|
|
|
Part 2 |
 |
A6.4(3): Analysis, evaluation and optimisation
of R.F. current-transformer bridges
Part 3
|
<<< Part 2. 18d. Phase-shift compensation. 18e. Herzog compensation. 18f. Load-side voltage sampling. |
18x. Collected results. 19. Bridge with no Faraday shield. 20. Final comments. A6.5 >>>. |
| One drawback with this technique is that it makes a noticeable contribution to the magnitude of Vi' as well as to the phase. This means that the adjustments for in-phase and quadrature bridge balance interact more than they do with other methods. The problem is not particularly serious in practice however, and it was possible to set up the test bridge (shown below) with only three rounds of iteration. |
|
A single-turn compensation winding was sufficient to neutralise
the test-bridge. A 100W Cermet pot. was used for Ra because very low-value non-inductive variable resistors are difficult to obtain (lower value multi-turn cermet trimpots are available, but these have a limited operational life and the worm-drive is too slow for good null-seeking). This meant that the required resistance was very close to the minimum setting, making the adjustment rather fierce. The solution was to calibrate the bridge roughly and then place a fixed resistor across the pot. to bring the adjustment closer to the middle of the range. Final calibration was accomplished with a 10W shunt resistor, as shown right, and the resistance of the parallel combination, measured after the test, was 7.49 W. This is 1.8 times the W/turn of the main secondary winding. The test results, summarised below, are comparable to to the effect of placing a capacitor across the load port (but without the power-factor penalty), and slightly better than for quadrature-voltage compensation. |
|
|
Gen. side shield gnd. 1.6 to 30MHz |
Phase-shift neutralisation. R0 = 50W, Ri = 50W. L-bal. coil. |
|Z| error. |
( ) ® best possible |
| testbrg61-13_5.ods | Ra = 7.49 W. |
|
|
| One peculiarity of the current transformer with phase-shift neutralisation is that it is not immediately obvious how to count the turns. In practice, it can be considered to behave either; like a transformer with N turns and an efficiency factor (k) of slightly greater than 1; or like a transformer with a non-integer number of turns between N and N-1. The reason that the effective number of turns is slightly less than N, rather than greater, is that the circuit gives more output than would be expected for a winding of N turns, whereas adding a turn to a normal current transformer makes the output level go down. The transformer constant is irrelevant to the perturbation analysis however (except it that can be adjusted to establish a realistic value for C1s); and the network is intended for use in working bridges, not for circuit parameter determination. |
|
18e. Herzog's HF compensation. The use of an inductance placed in series with Ri as a means for neutralising the phase error of a current-transformer bridge was patented by Will Herzog, K2LB, in 1988 [US Pat. 4739515]. Some additional qualitative discussion is given in ref. [45]. In section 13, it was indicated that the effective secondary capacitance has a large negative derivative with respect to the inductance of the secondary load resistor (here known as Herzog's inductance and given the symbol Lh). From equation (13.1): ¶Cieff/¶Lh = -1/Ri² Thus, for example, if Ri=50W, then ¶Cieff/¶Lh = -0.4pF/nH. The effect is so great that, for the test bridge, the apparent secondary capacitance could be made negative merely by soldering the secondary load resistor in place without trimming its wires. Achieving the 7pF or so of capacitance cancellation required for neutralisation was therefore not so much a matter of installing an inductor as of placing a slug of magnetic material in the vicinity of the resistor. To that end, and to provide a degree of adjustment, the resistor was bent into a U-shape over a 7mm diameter coil former with an M6 dust-iron slug as shown below. |
|
Gen. side shield gnd. 1.6 to 30MHz |
Herzog's neutralisation method. |
|Z| error. |
( ) ® best possible |
| testbrg61-1214.ods | R0 = 50W, Ri = 50W. L-bal. coil. |
|
|
|
The phase performance obtained from the test bridge is fairly
typical for a dispersion limited (2-point tracking) bridge. One slight drawback of the method is the inherent unpredictability of the final physical construction. In an early experiment, with the Faraday shield earthed on the load side, it was found that a single 49.9W helical-cut metal-film resistor, looped around a coil-former as shown above, provided too much inductance. The solution in that case was to use two 100W resistors in parallel. An unshielded bridge discussed in section 19, on the other hand, having a much larger Cieff, required a whole turn around around the dust-iron core. The method works well, but some experimentation is required in order to get it to work. In common with the other compensation methods tested, the amplitude error is largely attributable to the 8-48pF reference capacitor. An early experiment, unfortunately invalidated by failure to control reciprocity error, gave a maximum amplitude error of ±0.28% when using the 500pF reference capacitor. For those interested in the commercial manufacture of bridges, it should be noted that US Patent [4739515] remains current until April 2008. This author (DWK) has not conducted a search for Patents in other countries. |
| One problem with this configuration is that, in a calibration sequence which involves altering C2, all of the adjustments become highly interactive. Indeed, it is not possible to distinguish between the effects of adjusting C2 and L2 at high frequencies, which means that calibration by observation only of bridge balance is effectively a non-convergent process. The solution used in the case of the test bridge was to measure RV via the detector port in the same way as is done during evaluation, and to use this information as a guide for the setting of C2. The procedure was as follows: The bridge was balanced at 2MHz using C1 and RV, and at 26MHz (which was found to be about the optimum upper calibration frequency) by adjusting L2 and RV. An increase in RV on going to the upper frequency is indicative of a positive Cieff, which means that C2 needs to be increased. A decrease in RV indicates a negative Cieff, which means that C2 needs to be decreased. The cycle was repeated until the change in RV between the two calibration frequencies fell to less than 1%. Final adjustment required tiny changes to C2, and calibration would have been extremely difficult using a 180° trimmer capacitor. |
|
Gen. side shield gnd. 1.6 to 30MHz |
Load-side voltage sampling. R0 = 50W, Ri = 50W. L-bal. coil. |
|Z| error. |
( ) ® best possible |
| testbrg61-1215.ods | C2 = 8.7pF* |
|
|
| * The final value of C2 was determined by adjusting it in the least-squares fit to the RV data to reproduce the known value of Li (8.98mH). |
| Notwithstanding the tedious process of calibrating it; in the final analysis, the self-neutralising bridge is slightly inferior to to the other configurations tested (although it is still very good). The reason is probably that the voltage sampling network is not a pure capacitance, having a resistive component due to RV and an inductive component due to L2. It is likely therefore that the performance can be improved by minimising L1 (and hence L2) and by increasing Li to give an increase in the set value of RV. |
|
18x. Collected results. The collected results for sections 18a to 18f are summarised below. The common experimental conditions (unless stated otherwise) were:  Generator side
Faraday shield earth. Test frequency
range: 1.6 to 30MHz Inductance balance
coil in series with C2, adjusted before
test. C2
= 10pF, R0 = 50W,
Ri = 50W. N=12, Core type = FT50-61,
Lsec = 9.25mH. Reference capacitor:
8 - 48pF with 82pF in parallel. L1 = 85nH
approx. |
| Neutralisation method | Data and analysis. |
|Z| error. |
( ) ® best possible? |
| Load-port capacitor. | testbrg61-1212.ods |
|
|
| Quadrature current, 2-point tracking | testbrg61-13_2.ods |
|
|
| Quadrature current, 3-point tracking | testbrg61-13_4.ods |
|
|
|
Quadrature current, 3-point tracking C2 = 4.9pF. Lsec = 8.15mH. 3-30pF ref cap. |
see appendix 6.5 |
|
|
| Quadrature voltage. | testbrg61-14_2.ods |
|
|
| Quadrature voltage, 3-30pF ref cap. | testbrg61-14_3.ods |
|
|
| Phase shift. | testbrg61-13_5.ods |
|
|
| Herzog. | testbrg61-1214.ods |
|
|
|
Load-side voltage sampling. C2 = 8.7pF. 8-48pF ref. cap. with 56pF in parallel. |
testbrg61-1215.ods |
|
|
|
The best possible phase performance (indicated in brackets) is
an estimate of what can be achieved if the optimal upper calibration
frequency is chosen. Most of the 2-point tracking bridges are
similar in this respect, and there is effectively no practical
difference between a number of methods. Generally, those methods
which do not carry a generator power-factor penalty are to be
preferred when extreme accuracy is not required. 3-point tracking
gives the best results; but the additional circuit complexity
is unwarranted for typical working bridges, and particularly
if a conventional diode detector is to be used. The amplitude performance is limited by the characteristics of the reference capacitor. It is generally best when the inductance of the lower voltage-sampling arm is minimised. This is demonstrated by the results for bridges using a 3-30pF trimmer rather than a physically-large variable capacitor. The results tabulated above are a demonstration of what can be achieved, but not necessarily of what has to be achieved. In practice, anything better than about ±0.5° and ±0.5% (±0.25W in a 50W system) is fine for a working impedance-monitoring bridge. |
|
19. The utility of the Faraday shield. The application of the current transformer to the field of RF measurement was patented by Josef Stanek of Siemens and Halske (Germany), the US Patent [# 2134589] being awarded in 1938. The Faraday shield was included in this invention, and its purpose was explained as follows: "The metallic coating of the insulating sheath is electrically connected to one terminal of the measuring instrument, preferably to that terminal which .. is grounded. This arrangement serves the following purpose. In the case of high-frequency, a transformer is also to be regarded as a condenser, the primary conductor .. forming the one and the secondary conductor forming the other electrode.. Therefore a capacitive displacement current could flow from the primary conductor to the secondary circuit and impair the measurement. The grounded coating however carries off this displacement current and makes it ineffective." Since that passage was written, the need for the shield has become an article of faith among radio engineers; and others have gone on to say that the purpose of the shield is not to minimise the transmission-line mismatch but to provide electrostatic screening. In section 11 however, we showed that the shield upsets the forward and reverse symmetry of the transformer phase performance, and in section 18b we showed that it can slightly degrade the frequency-tracking; i.e., at a subtle level, perhaps not relevant in 1938, it can cause the type of problem that it is supposed to prevent. Moreover, it does help to minimise the through-line mismatch; but since that can also be done in other ways, the utility of the shield is open to question. |
| An unshielded version of the test bridge was put together as shown in the photographs below. The through-line is a 0.9mm diameter silver-plated wire, and the transformer core is spaced away from it by means of a stub of polythene honeycomb insulator as used in 75W UHF TV-antenna cable. To prevent the polythene from melting during the soldering to the BNC sockets, the heat was shunted away by attaching to the wire a small pair of pliers with a rubber band around the handles. Sliding the insulator away from the point being soldered also helped. |
|
In a jig designed for shielded transformers, it is obvious that
the the through-line is a lot longer than it needs to be. That
does not pose a problem analytically, but it raises the point
that doing without the shield is an aid to miniaturisation. In
fact, the transformer core could simply be placed on the back
of the load-port socket, in which case the mismatch of the load-side
line would be negligible. The investigation was begun by acquiring two datasets as listed below: |
| No Faraday shield | No compensation. 12 turns, FT50-61, Ri = 50W. |
|
| testbrg61-12_8.ods | 50W reference load. |
|
| testbrg61-12_9.ods | 75.5W reference load. |
|
|
An obvious feature of the data for the uncompensated bridge with
the 50W load connected is that that
the apparent secondary capacitance is higher than for a shielded
bridge. This might seem anti-intuitive to those who see the shield
as a distributed capacitance across the coil, but since the average
characteristic resistance of the through-line is now somewhere
around 300W, equations (14.3)
or (15.1) tell us that at
least some of this increase must be due to the mismatch. Other
features are that the stray capacitance across the voltage sampling
network is reduced, and the value of the upper voltage sampling
capacitor C2 has to be increased by about
0.7pF in order for the Rv data to reproduce
the known value of Li (8.97mH). The
increase is, of course, partly due to stray capacitance between
the line and the summing point (i.e., the voltage-sampling network
end of the transformer secondary). In the case of the uncompensated bridge with the 75W load, agreement with the model developed in section 2 is not so good. One reason for this is that the bridge with the 50W load has a phase crossover frequency of 15.3MHz and the data only go up to 14.5MHz, whereas with a 75W load load, the apparent secondary capacitance is reduced and the data go up to 30MHz. With this extra coverage, it can be seen that the quadrature balance data are somewhat more divergent from theory at high frequencies than for shielded bridges. Also, the value of L1 returned from the in-phase balance data is too large by about 30%, the reduced c² is about 6, and the graph of residuals shows a distinct curvature. In fact, we should expect the data do deviate from the model; and we should expect the deviation to be worse with a 75W load than it is with a 50W load because the ratio of voltage to current is increased with increasing load impedance, i.e., the relative output of the current transformer goes down. This gives greater influence to the principal suspect, which is the stray capacitance between the line and the detector port. If C2 appears to have increased by 0.7pF, then this capacitance must be of the same order. Hence a more accurate model for the unshielded bridge is as in the diagram below. |
|
Since the voltage at the detector is zero when the bridge is
balanced, the stray capacitance Cx passes
its current into a virtual earth. Therefore Cx
is effectively across the load, and as we know from equation
(14.1), one of its effects
will be to reduce the apparent secondary capacitance. In this
sense its presence is beneficial, because it reduces the mismatch
of the through-line, but there is a greater issue (N times greater
it will transpire) regarding the current injected into the detector
port. In the absence of Cx, the bridge balances when VV=Vi. When Cx is included however, the balance point is skewed such that the voltage VV-Vi produces a current Id which is equal and opposite to the injected current Ix. The magnitude of the voltage difference needed to counteract Ix depends on the source impedance of the Thévenin-equivalent generator producing it. This impedance is the sum of the output impedances of the voltage sampling network and the current sampling network, these being: Zv = Rv // jXC1 // jXC2 and Zi = Ri // jXLi // jXCi Thus we can draw an equivalent circuit which allows us to determine the balance condition. |
| The balance condition is Id = -Ix. Written explicitly this is: |
|
Zv + Zi |
|
jXCx |
|
| Expressions for VV and Vi were given in section 2. Modified to allow for the capacitance Cx in parallel with the primary load, these become: |
|
|
 |
|
N² (R0 // jXCx) |
 |
jXC2 |
| and |
|
|
N (R0 // jXCx) |
| Substituting these into equation (19.1), cancelling the voltages and rearranging gives the dimensionless balance relationship: |
|
jXC2 |
|
|
N² (R0 // jXCx) |
|
|
N (R0 // jXCx) |
|
jXCx |
| which can be regrouped: |
|
jXC2 |
|
|
N² (R0 // jXCx) |
|
|
jXCx |
|
N (R0 // jXCx) |
|
jXCx |
| Now, multiplying XC2 into top and bottom of the last term on the left-hand side, and noting that XC2/XCx = Cx/C2; and also factoring Zi/N from the right-hand side and noting that 1/(a//b)=(1/a)+(1/b): |
|
jXC2 |
|
|
N² (R0 // jXCx) |
|
C2 |
|
|
N |
|
R0 |
|
jXCx |
|
| Inverting this expression and rearranging gives: |
|
Zv |
|
Zi |
|
|
N² (R0 // jXCx) |
|
C2 |
|
| and hence: |
|
Zv |
|
|
|
C2 |
|
Zi |
|
N (R0 // jXCx) |
|
| Taking the right-most term on its own and expanding it gives: |
|
N (R0 // jXCx) |
|
N [ R0 + jXCx/(1-N) ] |
|
R0 |
|
jXCx |
|
| Now factoring 1/(1-N) from the denominator and cancelling, then multiplying numerator and denominator by the complex conjugate of the denominator, we obtain: |
|
N (R0 // jXCx) |
|
N [ (1-N)² R0² + XCx² ] |
|
R0 |
|
jXCx |
|
Here we can make an approximation by noting that for the test
bridge, with N=12, R0=50W
and Cx no greater than 1pF (actually 0.33pF as it turned out)
, the term (1-N)²R0²=302500,
whereas at XCx², at its lowest (at
say 2x10 radians/sec)
will be >25 million. Hence the error in deleting (1-N)²R0² from the denominator will be <1%
at the high end of the HF spectrum and negligible at the low
end. Hence: |
|
N (R0 // jXCx) |
|
N XCx |
|
R0 |
|
jXCx |
|
| And after putting the first factor into a+jb form (and noting that j²=-1): |
|
N (R0 // jXCx) |
|
N |
|
|
XCx |
|
|
R0 |
|
jXCx |
|
| Multiplying out: |
|
N (R0 // jXCx) |
|
N |
|
|
XCx² |
|
jXCx |
|
jXCx |
|
| We can now make another approximation, less serious than the previous one, which is that the second term of the series can be deleted because XCx²>>(1-N)R0². The error in this case is <0.05% for the test bridge at the high end of the HF spectrum, i.e., truly negligible. Now noting that 1-(1-N)=N and 1/j=-j, we get: |
|
N (R0 // jXCx) |
|
N |
|
XCx |
|
| Substituting this back into equation (19.2) and rearranging the terms gives: |
|
Zv |
|
N |
|
XCx |
|
|
|
C2 |
|
Zi |
|
|
The 1/N term is of course part of the transformer constant in
the in-phase balance condition. Since this is a small correction
to allow for the difference between V and V' caused
by the insertion impedance, and N is generally >10, the <1%
error introduced earlier is <1% of <10%, i.e., <0.1%,
and is therefore harmless. Now, taking the right-most factor on its own and expanding it (also recalling that Zi = Ri // jXLi // jXCi ) gives: |
|
Zi |
|
R0 + jXCx / (1-N) |
|
Ri |
|
jXLi |
|
jXCi |
|
| One again factoring 1/(1-N) from the denominator and cancelling, then multiplying numerator and denominator by the complex conjugate of the denominator, gives: |
|
Zi |
|
(1-N)² R0² + XCx² |
|
Ri |
|
jXLi |
|
jXCi |
|
| Again we can use the approximation that XCx²>>(1-N)²R0². This time however, the <1% error will slightly affect the goodness of fit to the model; potentially incurring a frequency-dependent curvature which will be small but possibly visible above the experimental noise. Thus: |
|
Zi |
|
|
|
XCx |
|
|
Ri |
|
jXLi |
|
jXCi |
|
| Multiplying out: |
|
Zi |
|
|
Ri |
|
XCx XLi |
|
XCx XCi |
|
jXLi |
|
jXCi |
|
jXCx Ri |
|
|
Now observe that XCxXLi=-Li/Cx. For the test bridge,
with N=12, R0=50W,
Cx<1pF and Li=9mH,
the second conductance term will be <0.00006 Siemens, i.e.,
it corresponds to a resistance of >16.7KW
in parallel with Ri. It represents a minor
frequency-independent adjustment to the apparent transformer
efficiency, which will be lost in the uncertainty of the efficiency
factor and can therefore be dropped. If we take Ci to be about 10pF, and the maximum frequency to be 2x10
radians/sec, the third conductance term will be <0.00022 Siemens,
i.e., it corresponds to a resistance of >4.5KW
in parallel with Ri and can be dropped
with the proviso that there will be an additional frequency dependent
deviation from the model of <1% in the in-phase response.Hence: |
|
Zi |
|
|
Ri |
|
jXLi |
|
jXCi |
|
jXCx Ri |
|
|
| The additional capacitive susceptance term, of course, gives rise to a shift in the apparent value of Ci in exactly the same way as did the other parasitic effects analysed in sections 11 to 14 and summarised in section 15. To quantify the effect we can introduce a new parameter Cix' to represent the apparent secondary capacitance (the parameter is given a prime because there is another small contribution to be added later). Noting that -(1-N)=N-1, Cix' is defined by the relationship: |
|
XCix' |
|
XCi |
|
XCx Ri |
|
i.e.: -2pf Cix' = -2pf Ci - 2pf (N-1) Cx R0 / Ri Thus:
Substituting equation (19.6) into (19.5): |
|
Zi |
|
|
Ri |
|
jXLi |
|
jXCix' |
|
| and substituting this back into equation (19.4): |
|
Zv |
|
N |
|
XCx |
|
|
|
C2 |
|
|
|
Ri |
|
jXLi |
|
jXCix' |
|
| Now we can see that the term jR0/XCx is also in a form which causes it to contribute to the apparent secondary capacitance. Moving it to the right hand side and moving j to the denominator shows that it is a positive contribution. It becomes part of the secondary admittance series when the denominator is multiplied by (1+Cx/C2)NR0. Hence: |
|
Zv |
|
N |
|
|
|
Ri |
|
jXLi |
|
jXCix' |
|
j (1+Cx/C2) N XCx |
|
|
|
Now we can define a new parameter Cix,
without the prime, such that: Cix = Cix' + Cx / [ (1+Cx/C2) N ] Also notice that (1+Cx/C2) = (Cx+C2)/C2. Hence: Cix = Cix' + Cx C2 / [ (Cx+C2) N ] The new contribution to the apparent secondary capacitance is 1/N times the capacitance of the series combination of Cx and C2. For the test bridge it amounts to about 0.03pF, and so it is not very important, but it might as well be included for completeness. Combining this result with equation (19.6) gives:
|
|
Zv |
|
N |
|
|
|
Ri |
|
jXLi |
|
jXCix |
|
| Zv is, of course, Rv // jXC1 // jXC2, and so, noting that XC2/XC1=C1/C2, the first term can be expanded giving: |
|
| Equating reals: |
|
|
Upon deletion of the factor (1+Cx/C2), this expression is the same as equation
(2.1) for the shielded bridge.
Hence, apart from the approximations made earlier, the introduction
of Cx merely causes a small reduction
in the apparent transformer efficiency (about 3% for the test
bridge). If we modify equation (19.10) by including the transformer efficiency factor k, we obtain a more general expression for the transformer constant introduced in section 5a: |
|
| Equating imaginaries: |
|
Rv |
|
|
|
XLi |
|
XCix |
|
|
The frequency dependent apparent secondary inductance is now
slightly different from the the quantity Li'
introduced in section 2, because
it involves Cix instead of Ci,
but since the apparent value of Ci is
disturbed by almost every parasitic reactance in the system,
there seems little point in generating a new definition. Hence,
by analogy with what was done previously, we will use: Xi = 2pf Li' = XLi // XCix Hence: |
|
2pf C2 Rv |
|
2pf Li' |
|
i.e.: Li' = C2 (1 + Cx/C2) N R0 Rv i.e.:
As can be noted from all of the derivations in the preceding sections, the secondary coupled inductance Li is a strongly conserved parameter; and so for a given transformer, the quantity (C2+Cx) can be determined from the original model as the value to which "C2" must be adjusted to reproduce the value of Li obtained when a Faraday shield is present. For the test bridge illustrated above, "C2" for the fit had to be increased by about 0.7pF, but this difference should not be identified as Cx. If removing the shield introduces stray capacitance between the through-line and the detector port, then there must be an almost equal amount of stray capacitance between the through-line and the summing point (i.e., the other end of the transformer winding). Hence we should expect the difference to be the sum of Cx and the additional stray capacitance across C2, i.e., if the shift is 0.7pF, then Cx is 0.35pF. And so, after the analytical challenge of finding the balance condition, it transpires that the spreadsheet template for data analysis can be modified to include unshielded bridges simply by re-labeling "C2" as "C2+Cx", multiplying the first term of the transformer constant by 1+Cx/C2 (equation 19.10a), and amending the series for apparent secondary capacitance. Deleting the shield-protrusion capacitance term from equation (15.1), then combining it with (19.8) gives: |
|
|
Note incidentally, that the introduction of the 'direct pickup'
capacitance Cx also provides a model for
very-near-field coupling between the generator and detector.
It explains an effect which was observed during the process of
experimental optimisation, which was that the phase crossover
frequency of the prototype bridge increased slightly when a nickel-plated
connector and an RG58 cable at the receiver input were replaced
by a silver-plated connector and a Belden 9880 cable (section
8). So now we return to the question of whether or not it is necessary to use a Faraday shield. To that, the answer is probably 'yes' if the circuit designer makes no attempt to correct for the apparent secondary capacitance, and probably 'yes' if the transformer has a large number of turns or a low value for Ri. This can be understood by differentiating the penultimate term of equation (19.12) above: ¶Cieff/¶Cx » (N-1) R0 / Ri Then again, there are effects, such as the inductance of the secondary load resistor which can make the apparent secondary capacitance negative, and given the practice of describing bijou inductorettes as 'resistors', we might be glad of something to push it the other way. An issue therefore is: 'how severe is the curvature introduced by the approximations used in deriving the model?', i.e., 'how correctable is the direct pickup effect?' An attempt to answer this question was made by correcting the bridge in various ways and analysing its performance using the method described in section 16. The datasets for experiments carried out on the test jig are given in the spreadsheets listed below. The results for a working bridge using quadrature-current-injection neutralisation are discussed in Appendix 6.5. |
|
No Faraday shield 1.6 to 30MHz |
R0 = 50W, Ri
= 50W. Inductance balance coil fitted. |
|Z| error. |
( ) ® best possible |
| testbrg61-1210.ods | Capacitor across load port. |
|
|
| testbrg61-1211.ods | Herzog compensation. |
|
|
| testbrg61-13_1.ods | Phase-shift compensation. |
|
|
| testbrg61-14_1.ods | Quadrature voltage compensation. |
|
|
| See Appendix 6.5 | Current injection, 3-point tracking. |
|
|
|
By placing a trimmer capacitor across the load port, adjusting
Rv and C2 at 2MHz
and adjusting the inductance balance coil and the trimmer at
24MHz, the bridge gave a phase error less than ±0.2°
and a magnitude error less than ±0.5% over the 1.6 to
30MHz range. Significant improvement could have been obtained
by moving the upper calibration frequency to 27 or 28MHz. The
amplitude performance is not as good as was obtained from the
shielded version of the bridge (section
18a) but is nevertheless perfectly acceptable. Using Herzog's compensation method, calibrating at 2 and 26MHz, the phase error was within ±0.11° and the magnitude error within ±0.14%. The amplitude performance is better than that obtained from the shielded version (section 18e), possibly indicating partial cancellation of system non-idealities. The phase shift compensation method (section 18d) did not work properly for this test. In particular, neutralisation could only be achieved by having less W/turn across the auxiliary winding than across the main secondary. This was not known until the resistance was measured at the end of the test run. Increasing the auxiliary winding to 2 turns would probably have fixed the problem, but the experiment was not repeated. The interesting point about this result however, is that neutralisation can be achieved with the secondary and auxiliary loading ratios reversed. The reason is that the vector sum Va+Vi has its smallest lead in relation to the primary current when both windings have the same W/turn. The phase swings positive again when if the resistance shunting the auxiliary winding is further reduced. Quadrature voltage compensation (section 18c), using a 2-turn winding with a 250W pot across it, gave a phase error within ±0.05° and magnitude error within ±0.35%. The remarkable phase performance in this case is possibly (but not necessarily) due to cancellation of system non-idealities, i.e., it might be difficult to reproduce (but then again, it might not). |
|
|
 |
| The greatest potential for phase accuracy is, of couse, given by the 3-point tracking scheme described in section 18b. Rather than evaluating this method on the test jig; a working bridge was constructed in order to improve the circuit layout and minimise the inductance of the lower voltage-sampling arm. The result was a bridge with no Faraday shield showing a maximum phase error of ±0.03° and a maximum amplitude error of ±0.04% over the 1.6 to 30MHz range; i.e., more than two orders of magnitude better than most of the impedance-monitoring bridges reported elsewhere prior to this work. The device is described in detail in Appendix 6.5. A Faraday shielded version gave practically identical results. |
 |
In summary we may note that, although the unshielded bridge has
a larger effective secondary capacitance than its shielded counterpart,
the correction process is no less effective. Whether the shield
is redundant however, must remain moot. The following points
should be considered when making a decision: The shield eliminates
the direct pickup signal, improving the phase performance of
uncorrected bridges and bridges with a low transresistance (i.e.,
a low value for Ri/N). The shield reduces
the average through-line characteristic resistance. The circuit model
for the unshielded bridge requires additional parameters, which
are difficult to determine unless measurements are also made
on a shielded version; i.e., the shielded bridge is easier to
simulate. The shield gives
rise to asymmetry in forward and reverse phase performance (a
problem which might be solvable by using a two-hole core and
earthing the shield at its mid point). Since forward-power bridges
do not normally require neutralisation however, this is probably
not important. The shield increases
the physical length of the transformer assembly.20. Final comments: The test procedures, theory and data-analysis techniques described in this article were developed without prior knowledge of the experimental conclusions. Consequently, the test-jig used was a Heath-Robinson affair and has plenty of scope for improvement. The temptation to redesign it from scratch part-way through the work was resisted in the interests of comparability between the various results. A particular suspicion arose during the course of the work, to the effect that the parasitic inductances of the components in the lower voltage-sampling arm were limiting the achievable magnitude performance. This network has several equivalent-series-inductances; that of the variable capacitor, that of the padding capacitor, and that of the pair of variable resistors in series. These do not combine into a single frequency-independent equivalent series inductance for the whole arm, and consequently cannot be balanced-out perfectly by a single coil in the upper voltage-sampling arm. In the working bridge described in appendix 6.5, this shortcoming is addressed by miniaturisation and attention to layout, and results in substantial improvement. A fairly obvious improvement to the experimental technique can also be had by building the bridge into a metal box and screwing the lid down tightly before calibrating and testing. This makes the bridge immune to the positions of the operators hands and the various tools and cables lying around, thereby improving the accuracy of calibration and reducing the amount of scatter in the evaluation data. The bridge described in appendix 6.5 is essentially a boxed version of the test bridge used here, and suggests an altogether better way of doing things in the event that anyone should wish to repeat or extend this work. Whenever a mathematical derivation is carried out for the first time, the resulting exposition is not usually the most elegant. During the work described here, the pressure was to deal with the insurgence of an ever-increasing number of subtle effects, all of which had somehow to be crammed into the model. The resulting analysis template is consequently a bit of a mess. It did its job however, which was to establish and quantify all of the factors which affect bridge resistance and reactance balance to a level of about 1 part in 1000 or better. With hindsight, i.e., by knowing in advance all of the things which are important, it is possible to simplify the analysis by factoring it into separate parts, and to dispense with the need for certain approximations and difficult-to-estimate parameters. The work stands however, there is little point in repeating the data analysis; and the true test of its worth is to see whether it can inform the design of future bridges. By following the working given in section 19, it is possible to write down a general balance condition for bridges which have been neutralised and corrected for voltage-sampling network inductance, i.e.: |
|
C2 |
|
RV |
|
N |
|
|
|
k Ri |
|
XLi |
|
|
|
This is obtained from equation (19.9)
by substituting Z0 in place of
R0, inserting the transformer efficiency
factor (k), and deleting the term for apparent transformer-secondary
capacitance. There is no need to include the inductances of the
voltage-sampling capacitors, because the inductance-balance correction
cancels them out. Likewise, there is no need to include a long
series of terms to account for the numerous causes of high-frequency
phase error, because a phantom neutralisation term cancels them
all out. The result, as is evident from the neutralisation experiments,
is an extremely accurate model; which even accounts for the differences
between Faraday-shielded and unshielded transformers (when a
Faraday shield is used, Cx=0). It separates
cleanly into frequency-independent expressions for resistance
and reactance balance provided that Z0
is purely resistive, and it can be used for circuit component-value
calculations insofar as k, Li and various
stray capacitances can be estimated. Equation (20.1) may look like an obvious conclusion of this work, or even the outcome of a naive derivation based on the idea that capacitors don't have inductance and coils don't have capacitance. It contains a subtle distinction however, which unravels the problem of bridge evaluation by perturbation analysis. The trick (being discovered in one of the author's 'why didn't I think of this before' moments) is that the bridge can perfectly well be balanced when the load impedance is not purely resistive. Hence we can generalise the balance condition by replacing R0 with Z0=R0+jX0 . The point of the perturbation analysis is to convert deviations of C1 and RV from their calibration settings, into equivalent deviations of load resistance and reactance in the event that the calibration is left untouched. All that is needed for that is the rate of change of R0 with respect to C1 (i.e., ¶R0/¶C1) and the rate of change of X0 with respect to RV (i.e., ¶X0/¶RV ). With the reactance error X0 appearing explicitly in the balance equation, the required derivatives are easily obtained without the need for further approximations; and it transpires that there is no need to try to estimate the strays across C1. This simplified method of bridge evaluation is developed and used in Appendix 6.5.  |
© D W Knight 2007.
David Knight asserts the right to be recognised as the author of this work.
|
|
|
|
Part 2 |
 |