TX to Ae

Magnetics

Self resonance

-


The self-resonance and self-capacitance of solenoid coils
by David Knight
Additional comments and further work

Since writing the article and publishing it on ResearchGate, a number of points have emerged from re-reading and correspondence. I also have some new ideas for measurements and test-piece 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 square-root bracket is not correct; but the following function, using the argument x as defined in the article, has the correct form and limits:

kw = √[ K1(x) I1(x) K0(x) I0(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 k1 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).


Non-uniform current distribution
Bart McGuyer's paper on nonreciprocal mutual inductance in systems having non-uniform 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


Modern coil measurements
The main point of the article was to take all of the solenoid coil resonance and self-capacitance 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 self-resonance and self-capacitance calculation would be greatly assisted by some new measurements. In particular, a series of high-precision air coil self-capacitance measurements would be more convincing than than using Medhurst's data corrected for coil-former dielectric.
     The problem is that this is difficult to do. Making self-supporting 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 accurately-spaced 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 PTFE-cored 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 coil-former 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 coil-former. 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 self-guiding 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 long-hole boring method opens the possibility of making threaded coil formers with close-fitting internal sleeves and rods. This is what is needed in order to produce data on the effect of varying coil-former 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[2], the dielectric constant of Ni-Zn ferrite material is in the 10 to 40 range at radio frequencies, but the figure for Mn-Zn materials is much higher.
     Resonance and propagation delay measurements using Ni-Zn 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.

[2] Soft Ferrites, properties and applications, E C Snelling. 2nd ed. Butterworth 1988.
ISBN 0-408-02760-6. See p127-129


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 conductively-separate 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.

DWK


TX to Ae

Magnetics

Self resonance

-