|
|
|
|
|
A6.4(2): Analysis, evaluation and optimisation
of R.F. current-transformer bridges
Part 2
|
12. Upper V-sampling arm inductance. 13. Inductance of secondary load resistor. 14. Through-line mismatch. 15. Correction series for self-capacitance. 16. Bridge evaluation. 16a. Uncorrected bridge performance. |
17. Inductance balance. 18. Frequency tracking. 18a. Load port capacitance. 18b. Quadrature current injection. 18c. Quadrature voltage compensation. . |
|
12. Inductance of the upper voltage-sampling arm. The quadrature balance condition for Douma's bridge was given in section 3 as:
As has been mentioned previously however, this expression contains the assumption that C2 does not vary with frequency, i.e., it assumes that there is no inductance in the upper voltage-sampling arm. More realistically, the wiring between the take-off point and the summing point has considerable inductance, as does the capacitor itself, and the total must amount to several tens of nano-Henries. This parasitic inductance is represented as a lumped component L2 in the diagram below. |
|
We can account for the existence
of L2 by replacing C2,
in equation (3.3) with a
new quantity C2'. A definition for C2' is obtained by working backwards from the
total reactance of the arm, i.e.: XC2' = XC2 + XL2 Hence: |
 |
|
-1/(2πf C2') = [ -1/(2πf C2) ] + 2πf L2 which, upon multiplying both sides by -2πf gives: 1/C2' = (1/C2) - (2πf )² L2 We will leave the expression in its reciprocal form, because that is how we will need to use it. |
Equation (3.3), with the
modified definition for C2 becomes:
This can be re-grouped:
Hence, comparing this with equation (3.3) reproduced above, we can see that L2 reduces the apparent or 'effective' value of the secondary parallel capacitance. The adjustment caused by L2 can be written:
Expression (12.3) above is perfectly valid for a working bridge, which has a fixed value for Rv. For the test bridge however, which is operated by adjusting Rv to track the changing balance point, it causes Cieff to become frequency dependent. The result will be a slight deviation from the model used for the data analysis technique developed in section 3. The solution is to correct the y-values used in the least-squares fit by subtracting the quantity (2πf )²L2/(NR0Rv); i.e., the linear regression formula now becomes:
In this way, a value for Ci which is already corrected for L2 is obtained from the fit. The correction has in fact been included in the spreadsheets used for the post-optimisation data analysis (version 1.00 and above) and a flag is provided so that it can be turned on and off. For the test bridges studied so far, putting in 50nH as a plausible value for L2 causes a slight decrease in the reduced χ² for the fit and increases Ci by about 0.06pF. Hence the small parasitic inductance of the upper voltage-sampling arm does not make much difference to the apparent value of Ci, but a more substantial difference will be obtained when we later insert an inductor in series with C2 to correct the in-phase balance condition for the inductance of the lower potential-divider capacitor. (see section 17). |
||||||||||||||||||||||||||||||||||||||||
|
13. Inductance of the secondary load resistor. In section 11, we wrote down an expression for the reciprocal transfer function of the 'ideal current transformer with parallel reactance' model:
This can be used as the basis for working-out all of the parasitic reactance corrections associated with the current transformer network; the trick being to include the component and drop the "eff(ective)" from the subscripts on Ri and Ci. Rearranging the new equation into the same form as (11.1) allows us to find the correction by comparing terms. |
|
In the circuit on the right, the
secondary load resistor has been allowed to have a finite inductance
Lh. Thus the impedance of the secondary
load resistor (temporarily taking the liberty of allowing the
symbol R in bold to represent a complex quantity) becomes: Rieff = Ri + jXLh Substituting this into equation (11.1) (and changing XCieff to XCi) gives: |
 |
Multiplying numerator and denominator of the first term in square brackets by the complex conjugate of its denominator then gives
But Lh is a very small inductance in HF radio engineering terns (<20nH if the resistor wires are kept short) and so we may reasonably apply the approximation Ri² >>XLh². This allows us to delete XLh² from the denominator of the first term. Hence:
The term containing XLh has been grouped with the capacitive admittances because an inductance divided by a resistance-squared has dimensions of capacitance. Comparing this with equation (11.1) reproduced above, and noting that j=-1/j, we get: -2πf Cieff = -2πf Ci + 2πf Lh / Ri² i.e.:
In the optimised test bridge, the secondary was terminated by a single 49.9Ω 0.6W metal-film resistor. It was thought initially that the use of such a component would help to minimise parasitic reactance, but suspicions began to arise in the process of reconciling the data from different experiments. Finally, the paint was scraped from one of the units to see what lay beneath, and the sorry outcome is shown below. |
 |
Low Q current-sheet inductor described in the sales literature
as a "resistor". N=4.5, l=4.2mm d=1.7mm, giving Lbody=11.6nH. R=49.9Ω. For C=0.4pF, SRF=2.3GHz. |
|
We may guess that cutting a helix is cheaper than depositing
a thinner film or making a number of offset slits. Whatever the
reason for the abomination however, the result is not optimal
for RF applications. A current-sheet formula gives the inductance
of the body as 11.6nH which, with an allowance for the end-caps
and the short connecting wires (the distance between the terminals
was 10.5mm), makes the total inductance up to about 15±2nH.
A properly designed resistor of the same body and lead dimensions
would have had an inductance of 6nH. Using equation (13.1), it can be determined that a 49.9Ω resistor with an inductance of 15±2nH reduces the apparent secondary capacitance by 6±0.8pF. Hence, the inductance of the secondary load resistor has a large effect on the apparent value of Ci. Equation (13.1) also tells us that the secondary capacitance can be neutralised by adjusting the series inductance of the secondary load resistor. This is the basis of an empirical high-frequency bridge compensation scheme patented by Will Herzog, K2LB, in 1988 [US Pat. 4739515]. Hence Lh is 'Herzog's inductance', and the subscript h is used here for that reason. For bridges with relatively few turns in the transformer secondary, such as the ones so-far studied, the additional inductance required is extremely small; and indeed, it is possible to over-compensate merely by leaving long wires on the resistor. The secondary parallel capacitance can be split into two components: Ci = Ci' + Cis Where Ci' is the 'self-capacitance' of the coil and Cis is the stray capacitance across the winding. Hence we can rewrite equation (13.1):
Notice here that if two resistors in parallel are used for the secondary load, the inductance will be halved and the capacitance doubled. The total contribution from the resistor in the example above is -6 +0.4 = -5.6pF, but for two ½W resistors of similar inductance in parallel it would be -3 +0.8 = -2.2pF. Observe also that the resistor provides a neutral termination when: Cis - Lh / Ri² = 0 i.e., when Ri = √(Lh / Cis) This is a transmission-line formula which defines the properties of a perfectly matched terminating resistor. The smaller the values of inductance and capacitance used to match the resistance, the higher the self-resonance frequency (SRF) of the resistor. |
The capacitance of the transmission line on the load side of
the transformer core is represented in the diagram on the right.
We can account for it by replacing R0
with R0//jXC0
in equation (11.1). There
is no need to bother with the rest of the working however, because
the situation is exactly analogous to the effect of the Faraday
shield protrusion capacitance as discussed in section
(11). Hence, taking the solution from equation (11.3):
|
 |
|
From the sign of the contribution, it is evident that the transformer
secondary capacitance can be neutralised by placing a capacitor
across the load port. C0 will also have a small effect on the in-phase response of the transformer. Taking the solution from equation(11.2) we get:
|
In order to account for the transmission-line inductance, we
can replace R0 in equation (11.1)
with R0+jXL0.
This gives:
Since we need to put this into a form comparable to equation (11.1), we will start by factoring out R0. Thus: |
 |
|
Vi |
|
 |
|
R0 |
 |
|
Ri |
|
jXLi |
|
jXCi |
|
| Multiplying-out the brackets gives: |
|
Vi |
|
|
Ri |
|
jXLi |
|
jXCi |
|
R0 Ri |
|
jR0 XLi |
|
jR0 XCi |
|
| and the terms can be regrouped into conductive, inductive and capacitive admittances: |
|
Vi |
|
|
Ri |
|
R0 Li |
|
R0 XCi |
|
jXLi |
|
jXCi |
|
jR0 Ri |
|
Hence we can make the identifications:
Now, to obtain the combined effect of the through-line inductance and capacitance on the apparent secondary capacitance, we combine equations (14.1) and (14.3): Cieff = Ci + [ L0 / (R0 Ri )] - C0 R0 / Ri i.e.:
L0 = C0 Rline² Hence: Cieff = Ci + (C0 / Ri) [ (Rline² / R0) - R0 ] or alternatively
One conclusion which we can draw here is that the stub of coaxial cable used for the Faraday-shielded primary should normally have a characteristic resistance which is the same as R0. It is also advisable to keep the stripped sections before and after the transformer as short as possible, since these will raise the average surge-resistance of the through-line and so lower the phase-crossover frequency of the transformer output. |
|
The effect of through-line mismatch on the in-phase transformer
response is obtained by combining equations (14.2)
and (14.4). Rieff = Ri // ( R0 Li / L0) // [ -Li / ( C0 R0 ) ] // (R0 XCi / XL0 ) // ( XC0 XCi / R0 ) This is easier to parse when expressed as a series of conductances: |
|
Rieff |
|
Ri |
|
R0 Li |
|
Li |
|
R0 XCi |
|
XC0 XCi |
| Which can be rearranged thus: |
|
Rieff |
|
Ri |
|
Li |
|
R0 |
|
|
|
|
R0 |
|
|
||||
| and then factorised: |
|
Rieff |
|
Ri |
|
|
Li |
|
|
|
R0 |
|
|
||||
| Once again we can use the substitution L0 = C0 Rline², hence: |
|
Rieff |
|
Ri |
|
|
Li |
|
|
R0 |
|
||||||
|
As before, the contribution is zero when Rline=R0, but a frequency dependence is introduced
when there is a mismatch. The change in the apparent efficiency
of the transformer is however very weak for realistic parameters
(<0.2%), as can be seen by examining the simulation in the
spreadsheet mismatch_sim.ods.
It is also similar in form to the effect of the inductance of
the lower voltage sampling network capacitor, and so its effect
on the in-phase balance condition will be absorbed into any inductance-balance
correction. Notice incidentally that the factor (1/Li)-(2πf )²Ci is 1/Li', the reciprocal of the apparent secondary inductance as used in the least-squares fitting procedure described in section (3). |
| One way to show the effect of the transmission-line mismatch is to measure (or attempt to measure) the apparent secondary capacitance with two different values of load resistor. This is done in the spreadsheets listed below: |
| testbrg61-12_6.ods | 75.5Ω reference load. Generator side shield earth. |
| testbrg61-12_7.ods | 50Ω reference load. Generator side shield earth. |
| The dominance of the line capacitance for the bridge with the 75.5Ω load is so great that the apparent secondary capacitance is negative. |
|
15. Correction series for effective transformer-secondary
capacitance. The full menagerie of parasitic inductances and capacitances, or at least the ones which make a significant contribution, are shown on the diagram below. |
| Combining equations (13.1), (12.3), (14.1), (14.3) and (11.3) gives an arithmetic series for the apparent secondary capacitance of the current transformer; i.e., the capacitance which, in combination with the secondary coupled inductance Li, determines the phase crossover frequency and the bridge phase-error at high frequencies. |
|
|
It is now possible to see why there has never been a consistent
view regarding the importance of transformer self-capacitance
in determining bridge performance. If the apparent secondary
capacitance included in the basic model (section
2) is attributed to self-capacitance alone; then bridge
designers (should they have a method for measuring it) will find
that self-capacitance can be either positive, negative, or accidentally
zero. It all depends on the physical layout, the parasitic reactances
of the components, and the lengths and diameters of the various
wires and cables. While such issues remain uncontrolled, it will
be impossible for one constructor to reproduce the results obtained
by another. It is perhaps relevant at this point to ask whether equation (15.1) is complete (at least with regard to effects which make a difference of more than about 0.1pF), and the answer is "not quite". There is nothing in the theory developed so-far which accounts for the fact that the effective velocity for a wave travelling in the line from the voltage sampling point to the current-transformer is less than c. The PTFE dielectric used in the test bridge has a velocity factor of 0.7. The small additional delay will increase Cieff slightly, but in terms of the model, it will merely decrease the effective inductance of the upper voltage sampling arm (i.e, a small delay in the current sampling path is equivalent to a small advance in the voltage sampling path). In equation (15.1), all of the terms with a (-) sign represent possible methods for neutralising the transformer self-capacitance, and there are others. Neutralisation techniques are explored in section 18. |
|
16. Bridge performance evaluation: In normal design practice, component tolerances and uncertainty in the known value of the transformer secondary coupled inductance dictate that the LF compensation resistance Rv should be made adjustable. Likewise, a lack of knowledge of the transformer efficiency and the amount of stray capacitance dictates that the voltage sampling network ratio should be made adjustable. As has been demonstrated by the previous experiments however, these two adjustments are only sufficient to allow the bridge to be calibrated at a single frequency, and there will be some error everywhere else. A question which needs to be answered therefore is: "If a bridge is calibrated at some specified frequency, how accurate will it be at other frequencies?" As was mentioned in the introduction, a direct answer to this question can only be had by using a dummy antenna and some kind of impedance analyser; but we can turn the problem on its head by choosing notional 'set' values for the adjustable components and then measuring how far the actual components need to differ from the set values in order to balance the bridge. These parameter shifts are 'perturbations', which can be applied to the circuit model in order to determine the extent to which the load impedance must differ from the target load impedance if the bridge is to balance using the set values. The situation which needs to be analysed here is that of a bridge which is calibrated at some frequency by adjustment of C1 and RV as shown in the diagram below. At the calibration frequency, the bridge balances when connected to a reference impedance R0; but in normal service C1 and RV cannot be changed, and so at frequencies other than the calibration frequency, the load impedance required to balance the bridge (Zbal) will differ from R0. Our objective is to determine Zbal at some arbitrary frequency by balancing the bridge with R0 connected, measuring C1 and RV, and comparing these values with the values (C1set and RVset say) which were required at the calibration frequency. The overall balance condition for a Douma bridge was given earlier as: [(jXC1 // jXC2 // RV) / jXC2] [1 + Zi / (R0 N²) ] = Zi / (N R0) . . . . (16.1) |
|
Where Zi = Ri // jXi and in this case Xi = 2πf Li' If the bridge is out of balance at the test frequency because C1set and RVset are used instead of the optimal values, then equation (16.1) becomes: |
 |
|
[(jXC1set // jXC2
// RVset) / jXC2]
[1 + Zi / (Zbal
N²) ] = Zi / (N Zbal) .
. . . (16.2) Hence, the relationship between Zbal and R0 can be determined by dividing equation (16.1) by equation (16.2):
Now, to make the problem mathematically tractable, we may note that the factors 1+Zi/(R0N²) and 1+Zi/(Zbal N²) are present only to account for the small difference between V and V'. Since these factors are already close to unity, and we do not expect Zbal to differ greatly from R0, we may reasonably use the approximation: 1 + Zi / (R0 N²) = 1 + Zi / (Zbal N²) This gives the simplification:
If we also define a new variable: CV = C1 + C2 and a new parameter CVset = C1set + C2 then XC1 // XC2 = XCV etc. Hence:
Expanding the parallel products gives:
and recalling that XC = -1/(2πfC)
Now we can multiply the numerator and denominator by the complex conjugate of the denominator with a view to putting the equation into a+jb form:
Multiplying out:
Thus we obtain the formula for bridge evaluation by perturbation analysis: |
|
Let U = RV CVset / [ RVset CV ( RV² + XCV² ) ] where CV = C1'0 + C1s + C2 CVset = C1set + C2 and let Zbal = Rbal + jXbal Then Rbal = R0 U ( RV RVset + XCV XCVset ) Xbal = R0 U ( RV XCVset - RVset XCV ) |Zbal| = √( Rbal² + Xbal² ) φbal = Arctan( Xbal / Rbal ) |
|
|
Note that the C1' required here is the
true value (corrected for inductance), i.e., C1'0; the point being that the frequency at which
the reference capacitor was calibrated should not have any bearing
on the outcome of the test. A definition for C1'0 is given by rearrangement of equation (4.2): C1'0 = 1/[(1/C1'm) + (2πfm)² L1] C1'm being the raw measured value and fm the measuring frequency. When the capacitance meter operates at 107 radians/sec this becomes: C1'0 = 1/[(1/C1'm) + 1014 L1] In practice, this correction is small, but since it can be computed easily it might as well be included. The value of C1s does not need to be known accurately. In the evaluation spreadsheet discussed below, C1s is calculated from the transformer constant, which is defined as: KT = C1/C2 = (N R0 / k Ri) + (1/N) - 1 where C1 = C1'0 + C1s but it can be put in directly if so desired. If the value used is accurate, then the value of C1set which gives optimal performance will be accurate; otherwise, the determined value for C1set will be nominal only. It is a good idea to ensure that C1s is plausible, but the chosen value makes little difference to the outcome of the analysis. Notice also that the formula (16.3) contains no current-transformer network parameters. The details of the current transformer are irrelevant as far as the evaluation procedure is concerned; the test merely determines how well the bridge stays in balance as the frequency is changed. |
|
16a. Performance of uncorrected bridges.. In sections (10) and (11) we extracted the lumped-component circuit parameters from four datasets, the raw data and mathematical analysis being given in the spreadsheets listed below: |
| Faraday shield earth ↓ |
|
|
| Load side |
|
|
| Generator side |
|
|
|
In each case, the data were also subjected to the perturbation
analysis developed above, the details appearing on sheet 5 of
the spreadsheet. The most striking outcome in every case is that the graph of phase error in degrees vs. frequency is always a straight line; so straight in fact that an additional least-squares routine was written to extract the gradient. The result for the bridge with the Faraday shield earthed on the generator side (taken from testbrg61-12_4.ods) is shown below: |
|
A measurement in degrees per MHz is a time. Specifically
it is only necessary to divide the gradient by 360 to get a time
in microseconds. The fit to the phase error returned a gradient
of -0.1461°/MHz, which corresponds to a time of -0.4059ns
(negative because it is a delay). The linearity of the graph is a clear indication that the phase error of the bridge is merely an error of time-coincidence.. This can best be understood by considering the bridge as an optical interferometer. An EM wave enters the bridge and is split into two components. The two components make separate journeys over electrical distances which do not have to be identical, one is inverted, and then the waves are combined. Any difference in the electrical paths results in incomplete cancellation. Much of the error can be accounted for merely by considering the distances involved, but the various inductances and capacitances encountered along the way alter the apparent propagation velocity in the regions in which they reside. A series inductance largely serves to convolute the path, and the delay can be deduced approximately by unravelling the wire and measuring it. A parallel reactance however acts as a scattering element, by storing energy and returning it to the system with a shift of phase. It is the combination of the incident and the scattered wave which appears to have a velocity which differs from c. What is remarkable is not that the bridge can be understood in terms of time and distance, but that the lumped component theory accounts for its behaviour so well. The only component which seems to come from nowhere is the 'self-capacitance' of the transformer Ci', and although it is difficult to control the experiment well enough to prove it exactly, a plausible value, definitely close to the true one, is obtained by taking the average distance which must be travelled by a wave propagating through the transformer. That distance is half the electrical length of the secondary winding wire, plus the electrical distance from the voltage sampling point to the transformer core. For the test bridge with the Faraday shield earthed on the generator side, the distance from the input port to the middle of the transformer was 28mm, 12mm of that being inside a cable with PTFE dielectric. The length of the secondary winding wire was 228mm. Assuming an effective propagation velocity of c within the transformer, this gives: Ci' = [ (228/2) + 1.6 + 1.2/0.7 ]×10-3/ (Ri c) = 0.1173 / (49.9 c) = 7.8pF The corresponding time delay is: 0.1173 / c = 0.3913ns. This is remarkably close to the gradient of the graph above, but from the discussion in sections 12 to 15, the agreement is obviously accidental. In practice, the effective velocity within the transformer will be somewhat less than c, and Ci' will be correspondingly increased. So, hopefully having laid to rest the idea that the self-capacitance of a coil is due to the proximity of adjacent turns, we may now admit that the graph of phase error vs. frequency is not perfectly linear. The deviation is however very small, as can be seen by noting the scale of the y-axis in the graph below. |
| There is a slight increase in the phase lag between about 6 and 17MHz. This, of course, corresponds to the increase in permeability at the onset of a dispersion region in type 61 ferrite material, as was discussed in Section 10. The reason why the peak is inverted in the graph above is that an increase in the inductance of a parallel LCR system (i.e., the transformer secondary) gives greater dominance to the capacitive arm. The deviation amounts to about 60 milli degrees, and so, if the system is adjusted to absorb it, it limits the ultimate phase performance of transformers using type 61 material to about ±0.03° (and other ferrites are similar). When the dispersion peak is excluded from the fit, the rest of the data (for any of the experiments) can be fitted with an RMS deviation of about ±0.0075° , i.e., 7.5 thousandths of a degree. This is the phase-resolution of the experimental method. |
| The perturbation method also produces magnitude information. Shown below are graphs of magnitude error, the two curves giving a comparison between the results obtained with the 40pF reference capacitor, which has an effective series inductance of about 80nH (neglecting L2) ; and the 500pF capacitor, which has an effective inductance of about 62nH. |
| The inductance of the 500pF reference capacitor gives rise to a magnitude error of about 10% at 21MHz, i.e., a bridge which should balance with a load of 50Ω, will actually balance when the load is 55Ω. In a working bridge, of course, we might minimise this error by keeping the wires short and using very small components for C1; but the test bridge had a divider ratio of only about 11.6:1 in the voltage sampling network. For a bridge with a larger transformer turns-ratio, or a lower value of Ri (i.e., a larger transformer constant), the asymmetry of the voltage network will be greater and so too will be the effect of the parasitic inductance. The uncorrected test bridge is barely good enough for crude SWR measurements using a diode detector. Many bridges described in the literature are a lot worse than this; but on the subject of accuracy, most writers have so far shown a tendency to avoid comment . |
|
17. Voltage sampling network inductance balance: In order to obtain a flat frequency response from a capacitive potential divider; it is not necessary that the arms be composed of pure capacitance, but only that the two impedances remain in constant proportion as the frequency is varied. Consequently, insofar as the non-ideality of the dominant (i.e., lowest impedance) arm can be represented as a series inductance, a flat frequency response can be obtained by placing a balancing inductance in the other arm. |
|
In the potential divider circuit
shown right, the impedance marked Z1
is dominant and is provided by a capacitor with a series parasitic
inductance L1. L2
is an adjustable inductance which includes the parasitic inductance
of the capacitor C2. The output of the
network is given by: VV = V' Z1 / (Z1 + Z2) Hence, working with the reciprocal transfer function: V' / VV = 1 + Z2 / Z1 i.e.
|
 |
Writing the reactances explicitly gives:
and multiplying top and bottom of the right-most term by -2πf gives:
This rearranges to: [(V' / VV) - 1] [ (1/C1) - (2πf)²L1 ] = (1/C2) - (2πf)²L2 Now grouping capacitance on one side and inductance on the other gives: [(V' / VV) - 1] (1/C1) - (1/C2) = [(V' / VV) - 1] (2πf)²L1 - (2πf)²L2 The frequency dependence of this expression is removed when both sides are made equal to zero, i.e. when: [(V' / VV) - 1] L1 = L2 in which case also: [(V' / VV) - 1] (1/C1) = 1/C2 i.e.: [(V' / VV) - 1] = C1 / C2 Hence:
The relationship between C1 and C2 for a Douma bridge was given in equation (2.1). Modified to allow for transformer efficiency it gives the transformer constant: KT = C1/C2 = (N R0 / k Ri) + (1/N) - 1 For a bridge with R0=Ri, N=12 and k=0.96, C1/C2=11.58. If the bridge is constructed so that the capacitor C1 has very short wires, the parasitic inductance might be kept down to about 20nH. In that case, the required balancing inductance will be 232nH. For the test bridge, due to the need to use large variable capacitors, the parasitic inductance is in the 60 to 80nH range, and the required compensating inductance is 695 to 926nH. Note that there will be a frequency at which the balancing inductance L2 resonates with C2. This, in fact, is the same as the frequency at which C1 resonates with L1, because L1C1=L2C2; but whereas resonance of the lower voltage sampling arm will merely result in inaccurate readings, resonance of the upper arm will present the generator with a short circuit. The frequency at which this short circuit occurs is:
|
|
|
The effect of the inductance balance coil on quadrature balance,
being an adjunct to the parasitic inductance of the upper voltage
sampling arm, is given by equation (12.3).
As was mentioned before, L2 partially
neutralises the 'self-capacitance' of the transformer by increasing
the length of the electrical path from the voltage sampling point
to the summing point. The contribution is not great however,
even for a relatively large inductance, as can be seen by differentiating
equation (15.1): ∂Cieff/∂L2 = -1/(N R0 Rv) For the test bridge, with N=12, R0=50Ω and Rv=1455Ω, ∂Cieff/∂L2 = -1.15 pF/μH. Inductance balance compensation is simple and boringly effective. A small adjustable inductor, the number of turns easily determined by experiment, is all that is required. For tracking over the HF spectrum, the capacitor C1 is adjusted at about 2MHz, and the balance coil L2 at a frequency a little below 30MHz. The adjustments are repeated two or three times until no further improvement can be obtained. Amplitude flatness of better than ±0.4% (±0.034dB) over the 1.6 to 30MHz range is easily achievable, even with the excessively inductive reference capacitors used with the test bridge. If the inductance of the lower voltage sampling arm is kept to a minimum, the flatness can be better than ±0.03% There is no point in evaluating the technique separately. The performance figures given above will be corroborated by the experimental data to follow. An inductance balance coil should not be used during model-parameter determinations, because L2 contributes to Cieff, but one should always be included in a working Douma bridge. |
|
18: Frequency tracking. In the next few sections we will look at techniques for neutralising bridge phase error. Several possibilities are obvious from the theoretical considerations of sections 11 to 14, and it is now a matter of seeing how well they work in practice. There are further less-obvious neutralisation methods however, and these will be introduced. With a balance coil to cancel the effect of the inductance of the lower voltage-sampling network capacitor, and some means for neutralising the apparent secondary capacitance, the bridge has 2-point frequency-tracking capability. Calibrating the bridge is then a matter of adjusting C1 and RV at some low frequency such as 2MHz, and adjusting the inductance balance and phase neutralisation at some frequency approaching to the upper limit of the desired working range. Generally, the low and high-frequency settings interact slightly, and so the adjustments have to be repeated a few times until no further improvement can be obtained. Two or three cycles of adjustment is usually sufficient. Achieving the maximum benefit from 2-point tracking is a matter of selecting the optimal upper calibration frequency. After calibration, the bridge is perfect at two frequencies but there will be some residual error elsewhere. Typically, the quadrature balance deviates slightly according to a curve which mimics the inverted real-part of the permeability dispersion in the core material. Apart from that, the phase error is proportional to frequency and eminently correctable; consistent with it being a timing error in a system which is free from any other significant dispersive effects. The residual error in the in-phase balance condition is a little more complicated however. It partially mimics the inverted imaginary part of the permeability dispersion, but there are other effects caused by the fact that the inductance of C1 is not perfectly described as a single series component, and by the minor approximations inherent in ignoring the frequency-dependent effects of parasitic reactances on the apparent efficiency of the transformer. Most of the experimental data given below was obtained using the 8 to 48pF reference capacitor described in section 6. This capacitor has an equivalent series inductance of about 85nH; and since it has to be used with a parallel padding capacitor, lumping all of its inductance into a single series component does not provide a perfect description. Consequently, the in-phase performance of a bridge using this capacitor is limited to about ±0.2%. The reference capacitor was identified as the cause of this limitation by performing a control experiment using a 2.5 to 30pF trimmer and padding capacitor with an equivalent series inductance of about 50nH, i.e., by reverting to the method used with the prototype bridge described in section 3. The business of unplugging a capacitor and measuring it on a separate bridge is too laborious for general adoption, and the data are not particularly precise (±0.25pF). The results are nevertheless accurate, and show that the in-phase balance tracking is good to within ±0.03% if the inductance in the lower voltage-sampling arm is kept to a minimum. Given that there are bound to be residual errors, the best upper calibration frequency is the one which gives approximately equal run-out above and below the target in-phase and quadrature balance points over the working range. For the bridges evaluated below, it lies somewhere in the 20 to 28MHz region and cannot always be determined prior to the first test-run. To avoid repeating experiments ad-nauseam without gaining any new information, most neutralisation methods are tested only once; which means that there may still be room for improvement. The ultimate performance obtainable from a particular neutralisation method is given by the least-squares fit to the phase error (see graph of obs-calc on sheet 5); and the optimal upper calibration frequency can usually be determined by adjusting the fitting weights for equal run-out above and below zero and noting the upper zero-crossing frequency for the error curve. |
|
18a: Phase neutralisation by load port capacitance. The most straightforward and obvious method for neutralising the apparent secondary capacitance of the transformer is to add some capacitance across the load side of the through-line. The effect of such a capacitance is given by differentiating equation (14.1): ∂Cieff/∂C0 = - R0 / Ri A bridge neutralised by this method is shown below. |
| The bridge was calibrated at 2 and 26MHz and then evaluated according to the procedure outlined in section 16. The spreadsheet file and a summary of the outcome is as follows. |
|
Gen. side shield earth. 1.6 to 30MHz |
R0 = 50Ω, Ri = 50Ω Inductance balance coil fitted. |
|Z| error. |
( ) → best possible |
| testbrg61-1212.ods | 6.6pF across load port |
|
|
| The magnitude error is mainly attributable to the reference capacitor. The choice of 26MHz as the upper calibration frequency was not optimal, and the regression line for the overall phase error indicates that the run-out can be reduced to ±0.09° by calibrating at 21MHz. |
| The result is, of course, at least an order of magnitude better than for any uncompensated bridge, but it is not the best that can be achieved. Placing capacitance across the through-line in addition to the voltage sampling network also adversely affects the generator power-factor; and although this problem is sometimes correctable, and does not matter if the bridge is to be taken out of circuit after use, loading the generator with extra capacitance is not generally desirable. |
|
18b: Quadrature current compensation. The effect of the Faraday shield protrusion capacitance in a bridge with the shield earthed on the load side was investigated in section 11. The theory developed there implies that the apparent secondary capacitance can be neutralised either by injecting a capacitive quadrature current into the shield itself, or into an additional compensation winding. Connecting the neutralising capacitor to the shield is the most economical method if the current is to be delivered from the generator terminal, and it is doubtful that the use of a separate winding will make any difference in that case. The initial test configuration is shown below; and the results and a link to the analysis spreadsheet are given below that. |
|
Load side shield earth. 1.6 to 30MHz |
R0 = 50Ω, Ri = 50Ω. Inductance balance coil fitted. |
|Z| error. |
( ) → best possible |
| testbrg61-1213.ods | 6.7pF, gen. to shield |
|
|
|
In the derivation given in section 11,
assumptions were made firstly: that no voltage is developed between
the ends of the Faraday shield; and secondly: that the voltage
drop across the transformer primary is negligible. With these
simplifications, it was found that the shield capacitance has
the same effect as placing a capacitance across the load port.
Evidently the two neutralisation effects are not the same however,
because placing a capacitor across the load port results in better
overall performance than injecting a current into the shield. In a current transformer, the flux-density in the core and hence the primary voltage Vii, is controlled by the main secondary load. Hence the shield, being effectively a secondary winding with the same number of turns as the primary, will develop a voltage which is of the same magnitude as Vii; but since the shield is loaded only by its own inductance, its voltage (Va say) will be shifted +90° in phase relative to Vii. The voltage Va must be subtracted from the voltage across the line-to-shield capacitance in order to establish the exact phase of the (not quite perfect) 'quadrature' current injected. In the original derivation, the shield protrusion displacement current was given approximately as: Ish = V / (jXCsh) Now however, with the notation altered to suit the general context of neutralising currents injected into auxiliary windings, we have to admit that it is better described by the expression: In = (V + Vii - Va ) / (jXCn) the reasoning being illustrated by the diagram below. Here the vector In (noting that j is on the bottom of the fraction and capacitive reactance is negative) lies at +90°, not to the voltage across the load, but to the vector V+Vii-Va. This is the actual neutralisation reference voltage, to which we will assign the symbol Vref. |
| We can obtain a qualitative idea of how Vref evolves with frequency by noting that Vii is a scaled down version of Vi; where Vi is the voltage across Zi, and the locus of Zi in the Z-plane is a constant-conductance circle mainly determined by the relative magnitudes of Ri and XLi. At low frequencies, Vii leads V, but moves into phase with it as XLi increases with frequency. Hence we can draw the sum of the three vectors which make up Vref and show that it is always slightly lagging; and that the lag increases with frequency. The result is that the neutralising current, (at +90° relative to Vref) is slightly out of true quadrature to V, with a lag that gets worse as the frequency increases. The error is in the same direction as that caused by dispersion in the ferrite. The result is that the neutralising current does not cancel Cieff to the limit permitted by the dispersion, but introduces an error in addition to the dispersive effect. This error moreover, permits the neutralising current to affect the in-phase balance condition, thereby also slightly degrading the bridge amplitude performance. |
 |
| One point which arises from these considerations is that we do not want the partial neutralising effect which results from earthing the shield on the load side of the transformer core. Hence the shield should be earthed on the generator side (or perhaps omitted altogether - see section 19). The subject of current-injection neutralisation is far from dead however; firstly because we can change the phase of Va by loading the auxiliary winding; and secondly because, if we use an actual winding instead of the shield itself, we can connect the neutralisation capacitor to the load terminal instead of the generator terminal. |
| Consider what happens when a 1 turn auxiliary winding is resistively loaded with the same number of Ohms per turn as the main secondary. If the secondary has (say) 12 turns, and Ri = 50Ω, then this involves placing a resistance of 50/12 = 4.2Ω across the auxiliary winding. Now, in order to calculate Vi, we can consider that the transformer has 13 turns loaded with 54.2Ω, and that Vi is obtained by tapping in at the 12th turn. From that electrical equivalence, it should be obvious that Va is now in phase with Vi and hence, to a very good approximation, identical in magnitude and phase to Vii. The result is that Vref becomes equal in magnitude and phase to V, as can be seen from the vector diagram on the right. Hence the neutralising current In will be in true quadrature to V, and the neutralisation errors will be eliminated. |
 |
|
There is a problem with this improved
neutralisation technique however, and it is this: If we make
Vref identical to V, then
we eliminate the approximations from the analysis given in section 11. This means that the
result, in phase performance terms, is identical to that of placing
a capacitor across the load port. Hence the circuit is made more
complicated for no benefit, and furthermore, the efficiency of
the transformer is reduced by a factor of N/(N+1). Consequently,
it is difficult to see why anyone would want to use this method,
except perhaps that placing a resistance of Ri/N
across the ends of a load-side earthed Faraday shield will eliminate
the deleterious effects of the displacement current. This might
be useful if the shield is earthed on the load side intentionally
(e.g., to allow neutralisation by sliding the core along the
shield -see note below). Let us now turn our attention to what happens when the auxiliary winding is connected to the load terminal. The test bridge so configured is shown below, and a summary of the results is given below that. |
|
Gen. side shield earth. 1.6 to 30MHz |
R0 = 50Ω, Ri = 50Ω. L-bal.
coil. 1-turn I-compensation winding. |
|Z| error. |
( ) → best possible |
| testbrg61-13_2.ods | 3.5pF to load terminal. |
|
|
| When the test bridge was neutralised by connecting a capacitor across the load port, or by injecting a quadrature current from the generator terminal, the neutralising capacitance required (measured after the test) was nearly 7pF. When the neutralising current is taken from the load terminal however, the required capacitance is approximately halved. The reason for that can be understood by inspecting the diagram below, which shows that In flows through the transformer twice, once through the primary winding and once through the auxiliary winding. This gives the circuit a definite advantage over previous configurations, which is that neutralisation can be achieved with a reduced penalty in terms of generator power-factor. It is, of course, possible to reduce Cn still further by adding more turns to the auxiliary winding, but that is not a good idea because it will increase the quadrature error. |
|
With regard to overall performance with the neutralisation current
taken from the load terminal; it is better than when the current
is taken from the generator terminal but not quite as good as
when a capacitor is placed across the load port. The reason for
that can be seen in the vector diagrams, which show a reference
voltage with a residual phase lag which increases with frequency.
This supplements the ferrite dispersion effect, as before, but
less severely. The obvious next experiment is to see what happens when the auxiliary winding is loaded with a resistance of about Ri/N to bring Va into phase with Vi. The test bridge with this addition is shown below and the test results are given below that.. |
|
Gen. side shield earth. 1.6 to 30MHz |
R0 = 50Ω, Ri = 50Ω. L-bal.
coil. 1-turn I-comp. winding // 4.26Ω |
|Z| error. |
( ) → best possible |
| testbrg61-13_3.ods | 3.4pF to load terminal. |
|
|
| The performance of this bridge is relatively poor, being barely an order of magnitude better than typical published designs; but it nevertheless has a very interesting property. The phase deviation curve has the opposite sense to that of the bridge with the unloaded auxiliary winding, and is also in the opposite sense to the deviation caused by the permeability dispersion. The reason can be understood by looking at the vector diagrams below, which show that the neutralisation reference voltage moves anti-clockwise with increasing frequency, whereas it moves clockwise when the auxiliary winding is unloaded. |
| If the phase deviation is negative in the middle of the working frequency range when the neutralisation winding is unloaded, and positive when it is loaded with a resistance of Ri/N; then logically, there must be an intermediate loading condition which brings the overall phase error to a minimum. To explore this possibility, the bridge was set up with a 100Ω Cermet variable resistor for the auxiliary load. With this modification, the bridge has 3-point frequency tracking for phase error and 2-point tracking for magnitude error. |
| It was thought initially, that this new arrangement would be difficult to calibrate, but the task proved to be surprisingly straightforward. C1 and RV were adjusted at 2MHz, Cn and L2 at 26MHz; and the loading resistance Rn was adjusted at 18MHz, using C1 for in-phase balance while the latter operation was being carried out. The adjustments were not strongly interactive, and only 3 rounds of iteration were required. It was also obvious that there was now considerable latitude in the choice of upper calibration frequency, since the bridge barely went out of balance during the search for maximum run-out. |
 |
|
Gen. side shield gnd. 1.6 to 30MHz |
R0 = 50Ω, Ri = 50Ω. L-bal.
coil. 1-turn I-comp. winding // 100Ω pot. |
|Z| error. |
( ) → best possible |
| testbrg61-13_4.ods | Measured: Cn = 2.95pF, Rn = 21.7Ω |
|
|
|
There was certainly no need to use the coarse adjustment for
RV during the evaluation of this bridge.
The reason is evident from the graph above. The chosen compromise
between the negative mid-band run-out given by an unloaded neutralisation
winding, and the positive run-out given by a heavily loaded winding
has reduced the phase error to better than ±0.044°.
Moving the mid-point adjustment from 18MHz to about 17MHz will
reduce it further to about ±0.038° (i.e., it will
drag the curve onto the zero line at 17MHz). This level of performance is easily two orders of magnitude better than that offered by typical current-transformer bridges; and led the author to the view that the quadrature-current neutralisation technique is a good candidate for the design of reference instruments. On this basis, a working bridge was constructed and, by paying careful attention to layout, grounding, and minimisation of inductance in the lower arm of the voltage-sampling network, it was found possible to get the maximum phase error down to ±0.025° and the maximum magnitude error down to ±0.04%. It is doubtful that such performance will hold for long periods without drift, but ±0.05° and ±0.1% is a perfectly reasonable expectation over a normal annual calibration interval (see appendix 6.5). Some additional comments on current-injection neutralisation: • Impedance bridges are often used reversed (i.e., with the generator and load connections swapped) for forward power measurement, or to obtain a forward transmission reference for the measurement of SWR. It should be noted therefore, that current injection neutralisation is asymmetric; i.e., the condition required for obtaining perfect balance (minimum reverse power reading) is not the same as the condition required for perfect anti-balance (maximum forward power reading). In practice however, any neutralisation error in a forward-power bridge will make negligible difference to the reading. The purpose of neutralisation is to produce the rigorous conditions required for a null. Forward reading bridges do not null and so (assuming that the phase error is no more than a few degrees) do not need to be neutralised. Consequently, in an SWR bridge or directional power meter, neutralisation can be applied to ensure accurate reverse readings, and the very minor effect on the forward reading can generally be ignored. • It was mentioned earlier that the Faraday shield should be earthed on the generator side in order to eliminate the uncontrolled neutralising effect of the shield protrusion capacitance. For practical purposes however, the extremes of accuracy demonstrated here are not usually needed. That being the case; there is a property of the load-side earthed shield configuration which might be used to eliminate an expensive trimmer capacitor in production bridges. In a private communication, David Stansfield, G0EVV, noted that it was possible to improve the frequency tracking of a bridge by sliding the transformer core along the shield. This action, of course, changes the shield protrusion capacitance; and so, with a little slack in the secondary connecting wires, sliding the core does the same job as adjusting a trimmer. There is a caveat however, which is that, once calibration has been accomplished, it must be ensured that the core will not move if the bridge is bumped or dropped. • The current-injection neutralisation technique developed here is comparable to a method for cancelling the parasitic capacitance of power-supply filter inductors described by Neugebauer and Perreault [49]. In a curent-transformer, the cancellation can be considered to be referred either to the pimary or to the secondary winding • There are other ways in which multiple frequency-tracking adjustments can be provided in bridges using load-port capacitance or current-injection neutralisation. For example: if a small adjustable inductor is placed in series with the neutralisation capacitor, the neutralisation capacitance will appear to increase with frequency. Such a combination can therefore compensate for a situation in which a bridge exhibits a lagging phase error at high frequencies but a leading phase error in mid band (such as can be arranged when the current-injection winding is loaded by a resistance). |
|
18c. Quadrature voltage compensation:
The neutralisation techniques described in the previous two sections all carry a possible penalty in terms of generator power-factor; and although it will be difficult to improve on the performance of the 3-point tracking method described above, extreme accuracy is not always required. Indeed, a bridge which is accurate to within about ±0.5° and ±0.5% is almost certainly good enough for any circuit which uses a diode detector. Hence there is a place for neutralisation techniques which do not noticeably alter the characteristics of the through-line. As was noted in the previous section, an unloaded auxiliary winding on a current transformer produces a voltage which lies at +90° relative to the main secondary voltage. Hence if a potentiometer of high-enough resistance not to constitute a significant load is placed across such a winding; it is possible to produce an adjustable quadrature voltage which can be added to the main output to cancel the high-frequency phase error. The principle of the technique is summarised in the diagrams below. |
| Ideally, the locus of the corrected output voltage should lie on a constant-conductance circle if the network is to give an accurate simulation of a transformer with no secondary capacitance. From the vector diagrams it can be seen that this does not happen in practice, but by adjusting RV at the low end of the working frequency range and |Vq| at the high end, a reasonable compromise can be reached. The test bridge set up for quadrature voltage neutralisation is shown below. |
|
|
For the first attempt at testing the technique, a single turn
compensation winding with a 250Ω Cermet potentiometer connected
across it was used. This worked for a bridge with the Faraday
shield earthed on the load side; but with the shield earthed
on the generator side according to later-discovered best practice,
Cieff is greater and neutralisation could
only just be achieved with the pot. set somewhere close to maximum.
Since reaching the end-stop of a control interferes with the
process of searching for nulls and makes calibration difficult
to achieve, the compensation winding was changed to 2-turns and
a 120Ω resistor was placed in series with the pot.. The
purpose of the additional resistor was to keep the loading light
and bring the adjustment roughly to the middle of the available
range. One idiosyncrasy of quadrature voltage compensation is that it places additional DC resistance in series with the detector output. This resistance, which is the parallel combination of the two sides of the adjustment pot., will make practically no difference to the performance of a working bridge, but it lies in the way of measuring RV via the detector port. Hence, designating the additional resistance Rq, the experimental quantity is RV+Rq. For the evaluation experiments listed below, Rq was measured at the end of the test run by shorting RV with a jumper lead. An extra column was then added to the analysis spreadsheet so that Rq could be subtracted from RV+Rq to obtain RV. |
|
Gen. side shield earth. 1.6 to 30MHz |
R0 = 50Ω, Ri = 50Ω. L balance
coil. 2-turn V-compensation winding with 250Ω pot + 120Ω. |
|Z| error |
( ) → best possible |
| testbrg61-14_2.ods | 8-48pF ref. cap. with 82pF padding |
|
|
| testbrg61-14_3.ods | 3-30pF trimmer with 82pF padding. |
|
|
|
Two experimental runs were performed in this case, one using
the 8-48pF reference capacitor for C1'
and one using a 3-30pF trimmer. Padding in both instances was
82pF. The trimmer had to be removed and measured at each frequency
point, considerably reducing the precision of the C1'
data, but the smaller assembly had an equivalent series inductance
(L1) of 50nH, as opposed to 85nH for the
larger reference capacitor. The object of the exercise was to
show that reducing L1 (and also L2) improves the magnitude performance without
significantly affecting the phase performance. The phase performance of the quadrature-voltage compensated bridge is very slightly inferior to that of the bridge compensated by means of a capacitance across the load port (section 18a); but of course, quadrature-voltage compensation has no generator power-factor penalty. |
© D W Knight 2007.
David Knight asserts the right to be recognised as the author of this work.
|
|
|
|
|
|
u