hp-an1304-2_time domain reflectometry theory

8/14/2019 HP-AN1304-2_Time Domain Reflectometry Theory

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Time Doma in Reflectom etryTheory

Applica t ion Note 1304-2

For Use withHP 54750A andHP 83480A Mainframe s


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The most general apr oach to evalua ting th e time doma in response

of an y electr omagn etic system is t o solve Maxwells equa tions in t he

time domain. Such a procedure would tak e into account a ll the

effects of the system geometry and electrical properties, includingtransmission line effects. However, this would be rather involved

for even a simple conn ector an d even m ore complicated for a

stru cture such as a m ultilayer high speed backplane. For this

reason, various test and measurement methods have been u sed to

assist the electrical engineer in an alyzing signal int egrity.

The most common met hod for evaluating a t ra nsmission line an d

its load h as t ra ditiona lly involved applying a sine wa ve to a system

an d measu ring waves resulting from discontinuities on the line.

From these measur ements, the stan ding wave ratio () iscalculat ed an d used a s a figur e of merit for the t ran smission

system. When th e system includes several discontinuities, however,

the st an ding wave rat io (SWR) measu remen t fails to isolate th em.

In ad dition, when t he broadban d quality of a tr an smission system

is to be determined, SWR measurements m ust be made a t ma ny

frequencies. This method soon becomes very time consuming and

tedious.

Anoth er common inst rum ent for evaluat ing a tra nsm ission line is

the n etwork an alyzer. In th is case, a signal genera tor produces a

sinusoid whose frequen cy is swept to stimu late t he device un der

test (DUT). The network an alyzer mea sur es the reflected and

tra nsm itted signals from the DU T. The reflected waveform can be

displayed in various formats, including SWR and reflection

coefficient . An equivalen t TDR form at can be d isplayed only if thenetwork a na lyzer is equipped with t he pr oper softwar e to perform

an Inverse Fa st F ourier Transform (IFF T). This method works well

if th e user is comforta ble working with s-par amet ers in th e

frequen cy domain. H owever, if th e user is not familiar with t hese

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

Fu rth ermore, most digital designers pr efer working in th e time

domain with logic an alyzers a nd h igh speed oscilloscopes.

When compa red to oth er measu remen t techniques, time domain

reflectomet ry pr ovides a more in tu itive and d irect look at t he DU Ts

chara cteristics. Using a step gener ator a nd an oscilloscope, a fast

edge is laun ched into the t ra nsm ission line un der investigation.

The incident an d reflected volta ge waves are m onitored by th eoscilloscope at a particular point on the line.

Introduct ion


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This echo technique (see Figur e 1) reveals at a glan ce th e

chara cteristic impedance of the line, an d it sh ows both t he position

and the nature (resistive, inductive, or capacitive) of each

discontinuity a long th e line. TDR also demonstr at es wheth er lossesin a tra nsm ission system ar e series losses or shu nt losses. All of

th is informa tion is im media tely a vailable from t he oscilloscopes

display. TDR also gives more mea nin gful informa tion concern ing

the broadband response of a transmission system than any other

measuring technique.

Since the basic principles of time domain reflectometry are easily

grasped, even t hose with limited experience in high frequency

measu remen ts can quickly mast er th is techn ique. This applicat ion

note att empts a concise present at ion of th e fun damen ta ls of TDR

and th en relates these fundam entals to the para meters that can be

measu red in actua l test situa tions. Before discussing th ese

principles further we will briefly review transmission line theory.

Figure 1. Voltage vs t ime a t a part icular point on a mismatch ed

transmiss ion l ine driven w ith 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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The classical t ran smission line is assu med t o consist of a

cont inu ous st ru ctur e of Rs, Ls a nd Cs, as shown in F igure 2. By

stu dying this equivalent circuit, several chara cteristics of the

tra nsm ission line can be determ ined.

If the line is infinitely long and R, L, G, and C ar e defined per u nit

length, t hen

R + j LZin = Zo

G + jC

where Zo is the characteristic impedance of the line. A voltage

introduced at t he genera tor will require a finite time to tra vel down

the line t o a point x. The ph ase of the volta ge moving down th e line

will lag behind t he volta ge intr oduced at th e generat or by an

amount per un it length. Fur th ermore, the voltage will beattenu ated by an amount per un it length by the series resistan cean d shun t condu ctance of th e line. The phase shift an d att enua tion

are defined by the propagation const an t , where

= + j = (R + jL) (G + jC)

and = attenu ation in n epers per unit length = phase shift in r adians per u nit length

Figure 2. The class ica l model for a transmiss ion l ine.

The velocity at which t he volta ge tra vels down th e line can be

defined in t erms of:

Where = Unit Length per Second

The velocity of propa gat ion a ppr oaches th e speed of light, c, fortra nsm ission lines with air dielectric. For t he genera l case, where

er is th e dielectric const an t:

c =

er

Propagat ion on aTransmission Line

ZS

ZLE

S

L R L R

C G C G


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The propagation constant can be used to define the volta ge and t hecurren t a t an y distance x down a n infinitely long line by th e relations

Ex = Ein ex

and Ix = I in ex

Since the volta ge and the curr ent a re related at an y point by the

chara cteristic impedan ce of the line

Ein ex Ein

Zo = = = ZinIin e

x Iin

where Ein = incident volta ge

Iin = incident curr ent

When the tra nsmission line is finite in length a nd is term inated in a load

whose impedan ce ma tches the characteristic impedance of the line, thevoltage and current r elationships are sa tisfied by the preceding equations.

If the load is differen t from Zo, these equa tions are n ot sa tisfied

un less a second wa ve is considered to origina te a t t he load an d to

propagate ba ck up t he line toward t he source. This reflected wa ve

is energy tha t is n ot delivered to the load. Therefore, th e qua lity of

the tra nsm ission system is indicat ed by the r atio of this reflected

wave to the incident wa ve origina ting at th e source. This ra tio is

called the voltage reflection coefficient, , and is related to thetra nsm ission line impedan ce by the equat ion:

Er ZL Zo

= = Ei ZL + Zo

The ma gnitude of the stead y-sta te sinu soidal voltage a long a line

terminated in a load other than Zo var ies periodically as a function

of distance between a maximum and minimum value. This variation,

called a st an ding wave, is caused by the ph ase relat ionsh ip between

incident a nd r eflected waves. The ra tio of the ma ximum a nd

minimum values of th is voltage is called th e voltage sta nding wa ve

ratio, , an d is relat ed to th e reflection coefficient by t he equ at ion

1 + =

1 As has been said, either of the above coefficients can be measured

with present ly available test equipment. But th e value of th e SWR

measurement is limited. Again, if a system consists of a connector,

a short t ran smission line and a load, the m easured stan ding wave

rat io indicates only the overall quality of th e system. It does n ot tell

which of the system components is causing the reflection. It does

not t ell if the r eflection from one componen t is of such a pha se as t o

cancel the reflection from another. The engineer must make detailed

measurements at many frequencies before he can know what must

be done to improve th e broadband tra nsm ission qua lity of th e system.


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A time doma in reflectometer set up is sh own in Figure 3.

The st ep genera tor produces a p ositive-going incident wave t ha t is

applied to th e tra nsmission system u nder t est. The step tra velsdown the tra nsm ission line a t t he velocity of propagat ion of the

line. If the load impedance is equal t o the char acteristic impedan ce

of the line, no wave is r eflected a nd a ll tha t will be seen on th e

oscilloscope is the incident voltage step recorded as the wave passes

the point on the line monitored by the oscilloscope. Refer to Figure 4.

If a mismat ch exists a t th e load, par t of th e incident wave is

reflected. Th e reflected volta ge wave will ap pear on th e oscilloscope

display algebraically added to th e incident wave. Refer t o Figure 5.

Figure 3. Funct ional block diagram for a t ime domain ref lectome ter

Figure 4. Oscil loscope display wh en Er = 0

Figure 5. Oscil loscope display wh en Er 0

TDR Step ReflectionTesting

SamplerCircuit

Device Under Test

E i E r

ZL

StepGenerator

High Speed Oscilloscope

E i

E i

T

Er


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The r eflected wa ve is readily ident ified since it is separ at ed in t ime

from th e incident wave. This time is also valuable in determinin g

the length of the t ra nsm ission system from t he monitoring point t o

the misma tch. Letting D denote this length:

T D = =

2 2

where = velocity of propagation

T = tran sit time from monitoring point to the m ismatch a nd

back again, as measured on the oscilloscope (Figure 5).

The velocity of propagat ion can be determ ined from an experiment

on a known length of the same type of cable (e.g., the time r equired

for the in cident wa ve to travel down an d th e reflected wa ve to

tra vel back from an open circuit t ermina tion at t he end of a 120 cm

piece of RG-9A/U is 11.4 ns giving = 2.1 x 10 cm/sec. Knowing an d rea ding T from t he oscilloscope determ ines D. The m ismatch is

th en locat ed down th e line. Most TDRs calcula te t his dist an ce

au tomatically for th e user.

The sh ape of th e reflected wave is also valua ble since it reveals both

the n atu re an d magnitu de of the mismatch. Figure 6 shows four typical

oscilloscope displays a nd t he load imped an ce responsible for each.

Figures 7a a nd 7b show actual screen capt ur es from the H P 54750A

Digitizing Oscilloscope. These displays are easily interpreted by recalling:

Er ZL Zo = =

Ei ZL + Zo

Knowledge of Ei and Er , as m easu red on th e oscilloscope, allows ZLto be determined in ter ms of Zo, or vice versa . In F igure 6, for

example, we may verify tha t t he r eflections are a ctually from th e

term inations specified.

LocatingMismatches

AnalyzingReflections


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Figure 6. TDR displays for typical loads.

Assuming Zo is real (approximat ely tr ue for h igh quality comm ercial

cable), it is seen tha t r esistive misma tches reflect a voltage of th e

same sh ape as th e driving voltage, with th e magnitu de and polar ity

of E r determ ined by the r elative values of Zo and RL.

Also of int erest ar e th e reflections pr oduced by complex load im ped-

ances. Four basic examples of these reflections are shown in Figure 8.

These wa veforms could be verified by writing t he expr ession for (s) in t erm s of the s pecific ZL for each example:

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

1 + RCs

Eimultiplying (s) by th e tra nsform of a step fun ction of 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

E13 i

ZL 2Z o

(D) Line Terminated in Z = ZL1

2 o

12

Z o

E i1

3

E iZL

(A) E = Er i Therefore = +1

Which is true as Z

Z = Open Circuit

L

ZL Z o

Z L + Z o

(B) E = Er i Therefore = 1

Which is only true for finite Z

When Z = 0

Z = Short Circuit

ZL Z o

Z L + Z o

o

L

(C) E = + Er i Therefore = +

and Z = 2Z

ZL Z o

Z L + Z o

oL

13

1

3

(D) E = Er i Therefore =

and Z = Z

ZL Z o

Z L + Z o

oL

1

3

1

3

12


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and then transforming this product back into the time domain to

find an expression for er (t). This procedure is useful, but a simpler

an alysis is possible with out resorting t o Laplace tr an sform s. The

more direct an alysis involves evaluat ing th e reflected volta ge at t =0 and at t = and a ssuming any tra nsition between these twovalues to be exponent ial. (For simplicity, time is chosen t o be zero

when the reflected wa ve arr ives back at the monitoring point.) In

the case of the series R-L combination, for example, at t = 0 the

reflected volta ge is +E i. This is because the inductor will not accept

a su dden chan ge in cur rent ; it initially looks like an infinite

impedance, and = +1 at t = 0. Then curr ent in L builds upexponent ially an d its impedan ce drops toward zero. At t = ,ther efore er (t) is deter min ed only by the valu e of R.

R Zo( = When = )

R + Zo

The exponential tr an sition of er (t) has a time const an t determ ined

by the effective resistan ce seen by t he indu ctor. Since th e outpu t

impedance of th e tra nsm ission line is Zo, the indu ctor sees Zo in

series with R, an d

L=

R + Zo

Figure 7b. Screen capture of shortcircuit terminat ion from HP 54750A

Figure 7a. Screen capture of opencircuit terminat ion from HP 54750A


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Figure 8. Oscil loscope displays for complex ZL.

ZL

ZL

ZL

ZL

SeriesRL

Shunt

RC

ShuntRL

SeriesRC

E i

Ei

0

Where = L

R+Zo

(1+ )E iRZo

R+Zo

t

E i (1+ )+(1 )eRZ o

R+Z o

RZo

R+Zo

t/

(1+ )E iRZoR+Zo

Where = CZoR

Z o+R

E i (1+ ) (1e )RZ o

R+Zo

t/

0t

E i E i

( )E iRZoR+Zo

Where = LoR+Z

RZo

E i (1+ )eRZo

R+Zo

t/

0

t

E i

( )E iRZoR+Zo

0t

E i

Where = (R+Z ) Co

2EiE i (2(1 )eRZ oR+Zo

t/

A

B

C

D

R

C

RL

R C

R

L


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A similar a na lysis is possible for th e case of th e par allel R-C

term ination. At t ime zero, the load appear s as a short circuit since

the capacitor will not a ccept a sudden change in voltage. Therefore,

= 1 when t = 0. After some t ime, however, volta ge builds up on Can d its impedan ce rises. At t = , th e capa citor is effectively anopen circuit :

R ZoZL = R an d =

R + Zo

The resista nce seen by the capacitor is Zo in par allel with R, and

ther efore the t ime consta nt of the exponent ial tra nsition of er (t) is:

Zo R

C

Zo + R

The two rema ining cases can be tr eated in exactly the sam e way.

The results of th is ana lysis are su mma rized in F igur e 8.

So far, ment ion h as been ma de only about th e effect of a misma tched

load at the end of a transmission line. Often, however, one is not

only concerned with wh at is happen ing at t he load, but also at

interm ediate positions a long th e line. Consider t he tr an smission

system in Figure 9.

The junction of the two lines (both of characteristic impedance Zo)

employs a connector of some sort. Let us assu me t ha t t he conn ector

adds a sma ll indu ctor in series with th e line. Analyzing this

discontinuity on t he line is n ot much different from a na lyzing a

mismat ched termina tion. In effect, one treat s everything to the

right of M in th e figur e as a n equivalent impeda nce in series with

the sm all inductor an d th en calls th is series combinat ion th e

effective load impedance for t he system at th e point M. Since th e

input impedance to the r ight of M is Zo, an equivalent representa-

tion is shown in F igure 10. The pa tt ern on th e oscilloscope is mer ely

a special case of Figure 8A and is shown on Figure 11.

Figure 9.

Discont inui t ieson the Line

ZLZo

M

Zo

Assume Z = ZL o


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Figure 10.

Figure 11.

Time domain reflectomet ry is also useful for compar ing losses in

tra nsm ission lines. Cables wher e series losses predominat e reflect a

voltage wave with an exponent ially rising char acteristic, while

those in which shu nt losses predominat e reflect a voltage wave

with an exponent ially-decaying char acteristic. This can be

un derstood by looking a t t he inpu t impeda nce of the lossy line.

Assuming th at th e lossy line is infinitely long, the input impedance

is given by:

R + jLZin = Zo =

G + jC

Treat ing first t he case wher e series losses predominat e, G is so

small compa red to C th at it can be neglected:

R + jL L R1/2

Zin = = ( 1 + )

jC C jL

Recalling the approximation (1 + x)a (I + ax) for x


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In t erms of an equivalent circuit valid at t = 0+, the transmission

line with series losses is shown in Figure 12.

Figure 12. A s imple model v alid at t = 0+ for a l ine with series losse s

The response to a step of height E a ppears a s Figure 13, where Zssource impedan ce, and a ssum ed resistive.

Figure 13.

In th e case where Zs = R, = 2Zs C an d t he in itial slope of ein (t) isgiven by:

E Emi = = R

4R C 8L

The ser ies resist an ce of the lossy line (R) is a fun ction of th e skin

depth of th e condu ctor an d th erefore is n ot const an t with frequency.

As a resu lt, it is difficult t o relate th e initial slope with a n a ctu al

value of R. However, the ma gnitu de of th e slope is us eful incompa rin g condu ctors of differen t loss.

A similar an alysis is possible for a conductor where shu nt losses

predominate. H ere th e input adm ittan ce of the lossy cable is given

by:

1 G + jC G + jCYin = = =

Zin R + jL jL

e in

Zs

R'

Z

C'

Zin

in = R' + 1

jC'

E

e in (t)

m

t

= (R' + Z ) C'sZ +R'

ER'

s

i


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Since R is assum ed sma ll, re-writing this expression for Yin :

C G 1/2Yin = ( 1 + )

L jC

Again ap proxima ting the polynominal un der th e squar e root sign:

C GYin ( 1 + ) When G


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A qualitative interpr etat ion of why ein (t) behaves a s it does is quite

simple in both these cases. For series losses, the line looks more

an d more like an open circuit a s time goes on because t he volta ge

wave tra veling down t he line accumu lates more an d more seriesresistan ce to force cur rent th rough. In th e case of shu nt losses, the

input eventua lly looks like a short circuit because t he curr ent

tra veling down th e line sees more and more accum ulat ed shun t

cond ucta nce to develop volta ge across.

One of th e adva nt ages of TDR is its ability to han dle cases in volving

more th an one discontinu ity. An exam ple of this is Figure 16.

Figure 16.

The oscilloscopes display for th is situ at ion would be s imilar to th e

diagram in Figure 17 (dra wn for th e case wher e ZLZ'


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Two things m ust be considered before t he a ppar ent reflection from

ZL, as sh own on th e oscilloscope, is us ed to deter min e 2. First , thevoltage step incident on ZL is (1 + 1) E i, not merely E i. Second, the

reflection from th e load is

[ 2 (1 + 1) E i ] = ErL

but t his is not equa l to Er 2since a re-reflection occurs at the

mismat ched jun ction of th e two tran smission lines. The wave tha t

retu rn s to the monitoring point is

Er 2= (1 + 1) E rL = (1 + 1) [ 2 (1 + 1) E i ]

Since 1 = 1, E r 2 ma y be re-written as:

Er2

Er2

= [ 2

(1 1

2 ) ] Ei

The par t of Er Lreflected from t he jun ction of

Er LZo and Zo (i.e., 1 Er L)

is again r eflected off the load and heads back to the m onitoring

point only to be partially reflected at the junction of Zo and Zo.This cont inues indefinitely, but after some t ime th e ma gnitude of

the reflections approaches zero.

In conclusion, th is applicat ion n ote h as described the funda ment al

th eory behind t ime domain reflectomet ry. Also covered wer e some

more pra ctical a spects of TDR, such a s r eflection an alysis andoscilloscope displa ys of basic loads . Th is cont ent should provide a

strong foun dat ion for the TDR neophyte, as well as a good brush -up

tut orial for th e more experienced TDR u ser.


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hp-an1304-2_time domain reflectometry theory

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