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Self resonance


Solenoid self-resonance measurements by Alex Pettit, KK4VB

Alex Pettit, KK4VB, has described two series of solenoid SRF measurements made using a fixed length of wire and a near-constant diameter and turns number. In both cases, a close-spaced coil of about 30 turns and 38 mm diameter was made using a 140.2" (3.561 m) length of #13 AWG wire (d = 1.83 mm). In the first instance, a dip meter was used as the excitation source, and an oscilloscope and a frequency counter were used to find the maximum response and measure the frequency. This result was confirmed using a loop connected to an Array Solutions AIM4170 single-port impedance analyser. Thereafter, SRF measurements were made using the impedance analyser as the coil was repeatedly stretched in length and allowed to settle. Note that as the coil is stretched-out, the turns number and the diameter are reduced slightly as coil circumference is traded for axial length. Coils of less than 8.5" in length were supported on a polyethylene (PE) rod, as shown in the photographs below. In the first run, the longer coils were supported on a length of Dacron string, and in the second run they were hung vertically above the analyser coil. The first series recorded the first (parallel resonant) SRF over a range of length/diameter ratios from 1.7 to 40. The second series recorded SRF and 1st and 2nd overtones over a length/diameter range from 1.7 to 22.

MFJ dip meter, HP frequency counter and Hitachi 'scope.
Coil length = 2.45" = 62.2 mm.
Length / diam = 1.678 , SRF = 37.0 MHz.

Array Solutions AIM 4170 single-port analyser.
Coil supported by polyethylene rod..

Coil length = 6.25" = 158.8 mm.
Length / diam. = 4.28 , SRF = 57.4 MHz

Coil length = 12.75" = 323.9 mm.
Length / diam. = 8.73 , SRF = 67.8 MHz

An obvious point of interest arising from SRF measurements is to convert them into self-capacitances by calculating the coil inductance. There are however some not-widely-understood difficulties in this respect when it comes to measurements made on long coils. The first issue is that practically all of the widely-used inductance calculation programs will fail because both the Rosa-Nagaoka and the Kirchhoff-Maxwell-summation methods are only applicable to coils of small pitch angle. The fact that standard models lack helicity and therefore have no axial inductance component was first addressed by Chester Snow of the NBS (see Snow, Sci 537, 1926 and Snow, RP 479, 1932). His helical inductance model however is much more complicated than standard methods, and working long before the advent of electronic computers made it necessary for him to make some unsatisfactory approximations to turn it into a practical method. This problem however has recently been addressed by Bob Weaver, who has produced program routines that evaluate the full helical inductance Neumann integral numerically (see Numerical methods for inductance calculation, sections 2c - 2d). The resulting program has been adapted here by changing it so that the the conductor self geometric mean distance (gmd) can be controlled in the procedure call (HelicoilSG). This allows the calculation to be performed for external inductance only, so that the internal inductance at radio frequencies can be calculated separately and added to the result (see Practical functions for internal impedance). Bob has also written a front-end routine (HelicoilX) that addresses a problem of filament offset that can occur when the pitch angle is small (see the macro function library in the accompanying spreadsheet kk4vb_srf.ods). Note that the calculation is slow (ca. 20 sec per call on a fast PC) when implemented in interpreted Basic, and the spreadsheet macro results are best pasted back into their cells as numbers after calculation so that the file can be re-opened without a long delay.

Length = 292 mm
Length/diam. = 7.72
SRF = 64.3 MHz

The graph left shows the difference between a helical inductance calculation (Helicoil) and a standard calculation (Rosa) for the ≈30 turn test coils. At a pitch angle (ψ) of about 3.5° the coil inductance passes through the straight wire value, i.e., the partial inductance of the same piece of wire pulled out straight. Thereafter it falls to a minimum and then comes back to the straight wire value at ψ=90°. This behaviour is due to the fact that the mutual inductance at a given point in the coil with points carrying an opposing current is significant for large pitch angles. The effect is correctly modelled by the Helicoil program, but a standard calculation procedure such as Rosa-Nagaoka has the inductance falling to zero at the straight wire limit.

With the ability to calculate a nominal inductance, L, a nominal self-capacitance can be calculated using a rearrangement of the standard resonance formula:

CL = 1 / [ (2π f0s )2 L ]

(where f0s is the lowest parallel-resonant SRF). The result for Alex's first series is shown below.

The nominal self capacitance, calculated from the SRF using the resonance formula, is apparently constant apart from experimental uncertainty.
     The DAE formula is a re-fitted and corrected version of Medhurst's formula. It gives the self capacitance that would be obtained by the Howe extrapolation method, i.e. by measuring the coil parallel resonance frequency with several different reference capacitances connected to it and then extrapolating the results to the zero-added-capacitance intercept.

Using a conventional inductance calculation program, Alex noted that the nominal self-capacitance was approximately constant up to length/diam. ratios of about 6. Using the helical inductance program however, we see that it is actually fairly constant for all pitch angles (1.22 ±0.14 pF, the scatter having a largely random appearance suggesting that it is measurement uncertainty).
     The self-capacitance calculated using the DAE formula is shown on the graph for comparison. This formula is based on data obtained by resonating coils with reference capacitors at frequencies well below the SRF and extrapolating to find the residual capacitance. It therefore calculates the self-capacitance on the assumption that the current flowing in the coil is uniform, i.e., it is the self-capacitance that needs to be put into the coil model for circuit-simulation purposes.
     The current in an isolated coil is not uniform. At first consideration, we might assume that it falls to zero at the two ends of the coil, but this view is too simple. The field associated with a coil is inhomogeneous at the ends. This inhomogeneity gives rise to a static component of self-capacitance that partially completes the circuit. In short coils, this component is large, because the ends are close together, and so the isolated coil 'self capacitance' converges with the DAE value (see graph above). In long coils however, the static component is small and the current falls-off at the ends.
     Strictly, lumped inductance is only defined for a closed circuit. It also only exists in the presence of current. Hence when we calculate inductance of a coil from magnetic considerations (i.e., by using either Helicoil or a standard method), we do so on the assumption of a uniform current as we move along the helix. If the current is not uniform, then the inductance is reduced. Hence a somewhat inverted explanation for the divergence between the DAE and the nominal self-capacitances, which is that the inductance is falling below the uniform-current value as the coil gets longer, and so the capacitance needed to bring the system to resonance is less than in the uniform-current case.
     The problem with attributing the behaviour of the nominal self-capacitance to falling inductance however is that it does not explain the constancy. It is better to say that the concept of lumped inductance is not valid for very long isolated coils; they can only have inductance per unit length. In other words, they need to be treated as transmission lines.
     If a coil is understood to be a transmission line, then its resonances in isolation are dictated by the propagation of waves along it and the standing waves that arise by internal reflection. A short coil however, is a transmission line terminated in a capacitance of its own making, so that it can to some extent be regarded as a lumped inductance in parallel with a capacitance. The surprise here is that we have been able to extract a constant quantity. This might provide a basis for determining one or more transmission line parameters, but a derivation of the result from theory will be needed to confirm that proposition.
     Ultimately however, the nominal self capacitance cannot remain constant for a series of coils based on a fixed length of wire. The reason is that when the wire is stretched out straight, the velocity factor for wave propagation along it will be close to c, as it is for a wire antenna. For a 3.56 m length of wire, the half-wave resonance will be at about 42.1 MHz. The partial inductance for this length of 1.83 mm diameter Cu wire at that frequency is 5.67 μH, which gives the 'self capacitance' as 2.52 pF, about twice that of the helix. Note however that we cannot excite wave propagation on a straight wire by using an induction loop in a plane perpendicular to the wire. Hence, as the self-capacitance increases due to the disappearance of the conductor helicity, the response to the induction field will fade away.

In his second series of measurements, Alex recorded the self resonance frequency at the fundamental and the first and second overtones. The impedance of the coupling loop is shown below for the coil at its shortest extension (length/diam. = 1.68).

Trace from the AIM 4170 impedance analyser. Coil length = 2.5" = 63.5 mm. Length / diam = 1.68.

Self-resonance occurs because an electromagnetic wave travelling along the conductor will reflect at the impedance discontinuities that occur at the ends. A resonance corresponds to the build-up of a standing-wave pattern at a frequency at which a wave arrives back at its starting point in phase with itself. Thus we expect to find resonances at frequencies at which a wave can undergo a whole number of cycles during the course of a go-and-return trip. It follows that the fundamental SRF occurs at the frequency at which the electrical length of the wire is λ/2 (one cycle per round trip); the first overtone occurs when the electrical length is λ (2 cycles per round trip); the second overtone occurs when the electrical length is 3λ/2 (3 cycles per round trip); and so on. With this understanding, we can convert every measurement into an average phase velocity for a wave travelling along the helix.

When an electromagnetic wave travels through free space, the relationship between frequency and wavelength is given by:

c = f λ0

Where λ0 is the free-space wavelength, and the speed of light c = 299 792 458 m/s. When the propagation environment is occupied by physical objects however, the wave undergoes scattering (absorption and re-emission with a change of phase). In this case, a new resultant wave is formed from the superposition (addition) of incident and scattered waves. Through the accumulation of phase shifts, the resultant wave will generally appear to have a velocity that differs from c, and so we write:

v = f λ

where v is the 'phase velocity'. The quantity v/c is, of course, the velocity factor (and its reciprocal is the refractive index) of the medium or environment in which propagation is taking place.
     The phase velocity can be, and often is, faster than the speed of light. It is however, only an apparent velocity. Information cannot travel faster than c because no scattered wave can outrun the incident wave that initiates it (a restriction that lies behind the principle of causality).
     To find the average velocity factor for propagation along the coil wire, we divide the apparent wavelength λ (calculated from the resonant frequency) by the actual distance travelled during a complete cycle. For the fundamental resonance, that distance is twice the conductor length. For the 1st overtone it is equal to the conductor length; and for the 2nd overtone it is 2/3 of the conductor length (etc.). The results for Alex's data are shown below as a function of the coil length / diameter ratio.

Slow-wave effect.
For very long coils there are two principal modes of electromagnetic propagation along the coil, the helical wave and the axial wave. This behaviour is described to a fair approximation by a model known as the 'Ollendorff sheet-helix', which was the the theoretical basis for the development of the travelling wave tube (see refs on the self-resonance page, especially Pierce 1950, Sichak 1954, and papers by the Corum brothers). The two waves remain in lock-step so that the helical wave has a phase velocity >>c while the axial wave (the slow wave) has a phase velocity <<c.

It must be stressed here that to make measurements on very long coils is extremely difficult because the dimensions are so easily disturbed. Coils of length/diameter ratio greater than 5 were also suspended vertically, which means that the pitch angle will not have been completely uniform from top to bottom. For that reason there is considerable scatter in the data for large length/diameter ratios, but the major trend is obvious enough.
     For the solenoid geometries that work best for making high-Q radio inductors, the average velocity factor for helical propagation is around unity. As the coil is stretched-out however, the helical phase velocity becomes much faster than the speed of light. This is known as the 'slow-wave effect'. We can explain it by resolving the propagation process along the coil into two waves in superposition. One wave follows the helix, and the other, the slow wave, travels in the axial direction. Coupling between the two depends on the coil length, and so as the coil is stretched out, the axial wave component becomes more significant. The phase velocity overall is a compromise between the phase velocities of the two partial modes. If the axial wave were to have vax=c, then the helical wave would have to have vhx>>c in order to keep up. What happens instead is that as the coupling between the two modes increases, the phase velocity of the axial wave decreases while the phase velocity of the helical wave increases. The ability to produce an axial slow-wave in this way is exploited in the travelling-wave tube (TWT) amplifier, where coupling between the slow-wave and an axial electron beam is used to produce gain in signals injected at one end of the helix and extracted at the other. The theoretical model used in the development of the TWT is known as the 'Ollendorff sheet-helix'. This model makes use of a type of current-sheet constrained to conduct only in the helical direction. It is somewhat approximate and does not necessarily show the correct limiting behaviour. Practical results however suggest that the helical phase velocity has either a limiting value or a broad peak in the vicinity of 2c. For this constant conductor-length experiment however, it will collapse back towards c as the wire is drawn out straight and helicity disappears.
     The scatter in the data makes it difficult to see whether or not the helical phase velocity has reached a peak value for large length/diameter; but it is possible that it has. A further point is that a helical transmission line is dispersive, which means that its velocity factor varies with frequency, which is why we see separate curves for the overtone series. They should converge (at least approximately) at the straight-wire limit.

Comparison with Drude's data
In 1902, a study of coil self-resonance was published by the German physicist Paul Drude. An English translation of this article has been produced by DWK and Bob Weaver (see 'On the construction of Tesla transformers . . . '). Drude made a large number of SRF measurements, using a resonant induction loop energised by means of an induction coil with a Tesla transformer and a spark gap. Resonance of the test coil was detected by placing a low-pressure sodium-vapour discharge tube close to it, and wavelength calibration to better than 1% was obtained by detecting standing waves on a resonant parallel-wire transmission line. Unfortunately, most of Drude's coils were wound on wooden or ebonite (hard rubber) cylinders, and these materials have fallen completely out of use because of their appalling dielectric losses. He did however, make some measurements on coils with air cores, and these are still of interest.
     On pages 319-320 of his paper, Drude describes winding coils of cotton-insulated copper wire onto solid formers and then removing them carefully and binding the turns together using three pieces of twine, so as to restore a good cylindrical shape. These coils were then suspended from a cotton thread during induction measurements, specifically because it was found that placing them on wood, ebonite or glass supports increased the resonance period (i.e., lowered the frequency). Three series were recorded, for pitch / wire diameter ratios of 1.09, 1.24 and 2.4, over a range of coil length / diameter ratios from 0.04 to 6 (see table on p 322). In each case, Drude recorded the ratio of the self-resonant half-wavelength to the wire length (he gives this quantity the symbol f ). This corresponds to the average refractive index for helical propagation ( f = nhx ), which is simply the reciprocal of the nowadays more familiar velocity factor.
     Unfortunately the paper contains numerous mistakes and signs of rushed compilation, and so the annotation at the head of the 'without core' columns on p322 suggests that the turns were bound closely together in all three cases. If that was so, we would expect the resonances of all three series to be reduced in frequency by the cotton wire insulation, not because of inter-turn capacitance (which is negligible for small turn-to-turn phase shifts), but because the effect is that of embedding the whole coil in dielectric material. While close binding is plausible for the p/d = 1.09 and 1.24 series however, it does not make sense for the series with p/d = 2.4. In that latter case, it seems more likely that the twine was crossed-over between turns, in which case the gap between turns would have been large relative to the insulation thickness and effect of dielectric would be expected to be small. To test that proposition, four additional air coil measurements using a 2-port VNA, two small antennas, and a suspension jig (see self-resonance experiments) were made for length / diameter ratios of 0.9, 1.2, 1.6 and 1.8. The comparison coils were all made from stiff uninsulated wire or tubing, and their nhx measurements were all found to fall on or extremely close to Drude's curve for p/d=2.4. Thus we conclude that there was almost certainly a large inter-turn air gap for Drude's wide-pitch series.
     Expressed as helical velocity factor (vhx = 1/ f ); the data from Drude's table, the four supplementary measurements (DWK), and Alex's data are shown below (click on the graph to expand it in a new window).

As might be expected, Drude's measurements on coils with a narrow inter-turn gap fall below the measurements for p/d=2.4. This is almost certainly due to the effect of the wire insulation. Also, we see that Alex's measurements made with coils supported on a polyethylene rod fall below the p/d=2.4 measurements; particularly in the case of short coils, but getting less so as the coils become longer. This behaviour is due to axial electric-field cancellation. Essentially, when there is a relatively small phase shift in going from one turn to the next, and given that the helical wave propagates with its electric field in the pitch direction, we expect the electric fields from opposite sides of the coil to be very nearly equal and opposite at the coil axis. This cancellation effect increases with increasing length/diameter ratio, to the extent that the effect of internal dielectric dies out almost completely in very long coils. Incidentally, it is found that extrapolation measurements for the self-capacitance of coils carrying uniform current show comparable behaviour, i.e., the effect of coil former dielectric is large in short coils but small in long coils (see Self-resonance and self-capacitance of solenoids).
     So now, for the strictly comparable air coil measurements, we have Drude's p/d=2.4 series, the DWK spot measurements, and the measurements made by Alex on coils without any significant amount of internal dielectric. What we then find is that Alex's measurements made with the coil suspended horizontally on a Dacron string are contiguous with Drude's measurements, whereas the measurements made with the coils in vertical suspension give the impression of being displaced. The obvious inference is that there is a systematic error in the vertical measurement case, but it is not good practice to disregard data without a sound scientific reason. One issue that might affect the vertical measurements however, is that it is not possible to obtain a uniform pitch distance. The reason for that is that the weight of the coil is in balance with its own elasticity, and the weight suspended at a given point decreases as we move from top to bottom. Drude (1902) found pronounced effects arising from uneven pitch (p 310-311), although his specific observations are not particularly relevant here.
     The graph below shows only the air-cored and air-spaced measurements of guaranteed near-uniform pitch.

Drude was a practitioner of smoothing, not in the modern sense of fitting the points to a cubic or higher order spline, but the dark art of sketching a curve among the scattered points and reading the smoothed values back from the graph paper. Hence there are at least two kinks in the curve he produced, obvious artifacts, and the DWK measurements (over a range for which self-supporting coils are easy to make) seem to bridge across one of the kinks. Alex's horizontal suspension measurements on the other hand, start where Drude left off and provide an obvious continuation. They do of course exhibit considerable experimental scatter; but to practice Drude's method of smoothing on them would take us into the realm of conjecture. In summary however, we can say with some confidence that the overall velocity-factor profile is a sigmoid, with a possible flattening of in the region of v/c=2 for long coils. This is broadly consistent with slow-wave theory for helices of constant diameter when the conductor length is allowed to increase as the coil is elongated. Coils of fixed conductor length can be expected to exhibit similar behaviour until the diameter shrinks to the point at which a helical wave can no-longer be induced by a loop perpendicular to the axis.

Neon lamp experiments
One of the shortcomings of the gas discharge tube method for visualising the electric field around a solenoid (see Inductor resonance experiments) is that a long tube cannot easily distinguish between the axial and radial field components . Alex has come up with an interesting solution to that problem, which is to visualise the fields using a linear array of neon bulbs. The setup is shown below.

The neon lamps are type NE2, hot glued to a plastic rod, with no electrical connection. Due to variations in the strike voltage between different bulbs, it was necessary to replace some bulbs on test in order to obtain consistent sensitivity.

(click on the image to enlarge)

The photograph below on the left shows the fundamantal-resonant voltage maxima in the radial field at the two ends of a coil, without interference from the overall axial field. The photograph on the right shows the end fringe-field and the axial node that results from internal radial-field cancellation.

The coil is 80 turns closewound on a phenolic former, 2.3" (58.4 mm) in diameter and 2.9" (73.7 mm) long. The inductance is about 215 μH and the SRF is 7.2 MHz.

kk4vb_srf.ods (open-document spreadsheet) shows the calculations and contains the Basic macro code used (opens with Open Office).

On the construction of Tesla Transformers, P Drude 1902 (English translation) gives Drude's discussion and measurements.

Inductor resonance and self-resonance experiments.

Version 1.02, 2016-03-08

© D W Knight 2016, © Alex Pettit Jr. 2015, 2016.

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Self resonance