TX to Ae

Ch6 Contents

A6.3

Part 2


A6.4(1): Evaluation and optimisation
of R.F. current-transformer bridges
Part 1

1. Bridge parameter perturbation method.
2. The Douma bridge: basic theory.
3. Prototype test bridge.
4. Parasitic inductance of the reference cap.
5. Stray capacitance.
5a. Transformer constant.
6. Test procedure optimisation.		 7. Reciprocity error due to common mode.
8. Reciprocity error due to radiation.
9. Capacitance of the reference load.
10. Post optimisation test data.
11. Faraday shield displacement current.

Part 2 >>>>.

Abstract:
An impedance monitoring bridge can be characterised by choosing two independent (or nearly independent) circuit parameters related to the magnitude and phase of the load impedance at balance. By adjusting the selected parameters to balance the bridge exactly with a reference load attached, the deviations of the parameters from their target values can be used to compute the bridge error at a given frequency. In a bridge which uses a capacitive potential divider for voltage sampling (Douma's bridge), suitable parameters are the lower voltage-sampling network capacitance and the LF-compensation resistance. Adjustment of the resistance can only compensate for leading phase error in the current transformer output, but the range of operation can be extended by shifting the phase in various ways. By suitable placement of a DC isolation capacitor, the resistance can be measured continuously while the RF generator is running. The bridge balance point may be located with great precision by using a communications receiver as the detector. Shielding and the use of common-mode chokes in the earth-loop between the signal generator and receiver prevents errors due to spurious signal injection. The optimised system can make relative phase measurements with an RMS uncertainty of about ±0.0075 degrees.
     The effect of the series inductance of the lower voltage sampling capacitor is clearly determined by the data. Compensation for this parasitic reactance can be obtained by inserting a small adjustable inductance in series with the upper voltage-sampling arm. Magnitude flatness of around ±0.03% is possible by this method.
     The parallel-equivalent secondary-inductance of the current transformer is a strongly conserved model parameter. The measurement of parallel secondary capacitance is however skewed by through-line mismatch and other parasitic reactances, to the extent that it may appear to be positive, negative, or accidentally zero. A perturbation series is derived to account for the various contributions, and includes a hitherto undocumented effect of Faraday shield displacement current. Control of parasitics is needed if bridges built by different individuals are to give comparable results.
     The data show a linear relationship between phase error and frequency except for a small deviation attributable to a dispersion region in the premeability of the ferrite transformer core. This supports the view that the phase error can be considered as a time delay ocurring primarily in the transformer.
     Various phase compensation schemes are proposed and evaluated. These lead to bridge designs with 2-point frequency tracking which can easily achieve a maximum phase error of better than ±0.2° and a maximum magnitude error of better than ±0.3% over the 1.6 to 30MHz range. A 3-point tracking scheme which gives a maximum phase error of ±0.04° is also demonstrated.
     The need for the transformer Faraday shield is investigated. Theory indicates that the effect of the parasitic capacitance from line to detector port is correctable depending on the coupling factor. An unshielded bridge with 2-point frequency tracking gave a maximum phase error of ±0.05°, close to the ±0.03° limit imposed by dispersion effects in the ferrite used.

Data and mathematical analysis relating to the following discussion is contained in Open Document Spreadsheet (.ods) files. These can be viewed and modified using the Open Office software suite, available without charge from OpenOffice.org.

1. Bridge parameter perturbation.
It is too much to expect that a current-transformer bridge designed using a simple idealised-component analysis will give perfect results. Therefore a practical circuit evaluation method is needed, but unfortunately, conventional frequency-response test procedures are not well suited to this task. The problem is that; in order to characterise an imperfect bridge, it is necessary to apply a generator to an input port and then determine the impedance which must be connected to an output port in order to bring the voltage at a detector port to zero. This implies the need for a variable impedance load and an auxiliary bridge or network analyser good enough to measure it accurately, i.e., it seems on first appraisal that bridges cannot be tested without access to exotic test equipment.
     There is however, an alternative approach which uses what might reasonably be called a 'perturbation method'. The basic idea here is that, provided that the theoretical model is fairly accurate, it should be possible to make measurements which are related to the extent to which a bridge is out of balance and then use that information to calculate the impedance which must be connected to the load port in order to restore balance.
     For a first try at devising a test procedure based on perturbations, we might note that; if a fixed and accurately characterised load (i.e., a UHF coaxial resistor) is connected to the output port, and the degree of imbalance is small; then the actual load required to balance the bridge can be calculated from the phase and magnitude of the error voltage appearing at the detector port. Unfortunately, this requires the use of a sensitive vector voltmeter or a very accurately phase-compensated oscilloscope, and so is neither a low-cost nor a potentially straightforward approach.
     There is however a much simpler solution: If a bridge has a minimum of two suitably chosen adjustable components (ideally analogs of R and X or |Z| and f), then it can be made to balance exactly at a single frequency when connected to the design load impedance. When the frequency is changed, the bridge, being imperfect, will go out of balance, but the adjustable components can be altered to restore it. If we can measure the changes in the component values, then these perturbations (being hopefully small) can be applied to the circuit model in order to deduce the load-impedance error which would have occurred at the new frequency had the adjustments not been made.
     A bridge which uses a potential divider as its voltage-sampling network can be re-balanced; by altering the basic ratio to correct for load-resistance errors, and by adjusting the LF-compensation component to correct for reactance errors. In the case of a bridge with a resistive potential divider, changing the ratio is straightforward; but the LF-compensation component is an inductor, and this presents a serious practical drawback. A significant non-ideality of current-transformers is the propagation-delay or 'self-capacitance' of the secondary winding. At some frequency, the phase-lag due to the delay will cancel the phase-lead due to the secondary inductance. The LF compensation inductance can, in-principle, be adjusted to track the deviation from the 'ideal transformer with secondary inductance' model, but at the phase-crossover (secondary network pseudo-resonance) frequency, the inductance required becomes infinite. Hence, in order to make measurements anywhere near the crossover point, a large variable inductance is required, and to produce such an inductance without introducing further non-idealities is all but impossible.
     The Douma bridge is a different proposition however. Here the potential divider is a pair of capacitors, and the LF-compensation component is a resistor. Reasonably well-behaved variable capacitors and resistors are realisable, and test-equipment for measuring resistance and capacitance is commonplace. Hence the Douma bridge is the obvious candidate for the perturbation method.
     An important aspect of basic experimental design is that of how to find out when the bridge is balanced. The issue here is that the null-point must be located accurately at each measurement frequency in order to minimise the scatter in the data. If a diode detector is used, the null will be difficult to locate because the forward-threshold effect makes such detectors insensitive to small voltages. Running the test with high power levels in an attempt to overcome this limitation will tend to make the bridge components hot and introduce the problem of thermal drift. The solution is to construct the bridge in such a way that any passive detector can be removed and a radio receiver substituted in its place. A laboratory signal generator or VFO with an output of around 0dBm (224mV) can then be used instead of a radio transmitter. In the experiments to be described here, null depths of 80 to 120dB were typical, with the signal at balance falling below the noise floor of a good HF communications receiver (<100nV).

2. The Douma bridge:
Shown below is the circuit of Douma's bridge (US Pat. No. 2808566) terminated by its target load resistance R0. The bridge is balanced when the current analog Vi is equal in magnitude and phase to the voltage analog Vv, the two voltages being arranged in series opposition. The current transformer is tightly-coupled and is therefore largely described by its turns-ratio N and its secondary inductance Li. It has a finite insertion impedance Zii , which causes a voltage-drop Vii in the path to the primary load. The secondary winding is terminated in a resistance Ri, which serves to damp, but cannot eliminate, the effects of transformer reactance. A component of the transformer output voltage in phase with the primary current is cancelled by choosing an appropriate ratio for the potential-divider capacitors C1 and C2. A leading quadrature component in the transformer output, due to the finite secondary inductance, is cancelled by introducing a corresponding quadrature component in the potential-divider output by appropriate choice of the resistor Rv. Also shown is a parasitic capacitance Ci, which notionally represents the transformer propagation-delay and the strays across the secondary winding; but is, as will be shown later, also a composite of the effects of the various parasitic reactances in the system. This capacitance (when positive) produces a lagging quadrature component in the transformer output, which increases in magnitude as the frequency of operation is increased, and will at some point overwhelm the effect of the inductance.



The Douma bridge, as shown, has no means of compensation for the parasitic capacitance Ci, save for the use of a relatively low value of resistance for Ri. Hence the design equations for the bridge have to be derived on the basis that Ci is negligibly small. The procedure is to write dimensionless expressions for the voltage and current analogs and set them to be equal when the bridge is terminated in its target load resistance R0. The resulting complex expression is then separated into its real and imaginary parts, the real part being the in-phase balance condition, and the imaginary part being the quadrature balance condition. Note that the transformer losses, which can be represented as a resistance (Rk say) in parallel with the secondary winding, have been neglected at this stage of the analysis. The slight shortfall in output can be compensated by a trivial adjustment of the voltage sampling potential-divider ratio, and may be quantified if so desired by replacing Ri with an effective value Rik=Ri//Rk

The output of the current transformer, by equating primary and secondary Ampere-turns, is given by:
Vi = I Zi / N     where:    Zi = (Ri // jXLi // jXCi)
but
I = V / R0
hence
Vi = V Zi / (N R0)
This gives the dimensionless current transfer function at balance:
Vi / V = Zi / (N R0)

The output of the voltage sampling network is derived as for any potential divider network:
Vv = V' (jXC1 // Rv) / [ jXC2 + (jXC1 // Rv) ]
and a more convenient form is obtained by multiplying top and bottom of this expression by jXC2:
Vv = V' (jXC1 // jXC2 // Rv) / jXC2
but note that V' is not the same as V.
V' = V + Vii  =  V + I Zii  =  V + V Zii / R0  = V (1 + Zii / R0 )
where the insertion impedance Zii is the secondary load impedance reflected back into the primary, i.e., it is the secondary load impedance divided by the square of the turns ratio, hence:
V' = V [1 + Zi / (R0 N²) ]
and the dimensionless voltage transfer function is:
Vv / V = [1 + Zi / (R0 N²) ] (jXC1 // jXC2 // Rv) / jXC2

The overall balance condition is given by equating the voltage and current transfer functions:
[1 + Zi / (R0 N²) ] (jXC1 // jXC2 // Rv) / jXC2 = Zi / (N R0)
This can be simplified by dividing both sides by 1+Zi/(R0N²) and inverting the whole expression:
jXC2 / (jXC1 // jXC2 // Rv) = [1 + Zi / (R0 N²) ] N R0 / Zi
Multiplying out the right-hand side gives:
jXC2 / (jXC1 // jXC2 // Rv) = ( N R0 / Zi ) + 1/N
and noting that Zi=(Ri // jXLi // jXCi):
jXC2 / (jXC1 // jXC2 // Rv) = [ N R0 / (Ri // jXLi // jXCi) ] + 1/N
This can now be separated into real and imaginary parts by observing that:
1/(a // b // c // . . . ) = (1/a) + (1/b) + (1/c) + . . .
(see section 1-14); but before we do that we will combine the transformer reactance into a single quantity, i.e.;
Xi = XLi // XCi
The reason for so doing is that the circuit has no provision for compensating for the effect of Ci, and so a frequency-independent set of balance conditions can only be obtained insofar as Xi can be treated as a purely inductive reactance (i.e., at low frequencies where XCi®-¥). When Xi is inductive, we can consider it to arise from an inductance Li', i.e.:
Xi = 2pf Li'
and bridge-balance depends on the approximation that Li' does not vary with frequency. Now the balance identity becomes:
jXC2 / (jXC1 // jXC2 // Rv) = [ N R0 / (Ri // jXi) ] + 1/N
and the reciprocal impedance (admittance) factors separate into arithmetic series:
jXC2 [ (1/jXC1) + (1/jXC2) + (1/Rv) ] = N R0 [ (1/Ri) + (1/jXi) ] + 1/N
Multiplying out, and noting that XC=-1/2pfC gives:
(C1/C2) + 1 + (jXC2 / Rv) = (N R0 / Ri) + (N R0 / jXi) + 1/N
Equating reals:
(C1/C2) + 1 = (N R0 / Ri) + 1/N
Hence, the in-phase balance condition is:
C1 = C2 [ (N R0 / Ri) + (1/N) - 1]

(2.1)
Equating imaginaries:
jXC2 / Rv = N R0 / jXi
and noting that capacitive reactance is negative and 1/j = -j :
-XC2 / Rv = N R0 / Xi
Expanding the reactances:
1 / (2pf C2 Rv ) = N R0 / (2pf Li' )
which rearranges to give the quadrature balance condition:
Rv = Li' / (N R0 C2)

(2.2) 

Now notice that both C1 and Rv depend on the choice of C2. Consequently, given that N and R0 are fixed parameters during any test; if C2 is chosen to be a fixed parameter (ignoring its parasitic inductance for the time being), then C1 is a measure of the effective value of Ri, and Rv is a measure of Li'. With C2 fixed, any adjustments of C1 and Rv are non-interactive, i.e., data relating to the in-phase and quadrature balance conditions can be separated and analysed independently (at least, to the approximation that C2 does not change with frequency).

3. Prototype test bridge:
On the basis of the preceding discussion, a Douma bridge was constructed in such a way that C1 and Rv could be both adjusted and measured. The bridge, in its earliest form, is shown in the photograph below. Some important initial considerations were that the transformer Faraday shield should be earthed at the detector socket (see Appendix A6.3), that the three ports should be reasonably shielded from each other, and that the voltage and current sampling networks should be mounted on opposite sides of a metal bulkhead. Experiments were conducted using a signal generator having an output of about 0dBm (224mV), and the detector was a Kenwood TS930s short-wave transceiver used in USB mode (3KHZ bandwidth, noise floor circa 100nV) with the AGC set to 'fast' . The basic arrangement proved to be adequate in that the bridge could always be nulled to the point where the detector signal fell below the receiver noise.
     In the first version, the resistor Rv was a 500W carbon skeleton potentiometer with a fixed resistor in series; measurement of its value being effected by connecting clip-on probes from a resistance meter across it, with the radio receiver unplugged. The problem of how to measure the capacitance C1 however, is a little less straightforward. An adjustable capacitance is shown mounted on a plug-in header so that it can be taken away and measured using a laboratory bridge. C1 however is composed of this capacitance plus strays; and so the plug-in capacitance is designated C1'. Hence:
C1 = C1' + C1s
C1s is the stray capacitance due to the socket, the wiring, ceramic insulating pillars, the capacitance of Rv and the capacitance to ground of the Faraday shield. It amounts to several pF and can only be determined by estimation using transformer loss data from other sources (see Appendix 6.1). Hence the experiment permits accurate relative measurements of C1, but the uncertainty increases somewhat for absolute measurements. This limitation is not serious for the present purpose however, because the measurement of transformer losses is not part of the experimental objective. All capacitors used were either silvered-mica or air-spaced, and had negligible ESR in comparison to external circuit impedances.




Silver-mica capacitor and
2.5 - 30pF trimmer.

To commission the bridge, a current transformer was made using an Amidon FT50-61 ½" ferrite bead (AL = 68.8nH/turn² nominal), with 12 turns of 0.9mm diameter enamelled copper wire for the secondary winding ,and a stub of URM108 (Ag-PTFE 50W) cable for the primary. The fixed circuit parameters are listed on the right, where Lsec is the measured secondary inductance, expected to be a few % higher than Li due to leakage inductance and neglect of self-capacitance, but nevertheless providing a first estimate.

N = 12
Lsec = 9.25 ±0.23 mH
Ri = 49.80 ±0.06 W
R0 = 49.96 ±0.06 W
C2 = 10.0 ±0.2 pF

     Resistances were measured using a Fluke 8060A 4½-digit multimeter. Inductance and capacitance were measured at 1.5915MHz using a Hatfield LE-300A/1 TRAB. The capacitor C2 was a silvered-mica type with a nominal tolerance of ±5%, but this was reduced to ±0.2pF or better by selection of the median sample from a batch of 5.
     According to equation (2.1), the expected value for C1, neglecting transformer losses, is:
C1 = C2 [ (N R0 / Ri) + (1/N) - 1] = 111.2pF
     According to equation (2.2), on the basis that Lsec is approximately equal to Li and the secondary capacitance is negligible, an initial estimate for Rv is:
Rv = Lsec / (N R0 C2) = 1543W
     The first dataset to be acquired is listed and analysed in the spreadsheet testbrg61-12_1.ods, measurements being made at points over a range from 1.4 to 15MHz. The value of Rv required to balance the bridge remained reasonably constant at around 1520W at low frequencies, allowing nulls to be found using a 500W variable in series with a 1.2KW fixed resistor. Rv started to increase rapidly above about 4MHz however, and at this stage the fixed resistor had to be changed at each new frequency. No resistance measurements were made above 13MHz because nulls became difficult to locate even with the variable resistor changed to 5KW, and the point at which balance was achieved with Rv disconnected was found to lie somewhere around 14MHz.
     Variation of Rv with frequency is, of course, expected in the event that the parasitic capacitance Ci is finite. Ci adds a lagging component to the transformer output in opposition the leading component associated with the inductance. The result is that the inductance appears to increase with frequency up to the phase crossover point, above which the secondary reactance becomes capacitive. Allowance for this was made in the initial theoretical considerations, where we allocated the symbol Li' to the apparent inductance and defined it in the relationship:
Xi = 2pf Li' = XLi // XCi

(3.1)
Experimentally, Li' is obtained from equation (2.2) rearranged thus:
Li' = N R0 C2 Rv

(3.2)
A graph of Li' vs frequency calculated from the experimental data is shown below superimposed on the same graph calculated from a regression analysis (discussed below)..



Note that the graph should not be interpreted to mean that the bridge becomes wildly inaccurate above 4MHz. On the contrary, the phase error due to Ci is only about 3° at 15MHz, and the graph simply shows that the experiment is extremely sensitive.
     The Li' vs frequency data were analysed in order to extract the 'true' secondary inductance Li and the nominal shunt capacitance Ci (see testbrg61-12_1.ods). The procedure used begins with the combination of equations (3.1) and (3.2).
2pf N R0 C2 Rv = XLi // XCi
Inverting this expression gives:
1 / (2pf N R0 C2 Rv) = [1 / (2pf Li) ] - 2pf Ci
Multiplying top and bottom of the right-most term by 2pf and cancelling through then gives:
1 / (N R0 C2 Rv) = (1 / Li) - (2pf)² Ci

(3.3) 
This is in the form y=a+bx and so can be subjected to a linear regression analysis (see Appendix 2). A suitable breakdown into variables and parameters is:
y=1/(NR0C2Rv),  a=1/Li,  b=Ci,  and  x= -(2pf)².
A weighted linear regression procedure was used (see Appendix 2, section 11) in order to allow for the non-linear scaling of the uncertainties in Rv due to its reciprocal relationship with y. The estimated standard deviation of a measurement of Rv (sRv) was taken to be 1%, not due to the uncertainty in the meter reading but on the basis of repeatability; the issue being that, for this crude early version of the experiment, there was some ambiguity regarding the exact balance point. The figure of 1% was arrived at by repeating a few measurements and noting the scatter of results. Observe also that the fitting weight for a y value is determined only by the effect of Rv on its precision. It does not depend on the overall accuracy of a y value because the uncertainty contributions from R0 and C2 are correlated over the whole dataset and so do not contribute to the weight (although they do, of course, contribute to the uncertainties in the derived parameters Li=1/a and Ci=b). The precision of a y value, sy, is given by:
sy = |y/Rv| sRv
where
y/Rv = -1/(NR0C2Rv²)
The fitting weight is the reciprocal of the square of the precision:
w = 1/sy²
The fit gave a reduced c² of 1.3 on 17 degrees of freedom; indicating that, within the precision of the data, the hypothesis that the transformer secondary reactance can be modelled as an inductance in parallel with a capacitance is validated (i.e., the shape of the graph of Li' vs f given above, is so-far accounted for by the theory). The transformer parameters obtained from the fit were:
Li = 9.006 ±0.026 mH
Ci = 14.11 ±0.06 pF
but note that the uncertainties from the fit represent only the precision of the data. The true uncertainty of a parameter is greater than the precision, because the contributions from to the uncertainties in R0 and C2 have not been taken into account; and also because these are the results of an experiment which has yet to be investigated for hidden sources of systematic error. It will transpire that the experiment is in need of improvement, which means that the results obtained at this stage are not trustworthy; but if we treat this as an exercise which serves to establish the basic technique (and the spreadsheet template for the data analysis), then it is sensible to determine the parameter accuracies. To do so analytically is somewhat difficult, because of the various correlations involved, but there is a simple solution to the problem. The spreadsheet (testbrg61-12_1.ods) was written in such a way that the input parameters R0 and C2 are taken from a single cell in each case. Hence it is possible to shift these parameters, one at a time, to the upper and lower limits of their standard deviations and note the changes in Li and Ci which occur. The RMS average deviation in (say) Li, which occurs when (say) R0 is shifted, is the error contribution to Li from the uncertainty in R0, i.e., it is, to a very good approximation, (Li/R0)sR0. The other error terms (Li/C2)sC2, (Ci/R0)sR0, and (Ci/C2)sC2 are collected in similar fashion. All we need then are the error contributions from the experimental uncertainties, i.e., (Li/Rv)sRv and (Ci/Rv)sRv, but these are already available as the estimated standard deviations (precision) obtained from the fit. The input parameter and fitting uncertainties sR0, sC2, and sRv are uncorrelated, and so their contributions to sLi and sCi are orthogonal (see Appendix 2 section 6). Hence:
sLi = Ö{ [(Li/R0)sR0]² + [(Li/C2)sC2]² + [(Li/Rv)sRv]² }
sCi = Ö{ [(Ci/R0)sR0]² + [(Ci/C2)sC2]² + [(Ci/Rv)sRv]² }
Using this approach, the determined transformer parameters become:
Li = 9.01 ±0.18 mH
Ci = 14.1 ±0.1 pF

4. Parasitic inductance of the adjustable capacitor:
Although the variation of Rv with frequency so-far supports the 'ideal transformer with parallel reactance' model for the transformer under test, the data for the adjustable capacitor C1' were not so well behaved. Evidence of deviation from the model can be seen in the graph of C1' vs frequency below (see sheet 2 of testbrg61-12_1.ods); C1' in this instance being the value of the detachable part of C1 as given by measurement on a laboratory bridge operating at 1.5915MHz.



Recall that C1 is given by equation (2.1) as:
C1 = C2 [ (N R0 / Ri) + (1/N) - 1]
It is apparently related only to fixed system parameters and so should not vary with frequency, and yet according to the graph it drops by about 4% over the range covered. One possible but unlikely explanation is that the apparent efficiency of the transformer could be increasing as the frequency rises (Recall from the earlier discussion that Ri can be replaced with Rik=Ri//Rk if efficiency needs to be taken into account). Such an effect is possible in the event of a severe mismatch in the primary transmission line (see section 14), but measurements of the relative amplitude vs frequency response of a similar transformer over a 1.6 to 30MHz frequency range did not show it (see Appendix 6.3). Hence that hypothesis has to be rejected. The only sensible conclusion is that the effective value of C1 (and by inference also C2) is changing with frequency, i.e., the inductance of the potential divider network is not negligible.
     When a capacitor has a significant parasitic series inductance, the inductive reactance cancels some of the capacitance reactance, causing the effective capacitance to increase with frequency. Hence, according to this second hypothesis, the setting of the capacitor designated C1' has to be backed-off as the frequency is increased, giving the impression that the capacitance required to balance the bridge has fallen. If C1 varies with frequency, then of course, so must C2, but there is reason to believe that the variation of C1 will be substantial, whereas the variation of C2 will not. The point is that C2 is a small capacitance (10pF) and therefore, over the frequency-range of measurement, will always have enough reactance to keep the inductance in its arm of the circuit at bay. An educated guess for the inductance in series with C2 for the test bridge is that it will be somewhere in the region of 50nH. At a frequency of 10 radians/sec (15.915 MHz) a 10pF capacitance has a reactance of -1000W, whereas an inductance of 50nH has a reactance of only +5W. Hence the deviation from pure capacitance for C2 over the range from 0 to 16MHz is only about 0.5%. C1 however, is a different matter. A 102pF capacitance has a reactance of -98W at 10 radians/sec, and so a series inductance of 40 to 50nH could well account for the observed 4 to 5% deviation from the model. The test of this reasoning however, is to see how well it agrees with the data; and to this end we must define the problem correctly. We will start by allocating the following symbols:
  C1' is the value of plug-in capacitance required to balance the bridge, and is assumed not to vary with frequency.
C1'm is the capacitance of the plug-in capacitor measured at 1.5915MHz (the raw data).
C1'0 is the true capacitance of the plug-in capacitor at zero frequency.
L1 is the inductance of the plug-in capacitor and its associated connector and wiring.
Recall that the total capacitance required is defined as:
C1 = C1' + C1s
Here we will make the reasonable assumption that the stray capacitance does not have significant inductance and is therefore constant. Hence if we can prove a condition which makes C1' invariant, then C1 is invariant and the transformer model requires no modification.
     When the plug-in capacitor is connected to the Douma bridge, its reactance is given by:
XC1' = XC1'0 + XL1
i.e.:
-1/(2pf C1') = -[1/(2pf C1'0)] + 2pf L1
Multiplying both sides by -2pf gives:
1/C1' = (1/C1'0) - (2pf)² L1      . . . . . . (4.1)
Similarly, when the plug-in capacitor is mounted on the measuring bridge, which operates at
10 radians/sec (1.5915MHz) we obtain:
1/C1'm = (1/C1'0) - 10 L1     . . . . . . . (4.2)
This assumes that the inductance L1 does not change when the capacitor is transferred, for which reason the measuring bridge was fitted with a socket identical to the one on the test bridge. Also note that the inductive reactance correction is small at 1.5915MHz, so that any error in this assumption will have negligible effect. Now, subtracting equation (4.1) from equation (4.2):
(1/C1'm) - (1/C1') = -10L1 + (2pf)² L1
i.e.:
(1/C1'm) = (1/C1') + [(2pf)² - 10] L1
This is in the form y=a+bx, with y=1/C1'm, a=1/C1', b=L1, and x=(2pf)² - 10. As before, a weighted linear regression procedure was used in order to cope with the non-linear scaling of uncertainties when computing y from C1'm. (see sheet 2 of testbrg61-12_1.ods). As may be noted from the data, the C1'm values were recorded to the nearest 0.5pF. Hence, due to the rounding error, this gives the precision of the data as ±0.25pF, i.e., sC1'm was taken to be 0.25pF. sy is given by:
sy = |dy/dC1'm| sC1'm
where
dy/dC1'm = -1/C1'm²
     The fit gave a reduced c² of 1.15 on 19 degrees of freedom, confirming that the apparent variation of C1' with frequency is an artifact. The graph below illustrates the point by showing the smooth curve underlying the rounded measurements. The derived circuit parameters were:
C1' =102.32 ±0.07 pF
L1 = 50 ±2 nH
the quoted uncertainties representing precision, not accuracy.



The inductance L1 obtained from the fit is, incidentally, slightly smaller than the true inductance of the capacitor because the inductance of the upper voltage sampling arm has been neglected.

5. Stray Capacitance:
Equation (2.1) gives C1 as:
C1 = C2 [ (N R0 / Ri) + (1/N) - 1]
but if we wish to take transformer efficiency into account, then Ri should be replaced by:
Rik = Ri // Rk = k Ri
Hence a refined prediction for C1 is given by the expression:
C1 = C2 [ (N R0 / k Ri) + (1/N) - 1]
with uncertainty
sC1 = Ö{ [(C1/C2)sc2]² + [(C1/R0)sR0]² + [(C1/k)sk]² + [(C1/Ri)sRi]²}
where
C1/C2 = [(N R0 / k Ri) + (1/N) - 1]
C1/R0 = C2 N / (k Ri)
C1/k = -C2 N R0 / (k² Ri)
C1/Ri = -C2 N R0 / (k Ri²)		 [dimensionless]
[Farads/W]
[Farads]
[Farads/W]
     An efficiency measurement on a 12-turn current transformer wound on a FT50-61 bead (see Appendix 6.1) gave k=0.959 ±0.011. Hence, using the formulae and derivatives given above, our best prediction for C1 is:
C1 = 116.4 ±2.8 pF
(for calculation, see sheet 3 of testbrg61-12_1.ods)
     The previous investigation gave C1' =102.318 ±0.07 pF. The uncertainty in this case is the precision from the fit. A more realistic standard deviation must also take the calibration uncertainty of the measuring bridge into account (about 1%), which gives C1' =102.3 ±1.0 pF. The estimated stray capacitance is the difference between the predicted C1 and this value, i.e.:
C1s = C1 - C1'
and
sC1s = Ö [sc1² + sc1'²]
giving:
C1s = 14.0 ±2.9pF
This value appears to be somewhat high in view of the bridge layout, and so raises the issue of a possible systematic error.

5a. Transformer constant.
The quantity [ (N R0 / k Ri) + (1/N) - 1] is a circuit constant which defines the ratio C1/C2 and also the derivative C1/C2. As was outlined above, it is used for error analysis and for estimating the stray capacitance C1s. Since it involves only parameters related to the transformer, it will be referred to from here on as the transformer constant.
KT = C1/C2 = (N R0 / k Ri) + (1/N) - 1

(5.1)
This version of the transformer constant is correct for Faraday-shielded bridges. A modified version for unshielded bridges is given in section 19 (equation 19.9a)

6. Test procedure optimisation:
In the matter of demonstrating the feasibility of the proposed bridge evaluation method, the prototype served its purpose. Experimentally however, it left a lot to be desired; and further investigation, triggered by a slightly high value for C1s, uncovered several sources of systematic error. The various problems and their solutions are discussed below.

The need to change resistors to track the variation of Rv with frequency was both tedious and potentially disruptive to the circuit layout. The solution was to replace the Rv arm with two plastic-bodied 2W Cermet potentiometers (Vishay 93R1A-R22-xxx) in series, one having a value of 250W (fine) and the other having a value of 25KW (coarse). The potentiometers were initially mounted on a metal plate, screwed to the jig; but it was found that adjustment of the fine control was accompanied by a small change in the capacitance of the arm, making nulls difficult to locate. The solution was to mount the pots away from any conducting surfaces.
     The need to disconnect the radio receiver and connect test-leads to measure Rv was once again tedious and potentially disruptive. In looking for a resolution to this issue, it was first noted that the resistance of the transformer secondary-winding is small in comparison to Rv; and so a more reliable connection can be made by fitting the resistance-meter with a BNC to 4mm adapter, and then connecting it to the detector port via a short length of coaxial cable. A further refinement resulted from the observation that; if the input to the radio receiver could be made a DC open-circuit, then the line to the receiver could be left connected and the resistance measured via a T-piece. The required DC-block was implemented by mounting a capacitor between two BNC connectors, as shown below, and inserting it into the coaxial line.


DC blocking capacitor.
10nF 1KV Ceramic.

The BNC connectors are joined using threaded Ni-plated brass pillars. The final assembly is wrapped in aluminium foil and covered by heat-shrink tubing.

Breaking the DC path to the receiver and measuring Rv via a T-piece lead to the discovery that the resistance could be measured continuously, i.e., with the RF on and while searching for nulls. This certainly works when using the Fluke 8060A, and will probably work with other meters. No sign of sampling noise could be heard in the receiver, and the only interference was a pure tone of less than 1mV at 16.00MHz and another at 3.20MHz (presumably sub-multiples of the 8060A's clock oscillator), which could be eliminated by switching the meter off whilst balancing the bridge at those frequencies. The resistance reading did not change at any of the measurement frequencies when the the signal-generator was switched off and on. The Fluke 8060A did not even care when the generator and receiver connections were swapped to test for bridge reciprocity (but some digital multimeters are known give misleading results when making DC readings in the presence of superimposed RF). There was, furthermore, no change in the balance settings on plugging-in and removing the meter.

Setup for continuous measurement of Rv.

Also tedious, and requiring considerable care, was the business of unplugging the reference capacitor and measuring it on a separate bridge. The solution in this case was to make up an assembly consisting of an air-spaced variable capacitor and a reduction-drive; the control knob and the capacitor being separated by insulating pillars and a non-conductive drive-shaft, so that adjustment could be accomplished without proximity of the operator's hand. The capacitor was fitted with a connector so that it could be taken away and measured as before, but the point in this instance was to take a series of readings of capacitance versus dial-setting and fit them to a regression function. From then on, measuring capacitance became a matter of noting the dial-readings during an experimental run; the nominal (1.5915MHz) capacitance being obtained later by entering the data into a spreadsheet and applying the formula. Details of the data reduction are given in the spreadsheet file: capcal_p500.ods.




Philips dual-gang 500pF variable capacitor and General Electric 50:1 anti-backlash worm drive (capcal_p500.ods). The drive-shaft is machined from 19mm diameter acrylic bar for stiffness and the assembly is clamped to a baseboard in use to prevent torsion of the mounting frame. Mounting plates and pillars are made from non-conductive materials. Only one gang of the capacitor is connected, but both can be used if a large capacitance is required. Wiring to the connector is via copper straps to provide rigidity and minimise inductance.

One serious problem with early versions of the 500pF variable capacitor assembly was backlash. The first version had two flexible couplers, but these permitted a twisting motion which made the backlash so severe that accurate capacitance measurement was impossible. The problem was reduced by using one straight coupler and one flexible coupler, but even then, due to the elasticity of the 6mm diameter plastic drive-shaft fitted at the time, the backlash remained at ±5 divisions of the vernier dial. Finally, the backlash was reduced to ±1 division by using a straight drive-shaft made from solid 19mm diameter acrylic bar. To allow for the residual backlash, all measurements were made by approaching the null from the clockwise direction of the control knob.
     The 500pF variable capacitor is useful for tracking the impedance meanderings of uncompensated bridges, but is necessarily limited in resolution on account of its large variation range. A second variable reference capacitor was therefore constructed, this time with a 40pF range and a 36:1 reduction drive, with a socket for a plug-in padding capacitor. This device fortuitously gave exactly 2.5pF per turn in its linear region, with a resolution of better than 0.025pF and no detectable backlash (capcal_8-48.ods).


40pF variable reference capacitor with 36:1 anti-backlash worm drive (capcal_8-48.ods). A socket is provided for a padding capacitor


7. Balance disturbances due to common-mode currents:
An impedance bridge is a linear reciprocal network. Therefore, if it is configured correctly, the balance condition should not change when the generator and detector are swapped. Note that when the generator is connected to the 'detector' port, it requires a terminating resistor, which can be provided on a T-piece near to the bridge.

Reciprocity test.

The cable to the DMM is kept short (ca. 100mm) to avoid excessive mismatch.

It was discovered, during the course of the optimisation work, that the bridge did not behave in an exactly reciprocal fashion. Significantly different results were obtained with the generator and radio-receiver connections swapped, and the results were different again if the generator was left un-terminated. The capacitance setting was seen to change by 3 to 4%; and the resistance setting changed by about 4% at low frequencies and considerably more around the phase-crossover frequency where Rv is heading for infinity. Such deviations are not acceptable if any physical meanings are to be attached to the derived parameters; and can seriously skew the values obtained for Ci and C1s because these are dependent on small changes in other parameters.
    After some deliberation it was concluded that the problem was occurring because the radio receiver does not have infinite common-mode rejection, i.e., it can pick up radio signals travelling on the outside of the antenna cable by virtue of the voltage developed between the antenna socket and the mains wiring. When this problem is related to the situation in which one piece of mains-powered equipment feeds signals into another, it is known, in old parlance, as an "earth loop". The traditional solution, before the invention of 'Health and Safety at Work', was to disconnect the earth-wires from the mains plugs; and although this is unlikely to work, because there is still plenty of capacitance between the chassis and the live and neutral wires, it does add excitement to otherwise dull activities by introducing the risk of lethal electric shock.
     The hypothesis for the case of a bridge which fails to obey the reciprocity rule is that the receiver is picking up a small signal from the generator via the common mode. Thus, when a null is obtained, it represents not exact balance, but a situation in which the desired differential-mode output signal is of the correct magnitude and phase to cancel the common-mode signal. The common mode signal changes depending on how the generator is terminated and the bridge connected, and so the balance-condition appears to change according to the external configuration. A test of this hypothesis is to see what happens when the common-mode current is disrupted; not by indulgence in illegal wiring practices, but by the insertion of one or more common-mode chokes into the loop.
     Two common-mode chokes (1:1 unun transformers) were constructed as shown on the right. They were made using ferrite toroids of unknown origin, the sleeves having an AL value of 2.5mH, and the rings having an AL value of 400nH, both determined at 1.5915MHz by winding a few turns of wire and measuring the inductance. Both chokes have three sections of differing inductance adding up to 47.5mH (X=+475W at 1.5915MHz), and were made deliberately different from each other; the point being that a series resonance in any section can never bring the total reactance to zero.


Common-mode chokes.
Left: 5+30+12.5mH
Right: 10+30+7.5mH
The cable is URM108
     It was found that the reciprocity error fell within the measurement repeatability at low frequencies with a single choke in either the lead from the generator or the lead to the receiver. It was also discovered that the balance point could be shifted by connecting a jumper lead across the choke (i.e., by shorting-together the outer bodies of the two BNC plugs), thereby directly confirming the common-mode signal hypothesis. Inserting chokes in both sides gave a lesser but still worthwhile improvement. Note that there is a general point here relating to the extraction of a signal from any bridge for processing by mains powered circuitry, or likewise for feeding a signal into a bridge for purposes such as quiet tuning [35][36]. If the auxiliary electronics is earthed in any way; then for best accuracy, a common-mode choke should be used to isolate it from the main transmission line.


Optimised experimental configuration. Common-mode chokes (47.5mH each) are connected on both sides. The inductance balance coil (discussed in Section 17) is used for bridge optimisation, but not during the determination of basic circuit parameters.


  Bridge with 500pF variable capacitor fitted.


Bridge with 40pF variable capacitor and padding capacitor fitted.



Close-up view of the voltage-sampling network. To minimise interaction between the in-phase and quadrature balance adjustments; the potentiometer mounting bracket is made from polypropylene and the network is earthed at the detector socket via a copper strap.

8. Reciprocity error due to radiated signals.
After elimination of the common-mode signal, and after changing the transformer secondary load resistor from two 100W resistors in parallel (about 1pF parallel capacitance) to a single 49.9W resistor (about 0.4pF), the apparent phase-crossover frequency for the 12-turn transformer went up from 14.1MHz to 22MHz. At 21MHz, with Rv still in range of the two Cermet pots, it was found that there was still some residual reciprocity error. The error was tiny in phase terms, but sufficient to affect Rv by several KW in this critical region.
     In the process of investigating this problem it was discovered that, having balanced the bridge to the point where the signal had disappeared into the receiver noise; a weak signal (not sufficient to move the S-meter) would reappear on unplugging the cable from the radio receiver. Hence the bridge balance point was being skewed slightly by a radiated signal picked up by the cable. The signal became stronger on moving the receiver cable close to the generator cable.
     A major part of this spurious sensitivity was traced to a nickel-plated connector at the receiver antenna socket (several milliOhms resistance at the shield connection). Replacing the offending connector with a silver-plated version reduced the signal by an order of magnitude. The very weak residual signal was then attributed to the imperfect shielding provided by the RG58 cable in use at the time. When the receiver cable was changed to quadruple-shielded Belden 9880 10Base5 Ethernet cable there was no detectable pickup from the generator with the cable unplugged from the bridge, regardless of proximity to the generator cable. The bridge showed perfect reciprocity within the measurement repeatability at all frequencies after this substitution.

9. Capacitance of the reference load resistor.
A coaxial resistor, for all its pretensions, is just a resistor in a tube; and low value resistors (presuming that they are properly designed for RF applications) tend to be capacitive at high frequencies. As will be demonstrated in Section 14, a correction for the secondary parallel capacitance of the current transformer can be had by connecting a capacitor across the load port. Therefore, any parasitic capacitance across the reference load gives the impression that the 'self capacitance' of the transformer is less than it really is.
     With the test bridge operating at a frequency just below the phase crossover point (21MHz), a variety of load resistors of differing sizes and power-ratings were tried in order to see the effect on Rv. Observe here that load resistors with residual capacitance have the effect of shifting the apparent phase crossover to higher frequency; i.e., the load with the least capacitance (the best reference load) gives the lowest measured crossover frequency. Rv was found to vary over the range from about 7KW to 16KW depending on the load, but note that this only corresponds to about 0.1° change in phase angle. Perhaps not surprisingly, the best resistor (the one giving Rv=16KW) was also physically the smallest.
     In the absence of evidence to the contrary, the best selected reference load resistor has to be assumed to be a pure resistance. Any residual reactance adds uncertainty to to the derived secondary parallel capacitance Ci.

10. Post optimisation test data:
After completion of the optimisation work, two new datasets for the 12-turn FT50-61 current transformer were obtained. The first experiment was conducted using the 500pF variable reference capacitor, and the second with the 40pF (8-48pF) variable capacitor and a 68pF padding capacitor to bring it into range. The data analyses were performed as described in previous sections, the only difference being that values for C1'm were obtained from capacitor dial-readings by means of the appropriate fitting function (see sheet 4 of spreadsheets testbrg61-12_2.ods and testbrg61-12_3.ods). The value for C2 was amended to 10.3±0.2pF for both of these fits on the basis of an estimated 0.3pF of strays across the upper voltage-sampling arm.
     In both experiments, the data for C1' were fitted perfectly on the assumption that the non-ideality of the reference capacitor can be modelled as a series inductance. In both cases, the graph of residuals (observed minus calculated) was chaotically scattered around zero, indicating that what is is left over after the fitting process is mainly noise. The 500pF reference capacitor turned out to have an effective series inductance of 62nH and a precision of ±0.14pF (obtained by adjusting the ESD of an observation to obtain a reduced c² of about 1). The 8 to 48pF variable capacitor with 68pF in parallel turned out to have a effective inductance of 81nH and a precision of 0.08pF. As mentioned previously, the inductances obtained from the fits are slightly lower than the true values due to the neglect of the inductance associated with C2.
     The extracted voltage-sampling network parameters are summarised below. Note that the accuracy of the capacitance measurement (about ±1.1pF) is a lot worse than the precision, the overall uncertainty being responsible for the differences between C1' and C1s for the two experiments; but for bridge evaluation purposes, it is the change in capacitance rather than the absolute capacitance which must be determined.

 

500pF ref. cap.

40pF ref. cap.
  
Data file

testbrg61-12_2.ods

testbrg61-12_3.ods
 See sheets 2 and 3
Parameter

Value ±ESD

Value ±ESD

.

C1'

109.0 ±1.1 pF

107.3 ±1.1 pF
 Corrected for L1.

L1

62.3 ±0.5 nH

80.8 ±0.3 nH
 Effective inductance of the ref. cap.

C1s

10.8±3 pF

12.5 ±3 pF
 Stray capacitance.

     The data for Rv were not so well behaved and show a clear deviation from the model in the 14 to 21MHz region, as can be seen in the graph of Li'=NR0C2Rv vs. frequency reproduced below.



It was found for both experiments, that if the uncertainty of an Rv measurement was taken to be 0.25%, then the datasets could be fitted up to 14MHz with a reduced c² (c²/n) of just less than 1. Trying to fit all of the data however, pushed the c²/n to about 60, indicating a failure of the model (or perhaps, given the faith nowadays placed in circuit simulations, a failure of reality). Interestingly, if the data were only fitted up to 5MHz, the c²/n went to 1 for an ESD of observation of about 0.06%, which happens to be the same as the stated accuracy of the Fluke 8060A used for the resistance measurements. This would seem to indicate that the balance-point determinations were very nearly exact, and that something not described by the model is happening above 5MHz.
     There are several possible sources for the anomaly, all of which may be active to some extent:
High-order effects due to circuit parasitic reactances.
Failure to include transformer leakage inductance and winding resistance in the model.
Breakdown of the lumped-component approximation (transmission-line effect).
Dispersion in the permittivity associated with the transformer 'self-capacitance' (variation of velocity factor with frequency).
Dispersion in the permeability, and hence AL, of the transformer core, causing Li to vary with frequency.
     In looking for evidence of dispersive effects, we can observe that every measurement of Rv can be regarded as an indirect measurement of Ci if Li is a constant, or as an indirect measurement of Li if Ci is a constant. Hence, if Li is set to an estimated value, we can use the observed value of Rv at a particular frequency to compute a corresponding value for Ci. Similarly, if Ci is set to an estimated value, we can use the observed value of Rv to compute a corresponding value for Li. The required relationship for such computations was given earlier as equation (3.3):
1 / (N R0 C2 Rv) = (1 / Li) - (2pf)² Ci
Rearranging this in favour of Ci gives:
Ci = [ (1 / Li) - 1 / (N R0 C2 Rv) ] / (2pf)²     . . . . . (10.1)
The result of computing Ci in this way is shown in the graph below. The upper curve uses Li=8.970mH, which is the inductance obtained by fitting all of the data up to 14MHz. The lower curve uses Li=8.982mH, which is the inductance obtained by fitting all the data up to 5MHz.



The capacitance graphs are chaotic below 3MHz because the small amount of information in each observation amplifies the measurement uncertainty. Ci begins to assert itself above 3MHz however, and soon becomes well defined. What we see then is a hump in the apparent capacitance between about 6 and 20MHz, and this is indeed characteristic of a dispersive effect. It is not necessarily indicative of a dispersion in permittivity however, as we may observe by differentiating equation (10.1) with respect to Li.
Ci/Li = -1 / (2pf Li)²
Ci is negatively correlated with Li; and we could flatten the line for Ci by allowing Li to rise in the 6 and 20MHz region. There is very good reason for wanting to do so, as can be seen by examining the manufacturer's graph of complex permeability for type 61 ferrite material.


The manufacturer's data show classic Kramers-Krönig behaviour, with the onset of a dispersion becoming evident in the real part of the permeability at about 8MHz. Note that the inductance of a coil is directly proportional to the permeability of the core material. Hence, viewing the graph, we would expect to see a rise in Li on moving above 8MHz, followed by a fall commencing somewhere around 18MHz..
     Rearranging equation (3.3) in favour of Li gives:
Li = 1 / [ (2pf)² Ci + 1 / (N R0 C2 Rv) ]      . . . . . (10.2)
Shown below are two graphs of Li computed from Rv via equation (10.2). The upper curve uses Ci=5.74pF, which is the value obtained by fitting the data up to 5MHz, and the lower curve uses Ci=6.29pF, which is the value obtained by fitting the data up to 14MHz.





The upper curve bears a striking similarity to the dispersion shown in the permeability data and must leave us in little doubt that one cause of deviation from the model has been identified. There is a little more to it than that however. Estimation from the manufacturer's graph shows that the permeability of the ferrite rises from 125 at low frequencies to about 160 at 18MHz. Hence we would expect the inductance to rise in proportion from 9mH at low frequencies to 11.5mH at 18MHz. Computed on the basis that Ci is constant however, the inductance starts to drop away at about 15MHz after a rise to only 9.4mH. This could be indicative of other unknown effects, but it could also merely indicate the increasing dominince of the capacitance, and inapplicability of the manufacturer's permeability graph. By modeling the complex permeability on the basis of a single relaxation process it was possible to produce curves consistent with the data, but this matter is too conjectural to be worth pursuing.
     Note, incidentally, that these investigations of model breakdown do not imply that the experiment has in some way failed. It is rather the fact that the data are somewhat better than expected. Had the Rv measurements been made with a meter having an accuracy of 2% instead of 0.06%, then all of the data could have been fitted with a c²/n of 1, and we would have remained blissfully unaware of the additional information lurking in the noise.
     Shown below is the spread of results obtained by fitting the quadrature-balance data in different ways. Given that we know that the AL value for the core varies with frequency above 5MHz, it seems sensible to take the low-frequency inductance value from the average of the RV readings up to 5MHz.

 

500pF ref. cap.

40pF ref. cap.
  
Data file

testbrg61-12_2.ods

testbrg61-12_3.ods
 See sheet 1*
Parameter

Value ±ESD

Value ±ESD
 ESD in grey is precision from the fit.

Li

8.98 ±0.18 mH

9.01 ±0.18 mH
 Fitting data up to 5MHz

8.97 ±0.18 mH

8.98 ±0.18 mH
 Fitting data up to 14MHz

9.08 ±0.18 mH

9.09±0.18 mH
 Fitting all data

Ci
(apparent)

5.74 ±0.07 pF

5.74 ±0.08 pF
 Fitting data up to 5MHz

6.29 ±0.13 pF

6.55 ±0.02 pF
 Fitting data up to 14MHz

5.81 ±0.03 pF

6.11 ±0.03 pF
 Fitting all data
* L2 fitting flag (Sheet 1, cell B7) must be set to zero to reproduce these results.

The uncertainties for the capacitance determinations represent precision from the fit, not accuracy. As we shall soon see moreover, there is no point in considering any of these results to represent a good value for Ci because there are a number of systematic error contributions which need to be taken into account first. With a view to understanding how large the residual systematic errors are however, it is interesting to compare the results obtained so far with the hypothesis that 'self capacitance' is a lumped-component alias for time delay. If the effective velocity for an electromagnetic wave travelling along the winding wire is taken to be exactly c (the speed of light), then the equivalent capacitance is given by:
Ci' = lw / (2 Ri c)
Where lw is the length of the secondary winding (see section 6.2-13). The wire length for the test transformer was 228mm, which predicts Ci'=7.6pF. Naively adding about 0.4pF to this, to allow for the capacitance of the secondary load resistor, we thus predict Ci=8pF. From this we might conclude; either, that the effective velocity is greater than c; or, that the data are skewed.  A velocity-factor of greater than 1 is by no means impossible, but for a transmission line system in proximity to a high-permeability medium it is only likely to occur on the high-frequency side of a major dispersion such as the half-wave line resonance (which for 0.228m of wire occurs at 657MHz). Hence we are looking for errors of at least 2 to 2.5pF.
     As mentioned previously, we are to some extent cursed by the quality of the data; and there has to be a cut-off point in the matter of how far we go in accounting for minor discrepancies. Engaging in a search for a missing 2pF might therefore seem unreasonable; but any effect which makes the coil self-capacitance seem smaller than it really is can provide the basis for a high-frequency phase-compensation scheme.

11. Faraday shield displacement current:
In the test bridge used during the previous experiments (shown below), the Faraday shield earth connection was made on the load-port side. The unconnected end of the coaxial-cable braid protruded from the transformer core to a distance of about 13.5mm when measured from the middle of the core. 50W coax. has a capacitance per unit length (C0) of about 100 ±5 pF/m (a 34.5mm length of the URM108 used here measured 33pF, i.e., 96pF/m), and so the capacitance between the protruding part of the braid and the central conductor, allowing a little extra for the fringing fields at the end, is about 1.6±0.1pF. The displacement current for this capacitance passes through the transformer core on its way to ground and will therefore add a quadrature component to the transformer output.



The situation is represented in the equivalent circuit shown below. Subject to the approximation that no voltage is developed across the shield (the consequenses of which will be discussed in section 18b) the primary current is I + Ish, where:
I = V / R0
and
Ish = V' / (jXCsh)
or, using the approximation that V'=V :
Ish = V / (jXCsh)
Now, if we use the definition:
Zi = (Ri // jXLi // jXCi)
Then, by the ampere-turns rule:
Vi = (I + Ish) Zi / N
i.e.:

Vi = 

V Zi
N

1
R0

+

1
jXCsh

 

Now notice that
( 1/R0 ) + ( 1/jXCsh ) = 1 / ( R0 // jXCsh )
Hence, in terms of its effect on the current-transformer output, the Faraday shield protrusion can be considered to act (to a fair approximation) as an extra capacitance in parallel with the primary load.
     Note incidentally, that it is only the unconnected portion of the Faraday shield protruding beyond the exact centre of the transformer core which concerns us here. The shield on the load side also has capacitance, but if the cable used has a characteristic resistance equal to the load resistance R0, then the inductance per unit length (L0) balances the capacitance per unit length such that Ö(L0/C0)=R0 and there is no net effect. It is the fact that the displacement current from the unterminated end of the shield must pass through the transformer core (and only that) which gives rise to the additional quadrature component in the transformer output.
     What we need to do now is to write an expression for the current transfer function and compare it with the transfer function for a model with no Faraday shield protrusion capacitance. The difference in the imaginary part between the new model and the old model will tell us the apparent or 'effective' secondary parallel capacitance (Cieff say) in the event that the Faraday shield protrusion displacement-current exists.
     For a transformer with no shield displacement-current, the transfer function is:
Vi / V = Zi / (N R0)
The apparent secondary capacitance is of course hidden within Zi, and since it is easier to expand this in reciprocal form, we will work with reciprocal transfer functions. Hence:

V
Vi

= N R0

 1
Rieff

 +

 1
jXLi

 +

 1
jXCieff

(11.1)

Notice here that Ri has also been changed to an effective value to allow for the fact that if the shield protrusion capacitance (Csh) changes the phase of the transformer output, then it will also affect the magnitude (and hence the apparent transfer efficiency).
     Now, to include the effect of Csh, we simply replace R0 with ( R0 // jXCsh ), and Ci and Ri change from their effective values to their 'true' (i.e., less-seriously skewed) values. Thus:

V
Vi

= N ( R0 // jXCsh )

 1
Ri

 +

 1
jXLi

 +

 1
jXCi

 

Expanding the parallel product gives:

V
Vi

= N

 R0 jXCsh
( R0 + jXCsh )

 1
Ri

 +

 1
jXLi

 +

 1
jXCi

 

and multiplying numerator and denominator by the complex conjugate of the denominator:

V
Vi

= N

 R0 jXCsh ( R0 - jXCsh )
( R0² + XCsh² )

 1
Ri

 +

 1
jXLi

 +

 1
jXCi

 

This is not a particularly tractable expression as it stands, but we may make the observation here that Csh is a small capacitance in HF radio engineering terms and therefore XCsh²>>R0². Hence we may delete R0² from the denominator without making a significant difference. Thus:

V
Vi

= N R0

 jXCsh ( R0 - jXCsh )
XCsh²

 1
Ri

 +

 1
jXLi

 +

 1
jXCi

 

Multiplying out the left-most bracket (and noting that j²=-1) gives:

V
Vi

= N R0 ( 1 + jR0 / XCsh)

 1
Ri

+

 1
jXLi

+

 1
jXCi

 

and multiplying out completely gives:

V
Vi

= N R0 

 1
Ri

+

 1
jXLi

+

 1
jXCi

+

 jR0
XCsh Ri

+

 jR0
jXCsh XLi

+

  jR0
jXCsh XCi


Now, noting that j= -1/j, the admittances in brackets can be regrouped thus :

V
Vi

= N R0 

 1
Ri

+

 R0
XCsh XLi

+

  R0
XCsh XCi

+

 1
jXLi

+

 1
jXCi

-

 R0
jXCsh Ri


Comparing this with equation (11.1), we get:

 1
Rieff

=

1
Ri

+

 R0
XCsh XLi

+

  R0
XCsh XCi
  

which is the same as:
Rieff = Ri // [ -Li / ( Csh R0 ) ] // ( XCsh XCi / R0 )

(11.2)
Thus the shield displacement-current does affect the in-phase output, but since the magnitudes of the additional parallel components will be much larger than Ri, we can expect the difference identified here to be lost in the uncertainty of the transformer efficiency.
     The capacitive susceptances tell a different story however:

 1
XCieff

=

 1
XCi

-

 R0
XCsh Ri
  

i.e.:
-2pf Cieff = -2pf Ci + 2pf Csh R0 / Ri
Hence:
Cieff = Ci - Csh R0 / Ri

(11.3)
This expression tells us that if the earth connection to the Faraday shield is made on the load side of the transformer core, then the capacitance of the braid protruding from the other side reduces the apparent secondary parallel capacitance. In the preceding experiments, with R0=Ri, we measured an effective secondary capacitance of about 6pF, expected at least 8pF on the basis of time delay, and the shield protrusion capacitance was about 1.6pF. So far, so good, but the true test of a hypothesis lies not in what it explains but in what it predicts.
  &