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Appendix 6.1. Current Transformer Coupling Factor.
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A6.1-1. Transfer efficiency of a current transformer: An good description of a tightly-coupled current transformer is given by using an ideal transformer model with parallel secondary reactance. The relationship between output voltage and input current for that model is given by the expression: Vi = I Zi / N where N is the turns ratio and Zi is the parallel combination of the secondary load resistance and the secondary reactance, i.e.: Zi = (Ri // jXLi // jXCi) A potential problem here however is that various transformer non-idealities, particularly core loss, winding resistance and leakage inductance, have been partially neglected. This means that the actual output voltage of the transformer is always slightly less than that predicted by the model. The discrepancy is often of little consequence because it is small in engineering terms; but it is necessary to keep track of it, and if possible to quantify it, when calculating the balance condition for a bridge, or when calibrating an accurate power meter or ammeter. One way of accounting for the output shortfall is to include an empirical frequency-independent coupling factor (or transfer efficiency factor) in the input-output relationship. The justification for dealing with the problem in this way is that the relative frequency response of the transformer is well described when only parallel reactance is included (see Appendix 6.3); the reason being that, although the losses and inductance leakages are frequency dependent, the dependence is weak in comparison to the major non-idealities and so can be absorbed into the parallel reactance parameters. Hence we might modify the transfer relationship in one of two ways: Either Vi = k I Zi / N Or Vi = I (kRi // jXLi // jXCi) / N In the first approach, we simply multiply the predicted output voltage by a factor k, which is slightly less than one. In the second approach, we multiply the load resistance by a factor k, in which case we can consider k to have come about as a result of a parasitic parallel resistance Rk (say) defined such that: k Ri = Ri // Rk = Ri Rk / (Ri + Rk) i.e: k = Rk / (Ri + Rk) and by rearrangement: Rk = k Ri / (1-k) There is little difference between the two approaches, except that the former slightly modifies the effective parallel reactances in the process of reducing the output voltage, whereas the latter does not. In other experiments performed by the author, it was found that the value of secondary inductance obtained by direct measurement was always slightly greater than the inductance obtained by linear regression analysis of a set of frequency response measurements. Multiplying the measured inductance by a realistic k value brought the two values into agreement, but since the frequency response method gives a very accurate inductance value there seemes to be little merit in linking its value to an amplitude correction factor. The difference between directly measured secondary inductance (total inductance) and the value of inductance required to fit the frequency response curve (coupled inductance) is of course the secondary leakage inductance (a few % of the total). Hence it seems sensible to define Li as the coupled secondary inductance. |
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Measuring transfer efficiency: Described below is a procedure for measuring the transfer efficiency of current transformers; followed by various measurements made using that procedure. For the purpose of these tests, transfer efficiency is defined as follows:
|Vi (theoretical)| = |I| |(Ri' // jXLi // jXCis)| / N and Ri' is the parallel combination of the secondary load resistance and the voltmeter (detector) input resistance. Notice that the secondary shunt capacitance is given as Cis. Only stray capacitance is included, and the 'self-capacitance' of the coil is ignored for the purpose of calculating magnitudes. The reason for this is that none of the transformers tested had more than 20 turns on the secondary, and amplitude frequency-response measurements indicated that all of the transformers were operating in a completely flat region of the frequency response at the measurement frequency of 30MHz. Had the coil 'self-capacitance' been taken into account, this would have predicted a slight roll-off at 30MHz, but no such roll-off occurs in practice. The explanation is that 'self-capacitance' is largely fictitious. It is mereley a lumped-component representation for the time delay which occurs due to the finite velocity of electromagnatic radiation propagating along the winding wire. The transformer secondary is effectively a transmission line, and although unlikely to be terminated in its characteristic impedance, the principal effect of the time delay is to shift the phase of the output without affecting the magnitude. Including a fictitious capacitance in the model would shift the phase of the theoretical output voltage and also reduce its magnitude, inflating the apparent efficiency in some cases to more than 100%. The fact that current transformers do not have gain must advise our choice of model in this case. The secondary inductance, on the other hand, is a perfectly good lumped parameter. Hence it has been included, even though its effect on the output amplitude at 30MHz is small. Note that, since the resulting correction is small, the effect of any uncertainty in the measured inductance is negligible. An expression for the magnitude of an impedance in parallel form is given in section 1-15b. Here it becomes: |Zi| = Ri' / Ö[1 + ( Ri'/Xi )²] where: Xi = XLi // XCis Hence (shortening "theoretical" to "theor"):
1/Ö[1 + ( Ri'/Xi )²] which is intentionally made close to unity by adopting 30MHz as the measurement frequency, i.e., by choosing a frequency at which the reactances of the secondary inductance Li and the 'true' parallel capacitance Cis approximately cancel. Measurement method: The main experimental problem is that of how to establish an accurately defined RF reference current in the transformer primary and make concomitantly accurate measurements of output voltage and load resistance. Since transformer efficiency is normally very high, systemmatic errors of a few percent in any of those quantities will make nonsense of the results. The author's solution, using the attenuator, load and diode detectors shown below, was to calibrate the current detector by connecting a DC power supply in place of the generator, setting the meter to read FSD for a direct current equal to the peak RF current which flows when a generator is delivering 10W into 50W. Tests were then conducted on a variety of current transformers, with several different secondary load resistors; each time setting the generator output to give FSD of the meter monitoring the input current, and then setting the output level meter to read FSD by adjusting its series resistor. After setting, the output detector was then transferred to a DC power supply, and the voltage giving FSD, being equivalent to the peak value of the RF output voltage, was determined. |
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| Fig 2. Establishing the reference current. |
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The initial current calibration step is illustrated in the circuit
diagram above. The point in delivering direct current into the
entire load assembly in this way is that it automatically takes
into account the diode forward voltage drop and the tolerances
of the resistors in the T-attenuator and the terminator. All
we are interested in is the input current, and we do not care
if the load presented to the transmitter is not exacly 50W. The RMS current which flows when a generator
is delivering exactly 10W into exactly 50W
is |I|=Ö(P/R)=447.2mA.
The equivalent peak current is 447.2Ö2=632.4mA,
and so this is the direct current which should be injected as
accurately as is possible when setting the detector to read full-scale.
Note however that typical bench power-supplies have a 0-30V output
range, whereas it requires 31.6V to give 632mA into a 50W resistance. The solution to this problem
is to put two power supplies in series (the author used a 13.8V
PSU in series with a 0-30V PSU), having first checked that the
terminals of the most positive power supply in the stack are
floating with respect to ground. A further trick is to cook the
whole assembly at an input current of 447mA for 10 minutes or
so, to raise its temperature to that which will be encountered
during the RF measurements, and then briefly ramp the current
to 632mA for the setting of the meter series resistor. In this
way the effect of thermal variations in resistance and diode
forward voltage drop are minimised. For the purpose of establishing the DC input current, the author had available two 3½ digit multimeters of different make but in a known good state of calibration. Consequently, although only one ammeter is shown in series with the PSU, both meters were placed in series so that the average could be taken. Fortuitously, the meters read 632mA simultaneously, which gave confidence in both an emotional and a statistical sense. Both meters had a stated accuracy of ±0.5% ±1mA on the ranges used, i.e., ±4.16mA (the numbers after the decimal place are not significant), but by averaging the readings the uncertainty is reduced by a factor of 1/Ö2. Hence the estimated standard deviation of the peak current setting was 4.16/Ö2=2.9mA. The peak current must of course then be divided by Ö2 to find the RMS current during the RF measurements, and the estimated standard deviation is scaled down accordingly. This gives an equivalent RMS input current of 446.9±2.1mA. Finally, with much tapping of the case to jog the bearings, and careful use of the anti-parallax mirror, it was estimated that the detector current meter could be set to within ±0.2% of FSD. This corresponds to ±0.9 parts in 447, and so it was established that RF measurements were made with a reference input current of 446.9±3mA RMS. |
| Fig 3. Setting the output detector meter series resistor. |
| The RF part of the procedure was conducted with the transformer under test installed as in the diagram above. In each case, the carier level of the generator (radio transmitter) was turned up to give FSD of the input current meter, the output meter was set roughly to FSD by adjusting its series resistor, and the whole assembly was allowed to cook for about 10 minutes. With the system in thermal equilibrium, and much tapping of both meters to jog the bearings, the carrier level was then set exactly, and the series resistance of the output level meter was given its final adjusment. Precise adjustment of the transmitter carrier control was made possible by the use of an auxiliary reduction drive as shown below. |
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The immediate priority after switching off the transmitter was
to measure the resistance of the transformer secondary load resistor
before it had a chance to cool down. This was accomplished by
having a DMM set to the correct range and ready with a good clean
silver-plated BNC to 4mm adapter installed. It was found possible
to obtain a resistance reading within 3 seconds of switching
off the RF input, and none of the resistances were seen to change
on that timescale. This step was necessitated by the discovery
that preliminary experiments had been invalidated by thermal
variation of about 2% in some of the load resistors used. The
meter used for the measurement had a stated accuracy of ±0.8%
±0.1W on the range used. It
gave the following readings when used to measure various 0.1%
precision resistors mounted on good quality BNC plugs:
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| Fig 5. Reading the detector sensitivity to determine Vi(peak). |
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The final step in making a measurement was to determine the DC
input voltage corresponding to FSD of the current transformer
output detector, as shown in the diagram above. The voltage obtained
corresponds to the peak value of the current transformer output
under RF conditions and will be given the symbol Vi(peak).
The DVM used for the measurement has an input resistance of 10MW and gave the voltage of a Weston Standard
Cell at 20°C to be 1.019V on its 2V range (exactly correct
for an instrument of its resolution), and flickered betwwen 1.01
and 1.02V on the 20V range used. Its accuracy was therefore also
better than specified, and the voltage measurements obtained
were assumed to have a standard deviation no worse than ±0.02V.
To this however, must be added the resetting uncertainty of the
detector meter, which was about ±0.2% for the RF setting
step and ±0.2% for the DC measurement step, giving 0.3%
overall. Hence detector voltage measurements were made with an
uncertainty of ±0.02V ±0.3%, the two sources of
error being uncorrelated (see sections 1-31
and A2-9).
This uncertainty is scaled down by a factor of 1/Ö2
when we convert from peak to RMS values. Some readers may question the need to read the output voltage by applying DC to the detector, since it is 'obvious' that the voltage can be read by connecting a DVM across the smoothing capacitor while the transmitter is running. It was found however, that DVM readings in the presence of an operating radio transmitter could not be trusted. The author saw readings which were in error by a factor of as much as 2.2 (reading 22V instead of 10V) when this was tried. This problem might be solved by feeding the DVM via an efficient low-pass filter, but apparent plausibility of the readings is the only measure of success in such a case. The DC method also takes the diode forward voltage drop into account automatically, whereas a direct reading must be corrected. Hence, any improvement in accuracy engendered by a direct reading will be partially negated by uncertainties in the diode model used. Any error in the diode correction will moreover be systematic (i.e., it will introduce bias into all of the results obtained), whereas the setting and resetting errors of a moving-coil meter are random. As mentioned previously, the need to control random errors very carefully arises because transformer efficiency is high. We must also be aware however, that there may be residual systemmatic errors; one being due to the possibility that there may be detector inefficiency beyond that associated with the diode forward voltage drop under static (DC) conditions. The issue here is that when a diode conducts under dynamic conditions, the current occurs in pulses which are considerably larger than the average current. Thus there may be a greater effective diode forward drop than has been allowed for by making DC settings and measurements. The author's partial solution to this problem was to load the detectors lightly by using 50mA FSD meters, and to choose some of the test transformer and load resistor combinations to give output voltages comparable to the voltage applied to the input current detector. The point in the latter case is that when the RMS voltages applied to the two detectors are about the same, the effects of detector inefficiency are cancelled. In the event that the voltages at the two detectors are widely different however, we must be aware of detector inefficiency as a possible additional source of error. Detector loading: The output voltage detector requires a small amount of power to drive it. Consequently, the effective current transformer load resistance is very slightly lower than that obtained by measuring the resistor. The power consumed by the detector is the same as that which is required to deflect the meter to full-scale when a DC supply is connected to it, i.e., it is given by: Pdet = Vi(peak) ´ Ifsd This power can be converted into an equivalent parallel load resistance Rdet using: Pdet = |Vi(meas)|² / Rdet where |Vi(meas)| is the 'measured' RMS transformer output and is given by: |Vi(meas)| = Vi(peak) / Ö2 Hence: Rdet = |Vi(meas)|² / ( Vi(peak) Ifsd ) = ( Vi(peak) / Ö2 )² / ( Vi(peak) Ifsd ) Rdet = Vi(peak) / (2 Ifsd ) The effective current transformer load resistance is thus: Ri' = Ri // Rdet Hence: Ri' = 1 / [ (1/Ri) + (2Ifsd/Vi(peak) ) ] For the nominal 50mA meter used, Ifsd was measured to be 52.0±0.36mA. Hence:
Data analysis: As was discussed above, the output voltage of a current transformer, according to the 'ideal transformer with secondary reactance' model, is given by the expression:
|Vi (theor)| = |I| Ri' / N The way in which the errors in two or more variables combine to determine the error in the output of a formula using those variables is explained in sections 1-31 and A2-6. Two methods for estimating the standard deviation of the output quantity are available, one (numerical) involving parameter shifting, the other (analytical) involving calculus. Here we will use the analytical method because it will simplify things greatly if we have the error function as an algebraic expression. Since the errors in |I| and Ri' are uncorrelated, the error in |Vi (theor)| is given by the orthogonal vector sum of the rate of change of the formula with respect to |I| multiplied by the uncertainty in |I| and the rate of change of the formula with respect to Ri' multiplied by the uncertainty in Ri', i.e.: svi theor = Ö{ [ ( ¶|Vi (theor)|/¶|I| ) s|I| ]² + [ ¶|Vi (theor)|/¶Ri' ) sRi ]² } where s represents the estimated standard deviation of the quantity indicated by its subscript, and ¶ indicates a partial differential (i.e., differentiation of one quantity with respect to another with all other variables held constant is implied). The differentiations are trivial in this case: ¶|Vi (theor)|/¶|I| = Ri' / N ¶|Vi (theor)|/¶Ri' = |I| / N Hence: svi theor = Ö[ ( s|I| Ri' / N )² + ( sRi |I| / N )² ] which simplifies to: svi theor = (1/N) Ö[ ( s|I| Ri' )² + ( sRi |I| )² ] Volts RMS From the previous discussion we have: |I|=0.4469A, s|I|=0.003A, and sRi=0.2W. The other quantities vary between experiments. Hence our error function is:
The object of the exercise is to measure the transformer efficiency, as defined by the expression: k = |Vi (measured)| / |Vi (theor)| Previously, we determined the uncertainty of the measured RMS output voltage to be ±0.02/Ö2 V ±0.3/Ö2 % from two uncorrelated error sources. To use these numbers, we must first convert the percentage into Volts, and then combine them as orthogonal vectors. Hence (abbreviating "measured" to "meas"):
Now, having estimated standard deviations for both the measured and the theoretical values of |Vi|, and assuming them to be uncorrelated, we have: sk = Ö{ [ ( ¶k/¶|Vi (meas)| ) svi meas ]² + [ ¶k/¶|Vi (theor)| ) svi theor ]² } where: ¶k/¶|Vi (meas)| = 1/|Vi (theor)| and: ¶k/¶|Vi (theor)| = -|Vi (meas)| / |Vi (theor)|² hence:
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Measurements: Shown in the spreadsheet below (Itr_k.ods) are the results of measurements made on five different current transformers with load resistances ranging between 90 and 25W. The generator frequency was 30MHz in all cases. Column A is the test transformer identified by its core material and turns (Npri:Nsec). Column B is the turns ratio, defined as N=Nsec/Npri. Column C is the secondary inductance in mH measured at 1.5915MHz. Column D is the secondary parallel capacitance, assumed to be about 2pF for the input capacitance of the 1N5711 detector diode, plus 2pF for the short length of unmatched transmission line leading to the detector. Altering this capacitance by ±3pF affects only the 4th decimal place of the calculated efficiency, and so its exact value is not important. Column E is the load resistance measured within 3 seconds of switching off the RF power. Column F is the effective load resistance given by equation A6.1-1. Column G is the theoretical output voltage given by equation A6.1-2, and column H is its estimated standard deviation (ESD) given by equation A6.1-3. Column I is the measured peak value of the output voltage obtained by applying DC to the detector. Column J is the 'measured' RMS value obtained by dividing the peak value by Ö2, and column K is its ESD as given by equation A6.1-4. Column L is the transfer efficiency k, and column M is its ESD calculated using equation A6.1-5. Note that there are two measurenemts on the 1:12 transformer with an 89.5W load. One of these was performed early in the experimental run, and one was performed towards the end as a test of reproducibility. |
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Interpretation: As mentioned earlier, one possible source of systematic error is detector inefficiency. If such an experimental defect were present, it would manifest itself as a tendancy to produce pessimistic estimates of transformer efficiency whenever the voltage at the output detector is low in comparison to the voltage at the input current detector. For the purpose of examining this possibility, a plot of transformer eficiency against measured secondary voltage is shown below: |
| It might appear at first glance that there is an upward trend in the efficiency as the output voltage increases. The behaviour is however, also chaotic, and were we to fit the graph to a regression line and use the resulting function to correct the data, it would have the unfortunate consequence of making some of the transformers appear to be more than 100% efficient. More reasonably we should note that it is the transformers with low turns ratio which give the greatest output, which means that a tendancy for the efficiency to fall as the turns ratio increases would produce a similar correlation. The scatter diagram shown below examines this alternative. |
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Here there is an ordered trend. There are also sound physical
reasons for expecting the efficiency to improve as the turns
ratio is reduced, which is that there will be a relative reduction
in primary leakage inductance. Hence we should reject the detector
inefficieny hypothesis and conclude that low-ratio current transformers
are more efficient than high-ratio ones. Now consider the two transformer models shown below in relation to the plot of transformer efficiency versus load resistance shown below them: |
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Unfortunately, there are insufficient data to permit a clear
statistical distinction to be made between the two models, but
if the plot is considered as a scatter diagram there is an apparent
downward trend as the load resistance increases. There is also
a sound physical reason for favouring the parallel loss model,
which is that the loss resistance is identifiable in part with
the core loss referred to the transformer secondary. Hence, bearing
in mind that any simplification applied to a component model
reduces its accuracy, we may conclude that it is reasonable to
account for the shortfall in output of an 'ideal current transformer
with secondary reactance' by invoking a parasitic parallel resistance
Rk, which is defined as: Rk = k Ri / (1-k) The experimental results produceed during this investigation are summarised below. The broad conclusion is that small RF current transformers wound on ferrite beads have transfer efficiencies in the region of 95 to 99%. Transformers of low turns ratio are more efficient than transformers of high turns ratio. Provided that the winding resistance is low, the transfer loss can be considered to be due to a parallel parasitic resistance. |
Measured efficiency (k) factors for a selection of RF current tranformers.
Standard deviations are expressed in brackets after the number as uncertainty in the last digit.
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© D W Knight 2007.
David Knight asserts the right to be recognised as the author of this work.
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