 114 years and explain them. That was done, but it might be said that the development of empirical methods for selfresonance and selfcapacitance calculation would be greatly assisted by some new measurements. In particular, a series of highprecision air coil selfcapacitance measurements would be more convincing than than using Medhurst's data corrected for coilformer dielectric. The problem is that this is difficult to do. Making selfsupporting coils with length / diameter ratios from about 0.5 to 2 is achievable, but outside that range the results are generally poor. This means that dielectric supports are required, and the problem is that of how to minimise the amount of material used and correct for the systematic errors that will result. A possible solution is to use strips of thin polystyrene sheet. Thin strips of brittle plastic can be made from sheet by scoring with a modelling knife and snapping with a box bender. A pile of strips can then be bolted together and drilled with a series of accuratelyspaced holes by using the feed screw of a milling machine for measurement. Using three or four of these strips to maintain the shape of a wire helix, it should be possible to occupy less than 1% of the coil circumference with plastic material. 
An interesting approach to boring The PTFEcored coil provided by Dr Duncan Cadd (G0UTY) came with an unfinished 35 mm diameter hole bored into one end. In order to make measurements that could be compared against tubular and solid coilformer models, it was necessary to complete this hole; but with an otherwise finished item, this was not an easy problem. The length of cylinder unoccupied by conductor was not sufficient for mounting in a lathe chuck, and drills and boring bars of sufficient length were neither available nor likely to give an acceptable result. The solution was to use a 35 mm diameter Forstner woodworking bit mounted in a long extension bar; the latter being made on the lathe to give an accurate fit to the shaft. The coil was placed on the table of a pillar drill, prevented from rotating by hand (wearing a cotton glove to prevent discolouration of the copper), and the hole was gradually extended, with withdrawal of the tool after every 3 to 4 mm to remove the swarf. A flat piece of wood was placed under the coil for the final breakthrough cut. The Forstner bit is guided by a central point, and remarkably, this kept the bore true for the entire length of the coilformer. Consequently, with a final diameter of about 35.3 mm, it was possible to insert a PTFE rod of 35.0 mm diameter without further machining or adjustment. The upshot is that Forstner bits, which are available in a wide variety of sizes, are suitable for making long bores in plastic material, and have a selfguiding facility. In the example shown, the work could not be mounted in a lathe chuck, but there is no reason why the bit and extension bar cannot be mounted in a lathe tailstock. 

Effective permittivity of a hollow coil former The availability of a good longhole boring method opens the possibility of making threaded coil formers with closefitting internal sleeves and rods. This is what is needed in order to produce data on the effect of varying coilformer wall thickness. A good material for making the outer tube seems to be polypropylene; which is cheap, machinable and has a dielectric constant of 2.1 with a low tanδ. The dielectric constant is also the same as that of paraffin wax (same alkane structure, shorter chain length), which can therefore be used for the inner slugs and sleeves. 
Ferrite rods The scattering jig lends itself perfectly well to the study of loopstick coils. Some preliminary measurements were made, but not a comprehensive study. This matter is of considerable interest because it relates to the problem of wave propagation in proximity to materials of extremely high permeability and permittivity. According to Snelling^{[4]}, the dielectric constant of NiZn ferrite material is in the 10 to 40 range at radio frequencies, but the figure for MnZn materials could be much higher. For such materials, dimensional resonance effects (due to the extremely low phase velocity for waves propagating in the ferrite) complicate matters greatly^{[5]} ^{[6]}. Resonance and propagation delay measurements using NiZn materials suggest that the phase velocity of the travelling wave on the helix is not greatly affected by the enormous refractive index of the ferrite. Around 0.85c is fairly typical. [4] Soft Ferrites, properties and applications, E C Snelling. 2^{nd} ed. Butterworth 1988. ISBN 0408027606. See p127129 [5] Determination of MnZn ferrite's dimensionindependent complex permeability and permittivity, Ruifeng Huang, 2008 Doctoral Thesis, Nanyang Technical Univ., School of Elec. Eng.. (Open Access: repository.ntu.edu.sg/handle/10356/13336) [6] Determination of dimensionindependent magnetic and dielectric properties of MnZn ferrite cores and its EMI applications, R Huang, D Zhang, KJ Tseng, IEEE Trans. EMC, 50(3), Aug 2008, p597602. 
Coupled resonators Drude, in his 1902 Tesla transformer paper, suggested that the overtone resonances of solenoids could be understood by imagining two separate coils magnetically coupled. This is wrong because there is no physical break between the supposedly conductivelyseparate sections. A current node is not the same as a physical disconnection (current waves still pass over the node, it is just that their superposition adds to zero). Two separate coils will show two separate resonances. If the coils are identical, the degenaracy of the resonace frequencies will be lifted depending on the degree of coupling, i.e., the peak will split into two peaks as the coils get closer together. It should be possible to demonstrate this effect on the scattering jig; and also to show that the double resonance disappears when the two coils make contact. The phenomenon of degeneracy lifting in identical coupled resonators is intriguingly analogous to the phenomenon of quantummechanical tunelling between the left and right handed forms (enantiomers) of molecules having a finite inversion barrier (see the author's PhD thesis). 
DWK LU 20161221 



