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6.4 Reflectometry
(Measurement of reflection coefficient, return loss, and SWR)
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Reflectometry: One of the principal applications of the impedance bridge is the measurement of forward and reflected power in transmission lines; this usually being expressed in terms of SWR or return loss (to be defined shortly). The bridge however, cannot measure reflections, it can only measure quantities which can be derived from voltage and current. The ratio of forward to reflected power in a line is a function of the load impedance and the characteristic impedance; and, in the case of a lossless line at least, remains constant throughout the length of the line. The actual input impedance of a mis-terminated line however, can vary radically depending on its length; and so impedance on its own is not sufficient to determine the required power ratio. The key to reflectometry therefore, is to devise bridges which can measure some quantity which is conserved regardless of the length of the line, and to derive the required information from that. Consider a wave traveling from generator to load in a lossless transmission line. As the wave moves along the line, it has no knowledge of the conditions it will meet when it arrives at the load; and so the relationship between voltage and current in the forward traveling wave is dictated entirely by the surge resistance (i.e., the characteristic resistance), R0, of the line. Thus we may write:
We cannot, of course, measure the forward and reflected voltages and currents independently. If we make measurements of voltage and current at various points on the line, we will always measure the sum of forward and reflected voltages and the sum of forward and reflected currents at each point. At the load however, we can determine the relative magnitudes of the forward and reflected voltages or currents, because the relationship between the total voltage and the total current at this point is defined by the load impedance Z. The ratio so determined is called the reflection coefficient, and is true of the magnitudes of the forward and reflected voltages and currents at any point in the line because the power (energy per unit of time) in the forward and reflected traveling waves is constant. An expression for the reflection coefficient can be determined as follows: If V is the voltage across the load, and I is the current flowing in the load, then we may write:
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From the relationships given above, we have IF=VF/R0 and IR=-VR/R0. Substituting these expressions into equation
(4) gives: I = (VF / R0) - (VR / R0) = (VF - VR) / R0 and using (3) gives: (VF + VR) / Z = (VF - VR) / R0 which can be rearranged as follows: (VF + VR) R0 = (VF - VR) Z VF R0 + VR R0 = VF Z - VR Z VR R0 + VR Z = VF Z - VF R0 VR (Z + R0) = VF (Z - R0) To give:
The quantity |VR|/|VF| is the desired reflection coefficient, and is variously given the symbol ρ (rho), Γ (capital Gamma), or k, depending on the commentator. Here we will prefer Γ, because ρ is already used elsewhere to denote both resistivity and density. Γ is sometimes called the "voltage reflection coefficient", but this is a peculiar affectation as we may see by using relationships (1) and (2) to eliminate all voltages from equation (3); and then using equation (4) to eliminate I. By so doing we obtain: IF Z + IR Z = IF R0 - IR R0 which rearranges to:
Thus, to summarise:
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>>> work in progress Lossy lines: Z0 = R0 + jX0 (X0 is negative) and Z0* Return loss = 20Log10( 1/Γ ) [dB] SWR S=Max peak voltage/min peak voltage = Max peak current/min peak current (Only strictly defined for lines of λ/4 and longer) S = ( 1+Γ ) / ( 1-Γ ) True for any length of line and for lossy lines. Γ = (S-1)/(S+1) Γ = √(PR/PF) |
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>>> Directional coupler: The transmission line between a radio transmitter and an antenna does not necessarily need to be matched; but the presence of standing waves on an unmatched line creates a problem of power measurement when the intrumentation (as is usually desirable) is located in the radio room. The line performs an impedance transformation such that, at a distance of λ/4 (an electrical quarter-wavelength, or 90 electrical degrees) from the load, the impedance looking into the line will be of high magnitude if the magnitude of the load impedance is less than the characteristic resistance of the line, and vice versa. The impedance varies cyclically as the distance from the load increases, and returns to the same value (neglecting losses) at intervals of λ/2. Since the length of the line is usually dictated by the physical installation rather than by electrical considerations; early attempts to monitor transmitter performance by measuring either the voltage or the current at the transmitter terminals were bound to produce inconsistent results. >> Invention of: (early patents). Overcoming misleading measurements by taking average of voltage and current analogs at a given point. Gothe, Buschbeck, Kautter 1939, US 2165848 . Alexander 1949 US 2467648 . the directional coupler is related to the telephone hybrid circuit, a device used for separating and recombining upstream and downstream signals in long telephone lines so that the separated signals can be passed through amplifiers. Also used to reduce side tone, so that callers are not deafened by their own voices. The name 'hybrid' probably derives from the fact that any particular transformer winding is neither a primary or a secondary, but performs both functions simultaneously. Determining Γ as an analog computation problem.
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>>>>> to be rewritten 6.4-x. The Reflectometer (or SWR) bridge: An SWR bridge is a reflectometer bridge with a meter scale calibrated in SWR. A reflectometer bridge is a set of two impedance bridges; one used normally, and one used with the generator and the load transposed. When an impedance bridge is designed to balance at the characteristic resistance (R0) of a transmission line into which it is inserted, the balance condition, by definition, occurs when the power reflected back from the load is zero; i.e., the bridge indicates whether or not the transmission line is correctly terminated, and it transpires that any imbalance reading is proportional to the square-root of the reflected power. If the load and the generator are transposed however, the bridge will no longer balance, because the output at the detector port is now Vv+Vi rather than Vv-Vi (the polarity of the current sample is reversed). The configuration is still useful however, because it solves the problem of how to measure the forward power in a transmission line in the presence of standing waves. If there is reflected energy travelling in a line, the voltage on the line will be the sum of the voltages due to forward and reflected power, and this will vary sinusoidally with the distance from the load. Hence, if the forward power is to be measured at an arbitrary point on the line, it cannot be calculated from |V|²/R0. Similarly, the current in the line will vary sinusoidally, and the forward power cannot be calculated from |I|²R0. Current maxima however will always correspond to voltage minima, and vice versa, and so the forward power can be deduced by making two determinations, one from the current and one from the voltage, and taking the average. Hence |Vv+Vi| is proportional to the square root of the forward power, provided that the current and voltage samples are taken at exactly the same electrical point on the line. In a current-transformer bridge with more than one turn in the current-transformer primary winding, an apparently sensible point at which to sample the voltage is at a primary centre tap. Since most bridges only have one primary turn however, the voltage must be taken from one side or the other, but the error which results is negligible provided that the distance from the middle of the current transformer is small in comparison to one wavelength at the highest frequency of operation. Such is the accepted practice, but in fact it is not always best to sample voltage and current at exactly the same physical point; because it takes time for the current sample to propagate out of the transformer coil. The best technique is to take voltage and current samples from points of equal time-delay ralative to the source, a matter which can be dealt with by neutralisation, or by moving the sampling point down-line by an electrical distance equal to half the electrical length of the current-transformer secondary winding. A prototype SWR bridge is shown below: where the magnitude of Vf is proportional to the square-root of the forward power and is independent of reflected power; and the magnitude of Vr is proportional to the square-root of the reflected power, and is independent of forward power. |
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Since the output of the current transformer is shared between
two bridges, the balance condition is now: Vv - Vi/2 = 0 This means that the voltage sampling network components must be chosen to give half the output required for a single bridge, but the bridge design procedure is otherwise the same. Caveat: Notice that the dual bridge circuit shown above has a serious flaw in one of its most popular implementations; which is that in which the voltage sampling network Z1, Z2 is a high-impedance capacitive potential divider, the forward and reflected output ports are terminated with diode detectors, and the circuit is used to drive two meters simultaneously. When the load ZL is correctly matched to the cable, the forward output will be large and will drive its detector hard; causing the output of the voltage sampling network to droop, especially at low frequencies. This will throw the balance condition for the reflected power bridge and give rise to a spurious reading. Essentially, the shared capacitive-divider version does not work properly at low frequencies when used to drive two separate meters, it being an ill-conceived extension of a circuit intended to have a single meter and a switch to select between forward and reverse readings. Warren Bruene's solution to this problem was to use separate voltage sampling networks for the two bridges [40]. Also, if we decide to compensate for current-transformer delay by moving the voltage sampling point along the line, we will need to move the reflected power voltage network towards the load and the forward power voltage network towards the generator, and so we will be forced to use two voltage sampling networks anyway. Another solutions are to use a voltage sampling network with a low output impedance. It was mentioned in the last paragraph, that in order to delay the voltage sample for reflected power, it is necessary to move the sampling point towards the load. This might seem anti-intuitive, but only if we subscribe to the view that the reflectometer can measure reflected power. It can do no such thing: it can only infer the existence of reflected power from the difference between the actual load impedance and the target load impedance. To understand this point, consider an SWR bridge designed to balance when the load is 50+j0Ω. If we connect this bridge directly to a 100Ω load resistor, it will declare an SWR of 2:1. The resistor is not reactive however, and so will absorb all of the power delivered to it and reflect none. The 2:1 SWR reading is only true when the bridge sees an impedance magnitude of 100Ω (or 25Ω) at the input to a 50Ω transmission line. The bridge is just an impedance bridge, it has no special psychic powers, and its readings are only true when it is inserted into a line having the same characteristic resistance. |
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6.4-x. The Bruene Directional Wattmeter: >>> To be rewritten. >>>> The Bruene bridge is discussed below because it is historically interesting. It is not recommended as a basis for modern designs. The "Directional Wattmeter" (i.e., SWR bridge) circuit shown below is based on a circuit devised by Warren Bruene (W0TTK, W5OLY, of the Collins Radio Company) which became popular among radio amateurs as a result of an article published in 1959 [40]. The original Bruene bridge was essentially the same as Douma's bridge, but without low-frequency compensation and rearranged to use the shunt-diode detector configuration (see section 6-7b). These changes were sufficient to circumvent Douma's 1957 patent [USP 2808566], but the circuit is also a logical development of the capacitor ratio-arm bridge (CRAB) used in earlier Collins designs and so may well have been invented independently. The CRAB used as a mismatch (SWR) indicator in the Collins 180L-3 (2-25MHz) automatic antenna tuner is discussed elsewhere. A "high-frequency iron" toroidal current transformer was however used for the phase and magnitude bridges of that unit, and so it was only a matter of time before the toroid migrated into the reflectometer. The CRAB was used down to 2MHz however (limited only by the output impedance of the ratio arms); whereas the lack of LF compensation in the original Bruene bridge raised the useful minimum frequency, and the designs discussed in references [40], [41], and [42] are only suitable for 3.5-30MHz. Douma's patent expired in 1977, and so, while home constructors never had valid reason to omit the compensation resistors (apart from lack of awareness of their importance), commercial designers now have no reason to omit them either (and probably never did, because Douma did not invent this compensation method - it was used by Korman in 1942 [US Pat. 2285211] and so was out-of-Patent by 1962). Consequently, the resistors (Rv) have been added to the circuit shown below, and their inclusion should be regarded as mandatory. The inclusion of compensation resistors also necessitates the inclusion of blocking capacitors (Cb) to prevent the DC outputs of the detectors from being shunted to ground. Cb merely needs to have a low reactance at the minimum frequency of operation (10nF ceramic will usually do the trick) but an excessively large capacitor (i.e., several μF) in this position will slow the rate at which the DC output can change and will damp the meter response. |
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Notice that the shunt-diode detector is connected directly between
the current and voltage sample outputs and the rectified signal
is extracted through an RF choke (RFC). This floating detector
configuration is a Hallmark of the Collins Radio Company from
the 1950s; but is perhaps nowadays somewhat archaic. One disadvantage
is that the detector 'ports' are not referenced to ground, and
so isolation transformers will be required if signals are to
be injected into them. Note also that the RF chokes must be carefully
chosen to have a very high impedance throughout the operation
frequency range, since reactive loading of the voltage sampling
network will introduce serious errors into the balance
condition (multi-segment RF chokes of 1mH or more are normally
used). If the drop in meter sensitivity which results from using
Rv as the detector DC return path can
be tolerated, it is a good idea to swap the detector connections,
i.e., connect the anode of the diode to the junction of C1 and C2, and connect
the blocking capacitor Cb to the current
transformer output. Due to the very low output impedance of the
current transformer circuit, self-capacitance effects in the
choke will then only affect the magnitude of the meter reading
and will have little effect on the balance condition. If the
detector is reversed in this way incidentally, attempting to
restore its sensitivity by connecting a choke across Rv
is not a good idea; but as we shall see by performing an actual
design calculation, the loss of sensitivity will not be particularly
large because Rv will be measured in hundreds
rather than thousands of ohms. The balance condition is: Vv - Vi/2 = 0 where, from the derivation given in section ?: Vv/V = ηv = (Rv // jXC1 // jXC2) / jXC2 and the current transfer function at balance (from section ?) is: Vi/V = η0 = (jXLi // Ri) / Ni R0 Since the current sample is shared by two bridges, we must divide it by two. Hence, at balance: ηv = η0/2 i.e., (Rv // jXC1 // jXC2) / jXC2 = (jXLi // Ri) / (2 Ni R0) which upon inversion of both sides gives: jXC2 [(1/jXC1) + (1/jXC2) + (1/Rv)] = 2Ni R0 [(1/Ri) + (1/jXLi)] Recalling that XC=-1/2πfC and 1/j=-j, this rearranges to: [(C1+C2)/C2] + jXC2/Rv = [2Ni R0 / Ri] - j2Ni R0/XLi Hence, equating the real parts:
Example: A Bruene bridge design is given in ref. [41]. This uses an Amidon T68-2 core with 35 turns of #26AWG enamelled wire, and the current transformer secondary load is 20Ω, i.e., the two resistors marked Ri/2 above are 10Ω each. The capacitors here designated C1 were originally 330pF, and capacitors C2 were 7pF trimmers. Low-frequency compensation resistors were not used in the original circuit, and the operating frequency range was stated to be 3.5 to 30MHz; but since we have data for the transformer core we can compute values for the compensation resistors and extend the frequency range to 1.8MHz. Using equation (x.1) we will first find the nominal capacitance of C2 when the bridge is balanced for 50Ω loads: (C1/C2) + 1 = 2Ni R0 / Ri = 2×35×50/20 = 175 C1/C2 = 174 330pF = 174C2 C2 = 1.90pF This has a reactance of -24KΩ at 3.5MHz and -46.6KΩ at 1.8MHz. To stay roughly in keeping with the original design intentions we should increase C2 to obtain a reactance of about -24KΩ at 1.8MHz, and so a candidate value for C2 is 3.68pF and C1=174C2=641pF. 680pF is the larger nearest preferred value, and 1% silvered-mica capacitors of this value are available, and so we end up with C1=680pF, C2=680/174=3.9pF (obtained by adjusting a 2-10pF trimmer). Since there are two voltage sampling networks, our choice will result in a fixed capacitance of nearly 8pF across the generator, but since one of the networks is on the load side, it will be absorbed into the load impedance after adjustment. Hence the mismatch seen by the generator when the bridge is balanced will be mainly due to the presence of only one of the voltage sampling networks, and the effect of 3.9pF is comfortably within normal load tolerance limits. For low frequency compensation, we note that the transformer has 35 turns, and the AL value of the T68-2 core is 5.7nH. Hence the secondary inductance is ALN²=7μH. Using equation (x.2): Rv = Li/2Ni R0C2 = 7×10-6 / ( 2 × 35 × 50 × 3.9×10-12 ) = 512Ω A measurement of the actual inductance of the transformer coil is, of course, a better criterion for the calculation of the compensation resistor. In the matter of setting the balance points and calibrating such a bridge, notice that the circuit is completely symmetric. Having connected a 50Ω resistor to the load port and adjusted the reflected-signal voltage-sampling network for a null reading on the corresponding meter, the generator and load connections can be swapped for adjustment of the other trimmer. For calibration of the meter scales, the voltage across the load can be measured (e.g., using a calibrated oscilloscope and a ×10 probe, provided that the 'scope input voltage rating is not exceeded), swapping the generator and load connections as before for adjustment of the two meter series resistors. For very high-power transmitters, the voltage should be measured at the output of a through-line attenuator (i.e., a tapped dummy-load resistor). Recall that forward power is proportional to the square root of Vv+Vi, and so the meters will have to be fitted with non-linear scales if calibrated in watts or relative power. It is quite common for the meter resistors to be switched for different FSD power readings. Dual-gang variable potentiometers are also used, but the tracking of inexpensive double potentiometers, particularly of the logarithmic variety, is notoriously bad. Since most transmitters have a drive-level control, infinite variability of the bridge sensitivity is not usually needed, and switched ranges calibrated in actual power are much more useful. Note that the accuracy of the calibration at low frequencies depends on the relationship between XLi and Ri as discussed in section ?. In the example given above XLi=4Ri at 1.8MHz, and so the error will be less than 5% (see table ?); but in that case the design is for use in conjunction with kilowatt transmitters, and more sensitive designs (less turns on the current transformer core or a larger value of secondary load resistance) will not be so good in this respect. >>>>> Use of transformers to take off |Vv±Vi|. Gets rid of the chokes and costs about the same. >>>> |
| 6.4-x. SWR Bridge with load-side voltage sampling. |
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>>>>> Same design considerations as Bruene. Inductance of half winding is Li/2 as explained in section 6-x. Gets rid of the chokes. Gets ride of the loading defect of shared VS. CVS networks give partial self-neutralisation. Ground referenced det. ports allow reciprocal cal. and stealth tuning (must short Lα during cal. reversal) Lα calc as per section 6-x, compensates C across line. Rv can be R in series with small pot. C1 can be fixed C // trimmer. >>>>> |
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Power measurement: . |
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6.4-x. Square-Law detectors: A section on the use of square-law detectors in power measurement. Large signal "linear" detector. Small signal. Output voltage is proportional to the square of the input voltage Agilent application notes [18]. The square root of a number can be obtained by halving its logarithm and taking the antilog. Log and antilog amplifiers. Temp compensation using dual transistors. >> |
© D W Knight 2008.
David Knight asserts the right to be recognised as the author of this work.
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