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6.1 Radio-Frequency Bridges.
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6-1. Christie's Bridge. 6-2. Arms. 6-3. Reciprocity. 6-4. Generalising the Bridge. |
6-5. Voltage splitting networks. 6-5a. Inductive voltage splitters. 6.1 Part 2 >>> . |
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Introduction: In this chapter we explore the subject of AC measuring and monitoring bridges; particularly with a view to understanding how they work and how to optimise them for use across the 1.6 - 30MHz HF radio spectrum. In the course of the discussion we will show how to devise bridges which can be used to make accurate measurements of any impedance-related quantity, including reflection-coefficient, impedance magnitude, resistance, conductance, and phase; thereby covering the design considerations for SWR bridges, Match Meters, and bridges for automatic antenna matching systems. Full comprehension of the discussion to follow requires familiarity with the material presented in chapters 1 to 5. 6-1. Christie's Bridge: The term 'bridge' came into the language of electrical engineering in connection with the "Wheatstone Bridge", which was actually invented by S. H. Christie of the Royal Military Academy in Woolwich in 1833. The bridge is usually attributed incorrectly to Sir Charles Wheatstone, who employed it extensively in his scientific experiments, although Wheatstone himself always credited the bridge to Christie [1]. The circuit is so called because the detector (a centre-zero galvanometer, i.e., a microammeter or voltmeter) occupies the position of a bridge between two potential-divider networks. Two possible implementations of Christie's bridge are shown below: |
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The bridge is balanced by adjusting a variable element until
the voltage across the meter (and hence the current through it)
is zero. In the left-hand version, the variable element is a
potentiometer R1+R2,
traditionally constructed by laying a length of resistance wire
against a measuring scale, and known as a "slide wire".
At balance the ratio R1/R2
is equal to R0/Rx,
where R0 is a standard (i.e., a laboratory
reference) resistor, and Rx is the unknown
resistance under test. In the right-hand version, the ratio R1/R2 is fixed, and R0 is a calibrated variable reference resistor.
In this case, it is quite common for R1
to be made equal to R2, because this situation
gives greatest meter sensitivity and hence the best accuracy
of measurement. When R1=R2,
then V1=V2, and
the bridge balances when R0=Rx. There are various ways of proving the balance relationship R1/R2=R0/Rx (see, for example, ref [2]), but here we are interested not so much in measuring resistors, as in generalising the bridge concept and showing that all bridges are direct descendants of Christie's. We will begin therefore by observing that the unknown resistor Rx must obey Ohm's law, i.e.,Vx=IxRx, or:
Ix = V0 / R0 If we now substitute this expression for Ix into equation (6-1.1) above, we obtain: Rx = R0 Vx / V0 Now at balance, V1=V0 and V2=Vx, and so we may write:
Rx = R0 R2 / R1 but the important point is that we do not need to know the supply voltage V, and we do not need to know the absolute values of R1 and R2, only their ratio. Indeed, we do not even need to use resistors R1 and R2, because what we really need is two voltage references V1 and V2, of known ratio; which we might, for example, obtain instead by using two electronically controlled voltage-sources in series. From the above, we may deduce that a bridge is a device which measures the ratio of two voltages, one being a voltage sample taken from across the device under test, the other being a voltage derived by sampling the current passing through the device under test. Put more formally: a bridge is a device which measures the ratio of a voltage analog and a current analog, an 'analog' being a quantity which is analogous to, i.e., proportional to, the quantity in which we are interested (the British spelling is "analogue"; but the American spelling is nowadays preferable in a technical context, and helps to avoid creating a world shortage of the letters u and e). By comparing two quantities which both vary in proportion to the power-supply voltage, we can make measurements which are independent of the power-supply voltage. 6-2. Arms: Because of their role in determining the ratio Rx/R0, the voltage splitting resistors R1 and R2, are known as the ratio arms of the bridge. Later on, when we generalise the bridge to AC, we will be able to use devices other than resistors in the voltage splitting network, but they will still be known as the ratio arms. The term "arm" hails from the 19th Century, and seems to have arisen in an ad hoc way for want of terminology to use while explaining bridges. It has since acquired a life of its own however; and nowadays (according to Langford-Smith [3]), an arm is defined as "a distinct set of elements, electrically isolated from all other conductors except at two points", i.e., an arm is a two-terminal network. On this basis, a bridge has six arms. 6-3. Reciprocity: In the diagram below, the positions of the meter and the battery in the bridge discussed earlier have been interchanged. |
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The swap has made rather a mess of our original voltage and current
definitions, and so the circuit is re-drawn on the right to make
the analysis more obvious. Using the same arguments as before,
we may observe that, at balance, Rx/R2=R0/R1,
i.e.: Rx = R0 R2 / R1 Which is exactly the same result as was obtained previously. This means that the battery and the meter can be swapped without affecting the balance of the bridge; or to put it formally: the bridge is a linear reciprocal network, its measuring function being unaffected by transposition of the generator and the detector. The term 'linear' refers to the fact that the elements within it obey Ohm's law, i.e., the graph of voltage against current is a straight line. If the devices used to make a bridge are non-linear (e.g., semiconductor diodes) then the bridge is not a reciprocal network. 6-4. Generalising the Bridge: We may just as well use Christie's bridge by replacing the battery with an AC generator and the galvanometer with a detector sensitive to alternating voltages. Analysis of the bridge however then entails treating all of the voltages and currents involved as phasors; but (as always) we can satisfy this requirement by writing impedances as complex numbers and deriving expressions for the voltages and currents from them. Here we will use the notation and various simplifying techniques developed in chapter 1. In particular, quantities which must be treated as vectors will be written in bold; and we are at liberty to nominate one vector in any complete set to be a reference vector, and by choosing its direction to be 0°, may treat it and any other vectors in phase with it as scalars (i.e., real numbers) (see theorem 1-21.5). |
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The AC version of the bridge, with attendant vectors is shown
on the right. In order to analyse this network, we first observe
that V1+V2
must be equal to V0+Vx, and that the voltage across the detector
D (Vdet) is equal to both V1-V0 and Vx-V2, i.e., V0 + Vx = V1 + V2 Vdet = V1 - V0 Vdet = Vx - V2 |
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Let us assume, for the time being, that the detector is a high
impedance device and that any current which flows through it
is insignificant in comparison to Ix
(In fact, provided that the objective is to balance the bridge
rather than read the error voltage, such an assumption will never
cause any measurement errors because, at balance, the current
flowing through the detector goes to zero). We may then observe
that the two impedances Zx=Rx+jXx and Z0=R0+jX0 carry the same current, Ix,
and that this establishes the phase relationship between V0, Vx, V0+Vx, and V1+V2. Also, we
may simplify the problem by choosing Ix
as the reference vector and setting its phase to be 0°, allowing
it to be treated as a scalar Ix = |Ix|. Hence: V0 = Ix Z0 = Ix (R0 + jX0 ) Vx = Ix Zx = Ix (Rx + jXx ) V0 + Vx = Ix [ R0 + Rx + j( X0 + Xx ) ] = V1 + V2 The only matter we have yet to establish is the relationship between V1 and V2, but to simplify the analysis, we will start by making the obvious choice of a voltage splitting network which produces V1 in phase with V2. Assuming that we also know the relative magnitudes of V1 and V2, we can then draw voltage phasor diagrams for the system, as below: |
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The main phasor diagram (left), which represents the relationship
V0+Vx=V1+V2, is constructed
as follows: A line representing V0
is constructed by plotting a point, moving right by a distance
IxR0 and upwards
by a distance IxX0
(or downwards if X0 is negative), then
plotting another point. A similar procedure is used to construct
a line representing Vx. The end
of V0 is placed against the beginning
of Vx, and the vector sum V0+Vx is obtained
by drawing a line from the beginning of V0
to the end of Vx. Now, since we
have chosen V1 and V2 to be in phase, we know that the associated
voltage phasors must lie in a straight line relative to each
other, and we know that the phasor representing their sum is
equal in magnitude and phase to V0+Vx. We therefore divide the line representing
V0+Vx
according to the relative magnitudes of V1
and V2 to establish the relationship
between the four principal voltages. The constructions on the
right of the diagram above merely serve to illustrate the point
that the detector voltage Vdet
is obtained by drawing a line from the junction of V0 and Vx to the
junction of V1 and V2. What is remarkable about the vector addition process which takes place, and which should be obvious by studying the phasor diagram, is that the bridge will not balance (i.e., Vdet will never reach zero) unless V0 and V1 (and also, by association, VX and V2) are equal in both phase and magnitude. This means that, despite the fact that there are two variable elements in the current-sampling arm of the bridge (R0 and X0), there is still only one way in which the bridge can be balanced once the relationship between V1 and V2 has been established. The balance condition for the bridge is:
What all of this means, of course, is that to balance the bridge, we must adjust R0 and X0 independently; and the series-equivalent impedance for Zx is given by:
6-5. Voltage Splitting Networks: In the case of Christie's original bridge, the voltage-splitting network used (i.e., the pair of ratio arms) was simply a resistive potential-divider or a potentiometer. In the process of generalising the bridge to work with alternating voltages and currents, we simply replaced the unknown and the reference resistances with impedances. We now take this process to its logical conclusion by also replacing the voltage-splitter resistors with impedances. The relationship between the two voltages obtained may be deduced by constructing phasor diagrams according to the scheme outlined below, where the current I is the reference vector, and I=|I|: |
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Fig. 6-5#1 |
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The magnitude of an impedance is obtained by using Pythagoras'
theorem, i.e.: |Z|=Ö(R²+X²);
and by Ohm's law, the magnitude of the voltage across an impedance
is given by: |V|=|I||Z|=I|Z|. The
relative magnitudes of V2 and V1 are therefore given by the expression: |V2| / |V1| = |Z2| / |Z1| The phase difference between V2 and V1 is simply f2-f1. where Tanf2=X2/R2 and Tanf1=X1/R1. The problem with this type of network is that reactance is frequency dependent, and so if we write down the general expression for f2-f1 we find that the phase difference between V2 and V1 will vary with frequency. This is acceptable if we are trying to design a bridge for operation on a single frequency, but such is usually not the case in the context of RF measurement. There is one important exception to this phase variability however, which occurs when we make f2=f1 with both impedances either capacitive or inductive (or purely resistive); i.e., we specify that the two voltages must be in phase at all frequencies, which means that the two impedances must follow the same frequency law. In this case, Tanf2=Tanf1 and so:
Perhaps what is less obvious, but most useful of all, is that the phase and magnitude relationships still hold if the resistive components of the two impedances are allowed to become arbitrarily small. If we put R2=0 and R1=0 into the equation above, the result amounts to ¥=¥, which may appear problematic until we realise that if the relationship between V2 and V1 holds good as X2/R2 and X1/R1 become arbitrarily close to infinity, then it must hold good upon reaching infinity. Furthermore, as the resistance terms disappear, so does the requirement that the resistance must remain in strict proportion to the reactance. In other words, as the resistance terms disappear, V2 and V1 are forced more and more accurately into phase at all frequencies. This means, in principle at least, that we can construct potential divider networks using pairs of capacitors, or pairs of high Q inductors, which can do as good a job as a pair of resistors. There is one very good reason for wanting to do so; which is that reactive potential dividers do not consume any power, and so will not get hot if we are, for example, trying to divide or sample the voltage output of a large radio transmitter. |
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If we opt for the resistive potential divider, it may be built
using metal-film or foil resistors. The resistive network is
therefore a possible choice for an ultra-wideband voltage splitter
which will work from DC to VHF, and can maintain an accuracy
of 1% or better without the need for initial adjustment or frequency
compensation, provided that the resistor values are chosen sensibly
and the the resistors themselves have low reactance. From the
discussion in section 2-7,
we know that film resistors have a turnover frequency above which
they become capacitive, and that for a typical wire-ended resistor
with a self-capacitance of around 0.5pF, the turnover frequency
will occur in the 0-30MHz range if the resistance value is greater
than about 1KW. There is also an appalling
tendency among manufacturers of cylindrical resistors to cut
a helix into the film to increase the length of the conduction
path. We can therefore only use higher value resistors if both
resistors are to be identical (in which case the impedances
of the two resistors will always be identical and V1 and V2 will
remain in phase), or if we adopt chip (surface-mount) resistors
which have very small parasitic reactances. In general therefore,
low value resistors, typically of around 50 to 100W,
are used; which means that the divider network will load the
generator quite heavily and may produce considerable heat. If
this is not acceptable, i.e., if high resistance values must
be chosen in order to minimise heating, then the voltage balance
will also become susceptible to stray capacitance effects, which
reduce accuracy at high frequencies, and detector loading effects,
which reduce the sensitivity of the bridge. A compromise is often
necessary, with resistor values chosen to be higher than required
for optimum swamping of stray capacitances, and compensation
for the effects of stay capacitance obtained by placing one or
more trimmer capacitors in parallel with the resistors. Capacitors are inherently high-Q devices when correctly chosen for the job in hand, and so capacitive potential dividers are capable of good phase accuracy. Component tolerances are generally broader than for resistors however, and any stray capacitances will be absorbed into the network. Consequently it may be necessary to place a trimmer capacitor across one of the capacitors so that the voltage ratio can be adjusted to the required value. Capacitors consume no power (neglecting losses) but the reactance of the network varies with frequency. At very low frequencies, the reactance becomes extremely high, leading to low output current capability (from the junction between the two capacitors) and consequent insensitivity of the bridge. As the frequency is increased, the reactance falls, and a point will be reached when the network presents an unacceptably reactive load across the generator. Care must therefore be taken in the choice of capacitor values, but a working range of 4 octaves or more is possible in RF applications. We may also, of course, place resistors in series with the capacitors (in appropriate proportion to the reactance, as mentioned previously) in order to limit the minimum impedance presented to the generator at high frequencies; or we may place suitably proportioned resistors in parallel with the capacitors in order to assist the output at lower frequencies. In the latter case, the network can be designed using the simple observation that if a reactive network designed to give a particular voltage output is placed in parallel with a resistive network designed to give the same output voltage; then the off-load voltage will be unchanged, but the output impedance will be lower (i.e., the network will be less susceptible to loading effects). If resistors are added to the network, of course, their parasitic reactances must be taken into account. The capacitors also must either have very small or proportionate parasitic inductances. A small compensating inductance is sometimes placed in series with one of the capacitors if the arms of the voltage divider are not physically identical. The use of capacitive ratio arms in RF impedance monitoring applications was adopted by the Collins Radio Company in the 1950s, and the resulting configuration is referred to in the manuals of that period as a "modified Schering bridge". The original Schering bridge which gave rise to the idea is a high-voltage test-set for measuring the the dielectric losses of capacitors [4]. Other authors [5], [6](p27-4), refer to the capacitive ratio-arm bridge as the "Micromatch", and the "De Sauty - Wein" Bridge. In fact, the configuration resembles the De Sauty bridge, but to name it so in this context would be to miss the point; which is that the need to replace the arms of Christie's bridge with impedances is obvious within a particular field of application, and just about every possible bridge configuration which can be obtained by so doing is named after someone [4], [7]. The author's preference is to refer to the device generically as a "CRAB" (capacitor ratio-arm bridge). |
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6-5a. Inductive voltage splitters: Although the capacitor ratio-arm bridge is attractive for monitoring the output of radio transmitters, progress in the design of bridges for accurate impedance measurement depended on a move away from the idea of discrete or non-interacting arms. The first step in this direction is to consider what happens when the ratio arms are replaced by inductors. Of the basic passive electronic components, inductors are the least well-behaved. They have substantial RF resistance, which varies with frequency, and they have substantial self-capacitance, which may easily give rise to self-resonance in the frequency-range of interest. Air-cored inductors are also sensitive to environmental factors, they are difficult to make accurately, and the introduction of adjustable magnetic cores (slugs) may give rise to further frequency dependent variations of reactance and resistance. Inductors might therefore seem to be the least prepossessing of candidates for construction of voltage dividers; that is, until we allow mutual inductance between the two coils and turn the assembly into an auto-transformer. If the sense of the windings of the two coils is the same, mutual inductance increases the inductance of the series combination and initiates transformer action. This allows us (in principle) to have the best of both worlds: a large inductance across the generator to minimise standing current, and a low impedance output from the tapping point to maximise detector sensitivity. If the magnetic coupling between the windings is tight moreover, then the output voltage ratio is precisely defined by the turns ratio, i.e., (referring to the left-most diagram below): |V2|/|V1|=N2/N1, and the two voltages are in phase. This configuration was invented in 1926 by Alan Blumlein, the coupling being achieved by means of twisted-bifilar winding. Blumlein named his invention "the closely-coupled inductor ratio-arm bridge" [8]. Further improvement in the coupling is nowadays achieved by using a closed magnetic path between the coils, e.g., by winding them (bifilar) on a high-permeability ferrite toroid. Extremely wide bandwidths are possible, the lower frequency limit being dictated by insufficient inductive reactance and consequently excessive standing current, and the upper frequency limit being dictated by self-capacitance effects and core losses. Design data for ferrite-cored toroidal coils is given in Section 6.2-12. |
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In the central diagram above, we take the inductive splitter
idea to its logical conclusion by using a true transformer. A
bridge based on such an arrangement is known as a 'transformer
ratio-arm bridge' (TRAB), and (according to the technical author
of Hatfield Instruments [See LE300/A
handbook]) this approach was developed in the 1940s by
Mayo of the BBC Research Department. A true transformer gives
the additional benefit of DC isolation from the generator, and
permits optimisation of the loading relationship between the
generator and bridge network. The auto transformer and the tapped isolation transformer have the advantage that any imperfections in behaviour manifest themselves in both outputs, and so tend to cancel. The third option (above right) therefore appears to be less promising, because any phase or magnitude errors in the transformer output V1 will be transferred into the relationship between V2 and V1. Such errors can be taken into account however, provided that the transformer is operating within a defined pass-band; and compensation for residual errors is possible. Later on moreover, we will return to the point that the purpose of the voltage splitter is not to provide a power supply, but to establish the ratio between a voltage sample and a current sample. If the ratio can be established in some other way (which it can) then only one voltage sample is required. The resistive, capacitive, and transformer voltage splitting networks introduced above should be regarded as the basic building blocks for the ratio arms of Christie or Blumlein-type bridges. Later on however, when we start to deviate from the Christie topology, we will return to the idea of using voltage sampling networks which have frequency dependent outputs; not so that we can make bridges which are unnecessarily complicated and difficult to understand, but so that we can tailor the frequency response of the voltage sampling network in order to compensate for the amplitude and phase errors in the output of the current sampling network.  |
© D W Knight 2007.
David Knight asserts the right to be recognised as the author of this work.
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