TDR Theory

7/29/2019 TDR Theory

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Time Domain Relectometry Theory

Application Note

For Use with Agilent 86100 Ininiium DCA


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The most general approach to evaluating the time domain response o any

electromagnetic system is to solve Maxwells equations in the time domain.

Such a procedure would take into account all the eects o the system geom-

etry and electrical properties, including transmission line eects. However,

this would be rather involved or even a simple connector and even more

complicated or a structure such as a multilayer high-speed backplane. For this

reason, various test and measurement methods have been used to assist the

electrical engineer in analyzing signal integrity.

The most common method or evaluating a transmission line and its load has

traditionally involved applying a sine wave to a system and measuring waves

resulting rom discontinuities on the line. From these measurements, the

standing wave ratio (s) is calculated and used as a igure o merit or the trans-

mission system. When the system includes several discontinuities, however,

the standing wave ratio (SWR) measurement ails to isolate them. In addition,

when the broadband quality o a transmission system is to be determined, SWR

measurements must be made at many requencies. This method soon becomes

very time consuming and tedious.

Another common instrument or evaluating a transmission line is the network

analyzer. In this case, a signal generator produces a sinusoid whose requency

is swept to stimulate the device under test (DUT). The network analyzer

measures the relected and transmitted signals rom the DUT. The relected

waveorm can be displayed in various ormats, including SWR and relection

coeicient. An equivalent TDR ormat can be displayed only i the network

analyzer is equipped with the proper sotware to perorm an Inverse Fast Fourier

Transorm (IFFT). This method works well i the user is comortable working

with s-parameters in the requency domain. However, i the user is not amiliar

with these microwave-oriented tools, the learning curve is quite steep.

Furthermore, most digital designers preer working in the time domain with

logic analyzers and high-speed oscilloscopes.

When compared to other measurement techniques, time domain relectometry

provides a more intuitive and direct look at the DUTs characteristics. Using a

step generator and an oscilloscope, a ast edge is launched into the transmis-

sion line under investigation. The incident and relected voltage waves are

monitored by the oscilloscope at a particular point on the line.

Introduction


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This echo technique (see Figure 1) reveals at a glance the characteristic

impedance o the line, and it shows both the position and the nature (resistive,

inductive, or capacitive) o each discontinuity along the line. TDR also dem-

onstrates whether losses in a transmission system are series losses or shunt

losses. All o this inormation is immediately available rom the oscilloscopes

display. TDR also gives more meaningul inormation concerning the broadband

response o a transmission system than any other measuring technique.

Since the basic principles o time domain relectometry are easily grasped, even

those with limited experience in high-requency measurements can quickly

master this technique. This application note attempts a concise presentation o

the undamentals o TDR and then relates these undamentals to the param-

eters that can be measured in actual test situations. Beore discussing these

principles urther we will briely review transmission line theory.

Figure 1. Voltage vs time at a particular point on a mismatched transmission line driven

with a step of height Ei

Eiex

Zo ZL

Ei

E i +E r

t

ex(t)

X

Zo ZL

Transmission line Load


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Propagation on a Transmission Line

The classical transmission line is assumed to consist o a continuous structure

o Rs, Ls and Cs, as shown in Figure 2. By studying this equivalent circuit,several characteristics o the transmission line can be determined.

I the line is ininitely long and R, L, G, and C are deined per unit length, then

R + j wLZin = Zo

G + jwC

where Zois the characteristic impedance o the line. A voltage introduced atthe generator will require a inite time to travel down the line to a point x. The

phase o the voltage moving down the line will lag behind the voltage intro-

duced at the generator by an amount b per unit length. Furthermore, the volt-age will be attenuated by an amount aper unit length by the series resistanceand shunt conductance o the line. The phase shit and attenuation are deined

by the propagation constant g, where

g = a + jb = (R + jwL) (G + jwC)

and a = attenuation in nepers per unit length b = phase shit in radians per unit length

Figure 2. The classical model for a transmission line.

The velocity at which the voltage travels down the line can be deined in terms

o b:

w

Wherenr= Unit Length per Second

b

The velocity o propagation approaches the speed o light, nc, or transmissionlines with air dielectric. For the general case, where er is the dielectric constant:

nc nr= er

ZS

ZL

ES

L R L R

C G C G


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The propagation constant g can be used to deine the voltage and the current atany distance x down an ininitely long line by the relations

Ex = Einegx and Ix = Iine

g x

Since the voltage and the current are related at any point by the characteristic

impedance o the line

Eineg x Ein

Zo = = = Zin Iine

g x Iin

where Ein = incident voltage

Iin = incident current

When the transmission line is inite in length and is terminated in a load whose

impedance matches the characteristic impedance o the line, the voltage and

current relationships are satisied by the preceding equations.

I the load is dierent rom Zo, these equations are not satisied unless asecond wave is considered to originate at the load and to propagate back up

the line toward the source. This relected wave is energy that is not delivered to

the load. Thereore, the quality o the transmission system is indicated by

the ratio o this relected wave to the incident wave originating at the source.

This ratio is called the voltage relection coeicient, r, and is related to thetransmission line impedance by the equation:

Er ZL Zo r = = Ei ZL + Zo

The magnitude o the steady-state sinusoidal voltage along a line terminatedin a load other than Zo varies periodically as a unction o distance between amaximum and minimum value. This variation, called a standing wave, is caused

by the phase relationship between incident and relected waves. The ratio o

the maximum and minimum values o this voltage is called the voltage standing

wave ratio,s, and is related to the relection coeicient by the equation

1 +r s= 1 rAs has been said, either o the above coeicients can be measured with

presently available test equipment. But the value o the SWR measurement is

limited. Again, i a system consists o a connector, a short transmission line anda load, the measured standing wave ratio indicates only the overall quality o

the system. It does not tell which o the system components is causing the

relection. It does not tell i the relection rom one component is o such a

phase as to cancel the relection rom another. The engineer must make

detailed measurements at many requencies beore he can know what must

be done to improve the broadband transmission quality o the system.


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A time domain relectometer setup is shown in Figure 3.

The step generator produces a positive-going incident wave that is applied to

the transmission system under test. The step travels down the transmission

line at the velocity o propagation o the line. I the load impedance is equal to

the characteristic impedance o the line, no wave is relected and all that will

be seen on the oscilloscope is the incident voltage step recorded as the wave

passes the point on the line monitored by the oscilloscope. Reer to Figure 4.

I a mismatch exists at the load, part o the incident wave is relected. The

relected voltage wave will appear on the oscilloscope display algebraically

added to the incident wave. Reer to Figure 5.

Figure 3. Functional block diagram for a time domain reflectometer

Figure 4. Oscilloscope display when Er= 0

Figure 5. Oscilloscope display when Er 0

TDR Step Relection Testing

Sampler

circuit

Device under test

E i E r

Z

L

Stepgenerator

High speed oscilloscope

E i

E i

T

Er


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The relected wave is readily identiied since it is separated in time rom the

incident wave. This time is also valuable in determining the length o the trans-

mission system rom the monitoring point to the mismatch. Letting D denotethis length:

T nrT

D = nr =

2 2

where nr = velocity o propagation

T = transit time rom monitoring point to the mismatch andback again, as measured on the oscilloscope (Figure 5).

The velocity o propagation can be determined rom an experiment on a known

length o the same type o cable (e.g., the time required or the incident wave

to travel down and the relected wave to travel back rom an open circuit

termination at the end o a 120 cm piece o RG-9A/U is 11.4 ns giving

nr = 2.1 x 1010

cm/sec. Knowing nr and reading T rom the oscilloscopedetermines D. The mismatch is then located down the line. Most TDRscalculate this distance automatically or the user.

The shape o the relected wave is also valuable since it reveals both the nature

and magnitude o the mismatch. Figure 6 shows our typical oscilloscope

displays and the load impedance responsible or each. Figures 7a and 7b

show actual screen captures rom the 86100x DCA. These displays are easily

interpreted by recalling:

Er ZL Zo r = = Ei ZL + Zo

Knowledge o Ei and Er, as measured on the oscilloscope, allows ZLto bedetermined in terms o Zo, or vice versa. In Figure 6, or example, we may veriythat the relections are actually rom the terminations speciied.

Locating mismatches

Analyzing refections


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Figure 6. TDR displays for typical loads.

Assuming Zois real (approximately true or high quality commercial cable), it isseen that resistive mismatches relect a voltage o the same shape as the driving

voltage, with the magnitude and polarity o Er determined by the relative valueso Zo and RL.

Also o interest are the relections produced by complex load imped-ances. Four

basic examples o these relections are shown in Figure 8.

These waveorms could be veriied by writing the expression or r(s) in termso the speciic Z

Lor each example:

R( i.e., ZL = R + sL , , etc. ) ,

1 + RCs

Eimultiplying r (s) by the transorm o a step unction o Ei,

s

(A) Open circuit termination (Z = )

L

ZL

E i

E i

E i

E iZL

(B) Short circuit termination (Z = 0)L

(C) Line terminated in Z = 2ZL o

E i

E1

3 i

ZL 2Z o

(D) Line terminated in Z = ZL1

2o

1

2Z o

Ei1

3

E iZL

(A) E = Er i Therefore = +1

Which is true as Z

Z = Open circuit

L

ZLZ o

Z L+Z o

(B) E = Er i Therefore = 1

Which is only true for finite Z

When Z = 0

Z = Short circuit

ZLZ o

Z L+Z o

o

L

(C) E = + Eri

Therefore = +

and Z = 2Z

ZLZ o

Z L+Z ooL

1

3

1

3

(D) E = Er i Therefore =

and Z = Z

ZLZ o

Z L+Z o

oL

1

3

1

3

1

2


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and then transorming this product back into the time domain to ind an

expression or er(t). This procedure is useul, but a simpler analysis ispossible without resorting to Laplace transorms. The more direct analysis

involves evaluating the relected voltage at t = 0 and at t = and assumingany transition between these two values to be exponential. (For simplicity,

time is chosen to be zero when the relected wave arrives back at the monitor-ing point.) In the case o the series R-L combination, or example, at t = 0 the

relected voltage is +Ei. This is because the inductor will not accept a suddenchange in current; it initially looks like an ininite impedance, and r = +1 att = 0. Then current in L builds up exponentially and its impedance drops towardzero. At t = , thereore er(t) is determined only by the value o R.

R Zo( r = When t = ) R + Zo

The exponential transition o er(t) has a time constant determined by theeective resistance seen by the inductor. Since the output impedance o the

transmission line is Zo, the inductor sees Zoin series with R, and

L

g = R + Zo

Figure 7b. Screen capture of short circuit termination from the 86100Figure 7a. Screen capture of open circuit termination from the 86100


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Figure 8. Oscilloscope displays for complex ZL.

ZL

ZL

ZL

ZL

SeriesRL

ShuntRC

ShuntRL

SeriesRC

E i

E i

0

Where 1 = L

R+Zo

(1+ )EiRZo

R+Zo

t

E i (1+ )+(1 )eRZo

R+Zo

RZo

R+Zo

t/1

(1+ )EiRZo

R+Zo

Where 1 = CZoR

Z o+R

E i (1+ ) (1e )RZo

R+Zo

t/1

0

t

E i E i

( )EiRZo

R+Zo

Where 1 = LoR+Z

RZo

E i (1+ )eRZo

R+Zo

t/1

0

t

E i

( )EiRZo

R+Zo

0t

E i

Where 1 = (R+Z ) Co

2EiE i (2(1 )eRZo

R+Zo

t/1

A

B

C

D

R

C

RL

R C

R

L


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A similar analysis is possible or the case o the parallel R-C termination. Attime zero, the load appears as a short circuit since the capacitor will not accept

a sudden change in voltage. Thereore, r = 1 when t = 0. Ater some time,however, voltage builds up on C and its impedance rises. At t = , thecapacitor is eectively an open circuit:

R ZoZL = R and = R + Zo

The resistance seen by the capacitor is Zoin parallel with R, and thereore thetime constant o the exponential transition o er(t) is:

Zo R

C Zo + R

The two remaining cases can be treated in exactly the same way.The results o

this analysis are summarized in Figure 8.

So ar, mention has been made only about the eect o a mismatched load at

the end o a transmission line. Oten, however, one is not only concerned with

what is happening at the load, but also at intermediate positions along the line.

Consider the transmission system in Figure 9.

The junction o the two lines (both o characteristic impedance Zo) employs aconnector o some sort. Let us assume that the connector adds a small inductor

in series with the line. Analyzing this discontinuity on the line is not much

dierent rom analyzing a mismatched termination. In eect, one treats every-

thing to the right o M in the igure as an equivalent impedance in series withthe small inductor and then calls this series combination the eective load

impedance or the system at the point M. Since the input impedance to the righto M is Zo, an equivalent representation is shown in Figure 10. The pattern on theoscilloscope is merely a special case o Figure 8A and is shown on Figure 11.

Figure 9. Intermediate positions along a transmission line

Discontinuities on the line

ZLZoZo

Assume Z = ZL o


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Figure 10. Equivalent representation

Figure 11. Special case of series R-L circuit

Time domain relectometry is also useul or comparing losses in transmissionlines. Cables where series losses predominate relect a voltage wave with an

exponentially rising characteristic, while those in which shunt losses predomi-

nate relect a voltage wave with an exponentially-decaying characteristic. This

can be understood by looking at the input impedance o the lossy line.

Assuming that the lossy line is ininitely long, the input impedance is given by:

R + jwLZin = Zo =

G + jwC

Treating irst the case where series losses predominate, G is so small comparedto wC that it can be neglected:

R + jwL L R1/2

Zin = = ( 1 + )jwC C jwL

Recalling the approximation (1 + x)a_ (I + ax) or x < 1, Zin can beapproximated by:

L RZin _ ( 1 + ) When R < wL

C j2wL

Since the leading edge o the incident step is made up almost entirely o high

requency components, R is certainly less than wL or t = 0+. Thereore theabove approximation or the lossy line, which looks like a simple series R-Cnetwork, is valid or a short time ater t = 0. It turns out that this model is all

that is necessary to determine the transmission lines loss.

ZoZo

L

Ei

Ei

1 = L

2Zo

Evaluating cable loss


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In terms o an equivalent circuit valid at t = 0+, the transmission line with

series losses is shown in Figure 12.

Figure 12. A simple model valid at t = 0+ for a line with series losses

The series resistance o the lossy line (R) is a unction o the skin depth o theconductor and thereore is not constant with requency. As a result, it is dii-

cult to relate the initial slope with an actual value o R. However, the magnitudeo the slope is useul in comparing conductors o dierent loss.

A similar analysis is possible or a conductor where shunt losses predominate.

Here the input admittance o the lossy cable is given by:

1 G + jwC G + jwCYin = = =

Zin R + jwL jwL

Since R is assumed small, re-writing this expression orYin:

ein

Zs

R'

Z

C'

Zin

in= R' + 1

jwC'

E

C G1/2

Yin = ( 1 + )L jwC

Again approximating the polynominal under the square root sign:

C GYin_ ( 1 + ) When G < wC

L j2wC


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A qualitative interpretation o why ein(t) behaves as it does is quite simplein both these cases. For series losses, the line looks more and more like anopen circuit as time goes on because the voltage wave traveling down the line

accumulates more and more series resistance to orce current through. In the

case o shunt losses, the input eventually looks like a short circuit because the

current traveling down the line sees more and more accumulated shunt conduc-

tance to develop voltage across.

One o the advantages o TDR is its ability to handle cases involving more than

one discontinuity. An example o this is Figure 14.

Figure 14. Cables with multiple discontinuities

The oscilloscopes display or this situation would be similar to the diagram in

Figure 15 (drawn or the case where ZL < Zo < Z&o):

Figure 15. Accuracy decreases as you look further down a line with multiple discontinuities

Multiple discontinuities

ZLZ'oZo

1

1

2

r

r'

r

Z Z' ZLo o

r = = r 'Z' Zo o

Z' + Zoo1 1

r = Z Z'

Z + Z'

L o

L o2

Ei

Er1

Er2

Z > Z' < ZLoo

Going to an equivalent circuit (Figure 13) valid at t = 0+,

Figure 13. A simple model valid at t = 0+ for a line with shunt losses

ein

Zs

L'

Y

G'Yin

in= G' + 1

jwL'

E


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It is seen that the two mismatches produce relections that can be analyzed

separately. The mismatch at the junction o the two transmission lines

generates a relected wave, Er , where

Z&o ZoE

r= r

1

Ei

= ( ) Ei Z&o + Zo

Similarly, the mismatch at the load also creates a relection due to

its relection coeicient

ZL Z&o

r2 = ZL + Z&oTwo things must be considered beore the apparent relection rom ZL, asshown on the oscilloscope, is used to determine r2. First, the voltage step inci-dent on ZL is (1 + r1) Ei, not merely Ei. Second, the relection rom the load is

[ r2

(1 + r1) E

i] = E

rL

but this is not equal to Er2since a re-relection occurs at the mismatched

junction o the two transmission lines. The wave that returns to the monitoring

point is

Er2= (1 + r1&) ErL

= (1 + r1&) [ r2 (1 + r1) Ei ]

Sincer1& = r1, Er2may be re-written as:

Er2Er2

= [ r2 (1 r12 ) ] Ei

The part oErLrelected rom the junction o

ErLZ&oand Zo (i.e., r1& ErL)

is again relected o the load and heads back to the monitoring point only to be

partially relected at the junction o Zo&and Zo. This continues indeinitely, butater some time the magnitude o the relections approaches zero.

In conclusion, this application note has described the undamental theory

behind time domain relectometry. Also covered were some more practical

aspects o TDR, such as relection analysis and oscilloscope displays o basic

loads. This content should provide a strong oundation or the TDR neophyte, as

well as a good brush-up tutorial or the more experienced TDR user.


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Application Note 1304-2

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TDR Theory