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A6.2 |
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A6.3: Amplitude response of conventional
and maximally-flat current transformers
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Abstract: Relative amplitude response measurements over the 1.5 to 30MHz range were performed on a Faraday-shielded radio-frequency current transformer. The transformer was tested in both conventional and maximally-flat circuit configurations and the results agreed with the 'ideal transformer with secondary inductance' model to within the measurement precision of 0.25% (0.02dB). It was found that data could only be made to fit the model when the Faraday shield was maintained at the secondary network reference potential. The effective transformer secondary inductance determined by least-squares fitting to the data was slightly lower than the inductance obtained by direct measurement, the difference being attributable to leakage inductance. |
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Experimental method: Instruments which measure voltage at radio frequencies (oscilloscopes and valve/transistor voltmeters) tend to have the flatness of their frequency responses quoted in dB. Thus, while some may be nearly perfect in the 1.6 to 30MHz region, there is often no way in which the end-user can find out how flat a particular instrument really is. This presents a difficulty when the task in hand is that of characterising a device which is expected to be flat or, at least, to agree with its theoretical model to within a fraction of a percent. A solution to this problem, which may appeal to those who have to pay for their test equipment from personal funds, is to use a simple diode detector. Shown right is one of a pair of detectors used in the experiments described below. It consists of a 1N5711 Schottky diode with a 0.1mF ceramic smoothing capacitor, mounted on the back of a BNC socket. When used to drive a sensitive moving-coil meter via a suitable series resistor, it gives a frequency response which is flat from audio frequencies to several hundreds of MHz (i.e., until the series resonant frequency of the capacitor is reached). To achieve similar performance with an electronic amplifier is difficult, the downside of the simple detector being that large signal levels and correction for the effects of diode non-linearity are required. |
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| Shown below is a version of the experimental setup used by the author to test the theory of the maximally-flat current transformer. The equipment required is inexpensive and readily available; but as we will see, with careful procedure and proper data analysis, it will verify the circuit models for both the conventional and the maximally-flat current transformer and, as as an added bonus, it will tell us how best to earth a Faraday shield. |
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| Fig 3. Test circuit for current transformer amplitude vs frequency response. |
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The basis of the experiment is that, if a precisely constant
average power level can be maintained in a load resistor connected
to a radio transmitter, then a current transformer in the line
to the load will carry a primary current having a precisely constant
RMS value. Determining the frequency response of the transformer
network is then a matter of measuring its relative output voltage
as accurately as possible over a range of frequencies. Using
the equipment and circuit shown above, measurements can be made
with a transmitter output power of about 10W, this being obtainable
continuously for long periods from a typical 100W HF transceiver.
The exact power is not important when making relative measurements,
as long as it can be reset accurately at each measurement frequency.
Most of the output power (90%) is dissipated in a -10dB coaxial
T-pad attenuator (shown in the picture is a narda model 766-10,
rated 20W max, DC - 4GHz), the remainder being delivered to a
50W 1.5W coaxial terminating resistor.
The purpose of the attenuator is to reduce the transmitter output
voltage to a level suitable for a diode half-wave detector. A
generator delivering 10W into a 50W
load has an output voltage of Ö(10´50) = 22.36V RMS. The ratio of the
output voltage to the input voltage for the attenuator is given
by rearrangement of the expression: h/dB = -10 = 20Log10(Vout/Vin) i.e., Log(Vout/Vin) = -½ Taking the antilog of both sides (i.e., raising both sides to the power of 10) gives: Vout/Vin = 1/Ö10 Since the input voltage is Ö(10´50), the RMS voltage at the detector anode is: (Ö500)/(Ö10) = Ö50 = 7.07V Thus, neglecting diode forward voltage drop, the output after perfect rectification is: 7.07Ö2 = 10V. In practice, a 1N5711 diode will reduce this by about 0.3V. Consequently, establishing a working power level of around 10W is simply a matter of adjusting the meter series resistor (68K fixed + 100K variable) so that there is 9.7V DC across the detector smoothing capacitor when the meter reads full-scale. This can be done by connecting the detector to a variable DC power-supply and using a digital multimeter (DMM) to measure the voltage. Under no circumstances however, should a DMM be used to measure DC voltages when the transmitter is active. Initial attempts to calibrate the equipment using a digital meter to measure the rectified voltage resulted in erroneous readings (some in error by a factor of 2.2); the problem being that the sampling technique used by some multi-meters is disrupted even by relatively low levels of RF ripple. This undocumented performance limitation of digital meters means that quoted DC-level measurements for RF circuits are nowadays not dependable, whereas the mechanical averaging provided by a moving-coil meter is inherently perfect. Having established the sensitivity of the detector used for input current setting, the next step is to adjust the series resistor for the meter which monitors the transformer output. This is simply a matter of ensuring that the meter gives a high reading but will not go off-scale at any of the measurement frequencies. The nature of the experiment is such that the true parasitic capacitance across the secondary winding (about 4pF) is not sufficient to affect the output level in the test frequency range (1.5 to 30MHz); and it was the author's initial (and as it turned out, correct) hypothesis that the so-called 'self capacitance' of the coil is in reality a propagation delay, and therefore should not show up in an amplitude-response measurement. Hence both the maximally-flat and the conventional current transformer (should) give maximum output at the highest measurement frequency, and a sensible approach is to set the generator to the highest working frequency and, with the input current meter reading full-scale, set the output level meter to read about 98% of full-scale deflection (FSD). The actual rectified output voltage corresponding to FSD can be determined later using a DC power supply, as can the diode forward voltage drop over the range of currents encountered during the experiment. Acquiring a data set involves making a series of measurements at different frequencies, each time setting the input current meter to FSD and reading the output level. Large meters with anti-parallax mirror scales and jewelled (or, better still, taut-suspension) bearings are best for this purpose, but even so there may be difficulties. Using an HF transceiver having a conventional potentiometer for the carrier level control, the author found it practically impossible to set the power level accurately by turning the bare knob. The solution was to improvise a temporary slow-motion drive as shown in Fig 4 below. Transceivers which use a digital method to set the carrier level can be expected to be even less cooperative; although if the transceiver uses a separate 13.8V power supply it should be possible to fine-tune the output by varying the supply voltage between 12 and 13.8V. It is incidentally, by no means guaranteed that a particular transceiver will be good enough for the job. The first transceiver tried by the author (a Kenwood TS-430S) exhibited jitter of several percent in its carrier output and could not be used. The transceiver used for the actual measurements (a TS-930S) showed output fluctuations of around ±1% due to mains-voltage variations, which proved annoying but did not prevent the experiment from being carried out. |
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A particular problem when making accurate level readings is meter
sticking. This pathology of conventional moving coil meters is
evident when different readings are obtained by approaching the
set power level from opposite directions. It is not normally
noticed, but becomes significant when trying to measure to less
than 1/100th of the scale. The author tried several meters before
settling on the best ones available. The photograph of Fig
2 shows an early version of the experiment. The best data-set
was obtained using an old AVO model 8 multimeter set to its 50mA scale for the input current meter, with
the series resistor changed to 150K fixed + 100K variable (nominally
194KW). Good data could only be obtained
by tapping the input current meter gently (with a pencil or screwdriver
handle) while setting the transmitter carrier control, and by
tapping the output level meter before taking a reading. The carrier
level tended to drift for about 1 minute after the transmitter
was switched on, and it was found best to keep it running during
the experiment, switching off only briefly to change bands. Keeping
the transmitter on also helps to keep the system in thermal equilibrium,
although the various coaxial resistors used did not show any
change between hot and cold at the 0.1W
level, and the detector diodes remained cool to the touch. With
the precautions described, it was possible to read the transformer
output level to within about ¼ of a division on a 0-100
scale, with a repeatability of about ½ of a division.
This is consistent with an estimated standard deviation (ESD)
of measurement of about 0.0025. For each run of the experiment, data were acquired for a maximally-flat current transformer, and then the compensation capacitor was removed and the readings repeated. In this way, both a conventional and a maximally-flat current transformer were characterised, and the direct relationship between the parameters of the two models (same secondary inductance, same load resistance) gave rise to an effective doubling of the size of the data-set. This gives greater statistical significance to the data for the purpose of evaluating the experimental method by comparing the observed and the expected variances (see section A2-12). From the foregoing discussion of measurement precision, an ESD of fit of about 0.0025 is to be expected if the data agree with the model. As has already been implied, the object of the exercise here is to make measurements of the relative output voltage of a current transformer, with and without flatness compensation, over a range of frequencies. The measurements are then compared with the relative output voltage predicted by the appropriate model. The relative output is defined as the transfer function magnitude at the measurement frequency divided by the transfer function magnitude at "infinite" frequency (i.e., when the reactance of the transformer secondary inductance is very large and the reactance of the compensation capacitor, when included, is very small). For the maximally-flat current transformer, the relative output is given by the expression (A6.2-2):
RD/(RD+Ri). There is also a very small additional reduction in output due to the equivalent series resistance (ESR) of the boost capacitor Ch. |
 Fig. 5: Conventional current transformer. |
 Fig. 6: Maximally flat when Ch = 2Li / Ri² |
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Correction for detector non linearity is obtained by connecting
the detector to a DC power supply (taking care not to disturb
the setting of meter the series resistance) and measuring the
diode forward voltage drop (Vf) at a few
points over the range of meter readings which were encountered
during the experiment. These values are then fitted to an expression
of the form: Vf = V1Log(If) + V0 and used to correct the actual voltage at the detector output to the voltage which would have been present had the diode been perfect. This means that the detector voltage corresponding to FSD of the meter must also be measured, so that the meter readings can be converted into actual voltages. Correction is then effected by adding the diode forward drop given by the fitting function to the actual measured voltage, then converting back to a relative measurement by dividing by the full-scale voltage. If Mk" is an actual relative meter measurement (a proportion of FSD, i.e., a value between 0 and 1), and Vfsd is the voltage corresponding to a full-scale reading, then the absolute voltage measured is Mk"´Vfsd . Had the diode been perfect, the measured voltage would have been (Mk"´Vfsd)+Vf. Hence, the relative measurement corrected for diode voltage drop is: Mk' = (Mk" Vfsd + Vf) / Vfsd i.e.: Mk' = (Mk" Vfsd + V1Log(If) + V0) / Vfsd but the current flowing in the diode is Mk"´Ifsd , where Ifsd is the FSD meter current. Hence: Mk' = (Mk" Vfsd + V1Log(Mk" Ifsd) + V0) / Vfsd Note incidentally, that it does not matter if Ifsd is only the nominal (uncalibrated) value rather than the true FSD meter current, provided that the If values used to produce the diode model were obtained from the same meter. Note also that the static average diode forward voltage drop, obtained by making measurements with a DC power supply, may not correspond exactly to the dynamic average forward drop which occurs when the detector has an AC input. In other measurements made by the author however (Appendix 6.1), it was possible to infer that this effect is negligible. Finally, the corrected observations are all multiplied by a common fitting parameter (p say) which scales the data set in order to obtain the best fit to the model. Thus, a corrected observation is given by:: Mk = p Mk' i.e.,
Diode correction: Several experimental runs were performed, because various problems had to be ironed out. These all involved slightly different working power levels and various different microammeters, and consequently required slightly different diode corrections. For the purpose of illustration it will be sufficient to describe how only one of these correction functions was obtained. For the experiment in question, the current transformer output detector was connected to a 100mA meter via a 100KW variable resistor which had been adjusted to give 98mA (nominal, read from the meter scale) at the working power level with the system operating at 30MHz. On removing the detector from the circuit and connecting it to a 0-30V bench power supply, it was found that the voltage across the smoothing capacitor was 2.57±0.02V (measured using a DVM) with the meter reading full-scale. The range of currents which had been encountered during the preceding experiment was 77.5 to 100mA. The corresponding diode forward voltage drops at four points enclosing this range were (±0.002V):
Vf = 0.026117601 Loge(If/mA) + 0.172947811 Details of the fitting process are shown below and the formulae used can be inspected in sheet 2 of the spreadsheet maxflat_test1.ods. |
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Hence, for this particular experimental run (putting the numbers
into equation (4)), a corrected
meter reading is given by the expression: Mk = p (2.57 Mk" + 0.026117601 Loge(100 Mk") + 0.172947811) / 2.57 Since any errors in the diode function are small in comparison to the measurement precision, and p is always close to 1, the estimated standard deviation (ESD) of a corrected measurement remains approximately the same as that of a raw measurement, i.e., about 0.0025. |
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Data acquisition: A number of experiments were carried out using a 1:12 Faraday shielded current transformer wound on an Amidon FT50-61 bead. The diameter of the enamelled copper secondary winding wire was 0.9mm, and the measured inductance at 1.5915MHz was 8.45±0.21mH. The current transformer load resistor Ri was 49.8±0.2W in all cases. The detector input resistance, of course, slightly reduces the effective value of Ri. but since the effect is small in these experiments, and identical for the conventional and maximally flat circuit configurations during an experimental run, it was eventually neglected (i.e., allowed to become part of the fitting parameter p). Only two of the experiments will be reported, the others being concerned with resolving practical problems such as meter sticking and variation in the transmitter output level. The first valid experiment, with the current transformer mounted as shown in Fig 9 below, was a failure in the sense that the data did not fit the models; but it provided important information nevertheless. The graphs of relative output vs frequency and deviation from theory are shown below. The data and details of the data adjustment procedure are given in the spreadsheet maxflat_test1.ods. |
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| The problem with the first experiment is that the data show a steadily rising frequency response. The extent of the deviation from theory is apparent in the graph of Fig. 8 above; which shows a rise of about 7% over the 1.5 to 30MHz range, with no statistically significant difference in this respect between the conventional and maximally flat configurations. This is obviously an artifact, a supposition confirmed by measuring the actual voltages involved against the input current using the method described in Appendix 6.1. By making absolute (rather than relative) measurements, it was found that the apparent transfer efficiency of the transformer exceeded 100% at 30MHz, which is of course impossible. The cause of the problem can be deduced by considering the earthing arrangement in Fig. 9 below. |
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| The experimental defect lies in the fact that the primary current flows in the aluminium bracket on which the transformer is mounted and the Faraday shield is grounded at the generator port. The inductance of the bracket is small, but nevertheless finite; and so a potential difference, which rises with frequency, exists between the ports. Hence the Faraday shield potential differs from the detector reference potential (i.e., the ground connection at the detector socket), and some of this difference is coupled to the detector by the capacitance between the Faraday shield and the transformer secondary winding. The reactance of this parasitic capacitance, of course, falls with frequency, and so the coupling also increases with frequency. The problem was cured by rewiring the jig as shown in Fig 10 below. Fortunately this issue was resolved at an early stage in the author's experiments because the corollary is that current transformers do not behave according to theory unless the Faraday shield is maintained at the secondary network reference potential. |
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| Final normalised data for the Faraday-shield earthing configuration shown above are plotted in Fig 11 below. It is obvious that the flatness compensation scheme has worked, and so a detailed analysis follows. |
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The data were first corrected for diode forward voltage drop
and normalised for minimum Chi². It was then found that
the fit to the data could be improved by varying Li
and Ch slightly. The compensarion capacitor
was removed from circuit and its capacitance was measured to
be 6.25 ±0.16nH at 1.5915MHz, in agreement with the experiment
but about 7% low of its value read earlier from a digital capacitance
meter. The optimal secondary inductance value was found to be
around 8.3mH, as opposed to the measured
inductance of 8.45±0.21mH at
1.5915MHz. The adjusted inductance value is, of course, still
in agreement with the measured value, but the bridge used for
the measurement was known to be more accurate than its specification
and the discrepancy was considered to be genuine. The cause of
the problem remained unknown until a technique was developed
for extracting the effective secondary inductance and 'self-capacitance'
of a current transformer from bridge-parameter measurements (Appendix 6.4). It then became apparent that
the effective inductance is always less than the measured inductance,
the difference being a few %. Steps were taken to eliminate all
causes of systematic error, but the difference persisted; leading
to the conclusion that it was due to leakage inductance; i.e.,
the effective inductance is the coupled inductance, whereas the
measured inductance is the sum of the coupled and the leakage
inductances. On the basis that that it is legitimate to adjust the fit for leakage inductance, the data were re-analysed in order to extract the effective or 'coupled inductance'. Working with the data for the conventional current transformer, this was done by noting that a normalised amplitude measurement (M) should be in agreement with equation (3), i.e., M = 1/Ö[ 1 + ( Ri/XLi )² ] but M = p M' i.e., a normalised measurement is a measurement corrected for diode forward drop (M') multiplied by a fitting parameter. Hence: M' p = 1/Ö[ 1 + ( Ri/XLi )² ] This rearranges to:
syk = |¶y/¶M'| sM' where ¶y/¶M' = -2 / (M')³ Hence: syk = 2 sM' / (M')³ and the fitting weight for an observation yk is 1/syk². How the fitting was done can be determined by examining sheet 2 of the spreadsheet Maxflat_test2.ods. The inductance is obtained from the fitting parameters, i.e.: b=(p Ri / Li)² and p=Öa. which give: Li = Ri Ö(a/b) . . . . . (5) The variance (i.e., the square of the uncertainty) in Li, on the assumption of minimal correlation between a and b, is given by: sL² = ( sRi ¶L/¶Ri )² + ( sa ¶L/¶a )² + ( sb ¶L/¶b )² where, by differentiating equation (5): ¶L/¶Ri = Ö(a/b) ¶L/¶a = (Ri /Öb)/(2Öa) = Ri / [2Ö(ab)] ¶L/¶b = -(Ri Öa)/(2b    )The uncertainty in Ri (sRi) was 0.2W. The uncertainties in a and b were obtained from the fit. The inductance measurement thereby obtained was Li = 8.328 ±0.034mH, the uncertainty in this value being due largely to the uncertainty in Ri. The 'leakage inductance' is of course the difference between this and the directly measured inductance (8.45±0.21mH), but since the uncertainty in the difference is Ö(0.034² + 0.21²), we get the result LL = 122 ±213nH i.e., the measurements are not accurate enough to determine it. Having determined Li from the fit, the value obtained was fed back into the comparison between the data and the model (see sheet 1 of Maxflat_test2.ods). The numer of degrees of freedom in the data was thereby reduced by 1, and the calculation of the ESD of the fit adjusted accordingly. The maximally flat transformer data gave c²/15=0.95, and the conventional transformer data gave c²/14=0.98, consistent with a reasonable initial estimate for the uncertainty of an observation (0.0025). The pattern of residuals (expressed as percentage deviation) is shown in Fig 12 below. Observe that the Y-axis covers an interval of 1%. The fluctuations are typical of rounding error, there being only 4 possible recorded values for a measurement in this interval (ending in 0, 2 or 3, 5, 7 or 8), the decision between 2 or 3 and 7 or 8 being made by tossing a coin. |
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Note that for a secondary inductance of 8.328mH
and a load resistance Ri of 49.8W, the optimum value for the boost capacitance,
given by Ch=2Li/Ri² is 6.72nH. The actual value installed
was slightly lower than this, at 6.25nH, resulting in a slight
over-boost in the theoretical value for hrel (see sheet 1 of Maxflat_test2.ods).
This over-boost amounted to 0.2%, indicating that the selection
of Ch for a maximally-flat current transformer
is not critical within about 10%. Summary:  The maximally-flat
current transformer is a viable solution to the problem of low-frequency
amplitude error in RF ammeters. Selection on test
for the boost capacitor is not required. When the 'true'
secondary network parallel capacitance is small, the output amplitude
vs frequency response (but not the phase response) of a current
transformer can be computed on the basis that the transformer
secondary reactance is purely inductive. The effective
secondary inductance of a current transformer is smaller that
the inductance obtained by direct measurement of the coil. The
difference is due to leakage inductance. Allocation of different
symbols to these to these two quantities is recommended, e.g.:Lsec = measured inductance of the secondary winding. Li = effective secondary parallel inductance or 'coupled inductance'. Lsec serves as an approximation for Li in initial design calculations, but allowance or adjustment for the difference may be required.. The Faraday shield
of a current transformer must be grounded at the secondary network
reference point in order to obtain conformance to the 'ideal
transformer with secondary inductance' frequency-response model.. |
© D W Knight 2005 - 2007.
David Knight asserts the right to be recognised as the author of this work.
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A6.2 |
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