NBSIR 73-330

FREQUENCY DOMAIN MEASUREMENT

OF BASEBAND INSTRUMENTATION

N. S. Nahman and R. M. Jickling

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

July 1973

Final Report

Prepared for:

Department of Defense

Calibration Coordination Group 71-52

c/o Mr. M. L. Fruechtenicht AMSMI-M, BIdg. 5435

U. S. Army Metrology & Calibration Center

Redstone Arsenal, Alabama 35809

NBSIR 73-330

FREQUENCY DOMAIN MEASUREMENT

OF BASEBAND INSTRUMENTATION

N.S. Nahman and R.M. Jickling

Electromagnetics Division

Institutefor Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

July 1973

Final Report

Prepared for:

Department of Defense

Calibration Coordination Group 71-52

c/o Mr. M.L Fruechtenicht AMSMI-M, BIdg. 5435

U.S. Army Metrology & Calibration Center

Redstone Arsenal, Alabama 35809

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS. Richard W Roberts Director

CONTENTS

Page

1. INTRODUCTION 1

1, 1 Objectives and Results 1

1. 2 The Need for Frequency Domain Measurements 2

1. 3 The Need for New Frequency Domain Measurement 3

Methods

1. 4 Organization of the Report 4

2. NETWORK PROPERTIES OF A SAMPLING HEAD 6

2. 1 Introduction 6

2. 2 Parameters Relevant to Time Domain Analysis 6

2. 3 Measurements Equivalent- Circuit Network 9

2.4 Observability of Measurement Quantities 12

2. 5 Generality of Measurement Methods 15

2. 6 A Lossless Canonical Measurement Model 16

2 . 7 Summa ry 19

3. SAMPLING HEAD VOLTAGE MEASUREMENTS 20

3. 1 Introduction 20

3. 2 Voltage Measurements 20

3, 3 Probe Indicator Voltage Calibration 22

3. 4 Experimental Results 25

4. SAMPLING HEAD INPUT IMPEDANCE MEASUREMENTS 29

4. 1 Introduction 29

4. 2 Impedance Measurement Data 29

4. 3 Sampling Head Physical Structure 29

5. IMPEDANCE PERTURBATION MEASUREMENTS 34

5. 1 Introduction 34

5. 2 Capacitance Perturbation 34

5. 3 Experimental Technique 34

5.4 Experimental Results 35

iii
Page

6. SAMPLING- LOOP IMPEDANCE MEASUREMENTS 37

6. 1 Introduction 37

6. 2 Sampling- Loop Impedance vs. Bias Current 40

6. 3 Experimental Impedance Measurements and Results 43

7. SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK 47

7. 1 Summary 47

7. 2 Recommendations for Future Work 47

REFERENCES 49

APPENDIX: THE ANALYSIS OF DISCRETELY LOADED 50

TRANSMISSION LINES IN TERMS OF

TRANSMITTED AND REFLECTED PULSE

TRAINS

iv
LIST OF FIGURES

Page

Figure 2. 1 A typical feed- through sarapling head. 7

Figure 2, 2 General measureraent network model of a 10

feed-through sampling head.

Figure 2. 3 A canonical form equivalent network for 10

representing N or N . Figure 2. 4 An equivalent circuit for the measurement 13

network (Fig. 2. 2) in terms of cascaded

admittance canonical forms.

Figure 2. 5 Reduced equivalent circuit for the measure- 14

ment network with sampled data network

included.

Figure 2. 6 Reduced equivalent circuit for the measurement 14

network with sampled data network included

assuming lossless transmission line elements.

Figure 2. 7 Block diagram of the basic instrumentation for 18

measurements on a sampling head.

Figure 3. 1 Voltage probe calibration and impedance 21

measurement system.

Figure 3. 2 Slotted- section terminated in a lossless matching 23

section M and a calibrated power meter PM,

is the probe loading impedance.

Figure 3. 3 Typical tuned probe equivalent circuit. 23

Figure 3. 4 Detected probe voltage vs. transmitted power. 26

Figure 3. 5 Typical results for an oscilloscope deflection 28

response measurement.

V
Page

Figure 4. 1 Typical results for the magnitude of the input 30

impedance, /Z/ , as a function of frequency.

Figure 4. 2 Typical results for the phase angle of the 31

input impedance, 9 , as a function of frequency . Figure 4. 3 Sampling head physical structure. 33

Figure 4. 4 Loaded-transmission line equivalent network 33

deduced from the physical structure of the

sampling head.

Figure 6. 1 Balanced sampling gate across an unbalanced 38

signal transmission line represented in a strip

line geometry.

Figure 6. Z (A) Sampling loop with loop switches open. 39

(B) Equivalent circuit when sampling loop

' switches are closed.

Figure 6. 3 Equivalent circuit reduction for the measurement 41

method of Section 2. 6 applied to the measurement

of the sampling loop impedance Z (s)/2 . s Figure 6, 4 Equivalent circuit for the sampling current loop. 41

Figure 6. 5 Physical structure of the sampling diode package. 42

Figure 6. 6 External bias supply connected to the diode sampling 42

gate . Figure 6.7 Real part of the 10 GHz sampling loop 45

impedance vs. sampling diode bias

current.

Figure 6. 8 Imaginary part of the 10 GHz sampling loop 45

impedance vs. sampling diode bias current.

Figure 6. 9 Static I-V characteristic for a sampling- 46

diode.

vi
Page

Figure A-1 Uniform transmission line connecting an arbitrary 52

generator and arbitrary load.

Figure A -2 The RT diagram for a uniform transmission line 52

connecting an arbitrary generator and load.

Figure A-3 The circuit and RT diagrams for a uniform 58

transmis sion line terminated at each end in its

characteristic impedance Zq(s) and discretely loaded by the impedances Zj^(s)

and Z ( s).

vii
ABSTRACT

Microwave measurement techniques were developed for characte

izing the wideband feed-through sampling heads associated with time

domain sampling oscilloscopes and frequency domain network analyzer

Such characterization or modeling is necessary for the removal of the

oscilloscope distortion from the observed waveform to yield the input

waveform; also, it is useful for extrapolation in estimating oscilloscope performance at higher frequencies.

The techniques were developed through measurements on a sampling oscilloscope having a 28 picosecond transition time (10% to 90%)

and a 12. 4 GHz baseband bandwidth. The major results of the work ar

the development of voltage and impedance measurement techniques

which provide the means for determining the sampling-head equivalent

circuit parameters. The techniques are based upon slotted-line

measurements and are not inherently limited to any particular frequency range. Experimental results were obtained for the samplinghead input impedance over the 7-12 GHz frequency range, and for the

10 GHz sampling loop impedance (vs. sampling -diode bias current).

Keywords: Bandwidth; diode; impedance; sampling;

slotted line.

viii
1. INTRODUCTION

1. 1 Objectives and Results

This is the final report on COG project 71-52. The objective of

this project was to deterraine how frequency domain measurements could

be employed to characterize time domain baseband pulse instrumentation.

The actual work was confined to the development of microwave measurement techniques for characterizing wideband feed- through sampling-heads.

Such sampling heads are used for picosecond pulse oscillographic

measurements and in automatic network analyzers.

The major results of the work are the development of voltage

and impedance measurement techniques which provide the means

for determining the parameters of equivalent- circuit networks when

applied over suitable frequency ranges. The techniques are based upon

slotted line measurements and are not inherently limited to any

particular frequency range.

Specifically, techniques were developed for:

1. Modeling the measurement system for

measurements to determine the sampling

head network parameters.

2. Calibration of a slotted-line SWR detector

for voltage measurements in conjunction with

impedance measurements,

3. Impedance perturbation of the sampling- gate plane

for equivalent network parameter determination.

4. Sampling-loop impedance measurements as a function of sampling- gate bias.

The techniques were applied in the frequency range of 7 to 12. 4 GHz.
The Need for Frequency Domain Measurements

The sampling-oscilloscope is a linear physical system operating

in a sampled-data mode. By making frequency domain measurements

on an operating sampling -oscilloscope and also on its component parts,

experimental data may be obtained which may be used to construct a mathematical model for the oscilloscope. Such a model would describe

the driving point and transfer properties of the oscilloscope in both the

time and frequency domains.

The step response of a linear physical system is related to its

high frequency bandwidth limitations. In principle, by Fourier Transformation methods the complete (amplitude and phase) frequency domain

response could be transformed to yield the time domain (step) response.

Conversely, the complete (o ^ t ^ °°) time domain step response could

be transformed to yield the frequency domain response.

In practice, the frequency or time domain responses are not

completely known due to such things as (1) the inability to measure

amplitude and/or phase characteristics over great enough frequency

ranges, or (2) the inability to generate and observe the true step re- sponse undistorted by the generator waveform and the oscilloscope

transient response.

The separation between principle and practice may be bridged

by modeling the physical system. Such models essentially provide the

mathematical means for extrapolating the limited measured data into

time and/or frequency ranges beyond that of the experimental data.

Of course, to be effective the selected model must closely represent

the physical system over suitable frequency and time ranges.

Finally, because the sampling-oscilloscope is a time domain instrixment, measurements in the frequency domain provide an independent

means for obtaining experimental data from which a mathematical model

2
could be deduced. The resultant model would have many uses.

Several important ones would be as follows:

1. The computation of numerical values which describe the

oscilloscope in terms of engineering and design parameters, e. g. , transition time, overshoot, tilt, high

frequency cut-off, bandwidth, phase- shift, etc,

2. The deconvolution of time domain waveforms, i, e. , the removal of the oscilloscope distortion from the

observed waveform to yield the input waveform,

3. Characterization of measurement errors.

1. 3 The Need for New Frequency Domain Measurement Methods

The high frequency bandwidth limitations of a sampling-oscilloscope reside in the sampling head which consists of two major parts:

(1) the electrical transmission line structure in which the sampling- gate

is imbedded, and (2) the sampling- gate itself with its associated sampling

current loop. Consequently, the circuit properties which must be

considered include (1) the transfer of the input signal from the input

terminals of the sampling-head to the sampling- gate input terminals

during gate closure, and (2) the sampling processes itself, i, e. , the

transient response and storage properties of the sampling loop.

Since the sampling operation is a transient process of brief

duration and low repetition rate, steady state measurements such as

microwave impedance measurements can not observe the loading produced by the sampling- gate. This point can be simply illustrated by

making a 10 GHz measurement of the sampling-head input impedance

with and without triggering the oscilloscope.

Because the sampling- gate input terminals are imbedded within

a nonuniform transmission line network, signal transfer from a 50 ohm

3
system to the sampling -head input terminals is different from that to the

sampling- gate input terminals. Furthermore, the transfer functions

from the sampling-head input and the sampling- gate input, respectively,

to the oscilloscope display are also different.

As a consequence of the above listed properties, new measurement methods are needed; specifically, techniques for determining the

following quantities are required:

1. voltage and impedance at the sampling-head input reference

plane.

2. voltage and impedance at the sampling- gate plane,

3. impedance of the sampling-loop as a function of the samplinggate biasing current.

1. 4 Organization of the Report

This report is divided into six chapters, the first chapter being

the present introduction. Chapters 2 through 6 report the technical

work while Chapter 7 presents a sumcmary and recommendations for

future work. The technical contents of Chapters 2 through 5 are briefly

described below.

Chapter 2 presents a general measurement network model for

measurements on the sampling- head. The measurement data presented

in chapters 3-5 illustrate the application of the measurement model.

Chapter 3 presents the basic voltage and impedance measurement system which consists of a slotted- line section and a calibrated

voltage probe. The theory and application of the measurement techniques are discussed along with experimental examples of measurements and probe calibration.

4
Chapter 4 presents the results of sampling-head input impedance

measurements. The results are correlated with the physical structure

of the sampling -head.

Chapter 5 presents the technique for immittance perturbation of

the sampling- gate reference plane and the corresponding effect at the

sampling-head input reference plane.

Chapter 6 presents the technique for impedance measurements

on the sampling-loop as a function of the sampling-gate bias. Experimental data are presented.

Chapter 7 is a summary and includes recommendations for

future work.

An appendix is included to explain the passage of transmitted and

reflected pulse trains in a discretely loaded transmission line.

5
2. NETWORK PROPERTIES OF A SAMPLING HEAD

2. 1 Introduction

The high frequency bandwidth limitations of a sampling oscilloscope reside in the sampling head which consists of two major parts:

(1) the electrical transmission line structure in which the (sampling)

gate is imbedded, and (2) the gate itself with its associated sampling

current loop. Consequently, the circuit properties being considered

here include (1) the transfer of the (sampling) head input signal to the

gate input terminals during the gate closure, and (2) the sampling

process itself. The former necessarily includes the gate input terminal admittance during gate closure.

2. 2 Parameters Relevant to Time Domain Analysis

A typical head circuit represented as a transmission line network is shown in Fig. 2. 1. The circuit elements shown in Fig. 2. 1 are defined as follows: = Input or output connector-pair discontinuity

admittance

Y = Gate input admittance (varies with gate state)

Y^ s Low frequency (blow- by) compensator probe

admittance

Yq = Characteristic admittance of the uniform input or

output transmission lines.

Typically, the transmission line lengths I , ^ , and ^ are different

and are each of the order of 5 cm.

6

The frequency domain parameters which are paramount for characterizing the time domain (pulse) response of a sampling head are different

from those terminal network parameters used in characterizing microwave networks. For time domain considerations, parameters internal

to the network (sampling head) terminals are paramount and convenient

in application. That such is the case arises from the fundamental fact

that a narrow pulse or impulse traveling through a discretely-loaded

uniform transmission line encounters each load (discontinuity) separately,

successively in increasing time. With each encounter, the transmitted

pulse waveform is altered and a reflected waveform is generated. Overall, the output from the discretely loaded line of length I will consist

of a train of pulses, the first of which having the Laplace Transform

(Appendix I)

m 2Y (s) e-'n''<^)

= ^I'^'rTil Y(.)°2Y(.) <^-"

n o

Initial Pulse

where Y (s) is the n-th discrete load and Y (s) the uniform line

n o characteristic admittance. The product of the m factors in (2-1)

corresponds to the cascaded effects of m discontinuities (discrete loads)

The expressions corresponding to pulses following the first output pulse

will contain reflection factors.

Equation (2-1) may be applied to the analysis of a sampling head.

In Fig. 2. 1, assuming that the tapered section maintains the characteristic admittance Y^ , then by (2-1) the initial pulse appearing at the

sampled output has the Laplace Transform

- jjj^s) - l^y{s)

^d(^^ = 4 ^i(^) Y ('s)+2G yJs)+2G ^(^^

C O g o (2-2)

8
where the impedances have been written as admittances. has been

assumed to be a real conductance G (i. e. , lossless line), and T(s) is o the sampler transfer function. The time domain response corresponding

to (2-2) is valid for the time before reflections from Y and Y, return

c b

to the plane. Since JI3 approximately 5 cm (air dielectric), the first reflections will arrive in 334 ps. Consequently, the

inverse Laplace Transform of (2-2) is valid for the time interval

0 ^ t ^ 334 ps.

The significance of (2-2) is now apparent. If Y (s), Y (s)

c g

andT(s) are known in the s domain, then e,(t) for 0<t< 334 ps

d

may be compared to (2-2) through ed(t) = E^(s). Furthermore, only

Y^ (at the input) and Y^ in conjunction with G^ govern the transfer of

the pulse from the uniform input line to the gate input plane, during the

time interval 0<t< 334 ps.

These results are in marked contrast to the steady state situation in which the cumulative effects of all reflections combine to produce

steady state transfer functions. Clearly, it is evident that microwave

measurements which directly determine Y (s), Y (s)andT(s) are

C o needed,

2. 3 Measurements Equivalent- Circuit Network

The circuit properties being considered here are those primarily

related to the transfer of a signal from a node on a uniform input transmission line to the sampling- gate input node. This necessarily includes

the effect of the sampling- gate input admittance during the gate closure.

For measurement purposes it is convenient to define three nodes

or reference planes of a general network. Fig. 2. 2:

9
I I

Uniform Trans- Lumped Equivalent Uniform Transmission Line Networks mission Line

Figure 2. 2 General measurement network model of a feed-through

sampling head.

10
1. ' ^ plane (or node E^^) in a uniform transmission line

before the sampling head input terminals.

X , a plane (or node E ) intei

the sampling- gate input node.

a plane (or node E^) in a uniform transmission line

following the sampling head (feed- through) output terminals,

2. X , a plane (or node E ) internal to the sampling-head at

The three defined planes are connected together by the lumped equivalent networks N,12^ and , and the equivalent admittance Y . That

g

is, in Fig. 2. 2 to left and right of the planes x ^ and respectively,

uniform transmission lines are defined, while between the planes

and x^ lumped equivalent networks are defined:

1. is an equivalent lumped network representing the network properties between the reference planes x and x

A. Cd includes any waveguide to transmission line transitions,

sampling-head input connectors, and networks internal to

the sampling-head leading to the reference plane x from

X . 1 2. Y is the sampling- gat e input admittance located at the

plane x

3. N is an equivalent lumped network representing the network properties between the reference planes x^ and x^,

N includes any waveguide to transmission line transitions,

sampling-head input connectors, and networks internal to

the sampling-head leading to the reference plane x from

"3-

n
The equivalent circuit networks N^^ and can be of the

canonical form shown in Fig. 2. 3 [ 1] . Consequently, the general

measurement network of Fig. 2. 2 in terms of the canonical form in

Fig. 2. 3 would contain a cascade of two such networks. In terms of

admittance elements Fig. 2. 4 results and may be reduced to that of

Fig. 2. 5 in which

j (2-3)

G . +n^ G ^ +G (2-4)

1 pl 2 p2 g

= n^ B . +n^ B , + B (2-5)

2 1 pi 2 p2 g

Also, beyond the considerations discussed in relation to Fig. 2, 4, the

sampled- data circuit has been included in Fig. 2. 5. Y2 contains the

driving point admittance of the sampled data network while T(s) is the

transfer function from the real time voltage to the sampled- data

voltage E . d

2. 4 Observability of Measurement Quantities

Referring to Fig. 2. 5 at a given frequency there are three fully

observable quantities; they are the driving point impedance and driving

point voltage at each of the reference planes and , and the

sampled-data display voltage E^.

E is not an observable quantity; generally, it may be deduced

from a measurement of a driving point voltage and a knowledge of those

equivalent circuit parameters relevant to the conditions of the voltage

measurement. However, it can be simply deduced when E^ is zero

because E^ is directly proportional to E^ ,

12
00

pq

<M

.1-4 OI

<u

C!

0)a

:3CO

.0

r—I a>

^T-^ <

(U

•t-< P4 oOu

PIa

0)

nJU

COUmO

13
I n :1 I l:n I

I '

I

I

Figure 2. 6 Reduced equivalent circuit for the measurement network

with sampled data network included assuming lossless

transmission line elements.

14
(2-6)

If E has been deduced from experimental data, then by virtue

of (2-6)»the magnitude of the sampling-gate voltage transfer function

|T

I can be deduced as E^ is observable.

Also, the admittance Y2 is not an observable quantity; it must

be deduced from driving point admittance measurements.

And finally, the equivalent circuit parameters are generally

not observable quantities but must be deduced from a set of driving

point immittance measurements.

2. 5 Generality of Measurement Methods

In light of the observability conditions delineated in Section 2. 4,

the equivalent circuit parameters of Fig. 2. 5 can be deduced by application of the general immittance measurement methods for canonical

networks [2] in combination with voltage measurements.

Because the sampling-gate is imbedded in a transmission line

structure which is tapered down to small dimensions, the sampling

head losses are not negligible at the upper frequencies of operation,

18 GHz. Consequently, at the upper frequencies the general measurement methods [2] must be used which include losses in the networks,

N and N , Fig. 2. 2. However, at 10 GHz the losses are relatively minor and, simpler (lossless) methods may be employed.

In the experimental work reported in this report (Chapters 5

and 6), only the lossless methods were used as the frequency was restricted to about 10 GHz. This was done because the emphasis of the

work was to develop the technique and demonstrate it in a simple

manner. In the next section measurement methods are discussed which

15
are based upon lossless canonical networks.

2. 6 A Lossless Canonical Measurement Model

When N and N in Fig. 2. 2 are lossless, then the canonical

network in Fig. 2. 3 reduces to a lossless one, i. e. , R vanishes and

s becomes infinite. Accordingly, the reduced equivalent circuit

becomes that of Fig, 2. 6 with the components of Y becoming

= G (2-7) ^ g

2 2

B2= n B +n B +B (2-8)

1 pi 2 p2 g

Note that here the transmission line elements (measurement system

components) have been assiimed to be lossless. However, the sampling

gate elements are lossy; consequently, Y2and T remain complex.

When the lossless transmission line conditions are satisfied,

the immitance seen at reference plane is simply related to that

at plane X2 through the turns ratio n^. Specifically, in Fig. 2. 6 the

admittance seen at the reference plane x^ would be given by

= Y^/n^ (2-9)

where Y is the total load admittance at reference plane x , and n

is the transformer turns- ratio. Also, E would be related to E by

J. Cd

E^ = E^/n^ . (2-10)

16
To establish the proper reference planes [3] so that the equivalent circuit in Fig. 2. 6 is valid, a slotted- transmission line section is

connected to the input terminals of the sampling head while the feedthrough output terminal is terminated in 5 0 ohms. With the slottedsection excited from an isolated CW source, the oscilloscope is placed

in normal operation so as to display the CW waveform. Caution, the

peak to peak value of the CW waveform should not exceed the maximum

ratings specified for the sampling head (usually about one volt p-p).

Then, the 50 ohm output termination is replaced by a sliding short.

Fig. 2. 7 shows the block diagram of the basic instrumentation system

for measurements on a sampling head, x , x , and x denote positions

corresponding to those in Fig. 2. 6,

The sliding- short is adjusted so that the oscilloscope display

goes to zero indicating a voltage zero (E =0) at the sampling-gate

reference plane x^ ; the sliding short position defines x^. The

slotted- section probe position is moved to a voltage zero, the position

of which defines x^. With x^ so determined the admittance and voltage

seen at x ^ will be related to those at through (2,9) and (2, 10),

respectively. Consequently, if the sliding short were to be shifted by a quarter-wavelength, then the admittance seen at x^ would be

^1

X3 +X/4

= Y^/nj (2-11)

because an open circuit would be established across Y^* Also, the short

could be shifted to a position so that would be real which would mean

that the residual susceptances at x were canceled out.

17
18
Using an immittance perturbation method, the turns ratio nj

may be determined. For example, referring to (2-9), if a. known

admittance perturbation A is introduced at x , then the admittance

seen at is

+6 = ^

(Y^ + A) (2-12)

"l

where 6 is the corresponding change in the admittance Y^^ . Consequently, n^ can be determined by

n^ = (A/6)^^^ (2-13)

because A is a known admittance perturbation while 6 is an observed

admittance change.

2. 7 Summary

A general measurement network model for measurements on a feed-through sampling-head has been presented. A reduced equivalent

circuit for the measurement network was developed and discussed in

some detail for the lossless case.

In the next chapter, the basic voltage and impedance measurement system will be discussed. Experimental results will be presented

for frequencies in which the lossless measurement system assumptions

are valid (7 to 12. 4 GHz. ).

19
3. SAMPLING-HEAD VOLTAGE MEASUREMENTS

3. 1 Introduction

The voltage and impedance measurement system is a slottedline system in which the detected probe voltage is calibrated in respect

to transmission line voltage at the probe position. The probe output

voltage is calibrated by establishing a unity standing- wave ratio in the

slotted section and computing the slotted- section line voltage from a measurement of the transmitted power. A typical measurement system

is shown in Fig. 3. 1.

3. 2 Voltage Measurements

At a given probe position in the slotted section I meters from

d, Z

is given by

the load, Z , at the reference plane, the voltage on the slotted section L

E (

Ji ) e"^ + E" (3. 1)

,)..,,,

where p is the voltage reflection coefficient

Z - Z

L o and E is the voltage at the load reference plane, y is the propagation

constant of the slotted section and Z is its characteristic impedance.

In (3. 2) the first term is the incident voltage E , while the second term

is the reflected voltage E .

20

For a lossless line y = j B ; if ^Jl = TI radians, i is equal to a half-wavelength \ /2 at the operating frequency; and (3.2) reduces to

E (X/2) = - (3. 4)

Consequently, the voltage at X/2 meters from the load reference

plane is equal to the negative of that across the load Z . In general,

the detected probe voltage E^ is proportional to the magnitude of the

line voltage E (;^/2) ,

E = K |E(X/2)| (3.5)

P

where K is a scalar quantity which may depend upon the magnitude of

|E(X/2)[. Therefore, if K is known, the slotted line measurements

provide a means for computing the incident voltage magnitude [ E^| ,

i. e. ,

p is computed from (3. 3) using the value of Z observed by the slottedsection using the null-shift method for impedance measurement [4].

3. 3 Probe Indicator Voltage Calibration

K is determined by the probe calibration procedure. The

probe indicator is calibrated by terminating the slotted- section in its

characteristic impedance Z using a lossless matching- section and a

o calibrated power meter as a load. The equivalent circuit for the calibration procedure is shown iii Tig, 3. 2. M is adjusted so that the

VSWR on the slotted section is unity. For negligible losses in the

slotted- section and matching section, the power transmitted along the

22
Slotted Section

X = 0

Figure 3. 2 Slotted- section terminated in a lossless matching section M

and a calibrated power meter PM. is the probe loading

impedance.

R E

Figure 3, 3 Typical tuned probe equivalent circuit.

23
slotted- section, through the matching- section, and absorbed in the

power meter load will each be the same. Accordingly, for power P

transmitted, the transmission line voltage magnitude [E(x)| everywhere in the interval 0 ^ x ^ ^ will be independent of position x and

given by

\k{1)\ = \/1pR (3.7) ' ' o and

|E(i)|^= P (3.8)

Now in (3. 5), K in general may depend upon the magnitude of the line

voltage. Commonly, K exhibits a linear dependence on E for a square-law detector, i. e. , for

K = K' E(^)

the detected probe voltage is

E K' |e(£)|^ (3.9)

ir or

E = K' R P (3. 10)

p o

which is proportional to power P.

For the probe calibration to be valid, care must be taken to

insure that the probe coupling, tuning, etc, are constant, i, e. , the

same values are present during a measurement run as were present

during the calibration.

Structural stability and uniformity will insure that the equivalent

probe coupling parameters are constant; however, small changes in

24
frequency will alter the probe tuning which in turn will change the

value of K in (3. 5) from that of the calibration. A typical probe

equivalent circuit is shown in Fig. 3. 3 in which

C = effective coupling capacitance to electric field

in slotted line

= capacitance of probe to transmission line slottedsurface

= diode by-pass capacitor

R = detector load resistor .

shifts cause corresponding changes of Ep ; the magnitude of such

changes depend upon the effective Q of the probe circuit.

A typical probe calibration curve is shown in Fig, 3. 4 ; here,

the detector had square-law characteristics as is seen by the linear

relation between transmitted power and detected probe voltage. The

transmission line voltage is related to the detected probe voltage

through (3. 9) ,

3. 4 Experimental Results

E( i) = ' E (3. 11)

where E

P

R P K ' = o

200 X 10

-6

-3

-3

= 1. 25 X 10

50 X 3. 2 X 10

25
400
Figure 3. 5 shows the typical results for an oscilloscope deflection response measurement vs. the incident voltage E

''^ at the samplinghead input port (curve A). Note that oscilloscope response saturates

and only remains linear up to about E =1,0 volts peak to peak which

d

corresponds to 0. 35 volts RMS (this particular sampling-head has a

maximum rating of 1. 0 V p-p ; consequently, the measurements . reported here exceed the safe range).

27
1.5-

1.0-

ft 0.5-

/

6»

o-o—

©

o

o

/

—

I f I I [ I 1 1 1

« 0.2 0.4 0.6

E ^ in volts (RMS)

Figure 3. 5 Typical results for an oscilloscope deflection response

measurement.

28
4. SAMPLING HEAD INPUT IMPEDANCE MEASUREMENTS

4. 1 Introduction

Using the sampling head input port as a reference plane, steadystate impedance measurements can be easily made. In this chapter,

typical results of slotted-line impedance measurements are presented

along with a brief discussion of the frequency dependence due to the

sampling head physical structure.

4. 2 Impedance Measurement Data

Input port impedance measurements were made on a sampling

head over the frequency range of 7. 0 to 12, 4 GHz. The output port was

terminated in a nominal 50 ohm load as provided by a 50 ohm termination

specified by the sampling-head manufacturer. The measurements were

made with the system shown in Fig. 3.1.

Typical magnitude and phase data for the input impedance are

shown in Figs. 4. 1 and 4. 2, respectively. The magnitude and phase data

indicate that discontinuities are present in the 50 ohm transmission line

structure which comprises the sampling head. Note that the oscillation

of the data contains a fine- structure ripple. The major oscillation has a period of about 2. 5 GHz measured between the zero crossings of the

phase angle curves. The ripple in the roughly periodic data suggests

that the physical structure of sampling-head is nearly periodic.

4. 3 Sampling-Head Physical Structure

As a physical structure, the sampling head consists of a cascade

of three transmission line sections and consequently four connection

planes. At each connection plane, discontinuities are present. Starting

29
100-

a

o

50-

\

0

\

\

-f——' f

» 8 9

-| 1 1 1 1 r

10 11 12

Frequency in GHz

Figure 4. 1 Typical results for the magnitude of the input impedance,

I Z |, as a function of frequency.

30
60-

40-

-40 -

60

Frequency in GHz

Figure 4. 2 Typical results for the phase angle of the input

impedance, 6 , as a function of frequency.

31
from the input plane, the discontinuities arise from four components.

Fig. 4. 3 ; in order, they are the input connector, the sampling gate,

the blow-by compensator probe, and the output connector. The discontinuities are predominately capacitive elements; accordingly, with

the transmission line physical lengths the equivalent circuit of Fig. 4. 4

is a reasonable one to assume. As indicated, the transmission line

lengths , &^ , and jj are 5. 1, 5. 1, and 5.6 cm, respectively. Y

denotes a connector discontinuity, Y that of the sampling gate, and

Y^ that of the blowby compensator probe.

Because the magnitudes of the discontinuity admittances are

small compared to the characteristic admittance of the 20

millimho coaxial lines which predominately comprise i , H , H , the electrical wavelength of the loaded transmission line should be

close to that of the physical line length. 5. 1 and 5. 6 cm represent free

space half-wavelengths corresponding to the frequencies 2. 94 and 2. 68

GHz, respectively. Consequently, when the sampling-head is terminated

in a 20 millimho load, the input admittance (or impedance) should

oscillate with a period somewhat less than 2. 94 or 2. 68 GHz due to the

capacitive loading. That such is the case is shown by the data in Figs.

4. 1 and 4. 2 where the oscillation is about 2. 5 GHz; also, the presence

of a ripple superimposed on the oscillation is consistent with the fact

the physical structure is not periodic though multiply loaded.

32
Input Sampling JL z Blow- by Output

Connector Gate Comp^ns^toj Connector

Figure 4. 3 Sampling head physical structure.

Input Port

Ref. Plane

Y = 20 X 10 mho.

o

Output Port

Ref. Plane

I I

5. 1 cm

~2

5. 1 cm.

Y

b

5. 6 cm

20 X 10

mho

Figure 4. 4 Loaded- transmission line equivalent network deduced

from the physical structure of the sampling head.

33
5. IMPEDANCE PERTURBATION MEASUREMENTS

5. 1 Introduction

In Chapter 2 it was shown that a known impedance perturbation

could be used to experimentally determine the turns ratio (2. 13) of the

transformer in the lossless canonical measurement model. In this

chapter typical experimental data will be presented for the observed

incremental capacitance due to the insertion of a capacitance perturbation.

5. 2 Capacitance Perturbation

Two dummy sampling-diodes were used for the capacitance perturbation experiments. A section of cross-linked polystyrene plastic

was machined to the same size as a sampling-diode and tipped with a tinned copper wire imbedded in the plastic. The wire was 0. 6 mm in

diameter and 4. 5 mm long and was imbedded in the end of the dummy

diode which contacted the inner conductor of the sampling head coaxial

line. The wire protruded from the plastic just enough to be in contact

with the inner conductor. The resultant dummy diodes were rugged

and stable and could be handled without fear of breakage,

5. 3 Experimental Technique

Low frequency capacitance measurements were used to determine

the value of the capacitance perturbation. First, the actual sampling

diodes and the output port termination were removed from the samplinghead. The unterminated sampling head appears as a capacitive load

at low frequencies. Next, the capacitance without diodes was measured

34
and recorded. The duramy diodes were inserted, and the capacitance

measurement repeated. The difference between the two observed

capacitances was equal to the value of perturbation.

Because the physical dimensions of the sampling-diodes are so small, their loading effect on the signal line is localized to a region

much less than a quarter wave up to 18 GHz. Consequently, the perturbation capacitance will be independent of frequency up to 18 GHz.

This in turn means that low frequency measurements may be used to

determine the capacitance of the perturbation.

Once the capacitance has been determined, then it may be used

as A in the microwave measurements relevant to determining the turns

ratio n, (2. 13).

5. 4 Experimental Results

Typical experimental results are shown in Table 5. 1. Low

frequency capacitance bridge measurements were made at 1 KHz and

30 MHz; introduction of the dummy diodes produced a capacitance

change AC of about 0. 035 ± 0. 005 pF at both frequencies. Microwave

slotted line measurements at 10 GHz using the perturbation method

encompassed in (2-11) and (2-12) gave a 6 (0. 028 ± 0. 005 pF) which was

essentially equal to A ^35 ± 0. 005 pF) because the uncertainty was

within the attainable accuracy of the experimental measurements. For

these results at 10 GHz n^ = 1 which in turn says that the transformer

effects in the canonical network are negligible at 10 GHz.

35
TABLE 5. 1 LOW AND MICROWAVE FREQUENCY MEASUREMENTS

OF CAPACITANCE PERTURBATION AT SAMPLING POINT

WITHIN THE SAMPLING HEAD

Frequency Capacitance Perturbation AC AC Uncertainty

1 kHz 0.035 pF ± 0.002 pF

3 0 MHz 0.036 pF ± 0. 005 pF

10 GHz 0. 028 pF ± 0. 005 pF

36
6. SAMPLING-LOOP IMPEDANCE MEASUREMENTS

6. 1 Introduction

The sampling-head consists of a signal transmission line (coaxial

line) shunt-loaded at some point by a balanced sampling gate. Qualitatively, if the outer conductor of the coaxial transmission line were pictured as being longitudinally slit and folded out into a plane surface, the

sampling head network would appear as a strip line structure, Fig. 6.1.

The input signal to the sampling head is e^{t) while the balanced samplingpulse is ± e (t - r) which drives the sampling gate. The diodes D and

s 1 D comprise the sampling gate and are hot carrier diodes which are normally biased to cutoff. When the sampling pulse occurs at ^ -

^ diodes conduct strongly and provide two equal paths for signal currents

from the signal line. Each path essentially consists of a diode in series

with a sampling (storage) capacitor and a sampling pulse generator.

Ideally, the sampling path or current loop should be such that the loop

current ig(t) should have a transition time of zero, i, e. , it should jump

to its final value in zero time at the instant that the loop switches are

closed. In practice, the impedance of the sampling current loop is not

purely resistive; consequently, the current transition time will not be

zero. If the impedance of the conducting diodes and all other elements

appearing in the sampling-loop are lumped together into a single impedance

Z (s) , the sampling loop can be represented as shown in Fig. 6. 2A where

s the diode gating or switching function is represented by two synchronized

ideal switches. When the switches are closed, the equivalent circuit of

Fig. 6. 2B results.

Assuming that the signal line is lossless, then the measurement

methods described in Section 2. 6 may be applied so that the sampling

loop impedance may be observed by slotted-line impedance measurements.

37
Figure 6. 1 Balanced sampling gate across an unbalanced signal

transmission line represented in a strip line geometry.

38
Figure 6. 2 (A) Sampling loop with loop switches open.

1 P 2

E

1

Figure 6. 2 (B) Equivalent circuit when sampling loop switches are

closed.

39
Fig. 6. 3 shows the equivalent circuit reduction for the measurement

method. Briefly, the position of the sliding short on port 2 is adjusted to

be x/4 or (2N-l)x/4 from sampling point P, thus presenting an open circuit to the right of P. If the position of the slotted line probe, point 1, is

adjusted to be \/Z or N \/2 then the impedance at 1 is the same as at P.

Fig. 6. 4 gives an equivalent circuit for the sampling loop [5,6].

R. and C. are the resistance and capacitance of a diode junction, respectively; they are equivalent circuit elements for the diode depletion region

electronic phenomena and are functions of the diode dc bias current, I, . b

Cs is a sampling capacitor which stores the sampled charge. R is principally due to the bulk resistivity of the metals and semi-conductors on

each side of the depletion region in a diode; it also includes the resistance

of the sampling loop conductors. L is principally due to the diode catwhisker conductor but also includes the inductance of the sampling loop

conductors. C is the package capacitance which physically bridges the

diode terminals. Fig. 6. 5 shows the physical structure of a sampling

diode package. Note that C is built into the package. Finally, when

the diodes are conducting, the two diode -sampling paths are in parallel;

hence, the overall equivalent circuit contains the individual equivalent

circuit parameters multiplied or divided by a factor of two.

6 . 2 Sampling -Loop Impedance vs. Bias Current

At this point it is instructive to qualitatively discuss the behavior

of the sampling-loop impedance as the bias current is varied. Typically,

when the bias current is zero, R^ is very large and the diode is cut off.

The sampling loop impedance Z^(s)/2, Fig. 6. 4, is primarily a capacitive

reactance. When the bias current is applied R^ is small and a high frequency signal current flov^'s through R/2, L/2 and the diode junction impedance. For a large enough bias current, most of the signal current

flows through the diode branch, and the impedance appears to be inductive.

40
X/2

X/2

X/4—>J

P

Figure 6. 3 Equivalent circuit reduction for the measurement method

of Section 2. 6 applied to the measurement of the sampling

loop impedance Z (s)/2 . s 2C

R

Z^(s)

l-^yyV—'TP —WV— —

Rj{I)/2

C.{V)/2

2 C

Figure 6. 4 Equivalent circuit for the sampling current loop.

41
Figure 6. 5 Physical structure of the sampling diode package.

10

Figure 6. 6 External bias supply connected to the diode sampling gate.

42
By measuring the sampling loop impedance as a function of frequency and bias current, it is possible to deduce the element values for

the equivalent circuit of Fig. 6. 4. In the present study, impedance

measurements were made at one frequency, 10 GHz, over a bias range

of 0 to 10 ma. The impedance varied from about 65 - j95 to 14. 5 + j 5

ohms. The 10 GHz resistance seen for 5 ma bias was equal to the 5 ma. dc dynamic resistance of the sampling diode. Thus, it appeared that at

10 GHz the hot carrier diodes were still functioning efficiently.

6. 3 Experimental Impedance Measurements and Results

Using the microwave impedance measurement system shown in

Fig. 2. 7, the impedance measurement reference planes are established

by the procedure described in Section 2. 6. The 50 ohm termination on the

sampling head is replaced by a sliding short. The oscilloscope is then

shut-off, and an external biasing supply connected to the diode samplinggate. Fig. 6.6. Slotted-line impedance measurements are then made

as a function of the sampling- gate bias current.

Figs. 6. 7 and 6. 8 present typical 10 GHz data for the real part

of the sampling-loop impedance R and the imaginary part of the samplings loop impedance X , each as a function of the sampling bias current, I . s b

When the sampling-diodes are conducting strongly (I^ = 5 ma) , Fig. 6.7 shows a 10 GHz resistance of 14. 5 ohms for the two diodes in

parallel. This value correlates very well with the slope of the static

I-V curve for the sampling diodes. Fig. 6. 9 shows the static I-V

curve for one sampling diode. At 1 = 5 ma the slope A V/ ^ equals

28. 5 ohms. For two diodes in parallel the value would be 14. 25 ohms

which is very close to the 10 GHz value. Because the diodes are hot

carrier devices, it is expected that their conduction process should be

relatively independent of frequency to well above 20 GHz,

43
Fig. 6, 8 shows a 10 GHz reactance of + 5 ohms at = 5 ma

Referring to Fig. 6.4, if the sampling- capacitor ' s reactance, CO 2C , is

s negligible and the current all flows through the inductive branch, then

the observed inductive reactance should be ~ Typically 2C

s is 4 pF, consequently at 10 GHz the sampling capacitor reactance would

be about -0. 4 ohms ; accordingly, the observed reactance would be

- 0. 4 + COL/2 , and C0L./2 would equal 5. 4 ohms. This in turn would

give an L of about 17 0 pH which is of the same order as reported

earlier [ 5, 6]

.

44
o

CMN

70

60

50

40 .

30 -

20-

10-

-©—O- •O-^O—O —ft ——* —ft

\

e

ft

\o

0

10^ ,1 10" lO'^ 10" 10- 10'

Bias Current in jLiA

Figure 6. 7 Real part of the 10 GHz sampling loop irapedance vs. sampling diode bias current.

+20

o

a

•i-i

CM

I

0-

-20-

-40

-60

-80

100 ^

120

.0 Q

/

iv—

.0

,1 —I

10^ 10" 10" 103^ 10

Bias Current in

Figure 6. 8 Imaginary part of the 10 GHz sampling loop impedance vs. sampling diode bias current.

45
0 0.2 0.4 0.6 0.8 1. Diode Voltage in V

gure 6.9 Static I-V characteristic for a sampling diode.

46
7. SUMMARY AND RECOMMENDATIONS FOR FUTURE WORK

7. 1 Summary

The major results of this work have been the development of microwave voltage and impedance measurement techniques which provide the

means for determining the equivalent circuit parameters for the sampling

head when applied over suitable frequency ranges. The techniques are

based upon slotted-line measurements and are not inherently limited to

any particular frequency range. Experimental data were obtained in the

7 to 12. 4 GHz range.

7. 2 Recommendations for Future Work

Additional work should proceed in two phases:

1. Coaxial slotted-line measurements should be extended up

to 18 GHz; such measurements would require the use of

the lossy canonical form measurement network. However,

the reduced data would provide additional information

about the sampling-loop impedance. This in turn would

establish the sampling- gate equivalent network and in

conjunction with sampling pulse shape estimates allow an

extrapolation to estimate higher frequency performance

(above 18 GHz).

2. Waveguide slotted-line measurements should be applied

over the 18 to 26, 5 GHz band; such measurements would

provide data on the frequency response cut-off characteristics of the sampling head due to sampling- gate frequency

limitations and transmission line multimode effects.

47
Results would provide an examination of the effectiveness

of the extrapolation methods in the first phase and serve

as a basis for further extrapolation to the 40 GHz frequency range.

48
REFERENCES

Ginzton, E. L. , Microwave Measurements , McGraw-Hill, Inc.,

New York, 1957, pp. 323-326.

Ginzton, E. L. , ibid., pp. 343-345.

Ginzton, E. L. , ibid., pp. 277-283.

Ginzton, E. L. , ibid. , pp. 274, 278-285,

Howard, D. , Best, A. , Hvunphrey, J. , "The Wideband Sampling

Gate, an Analysis, Characterization and Discussion". WESCON

Proceedings, Session 23, Aug. 23-26, 1966, Paper 23/1.

Grove, Wayne, "Sampling for Oscilloscopes and Other R. F.

Systems: DC through X-band". IEEE Trans. Microwave Theory

and Techniques, Vol. MTT-14, No. 12, pp. 629-635,

December 1966.

Johnson, Walter C, Transmission Lines and Networks , McGraw-Hill, N. Y. , 1950, chp. 4, in particular s ee p. 100.

Nahman, N. S. , "A Discussion on the Transient Analysis of

Coaxial Cables Considering High-Frequency Losses", IRE

Trans. Professional Group on Circuit Theory, Vol. CT-9, No. 2,

June 1962, pp. 144-152.

49
APPENDIX

THE ANALYSIS OF DISCRETELY LOADED TRANSMISSION LINES

IN TERMS OF TRANSMITTED AND REFLECTED PULSE TRAINS

In the classical frequency domain analysis of transmission lines

containing reflections, the assumption is implicit that the sinusoidal

signals have been present since minus infinity. In time domain instruments for the measurement of transient phenomena, such as a sampling

oscilloscope, such is not the case. Prior to some time, say t=0, there

was no signal present. When the 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 observation point 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 instruments it is very important to comprehend this

time isolation of reflections. As an aid to developing an understanding

of this concept this appendix develops the analysis of 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.

50
In general, the generator to transmission line terminals transfer

functions, Fig. A-1, can be written [7] as

E^(s) Z (s)

o Z (s) + Z (s)

s o

1 + p (s) e

r

2 j^y(s)

s r

(A-1)

and

E^(s)

= 2

Z (s)+ Z (s)

_ s o

Z (s)

r Z (s) + Z (s)

o r _ 1- p (s) p (s) e

s r

2;^y(s)

(A-2)

where Z (s) and Z (s) are the sending and receiving end impedances,

s r respectively." p (s) and p (s) are the sending and receiving reflection

coefficients, respectively.

Z (s) - Z (s)

_5 2

Z (s) + Z (s)

s o (A-3)

Z (s) - Z (s)

r o Z (s) + Z (s)

r o (A-4)

=!^To obtain equation A-2 from Johnson's [?] equation 4.23 the relationship

2 Z (s)

l+p_(s)-

Z (s) + Z (s) o r

is used.

51
Z (s)

EJs) E (s) Z (s), y(s)

Figure A-1 Uniform transmission line connecting an arbitrarygenerator and arbitrary load.

H— ^ H

E^(s) E^(s)

Figure A-2 The RT diagram for a uniform transmission line

connecting an arbitrary generator and load.

52
I is the transmission line length while y{s) and ^^(s) are the line

propagation function and characteristic impedance, respectively.

y(s) =

Z (s) o

[Z(s) Y(s)]

Z(s)

Y(s)

1/2 (A-5)

(A-6)

Z(s) and Y(s) are the transmission line equivalent circuit parameters:

series impedance/meter and shunt admittance/ meter. y(s) is of the

form sT'+'y'(s), where x'is the constant J LC; the time delay per unit

length (see discussion following equation A- 16).

Now consider a step generator of impedance Z^(s) applied to the

system described by (A-1) and (A-2), i. e. , E (s) = 1/s and Z (s) = Z (s).

§ SO

Then p (s) = 0, and (A-1) and (A-2) reduce to

s

and

E (s)

s

E^(s)

2 s l4-p(s)e-^^t^^'+^'(^^^'

Z^(s)

Z (s) + Z (s) o r

jj[sT' + y '(s)]

(A-7)

(A-8)

Rewriting,

and

E^(s)

E^(s)

1 ^ -2sj^T

+ e 2s

P^(s)e-^^>''<^>]

ay '(s)

-s^Tlfri!i___

^

\ s [Z (s) + Z (s)]

(A-9)

(A-10)

53
Consideration of (A-9) and (A- 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

o

/ 0<t<2j2,T; i. e. , the load impedance is not seen at the

generator 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 < jj T also, its shape

is controlled by the characteristic impedance to load impedance voltage divider and the dispersive part of the

propagation function, y '(s).

Generally, for arbitrary terminations, E (s) and E (s) each

s r consist of an infinite sequence of terms similar to those in (A-9) and

(A- 10) and having their origins in the conclusions delineated above.

By using the geometric series expansion

fi / ^ / \ -2jiy(s)"l-l , , -2j^y(s)1'^

I 1 - p^(s)p^(s) e M =J2\p^{s)P^{s)e '''j

n=o L

Ps(s)p^(s) e

in the second and third factors of (A-1) and (A-2), respectively, the

transfer functions can be expressed as infinite sequences which represent impulse responses consisting of pulse trains. Upon putting (A- 11)

into (A-1) and (A-2) obtains the sending end pulse train

54
E (s) = E (s)

Z (s) o Z (s) + Z (s)

s o

P>)p>) = ' + p>)p("+l>(.)e-<"+'' ^ =

'

(A-12)

and the receiving end pulse train,

E (s) = 2 E (s)

r 2

Z (s) o Z (s) + Z (s)

s o

Z^(s)

Z (s) + Z (s)

r o

n=o

(A-13)

where (A-12) and (A-13) are both subject to the condition specified in

(A- 11), The physical situation represented by the pulse trains (A-12)

and (A-13) is lucidly shown by a reflection-transmission (RT) diagram,

Fig, A- 2. In fact, the RT diagram, itself, provides an intuitive (but

exact) means for writing out the terms of each pulse train.

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

s node to a line segment represents multiplication by the

transfer function Z (s)/[Z (s) + Z (s)].

o s o

2. Passage along a line segment represents multiplication by

the transfer function exp [ - i, y ( s) ] .

55
3. Passage through a node to a load Z (s) represents multiplication by the transfer function 2Z^(s)/[Z (s) + Z (s)] .

4. Reflection from a node represents multiplication by the

reflection coefficient n (s) \ Z (s) - Z (s)l [Z (s) +

, ^r r o r Z^(s)]- .

Application of the rules to the diagram in Fig, A -2 obtains the

following results:

E (s) =

s E

g Z so+ Z

+ E

Z + Z

s o Gen to A

2 Z

^r Z + Z

s o ABC

+ Z . 2 Z

TP , \ Q - ly - iy - ly n

^ E s)—— e e e ^ P e

g Z + Z ^r '^s ^ o s Z + Z

s o ABCDE

+

g Z so+ Z

1 , ^ ^r^s -2 iy , ^ ^r ^s ^s -4 j^y

s o

2 p. P.

z + z

s o

+ .

(A-14)

Since

Z, - Z

1 o

^i ~

z. + z

1 o

the voltage divider may be written as

1 +

Pi

Z. + Z

1 o

56
Putting (1 +p )/2 into (A-14) for the factor Z /(Z + Z ) obtains the

S S s o result

^ =

I

1 + P (1 +P p% (1 + p )e-^^^+..

s g Z + Z I "^r "^s '^r s s

s o

which is identical to that given by expansion of (A- 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

E^(s) =

Gen to A AB

E

o Z + Z

s o

2 Z

z + z

r o

Z 2 Z

+ Z + Z ^ Pr ^ ^s ^ Z + Z

^ - s o r o ABCD

2 E

g Z + Z Z + Z

" s o r o

- ly ^ -3 jiy e ^ +p p e ^ +. ^r '^s

(A-16)

which is identical to that given by expanding (A- 13).

The RT diagram is a powerful tool for analyzing a discretely

loaded uniform transmission line, Consider Fig. A- 3 in which a uniform transmission line, characterized by Z^(s)and y(s), is
Figure A-3 The circuit and RT diagrams for a uniform transmission

line terminated at each end in its characteristic impedance

Zq(s) and discretely loaded by the impedances Z^(s) and

58
discretely loaded by the shunt impedances Z^(s) and ^^(s) and doubly

terminated in the characteristic impedance Z ( s ). Consequently, there

o will be no reflections from the sending and receiving end terminations

while internal reflections will exist due to the loading by Z^{s) and

Now it will be instructive to write out the initial voltage transmitted to node 2, [^^{s)] , using the R-T diagram rules. However,

initial

this time admittances will be used as their use is convenient for shunt

loading. The change-over to admittances only requires 1/Y substitution,

e. g. , rule 1 becomes

Z (s) + Z (s)

s o

Y (s)

_ o

^-1

Y (s; o Y (s)

s Y (s) + Y (s)

s o

Similarly, rules 3 and 4 transform to the factors 2Y (s)[Y (s)+Y (s)]

^ o o r and [Y(s)-Y(s)][Y(s)+Y(s)]" , respectively. Accordingly,

o r o r o J

tracing the initial path from node g to node 2, the initial voltage at

node 2 is given by

initial

'^g^^'' Y (s) + Y (s)

2 Y (s) o

s)

Y^{s) + Y^(s)

g to s to 1

2 Y (s) o Y^ (s) + Y^(s)

1 to 2

(A-14)

59
where

y/(s) = Y .(s) + Y (s)

1

^ r o

because the total admittance at node i seen by the advancing voltage

wave is the sum of the discrete admittance Y^(s) and the characteristic

admittance Y (s) of the following transmission line segment. (A-14)

o reduces to

E (s) 2 Y (s) 2 Y (s) -( l,+ l^)y{s)

g O O L C

' E^(s)

2 Y.(s) + 2 Y (s) Y (s) + 2 Y (s)

initial f ° ^ ° (A-15)

The initial voltage (A-15) exists for the time interval 0 t ^ 2 T where

- T = lim —^ = i ^JTC (A-16)

where L and C are the external inductance/m and capacitance/m of

the uniform transmission line. As mentioned earlier y(s) can be

written in the form sT' + y

' (s); consequently, in the light of (A-16)

the limit of y (s) must vanish as | s | This is the case as y (s)

is the minimum phase part of y{s).

60
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. 2. Gov't Accession BIBLIOGRAPHIC DATA NBSIR 73-330

SH c c T

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Frequency Domain Measurement of Baseband Instrumentation

5. Publication Date July 1 i 1 5

6. Performing Organization Code

7. AUTHOR(s) s. Nahman and R. M. Jickling 8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS , Boulder Labs.

DEPARTMENT OF COMMERCE

wxsa«tCTi3iHi3Dfl23<23a3S34 Boulder, Colorado 80302

10. Project/Task/Work Unit No. 2720513 -FY 71

11. Contract/Grant No. DoD CCG 71-52

12. Sponsoring Organization Name and Address Dept. of Defense, Calibration Coordination Group

c/o Mr. M. L. Fruechtenicht AMSMI-M, Bldg. 5435

U.S. Army Metrology 5 Calibration Center

Redstone Arsenal, Alabama 35809

13. Type of Report & Period Covered Final Report FY 72

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Microwave measurement techniques were developed for characterizing

the wideband feed-through sampling heads associated with time domain

sampling oscilloscopes and frequency domain network analyzers. Such

characterization or modeling is necessary for the removal of the

oscilloscope distortion from the observed waveform to yield the input

waveform; also, it is useful for extrapolation in estimating oscilloscope

performance at higher frequencies.

The techniques were developed through measurements on a sampling oscilloscope having a 28 picosecond transition time (101 to 90%) and a 12.4 GHz baseband bandwidth. The major results of the work are the

development of voltage and impedance measurement techniques which provide

the means for determining the sampling-head equivalent circuit parameters.

The techniques are based upon slotted-line measurements and are not

inherently limited to any particular frequency range. Experimental

results were obtained for the sampling-head input impedance over the

7-12 GHz frequency range, and for the 10 GHz sampling loop impedance

(vs. sampling-diode bias current).

17. KEY WORDS (Alphabetical order, separated by semicolons)

Bandwidth; diode; impedance; sampling; slotted line.

18. AVAILABILITY STATEMENT

fxl UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

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UNCL ASSIFIED

20. SECURITY CLASS

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UNCLASSIFIED

21. NO. OF PAGES

22. Price

USCOMM-DC 66244.P7I

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