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A6.4.3 |
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A6.5: A Self-evaluating Precision Reference Bridge.
Max.
f error of ±<0.1° and
max. |Z| error of ±<0.2% over the 1.6 to 30MHz
range |
1. Introduction. 2. Circuit, components, and layout. 3. Setup and calibration. 4. Theory of the evaluation process. 5. Evaluation procedure and final calibration. |
6. The effect of temperature on accuracy. 7. Input power-factor. 8. Test results. 9. Using the bridge. . |
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Abstract: This article describes the construction and evaluation of an accurate 50W impedance-matching-reference bridge operating over the 1.6 to 30MHz range. The bridge is based on a 1:12 toroidal current transformer and has a nominal mid-band insertion loss of 0.06dB. The maximum recommended continuous power throughput is 100W. |
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Voltage sampling is by means of
a capacitive potential divider. Two-point magnitude-vs.-frequency
tracking is achieved by adjustment of the potential-divider ratio
at the LF end of the operating range and by adjustment of an
inductance compensation coil at the HF end. LF phase compensation
is by means of an adjustable resistance in parallel with the
lower voltage-sampling arm. A wire passing through the transformer core is used for HF phase compensation. A quadrature current is injected into this auxiliary winding by means of an adjustable capacitor. Fine control of the phase of the correction signal is achieved by means of an adjustable resistance connected across the extra winding. This arrangement gives three-point phase-vs.-frequency tracking. |
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The bridge is rendered self-evaluating
by choice of orthogonal adjustments for resistance and reactance
balance, and by including provision for the associated adjustment
parameters to be measured. The theoretical basis on which frequency-dependent
deviations of the parameters from their calibration settings
can be converted into performance data is given. Both Faraday-shielded and unshielded versions of the current transformer were tried. In both cases, with the bridge maintained at constant temperature; the maximum maximum peak-to-peak phase runout was about ±0.03°; and the maximum magnitude runout was about ±0.04%. The ultimate magnitude accuracy is limited by the uncertainty in the value of the reference load resistance used during calibration. Using a 4½-digit Fluke 8060A multimeter for the resistance measurement gave a final magnitude accuracy of about ±0.13%. The results obtained constitute a two orders-of-magnitude improvement over typical Douma (CVS) transmission-bridge designs. The effect of temperature variation on phase accuracy is investigated. This issue is related to the temperature coefficient of the permeability of the transformer core. The effect is only significant at low frequencies. A compensation scheme using a Linear PTC thermistor is proposed but not tested.  This
is an extension of the work described in appendix
6.4. For a full discussion of the background theory,
readers should refer to that article. |
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1. Introduction: A study of the various causes-of and remedies-for phase and magnitude error in RF current-transformer bridges was made in appendix 6.4. A summary of the relevant findings can be given as follows:  For a bridge with
a capacitive potential-divider for the voltage-sampling network;
the major cause of amplitude error is the inductance of the lower
voltage-sampling arm. Best results are obtained when:a) The inductance of the lower voltage sampling arm is kept to a minimum. b) The residual inductance is balanced-out by means of an adjustable inductor in series with the upper voltage-sampling arm. Of the various
HF phase compensation schemes tried, by far the best was a quadrature
current injection technique with two adjustable parameters. The
method involves fitting the transformer with an auxiliary 1-turn
winding. The injected current is set by means of a trimmer capacitor,
and fine-tuning of the phase of the correcting signal is accomplished
by means of an adjustable resistor placed across the winding.
The system permits the phase error to be brought to zero at two
points in the mid-to-upper frequency range. For a bridge corrected
to give near-perfect phase and amplitude performance and having
a relatively high coupling factor; there is little basis on which
to choose between current-transformers with or without a Faraday
shield.This article describes the construction of a 1.6 - 30MHz 50W impedance reference (match monitoring) bridge employing the corrections outlined above. During the optimisation work, the opportunity was taken to test for performance differences between a Faraday-shielded and an un-shielded version. No significant differences were found. A problem with the production of current-transformer bridges is that of how to determine the accuracy of the finished unit. Traditionally, this has been such an awkward matter that most builders take the performance as a matter of trust. This kind of trust is rarely warranted; and underlies such ludicrous comments as: "I get a perfect 1:1 SWR on [some frequency at the low or high end of my bridge's working range] but my transmitter refuses to give full output". Fortunately however, such privations can be avoided without recourse to exotic test-equipment. This is because a bridge can be made to reveal its own shortcomings; provided that there is a way of determining the extent to which it is out of balance at a particular frequency when connected to a reference load resistance. The trick is to design the bridge in such a way that it has a pair of adjustments capable of bringing it to balance at any frequency in its operating range, and to arrange things so that relative measurements of the values of the two adjustable components can be made without causing disruption. A method of bridge evaluation by parameter-shift measurement was developed in appendix 6.4 section 16. Here we refine that method. An ideal candidate for the technique is Douma's bridge; which has a capacitive potential divider, and uses a resistance placed across the lower voltage-sampling arm to compensate for the finite inductance of the current-transformer secondary winding. Normally the bridge is calibrated at the low end of its operating frequency range by adjusting the resistance and one of the potential-divider capacitors; but provided that the phase error at high frequencies is not too great, these adjustments can be used to balance the bridge at any frequency. That being the case, if the bridge re-balanced at some test frequency, the deviations of the new potential divider settings from the calibration settings can be used to calculate the phase and magnitude error which would have occurred had the settings been left alone. Evaluating the bridge is then a matter of reading the resistance and capacitance values at a representative set of spot frequencies and analysing the data. The potential divider resistance is easily determined by measuring the DC resistance looking into the detector port. This presumes that any passive detector (if used) can be unplugged. A diode detector is however, far too insensitive for accurate balance-point determination; and so, at least during the evaluation process, the detector port is connected to a radio receiver. When this is done, a DC blocking capacitor can be placed in the line to the receiver input, allowing a resistance meter to be left connected via a T-piece during the test procedure. Measuring the potential divider capacitance is a little more difficult. For the experiments which preceded this work, a plug-in variable reference capacitor was used; but such an approach is not suitable for a working unit. The solution adopted here is to scribe logging scales onto a rotary-turret-type ("beehive") trimmer capacitor. The author used a lathe with an angle-finding attachment for that purpose, but it is also possible to improvise a suitable jig using a 360° protractor and a drill-press. Measurements of scale reading vs. capacitance are taken prior to installation and fitted to a regression function. The regression formula is used to convert readings into capacitances for the data analysis. |
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2. Circuit, components, and layout. The circuit and layout of the bridge is shown below. This is based on Douma's bridge [US Pat. # 2808566 (1953) ], with the addition of a small inductor L2 in series with the upper voltage-sampling arm, and a phase-neutralising network comprising Cn, Rn and a 1-turn auxiliary winding on the current transformer. |
| The current transformer is made by winding 12 turns of 0.9mm diameter enamelled copper wire onto an Amidon FT50-61 ½" ferrite toroid. The AL value of the core is 69nH/turn² ±25%, predicting a secondary inductance of 9.9 ±2.3mH. The measured secondary inductance of the author's transformer (Lsec) was 8.15mH at 1.5915MHz. In the author's experience, the inductances of coils wound on FT50-61 beads always come out lower than predicted by the nominal AL value, and so the published value may be optimistic. Due to leakage inductance, the effective or 'coupled' secondary inductance (Li) is about 1% less than the measured value. |
| The bridge was tested both with and without a Faraday shield. The Faraday shielded primary was made from a stub of UR-M108 (Ag-PTFE, FEP-coated, 50W) coaxial cable. The unshielded primary was a piece of 1mm diameter silver-plated copper wire. An insulator to keep the wire centred in the transformer was cut from a short length of polythene honeycomb insulator as used in 75W UHF-TV downlead cable (the point being to keep the dielectric constant as low as possible to minimise stray capacitance). |
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Note that the Faraday shield earth
is taken from the generator (TX) side of the transformer core.
This arrangement prevents capacitive currents from the through-line
to the shield from affecting the high-frequency phase performance
of the transformer [A6.4-11]. The effective secondary inductance (Li) determines the required value of the LF compensation resistance (Rv=RVa+RVb). For the voltage-sampling divider ratio used and Li=8mH, it was found that 2.2 to 2.4KW was a suitable choice for RVb when the transformer was fitted with an unshielded primary, and 2.4 to 2.7KW when a Faraday shield was used (an unshielded bridge has additional stray capacitance across C2 and stray coupling from the through-line to the detector port, both of which affect the balance condition). On commissioning, it is a simple matter to set the RVa to the middle of its travel and connect a 5KW pot. in place of RVb. Then, once the bridge has been roughly set up, the pot. can be removed, measured, and replaced by the nearest fixed resistor. The power rating of the resistor is not critical, the total dissipation in the voltage sampling network being less than 20mW when the input to the bridge is 100W. A low value for C2 was chosen in order to keep the generator (transmitter) power-factor reasonably close to unity. For the layout used, with a Faraday shielded transformer, the effective value for C2 is about 0.2pF greater than the measured value of the capacitor itself, due to strays. When the bridge was tested with no Faraday shield, changes noted in the balance condition indicated additional strays, from the primary to the summing point (s), of about 0.23pF. The transformer core is mounted close to the load socket. This keeps the load-side line mismatch to a minimum. Mismatch on the generator (TX) side does not affect the balance condition, and it is beneficial to leave a short length of this part of the line unshielded. For a 50W match, the through-line should have an average capacitance of about 94pF/m. Removing most of the shield on the TX side subtracts about 2pF, offsetting some of the 4.5pF or so of voltage-sampling-network capacitance. Note that the neutralisation network also places capacitance across the generator, giving a total burden of about 7pF. This capacitance, which gives rise to an input SWR of about 1.1 at 30MHz, can be cancelled-out if so desired by placing additional inductance between the TX socket and the bridge circuitry (see section 7). The transformer secondary is terminated by two 0.5W metal-film resistors with the leads clipped to a length just sufficient for connection. The point of the arrangement is that the inductance of the termination gives rise to a positive (leading) phase error in the transformer output at high frequencies [A6.4-13], whereas the neutralising network can only compensate for lagging phase errors. The total power rating of 1W allows for a maximum secondary current of 141mA, i.e. a maximum primary current of 0.141x12=1.7A, corresponding to 144Watts in a 50W load. This gives a burnout safety-factor of 1.44 when using a standard 100W transmitter. Carbon-film resistors, being liable to change in value when overheated, are not recommended for this position. In construction and in the choice of components, the inductance of the lower voltage sampling arm (C1a, C1b, RVa, RVb) must be kept to a minimum. The point is that the total inductance of the arm must resolve accurately into a frequency-independent equivalent-series-inductance (ESL) if the frequency response is to be corrected by a single inductance in series with C2. This requirement is also assisted by star-wiring the components to the summing point (s), rather than distributing the connections along a rail or circuit-track. If L1 is the ESL of the lower voltage-sampling arm, the compensating inductance is given by: L2 = L1 C1 / C2 [A6.4-17]. Since C1 is about 12 times larger than C2, there will still be a need for a compensating inductance of about 250nH after careful minimisation of L1. Consequently, since the inductance of C2 is part of the total compensating inductance, there is no particular need for C2 to have short leads. The leads simply need to be short enough to allow a positive inductance value for the coil L2, so that there will be some latitude for inductance adjustment. For the author's layout, 3 turns of 0.9mm diameter wire on a 7mm former, spaced at a little more than the wire diameter, did the trick (the spacing can be adjusted to vary the inductance slightly). The phase-neutralising trimmer Cn causes a quadrature current to flow in a 1-turn auxiliary winding. The extra winding is a piece of 0.7mm diameter enamelled copper wire pushed through the core in the gap between the secondary lead-out wires. Since this wire is effectively at ground potential, it causes the minimum of disturbance to the secondary winding when threaded through in that position. The phasing is chosen so the the quadrature current cancels the HF phase-lag in the transformer output caused by propagation delay and load-side through-line mismatch. The auxiliary winding produces a small voltage, the phase of which can be adjusted by means of the variable shunt resistance Rn. This gives a minor adjustment of the phase of the neutralising current. When designing for accuracy, it is important to be aware that the earth conduction path between the transmitter and load sockets is part of the main transmission line. This means that there will be a potential difference between the outsides of the two sockets and a corresponding potential gradient along the earth-plane. It follows that the lower voltage-sampling components and the Faraday shield should be either; star-wired to the detector socket ground; or connected along a line of equipotential (i.e., a line on the ground-plane drawn at an approximate right-angle to the through-line). Socket ground connections of extremely low resistance are essential for accuracy. This can be achieved by fitting a copper sub-chassis to the inside of the box, and by using 4-bolt silver-plated BNC sockets with the flanges mounted on the inside. |
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Parts List (neglecting fasteners and sundries)
| Part: | Description: | Source or RS stock code. |
| Transformer core |
Fair-Rite
5961000301 (Amidon FT50-61), ½" OD, AL = 69 ±17 nH/turn² |
Amidon
Sycom |
| Through-line & Faraday shield | Uniradio M108, 50W, Ag-plated Cu conductors, PTFE dielectric, FEP coat, 4.5mm OD, 94pF/m. (or similar). | |
| L2 former | 7mm dia., polystyrene, with M6 dust-iron slug. | |
| C1a | Philips 3-30pF beehive trimmer 5910-99-0167006 MPH/YL (modified, see text). | Garex (NOS) |
| C1b | 39pF Ag-mica. | RS 495-694 |
| C2 | 4.7pF, 500V, Ag-mica. | RS 495-616 |
| Cn | Philips 2-8pF beehive trimmer. | Garex (NOS) |
| Ria, Rib. | 100W, 0.5W, metal film. | RS 149-644 |
| Rva | Vishay 3/4" Cermet trimpot, 20 turn, type 43p, 500W. | RS 162-192 |
| Rvb |
2 - 3 KW (select on test), carbon
or metal film.  |
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| Rn | Vishay 3/4" Cermet trimpot, 20 turn, type 43p, 50W. | RS 162-653 |
| Trimpot mounts | Bourns H-38P panel mounting adapter for 3/4" trimpots. | RS 107-6878 |
| Box | Eddystone 7134p. 111 x 60 x 30mm OD. (or similar). | RS 343-9524 |
| Sockets | BNC 50W, Ag-plated, PTFE insulated, 4-hole flange. |
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Beware the guilty
secret that often lies beneath the smooth exterior. Perform the
scrape test! If helical resistors are all that is available,
there may be some advantage in using 3 or 4 resistors in parallel. |
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3. Setup and calibration: Equipment required: A Laboratory signal
generator covering at least 1.6 to 30MHz; output level Ca. 0dBm
(224mV RMS). Digital readout is desirable, but a VFO-type generator
can be used with a digital frequency meter (DFM). A continuous coverage
HF communications receiver with digital frequency readout; sensitivity
ca. 100nV for 6dB s/n in 3KHz BW; with Fast AGC facility (the
author used a Kenwood TS-930s). A 50W
±1% coaxial load resistor, rated to several GHz (the reference
resistance, R0cal) A Resistance meter
or multimeter, in a good state of calibration, preferably 4½-digit
(e.g. Fluke 8060A). An adapter, with
silver-plated conductors, for connecting the resistance meter
to BNC plugs. A standard BNC to 4mm adapter with 19mm pin spacing
fits most multimeters. A shorting plug,
made from a silver-plated BNC plug, for zeroing the meter prior
to measuring the reference resistance. A 10cm BNC to
BNC patch cable and a BNC T-piece; all preferably silver-plated. An in-line DC
blocking capacitor (see appendix 6.4 section
6). 2x
common-mode chokes (see appendix 6.4 section
7). Double or triple
shielded 50W test leads, with silver-plated
connectors. Trimming tools
and heat-shunt pliers (see below). A watchmaker's
eyeglass. A small 5KW 270° carbon or Cermet preset pot. A selection of
0.25 or 0.5W (or thereabouts) carbon-film resistors covering
the range 2.0, 2.2, 2.4, 2.7 and 3.0 KW,
OR a selection of metal-film resistors (for connection
in parallel in sets of 3 of the same value) covering the range
6.2, 6.8, 7.5, 8.2 and 9.1 KW. |
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Optimising the test setup: The balance point of the bridge can be determined with extreme accuracy when using a sensitive radio receiver as the detector. The operator can however be fooled if the receiver has any sensitivity to common-mode signals. If the generator can be picked-up by a route other than through the detector port; the apparent balance point will be skewed because the null will occur, not at true balance, but at the point where the signal from the bridge is of the correct magnitude and phase to cancel the spurious signal. There are three main causes of this spurious sensitivity: Poor shielding
of the antenna input cable.- This problem can be solved by using
braided and taped cable (such as Belden 9880) for the line to
the receiver. High-resistance
shield connections at the antenna socket and at joints in
the cable. - This problem can be solved by using silver-plated
connectors throughout. If however, the receiver has that most
execrable of input connectors, the cheap nickel-plated SO239,
there is much to be said for replacing it with a 4-hole silver-plated
N-type connector, bolted to the chassis with extreme prejudice.
It takes only a few milli-Ohms at the antenna socket ground to
let the receiver pick-up broadcast stations on the outside of
a 1.5m length of coaxial cable. Earth-loop
currents due to electrical continuity between the RF terminals
and the mains earth of most items of test equipment. - This problem
can be solved by installing a common-mode choke (unun) in the
line from the generator to the bridge, or in the line from the
bridge to the receiver, or preferably both. Suitable chokes are
described in appendix 6.4 section 7.When the bridge is balanced, the signal at the detector port (when using a generator output of around 0dBm) will fall below the noise level of a very good communications receiver. Also, during the process of commissioning; when the bridge begins to get close to its final working condition; it will, to all intents and purposes, never go out of balance when the operating frequency is changed. Using a VFO-type signal generator, it then becomes practically impossible to hear the detector signal while sweeping through, unless the bridge is deliberately thrown out of balance first. When testing with the lid off, the balance can be thrown temporarily by inserting a finger into the works, but this cannot be done once the lid is on. This is a nuisance unless the frequencies of the generator and receiver can be set accurately, because the process of calibrating the bridge is not assisted by the need to keep disturbing the adjustments. Hence both generator and receiver should have reasonably accurate digital frequency readouts. Using USB mode, the receiver is set about 800Hz lower than the generator at each spot-test frequency. |
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Commissioning Procedure: Before the trimpots are connected to the circuit, they should be set to the middle of travel with the aid of a resistance meter. A small 5KW 270° preset pot (on the shortest possible wires) can be connected in place of RVb in order to determine the best value for the intended fixed resistor. This determination can be done to good-enough accuracy with the lid off. Start with about 4-turns on L2. When the temporary pot. is replaced by a fixed resistor, this may be reduced to 3 turns, provided that the resistor is of low inductance. Carbon-film resistors tend to be less inductive than metal-film resistors, for reasons hidden beneath the paint. There is an issue that CF resistors may have inferior long-term stability compared to MF, but the ones used by the author had been kept in an unheated store-room for more than 20 years and were still well within spec.. Some old carbon-slug resistors tested however were all well out of spec., and therefore highly unsuitable for use in accurate instruments (and also for just about any other conceivable purpose). If helical-cut MF resistors must be used, there may be an advantage in making the required resistance by connecting 3 or 4 identical resistors in parallel. Having attached the reference load resistor to the load port and connected the test equipment; the alignment procedure is as follows: At 2MHz: balance
the bridge using C1 and RV. At 28MHz: balance
the bridge using L2 and Cn. At 18MHz: minimise
the detector signal by adjusting Rn. Repeat the above
three steps in the given order until no further improvement can
be obtained.After the first try at alignment, it is worthwhile to disconnect one end of Rn and check that it is somewhere in the region of 20 to 25W. A low setting for Rn, particularly if it is not considerably greater than 4.2W (Ri/N=50/12), is strongly indicative of a defect in some other part of the circuit. Once a fixed resistor is in place for RVb, the lid of the box should be screwed down tightly and the alignment procedure repeated. It will be found that the inductance of L2 needs to be reduced when the lid is put on, so it is a good idea to arrange things so that the slug is fairly well into the coil after alignment with the lid off. The number of turns and the turn-spacing can be adjusted for that purpose. The bridge is then evaluated to see if it has achieved its performance targets; and if not, the circuit is investigated for faults and the whole process is repeated |
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SPICE simulation: For those who would like to experiment with the circuit in simulation, the circuit file " refbrg_fs.asc " can be opened using the simulator LTspice / SwitcherCAD III, available free from the Linear Technology website. LTspice will convert the schematic into a pspice compatible netlist for its own or for other simulators. C1, Rv, L2, Cn and Rn can be adjusted according to the commissioning procedure described above to minimise the detector output over the chosen frequency range. |
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| The parasitic elements; C1s (strays from summing point to ground); L1 (inductance of lower voltage-sampling arm); and Ci (equivalent secondary parallel capacitance), have been put into the circuit as separate components. Transformer leakage inductance is not included. Note that this simulation will not lead to a realistic value for Rn because the model does not account for dispersion effects in the type 61 ferrite material used for the transformer core. The value placed against Lii and Ln is the core inductance factor (AL). Li = AL N² (and N=12). Transformer coupling coefficients Kn, Kni and Ki have all been set to 1. |
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4. Theory of the evaluation process: During the calibration process, the low-frequency balance-point is set by connecting a reference resistor to the load port and adjusting C1a and RVa. C1a adjusts the in-phase (resistance) balance, and RVa adjusts the quadrature (reactance) balance. The lower voltage-sampling capacitance (C1) is chosen as the variable parameter because this makes the in-phase and quadrature balance adjustments non-interactive. A further advantage of adjusting C1 rather than C2, is that one end of the trimmer capacitor is grounded. Provided that the high-frequency balance errors are not too great, C1a and RVa can be used to balance the bridge exactly at any frequency in the working range. Small deviations from the low-frequency calibration values can then be used to compute bridge performance figures. As mentioned previously; RV=RVa+RVb is measured via the detector port, and relative capacitance values are obtained by peering through a small inspection window with an eyeglass and converting the logging-scale readings by means of a regression formula. When a bridge is corrected for resistance-balance errors, the inductance of the lower voltage sampling arm is cancelled out. For this reason, the compensation coil L2 does not need to be included in the balance equation because it exists only to make C1 look like a pure capacitance. Similarly, when the bridge is corrected for reactance-balance errors, we do not need to include the transformer self-capacitance in the balance equation, because the purpose of the neutralisation network is to make it appear that the reactance of the current transformer secondary winding is purely inductive. On that basis (using the derivation given in appendix 6.4 section 19), an accurate expression for the balance condition for a doubly-corrected bridge is: |
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C2 |
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RV |
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N |
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k Ri |
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XLi |
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where the values inserted for C1 and C2 must include strays; XC2=-1/(2pfC2), Cx
is the stray capacitance from the through-line to the detector
port (Cx is negligible for a well laid-out
bridge with a Faraday shield); N is the number of turns in the
transformer secondary; Z0 is the
impedance connected to the load port (defined as Z0=R0+jX0); k is the transformer efficiency factor
(about 0.96 for the transformer used - see appendix
6.1); Ri is the transformer secondary
load resistance; and XLi is the inductive
reactance of the transformer secondary winding. One difference between equation (1) and the original derivation is that we have substituted Z0 in place of R0. This has been done to allow us to analyse the the case when there is some residual load reactance when the bridge is at balance. Now, multiplying Z0=R0+jX0 into the square bracket, we get the explicit balance condition for an arbitrary load: |
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C2 |
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RV |
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N |
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kRi |
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kRi |
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XLi |
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XLi |
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Resistance balance error: The resistance balance condition is obtained by dropping all of the imaginary terms from equation (2): |
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C2 |
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N |
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(1+Cx/C2) N |
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kRi |
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XLi |
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| This expression can be rearranged in favour of R0: |
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(1 + Cx/C2) N |
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C2 |
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N |
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XLi |
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| The evaluation method is a type of perturbation analysis. The logic of it is that, if the capacitance C1 has to be changed to rebalance the bridge on moving away from the calibration frequency, then the same effect can be had by leaving the capacitor alone and making a suitably proportioned change to the load resistance. Hence the deviation of the capacitance from its calibration setting is a measure of the resistance error which will occur if the calibration is left untouched. The amount by which R0 must change for a given change in C1 is given by differentiating R0 with respect to C1, i.e.: |
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¶C1 |
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(C2 + Cx) N |
[ Ohms per Farad ] |
| Hence, if dR0 represents a change in R0, and dC1 represents a change in C1, we can determine the relationship which exists between these changes by equating their ratio to the derivative (5), i.e.: |
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dC1 |
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¶C1 |
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(C2 + Cx) N |
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For reasons of mathematical consistency, changes or errors are
always defined as "obs. minus calc.", i.e., the observed
or actual value minus the expected or calculated value. Hence
the shift in R0 is defined as dR0=R0-R0cal,
where R0 is the actual load resistance
and R0cal is the reference load resistance;
and the shift in C1 is defined as dC1=C1-C1cal, where C1cal is
the calibration setting and C1 is the
setting required when the load resistance is R0. To get the correct sign of the resistance error when the change in C1 is due to a defect of the bridge however requires a little careful thought. The derivative (5) is positive, which means that an increase in R0 mandates an increase in C1. If however, C1 should increase spontaneously due to a defect in the system, then R0 must be reduced to restore the capacitor to its calibration setting. Hence we translate spontaneous changes in C1 into resistance balance errors by taking the negative of the derivative. Thus, if dR0 represents the predicted error in R0: |
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C1-C1cal |
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¶C1 |
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(C2 + Cx) N |
| Hence, resistance balance error is given by: |
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Where the shift in C1 has been changed
to 'cal.-obs.' to get rid of the minus sign, i.e., C1-C1cal = -(C1cal-C1). Notice that (6) does not require that we have to know C1. We only need to know the amount by which it has changed. The parameters k, Ri, and C2+Cx, all remain substantially constant over the operating frequency range; which is another way of saying that the derivative in this case is a constant. Hence the percentage deviation of C1 away from its calibration setting translates directly into a resistance error expressed as a percentage. Hence, if we do happen to have a rough estimate for C1 (including strays), we can gain a fair idea of how good the bridge will be before subjecting the shift measurements to a formal analysis. For the bridge under discussion, the circuit parameters were chosen to make C1=C1a+C1b+strays to be about 57pF, and the beehive trimmer used was found to have an adjustment gradient of -3.32pF/turn. This means that an adjustment of 0.066pF (1/50 of a turn) corresponds to a shift in the resistance balance of about 0.12%. Hence, if the bridge is to be accurate to better than ±0.12%, the trimmer must be shifted from its calibration position by no more than ±1/50th of a turn over the 1.6 to 30MHz range. With a practiced eye, the trimmer position can be logged to 1/200th of a turn, with an uncertainty of ±1/200th of a turn. Hence the resolution for the in-phase performance evaluation process lies at about 0.029%. Typically, during the various tests carried out by the author, the shift was about ± 1/100th of a turn, i.e., about ±0.058%. We can also get an idea of how well the commissioning process is going by estimating the derivative explicitly. For the bridge described here: k=0.96, Ri=50W, C2=4.9pF (including strays), Cx=0 when a Faraday shield is used, and N=12. This gives: ¶R0/¶C1 = 0.8163 W/pF. A shift of 1/200th of a turn corresponds to a shift of 0.0166pF, which corresponds to a deviation of the resistance balance by 0.8163 x 0.0166 = 0.0135W, this being 0.027% of 50W. |
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Reactance balance error: The reactance balance condition is obtained by dropping all of the real terms from equation (2) and cancelling-out the js: |
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RV |
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kRi |
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XLi |
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| Using the substitutions; XC2=-1/(2pfC2) and XLi=2pfLi gives: |
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2pfC2 RV |
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kRi |
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2pfLi |
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| Multiplying top and bottom of the first term in square brackets by 2pf, cancelling through and multiplying both sides by -1 gives: |
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C2 RV |
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Li |
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kRi |
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| If the bridge is perfect, X0 is always 0, and so X0 itself is a measure of the reactance error. Interestingly, if we treat X0 as a capacitive reactance defined as X0=-1/(2pfC0s), we get: |
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C2 RV |
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Li |
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C0s kRi |
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| This (recalling that Z0=R0+jX0) shows that reactance error can be represented as a hypothetical frequency-independent capacitance (C0s) (which can be positive or negative) in series with the load impedance. Here however, we will prefer series reactance, because it can be most readily converted into phase (using Tanf=X0/R0). Rearranging (8) in favour of X0 gives: |
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2pf |
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Li |
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N (C2+Cx) RV |
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The relationship between reactance error (dX0) and deviation of RV
from its low-frequency calibration setting is given by differentiation
as before. In this case, the power of RV
is -1, and if y=x ,
then dy/dx=-x ².
Hence: |
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¶RV |
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2pf N (C2+Cx) RV² |
[ Dimensionless ] |
| In this case, we define dX0=X0 (because X0cal=0) and dRV=RV-RVcal, where RV is the actual setting and RVcal is the calibration setting. Once again, we take the negative of the derivative in order to find an expression for the reactance error: |
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dRV |
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¶RV |
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2pf N (C2+Cx) RV² |
| Hence, reactance balance error is given by: |
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Here the derivative is frequency dependent in such a way that
the shift in RV corresponding to a given
phase error is much larger at high frequencies than it is at
low frequencies. The derivative is also proportional to the reciprocal
of RV², which means the error expression
is only true for small changes in RV;
but since the shifts encountered in practice are <1%, this
is not a problem. The resistance RV is found by measurement, being dependent on the secondary inductance Li. For the Faraday shielded version of the bridge described here (Li=8.06mH) the calibration setting was about 2748W. Strictly, we should use the actual value of RV rather than the calibration value when calculating the derivative, but for the purpose of illustration it doesn't make much difference. Including the other values: k=0.96, Ri=50W, C2=4.9pF, Cx=0, N=12, we get: ¶X0/¶RV = 17205 / f It is the precision rather than the accuracy of a measurement which must be taken into account when determining differences. The Fluke 8060A gives a readout to the nearest 1W on the 20KW range, giving a precision of ±0.5W for a single reading and an RMS uncertainty of ±0.7W for a difference. Hence using a 4½ digit meter gives the system a phenomenal RMS resolution for the measurement of reactance or phase error; as shown in the table below. |
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(dX0/50) |
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The RMS phase resolution is better than ±9 milli-degrees
at 1.6MHZ, reaching ±10³
degrees at 13.8MHz. Hence we need not concern ourselves with
the issue of whether we can get better information using impedance
analysers or vector voltmeters. For guidance whilst testing the bridge, it is useful to produce the kind of information shown in the right-most column. The figures tell us the maximum deviation from RVcal which can be allowed if the bridge is to have an overall phase accuracy of better than ±0.1°. The calculation formula is derived as follows: dX0 = -dRV ¶X0/¶RV and Tan(df) = dX0 / R0 Hence R0 Tan(df) = -dRV ¶X0/¶RV i.e.,
|dRV| = 50 f Tan(df) / 17205 If we put in a phase performance criterion as df, the formula returns a maximum allowable resistance deviation as |dRV|. The number 17205 applies of course, only to the example. The correct value should be calculated from the parameters of the actual bridge under test. The formula is practically linear for small phase errors. Hence it is possible to estimate the phase error at a particular frequency as a proportion of the ±0.1° criterion. For example, referring to the table above; If the bridge balances when RV is shifted away from RVcal by 30W at 12MHz, the phase error at 12MHz is within ±0.05°. If the guidance given here is followed, and provided that the temperature of the transformer core remains reasonably constant during the evaluation procedure, the bridge should have a phase performance no worse than ±0.03° at any frequency in the 1.6 to 30MHz range (although it may take some practice with the alignment procedure before this is achieved). If there are any peculiar kinks or bumps in the phase response, these are most likely to be due to resonances in the common-mode chokes. Hence, if the bridge is free from obvious faults but refuses to meet its performance targets, it is worth investigating and possibly redesigning the chokes. |
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5. Evaluation procedure and final alignment: The evaluation procedure is based on the application of two formulae; equations (6) and (12). The author's preference is to use the Open Office spreadsheet program (available free from OpenOffice.org) for the calculations, and test results given below are consequently in the form of open-document spreadsheet (.ods) files. The notionally separate evaluations for resistance and reactance balance can be combined to produce results expressed as magnitude and phase error. The magnitude error can also be expressed relative to the design load resistance (R0d=50W usually) rather than the actual reference load resistance (which is probably not exactly 50W). By adjusting (in the spreadsheet) the initially used calibration setting of the capacitor scale, a setting can be found which distributes the errors evenly above and below the design load resistance. Final alignment (equivalent to having used a reference resistance which was exactly on target) can then be obtained by applying this setting to the bridge A further refinement to the evaluation procedure is to use the required input parameters to calculate the non-essential parameters Li and C1. Realistic values for these quantities show that the circuit is free from physical defects. The fixed input parameters used in the full analysis are: |
| R0cal | Ref. load resistance (preferably within 1% of R0d). | Measured accurately |
| sR0 | Uncertainty in the measurement of R0cal. | From resistance meter manual. |
| R0d | Design load resistance | Usually 50W |
| k | Transformer transfer-efficiency factor. | about 0.96 ±1% |
| Ri | Secondary load = Ria // Rib | preferably measured |
| N | Number of secondary turns on the transformer. | |
| C2 | Upper voltage sampling capacitance, including strays. | |
| Cx | Strays from through-line to detector port.* | = 0 when Faraday shield is used |
| xcal | Cia scale reading at 2MHz | Starting value. Optimised later |
| sx | Uncertainty in the reading of the capacitor scale. | 0.005 of a turn |
| RVcal | Rv reading at 2MHz | Starting value. Optimised later. |
| * The inscrutable nature of Cx would seem to mandate the use of a Faraday shield; but a sensible estimate will do (ca. 0.23pF), and for those prepared to run tests on both shielded and unshielded versions of the bridge, it can be determined accurately. |
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Before the bridge is built, measurements of capacitor scale-reading
(x) vs capacitance (C1a) are fitted to
a regression line (see above). The spreadsheet
used for that calculation is pasted into a separate sheet of
the bridge evaluation file. The fit returns two parameters, a
and b, which convert scale readings into capacitances according
to the formula: C1a = a + bx Notice that we do not need to know the actual value of C1=C1a+C1b+strays, because the analysis (6) requires only differences, i.e: -dC1 = C1cal - C1 = C1acal - C1a strictly, we do net even need to know the fitting parameter a, but excessive elimination of redundancy makes it difficult to spot scale interpretation errors and other mistakes. A datum for the evaluation is a set of three numbers (f, x, Rv). Measurements are made at a representative set of spot frequencies, with the readings more crowded at low frequencies than at high frequencies (i.e., very crudely, at approximately equal intervals on a logarithmic scale). The author used a set of 29 frequencies in the 1.6 to 30MHz range. It is easiest to start taking readings at 30MHz and work downwards. Fewer data are required at high frequencies because the evolution of the phase error is very smooth in that region. The reason is that the experimental resolution is extremely good at high frequencies and, apart from a very low-Q dispersion effect in the type 61 ferrite used in the transformer, there is nothing in the box that can resonate at less than 30MHz. As mentioned before, if glitches are seen, they are almost certainly due to the external test setup. The calculated phase errors will appear more chaotic at low frequencies. This (presuming that the transformer core temperature remains constant) is not due to imperfections in the bridge, but due to reduced resolution and increasing difficulty in finding the exact balance point. In the 1.6 to 3 MHz region, the detector signal disappears below the receiver noise for about ±1 turn of the trimpot RVa, meaning that the mid point has to be found by estimation. A setting error of ±2W at 1.6MHz looks like a phase error of about ±0.025°, so the apparent scatter is an artifact. The solution is to take more closely-spaced readings at low frequencies, so that the scatter can be averaged to find a good final value for RVcal. |
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Magnitude and phase error. Final calibration: The resistance and reactance errors are increments to an impedance expressed in the normal series form: Z0 = R0 + jX0 where: R0 = R0cal + dR0 and X0 = dX0 because X0cal = 0. The magnitude of the predicted load impedance for perfect balance (i.e, the load impedance including the errors) is: |Z0| = Ö(R0² + X0²) where, strictly, X0 is the error returned from the reactance balance evaluation (but setting it to zero makes very little difference if X0 is small). Hence the magnitude error is: d|Z0| = |Z0| - R0d If the design load resistance R0d is used as the comparison reference, the initial capacitor calibration setting xcal can be tweaked to distribute the errors around R0d, thereby determining the optimal final setting for the capacitor. The phase angle of the predicted load impedance is given by: f = Tan (X0/R0)where, strictly, R0 is the load resistance obtained from the resistance balance evaluation (but setting it to 50W makes practically no difference). Since the target phase angle is zero, df = f The value of RVcal in the spreadsheet can be tweaked to get the best distribution of errors above and below zero. This is the optimal final setting for RV. Final accuracy and realistic performance claims: Since the experimental phase resolution is much better than the phase accuracy of the bridge over most of the frequency range, the final phase accuracy at the evaluation temperature is given by the maximum or minimum excursion of the phase error away from zero (i.e., ±|df|max ). The objective, in calibration and in choice of the final value for RVcal, is to make the maximum positive and negative excursions about equal. At low frequencies, small excursions outside the high-frequency phase-performance limits can be discounted. These are artifacts and can be reduced by obtaining more practice at balancing the bridge. The phase accuracy of the bridge is affected by temperature. This issue is investigated in section 6. Although the experimental phase resolution is very good; the experimental magnitude resolution is limited by the scale resolution of the cheap and cheerful modified beehive trimmer. There will also be uncertainty contributions due to the uncertainty in the measurement of the reference load resistance (R0cal), and due to the uncertainty in making the final setting of the trimmer. Hence there will be a statistically significant difference between the experimentally-determined precision of the bridge (i.e., the maximum RMS magnitude runout obtained from the raw data) and the claimable magnitude accuracy. Defining the true or claimable magnitude accuracy of the bridge as sZ, we can write an expression for this quantity on the basis that various errors which contribute to it are uncorrelated [see Data analysis A2-6]; i.e., we add the various uncertainty contributions as though they are a set of orthogonal vectors. Thus: sZ = Ö [ ( |dZ0|max )² + ( dZ,x )² + ( dZ,R0 )² + ( dZ,xcal )² ] (14) where: |dZ0|max is the maximum RMS magnitude runout (the precision of the bridge); dZ,x is the uncertainty contribution due to uncertainty in reading the capacitor scale; dZ,R0 is the uncertainty contribution due to uncertainty in the measurement of R0cal; dZ,xcal is the uncertainty contribution due to uncertainty in the final capacitor setting. For the three additional uncertainty contributions, we apply the rule: The uncertainty contribution due to a particular parameter (or variable) is equal to the rate of change of the desired quantity with respect to the parameter multiplied by the uncertainty in the parameter. Hence: dZ,x = ( ¶|Z0| / ¶x ) sx dZ,R0 = ( ¶|Z0| / ¶R0 ) sR0 dZ,xcal = ( ¶|Z0| / ¶x ) sx Note that two of the uncertainty contributions in this case are the same, so we only have two derivatives to find.Moreover, we can simplify the problem enormously by making a very minor approximation (the point is to find an estimated standard deviation, we do not require large numbers of significant figures). We first note that |Z0|=Ö(R0² + X0²), and that by proper application of the 'function of a function' rule, we get: ¶|Z0| / ¶R0 = R0 / [ Ö(R0² + X0²) ] but this derivative is so close to unity that we might as well take it to be unity. Hence: ¶|Z0| = ¶R0 Now, feeding this back into our error contributions we get: dZ,x = ( ¶R0 / ¶x ) sx dZ,R0 = sR0 dZ,xcal = ( ¶R0 / ¶x ) sx We already have an expression for R0 in terms of C1 in the form of equation (4). We can also express C1 as: C1 = C1a + C1b + C1s where C1s is the contribution from strays. C1a is given by the regression formula as: C1a = a + bx Hence: C1 = a + bx + C1b + C1s Substituting this into equation (4) gives: |
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(1 + Cx/C2) N |
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C2 |
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N |
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XLi |
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¶x |
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(C2 + Cx) N |
[ Ohms ] |
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But this is just b times the derivative ¶R0/¶C1
(5), which we have already
had to calculate for the precision part of the evaluation. Hence: ¶R0/¶x = b ¶R0/¶C1 Hence we are in a position to substitute for all the error contributions in equation (14) and write our final error function:
From various tests carried out on both versions of the author's bridge; the typical maximum magnitude runout was about ±0.02W, whereas the accuracy with which the reference load resistor can be measured using the Fluke 8060A is ±0.06W. The scale reading and resetting error of the trimmer capacitor makes only a small contribution. This means that the final claimable accuracy is primarily limited by the uncertainty in the reference load resistance. The bridge is a lot better than the resistance meter used, but can only be calibrated by reference to the resistance meter. |
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Realistic circuit parameters: The bridge evaluation method used here is most accurate when the circuit parameters used for the various calculations are also accurate. Fortunately however, the technique is fairly forgiving, and will not be greatly affected by the differences between nominal and actual component values, provided that reasonable allowances for stray capacitance are made. This circumvents a problem which will confront some constructors, which is that it is not easy to make accurate absolute measurements of capacitances of less than 100pF. Standard 1KHz component bridges are far too insensitive, and digital multimeters with capacitance ranges display numbers with a disarming certainty which belies the lack of any relationship with reality. For those who possess the ability to measure inductances of around 10mH however (e.g., using an SWR analyser such as the MFJ-269 [47]); there is a way to determine the voltage-sampling network capacitances accurately, and thereby check that all of the other components are within spec. and that everything is working properly. Equation (8) gives the relationship between RV and the coupled secondary inductance Li. Here we reproduce it with X0 set to zero, because that represents the situation when the bridge is balanced. |
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C2 RV |
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Li |
This can be rearranged, and at the same time we can observe that
the inductance is best determined by the average value of RV at low frequencies, i.e., RVcal.
Thus:
We can use this expression to calculate Li from the parameters already available. It should be about 1% less than the measured inductance of the transformer secondary (Lsec). Since we have to have a good resistance meter on hand to perform the evaluation, the only dubious parameter here is the quantity C2+Cx. Hence we can improve the accuracy of C2+Cx by tweaking its value in the spreadsheet to make Li come out about 1% lower than Lsec. For the author's bridge, Lsec=8.15mH. The measured value of the upper voltage sampling capacitor was 4.70pF. For the Faraday shielded version (i.e., with Cx=0), it was found that setting C2 in the spreadsheet to 4.90pF gave Li as 8.06mH (i.e., 99% of Lsec). Hence, with the layout used, the strays across the upper voltage sampling arm are about 0.2pF. The coupled inductance of the current transformer is a strongly conserved parameter. It is not affected by stray capacitance and therefore remains the same when the Faraday shield is removed. Hence, by making measurements on both shielded and unshielded versions of the same bridge, we can determine Cx. For the unshielded version of the author's bridge, it was found that C2+Cx had to be set to 5.35pF to give Li=8.06mH as in the shielded case. This is an increase of 0.45pF, but not all of it can be attributed to Cx. Removing the shield not only gives rise to stray capacitance from the line to the detector port (Cx), it also gives rise to additional stray capacitance from the line to the summing point (s). We would expect these two additional stray capacitances to be about equal. Hence an increase in C2+Cx of 0.45pF is half due to Cx becoming greater than zero, and half due to an additional contribution to C2 from strays. Hence our best estimate for Cx for the unshielded bridge is 0.225pF. |
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For the shielded bridge (Cx=0), determination
of C2 also allows us to determine C1. For the unshielded bridge, the ability to
determine C1 also depends on a knowledge
of Cx. Hence, in order to make a full
characterisation of an unshielded bridge, it is necessary to
install a shield at some point and make additional measurements.
This, as it will transpire, is the only notable difference between
the two versions of the instrument: omitting the shield makes
it difficult to estimate a parameter which we don't really need
to know. The resistance balance condition was given earlier as equation (3). Here we reproduce it with X0 set to zero as before: |
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C2 |
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N |
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k Ri |
| This can be rearranged into an expression for C1. At this point we can also note that if the value of R0 inserted is the design load resistance R0d, the equation should return an estimate for the final calibration value for C1. Thus: |
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In appendix 6.4, the quantity in square brackets was referred
to as the transformer constant. The most diagnostically helpful application of equation (17) is to use it to estimate the stray capacitance component of C1, i.e.:
C1s needs to be positive and sensible. If it is not, then there is something wrong with the other bridge parameters. For the Faraday shielded version of the author's bridge, C1s was calculated to be about 5pF. For the unshielded version, based on a good estimate for Cx, C1s was about 4pF. These values are not particularly accurate because they are affected by the uncertainties in the other circuit parameters, but they are reasonable. If the analysis shows up an unrealistic value for C1s, it could be simply that some of the circuit components, particularly the fixed capacitors, are out of spec.. If everything appears to be in order however, then there could be a problem with the transformer efficiency factor k. The point is that k is affected by resistive loading of the neutralisation winding. The effect is very small provided that Rn>>Ri/N. Hence if C1s looks wrong, disconnect one end of the neutralisation trimpot and measure the resistance. Rn should be >20W, and certainly >>Ri/N=4.2W. Estimating the compensating inductance: At very high-frequencies (well above the operating range), the impedance seen at the input (TX) port is essentially the impedance of the upper voltage-sampling arm in parallel with the load resistance. By connecting an MFJ-269 antenna analyser to the TX port; it was found that the unshielded version of the author's bridge exhibited one resonance in the 1.7 to 170MHz range, this being a series resonance at 145MHz. Leaving the detector port open-circuit or applying a shorting-plug made no difference, so it must be assumed that the impedance from the summing point (s) to ground is very low at 145MHz. This means that the resonance is almost entirely governed by L2 in series with C2. For the unshielded bridge, the effective C2 is 5.13pF, but the part of it in series with L2 is about 4.7pF. Hence: L2 = 1 / [ (2pf0)² C ] = 256nH The ratio C1/C2 is given by equation (17). For the unshielded br |