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Part 2 |
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3. Inductors and Transformers: Part 3.
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Some preliminary notes on the self-capacitance and self-resonance
of coils. In 1947, R G Medhurst published a semi-empirical formula for calculation of the self-capacitance of solenoids. Expressed in SI units this is:
  F/m,
and 
is in metres . This formula is accurate to within about 4%, despite the fact that all of Medhurst's test coils were wound on polystyrene formers, and no allowance was made for the dielectric constant of the former. Medhurst used a modified version of the G W O Howe method for determining coil self-capacitance. The resonance formula is first written: (2pf0)² = 1/[ L (CL+Cref) ] and then rearranged to give the regression line: Cref = -CL + 1/ [ (2pf0)² L ] A coil is resonated at several frequencies and, on the assumption that the inductance is a constant, the regression line is extrapolated back to the y-axis to find the self capacitance CL. The self capacitance can then be used to calculate the self-resonant frequency (SRF) of the coil. One particular feature of both Howe's and Medhurst's methods, is that the actual measurements are made at frequencies much lower than the SRF, in which case a well-defined and apparently constant self-capacitance is found to exist. It is evident that Medhurst knew that a coil is a transmission line, even though he did not stress the point in his paper. He simply chose his boundary conditions wisely in recognition of a well-known curious fact, which is that when a coil is long and thin, the SRF occurs at the frequency at which the conductor length is approximately l/2. This again is true to within about 4%, and implies that the energy storage process is not as envisaged by considering static magnetic fields, but involves an EM wave trapped by reflection between the impedance discontinuities which occur at the ends of the helix. The wire is approximately l/2 long at the SRF because that corresponds to a total distance of l for a wave to arrive back at its starting point in phase with itself. The wire is not exactly l/2 long because an EM field cannot decay to zero instantaneously at the end of the conductor, i.e., there is an end effect just as there is with wire antennas; and it is reasonable to expect that the phase velocity may differ from c on account of the curvature of the helix and the electric and magnetic environment created by adjacent turns. Medhurst's data are shown plotted below with his fitting function. The formula converges with the line: CL = 4 e0 / p when the coil becomes long and thin, the latter corresponding to the case when the conductor length is l/2 at the SRF and the number of turns is relatively large. |
] Farads
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One question which is left unanswered by Medhurst's work is:
"does the dielectric constant of the coil former make any
difference to the self-capacitance?" It was with a view
to investigating this issue that the author (DWK) decided to
repeat some of Medhurst's experiments to see if there was any
difference between self-supporting coils and coils on formers.
The answer is that the difference is small, the likely reason
being that the electric field due to the travelling wave is largely
cancelled on the inside of the solenoid. A major difference only
occurs if the coil is immersed in oil or some other dielectric,
in which case the self-capacitance is increased by a factor er', this being
the dielectric constant of the surrounding medium. It was during the course of these supplementary studies that certain anomalies were noted. Instead of taking data at frequencies a long way from the SRF as Medhurst did; the decision was made to carry on taking data as close as possible to the SRF in order to obtain the best value for the self capacitance from the intercept. It was then found that the graph always deviated from linearity as the SRF was approached. The first thought was that there must be something wrong with the experimental method, particularly that there must have been an error in the calibration of the smallest-value reference capacitors and in the determination of the jig capacitance. No amount of careful re-calibration would make the problem go away however, and a radical solution was obviously needed. Eventually, it was realised that there is a way to measure the SRF of a coil without connecting it to a jig. This involves scattering radiation from the coil and picking up the scattered signal using a small antenna connected to an oscilloscope. The basic setup is shown below. |
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For this experiment, current from a radio transmitter is passed
through a small loop antenna on its way to a dummy load resistor.
In this way, the transmitter sees a reasonably good match, and
a weak radio signal is radiated when the transmitter is delivering
something in the region of 2 to 10W. A similar loop antenna just
under 1m away is connected to an oscilloscope. When a coil is
placed between the antennas and the frequency is swept, a huge
increase in the received signal is seen at the coil SRF. The remarkable outcome of this experiment is that the coils tested did not agree with Medhurst's formula when measured at the actual SRF. They only agree with Medhurst when the self-capacitance is determined by extrapolation from measurements made at frequencies well below the SRF. For the coil shown above, the true SRF occurs when the conductor length is 97% of l/2. A variety of other coils were all within 20% of l/2, the most deviant cases having closely-spaced turns. In other words, all of the test coils were found to self-resonate when the conductor is approximately l/2 long, not just long thin ones. The disparity between the Medhurst / Howe self-capacitance and the true SRF is explained by partial solutions for Maxwell's equations which show that the coil is a dispersive transmission line, i.e., the phase velocity for a wave travelling along the wire changes with frequency. Medhurst's formula relates to a region where the phase velocity is changing smoothly, and leads to an extrapolated self-capacitance which is perfectly valid for use in situations in which the lumped-inductance approximation applies, but does not predict the true SRF (see [TA3.3] ) With hindsight of course, the coil is bound to be dispersive. It is a macroscopic scattering object and must obey the Kramers-Krönig relations like everything else. The non-linearity in the self-capacitance data is due to the rising refractive index on the low-frequency side of a strong natural resonance. Coils have a variety of scattering resonances. The l/2 resonance which corresponds to the lumped component SRF is the strongest, but resonances at l/4, l, and 3l/2 are usually not difficult to detect. |
 An online coil
parameter calculation program based on Rosa and Grover's method
has been written by Serge Stroobandt, ON4AA: hamwaves.com/antennas/inductance.html
.The formulae used, except for the determination of effective current sheet diameter, are taken from a 2005 draft of this chapter. The methods available at that time remain valid within their limitations; but improvements introduced in the July 2008 revision have not, so far, been implemented. Note, in particular, that the program uses thick conductor approximations for AC resistance. Hence the Q calculation is only valid for cases where skin depth is less than the wire radius. The program accepts input data regardless of whether it meets this criterion. The method used to predict SRF and self capacitance was not taken from this work and does not accord with theory and measurements presented here. A more accurate predictor of SRF is to find the frequency at which the total wire length is equal to half the free-space wavelength (see [TA3.3] ). DWK. 20th July 2008 |
© D W Knight 2008.
David Knight asserts the right to be recognised as the author of this work.
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Part 2 |
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