 TX to Ae Magnetics Self-resonance -

Capacitor standardisation using a reference inductor.
David W Knight

 1. Introduction. 2. Background. 3. Initial capacitor calibration. >>> Not finished.

 Abstract: A solenoid wound with thin wire and having a large pitch / wire-diameter ratio can be used as a radio frequency reference inductor, provided that the operating frequency range is such that the length of the winding wire never exceeds 60 electrical degrees. When the wire diameter is small relative to the solenoid diameter, and the spacing between turns is relatively large, dispersion due to the proximity effect is negligible. Dispersion due to the skin effect is however significant for thin wire, but this can be modelled accurately. Hence, presuming that there is no magnetic core, that losses in the dielectric of the coil-former are small, and that the dispersion region around the SRF is avoided; it becomes possible to correct for all known non-idealities of the inductor to an accuracy of better than 1 part in 104.      A quasi-ideal lumped-element inductor, realised by applying mathematical corrections to the behaviour of an actual inductor, can be used to standardise small capacitors to a precision of a few femto Farads. A method is demonstrated whereby the uncertainties in a set of reference capacitors are aggregated in a manner which produces new nominal values of greatly increased precision. Background to the statistical method used in this article is given in the article 'Data Analysis'.

 1. Introduction: If a measurement procedure produces a result in such a way that the statistical noise is apparent (i.e., if there is some variability in the result beyond a certain number of significant digits), then it is possible to improve the precision of the measurement by repeating it several times and taking the average. If the number of measurements is n, then the standard deviation of the average is given by the standard deviation of a single measurement divided by √n.      When the same measurement is repeated several times, the individual results can be regarded as samples taken from a single parent population. The averaging process improves the precision because the sample mean tends towards the parent mean as the number of samples increases.      It is possible to treat samples from different parent populations as though they belong to the same parent population, provided that some rule which relates the various populations is known. Such is the case when data are subjected to a weighted regression procedure, the weighting factor serving to scale the individual error distributions so that they become identical. In this way, the statistical advantage which accrues from repeating a single measurement can be obtained from a set of disparate measurements, provided that the scaling rule can be accurately defined.      It is well known that the inductance and the so-called 'self-capacitance' of an inductor can be determined by measuring the parallel resonant frequency against a series of reference capacitances. For such an experiment, to a degree of approximation to be discussed in detail shortly, the resonant frequency is given by the expression: f0 = 1 / { 2π√[L (C0 + Cref)] } Where Cref is the applied reference capacitance, and C0 is the sum of the strays and the self-capacitance of the coil. After rearrangement, the resonance formula becomes: Cref = -C0 + 1/[ (2πf0)² L ] This is a straight-line graph of the form y=a+bx, where the independent variable x might be chosen as (say) 1/(2πf0)², to find the gradient b=1/L and intercept a=-C0.      The problem with the linear regression procedure for finding L and C0 however, is that it only gives a statistically convincing outcome when the accuracy of measurement is relatively poor. If we assume that all of the experimental error is confined to the determination of the reference capacitances; a satisfactory outcome of a reduced χ² test applied to the data will typically only be obtained if the uncertainty in the capacitance measurements is of the order of several percent. An improvement in accuracy will show that the straight-line graph is not straight after all, the discrepancy being due to the assumption that L and C0 are constants.      An inductor operating at radio frequencies can only be fully understood as a transmission-line device. It stores energy by virtue of the fact that it takes a finite time for electromagnetic radiation to travel the length of the winding wire. This is a very different picture to that given by lumped-component theory, which is based on the conception that the dimensions of electrical devices are negligible in comparison to the wavelength. That the inductor, in particular of all circuit components, does not fit into the latter regime should be obvious; and so self-capacitance is primarily a model parameter, which serves to extend lumped element theory to moderately high frequencies; but does not wholly exist for static electric fields, and does not necessarily predict the self-resonant frequency (SRF).      Hence the first requirement, prior to proposing that an inductor can be used to provide a scaling rule for the comparison of capacitors of differing values, is to establish the extent to which the lumped component model is valid. Part of the answer to this question lies in the solution of Maxwell's equations for the Ollendorf sheath-helix model, which has received much discussion over the years (see, for example ,  and references for Inductor self-resonance . . . ). The theory shows that the apparent velocity for electromagnetic propagation along the helix varies with frequency; and that it does so in a particularly non-linear manner around the SRF. The lumped component model is only accurate insofar as the apparent inductance vs. frequency curve that it produces corresponds to that of a dispersive short-circuited transmission line. This observation might seem to put an end to the idea of making use of the 'linear' relationship mentioned above, except for an ideosyncracy of the mathematics. The impedance-related properties of cylindrical objects are given by combinations of Bessel functions (also known as 'cylinder functions'). The resulting mathematical description is that of systems for which changes of behaviour can have a relatively rapid onset. For an inductor, it transpires that the lumped component model is accurate over a large interval and then suddenly breaks down. Theory and data indicate that the representation of a solenoid as a lumped inductance in parallel with a lumped capacitance works until the length of the conductor exceeds about 60 electrical degrees.      Further difficulties exist however, the problem being that the idealised sheath helix on which the transmission-line theory is based is not physically realisable. Thus, presuming that we have the good sense to eliminate permeability dispersion by not using a magnetic core, and that we eliminate permittivity dispersion by using a low-loss coil-former; we must still take into account two remaining dispersion phenomena, these being the skin effect and the proximity effect.      Most people interested in electromagnetism are aware of the proximity effect as a frequency-dependent increase in the electrical resistance of coils beyond that due to the skin effect. The principle of causality demands however, that an increase in losses is always followed by a reduction of reactance; which means that the proximity effect must be associated with inductance variation. It is possible to rationalise this decline in inductance as a reduction in the effective diameter of the solenoid; there being a tendency, in wire of finite diameter, for the current to crowd in the region closest to the coil axis at high frequencies. It follows that the amount of variation can be controlled by choosing a wire diameter which is small relative to the diameter of the solenoid. Thin wire, of course, is not optimal when designing inductors for high Q, but the issue here is to obtain constant inductance. For the standardisation technique to be described; a solenoid of 48.7mm diameter wound with 0.15mm diameter wire was used. This limits the inductance variation due to the proximity effect to a factor of: (48.7 - 0.15) / 48.7 = 0.9969 i.e., the maximum possible variation, neglecting end effects which will limit it further, is about 3 parts in 1000 (0.3%).      Although a constancy of external inductance to within 0.3% is good, it is still not good enough for the purposes of this study. Hence some additional measure which precludes the maximum variation from occurring is required. The solution is to use a large winding-pitch to wire-diameter ratio; i.e., to minimise the proximity effect by maintaining a relatively large distance between adjacent turns.      An empirical study of the proximity effect in solenoids was carried out in 1947 by R G Medhurst [Medhurst 1947]. Results were given in the form of a table of proximity factors, in the high-frequency limiting case, for solenoids of varying length / diameter and pitch / wire-diameter ratio ( /D and p/d). Shown below is a graph of the data for the case when /D = 0.8, this geometry being close to that of the coils to be discussed. The proximity factor, Ψ, is defined as the amount by which the AC resistance of the wire is increased by winding it into a coil. 2. Background: The experiment to be described was devised in order to exploit an opportunity created by the purchase of about 100 close-tolerance capacitors. The point is to make use of the fact that a set of n electrical components each having a known relative uncertainty δ in its nominal value posesses a collective uncertainty δ/√n; provided that the errors are normally distributed, and provided that it is possible to scale the individual measurements in such a way as to normalise their means and standard deviations.      The capacitors were obtained for the purpose of making plug-in reference capacitances, to be used primarily for the determination of inductor self-capacitance by the linear regression method described earlier. The reason for adopting a set of fixed capacitors, rather than using a variable reference, was that writing down the value of a previously measured capacitor would be much quicker than an earlier used method; which involved adjusting a trimmer capacitor and then unplugging it and transferring it to a measuring bridge. The fixed capacitors moreover, all being physically small and of similar dimensions, have parasitic inductance which is minimal, relatively easy to calculate, and fairly consistent across the whole set.      Since the capacitors were to be used in the production of data to be fitted to a regression line; it seemed natural to use the same regression procedure to standardise them. That this could be done (and perhaps had to be done) became obvious after performing a few self-capacitance determinations using their initial calibration values.      When evaluating a linear regression procedure, there is usually little point in plotting a graph of the regression line itself. It will look like a straight line wilth data points adhering closely to it, and little insight will be gained. Far more interesting is the graph of residuals (nominal value minus calculated value), which shows the experimental noise. The reduced χ² test, which tells whether the linear relationship is plausible, is well and good; but it is the overall appearance of the noise graph which tells the experimenter whether of not the scatter is truly random. Any curvature due to systematic error is usually immediately obvious; but for the experiments in question, it was the appearance of pseudo-randomness which attracted attention. Apart from known sources of curvature, such as internal inductance and velocity dispersion; the graphs produced did appear to be random when considered individually. When compared however, they all looked similar; i.e., the same 'random' pattern was being produced by every experiment. Hence the graphs were not showing experimental noise, but a set of points corresponding to the deviation of the known value of a given reference capacitor from the far better estimate given by combining the uncertainties of of all of the capacitors.      Thus it was discovered that the somewhat high level of experimental noise associated with using variable reference capacitors had been suppressed, only to show up the calibration errors in the fixed capacitors. This led to the idea of recalibrating the fixed capacitors by comparing each one against the whole set, and thereby refining the experimental procedure. To do that however, would require a quasi-perfect inductor for reference. Hence the deliberations relating to the design of RF reference inductors, as outlined in the previous section.

 4. Apparatus. After the initial calibration, the capacitors were used to make a number of self-capacitance determinations on small toroidal inductors [see 'Self-capacitance of toroidal inductors']. It was during the course of that work that the pseudo-random pattern of residuals was noticed; superimposed on a background of minor non-linearities attributable primarily to the dispersive nature of the magnetic core material, skin and proximity effects, and the transition to the limiting helical phase velocity on the approach to the SRF. Thus, although it was evident that the capacitor calibration could be improved, it was also recognised that it would not be possible to do so on the basis of the data then available.      Hence a solenoid inductor, of thin wire, with wide spacing between the turns, was wound on a ribbed and grooved ceramic coil-former of World War II vintage made by the Stupakoff Ceramic & Manufacturing company of Pennsylvania. This was mounted on plastic stand-off pillars, 10cm above a wooden bench and connected to the test jig used for the toroidal inductor measurements. The connection was by means of a pair of 10cm long, 1.6mm diameter copper wires.      Shown below is the coil-former fully populated with 15 turns of wire. . spreadsheet: refcoil_1.ods . spreadsheet: refcoil_Cstd.ods . Correlation. The error in a calibrated value is always correlated with the error in the standard against which the calibration is made. Hence, in a mutual calibration of this type, although the accuracy is greatly increased, there will always be some correleation between the uncertainties in the various capacitor values.

References
  "Numerical calculations of internal impedance of solid and tubular cylindrical conductors under large parameters" W. Mingli and F. Yu (Northern Jiaotong University, School of Electrical Engineering, Beijing, China). IEE Proceedings, Generation, Transmission and Distribution. January 2004, Vol 151, Issue 1, p. 67-72.  "Theory of the Beam-Type Traveling-Wave Tube". J R Pierce. Proc. IRE. Feb. 1947. p111-123. See Appendix B, p121-123, "Propagation of a wave along a helix", which gives Schelkunoff's derivation of propagation parameters for the Ollendorf sheath-helix.  "Coaxial Line with Helical Inner Conductor". W Sichak. Proc. IRE. Aug. 1954. p1315-1319. Correction Feb. 1955, p148.  Radio Frequency Transistors, Norm Dye and Helge Granberg. Motorola inc. / Butterworth Heinemann, Newton MA. 1993. ISBN 0-7506-9059-3 Line-length resonance: p142. TX to Ae Magnetics Self-resonance -