TX to Ae Inductors Self-resonance Discharge Tubes

Inductor resonance and self-resonance experiments

(Including The Use of Low-Pressure Gas Ionisation for RF Field Visualisation).
by David W Knight

The following article describes methods for determining the resonant and self-resonant properties of coils. The associated electric field patterns are also demonstrated by using low-pressure gas discharge tubes.

The theoretical background to this material and the physical conclusions arising from it are discussed in the article: The self-resonance and self-capacitance of solenoid coils .

Shown below is apparatus for finding and measuring the resonances of coils by scattering radiation from them. It was constructed because, after much experimentation, I came to the conclusion that simple corrections for jig inductance and stray capacitance were not sufficient to permit the accurate determination of inductor self-capacitance. In other words, different jigs seemed to give different results; implying that there might be some electromagnetic effects yet to be accounted for. This also suggests that measurements obtained from diverse sources might be inconsistent; and to compound matters, the most widely used formula for self-capacitance (that of R G Medhurst) assumes that the permittivity of the coil former has no effect. That cannot be true, but it encourages experimenters to ignore the presence of dielectric material. These problems led to the idea, originally voiced as a joke, that the only way to get sensible results would be to suspend formerless coils in mid air and measure their properties without making any electrical connection.

inductor self-resonance - scattering experiment
DWK 3rd July 2013

The connectionless measurement method turns out to be straightforward because the electromagnetic field around a resonating inductor is extensive, and the E-field magnification effect is so great that the scattered signal tends to swamp the electric component of the excitation field. Complete electrical isolation is, of course, impossible; but situations involving either minimal disturbance or quantifiable disturbance are not difficult to achieve. The basic technique is that of exciting the coil using an induction loop, and sampling the field using either a small dipole or another loop.
     The amount of information that can be obtained is considerable. Note however, that the phenomena observed are not readily explicable using lumped-element theory, and the coil is more sensibly regarded as a transmission line. For any given coil; it is a straightforward matter to measure the frequencies of the transmission-line overtone resonances, and to measure the Q of each resonance from the bandwidth. If a high-power source is available, the field patterns due to EM wave propagation in the coil can be visualised using glow discharge tubes. Solid cylinders and tubes of dielectric material can be inserted into the coil to demonstrate the coil-former effect, and by changing coils, it is possible to determine the effect of shape-factor and other parameters. Also, of course, we are still at liberty to make connections; so that we can add parallel capacitance, measure voltage, etc., and otherwise see what it is that gives rise to inconsistencies in conventional impedance measurements.
     The unattenuated RF output of a signal generator (ca. +6 dBm) is suitable for field-strength measurements; but for glow discharge tube experiments, a power amplifier or a radio transmitter is required. The high-power source shown in the photograph above is a 1.6 MHz to 30 MHz radio transceiver (Kenwood TS430s modified for continuous transmitter coverage), which passes current through a 2-turn induction loop. The loop is inserted in series with a 50 Ω terminating resistance, this arrangement being used so that the transmitter operates without serious load mismatch. The load resistance is a 10 dB attenuator followed by a terminating resistor. This allows a diode detector to be connected by tapping at the attenuator output, giving a reading proportional to loop current. The receiving antenna shown is a short dipole with a ferrite-bead balun, monitored by means of a DFM and an oscilloscope.
     The coil is suspended from a length of multi-strand UHMWPE (Dyneema) fishing line, which is tensioned by means of a lead weight. Polyethylene is a non-polar dielectric, and the amount placed inside the coil by this support method is negligible. The support string passes over two SRBF (Whale Tufnol) pegs, which can be raised or lowered in 1cm increments by means of a series of holes. The antennas are mounted on clamp stands made from SRBF and Nylon and their heights are continuously adjustable. Note that non-conducting materials are used wherever possible, especially in the vicinity of the test coil, the point being to minimise eddy-currents.
     The circuit for the basic high-power experiment is shown below. This however, is only one possible configuration, and the arrangement can be altered depending on circumstances.

The simple high-pass filter shown after the pickup antenna is located in the small box shown connected to the oscilloscope Y1 input in the photograph given earlier. The interface box provides a choice between the HPF and a broadband 1:10 step-up transformer with an input impedance ( |Z| ) of about 10 Ω at radio frequencies. The HPF rejects mains hum when using a high impedance pick-up device. The step-up transformer was originally intended for use with loop sensors, but was also found to provide a useful voltage boost when using dipole E-field sensors. The transformer works well at HF radio frequencies, but becomes lossy at VHF, where the HPF is generally preferable.

Scope interface Scope interface

Oscilloscope interface box

Note incidentally that, when using a transceiver rather than a transmitter or signal generator, the SRF of a coil (and the parallel resonance frequency of an LC network) can be found initially by placing the loop near the coil under test and tuning the receiver to find the point at which received signals are strongest. This is because the system can operate as a link-coupled resonant loop-antenna. If a variable capacitor is connected across the test coil, the receiver response can, of course, be peaked by adjusting the capacitor.
     Also note that close-spaced air variable capacitors connected across the coil tend to flash over when the radio transmitter is running and resonance is found. Unless rated at several kV, solid dielectric capacitors should only be used with low-power sources.

First version of the author's scattering experiment (27th June 2007) prior to construction of the suspension frame. The test coil is supported on a plastic metre rule. The RX antenna is a loop with a 1:10 voltage transformer.

Early induction measurements

The study of coil resonance using induction and field detection is not new. In fact, a detailed investigation of the resonant properties of Tesla transformers: "Zur construction von Teslatransformatoren. Schwingungsdauer und Selbstinduction von Drahtspulen", was published by Paul Drude in 1902. This work was largely ignored by English speakers however, so that, for example, just after the end of World War II, we have Medhurst's work being interpreted to mean that the self-resonance of coils is not affected by the coil former dielectric; despite Drude's careful experimental confirmation of the opposite some 45 years earlier.
     Drude's 1902 paper has now been translated by myself and Bob Weaver (June 2015) and has the English title: "On the construction of Tesla transformers: Period of oscillation and self-inductance of the coil." It is also available from arXiv.org for those who wish to cite it.
     Drude excited his coils using a tuned induction loop energised by an induction coil, and detected resonance and various field patterns using an electrodeless sodium-vapour discharge tube. This technique might appear to have provided the basis for the glow-discharge experiments described below; except that I had no idea of the content of Drude's paper when the original investigations took place. Instead, my experiments stemmed from the amusing 'magic' trick of lighting neon bulbs and fluorescent tubes using an inductively-loaded HF mobile antenna.
     Drude investigated the relationship between resonant half-wavelength and conductor length and produced a graph of how the ratio of these two quantities varies with the coil length-to-diameter ratio (this effect being found by Drude to be first-order, whereas other effects were second-order). This is related to Medhurst's approach, although the latter resulted in a graph of self-capacitance / diameter vs. length / diameter. We nevertheless see a common effect in both cases; which is that, in addition to propagation delay represented as a capacitance, short solenoids seem to provide themselves with a static capacitance across the terminals, whereas long coils do not. Drude correctly attributed this to the proximity of the two ends.
     When Drude documented the effect of coil-former dielectric, he determined that it had greater effect on short coils than long coils. This he correctly attributed to axial field cancellation. He also observed overtone resonances and the associated nodes, and made an observation relating to the way in which nodes shift when an end electrode is attached that relates to what is now referred to as the 'ground mirror effect' (see below).
     In the theory article: The self-resonance and self-capacitance of solenoid coils, free coil resonance measurements including Drude's are compared with in-circuit (uniform current) measurements such as Medhurst's. It is thereby established that there is a fundamental difference between the two types of data due to a difference in the phase velocities for helical wave propagation in the two cases. This difference turns out to be responsible for the discrepancies between the many studies of self-capacitance and self-resonance.

Glow discharge (plasma) experiments

The glow discharges shown in the photographs to follow were all produced with a transmitter output power of about 10 W, i.e., about 0.45 A passing through the induction loop. The pictures generally show the position of the loop, to give an idea of the amount of coupling needed to get a particular configuration to light up. Sometimes it was necessary to put the loop or the discharge tube close to the coil to get ignition before adopting the arrangement seen in the picture.

Integer multiples of a half-wavelength
A coil exhibits self-resonance because a wave travelling along the helix is reflected at the impedance discontinuities that occur at the ends of the wire. Resonance occurs when the wave gets back to its starting point in phase with itself, and a corresponding standing-wave pattern develops. A very strong response is obtained when the wire-length approaches an electrical half-wavelength. This is the fundamental self-resonance frequency (SRF), generally simulated, with moderate accuracy, by representing the coil as a lumped inductance in parallel with a capacitance (the 'self-capacitance'). That this is just a representation, with little to do with the physics of the processes occurring in the coil, becomes obvious when we note that there is also a series of overtone resonances. Overtones occur whenever the wire length is an integer number of electrical half wavelengths. They are however, not in an exactly harmonic sequence because the phase velocity for wave propagation along the wire is frequency dependent (the propagation environment is dispersive).

Self-resonating inductor - node patterns
DWK 4th July 2013

The images above show a coil operating at each of the first four nλ/2 conductor-length resonances. The nodes in the E-field standing-wave pattern (of which there are n-1) are revealed by placing a clear silica-glass mercury-vapour lamp tube alongside.
The lamp used here is an Osram HNS 30W G13 2ft 1" diameter UVC germicidal tube. Note that the visible blue-cyan glow of the discharge is due to a minor spectral line. The main output is invisible and harmful to living creatures. When used near people, this type of lamp should either be fitted with a UV filter sleeve, or operatives and bystanders should wear UV-opaque protective glasses and avoid any skin exposure. The use of a Lee type 226 filter sleeve to make the germicidal lamp safe for experiments and demonstrations is discussed in a separate article. see: Glow-start Hg vapour lamps for more information.

When the length of the conductor is an odd number of electrical half-wavelengths, the coil exhibits a high impedance across its end terminals. Hence the fundamental SRF (n=1), is a parallel (voltage magnifying) resonance, and so are the overtones resonances for n = 3, 5, 7, etc.. When n is even, the coil exhibits a low impedance across its terminals, i.e., it behaves as a series resonator.
     The overall behaviour, apart from the somewhat non-harmonic series of overtone frequencies, is strongly reminiscent of the input-impedance characteristic of a length transmission-line that has been terminated in a short-circuit. The principal difference is that a short-circuited conventional line resonates when the electrical length is nλ/4, whereas the coil resonates when the wire length is nλ/2. This conundrum is resolved however, when we note that a short-circuited line of length nλ/4 is actually a hairpin loop of wire of length nλ/2. The conductor length resonance is the same in both cases, except that configuring the wire as a helix instead of a hairpin gives rise to dispersion effects. An important example of a linear single-conductor transmission-line is incidentally, the simple wire antenna, which also resonates at nλ/2.
     Note that some observations of overtone resonances were also made by Drude [1902, loc. cit.]. He regarded the coil as behaving like a pair of coils at the first overtone, the non-harmonic relationship between this resonance and the fundamental being attributed to the strong magnetic coupling between the<hr noshade="noshade">
 two half-coils. Unfortunately however, this attempt at a lumped element explanation is untenable because, in a system of n identical coupled oscillators (either classical or quantum mechanical), the degeneracy of the oscillations is lifted and we see a splitting of the resonance spectrum into n separate components. The degree of splitting is dependent on the strength of coupling, and the frequencies of the components are such as to conserve the trace (diagonal sum) of the energy matrix (according to the principle of local energy conservation) as the coupling coefficient is varied. Such splitting does not occur for solenoid overtones. The resonance instead undergoes a unilateral shift; and so the coil remains a single system, which exhibits behaviour analogous to that of a shorted transmission line in a dispersive medium. Splitting would occur however, if the two half-coils were each given a separate resonating capacitor, thus converting the system into a double-tuned transformer.
     Of practical interest in Drude's paper is the observation that if the coupling between the exciter loop and the coil is sufficiently strong, and the excitation sufficiently vigorous, then the coil wire itself will light up to show the pattern of nodes. This, of course, is due to the ionisation of the air around the coil. It suggests further experiments involving an evacuated bell jar and a low-pressure gas mixture more easily ionisable than air at atmospheric pressure [pending, see below].

The fundamental SRF
self-resonating coil with xenon tube
The high impedance of the coil at the λ/2 conductor-length resonance (the fundamental parallel resonant SRF) can be demonstrated by direct connection of a gas discharge tube. Shown above is a coil with a Xenon strobotron arc tube attached. The tube is a Maplin FS79L (discontinued) with the external trigger electrode removed. This tube has a strike voltage in the kilovolt range (just in case you were thinking of connecting expensive reference capacitors when using a radio transmitter as the source). Note that the connecting wires increase the transmission-line length and add some stray capacitance. Both of these changes reduce the SRF, but not sufficiently to confuse the resonance assignment.

Self-resonating coil with strobotron
Shown above is a coil resonating close to its parallel resonant SRF, with a Ferranti NSP2 neon strobotron tube (plugged into an octal valve base) attached .

self resonating coil with nearby neon tube
The gas discharge tube does not have to be directly connected. Here the strobotron is placed close to the end of the coil. Grounding the cathode pin causes the tube to light-up brightly around that electrode, whereas leaving the tube completely disconnected gives a more diffuse glow.

Self-resonating coil with nearby neon lamp.
Here the lamp is a 1940s vintage GEC Osglim 5W beehive neon (99% Ne, 1% Ar). An earthing wire is connected to the outer electrode.

Hg vapour tube shows axial node of helical transmission line
All helical waveguide theories agree that a wave propagates on the conductor of an infinitely-long helix with its E-field in the pitch direction (i.e., almost perpendicular to the axis). Also, the E-field is continuous across the helix wall. For a short coil, the field tilts-over at the ends, giving rise to an axial field-component (shown in later photographs) and a component of the self-capacitance; but placing a discharge tube perpendicular to the axis shows the field pattern of the dominant propagation mode. Also, we see the axial node; which arises because the phase-shift around a turn is small, so the E-field for the helical propagation process cancels at the axis.

When a good 2-stage rotary vacuum pump is available, low-pressure air can be used for field visualisation (although it requires stronger fields than the Hg-Ar mix, as can be seen from the proximity of the induction loop). Here the pressure in the tube is a little over 1 Torr (mm Hg). The tube has an outside diameter of 38 mm, and the I.D is probably about 32 mm (it is actually a chromatography column, hence the sintered-glass disk at the end with the stop-cock. The glass is borosilicate.). Note that the large diameter allows structure to be seen in the gas discharge perpendicular to the tube axis. It appears to confirm that the e-field is tilted, i.e., the low-field region at the end of the coil is funnel-shaped.

Hg vapour tube shows external E-field of resonating coil
In this photograph, the coil is shunted with an air variable capacitor, making a conventional parallel LC resonator. The Hg tube is placed parallel to the axis in the external field. The glow discharge is intense because this is where most of the energy is concentrated. Notice that there are three distinct regions in the discharge; the helical propagation region adjacent to the coil, and the two sprawling fringe-field regions. Also notice that the glow intensity is relatively uniform along the length of the coil. This is because the resonating capacitor makes the current distribution along the coil more uniform than it would be in its absence (there are current nodes at the ends when nothing is connected).
     Note incidentally, that the capacitor shown has a plate spacing of 0.5 mm and tends to flash over when resonance is obtained using a high-power source and a high Q coil. 0.5 mm air gap corresponds to about 2.4 kV peak breakdown voltage (assuming no rough edges). When the gas tube strikes, the overall Q is lowered and the arcing stops.
     Assuming lumped element theory, a parallel capacitance of 3× the self-capacitance should cause a coil to resonate at half its SRF. The SRF of this coil is 26.6 MHz, but here it is resonating at 13 MHz with a parallel capacitance of about 9 pF. This gives us a first estimate for CL at about 3 pF (my DAE formula predicts it to be 3.24 pF, and a Howe extrapolation measurement gives it as 3.2 pF).

self-resonating coil with nearby fluorescent tube
The fields around a coil can also be visualised using a conventional fluorescent tube. The results obtained with clear glass tubes are however superior, because the internal phosphor coating hides the discharge density-variation perpendicular to the tube axis. Also, although not relevant for this photograph, it reduces the intensity variation in the vicinity of nodes.

Here the 32 mm ID low-pressure air-filled tube is used to show the external field. The pressure is about 0.5 Torr. In this case, structure can be seen in the gas discharge adjacent to the coil. The pattern appears to confirm that the e-field is in the radial direction in the middle of the coil, but tilts over towards the axis at the ends. In the absence of a resonating capacitor, the current distribution is non-uniform, hence the bump in the brightness profile. This however is not complete interpretation. There is also a high voltage from end to end, and the two effects are combined. It is possible to separate the end-to-end field from the radial; field by using a linear array of neon lamps, as has been demonstrated by Alex Pettit (see KK4VB Solenoid self-resonance experiments).

Resonating coil with axial Hg vapour tube
In this photograph, the Hg tube shows the electric field along the coil axis. The receiving antenna has been changed to a loop, so that the tube can pass through. The striations in the gas discharge outside the coil are not nodes. They are visible in the plasma discharges of most gases and are sometimes affected by RF excitation. They vary with field intensity and excitation frequency and are associated with the mean-free-path of moving particles.
     The axial E-field should be zero for an infinitely long coil with conductor-length-per-turn <<λ , but arises from the loss of translation symmetry in short coils. In this case also, the translation symmetry is further disrupted by a parallel capacitor. The weaker (banded) gas discharge outside the coil is due to the capacitance of the two loop antennas.

Resonating coil with axial Hg-vapour tube
Another view of the axial-tube setup.

Above, a coil is wound directly on the 38mm OD borosilicate tube (71.8 turns of 2 mm diameter wire. Overall solenoid length = 174mm. Average coil diam. = 40 mm. Wire length = 9.024 m). Here the coil is shown with RF excitation at 19.9 MHz, which is close to its first SRF (this is a little above the free-space resonance for the straightened wire because the coil has a length/diameter ratio of 4.4, which results in a helical phase velocity >c). With a fairly high pressure in the tube (ca. 2 Torr), the discharge tends to localise and can be persuaded to occur at either end or in the middle by varying the conditions and by placing fingers near the coil to encourage the plasma to relocate. This photograph shows that the field is strongest near the helical conductor, and weak along the solenoid axis. The point is that there is an axial node for the helical wave, and the discharge in a relatively thin tube placed along the axis of a large diameter coil of finite length (previous photographs) is due to the field from end-to-end.
     This experiment, incidentally, might also be regarded as a demonstration of skin-effect in a straight cylindrical conductor. There is nothing in Heaviside's derivation to dictate how a wave has to propagate along the outer surface of a conductor (in this case a gas plasma) in order to induce a conduction current. Hence the decaying current density from surface to middle is analogous to the decay that occurs in wires operating at high frequencies. Unfortunately however, the quantitative usefulness of the analogy is weakened by the highly non-linear conduction behaviour of plasmas.

Ground-plane effect
coil with ground plane
DWK 7th July 2013
The picture above shows the setup for investigating the effect of ground-plane proximity. The coil is suspended above an anodised aluminium plate, which has various binding posts and 4 mm sockets for making electrical connection. The plate is shown earthed to the outer body of one of the BNC connectors attached to the induction loop.
     The coil is here suspended at a height that reduces its resonant frequency (26.6 MHz) by about 10 kHz. In other words, the ground plane is having practically no effect.

self-resonant coil with ground plane. Hg vapour tube shows max E-field
With the coil and the metal plate moved closer together, an Hg tube laid on the plate indicates the interaction. Here, the resonant frequency is reduced from 26.6 MHz to 25.9 MHz, so the effect is still relatively small even though the plate is less than a diameter away from the coil. The reason for the weak interaction is that the coil is not disposed in a way that allows it to induce significant eddy currents in the plate, so the effect is merely capacitive.

Hg vapour tube shows E-field between self-resonant coil and ground plane
Here the coil is placed close to the plate, and the resonant frequency is reduced from 26.6 MHz to 24.7 MHz. The Hg vapour discharge shows that the interaction is much the same as placing a small static capacitance across the coil.
    According to the lumped element theory, if a capacitance equal to the self-capacitance is placed in parallel with a coil, the resonant frequency will be reduced by a factor of 1/√2. In this case, the self capacitance is 3.2 pF, and so placing that much again in parallel should reduce the resonant frequency to about 18.8 MHz. Evidently, the ground plane is not having much effect.
     Various commentators, including Medhurst, attribute the self-capacitance of a coil mainly to the proximity of a groundplane. In fact, a groundplane presented broadside to the coil just adds a small ordinary stray capacitance. A groundplane presented perpendicular to the coil axis of course reduces the inductance by acting as a shorted-turn; but that is a well-understood magnetic effect, not a capacitive one.

Integer multiples of a quarter-wavelength - the ground mirror effect
There is, however, a very pronounced ground effect that can neither be attributed to stray capacitance nor magnetic induction. This can seem highly paradoxical when performing scattering experiments (and is inexplicable using lumped-element theory), but it makes perfect sense if we refer to it as the 'transmission-line extension effect'.
     If a wire is attached to one end of a coil; the fundamental scattering resonance frequency drops. The response also becomes less pronounced until the wire is long-enough, or any counterpoise to which the wire leads is large enough, to reduce the resonance frequency to roughly half its original value. It does not particularly matter how the auxiliary conductor is arranged (presuming that we are not trying to maximise radiation resistance), because what is happening is that its electrical length is being added to the electrical length of the coil. In antenna theory, this is known as the 'ground-plane mirror effect', i.e., adding a sufficiently-large counterpoise doubles the effective length of the antenna. This effect, incidentally, was noticed by Drude in 1902, when he added spherical electrodes to one end of an otherwise free coil. He attributed the effect to the shift in the voltage node and found that the SRF reduction could never be more than half its original value.
     The original half-wave conductor-length resonance is nevertheless still present, it is just that it can no longer be strongly excited by scattering radiation from the coil (there is no-longer a sharp impedance discontintinuity at the grounded end). To see the original resonance, it is now necessary to measure the impedance by direct connection across the two ends of the coil. There is, of course, a line-extension effect due to connecting the coil to a circuit, and stray capacitance is added, but the direct-connection effect is much less pronounced than the ground-mirror effect. Note incidentally, that because the λ/2 wire-length resonance is the one relevant to the circuit-applications of coils, it is the theory of this (λ/2) resonance that lies behind the theory of self-capacitance prediction for the purpose of lumped-element analysis.

Normal mode helical resonator with nearby Hg vapour tube. Quarter-wave resonance.
DWK 6th July 2013
If a coil is mounted above a groundplane and some top capacitance is added, it becomes a normal-mode vertical antenna, or a Tesla coil, or a helical resonator (depending on your preference). It then exhibits an nλ/4 conductor-length series of scattering resonances.
     Here the coil is energised at the λ/4 frequency (2.5 MHz) by connecting the output of a radio transmitter between the black-anodised aluminium plate on which the equipment sits and the bottom end of the coil (it is an end-fed quarter-wave vertical antenna, albeit an electrically very short one). The effective counterpoise is however, mainly provided by the mains wiring of the building and the actual earth. Impedance matching is accomplished by means of a modified MFJ989C T-network (the black box undernearh the antenna base)

Normal mode helical resonator with nearby Hg vapour tube. 3/4 wave resonance with node.
That the ground-reflection effect is operative is evident from the peculiar series of overtones. Searching for resonance at around twice the fundamental frequency yields nothing of appreciable Q, but odd multiples give a good response. Here the antenna is operating at its 3λ/4 frequency (8.4 MHz). Note that the overtones are not necessarily exact integer multiples of the fundamental, because the helical transmission line is dispersive, but the response is anyway rather broad with a gas discharge tube absorbing energy from the field.

Normal mode helical resonator with nearby Hg vapour tube. 5/4 wave resonance with nodes.
For the 5λ/4 resonance (at 12.6 MHz) an induction loop was used to energise the antenna because a good match for end-feeding using the T-network could not be found. The loop is placed halfway along the coil to avoid obscuring the node pattern (the strong magnetic field from the loop does not affect the un-ionised gas, but its proximity can give rise to capacitive currents).

Note, incidentally, since we have mentioned Tesla coils: it is not advisable to let a Tesla coil spark onto the wall of a glass low-pressure vessel (such as a gas discharge tube). In the lab where I did my PhD research, there was a hand-held mains-powered vacuum testing device of ancient origin known as 'the Tesla coil'. It was, as was said by old hands, 'good for leaks' (in the same sense that gargling with trichlorophenol is 'good for sore throats', and swallowing medicinal paraffin is 'good for vitamin deficiency'). The device was meant to (and sometimes did) find leaks in glass vacuum systems because sparks preferentially track through any cracks or holes. Just as likely however, was that it would punch micro-pores into the glassware, causing the vacuum to degrade and suspending work for days while the glass workshops re-made the ruined sections.
     The Tesla coil leak detector is used by tube makers during the pumping and sealing phase of production. It should not be used on working tubes.

Quantitative experiments

VFO sources
For quantitative measurements on coils, a high powered radio transmitter is not needed and can in fact damage components (such as capacitors) that might be connected across the coil. For measurements using an oscilloscope and a frequency counter, sources with an output of around 0 to +6 dBm into 50 Ω are adequate. A conventional RF signal generator is an obvious choice, but simple analogue VFO sources such as antenna bridges and grid-dip oscillators work very well. Some configurations that I have used successfully are shown below.

MFJ269 antenna analyser, with extended tuning control, used as +6 dBm signal source.
Antenna bridge
Here the RF source is an MFJ-269 antenna analyser, which is connected to the induction loop via a 3-stage 47.5 μH unun cable. The unun serves to maximise the coil SRF by suppressing ground currents (see below). A ferrite bead in the 12 V power cable also helps. This early model MFJ-269 is tunable from 1.7 MHz to 170 MHz by means of a tuning capacitor with a built-in slow motion drive. On this unit an acrylic extension spindle with an additional slow motion drive has been added. A fibreglass and plastic bracket anchored to the case screws fixes the stator. Isolating the control knob in this way eliminates hand-capacitance effects and facilitates accurate resonance finding.
Common mode chokes

Common-mode chokes (serving as 1:1 balun or unun transformers) are useful in suppressing the ground-plane effect caused by the proximity of induction loops and pick-up antennas. The chokes shown above were originally made for the purpose of forcing experimental RF bridges to obey the principle of reciprocity. They are made in unequal sections so that they cannot exhibit a low common-mode impedance due to self-resonance in any of the sections. They are also made deliberately different, so that when used as a pair on the input and output of a system under test, they cannot both exhibit the same resonance pattern.

Grid-dip oscillator
The signal source shown below is a 1960s vintage Heathkit GD-1U grid-dip oscillator. This is based on an EC92 / 6AB4 hard-vacuum triode and so produces a much stronger induction field than more modern FET instruments. It is therefore still one of the best-performing simple GDOs in existence. The relatively large power requirement however means that it needs a mains power supply. It is therefore necessary to minimise the ground effect by adding a large amount of common-mode choking inductance in the mains lead (an addition that is advisable regardless of the application).
     The first picture below shows the GDO fitted with the 90 MHz to 230 MHz U-shaped loop inductor. In that case the GDO is presented side-on to the coil under test. In the picture below that, the GDO is laid on its side to give best coupling when the plug-in coils for 1.8 MHz to 90 MHz are used.

Note that the tuning scales on the GD-1U are for comic relief only. Frequency is measured by using a pick-up antenna connected to a DFM. The pick-up antenna is also connected to an oscilloscope for finding coil resonances, since if the GDO is close-enough for a dip to be seen on its meter, it will also be affecting the resonant frequency.
     The coil-under-test incidentally becomes part of the oscillator tank circuit. Consequently, there is a tendency for the oscillator to lock to the resonant frequency of the coil. When sweeping the tuning dial, this gives rise to a jump in frequency and a dead-band in which the frequency does not seem to change much. The actual resonant frequency is in the middle of such regions, but best measurement accuracy is obtained by moving the GDO away from the coil until the effect is no-longer noticeable.

Extrapolated self-capacitance measurements using Howe's method
In the G W O Howe method for measuring self capacitance, a coil is resonated against a series of known capacitances placed in parallel with it, and the data are extrapolated back to zero additional capacitance. In this way, a phantom capacitance that appears to be always in parallel with the coil is found and quantified, and this is the self-capacitance. Note incidentally, that the self-capacitance obtained by extrapolation is not the same as might be predicted by calculating the coil inductance and measuring the SRF of the free coil. This discrepancy arises because a free-coil has a non-uniform current distribution, and this gives rise to a higher phase velocity for helical propagation than occurs when the coil is in circuit. The Howe self-capacitance is thus not a good predictor of the SRF of a coil; but it is nevertheless the appropriate quantity for modelling the coil when it is connected to an electrical circuit.
     Although Howe's method is simple in principle, carrying it out using conventional impedance measurement techniques is not straightforward. The problem is that an impedance measurement performed on a parallel resonator is actually a simultaneous comparative measurement of the impedances of a capacitive and an inductive arm. Thus, presuming that the Q is high enough to allow resistance to be ignored, we must determine the series parasitic inductances and parallel parasitic capacitances of both arms and make corrections to the measured resonance frequency. The model then becomes infested with parameters that are difficult to estimate. Self-capacitance measurements found in the literature are therefore often unreliable due to botched or non-existent corrections. The work of R G Medhurst, the author of the most widely-used empirical formula, is one notable exception; except that he (as mentioned earlier) assumed that the coil-former material makes no contribution to the result.
     The scattering method greatly simplifies the correction process. A VFO source is adequate, and actually easier to use than a programmable instrument such as a VNA. A VNA however allows peaks to be found very accurately (but somewhat slowly) by use of high resolution scans. A good way to proceed is to connect a socket to the ends of the coil so that pre-calibrated reference capacitors can be plugged-in. In the illustration below, it will also be seen that the socket is screwed to a non-conducting bar and clamped to a stand. This prevents the coil from swinging on the support string after a capacitor has been changed.

Coil with capacitor socket. Allows parallel connection of fixed and adjustable capacitors, small neon lamps, etc.. Also shown are an induction loop, an E-field probe with ferrite-bead balun, and some non-conducting mounting devices. The support rods are made from Whale Tufnol (phenolic-resin-bonded fabric), and the clamp boss and antenna mounts are Nylon 6-6. This is an arrangement suitable for self-capacitance measurement by the G W O Howe extrapolation method. The least-squares fitting procedure also yields an accurate value for the effective equivalent lumped inductance.

Plug-in capacitors. The fixed capacitors, mostly silvered-mica types, have all been accurately characterised by bridge measurements and have very high Q. They are mounted on plug-in headers with 1" (25.4 mm) pin spacing, all other pins having been removed.

With the coil in parallel with a fairly small capacitance, it will be found that the resonant frequency increases slightly as the induction loop and pick-up antenna are moved away. With sufficient excitation however, a point will be found at which the increase is negligible and the signal at the oscilloscope is still good. When that point is reached, the resonator is effectively un-loaded; and the apparatus is measuring the scattering resonance, rather than making a simultaneous parallel comparison of two impedances. We can therefore allocate the parasitic reactance either entirely to the attached capacitor or entirely to the inductor. The fitting of the data is pretty-much the same either way, but if we allocate all of the parasitics to the capacitor (so that the effective capacitance varies with frequency), then the fit yields the inductance and self-capacitance of the coil.
    A downside of the method is that there is no movable calibration plane for eliminating the lead parasitics. Therefore we have to indulge in the old-fashioned practice of making lead corrections. A good way to make the stray capacitance correction is to take the socket-and-lead-wire assembly away and measure it. This works because the tiny capacitance has a reactance far in excess of that of the lead partial-inductance. The setup shown has a measured capacitance (Cstray) of around 0.1 pF. Measuring the reduction in SRF resulting from the presence of the socket-and-lead assembly unfortunately isn't an alternative way of obtaining the stray capacitance, because the reduction due to the transmission-line extension effect is much greater.
     The main correction involves determining the lead partial-inductance and including it as a parasitic element in series with the added capacitance. The necessary formulae for calculation (assuming a rectangular configuration of connecting wires) are given by Rosa and Snow. Note however, that for RF measurements involving reasonably thick wires, in the calculation of wire self-inductance it is necessary to drop the contribution due to internal inductance. If we define mutual inductance as positive; the total lead partial-inductance is given as the sum of straight conductor-segment inductances, minus twice the mutual inductance between the solenoid and conductors parallel to its axis, minus twice the mutual inductance between the straight segments perpendicular to the coil axis).
     Note that it is also possible to determine the lead-inductance correction from the effect of the capacitor holder on an even-order conductor-length resonance. The transmission-line extension due to the addition of the wires is isolated because the actual terminating impedance has no effect on an even-order resonance (the first overtone resonance is the best choice). It is also possible to determine the exact electrical length each of the reference capacitors by comparing its effect on an even order resonance with that of an open or a short circuit.

Network analyser measurements
A vector network analyser can, of course, be used to make conventional impedance-related coil measurements. Such instruments however also provide a signal source and a sensitive radio receiver and are therefore ideal for locating and identifying scattering resonances.
    In the photograph below, a DG8SAQ VNWA3E USB-controlled VNA is connected to two loop antennas, with the large copper tubing coil that was used in earlier experiments in between. Note that multi-stage unun chokes are used on both sides. Scattering resonances are shifted noticeably to lower frequency when the ununs are not used; and the chokes help to suppress system resonances, i.e., artifacts that are present in the background signal when the coil-under-test is removed.

The graph above was obtained using the setup shown. Two S21 (transmission) measurements were made, one with the coil in place and one with the coil removed. 2000 points were used in each case, and so the resolution is 100 kHz. The data in each case were exported as touchstone files, but with the filename type *.csv (comma separated variables) instead of *.s1p. This allowed the files to be imported into Open Office as spreadsheets, so that the background signal could be subtracted from the signal obtained with the coil in place (both quantities in dB). Hence the graph shows how much the signal increases on sweeping through a resonance.
    Note that the undulations between the scattering-resonance peaks in the 130 MHz to 220 MHz range vary depending on the cable configuration and are therefore artifacts. They occur, even though the background signal has been subtracted, because installing the coil changes the resonances of the measuring system.
    Note also that antenna measurements can be subject to interference from radio transmitters, and from RF welding and diathermy equipment operating in the ISM bands (13.6 MHz, 27 MHz, etc.). Background subtraction can help to remove steady signals; but when a test coil is near resonance it enhances radio reception, and this can increase the strength of interfering signals relative to the background.

A little proof of concept
The photograph shows a coil wound on a polypropylene tube and placed inside a vacuum desiccator (the perforated Zinc shelf that normally sits above the desiccant has been removed). The desiccator sits on an an upturned polypropylene storage box, and a 2-turn RF induction coil has been placed below it. The excitation frequency is 17.4 MHz, which is the 1st SRF of the coil.
    The pressure inside the desiccator is about 2 Torr (2.7 mbar). At this relatively high pressure, the glow is localised but unstable. It flips to new regions as localised heating increases the breakdown field-strength of the presently ionised region. The glow is confined to locations at either end of the coil.
     As the pressure is reduced, the glow becomes more diffuse, stabilises and occupies more of the chamber.

Glow-visualised field experiments, on self-supporting coils suspended well away from the chamber walls, can be carried out by construction of an RF scattering jig inside a bell jar. The use of easily ionisable gas mixtures such as Ne-Ar would also help.
Note that large evacuated glass chambers present an implosion hazard and must be operated behind a safety shield.

Version history:
v1.05, 20th July 2016. Deleted passage saying that Annalen der Physik was free to download - sadly no longer true.
v1.04, 9th Mar 2016. Links to Alex Pettit's measurements. Small changes in interpretation due to findings in new version of Self-resonance & self capacitance theory article.
v1..03, 16th Nov 2015. - Comments on current distribution added.
v 1.02, 5th Nov. 2015. - Drude 1902 translated. Prior art section added. Comments on Drude's work added in text.
v 1.01, 2nd Feb. 2015. - VNA purchased. Section added.
v 1.00, 6th Oct. 2014.

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