A method for precision measurements of resonant frequency, Q factor, input impedance, energy storage inductance, and coupling coefficients.
When a solenoid is struck into resonance by means of a sharp transient, the base current contains the full spectrum of mode frequencies, with each component undergoing an exponential ringdown according to its own particular Q factor. The base current waveform can be captured by a digital scope and the resulting data can be analysed by software to extract the mode frequencies and their Q factors.
With reasonably clean waveforms captured by the scope, a repeatable accuracy of 0.01% on frequency and 0.1% on Q factor can be readily achieved. This method has the following advantages over more traditional methods of Q measurement:
The remainder of this document describes the measurement circuit and procedure, the analysis software, and we show examples of some of the applications for this technique.
An effective test setup, which we call a pinger, was devised by Terry Fritz and is shown below.
C around 27nF, R around 10Megohm. Supply circa 2.5 kV, Victoreen gap, Pearson CT.
Both the capacitance C, and the coil's DC capacitance Cdc, are charged up towards the supply voltage V via the resistance R. At some point the spark gap fires, thus placing a negative going step voltage transient onto the coil base, and the coil is excited with a total energy approaching 0.5 * Cdc * V2. The reservoir capacitance C provides a sufficient background current to ensure that the gap stays conducting at low resistance during the subsequent ringdown. The ping repetition rate is determined by the RC time constant.
The resistance across the CT output terminals should be chosen to match the specified output impedance of the CT.
The test setup is not sensitive to stray reactances introduced into the base circuit by the pinger, and therefore layout is not at all critical. However, for best results, care should be taken not to introduce any significant extra series resistance into the coil base circuit.
Waveforms should be captured with a small amount of pre-trigger to
ensure that the front of the event is acquired. Care should be taken
to ensure that at no point does the signal become clipped in the scope Y
amplifier. Timebase should be chosen so that the highest mode of interest
is represented by at least 2 or 3 samples per cycle. For best results the
captured trace should be averaged over say 128 shots of the ping circuit.
The resulting waveform has a typical spectrum
We see the lowest four resonant modes of the coil - further modes are obscured by the noise floor, the level of which is set by the quantisation noise due to the scope trace digitisation.
The maximum possible accuracy of the subsequent Q and F extraction is limited by the information content of the captured trace, such that the errors are at best
Q Error = 100 / sqrt( NY * NT) percent F Error = 100 / NT percent
for each component mode, where
NY is the number of Y-axis quantisation steps spanned by the particular mode, NT is the number of T-axis steps spanned by the ringdown.
NY should be maximised by turning up the Y gain as far as possible without
clipping occuring. NT should be maximised by using the shortest possible
time step consistent with capturing the whole or most of the ringdown.
Typical values are NY=400, NT=8000, giving an accuracy of 0.06% on Q and 0.01% on F, in the absence of noise. Random trace noise, which can be easily reduced by averaging over many pings, will degrade the actual accuracy below these theoretical limits. In practice, the theoretical accuracy can be approached quite closely when using say 128 pings per trace.
Waveform data should be stored as a two-column ASCII data file with either a comma or a space separating the two fields. The program will accept either DOS or UNIX line terminators. The first column should be the time step, in seconds, and the second column should be the relative current amplitude. The time interval between steps should be uniform. The program will ignore any frequency components below 30kHz.
The example waveform above, which consists of a 16-ping average, when analysed gives the following output:
PK FREQ kHz (Error +/-) Q FACTOR (Error +/-) LEVEL 1 229.956 (0.01%,21Hz) 465.35 (0.12%, 0.6) -0.1dB 2 578.100 (0.01%,53Hz) 184.46 (1.43%, 2.6) -17.3dB 3 904.162 (0.01%,83Hz) 106.52 (14.33%,15.3) -24.6dB 4 1237.308 (0.01%,113Hz) 73.92 (107.4%,79.4) -30.3dB Accounted for 99.84% of input signal
In this example, the deterioration of Q determination with higher modes is almost entirely due to Y axis quantisation noise and the extraction is working close to the theoretical limits.
|Analysis Program: TCMA|
The source code for the program (current version: 0.1g) can be download from tcma.c and compiled according to your local C compiling conventions. A typical compiler command would be
cc -o tcma tcma.c -lm
and you will benefit from turning on full optimisation. The program will use up to 5 Mbytes or so of RAM, depending on how much work it is told to do.
Run the program by feeding the waveform datafile into the program's standard input, eg
./tcma < TEK00001.CSV
and the results and progress messages go the the stderr stream. By default, the program will go to work on the strongest four components found in the input file. This behaviour can be controlled by specifying the number of peaks to look at on the command line, eg
./tcma -npeaks 12 < TEK00001.CSV
will generate output for the strongest 12 peaks. Analysis of a particular peak can be selected with
./tcma -peak 5 < TEK00001.CSV
which will report results for only the 5th strongest peak in the input file. You can find out what signals the program has detected by using
./tcma -signals < TEK00001.CSV
which will make tcma print out approximate frequency values, but not go on to analyse them in detail.
The program is able to make an estimate of what the final accuracy of each F and Q result actually is, and these are printed as +/- limits following each value.
The LEVEL column shown in the results indicates the total RMS signal level of each mode, taken over the whole ping event.
The stdout stream is reserved for various other optional functions of the program. For example, the fourier transform (FT) of the input waveform can be generated by
./tcma -ft < TEK00001.CSV > tek00001.ft.csv
which generates a two column FT datafile.
Inspection of the code will reveal a number of other command line options which are used only for program testing. These are not yet documented.
The program works by synthesizing a waveform with the equation
V = A1 sin( 2 pi F1 t + P1) exp( -pi F1t/Q1) + A2 sin( 2 pi F2 t + P2) exp( -pi F2t/Q2) + ... + An sin( 2 pi Fn t + Pn) exp( -pi Fnt/Qn)
Aj is the amplitude,
Fj is the frequency,
Qj is the Q-factor,
and Pj is the phase,
of the jth component,
and n is the highest component number under consideration.
Initial values for the parameters are estimated from a fourier transform of the waveform, and then the parameters are systematically and deterministically refined to minimise the error when compared with the captured scope trace. To avoid the computationally intensive problem of optimising 4n parameters simultaneously, the program applies a narrow band filter to the waveform so that it can obtain a match on each component separately. The band-limited exponential ringdown waveshape is highly modified by the filtering process, so the program must apply the same narrow band filter to the synthesized waveform before making comparison. By this means, only 4 parameters need be matched simultaneously.
As a further optimisation, the program uses a set of three different correlation functions, each of which is sensitive separately to F, Q, or P errors, and each of which is insensitive to overall amplitude A. Thus three parameters of each mode can be refined more or less independently, and the amplitude coefficient A is determined at the end by a standard correlation function.
The actual error in any particular measurement is determined by constructing a second synthesized version of the waveform but with slight errors in the parameters. This synthesised waveform is compared with the waveform that was synthesized to match the input trace, and the artificial error is adjusted to achieve the same values of the three correlation functions that the original synthesized waveform has when compared with the input signal.
An estimate of overall signal quality is given at the end by synthesizing the whole waveform and subtracting this from the input signal, without any filtering. The total amount of input signal accounted for in this way is indicated as a final percentage.
High Q factor is of crucial concern to the CW coiler, and tcma can be used as an investigative tool to track down sources of dissipation in the coil. The effect of changes to toploads, ground arrangements, etc, can be accurately and repeatably assessed. By a careful and thoughtful process of elimination, it should be possible to construct an energy loss budget for a given resonator, although this has not yet been achieved.
The program may also be of use in investigating the relative importance of gap loss and proximity loss in traditional TC primary windings. There is evidence that both these factors may be significantly involved in setting the overall efficiency, but the choice of optimum primary design requires a trade-off between the two. Therefore it would be helpful to know the relative contribution of these two sources of loss. tcma may provide a useful tool for this kind of investigation.
The program is able to analyse the beat envelope of a
traditional Tesla coil. An example waveform from such a coil (tfss270501) has the spectrum
and tcma reports the following for the two strongest component signals,
PK FREQ kHz (Error +/-) Q FACTOR (Error +/-) LEVEL 1 257.060 (0.49%,1252Hz) 101.50 (6.98%, 7.1) -1.7dB 2 226.677 (0.21%,465Hz) 54.19 (5.23%, 2.8) -4.8dB Accounted for 98.96% of input signal
The closeness of the two modes has reduced the accuracy of the individual Q determinations, but further work on tcma will resolve this problem. The program should eventually able to supply the essential information from which accurate values of the effective coupling coefficient can be determined.
More than one coil can be pinged at a time, by simply connecting their
bases together. The low resistance of the spark gap (relative to coil
base impedance) ensures that the coils are not mutually coupled through
the ping circuit. An example FT for a two-coil ping is shown below,
and the tcma output for this waveform is,
PK FREQ kHz (Error +/-) Q FACTOR (Error +/-) LEVEL 1 351.208 (0.01%,29Hz) 216.04 (1.65%, 3.6) -2.0dB 2 143.476 (0.06%,93Hz) 241.10 (0.11%, 0.3) -4.9dB 3 873.002 (0.01%,71Hz) 243.24 (0.45%, 1.1) -16.7dB 4 511.845 (0.01%,42Hz) 112.90 (1.43%, 1.6) -23.8dB 5 1265.541 (0.01%,103Hz) 191.11 (3.92%, 7.5) -22.8dB 6 666.399 (0.01%,54Hz) 91.25 (1.65%, 1.5) -27.7dB 7 1602.539 (0.01%,131Hz) 138.07 (6.98%, 9.6) -26.1dB 8 819.766 (0.01%,67Hz) 78.88 (6.98%, 5.5) -28.4dB 9 977.371 (0.01%,80Hz) 67.92 (9.31%, 6.3) -32.8dB 10 1911.548 (0.01%,156Hz) 105.95 (14.33%,15.2) -28.9dB 11 1133.085 (0.01%,93Hz) 61.56 (16.55%,10.2) -33.3dB Accounted for 93.63% of input signal
The poor error of the first peak at 351kHz is due to the collision of the 1/4 wave mode of one coil, and the 3/4 wave mode of the other coil, which happen to be within 2kHz of one another. tcma has locked onto the strongest component, but the Q extraction is upset by the presence of the other signal. Therefore, this method may only give useful results when the coils involved have suitably distinct resonances. Ideally, each component frequency should be more that 6% away from any other.
This method can be used to accurately determine the current profile of a suitably modified coil, by means of a resistance insertion method. The method requires high precision Q measurements, and tcma, along with the pinger circuit, works very well in this application. See tfcp260302 for an experiment which employs this technique.
Both of these can be measured to high precision using tcma, the pinger circuit, and an additional resistance. The Q of the coil is measured twice, once with a known resistance R inserted into the base circuit to give the loaded Q factor Qr, and again without the extra resistance, giving the unloaded Q factor Qu. The resistance should be chosen so that Qr is less than half that of Qu. From these measurements, we can calculate
Rin = Qr R / (Qu - Qr) Lee = Qr Qu R / (Qu - Qr) / (2 pi F)
for each resonant frequency in the ping waveform.
A similar procedure to the above can be carried out to obtain the output admittance and the energy storage capacitance, by applying a known shunt loading resistance between the top of the coil and ground. This is inevitably a less accurate measurement, due to the sensitivity of the resonator to stray reactances applied to the coil top.
The plan is to extend tcma so that it can work from a waveform captured by a PC soundcard. This would normally require a frequency translation of the base current signal before it can be applied to the soundcard input, which means that the program could in most cases only analyse one resonant mode at a time, and this would be at reduced accuracy. Nevertheless, it may be possible to obtain useful results and in view of the fact that more coilers possess soundcards than digital scopes, the approach may be more widely usable.
The present version of the program is severely limited in its ability to separate closely spaced resonant modes, which limits the dual resonator analysis to situations involving fairly high coupling coefficients. At low k factor the program cannot separate the two modes sufficiently to match each one separately, and it is therefore necessary to iteratively match both simultaneously. It is hoped that eventually it will be possible to compare k factors measured at the resonant frequencies with the k factor determined from low frequency mutual inductance measurements.
Many thanks to Terry Fritz for providing the high quality scope waveforms used in the above examples.