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How to find faults
in coaxial cables
using TDR
TDR or time-domain reflectometry is a technique used to track
down faults in cables – mainly coaxial cables but other types as
well. But do you know how TDR actually works? This article is
a primer on TDR. It’s a lot easier to understand than you may
think and we also explain what the terms “velocity factor” and
“characteristic impedance” mean.
By JIM ROWE
D
ON’T BE PUT OFF by that complex sounding term “time-domain
reflectometry” or its cryptic acronym
“TDR”. They’re just techno jargon for
a fault-finding technique that’s simpler
than it sounds – at least in principle.
First off, we need to explain that the
main use for TDR is for finding faults
and discontinuities in cables – primarily coaxial cables. These are the cables
used to carry RF signals between antennas, receivers and transmitters and
also to carry RF, video and high-speed
digital signals between professional
and domestic equipment.
In essence, coaxial cables behave
as transmission lines, in that when
electrical energy is fed into one end
of the cable, it takes a finite time for
that energy to travel along the cable to
HIGH SPEED
OSCILLOSCOPE
the other end. That’s because the distributed inductance and capacitance
inside the cable force the energy to
propagate along it in the form of an
electromagnetic wave (a combination
of electric and magnetic energy).
This is very similar to light energy
travelling along a fibre-optic cable –
which is not surprising, because light
is simply electromagnetic (EM) energy
of a much higher frequency. When
any kind of EM energy is propagating
through empty space (ie, a vacuum),
it does so at the speed of light, equal
to 299,724,580 metres per second, or
near enough to 300,000km/s.
By the way, this equates to 300
metres per microsecond (300m/μs)
and also to 300 millimetres per nano
second (300mm/ns). Both of these
figures are worth remembering.
When EM energy is propagating
through a more substantial medium
like a coaxial cable (or a fibre-optic
cable in the case of light), it moves at
a slower speed; still very fast but not
quite as fast as light in a vacuum.
Velocity factor
In the case of EM energy propagating along a coaxial cable or similar
transmission line, its speed or velocity (Vp) is related to the speed of light
in a vacuum by a factor known as the
“velocity factor” (Vf) of the cable. In
other words:
Vp = Vf.c
. . . (1)
where c is the speed of light in a
vacuum.
Fig.1: the basic circuit for a Step TDR. It uses a step generator with a
source resistance of Rsource, while the load at the end of the coaxial
cable has a resistance of Rterm. A high-speed oscilloscope is used to
monitor the voltage at the input end of the cable.
INPUT
Rsource
STEP
GENERATOR
76 Silicon Chip
TRANSMISSION LINE (COAXIAL CABLE OR SIMILAR)
CHARACTERISTIC IMPEDANCE = Zo
LOAD
(Rterm)
siliconchip.com.au
As you might expect, the value of
Vf is closely related to the dielectric
constant Er of the dielectric material
used in the cable itself – between the
centre conductor and the outer screen
conductor. In fact:
Vf = 1/√
√Er
. . . (2)
Most commonly available coaxial
cables use a polyethylene (PE) dielectric in either solid or cellular
foam form – or as small discs of solid
PE spaced apart (“air-spaced” PE). A
small number of cables for specialised
applications use dielectric materials
like fluorinated ethylene propylene
(FEP), poly tetrafluoroethylene (PTFE)
or polyvinyl chloride (PVC).
Table 1 shows the dielectric constant Er, the velocity factor Vf and
the propagation velocity Vp of some
common types of coaxial cable dielectric, along with the figures for
air or a vacuum for comparison. This
information comes in handy when
you’re using a TDR adaptor with a
scope to locate the position of faults
or discontinuities in cables.
Characteristic impedance
Now let’s consider another important aspect of coaxial cables and other
transmission lines: their characteristic
impedance. Just as the distributed
capacitance and inductance of a cable
forces EM energy to propagate along
it at a specific velocity, they also force
the energy to adopt a specific voltageto-current ratio. This V/I ratio is called
the cable’s characteristic impedance,
and is usually represented as “Zo”.
The value of Zo for any particular
cable depends mainly on the ratio of
the outer conductor’s inside diameter
(D) to the inner conductor’s outside
diameter (d), together with the dielectric constant of the insulating material
between them (Er). In fact, if you neglect the series resistance of the inner
and outer conductors per unit length,
Table 1: Common Coaxial Cable Dielectrics
DIELECTRIC
MATERIAL
(DIELECTRIC
Er CONSTANT)
AIR (OR VACUUM)
1.00
1.00
300mm/ns
SOLID PE
2.3
0.66
198mm/ns
CELLULAR FOAM PE
1.4 – 2.1
~0.87
261mm/ns
Vf
(VELOCITY
FACTOR)
Vp
(VELOCITY OF
PROPAGATION)
AIR SPACED PE
~1.1
0.95
285mm/ns
SOLID PTFE
2.1
0.69
207mm/ns
CELLULAR FOAM PTFE
1.4
0.84
252mm/ns
SOLID FEP
2.1
0.69
207mm/ns
CELLULAR FOAM FEP
1.5
0.82
246mm/ns
-9
Note: velocities shown in millimetres per nanosecond (10 s)
the Zo of a coaxial cable can be found
from this simple formula:
Zo = (138Ω/√
√Er) x log(D/d) . . . (3)
which can be simplified to:
Zo = 138Ω x Vf x log(D/d)
. . . (4)
Although you can calculate the Zo of
any particular cable with this formula,
it’s generally not necessary because
cable manufacturers usually provide
this information. Table 2 shows the
relevant details for some common
coaxial cables. All of them have a Zo
of either 50Ω or 75Ω.
Knowing the Zo of a cable is important because when the cable is used
to transfer electrical energy from a
source or “generator” to a load, you
only get maximum power transfer
when the generator’s source resistance/
impedance and the load’s resistance/
impedance are both matched to the Zo
of the cable.
If the load resistance is not matched
to the cable impedance (Zo), some of
the energy reaching the load end of the
cable is reflected back along the cable
to the generator (with a polarity which
may be the opposite of the ‘incident’
energy). If the source resistance of
the generator is not matched to the
cable Zo either, some of this returned
energy is reflected back towards the
load again. The net result is that some
of the energy bounces back and forth
along the cable and is wasted as heat.
So a cable’s Zo or characteristic
impedance is most important in that
it allows you to match the resistance
of the load and generator to it, in order
to achieve the most efficient transfer
of energy/power (and preserve signal
integrity).
TDR basics
Having explained coaxial cable
operation and the significance of Vp
(velocity of propagation) and Zo (characteristic impedance), we are primed
to understand the basics of TDR.
First, there are actually three versions of TDR, known as “Step TDR”,
“Pulse TDR” and “Spread-Spectrum
TDR”.
We’re going to be dealing mainly
with Step TDR because it’s the version
most commonly used nowadays and
it’s the easiest to understand.
Now take a look at the simple circuit
in Fig.1. It shows a length of coaxial
cable connected between a voltage
step generator and a load resistor.
The step generator has a source resist-
Table 2: Typical Characteristics Of Some Common Coaxial Cables
(VELOCITY OF
PROPAGATION)
D/d (mm/mm) Zo (CHIMAPREADCATNERCISET) IC Loss (dB/m <at> F)
COMMON USES
4.7/1.0
75W
0.2 <at> 1GHz
CABLE TV & SATELLITE ANTENNA INSTALLATIONS
198mm/ns
2.9/0.81
50W
1.056 <at> 2.4GHz
THIN ETHERNET, RF & INSTRUMENTATION
SOLID PE
198mm/ns
3.7/0.64
75W
0.39 <at> 1GHz
BASEBAND VIDEO, DOMESTIC TV ANTENNAS
RG-174/U
SOLID PE
198mm/ns
1.5/(7 x 0.16)
50W
2.46 <at> 2.4GHz
WI-FI PIGTAILS, GPS & INSTRUMENTATION
RG-213/U
SOLID PE
198mm/ns
7.2/(7 x 0.75)
50W
0.27 <at> 1GHz
LOW LOSS UHF ANTENNA CABLES
Vp
CABLE TYPE
DIELECTRIC
RG-6/U
CELLULAR FOAM
PE
261mm/ns
RG-58/U
SOLID PE
RG-59/U
Note: RG-XX/U type numbers are a carr y-over from US militar y specs during WW2. They are nowadays used mainly to identify matching connectors.
siliconchip.com.au
November 2014 77
Table 3: Scope Displays With Step Generator
CONDITIONS
RESULT
Zo = Rsource
Rterm = Zo
(CORRECT MATCHING)
ALL ENERGY IS CARRIED
TO THE LOAD, WITH NO
REFLECTED ENERGY
Ei
Zo = Rsource
Rterm = ZERO
(SHORT CIRCUIT
AT LOAD END)
ALL ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH REVERSED POLARITY
(Tr = 2 x cable transit time)
Ei
Zo = Rsource
Rterm = INFINITY
(OPEN CIRCUIT
AT LOAD END)
ALL ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH THE SAME POLARITY
(Tr = 2 x cable transit time)
OSCILLOSCOPE DISPLAY
Er (= –Ei)
Tr
Er (= +Ei)
Ei
Tr
Zo = Rsource
Rterm = 2 x Zo
(TWICE Zo
& Rsource)
ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH THE SAME POLARITY
(Tr = 2 x cable transit time)
Ttransit = L/Vp
Ei
Er (= +Ei/3)
Tr
Zo = Rsource
Rterm = Zo/2
(HALF Zo
& Rsource)
ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH REVERSED POLARITY
(Tr = 2 x cable transit time)
Er (= –Ei/3)
Ei
Tr
ance of Rsource, while the load has a
resistance of Rterm (shortened from
Rterminating). The only other item in
the circuit is a high-speed oscilloscope
with its input being used to monitor the
voltage at the input end of the cable.
To begin, consider the situation
where both Rterm and Rsource are
correctly matched to the Zo of the
cable. What would you expect to see
on the scope?
In this case, you would see a single
voltage step as shown in the top trace
of Table 3. The step would have an amplitude of Ei volts and would continue
at that level indefinitely – or at least
until the generator output falls again.
Note that the value of Ei will be equal
to half of the generator’s open-circuit
output voltage, because the effective
input resistance of the cable will be
equal to Rsource and together they will
constitute a 2:1 voltage divider.
Shorting the cable
Now consider what the scope would
show if the load resistor Rterm were
replaced with a short circuit – in effect,
a resistor of zero ohms. This would be
an extreme mismatch at the load end
of the cable and as a result all of the
voltage step energy would be reflected
Scheduled for publication in the December issue, this do-it-yourself adaptor lets
you use any reasonably “fast” scope to perform step TDR on your own cables.
78 Silicon Chip
back towards the generator as another
voltage step Er – with the same amplitude as Ei but of opposite polarity.
So the scope display would look
like the second trace in Table 3, with
the voltage at the cable input dropping
to zero as soon as the reflected energy
arrived back.
Note the significance of Tr. It is the
time between the start of the voltage
step and its sudden end. It represents
the time taken for the incident step
to travel to the end of the cable, plus
the time taken for the reflected step
to travel back to the start. In other
words, it will be equal to twice the
cable transit time. And we can work
out the transit time. It’s equal to:
. . . (5)
where Vp is the velocity of propagation in the cable (as before, measured
in mm/ns), while L is the cable length
in millimetres.
So Tr will equal twice this value and
if we measure Tr using the scope we
can calculate the effective length of the
cable using this rearranged equation:
L = (Tr x Vp)/2 . . . (6)
where L is the cable length in millimetres, Tr is the step “length” in
nanoseconds and Vp is the velocity of
propagation in mm/ns. So by measuring Tr, we can quite easily work out
the cable length – or more precisely
the distance to the short circuit.
Disconnecting the cable
Next consider what would happen
if we removed the short circuit from
the load end of the cable and left it
without any termination at all – an
open-circuit. This will again represent
an extreme mismatch but of the opposite kind to a short circuit.
All of the voltage step energy will
again be reflected back to the generator
as a voltage step Er, but this time with
the same polarity as Ei.
When the reflected step reaches the
start of the cable, the scope will show
the voltage suddenly rising to twice
its initial value, as shown in the third
trace of Table 3. The Tr time will still
have the same significance as before,
in this case allowing us to work out
the cable length to the open circuit.
Get the idea?
Now let’s consider what would
happen if we don’t connect a short
circuit or an open circuit to the end
of the cable but instead connect a load
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Voltage reflection coefficient
Perhaps you’re wondering why the
value of Er is only equal to a third of Ei,
when the load resistance is twice the
value of Zo? That’s because Er and Ei
are related by a factor called the voltage reflection coefficient (Rho), which
has a value given by this expression:
Er/Ei = Rho = (Zload - Zo)/(Zload + Zo) ...(7)
where Zload is the load impedance,
which in this case is equal to 2Zo.
Rearranging this and substituting for
Zload, we find that the value of Er
becomes:
Er = Ei(2Zo - Zo)/(2Zo + Zo)
= Ei.Zo/3Zo = Ei/3
2.00
1 .00
0.80
0.60
0.40
0.30
RG-174/U
0.20
RG-213/U
0.10
0.08
0.06
RG-58/U
0.05
0.04
0.03
0.02
0.01
. . . (8)
Next consider what will happen if
we again connect a load resistor to the
end of the cable but this time with a
value which is HALF the value of Zo
and Rsource. Again this is a mismatch,
so some of the voltage step energy will
be reflected back towards the generator as before. This time though, the
reflected voltage Er will be reversed
in polarity compared with Ei, because
the load resistance is lower than Zo.
You can see the resulting downward
step in the fifth (lowest) trace in Table
3. You’ll also see that the value of Er is
again equal to one third of Ei, which
is confirmed thus:
Er = Ei(Zo/2 - Zo)/(Zo/2 + Zo)
= Ei(-Zo/2)/(3Zo/2) = -Ei/3 . . . (9)
From these five examples you’ll be
starting to see how TDR works and
how it allows us to calculate some
important details about the operation
of a cable and transmission line:
(1) Whether the cable is correctly
terminated in a matched load, which
means no reflected energy. This is
shown by the voltage step extending
indefinitely.
(2) If there is a further step in the scope
display, indicating some kind of mismatch, then the Tr time between the
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3.00
CABLE LOSS IN DECIBELS PER METRE (dB/m)
resistor with a value Rterm which is
TWICE the value of Zo and Rsource.
This is again a mismatch, although not
as severe as a short or open circuit.
Some of the voltage step energy will
be reflected back towards the generator
but not as much as before – and with
the same polarity as Ei.
So the scope will show an upward
step after time Tr, with a step height
Er in this case equal to Ei/3 as shown
in the fourth trace of Table 3.
0.006
1
2
4
6
8 10
20
40
60
100
200
400 600 1GHz
2
3
FREQUENCY IN MEGAHERTZ & GIGAHERTZ
Fig.2: these curves show the losses in three common types of 50Ω coaxial cable
as a function of frequency. RG-174/U cable ranges from 0.06dB/m at 1MHz up to
almost 2.5dB/m at 2.4GHz, while RG-213/U cable ranges from just 0.006dB/m at
1MHz up to 0.49dB/m at 2.4GHz. RG-58/U cable is midway between these two.
initial step and the “reflection” step
can be used to work out the length of
cable L between the generator end and
the mismatch.
(3) The amplitude and polarity of the
reflected voltage step Er can be used
to work out the effective resistance of
the mismatched load.
Cable losses
There’s one complication we need to
consider before moving on: the effect
of cable losses.
In the discussion so far, we’ve made
the assumption that the cables being tested are “perfect”, in the sense
that when the generator and load resistances are properly matched to the
cable’s Zo, all of the energy fed into
the cable at one end emerges from the
other end and passes into the load. In
other words, we’ve assumed that the
cables are lossless.
But in the real world, nothing is
perfect. As shown in column six in
Table 2, all cables have a loss which is
usually listed in terms of decibels per
metre (dB/m), or decibels per 100 feet
(dB/100ft) in countries like the USA
which still use the Imperial system.
Because cable losses rise with increasing frequency, the loss figure is
usually qualified with a frequency
figure, as shown.
To put things into perspective, look
at the curves in Fig.2. These show
the losses in three common types of
50Ω coaxial cable, all plotted against
frequency.
As you can see, the small diameter RG-174/U cable has a loss figure
ranging from 0.06dB/m at 1MHz up
to almost 2.5dB/m at 2.4GHz, while
RG-213/U cable with its much larger
diameter has a loss figure ranging
from only 0.006dB/m at 1MHz up to
0.49dB/m at 2.4GHz.
The common RG-58/U cable is
midway between the other two in
terms of its loss factor – ranging from
November 2014 79
Fig.3: a Pulse TDR is almost idential to a Step TDR, the difference
being that the stimulus generator delivers a narrow voltage pulse
rather than a DC voltage step.
HIGH SPEED
OSCILLOSCOPE
INPUT
TRANSMISSION LINE (COAXIAL CABLE OR SIMILAR)
Rsource
NARROW
PULSE
GENERATOR
CHARACTERISTIC IMPEDANCE = Zo
0.013dB/m at 1MHz up to just over
1.00dB/m at 2.4GHz.
So real cables do lose some of the
input EM energy (as heat), even when
the generator and load are correctly
matched to their Zo. But what effect
does this have when you are checking
a cable using Step TDR?
This depends on things like the cable loss factor and the cable’s length.
These are not likely to have much effect on a fairly short cable but when
you’re checking a fairly long run of a
relatively lossy cable, the cable loss
will tend to attenuate the indicated
level of reflected step Er. So any mismatch will appear to be less serious
than it should.
Pulse TDR
Remember that the version of TDR
we’ve been discussing so far is Step
TDR – the name referring to the way
it uses a voltage step waveform as the
incident “stimulus” being fed into
the cable to be tested. But we’re now
going to look briefly at the other basic
version: Pulse TDR, where a short
voltage pulse is used as the stimulus
rather than a step.
Fig.3 shows the basic circuit for
a Pulse TDR. It’s almost identical to
the Step TDR circuit of Fig.1, the only
difference being that the stimulus generator is now labelled ‘Narrow Pulse
Generator’; it generates a narrow voltage pulse rather than a step.
In effect, Pulse TDR works in much
the same way as Step TDR. If you compare the traces shown in Table 4 with
those for Step TDR in Table 3, you’ll
see that the only differences are that
Table 4: Scope Displays With Pulse Generator
CONDITIONS
RESULT
Zo = Rsource
Rterm = Zo
(CORRECT MATCHING)
ALL ENERGY IS CARRIED
TO THE LOAD, WITH NO
REFLECTED ENERGY
Zo = Rsource
Rterm = ZERO
(SHORT CIRCUIT
AT LOAD END)
ALL ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH REVERSED POLARITY
(Tr = 2 x cable transit time)
OSCILLOSCOPE DISPLAY
Ei
Ei
Er (= –Ei)
Tr
Zo = Rsource
Rterm = INFINITY
(OPEN CIRCUIT
AT LOAD END)
ALL ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH THE SAME POLARITY
(Tr = 2 x cable transit time)
Ei
Er (= +Ei)
ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH THE SAME POLARITY
(Tr = 2 x cable transit time)
Er (= +Ei/3)
Ei
Tr
Zo = Rsource
Rterm = Zo/2
(HALF Zo
& Rsource)
ENERGY REFLECTED BACK
TO THE GENERATOR, BUT
WITH REVERSED POLARITY
(Tr = 2 x cable transit time)
Ei
Tr
80 Silicon Chip
each voltage step of Table 3 is now
replaced with a voltage pulse. The
basic behaviour is unchanged, because
we’re still looking at the effects caused
by the interaction between cable parameters Vp and Zo and changes in
load resistance.
Step TDR is more popular
But if there’s so little difference between the two, why is Step TDR more
popular than Pulse TDR? For a couple
of reasons, one being that during the
stimulus pulse in Pulse TDR, the scope
can’t be allowed to monitor the cable
input voltage Ei because it would be
overloaded. So with this approach, the
stimulus pulse creates a ‘dead zone’,
during which the scope can’t look for
reflections.
But when the pulse width is made
very narrow to reduce the dead zone,
this also reduces the TDR’s range. So
Pulse TDRs generally need to provide
a number of different pulse widths, to
achieve different trade-offs between
dead zone and range.
Another allied problem with Pulse
TDRs is that because a pulse stimulus
carries much less energy than the step
stimulus, the technique is not capable
of delivering the same signal to noise
ratio. So with real-world (read “lossy”)
cables, Pulse TDR can’t reveal cable
faults or discontinuities as clearly as
Step TDR.
Summarising
Tr
Zo = Rsource
Rterm = 2 x Zo
(TWICE Zo
& Rsource)
LOAD
(Rterm)
Er (= –Ei/3)
You should now have a reasonable
understanding of what TDR is, how it
works and how it’s used for checking
coaxial cables in particular. Think of
it as “echo location for cable faults”
if you like.
And as you may have guessed, this
article is a prelude for a planned lowcost adaptor which lets you use any
reasonably “fast” scope to perform
Step TDR on your own cables. Look
SC
for it in the December issue.
siliconchip.com.au
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