Reference

NAT L INST. OF STAND & TECH

NBS

Publications

AlllDS T777S7

^;^A f ' *^ I^

o

ft NBS TECHNICAL NOTE 1042

U.S. DEPARTMENT OF COMMERCE /National Bureau of Standards

Shielded Balanced and Coaxial

Transmission Lines — Parametric

Measurements and Instrumentation

Relevant to Signal Waveform Transmission

in Digital Service

-QC

100

.05753

NO. 1042

1981
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

'physical and chemical measurement throughout the Nation's scientific community, industry,

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standards, and data on the properties of materials needed by industry, commerce, educational

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agencies; develops, produces, and distributes Standard Reference Materials; and provides

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provides engineering measurement traceability services; develops test methods and proposes

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research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

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'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. ^Some divisions within the center are located at Boulder, CO 80303.
Shielded Balanced and Coaxial Transmission Lines

Parametric Measurements and Instrumentation

Relevant to Signal Waveform Transmission in

Digital Service

MATIOMAL BUBEAU or 8TANDAEUS

UBBaAT

SEP 16 1981

W. L. Gans ' '^

N. S. Nahman -U^l^l

no /ot/s^

Electromagnetic Technology Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Final report prepared for: Department of the Army

U.S. Arniy Communications Command

Communications - Electronic Engineering and Installations Agency

Fort Huachuca, Arizona 85613

fO^S 4fr

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

Issued June 1981
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1042

Nat. Bur. Stand. (U.S.), Tech. Note 1042, 260 pages (June 1981)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTJNG OFFICE

WASHINGTON: 1981
TABLE OF CONTENTS

Page

1. INTRODUCTION .' 1 1.1 Background 1 1.2 Project Objective 2 1.3 Report Organization 2

2. DESCRIPTION OF THE TRANSMISSION LINE MODEL AND THE DEFINITION OF THE MODEL PARAMETERS. . . 4 2.1 The General Transmission Line Equations 4 2.2 The s Model and Its Parameters 6 2.3 s-Domain Circuit Properties of the Model 8 2.4 High Frequency Attenuation and Phase Shift 10

3. MEASUREMENT METHODS FOR THE DETERMINATION OF THE TRANSMISSION LINE PARAMETERS 19

3.1 The Need for Measurements 19 3.2 Measurement of the per Unit Length Resistance and Capacitance, R and C 20

3.3 Measurement of the Nominal Characteristic Impedance, R 20

3.4 Insertion Measurements for the Determination of the High Frequency Loss Parameter and the Exponent m 22

4. AN OVERVIEW OF THE COMPUTER CALCULATION FOR THE TRANSMISSION LINE IMPULSE AND STEP RESPONSES 3^

4.1 Data Acquisition Method 34 4.2 Fourier Transformation of e,. (nT) and e,„(nT) 34 dl ^ dz 4.3 The Calculation of S (k/2NT) and a (k/2NT) 35 4.4 The Extraction of the Model Parameters K and m 36 4.5 Simulation of the Impulse Response 38 4.6 Determination of the do Level for the Impulse Response, h(nT) 38 4.7 Simulation of the Step Response 39

5. TYPICAL RESULTS FOR TRANSMISSION CHARACTERIZATION AND RESPONSE 44 5.1 Description of the Cable 44 5.2 The Results 44

6. USE OF TIME DOMAIN MEASUREMENTS TO ESTIMATE MAXIMUM CABLE BIT RATE

AND BIT-ERROR-RATE DEGRADATION . . . 58 6.1 Digital Communications System Model 58 6.2 Maximum Bit Rate Estimation 58 6.3 Code Dependence 60 6.4 Bit-Error-Rate Degradation 61

7. CABLE TESTING USING TIME DOMAIN REFLECTOMETRY 81 7.1 The General Theory of Time Domain Reflection and Transmission 81 7.2 The Step Response of the Sending-End Voltage 87 7.3 A Practical Method for Measuring the Step Response of the Sending-End Voltage ... 92 REFERENCES 104
page APPENDIX A — MEASURED AND MODELED CABLE DATA 105

A.l Cable Parameter Tables ^05

A. 2 Cable Graphical Data 108 APPENDIX B — CABLE MEASUREMENT AND ANALYSIS COMPUTER PROGRAMS I85

B.l APMS System Calibration Programs and Subroutines 185

B.1.1 Y-Axis Calibration Subroutine .... 185

B.l. 2 X-Axis Calibration Subroutine 187

B.l. 3 Calibration Data Acquisition Subroutine I9O

B.l. 4 Linear Least Squares Curve Fit Subroutine I95

B.l. 5 General Matrix Multiplication Subroutine ig^

B.2 Cable Measurement Programs and Subroutines 2.97 B.2.1 Main Cable Time Domain Transfer Function Measurement Program I97

B.2. 2 Sweep-Sequential Waveform Acquisition Subroutine 203

B.2. 3 Point-Sequential Waveform Acquisition Subroutine 208

B.2. 4 Assembly Language Fast Fourier Transform Subroutine 213

B.2. 5 General Time Domain/Frequency Domain Plot Program 221 B.3 Cable Modeling and Analysis Programs 231

B.3.1 Main Cable Model Program for Calculations of "m", "K", and

Cable Impulse Response 231

B.3. 2 Main Cable Model Program for Calculation of Cable Step Response,

Bit Error Waveform and Cable Square Wave Response 239

B.3. 3 Main Cable Model Program for Calculation of Unit Step TDR Waveform 245 B.3.4 Main Program for Calculation of Cable Matching Network Resistors 247 B.3.5 Main Program for Calculation of the Error Function 249

B.3. 6 Function Subroutine for Calculation of the Gamma Function 250

B.3. 7 Subroutine for Accurate Calculations of Cable Time Domain

Impulse Response dc Level 251 APPENDIX C ~ TIME DOMAIN DATA ACQUISITION SYSTEM DESCRIPTION 254
SHIELDED BALANCED AND COAXIAL TRANSMISSION LINES - PARAMETRIC MEASUREMENTS AND INSTRUMENTATION RELEVANT TO SIGNAL WAVEFORM TRANSMISSION IN DIGITAL SERVICE

W. L. Cans and N. S. Nahman

A method is presented for determining the impulse and step responses of a shielded cable using time domain terminal measurements and a physically

based mathematical model for the transmission line properties of the cable.

The method requires a computer controlled time domain measurement system

and was implemented using the NBS Automatic Pulse Measurement System (APMS)

Data are also developed for the frequency domain complex propagation function

(attenuation and its related minimum-phase shift). The method is applied to

12 shielded paired-conductor (balanced) cables and 5 coaxial cables. Time

domain responses are presented for three nominal cable lengths, 60 m (200 ft),

150m (500 ft) , and 300m (1000 ft) . The time domain responses are applied to the estimation of bit error rate increases due to the insertion of the cables

into a digital signaling system employing a balanced polar NRZ waveform.

Also discussed is the application of the time domain responses to time domain

ref lectometry techniques for cable acceptance tests and field-site testing of installed cables.

Key words: bit error rate; coaxial transmission lines; digital communication

impulse response; instrumentation; measurements; shielded balanced transmission lines; time domain; transmission line model; waveform.

1. INTRODUCTION

1.1 Background

Nominal frequency response data supplied by cable manufacturers are not sufficient for the complete

evaluation or simulation of cables (transmission lines) in digital systems. Data which must be acquired

prior to engineering analysis for digital transmission are the driving-point and transfer impulse

responses. These are most easily obtained through pulse or time domain testing.

Furthermore, time domain measurement methods and performance data for the characterization of

balanced digital-service, shielded transmission lines [60m (200 ft) to 300 m (1000 ft) lengths] are required for the design, signal analysis, and circuit monitoring relevant to the digital-service transr mission line interconnections between the Technical Control Facility (TCF) and the Radio/Microwave

Exchange (MUX) as encountered in U.S. Army telecommunication installations. Also, time domain measurement methods are required for the identification and location of installation and/or operating faults

which can impair the digital waveform fidelity, which in turn can cause a degradation of the digital

signaling capacity. Such measurement methods and their interpretation fundamentally depend upon a knowledge of typical impulse responses. Because response data suitable for digital cable simulation was not available, the U.S. Army

Communications - Electronics Engineering Installation Agency (USACEEIA) sought assistance from the

National Bureau of Standards (NBS) to develop the required data for a specified group of commercially

available cables. The NBS Time Domain Metrology Group was uniquely qualified to render such assistance
by virtue of their experience in (1) the development and use of the NBS Automatic Pulse Measurement

System and (2) the characterization of transient responses for shielded cables. The subject report

summarizes the work performed by the NBS Time Domain Metrology Group for USACEEIA and is for the most

part a catalog of cable impulse responses and associated transfer functions.

1.2 Project Objective

The overall project objective contained three components; listed in order of their relative technical

priority, they were:

1. Determine and compile from measurements the parametric data necessary for engineering

design and computer simulation of typical shielded cables for digital service. Demonstrate

the use of the data to provide time domain and frequency domain simulation.

2. Simulate time domain reflectometry reference signatures suitable for cable field testing

and fault location.

3. Demonstrate a time domain data acquisition system capable of (a) acquiring the signal waveforms of a transmission line under test conditions and (b) provide measured cable parameters

suitable for subsequent batch processing in a central computer system for engineering design

and simulation studies.

Originally, it was proposed to accomplish each one of the above components individually in a specific

phase of the work during the contract duration. However, to allow for cable and component delivery delays,

and the natural interaction between the objective components, the work was not rigidly accomplished in

specific phases. The subject report presents the results of the work in accomplishing objectivecomponents (1), (2), and (3) above.

1.3 Report Organization

This report is divided into eight chapters and two appendices, the first chapter being the present

introduction. Chapter 2 presents the transmission line model, the definition of parameters, and a discussion of the model frequency dependence.

Chapter 3 describes the measurement methods used to obtain the frequency and time domain data

required for the calculation of the transmission line parameters.

Chapter 4 provides a qualitative overview of the computer calculations for (1) the extraction or determination of the transmission line parameters, (2) the impulse response, and (3) the step response. Chapter 5 presents an example of the typical results for the determination of the transmission line

parameters and the corresponding frequency and time domain responses for a given cable.

Chapter 6 describes a method of estimating the bit-error-rate degradation of a balanced polar NRZ waveform produced by a given length of cable.

Chapter 7 discusses the time domain reflectometry method for testing cables which is suitable for acceptance and field-site tests. Typical results are given for the worst case situations of open and

short-circuit discontinuities.
Appendix A is a catalog of complex phasor transfer functions, their associated time domain responses

(both reflection and transmission) for selected commercially available shielded cables of different lengths , usually 60 m (200 ft), 150 m (500 ft), and 300 m (1000 ft). The cables included are:

A. Shielded Paired-Conductor Transmission Lines: RG 22, U.S. Army WD-37, and ten commercially available cables designated as cables A through J.

B. Coaxial-conductor Transmission Lines: RG58C, RG59B, RG214, RG223, and a commercially

available triaxial cable designated as cable K,

Appendix B contains the computer programs which were used in the NBS APMS to accomplish the cable

measurements, characterization and simulation. Appendix C is a description of one design for a time

domain data acquisition system (NBS APMS)

.
2. DESCRIPTION OF THE TRANSMISSION LINE MODEL

AND THE DEFINITION OF THE MODEL PARAMETERS

In this chapter a brief discussion of the general transmission line equations is presented so as to provide a basis for the formulation of the transmission line simulation model. Following the introductory

discussion, the simulation model parameters are defined and the transmission line equations are expressed

in terms of the model parameters. Finally, various limiting cases of the model versus frequency are discussed to provide insight as to the correlation of the model with experiment.

2.1 The General Transmission Line Equations

A uniform transmission line can be characterized in terms of a series impedance per unit length,

Z(s), and a shunt admittance per unit length, Y(s), fig. 2.1. This follows from the uniformity property

which simply means that the transmission line geometric form and material composition does not change

with length, i.e., any segment of the transmission line possesses the same values of Z(3) and Y(s). The

specific functional forms of Z(s) and Y(s) depend upon geometric form and material parameters. In practice Z(s) and Y(s) are combined to form two other functions which naturally appear in the transmission

line equations and are called the characteristic impedance Z (s) and the propagation function y(s).

z„(s, ./IW

Y(s) , ohms

Y

(s) = -y Z(s)Y(s) , per unit length

(2-1)

(2-2)

For a transmission line of length f. connected between a generator E (s) of impedance Z (s) and a load impedance Z (s), fig. 2.2, the transfer and driving point equations are given by [1], respectively.

E^(s)

E (s)

Z (s)

o

Z (s)+Z (s) R

Z,(s)

Z,(s)+Z^(s)

I o

IlT(s)

1-0 (s) p (s) e

s r -2rf(s) (2-3)

E (s)

s I (s)

s

= Z (s)

1 + p (s) e -2£y(s)

o

" ' , / N -2?.y(s) 1 - p_(s) e

(2-4)

where the reflection coefficients p (s) p (s) are obtained from

s r

P^(s)

Z.(s) - Z^(s)

1

Z, (s) + Z (s) i o

(2-5)
These equations describe a transmission line network which is not terminated at either end in the trans- mission line characteristic impedance.

When a termination Z. (s) is equal to Z (s) , then the corresponding p.(s) is zero, and the trans- mission line is said to be matched at that end. There are three possible situations for matched terminations :

1. Doubly matched Z (s) = Z„(s) = Z (s)

g 2. O

\(s)

E^(s)

-2.Y(s)

p = p =

K r

(2-6)

E^(s)'

I (s)

s

= Z^(s)

'k

•= P^ =

(2-7)

2. Matched Load Z. (s) = Z (s)

V- o

E^(s)

= Zjs)

E (s) g Zg(s) + Z^(s)

Eg(s)

I^(s)_ = Z^(s)

'l

=

-JlY(s)

(2-8)

(2-9)

3. Matched Generator Z (s) = Z (s) g o

E (s) rE^(s)

Z,(s)

ZJs) + Z^(s)

p^=0

-^Y(s)

(2-10)

^^^^1

=

1 + p (s) e-2^^(-)

7 (-) IJs) "°

1 - p^(s) e-2^^(^)

p =o

(2-11)
Equations (2-3) through (2-11) reduce to the lossless or ideal cases when

Z(s) = sL (2-12)

and

Y(s) = sC (2-13)

which in turn yield a delay propagation function and a resistive characteristic impedance, respectively,

Y (s) = s-^Lc" (2-14)

and

Z^(s) = R = Y^ . (2-15)

o o ^ C

Under such conditions, there is no distortion with propagation, and also the transmission line can be

terminated with a resistor, R , for reflection-free transmission.

o For practical shielded cables, there are losses present, which in turn, means that both y(s) and

Z (s) will be complicated functions of s. The resultant propagation along the line will distort pulses

due to the nature of yC^)- Also, because Z (s) is s dependent, resistive terminations R , R will not

o g * match (equal) the characteristic impedance; under such conditions in equations (2-3) through (2-11), the

voltage divider terms and their combinations which form the reflection coefficients.

Z„(s) Z.(s) R. - Z (s)

o SL i o Rgo+ Z (s) ' R + Z (s) ' R^ + Z (s)

y, o i o

contribute to the overall distortion of the transmitted pulse as they are not independent of s.

2.2 The s Model and Its Parameters

The physically-based mathematical model used in this work was originally developed by Nahman in

1962 for the sub-nanosecond to nanosecond domain transient analysis of coaxial-conductor cables [2].

Subsequent investigation by Nahman. and Riad using the NBS APMS showed that the model could also be used

to simulate shielded paired-conductor cables of parallel or twisted conductor construction [3].

The model specifies Z(s) and Y(s) as

Z(s) = R + sL + KsTM , 0< m <1 (2-16)

and

Y(s) = sC (2-17)

where

Re dc resistance/unit length (2-18)

L s Inductance/unit length (2-19) K E High frequency loss coefficient/unit length (2-20) m = High frequency loss exponent (2-21)

C E Capacitance/unit length (2-22)

Consequently, the model equivalent circuit per unit length is as shown in fig. 2.3. The propagation

function and the characteristic impedance are then given by
y(s) =

I (R + sL + KsTM)sC r-'^ , < m < 1 (2-23)

and

= [(R + sL + KsTM)sC

Z (s) = [(R + sL + Ks°')/sc]-'''^ , < m < 1

o u J In the high frequency region, i.e., when s is such that

the model series impedance is closely approximated by

which in turn gives for y(s) and Z (s)

and

Because

then

.m . 1/2

^

(-ff) ' - 2sL

Defining

then the high frequency y(s) and Z (s) may be expressed as

HF

and

(2-24)

sL » KsTM » R (2-25)

Z(s)] = sL + KsTM (2-26) HF

Y(s)j = sVlc(i+H-) (2-27)

^o(^l=Vt(i^fM • (2-2«>

^

« 1 (2-29) sL

(2-30)

6(s) = fj^ (2-31)

y(.s)J = s Vlc" r 1 + «(s)l , h (s) 1« 1 (2-32) HF >- -I

Z^(s)J = V^ [l+ 6(s)J , l6(s)| « 1 . (2-33)

In the limit as |s| approaches infinity, (2-32) and (2-33) yield the time delay per unit length and the

high frequency or geometric or nominal characteristic impedance, respectively,

_

lim [ y(s) ] = s VlC (2-34)

ls|-> " HF

lim [ Z (s) ] = *. F

(2-35)

Isl -^ -° HF ^
which defines the time delay per unit length

T = V LC (2-36)

and the high frequency characteristic impedance.

R = -% ^, . (2-37)

2.3 s-Domain Circuit Properties of the Model

Consider the circuit of fig. 2.4 which is the circuit of fig. 2.2 with

Z (s) = Z (s) = R (2-38)

g X, O

Y(s) = [(R + sL + Ks")sc]

^^'^ (2-23)

Z (s) = [(R + sL + Ks") /sc]"""^^. (2-24)

Here, the transmission line is terminated at each end in the nominal or high frequency characteristic

impedance, R . This is a doubly mismatched situation for low frequencies, but this is the usual case encountered in practice. In all that follows in this report , the generator and load Impedances will

be equal to R . Consequently, equations (2-3) through (2-5) become

\\ + Zo(^>J L^o + ^o(^> J [ 1 - [p(s)]V2^^^^>J

E^(s)

fj^ = 2| „- ^ , ,„, I U ^: .„^ I I o _....,

(2-39)

^s^^) 1 +p(s)e-2^^<^>

-^-r^ = Z (s) ^ nn / N (2-40)

^s(^) ° l-p(s)e-2^^(^)

g

p^(s)=p^(s) =p(s) = ^ ^ Yl) (2-41)

o o

For the purposes of the following discussion, it is convenient to express equations (2-38) and (2-39) in terms of the hyperbolic functions

8.y(s) -aY(s)

cosh JIy(s) - ^ Y^ (2-42)

5,y(s) _ -)It(s) sinh )Iy(s) = ^

. (2-43)
Doing so, there results

2 Z (s) R

o o 2[R + Z^(s)] sinh )Iy(s) + 4 R Z (s) cosh JIy(s) o o o o

(2-44)

E (s) R cosh ly(s) + Z (s) sinh ly (s) -S = 7 (s) —

I (s) o^ '^ Z (s) cosh JIy(s) + R sinh JIy(s)

(2-45)

Consideration of fig. 2.4 for s = (dc) leads to the network shown in fig. 2.5 and the following

conclusions

1. The transfer function E (o)/E (o), is equal to R /(2 R + )IR)

.

r g o o

2. The impedance looking into the line, E (o)/l (o), is equal to ilR + R •

To demonstrate that these two conclusions are consistent with the mathematical model, consider (2-44) and

(2-45) when |s| -> o. First of all,

Z^(s)

s U-o 1 sC (2-46)

Y (s) -> V scR

I I '

S -K) (2-47)

and in the limit

,

lim Z (s) -^ o (2-48)

lim Y (s) -> o

Is I ->-o (2-49)

Putting (2-46) and (2-49) into (2-44) and (2-45) gives for |s|->o

E^(s)

E„(s)

2 R -v/sRC

o ' 2(R sC + R) sinh i \ scR + 4 R -J scR cosh l ^ scR(2-50)

_I^(s)

'sCR R Cosh a VscR + R sinh I "V/scR

r o

' ' 'scR cosh i \/scR + R sc sinh I J scR

o

V

s ->-o

(2-51)
When |s| = o, (2-50) and (2-51) both yield o/o and are indeterminate. After differentiating the numerators and the denominators, and again applying the limit, the results are

lim "E^(s)" R

s |->-o 2 R + I R

r^js)!

lim

s |->o

= R + JIR I^(s)

(2-52)

(2-53)

which agrees with the earlier conclusions.

Next, consider the high frequency behavior of the circuit. Putting (2-32) and (2-33) into (2-44)

and (2-45) yields

E^(s)

E^(s)_ HF

sTJ,[l + 6(s)] _^ [5(s)] , , . „.ri j. x/ m\ i ^ ^ '" + 2ri + 6(s)1 isT^[l + 6(s)]}

-sT£[l + 5(s)] (2-54)

and

^sT£[l + 5(s)]

_^ 6(s) sinh {sT£[l + 5(s)]}

Z (s)l = R [1 + 6(s)] „.„ , .. .^

' ^p

° ^sT£[l + 6(s)] ^ 6(s) cosh {sTm + 6(s)]}

R^[l + 6(s)] (2-55)

as l5(s)| << 1. Note that Z (s) is equal to Z (s)]HF

2.4 High Frequency Attenuation and Phase Shift

Equation (2-5 4) can be written as

E^(s) -sTl -sT£5(s)

e e (2-56)

where the first factor represents the time delay of the cable (linear phase shift in the frequency domain)

,

while the second contains the distortion due to high frequency losses. The inverse Laplace transform of

(2-56) would yield the initial or early time response of the circuit impulse response. Consequently,

(2-55) exhibits the time delay and then the initial impulse response.

In the frequency domain the transfer function is

E^(jw)

Eg(joj)

-£TK ,. .m

-ja3£T "21 ^^")

e

-^ e

10
or from (2-36) and (2-37),

TT / • \ £ K , . -m

T. ^. s = 6-' e o , o<in<l (2-57)

E (J W

The attenuation a, (w) and the phase 6(u) of the transfer function are defined by

ri^ = e-"!^"^ - J^(") (2-58) E (ju)

and are

a^(w) = Real Part of 2!^ (ju)" (2-60)

o

B(oo) = uilT + Imaginary Part of 2|^(ju))TM . (2-61)

o

In the phase S(w), the first term is the linear phase shift associated with the transmission line time

delay. In network terminology exp(-ja)JlT) is an all-pass function possessing a linear phase shift. The

second term in the phase is uniquely related to the attenuation because the two terms are the real and

imaginary parts of an analytic function along the frequency axis, jw, of the complex frequency plane

s = a + j(jo; thus the factor in (2-57)

i^K ,. .m

- 2r(J")

F^(ja)) =e ° 0<m<l (2-61)

is a minimum phase function. Hence, it is seen that the frequency dependent attenuation has associated

with it a frequency dependent (non-linear) phase function.

Defining

3(0)) = B (w) + B,(a)) (2-62)

o 1

where

B (u) = iMiT (2-63)

o

B-. (u) = Imaginary Part of :r-— (ju) (2-64)

o

the minimum phase complex exponent is then

A(ja)) = a^(w) + j B^(oo) (2-65)

it is equal to the exponent of (2-57) with the linear phase shift removed, i.e..

{,n

!E (ju) ) E^(:a))

j 1 1

a^(w) + j 33^(10) = II (jo))"* (2-66)

o

11
a^Xu) + j B^(u) =2^ <^^ (cos 2- + j Sin 2" ) (2-67)

o

.Hence

,

''1^"' " 2R

and

a. (w) = -;rr— oj COS nnr/2, nepers/unit length (2-68)

o

6^ (u) = -r^ 0) sin imr/2, radians/unit length (2-69)

o

which shows that both the attenuation and phase vary as uTM. Their ratio is

a, (u) •r—r-T- = cot imr/2. (2-70)

Figure 2.6 shows a log-log graph illustrating equations (2-68) through (2-70). Note that the linear slope

(i.e., the measured slope using a linear scale) is equal to the fractional power, m. Similarly, a graph

of the attenuation a^ (to) in dB (8.68 nepers equals one dB) or the minimum phase 6, (oj) in degrees (0.0175

radians equals one degree) would have a linear slope of m. For frequencies ranging from DC to very high frequencies, the log-log attenuation graph for the model

has two asymptotes, fig. 2.7. For very low frequencies the attenuation asymptotically approaches the

value set by the transfer ratio (2-52), i.e.,

2 R + X,R a(o) = 20 log ^ . (2-71)

o

On the other hand, for very high frequencies a(u) asymptotically approaches the fractional power response

(2-68); in dB, the asymptotic response is

a^(a;)l = 8.68 -^ \m\^ cos mTr/2, dB . (2-72)

J o dB

(Consequently, the model high frequency loss parameter K and the exponent m can be determined from experimental data in the asymptotic high frequency region, provided, of course, that the model is consistent

with the physical situation.

In practice, some care must be taken in fitting the model to the experimental data. Specifically,

the slope of log-log plot in the high frequency region may not be constant, fig. 2.8; this in turn requires

a judgement as to how to select a suitable constant slope curve which is tangent to the log-log attenuation

curve in the high frequency region, fig. 2.8. Because the NBS measurement system has a dynamic range of

better than 40 dB, the experimental data in the to 30 dB range will possess a good signal to noise ratio.

Consequently, the determination of m and K is accomplished by fitting in the 30 dB range. The effectiveness of the fit can only be evaluated by comparing the simulated output waveform based upon the modeled

impulse response with that of the observed waveform as will be demonstrated in Chapters A and 5.

Finally, one other comment should be made regarding the s model. The shunt admittance/unit length

Y(s) is specified as being a pure susceptance sC, (2-17); thus Y(s) contains no conductance G which would

represent one particular form of dielectric loss. However, in the computer program written for this work

Y(s) has the provision for the form

Y(s) = G + sC (2-73)

12
so that such a loss form can be easily Included if needed. In the cables considered here, G was negligible (as it was infinitesimal, or for all practical purposes, zero). This is not to say that the

s model excludes all dielectric loss effects when G = 0. To the contrary, the slope m and the

coefficient K can be effected by high frequency losses due to small dielectric losses in combination

with conductor losses.

13
Z(s)

Y(s)

O

Unit lengthFigure 2.1 The transmission line equivalent circuit per unit length.

z^ I^(s).

E^(s)

t

Eg(s) Y(s), Z^(s)

(

E (s)

r

I

Z,(s)

K

Figure 2.2 The basic transmission circuit consisting of a generator,

a transmission line, and a load.

14
sL

Ks

m

sC

K Unit length'

Figute 2.3 The equivalent circuit per unit length for the physicallybased mathematical model.

R I (s)

E (s)(A/

t

E^(s) y(s), Z (s)

I

t

E (s) r

i

R

Figure 2.4 The transmission line circuit terminated at each end in

R , the nominal characteristic impedance. This is a doubly

mismatched transmission line because Z (s) is not equal to

R for all s. o

15
1^(0)

Eg(o) [rj

-VnA

f

E^(0)

i

AAA

E (0) r

I

Figure 2.5 The transmission line circuit when s = (dc)

COu

exd

o

i

* 3-, (w ) = a^(a)^)tan m7T/2

f^-^"^^

k"^

^^'^

^^^^^^"^

^^^^

-1

y

>—a^(oo^) = B^(oo^)cotm7T/2

^^^-"^""^ 1 m=^

X

o

00CD

H

3

CO

log CO

Figure 2.6 Log-log graph of the minimum phase attenuation a^(w) and

phase 3 (w) . The slope y/x as measured with a linear

scale is equal to m.

16
i

i—

TJC

cd iH

nJ

CJ •Hu

V4

0) >

• >sO

ti iH (U 3 d

l4-(

3

bO

^"^ 00

• d

Cfi •H > CJCO

PI CX o CO •H

4J 0)

nJ T3 3 cd

fl o

QJ (U 3

4-1 TJ 3 00 rt rH o CO rH 00 a

o •ri

1

4J

00 (U o ta

rH •H

d) QJ ^ >

+J CO

>-l o CO

M-l Q) H

CO CO

QJ O

4-1 CO O

4-1 H

ft cd &

4J

CO o

a

•ri d) U ^ O H X!

gp '(m)ro§ox +

c PiCM

Pi

00oo

CM

(Uu

3

00 •H

fX4

17
I

30 dB

133

00o

log CO

Figure 2.8 Two constant slope curves tangent to a log-log attenuation

vs. (angular) frequency curve.

18
3. MEASUREMENT METHODS FOR THE DETERMINATION

OF THE TRANSMISSION LINE PARAMETERS

In this chapter the measurement methods required to determine the transmission line model parameters

are presented. A brief discussion is presented on the need for measurements and some relations between

the basic parameters. Then specific measurement methods are described for the determination of R, C, R , m and K. L is determined from C and R . o o

3.1 The Need for Measurements

A commercially available or military cable may be characterized in terms of a set of nominal

frequency domain transmission line parameters. Typically, they are

R - resistance/unit length

C - capacitance/unit length

R - high frequency or nominal characteristic impedance (also called the

o geometric characteristic impedance)

.

a (f)- attenuation frequency, f

.

These parameters are representative of the particular cable type and have been derived from measurements

on many different runs of cable. The values can be significantly different from those of a single cable

sample and also can vary from manufacturer to manufacturer. Consequently, for precise estimates of the

frequency domain performance of a given length of cable, the parameter values should be experimentally

determined, i.e., measured. For system design applications the nominal parameter values are used with

the hope that they are representative design center values. The main point being stressed here is that

the manufacturer's technical specifications are always representative values; individual samples may

vary.For time domain applications, knowledge of the parameters R, C, R and a (f) are also required along

with the phase shift, 3 (f ) , which is uniquely related to a (f ) . 3^ (f ) is the minimum phase shift

component of the total cable phase shift, B(f), (2-62) written explicitly in terms of frequency, f, i.e.,

f = a)/2TT.

g(f) = &(f) + 3,(f) (3-1)

o 1

= 2TTfT£ + B-,^(f) (3-2)

where ^^^(f) is the phase shift due to the inherent transmission delay/unit length, T. The delay/unit length is related to the velocity of propagation, v, and therefor, related to the

inductance/unit length, L, and the capacitance/unit length, C.

V = -rr = —=::^ , unit length/sec. (3-3)

Also, the nominal or high frequency characteristic impedance, R , is dependent upon L and C,

h- 4T- ohms . (3-4)

19
Consequently, if C is specified, then a knowledge of v or R determines L. Alternately, v and R may

be specified which in turn determine L and C,

R

L = ^ (3-5)

and

^ = i- • (3-6)

o

The relations (3-3) through (3-5) are useful in both the time domain and frequency domain modeling;

furthermore, some manufacturers specify v and R rather than C and R .

3.2 Measurement of the per Unit Length Resistance

and Capacitance, R and C

Normally, R and C are specified by the manufacturer. If for some reason they must be determined

experimentally, bridge measurements may be used. R is determined by measuring the loop resistance,

IR using a D.C. resistance bridge, figs. 3.1 and 3.2. Similarly, C is determined using a 1 kHz

capacitance bridge. With a bridge having one terminal grounded, the measurement is straight-forward

for a coaxial conductor unbalanced cable, fig. 3.3. However, for a shielded paired-conductor cable, in

general, three capacitance measurements have to be made to determine the total cable capacitance.

''AB'

^^AB = ' ]

C C S ^ c^T I (3-7)

The terminal designations are defined in fig. 3.4 while the equivalent circuit capacitances are defined

in fig. 3.5. The measurement technique follows the procedure in [4] and consists of the following

capacitance measurements: (1) C, A to ground with B grounded, (2) C", A to ground with A and B shorted

together, (3) C'', B to ground with A grounded. The values of IC^, JIC„, JIC. are then given by

iC

iC^

IC^

C - C" +c"' (3-8)

C' + C" - C" (3-9)

-C' + C" + C" (3-10)

3.3 Measurement of the Nominal Characteristic

Impedance, R

The nominal characteristic impedance R is actually the limit of the high frequency impedance

(2-33, 2-37) and is conceptually viewed as the characteristic impedance which depends solely upon the

inductance and capacitance per unit length. Furthermore, R is resistive, i.e., independent of s.

20
For large values |s|, the characteristic impedance is

Z^(s) = y| [1+ 6(s)] , l6(s)| « I (2-33) HF

= R^ [ 1 + 6(s)]. (3-11)

Now consider for the moment, that a step of current I /s is applied to the impedance (3-11). The

voltage across Z (s) at t = would jump up to I R volts and then commence to vary with time as dictated by the inverse Laplace transform of I R 6(s)/s, fig. 3.6. By applying a step of voltage from

a resistive source R to the input of a long length of cable, it is possible to calculate the value

of R from the initial response of the sending end voltage e (t), fig. 3.7. First of all, when a signal is abruptly applied to a transmission line, the transmission line

appears as an impedance equal to the characteristic impedance Z (s) . This is why the term surge impedance is also used in place of characteristic impedance. The impedance continues to appear as Z (s) until reflections from the receiving end arrive back at the sending end. If the line is terminated in

Z (s) then there will be no reflections, and the line will continue to appear as an impedance equal to

Z (s) . More will be said about such phenomena in the chapter on time domain ref lectometry (Chapter 7).

Furthermore, for the initial response, only the high frequency components of Z (s) contribute to

o the response. Consequently, for some time interval, say t, , the high frequency approximation to Z (s) (3-11) would be valid for predicting the initial response of the transmission line, as shown by the

solid-line portion of e (t) between and t, in fig. 3.7. Also, in fig. 3.7 t^ denotes qualitatively

the time required for reflections to return to the sending end from the open circuited receiving end. The value of t„ is set by I and must be large enough so that reflections do not interfere with the

observation of the abrupt rise in the neighborhood of t=0 . Consequently, the initial response of the cable to a voltage step of magnitude E is given by

E (s) = -^ — -^ "

T (3-12)

s \

"

^o(^)JhF

E

s

R + R 6(s)

o R + R + R 6(s)

g o ^

'

or approximated by

E

E (s) * -^

s s g o

(3-13)

(3-14)

because R » R |6(s)| and R„ + R is larger than R . In the time domain, the initial response of e (t) O O Oo K O Q consists of an abrupt jump E,

E R _S—2_ (3-15) R + R

g o

21
and then a slower upward trailing, fig. 3.7. From (3-15) R^ is obtained as

R = —^ R^ . (3-16)

o E - E g

If Rgohappens to equal R , then E would equal E /2 . An example of an initial response expressible

8 mathematically in closed form is given for m = 0.5 in [5].

The measurement of R can be implemented by the methods shown in figs. 3.8 and 3.9 for the coaxialo conductor cable and the shielded paired-conductor cable, respectively. For the coaxial-conductor cable

measurement, the effective generator impedance is 50 fl. For the paired-conductor case the generator impedance is 100 Q and the sampling oscilloscope

display is set for channel A minus channel B which yields a display voltage

e^(t) = e^(t) - eg(t) = e^(t). (3-17)

Also, remember that the ground connection must be connected to the signal -ground, i.e., the zero potential for the balanced-transmission line (refer to fig. 3.5). For both cases, the cable length i should

be long enough so that reflections do not occur in the region of the initial response, fig. 3.7. Closely related to the measurement of R is the use of time domain reflectometry (TDR) to evaluate

or test cables. The reader is referred to Chapter 7 of this report and in particular. Section 7.3 on

a practical measurement method for observing the sending-end voltage.

3.4 Insertion Measurements for the Determination

of the High Frequency Loss Parameter and the Exponent m

The loss parameter K and the exponent m are determined by model-fitting to data obtained from a time

domain (pulse) insertion measurement, the basic method being shown in fig. 3.10. e,^ (t) and e,„(t) are

the observable quantities displayed on an oscilloscope before and after the insertion of the cable of length I between a generator and matched load. The generator and load impedances are both equal to the

nominal transmission line characteristic impedance. lienoting the inverse Laplace transform as £ { }, Ej-, (t) and e 2(t) are given by

1 ( E (s) i e^^(t) = x'^JT(s) -^J (3-18)

(3-19)

where T(s) is the oscilloscope transfer function (input to display), and T(t) Its inverse transform or impulse response. Also,

R - Z (s) ^(^) = R° ^ Z°(s) • ''-'''

o o

22
Aside from the factor T(s) (3-19) is taken from (2-39), while (3-20) is from (2-41). The ratio,

E^2(s)/E^^(s), is equal to E2(s)/Ej^(s)

,

E2(s)

E^liT

= 4

(s)

-X.y(s)

r / m2 -25.Y L 1 - [p(s)] e (s)J

(3-21)

because T(s) cancels out along with the pulse dependence, E (s) . The ratio E„(s)/E^(s) is equal to the

scattering network parameter S„^(s).

Briefly, the transmission line can be represented by the scattering parameters S^^(s), S„„(s), and

S^„(s) = S„^(s). Figure 3.11 shows a transmission line inserted into another uniform transmission

system of characteristic impedance Z (s). In general, the incident and reflected voltages on the Z (s) transmission line system are

and

V^3^(s) = V^^(s) S^^(s) + V^2(s) S^2(s)

Vj.2(s) = V^^(s) S2^(s) + V^2(s) S22(s)

(3-22)

(3-23)

The terminal voltages of the inserted transmission line are

and

V^is) = V^^(s) + V^^(s)

V2(s) = V^2(^> + Vr2^^^

(3-24)

(3-25)

When the uniform transmission line is terminated in Z (s), fig. 3.12, V. „(s) is zero, and the voltages

become

V^^(s) = V^^(s) S^^(s)

V^2(s) = V^i(s) S23^(s)

V3^(s) = V^^(s) + V^^(s)

V2(s) = V^2(s)

(3-26)

(3-27)

(3-28)

(3-29)

The voltage V„(s) corresponds to E»(s) in (3-21). If the inserted transmission line in fig. 3.12 is removed from the circuit and terminals 1 and 2 connected together, then V (s) is zero, and

V^is) = V^^(s) (3-30)

which corresponds to E (s) in (3-21). Then from (3-27), (3-29), and (3-30), there results

E2(s)

E^ S2i(s) (3-31)

23
Returning now to consideration of (3-21), in terms of the model for high frequencies y(s) and

Z (s) become

o

and

y(s)1HF

s T[l + 6(s)] |6(s)|« 1

^o^^^] = \^^ + *(^)] l«(s)|« 1 HF

(2-32)

(2-33)

which reduces (3-21) to

£2(3)

E^(s)HF

-sTJ. - TJl6(s)

or in terms of the Fourier transform

(3-32)

E2(ja))

E-l(Ju)

-juTJl - TJ,6(ja3)

HF

Thus, the model attenuation at high frequencies

a^(oj)

J = 20 log

dB

E^(joj)

E^Tj^

HF

(3-33)

8.68 - [oi

I cos miT/2, /R

O (3-34)

can be compared with the Fourier transform attenuation data computed from the experimentally observed

waveforms, e (t) and e-(t), in order to determine K and m. As discussed in section 2,3 and illustrate

in figs. 2.6 - 2.8, this is accomplished in a tractable way using a log-log graph, log a. (u) vs. log

In practice, the insertion measurements are carried out using a 50 ohm sampling oscilloscope

system which has two channels. Because R may not be equal to 50 ohms or 100 ohms (for coaxial-conductor

or paired-conductor cable, respectively) resistive impedance matching networks may be required,

figs. 3.13 and 3.14. The matching networks are in an "L" configuration, each different value of R

requiring its own matching network. The networks used in this study are summarized in fig. 3.15.

24
dc

resistance

bridge

r

inner conductor

e

L

>

outer conductor

short

circuit

Figure 3.1 Measurement of the total coaxial-conductor cable resistance, £R.

Paired-conductors

short

circuit

Figure 3.2 Measurement of the total shielded paired-conductor cable

resistance, £R.

25
1 kHz

capacitance

bridge

r

inner conductor

Q

T

outer conductor

open

circuit

Figure 3.3 Measurement of the total coaxial-conductor cable capacitance, £C.

Paired conductors

L k:

Shield

Ground \ Ground

Figure 3.4 Terminal designations for measuring the total shielded pairedconductor cable capacitance, £C.

26
£C,

.£C,

X

1

J

iC,

Shield , ground

Individual coaxial

shields on each pairedconductor; ground terminal is the zero potential for the balanced

transmission line

A £C =0 B

1 1

T \ T

iC,

Outer shield

Figure 3.5 Shielded paired-conductor configurations and their equivalent capacitance circuits.

27
lg(t) i

I R

g o

(t) 1

t=0

.[l^(t )]

8,s

I

s — 1 t ^o^^^^HF

1

rr ^ rt- ^ 1 I R

- § "1-i-f^rcM

t=0

Figure 3.6 The response of Z (s)],_ to a current step of magnitude I . O HF ir o g

E

t=0

e (t) s I i

t=0

r

<

I

Figure 3.7 The initial response (solid line) of the sending end voltage

e (t) due to a vc

s source impedance.

e (t) due to a voltage step from a generator having a resistive

28
50 n 1

e (t) = e (t) a s

50 fi, coaxial

feed-through

sampling head

T:

a

%<''

i

3-

Figure 3.8 R measurement set-up for a coaxial-conductor cable

(unbalanced lines, generally).

g e^(t) = e^(t) - e^(t)

50 n Two channel

50 Q. coaxial

feed-through

sampling head

channels A

and B.

50 fi

Figure 3.9 R measurement set-up for a shielded paired-conductor cable

(balanced lines, generally).

29
%'

R

.. <±)

T

^i(^) < %

\

e^(t)

K

Figure 3.10 The time domain insertion method; e.^Ct) and ej„(t) are the di dZ

observables corresponding to e (t) and e (t) , respectively, before

and after the insertion of the cable of length £. T(t) is the

impulse response of the oscilloscope.

30
V.,(s)

V,(^)

Inserted Transmission

S-l-l(s) 5^2^^)

S^-^Cs) S22(s)

Line

1

V.,(s)

VjCs)

Figure 3.11 The scattering parameter representation of a transmission line

inserted into a uniform transmission line system of characteristic

impedance Z (s)

.

V,,(s) V^^(s) =

V^l(s) 1

Inserted

Transmission

Line

2 V^2(=) ».

1

V^(s)

I

V2(s) Z„(s)

1 1 1 2

Figure 3.12 Uniform transmission line system terminated in its characteristic

impedance Z (s)

.

31
50 Q. sampling

oscilloscope

1

^di('>

_l

Insertion plane

Figure 3.13 Practical implementation of the time domain insertion method for

a coaxial-conductor cable.

e (t)/jL

-e (t){"\r

50 $7 ^

50 ^'

50 Q.

50 Q

100 fi to

R

i>

R

2

R

R

R

oto

100 n

A

-O50 $>

50 Q^

2-channel

50 i^ sampling

oscilloscope

A-B

Display

a,,(t)]

A-B

Insertion plane

Figure 3.14 Practical implementation of the time domain insertion method for

a shielded-paired-conductor cable.

32
R,

->\A/^

50 fi

50 n

AA/^—

o

R

R

2 R

50 ^, R

(b)

R

o h ^2 h

75 25 75

78 23.45 83.1

95 11.18 212.25

98 7.07 692.00

124

1

113.6 27.27

Figure 3.15 The resistive impedance matching networks, 50 ohms to R . o

(a) Shielded paired-conductor cables,

(b) Coaxial-conductor cables.

33
A. AN OVERVIEW OF THE COMPUTER CALCULATION FOR THE TRANSMISSION

LINE IMPULSE AND STEP RESPONSES

In this chapter a qualitative overview is presented which describes the computer calculations for

the extraction of model parameters from measured data and the simulation of the impulse and step responses,

The calculations are accomplished by the NBS Automatic Pulse Measurement System (APMS) [6,7].

4.1 Data Acquisition Method

The data is generated by repetitive pulse excitation and is displayed by a sampling oscilloscope

interfaced to and controlled by a minicomputer system, i.e., the NBS Automatic Pulse Measurement System

(APMS). For a given display waveform, say e,„(t), the data is acquired by making 500 observations at

d-i a fixed time position t of 500 repetitions of the signal; the mean value [8,9] of the observations

e, ~(t^) is calculated by the computer,

500 -, E [^d2(h>J

^d2(^l) = 500 • , . (^-1)

The time window, T , is represented by 1024 uniformly spaced points; consequently, a set of 1024 mean

values represent the waveform, fig. 4.1,

1023 1

6^2 (""^^ " 23 1 «(t-nT) e^^inl) \ (4-2)

n=0 I

where 6(t) is the unit delta or impulse function multiplied by the strength e,„(nT) and where

T = T^/1024. (4-3)

Refer to figs. 4.5 and 4.6 for a graphical explanation of the delta function train (4.2). The waveforms e (t) and e .(t) of the time domain insertion method, fig. 3.10, are both acquired by the

procedure just described and will be denoted by e ,. (nT) and e,„(nT), corresponding to the form of

dl d/

(4.2).

4.2 Fourier Transformation of e ,, (nT) and e.-(nT)

dl aZ

The discrete acquired waveforms e (nT) and e (nt) are each transformed to the frequency domain

using the Fast Fourier Transform (FFT) algorithm for the discrete Fourier transformation [10]. To do

so, for each waveform a 2048 point time window is used which includes 1024 points of zero value, fig. 4.2. The extension of the waveform to 2048 points increases the frequency resolution by decreasing

the spacing between the discrete frequencies to one half of that available from the unaugmented waveform, fig. 4.3 and 4.4. The frequency domain spacing Af is equal to 1/T and the frequency window,

F remains unchanged when both N and T,, are doubled, but the number of points in F„ are doubled.

\, ^ h- The result of the FFT operation yields the discrete functions E^^iir—-) and Ej„(Trr—) which

dl zNl dz zNT represent complex values at the discrete frequencies k/2NT, e.g., the discrete waveform, e,„ (nT)

dz

of fig. 4.2 transforms to ^1 -j2TTnk/2N

^d2^2NT^ = S e^2^''^^^ ' k = 0,1,... 2N-1 (4-4)

n=0

34
where N equals 1024; consequently, Ej-(k/2NT) consists of 2048 complex values, one value for each of

the 2048 discrete frequencies, k/2NT. Furthermore, Ej„(k/2NT) is a symmetrical periodic function and

as such contains only 1024 distinct values.

4.3 The Calculation of S2,(k/2NT) and a (k/2NT)

The discrete insertion ratio S„^(k,2NT) is computed in the same manner as its continuous counterE,,(ju)) ^21 (^"> = 1^ ' (3-31)

part.

hence

E (k/2NT)

S (k/2NT) = T^^ (4-5) E (k/2NT)

. . . = |S2^(k/2NT)| /B^(k/2NT) (4-6)

where k=0,l, . . .2N-1, and | S. | and 3 are the magnitude and phase of the discrete function S. at the

discrete frequencies k/2NT, respectively.

However, an averaging process is used in arriving at S„. (k/2NT) . The process is repeated using

six different acquisitions for E,„(k/2NT) and a single acquisition for E,. (k/2NT) . Then, the resultant

S„ (k/2NT) data sets are expressed in terms of the discrete attenuation a^ (k/2NT) and phase B,(k/2NT),

Attenuation, a (k/2NT) e 20 log | S2^(k/2NT) I"""", dB

_ ( Imag. Pt. S (k/2NT) 1 Phase, & (k/2NT) = -Tan / ;; ? , degrees

^

{ Real Pt. S (k/2NT) }

and the mean values are determined by

Zl 20 lo g |S2^(k/2NT)|^^

a^ (k/2NT) = -^ g-^± '—, dB ^ (4-7)

J2

[§j^(k/2NT)]^

r 3^(k/2NT) = —

, degrees (4-8)

where

.3^ [E^^(k/2NT)]

[S (k/2NT] ] = -::^ (4-9)

[E^2(k/2NT)]^

|S2j^(k/2NT)rJ /[e^(k/2NT)]^ . (4-10)

35
and [E,„(k/2NT)] corresponds to the r-th acquisition from e „(t), i.e., [e,„(nT)] . The discrete values a (k/2NT) and g^ (k/2NT) , (4-7) and (4-8), are computed for the first three

hundred harmonics, because in most cases 300 harmonics cover a magnitude range of about to -30 dB. The capability for any other number of harmonics is retained in the computer program. At each value

of frequency, the standard deviation [8] for (4-7) and (4-8) are computed as a measure of the consistency

of the data. In particular, the standard deviation of the attenuation, a , is given by

6

a^(k/2NT) =r

?.[^(^^/2NT) -/a^Wim ^Y 1^^\

(,_,,)

Past experience has shown that consistent data yields a values in a typical range of 0.02 to 0.2 dB for an attenuation range of one to 30 dB.

4.4 The Extraction of the Model Parameters K and m

The parameters K and m are obtained by fitting the model at high frequencies to the (mean) attenuation data [a^(k/2NT)] as was pointed out in the discussion of (3-34) and figs. 2.6 - 2.8. The values

of K and m of the equation

, , I 8.68 I cos miT/2 \ „| im ,. ^,

.

a^^Ci^) =1 2"^^ —

I K|coI (3-34)

can be determined from the straight-line parameters of the equation resulting from the log of (3-34),

1 ~ / \ if 8.68 a cos mTr/2 i ,,,,,, i i ,, ,os log a^(co) = logl 2"^^ '—I + log K + m log|(D.| ; (4-12)

that is,

log 5^(k/2NT) = log / 8^6|e_cos_miT/2\ + ^^g j. + ^ log|k/2NT|. (4-13)

The parameters log K and m can be determined from the straight-line parameters of the logarithm of the mean attenuation (4-7), log [a (k/2NT)].

The slope m is determined by calculating the log-log graph slope from two values of the graph

rated by 50 abscissa points, e.g., k K is calculated using m, in (4-13) fromseparated by 50 abscissa points, e.g., k=50, and k=100. The slope so determined is designated m^ . Next

og a^(k/2NT) = iog/8.68 £ cos m^n/2 x + ^^g ^ + m^ log |k/2NT| (4-14)

o by an iterative solution method; the result is designated K^

.

The process for calculating m and K is repeated using the next two points k=51 and k=101; the

results are designated m^ and K2 . The process is repeated 150 times ending with k=200 and k=250. The mean values of m and K are then determined as

r r

150

E m /150 (4-15)

36
and

150

£ K^/150

K = ^~^

(4-16)

The calculation process is started at k=50, rather than k=0 to avoid the small errors which appear at the beginning of the frequency window due to system errors, noise, etc. Finally, the entire process is repeated three times; starting from e., (t) and e,„(t) the complete Qi al

operations of sections 4.1, 4.2, and 4.3 are repeated to yield three independent functions

[a^(k/2NT)]j^, [a^(k/2NT)]2, and [a^(k/2NT)] from which three sets of mean values for m and K are determined

Laj^(k/2NT) J^ . [m] ^ , [^].

fa^ (k/2NT)l, ^2 . [K]2

p^ (k/2NT)1 ,. [S]^ , [K],

Then, the mean values for [m] and [K] are calculated, rwhere

rm]^ + [S]„ + [m] [m] =

i H (^-17)

= [K]^ + [K] + m [K] = (4-18)

and the resultant values are the ones used for the model parameters m and K. The standard deviations,

and a are also calculated for control purposes to insure that the data are consistent. They are

3 J [en - m^i^-"^^^

a^ = I E -. '-^ I (A-19) r

3

and

r 3 J [K] - ml ^-| ^/^

37
4.5 Simulation of the Impulse Response

The impulse response of a cable of length i doubly terminated in its nominal characteristic impedance

R is obtained from (2-39) with E (s) = 1,

o g

H(s) = 2 -2——^

--^V-TTT 5-TT^T7rT (^-21) r V!i__ 1 r '^o '

I r__^:!!i!L___l U^ + Z,(s)

J

I^R, + Z,(s)J

1^, _ ^p(3)^2 ^-2^y(s)J

where

Z = [(R + sL + KsTM)/sC]^^^

o

y(s) = [(R + sL + Ks'°)sc]-'-^^

R - Z (s)

"^*- P(^) = K ^ Z°(s) • <2-^l)

o o

The received voltage (A-21) due to a unit impulse for e (t), E (s) = 1, is equal to one half of

E (s)/E (s), (3-21); consequently, from (3-31) the impulse response is

H(s) = JS2j^(s) . (4-22)

Using the model parameters (R, L, K, m, and C) which are now .known, a discrete 1024 point representation

of the insertion function -rS^, (s) is constructed. Next, the inverse discrete Fourier transformation of

1

' y S„^ (k/NT) will yield the discrete time-domain impulse response.

N-1

^(^T) = I Y. I

S2^(k/NT)eJ2^''''^^; n = 0,1, . . . N-1. (4-23)

A typical display for h(nT) would be similar to e „ (nT) , fig. 4.1.

4.6 Determination of the dc Level

for the Impulse Response, h(nT)

The dc level of the impulse response must be established before the step response can be

calculated. Theoretically, the point n = should yield h(o) = 0; if such were the case, then by setting

the level of h(nT) equal to zero at n = 0, the dc level would be properly located. However, due to the system errors in the region of o £ n < 10, the zero level can not be determined. Similar effects are encountered at the end of the time window near n = 1024. Physically, the impulse response of the cable at the end of the time window is not equal to zero;

-4 -3

it will be a small value, 10 , (typically 10 of the peak value of the impulse response). Consequently,

to set the dc level, the voltage at a specified time, say n- , is calculated using the model and the identified model paran^ters. To do so, the Fourier components of the transfer function (impulse response)

are summed for a specified time, typically, at 90% of the time window, 0.9 T . At the present time on w

the APMS this summation of the Fourier series requires considerable computation time, e.g.,

38
several hours, because a large number of harmonics must be summed without recourse to array processing

computation. Typically, for a 1 KHz separation of harmonics 5.12 X 10 harmonics are required for a one microsecond time window containing 1024 data points (harmonics from 1 kHz to 512 MHz). The typical

-3

convergence at 0.9 T is such that the voltage converges to about 10 V with a peak oscillatory

variation in the range of 10 V to 10 V . max max

Once the voltage for n. has been computed, then the dc level of the Impulse response waveform,

h(nT), is shifted so that the value of h(n^T) is equal to the computed voltage at n.

.

4.7 Simulation of the Step Response

The impulse response (4-23) may be expressed in the form

1023

h(nT) = ^ h(nT)6(t-nT) (4-24) n=o

where

T = T /1024 (4-25) w

and each term for the product h(nT)6(t-nT) is given by a corresponding sum over k in (4-23).

The discrete step response is obtained by integrating (4-24) and then sampling the result to provide discrete data values. Integrating the impulse train (4-24) gives

/h(»,

n

f(t) = / h(nT) dt = T V h(mT)u(t-mT) (4-26)

m^O

where u(t) is the unit step function. (4-26) is a staircase waveform whose envelope is the integral of

the impulse train. By sampling f(t) at the points t = nT, the discrete waveform for the step response

is obtained as 1023

f(nT) = T Yi f(nT)6(t-nT) (4-27)

n«»0

1023 n

T 22 5(t-nT) Y, h(mT). (4-28)

n=0 m=0

The integration of the impulse response (4-24) to yield the step response (4-28) is illustrated

graphically in figs. 4-5 through 4-8.

39
g,2(-T)

e^,(nT)

/

/I

• • 1 -i i I 4 i, etc.,

Time Window, T ,

i. nT

w

1024 equally spaced abscissa values

Figure 4.1 A given display waveform is represented by 1024 data points,

each point being the mean value of 500 observations.

i

/I

/« I < I

I

r ; I

I ^ 6 e ^ i etc.,,

1024

h 6

etc. ,

•1024-

nT

•Time Window

2Tw *l

Figure 4.2 A given display waveform of 1024 points augmented by 1024 zeros

to yield a total time window of 2048 points.

40
k

•*. *

1^ 512 points ^1*^ 312 points i^l

H -I

1024

w

Jk

NT

Figure 4.3 One cycle of the periodic 1024 discrete Fourier transform

corresponding to e, (nT) , fig. 4.1.

Edr2NT^'

A

rv

|.^1024 pts.pw) ^ 1024 pts

h- 1024

2NT

w

Figure 4.4 One cycle of the periodic 2048 point discrete Fourier transform

corresponding to the augmented-zero waveform, fig. 4.2.

41
k

h(nT)

+ ^

3 . N-1

Figure 4.5 The sequence of values representing the discrete impulse

response h(nT)

.

h(nT)

A

A

h(nT-T)

1 2 3 . . N-1

Figure 4.6 The sequence h(nT) represented as a train of delta functions of

varying strengths. Keep in mind that each delta function is

infinite in amplitude while its strength is given by its integral,

This representation corresponds to (4.24) and is the mathematical

representation for a sequence of values.

42
f(t)

1 2 3 . . .

Figure 4.7 The integral of the delta function train, fig. 4.6; yields a staircase function. Each step along the staircase corresponds

to the strength of the corresponding impulse in fig. 4.6.

f(t) is a piece-wise continuous function, (4-26).

h

f(uT)

4 1 h

• •

1 N-1

Figure 4.8 The sequence of values representing the discrete step response

f(uT). This is the discrete function corresponding to the

piece-wise continuous function (4-28) shown in Fig. 4.7.

43
5. TYPICAL RESULTS FOR TRANSMISSION CHARACTERIZATION AND RESPONSE

In this chapter typical results are presented for the determination of the transmission line

parameters and the corresponding frequency and time domain responses. These results are typical of the

data obtained for all of the cables considered in this report; data on specific cable types vs. length

are given in Appendix A. Here, a single cable type of fixed length is used to illustrate results for

parameter characterization and response simulation.

5.1 Description of the Cable

The cable results reported below were obtained using a 320 meter (1050 ft.) length of shielded

paired-conductor transmission line. This cable is commercially available and for the purposes of this

report is designated by the letter I. The nominal specifications for the cable are as follows:

Nominal characteristic impedance - 124 ohms

Overall diameter - 1.07 cm (0.420 in.)

Inductance/unit length - 620 nH/m (189 nH/ft)

Capacitance/unit length - 40.3 pf/m (12.3 pf/ft)

Resistance/unit length - 61.7 x 10 ohms/m (18 ohms/ft.)

5.2 The Results

The results are divided into three groups. The first one illustrates the acquired time domain

insertion voltages and the subsequent computed (FFT) complex frequency domain data for the insertion

ratio S„^ (ju)) . The second section shows the model simulated responses obtained. In the third section

the simulation results for the impulse and step responses are shown.

Figures 5-1 and 5-2 show the acquired waveforms e,. (nT) and e,„(nT) which are the discrete insertion

dl dz waveforms corresponding to equations (3-18) and (3-19); they were acquired in the manner discussed in

Section 4.1. Next, the discrete Fourier transforms of e ,, (nT) and e,„(nT) are computed and are used to

,^ „ di al , calculate the complex ratio [E ,. (k/2NT) ]/ [E ,, (k/2NT) ] which is the insertion ratio S„,(k/2NT), Sections

aZ dl ^ zl 4.2 and 4.3. Table 5-1 lists the complex data for S„^ (k/2NT) , (4.10), along with the standard deviations

of each value of | S. | and the phase B . Figure 5-3 shows a log-log graph of the attenuation a(k/2NT),

(4.7). The phase data in Table 5-1 is used only to ascertain that the data for S„. (k/2NT) possesses a small enough standard deviation in the phase data, which in conjunction with a small standard deviation

for the magnitude | S.^ | , indicates that the data is consistent.

The parameters m and K are extracted from the data shown in fig. 5-3 according to the theory in

Section 3.4 and by the method of Section 4.4. Using the extracted values of m and K, the simulated cable

attenuation a (k/2NT) and its associated minimum phase shift 3, (k/2NT) are calculated, figs. 5-4 and 5-5. By taking the inverse discrete Fourier transform, using the inverse FFT algorithm, of the frequency

domain simulation data, the simulated impulse response of the cable, h(nT), is obtained, fig. 5-6. The

D. C. level of the impulse response has been carefully established by the procedure described in section

4.6. To validate the (simulated) impulse response, the acquired reference waveform e, . (nT) is convolved

with the impulse response and the result is compared to the acquired insertion waveform e , (nT)

.

Figure 5-7 shows e (nT) , fig. 5-2, with an expanded time scale while fig. 5-8 shows the result of con- volving e^ (nT), fig. 5-1 with the simulated impulse response h(nT), fig. 5-6. Comparison of figs. 5- and 5-8 shows no significant differences in shape, hence the model parameters accurately characterize

the cable. '

44
The step response of the cable, f(nT), is obtained by integration of the impulse response h(nT),

fig. 5-6, as discussed in Section 4.7; the result is shown in fig. 5-9. Note that step response f(nT)

has reached the normalized value of about 0.86 at the end of the time window, 999 nanoseconds. According

to eq. (2-52) the final value referred to the sending end generator voltage E would be R /(2R + iB.) or 0.463 for this particular cable (parameter values given in Section 5.1). Without losses at dc (R = 0),

this value would be 0.5. Normalizing the lossless value to 1.0, gives a final value dc value, i.e.,

the value for t = «>) of 0.926. Consequently, it is seen that the cable response has not reached the true final value of 0.926 in one microsecond. Consequently, the transmission line could not accurately transmit pulses changing state every microsecond.

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57
6. USE OF TIME DOMAIN MEASUREMENTS TO ESTIMATE MAXIMUM CABLE

BIT RATE AND BIT-ERROR-RATE DEGRADATION

This chapter is concerned with developing a method for using the time domain measurement techniques

and data presented in the previous five chapters to obtain estimates of two parameters of great interest

to the digital communication system designer. These are (1) the maximum bit rate that may be transmitted

through a given length and type of cable, and (2) the bit-error-rate that may be expected for a given

system signal-to-noise ratio. It will be shown that a rather unusual plot of the cable step response

may be used to estimate these two parameters.

6.1 Digital Communications System Model

Since the purpose of this study is to characterize certain transmission lines (as opposed to complete communication systems) for digital communication applications, a simple idealized transmitter/

receiver model has been chosen. By avoiding more complex "realistic" system models it becomes simpler

to observe the deleterious effects of the transmission line, alone, in an otherwise perfect system.

The chosen system model is shown in fig. 6.1. The following important assumptions have been made

with respect to this transmitter/receiver model:

a. The transmitter generates a binary non-return-to-zero (NRZ) signal of perfect

rectangular pulse shape. That is, a "zero" is represented by a constant zero- volts output, a "one" by a constant V-volts output and the transitions from a "zero" to a "one" or vice versa are instantaneous. The transmitter bit rate is controlled by the trigger generator.

b. The receiver is a perfect, triggerable threshold detector such that at any time

t it may be triggered to sample the input signal and decide if the input voltage

is less than or greater than some voltage V . If the receiver decision is "less

in than V ", then a "zero" is assumed to have been received; and if the decision is in "greater than V ", then a "one" is assumed to have been received.

c. The trigger generator and variable delay are assumed to allow perfect timing

control between the transmitter and receiver for any bit rate. Thus, compensation for any cable delay is provided and the delay may be "fine tuned" to allow a receiver sample to be taken at any chosen time within a bit period.

d. The transmitter output impedance, the receiver input impedance and the

summing circuit's input and output impedance are matched to the high frequency

limit of the cable characteristic impedance; i.e., \ L/C (see eqn. 2-35).

(The noise source will be discussed in Section 6.4; for the discussion in this section switch S^ is assumed

to be open.

)

6.2 Maximum Bit Rate Estimation

From the model asstimptions it is evident that the transmitter/receiver pair, connected directly

together, represents a system with no bandwidth (or, equivalently , bit rate) limitation. That is, since

the transmitter voltage transitions are assumed to be instantaneous, and the receiver has no speed of sampling restrictions, then the system can operate perfectly at any bit rate. In addition, it should be

noted that this system is not code dependent; it will work perfectly for any particular sequence of "ones"

and "zeros" generated by the transmitter.

58
If a section of transmission line is now connected into the system at the insertion plane indicated

in fig. 6.1, two changes in the operation of the system occur; the system becomes bit rate limited and

code dependent. In order to analyze and quantize these cable-induced system limitations, the example measurements on 1050 ft. of cable I presented in Chapter 5 will be extended

The step response for cable I for <^ t < 1 ys is shown in fig. 6.2. This, of course, is the

voltage that would be observed at the receiver if a "one" were transmitted preceded by an infinite (in

theory) string of "zeros". At the end of the time window the voltage is only about 0.86 V and, as was shown in Chapter 5, will rise to only 0.925V as t— «>. The fact that the received voltage never

reaches V volts is due to the DC resistive losses in the cable.

If the situation were reversed such that a "zero" were sent at t = 0, preceded by an infinite

string of "ones", the response voltage seen at the receiver would be as shown in fig. 6.3. That is, it would be 0.926 V at t = 0, decay to 0.066 V at t = 1 ys (0.926 V -.0.86 V) and finally reach V as

t -> 0°. From this information the receiver may now be adjusted for optional operation. That is, a threshold level, V , and a sampling time t may be chosen to maximize the receiver's ability to accurately

detect whether a "one" or a "zero" was transmitted. In both the "zero"-to-"one" and "one"-to-"zero"

transitions mentioned above the received voltage varied between the limits of V and 0.926 V in the

interval < t < >». Thus a logical choice for V is V /2 where V^ is a function of the type and

TH I t length of cable and, as was discussed in Chapter 5, is calculated as

^f = ^2 R 1\k - (^-^)

o

For the cable I parameters listed in Chapter 5 (R = 12A n, Jl = 1050 ft. and R = 18.8 m fi/ft) substitution

into eqn. 6-1 yields V^ = 0.926 V and thus V = 0.463 V. Any other choice of V will bias the receiver

t TH in such that it no longer would have an equal ability to detect a "one" and a "zero".

The logical choice for a receiver sampling time is at the end of a bit period (note that this choice

applies only to this model, e.g., no time jitter). From fig. 6.2 it is observed that in the time window

< t < 1 ys the "one" voltage is maximum at t = 1 ps. Also, from fig. 6.3, the "zero" voltage is minimum

at t - 1 ys. Therefore, since the received "one" or "zero" voltage is either rising or decaying monotonically, the best time to sample is always at the end of a bit period, regardless of tl.e actual value

of the bit period.

Figure 6.4 is a superposition of the positive and negative step responses shown in figs. 6.2 and

6.3. As expected, the two curves intersect at V = V^/2 which occurs at t si 50 ns. If the system were

TH I operated at the bit rate corresponding to this intersect time; i.e., BR = 1/50 ns = 20 Mb/s, then the

receiver voltage at the sample time would be V regardless of whether a "one" or a "zero" were sent. TH Furthermore, if the system were operated at any faster rate, say at BR = 1/10 ns = 100 MB/s, then the

receiver would always be wrong because a "one" signal (or a "zero" signal) would not have had sufficient

time to reach a value above (or below) V . Also, if the system were operated at any slower rate, say

TH

at BR = 1/100 ns = 10 Mb, the receiver would always be correct because a "one" signal (or a "zero" signal)

would have had sufficient time to reach a value above (or below) V . in Thus it is observed that the insertion of the cable into the otherwise unlimited-bit-rate system

imposes a maximum bit rate limitation, and also that the value of this maximum bit rate may be obtained

59
as the reciprocal of the intersect time of the positive and negative cable step responses.

It is admittedly difficult to extract receiver voltage versus bit rate information from fig. 6.4, But by redrawing the positive and negative cable step response voltages on an X-axis scaling of log,

„

(bit rate) instead of time it becomes a simple matter to relate the intersect point to a maximum bit

rate. This plot is shown in fig. 6.5 and represents the desired objective of this section. From it the

maximum permissible bit rate for correct system operation may be read directly by observing the point

on the X-axis, or bit rate axis, at which the two curves intersect, in this case about 20 Mb/s.

Since this plot (fig. 6.5) is central to the remaining developments of this chapter, a few

additional remarks about it are in order. The data from which it is derived, the positive and negative

cable step responses of fig. 6.4, consist of two discrete time signals, each of 1024 points in a 1 ys window. That is, the step responses are really a set of 1024 voltage points spaced 1 us/1024 points

or about 1 ns apart. Therefore, when plotted versus reciprocal time, or frequency, they range from

1 MHz (1/1 ys) to about 1 GHz (1/1 ns). As shown in fig. 6.5, then, the signals of fig. 6.4 are con- tained in the frequency interval from 10 b/s to 10 b/s.

Since it is economically unfeasible to calculate these step responses beyond the order of 1 ys,

(or equivalently , below 1 Mt/s), due to computer memory and speed limitations, a straight line approximation has been used for the step response voltages between 1 ys and 1 s or 1 Mb/s and 1 b/s. Because

the actual "one" voltage lies above and the actual "zero" voltage lies below their corresponding straight

lines in this time or frequency interval, this straight line approximation is a conservative or "worst

case" approximation.

6.3 Code Dependence

From the shapes of the cable step responses of figs. 6.2 and 6.3 it is evident that intersymbol

interference will occur at the receiver when a cable is placed in the system. Just as in the case of an RC exponential response, the positive modeled cable step response reaches V only as t —> °°. Therefore, the voltage sampled at the receiver at the end of a bit period will depend not only on whether a "one" or a "zero" was sent in that period but also on the particular sequence of "ones" and "zeros"

sent for all time before the bit period in question. In other words, the presence of this intersymbol

interference, caused by the infinitely long cable step response "tails", implies that the system is code dependent.

One approach to the analysis of this code dependence on system operation would be to analyze

separately the effects of all possible code sequences. However, a much simpler method exists which

is to find the worst case and best case code sequences as lower and upper bounds for correct system

operation.

The worst case is exactly the one referred to in the last section; an infinite string of "zeros"

for all t < 0, a transition to "one" at t = and an infinite string of "ones" for t > 0. In other

words it is represented by the (positive or negative) cable step response. If even a single "one" had occurred at some time t < 0, then the voltage at the receiver would not be exactly V at t = 0. Rather, it would be slightly greater than V due to the fact that the decay back to V, even from a single "one" only occurs as t — <». Therefore, if one or more "ones" occurred before t = and if a "one" is transmitted at t = as described above, then the step response commencing at t = will begin

from some voltage greater than V. The received voltage will thus have a "head start" towards V

TH and will rise to that value sooner than in the case of all "zeros" preceding t = 0. Of all possible

code sequences preceding t = 0, then, the one consisting of all "zeros" will yield the worst case because it is the only one that forces the transition to a "one" to begin at V. The same argument

may be used to show that the worst case for a "one"-to "zero" transition is the negative cable step response.

60
The best case is the sequence "one"- "zero"-"one"-"zero" occurring over all time. Fig. 6.6 (a through f) shows that this is true. Figure 6.6a is a plot of the transmitter signal for 64 bits of a "one"-"zero"-"one"-"zero" code sequence. (For all t < the transmitter output is V.) Figure 6.6b is the time domain convolution of this transmitter signal with the impulse response of 1050 feet of cable I, that is, figure 6.6b is a plot of the receiver signal that results from the transmitter signal passing

through the above cable at a 4 Mb/s rate. The plots shown in fig. 6.6c through f are of the receiver

signal that would be observed if the transmitter bit rate were increased to 8, 16, 32 and 64 Mb/s

respectively.

All of these responses may be thought of as transient responses starting at V and eventually

oscillating symmetrically about V as t —> °°. The worst case mentioned above is graphically illustrated

TH

by observing the first "zero" to "one" transition occurring at t = 0. For the responses shown in

fig. 6.6b, c and d, corresponding to 4, 8 and 16 Mb/s, respectively, the first "one" rises to a voltage

greater than V thus implying that no errors will be made. For the 32 and 64 Mb/s responses, however,

TH

the first "one" does not reach V ; an error in detection of this "one" is certain. This tends to con- ~" in

firm the .conclusion drawn from fig. 6.5 that the cutoff bit rate for proper system operation is about

2D Mb/s.

The fact that the "one"-"zero"-"one"-"zero" sequence over all time is the best case may be deduced

from the five responses shown in fig. 6.6 by observing that, regardless of the bit rate, all of the

responses end up oscillating symmetrically about V as t —» =°. Therefore, even for very high bit rates,

in when the receiver response signal reaches steady state the detector will properly discriminate between

a "zero" and a "one". Neither of these code sequences, the single step over all time or the square wave over all time,

contain any information, of course, but they do represent the code boundaries. That is, for a given

bit rate, they represent the minimum and maximum number of transitions allowable per unit of time.

Therefore, any arbitrary "real" code sequence will have a receiver response somewhere between these worst and best case responses. Consequently, the worst case plot shown in fig. 6.5 may be used to prudently estimate the maximum bit rate that may be transmitted over a given type and length of cable.

6.4 Bit-Error-Rate Degradation

In the previous sections in this chapter it was shown that the overlay plot of the positive and

negative cable step responses versus bit rate may be used to estimate the maximum bit rate that may be

transmitted over a given type and length of cable under ideal conditions. Quite often, though, there

is noise present in communications systems such that even if the system were operating at a bit rate

less than the cutoff bit rate, there is a nonzero probability of error simply due to the noise. This error probability is usually quantitatively characterized by the parameter "bit error rate" (BER)

which is the probability that an error will be made in the detection of any given bit. A BER of 10 , for example, means that the probability of misinterpreting any given bit is one in one million, or, stated another way, it can be expected on the average that for every one million bits sent, one will

be misinterpreted.

The goal of this section is to develop a method for determining the change in BER of an idealized

communication system by the addition of a given type and length of cable. The assumed communication

system is the same as that defined in section 6.1 and shown in fig. 6.1 but with the noise source switch

61
now closed. It will be further assumed that the noise source voltage output is Gausfiian white noise;

that is, uncorrelated noise with a Gaussian density function given as.

P(v) =

^ ^-(v-y)2/2a2^

^^_^^

4

2

2TTa

where y is the noise mean, assumed to be zero, and a is the noise standard deviation or rms value.

In order to determine the BER degradation due to the insertion of a cable into the system it is first necessary to calculate the BER of the system with the transmitter connected directly to the

receiver. In this case a "one" is represented by a transmitter output of V volts and a "zero" by the

volts. As before, the receiver threshold voltage is set to the midpoint of the received signal range, in this case V/2 volts. To this signal, then, is added a Gaussian noise voltage with zero mean and

rms value a .

If a "zero" is assumed to be sent then the probability of an error in detection is simply the

probability that the received voltage (signal plus noise) will exceed V/2 volts. From the assumed

Gaussian density function this is.

(6-3)

Likewise, if a "one" is assumed to be sent then the probability of detection error is the probability

that the received voltage is less than V/2 and is given as

V/2

P(v < V/2) = f

—^— e-(^-^)'/2a2 ^^ _ ^^_^^

Due to the symmetry of the Gaussian density function the probabilities of mistaking a "one"

for a "zero" (eqn. 6-3) or a "zero" for a "one" (eqn. 6-4) are the same so either equation may be used

to calculate the probability of error per bit regardless of whether a "one" or a "zero" was sent. The actual calculation of the error probability from eqn (6-3) or (6-4) is quite difficult and

fortunately unnecessary. Tables exist for the error function , defined as,

X 2 erf(x) = -^

f e"^ dy (6-5)

' o

and it is easily shown [12] that the error probability above may be expressed in terms of the error function as

P(E) = I

/ V \ 1

1 - erf / y \2yJTa I.(6-6)

62
Furthermore, if the voltage peak signal to rms noise ratio is defined as

VSNR = - (6-7)

a

then the error probability per bit, or equivalently, the BER may be expressed as a function of VSNR as,

P(E) ![--( 2

VSNR \ (6-8)

A computer-generated plot of BER versus VSNR is given in fig. 6.7a through e. As an example of using this plot assume that the transmitter output for a "zero" is volts, a "one"

is 24 volts and the rms noise value is 2 volts. Tl-ien, from (6-7) the VSNR is 12. From the plot in

-9

fig. 6.7c it is observed that a VSNR of 12 yields a BER of 10 . If the transmitter-receiver pair VSNR is known, it becomes possible to determine the new BER that

would be observed by inserting of a given type and length of cable. This may be accomplished by use

of the overlay step response versus bit rate plot of fig. 6.5 in conjunction with the BER versus VSNR

plot of fig. 6.7 in the following manner. For a desired bit rate, the new peak signal voltage is obtained from fig. 6.5 as

V (BR) = V (BR) - V (BR). (6-9)

That is, the new peak signal voltage is the difference between the "one" voltage and the "zero" voltage

that are present at the receiver at the end of a bit period for the assumed bit rate. From fig. 6.5,

for example, v for an assumed bit rate of 1 Mb/s is 0.86 V - 0.07 V = 0.79 V. Then, from eqn. (6-7),

P a new VSNR is obtained as

V

VSNR„ ^ . p 1,1 = -^ = -^^^^ = 0.79V X VSNR„ ^ . (6-10) System + Cable a a System

Continuing the above example, if the transmitter receiver pair VSNR were 12, implying a system BER

6

-9 of 10 , and then 1050 ft. of cable I were placed in the system operating at 1 Mb/s, the new system VSNR

would be 0.79 x 12 or about 9.5. From fig. 6.7b a VSNR of 9.5 yields a new BER of about 10 These two graphs, figs. 6.5 and 6.7a-e, may be used for a variety of system design purposes. As another example suppose that it is desired to know the maximum permissible bit rate for the system in

the previous example such that the new BER (after insertion of the cable) will not exceed 10 . From

fig. 6.7b this BER implies a minimum allowable VSNR of about 10.4. Then, from eqn. (6-10)

VSNR„ ^ . p t,i System + Cable ,, ,,^ V = V (6-11)

p VSNR„ ^

>.o -L-l; System

or

10.4 V = V —— = 0.87 V

p 12

where in fig. 6.5 it is seen that this difference voltage v occurs for a bit rate of 200 b/s.

However, rather than using the difference voltage of eqn. 6-9, one can proceed using a single curve,

e.g., the one curve, fig. 6.2. Due to the symmetry of the negative and positive step response, it is

evident that, for any bit rate

63
Vq(BR) = V^ - v^(BR), (6-12)

and substitution of this expression for v (BR) into (6-9) yields

For the example,

V (BR) = 2 V, (BR) - V, (6-13)

p 1 f

V, (BR) = ^ [v (BR) + V,]. (6-14)

1 z p r

V (BR) = ^ [0.87V + 0.926V] = 0.898V

0.9.

Observation of the point where the v^ (BR) curve (the "zero"-to-"one" step response curve) passes through

the 0.9 V line again yields a bit rate of approximately 200 b/s. The same answer has been obtained by

reading the value of only one curve rather than by reading the difference between two curves.

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cS » fl CO o 60 rH

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(U }-J ao IW

CO

4-J Pd tH s

o en > >

u

o

en>

m

3

in 0}

!-i ro 0) tH >w

Ph

•

• vO

(UH

bO

•H

CO

rijq aad j:oaaa jo A^-fXT'^^qo^d:

79
o

I— v£)

•H o

4-1 • CO v£> U rH

0)

CO

O

•H •u o1 o o • u -3- o 1 iH

tH

LO «J iH c g M O

•H S-i CO M-l

(1) Pi 00 S

nj cn

4-J >

iHo S-i > o

14-1

PiCO>

in CO3

<a- en

T-i >w

vO

<U

00 o

80
7. CABLE TESTING USING TIME DOMAIN REFLECTOMETRY

In this chapter the general principles of time domain reflection and transmission are developed and

then are applied to cable testing. It is shown how the cable sending-end step responses are employed to calculate (simulate) the reference echoes (TDK responses) which characterize an undamaged cable. Departures from the reference response waveforms indicate uniform degradation of the cable and physical damage,

or some combination of the two factors.

7.1 The General Theory of Time Domain Reflection and Transmission

In the classical frequency domain analysis of transmission lines containing reflections, the assumption is implicit that the sinusoidal signals are steady state signals, i.e., they have been present since

the time minus infinity. In time domain measurements of transient or pulse phenomena, such is not the

case. Prior to some time, say t=0, there was no signal present. When the pulse signal does enter the

transmission line, It propagates along the line at a finite velocity arriving at observation points displaced from the input later in time. When this propagating signal encounters a discontinuity, reflected

and transmitted signals are created. An observer remote from the point of reflection will not observe the reflected signal until still later in time due to the time isolation (delay) inherent in the

transmission line. For people working with time domain measurements and instruments it is very important

to comprehend this time isolation of reflections. As an aid to developing an understanding of this concept this section develops the analysis of a discretely loaded transmission line in terms of transmitted

and reflected pulse trains. A reflection-transmission (RT) diagram is used. The resultant transmission

line equations are then shown to be identical to those derived in the classical frequency domain manner.

The starting point for this discussion is based upon fig. 2.2 and the general equations (2-3) and

(2-4). First, equation (2-4) is expressed in terms of E (s) rather than I (s) through the relation

g s

I^(s) = E (s) - E (s) _S § Z (s) (7-1)

with the result

E_(s)

Z (s)

o

Z (s) + Z (s) g o |_l-p^(s)p^(s)e 2ilY(s) |ri.p,(s)e-2^^(^)l (7-2)

For convenience, (2-3) is repeated here.

E^(s)

= 2 Z^(s)

Z (s) + Z (s) g o

Z,(s)

Zj^(s) + Z^(s) L

1-p (s)p (s)e J

(7-3)

p (s) and p (s) are the reflection coefficients (2-5) produced by the sending and receiving end

S 1^ impedances, Z (s) and Zj^(s), respectively. I is the transmission line length while y(s) and Z^(s) are

the line propaga'.ion function and characteristic impedance, respectively.
The latter are given as

Z^(s) = Z(s)

Y(s)

1/2

(2-1)

y(s) = [Z(s) Y(s)] 1/2(2-2)

where Z(s) and Y(s) are the transmission line equivalent circuit parameters: series impedance/unit length

and shunt admittance/unit length. In general, for wave transmission on a transmission line Y(s) is such

that the limit

lim

|s|->.

y(s)

s

is equal to the time delay per unit length, T, i.e..

T = lim r y(s) 1

I s I —» °= [ s J (7-4)

For uniform transmission lines, the time delay/unit length is expressed in terms of the inductance and

capacitance per unit length.

1=^10 (7-5)

Also, y(s) can be written in the form

Y(s) = sT + y'(s) (7-6)

where sT is an all-pass function (linear phase function) and y'(s) is the minimum phase function (see

section 2.4 for some discussion of minimum phase). For y'(s) to be a minimum phase function, the ratio

y'(s)/s vanishes as |s| becomes infinitely large [2]. Now consider a step generator of impedance Z (s) applied to the system described by (7-2) and (7-3),

i.e., E (s) = 1/s and Z (s) = Z (s) . Then p (s) = 0, and (7-2) and (7-3) reduce to

g go s

%^^^ =2^ [l+P/s) e 2£[sT + y'(s)]1

(7-7)

and

E^(s)

Z,(s)

Z^(s) + Z.(s)

O X.

-£[sT + y'(s)]

(7-8)

Rewriting

E (s) =

s .2s + e

-2s!lT |'p^(s)< -2JIy'(s) (7-9)

and

E^(s) -s£T Z„(s) e

-itY'(s)

I s [Z^(s) + Zj^(s)]

j (7-10)

82
Consideration of (7-9) and (7-10) leads to the following conclusions:

1. The input terminals of the line presents to the generator the characteristic impedance

Z (s) for the time interval 0<t< 2S-T; i.e., the load impedance is not seen at the generator

o terminals until the incident wave travels to the load and the subsequent reflection returns

to the generator terminals.

2. The output waveform is zero for t<2.T; also, its shape is controlled by the characteristic

impedance to load impedance voltage divider and the minimum phase (distorting) part of the

propagation function, y'(s).

Generally, for arbitrary terminations, E (s) and E (s) each consist of an infinite sequence of terms similar to those in (7-9) and (7-10) and have their origins in the conclusions delineated above. By

using the geometric series expansion

1 - p^(s)p^(s)e 2£y(s) -I -1

where

p (s)p (s)e

s r -2JlY(s)

lpg(s)p^(s)e

-2iy(s) I 2

< 1

(7-lla)

(7-llb)

in the second and third factors of (7-2) and (7-3), respectively, the transfer functions can be expressed

as infinite sequences which represent impulse responses consisting of pulse trains. A note of caution

should be entered here; for a lossless line the inequality condition in (7-11) may not be satisfied and

hold for all values of s in the complex plane. Under such conditions an alternate mathematical pro- cedure must be used [13]

.

Putting (7-11) into (7-2) and (7-3) yields the sending end pulse train

E^(s) = E„(s)

s g

Z^(s)

Z (s) + Z (s) g o

^ - , n ,

. n. . -2nilY(s) , n. . (n+1) , . -(n+1) 2£y(s) X \ p (s)d (s) e ' + p (s)p^ (s)e V* p^' (s)p^'(s) e

'^ ' n=o

and the receiving end pulse train.

(7-12)

E (s) = 2E (s) r g [zg(s) +\(s)j[

Z,(s)

Zj^(s) + Z^(s)

X J^ p^(s) p^^^^ ^-(2n+l)JlY(s)

(7-13)

where (7-12) and (7-13) are both subject to the condition specified in (7-llb). The physical situation

represented by the pulse trains (7-12) and (7-13) is lucidly shown by a reflection-transmission (RT)

diagram, fig. 7.1. In fact, the RT diagram itself provides an intuitive (but exact) means for writing

out the terms of each pulse train.

83
The RT diagram is a signal flow chart embodying the reflection and transmission properties of the

terminal impedances, and the wave propagation property of the connecting transmission line. To interpret the diagram the following rules are used:

1. Passage from a generator of impedance Z (s) through a node to a line segment represents

multiplication by the transfer function Z (s)/[Z (s) + Z (s)].

o g o

2. Passage along a line segment represents multiplication by the transfer function exp[-J!.Y(s) ]

.

3. Passage through a node to a load Z (s) represents multiplication by the transfer function

2 Z^(s)/[Z^(s) + Z^(s)].

4. Reflection from a node represents multiplication by the reflection coefficient

p (s) = [Z,(s) - Z (s)] [Z rs) + Z (s)]"^.

r X, o >l o Application of the rules to the diagram in fig. 7.1 obtains the following results:

E^(s)

Z + Z

Gen to A

Z 2Z

Z + Z ^r Z + Zo

ABc"

E„(s)

Z 2 Z o -iy -Hy -ly -iy e Z + Z

^ p^e p^e p^e

^ + Z

o g go

ABCDE

Z + Z

2p Z „ 2p p Z

Z +Z

g o

Z + Z e-^^Y ^ (7-14)

Since

Pi =

Z. + Z

1 o

the voltage divider may be written as

Z.1

z. + z

1 o

1 + P

Putting (1 + p )/2 into (7-14) for the factor Z„/ (Z„ + Z ) obtains the result

E^(s)

g z^ ^

[l ^ p^(l + p^)e-2^^ + p2p^(l + p^)e-^^^ + (7-15)

which is identical to that given by expansion of (7-12). Therefore, it has been demonstrated that the RT diagram allows one to write down the equation for E (s) directly by tracing out the signal flow while

invoking the rules 1-4. Similarly, the received voltage E (s) is obtained from the RT diagram as

84
E^Cs)

z + z g o Gen to A"

-iy 'h

Z„ + Z

i oAB

2Z„

-Jly -^Y -^y 1—rr e P^ e p^ e Z^ + Z^

g o ABCD

\ \_ r

g ....] (7-16)

which is identical to that found by expanding (7-13). In this report the transmission lines are lossy and are normally operated in a doubly mismatched

condition due to the terminations being equal to the nominal high frequency characteristic impedance, R

Under such conditions, (7-15) and (7-16) become

E^(s) = E„(s)

s e

Z (s) R + Z (s) o o L

1 + p_(l+p )e ^^^^^^ + p_p (1+p )e ^ ^^^ + . . .1 (7-17)

and

E (s) = 2 E (s)

Z (s) R

o [R

y^l ^ Vol

+ Z (s)]^ L

s) ^^2 ^-3.y(s) ^ (7-18)

respectively, where

p = p^ = p^

R - Z (s) o o R + Z (s)

o o (7-19)

Briefly, consider the received voltage E (s). Inspection of (7-18) shows that the initial signal

transfer function is due to the product of the RT input voltage transfer function, the exponential

transmission factor, and the output RT voltage transfer function, i.e..

S^(s)j = 2Eg(s)r^

Z^(s)

(s) + R„

-5.y(s) Z^(s) + R^

o o (7-20)

This is in agreement with the RT diagram transmission/reflection rules. The condition 0<t<32.T describes

the time interval of the initial response, which is to say that the inverse Laplace transform of (7-20)

represents the time domain response only over the interval 0<t<3!?.T. Refer to fig. 7-2 and the brief disdiscussion in its caption.

Z (s) R

e^(t) -12E_(s)

Z (s) + R

o o

-£Y ^^)] ; <t<3)iT. (7-21)

It is important to keep in mind that (7-21) is the exact response for e (t) over the interval 0<t<3J.T. What is unusual about this method of analysis is that even though a series expansion (7-11) has been used,

a finite sequence of terms starting from the initial term is always an exact result out to some multiple of

85
the time ^T. Contrast this with other series expansions where an infinite number of terms must be present

for an exact result, e.g., Fourier Series, etc. The discussion now turns to the sending-end response.

The main purpose of this chapter is to develop the sending end response so that one would have a reference response or waveform for a given cable type; this response would then be used to evaluate the

measured test response of samples of the given cable type. To conclude this section, consider (7-17).

It is clear that for the time interval 0<t<2£T the response is only dependent upon the transmission line

through the characteristic impedance, Z^(s), and is given by the first term from the sequence (7-17)

E^(s) = Eg(s)

0<t<2X.T

Z (s)

o R + Z (s)

• o o (7-22)

In the next section this relation will be used to derive the reference responses. After the passage of

25.T seconds, the second term in the sequence must be added to the first to provide the temporal response

out to 4^T seconds

0<t<4£T

E (s)

Z (s) o R + Z (s)

o o

^o^^'' ,,. . -2JiY(s)

, 7 f^^ P (1+P )e + Z (s) r s (7-23)

Now consider the second term in detail; the term (1 + p ) is just 2 R /[R + Z (s)] which is the RT

s o o o transfer function from the transmission line to the load R ; p is the reflection coefficient of the

o r receiving end load. Z (s)/[R + Z (s)] is the RT transfer function from the generator to the sending end,

and exp. [-2£y (s) ] represents transmission over a length 2£. Consequently, the second term has a physical picture which correlates with the RT diagram as follows:

Z (s)

o R + Z (s)

o o

v_

-iY(s)

2 R

R + Z (s)

o o

"V^ y

Initial voltage at receiving end (7-20)

X r,j -£y(s)

V

Reflection factor

and transmission

back to sending end.

In other words, the initial sending end voltage is transmitted to the load; the received voltage is then

scaled by the reflection coefficient and then transmitted back to the sending end. Figure 7-3 qualitatively illustrates the time domain unit step response corresponding to (7-23) . In the next section the

time domain expressions for unit step excitation will be derived.

86
7.2 The Step Response of the Sending-End Voltage

The sending-end voltage E (s) for <_ t £ lit due to a unit step is obtained from (7-22) by letting

E^(s) (7-24)

which obtains for the sending-end voltage

s^^^ s R + Z (s) ' < t < 22.T . (7-25)

The time domain sending-end voltage is given by

e^(t) = £

Z^(s)

s R + Z (s)

o o

< t < 25,T. (7-26)

This expression is also valid for all time t when the receiving-end is terminated in the characteristic

Impedance, Z (s) , that is; when there are no reflections from the receiving-end, the line appears to be

infinitely long. For a finite length, i., e (t) is only valid for _< t <^ 25,T. In terms of a circuit

model, e (t) represents a voltage division of the generator voltage, e (t), fig. 7-4A. Qualitatively,

s g the response is shown in fig. 7.3. If there were no reflection from the load (receiving-end), the

waveform would continue to rise monotonically to unity, fig. 7.4B. From the final value theorem of the

Laplace transformation, the final value of (7-26) is

e (<») = lim s fl ^o(^>

s |s|^ o s R + Z (s)

Q

(7-27)

Note that (7-26) is independent of i; that is to say, the step response of any length of cable will

follow (7-26) until the first reflection arrives at the sending-end (t = 25.T)

Expressing (7-25) in terms of the model parameters gives

E^(s)

R + sL + Ks

sC

1/2

R +

o R + sL + Ks

sC

1/2

1/2

1 +

R + Ks

sL 1/2 R + R

o o

1 +

R + Ks

sL

z^M . [- ^^]

1/2

1/2

1 + 1 +

R + Ks

sL

87

(7-28)
For large s (small t) (7-28) can be expanded as the ratio of two power series.

E^(s)

1 + X X_ + 3 V _ 3.5 V

1 2 " 2-4 2-A-6 2'4'6-8

s 2 - 3 . . 4

- , V _ V 3 V _ 3.5 V

2 2-4 2«4-6 2-4-6'8

(7-29)

where

R + Ks

sL

(7-30)

and

R + Ks sL

< 1 (7-31)

If the condition (7-31) is violated, the expansion (7-29) is not valid. By dividing the denominator of

(7-29) into its numerator a single series in v can be obtained. This is easily done using the following method. If the product of two polynominals in v is given by

Y = HX

then the k-th coefficient of Y is given by

For example.

y(k) = £ h(i) x(k+l-i).

i=l

yd) = h(l) x(l)

y(2) = h(l) x(2) + h(2)x(l)

(7-32)

and

h(l) + h(2)v + h(3) V +

By writing out a few terms in the product, and starting with x(l) 5^ 0, it is possible to solve for h(k),

i.e., the quotient of Y/X. The result is

h(k) = x(l)

k-1

y(k) - ^ h(i) x(k+l-i)

i=l

(7-33)

where

h(l) = ^

x(l)

h(2) = ^^ [y(2) - h(l) x(2)]

88
Consequently (7-29) can be replaced by „.. 1.1 1 2, 5 3 14 4, (1 1/^ E (s) = ;r" + "5~ V - -TTT— V + T^Q V - i.T„ V + ••• (7-34)

s 2s 8s 16s 128s 512s

The series (7-34) is also convergent for

v(s) R + Ks°

sL

< 1. (7-35)

Upon substituting v(s) into the series (7-34) there results a series having terms of the form As ^ and

Bs where q is an integer 1,2,3..., and r is not an integer. The time domain response is obtained by

taking the inverse Laplace transform of each term using the appropriate transform pair

-^--^Af^u(t) (7-36)

gq q-1

^ - ^^"'

u(t) (7-37) r(r)

where u(t) is the unit step and r(r) the gamma function. The non integers r result from the fractional

power m, < m < 1, domain response ispower m, < m < 1, which appears in the high frequency loss term of the model, Ks . The resultant time

e^(t) = I

u(t) + \ w^(t) - ^ W2(t) + ^W3(t) - 2^ w^(t) + •.• (7-38)

or in terms of the coefficient b , n

where

e^(t) = I u(t) + E b^ w^(t) (7-39)

s z , n n

n=l

a . R^-^ K^ w (t) = T SI t^-ini u(t) (7-40)

tMi l" r [(n+l) - im]

^ni = i I ^;-i):

^^-""^^

nn (2i-i)

b = ^^ (-l)""^-"- (7-42) n n

(2n+2) n (2i)

i=l

nl = r(n+l) (7-43)

r (n+l) = n r (n) . (7-44)

89
For examples of the expansion of w (t) , (7-40), w (t) through w. (t) are given below.

^1^^^ = l1^) t u(t) +

L r (2-m)

t^-" u(t) (7-45)

/ N R^ 2 , , ^ 2 RK 2-m . . ^ K^ ^2-m , . W2(t) = —2

t ^(t) "'' "~2 ^ ^^^ ~2 "^^

L^ r (3) L r (3m) L r (3-2m)

(7-46)

R^ 3 , . , 3R^K 3-m , . ^ 3 RK^ ^3-2m ..

w^(t) = -^ t u(t) + -^ t u(t) + —^ t u(t)

L-^ r (4) L r (4-m) L r (4-2m)

+ -r^ t3-3- u(t) L^ r (4-3m)

(7-47)

w^(t) R

l" r (5)

t u(t) +

4R^K

L r (5-m)

t^-"u(t) +

6R^ K^ ^4-2m ,^.

t u(t)

L r (5-2m)

4 RK" L^ r (5-3m)

t^-3'°u(t) +

K

L r (5-4m)

t^-^"" u(t) (7-48)

For a given cable, certain precautions must be taken before using the time domain series (7-38) to calculate the sending-end voltage e (t) . First of all, the condition (7-35) must be satisfied for all

s frequencies comprising the sending-end step response voltage e (t) . As can be seen from (7-35), there

is a lower limit on |s| , i.e., a lower frequency limit, f . Letting s = j u in (7-35) obtains

R/L _^ (K/L)

< 1 (7-49)

which can be expressed as

2 2 1/2 K R 2RK (1-m) tt ' 2(l-m) ,2 2,2 2-m ,2

^^"^ 2

< 1. (7-50)

The frequency, f=w/2TT, at which the left hand side of (7-50) just equals unity is defined as f . For frequencies equal to and less than f , the series (7-34) will not converge to E (s) . Consequently, the

corresponding time domain series (7-38) will diverge for time greater than some valute T , which should

be of the order of the reciprocal of f , i.e.,

T = 1/f . o o

T can not be exactly specified due to the uncertainty properties of the Fourier (and Laplace)

90
transformation which prohibits a one to one correspondence between points in the time and frequency

domain

.

To demonstrate the application of the time domain step response, equation (7-38), and also its con- vergence properties, the sending-end step response of Cable I has been simulated using (7-38); the cable

parameters are given in section 5.1. Figures 7-5 and 7-6 show the responses for the time windows of 5 and 10 microseconds, respectively. Note the initial jump to 0.5 and the subsequent slow rise thereafter

increasing only by 10 to 12% of the final value (1.0). For each response the largest error occurs at the

-4 -2 -3

last point; they are 1.5x10 volts (1.5x10 %) and 3.3 x 10 volts (0.33%) for figs. 7.5 and 7.6,

respectively. The errors occur due to the use of a finite number of terms of the series e (t), (7-38).

s For figs. 7.5 and 7.5 the truncated series for (7-38) consisted of the first term 0.5 u(t) plus the

weighted values of w^ (t) through w, (t), a total of 28 terms. The error was computed by taking the

i magnitude of the difference between results for the truncated series and a second truncated series

containing one more weighted value of w (t) . The second truncated series contained w (t) and thus

1 o contained a total of 35 terms. Turning now to the divergent properties of the series, it is known that (7-34) will not converge for

values of |s] along the real frequency axis, i.e., |a)|, when u is equal to or less than 2tt x 28 x 10 , f = 28 KHz, (7-50). Consequently, in the neighborhood of t=T =l/f = 35.7 microseconds, the time domain

O

I y o

o o response (7-38) should diverge even when an infinite number of its terms are used. Truncation of the

series should lead to a divergent result well before 35.7 microseconds; that such is the case is shown

in fig. 7-7. Figure 7-7 presents the step responses for the time interval of 40 microseconds as a function ot the

number of terms of (7-38). The label 10 corresponds to 10 terms comprised of 0.5 u(t) plus the algebraic

sum of the weighted values of w^(t) through w (t). Similarly, 15 corresponds to 15 terms obtained from

the algebraic sum of the weighted terms through w (t). The largest number of terms, 36 yields the most accurate result. For example, at t=10 microseconds the responses for 28 and 36 terms are virtually

identical while those for 21, 15, and 10 terms show increasing error, respectively. Also, the divergence

increases as t approaches the neighborhood of T^, 35.7 microseconds. These curves show that the divergent

properties of the series (7-38) and its truncation to 28 or 36 terms have negligible effect on the com- putation of e (t) out to 10 microseconds (no greater than 0.33 % error at 10 ns)

.

For the cables considered here, the maximum physical length is typically about 300m (1000 ft).

Accordingly, for a typical relative dielectric constant of about 2,5, the maximum cable delay to be

encountered would be about 1.69 microseconds (/2T5 x 320/3x10^) and the corresponding 2JIT value would

be 3.38 microseconds (2xl.69xl0~ ). In the particular case of Cable I, )iT is equal to 1.6 x lO"^

(320 X V LC) and 2JIT is 3.2 microseconds. Accordingly, the observed response would depart from (7-38) at 3 ns in the manner shown in fig. 7-3; however, figs. 7-5 and 7-6 do accurately describe the observed

response out to 3.2 ns, i.e., for the time interval before the receiving-end reflection returns to the

sending-end

.

91
7.3 A Practical Method for Measuring the Step Response

of the Sending-End Voltage

When the generator impedance R is not equal to the high frequency characteristic impedance R of

g o the cable under test, (7-28) is not valid. Specifically, in the first form of (7-28) note the R term;

o for a generator of impedance R , the R term has to be replaced by R . Consequently, for a generator

impedance R all equations based on (7-28) are also invalid.

However, all is not lost because the principal result of Section 7.2, e (t),.i.e., eq. (7-38),

transforms as follows:

2 R

e^(t) R + R s o g

e_(t). (7-51)

The derivation of this result proceeds in the following manner. Consider fig. 7-4A in which R is

replaced by R . Then, the voltage across Z (s) is given by

R

Z^(s)

s R + Z (s) g o^

g

(7-52)

Upon expressing Z (s) in terms of the cable parameters obtains

o

e,(t)

] =^^

R +

R + sL + Ks

sL

R + sL + Ks'"

1/2

1/2

sL

1 +

R + Ks

sL

m

1/2

"1/2 R +

g

1 +

R + Ks sL

. [-H^]

1/2

1/2

g [ sL J

(7-53)

92
where R denotes R /R . Expanding the radical terms in a power series as was done in (7-28) yields

g go

2 - 3 V V 3v

1 + ^ - ^-^ +

3-5-v

e^(t)

2 2*4 2'4'6 2«4«6'

2 .3 T r ^

g 2 2-4 2-4-6 2-4-6-8

(7-54)

Because the quotient of polynomials, (7-54), can be expressed as

1 + a^ V + a V + • -

(1 + R ) + a, V + a„ V

S 1 2

the k-th coefficient of H is given by

h (k) = x(l)

k-1

y(k) - X) h(i) X (k + 1-i)

i=l

(7-33)

For eg(t)1 , i.e., (7-54) R

x(l) = 1 + R (7-56)

while for e (t), (7-38)

s

x(l) = 2

Because x(l) occurs in (7-33) only as the factor l/x(l) external to the brackets, the two series

expansions are related by a simple scale factor.

e3(t)]R

g

1 + R

e^(t)

2 R

O , s

o g which is the result given above in (7-51)

.

An experimental run on Cable I using a two channel 50 ohm feed-through sampling system produced

the data shown in fig. 7-8. Note that the initial step is given by

e^(0)]

100

100

I X 124 /l\

+ 124 \^ 2J

0.55 (7-57)

93
where R and R are 100 and 124 ohms, respectively. The measurement system is shown in fig. 7-9. When

the voltage of one of the balanced generators was reduced to zero with the 50 source impedance being

maintained, no significant change (none was discerned) in the energized channel's sending-end voltage,

(1/2) e^(t)j . 124

94
ZJs)

E (s) Z (s), y(s) Z,(s)

£r M

E (s), Z (s)

g B

^s<^'

Zgts)

Figure 7-1. The RT diagram for a uniform transmission line

connecting an arbitrary generator and load.

95
1.0

e^(t)

[e^(t)]

0<t<3JlT

[e^(t)]

3£T<t<°°

^—»—I

y

3J2.T

Figure 7.2 The initial received voltage [e (t)] assuming e (t) to be the

r ° g

0<t<3£T

unit-step. Equation (7-21) is the Laplace transform of this

initial voltage. Note the transmission delay of IT seconds

and then the emergence of the distorted step-like signal from

the transmission line, ZT to 3£T seconds.

96
e^^t)

0<t<2JlT

[e^(t)]

2£T<t<°°

il 2£T 3£T 4£T

Figure 7-3. The initial sending-end voltage [e (t)] assuming e (t) to be

s g

0<t<2£T

the initial step. The initial voltage jumps to 0.5 volts and

then rises slowly until the first reflection returns from the

load.

97
R

e (t)

g t

1

Figure 1-hk. The voltage divider circuit representing the voltage

transfer from e (t) to e (t) for 0<t<2£T or for all t when

g s £ = «>, or the line is terminated in Z (s)

o

1.0

e^(t)

0.5

Figure 7-4B. The sending-end voltage step response when the line is

infinitely long or terminated in Z (s).

98
' 11\ •

3.

CO3.

CM

0)

COGou

0)

COouomVo oo

CO>

0)uo

a)

COGO

CO<uu

Q*

0) 4J

CO T>cIoo

•H

a>

CO mm

•H

ft* >O o >oo

s:noA

99
1 11A

0)3. Oo

3. moo cou

0)COouu

tIBotHV

4JVo

to>u

0) uc

•H

0)6

0)o

(UCO

c:o

a,

00

0)

i-i a.

0) 4J

00 •d

0)I00 •H

00

(U3

00

•H >O >o oo

S3TOA

100
1.0

O^ 0.5

Time, microseconds

Figure 7-7. The divergent properties of the series (7-38) using the

parameters for Cable I. The parametric variable 10

corresponds to 10 terms comprised of 0.5 u(t) plus the

algebraic sum of the weighted values of w. (t) through w-(t).

Similarilarly, 15 corresponds to 15 terms obtained from the

sum through w, (t); and so on through w_(t) for 36 terms.

101
I

CO3.

(0BOOo

bO

CO Jo

-apei H

<U iH

jO

nJu ^ M-lo

(U CO CO 3. co O a

0) en

m S 0) •H t-i H a

0) p

c» • in

<u

m

00 • CO o

4-1 tH en o •H >

a- TJ (U c 4J

(U CO 60 iH c CO •H •H 13 U C •H

(U c

CO •H

0) <U X. ,c H H oo

c»

(1)

P"4 o

iH o oo

S3TOA

102
50 fi AAA—

^e(t 2 ^g^^(J- fe (t)]

2 s I2it 62 ^

;

i •."© 62 fi

^\/V^ O" 50 f^

Figure 7-9 The measurement system for observing the sending-end step

response. e (t) is a unit s^tep generator.

103
REFERENCES

[1] Johnson, W. C, Transmission Lines and Networks, McGraw-Hill, New York 1950,

[2] Nahman, N. S., A discussion on the transient analysis of coaxial cables considering high-frequency

losses, IRE Trans. PGCT, CT-9 , No. 2, 144-152, June, 1962.

[3] Nahman, N. S. , Signal waveform metrology at NBS, WESCON 77, Western Electronics Conference,

Sept. 19-21, 1977, San Francisco, California (10 pages).

[4] Terman, F. E. and Pettit, J. M. , Electronic Measurements, (McGraw-Hill, New York, NY, 1952), p. 113. Also, note discussion of capacitance measurements, p. 102.

[5] McCaa, Jr., W. D. and Nahman, N. S., Generation of reference waveforms by uniform lossy transmission

lines, IEEE Trans. Instru. and Meas. , Vol. IM-14, Nov. 1969, pp. 382-390.

[6] Cans, W. L., Present capabilities of the NBS automatic pulse measurement system, IEEE Trans. I&M,

No. 4, 384-388, December 1976.

[7] W. L. Cans and Andrews, J. R., Time domain automatic network analyzer for the measurement of RF

and microwave parameters, NBS Tech. Note 672, September 1975.

[8] Baird, D. C, Experimentation: An Introduction to Measurement Theory and Experiment Design,

Prentice-Hall, Englewood Cliffs, N.J. 1962.

[9] Stout, M. B., Basic Electrical Measurements, 2nd Ed., Prentice-Hall, Englewood Cliffs, N. J., 1960.

[10] Brigham, E. 0., The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, N. J., 1974. [n] Tables of the Error Function and its Derivative, National Bureau of Standards, Applied Math Series,

No. 41, 1954.

[12] Schwartz, M. , Information, Transmission, Modulation, and Noise, McGraw-Hill, New York, N.Y., 1959,

pp. 381-384.

[13] Tai, C. T. , Transients on lossless terminated transmission lines. IEEE Trans. Ant. and Prop.,

AP-26 , No. 4, 556-561, July 1978.

104
APPENDIX A. MEASURED AND MODELED CABLE DATA

A.l Cable Parameter Tables

i.<u

!-> OJE

&.(O

5^

I

<U jD

to

Q

' O ^^ 1— ^_^ .— ^_ 1— ^^ ^ ^^ ^ ^__ ^^^ ^_^ .— ,^

Jacket

idl

and

i

>>c

•r->O

E

E

Uf>

•

c

LO

. o

>>c

>o

10.80

mm

0.425

in

>>c

>

E

E

CO

CM

lO

c

•rLOO

CM

• O

lack

viny

6.15

mm

0.242

in

black

pvc

6.22

mm c

•t—

LOCM

• o

black

pvc

10.67

mm

(0.420

in >>c

•r->

3

E

E

lO

.-H

. VO

c

•f- CM

CM

• o

>

o.

0)3

3.81

mm

0.150

in >

o.

a;3

6.22

mm

0.245

in

lack

viny

10.67

mm

0.420

in

X}

"-^ Xi

"-' ^

—

£i *^ *-' £l "-^ "^ *-' £i "^

+i

<o S

Q •o T3 S- E i- E i. E E S. en 0) a; (. <U E ^_^ *» E^ (U s~ E^ i~ E^ <U

;hield(s) i•a

1

and

i

o.

o.o

o

o

a;c

•a

•r- 1

1

1

!-> <oO

O

a;>

i-o

o.

o.o

o

1

«so

o

>

i.Ol

o.O

1

1

are

coppe

braid

nned

copp

aid

4.70 c

•1—

LO00

t—

t

. o

nned

copp

aid

8.07

0.318

in

nned

copp

braid 1

1

are

coppe

aid

2.74

0.108

in

are

coppe

aid

4.95

0.195

in

nned

copp

braid

4-> c •r- .,- XI •1- S-

'~-' •r- t- •1— £i i-^-" x> s- "^

•I— •M 1/) (A +J ^ •»-> JH 4-> £i ^ •i->

rtJ s:

Q

O 0) 0) O) <u Q) 0) O) Q) <D O 0) c ^— c ^~ C '— c *— C —

•

C *—

»

c '—^ C ^— c — •r- T3

4-> «0 c

E c

0) E c <u E c 0) g c <^ E

c <u c <u E

c 0) E.'=<u E = (U e C

<u E •r— >> E

•^

>> E

•r- "^^'^ 7.^

•^ 7,^^ >> E

*"

>> E*^ >, E

" •^^•^

U

O) 1—

f— <o 4-> LO

cri U3

1—1 «—

1

o

CM

LO

COCM a;

LO «—

1

CM O»—

1

o<*CM a; :^-

.^

•r- $- >, CM >^ t^

• o

>> CM

• o ^-o ^^ o ^-o >> lO

• o

>i ^O >> -"o ^-o O Ol 1— O —^ o %_« o »— o '«_^ O ^-' o •_^ o < o •^^ O '-^ M o o. o. a. o. o. o. Q. o. D.

<o o. S

in •

tor(

O.D XJ E

E o

a; E

E

E

E |i £i ST3c

E

E

T3c

E

E

XJc

E

E si c i- *"" 3 C

T3 «0 c4^ 00

• c

4-> o

LO

CVJ

• c

o • c -S Sic

•f— en c

•1— LO C

lO •!- c

•r4-> cy> •!-

• ^2 =

gnal

Con^

laterial

"Oc

4->

copper

(0.032

o

>

1/)

copper

2

(0.089

o

>

copper

(0.036

copper

c(

steel

0.

(0.022

Stranded

copper

(0.013

bare

co

1.70

(0.067 o

c

copper

(0.038 c:

•M

copper

(0.022

x>

0) o

c

<o

scopper

(0.039

stranded

copper

1

(0.046

•f- SSL (/) </l

=> Z3 =3 =3 =3

-^. ^^ -^» *«s. >«.^ o ^ ro CQ 00

<u 00 .-1 CM O^ CM

I— LO CM CM lO CM ^ 1 1 1 1 1 <o C3 C9 cs o CD o • C^ OC a£ on <: CO U o LU Of

105
Q)

• C Q OJ

• r^ O ««•—>.. ^>«k ^^-«. ^.^-^ ^ ^ E P E

QJ

^mm^

4J C

a. E

"^ > ^ «= a E

"^ > E «= |i- c

•r- u

>

a.

Jacke

rial

a

black 6.22 0.245 bl

ack

10.67 0.420 bl

ack

6.22 0.245 black

10.67 0.420 bl

ue

6.22 0.245 poly

12.19 0.480

1 O 1

^-^ •*—

*»-«^ '-•"^ •^-^ .^ •i^-* <D o

4-> <o

(O ^ 2: ^

Q s. E S- E s- S- E E E

C OJ E^ ^ £^ QJ ^_^ O E^ i~ E^ S- S- E^

—- c

So- O-OC .^ Se- O-CC .^ O-LD *- QJ ^-^ Q.CVJ or-.'" OO ^ O E "^ OO -^ O.CT1 '" O.—

Qcr*

'"

<LI 1—

o

oc —. "—1 :"2

T f^

H- ^ CO "co--

o • O 1

1 ucv.;:^

•r- (0 O) X) 0) -o . q; . QJ -O . QJ -O 0) TO QJ T3

OO S- ^5 --S "£ --S 03 «

'-' •^ t. ' •I- S- ' •^ •r- S- ^ S- ^^

XI J2 S_ ""^

•o

4-> JE +-> ^ P P Xi ^ ^

3

c

+j

c Q c; o c

u O cu QJ QJ OJ <u

a; 0) o c ,—

c -^ c -

c —> c ^- ^~ ,,-.

.—

1

•1- TJ

+J TO

QJ E =

E "^ ii- -:ipolyethy

7.2

mm c

ypropyle

2.29

mm

0.09

in)

OJ

1— i

e

1

e

c

rial QJ 4.11 0.162 lyeth 7.21 0.284 lyeth 4.11 0.162 lyeth 7.21 0.284 lyeth 2.06 0.081

Lf) CO

CVI

• o Q <U O > - O >~' o ^ o *^ O w- '~~ ^-^ o. a D. Q. Q. E o

s: O

to • ^^ Q

o o

p

E

E Si £i £i UlUl pau ti

E

QJ E

3 C

(0

.-1 c

CO •.- ^^5 -s3 -£ =

7\ Lr> •!- -s c

TO CVJ "TT

"C fO • • +^ r^

'"" co ^

c_) m

QJc er

.032 inded er 1.053 inded er

.022 nded er.038 ded er

.022

nded er

2,

.108

nded r

0.

.030

i

g

n

a

1

Mater +-> Q-O

O

O

stra

coppi (0 stra

coppi (0

stran coppi (0 "1° stra

coppe (0

OO

O) t^

r— CO

JD

<o Q CJ) u_ to 3: »—

•-D :^ 3

106
i.Q)

O)E

<ouu

a;

CVJI

/-^ ^ CO LO ^-^ 1 o m VO CO O r- CM CO LO LO CO y—t CM CD o CD CO c 1 1—{ o r^ as LO 1—4 LO CO t^ CM "(d- r- ^ CO CO LO CD. to <o O r^ CO VO <T> P>. i-H CM CM LO CTi LO CM LO LO CD cr. LO

«/)p I—1

vo in VD 1—1 CM CM 1—4 LO 1—4 CO 1—4 P-. 00 LO <D CO 00 o t/1 X .—t O 1—1 CM LO CM 1^ ^ m CM LO 1—4 r-^ CM LO o O

_i c ^.^ • • • • • • • • • • • • • . • • • o o o O c o o o o o o o o o o o o o u '^

ir> 1—1 «d- «* 1—1 ^ Lf) r^ <i- CO LO CD CM CM CD CD CM o

<1> co

r«~ r^ 1—1 00 00 CM a\ o 1

o LC 1—4 LO LO 1—4 CM CO

criO CO OJ

a r- CO CO 1—1 1—4 LO 00 CM a-, LO 00 CM _E

CM LT CM CM c CM i-< Lf) CM CO 1—4 «* c CO 1—4 LO r^ • LT in• • LO Lf)•

Lf)• Lf) Lf)• LO LO• lO LO LO LO lO LO

W) o O o o o o o O o O o c o o c O

C

*-^ ID *— *—.

^-^ *-^ <-~ *-~. *—>

,—.

,»—>««-. -.-~ ^-^ <—« r^ -—. a

J- ^

CO• «—1• ^C3• o•<3~i 1-H 00• o o 1—4• ^ LC• CO oc• 00• CM• o c +J 1-* ro O Lf) P-- LO o o 1—4 CO LO CD CO 00 LO CM CM

<o <v c t—t

v..^ 1—4 *^ 1—(«_' CM LC CM 1—4 CM '>—>

«d- 1—4 LO •_^ ^

•M E •^ «— >w '«^ '•.i^ •—'

._« •.^ «_^ •..^ •-^ .^ <_«

«/) LO

•r- s. *—.

.—1 CO 00 LO 1^ o CM CT> CTi o LO LO 1—4 Px- «^ '^l- CO

«/l O) *-> • • • • • • • • • • • • • • • . • <U DuU- r^ o <;f t^ 00 o 00 LC 00 CO LO 1—4 o 1—4 ^ r^ r^

a: ^ n T-H CO

1—1

Lf) CM LO

1—4 LO ^ 00 CO LO

1—4 LO 00

1—4 ro1-4 o o o <T> CM CM a-' a r^ ^ 00 00 1—4 1—4 1—4 o o

0)uc

a;

4->

i^ r*. p^ un CT^ cri en CTi r»~. •;t cy> CD. CD CD CD LO CD

r^ r^ r^ 1—1 1—1 1—4 1—1 1—4 CM ^ ^ «d- 00 CO 00 LO P>- «o a; c: 1—t

1—t

1—4 1—4 *"• 1—4 1—4 t—41—4 1-H 1—4 1—4 1—1 +J E •- ^^ ^ ^"^ ^""^ *"' ^^ '"' **^ '^ "^"^ "^ ^^ "^ ~^ '"' o3 i. ,—.

^o <x> ^o CM 1—1 1—4 ^ ^ o 00 LO LO ^ ^ «d- CO CM o O " CSJ CM CM o 1—1 1—4 CO CO <T) CO 1—4 1—1 o o o CO r-. tn LO LO 00 CTi Oi CTi CTi 1—4 t^ CT) CD CM CM CM 1—4 00

CM C\J CM CO CO CO CO CO < -d- ^ «3- LO LO LO CM LO ,

(U «<- CO• oc• 00 vo• LO LO r- 1^ o o LO LO CO CO CO

O CO

Q.C o o

CO o

CO oCM •

1—4

<T> 1—4 CD

1—4

1—1 LO

1—4 • LO

1—4 • LO

1—4 CM

1—4 • CM

1—4 CM

1—4 • LO

CM • CO

1—4

4-> (1) •1-

"*"'' '—' ^"^ '—^ "** *~^ """' '*"' ^*^ *~^ "^^ ^^ """^

U E

„^ o o o• • CO CO LO LO CD LO CM CM •d- «d- «d- CO LO

.—4 o

t—1 o1—t

1—t c LO

«3- LO vo LO LO 00

LO

CM

LO

1—4 LO

i-< LO o

«d- o

«d- o LO00

CO

O

=3 =3 Z2 =3 =3 O «3- CO cc CO

<U 00 1—1 CM CT^ CM r^

p— LO CM CM IT) CM CO ^ 1 1 1 1 1 1 <o ts o o C3 cs Q o en a: cc Q^ «c CD <_) O LU ai U. iD 3: 1—•

•-3 ^ 3

107
A.l Cable Graphical Data

100

,-> ^-l-.\v. ^

,.-'

*- ^

-7-='

_^

^

^"^

10 100 1000

Figure A-1. Modeled attenuation plot for 304.8 meters (1000 feet) of RG-58C/U. Ordinate units are decibels and abscissa units are megahertz.

)Ul/ - - -hH- —

^ ^^^

»" ^ ——

^

^^'^ ^

00 - ^ ^^

" .^

,^

s^ ^^

10 -

10 100 1000

Figure A-2. Modeled minimum-phase shift plot for 304.8 meters (1000 feet) of RG-58C/U. Ordinate units

are degrees and abscissa units are megahertz.

108
1.09x10

-2

0.0 - ' r--^^ 1 I I I I

500 1000

1.09x10

Figure A-3. Modeled and computed time domain impulse response for 304.8 meters (1000 feet) of

RG-58C/U. Ordinate units are seconds , abscissa units are nanoseconds and time

spacing between points is 0.9766 ns. -2

0.0

125 250

Figure A-4. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000 feet)

of RG-58C/U. Ordinate units are seconds , spacing between points is 0.9766 ns.

abscissa units are nanoseconds and time

109
1.0 -

0.5

0.0

500 1000

Figure A-5. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of RG-58C/U.

Ordinate units are volts and abscissa units are nanoseconds.

1.0 -

0.5 -

0.0

10°

Figure A-6.

"

k^ 1 \ / \ /

N /

i

/\ / \ /f \\

^w.,.-

I I

10= iqIO

Plots of "zero'V'one" cable unit step response voltages versus logig frequency for 304. meters (1000 ft) of cable RG-58C/U. Ordinate units are volts and abscissa units are hertz.

110
1.0

0.5

0.0 -

^

.

1 0.0 2.5

I 5.0

Figure A-7. TDR unit step response for cable RG-58C/U. Source step generator has assumed 50 Q. resistive output impedance. Ordinate units are volts and abscissa units are microsecaads100

10

' 1 1 II

1 1 II

1 1 1

r itn

yy itll

y

y^

/" X y

y^

^ ^

y^

y

,,-'

100 1000 10000

Figure A-8. Modeled attenuation plot for 304.8 meters (1000 ft) of RG-214/U. Ordinate units are decibels and abscissa units are megahertz.

Ill
1000 - i

1 "^ "" T" '

'

-' L^

/^

X

>X

100 - y'

.^

-/ f^

> /

,y y7^

10 -

1 1 10 100 1000 10000

Figure A-9. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of RG-214/U. Ordinate

units are degrees and abscissa units are megahertz.

2.46x10"

0.0

1

1

\V\^

250 500

Figure A-10. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of RG-214/U. Ordinate units are seconds , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns

.

112
2.46x10

-2

0.0

Figure A-11.

1.0

A

\

\

\

\\

kV , \

, ^

\^

1 — _.,

0.5 -

0.0

Figure A-12.

62,5 125

Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of RG-214/U. Ordinate units are seconds" , abscissa units are nanoseconds and time

spacing between points is 0.9766 ns.

,

- — ^

/

/

/

/

250

Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of RG-214/U. Ordinate units are volts and abscissa units are nanoseconds.

500

113
1.0 -

0.5 -

0.0 -

Figure A-13. Plots of "zero"/"one" cable unit step response voltages versus login frequency for 304. meters (1000 ft) of cable RG-214/U. Ordinate units are volts and abscissa units are hertz,

1.0 -

0.5

0.0 - __ _

0.0 2.5 5.0

Figure A-14. TDR unit step response for cable RG-214/U. Source step generator has assumed 50 52 resistive output impedance. Ordinate units are volts and abscissa units are microseconds

.

114
100 -

I prj

^X

^

::

^-

^''

V'

10 - Z-

^

,^

^ -

1 -

1 ^° 100 1000

Figure A-15. Modeled attenuation plot for 304.8 meters (1000 ft) of RG-223/U. Ordinate units are decibels and abscissa units are megahertz.

1000

100

10 -

V'

^y^

,-^'^

-y.-^^

^^-

^'-

\~2'^

y^

1 10 100 1000

Figure A-16. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of RG-223/U. Ordinate

units are degrees and abscissa units are megahertz.

115
1.35x10

-2

0.0

1

1

\

\\

500 1000

Figure A-17. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of RG-223/U. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns. 1.35x10

.J I

125 250

Figure A-18. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of RG-223/U Ordinate units are seconds , abscissa units are nanoseconds and time

spacing between points is 0.9766 ns.

116
1.0

0.5 -

0.0

500 1000

Figure A-19. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of

RG-223/U. Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5 -

0.0

-—

,

/

/

/

/J

X

\

\

\\

.0 10

Figure A-20.

10" 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus log.Q frequency for 304. meters (1000 ft) of cable RG-223/U. Ordinate units are volts and abscissa units are

hertz.

117
100 -

10

1 -

Figure A-21,

100 1000

Modeled attenuation plot for 304.8 meters (1000 ft) of RG-59B/U. Ordinate units are decibels and abscissa units are megahertz.

.KJ\J\J —

~ "" ^mm» ~ ~ " I

~

"" "

~ "" ^ " " "

^_ A ^

^

^

X

X

^^

100 ^ " ^

,^^

^ ^' X

> ^

^

^

1 n

.

I 1 1 10 100 1000

Figure A-22. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of RG-59B/U. Ordinate

units are degrees and abscissa units are megahertz.

118
2.18x10

1

1\

V^—

2.18x10

0.0 -I

I 500 1000

Figure A-23. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of RG-59B/U. Ordinate units are seconds , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

2

ft

\

\

4\

..^

\V^

I 1 0.0 -I

Figure A-24.

125 250

Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of RG-59B/U. Ordinate units are seconds" , abscissa units are nanoseconds and time

spacing between points is 0.9766 ns.

119
1.0 -

0.5 -

0.0

500 1000

Figure A-25. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of

RG-59B/U. Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5 -

0.0

k /

.—

_

\ / \ /

v

A

/ \

// \^ \

^0 10

Figure A-26.

10" 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus login frequency for 304. meters (1000 ft) of cable RG-59B/U. Ordinate units are volts and abscissa units are

hertz.

120
100

10

^ ~" ^< - n n

1 ' ^"^ 1 ^

>

'^

Z' ^ y y

y" y

y

^^'

y^

, '^

,-'

' ^

^

L- ...

0.1

Figure A-27

.

10 100 1000

Modeled attenuation plot for 304.8 meters (1000 ft) of A. Ordinate units are decibels

and abscissa units are megahertz.

1000

100 -

10

~~ ~~~ ""^ ^

"""

,'

-« ^^'

> /

*'^

^

.-' y

~?

r*

>

y

_ - ^

.'' ^ y

J 0.1 10 100 1000

Figure A-28. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of A. Ordinate units

are degrees and abscissa units are megahertz.

121
1.96xl0x"

0.0

1

\

\^-—,

Figure A-29.

2500 5000

Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of A. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

1.96x10

0.0

A

\

\

\

\

\

\\

625 1250

Figure A-30. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of A. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns

.

122
1.0

0.5

0.0

I 7 I I

2500 5000

Figure A-31. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of A. Ordinate units are volts and abscissa units are nanoseconds.

1.0 -

0.5

0.0

""*- ^ r

\I /

\ ,

/

\

YA

/ i

y

f

\ [ y \

.0 10

Figure A-32.

10" 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus logiQ frequency for 304. meters (1000 ft) of cable A. Ordinate units are volts and abscissa units are hertz.

123
JOU -

^_ — —

~ "

100^ —

,

?:;

:::

" "

10. - ^

— . .

.

-,^-. — ^^

. .

.

"

1 - _.

10 100 1000 10000

Figure A-33. Modeled attenuation plot for 60.96 meters (200 ft) of B. Ordinate units are decibels and

abscissa units are megahertz.

10000-

1000

100

10

"^ZL — _= = — ; ;

-r- :: ^

rf-'i^

. ::=^=ar^

^^^

h—^— --?'^^ — — H *"

^ ^

I 10 100 1000 10000

Figure A-34. Modeled minimum-phase phase shift plot for 60.96 meters (200 ft) of B. Ordinate units

are degrees and abscissa units are megahertz.

124
1.409x10

-2

1

\

\

• V . '^

"•" I I I 25 50

Figure A-35. Modeled and computed time domain Impulse response for 60.96 meters (200 ft) of B. Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns. 1.409x10

0.0

r1 V

1

' \

\

\

\ 1 > V -—

1

J \sk^

1 V.

/

6.25 12.50

Figure A-36. Time-expanded modeled and computed time domain impulse response for 60.96 meters (200 ft)

of B. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

125
1.0

0.5 -

0.0 -

—

A

/

/

/

/

/

/

/

I 25 50

Figure A-37. Modeled and computed time domain unit step response for 60.95 meters (200 ft) of B. Ordinate units are volts and abscissa units are nanoseconds.

1.0 -

0.5

0.0

-^^ \r

\

-/J

,0 10

Figure A-38.

10- 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus logi^ frequency for 60.96 meters (200 ft) of cable B. Ordinate units are volts and abscissa units are hertz.

126
UUU - = = Fffll

1

100,

::::^

^ - . ,

. . 1

10,

,--<=—

1 _ ^ 1

1 1 1 ^mAMS

10 100 1000 10000

Figure A-39. Modeled attenuation plot for 152.4 meters (500 ft) of B. Ordinate units are decibels and

abscissa units are megahertz.

10000

1000

100

10 -

—~. —=

II 1 — NmtH „ -f §.• Ill

Vlllll

. — —

^^^^

,

1 ^

^ ^

^ ^

.'-

" " ^

.^

w^ ^*

*^

I 10 100 1000

I 10000

Figure A-40. Modeled minimum-phase phase shift plot for 152.4 meters (500 ft) of B. Ordinate units

are degrees and abscissa units are megahertz.

127
1.147x10

0.0

1

N.

1

'

100 200 Figure A-41. Modeled and computed time domain impulse response for 152.4 meters (500 ft) of B. Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

1.147x10

0.0

l\

\

\

\

\

\\V.

I25

I 50 Figure A-42. Time-expanded modeled and computed time domain impulse response for 152.4 meters (500 ft)

of B. Ordinate units are seconds , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

128
1.0

0.5

0.0 -

Figure A-43.

200

Modeled and computed time domain unit step response for 152.4 meters (500 ft) of B. Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5 -

0.0 -

10

Figure A-44,

10' 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus log,o frequency for 152.4 meters (500 ft) of cable B. Ordinate units are volts and abscissa units are hertz.

129
uu - ^

^-- -

X"^

^^*^

,-^

LO- ?"-

^"-^ —

^ Mi ^ * " "

1- 1 1 ^ 10 100 1000

Figure A-45. Modeled attenuation plot for 333.76 meters (1095 ft) of B. Ordinate units are decibels

and abscissa units are megahertz.

1000 -

100 -

10

^ ^ ^

^

^^ /

^'^^

-"^

^ >'

^^^<

^

I 10 100 1000

Figure A-46. Modeled minimum-phase phase shift plot for 333.76 meters (1095 ft) of B. Ordinate units are degrees and abscissa units are megahertz.

130
1.71x10

-2

0.0

\

\\

1.71x10

500 1000

Figure k-kl . Modeled and computed time domain impulse response for 333.76 meters (1095 ft) of B. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

2

0.0 -

Figure A-48.

A

'\

\

\

\

\\\k.^

125 250

Time-expanded modeled and computed time domain impulse response for 333.76 meters (1095

ft) of B. Ordinate units are seconds , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

131
1.0 -

0.5

0.0

. .

^^^

/^

/

/

/

/

500 1000

Figure A-49. Modeled and computed time domain unit step response for 333.76 meters (1095 ft) of Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5 -

0.0 -

— r

\\

\

/

/

" /

\ 1

.0 10

Figure A-50.

10" 10

10

Plots of "zero'V'one" cable unit step response voltages versus logiQ frequency for 333.76 meters (1095 ft) of cable B. Ordinate units are volts and abscissa units are hertz.

132
uu - ..L ^--'- _..,

,''

y^m

10 • ------A

-^

r- E-^' ^

- . .

1 - ....

0.1

Figure A-51,

10 100 1000

Modeled attenuation plot for 304.8 meters (1000 ft) of C. Ordinate units are decibels

and abscissa units are megahertz.

1000

100

10

0.1

Figure A-52.

"^^^^

, • ^' .''

y

y

/^

,x' r* y

.'' ^

?^^y

1

1 10 100 1000

Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of C. Ordinate units

are degrees and abscissa units are megahertz.

133
1.37x10

0.0 -

^^^

2500 5000

Figure A-53. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of C. Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

1.37x10

0.0 -

625 1250

Figure A-54. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of C. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

134
1.0 -

0.5

0.0 -i

I ]

I I

2500 5000

Figure A-55. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of C. Ordinate units are volts and abscissa units are nanoseconds.

1.0 -

0.5

0.0 -

10

Figure A-56. Plots of "zero'V'one" cable unit step response voltages versus logiQ frequency for 304. meters (1000 ft) of cable C. Ordinate units are volts and abscissa units are hertz.

135
1000-

100

— 1 — — ii

m 1

— — 9^ " ^r^

-.'^

L^

^ - - ' —

., 10 -

0.1

Figure A-57.

1000

100

10 -

0.1

Figure A-58.

1 10 100 1000

Modeled attenuation plot for 304.8 meters (1000 ft) of D. Ordinate units are decibels

and abscissa units are megahertz.

ssaB 1

1

^Bs

I11 ss w

i 1!§s 5

II11 MHHHMHa| IHiimn 1 II m'd 1 i §1^

I III

HHI1^ imm 1 II ^HBlPSm 11 HHH^1 1 III

mnnn 1 HHm I1 1 II PQ I1 11 1 1 1 III

l^g 1 mn 11 1 II PQ 1 1 1111

I11 mi 111 1I

Sfll 11 I1111

111 esi111 111 B H| 111111

'^^^Sus222

1\m0SmiJ52SI2 I iBB^^^^ 1!wim ^

i^^^lH^^1^p^ III'^MHBi 11 1li^BHI ^1 11

HHH11 K

'4 mmHiB I 11 ^H i i 1 II

HHIm 1 ^Hi I 1 1 1 1 IHH1 1 III

mmim a1 1 imiIB I1 ^^Hn I11 1III m 1 1 m 1 ^H 11 1 HH 11III

mi 11 n 11 1111 1111

1111 1111 11111 11111

HM|^M m ^iHH I 1 1 JI^M 10 100 1000

Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of D. Ordinate units

are degrees and abscissa units are megahertz.

136
8.42x10

\

iV,^% -

. 0.0

Q 2500 5000

Figure A-59. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of D. Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 4.883 ns.

8.42x10

0.0 - i i I —^^^

I

Figure A-60.

625 1250

Time-expanded modeled and co.nputed time domain impulse response for 304.8 meters (1000

ft) of D. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 4.883 ns.

137
1.0

0.5 -

0.0

2500 5000

Figure A-61,

1.0 -

Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of D. Ordinate units are volts and abscissa units are nanoseconds.

0.5

0.0 -

\ f

\

J J ,0 10

Figure A-62.

10" 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus logiQ frequency for 304, meters (1000 ft) of cable D. Ordinate units are volts and abscissa units are hertz.

138
100

10

1 -

"^ -^"1 " ^1§>

, *'

' III

y'

y

v* y'

y

y

y y

^_^

'^

_ ^

*

r^

I 0.1

Figure A-63.

10 100 1000

Modeled attenuation plot for 304.8 meters (1000 ft) of E. Ordinate units are decibels

and abscissa units are megahertz.

1000

100

10

:

-i

— y

v^ y

.'•

.-^

^__ — -y — y

1 1 \ I 0.1 10 100 1000

Figure A-64. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of E. Ordinate units

are degrees and abscissa units are megahertz.

139
1.93x10

0.0

1

,

\

v"^ "

—.

Figure A-65.

2500 5000

Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of E. Ordinate units are seconds

points is 4.883 ns

.

-1 abscissa units are nanoseconds and time spacing between

1.93x10

-2

0.0

A

\

i ^\ i \\

V\

\

^ 1 ~"~~-

625 1250

Figure A-66. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of E Ordinate units are seconds"-'- , abscissa units are nanoseconds and time spacing

between points is 4.883 ns.

140
1.0 -

0.5

0.0

^

4 ^

/

/

/

2500 5000

Figure A-67 . Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of E. Ordinate units are volts and abscissa units are nanoseconds.

x.u - ^\ r

\\ \ , <

0.5 - \/Y

A

/ \

J

f

y \

0.0 - V 10 10- 10

10

Figure A-68. Plots of "zero"/"one" cable unit step response voltages versus logj^Q frequency for 304. meters (1000 ft) of cable E. Ordinate units are volts and abscissa units are hertz.

141
100

10

—

-E=i:! —I^

•^

^

y^

^-'

^^-^ —

^ -^' .^ ^

I 10 100 1000 10000

Figure A-69. Modeled attenuation plot for 60.96 meters (200 ft) of RG-22B/U. Ordinate units are decibels and abscissa units are megahertz.

1000

100

10 -

=H-H - + — —

n

Z~''t\

" ^^l..

^d."^ :

,-'•

^

7^ ^

.'^

^

U^-''

*<

^^^

I 100 1000 1000

Figure A-70. Modeled minimum-phase phase shift plot for 60.96 meters (200 ft) of RG-22B/U. Ordinate

units are degrees and abscissa units are megahertz.

142
1.847x10

0.0 -

^^____^_ ^^^_^_^_ ^^_^^_^

125 50

Figure A-71. Modeled and computed time domain Impulse response for 60.96 meters (200 ft) of RG-22B/U. Ordinate units are seconds"'^, abscissa units are nanoseconds and time spacing

between points is 0.9766 ns. 1.847x10

-I

0.0 -

Figure k-11.

A

\

\\

\

\

, \\

\^^

i 1

II III . .

6.25 12.50

Time-expanded modeled and computed time domain impulse response for 60.96 meters (200 ft)

of RG-22B/U Ordinate units are seconds" , abscissa units are nanoseconds and time

spacing between points is 0.9766 ns.

143
1.0

0.5 -

0.0

. _ —^^—

II mmmesa

^^..aMwB^M —.^—»*«H~»^ ^^^—^m^m^^. .^^^-m^mm^.^m^ I ^_^MMi.^_«Mi M.a.^wMMM^ mm^^mamim^m^^ ^^^m^^.wm ^b^^^^^^^h

I 25 50

Figure A-73.

1.0

Modeled and computed time domain unit step response for 60.96 meters (200 ft) of RG-22B/U. Ordinate units are volts and abscissa units are nanoseconds.

0.5 -

— —-

—». r A>

\

\Y

A

/\

1 / \

y \

1 1 ^

\

.0

0.0

10 Figure A-74.

10^ 10^0

Plots of "zero'V'one" cable unit step response voltages versus logiQ frequency for 60.96 meters C200 ft) of cable RG-22B/U. Ordinate units are volts and abscissa units are hertz.

144
100 -

10 -

1 -

1000

100

10

100 1000 10000

Figure A-75. Modeled attenuation plot for 152.4 meters (500 ft) of RG-22B/U. Ordinate units are decibels and abscissa units are megahertz.

rrm

>

"^""^

y

>^^ m

^t

^' y

^^ U-7^-

>

>

^

/

.

10 100 1000

I 10000

Figure A-76. Modeled minimum-phase phase shift plot for 152.4 meters (500 ft) of RG-22B/U. Ordinate

units are degrees and abscissa units are megahertz.

145
1.367x10

-2

0.0

—^^^—^^^ ^^——^^^ ^—^—^—^ ———^^— .^————^ ——^i^^— ^^^—^^-.

^—-Li III

Figure k-11

,

100 200 Modeled and computed time domain impulse response for 152.4 meters (500 ft) of RG-22B/U. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.1953 ns.

1.367x10

0.0 -

A

\

' I\

\\\

k.\•v^

1 -^^

I 25 50

Figure A-78. Time-expanded modeled and computed time domain impulse response for 152.4 meters (500 ft)

of RG-22B/U. Ordinate units are seconds"^, abscissa units are nanoseconds and time

spacing between points is 0.1953 ns

.

146
1.0

0,5 -

0.0

-^ ^^

^^

z'

/

/

/

/

100 200

Figure A-79.

1.0

Modeled and computed time domain unit step response for 152.4 meters (500 ft) of RG-22B/U. Ordinate units are volts and abscissa units are nanoseconds.

0.5 -

0.0

/

/

/

/

\

\

\\

,0 10

Figure A-80.

10- 10

Plots of "zero'V'one" cable unit step response voltages versus logiQ frequency for 152.4 meters (500 ft) of cable RG-22B/U. Ordinate units are volts and abscissa units are hertz.

147
uu - ^' ^

^'-'^

LO ~ ^

^2 - ^^

,*^

L ^

'

1 - 1

1 1 10 100 1000

Figure A-81. Modeled attenuation plot for 304.8 meters (1000 ft) of RG-22B/U. Ordinate units are decibels and abscissa units are megahertz.

1000

100

10

—~ """ ^ " >

^

^

^^

..^

^

^

y

^^ 1 "

^ <^ ^'

k^

•^ ^

^

^

I1

Figure A-82.

I 10 100 1000

Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of RG-22B/U. Ordinate

units are degrees and abscissa units are megahertz.

148
1.84x10

0.0

500

Figure A-83. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of RG-22B/U. Ordinate units are seconds , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

1 . 84x10

0.0

^

\

\

\1

I\\\V^

..^

125 250

Figure A-84. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of RG-22B/U. Ordinate units are seconds" , abscissa units are nanoseconds and time

spacing between points is 0.9766 ns.

149
1.0 _

^

^-^

J 1

0.5 - /

/

/

/

0.0 - 500

I 1000

Figure A-85. Modeled and computed time domain unit step response for 30A.8 meters (1000 ft) of RG- 22B/U. Ordinate units are volts and abscissa units are nanoseconds.

1.0 -

0.5

0.0

Plots of "zero'V'one" cable unit step response voltages versus logig frequency for 304. meters (1000 ft) of cable RG-22B/U. Ordinate units are volts and abscissa units are hertz.

150
1.0 -

0.5 -

,o.i

^ - -

1 I 0.0 2.5 5.0

Figure A-87. TDR unit step response for cable RG-22B/U. Source step generator has assumed 50 fi resistive output impedance. Ordinate units are volts and abscissa units are microseconds.

100 -

10

^^^

— I I j _(^

^§^1

y^ III t

* * nil

y y

y

y'

y

.-'

,••'

I II 1 ^' ^

I 0.1 10 100 1000

Figure A-88. Modeled attenuation plot for 320.04 meters (1050 ft) of F. Ordinate units are decibels

and abscissa units are megahertz.

151
1000

100

10

^"*" ^ ^a... laaaiil

11

11 III

_r> III /^

<-'

x*

x''

? / " * ^ ^

^ /^ ^'

^

0.1 10 100 1000

1.29x10

Figure A-89. Modeled minimum-phase phase shift plot for 320.04 meters (1050 ft) of F. Ordinate units

are degrees and abscissa units are megahertz.

-2

0.0

Figure A-90.

1000 2000

Modeled and computed time domain Impulse response for 320.04 meters (1050 ft) of F.

Ordinate units are seconds"^, abscissa units are nanoseconds and time spacing between

points is 1.953 ns.

152
1.29x10

-2

0.0

Figure A-91. Time-expanded modeled and computed time domain impulse response for 320.04 meters (1050

ft) of F. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 1.953 ns.

1.0 -

0.5

-f-

"7

'

(\ C\ - [ I

1000 2000

Figure A-92. Modeled and computed time domain unit step response for 320.04 meters (1050 ft) of F.

Ordinate units are volts and abscissa units are nanoseconds.

153
1.0

0.5 -

0.0

/

\

\ / \ /

\ /

)

/\

/ \ / — \

\L_

^0 10

Figure A-93.

10- 10

I 10

Plots of "zero'V'one" cable unit step response voltages versus log.Q frequency for 320.04 meters (1050 ft) of cable F. Ordinate units are volts and abscissa units are hertz.

100 -

I — ~~~ """T "^ ^

^^^ ^

^

x*

y

^'' ^

^

y^

10 ~ ^

' ^

^

0^ ^ ^

^

-I

10 100 1000

Figure A-94. Modeled attenuation plot for 323.09 meters (1060 ft) of G. Ordinate units are decibels

and abscissa units are megahertz.

154
1000 -

100

10

, 5^

,y^ ^

^

^^"

-^_^

^ -"' .^-^ ^^ >

^

_.

1 10 100 1000

Figure A-95. Modeled minimum-phase phase shift plot for 323.09 meters (1060 ft) of G. Ordinate units

are degrees and abscissa units are megahertz.

2.10x10

-2

1

1

I

\

\

'

1

1 0.0

Figure A-96.

500 1000

Modeled and computed time domain impulse response for 323.09 meters (1060 ft) of G.

Ordinate units are seconds'^, abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

155
2.10x10

0.0

Figure A-97

125 250

Time-expanded modeled and computed time domain impulse response for 323.09 meters (1060

ft) of G. Ordinate units are seconds , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

1.0 -_

0.5

,^

^^

/

/

/

/

/

0.0 1

Figure A-98

,

I 500 1000

Modeled and computed time domain unit step response for 323.09 meters (1060 ft) of G. Ordinate units are volts and abscissa units are nanoseconds.

156
1.0 -

0.5

0.0

Figure A-99 • Plots of "zero"/"one" cable unit step response voltages versus logiQ frequency for 323.09 meters (1060 ft) of cable G. Ordinate units are volts and abscissa units are hertz.

157
100

10

' 1^wmwkmm%%\

"I

/"

^11

/ y y

.'

.'

^yy

^

:^

y^ '

.^

• ^

1

1 10 100 inno 10000 Figure A- iqO. Modeled attenuation plot for 152.4 meters (500 ft) of H. Ordinate units are decibels and

abscissa units are megahertz.

1000

100

10

' '""'

'

~" ^1aaaiii III

-

- -iS ...

^ y

A

i^

y^

X

y'

''

^^

' 1 ^ y^

^

-'' ,.-'

1

10 100 1000 10000

Figure A- 101. Modeled minimum-phase phase shift plot for 152.4 meters (500 ft) of H. Ordinate units

are degrees and abscissa units are megahertz.

158
1.315x10

0.0

1 —i—1

1.315x10

250 500

Figure A-102. Modeled and computed time domain Impulse response for 152,4 meters (500 ft) of H. Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points Is 4.883 ns. -2

0.0

A

" "

' \\

\

\\\

s.N

—

62.5 125

Figure A-103. Time-expanded modeled and computed time domain impulse response for 152.4 meters (500 ft) of H. Ordinate units are seconds"^, abscissa units are nanoseconds and time spacing

between points Is 4.883 ns.

159
i.U -^

y^

0.5 /

/

\

\

0.0

250 500

Figure A-104. Modeled and computed time domain unit step response for 152.4 meters (500 ft) of H. Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5

0.0

Figure A-105. Plots of "zero'V'one" cable unit step response voltages versus log,Q frequency for 152.4 meters (500 ft) of cable H. Ordinate units are volts and abscissa units are hertz.

160
100 -

10

1 -

•

• "~ §.1 IB «->•

11

II III

.'' III ^

* y

^ .^

y

.-'

,-'

' ^

-'' ^

,y

L0.1 10 100 1000

Figure A-106. Modeled attenuation plot for 326.14 meters (1070 ft) of H. Ordinate units are decibels

and abscissa units are megahertz.

161
1000

100

10 -

1 1 ^"^v "" — —

^^""

i^^M laaaii

III 1

y

y'

y

—

^^ y

- T^ —

IIIII 1

— -- < ••'y

y

0.1 10 100 1000

Figure A-107. Modeled minlnmm-phase phase shift plot for 326.14 meters (1070 ft) of H. Ordinate units

are degrees and abscissa units are megahertz.

3.03x10

0.0 -' 2500 5000

Figure A-108. Modeled and computed time domain impulse response for 326.14 meters (1070 ft) of H. Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 4.883 ns.

162
3.03x10

-2

VV

0.0 -

I I

I 625 1250

Figure A-109. Time-expanded modeled and computed time domain impulse tesponse for 326.14 meters (1070

ft) of H. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 4.883 ns.

1.0 -

0.5 -

0.0

y

^

/

/

/

2000 5000

Figure A-110. Modeled and computed time domain unit step response for 326.14 meters (1070 ft) of H. Ordinate units are volts and abscissa units are nanoseconds.

163
1.0 -

0.5

0.0

10 10- 10'

Figure A-111. Plots of "zero'7"one" cable unit step response voltages versus logj^Q frequency for 326.14

meters (1070 ft) of cable H. Ordinate units are volts and abscissa units are hertz.

1.0

0.5

0.0 I I 1 I ________ —^^— I I I 2.5 5.0

Figure A-112. TDR unit step response for cable H. Source step generator has assumed 50 ^ resistive

output Impedance. Ordinate units are volts and abscissa are microseconds.

164
100

10

-H-- — ?'

!^.: ."?* ^ ^ ,X ^

.'^

y

y^

_^

y-

^

_ ^ ^ y

10

( 100 1000 10000

Figure A-113. Modeled attenuation plot for 60.96 meters (200 ft) of I, abscissa units are megahertz.

Ordinate units are decibels and

165
1000

100

10 -

"^ -r " ^^" r ""*

1 2 -.-^

^ ^' ^ / ^

y"

X

_^ " ^ f^ ^,.- ,-^

^

^

1 10 100 1000 10000

Figure A- 114 Modeled minimum-phase phase shift plot for 60.96 meters (200 ft) of I. Ordinate units

are degrees and abscissa units are megahertz.

1.886x10

-2

0.0

Figure A-115. Modeled and computed time domain impulse response for 60.96 meters (200 ft) of I. ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

166
1.886x10

0.0

A

\'\

)

\

\\\ k

, V>^^^

'

t

1

6.25 12.50

Figure A- 116. Time-expanded modeled and computed time domain impulse response for 60.96 meters (200 ft)

of I. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

1.0 -

0.5 -

0.0

/

/

/

/

25 50

Figure A- 117. Modeled and computed time domain unit step response for 60.96 meters (200 ft) of I. Ordinate units are volts and abscissa units are nanoseconds.

167
1.0

0.5

0.0 -

~ -V r

\ 1

\ /

Y

A

/\

/ '

L y \ —

-/ V

10

I 10- 10'ao

Figure A- 118. Plots of "zero"/"one" cable unit step response voltages versus logj^Q frequency for 60.96 meters (200 ft) of cable I. Ordinate units are volts and abscissa units are hertz.

100 -

10

1 -

- = :: —

j

. .-—^ 'El

>^

*' y

' y

y y

.''

r

>

;z —^

^

y^y

I 10

I 100 1000 10000

Figure A-119. Modeled attenuation plot for 152.4 meters (500 ft) of I. Ordinate units are decibels and abscissa units are megahertz.

168
1000

100

10

— 1 1 [i[||[

_> j^

yy

' yy^

4. _ ^-- ^^ _ .^ ^ ^ yX

10 100 1000 10000

Figure A-12G. Modeled minimum-phase phase shift plot for 152.4 meters (500 ft) of I. Ordinate units

are degrees and abscissa units are megahertz.

1.526x10

0.0 -I

I

1

\ I

100 200

Figure A-121. Modeled and computed time domain impulse response for 152.4 meters (500 ft) of 1. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

169
1.526x10

-2

0.0

A

\

\

\\\

V

X.

^"-^

J

I 25 50

Figure A-122. Time- expanded modeled and computed time domain impulse response for 152.4 meters (500 ft)

of I, Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

1.0

0.5 -

0.0 -

.—

_^

^^

/

/

/

/

/

100 200

Figure A-123. Modeled and computed time domain unit step response for 152.4 meters (500 ft) of I. Ordinate units are volts and abscissa units are nanoseconds.

170
1.0 -

0.5 -

0.0

Figure A-124. Plots of "zero"/"one" cable unit step response voltages versus logj^Q frequency for 152.4 meters (500 ft) of cable I. Ordinate units are volts and abscissa units are hertz.

100

10 -

1 -

^ ""^ ^^^~

^ 1 ^^^ ^

^

^'•'

— ^

,^

10 100 1000

Figure A-125. Modeled attenuation plot for 320.04 meters (1050 ft) of I. Ordinate units are decibels

and abscissa units are megahertz.

171
.\JiJ\J - ~n ^^ JILL T

, ^

''^

_s^

^

^

^

100

" ^^ ^ ''^ ^i. ^' ^

^

^

L1 10 100 1000

Figure A-126. Modeled minimum-phase phase shift plot for 320.04 meters (1050 ft) of I. Ordinate units

are degrees and abscissa units are megahertz.

1.831x10

O.C -

L

500 1000

Figure A-127. Modeled and computed time domain impulse response for 320.04 meters (1050 ft) of I. Ordinate units are seconds" '^j abscissa units are nanoseconds and time spacing between

points is 0.9766 ns.

172
1.831x10

0.0

1 I

\

\

\\V'^

' -

125 250

Figure A-128., Time-expanded modeled and computed time domain impulse response for 320.04 meters (1050

ft) of I. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

1.0 -

0.5

0.0

!! ^ ^^^m^^m^m,mm^^ » ^» »» »» I i — » m^^^^im^^^^ ^^mm^mmm^^m^ H^^^^^MHIH HMaHMMHM^

W .-—^— ...^....i..— ^...........^^ ^^_^_.,^ ^^^^.^^m^^, ^^^^m^^mm^^ \ mn^^^^m^pm M^B^^^^^^

^m^L^^m^a^t iii»M« ! I—.. I II II I —^—i WMM—_>M>a^_ m^^m^i^^mmmmm ^^^^^^^^^^ ^^^^HIH^^BB L ^

I 500

I 1000

Figure A-129. Modeled and computed time domain unit step response for 320.04 meters (1050 ft) of I. Ordinate units are volts and abscissa units are nanoseconds.

173
1.0

0.5 -

0.0

/

" \

/

\N

}y \

1 / \

10 10- 10

10

Figure A-130. Plots of "zero'V'one" cable unit step response voltages versus log,(-, frequency for 320.04 meters (1050 ft) of cable I. Ordinate units are volts and abscissa units are hertz.

1.0 -

0.5

0.0 -

0.0 2.5 5.0

Figure A-131. TDR unit step response for cable I. Source step generator has assumed 50 fi resistive

output impedance. Ordinate units are volts and abscissa units are microseconds.

174
100

10

1

-

mmm I-1 "" " "" §

I *

• ^

^' ^

y ^

/^

^

y" y

.-'

.•'

f^

_^ ^^

•>

t^

0.1 10 100 1000

Figure A-132, Modeled attenuation plot for 304.8 meters (1000 ft) of J. Ordinate units are decibels

and abscissa units are megahertz.

1000

100

10 -

1

1—

— __

—

- . B^n " X — — - — —

I 1

,.'' * y

^

X

-y- .« _ - > — —

>

^

^ ^ - y _ 1

.'' y

X

7^-

1 ^m^md

1 1 I I 0.1 10 100 1000

Figure A-133. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of J. Ordinate units

are degrees and abscissa units are megahertz.

175
2.80x10

0.0

I

[

2500 5000

Figure A-134. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of J.

-1 Ordinate units are seconds , abscissa units are nanoseconds and time spacing between

points is 0.9766 ns. 2.80x10

0.0 -I 625 1250

Figure A-135. Time-expanded modeled and computed time domain impulse response for 304.8 meters (1000

ft) of J. Ordinate units are seconds"^, abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

176
1.0 -

y

^

0.5 - /

/

/

n.n -I

I I I 2000 5000

Figure A-136. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of J. Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5

0.0 -

^ \J /

\ / \ /

\

f

/,

/ \ / \

,TM _y V

10 10- 10

10

Figure A-137. Plots of "zero'V'one" cable unit step response voltages versus log^Q frequency for 304. meters (1000 ft) of cable J. Ordinate units are volts and abscissa units are hertz.

177
100

10

ttH

*' y'

y

/>

,^

/^

-'^

^

. y^ K

y^

.^

^' y^

10 100 1000 10000

Figure A-138. Modeled attenuation plot for 304.8 meters (1000 ft) of K. Ordinate units are decibels

and abscissa units are megahertz.

1000

100

10

.....

.-• yy

_*«

^^. _^?^ ^

y y yy

10 100 1000 10000

Figure A-139. Modeled minimum-phase phase shift plot for 304.8 meters (1000 ft) of K. Ordinate units

are degrees and abscissa units are megahertz.

178
3.26x10

-2

0.0

,

\

Vv

250 500

Figure A-140. Modeled and computed time domain impulse response for 304.8 meters (1000 ft) of K. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing between

points is 0.4883 ns.

3.26x10

-2

\

\

\

\

\V

62 .5 125 0.0

Figure A-141. Time-expanded modeled and computed time domain impulse response for 304i8 meters (1000

ft) of K. Ordinate units are seconds" , abscissa units are nanoseconds and time spacing

between points is 0.9766 ns.

179
i.U

^

^

/

^

/

n.s /

/

0.0

250 500

Figure A-142. Modeled and computed time domain unit step response for 304.8 meters (1000 ft) of K. Ordinate units are volts and abscissa units are nanoseconds.

1.0

0.5

0.0

., J

->_ r

"" \ k\ \

\ /

v

/

/ \

/ \

y

/

\ '— _/

loO in 5 \ 10^^

Figure A-143. Plots of "zero'V'one" cable unit step response voltages versus logj^Q frequency for 304.

meters (1000 ft) of cable K. Ordinate units are volts and abscissa units are hertz.

180
00 -J _I "TX

HMM ^

^ §^1

III 1 4

' ,y

V

>

^

^

10 ^

,•-'

_^ ^^

1

0.1 10 100 1000

Figure A-144. Modeled attenuation plot for 402.34 meters (1320 ft) of WD-37. Ordinate units are decibels and abscissa units are megahertz.

1000

100

1 ^^~ 1 1 1

^ 1 1 1 ^ 1 1 1

y 1 1 ii jT nil

y

/

y

y

^

5

*. ^-l ^ i:

> ^

z'y

. 10 0.1 1 10 100 1000 Figure A-145. Modeled minimum-phase phase shift plot for 402.34 meters (1320 ft) of WD-37. Ordinate

units are degrees and abscissa units are megahertz.

181
1.161x10

1

\

\\

0.0

^^^

0.0 1000 2000

1.161x10

Figure A-146. Modeled and computed time domain impulse response for 402.34 meters (1320 ft) of WD-37,

Ordinate units are seconds" , abscissa units are nanoseconds and time spacing between

points is 1.953 ns.

-2 A

' \

\

\

\\\

\,^

^^

} —

, 0.0

0.0 250 500

Figure A-147. Time-expanded modeled and computed time domain impulse response for 402.34 meters (1320

ft) of WD-37. Ordinate units are seconds" , abscissa units are nanoseconds and time

spacing between points is 1.953 ns.

182
1.0

0.5

0.0

....^.^.^.^.^ ^_^..w..._^ M^H^MMB^^ w^m^ti^^^^m^m ^a^^^mmmmm^m mam^mim^^^i^m ^t^^^m^immmm ^mmm^m^^mmm

0.0 1000 2000

Figure A-148. Modeled and computed time domain unit step response for 402.34 meters (1320 ft) of WD37. Ordinate units are volts and abscissa units are nanoseconds.

1.0-

0.5-

0.0-

s. /

\

\ / \/

) (

/\ / \ —/.

/ \ 10 10" 10

10

Figure A-149. Plots of "zero'V'one" cable unit step response voltages versus log^Q frequency for 402.34 meters (1320 ft) of WD-37. Ordinate units are volts and abscissa units are hertz.

183
l.U

'

,

0.5 -—

n.n

0.5 1.0

Figure A-150. TDR unit step response for cable WD-37. Source step generator has assumed 50 fl resistive

output impedance. Ordinate units are volts and abscissa units are microseconds.

184
APPENDIX B. CABLE MEASUREMENT AND ANALYSIS COMPUTER PROGRAMS

B.l. APMS System Calibration Programs and Subroutines

B.1.1. Y-Axis Calibration Subroutine

a u

I . 8 I

0«^ I-. E

E (00) -tlJ:

<t Q<rz«-«l

Q.Q^(rjQKO

!*-•»-» s a s s h-H

£

a.

SI

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S5

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KBIiTTl I

allziD -^ \xZ

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:?:£ SfeSX

2 8

f I VSt i822fc

^38i

K R

185
(D<Z w -

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s s

186
B.1.2. X-Axis Calibration Subroutine

^ ^

p ^m zoiij

LUI

•-•oRaujozoos OS P kTT

"i(Vl«-is s : s s h>

? s DO ^^

«4 V

S'd^ ®+ 38 9 -*? 83 Ho 9

"8 PS is e

-) • 03 •^'^ O

^ ^(h ^'*' +K ^

S®c3a> 2j JO) ¥

3JCO OCO CO •-• *-• -I »-• CO O0) •- -I•-• CO

S |3i R 9

187
>

u

(J

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ro

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u

o

o

" 8

ui ^• ro

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UJ ^Z II

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o

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¥ or Q

'^U.I-OZLl.

8 e

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<^_I BOH (/)-^_io-j:

¥ pi Q n pi

-J -• to l5 35 •-• -J •-• oi^

» g 2 ? ^ 3

188
5

^5_

2 t,

h\

§ s §§ g i ss

189
B. 1.3. Calibration Data Acquisition Subroutine

fiH!

If) I >

(Oi-«

fcguiS-'" ^"^

PPQttQffi*-^^ptL^U

M >£zz«-i ^23 -z <zpSQz8aQi-oiij^9:

r'i-(

mx

•-•i3?ZZU£Kf-QO( •-iSSZ

>4 UJ<

''^ C/)K!D(-iK1iJi-43(0Z3I)<E<C<Z4[<C<IS^& *--9;AQoHs!jzpzi-iooqH-qi--Qi-qqS

Ai-OOOOOOOOOOOUUOUOOU a««a^a«a««««<z

190
^ ^S- St Si

» fegxgJS* T(i,a,fc:fc:QQlEK_PPP^\R5- «3t<rr^.K

% « ^t tip, t 9

i<ioQroc<jroep05»-«»^GQS3auo4

i;giil^iig^^ii;ig§g§gigigggi§§liib8^§s

«<caa4:«<i«a<taa:a<c«c<ca«««<ca«<I««<E«aa««aa<

191
s IJ

opp o oK

*->UZQQtjQSQ*--iZSZQZQzQzT^Q5Q ffiNi^SZ2SSCN^Zi-4^i-^St-4Zi-iZi->C»J<CZOZOO)Z^Z

gg&

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§gii§§§§iii§»§»i^g»i»§i^gs;ii^^iiggi

a«a<c«««<caa<caa«<sa««<E«aa««3<i^<c^aaa««<i

192
s

in

fS.

,. E

ail , ^ „

>-«»-QettzoKzp»-»wcoo

c!)S)J(0..»-U(0!-i£S_ co£

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193
^ ^i

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lgfi:x2358^g85£32CSW ^

• ^ <*- "fcH" -^I "* ^ ^K- ^t~CLUJ ^

^-j-j«-»-j •-• anC<c w•-• w•-•Q ^

194
B.I .4. Linear Least Squares Curve Fit Subroutine

V^

gSg

jazz£ 2 1 ^

C<JZ

cup

•vwCVJ ^ ^^^r^ ^

IZP

CJ(

OO

I N II -^lUPw^^

8

i-i|-CLQ^«^U.p-FwPQQFP ooSBou^FCO>oBooo lOOUU

195
B.1.5. General Matrix Multiplication Subroutine

no

£

o<

CD(

a:

azz z-^l-- ^

zz«^«^ ^^Q. Q

OQOKOHPOOOSb]

2 8

196
B.2. Cable Measurement Programs and Subroutines

B.2.1. Main Cable Time Domain Transfer Function Measurement Program

b 88 8

zap] oS(Q(o

0.01(0 (OF»->ilJ

§(0K z <C(->(o|: ^

OU Oct O^N^CO (Vl'^ ZQUJ o5 & ug

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I

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U-^ oyj 111 ^1 ^cDs

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Uj ^S ffiU* *^3 Q>uJ ^^S VP ^X

s

202
B.2.2. Sweep-Sequential Waveform Acquisition Subroutine

•= 4-

0)

SO) III ^1 II *^m.

m in*

?'

O)

^i^ZlA.h'plCQUJli.i^^ Q^O

CO I >-l ^ fc-Slc53£o -l^t ujQ.mo)ujcouji^z:zz»-i oSii.-|oDQZ

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Ml zo

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APPENDIX C — TIME DOMAIN DATA ACQUISITION SYSTEM DESCRIPTION

Figure C.l is a simplified block diagram of the Automatic Pulse Measurement System (APMS) [6,7]. The

sampling oscilloscope is a commercially available unit that has been modified to allow either stand-alone

or computer-controlled operation. It should be noted that this type of oscilloscope is not a real time

instrument. Rather, it acquires one voltage versus time point of the measured waveform per waveform

occurrence. Thus only repetitive v/aveforms may be observed. The advantage of this sampling scheme over

conventional real time oscilloscopes is the large amount of available bandwidth. Whereas conventional

oscilloscopes are presently limited in bandwidth to below 1 GHz, the effective bandwidth of the sampling

oscilloscope used in the APMS is about 18 GHz. When the sampling oscilloscope is switched to computer-controlled operation, the oscilloscope is con- nected to the system minicomputer via an A/D-D/A unit and a digital control interface. Under program

control, the computer sends the oscilloscope the desired time location at which a voltage sample is to be taken via the 14-bit D/A converter. After the waveform voltage sample has been taken it is recorded in the computer memory via the 14-bit A/D converter. The digital interface is designed to allow asynchronous (independent) clocking of the sampling oscilloscope with respect to the minicomputer.

The system minicomputer is a 16-bit machine with 32k words of core memory. With the peripherals

shown in the block diagram, i.e., the flexible disk drive, the CRT graphics terminal, hard copy unit, paper tape punch (PTP) and paper tape reader (PTR), the system can generate either alpha-numeric or graphical data in either soft or hard copy. In addition, a hardware floating point number processor has been added to the computer to speed arithmetic calculations by an approximate factor of 10. The APMS is capable of running at a maximum sampling rate of approximately 8 kHz. The useful signal

voltage measurement range is +800 mV into the 50 V. sampling head. The available real time equivalent

sweep speeds range from 10 ps/cm to 500 ys/cm in a 1-2-5 sequence.

The system software consists of a commercially available flexible disk operating system with such

features as Basic and Fortran language support, disk files, text editor, and macro-command capability.

The sampling oscilloscope v/aveform acquisition is done with assembly language programming while virtually

all signal processing of the waveform is done in either Basic or Fortran. The software has been designed

to achieve a happy medium between system versatility and ease of operation.

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HEAD

Figure C.l Simplified block diagram of APMS

254
NBS-114A (REV. s-78)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS TN-1042

4. TITLE AND SUBTITLE

Shielded Balanced and Coaxial Transmission Lines - Parametric Measurements and Instrumentation Relevant to

Signal Waveform Transmission in Digital Services

! fiteelgtfettC'u ',c; ,-i>ii' n N9»

5. PubllCdiiuii udLc June 1981

*. PiK'fei^j^OrpRijEafioft C^4e

7. AUTHOR(S)

W. L. Gans and N. S. Nahman

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

km^ ftjtt No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rsfr«<. City, stste. zjp)

Dept. of the Army, US Army Communications Command

Communications - Electronics Engineering & Installations Ag

Fort Huachuca, AZ 85613

13. Type of Report & Period Covered

14. ^^p^OTJisg Ag^cy Coife

^„^<S;.^^>^^^. .,\»\-i!;> - ^^- ''^^ 15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, Fl PS Software Summary, is attached.

16. ABSTRACT (A 200-word or leas factual summary o{ mo3t significant information. If document includes a significant bibliography or

literature survey, mention it here.)

p^ method IS presented for determining the impulse and step responses of a shielded cable using time domain terminal measurements and a physically

based mathematical model for the tranmission line properties of the cable. The method

requires a computer controlled time domain measurement system and was implemented using

the NBS Automatic Pulse Measurement System (APMS). Data are also developed for the frequency domain complex propagation function (attenuation and its related minimum-phase

shift). The method is applied to 12 shielded paired-conductor (balanced) cables and

5 coaxial cables. Time domain responses are presented for three nominal cable lengths,

60 m (200 ft), 150 m (500 ft), and 300 m (1000 ft). The time domain responses are applied to the estimation of bit error rate increases due to the insertion of the cables

into a digital signaling system employing a balanced polar NRZ waveform. Also discussec

is the application of the time domain responses to time domain reflectometry techniques

for cable acceptance tests and field-site testing of installed cables.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Bit error rate; coaxial transmission lines; digital communication impulse response;

instrumentation; measurements; shielded balanced transmission lines; time domain;

transmission line model; waveform.

18. AVAILABILITY [XD Unlimited

I I For Official Distribution. Do Not Release to NTIS

r2 Order From Sup. of Doc, U.S. Government Printing Office, Washington, DC

20402.

[t] Order From National Technical Information Service (NTIS), Springfield, VA. 22161 ._

19. SECURITY CLASS

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PRINTED PAGES

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USCOMM-DC

V- U. S. GOVERNhENT PRINTING OFFICE 1981 - 777-002/1259 Reg. 8

NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research of the National Bureau of Standards reports NBS research and develop- ment in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry,

engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement

methodology and the basic technology underlying standardization. Also included from time to time are survey articles on topics

closely related to the Bureau's technical and scientific programs. As a special service to subscribers each issue contains complete

citations to allrecent Bureau publications in both NBS and non- NBS media. Issued six times a year. Annual subscription: domestic

$13; foreign SI 6.25. Single copy, $3 domestic; $3.75 foreign. NOTE: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical

Sciences." DIMENSIONS/NBS—This monthly magazine is published to in- form scientists, engineers, business and industry leaders, teachers, students, and consumers of the latest advances in science and

technology, with primary emphasis on work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution abatement,

health and safety, and consumer product performance. In addition, it reports the results of Bureau programs in measurement

standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and

automatic data processing. Annual subscription: domestic $11;

foreign $13.75. NONPERIODICALS

Monographs—Major contributions to the technical literature on

various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with in- terested industries, professional organizations, and regulatory

bodies. Special Publications—include proceedings of conferences sponsored by NBS, NBS-annual reports, and other special publications

appropriate to this grouping such as wall charts, pocket cards, and

bibliographies.

Applied Mathematics Series—Mathematical tables, manuals, and

studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE: The principal publication outlet for the foregoing data is the Journal of Physical and. Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American Institute of Physics (AlP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's tech- nological marketplace. Order the above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, DC 20402.

Order, the following NBS publications—FIPS and NBSIR's—from

the National Technical Information Services, Springfield, VA 22161

.

Federal Information Processing Standards Publications (FIPS

PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended.

Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717(38 FR 12315, dated May II, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Services, Springfield, VA 22161,

in paper copy or microfiche form.
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