6.4 Reflectometry
(Measurement of reflection coefficient, return loss, and SWR)
>>>
work in progress.
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 misterminated 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), R
_{0}, of the line. Thus we
may write:
If the line is not terminated in its characteristic resistance,
only part of the energy in the wave will be absorbed (or perhaps
none if the line is open or shortcircuit), and a reflected wave
will be launched on a journey from the load to the generator.
This reflected wave will also have no knowledge of the conditions
which prevail at its destination, and so the relationship between
voltage and current will also be determined solely by R
_{0}, i.e.,
In this case, a minus sign is required in order to maintain
consistency in the definitions of the forward and reflected waves;
i.e., we must assume that either the voltage or the current is
reversed relative to the forward wave at the reflection boundary.
To understand this point, note that if the line is open circuit,
a voltage may exist at the discontinuity, but the sum of forward
and reflected currents must be zero, i.e., the sign of the current
must reverse as the wave turns around and heads back towards
the generator. Conversely, if the line is short circuit, then
a finite current may exist at the end of the line, but the sum
of the forward and reflected voltages must be zero, i.e., the
sign of the voltage must reverse. When partial reflection occurs,
the voltage reverses relative to the forward wave if the magnitude
of the load impedance is less than R
_{0},
and the current reverses if the magnitude of the load impedance
is greater than R
_{0}; but the minus sign
in the expression above maintains consistency in either case,
and allows the forward and reverse wave definitions to be combined
(i.e., to be used as simultaneous equations).
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 through the load, then we may write:
V = I Z = V_{F}
+ V_{R} . . . (3) 
i.e., the voltage at the load is the sum of the voltages
in the forward and reflected waves. Similarly, the current flowing
in the load is the sum of the forward and reflected currents,
and we may also write:
I = V / Z = I_{F}
+ I_{R} . . . (4) 
The observations made so far are summarised in the diagram
below:
From the relationships given above, we have
I_{F}=
V_{F}/R
_{0} and
I_{R}=
V_{R}/R
_{0}. Substituting these expressions into equation
(
4) gives:
I = (
V_{F} / R
_{0})  (
V_{R} / R
_{0}) = (
V_{F} 
V_{R}) / R
_{0}
and using (
3) gives:
(
VF +
VR)
/
Z = (
VF 
VR) / R
0
which can be rearranged as follows:
(
VF +
VR)
R
0 = (
VF

VR)
Z
VF R
0 +
VR R
0 =
VF Z 
VR
Z
VR R
0 +
VR Z =
VF
Z 
VF R
0
VR (
Z + R
0)
=
VF (
Z  R
0)
To give:
VR / VF
= (Z  R0) / (Z + R0) 
which says that the ratio of the voltages in the reflected
and forward waves is a system constant which depends only on
Z and R
0. Recall however, that
while this expression is true at the load, it is only true of
voltage
magnitudes at other points in the line, and so
to generalise it we must write:
VR / VF
= Z  R0 / Z + R0 
which is true everywhere in the line.
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 R
0

IR R
0
which rearranges to:
IR / IF
= (Z  R0) / (Z + R0) 
The minus sign is lost in generalising to all points
on the line (i.e., by taking magnitudes), because 
IR= 
IR , and
so:
IR / IF
= Z  R0 / Z + R0 
Thus, to summarise:
Γ = 
VR VF 
= 
IR IF 
= 
 Z  R0  
Z + R0  

>>> work in progress
Lossy lines:
Z0 = R
0
+
jX
0 (X
0
is negative) and
Z0*
Return loss = 20Log
10( 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 ) / ( 1Γ )
True for any length of line and for lossy lines.
Γ = (S1)/(S+1)
Γ = √(P
R/P
F)
>>>
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
quarterwavelength, 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.
Γ = 
VR VF 
= 
IR IF 
= 
 Z  R0  
Z + R0  
= 
 Vv  Vi   Vv
+ Vi  

>>>>> to be rewritten
6.4x. 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 (R
0)
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 squareroot
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 
V²/R
0. Similarly, the current in the line will
vary sinusoidally, and the forward power cannot be calculated
from 
I²R
0. 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 currenttransformer bridge
with more than one turn in the currenttransformer 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 timedelay ralative
to the source, a matter which can be dealt with by neutralisation,
or by moving the sampling point downline by an electrical distance
equal to half the electrical length of the currenttransformer
secondary winding.
A prototype SWR bridge is shown
below: where the magnitude of
Vf
is proportional to the squareroot of the forward power and is
independent of reflected power; and the magnitude of
Vr is proportional to the squareroot of the
reflected power, and is independent of forward power.
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 highimpedance 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 capacitivedivider
version does not work properly at low frequencies when
used to drive two separate meters, it being an illconceived
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 currenttransformer 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 antiintuitive, 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.
6.4x. 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 lowfrequency compensation and rearranged
to use the shuntdiode detector configuration [see Detectors
for RF meas.]. These changes were sufficient to circumvent Douma's
1957 patent [USP 2808566], but the circuit is also a logical
development of the capacitor ratioarm 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 180L3
(2 MHz  25 MHz) automatic antenna tuner is discussed elsewhere
[ /zdocs/zmatching/ ]. A "highfrequency 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 2 MHz 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 3.530 MHz. 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 outofPatent by 1962). Consequently,
the resistors (R
v) 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 (C
b)
to prevent the DC outputs of the detectors from being shunted
to ground. C
b merely needs to have a low
reactance at the minimum frequency of operation (10 nF 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.
Notice that the shuntdiode 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 (multisegment RF chokes of 1mH or more are normally
used). If the drop in meter sensitivity which results from using
R
v 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 C
1 and C
2, and connect
the blocking capacitor C
b to the current
transformer output. Due to the very low output impedance of the
current transformer circuit, selfcapacitance 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 R
v
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 R
v 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 = (R
v //
jX
C1 //
jX
C2) /
jX
C2
and the current transfer function at balance (from
section
?) is:
Vi/
V =
η0 = (
jX
Li //
R
i) / N
i R
0
Since the current sample is shared by two bridges, we must divide
it by two. Hence, at balance:
ηv =
η0/2
i.e.,
(R
v //
jX
C1
//
jX
C2) /
jX
C2
= (
jX
Li // R
i)
/ (2 N
i R
0)
which upon inversion of both sides gives:
jX
C2 [(1/
jX
C1)
+ (1/
jX
C2) + (1/R
v)]
= 2N
i R
0 [(1/R
i) + (1/
jX
Li)]
Recalling that X
C=1/2πfC and 1/
j=
j,
this rearranges to:
[(C
1+C
2)/C
2] +
jX
C2/R
v = [2N
i R
0
/ R
i] 
j2N
i R
0/X
Li
Hence, equating the real parts:
(C1+C2)/C2 = 2Ni R0/Ri 
x.1

and, equating the imaginary parts:
Example:
A Bruene bridge design is given in ref. [
41]. This uses
an Amidon T682 core with 35 turns of #26AWG enamelled wire,
and the current transformer secondary load is 20 Ω, i.e.,
the two resistors marked R
i/2 above are
10 Ω each. The capacitors here designated C
1
were originally 330 pF, and capacitors C
2
were 7 pF trimmers. Lowfrequency compensation resistors were
not used in the original circuit, and the operating frequency
range was stated to be 3.5 to 30 MHz; but since we have data
for the transformer core we can compute values for the compensation
resistors and extend the frequency range to 1.8 MHz.
Using equation (
x.1) we will
first find the nominal capacitance of C
2
when the bridge is balanced for 50 Ω loads:
(C
1/C
2) + 1 = 2N
i R
0 / R
i
= 2×35×50/20 = 175
C
1/C
2 = 174
330 pF = 174C
2
C
2 = 1.90 pF
This has a reactance of 24 kΩ at 3.5 MHz and 46.6 kΩ
at 1.8MHz. To stay roughly in keeping with the original design
intentions we should increase C
2 to obtain
a reactance of about 24 kΩ at 1.8 MHz, and so a candidate
value for C
2 is 3.68 pF and C
1=174C
2=641 pF. 680 pF is the larger nearest preferred
value, and 1% silveredmica capacitors of this value are available,
and so we end up with C
1=680 pF, C
2=680/174=3.9 pF (obtained by adjusting a 210pF
trimmer). Since there are two voltage sampling networks, our
choice will result in a fixed capacitance of nearly 8 pF 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 3.9 pF is comfortably within
normal load tolerance limits.
For low frequency compensation, we note that the transformer
has 35 turns, and the A
L value of the
T682 core is 5.7 nH. Hence the secondary inductance is A
LN²=7 μH. Using equation (
x.2):
R
v = L
i/2N
i R
0C
2
= 7×10
^{6} / ( 2 × 35 × 50 ×
3.9×10
^{12} ) = 512 Ω
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 50 Ω resistor to the load port and adjusted
the reflectedsignal voltagesampling 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 highpower transmitters,
the voltage should be measured at the output of a throughline
attenuator (i.e., a tapped dummyload resistor). Recall that
forward power is proportional to the square root of
Vv+
Vi, and so
the meters will have to be fitted with nonlinear scales if calibrated
in watts or relative power. It is quite common for the meter
resistors to be switched for different FSD power readings. Dualgang
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 drivelevel
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 X
Li and R
i as discussed
in
section ?. In the example
given above X
Li=4R
i
at 1.8 MHz, 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 
V_{v}±
V_{i} . Gets rid of the chokes and costs about the same.
>>>>
Refs
[
40]
"An Inside Picture
of Directional Wattmeters", Warren B Bruene W0TTK [W5OLY],
QST April 1959 p2428.
[
41]
"InLine RF Power
Metering" Doug DeMaw W1CER [W1FB], QST Dec 1969 p11
16.
Construction article for Bruenetype bridges for 3.530MHz.
[
42]
Ferromagnetic Core
Design & Application Handbook, M F "Doug" DeMaw
W1FB, 1st edition, 2nd printing 1996. MFJ publishing co. http://www.mfjenterprises.com/ .
Bruene bridge: p9495.
6.4x. SWR Bridge with loadside voltage sampling.
>>>>>
Same basic design considerations as Bruene.
Inductance of half winding is L
_{i} /2 as
explained in section 6x.
Gets rid of the chokes.
Gets rid of the loading defect of shared voltagesampling networks.
CVS network across load allows neutralisation by the loadcapacitance
method. HF phase neutralisation can be effected by arranging
CVS network capacitance to be larger than required, then adding
a small amount of capacitance across the transformer secondary.
HF amplitude correction (compensation for inductance of lower
voltagesampling arm) can be effected by placing a small inductance
in series with C2.
Ground referenced det. ports allow reciprocal calibration and
stealth tuning (must short L
_{α} during the calibration
reversal)
L
_{α} calculated as per section 6x, compensates
for capacitance across the line.
Rv can be a resistor in series with small pot.
C1 can be a fixed capacitor in parallel with a trimmer.
Power Measurement
6.4x. SquareLaw detectors:
A section on the use of squarelaw detectors in power measurement.
Large signal "linear" detector.
Small signal. Output voltage is proportional to the square of the input voltage
Agilent application notes [see
diode detectors].
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, 2013.
David Knight asserts the right to be recognised as the author of this work.
Last edited:2021 Sept 12
^{th} (but still not conforming to HTML 5)