calibration of a measurement system for high frequency...

Calibration of a Measurement System for High Frequency Nonlinear Devices

Jan Verspecht

Jan Verspecht bvba

Gertrudeveld 151840 SteenhuffelBelgium

email: [email protected]: http://www.janverspecht.com

Doctoral Dissertation - Vrije Universiteit Brussel, November 1995

September 1995

Promotor Prof. dr. ir. A. Barel

Copromotor dr. ir. M. Vanden Bossche

Proefschrift ingediend tot het behalen

van de academische graad van doctor

in de toegepaste wetenschappen

Calibration of a Measurement System for

High Frequency Nonlinear Devices

Jan Verspecht

VR

IJE

UNIVERSITEIT BRUS

SE

L

SC

IEN

TIA VINCERE TENEBRA

S

VRIJE UNIVERSITEIT BRUSSEL

FACULTEIT TOEGEPASTE WETENSCHAPPEN

Department ELEC

Pleinlaan 2, B-1050 Brussels, Belgium

September 1995

Calibration of a Measurement System forHigh Frequency Nonlinear Devices

Jan Verspecht

VR

IJE

UNIVERSITEIT BRUS

SE

L

SC

IEN

TIA VINCERE TENEBRA

S

VRIJE UNIVERSITEIT BRUSSEL

FACULTEIT TOEGEPASTE WETENSCHAPPEN

Department ELEC

Pleinlaan 2, B-1050 Brussels, Belgium

Jury: prof. dr. A. Barel (Vrije Universiteit Brussel)

prof. dr. A. Cardon (Vrije Universiteit Brussel)

dr. U. Lott (Eidgenössische Technische Hochschule Zürich)

prof. dr. G. Maggetto (Vrije Universiteit Brussel)

prof. dr. L. Martens (Universiteit Gent)

prof. dr. B. Nauwelaers (Katholieke Universiteit Leuven)

prof. dr. R. Pintelon (Vrije Universiteit Brussel)

dr. A. Roddie (National Physical Laboratory - United Kingdom)

prof. dr. J. Schoukens (Vrije Universiteit Brussel)

dr. M. Vanden Bossche (Hewlett-Packard Company)

prof. dr. I. Veretennicoff (Vrije Universiteit Brussel)

Table of Contents

. 2 . 2. . 34

. 6. . 6 . 6

. . . 7. . . 9

. 11

14

14 . 1415

6 . 1616

. 18. 21. 23

24. 24 . 25

Preface

Notations and Abbreviations

Chapter 1 Introduction1.1 Why do we need a measurement system for high frequency

nonlinear devices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1.1 Computing, prototyping and measuring . . . . . . . . . . . . . . . . . . .1.1.2 Tools for the electronic engineer. . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Vectorial “Nonlinear Network” Analyzers . . . . . . . . . . . . . . . . . .

1.2 Why do we need to calibrate? . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Metrology and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2.3 TraceablecalibrationofaVectorial “NonlinearNetwork”Analyzer

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Consistency checks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Hardware Implementation of a Vectorial “Nonlinear Net-work” Analyzer2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.1 On the use of a voltage wave description . . . . . . . . . . . . . . . . .2.2.2 The VNNA as a modular instrument . . . . . . . . . . . . . . . . . . . . .

2.3 Pre-existent Hardware Prototypes . . . . . . . . . . . . . . . . . . . 12.3.1 General remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.2 Prototype of Markku Sipilä et al. . . . . . . . . . . . . . . . . . . . . . . . .2.3.3 Prototype of Urs Lott . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.4 Prototype of Gunther Kompa et al. . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Prototype of Markus Demmler et al. . . . . . . . . . . . . . . . . . . . . .

2.4 Prototype of Hewlett-Packard Network Measurement andDescription Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.4.1 A short historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Simplified schematic and comments . . . . . . . . . . . . . . . . . . . . .

i

2628 . 31

. 34

. 35

38

38. 3838

0

41. 41 . 41 . 51 . 54

54. 54d

. . 55 . 575966

. 67 . 70

70. 7071

. 72

. 73

. 74

78

78 . 7880

2.4.3 The broadband downconvertor: hardware . . . . . . . . . . . . . . . . .2.4.4 The downconverting process: mathematical description . . . . . .2.4.5 The signal source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Accuracy of Broadband Sampling Oscilloscope Measure-ments3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 The HP-54120 sampling oscilloscope . . . . . . . . . . . . . . . . .3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The instrument working principles . . . . . . . . . . . . . . . . . . . . . . .

3.3 Accuracy aspects of a broadband sampling oscilloscope . 4

3.4 Timebase distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Timebase distortion measurement. . . . . . . . . . . . . . . . . . . . . . .3.4.3 Spectral estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5 Timebase jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Mathematicalequationsoftheextended“PDFdeconvolution”metho

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 The use of a parametric model . . . . . . . . . . . . . . . . . . . . . . . . .3.5.4 Comparison versus “median” method . . . . . . . . . . . . . . . . . . . .3.5.5 Asymptotic bias of the classical way to estimate the jitter PDF .3.5.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6 Timebase drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Logarithmic Spectral Averaging . . . . . . . . . . . . . . . . . . . . . . . . .

3.7 The oscilloscope impulse response . . . . . . . . . . . . . . . . . .

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 The “Nose-to-Nose” Calibration Procedure4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.1 Sampling head equivalent scheme . . . . . . . . . . . . . . . . . . . . . .4.2.2 Derivation of the “sampler drive” equivalent scheme . . . . . . . .

ii

1845

. 88

0

92. 92 . 92

93. 93se. . 93106. 108109111112112

3

115

116

18

n

118119

0

122. 122. 126

130. 13042

4.2.3 Derivation of the “signal sampling” equivalent scheme . . . . . . 84.2.4 The effect of sampler circuitry asymmetry . . . . . . . . . . . . . . . . .4.2.5 “The kick-out is proportional to the impulse response”. . . . . . . 84.2.6 Measuring a mismatched signal source . . . . . . . . . . . . . . . . . .

4.3 Determination of the oscilloscope impulse response . . . . . 9

4.4 Practical measurement set-up . . . . . . . . . . . . . . . . . . . . . . .4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5 Accuracy and precision aspects of the “nose-to-nose”calibration procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Error due to possible asymmetricity of the sampling aperture pul

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Correspondence with other measurement techniques . . . . . . .4.5.4 Timebase errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Sampling head linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5.6 Repeatability and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5.7 Check on sampling aperture asymmetry. . . . . . . . . . . . . . . . . .4.5.8 Kick-out drop-outs due to pulse creator screen updates. . . . . .

4.6 Oscilloscope transfer function measurements . . . . . . . . . 11

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5 The Absolute Calibration of a Vectorial “Nonlinear Net-work” Analyzer5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

5.2 Consistencyversuscascadingindependentofabsolutecalibratio1185.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2.2 Mathematical prove . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Classical absolute calibration approaches . . . . . . . . . . . . 12

5.4 Absolutecalibrationprocedureforconnectoreddevicemeasurements1225.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.4.2 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5 Absolute calibration for on wafer measurements . . . . . . 1305.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5.2 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Hardware and software implementation. . . . . . . . . . . . . . . . . . 1

iii

45148

491495055

158

160

64

64. 164. 169

70170170171172

7717717817881

183

192

. 194

194

19519595196e

. 19697

199

5.5.4 Quality of the calibration procedure . . . . . . . . . . . . . . . . . . . . . 15.5.5 Derivation of test-set characteristics . . . . . . . . . . . . . . . . . . . . .

5.6 The reference generator . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.1 Construction of the reference generator . . . . . . . . . . . . . . . . . .5.6.2 Characterization of the reference generator. . . . . . . . . . . . . . . 15.6.3 On the optimization of the reference generator . . . . . . . . . . . . 1

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6 Consistency of the Absolute Calibration versus Large SignalModels6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

6.2 Large Signal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 The Root-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The model developed by Ph. Jansen et al.. . . . . . . . . . . . . . . .

6.3 Early consistency measurements . . . . . . . . . . . . . . . . . . . . 16.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3.2 Experiment description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3.3 Comparison in the frequency domain. . . . . . . . . . . . . . . . . . . .6.3.4 Comparison in the time domain . . . . . . . . . . . . . . . . . . . . . . . .

6.4 More advanced consistency measurements . . . . . . . . . . . 16.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4.2 Short description of the experiments. . . . . . . . . . . . . . . . . . . . .6.4.3 Small-signal s-parameter measurements . . . . . . . . . . . . . . . . .6.4.4 The harmonic distortion measurement . . . . . . . . . . . . . . . . . . . 1

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 7 Conclusions and ideas for further research7.1 Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2 Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.3.2 Full traceability towards national standards labs. . . . . . . . . . . 17.3.3 Extending the sampler topological model . . . . . . . . . . . . . . . . .7.3.4 Adapting the calibration procedure towards commercial VNNA us

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Putting error flags on VNNA measurements . . . . . . . . . . . . . . 1

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv

Preface

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When I first met Mark Vanden Bossche in 1990, it was one of those scarce “right timeplace” happenings. The whole of this thesis is a logical consequence of that meeting.

Mark was finishing his Ph. D. thesis “Measuring Nonlinear Systems: A Black BApproach for Instrument Implementation”, sponsored by the Hewlett-Packard compsupported by the people of the ELEC department of the VUB (university of Brussels). Heinvestigated the feasibility of mainly two things: firstly the use of mathematical modelsdescribe the nonlinear behavior of high-frequency electronic circuits (such a model was cVIOMAP), and secondly the construction of the associated instrumentation. His work hadthat persuasive that Hewlett-Packard had asked him to put up a small research group to conhis research work. Mark called his group the Hewlett-Packard Network MeasurementDescription Group (HP-NMDG). As such Mark was looking for people interested in continuwhat he had started. At the same time I was graduating and, finding the combinatio“nonlinear” and “high-frequency” quite a challenge, I applied for the job (with success).

Starting in October 1990, my goal was to develop the calibration procedure for the newof instrumentation under development, which was later called “vectorial nonlinear-netwanalyzer” (VNNA). It was found out very soon that developing a traceable phase distocalibration for VNNA’s was very essential but far from obvious. An introduction to the conceof the VNNA and comments on the calibration aspects are given in Chapter 1,”Introduction

A possible solution to the phase distortion problem was found in the article “CharacterHigh-Speed Oscilloscopes”, appearing in the September 1990 issue of IEEE Spectrum, auby Ken Rush, Steve Draving and John Kerley. Their article described a new and accurate mto measure the impulse response of broadband oscilloscopes (50GHz bandwidth). This mwas later called the “nose-to-nose” calibration procedure. Realizing the important roleinstruments could play in a traceable VNNA phase calibration procedure, I started to stukinds of broadband oscilloscope accuracy aspects. During a three week visit to the HP CoSprings Division, March 1992, I was able to hang around the workbenches of the broadoscilloscope designers themselves (namely the aforementioned authors) and I was able tolot about how those instruments work and particularly about the practical details“nose-to-nose” measurements. The results of my research concerning broadband oscilloare described in Chapter 3,”Accuracy of Broadband Sampling Oscilloscope Measurementsin Chapter 4,”The “Nose-to-Nose” Calibration Procedure”. These chapters deal with all kinbroadband oscilloscope errors like timebase distortions, timebase drifts, timebaseoscilloscope vertical nonlinearities and the determination of the impulse response by usin“nose-to-nose” calibration procedure.

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In the meanwhile I was also involved with the different VNNA prototypes constructedHP-NMDG. A first prototype was build in 1992. It was based on the use of a broadboscilloscope for the data acquisition. Although very useful to prove the concepts of the abscalibration procedure, this prototype was much too slow to be of any real practical use (it toleast a few hours to calibrate and about three minutes for each data acquisition). The low spthis prototype was mainly due to the slow data acquisition speed of the broadband oscilloscsolution was found in 1993 by replacing the oscilloscope by two much faster “MicrowTransition Analyzers” (MTA). The use of these instruments allowed the calibration to be doless than half an hour, with one data acquisition taking only a few seconds. An important fustep was the availability of a wafer probing station in September 1994. Although not plaoriginally, this allowed to measure nonlinear microwave devices on wafer. At the beginnin1995 the latest VNNA prototype was ready. It originated from using only those MTA partswere really needed for our measurements and using a high-precision analog-to-digital conto replace the MTA’s internal ones. This resulted in a VNNA which, compared with the Mbased prototype, is somewhat faster, much cheaper, has more dynamic range, has inlinearity, occupies much less volume,... . The construction of these prototypes, togetheprototypes of other people, is discussed in Chapter 2,”Hardware Implementation of a Vec“Nonlinear Network” Analyzer” (for readability reasons this chapter is put before the ones onoscilloscope mentioned above, note that this does not correspond with the chronologyresearch work).

The availability of the different prototypes finally allowed to investigate the practiimplementation aspects of the VNNA calibration procedures, both for connectored devices awafer devices. Note that a so-called reference generator is used as a transfer standard betw“nose-to-nose” calibrated oscilloscope and the VNNA. The theoretical and practical aspethese calibration procedures, including the use and construction of the reference generadiscussed in detail in Chapter 5,”The Absolute Calibration of a Vectorial “Nonlinear NetwoAnalyzer”.

Already from the very start it was clear that at least one experiment was needed, carrieindependently from the VNNA measurements, which proved the validity of the VNNA absocalibration procedure. Without such a scientific “consistency check” it would be difficult, ifimpossible, to convince the scientific world of the value of the calibration proceduredeveloped. The only possible experiment we could think of was the comparison between asignal model and actual VNNA measurements of the behavior of an on wafer transistor. Theof large signal model we refer to is known as the Root-model. It is a model whose parametebe extracted out of a lot of small signal s-parameter measurements in a lot of differenvoltages. This kind of models are usually considered to be the most accurate nonlarge-signal models presently available. The first consistency measurement was doSeptember 1994. The device-under-test was a MESFET transistor provided by the depaINTEC of the university of Ghent (UG), in cooperation with IMEC. The correspondence betwthe Root-model, extracted in the lab of the UG, and the VNNA measurements, performed

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lab at the VUB, were as good as other results found in nowadays literature. After tmeasurements the VNNA on wafer prototype was optimized, and in January 1995 aconsistency measurement took place. This time the device-under-test was a HEMT tranprovided by the department ESAT of the university of Leuven (KUL), also in cooperation wIMEC. The model used for the comparison was one developed at the KUL, very similar toRoot-model. This time the correspondence between VNNA measurements and model wasbetter than the results achieved with the first measurements. As far as I know of, such resunowhere be found in present literature. The whole story of these consistency measuremgiven in Chapter 6,”Consistency of the Absolute Calibration versus Large Signal Models”.

The successful consistency measurements are considered as the conclusion of my rwork as far as the contents of this Ph. D. thesis concerns. Conclusions together with stufurther research can be found in Chapter 7,”Conclusions and ideas for further research”.

Undoubtedly, many people and organizations significantly contributed to this thesis. Bfollowing, I would like to express my gratitude to• Mark Vanden Bossche (HP-NMDG), for a zillion of things• Doug Rytting (HP-SRSD) for the continuous Californian support and inspiration• Alain Barel (ELEC-VUB) for being an inspiring host• Rik Pintelon, Yves Rolain and Johan Schoukens (ELEC-VUB) for the many useful discus• Ann Cumps, Pascal Geerinck, Tom Van den Broeck (ELEC-VUB), Beya Kam

Luc Van den Torren (former HP-NMDG) and Frans Verbeyst (HP-NMDG) for a fruitful coopation as my colleagues on the 7th floor

• Peter Monsieurs, Jean Lonnay, Jean Temmerman (HP-Brussels), Stefan Sercu (UG) aCascade Microtech, Inc. for the free lending of necessary instrumentation

• Ken Rush, Steve Draving and John Kerley (HP-Colorado Springs Division) for turning myin Colorado into a valuable and unforgettable experience

• Peter Debie (UG, IMEC) and Dominique Schreurs (KUL, IMEC) for providing me withwafer components and existing large signal models

• Dave Sharrit (HP-SRSD) for the demystification of the MTA• Fred Delbaen (Mathematics Dpt.-VUB) for helping me with mathematical formulations• all people of HP, the VUB, the UG, the KUL, IMEC and the rest of the world, not mention

above, who contributed to my thesis• Andrea and “vaken en moeken” for their encouragements without cease.

Jan Verspecht, September 11th 199

vii

s
Notations and Abbreviation
rsier)

Mathematical notations

• j: complex number equal to the square root of -1• x: a vector or matrix (noted in bold when no indices are present)

• or : element with index i of vectorx

• co(X): convex hull of the set X• det(X): determinant of the matrixX

• : derivative of the function x(u)

• : the convolution of x and y (circular convolution ifx andy are vectors)

• : vector with the i indexed element equal to 1, and all other elements equal to 0

• : equal to 1 if i equals 0, equal to 0 otherwise

• : phase of the complex number x

• : the expected value of a stochastic variable x

• : Fourier transform of x (discete Fourier transform ifx is a vector)

• : inverse Fourier transform of x (inverse discrete Fourier transform ifx is a vector)

• : Heaviside function, equal to 0 for strictly negative x, equal to 1 for positive x

• Xt: transpose of the matrixX

• XH: transpose conjugate of the matrixX

• x*: conjugate of x• Re(x): real part of x• Im(x): imaginary part of x• Prob(x): the statistical probability that x (a set of realizations of a stochastic variable) occu• x dBc: power (in dB) of x relative to the power of the exciting spectral component (dB carr

Abbreviations

• ADC: analog -to-digital convertor• BIPM: International Bureau of Weights and Measures• BJT: bipolar junction transistor• CDF: cumulative distribution function• CIPM: International Committee for Weights and Measures

x i[ ] xi

x' u( )x* y

δi

δi0

ϕ x( )x⟨ ⟩

F x( )

F 1– x( )U x( )

ix

nts)

• DC: direct current• DFT: discrete Fourier transform• DUT: device-under-test• FET: field effect transistor• FFT: fast Fourier transform• FWHM: full-width-half-maximum• HEMT: high electron mobility transistor• HP: Hewlett-Packard Company• HP-NMDG: Hewlett-Packard Network Measurement and Description Group• IDFT: inverse Discrete Fourier transform• IF: intermediate frequency• IMEC: Interuniversitair Micro-Electronica Centrum• KUL: Katholieke Universiteit Leuven• LRM: line-reflect-match• MESFET: metal semiconductor field effect transistor• MMIC: microwave monolithic integrated circuit• MTA: microwave transition analyzer• NIST: National Institute of Standards and Technology• NPL: National Physical Laboratory• UG: Universiteit Gent• RF: radio frequency• SI: International System of Units• SNR: signal-to-noise ratio• SOLT: short-open-load-thru• SRD: step recovery diode• TDR: time domain reflectometry• VIOMAP: Volterra input output map• VNA: vectorial network analyzer (classical, for linear networks or small signal measureme• VNNA: vectorial “nonlinear network” analyzer• VUB: Vrije Universiteit Brussel

x

Chapter 1

Introduction

Abstract - The research work described in the thesis is situated in a broadtechnological context. It is explained what vectorial “nonlinear network” analyzersare and why they are needed. The importance of traceable calibrations is highlighted,in general as well as applied to the vectorial “nonlinear network” analyzer.

1

Chapter 1 Introduction

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1.1 Why do we need a measurement system for high frequency nonlineadevices?

1.1.1 Computing, prototyping and measuring

We can talk and listen to people that are at a distance of ten thousand kilometers; danight we can find out exactly where we are on sea as well as on land; we can watch a bagame taking place seven thousand kilometers away; we can record whatever happens in oand watch it ever after; we can see what happens inside our body without carving; we can liswhatever music we want at any time, on any place; we can see beautiful pictures of planetsbillion kilometers away; we can predict the weather it is going to be within three days from nand this all over the planet; our car will be able to see obstacles and hit the brakes beforeeven aware of any danger; we will be able to watch any moving pictures we like at any time oday;....

As far as I am concerned I think all these things are marvelous, amazing, astounmiraculous, staggering, wonderful. They all transmute from being plain science-fiction to bcommon everyday-things thanks to the efforts of skilled engineers, who design and concomplex high-technological systems. Like craftsmen, however, these engineers can notwithout the appropriate tools. Indeed, when an engineer desperately states “I have a probleoften really means “Please provide me with the right tool.” Mathematics and physicsundoubtedly the two most important engineering tools. Many other tools are indispensablemodern engineer needs for example powerful computing tools. Powerful computers togethespecialized programs allow him to exploit mathematics and physics optimally. Using comsimulations complex designs can easily be optimized versus specific parameters, superformance, cost, volume, weight,.... And all this can be done just by hitting a keyboardshuffling with a computer mouse. When the computer design is ready hardware tools are nin order to build the system. At that time the system is not ready for being used in pracExtensive tests are needed in order to check whether the prototype system is behaving exaexpected. These necessary tests bring us to another kind of indispensable engineerinnamely measurement instruments. Without these instruments satellites cannot be testedbeing launched, consumer products can not be tested before being sold.

The design process in three phases, namely computing, prototyping and measuringideal one. Engineers will immediately reply that it is seldom met in practice, except for somesimple designs. Most of the time the measurement of a prototype reveals that it doeimmediately meets all specifications, pointing out that there is an inconsistency betweemathematical model used by the computer and the physical behavior of the system. The enwill have to use the valuable information he gets from these or supplementary measuremeimprove the consistency between model and physical behavior, which implies that the systeactually perform as expected. He can achieve this goal by a combination of two things. Hchange the prototyping process such that the designed system is closer to the model usexample trying to eliminate parasitic hardware effects, or he can improve the mathem

2

1.1 Why do we need a measurement system for high frequency nonlinear devices?

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models such that they better reflect the physical behavior, for example modeling parasitic eand compensating them by altering the initial design. The above reveals that an impfeedback process usually exists between the measurements on one side and the prototypcomputing at the other side.

It is important to note that this feedback design cycle is present at all design layerscomplex system. For a satellite communication system for example, the cycle is present onlayer where people are concerned about the quality of uplinks and downlinks, as well as onlayer were people are worried about the quality of the transistors on an integrated circuitfeedback in the design process combined with the fact that this process is present at all layecomplex design, results in an incredible amount of computing, prototyping and measuring ba system is ready for use. I wonder how many people are aware of the fact that it took thouof computings, prototypings and measurements before they were able to say “hello” topartner who was at the other side of the ocean! The important conclusion here is thafeasibility of building complex technological systems is highly correlated with the quality ofcomputing, prototyping and measuring tools that are provided to the engineers, and the eawhich these tools can be combined.

1.1.2 Tools for the electronic engineer

As one might have noted electronic systems, both analog and digital, play a very imporole in all of the marvelous applications mentioned above. The question then rises what tooalready available for the electronics engineer and what tools should still be developed?

A lot of tools exist for digital system design. Computer programs can simulate advamicroprocessors and can automatically generate the masks needed for the integrated circulogic state analyzers are able to measure the behavior of complex digital systems. The challproblem of many digital electronic engineers today lies in the very high speed of some prdigital circuits. With clock rates exceeding 100MHz all digital signals can no longer hide tanalog nature, which is in essence electromagnetic. Things like electromagnetic interfebetween adjacent lines, ground bounce and impedance mismatches can no longer be neWhen these effects are mixed with the highly nonlinear behavior of transistors the performanhigh-speed digital circuits can easily be degraded. The technology for building these highdigital electronic circuits as well as software tools for simulating the behavior of these circureadily available. Measurement instruments that are able to measure the electromaproblems as such (when isolated from the rest of the circuit) are also commercially avaiMeasurements of that kind can be done with so-called “Time DomReflectometer”-oscilloscopes (TDR-oscilloscope) or with so-called “Vectorial NetwAnalyzers” (VNA). There is a lack, however, of adequate instrumentation to measurecomplete behavior of those high-speed digital circuits easily and accurately when electromaphenomena are combined with the highly nonlinear transistor behavior. Just recently logicanalyzers showed up that are able to measure also the analog nature of all digital signals presuch a circuit. Unfortunately, these instruments will primarily measure the voltages presen

3


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device port, and not the currents. The combined electromagnetic information of both voltagcurrent is important to know accurately what is actually going on. An ideal measurement tothe high-speed digital designer would be an instrument that is able to measure both voltagcurrent waveforms at all ports of a circuit. This measurement tool would be even more poweit can be combined with an advanced and easy to use signal generator. Such an instrumentwe will call a “Nonlinear Network” Analyzer, does not, however, exist at this moment.

For analog systems design the situation is similar. An analog electronic system conmany parts, which all have their own specific functionality such as mixing, filtering, amplifymodulating, demodulating. A lot of software tools are available for the design of such systemsystem parts. For time domain representations we hereby think of programs like SPICE [1for frequency domain representations we think of harmonic balance simulators [2] and simubased on Volterra series analysis [3]. Concerning instrumentation, TDR-oscilloscopes and Vare commercially available instruments to characterize accurately all kinds of linear elecnetworks, typically used to measure the characteristics of small-signal amplifiers, filtersinterconnections. Instruments with bandwidths up to 50GHz are no exception. In many chowever, engineers have to deal with nonlinear networks. For some applications, like mixefrequency multipliers, the functionality is based upon the nonlinear behavior of diodestransistors, for other applications, like power amplifiers, the nonlinear behavior of the compois an annoyance and should be avoided. Two instruments are primarily used to measunonlinear behavior of circuits: oscilloscopes and spectrum analyzers. Oscilloscopes are utake a look at all kinds of time domain voltage waveforms. They typically visualize effectsclipping, slew rate,.... Spectrum analyzers are able to measure the amplitude of all spcomponents of signals present in a circuit. They are typically used to measure intermoduproducts, harmonics,.... Unfortunately, both instruments can never accurately measure bocurrent and the voltage that is present at the signal port of a nonlinear electronic circuihigh-speed nonlinear circuits both quantities are needed to get a good picture of what is acgoing on, especially when electromagnetic wave phenomena occur. An ideal instrumentengineer dealing with this kind of design is actually the same “Nonlinear Network” Analyzethe one that is needed by the high-speed digital designer.

One can conclude that engineers dealing with high-speed nonlinear electronic circuitsa need for a new measurement instrument, a “Nonlinear Network” Analyzer. This new instrushould be able to accurately measure both the current and voltage waveforms appearingsignal ports of a nonlinear electrical network, thereby measuring all physical quantities thaneeded to characterize the behavior of the circuit.

1.1.3 Vectorial “Nonlinear Network” Analyzers

This thesis concerns a subclass of this general “Nonlinear Network” Analyzer, naso-called Vectorial “Nonlinear Network” Analyzers (VNNA). This name was chosen in analwith the classical VNAs that are presently used to analyze linear broadband electrical netwothe frequency domain. A classical VNA excites the linear device-under-test (DUT) with sinew

4

1.1 Why do we need a measurement system for high frequency nonlinear devices?

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signals (one frequency component at the time) and detects the response at the DUT signarelative to these sinusoidal excitations. Both output and input signals are sinewaves (allsame frequency because of the DUT linearity) and can be described by complex numberinput-output relationships are then described by ratios of complex numbers, resulting in a nof complex numbers. This set of complex numbers is measured by the VNA and fully descthe behavior of the linear DUT when excited by sinewaves with that particular frequency. Scomplex numbers are often represented as vectors, the instrument was called a Vectorial NAnalyzer.

A Vectorial “Nonlinear Network” Analyzer is similar. It excites a nonlinear DUT withcombination of sinewave signals (more than one frequency component can now be presensame time) and detects the response at the DUT signal ports. Excluding DUTs showing chasubharmonic behavior, both output and input signals will be combinations of sinewave si(new frequency components show up not present in the input signals because of the creaharmonics and intermodulation products by the nonlinearity) and can be described by conumbers. This set of complex numbers is measured by the VNNA.

Although not apparent at first one has to note though that there is an important concedifference between a VNA and a VNNA. A classical microwave network analyzer willmeasure the absolute value of input and output, but will measure the ratio between the twothat it is sufficient to measure input and output up to a scale factor. The measured ratio compcharacterizes the linear DUT, if it is excited with a sinewave with that particular frequency. Foamplitudes and phases of input sinewaves, it will suffice to multiply the complex numdescribing the inputs with the measured associated ratios in order to find the complex nudescribing the output sinewaves. For a VNNA the situation is different. Constant ratios betoutput and input can not model nonlinear DUT behavior as is the case for linear DUTs.VNNA will not do a relative ratio measurement, but will measure input as well as output inabsolute way. The only conclusion concerning DUT behavior that can be drawn from one Vmeasurement is that for one particular input signal (same frequencies, amplitudes and phaall spectral components) there will be one particular output signal. It will be the responsibilithe instrument user to interpret a lot of measured data in order to end up with more generallymodels. To do so, a priori information about the DUT will have to be used. If one for examassumes that the DUT is linear, one can calculate out of the measured data the same ratiosdirectly measured by a VNA.

The fact that a VNA immediately comes up with a rather general model, valid foramplitudes and phases of the input signals, seems very interesting. There is, howevimportant drawback. One notes that DUT linearity is actual necessary a priori informaimplicitly used by a VNA. The measurements of a VNA are worthless when the a pinformation is wrong. This is for example the case when a small-signal amplifier is drivencompression during a VNA measurement, resulting in a measured small-signal gain which cconsiderably lower than the actual value.

5


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1.2 Why do we need to calibrate?

1.2.1 Introduction

As one might notice after reading this thesis, the research work described is dedicatedto the calibration aspects of VNNAs than to the actual hardware design of these instrumentfollowing paragraphs try to justify this dedication towards the calibration aspects ofinstrument.

1.2.2 Metrology and calibration

Metrology, or the science of measurement, plays a vital role in the present societyimportance of measurements for engineering purposes was already explained in the prparagraphs. But not only engineers use metrology. Hundreds of millions of people have towith measurements over and over again. National as well as international trade would be himagine without metrology. A huge amount of watches, from atomic clocks to wrist watchescontinuously measuring time, thereby synchronizing happenings all over the world. Measrods are used to make sure that different parts match, even if they are produced on differenBalances assure that one kilogram of gold represents the same amount of the valuableallover the world. And a lot of other examples can easily be found. Most people takeworldwide consistency between measurements for granted. As stated in [5] “Metrologyingrained in everything, most people fail to give it any notice.” But the mechanisms insuconsistency between instruments all over the world is actually very fascinating.

The foundation of modern metrology is the worldwide acceptance to use the InternatSystem of Units (SI), established in 1960 by the General Conference of Weights and MeaThe SI defines all units that have to be used to express measured quantities. At the presentscience, there are only seven independent SI units. They are called the base units. Theunits are the meter (m) as a measure of length, the kilogram (kg) as a measure for masecond (s) as a measure for time, the ampere (A) as a measure for electric current, the Kelas a measure for thermodynamic temperature, the candela (cd) as a measure for luintensity and finally the mole (m) as a measure for the amount of substance. The metekilogram, the second, the ampere and the Kelvin are the base units needed to descrmeasurements in this thesis, they are defined as follows.• The meter is the length of path traveled by light in a vacuum during a time interval of

1/299.792.458 second.• The kilogram is the mass of the artifact cylinder of platinum iridium alloy kept by BIPM.• The second is the duration of 9.192.631.770 cycles of radiation corresponding to the tran

between the two hyperfine levels of the ground state of the cesium-133 atom.

• The ampere is the electric current producing a force of 2 x 10-7 newtons per meter of lengthbetween two long wires, one meter apart in free space (the newton is a derived unit, bridefined later).

• The Kelvin is 1/273.16 of the thermodynamic temperature of the triple point of water.

6


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All units which are not base units like the Volt, the Ohm, the Henry, the Farad and mothers can be derived from the base units. Note that there are a few remarkable consequethe fact that the base units are defined this way. Many people are for example surprised if ythem that, when using the SI-units, you can not measure the speed of light in vacuum. Sothose people will even assure you that they actually did it as a student. This can be expbecause the speed of light in vacuum was defined no earlier than 1983 to be e299.792.458m/s. Ever since the speed of light itself is reference for all other measuremenno reference can be used to measure the a priori defined constant. This implies that it was apossible to measure the speed of light before 1983. People that claim to have measured thof light after 1983 have actually measured the accuracy of their chronometers and their mearods.

The mechanism that matches a certain instrument to all other instruments of the wocalled traceable calibration. Calibration of an instrument implies the removal of systemmeasurement errors. This is achieved by measuring a device, called standard, from whicpriori know the value to be measured, with an accuracy that is higher than that of the instrumbe calibrated. A comparison between the measured value and the a priori known value revesystematic error of the instrument, which can then be eliminated during further measuremewhatever devices. But how can we be so sure about the value of such a standard? Thprocedure is then repeated at a higher level and the standard element is compared with abetter standard. At the end of the chain are only a very limited set of standards. These staare called the primary standards. These standards can be found in national standards labs,United States National Institute of Standards and Technology (NIST), the British NatiPhysical Laboratory (NPL), or in the one and only international standards lab, the InternaBureau of Weights and Standards (BIPM) located in Paris. The organization that bringsstandards labs together and that assures that all their primary standards are matched, whicbeing conform to the SI, is the International Committee for Weights and Measures (CIPM).

The calibration of an instrument is called traceable if there exists a record containformation about the calibration procedure that was used, such as identification of the staused, time and date of the calibration, who performed it,.... A similar traceability concept efor the standards themselves. This way the traceability records form a chain, where each stis traceable to a higher quality standard. One can only claim that an instrument is usingunits if this chain ends at a primary standard, approved by the CIPM. Recently, the interemany companies in continuously performing instrument calibrations traceable to natstandards labs has been increased because of the ISO-9000 standard. One of the econditions for a company to comply with the ISO-9000 standard, which is an internationrecognized standard for product quality, is that all measurement systems used are recalibrated with a procedure that allows traceability to a CIPM recognized primary standard.

1.2.3 Traceable calibration of a Vectorial “Nonlinear Network” Analyzer

What about the calibration aspects of a VNNA? This instrument measures the com

7


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values representing the spectral components of electromagnetic signals appearing at theports of nonlinear electronic devices, with the SI volt as the unit of measure. It is necessaestablish in one way or another a traceability path that goes up to a national standards lab.the only way to make sure that all VNNA measurements are consistent with the rest oscientific and technological world.

Establishing this path is more difficult than apparent at first sight. The problem wiVNNA is that it is not measuring scalars representing DC voltages, but complex numrepresenting high-frequency electromagnetic voltage waves. The complex numbers thmeasured have an amplitude and a phase. The calibration procedure has to take carequantities.

The amplitude calibration procedure described in this thesis is based on radio-freq(RF) powermeters. A sinusoidal signal is thereby applied to the VNNA. By choosing theconfiguration the amplitude of this signal is measured at the same time by the VNNA and bpowermeter (the power of a voltage wave can be converted into an amplitude by making usevoltage wave characteristic impedance). The measurement result of the VNNA is then comwith the measurement result of the powermeter, which is used as the reference standarresults in the knowledge of the systematic amplitude error of the VNNA. The whole VNamplitude calibration procedure will then be traceable to national standards under the conthat the powermeter used is traceable to these standards. Fortunately this is the casecommercially available powermeters. It might be interesting to point out that national stanlabs calibrate their powermeters through high precision calorimetric measurements [4].temperature rises caused by the heating effect of the RF waves are thereby measurecalorimetric measurement is essentially a thermodynamic experiment based upon the definthe Kelvin. This surprisingly implies that the determination of the triple point of water isessential part of the final VNNA amplitude calibration!

The phase calibration procedure of the VNNA is another story. To illustrate thisconsiders a typical so called harmonic distortion analysis of an amplifier. With this experimesinusoidal signal is exciting the amplifier input. At the output several spectral componentsappear, with frequencies that are integer multiples of the input signal frequency, they areharmonics. The VNNA will measure the amplitude and phase of all these harmonicsexplained above there is no real problem to eliminate the systematic amplitude errorsVNNA, but what about the systematic phase errors? In order to find out the systematic Vphase error a phase reference standard is needed. This can be a signal generator genefundamental and harmonics with a precisely known phase relationship, or an instrumentable to precisely measure the fundamental and harmonic phase relationship of a multi-harsignal generator. This instrument can then be used as a reference for the phase calibratisimilar way as the powermeter is used for the amplitude calibration. Unfortunately no preference exists at present allowing traceability to a national standards lab.

The best thing to do when there is no national standard available is to build a “consestandard” [5]. This is the technical name for a standard that is not traceable to national stan

8


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but which is mutually acceptable to both the supplier and the customer. If the “consestandard” is supported by enough scientific evidence and economical interest it even has aof getting accepted one day by the CIPM. Of course this is not a one day achievement.

A large part of this thesis is dedicated to the construction of such a “consensus preference standard” for the characterization of high speed nonlinear electronic circuitsconsensus standard that is developed is based on the so-called “nose-to-nose” caliprocedure for broadband sampling oscilloscopes. This procedure, using no other instrumebesides sampling oscilloscopes themselves, can accurately characterize the phase distothese instruments. A calibrated sampling oscilloscope can then be used to accurately measphase of the fundamental and all harmonics of a multi-harmonic generator. This multi-harmgenerator can then be connected to the VNNA in order to find out the systematic phase ethe instrument. This way the calibration of the VNNA is made traceable to the “nose-to-ncalibration with the multi-harmonic generator being used as a so-called transfer standard.

1.2.4 Consistency checks

Even with a lot of theoretical evidence, people may still have a sound scientific doubt athe actual accuracy of the “nose-to-nose” procedure. They will ask for measurements provinthe procedure succeeds as well as theoretically predicted in determining the phase discharacteristic of a sampling oscilloscope. But exactly because of the lack of another prphase reference standard this is scientifically impossible. In the days where the meter wasartifact standard this would correspond to the request of determining the length of the aprimary standard meter, kept at the BIPM!

One might then ask what exactly is the characteristic of a standard that makes himconsidered as primary. Regardless of the uncertainty issue, the answer is that no experimenstandard may have been done resulting in a scientific inconsistency. Take for example tartifact meter standard. The days during which this rod was the primary meter standard, a sexperiment would show that the speed of light in vacuum varies with the temperature oprimary rod standard. This sounds of course as scientific nonsense, and immediately revealis better to use the speed of light in vacuum as the base for establishing a primary meter staAs long as the “inconsistency experiment” is not performed, however, there is no reasonconsider the artifact standard as primary.

Consider now the “nose-to-nose” calibration procedure as a potential primary preference standard. What experiments can one think of, able to check a “scientific consistenthis standard? One possibility is the comparison between calibrated VNNA measurement rand large-signal nonlinear models of a DUT which can accurately be constructed independVNNA measurements. Large-signal nonlinear models of this kind are described in literaturfor certain devices, typically FET-transistors. Using some a priori assumptions the parametthese models are determined by means of a large amount of small signal measureperformed at many DC bias points of the transistor. For these small signals the DUT behavibe assumed linear and measurements can be performed with a classical VNA. Results of

9


s ofniqueswouldthe

consistency experiment [7] are described in Chapter 6. In the future, other possibilitieconsistency checks could rely on comparisons with even more broadband sampling techbased upon electro-optics and photo-conductivity. Only in the case that such an experimentbe able to show some at this moment still undefined “scientific inconsistency” will“nose-to-nose” calibration procedure have to be replaced by a better standard.

10

1.3 References

ron.5.

its

is ofst

05,

Nonvice

mentltage

1.3 References

[1] L. W. Nagel, “SPICE2: A computer program to simulate semiconductor circuits,” ElectRes. Lab, Univ. California, Berkeley, California, Technical Report ERL-M520, May 197

[2] Kenneth S. Kundert and Alberto Sangiovanni-Vincentelli,”Simulation of Nonlinear Circuin the Frequency Domain,”IEEE Transactions on Computer-Aided Design, Vol.5, No.4,October 1986.

[3] Maas, S. A.,”A General-Purpose Computer Program for the Volterra-Series AnalysNonlinear Microwave Circuits,”IEEE MTT-S International Microwave Symposium Dige,pp.311-314, 1988.

[4] J. P. Ide,”The United Kingdom Power Standards Above 40 GHz,” NPL Report DES 1November 1990.

[5] Fluke Corporation,”Calibration: Philosophy in Practice,” Second Edition, May 1994.

[6] David E. Root, Siqi Fan and Jeff Meyer,”Technology Independent Large SignalQuasi-Static FET Models by Direct Construction from Automatically Characterized DeData,”21st European Microwave Conference Proceedings, pp.927-932, September 1992.

[7] Jan Verspecht, Peter Debie, Alain Barel and Luc Martens,”Accurate On Wafer MeasureOf Phase And Amplitude Of The Spectral Components Of Incident And Scattered VoWaves At The Signal Ports Of A Nonlinear Microwave Device,”Conference Record of theIEEE Microwave Theory and Techniques Symposium 1995, Orlando, Florida, USA,pp.1029-1032, May 1995.

11


12

Chapter 2

l
Hardware Implementation of a Vectoria“Nonlinear Network” Analyzer
Abstract - Talking about VNNAs is not difficult, effectively building one is, however,quite a challenge. In this chapter one will first read about some generalconsiderations that are valid for all VNNAs, next a quick look will be taken to VNNAprototypes that have already been build by other people in the past and finally theprototype developed by HP-NMDG will be described in more detail.

13

Chapter 2 Hardware Implementation of a Vectorial “Nonlinear Network” Analyzer

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2.1 Introduction

In this chapter one will first read about some general considerations that are valid fVNNAs. Next several VNNA prototypes that were build in the past will be shortly presentedtheir primary characteristics will be discussed, especially the aspects relating to calibrationinstrument. Please note that only instruments are considered allowing the measurementphase and amplitude of generated harmonics. In order to honor the pioneering peopleconstructed these prototypes they are named after the authors that, by the knowledge of theof this thesis, first reported of a certain VNNA construction. Finally the prototype constructeHP-NMDG will be discussed in detail. This way it is possible for the reader to situateHP-NMDG prototype relative to prior work.

2.2 General considerations

2.2.1 On the use of a voltage wave description

The typical purpose of a VNNA is to accurately measure the behavior of high-frequnonlinear electronic circuits or components. In order to do this excitation signals haveprovided and the response of the DUT has to be measured. One can ask which quantitiesmost useful to describe the excitation and response signals. At low frequencies the excitatioresponse signals are easily described by voltage and current, where one of the two wconsidered the excitation or input signal and the other the response or output signaldescription is useful because, at low frequencies, good voltage meters and current metavailable, as well as voltage or current sources to provide the excitation signals. Atfrequencies (in practice all frequencies higher than about 10MHz) those near ideal voltagcurrent meters, as well as voltage and current sources, are no longer available. The main pis the connection between signal source, DUT and the actual data acquisition samplers. Fohigher frequencies every cable used will have a length which is no longer negligible compathe electromagnetic wavelength of the signals and it will behave like a transmission line wcertain delay and characteristic impedance. If the signal sources and the data acquisition sahave output and input impedances, respectively, which are not closely matched tocharacteristic impedance of the interconnecting cables that are used, standing wavescreated. Those standing waves correspond to very poor energy transfer through the cacertain frequencies, resulting in very bad measurements. The only way to solve this prowhich occurs for all high-frequency measurement systems, is to choose a fixed charact

impedance for all interconnecting cables, noted , and to make sure that the output and

impedances of all signal sources and data acquisition samplers are matched to . Fo

commercially available instruments at this moment, and also for the VNNA prototypescharacteristic impedance of 50Ohms is chosen, although instruments with other characimpedances, like for example 75Ohms, also exist. Practical descriptions of excitation

Zc

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14

2.2 General considerations

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called the incident voltage wave ( ) and the scattered voltage wave ( ). These are related

signal port current and voltage by the following relationships:

, and (2.2-1)

. (2.2-2)

One notes that (2.2-1) completely describes the port behavior of a signal source hav

output impedance and containing a voltage source with value (cf. the equivalent sc

depicted in Figure2.2-1). This means that the quantity will be independent from the D

behavior, making it the ideal quantity for describing the excitation during an experiment.

that replacing in (2.2-1) and (2.2-2) by an impedance different from the output impedan

the signal source, would result in an input quantity depending on both signal source charactas well as DUT behavior. Since you do not know a priori the DUT behavior this would result iunrealistically complex experiment design.

2.2.2 The VNNA as a modular instrument

Each VNNA can be considered as being constructed by combining modules with ceprimary functionalities. These modules are named the source module, which takes caregeneration of the excitation signals, the test-set, which guides the RF-energy to the DUwhich detects all spectral components of incident and scattered voltage waves and finadata-acquisition, which digitizes the detected waves. A general schematic of a two-port VNNillustrated in Figure2.2-2. Until now only two-port VNNAs have been constructed,HP-NMDG has concrete plans to build a three-port version in the near future. This threeVNNA will enable the accurate measurement of three-port mixers. Note the definition o

different quantity variables. All variables refer to voltage waves incident to the DUT, while

variables refer to voltage waves scattered by the DUT, the letters , and in a sub

refer to the generator, measured and DUT voltage waves respectively, while the numbesubscript refers to the test port number.

Figure2.2-1

i v

a b

av Zci+

2-----------------=

bv Zci–

2----------------=

Zc 2a

a

Zc

2a v

i

a

b

Zc

a

b g m d

15


inrs takeurier

ing of

bythe

of thea biasith the

or thes thesignalupling

2.3 Pre-existent Hardware Prototypes

2.3.1 General remark

Although it will not be explicitly shown or mentioned, all prototypes that are describedthe following paragraphs use computer controllers to do the measurements. These computecare of things like the calculations needed for the calibration, the execution of discrete Fotransforms to transform time domain data into the frequency domain, the appropriate settsources, switches, attenuators,....

2.3.2 Prototype of Markku Sipilä et al.

By the knowledge of the author, the first report of a VNNA was published back in 1988Markku Sipilä, Kari Lehtinen and Veikko Porra [1]. The typical use of the instrument wasmeasurement of the large-signal behavior of microwave transistors. A simplified schematicset-up is given in Figure2.3-1. As a test-set a coupler and a bias tee are used at port 1 andtee and an attenuator at port 2. The bias tees are necessary to provide the transistors wappropriate DC-bias voltages. A two-channel broadband sampling oscilloscope is used fdata acquisition. This sampling oscilloscope has a bandwidth of 14GHz and determinebandwidth of the whole measurement system. One microwave synthesizer is used for thegeneration. One part of the generator signal is directed towards the transistor through the conetwork 1, the other part is directed to the oscilloscope trigger input.

Figure2.2-1 VNNA General Schematic

DETECTOR PORTS

RF-INPUT 1

RF-INPUT 2

TEST PORT 1

TEST PORT 2

TEST-SET MODULE DUTSOURCEMODULE

DATA ACQUISITION MODULE

ag1

bg1

ag2

bg2

ad1

bd1

ad2

bd2

am1 bm1 am2 bm2

16


he

nts are

e out-

ponse.sider-

using

ccom-erator

harac-

t 1 isshortshownome

In the view of Markku Sipiläet al.a number of assumptions need to be valid in order for tsystem to make correct measurements.1. The generator and the sampling heads are perfectly matched or their reflection coefficie

accurately known.2. The generator harmonic output is negligible, or one has to use a low-pass-type filter at th

put of the generator.3. The gain of the sampling head is independent of frequency and it has a linear phase res

This is considered to be a safe assumption if the highest harmonic frequency under conation is well below the upper frequency limit of the sampling head.

4. The sampling heads are operated in their linear region. This can be accomplished bysuitable attenuators.

5. The load presented by the DUT does not affect the generator output. This can be aplished by applying so-called padding. This implies the use of an attenuator at the genoutput to improve the match and isolation of the generator.

6. The s-parameters of the coupling networks are accurately known. This is achieved by cterizing these networks with a commercial vectorial network analyzer.

Taking a look at the schematic one notes that only the reflected voltage wave at pordetected. In order to measure the incident voltage wave one typically replaces the DUT by aand one performs a measurement. When the above assumptions are valid, it can readily bethat enough information is available in order to calculate the incident voltage wave. Scomments on this oldest prototype are given in what follows.

Figure2.3-1 Markku Sipilä et al. VNNA Prototype: Simplified Schematic

DUT

SAMPLING OSCILLOSCOPE

CH1 CH2

TRIGGER

BIAS2BIAS1

COUPLINGNETWORK 1

COUPLINGNETWORK 2

17


ok it

possi-DUT.

ated.

onnectlinearially). Theion ofmpling

in thea fewmea-ary tog fac-

use

tic

ms ofonic

input aase ofn andplersssicalental

ion is

ourceeratesignalrangeat the

An advantage of this measurement set-up is the simplicity. If one takes a close lobecomes clear however that the set-up has several limitations.1. The measurement set-up only allows harmonic distortion type measurements. It is not

ble to provide excitation at both ports simultaneously and to measure the response of theExciting with more than one frequency component would also become very complicThese arguments limit the general applicability of the set-up as a VNNA.

2. The calibration of the set-up relies on many assumptions and makes it necessary to cand disconnect the coupling networks. Note that it is necessary to characterizethree-port networks, which is nontrivial (only two-port network analyzers are commercavailable, special procedures need to be applied for characterizing three-port networkstraceability of the calibration procedure is an issue, especially concerning the assumptthe perfect sampling heads. How can one be so sure that the phase distortion of the sahead is negligible?

3. The measurement speed is probably rather slow. Although no numbers are mentionedtext, sampling oscilloscopes are rather slow instruments, with actual sampling rates ofkHz and with limited dynamic range. In order to achieve dynamic ranges of about 50dBsurement times of about one minute are typically needed. The fact that it is necessreplace the DUT by a short in order to measure the incident wave is also a speed reducintor.

4. As reported in [4] the trigger stability of the sampling oscilloscope makes it difficult tofundamental frequencies higher than 5GHz.

2.3.3 Prototype of Urs Lott

A different approach is found in a 1989 publication by Urs Lott [2]. A simplified schemaof his set-up can be found in Figure2.3-2.

The goal of this instrument set-up is the accurate measurement of the output waveformicrowave transistors under large signal excitation. All measurements are of the harmdistortion type. Since no reflected waves have to be measured the test set is simple. At thecoupler detects the incident power, which is measured by a power meter. Note that the phthe incident fundamental wave is not measured. Attenuators take care of generator isolatiomatching at the input and of signal attenuation at the output (in order not to drive the saminto compression). Two bias tees provide the component with the appropriate DC bias. A clavectorial network analyzer is used for the detection of the amplitude and phase of the fundamand the harmonics created by the DUT. The ingenious principle of this coherent detectexplained in what follows.

The choice of an appropriate signal source is very important. Urs Lott needs a signal swhich not only generates a fundamental frequency component but which also genharmonics of this fundamental. For this purpose an HP83595A RF plug-in is used. This ssource is based on the principle of generating a fundamental signal with a frequencybetween 2.3GHz and 7GHz and using multipliers in order to extend the frequency range

18


urce,on ist theh asse of

r at thephase

t thefed tol thed the

dure isrt. Thissignallyzersionis the

output to 26.5GHz. Although only the multiplied signal appears at the RF output of the sothe fundamental signal is available at an auxiliary output. In the schematic this multiplicatidepicted by the “xn” box. The idea of Urs Lott is to use the fundamental signal appearing aauxiliary output as the excitation signal for his experiment. Some signal conditioning sucamplification and filtering is used in order to get a clean and powerful enough signal. Becauthe principle of the signal source up to four harmonics (one at a time) can be made to appeaactual generator output. Neglecting temperature effects, these output signals have a fixedrelationship versus the fundamental signal, with a frequency f.

Let us now take a look at the vectorial network analyzer that is used to detecfundamental and harmonics. The sinusoidal signal (only one harmonic at the time) that isthe RF input will be directed towards port1. The output of the network analyzer will equacomplex ratio of the corresponding frequency component of the signal received at port2 ansignal coming out of port 1 (which is the reference signal for the network analyzer).

The measurement procedure is then the following. For each harmonic the same proceapplied. First the generator is made to generate the corresponding harmonic at its output poharmonic signal appears at port1 of the network analyzer and is added by a coupler to thecoming from the DUT. During the standard forward response calibration of the network anathe DUT is replaced by two matched loads. After this calibration, a forward transmismeasurement of the signal coming from port1 of the analyzer has the result 1.0. Thisreference signal vectorrn in Figure2.3-2 and Figure2.3-3.

Figure2.3-1 Urs Lott VNNA Prototype: Simplified Schematic

x n

BIAS1 BIAS2

DUT

power meter

SIGNAL GENERATOR

PORT 1 PORT 2

RF INPUT

NETWORK ANALYZER

nf

nf

f

snrn

dn

19


r

remente

d

rt2 ofresultsled

mentat itsivelyible tott is a

andvior,

moree

The nonlinear DUT produces the componentdn at the frequency nf. The analyze

measurement with the DUT in place gives the sum vectorsn. Therefore the vectordn is calculated

fromdn = sn - rn = sn - 1.0 (2.3-1)

This is only the first step of the measurement procedure, because the raw measuvectors dn still are relative values (referred to thern vectors). In order to get the absolut

amplitudes and the phases of the spectral componentsdn the rn vectors have to be denormalize

by a calibration procedure. For the amplitude calibration the DUT is disconnected and pothe analyzer is replaced by a power meter. For each harmonic the power is measured. Thisin the denormalization of the length of thern. The phase calibration is based upon the so-cal

“golden diode” approach. With this is meant that the DUT is replaced by a nonlinear elefrom which one assumes to know a priori what the phases are of all harmonics it will createoutput when excited at the input with a signal of a certain frequency and power. By effectmeasuring the harmonics and comparing the result with the modeled harmonics, it is possdenormalize the phase ofrn. The nonlinear device used as a phase reference standard by Lo

fast diode connected in parallel to a 50Ω microstrip line, illustrated in Figure2.3-4. An

equivalent model of the diode (provided by the supplier?), containing parasitic inductivecapacitive effects, is used to do SPICE [3] simulations of the diode circuit large signal beharesulting in the knowledge of the phases of the different harmonics at the circuit output.

Some advantages of the implementation by Lott:1. The use of the network analyzer undoubtedly results in a faster data acquisition with

dynamic range as is the case for the prototype of Sipiläet al.. This can be achieved becaus

Figure2.3-2 The signal vectors at port2 of the measurement system (with frequency nf)

Figure2.3-3 The diode circuit for generating the reference harmonics

rn

sn dn

50Ω through line

reference diode

OUTIN

20


in anough

with a

abilityused

ciplee accu-ts done

possi-ugh”count.l fre-mon-

es the

large.

n aof the

plifiersnship

rature

et-up

of theltagesignal

wavetion orerminescope,

mpling

the network analyzer is inherently a smallband measurement instrument, filtering outanalog way all the noise not close to the harmonic frequencies one is interested in. Althno numeric values are available, the dynamic range can probably reach about 70dBmeasurement time of a few seconds.

2. The calibration procedure that is used has interesting properties. It easily allows a tracepath for the amplitude calibration by using a power meter. The reference diodes that arefor phase calibration are conceptually also interesting for establishing traceability. In prinstandard labs can try to establish measurement techniques to measure and specify thracy of these phase reference “golden diodes”. Consistency between the measuremenby different people can be easily be achieved if they use the same reference element.

Some disadvantages are:1. The use of the set-up is rather limited. Only harmonic distortion type measurements are

ble. It is impossible to accurately measure what happens at the DUT input, only “thromeasurements are possible. It is impossible to take the effects of mismatches into acDue to the use of the multiplying principle inside the signal generator the fundamentaquency is practically limited from 3.5GHz to 6.5GHz, and the number of measurable harics is limited to four.

2. The accuracy of the “golden diode” approach is questionable. In his paper Lott estimaterror between the model and the real device to be smaller than 10° at 15GHz and less than 16°at 20GHz for a fundamental frequency of 5GHz. These uncertainty bounds are ratherThey are due to significant contributions of unknown parasitical effects.

3. Although there is no problem of “oscilloscope trigger drift” the method of Lott relies oconstant phase relationship between the fundamental appearing at the auxiliary outputgenerator and the harmonic appearing at the generator RF output. Since filters and amare used inside the generator in order to condition the RF output, the phase relatiobetween fundamental and harmonic output is probably rather sensitive to tempechanges.

2.3.4 Prototype of Gunther Kompa et al.

In 1990 Günther Kompa and Friedbert van Raay [4] reported on the construction of a swhich can be considered as a significant extension of the set-up of Sipiläet al.. A simplifiedschematic of his set-up is depicted in Figure2.3-5. The RF signal injected into the RF inputnetwork analyzer test set is guided towards port1 in order to excite the DUT. The incident vowave at port1 is detected and appears at the REF output of the test set. The TEST outputwill be the detected transmitted voltage wave (going into port2) or the reflected voltage(reflected towards port1), depending on the chosen mode of operation of the test set (reflectransmission measurement). Note that all signals may contain harmonics. Two switches detwhether the test set REF and TEST signals are detected by a broadband sampling oscillotriggered by the fundamental, or are detected by a network analyzer. The broadband sa

21


easurerated

ssicall [5],in whatected

t is leftof theput isest setient to

trans-harac-

n the

oltage-up of

oscilloscope is used to measure the harmonics coherently, the network analyzer is used to maccurately the fundamental behavior and is used for calibration purposes. The calibbandwidth of the system is 20GHz.

An ingenious calibration procedure was developed by Kompaet al. were the networkanalyzer is effectively used to fully characterize the test set. For this purpose first a clarelative calibration is done using open, short, load and making a through at the DUT levenext three extra measurements are done. These measurements are shortly describedfollows. For the first of these three measurements the network analyzer TEST input is connto the test set REF input and vice versa, with a short connected at port1. Secondly the shoron port1 but the oscilloscope inputs are connected to the test set where channel2oscilloscope is once replaced by a power meter, and finally the network analyzer TEST inconnected to port1 of the test set, the network analyzer REF input to the REF output of the tand a load is put on the test set REF output. As shown in [4] these measurements are sufficfully characterize the test set and to perform an accurate amplitude calibration.

The measurement method of Kompa et al has several advantages.1. It can measure the fundamental and the harmonics of the incident, the reflected and the

mitted voltage waves. Because of the calibration procedure mismatches are accurately cterized and their effect is taking into account. Note that harmonics may be present iexcitation signal.

2. Because the excitation signal is measured together with the reflected or transmitted vwave, the oscilloscope trigger drift is no longer a problem as was the case with the set

Figure2.3-1 Günther Kompaet al. VNNA Prototype: Simplified Schematic

signal source

CH2

CH1

samplingoscilloscope

TEST

REF

networkanalyzer

trigger

RF input

REF TEST

PORT 1 PORT 2

DUT

network analyzertest set

22


all

by theedure

ion ele-

scope.intro-mea-ead is

d byalso

to

ty ofnmain

Sipilä et al. [1]. In fact the excitation signal has the role of being a timing reference sincetrigger drifts will delay as well the input as the output channel by the same amount.

3. The calibration procedure allows the accurate correction of all signal distortions causednon ideal test set and allows an accurate power calibration of the whole system. The procis based on traceable techniques such as the use of classical network analyzer calibratments and a power meter.

Some problems are common with the Sipilä et al. set-up.1. Only excitations at port1 are possible.2. The measurement speed is probably rather slow because of the use of a sampling oscillo3. The phase calibration still relies on the assumption that the oscilloscope sampling heads

duce no phase distortion (note that only the phase distortion introduced by the test set issured by the calibration procedure, any phase distortion appearing inside the sampling hnot taken into account).

2.3.5 Prototype of Markus Demmler et al.

About coinstantaneously with HP-NMDG a more advanced set-up was developeMarkus Demmler, P. J. Tasker and M. Schlechtweg in 1994 [6]. A similar set-up was laterused by J. G. Leckeyet al. [7] and C. J. Weiet al. [8] [9] in 1995. A simplified schematic of theset-up of Demmleret al. is depicted in Figure2.3-6, it allows harmonic measurements up

40GHz.The functionality of the couplers and the switch is actually the same as the functionali

the network analyzer test set used by Kompaet al.. Besides the use of wafer probes to allow owafer measurement of devices (a technology which was not readily available in 1990), the

Figure2.3-1 Markus Demmler et al. VNNA Prototype: Simplified Schematic

BIAS1 BIAS2DUT(on wafer)

TransitionAnalyzer

CH1 CH2

signal source

23


hiss andis forn onwhat

ically

self istest set.d themeter

ole on

aininginingduceson can

urdwidthwere

madetion1994ch ofsulted

thewhich. Thisphase

sential

difference is the use of a microwave “Transition Analyzer” [10] for the data acquisition. Tcommercial instrument was introduced in 1991. It allows the measurement of the phaseamplitudes of the fundamental and harmonics present at both input channels, and thfrequencies up to 40GHz. Since it is based on the harmonic mixing principle, rather thaequivalent time sampling, the instrument allows data acquisition about 100 times faster thanis possible with a sampling oscilloscope, for the same or even a better dynamic range (typbetter than 50dB).

The basic principles of the calibration are the same as those used by Kompaet al.. In steadof using a separate network analyzer as in Figure2.3-5, however, the transition analyzer itused to do the network analyzer measurements that are needed in order to characterize theTo avoid the use of on wafer power sensors, the principle of reciprocity between probe tip another connectors is used. This allows the power calibration to be transferred from the powerstandard connector technology to the probe tip co-planar waveguide technology. The whwafer calibration procedure is very similar to the technique used in [11].

The advantages of this set-up relative to the set-up of Kompaet al. is undoubtedly the fastdata acquisition, which probably takes just about one second per measurement. One remdisadvantage is the fact that still only one excitation signal can be used. Another remaproblem is that the calibration is based on the assumption that the transition analyzer introno phase distortion when measuring the harmonics. The big question is how this assumptibe verified by a traceable procedure?

2.4 Prototype of Hewlett-Packard Network Measurement and DescriptionGroup

2.4.1 A short historical overview

The first VNNA prototype of HP-NMDG was built in 1992 [13]. It was based on focouplers to detect the incident and scattered waves at both ports and it used a 20GHz bansampling oscilloscope for the data acquisition. Only connector-based measurementspossible. A slow data acquisition (3minutes per measurement) and trigger drift problemsthis system rather impractical. The solution was found in 1993 by modifying two “transianalyzers”, such that four fully synchronized RF data acquisition channels were available. Inthe measurement possibilities were extended to on wafer devices [14]. Finally, the approausing only those “transition analyzer” parts that are really needed for our measurements rein the present prototype. This prototype will be described in more detail in what follows.

A big part of the calibration procedure that is used is similar to the one used by Kompaet al.and Demmleret al.. There is however one significant extension, namely the fact thatcalibration procedure also takes into account the phase distortion of the transition analyzer,is analog to the approach of Lott. This is achieved by the use of a “reference generator”reference generator generates a fundamental and harmonics with a precisely knownrelationship. In a way this reference generator can be viewed as a “golden diode”. The es

24

2.4 Prototype of Hewlett-Packard Network Measurement and Description Group

plingationatablesensusst setoidingup isd the

“on

e biasent andouplersplingf thesignal

difference is, however, that the reference generator itself is characterized by a samoscilloscope. This oscilloscope in its turn is characterized by the “nose-to-nose” calibrprocedure [15]. This makes the phase calibration traceable to the accurate and repe“nose-to-nose” calibration procedure, which is used as a microwave phase reference constandard. Another advantage of the calibration procedure used by HP-NMDG is that the tenever needs to be disconnected from the data acquisition RF input channels, thereby avconnector repeatability problems. The useful calibrated bandwidth of the present set-18GHz. This bandwidth is limited primarily by the calibration elements, the power meter anreference generator. The couplers and the data acquisition have a bandwidth of 40GHz.

2.4.2 Simplified schematic and comments

A simplified schematic of the prototype is depicted in Figure2.4-1. Note that only the

wafer” set-up is illustrated in the figure. For the set-up to measure connectored devices thtees are usually omitted and the wafer probes are replaced by classical test-set cables. Incidreflected voltage waves at the signal ports are detected by four broadband couplers. The cused by HP-NMDG have a useful bandwidth from about 500MHz to 50GHz and have a coufactor of 14dB. Attenuators may be used in order to bring the signal level at the input obroadband downconvertor below -10dBm, necessary to assure linearity of the convertor

Figure2.4-1 HP-NMDG VNNA prototype: simplified schematic

BIAS1 BIAS2DUT(on wafer)

signal source

4 channel broadband downconvertor

precision analog-to-digital convertor

25


e RFill begnalsfourigitaltheidth

oneer to

one

tivelyes:DUT

f inci-amplelers,ptions

takeses allency

rated

ced bycali-

uturean in

rationp tohave

tandard

or. Itsare

hat ournalyzer

samplers. The broadband downconvertor uses the harmonic mixing principle to convert thfundamental and harmonics into IF fundamental and harmonics. The downconvertor wdescribed in more detail in what follows. its RF bandwidth goes up to 40GHz, and these sican be converted to frequencies below 10MHz. The final data acquisition is done byHP-E1430 data acquisition modules. These modules are precision (23-bit) analog-to-dconvertors (ADC) which digitize the four IF signals appearing at the outputs ofdownconvertor. At present the sampling rate of the ADC’s used is 10MHz, their useful bandwis about 4MHz. A dual signal source was built, allowing to excite both DUT ports with a two-tsignal. This dual signal source will also be described in more detail in what follows. In ordsynchronize the downconvertor, the ADC and the signal sources they are all locked toprecision ovenized 10MHz reference clock.

Despite the disadvantage of using some non commercially available parts and the relahigh price, this set-up has some significant advantages compared with preexistent prototyp1. The use of four couplers and a four channel data acquisition system allows to excite the

at both ports simultaneously and allows to accurately detect all spectral components odent and reflected voltage waves. This allows the accurate measurement of for extwo-port mixers and measurements of the “load-pull” type [11]. Because of the four coupall mismatches can easily be taken into account during the measurements, no assumconcerning perfect matches are needed.

2. The data acquisition is fast with a good dynamic range. One measurement typicallyabout two seconds with a dynamic range of about 55 dB. This measurement time includcomputer calculations needed to convert the data from the time domain into the frequdomain and to perform the calculations needed for converting the raw data into calibdata.

3. The use of a reference generator has several advantages. At first all distortions introduthe sampling heads and IF circuitry of the downconvertors are taken into account duringbration, and the procedure allows a relatively easy calibration traceability path to a fmicrowave phase reference primary standard. At present this traceability path goes up tohouse performed “nose-to-nose” calibrated sampling oscilloscope. In the future coopewith national standards labs (NIST, NPL) can hopefully lead to a traceability path going unational and international (BIPM) standard labs. It is sufficient for these standard labs toan accurate fast pulse characterization system in order to establish a national phase straceability path.

2.4.3 The broadband downconvertor: hardware

The key component of the whole system is undoubtedly the four channel downconvertworking principles will be explained in some more detail in what follows. At present the hardwthat is used is the same as the one used in a transition analyzer. The main difference is tset-up enables the use of four perfectly synchronized channels, where as the transition aonly has two channels available.

26


Hz

esizeHz,soidalingl inputfournnel1headllatort 10ps.lingIf theC isill be-off

A hardware schematic of the downconvertor is depicted in Figure2.4-2. The 10M

reference clock is fed into a so-called “fracn synthesizer”. This component is able to synthany local oscillator frequency between 10MHz and 20MHz with a frequency resolution of 1and with an accuracy determined by the accuracy of the reference input. The near sinuoutput of the fracn is send into a pulse forming circuit with four outputs. The pulse formnetwork is based on the use of a so-called “step recovery diode” and converts the sinusoidasignal into a narrow pulse train, with the same repetition rate as the input signal. Thesynchronized pulse trains are then guided towards the four downconvertor circuits. The chadownconvertor circuit is given in more detail. The pulse train arrives at the actual samplingwhich is the heart of the downconvertor circuit. Inside the sampling head the local oscipulses are first properly shaped into near ideal square wave pulses, with a duration of abouThis signal is then multiplied with the RF signal, which is injected directly into the samphead. At the IF output of the sampling head a lot of harmonic mixing products are present.local oscillator frequency is properly chosen the mixing product with the lowest frequency (Dnot considered) will correspond to the image of the RF fundamental, and all RF harmonics wconverted into harmonics of this fundamental mixing product. A low pass filter with a cut

Figure2.4-1 four channel downconvertor: simplified schematic

low passfilter buffer

sampling head

channel 1 downconvertor circuit

RF input IF output

pulseshaping

channel 3downconvertor board



10MHzreference clock

fracNsynthesizer

pulse former

27


. Theector.

ningthe

s the

n be

he

will

qual

nds to

tion

form

frequency of about half the local oscillator frequency only retains the useful mixing productsoutput of this low pass filter is buffered before being send to the downconvertor output conn

2.4.4 The downconverting process: mathematical description

A simple mathematical description which already gives a good feeling of what is happeis given in what follows. Note however that there exist actual implementations ofdownconversion process which are much more compliant with the following model than icase for the relatively complex transition analyzer downconvertor board.

Consider a real multiharmonic signal going into the sampling head. The signal ca

written as

, with and . (2.4-1)

In (2.4-1) is the ith complex harmonic represented by the complex number . T

fundamental RF frequency equals . The signal at the output of the sampling head

equal

, with (2.4-2)

. (2.4-3)

In (2.4-2) represents one pulse realization of the pulse train with a repetition rate e

to the local oscillator frequency . The mixing process inside the sampling head correspo

the multiplication of the RF signal with the pulse train, which is described as a summa

of an infinite number of delayed versions of . We are interested in the Fourier trans

of . This is given by

, with (2.4-4)

. (2.4-5)

In what follows (2.4-5) will be simplified.

From (2.4-5) follows , and substituting

x t( )

x t( ) xi t( )i N–=

N

∑= xi t( ) Aiej2πif RFt= A i– Ai

*=

xi t( ) Ai

f RF y t( )

y t( ) x t( ) p t nf LO--------–

n ∞–=

∞

∑

yi t( )i N–=

N

∑= =

yi t( ) xi t( ) p t nf LO--------–

n ∞–=

∞

∑

Aiej2πif RFt p t n

f LO--------–

n ∞–=

∞

∑

= =

p t( )f LO

x t( )p t( )

Y f( ) y t( )

Y f( ) Yi f( )i N–=

N

∑=

Yi f( ) yi t( )e j2πft– td

∞–

∞

∫ Aiej2πif RFt p t n

f LO--------–

n ∞–=

∞

∑

e j2πft– td

∞–

∞

∫= =

Yi f( ) Ai p t nf LO--------–

ej2π if RF f–( )t td

∞–

∞

∫n ∞–=

∞

∑=

28


,

as a

at the

t the

, also

sed one,

choose

)

eingthat

illator

nsfer

n be

results in . (2.4-6)

This expression can further be simplified in to

; (2.4-7)

and based upon the formula , proven in [12], in to

and finally (2.4-8)

, (2.4-9)

with equal to the Fourier transform of .

The result of (2.4-9) can now be interpreted. The ith complex harmonic of the input signal

characterized by the complex number , will appear at the output of the harmonic mixer

series of intermodulation products. Every intermodulation product, indexed by , appears

frequency and has a complex conversion factor equal to . Note tha

intermodulation products are located on an equidistant frequency grid with distance

note that for every positive index i a conjugate component with index -i is present. Thecomponents appear on the frequency grid which is the mirrored image of the positive indexe

with frequencies .

Suppose now one wants to do a measurement. and N are given and one needs to

an appropriate value for . If one chooses and m as follows, defining ,

, one automatically gets that (2.4-10

. (2.4-11)

As shown in Figure2.4-2 the output of the harmonic mixer is low-pass filtered before bbuffered and send to the output of the downconvertor board. Suppose that this filter isperforming that all spectral components with frequencies greater than half of the local osc

frequency can be neglected at the filter output. If we define as the filter complex tra

function (assumed to be zero for ), the downconvertor output spectrum ca

written as

t' t nf LO--------–= Yi f( ) Ai p t'( )e

j2π if RF f–( ) t' nf LO--------+

t'd

∞–

∞

∫n ∞–=

∞

∑=

Yi f( ) Ai ej2πn

if RF f–( )f LO

----------------------

n ∞–=

∞

∑

p t'( )ej2π if RF f–( )t' t'd

∞–

∞

∫

=

ej2πng

n ∞–=

∞

∑ δ g k–( )k ∞–=

∞

∑=

Yi f( ) Aif LO δ f if RF– kfLO+( )k ∞–=

∞

∑

P f if RF–( )=

Yi f( ) Aif LO δ f if RF– kfLO+( )P* kf LO( )k ∞–=

∞

∑

=

P f( ) p t( )

Ai

k

if RF kf LO– fLOP* kf LO( )

f LO

if RF– kfLO+

f RF

f LO f LO f IF

f RF mfLO– fIF

f LO

2N--------<=

if RF imf LO– if IF

f LO

2--------<=

H f( )2f f LO> Z f( )

29


vely

has

uals

the

uency

and m

kHzRF

et usmes astate

.

,

. (2.4-12)

This formula proves that the spectrum at the downconvertor output is effecti

representing an image of the input spectrum, where every RF harmonic with frequency

been mapped into a component with frequency . The conversion factor eq

.

A practical example together with Figure2.4-3 will illustrate the above theory. Suppose

input signal has a fundamental frequency of 1GHz and has 20 harmonics, the highest freq

component present having a frequency of 20GHz. If one chooses equal to 19.998MHz

equal to 50, one finds corresponding to (2.4-10):

. (2.4-13)

This implies that the fundamental 1GHz RF component will be converted into a 100component and all harmonics of this RF signal will be converted into correspondingharmonics. The 20GHz RF component will thus be converted into a 2MHz IF frequency. Lnow take a look at the conversion factors for the different frequency components. If one assuflat frequency response low-pass IF filter with a cut-off frequency of about 10MHz, one can

that , for . The conversion factor will thus be approximated by

The pulse can very well be approximated by a function that equals 1 for

Figure2.4-1 Illustration of the downconversion process.

Z f( ) Aif LOδ f if IF–( )H if IF( )P* imf LO( )i N–=

N

∑=

if RF

if IF

f LOH if IF( )P* imf LO( )

...

...

f (GHz)1 2 3 18 19 20

f (MHz)0.1 0.2 0.3 1.8 1.9 2.0downconvertor output spectrum

downconvertor input spectrum

f LO

f IF 19×10 Hz 50 19.998

6×10× Hz– 1003×10 Hz= =

H if IF( ) 1≈ i 20≤ f LOP* ilf LO( )

p t( ) T 2⁄– t T 2⁄< <

30


qual

thatich

. Noteactual

takingave a

of aated

ble

izerstheeachrol the

sible.ugh nots for a

with , and which equals 0 for all other . This results in

, (2.4-14)

and results in a conversion factor for the ith harmonic given by:

(2.4-15)

,where D stands for the duty cycle of the pulse train. For our case D is approximately e

to . Some practical values of are given in Table2.4-1.

One notes that the conversion factor of the 20GHz signal is about 600mdB lower thanof the 1GHz signal. At 44GHz the conversion factor would reach the -3dB point, whcorresponds to the specified bandwidth of the downconvertor board equal to about 40GHzthat to be accurate one also needs to add the losses from RF input connector to theharmonic mixer which urges to take some reserve versus the bandwidth specified solely byinto account the harmonic mixing process. At 100GHz the harmonic mixing process would hzero output. This is logic since this corresponds to a signal with a period of exactly 10ps.

2.4.5 The signal source

The dual signal source appearing in Figure2.4-1 allows to excite both port1 and port2DUT with a two-tone signal. An example of a construction of the dual signal source is illustrin Figure2.4-4. One synthesizer generates a signal with a frequency f1, the other a signal with

frequency f2. The use of the two by two combination of the switches of the “single pole dou

throw” (SPDT) type allows the controllable attenuators to be in the path of the synthestowards port1 or towards port2. If both switches of an SPDT-pair are in the “1” modeattenuator is in the path towards port1 and vice versa. By controlling the power output ofsynthesizer, the SPDT switches and the attenuators, it is possible to independently contpower of each frequency component at each port.

In contrary to the powers, controlling the phases between the different signals is imposFor the phase relationship between the two synthesizers there is a way out because althocontrolled it can be randomized. This can be achieved by switching one of the synthesizer

Table2.4-1

1GHz -0.001dB

10GHz -0.143dB

20GHz -0.579dB

44GHz -2.967dB

100GHz dB

T 10ps= t

P f( ) P* f( ) Tsinc πfT( )= =

Ci

Ci f LOTsinc πilf LOT( ) Dsinc πif RFT( )≈=

2006–×10 sincπif RFT( )

if RF sinc πif RFT( )

∞–

31


telyizers

e othering the

rawlyphases.

inll 300

y theasured

drifts.n twoperiodfound

haseability

nt

ips willndent

very short period to a random frequency after which the original frequency is immediarestored. This is experimentally verified as follows. For a first experiment, the two synthesare set at 1GHz, with one synthesizer connected to channel 1 of the downconvertor and thsynthesizer connected to channel 3. Next, 300 data acquisitions are done, without changsynthesizer frequency, and the phase relationship between the two synthesizers, asmeasured by the system, is each time calculated. This results in a record of 300 measuredThe experimental cumulative distribution function (CDF) of this experiment is illustratedFigure2.4-5. This figure reveals that the relative phase between the two synthesizers, for aexperiments, is about a constant 2° with a maximum deviation of only about 1.5°. Also note thatthe experimental CDF is similar to the CDF of a normal distribution. This can be explained bfact that both synthesizers are locked to the same 10MHz mother clock, such that the mephase relationship between the two can only change due to noise and temperature relatedNext, a second experiment is performed, similar to the previous one, but where, in betweedata acquisitions, one of the sources is switched to a random frequency for a very short(after which the source is reset to 1GHz). The experimental CDF of this experiment can bein Figure2.4-6. This figure reveals that the relative phase now ranges from -180° to +180°, withthe experimental CDF practically equal to a straight line. This indicates that the relative pbetween the two synthesizers has been successfully randomized and has a uniform probdensity function.

The remaining problem is the phase relationship between the f1 component at port1 and the

f1 component at port2 and the analog problem for f2. These phase relationships will be depende

on the controllable attenuator setting. For a fixed attenuator setting these phase relationshnot change. The only way out would be the use of controllable phase shifters or of indepe

Figure2.4-1 The dual signal source

1

1

1

122

1

1

1

12 2

to port1 to port2

SPDT switch

f1 f2

32


moreent oration.

synthesizers.Depending on the measurement type other source configurations can however be

interesting, such as the use of only one source for a simple harmonic distortion measuremthe use of one source at port1 and another source at port 2 for a two-port mixer characteriz

Figure2.4-2 Experimental relative phase CDF (without changing the frequency)

Figure2.4-3 Experimental relative phase CDF (with changing the frequency)

0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Phase (degrees)

CDF

-135 -90 -45 0 45 90 135 1800

0.2

0.4

0.6

0.8

1

Phase (degrees)

CDF

33


fterwasin thees of

s and. Thisof anIt alsologicalsed bytor. Byure can

re for

2.5 Conclusions

In this chapter is shown what the VNNA hardware implementations look like. Ashowing and commenting on the work of others, the implementation by HP-NMDGconsidered in more detail and a simple mathematical model was used in order to expladownconversion process, which is the key element of the VNNA. The technological advantagthe NMDG prototype are the use of four couplers, four synchronized downconvertor boardfour digitizing channels to detect and digitize the incident and reflected waves at both portsresults in fast measurements with a good dynamic range, not possible with the useoscilloscope, and allows excitation signals to be present at both ports at the same time.allows to take easily into account all possible test set mismatches. Besides these technoadvantages, the calibration procedure used allows one to take into account distortions cauthe non-perfect test set, as well as those caused by the non-idealities of the downconverusing a reference generator as a phase reference transfer standard, the calibration procedbe made traceable to the accuracy of the so-called “nose-to-nose” calibration procedusampling oscilloscopes.

34

2.6 References

aind

inear

ron.

ormope

es,”

eter

its

oad

tion

ode

ses,”

aferand

2.6 References

[1] Markku Sipilä, Kari Lehtinen and Veikko Porra,”High-Frequency Periodic Time-DomWaveform Measurement System,”IEEE Transactions on Microwave Theory anTechniques, Vol.36, No.10, pp.1397-1405, October 1988.

[2] Urs Lott,”Measurement of Magnitude and Phase of Harmonics Generated in NonlMicrowave Two-Ports,”IEEE Transactions on Microwave Theory and Techniques, Vol.37,No.10, pp.1506-1511, October 1989.

[3] L. W. Nagel,”SPICE2: A computer program to simulate semiconductor circuits,” ElectRes. Lab, Univ. Calif. Berkeley, Ca, Tech. Rep. ERL-M520, May 1975.

[4] Gunther Kompa and Friedbert Van Raay,”Error-Corrected Large-Signal WavefMeasurement System Combining Network Analyzer and Sampling OscilloscCapabilities,” IEEE Transactions on Microwave Theory and Techniques, Vol.38, No.4,pp.358-365, April 1990.

[5] D. Rytting,”An Analysis of Vector Measurement Accuracy Enhancement TechniquProc. Hewlett-Packard RF & Microwave Symposium, pp.16-20, March 1982.

[6] M. Demmler, P. J. Tasker and M. Schlechtweg,”On-Wafer Large Signal Power, S-Paramand Waveform Measurement System,”Conference Record of the INMMC‘94 - ThirdInternational Workshop on Integrated Nonlinear Microwave and Millimeterwave Circu(Duisburg-Germany), pp.153-158, October 1994.

[7] J. G. Leckey, A. D. Patterson and J. A. C. Stewart,”A Vector Corrected Waveform and LLine Measurement System for Large Signal Transistor Characterisation,”ConferenceRecord of the IEEE Microwave Theory and Techniques Symposium 1995, Orlando, Florida,USA, pp.1243-1246, May 1995.

[8] C. J. Wei, Y. Lan, J. C. M. Hwang, W. J. Ho and J. A. Higgins,”Waveform Characterizaof Microwave Power Heterojunction Bipolar Transistors,”Conference Record of the IEEEMicrowave Theory and Techniques Symposium 1995, Orlando, Florida, USA,pp.1239-1242, May 1995.

[9] C. J. Wei, Y. A. Tkachenko, J. C. M. Hwang, K. E. Smith and A. H. Peake,”Internal-nWaveform Probing of MMIC Power Amplifiers,”Conference Record of the IEEEMicrowave and Millimeter-Wave Monolithic Circuits Symposium 1995, Orlando, Florida,USA, pp.127-130, May 1995.

[10] Jack Browne,”Transition Analyzer scans amplitude and phase of 40-GHz pulMicrowaves & RF, March 1991.

[11] Andrea Ferrero and Umberto Pisani,”An Improved Calibration Technique for On-WLarge-Signal Transistor Characterization,” IEEE Transactions on Instrumentation

35


ny,

rk”sium

mentltage

pling

Measurement, Vol.42, No.2, pp.360-364, April 1993.

[12] A. Papoulis,”The Fourier Integral and its applications,” McGraw-Hill Book CompaChapter3, 1962.

[13] Tom Van den Broeck and Jan Verspecht, “Calibrated Vectorial “Nonlinear NetwoAnalyzers,”Conference Record of the IEEE Microwave Theory and Techniques Sympo1994, San Diego, California, USA, pp.1069-1072, May 1994.

[14] Jan Verspecht, Peter Debie, Alain Barel and Luc Martens,”Accurate On Wafer MeasureOf Phase And Amplitude Of The Spectral Components Of Incident And Scattered VoWaves At The Signal Ports Of A Nonlinear Microwave Device,”Conference Record of theIEEE Microwave Theory and Techniques Symposium 1995, Orlando, Florida, USA,pp.1029-1032, May 1995.

[15] Jan Verspecht and Ken Rush,”Individual Characterization of Broadband SamOscilloscopes with a “Nose-to-Nose” Calibration Procedure,”IEEE Transactions onInstrumentation and Measurement, Vol.IM-43, No.2, pp.347-354, April 1994.

36

Chapter 3

Accuracy of Broadband SamplingOscilloscope Measurements

Abstract - Oscilloscopes are usually used to look at signals more in a qualitative thanin a quantitative way. Although modern sampling oscilloscopes are essentiallysampling digitizers, they still have a bad reputation concerning measurementaccuracy. In this chapter is shown that many sampling errors can be characterizedand their effects eliminated. If done properly, this results in accurate measurements.Both timing and amplitude errors (horizontal and vertical errors) are considered.Understanding the accuracy of sampling measurements is necessary if one wants touse a broadband sampling oscilloscope as a phase reference consensus standard.

37

Chapter 3 Accuracy of Broadband Sampling Oscilloscope Measurements

sensusr thes of theFirst athee will

tion of

pulseith thet thethisonly

Rushndedain

digitaldualthat aatable

ose ofl1. Ineded.of the

rsionthe

s 10pslingin the

3.1 Introduction

As already mentioned in the previous chapters the actual phase reference constandard in our work will be a “nose-to-nose” calibrated sampling oscilloscope. In order fosampling oscilloscope to be a standard it is necessary that all possible measurement errorinstrument can theoretically be described and can be corrected or quantized in practice.simplified explanation will be given on the working principles of a sampling oscilloscope ofHP-54120 type. Next an overview of the errors of a general broadband sampling oscilloscopbe given. In this chapter one will specifically read about the characterization and compensatime base measurement errors.

One of the oscilloscope characteristics is of special interest to us, namely the imresponse of the oscilloscope. This impulse response can accurately be determined w“nose-to-nose” calibration procedure [1], which will be explained in Chapter 4. The fact thaHP-54120 is the only sampling oscilloscope type which allows the implementation ofprocedure explains our special interest in this instrument (besides the fact that it is theoscilloscope type available in our lab).

3.2 The HP-54120 sampling oscilloscope

3.2.1 Introduction

The HP-54120 series sampling oscilloscope was designed in the late eighties by Kenand his colleagues of the Hewlett-Packard Colorado Springs Division. The instrument is intefor the characterization of high speed digital circuits at one hand, and “time domreflectometer”-measurements (TDR) on the other hand. The instrument is a four channeloscilloscope based on the “equivalent time” sampling principle. This implies that every indivisample is taken by the oscilloscope at the corresponding delay after a trigger instant. Notetriggering is done for each individual sample. This means that the oscilloscope needs a repesignal and an appropriate trigger signal (this may be one of the signals itself). For the purpTDR-measurements a fast step generator is included into the oscilloscope channeTDR-mode the oscilloscope is internally triggered, such that no external trigger signal is neThe actual maximum sample rate of the instrument is a few kHz. Nevertheless, becauseequivalent time sampling principle, the instrument has a bandwidth of 20GHz for a first ve(with the HP-54121T test-set) and a bandwidths up to 50GHz for the latest version (withHP-54124T test-set). All channel inputs are terminated in 50Ω, and the full scale input range is+/-320mV, which corresponds to a sinusoidal signal with a power of 0dBm.

3.2.2 The instrument working principles

The timebase has a very high dynamic range. With the smallest time range one seeper division, with 100ps for the whole time range. The oscilloscope will display 400 samppoints with a time base resolution of 0.25ps. On the other hand the maximum time range is

38

3.2 The HP-54120 sampling oscilloscope

ge is

dingl to annts toampleggerator ismumnected

gnedsuch

gersof the

afterhargeat theally

millisecond range. The hardware structure that allows this large timebase dynamic randepicted in Figure3.2-1. More technical detail can be found in [3].

Two timing mechanisms exist within the oscilloscope: a fine delay mechanism proviany delay between zero and 4ns, and a course delay mechanism providing a delay equainteger times 4ns. An example will clarify what happens. Suppose that the oscilloscope wasample the input signal 62.450ns after the trigger signal reaches 50mV. Before taking the sthe trigger level DAC will receive the binary code to set its output to 50mV. As soon as the trisignal reaches this setting, the comparator will start a ramp generator. This ramp generinitialized by a DAC such that its output reaches zero volts at exactly 2.450ns. The maxidelay that can be generated by the ramp generator is 4ns. The output of the comparator conto the ramp generator output will start a 250MHz oscillator. This oscillator is especially desifor a quick start up. The output of this oscillator goes into a counter. This counter is presetthat it will produce a pulse at its output after 15 oscillator cycles. This output pulse finally trigthe signal sampling switches. These switches will close during about 10ps. Becausesettings of the fine delay and the coarse delay the switching will occur at precisely 62.450nsthe trigger event. The current that flows through the switch is integrated and goes into a camplifier. Note that the charge collected is proportional to the instantaneous signal levelsampling instant. The output of the charge amplifier is send to an ADC. This ADC will fin

Figure3.2-1 Simplified structure of the sampling oscilloscope hardware

DAC

4ns fine delay

250MHzstartableoscillator

digitalcounter

DAC

ADC

screenswitch

CH1input

CH2input

CH3input

CH4input

triggerinput

trigger levelbinary input

fine delay ramp coarse delaybinary input

ramp generator

initializationbinary input

∫∫∫∫

39


tant,. Thisnd finetually

ate

ith thee the

e theo thesed.ebaseus thend aning

timinghavelleds. It

pliesmplingnsatedd gainoise. Itthe

abouterrorThisy theristicse anderencesponsewith

digitize the collected charge. After being multiplied with the appropriate proportionality consthe sampled value is send to the oscilloscope screen. After the digitizing the circuit is resetwhole process is repeated for every sample taken, each time with the appropriate coarse adelay settings. Note that only one channel at the time is selected by the switch and is acmeasured. It takes about 300µs to complete one sampling cycle, limiting the actual sampling rto about 3kHz.

The sampling process itself will be explained in more detail in Chapter 4.

3.3 Accuracy aspects of a broadband sampling oscilloscope

Since the instrument is rather complex, it is not easy to define what one understands waccuracy of a sampling oscilloscope. A first thing to do is to classify the possible errors. Sincinstruments returns two-dimensional data, voltage versus time, one kind of errors will bhorizontal errors, related to the time axis, and the other will be the vertical errors related tvoltage axis. In what follows all recognized errors which are significant to our work are discus

Several errors can be recognized on the time axis. A first error is called systematic timdistortion. This expresses a systematic deviation of the time instant as it is programmed versactual physical sampling instant. This is mainly caused by an imperfect ramp generator aimperfect 250MHz oscillator. A second error is called timebase jitter. This is a stochastic timerror. Suppose the oscilloscope is programmed to sample many times at only one particularinstant. Mainly due to noise present at the trigger input the physical timing instants will belike a stochastic variable characterized by a probability density function. A third error is catimebase drift. It is mainly caused by the sensitivity of the trigger to temperature variationappears as a systematic shift of the time base correlated with temperature.

Several vertical errors are present. First there is a systematic voltage distortion. This imas well the presence of an offset and gain error as a nonlinear distortion caused by the sahead itself. The main part of the nonlinear distortion caused by the sampling head is compeby the use of a conversion look-up table, measured at manufacturing time. The offset anerror are easily characterized and compensated by a DC measurement. A second error is ncan be modelled as additive white noise with a normal probability density function. For20GHz version, the rms value of this noise is about 3mV. Note that the quantization level is20µV, which makes all quantization errors insignificant versus the additive noise errors. Thewhich is of most interest to our work is undoubtedly the linear distortion of the oscilloscope.is the error characterized by the instrument’s impulse response in the time domain or binstrument’s complex transfer characteristic in the frequency domain. This charactedetermines the error made by the instrument when one wants to characterize the phaamplitude of the spectral components of a fast repetitive pulse generator, like a phase refgenerator. In 1990 Ken Rush came on the idea of an accurate method to find the impulse reof the instrument. It is called the “nose-to-nose” calibration procedure. Chapter 4 will dealthis method, which is used as the phase reference consensus standard.

40

3.4 Timebase distortion

singatal waspearedtudeis mains was aed toound. Thation ofctrumon thethe

basethe

of theitalulatedeffect

easer with awelld an

te the. Aof annovelortion

tione error


3.4.1 Introduction

The timebase distortion of the oscilloscope got our attention in 1992 as it was cautrouble with the first HP-NMDG VNNA prototype which used the oscilloscope for the dacquisition. When a discrete Fourier transform of a data record of a sinusoidal signacalculated with no theoretical leakage present, some spurious spectral components apwhich were an integer multiple of 250MHz distant from the main component. Their magniwas largest close to the main frequency component and increased when the frequency of thcomponent also increased. It was soon found out that the cause of these spurious signalsystematic timebase distortion with a repetition rate equal to 4ns, which correspondnonidealities of the 4ns fine delay ramp generator of Figure3.2-1. Very little literature was fconcerning the subject of timebase distortions and its effects on discrete Fourier transformswas the motivation to develop a method ourselves [2] which allows the accurate determinatthe timebase distortion and uses this information to accurately determine the signal spe(without spurious responses), which was the final goal. Because the method is basedcalculation of the analytical signal [5] associated with a digitized signal, it is called“Analytical signal”-method.

As mentioned above, two parts can be distinguished in the method. At first the timedeviation from ideality is measured by digitizing a sinusoidal signal. The method is similar tomethod mentioned in [4], and is based on the calculation of the instantaneous phaseanalytic signal [5] of the digitized waveform, which in fact corresponds to performing a digphase demodulation. Since the digitized sinusoidal signal appears to be slightly phase modby the timebase distortion, the demodulated phase is a good measure for this distortion. Theof additive noise is studied, and it is explained how windowing techniques [6] can incraccuracy. Three other methods to measure the timebase error are also discussed, togetheshort description of their main characteristics. It is explained why the one actually used issuited for our specific application. The use of the method is illustrated by a simulation anexperiment.

Once the timebase distortion is characterized, this information can be used to estimavalues of the spectral components in a microwave signal with good accuracyleast-squares-error approach is used to actually perform the estimation. The resultsexperimental verification are reported in the last section. These results prove that themethod proposed effectively removes the erroneous effects which a digitizer timebase distmay have on the estimation of the values of spectral components.

3.4.2 Timebase distortion measurement

Mathematical theory of the newly developed “Analytical signal”-method

In this section will theoretically be explained how an a priori unknown timebase distorcan be calculated out of a digitized sinusoidal signal. It is hereby assumed that the timebas

41


is the

priori

will

r,

It canal is

rite

rtion,

hat

h

ment,

-3),

is bandlimited, which is practically valid for a broadband sampling oscilloscope.Suppose that there are N sampling points and that the ideal sampling period, this

sampling period of the digitizer if it would have no timebase error, equals Ts. We then defineωbase

by (3.4-1):

. (3.4-1)

If t denotes the vector of the actual sampling instants, then it is possible to define the aunknown timebase distortion, calleddis, by the following equation:

. (3.4-2)

Let ωcal be the angular frequency of the cosine waveform, called calibration signal, that

be used to measure the timebase distortion.ωcal will be chosen such that it equals an intege

called L, timesωbase:

. (3.4-3)

We then define the complex vectorw andd by (3.4-4) and (3.4-5):

and (3.4-4)

. (3.4-5)For ease of notation we will assume that the calibration signal has an amplitude of 2.

however easily be verified that the algorithm is insensitive to this amplitude. When this signapplied at the input of the digitizer, the vector of sampled values, calleds, will be given by (3.4-6):

. (3.4-6)If (3.4-2), (3.4-4) and (3.4-5) are substituted in (3.4-6) the following results:

. (3.4-7)Next the discrete Fourier transform (DFT) ofs is calculated, calledf, this results in (3.4-8):

. (3.4-8)If we define i as being (L modulo N) and using (3.4-1), (3.4-3) and (3.4-4), we can w

(3.4-8) as:

. (3.4-9)

In practice, with L not exceedingly large and because of the bandlimited timebase disto

will have about all of its energy concentrated around DC. This is explained in w

follows.First approximated, defined by (3.4-5), by the first two terms of its Taylor series:

. (3.4-10)

The approximation of d[k] as a linear function of dis[k] will be valid if is muc

smaller than 1. In practice dis[k] represents the residual error of the timebase of an instru

such that the above condition will be valid if is not exceedingly large. Looking at (3.4

ωbase2π

TsN----------=

dis k[ ] t k[ ] kTs–=

ωcal Lωbase=

w k[ ] ejωcalkTs=

d k[ ] ejωcaldis k[ ]

=

s k[ ] ejωcalt k[ ]

ejωcalt k[ ]–

+=

s wd w∗d∗+=

f F w( )* F d( ) F w∗( )* F d∗( )+=

f δi * F d( ) δ N i–( )* F d∗( )+=

F d( )

d k[ ] 1 jωcaldis k[ ] ...+ +=

ωcaldis k[ ]

ωcal

42


t-timerors to

tions

of L

n of

f

he

zero

tion of

the

terms

f

this means that (3.4-10) will hold for all values of L smaller than a certain value LLIN. For these

values of L, can be approximated as follows:

. (3.4-11)

Because of the assumption that the timebase error is bandlimited (in an equivalensampling oscilloscope there will be enough correlation between successive timebase er

assume this) will have significant components only at indices corresponding to pulsa

lower than . It appears from simulations that this statement even holds for values

significantly larger than LLIN.

As mentioned above we may thus assume, looking at (3.4-9), that the contributio

will be concentrated in components off with an index close to i, and the contribution o

in components with an index close to N-i. A good choice for L, to avoid t

overlap of the spectra of the two terms in (3.4-9), is given by:

, (3.4-12)

with q integer. In this case i will equal or and the distance between the non-

component ofδi andδ(N-i) will equal , which is the largest possible distance for a fixed N.

Suppose now that L has been chosen and that q is even. In this case the contribu

in (3.4-9) will be concentrated in the components off with an index smaller than (in

the case of q being odd we would read larger than ). We will now constructa, with

for and for . In the case of a good separation between

two terms of the sum in (3.4-9) we can write:

, (3.4-13)

with ε being a small error vector caused by a non perfect separation between the twoof the sum in (3.4-9). An estimatede for the unknownd can now be calculated by applying

(3.4-14).

(3.4-14)

Using (3.4-3) until (3.4-8) it can be verified thatd andde would be identical in the caseεequals zero. Finally an estimatedise of the timebase distortiondis can be calculated by the use o

(3.4-15).

F d( )

F d( ) δ0 jωcalF dis( )+=

F dis( )N4----ωbase

δi * F d( )

δ N i–( )* F d∗( )

L N4---- q

N2----+=

N4---- 3N

4-------

N2----

δi * F d( ) N2----

N2----

a k[ ] f k[ ]= kN2----< a k[ ] 0= k

N2----≥

a δi * F d( ) ε+=

de F 1–a( )w∗=

43


ithout

h that

eters

g theon thethe

onenteticalrence

(3.4-15)

An important remark on (3.4-15) is that the phase information ofde needs to be unwrapped

beforedise can be calculated. In practice it appears that this unwrapping can be executed w

any ambiguity if the signal-to-noise ratio is higher than 10dB and if L has been chosen sucthe ratio of the total signal energy to the energy ofε is larger than 10dB.

Computer simulation

A computer simulation is performed to illustrate the theory explained above. The paramof the simulations are:• Ts = 40 picoseconds• N = 500

• ωbase = 2π x 50 106 rad/s• timebase distortion (dis) : see Figure3.4-1

• q = 2, L = 625, i = 125,ωcal = 2π x 31.25 109 rad/s

The discrete Fourier transform of the simulated calibration signal is calculated, applyintimebase distortion. The result can be seen on Figure3.4-2. Remark that the spectrumfigure is only given until the Nyquist frequency of 12.5GHz. We see that most of the power inspectrum is concentrated around the component with frequency 6.25GHz. This compcorresponds to a frequency index equal to 125, which is consistent with the theorpredictions. On the figure this component has an x-value of 6.25GHz. The level of interfe

Figure3.4-1 Simulated timebase distortion (dis)

dise k[ ]ϕ de k[ ]( )

ωcal----------------------=

0 5 10 15 20

0

20

40

60

Time (ns)

Tim

ebas

e er

ror

(ps)

44


and atut 40

and

between the two terms of (3.4-9) can be seen by looking at the values of the spectrum at DCthe Nyquist frequency. This reveals an overlap between the two contributions at a level abodB lower than the main components. This effect, causing thatε in (3.4-13) is not equal to zero,causes a small error in the estimation ofdis. Next (3.4-14) is applied and the phase ofde is

calculated. The result of this calculation is illustrated in Figure3.4-3. Unwrapping this phase

Figure3.4-2 Spectrum of digitized calibration signal (no window)

Figure3.4-3 Phase of de.

2 4 6 8 10 12

-20

-10

0

10

20

Frequency (GHz)

Am

plitu

de (

dBV

)

0 5 10 15 20-3

-2

-1

0

1

2

3

Frequency (GHz)

Pha

se (

rad)

45


oned

(notory is

startnts in

the

re

5. In

h all[6]

at a

applying (3.4-15) results in an estimation ofdis. The absolute value of the difference betweendisanddise is given in Figure3.4-4. This difference is not zero because of the previously menti

overlap. As can be seen this error is only significant for the first few time instants, and alsoshown on the figure) for the last few time instants. This is caused by the fact that the thebased on circular convolution techniques, meaning that there is a discontinuity effect at theand at the end of the sampling window. This discontinuity causes non-zero valued compone

which are no longer close to DC, hereby causingε to be different from zero.

Windowing to increase accuracy

Applying a windowing technique [6] on the digitized calibration signal, can increaseaccuracy. Suppose that we use a windowm. We definesw by (3.4-16):

. (3.4-16)

The DFT ofsw will be given byfw, defined in (3.4-17):

. (3.4-17)

With an appropriate window [6], the energy of will be much mo

concentrated around DC than it is the case for . This fact is illustrated in Figure3.4-

this graph we see the discrete Fourier transform of the digitized calibration waveform witparameters identical to those used for the simulation above, but with a Hanning windowapplied prior to calculating the spectrum. Now the overlap between the two contributions islevel about 120dB lower than the main components. If we then constructaw out of fw as we

Figure3.4-4 Absolute value of (dise-dis).

0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.2

0.4

0.6

0.8

1E

stim

atio

n er

ror

(ps)

Time (ns)

F d( )

sw ms=

f w δi * F d( )* F m( )( ) δ N i–( )* F d∗( )* F m∗( )( )+=

F d( )* F m( )

F d( )

46


lied

antlyhas

ned.

e

zero

constructeda out off, we can write (3.4-18):

. (3.4-18)

Note thatεw is much smaller thanε. It is easily verified that the estimatedwe, defined by

(3.4-19), will equal ifεw equals zero.

(3.4-19)

The estimatordiswe for dis will be calculated in (3.4-20):

. (3.4-20)

Note that we hereby use the fact that a multiplication of a complex vector withm does notaffect the phase of the vector. In this we assume implicitly that all elements ofm are real andpositive. As can be found in [6] this assumption is generally true for all practically appwindows. The absolute value of the difference betweendis and diswe is given in Figure3.4-6.

Comparing Figure3.4-6 with Figure3.4-4 shows that the accuracy has been significimproved by using the window. One important remark however is that the value with index 0

been removed from Figure3.4-6. Because m[0] equals 0, will be undetermi

Looking at (3.4-20), this means that disew[0] can have any value between and . Th

use of other windows, such as for example a Hamming window [6], which contains novalues, solves this problem.

Figure3.4-5 Spectrum of digitized calibration signal (with window).

2 4 6 8 10 12-120

-100

-80

-60

-40

-20

0

Frequency (GHz)

Am

plitu

de (

dBV

)

aw δi * F d( )* F m( )( ) εw+=

md

dwe F 1–aw( ) w∗⋅=

diswe k[ ]ϕ dwe k[ ]( )

ωcal-------------------------=

ϕ dwe 0[ ]( )

πωcal----------–

πωcal----------

47


ndard

real

, by

se ofthodscisely

f the

Sensitivity to noise:

Suppose that the calibration signal has a peak amplitude A and an additive noise stadeviationσn. We assume thatσn is much smaller than A, and thatε equals zero. We can then

write:

, (3.4-21)

with and two vectors, with elements that are mutually independent stochastic

variables with a standard deviation equal to . (3.4-21) can be approximated as:

. (3.4-22)

If we defineστ as being the standard deviation on our estimate of dis[k] we can write

using (3.4-15) and (3.4-22):

. (3.4-23)

Other methods to measure timebase distortion:

By the knowledge of the author three other main methods exist to calibrate the timebaan equivalent-time sampling oscilloscope. In what follows a short description of all three meis given. All three methods are based on the use of a sinusoidal excitation signal, with a preknown angular frequency, calledωcal, and a peak amplitude A.

1. Zero-crossings: With this method [7] an estimate is calculated of the timing instants o

Figure3.4-6 Absolute value of (dise-dis) (with window).

0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.02

0.04

0.06

0.08

0.1

Time (ns)

Est

imat

ion

erro

r (p

s)

de k[ ] ejωcaldis k[ ] 1 nre k[ ] jnim k[ ]+ +( )=

nre nim

σn

A------

de k[ ] enre k[ ]ej ωcaldis k[ ] nim k[ ]+( )=

στσn

ωcalA--------------=

48


of thenceignalssingaddi-

ings.

d atphase

r equal

l will

an

ituder errore sineerning

stan-:

zero-crossings of a digitized sinusoidal signal, and sometimes also of the zero crossingsderivative of the sinusoidal signal, by the use of interpolation. By looking at the distabetween the zero crossings and comparing this with the known period of the calibration s(which is accurately known), an estimate of the timebase distortion exists at the zero croinstants. Concerning the sensitivity to noise, suppose that the standard deviation of thetive noise equalsσn. The resulting standard deviation of dis[k], calledστ, will equal σn

divided by the absolute value of the derivative of the digitized waveform at the zero crossThis results in (3.4-24):

. (3.4-24)

2. Nearzero-crossingdetection: Now, the values of the digitized sinusoidal signal are useinstants near the zero crossings. A sinusoidal signal is chosen with both frequency andsuch that the zero crossings would correspond to sampling instants with a timebase erroto zero. Lets be the digitized signal, we can write, using (3.4-2):

. (3.4-25)

The k-values for which the time instant is close to a zero crossing of the sinusoidal signabe given by:

, with N integer. (3.4-26)

For these k’s we can write:

. (3.4-27)

If the values of are small enough we can write (3.4-27) as follows:

. (3.4-28)

From this we can calculate dis[k]:

. (3.4-29)

If additive noise is present,σn andστ being defined as above, and using (3.4-29), we c

write:

. (3.4-30)

3. Sinefitting: With this method a sine is fitted on the measured data. First frequency, ampland phase are determined such that there is a minimum error (least-squares or othefunction) between the fitted sine and the measured data. Next an “unwrapped” inversfunction is applied to all sampled values to determine the associated time instant. Concthe noise sensitivity, using the fact that we use the inverse sine function, we see that thedard deviation will no longer be a constant, but will be given by the following expression

στσn

ωcalA--------------=

s k[ ] A ωcal kTs dis k[ ]+( )( )sin=

k πNωcalTs----------------=

s k[ ] A 1–( )N ωcaldis k[ ]( )sin=

ωcaldis k[ ]π

--------------------------

s k[ ] A 1–( )N ωcaldis k[ ]( )=

dis k[ ] s k[ ]A 1–( )Nωcal

-----------------------------=

στσn

ωcalA--------------=

49


newcy of

Nearis thatar zero

ationphase ofaybelitudeof the

ebasewith

se time

imeuals

note,mingto be

of thehat alle weebase120).

:

, (3.4-31)

with t as defined by (3.4-2).

Comparison between the methods

A conclusion that holds for all three methods mentioned above, as well as for themethod proposed, is that the sensitivity is proportional to the amplitude and to the frequenthe calibration signal ((3.4-23), (3.4-24), (3.4-30) and (3.4-31)). The “Zero crossings” and “zero crossings” methods are very straightforward and easy to implement. A disadvantagethe methods only succeed in an accurate estimation of the timebase error at time instants necrossings of the calibration signal. To find the timebase error at other points interpoltechniques are to be used. Other disadvantages are that the methods are sensitive to thethe calibration signal. In microwave applications the phase of the calibration signal mdifficult to adjust. The “Near zero crossings” method has also as disadvantage that the ampof the calibration signal need to be known accurately in order to have accurate estimatestimebase distortion (3.4-29). The “Sine fitting”-method has as advantage that it can find timdeviations in all points of the time axis, even if these timebase errors are totally uncorrelatedeach other. A disadvantage is the fact that the sensitivity of the method becomes zero at thoinstants where the derivative of the calibration signal becomes zero (3.4-31).

The “Analytical signal” method determines timebase deviations in all points of the taxis, like the “Sine fitting” method, but has a constant sensitivity along the time axis which eqthe maximum sensitivity that can be achieved by the other methods. It is important tohowever, that this can only be achieved by introducing the assumption of a bandlimited tierror. If this assumption is violated a biased estimate will result. It does have the advantageinsensitive to the phase and knowledge of the amplitude of the calibration signal.

We can conclude that the new method proposed incorporates most of the advantagesmethods mentioned above, without the disadvantages. It is important to mention, however, tthis can only be achieved with the assumption that the timing error is bandlimited. In practicnote that, except for the start and ending of our sampling window, the assumption of the timerror being bandlimited holds very well for the sampling oscilloscope we are using (HP-54All this explains why the “Analytical signal”-method is best suited for our application.

Experimental verification

For an experimental test, an HP-54120 oscilloscope is used with the following settings• first sample taken at 64 nanoseconds relative to trigger event• Ts : 20 picoseconds• N : 1024

• ωbase = 2π x 48.828125 106 rad/s

• q = 0, L = 256, i = 256,ωcal = 2π x 12.5 109 rad/s• power of calibration signal: -10 dBm

στ k[ ]σn

ωcalA ωcalt k[ ]( )cos-------------------------------------------------=

50


r is

ausewitchelay.ns, itration,aller

s thatuse offind

mptioninedl withs-error

• number of averages used: 1000

The estimate of the timebase distortion is illustrated in Figure3.4-7. This kind of erro

easily understood when looking at Figure3.2-1. A non ideal fine delay ramp generator will cthe same error to appear every 4ns. At the integer multiples of 4ns the ramp generator will sfrom a near 4ns delay to a near zero delay, it is to say from its maximum to its minimum dThis causes the discontinuities. Although the timebase error seen is still within specificationeeds to be said that it can be made much smaller with the appropriate hardware calibdescribed in [3]. After performing the calibration procedure the discontinuity can be made smthan 1ps, with an overall peak-to-peak error of about 4ps.

3.4.3 Spectral estimation

Introduction

In many cases one measures periodic signals with a sampling oscilloscope. This implieone a priori knows the frequencies of the spectral components present in the signals (cf. thethe sampling oscilloscope to acquire the data in a VNNA prototype). A classical approach tothe values of these spectral components is to use a DFT. This method is based on the assuof equidistant sampling. If this assumption is violated, errors result. In what follows is explahow it is still possible to estimate the values of the spectral components of a digitized signagood accuracy by using the knowledge of the timebase distortion to construct a least-squareestimator.

Figure3.4-7 Measured timebase distortion.

65 70 75 80

-4

-2

0

2

4

6T

imeb

ase

erro

r (p

s)

Time (ns)

51


fe and

es

of

y thesually

ants

s

in thesoidalcope.l, withn ofeen inue can

The construction of a least-squares-error estimator:

We assume that we know a priori all angular frequencies,w, of the C spectral components othe signal x(t). The remaining unknowns are the amplitude and phase, or equivalent the sincosine contributions,A, of these components.A is defined such that (3.4-32) is valid:

. (3.4-32)

Remark that there are 2C real unknowns. The measured vector of sampled valux,containing N elements, will be given by:

. (3.4-33)In this equationn represents a sampled additive noise vector andF is the matrix defined by:

, for ,

, for . (3.4-34)

The vector of sampling instantst is defined by (3.4-35),dis is the previously measuredtimebase distortion vector (cf. the previous section), and Ts is equal to the sampling period.

(3.4-35)

Looking at the structure of (3.4-33) we see that we can apply the theoryleast-squares-error estimation [8] to construct an estimatorAest for the unknown vectorA. The

estimator is a linear function ofx. Its definition is given by:

and (3.4-36)

. (3.4-37)

This linear estimator is by construction unbiased and, if the noise is white (as is mostlcase with sampling oscilloscope measurements where the white instrument noise is udominant) the estimator even has minimum variance [8].

Implementation of the estimator

In practice,dis andw often do not change during several measurements, and one just wto monitor the change ofA. If this is the case, it is sufficient to calculate once the matrixE, and touse it for all measurements (cf. (3.4-37)). OnceE is calculated, it takes 2.C.N real multiplicationand additions to calculateAest.

Experimental verification

For an experimental verification, the same experiment parameters are used asexperiment concerning the timebase distortion measurement (see Figure3.4-7). A sinusignal generator with a frequency of 7GHz is connected to the input of the sampling oscillosAn estimation is then performed on the value of all spectral components present in the signafrequencies ranging from 6GHz until 8GHz, with a step of 50MHz. The result of an estimatiothe spectral content, with and without taking the timebase distortion into account, can be sFigure3.4-8. With the assumption of the timebase error being zero, a significant spectral val

x t( ) A i[ ] w i[ ]t( )cosi 1=

C

∑ A i C+[ ] w i[ ]t( )sini 1=

C

∑+=

x F A⋅ n+=

Fki w i[ ]t k[ ]( )cos= 0 i C≤<

Fki w i[ ]t k[ ]( )sin= C i 2C≤<

t k[ ] kTs dis k[ ]+=

E Ft F⋅( )1–

Ft⋅=

Aest E x⋅=

52


tantare a4-7). Iftor, the

Thispling

ethodhat thee use

uaredersionrted

e newncies inld beectral. Withis wetions

be found not only at 7 GHz, but also at all frequencies which are a multiple of 250 MHz disfrom 7 GHz. These additional spectral components are not physically present, butmeasurement error caused by the 4ns periodic character of the timebase error (cf. Figure3.we use the measurement of the timebase error and use the least-squares-error estimaspectrum no longer has these erroneous peaks.

Comparison with a classical approach:

Another method which can solve the problem of compensation is described in [9].method is based on the conversion of the sampled values on a non-uniform grid of saminstants to the exact values that would have been found with a uniform grid. Note that this mis of more general use than the newly developed method, which requires the assumption tfrequencies of all spectral components are exactly known. The method in [9] is based on thof interpolating functions, called kernels. If there are N sampling instants, there are N-sqkernels. These kernels can be calculated if the timebase distortion is known. Once the convfrom a nonuniform grid to a uniform grid is done, a DFT is used to calculate the undistospectrum. When comparing this method with the newly developed method, it is clear that thmethod needs much less calculations. Indeed, suppose for example that there are 3 frequethe input signal, and that there are 1024 sampling instants. With the new method it wousufficient to calculate three complex inner products to find the correct values of the three spcomponents. This means a total of 3072 complex calculations and 3072 complex additionsthe method described in [9] we first need to calculate the values on the uniform grid. To do thalready need more than one million real multiplications and more than one million real addi

Figure3.4-8 Estimated spectrum of input signal: timebase error assumed to be zero: compensation technique for timebase error applied.

6 6.5 7 7.5 8-100

-90

-80

-70

-60

-50

-40

-30

Am

plitu

de (

dBm

)

Frequency (GHz)

53


at thets of

me is

f real

thodnewbe ons not

errors.effect.

ion ofethod,at thetents of

DFT,ion iseffectsts.

withtakening

ard tocope as

rentds arethodthe

rticlesedf thetheeform.dditive

(1024-squared = 1.048.576). Next a 1024-point DFT needs to be calculated. This shows thnew method is much more suited to calculate the values of the spectral componenmultispectral signals. The difference between the two methods concerning calculation ti

minimum in the case that there are spectral components. In this case the number o

additions and multiplications would be the same for both methods, but with the mementioned in [9] we would still need to perform a DFT. Another important advantage of themethod is the fact that it is not required for the sine wave to have a frequency which wouldthe grid used with a DFT. If the method in [9] is used and the frequency of the sine wave doecorrespond to the grid of the DFT applied, leakage [10] will appear. This leakage can causeWith the new method this problem does not exist at all, it is insensitive to the DFT leakage

3.4.4 Conclusion

A novel method is developed for the accurate measurement of the timebase distortequivalent-time sampling oscilloscopes. It is explained what the advantages are of this mcompared to other methods, and why the novel method is preferred. Next it is shown thinformation on the timebase distortion can be used to accurately measure the spectral cona digitized signal, which measurement would be erroneous if techniques are used, like awhich assume the timebase of the oscilloscope to be perfect. An experimental verificatperformed. The results show that the method proposed effectively removes the erroneousthat the timebase distortion may have on the estimation of the values of spectral componen

3.5 Timebase jitter

3.5.1 Introduction

“Time jitter” can cause significant systematic errors when waveforms are recordedequivalent-time sampling oscilloscopes. The presence of time jitter means that every sampleby the oscilloscope can only be situated on the time axis with a certain probability. This timjitter causes systematic errors in the estimate of the measured waveform which are very hcompensate because the jitter is a stochastic process very much dependent on the oscilloswell as the quality of the trigger signal. This means that the effect of the jitter will be diffewhenever we use another oscilloscope or another trigger signal! In the literature two methoencountered that deal with waveform distortions due to time jitter: the so-called “median” me[11] and the so-called “PDF deconvolution” method [13]. The “median” method is based oncalculation of the point-by-point median of a large set of waveform measurements. In their a[11] Souderset al. state that the estimate of the true waveform will be asymptotically unbiawhere the waveform is monotonic. It is mentioned, though, that nonmonotonic parts owaveform (local maxima and minima) will be distorted. In what follows it is explained thatpresence of additive noise also introduces a bias, regardless of the monotonicity of the wavThese two sources of bias make the method less suited for accurate estimation if a lot of a

N2----

54

3.5 Timebase jitter

a.sity

of allof thethedditiveases,ll by aative toin an“PDFer beof therrorforms

thminedthe

risingg thedata,

d ish these

with athe

e thef allns willoint itn timerm is

noise is present or if one wants an accurate estimation of the waveform maxima and minimThe “PDF deconvolution” method is based on an estimation of the jitter probability den

function (PDF) and on a technique to deconvolve this density function from the averagerecorded waveforms. The accuracy of this method is very much dependent on the qualityestimate of the jitter probability density function. In his article [13] Gans uses a portion ofwaveform that can be approximated by an ideal ramp and a measurement of the apparent anoise PDF at one time instant to estimate the jitter PDF. Although useful in many practical can asymptotic bias is present when no part of the waveform can be approximated very weramp, as is the case for pulses when the time jitter standard deviation becomes too large relthe pulse transition duration. In this article a novel method is proposed which resultsasymptotic unbiased estimate of the jitter PDF. The method extends the use of thedeconvolution” method to cases where the “ideal ramp approximation” approach can no longused. The method is based on the identification of the parameters of a parametric modeljitter probability density function. For the purpose of the parameter identification, an efunction is minimized which is a function of the parameters, the mean of the measured waveand the mean of the square of the measured waveforms.

In the following, the method [14] is derived mathematically, and it is shown what algorimay be used to identify the parameters of a parametric model for the jitter PDF. It is explahow the effect of additive noise is effectively removed. A comparison is made between“median” method and the novel method, based upon simulated data. Next the problems awhen applying the method of Gans [13] to estimate the jitter PDF are explained by illustratinasymptotic bias of the method. Finally, the new method is applied to experimentalillustrating its performance when the jitter standard deviation is large.

3.5.2 Mathematical equations of the extended “PDF deconvolution” method

For simplicity it is assumed that the sample rate with which the waveform is digitizehigh enough to avoid aliasing and that care has been taken to avoid leakage problems. Witassumptions all continuous equations derived can easily be used for a digitized equivalent.

It is assumed that two stationary noise sources are present, namely additive noise (finite variance) and jitter noise (with a finite characteristic function). The jitter PDF andundistorted waveform are the two unknowns. The two things that will be measured arpoint-by-point average of all measured waveforms, and the point-by-point average omeasured waveforms squared (first squaring and then averaging). In fact, these two functiobe equivalent to using the mean and the variance of a set of recorded waveforms. At this pshould be clear to the reader that this variance is not a constant at all sampling instants whejitter is present, but will be larger at those sampling instants where the slope of the wavefolarger.

Mathematical notations:

• x(t): undistorted true signal• a(t): expectation of the signal measurement (infinite number of averages)

55


ateby

. It isd A(n

nlation

-mean:

.the

eform

ead

as a

• s(t): expectation of the square of the signal measurement (infinite number of averages)• p(t): time jitter probability density function• n(t): additive noise• X(ω), A(ω), S(ω), P(ω): Fourier transforms of x, a, s and p

The general idea of the method is the following. In practice it will be possible to estimA(ω) and S(ω) in an asymptotic unbiased manner by recording many waveforms andcalculating the Fourier transforms of the mean and the mean of the waveforms squaredpossible to construct a set of equations describing the relationship between the estimateω)and S(ω) on one hand and the unknowns X(ω) and P(ω) on the other hand. This relationship cathen be used to estimate X(ω) and P(ω). The theory is explained in what follows.

One realization of the measured signal can be described as xm(t) = x(t-τ) + n(t). In this

equationτ is a realization of the stochastic jitter variable (τ has a different value for each t), andstands for a realization of the stochastic zero-mean additive noise variable. Now the rebetween x(t), p(t) and a(t), s(t) can be calculated.

The expectation of the measured signal, a(t), will equal the expectation of x(t-τ) plus theexpectation of n(t), because of the linearity of the expectation operator. Because n(t) is zeroadditive noise, its expectation will equal zero, which means that a(t) will be given by (3.5-1)

. (3.5-1)

In the following s(t) is calculated as a function of p(t), x(t) andσn, the standard deviation of

the additive noise source n(t):

, such that (3.5-2)

and finally (3.5-3)

. (3.5-4)

When deriving (3.5-3) from (3.5-2) the fact is used that n andτ are statistically independentAs can be seen from (3.5-4)σn is squared and added as a constant. Transformed in to

frequency domain (3.5-1) and (3.5-4) become

and (3.5-5)

. (3.5-6)

Equation (3.5-5) reveals that the expectation of the spectrum of the averaged wavequals the original signal spectrum X(ω) multiplied by the characteristic function P(ω) of thejitter noise (P(ω) is the Fourier transform of the jitter noise probability density function). The iddeveloped in the article by Gans [13] is to measure p(τ) and to deconvolve it from the measuresignal a(t). The main problem of course is how to measure p(τ). Instead of trying to measure p(τ)in a direct manner, which in some cases is hard to accomplish, (3.5-6) can be used

a t( ) xm t( )⟨ ⟩ x t τ–( )⟨ ⟩ n t( )⟨ ⟩+ x t τ–( )p τ( )dτ∞–

∞

∫= = =

s t( ) xm2

t( )⟨ ⟩ x2

t τ–( ) 2x t τ–( )n t( ) n2

t( )+ +⟨ ⟩= =

s t( ) x2

t τ–( )⟨ ⟩ 2 x t τ–( )⟨ ⟩ n t( )⟨ ⟩ n2

t( )⟨ ⟩+ +=

s t( ) x2

t τ–( )p τ( )dτ∞–

∞

∫ σn2

+=

A ω( ) X ω( )P ω( )=

S ω( ) X* X( ) ω( )P ω( ) σn2δ ω( )+=

56

3.5 Timebase jitter

d

olve

8]abilityr theodel

ion ithen acanthelay

)

gn of a

l to

s

timeill be

ll be

ill beDFT

d

supplementary equation allowing to indirectly determine p(τ). How this can be done is explainein what follows.

3.5.3 The use of a parametric model

According to the knowledge of the author, no algorithm is available in the literature to sthe set of equations (3.5-5) and (3.5-6) for a general X(ω) and P(ω). It will be possible, however,to use a parametric model for P(ω) and to identify the parameters of this model. In literature [several parametric models are used to model or at least sufficiently approximate most probdensity functions and associated characteristic functions appearing in practice. Fosimulations and the experiments mentioned in this article the use of a normal distribution mfor p(τ) appeared to be sufficient (cf. the experimental results of Figure3.5-9). In this sectwill be explained in a more general way, however, how the parameters can be identified wmodel is used for p(τ). Whatever parametric model is used, it is important to know that wealways assume that the expectation of p(τ) is equal to zero. It can theoretically be assumed thatexpectation of p(τ) is different from zero, but this assumption will correspond to a pure deapplied on the digitized signal. One example of a parametric model for p(τ) corresponds with a

so-called Edgeworth’s form of the type A series ([8], [16]). For M parameters (called until

the model for the corresponding characteristic function P(ω) is given by (3.5-7). This model isequivalent to a truncation of the Taylor’s series of log(P(ω)), the so-called cumulant-generatinfunction and has several interesting characteristics [8]. Note that the characteristic functio

normal distribution model for p(τ) corresponds to (3.5-7) with equal to the mean, equa

the variance and all other ’s equal to zero:

. (3.5-7)

In general it will now be explained how the parameters of a parametric model for P(ω) canbe identified when A and S are known. The parametric model for P(ω) will be noted as P(ω, λi),

whereλi refers to M parameters.

In order to explain the algorithm to identify the parametersλi, some mathematical notation

will first be introduced. In practice digitized waveforms are used. The equivalent samplingwill be noted as Ts, and the number of points on the time axis used for the measurements w

called N. The fundamental angular frequency of the discrete Fourier transform (DFT) winoted asωbase, and is given by

. (3.5-8)

To avoid confusion a single-sided DFT is used; this means that only components wconsidered with an index smaller than N/2. A subscript refers to the corresponding

component. and refer to the value of the ith component of the DFT of the measure

λ1 λM

λ1 λ2

λi

P ω λ1 ...,λM, ,( ) expλk

k!----- jω( )k

k 1=

M

∑

=

ωbase2π

NTs----------=

AiM Si

M

57


er of

nd

n by

ular

, fastT’s.

y

:

ertain

the

ys bef the

0 is

ion is

rs

average of the digitized waveform and the average of the waveform squared. If the numb

averages is increased, and will asymptotically tend to A(iωbase) and S(iωbase). The

notation will refer to P(iωbase,λj). With these notations and conventions (3.5-5) a

(3.5-6) can be applied by filling in the measured spectra and . The result is give

(3.5-9) and (3.5-10):

(3.5-9)

. (3.5-10)

The “ ” denotes the single-sided DFT equivalent of a double-sided DFT circ

convolution in the frequency domain. To calculate the convolution of two complex spectraconvolution techniques are used, performing inverse-DFT’s (IDFT’s), multiplications and DFThe fast convolution algorithm calculates the convolution as shown in (3.5-11):

. (3.5-11)In a first step Xk is eliminated by substituting Xk in the second equation of (3.5-10) b

. The result is written in (3.5-12):

. (3.5-12)

An error vector is then introduced based upon (3.5-12). It is defined by (3.5-13)

. (3.5-13)

To reduce the effects of noise, all components of that have an index larger than a c

iMAX , corresponding to the highest frequency component that can be distinguished from

noise, will not be considered. Since a large amount of oversampling is applied, this will alwapossible. Based upon this error vector an error function is introduced which is a function o

measured values , and of the unknown parameters . This error function r(λj) is defined

by (3.5-14):

. (3.5-14)

Note that this error function is not a function of the unknown because the index i=

eliminated from the summation. Also note that the largest component index in the summat

equal to 2iMAX . Considering components of with an index larger than 2iMAX is

theoretically possible, but will only add a constant independent ofλj to the error function, which

means that the final estimate will not be influenced. An estimate for the true paramete

AiM Si

M

Pi λ j( )

AiM Si

M

AiM

XiPi λ j( )=

SiM

Xk* Xk( )iPi λ j( ) σn2δi0+=

*

F* G F F 1– F( )F 1– G( )( )=

AkM

Pk1– λ j( )

SiM

AkM

Pk1– λ j( )( )* Ak

MPk

1– λ j( )( )( )iPi λ j( ) σn2δi0+=

ei λ j( )

ei λ j( ) SiM

AkM

Pk1– λ j( )( )* Ak

MPk

1– λ j( )( )( )iPi λ j( )– σn2δi0–=

AiM

AiM

SiM λ j

r λ j( ) ei λ j( ) 2

i 1=

2iMAX

∑=

σn2

ei λ j( )

λ jE λ j

T

58

3.5 Timebase jitter

one

ual to

.

uardt

f the

for

wasth the-15):

or

cally

r the

t outresult

iven

“PDFa ofinge

e

will be given by the value ofλj which minimizes the function .

By construction, will be an asymptotic unbiased estimator for . Indeed, suppose

could do an infinite amount of averages. In that case and would be become eq

A(iωbase) and S(iωbase), such that one would find a residue equal to zero for equal to

Simulations illustrating this asymptotic unbiasedness can be found in [16].For the experiments and simulations mentioned in this article a Levenberg-Marq

algorithm [15] was used to minimize . The method is based on an initial guess o

parameters and an iterative process that converges to values forλj corresponding to a local

minimum. Although there is no theoretical guarantee that the method finds the valuesλj

corresponding to a global minimum (which is important for finding the right estimate) thisalways the case for the measurements and the simulations performed (cf. also [16]), wiinitial parameters corresponding to the absence of jitter. The algorithm is summarized in (3.5

. (3.5-15)

In this equationλN represents the Nth approximation of , J is the Jacobian of the err

vector e, I is the identity matrix, and is an algorithmic scalar parameter which systemati

decreases when the algorithm is converging towards a solution. A typical stop criterion foiteration is the convergence level reaching the computer machine precision.

Concerning the calculation of the Jacobian of the error vector, it may be useful to pointhat it can easily be calculated despite the complex functional form of this error vector. Theof the calculation of the Jacobian is given in (3.5-16):

. (3.5-16)

As can be deduced from (3.5-12) an asymptotic unbiased estimate for is g

by (3.5-17):

. (3.5-17)

3.5.4 Comparison versus “median” method

Before going into the comparison between the “median” method and the extendeddeconvolution” method, a short overview of the “median” method [11] will be given. The idethe “median” method is the following. Assume a strictly monotonic waveform x(t) which is besampled at a nominal time instant Ts relative to the trigger event. Due to the timing jitter the valu

of our sample will not equal x(Ts) but will equal x(Ts-τ), with τ being a stochastic variable. In a

set of many samples taken at the nominal time Ts, about half of the samples will be taken at tim

r λ j( )

λ jE λ j

T

AiM Si

M

λ jE λ j

T

r λ j( )

λN 1+ λN Re J λN( )J+ λN( ) ΛI+( )( )1–Re J λN( )e λN( )( )+=

λ jE

Λ

λ j∂∂e A

MP 1–( )* A

MP 1–( )( ) λ j∂

∂P– 2P A

MP 1–( )* A

MP 2–

λ j∂∂P

+=

σnEST2 σn

2

σnEST2 S0

MAk

MPk

1– λ j( )( )* AkM

Pk1– λ j( )( )( )0–=

59


than

ed forat are[11],

nsated.mean

ed by

ditivens ifrmalre thelarify

ent isy theplingof

r

iasedthat

ble n,

ng to

v,T

)

instants earlier than Ts and the other half at instants later than Ts. Because of the strict

monotonicity of x(t), this also means that the value of about half of the samples will be lowerx(Ts) and the other half higher. To have a good estimation of x(Ts) it is sufficient to calculate the

median of all sample values. It is easy to prove that this estimator is asymptotically unbiasmonotonic waveforms when no additive noise is present. Unfortunately most waveforms thmeasured are not monotonic. When this is the case all maxima and minima will be clippedand no matter how many measurements are performed, this distortion can never be compe

The inevitable presence of additive noise, presumed to be stationary and to have a zeroand a finite variance, will also cause a bias. This fact, not mentioned in [11], can be explainthe fact that the relation between the PDF of the sampled values at a certain Ts, when additive

noise is present, equals the ideal additive-noise-free PDF convolved with the PDF of the adnoise. Since the median of a distribution is by no means invariant with respect to convolutiothe distribution is asymmetric (even when the distribution to convolve with is a zero mean nodistribution), the presence of the additive noise may cause a bias at all time instants whetheoretical additive-noise-free PDF of the sampled values is asymmetric. An example will cthe above reasoning. Consider the following monotonic signal, noted k(t):

, (3.5-18)with U(t) denoting the Heaviside function. First, consider that each signal measurem

distorted by normally distributed jitter noise with a standard deviation of 1s, represented bstochastic variableτ. The stochastic variable describing the samples taken at a certain saminstant Ts will then be equal to k(Ts-τ). Because of the monotonicity half of the realizations

k(Ts−τ) will be smaller than or equal to k(Ts) and half of the realizations will be larger than o

equal to k(Ts). This implies that the median of k(Ts−τ) will equal k(Ts), such that every

asymptotic unbiased estimator for the median will automatically be an asymptotic unbestimator for k(Ts). When additive noise is added this is no longer true. Suppose for example

normally distributed noise with a standard deviation of 1s, described by the stochastic variais added to the measurements. One then has to calculate the median of (k(Ts-τ)+n). This is done

by means of the cumulative distribution function (CDF). Considering the rules correspondithe addition of stochastic variables, the CDF of (k(Ts-τ)+n), noted H(v,Ts), will be equal to the

convolution of the CDF of k(Ts-τ), noted G(v,Ts), and the PDF of n. First, G(v,Ts) will be

calculated. Using the definition of the CDF one can write:

. (3.5-19)

Since k(t) is a positive function one can immediately conclude that, for negative v, G(s)

equals 0. Consider now a positive v. One can then say the following:

, such that, for v positive, (3.5-20

. (3.5-21)

k t( ) U t( )t=

G v TS,( ) Prob k TS τ–( ) v≤( )=

G v TS,( ) Prob TS τ– v≤( ) Probτ TS v–≥( )= =

G v TS,( ) 1

2π----------e

u2

2-----–

ud

TS v–

∞

∫=

60

3.5 Timebase jitter

s are

the

The

can

to 0s.ditive

pedtically-pass

ly be

For all v, one can then finally write:

, (3.5-22)

with erf denoting the error function [12]. H(v,Ts) will then be given by the following

convolution integral:

. (3.5-23)

To illustrate the above, G(v,0) and H(v,0) were numerically evaluated and the resultfound in Figure3.5-1.

The median of (k(Ts-τ)+n) is a function of Ts, noted kE(Ts), and will be equal to that value

of v for which H(v,Ts) equals 0.5. This function is equal to the waveform corresponding to

asymptotic value of the median estimator of k(t) with the presence of additive noise.

functions k(Ts) and kE(Ts) were numerically evaluated and are illustrated in Figure3.5-2. As

be seen on this figure, a noticeable difference between the two functions is present closeThis clearly illustrates that the median estimator becomes asymptotically biased when adnoise is present.

In order to compare the performance of the “median method” with the newly develomethod, software was written simulating the jitter process. The idea was to define an analydescribed pulse. For convenience the impulse response of a third-order, Butterworth, lowfilter was chosen. From this analytically defined waveform sampled versions could easi

Figure3.5-1 G(v,0) ( ) and H(v,0) ( ).

G v TS,( ) 12---U v( ) 1 erf

v TS–

2---------------

+ =

H v TS,( ) 12--- 1 erf

u TS–

2---------------

+ e

v u–( )2

2-------------------–

2π------------------- ud

0

∞

∫=

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1CDF

v (Volt)

61


everal

. Thersionsto the

d

alitysen:

lated

calculated. This pulse corresponds to the theoretical x(t) in the previous derivation. Then stime-jittered versions were constructed by evaluating x(t), not at instants nTs, but at instants nTs-τ,

with τ being a random number, different for each different version as well as for each sampleaverage of all jittered versions was then calculated as well as the average of all jittered vesquared. These two functions are the estimates for a(t) and s(t) and are transformed infrequency domain by an FFT, resulting in A(ω) and S(ω). The algorithm of 3.5.0 was then appliewith a normal distribution model for p(τ). This resulted in an estimate for P(ω). Using the methodas described by Gans [13] to deconvolve this P(ω) from A(ω) resulted in an estimate for X(ω).The “median method” estimator was also applied to the jittered waveforms, so finally the quof both methods could be compared. For the simulation the following parameters were cho• x(t): impulse response of a third-order, low-pass, Butterworth filter withωcut=1rad/s or,

. (3.5-24)

• Applied jitter noise: Gaussian withσjitter = .75 seconds• Applied additive noise: Gaussian withσn= 0.03 volts• Ts: 0.3 seconds (sampling time)• Number of samples: 256• Start time: -10 seconds (stop time = 66.5 s)• Nyquist frequency =5/3 Hz• Number of simulated waveforms used for averaging : 3000• Number of simulated waveforms used for median estimate: 3000.

In order to have an idea about the effect of the jitter, Figure3.5-3 shows 10 simu

Figure3.5-2 k(Ts) ( ) and kE(Ts) ( ).

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

2.5

3

Am

plitu

de (

V)

Time (s)

x t( ) U t( ) et– 1

3------- 3

2-------t

sin3

2-------t

cos– e

t2---–

+

=

62

3.5 Timebase jitter

n beare

lows.

waveforms, without the additive noise (for clarity only 81 time instants are shown). As caseen on this figure the jitter noise is very significant. The results of the whole simulationshown in Figure3.5-4 through Figure3.5-7. A discussion on these results is noted in what fol

Figure3.5-3 10 simulated waveforms without additive noise ( ), together withanalytical waveform ( ).

Figure3.5-4 Analytical versus reconstructed pulses.: analytical; : extended PDF deconvolution method;: median method.

0 5 10 15 20

0

0.1

0.2

0.3

0.4

Am

plitu

de (

V)

Time (s)

0 5 10 15 20

0

0.1

0.2

0.3

0.4

Am

plitu

de (

V)

Time (s)

63


againge of

Figure3.5-4 and Figure3.5-5 show the results of both methods in the time domain (only 81 time instants are shown). The “median” method underestimates the maximum volta

Figure3.5-5 Difference between analytical and reconstructed pulses.: extended PDF deconvolution method minus analytical;: median method minus analytical.

Figure3.5-6 Spectra of analytical and reconstructed waveforms.: analytical; : extended PDF deconvolution method;: median method.

-5 0 5 10 15 20

-0.04

-0.02

0

0.02

Time (s)

Am

plitu

de d

iffer

ence

(V

)

0.1 0.2 0.3 0.4 0.5-40

-35

-30

-25

-20

-15

-10

Am

plitu

de (

dBV

)

Frequency (Hz)

64

3.5 Timebase jitter

25%dueand

e, butian”esultsnoisends. Ais in

. They beratevery

re theoor.

f thecy”

the true waveform by about 12%, while the error for this maximum voltage is only about 0.with the extended “PDF deconvolution” method. The large error for the “median” method isto the previously mentioned clipping effect [11]. As can be seen from Figure3.5-4Figure3.5-5 the “PDF deconvolution” method can accurately estimate the maximum voltagit has the disadvantage of introducing a ringing effect which is not present with the “medmethod. This ringing is typical for the deconvolution process, and can also be noted in the rof Gans [13]. The bias introduced by the “median” method due to the presence of additivecan be seen when looking at the voltage values for negative time instants close to 0 secobias is present, although the waveform is perfectly monotonic at this time instant. Thisagreement with the theory as explained in the beginning of this section.

Figure3.5-6 and Figure3.5-7 show the results of both methods in the frequency domainresult of the “median” method is significantly biased at all frequencies. This can easilexplained by the clipping effect. The result of the “PDF deconvolution” method is very accufor frequencies smaller than 0.2Hz, but the results for frequencies higher than 0.3Hz arenoisy. This can be explained as follows. The 0.3Hz corresponds to the frequency whespectral components of A(ω) have amplitude values which become comparable to the noise fl(note that A(ω) always represents a low pass filtered version of X(ω), the signal to be measured)The estimate of X(w) is then found by multiplying the measured estimate of A(ω) with the inverseof the estimated P(ω). For the example shown, this means that the frequency components oestimated X(ω) above 0.3Hz will correspond with amplified noise. Such a “corner frequen

Figure3.5-7 Spectral difference between reconstructed waveform and analyticalwaveform.

: extended PDF deconvolution method minus analytical;: median method minus analytical.

0.1 0.2 0.3 0.4 0.5-4

-3

-2

-1

0

1

2

Frequency (Hz)

Am

plitu

de d

iffer

ence

(dB

)

65


t is anal tohen

inated,” willnoisele toan the

ill be

on thetra, thenoiseable)

od to

DF asibed instant,

ant,d

n by

ationthe

rted

is

ieved.of theform iswing

(here corresponding to 0.3Hz) will always be present for the “PDF deconvolution” method. Ilimit for the method, which causes the estimates of the spectral components of the true sigbe accurate only for frequency components smaller than this “corner frequency”. Wtransformed in to the time domain the erroneous spectral components need to be elimcausing the ringing effect mentioned above. Note that the value of the “corner frequencyincrease when more averaging is applied because this will result in a lower value for thefloor. Such a limit frequency does not exist for the “median” method. This method will be abestimate the values of spectral components with frequencies that are considerably higher th“corner frequency” of the “PDF deconvolution” method. Note, however, that the estimates wbiased.

To conclude it can be said that the decision of which method to choose depends upapplication. When ringing can be allowed and an accurate estimate is needed of signal specextended “PDF deconvolution” method gives good results. If there is not too much additivepresent and if the waveform is monotonic (or the clipping of maxima and minima is acceptthe “median” method is preferable.

3.5.5 Asymptotic bias of the classical way to estimate the jitter PDF

In this paragraph the problems mentioned in the introduction, concerning the methdetermine the jitter PDF as mentioned by Gans [13], will be illustrated.

Suppose we are sampling the waveform described by (3.5-24), with the same jitter Pin the previous section, but with no additive noise present. In this case the method as descr[13] would use the measurement of the slope of the averaged waveform at a certain time inwhich will be called tanθ, together with the standard deviation of the vertical noise at this instcalledσM, to estimate the jitter standard deviationσjitter. The relation between the two measure

quantities and the estimate of the jitter standard deviation, denoted by , is then give

(3.5-25):

. (3.5-25)

It is easy to show [13] that this method gives an asymptotic unbiased estimate forσjitter if

applied to an ideal ramp. If applied to pulselike signals, however, with a jitter standard deviwhich is significant relative to the pulse transition duration, an asymptotic bias will result onestimation ofσjitter. This will cause an error on the estimation of the spectrum of the undisto

signal. We will illustrate this on the waveform as described by (3.5-24).We choose the time instant to measureσM and tanθ equal to 0.8 seconds. This instant

chosen such that the value of the averaged waveform is about half the maximum value achAt this instant the pulselike signal can best be approximated by a ramp. Next the conditionsprevious section are used and 3000 jittered waveforms are simulated. The averaged wavecalculated for time instants equal to 0.5, 0.8 and 1.1 seconds. The result is given in the follotable.

σ jitterest

σ jitterest σM

θtan-----------=

66

3.5 Timebase jitter

d.

ts in a

n

cy ofed asctioner ant the

plishmplingountthe

le toan bewell.wastime

d thedB and

to ancan

gger

From these three values, the value of tanθ is estimated to be equal to 0.1763 volts/seconThis value is calculated by using linear regression. Next the standard deviationσM is calculated

for a time instant equal to 0.8 seconds. The result is 0.1450 volts. Using (3.5-25) this resul

value for equal to 0.8224 seconds. The exact value ofσjitter is equal to 0.75 seconds. We ca

conclude that there will be an asymptotic bias of about 10% on the estimate ofσjitter, due to the

fact that the waveform is not an ideal ramp. When this biased value ofσjitter is used in the

deconvolution algorithm the error in the estimation of the spectral component with a frequen0.2Hz would be about 0.8dB. This asymptotic bias is not present when the method is usdescribed in 3.5.0. It is important to note, however, that the bias mentioned in this sebecomes negligibly small when the waveform can be approximated very well by a ramp ovinterval large compared toσjitter. In such a case both methods have the same accuracy, bu

method described by Gans [13] is much simpler to implement.

3.5.6 Experimental results

Finally an experimental setup was made to check the new method in practice. To accomthis, a pulse generated by a step recovery diode (SRD) was measured by an HP-54120 saoscilloscope, and the extended “PDF deconvolution” method was applied. To control the amof jitter a controllable attenuator was introduced in the trigger signal path. By attenuatingtrigger signal the amount of jitter could artificially be enlarged. Then it became possibcompare the reconstructed waveforms with less and the ones with more jitter. This way it cshown that the reconstructed waveforms are insensitive to jitter and that the method worksThe results of the experiment are shown in Figure3.5-8 through Figure3.5-10. The SRDexcited by a 97.65625MHz sine wave; the oscilloscope took 1024 samples with a samplingequal to 10 picoseconds. For calculating the point-by-point average of the waveforms anwaveforms squared, 1000, 2000 and 4000 averages were used corresponding to 0dB, 1015dB of attenuation in the trigger path, respectively.

Figure3.5-8 shows the spectra of the three averaged waveforms, correspondingattenuation of 0dB, 10dB and 15dB respectively. The low-pass filtering effect of the jitterclearly be distinguished. At 7GHz, for example, the third spectrum (15dB of attenuation in tri

Table3.5-1 Averaged waveform values

time instant(seconds)

averaged value(volts)

0.5 0.1345

0.8 0.1821

1.1 0.2403

σ jitterest

67


Figure3.5-8 Measured spectra of averaged SRD pulse with different attenuation intrigger path.

: 0dB of attenuation; : 10dB of attenuation;: 15dB of attenuation.

Figure3.5-9 Reconstructed spectra with different attenuation in trigger path.: 0dB of attenuation; : 10dB of attenuation;: 15dB of attenuation.

2 4 6 8 10 12 14-60

-50

-40

-30

-20

-10

0

Am

plitu

de (

dBV

)

Frequency (GHz)

2 4 6 8 10 12 14-60

-50

-40

-30

-20

-10

0

Frequency (GHz)

Am

plitu

de (

dBV

)

68

3.5 Timebase jitter

path).s beenution.nger

of theg filternceslative

pectrat reach

beained) andcondis is

ich is.5-9).

itablege ofcause

path) has an attenuation of 30dB relative to the first spectrum (0dB of attenuation in triggerFigure3.5-9 shows the same spectra when the extended “PDF deconvolution” method haapplied. The parametric model that was used for the PDF corresponds to a normal distribNow the three spectra are practically coincidental. The fact that the third spectrum no locorresponds to the other two above 7GHz is due to the fact that at 7GHz the spectrumaveraged third waveform goes down into the noise floor. This means that the compensatinwill be amplifying noise for this spectrum once above 7GHz. Figure3.5-10 shows the differebetween the reconstructed spectra with 10dB and 15dB of attenuation in the trigger path reto the reconstructed spectrum with 0dB in the trigger path. This shows that the three scorrespond very well for those frequencies where the averaged waveform spectrum does nothe noise floor (correspondence within about 200mdB). Two parasitic effects candistinguished. First an offset of about 200mdB is noticed in Figure3.5-10. This can be explby the fact that the different experiments took place over a very long period (a whole nightthat there might have been a small gain drift of the oscilloscope during the experiment. A seeffect is the difference being substantially larger for the first 5 frequency components. Thprobably caused by a leakage effect of the periodic excitation component of the SRD whdominantly present in the measurement (cf. the peak near DC in Figure3.5-8 and Figure3Although the frequency of this component was chosen carefully to avoid leakage, an inevsmall error in the oscilloscope’s timebase accuracy will still cause a small amount of leakathis component, having a power about 11dB higher than the surrounding components. Be

Figure3.5-10 Spectral difference between reconstructed waveforms with differentattenuation in trigger path. x-axis scale in GHz, y-axis scale in dB.

: 10dB of attenuation minus 0dB of attenuation: 15dB of attenuation minus 0dB of attenuation

2 4 6 8 10 12 14-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Frequency (GHz)

Am

plitu

de d

iffer

ence

(dB

)

69


othert to

5.8ps

of theed bynstant,small

of theecially

shopwas tostomersd outt ofntrast

is ownthe

e (aboutions,beingneverle of a). Theitionedereby

stablenique,

the phase of this leaky component is different relative to the sampling window whenever anattenuation level is used in the trigger path, its effect will be different from experimenexperiment and this will show up as it does in Figure3.5-10.

The estimated values for the jitter standard deviation for the three cases are 6.4ps, 1and 57.6ps respectively.

3.5.7 Conclusion

A new method has been described for the determination of the characteristic functionjitter PDF. This method can be used to extend the “PDF deconvolution” method as describGans [13] to cases where the waveform can not be approximated by a ramp at any time iwhich is the case for pulselike signals when the standard deviation of the jitter is no longercompared to the pulse transition duration. The method uses the average and the averagesquare of a large set of recorded waveforms. It has been shown that the method is espuseful when a correct reconstruction of the spectrum is required.

3.6 Timebase drift

3.6.1 Introduction

The significance of timebase drift was first noticed in september 1992 at a workorganized by Ken Rush and his colleagues in Colorado Springs. The goal of the workshoppresent the “nose-to-nose” calibration procedures to some interested customers. These cuwere also invited to bring their test-set to the workshop in order to get it calibrated. It turnethat it was very difficult to get meaningful results out of the “nose-to-nose” calibrations. Mosthe time the measured bandwidth of the instrument just did not make sense. This was in cowith the many times Ken Rush himself had done these measurements successfully on hworkbench. It finally turned out that the problem was caused by timebase drift duringmeasurements. These measurements need a lot of averaging and therefore take a lot of tim20 minutes). This timebase drift turned out to be highly correlated with temperature variathaving a value of typically a few picoseconds per degree Celsius. It was later identified ascaused by the temperature sensitivity of the trigger circuitry. The fact that this problem wasseen before was simply because the originally used workbenches were situated in the middlarge open-space architecture (which acts like a good buffer versus temperature variationsmeasurements for the customer test-sets were, however, done in a rather small air condroom. In this room the temperature could change significantly over a short period of time, hcausing significant time drifts during one measurement.

In order to solve this annoying problem, besides trying to do the measurements in atemperature environment, I implemented the so-called logarithmic spectral averaging techwhich is described in what follows.

70

3.6 Timebase drift

ily be.25psuringriod,argin

of theThiscalledge isuredulse

:

i

to the

the

atio,

ven

y

3.6.2 Logarithmic Spectral Averaging

As was done for the jitter one assumes that aliasing effects do not occur (this can easachieved with an equivalent time sampling oscilloscope since it has a time resolution of 0and a bandwidth of “only” 50GHz) and that leakage is not present. This is achieved by measa periodic waveform such that the range is an integer multiple of the waveform repetition peor by measuring a repetitive pulse such that the whole pulse will each time be digitized (a mfor the possible maximum drift needs to be present).

When regular averaging is applied, the effect of the drift can be compared to the effecttiming jitter, meaning that it will appear as if an additional low-pass filtering had taken place.distortion can however be removed if, instead of an arithmetic averaging technique, a so-logarithmic spectral averaging technique ([1], [17]) is applied. With this technique, the averacalculated of the complex logarithm of the discrete Fourier transform (DFT) of each measpulse. By doing this, the timebase drift will not affect the estimate of the measured p

spectrum. This can be explained by modelling the ith pulse measurement, called m(t) as follows

, (3.6-1)

with being the stochastic time drift and being the additive noise of theth

measurement. Calculating the Fourier transform of (3.6-1) results in:

. (3.6-2)

The logarithmic spectral average will be called Alog(ω), it is defined as follows:

, (3.6-3)

where it has been assumed implicitly that phase unwrapping has been applied priorsummation. This can be written as follows:

, (3.6-4)

with having the same stochastic properties as . Next, we calculate

expectation of . This results in the following expression:

, (3.6-5)

where Bias(σn) denotes a bias which is a function of the inverse of the signal-to-noise r

calledσn, and the probability density function of the additive noise. As was theoretically pro

in [17] the analytical expression of Bias(σn) for noise with a Gaussian probability densit

function is the following:

mi t( ) x t τi–( ) ni t( )+=

τi ni t( )

Mi ω( ) X ω( )e jωτi– Ni ω( )+=

Alog ω( )

ln X ω( )e jωτi– Ni ω( )+( )i 1=

K

∑K

--------------------------------------------------------------------=

Alog ω( ) ln X ω( )( ) jω

τii 1=

K

∑K

-------------– ln 1Qi ω( )X ω( )---------------+

i 1=

K

∑+=

Qi ω( ) Ni ω( )

Alog ω( )

<Alog ω( )> ln X ω( )( ) jω<τi> Bias σn( )+ +=

71


of

ly be

on thehigherat the

an

tisturb

scopeariseame on

wereplingplingasiticmpler

, (3.6-6)

where Ei denotes the “exponential-integral”-function [18]. A plot of the bias as a function

the SNR is given in Figure3.6-1. Note that this bias is a real function, such that there will on

a bias on the amplitude estimation and not on the estimation of the phase. As can be seengraph, the bias will have a value of less than 5mdB as soon as the signal-to-noise ratio isthan 10dB, which is the case in a practical measurement. Looking at (3.6-5) we see th

expectation of the logarithmic spectral averager will actually equal , plus

imaginary part which is a linear function ofω. For our purpose this linear imaginary part will nobe of any importance since it corresponds to a delay of the measured pulse, which will not dthe measurement.

3.7 The oscilloscope impulse response

As mentioned in 3.1 measuring the impulse response of a broadband sampling oscillois an essential part in the traceability path of the VNNA phase calibration. Many problemswhen one wants to measure the impulse response of these instruments. Before Ken Rush cthe idea of the “nose-to-nose” calibration procedure [22], mainly two other methodsmentioned in the literature to measure the linear characteristics of broadband samoscilloscopes. A first method is based on a detailed modelling of the oscilloscope’s samcircuitry [19]. To be useful, an accurate knowledge is needed about the values of parelements. For the latest generation of sampling oscilloscopes, with extremely small sa

Figure3.6-1 Bias of logarithmic spectral average versus signal-to-noise ratio

Bias σn( ) 12---Ei

1–2σn

2---------

–=

10 11 12 13 14 15 160

1

2

3

4

5

S/N (dB)S/N (dB)

Bia

s (m

dB)

SNR (dB)

ln X ω( )( )

72

3.8 Conclusion

at then thehape.e thecs can. Thethe

e of amplinga kindard”,theby theof theith a

opeationpling

plingd. It hascan bee the

dimensions, the values of parasitic elements are very hard to acquire. This means thmodelling approach can hardly be used. A second method ([20], [21] and [23]) is based oavailability of a “pulse standard”, this is a pulse generator with an accurately known pulse sThe idea is to connect the “pulse standard” generator to the oscilloscope and to digitizmeasured waveform. Using deconvolution techniques the oscilloscope’s linear characteristibe calculated out of the knowledge of the “standard pulse” shape and the digitized waveformmain problem with this approach is the availability of a “pulse standard”. In [20] and [21]“pulse standard” itself is characterized by means of electro-optic sampling, in [23] by the ussuperconducting sampling oscilloscope. In both cases the assumption is made that the sacircuitry with which the “pulse standard” was measured was perfect, and could be used asof “oscilloscope standard”. The problem is how to characterize your “oscilloscope standwhich actually returns us to the initial problem. For the electro-optic as well as forsuperconducting systems, the assumption that the sampling system is perfect is justifiedauthors by stating that the sampling system bandwidth is much higher than the bandwidth“pulse standard”. This is however a questionable assumption if one talks about signals wbandwidth of 50GHz. A way out of the vicious circle “pulse standard” versus “oscilloscstandard” was found in 1990 by Ken Rush. The method is called the “nose-to-nose” calibrprocedure which will be explained in Chapter 4. With this method the 54120 broadband samsystem is used once as a signal sampler and once as a pulse generator.

3.8 Conclusion

In this chapter was explained what errors influence the accuracy of broadband samoscilloscope measurements. Both timebase errors as well as vertical errors were considerebeen shown how the effect of these errors can be compensated or how these errorsquantified. This is very important since a calibrated broadband sampling oscilloscope will bphase reference consensus standard for our VNNA measurements.

73


ling

h ant

ice

lingg

-Hill

rier

ain

ol.

formd

ntice

ts of

nc.,

form

3.9 References

[1] Jan Verspecht and Ken Rush,”Individual Characterization of Broadband SampOscilloscopes with a “Nose-to-Nose” Calibration Procedure,”IEEE Transactions onInstrumentation and Measurement, Vol.IM-43, No.2, pp.347-354, April 1994.

[2] Jan Verspecht,”Accurate Spectral Estimation Based on Measurements witDistorted-Timebase Digitizer,”IEEE Transactions on Instrumentation and Measureme,Vol.IM-43, No.2, pp.210-215, April 1994.

[3] Hewlett-Packard Company,”HP-54120B Digitizing Oscilloscope Mainframe - ServManual,” HP Part No.54120-90908, 1989.

[4] Yih-Chyun Jenq, “Digital Spectra of Nonuniformly Sampled Signals: A Robust SampTime Offset Estimation Algorithm for Ultra High-Speed Waveform Digitizers UsinInterleaving,” IEEE Transactions on Instrumentation and Measurement, vol.IM-39, No.1,pp.71 - 75, Feb. 1990.

[5] A. Papoulis,”Probability, Random Variables, and Stochastic Processes,” McGrawSeries in Systems Science, McGraw-Hill, Inc., pp.357, 1981.

[6] Fredric J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete FouTransform,”Proceedings of the IEEE, Vol. 66, No.1, pp.51 - 83, January 1978.

[7] W. R. Scott, Jr., and G. S. Smith,”Error Corrections for an Automated Time-DomNetwork Analyzer,”IEEE Transactions on Instrumentation and Measurement, vol. IM-35,No.3, pp.300 - 303, September 1986.

[8] M. Kendall and A. Stuart,”The advanced theory of statistics,” Charles Griffin & Co Ltd, V2, Fourth Edition, pp.82 -89, 1979.

[9] E. Van der Oudera, J. Renneboog,”Some Formulas and Applications of NonuniSampling of Bandwidth-Limited Signals,”IEEE Transactions on Instrumentation anMeasurement, Vol.37, No.3, pp.353 - 357, September 1988.

[10] S. M. Kay,”Modern Spectral Estimation,” Prentice-Hall Signal Processing Series, PreHall, pp.70, 1987.

[11] T. Michael Souders, Donald R. Flach, Charles Hagwood and Grace L. Yang,“The effectiming jitter in sampling systems,”IEEE Trans. Instrum. Meas., vol. IM-39,pp.80-85,February 1991.

[12] W. H. Beyer,”CRC Standard Mathematical Tables,” 27th Edition, CRC Press, Ipp.526-527, 1984.

[13] W.L. Gans,”The measurement and deconvolution of time jitter in equivalent-time wave

74

3.9 References

ms,”

ers,”

igevaniteit

nction

and

ling

n of

ing

samplers,”IEEE Trans. Instrum. Meas., vol. IM-32, pp.126-133, March 1983.

[14] Jan Verspecht,”Compensation of Timing Jitter-Induced Distortion of Sampled WaveforIEEE Trans. Instrum. Meas., vol.IM-43, No.5, pp.726-732, October 1994.

[15] D.W. Marquardt,”An algorithm for least-squares estimation of nonlinear parametSIAM J., Vol.11, pp.431-441, 1963.

[16] Christel Knops, ”Studie en compensatie van “timing jitter” in breedband“sampling”-oscilloscopen,” Afstudeerwerk ingediend tot het behalen van de graadburgerlijk electrotechnisch ingenieur, richting toegepaste natuurkunde, Vrije UniversBrussel, Academiejaar 1993-1994.

[17] P. Guillaume, R. Pintelon, J. Schoukens,”Nonparametric Frequency Response FuEstimators Based on Nonlinear Averaging Techniques,”IEEE Transactions onInstrumentation and Measurement, vol. IM-41, no.6, pp.739-746, 1992.

[18] I. S. Gradshteyn, I. M. Ryzhik,”Table of Integrals, Series, and Products. CorrectedEnlarged Edition,” New York: Academic Press, 1980.

[19] Sedki M. Riad,”Modeling of the HP-1430A Feedthrough Wide-Band (28-ps) SampHead,” IEEE Transactions on Instrumentation and Measurement, vol. IM-31, no. 2,pp.110-115, June 1982.

[20] D. Henderson, A. G. Roddie,”Calibration of fast sampling oscilloscopes,”Meas. Sci.Technol., no. 1, pp. 673-679, 1990.

[21] D. Henderson, A.G. Roddie and A.J.A. Smith,”Recent developments in the calibratiofast sampling oscilloscopes,”IEE Proceedings-A, Vol.139, No.5, September 1992.

[22] Ken Rush, Steve Draving, John Kerley,”Characterizing high-speed oscilloscopes,”IEEESpectrum, pp. 38-39, September 1990.

[23] William L. Gans,”Dynamic Calibration of Waveform Recorders and Oscilloscopes UsPulse Standards,”IEEE Transactions on Instrumentation and Measurement, vol. IM-39,no.6, pp. 952-957, December 1990.

75


76

Chapter 4

The “Nose-to-Nose” CalibrationProcedure

Abstract - The “nose-to-nose” calibration procedure is introduced as an accuratemethod to determine the impulse response of broadband sampling oscilloscopes. Themethod is based on the assumption that “the sampler kick-out is proportional to theoscilloscope’s impulse response”. The validity of this principle is proven by a detailedtheoretical study. Conclusions concerning accuracy and precision aspects of themethod are drawn, based upon theoretical considerations as well as experimentalvalidation. All this shows why the “nose-to-nose” can be used as a phase referenceconsensus standard for VNNA measurements.

77

Chapter 4 The “Nose-to-Nose” Calibration Procedure

sensusthod-nose”

waserator

f theto use

at comesationecondscope

plerticalse with

es on

the, and a

4.2-1.ith theistors not” thetor arematrixs arethese

e) andcurrent).rrentt

4.1 Introduction

As already mentioned in the previous chapters the actual phase reference constandard in our work will be a “nose-to-nose” calibrated sampling oscilloscope. With this metwo oscilloscope input channels are connected together, which explains the name “nose-tocalibration method. Ken Rush came on the ingenious idea behind the method while hedesigning a fast pulse generator. While he was drawing the schematics of the pulse gencircuitry he noted that the final circuit was nearly identical to that of the sampling head osampling oscilloscope he had designed some years earlier. This fact put the idea in his headone of the oscilloscope test set channels as a pulse generator and to measure the pulse thout of the input connector with a second oscilloscope. This is what the “nose-to-nose” calibris all about: one oscilloscope is used in such a way that it is creating pulses and a soscilloscope is used as a waveform digitizer. One such a pulse coming out of the oscilloinput connector is called a “kick-out”. On a rather intuitive base Ken Rush stated that “the samkick-out is proportional to the oscilloscope’s impulse response”. Assuming two idenoscilloscopes, this statement implies that one can measure the oscilloscope impulse respona “nose-to-nose” calibration procedure by applying mathematical deconvolution techniquthe measured waveform.

In what follows a mathematical model will be presented, with special interest going tophase distortion characteristic accuracy, the practical measurement set-up will be describeddiscussion concerning the accuracy and precision of the method will be given.

4.2 Mathematical model

4.2.1 Sampling head equivalent scheme

The general sampler model on which all calculations are based, is depicted in FigureThis model describes the transmission line structure which connects the input connector wsampling diodes by a generic s-parameter [7] network, with a perfect terminating resconnected to the node. It is important to note that this perfect terminating resistor doecorrespond with the physical termination structure, but is only present to theoretically “definevalues of the s-parameters, such that all parasitical dynamic effects of the terminating resisautomatically included in the s-parameter description. For a good sampler, this s-parameterwill be close to the s-parameter description of a pure delay line. The sampling diodemodelled by two nonlinear conductances connected to the sampling node. In series withnonlinear resistors one finds the so-called hold capacitors (which hold the sampled chargtwo voltage sources representing the so-called sampling strobe generators (these drive athrough the diodes for a very short time span during which the actual sampling takes place

Filled arrow heads indicate voltage definitions and unfilled arrow heads indicate cudefinitions. The source with an output impedance of 50Ω models the signal source at the inpuconnector of the sampler. The incident and scattered voltage waves are indicated by a1(t), b1(t),

78


that

iodes

rse

v

verse

otherTo

rfectlyditional

se anill be

a2(t) and b2(t). In practice the output of the signal source will be limited to a magnitude such

no significant nonlinear distortions of the sampled signal are introduced. The sampling dcurrent-voltage relationships are defined by the functions G1 and G2. Before the actual sampling

takes place, both hold capacitors C1 and C2 are charged such that the sampling diodes are reve

biased. This is expressed in (4.2-1) and (4.2-2), where vB1 and vB2 stand for the reverse bias

voltages (it is assumed that no strobe pulse is present before a time instant equal to 0s.

(4.2-1)

(4.2-2)

Sampling is performed by applying the strobe pulse, modelled by the voltage sourcesS1(t)

and vS2(t), to the circuit. This strobe pulse has a large negative amplitude which turns the in

bias voltage into a forward bias condition during a small time span (small compared to alltime constants involved, like for example the inverse of the input signal bandwidth).understand the sampling mechanism, first assume that the sampling structure is pesymmetric and that there is no signal present at the input. During the forward bias concharge will be transferred from C1 to C2, such that the sum of the capacitor charges will equ

zero after the sampling has occurred. When one applies a signal at the input this will cauinbalance of the diode currents during sampling, such that this time the sum of the charges w

Figure4.2-1 Model of sampler circuitry

2 a1(t)

50Ωs11 s12

s21 s22

C1

C2

D1

D2vD2(t)

vC2(t)

vD1(t)

vS1(t)

vC1(t)

vS2(t)

b2(t)

a2(t)

b1(t)

a1(t)

i3(t)

v2(t)

iD1(t)

iD2(t)

iD1 G1 vD1( )=

iD2 G2 vD2( )=

50Ω

vC1 0( ) vB1 0>=

vC2 0( ) vB2 0>=

79


ignalmming

theatical

ior of

ts A

nce of

tions

is,

uit is

different from zero. For small signals, the sum of the charges will be a linear function of the svalue during the sampling instant. This net charge is detected by charge amplifiers and a sunetwork. Finally the detected charge is digitized and its value, after being mulitplied byappropriate proportionality constant, is send to the screen. In the following sections mathemexpressions will be derived describing the sampling mechanism explained above.

4.2.2 Derivation of the “sampler drive” equivalent scheme

First some basic network equations will be derived, describing the large signal behavthe sampling diodes. The equations are given by (4.2-3) until (4.2-8).

(4.2-3)

(4.2-4)

(4.2-5)

(4.2-6)

(4.2-7)

(4.2-8)

In what follows 4 new functions and 2 new constants are introduced, with subscrip(“average”) and∆ (“delta”) replacing the subscripts “1” and “2” for vS1(t), vS2(t), vD1(t), vD2(t), vB1

and vB2. The new functions and constants are defined as the average and half of the differe

the original functions and constants. The idea is illustrated for vDA(t) and vD∆(t) by (4.2-9) and

(4.2-10), and has to be applied to all functions and constants mentioned.

(4.2-9)

(4.2-10)

Summing (4.2-5) and (4.2-6), substituting (4.2-3) and (4.2-4) and using the new funcresults in (4.2-11).

(4.2-11)

In practice vD∆(t) will be much smaller than vDA(t). This is the case because the sampler

by design, very symmetric and because the input signal is limited to a level where the circbehaving as a linear sampler. It will then be possible to expand G1 and G2 into a Taylor series

around vDA(t), and to neglect second and higher order terms of vD∆(t) without causing any

iD1 t( ) G1 vD1 t( )( )=

iD2 t( ) G2 vD2 t( )( )=

vD1 t( ) vC1 t( ) vS1 t( )+ + v2 t( )–=

vD2 t( ) vC2 t( ) vS2 t( )+ + v2 t( )=

vC1 t( ) 1C1------ iD1 u( )du

0

t

∫ vB1+=

vC2 t( ) 1C2------ iD2 u( )du

0

t

∫ vB2+=

vDA t( )vD1 t( ) vD2 t( )+

2--------------------------------------=

vD∆ t( )vD1 t( ) vD2 t( )–

2-------------------------------------=

vDA t( )G1 vDA u( ) vD∆ u( )+( )

2C1------------------------------------------------------

G2 vDA u( ) vD∆ u( )–( )2C2

-----------------------------------------------------+ du

0

t

∫+

v+ BA vSA t( )+ 0=

80


glectlinear

v

mallan berive”

l

willasic

and

significant error. There is in fact an equivalence between the conditions which allow to nethese higher order terms and the conditions needed to insure that the circuit acts as asampler. This results in (4.2-12).

(4.2-12)

In (4.2-12) the last term of the left hand side can be neglected because of two reasons:D∆(t)

is small and the circuit is practically symmetric, such that the integrand is the product of two squantities, making it a second order effect. When this term is neglected, the equation cassociated with the equivalent scheme of Figure4.2-2, which is called the “sampler d

equivalent scheme. The diode D has the following voltage-current characteristic:

(4.2-13)

Note the definition of CH:

, (4.2-14)

and that the initial charge on the capacitor CH equals 0. The solution of this electrica

network results in vDA(t).

4.2.3 Derivation of the “signal sampling” equivalent scheme

In what follows, an equivalent scheme will be derived describing how the input signalactually be sampled and how a kick-out is generated. A prove of the validity of the bnose-to-nose principle “the kick-out is proportional to the impulse response” [3] results.

To simplify notations, two time-variant conductances are defined by (4.2-15)

(4.2-16):

Figure4.2-2 “Sampler drive” equivalent scheme

12--- 1

C1------ 1

C2------+

C2G1 vDA u( )( ) C1G2 vDA u( )( )+

C1 C2+--------------------------------------------------------------------------------

du

0

t

∫ vDA t( ) vBA+ +

vSA t( )G1′ vDA u( )( )

2C1--------------------------------

G2′ vDA u( )( )2C2

--------------------------------– vD∆ u( )du

0

t

∫+ + 0=

DvSA(t) + vBA

CH12--- 1

C1------ 1

C2------+

1–

=

vDA(t)

iC2G1 v( ) C1G2 v( )+

C1 C2+--------------------------------------------------=

CH12--- 1

C1------ 1

C2------+

1–

=

81


rms

0open

tput

o the

to

, (4.2-15)

. (4.2-16)

Subtracting (4.2-6) from (4.2-5), dividing by 2, expanding G1 and G2 in a first order Taylor

series around vDA(t) results in (4.2-17).

. (4.2-17)

Note that the third term of the left hand side of (4.2-17) is the product of two small tebecause of symmetry, and may be neglected compared to the second term.

Next, the Thévenin equivalent of the input circuitry will be calculated, including the 5Ωtermination. The input circuitry is depicted in Figure4.2-3. It can readily be shown that the

circuit voltage of the Thévenin equivalent will be equal to . What about the ou

impedance? The description of this output impedance in the time domain will be equal t

value of , with the voltage source short circuited ( ) and with equal

. Since equals zero, one can write:

. (4.2-18)

This can be written as:

. (4.2-19)

Combining this with the following equation:

, (4.2-20)

with equal to , results in:

Figure4.2-3 Schematic of the input circuitry.

gA t( )G1′ vDA t( )( ) G2′ vDA t( )( )+

2-------------------------------------------------------------------=

g∆ t( )G1′ vDA t( )( ) G2′ vDA t( )( )–

2-------------------------------------------------------------------=

vD∆ t( ) 1C1------ 1

C2------+

gA u( )2

--------------vD∆ u( )du

0

t

∫ 1C1------ 1

C2------–

g∆ u( )2

--------------vD∆ u( )du

0

t

∫+ +

v2 t( )– vS∆– t( ) vB∆–G1 vDA u( )( )

2C1------------------------------

G2 vDA u( )( )2C2

------------------------------– du

0

t

∫–=

2 a1(t) 50Ωs11 s12

s21 s22 b2(t)

a2(t)

b1(t)

a1(t)

i3(t)

v2(t)

50Ω

i2(t)

s21 t( )* a1 t( )

v2 t( ) a1 t( ) 0= i3 t( )

δ t( )– a1 t( )

b2 t( ) s22 t( )* a2 t( )=

v2 t( ) 50Ωi2 t( )– s22 t( )* v2 t( ) 50Ωi2 t( )+( )=

i3 t( ) i2– t( )v2 t( )50Ω------------–=

i3 t( ) δ t( )–

82


inputation

left

.2-5.

are

ic

, (4.2-21)

where z(t) represents the time domain representation of the output impedance of thecircuitry. Finally, the Thévenin equivalent is depicted in Figure4.2-4. The corresponding equis:

. (4.2-22)

Next, i3(t) is written as a function of vDA(t) and vD∆(t):

. (4.2-23)

Taking into account the same arguments as above, G1 and G2 are very well approximated by

a first order Taylor series, resulting in

, (4.2-24)

with

. (4.2-25)

If (4.2-24) is substituted in (4.2-21), and this into (4.2-17) with the third term of thehand side eliminated, an equation in the unknown vD∆(t) results:

, (4.2-26)

with vAS(t) independent from the input signal a1(t) and vD∆(t) and defined as follows:

. (4.2-27)

The equivalent scheme corresponding to (4.2-26) and (4.2-24) is illustrated in Figure4

The initial charge on the capacitor is equal to zero. In this schematic, and

considered as the unknowns. Note that vAS(t) and iAS(t) are equal to zero for a perfect symmetr

sampler with a perfect symmetric strobe pulse. As such, vAS(t) and iAS(t) are the mathematical

representation of the effects of the asymmetry of the sampler.

Figure4.2-4 Thévenin equivalent scheme of input circuitry.

z t( ) v2 t( ) 25Ω δ t( ) s22 t( )+( )= =

v2 t( ) s21 t( )* a1 t( ) z t( )* i3 t( )–=

z t( )

s21 t( )* a1 t( )

i3 t( )

v2 t( )

i3 t( ) G2 vDA t( ) vD∆ t( )–( ) G1 vDA t( ) vD∆ t( )+( )–=

i3 t( ) 2 iAS t( ) gA t( )vD∆ t( )–( )=

iAS t( )G2 vDA t( )( ) G1 vDA t( )( )–

2---------------------------------------------------------------=

vD∆ t( )gA u( )

CH--------------vD∆ u( )du

0

t

∫ 2z t( )* gA t( )vD∆ t( )( )+ +

vAS t( ) 2z t( )* iAS t( ) s21 t( )* a1 t( )–+=

vAS t( ) vS∆ t( )– vB∆G1 vDA u( )( )

2C1------------------------------

G2 vDA u( )( )2C2

------------------------------– du

0

t

∫––=

i3 t( ) vD∆ t( )

83


ing ameterling,

copege isgiven

ling. This

eals

trobealue,ily

ect ofch

4.2.4 The effect of sampler circuitry asymmetry

The description of the sampling process and the creation of a kick-out when samplDC-voltage is based on the equivalent scheme of Figure4.2-5 and the scattering-paradescription of the input circuitry. The pulse coming out of the input connector when sampwill equal b1(t). It can easily be verified that it is given by

. (4.2-28)

When sampling occurs, the value that will be displayed on the screen of the oscilloswill be proportional to the sum of the charges stored on the hold capacitors. This chardetected by charge amplifiers and a summing network and the corresponding output value isby

, (4.2-29)

with L a constant. The value of L is determined by a DC measurement.First the effects of asymmetry of the circuit on the creation of a kick-out and the samp

process will be described. The equivalent scheme of Figure4.2-5 represents a linear systemmeans that i3(t) can always be written as the sum of a linear functional of the input signal a1(t),

this will be noted 2.iDI(t), and of a linear functional of the two sources vAS(t) and iAS(t), noted

2.iCO(t), which represents the effect of the asymmetry of the circuit. Looking at (4.2-28) rev

that this iCO(t) will cause a pulse to come out of the input connector, not correlated with a1(t), the

signal applied at this input connector. This pulse is mentioned in [1] as “a portion of the spulse which is coupled onto the input channel”. Concerning the effect on the sampled v(4.2-29) shows that iCO(t) introduces an offset. All asymmetry effects are in practice eas

eliminated by doing two measurements, once with -a1(t) applied and once with a1(t) applied.

Subtracting the two measurements from each other and dividing by two eliminates the effvAS(t) and iAS(t). In what follows will be assumed that this elimination always took place, su

that the equivalent scheme used is the one shown in Figure4.2-6. Note the definition of vDI(t) in

Figure4.2-6:

Figure4.2-5 “Signal sampling” equivalent scheme

-vAS(t)+s21(t)*a1(t) CH

gA(t)

i3 t( )2

-----------

vD∆(t)

iAS(t)

2z(t)

b1 t( ) s11 t( )* a1 t( ) s12 t( )* 25Ω.i3 t( )( )–=

Q Li3 t( )

2-----------dt

0

∞

∫=

84


pling

rated

it isp

iante willormnput.

lifyand

. (4.2-30)

The pulse coming out of the input connector and the charge collected when samoccurs, both measured with elimination of the asymmetry effects, will be noted as bDI(t) and QDI,

they are related to iDI(t) by the following equations:

and (4.2-31)

. (4.2-32)

The elimination of the asymmetry effects applied on a practical measurement is illustby Figure4.2-7.

4.2.5 “The kick-out is proportional to the impulse response”

Before the kick-out waveform and the oscilloscope impulse response are defined,interesting to note that iDI(t) is a linear functional of vDI(t). This means that the relationshi

between iDI(t) and vDI(t) can be written as follows:

. (4.2-33)

In this equation K(u,t) represents a kernel function fully characterizing the time-varlinear system of Figure4.2-6. First the kick-out waveform and the sampler impulse responsbe defined by means of K(u,t). The kick-out waveform k(t) will be equal to the pulse wavefcoming out of the oscilloscope input connector when a DC-voltage is present at the iMathematically expressed, k(t) will be equal to the value of bDI(t) with a1(t) equal to a

DC-voltage. Note that the kick-out waveform will be proportional to this DC-voltage. To simpthe calculations, an offset voltage of -20mV will be assumed. Using (4.2-33), (4.2-30)(4.2-31) results in the mathematical relation between K(u,t) and k(t):

Figure4.2-6 “Signal sampling” equivalent scheme with elimination of asymmetry effects.

vDI(t) = s21(t)*a1(t)

2z(t)

CH

gA(t)

iDI(t)

vDI t( ) s21 t( )* a1 t( )=

bDI t( ) s11 t( )* a1 t( ) s12 t( )* 50Ω.iDI t( )( )–=

QDI L i DI t( )dt

0

∞

∫=

iDI t( ) K u t,( )vDI u( )du

∞–

∞

∫=

85


t

input

ppearsirac

and

nthe

. (4.2-34)

It is implicitly assumed that the DC component of s21(t) equals 1 and that the DC componen

of s11(t) equals zero. This assumption is valid because of the physical structure of the

circuitry which mainly consists out of an input connector and a transmission line.The oscilloscope impulse response, noted h(t), is defined as the sampled value that a

on the oscilloscope’s screen at time instant t, with the input signal being equal to a Ddelta-function. Mathematically expressed, h(t) will be equal to the value of QDI with a1(u) being

equal toδ(u+t). Using (4.2-33), (4.2-30), (4.2-31) and some trivial substitutions of the integrvariables, the following expression can be derived:

. (4.2-35)

In what follows the function K(u,t) will be calculated. K(u,t) will be equal to the currewaveform iDI(t) when vDI(t) equalsδ(t-u). Using the equivalent scheme of Figure4.2-6, t

following equation can be derived (U(t) being the Heaviside function):

Figure4.2-7 Measured pulse coming out of the input connector without (above) and with(below) asymmetry compensation as measured by a second oscilloscope.

time (picoseconds)

ampl

itude

(no

rmal

ized

)

k t( ) K u v,( )du

∞–

∞

∫

s12 t v–( )dv

∞–

∞

∫=

h t( ) L K v– u,( )du

∞–

∞

∫

s21 t v–( )dv

∞–

∞

∫=

86


)

of

small

, such

for

hereA

e C

fect

k(t)

4)

40),

ments

. (4.2-36)

In (4.2-36) p(t) is defined by: . (4.2-37

Note that p(t) is dimensionless, has an asymptotic value of 1 for gA(t) going to plus infinity

and that gA(t) will always be positive, since it is the derivative of a voltage-current relationship

a stable diode. For a good sampler, the second term on the right hand side of (4.2-36) iscompared to the first term of the right hand side. For a perfect sampler, with CH equal to infinity

and s22 equal to 0, the second term of the right hand side of (4.2-36) disappears completely

that one could state that in a first order approximation:

. (4.2-38)

Substitution of (4.2-38) in to (4.2-34) and (4.2-35) results in first order approximationsh(t) and k(t):

and (4.2-39)

. (4.2-40)

These simple solutions correspond to the early modelling results as described in [1], wthe effect of CH was not taking into account and where s22 was assumed to be equal to zero.

much better second order approximation to K(u,t), taking into account the effects of a finitH

and an s22 different from zero, can be constructed by substituting the solution for a per

sampler (4.2-38) in to the right hand side of (4.2-36). The result is (4.2-41):

. (4.2-41)

Substitution of (4.2-41) into (4.2-34) and (4.2-35) results into the kick-out waveformand the sampler impulse response h(t):

and (4.2-42)

. (4.2-43)

In these equations f(t) is defined by: . (4.2-4

The quantitative determination of the quality of the approximations (4.2-39), (4.2-(4.2-42) and (4.2-43) is given by the following rather intuitive reasoning. First consider s22. Since

the input circuitry has practically no losses, the amplitude of s22 will be very close to the

amplitude of s11, which can be measured by a classical network analyzer. Such measure

K u t,( ) p t( )50Ω----------δ t u–( ) p t( ) s22 t v–( ) U t v–( )

50Ω.CH--------------------+

K u v,( )dv

0

∞

∫–=

p t( )50Ω.gA t( )

1 50Ω.gA t( )+-----------------------------------=

K u t,( ) p t( )50Ω----------δ t u–( )≈

k t( ) p t( )50Ω----------* s12 t( )≈

h t( ) Lp t–( )50Ω-------------1* s21 t( )≈

K u t,( ) p t( )50Ω----------δ t u–( ) p t( )

50Ω----------p u( ) s22 t u–( ) U t u–( )

50Ω.CH--------------------+

–≈

k t( ) p t( )50Ω---------- 1 p t( )* f t( )–( )

* s12 t( )≈

h t( ) Lp t–( )50Ω------------- 1 p t–( )* f t( )–( )

* s21 t( )≈

f t( ) s22 t( ) U t( )50Ω.CH--------------------+=

87


ast

an

er

dera

dingnds to

putal

p(-t)are

ressedonly

. Aentual

en aske waveIf one

s the

s:

reveal that the value of s22 (in the frequency domain) is smaller than 0.05 from DC to at le

20GHz (corresponding to a level of -26dB). Next, consider the effect of CH. An estimate of the

maximum relative contribution of this finite capacitance, notedε, is given by:

, (4.2-45)

with Ta equal to the time span during which p(t) is significantly different from 0. For

HP54124 Ta (called the sampling aperture time) is equal to about 10ps and CH is equal to 2pF,

such thatε equals about 0.1. Since s22 andε are totally neglected in (4.2-39) and (4.2-40), a rath

large relative error bound of about 0.15 will exist for the corresponding first orapproximations. In (4.2-42) and (4.2-43) s22 and ε are considered in a linear way, such that

much smaller relative error bound of about 0.022 (0.15 squared) will exist for the corresponsecond order approximations. The error bound of the second order approximating correspoamplitude errors smaller than 200mdB and phase errors smaller than 1.3°.

The interpretation of (4.2-42) - (4.2-44) is the following. Because of reciprocity of the incircuitry s12(t) will equal s21(t), such that the validity of the statement “the kick-out is proportion

to the impulse response” will be determined by the symmetry of p(t). Indeed, if p(t) equalsthe statement is true. In practice this situation will be approximated very well. Argumentsbased on the approximate knowledge of the waveforms vS1(t) and vS2(t), which are the sampler

“strobe pulses” and on computer simulations. Because (4.2-37) shows that p(t) is a compversion of g(t), since during the diode forward bias the differential resistance of the diode isabout 10Ω, a large part of the asymmetry of g(t) (if already existing) will be masked in p(t)detailed discussion and estimation of the maximum error that can be caused by evasymmetries of p(t) is given in 4.5.0.

4.2.6 Measuring a mismatched signal source

In the above was assumed that the signal source is perfectly matched. One might thwhat happens if one assumes a mismatched signal source. Consider therefore a voltagsignal source modelled as depicted in Figure4.2-8 (frequency domain representation).

calculates the Thévenin equivalent of the input circuitry, analog to Figure4.2-4, one find

schematic as depicted in Figure4.2-9. In this figure and are defined as follow

Figure4.2-8 Model of mismatched signal source.

ε 150Ω.CH--------------------dv

0

Ta

∫=

ΓG

AG

zM t( ) aM t( )

88


e can

qualzero

or ao theabout

hichLOWt the

ally:

t isf it ise of thet willipated

and (4.2-46)

, with r(t) given by: (4.2-47)

, (4.2-48)

and with , and referring to the Fourier transforms of s12(t), s21(t)

and s11(t) respectively. Using the same calculations as were done to derive (4.2-43), on

derive the description of the waveform appearing on the oscilloscope’s screen, noted aM(t):

, with (4.2-49)

. (4.2-50)

In what follows will be shown that the second term of the right hand side of (4.2-50) is eto zero. Consider therefore the waveform r(t). Looking at (4.2-48) one sees that r(t) equalsfor all t smaller than twice the electrical delay from input connector to the sampling node. Fsampling oscilloscope of the HP54120 series, the coaxial cable from input connector tsampling head is equal to about 6.5cm, such that twice the electrical delay corresponds with650ps. On the other hand, p(-t) will be equal to zero for all t smaller than -30ps, wcorresponds to the sampling aperture time of the HP54120 series oscilloscope in “BANDWIDTH” mode (which is the worst case for the sampling aperture). This means tha

function will be equal to zero for all t smaller than (650ps - 30ps), such that fin

, and (4.2-51)

. (4.2-52)

Intuitively explained, this means the following. When a signal is sampled, a kick-oualways generated. This kick-out is launched towards the input connector, where a part oreflected back towards the sampling node because of the signal source mismatch. Becauselectrical delay between the input connector and the sampling head, this reflected kick-oureach the sampling node long after the sampling took place, such that the reflection is diss

Figure4.2-9 Thévenin equivalent of input circuitry with a mismatched signal source.

zM t( )

v2 t( )

i3 t( )

s21 t( )* aM t( )

aM t( ) aG t( )* F 1– 11 ΓG ω( )S11 ω( )–-------------------------------------------

=

zM t( ) 25Ω δ t( ) s22 t( ) r t( )+ +( )=

r t( ) F 1–S12 ω( )S21 ω( )ΓG ω( )

1 ΓG ω( )S11 ω( )–---------------------------------------------------

=

S12 ω( ) S21 ω( ) S11 ω( )

aM t( ) aG t( )* F 1– 11 ΓG ω( )S11 ω( )–-------------------------------------------

* hM t( )=

hM t( ) h t( ) s21 t( )*p t–( )50Ω------------- p t–( )* r t( )( )

–=

p t–( )* r t( )p t–( )50Ω------------- p t–( )* r t( )( ) 0=

hM t( ) h t( )=

89


plingce, oneich is

he.eiving

ollows.se ofscope

this

up

eed,

n be

thebleo 1ns.uistmber.ssary.early

in the termination load and can never contribute to the charge collected during the samprocess. The practical consequence of all this is that, when measuring a mismatched sourcan simply replace the signal source of (4.2-8) by the signal source of Figure4.2-10, whperfectly matched.

4.3 Determination of the oscilloscope impulse response

In what follows will be explained how the principle “sampler kick-out is proportional to toscilloscope’s impulse response” can effectively be exploited in a measurement procedure

With the first “nose-to-nose” measurements it was assumed that the sending and recoscilloscope were identical and that the oscilloscope inputs were perfectly matched (s11 equal to

zero). With these assumptions, the measurement of the impulse response is done as fAssume that the kick-out waveform of both oscilloscopes is k(t) and that the impulse responthese oscilloscopes equals h(t). The kick-out waveform as measured by the receiving oscillo(including the distortion of this oscilloscope), noted m(t), will be given by:

, (4.3-1)With C being a proportionality constant. Transformed into the frequency domain

becomes

. (4.3-2)

Because “the kick-out is proportional to the impulse response” one can calculate

to a proportionality constant by taking the square root of the measured quantity . Ind

one can write:

, (4.3-3)with D equal to a proportionality constant. It is important to note that this constant D ca

determined by a simple DC measurement.In practice the transformation into the frequency domain is done by applying a DFT on

digitized waveform. Since the kick-out waveform is limited in time (no contribution noticeaafter about 600ps) leakage is easily avoided by using a time span larger than or equal tAliasing is also easily avoided by using a time resolution of only a few ps, resulting in a Nyqfrequency higher than 100GHz. Note that the square root in (4.3-3) is taken on a complex nuIn order to choose the right sign for every frequency component phase unwrapping is neceSince we are dealing with a pulse like waveform in the time domain, corresponding to a n

Figure4.2-10 Equivalent matched signal source.

AG ω( )1 Γ ω( )S11 ω( )–---------------------------------------

ΓG 0=

m t( ) C k t( )* h t( )( )=

M ω( ) C K ω( )H ω( )( )=

H ω( )M ω( )

M ω( ) C K ω( )H ω( )( ) DH ω( )= =

90

4.3 Determination of the oscilloscope impulse response

therean be

e

a

and

by

e.

thatd were

can bepulse

xt

air of

ope

uency

the

f the

hase

linear phase in the frequency domain, and since the signal-to-noise ratio is sufficiently high,is no problem to perform this phase unwrapping. In order to calculate h(t) an inverse DFT c

performed on .

An estimate of the error due to the assumption that s11 equals zero can be made using th

results of 4.2.0. Taking in to account the mismatch of both oscilloscopes, one can write:

, (4.3-4)

with and referring to the S11 of oscilloscope A and oscilloscope B

respectively. For both oscilloscopes S11 is typically smaller than -25dB, which results in

maximum error for M(ω) of about 27mdB for the amplitude and 0.18° for the phase, which can be

neglected. Note that this error can eventually be eliminated by effectively measuring

with a network analyzer and by multiplying the measured value of

. Until now, however, this correction has never been applied in practic

A natural question when implementing this procedure is the validity of the assumptionboth oscilloscopes are identical. In order to avoid this assumption a method was developethree oscilloscopes are used, such that the individual impulse response of each oscilloscopefound. The method is the following. Assume that one has three oscilloscopes with im

responses , , and with kick-out waveforms , and . Ne

one performs three different “nose-to-nose” measurements with each time a different p

oscilloscopes connected together. The results of these measurements are called ,

and , with the first number in the subscript referring to the kick-out receiving oscillosc

and the second number to the kick-out creating oscilloscope. When transformed in the freqdomain one can write (with the C’s equal to the proportionality constants):

, (4.3-5)

, (4.3-6)

. (4.3-7)

One can then calculate , an estimate proportional to as follows:

, (4.3-8)

with D1 equal to a proportionality constant. Since “the kick-out is proportional to

impulse response” one sees that is proportional to the Fourier transform o

individual impulse response of oscilloscope 1 . The same notes concerning p

H ω( )

M ω( ) C K ω( )H ω( )( )1 S11

A ω( )S11B ω( )–

--------------------------------------------=

S11A ω( ) S11

B ω( )

S11A ω( )

S11B ω( ) M ω( )

1 S11A ω( )S11

B ω( )–( )

h1 t( ) h2 t( ) h3 t( ) k1 t( ) k2 t( ) k3 t( )

m12 t( ) m13 t( )

m23 t( )

M12 ω( ) C12H1 ω( )K2 ω( )=

M13 ω( ) C13H1 ω( )K3 ω( )=

M23 ω( ) C23H2 ω( )K3 ω( )=

H1est ω( ) H1 ω( )

H1est ω( )

M12 ω( )M13 ω( )M23 ω( )

--------------------------------------- D1H1 ω( )K2 ω( )H2 ω( )----------------= =

H1est ω( )

H1 ω( )

91


.

cticaltricksllows.

ted in. One

copetimeat theput in

tainld behold-outnt toholdmplee with

unwrapping and taking the inverse DFT in order to get the time domain waveform are valid

4.4 Practical measurement set-up

4.4.1 Introduction

When designing the 54120 sampling oscilloscope, one did not think about the praimplementation of the “nose-to-nose” calibration procedure. This implies that some specialneed to be used in order to make the procedure actually work. This is discussed in what fo

4.4.2 Experimental set-up

The experimental setup for actually performing a nose-to-nose measurement is illustraFigure4.4-1. Both oscilloscopes are connected to a controller with an IEEE488-interface bus

input connector of the scope A (kick-out creating oscilloscope) is connected to one input of sB. Channel1 of scope B is connected to the trigger input of scope A. Scope B is put in “domain reflectometry”-mode (TDR-mode), this means that scope B will generate stepsconnector of channel1, and will take samples relative to the instant of the steps. Scope A is“HISTOGRAMMING”-mode such that it will take a sample (=create a kick-out pulse) a cerfixed time after it is triggered by the steps created by scope B. In theory, scope A shousampling a DC voltage. In practice this is achieved by applying an offset voltage on thecapacitors of scope A. This can be done by use of the build in “OFFSET” function. The kickpulse created when sampling a DC-voltage of 100mV without any offset-voltage, is equivalethe kick-out pulse created when sampling a zero-voltage with an offset-voltage on thecapacitors of -100mV. To remove the asymmetry effects mentioned in 4.2, scope B will satwo kick-out pulses, one created with a negative offset voltage on scope A and another on

Figure4.4-1 Practical measurement setup

HP 54120Mainframe

B

HP 54120Mainframe

A Trig

CH2A

CH1

CH2B

ComputerController

Single f-f Adapter

1m SMA Cable

IEEE-488

92

4.5 Accuracy and precision aspects of the “nose-to-nose” calibration procedure

bothd onceoffset

ents,cisionnoise

pecialandard.pulse

theror of

theetric.erifyis tore and

f the

ns inphase

of

akes

the opposite positive offset voltage on scope A. The asymmetry effects will be present inmeasurements with the same sign, the kick-out pulse will be present once with a positive anwith a negative sign. By subtracting the positive-offset measurement from the negative-measurement, the effect of the strobe pulse is successfully removed (see Figure4.2-7).

4.5 Accuracy and precision aspects of the “nose-to-nose” calibrationprocedure

4.5.1 Introduction

Since the “nose-to-nose” calibration procedure involves many oscilloscope measuremseveral topics of Chapter 3 will be very useful in order to discuss about the accuracy and preof the method. Typical effects that are involved and will be discussed are timebase errors,and nonlinearities. The repeatability will also be investigated. There exists, however, one serror, considered as being very fundamental to the establishment of a phase reference stThis is the error caused by any possible asymmetricity (in time) of the sampling aperturep(t). This error is so important since it is present even when one would be able to domeasurements with three perfect oscilloscopes, with other words, it really is an ingrained erthe “nose-to-nose” calibration procedure.

4.5.2 Error due to possible asymmetricity of the sampling aperture pulse

Introduction

As shown above the basic principle that “sampler kick-out is proportional tooscilloscope impulse response” is only valid if the sampling aperture waveform p(t) is symmAlthough there are good reasons to believe that this is practically true, it is impossible to vsince p(t) itself can not be accurately measured. The only way to deal with this problemrecognize this fact as a possible systematic error of the “nose-to-nose” calibration proceduto quantify the maximum amount of error possible. This is done in what follows.

Phase distortion error

In order to quantify the effects that an asymmetric p(t) might have on the estimation oFourier transform of the impulse response of an oscilloscope, the effect of s22 and CH will be

neglected. Referring to the calculations concerning the quality of the different approximatio4.2.0, one can expect that neglecting this can result in a worst case error on the estimatederror bound of about 15%. Using (4.3-8) together with (4.2-39) and (4.2-40) results in:

, (4.5-1)

with equal to the Fourier transform of the sampling aperture

oscilloscope 2, and D1 a proportionality constant. This reveals that the systematic error one m

when applying the “nose-to-nose” calibration procedure, once D1 is determined by a DC

H1est ω( ) D1H1 ω( )

K2 ω( )H2 ω( )---------------- D1H1 ω( )ejϕ P2 ω( )( )= =

P2 ω( ) p2 t( )

93


ourier

hase

elay inform

useded too findhas to

we doimizein theithle to

g toundveryturned

ians)

theSince

tiverteden

as

traint

ed.

,

ed

measurement, is an erroneous phase contribution which is equal to the phase of the F

transform of . It is important to note here that there is also an unknown linear p

component present with a “nose-to-nose” measurement, corresponding to an unknown dtime domain. This delay is not of interest to us since one only wants to know about wavedistortions.

Mathematical formulation of the problem

In order to quantify the maximum amount of error on the phase distortion that can be caby the fact that the phase contribution of the sampling aperture p(t) is unknown, we nedescribe in a mathematical way the set of all admissible p(t)’s. Within this set we have then tthe sampling aperture waveform that maximizes the phase distortion. Note that this processbe repeated for all frequencies since the phase distortion is a function of frequency, f, andnot know whether the element that maximizes the distortion at one frequency will also maxthe phase distortion at another frequency. Two sets of admissible p(t)’s will be consideredfollowing. A first set is the collection of all positive functions defined on a time interval wduration Ta, Ta being the sampling diodes aperture time. It is important to note that we are ab

consider positive functions only since p(t) is related to the differential conductance gA(t) by

(4.2-37). Since the diodes that are used are locally and globally passive devices gA(t) will always

be positive and so is p(t). A second set of admissible functions will be all functions belonginthe previous set but which have only one local maximum. This will result in a lower upperbofor the possible phase distortion error. The fact that p(t) has only one local maximum isreasonable since the diodes are first turned on with a typical increasing current and are thenoff with a typical decreasing current waveform.

The phase distortion of a function p(t), noted , is defined as the phase (rad

versus frequency of the Fourier transform of the function p(t) when a delay is applied tofunction such that the derivative of the phase versus frequency is zero for a zero frequency.this phase distortion is invariant versus multiplication of the function with any strictly posireal and versus delaying the function, it will be sufficient for us to consider all functions suppoby the open interval (0,Ta) with their integral value on the interval equal to 1. Note that an op

interval is prefered since this implies that . The first set described, with

only constraint that the functions are positive, will be called , the second set, with the cons

added of only one local maximum, will be called . Note that Dirac-delta functions are allow

The problem can then be formulated as follows.

For each f, find the element in the set of functions , with equal to respectively

called , which maximizes the absolute value of , with defin

by

p2 t( )

Φ p[ ] f( )

p 0( ) p Ta( ) 0= =

ΠΠL

∆ ∆ Π ΠL

pMAX ∆[ ] f( ) Φ p[ ] f( ) Φ p[ ] f( )

94


ith a

ted

d as

-6)

the

,

n

. (4.5-2)

Note that represents the phase of the Fourier transform of the function p(t) w

component added, linear in f (corresponding to the appliance of a delay), such that

. (4.5-3)

An upperbound for the maximum phase distortion error possible within will be no

, and will be given by

. (4.5-4)

In order to simplify mathematics, we will use normalized variables and u, define

follows:

, and (4.5-5)

. (4.5-6)

With these variables the problem becomes the following.

Define as the set of all positive functions belonging to with the normalization (4.5

applied. For each , find the element in , called , which maximizes

absolute value of , with defined by

. (4.5-7)

An upperbound for the maximum phase distortion error possible, noted

will then be given by

. (4.5-8)

will be called the extremal function.

Calculation of the extremal function inΠN.

Consider a fixed and define the function . This functio

maps the three dimensional vector space into . We immediately see that

Φ p[ ] f( ) ArcTan

p t( ) 2πft( )sin td

0

Ta

∫

p t( ) 2πft( )cos td

0

Ta

∫--------------------------------------------

– p t( )2πft td

0

Ta

∫+=

Φ p[ ] f( )

fdd Φ p[ ] f( )( )

f 0=

0=

∆ΦMAX ∆[ ] f( )

ΦMAX ∆[ ] f( ) Φ pMAX ∆[ ] f( )[ ] f( )=

θ

θ 2πfTa=

u tTa-----=

∆N ∆

θ ∆N pMAX ∆N[ ] θ( )

Φ p[ ] θ( ) Φ p[ ] θ( )

Φ p[ ] θ( ) ArcTan

p u( ) θu( )sin ud

0

1

∫

p u( ) θu( )cos ud

0

1

∫-----------------------------------------

– p u( )θu ud

0

1

∫+=

ΦMAX ∆N[ ] θ( )

ΦMAX ∆N[ ] θ( ) Φ pMAX ∆N[ ] θ( )[ ] θ( )=

pMAX ∆N[ ] θ( )

θ Γ x y z, ,( ) ArcTanxy---

– z+=

|R3

|R

95


the

me

mal

izes

h an

rages

.

an be

. (4.5-9)

What we have to do next is to map the set of functions onto through

transformation:

. (4.5-10)

This transformation will map the infinite dimensional onto a three dimensional volu

defined in , which will be noted V. Suppose we know V, the problem of finding an extre

function is then transformed into: find the vector, called , element of V, which maxim

the absolute value of the function . We can then write:

. (4.5-11)

Next V will be constructed. It can be expressed as:

. (4.5-12)

Because of the definition of as the set of all positive functions defined on (0,1) wit

integral value equal to 1, (4.5-12) reveals that V is the collection of all possible weighted aveof all the vectors belonging to the one dimensional curve, called C, defined as follows:

. (4.5-13)

In mathematical terms it is said that V is the convex hull of C, noted

Carathéodory’s theorem [9] states that co(C) is the union of all possible tetrahedra that c

defined on C. In order to find we calculate the gradient of :

Φ p[ ] θ( ) Γ p u( ) θu( )sin ud

0

1

∫ p u( ) θu( )cos ud

0

1

∫ p u( )θu ud

0

1

∫, ,

=

ΠN |R3

p u( )

p u( ) θu( )sin ud

0

1

∫

p u( ) θu( )cos ud

0

1

∫

p u( )θu ud

0

1

∫

→

ΠN

|R3

vMAX

ΓΦMAX θ( ) Γ vMAX( )=

V p u( )θu( )sin

θu( )cos

θu

ud

0

1

∫p u( ) ΠN∈

∪=

ΠN

Cθα( )sin

θα( )cos

θα0 α 1≤ ≤

∪=

V co C( )=

vMAX Γ

96


that

n be

atedk for

terized

l be

ce

mum

rtant

for

line

an be

ction

is

mal

and

can

eter

nce

. (4.5-14)

We immediately see that the function has no stationary points in . This means

is situated on the boundary of V. Since V is the union of all possible tetrahedra that ca

defined on C, every point on the boundary of V belongs to a triangle with its corner points situon the curve C (the boundary of one such a tetrahedron). It will thus be sufficient to loo

within this collection of triangles.

Consider now one such triangle, characterized by its three corner points:

. (4.5-15)

Take the intersection between this triangle and a plane that contains the z-axis (charac

by x/y is constant). This intersection will be a line segment. On this line segment wil

maximum at one of the end points (belonging to the borders of the triangle). This is true sin

is just a linear function of z on this line segment since x/y is constant, such that the maxi

is reached for the point on the line segment with lowest or the highest z. The impo

conclusion is that is on the border of such a triangle, such that it is sufficient to look

within the set of all line segments with their two end points belonging to C. This set of

segments is the image of the transformation (4.5-10) applied to all elements of which c

written as the sum of two Dirac-delta functions. We can thus conclude that the extremal fun

we are looking for is a weighted average of two Dirac-delta functions. Since

invariant versus applying a delay on the function p, it will be sufficient to look for the extre

functions within the set of functions that is described as , where

are two parameters ranging from 0 to 1, and stands for a Dirac-delta function. We

then define the phase distortion as a function of , and :

. (4.5-16)

The problem is now reduced to the search of the point in the two-dimensional param

space and , both ranging from 0 to 1, where is extremal. Si

grad Γ( )

y

x2

y2

+-----------------–

x

x2

y2

+-----------------

1–

=

|R3

vMAX

vMAX

θα1( )sin

θα1( )cos

θα1

θα2( )sin

θα2( )cos

θα2

θα3( )sin

θα3( )cos

θα3

, ,

ΓΓ

Γ

vMAX

vMAX

ΠN

Φ p[ ] θ( )

βδ u( ) 1 β–( )δ u γ–( )+ βγ δ u( )

θ β γ

Φ β γ θ, ,( ) ArcTan1 β–( ) γθ( )sin

β 1 β–( ) γθ( )cos+----------------------------------------------

– 1 β–( )γθ+=

β γ Φ

97


, it

e

ways

ays

will

e

,

ion:

f the

in

localp(u) iscludequal toerval,

us

and since we are only interested in the absolute value of

will be sufficient to confine the search to values of ranging from 0 to 0.5.

First we will investigate the dependency of on . It is clear that . W

then calculate the partial derivative of versus :

. (4.5-17)

We immediately see that the numerator is always positive and the nominator al

negative for ranging from 0 to 0.5, such that the partial derivative of versus is alw

negative. From this we can conclude that, for ranging from 0 to 1, the absolute value of

be maximum for equal to 1.

Next we will investigate the dependency of on . Sinc

there will exist a value between 0 and 0.5, called

which maximizes the absolute value of . is found by solving the equat

. (4.5-18)

This equation has two roots, we choose the one that belongs to the interval [0,0.5]:

. (4.5-19)

We can then finally write:

(4.5-20)

. (4.5-21)

For ease of interpretation we will express in degrees and as a function o

variable , which is equal to (cf. (4.5-5)). as a function of is illustrated

Table4.5-1.

Adding the constraint of only one local maximum

A smaller upperbound can be found if the constraint is added that p(u) has only onemaximum. Since p(u) has an integral value on the interval (0,1) equal to 1, one can say thatconstrained to all possible unimodal distributions supported by (0,1). Note that we hereby inthe cases where p(u) reaches this maximum value on an interval of the u-axis (p(u) being ea constant function on this interval). Note that, because p(u) is defined on an open int

. The formulation of the problem is identical to the case of the previo

Φ 1 β– γ θ, ,( ) Φ β γ θ, ,( )–= ΦβΦ γ Φ β 0 θ, ,( ) 0=

Φ γ

γ∂∂ Φ β γ θ, ,( )

4θ γθ2-----

sin2 β 12--- β–

1 β–( )

1– 2 1 γθ( )cos–( )β 1 β–( )+------------------------------------------------------------------------=

β Φ γγ Φ

γΦ β 1 θ, ,( ) β

Φ 0 1 θ, ,( ) Φ 0.5 1 θ, ,( ) 0= = βMAX θ( )

Φ β 1 θ, ,( ) βMAX θ( )

β∂∂ Φ β 1 θ, ,( )

βMAX

θsin1 2 1 θcos–( )βMAX 1 βMAX–( )–---------------------------------------------------------------------------------- θ– 0= =

βMAX θ( ) 12--- 1

4---

1 θsinθ

-----------–

2 1 θcos–( )-----------------------------––=

ΦMAX ΠN[ ] θ( ) Φ βMAX θ( ) 1 θ, ,( )=

ArcTan1 βMAX θ( )–( ) θsin

βMAX θ( ) 1 βMAX θ( )–( ) θcos+------------------------------------------------------------------------------

1 βMAX θ( )–( )θ–=

ΦMAX ΠN[ ]

θ2π------ fTa ΦMAX ΠN[ ] fTa

p 0( ) p 1( ) 0= =

98


called

s is

ugh

r

)

paragraph except that the search for the extremal function is now limited to a subset of ,

, which contains all elements of with only one local maximum. The mathematic

however more subtle. First one will look for the image in associated with thro

transformation (4.5-10), this image will be called VL.

In order to find VL it will be shown that:

, (4.5-22)

with equal to for and for and equal to 0 in all othe

cases.(4.5-22) is proven in two steps. First one shows that:

. (4.5-23)

Consider therefore an element . Choose as follows:

, with equal to the modus of p(u). (4.5-24One can then perform the following calculation:

. (4.5-25)

For this becomes:

Table4.5-1 Upperbound for the phase distortion error based uponΠN

(degrees)

0.0 0.00

0.1 0.23

0.2 1.98

0.3 7.54

0.4 22.1

0.5 90.0

fTaΦMAX ΠN[ ]

ΠN

ΠLN ΠN

|R3 ΠLN

ΠLN W α( )K α γ u, ,( ) αd

0

1

∫

W α( ) ΠN∈∪

0 γ 1≤ ≤∪=

K α γ u, ,( ) 1α γ–--------------- α u γ< < γ u α< <


0

1

∫

W α( ) ΠN∈∪

0 γ 1≤ ≤∪⊂

p u( ) ΠLN∈ W α( )

W α( ) p' α( ) α γ–( )= γ

W α( )K α γ u, ,( ) αd

0

1

∫ =

p' α( ) γ α–( )K α γ u, ,( ) αd

0

γ

∫ p' α( ) α γ–( )K α γ u, ,( ) αd

γ

1

∫–

u γ<

99


is

This

that

, (4.5-26)

and for this becomes:

. (4.5-27)

Next will be shown that . Two conditions need to be fulfilled. The first one

that , which is trivial, the second one is that the integral value on [0,1] equals 1.

integral is calculated as follows:

. (4.5-28)

Using partial integration this becomes:

. (4.5-29)

(4.5-25) until (4.5-29) show that:

, (4.5-30)

which is a prove of (4.5-23).Secondly one proves that:

. (4.5-31)

Consider therefore one and one . Define p(u) as:

. (4.5-32)

One then has to prove that the integral value of p(u) on the interval [0,1] equals 1 andp(u) is unimodal. The first conjecture is easy:

p' α( ) αd

0

u

∫ p u( ) p 0( )– p u( )= =

u γ>

p' α( ) αd

u

1

∫– p u( ) p 1( )– p u( )= =

W α( ) ΠN∈

W α( ) 0≥

p' α( ) γ α–( ) αd

0

1

∫ p' α( ) γ α–( ) αd

0

γ

∫ p' α( ) α γ–( ) αd

γ

1

∫–=

p α( ) γ α–( )[ ]0γ

p α( ) αd

0

γ

∫ p α( ) γ α–( )[ ]γ1

p α( ) αd

γ

1

∫+ + + 1=

p u( ) W α( )K α γ u, ,( ) αd

0

1

∫

W α( ) ΠN∈∪

0 γ 1≤ ≤∪∈


0

1

∫

W α( ) ΠN∈∪

0 γ 1≤ ≤∪⊃

γ W α( ) ΠN∈

W α( )K α γ u, ,( ) αd

0

1

∫

100


r two

t if

th

. (4.5-33)

The second conjecture, namely that p(u) is unimodal is proven as follows. Conside

values and such that . One then finds:

. (4.5-34)

Since it follows that . In an analog way it can also be shown tha

then . This proves that p(u) is an unimodal distribution wi

modus such that one can conclude that which proves (4.5-31).

To find VL transformation (4.5-10) is applied to (4.5-22). One can then write:

p u( ) ud

0

1

∫ W α( )K α γ u, ,( ) αd

0

1

∫

ud

0

1

∫=

W α( ) K α γ u, ,( ) ud

0

1

∫

αd

0

1

∫=

W α( ) αd

0

1

∫=

1=

u1 u2 0 u1≤ u2 γ≤<

p u1( ) W α( )K α γ u1, ,( ) αd

0

1

∫=

W α( )α γ–--------------- αd

0

u1

∫=

p u2( ) W α( )K α γ u2, ,( ) αd

0

1

∫=

W α( )α γ–--------------- αd

0

u2

∫=

W α( )α γ–--------------- αd

0

u1

∫ W α( )α γ–--------------- αd

u1

u2

∫+=

p u1( ) ε+=

ε 0≥ p u2( ) p u1( )≥

γ u1 u2< 1≤ ≤ p u1( ) p u2( )≥

γ p u( ) ΠLN∈

101


V.

nvex

m 0

ing as

n be

lue

. (4.5-35)

Exchanging the order of integration results into:

, (4.5-36)

and calculating the integral over u results into:

, with (4.5-37)

, element of . (4.5-38)

The problem of finding is now transformed into a problem very similar to finding

Indeed, (4.5-37) reveals that is equal to the union over (ranging from 0 to 1) of all co

hulls associated with the one dimensional curves , with parameter ranging fro

to 1. This can be written as:

. (4.5-39)

Consider now one such a curve with a particular value for and . The same reason

done previously holds. The convex hull of is the union of all tetrahedra that ca

defined on . Since the gradient of is never 0 in , will reach its extremal va

VL

W α( )K α γ u, ,( ) αd

0

1

∫

θu( )sin ud

0

1

∫

W α( )K α γ u, ,( ) αd

0

1

∫

θu( )cos ud

0

1

∫

W α( )K α γ u, ,( ) αd

0

1

∫

θu ud

0

1

∫

W α( ) ΠN∈∪

0 γ 1≤ ≤∪=

VL

W α( ) K α γ u, ,( ) θu( )sin ud

0

1

∫

αd

0

1

∫

W α( ) K α γ u, ,( ) θu( )cos ud

0

1

∫

αd

0

1

∫

W α( ) K α γ u, ,( )θu ud

0

1

∫

αd

0

1

∫

W α( ) ΠN∈∪

0 γ 1≤ ≤∪=

VL W α( )Q α γ θ, ,( ) αd

0

1

∫W α( ) ΠN∈

∪

0 γ 1≤ ≤∪=

Q α γ θ, ,( )

θα( )cos– θγ( )cos+θα θγ–

----------------------------------------------------

θα( )sin θγ( )sin–θα θγ–

---------------------------------------------

θα θγ+2

--------------------

= |R3

VL

VL γ

Q α γ θ, ,( ) α

VL co Q α γ θ, ,( )0 α 1≤ ≤

∪( )0 γ 1≤ ≤

∪=

γ θQ α γ θ, ,( )

Q α γ θ, ,( ) Γ |R3 Γ

102


the

curs

lane

such

pointremal

g to

d as

s

d by

the

centt thatto

ters

on the border of this possibility volume, which implies that this extremal value occurs on

border of a tetrahedron defined on . This further implies that the extremal value oc

on a triangle defined on . Taking the intersection between this triangle and a p

containing the z-axis results in a line segment. On this line segment is only function of z,

that the extremal value will be reached at one of the end points of this line segments (thewith the highest respectively lowest z-coordinate). The important conclusion is that the ext

value of will be reached on a line segment which has its two end points belongin

. This means that one only has to consider a limited set of ‘s, describe

. This further implies that the extremal function in belong

to a subset which can be written as follows:

, or (4.5-40)

. (4.5-41)

with parameters , and all ranging from 0 to 1. Within the set of functions describe

(4.5-41) one has then to look for the element that maximizes . Considering

definition of K, this set of functions can easily be interpreted as existing out of two adjarectangles situated on the interval [0,1], with a total area of 1. Taking into account the fac

is invariant towards delaying the function p(u), it will be sufficient to take in

consideration all functions of the form illustrated in Figure4.5-1, with a, b and c parameranging from 0 to 1.

One then writes as a function of a, b and c:

Figure4.5-1 Set of functions to which belongs the extremal function inΠNL.

Q α γ θ, ,( )Q α γ θ, ,( )

Γ

ΓQ α γ θ, ,( ) W α( )µδ α η–( ) 1 µ–( )δ α ν–( )+ ΠLN

p u( ) µδ α η–( ) 1 µ–( )δ α ν–( )+( )K α γ u, ,( ) αd

0

1

∫=

p u( ) µK η γ u, ,( ) 1 µ–( )+ K ν γ u, ,( )=

µ γ ηΦ p[ ] θ( )

Φ p[ ] θ( )

u

p(u)

10 cb c

1 a–1 b–( )c

-------------------

abc------

Φ p[ ] θ( )

103


the

ne

e

=b

tric,that

ime

ne is

l be

ents

s that

n 0

that

ach

es the

:

ce

.

g the

. (4.5-42)

The problem of finding the extremal function is then reduced to: for a given , find

values for a, b and c ranging from 0 to 1 maximizing . To simplify the problem o

will use the fact that , such that it will be sufficient to study th

function

, (4.5-43)

with .

This function has some interesting properties. will equal zero if a

or if a=0 or if a=1. The proof of this is trivial since the corresponding functions are symmesuch that there is no phase distortion present. Another property is

since replacing a and b by 1-a and 1-b corresponds to t

inverting the function which results in changing the sign of the phase distortion. Because o

only interested in the maximum of the absolute value of , this property implies that it wil

sufficient to look for this maximum in the triangle in the (a,b)-plane defined by the line segm

a=1, a=b and b=0. Numerical evaluation [8] in the (a,b)-plane for several values of reveal

for all values of a and b between 0 and 1, and for all values of betwee

and . Combining the fact that the partial derivative versus b is negative with the knowledge

the function equals zero for a=b and for a=1 results in the conclusion that will re

its maximum at the line segment b=0 of the triangle described above. This fact further reduc

problem since one will now be able to work with the function defined as follows

. (4.5-44)

Numerical evaluation shows that for a ranging from 0 to 1 and for

ranging from 0 to . will thus reach its maximum for the largest value of . Sin

and c ranges from 0 to 1, this largest value for a certain a will be equal to

The function has then to be maximized versus a. This can be done by solvin

following equation:

Φ a b c θ, , ,( ) ArcTana 1 b–( ) a 1–( )b cθ( )cos b a–( ) bcθ( )cos+ +

a 1–( )b cθ( )sin– b a–( ) bcθ( )sin–-----------------------------------------------------------------------------------------------------------------

– +=

cθ2------ 1 a– b+( )

θΦ a b c θ, , ,( )

Φ a b c θ, , ,( ) Φ a b cθ 1, , ,( )=

ψ a b θ', ,( ) ArcTana 1 b–( ) a 1–( )b θ'cos b a–( ) bθ'( )cos+ +

a 1–( )b θ'sin– b a–( ) bθ'( )sin–--------------------------------------------------------------------------------------------------------

– +=

θ'2---- 1 a– b+( )

Φ a b c θ, , ,( ) ψ a b cθ, ,( )=

ψ a b θ', ,( ) ψ a b θ', ,( )

ψ a b θ', ,( ) ψ 1 a– 1 b– θ', ,( )–=

ψ

θ'

b∂∂ ψ a b θ', ,( ) 0≤ θ'

πψ a b θ', ,( )

ψRED a θ',( )

ψRED a θ',( ) ψ a b θ', ,( )b 0→lim ArcTan

1 θ'cos–

a1 a–-----------

θ' θ'sin+

----------------------------------------

–θ'2---- 1 a–( )+= =

θ'∂∂ ψRED a θ',( ) 0≥ θ'

π ψRED a θ',( ) θ'

θ' cθ= ψRED a θ,( )

ψRED a θ,( )

104


ase

and

in

One

found

theneral

it has

, which results in (4.5-45)

, (4.5-46)

provided that the trivial solution has been omitted. The maximum ph

distortion in for a given is then finally given by:

, (4.5-47)

which can also be written as

. (4.5-48)

The corresponding extremal functions are the sum of a Dirac-delta function at the origin

a rectangular function on the interval [0,1]. Some values of are given

Table4.5-2, note the use of the variable for ease of physical interpretation (cf. (4.5-5)).

notes that the corresponding upperbound for is about three times lower than the one

for .

Conclusion

An important conclusion is that an upperbound can be found for the error due toasymmetricity of the sampling pulse. This upperbound is based on two simple ge

assumptions concerning this sample pulse, namely that it is of a finite duration and that

Table4.5-2 upperbound for the phase distortion error based uponΠNL

(degrees)

0.0 0.00

0.1 0.09

0.2 0.72

0.3 2.52

0.4 6.26

0.5 13.0

a∂∂ ψ a θ,( )

aMAX

0=

aMAX θ( ) 4 4 θcos– 2θ θsin–2 2 θcos– 2θ θ θ2+sin–------------------------------------------------------------=

aMAX 0=

ΠLN θ

ΦMAX ΠLN[ ] θ( ) ψRED aMAX θ( ) θ,( )=

ΦMAX ΠLN[ ] θ( ) ArcTan2 θ

2---sin2

θsin θ 4 4 θcos– 2θ θsin–( )2– θ2 2 θcos+ +

--------------------------------------------------------+--------------------------------------------------------------------------

– +=

θ 2– θ2 2 θcos+ +( )2 2 θ2 2 θcos– 2θ θsin–+( )--------------------------------------------------------------------

ΦMAX ΠLN[ ] θ( )

fTaΦMAX ΠLN[ ]

fTa

ΠLN

ΠN

Ta

105


0.2

hase

y rosee beenby aed withred thepticalnsferith the

o theope isresultsencywhole

rd toe takenows.output.orquemuche theal atbration

t-sine”called

heg afound

only one local maximum. For an HP54124, with being approximately 10ps, equals

for f equal to 20GHz, and 0.5 for f equal to 50GHz. Table4.5-2 then reveals that the pdistortion error is smaller than 0.72° at 20GHz and smaller than 13° at 50GHz.

4.5.3 Correspondence with other measurement techniques

Introduction

As soon as the first “nose-to-nose” experiments were elaborated the question naturallhow the validity could be verified. Two successful comparisons versus other techniques havreported. In [2] the amplitude of the oscilloscope transfer function as measured“nose-to-nose” procedure has been compared with the amplitude characteristic as measurclassical swept-sine power measurements [3]. In [10] Henderson, Roddie and Smith comparesults of the “nose-to-nose” with swept-sine power measurements as well as electro-osampling techniques. They reported about the comparison of the amplitude of the tracharacteristic and of the time domain representation of the impulse response as measured w“nose-to-nose” and with a very broadband electro-optical sampling system.

Swept-sine power measurements

What happens with this method is that a sinewave with a certain power is injected intchannel of the sampling oscilloscope. The power of the signal as measured by the oscilloscthen compared to the power of the signal as measured by a classical RF power meter. Thisin the value of the amplitude of the transfer function of the oscilloscope at the particular frequof the used sinewave. In order to get the values of the transfer function amplitude over theband of interest, it will then be sufficient to sweep the frequency of the signal source.

Although this swept-sine measurement seems rather simple at first sight, it is very haget accurate measurements. The whole procedure is explained in [1]. Special care has to bconcerning typical microwave measurement issues. An overview is given in what follPadding attenuators are used in order to minimize the effects of mismatches at the sourceHigh quality microwave cables have to be used, connectors are torqued with calibrated twrenches. The trigger signal of the oscilloscope has to be very good, in order to avoid toojitter. As trigger signal the 10MHz reference output of the synthesizer is used. To minimizjitter this 10MHz signal is amplified by a fully saturated amplifier, such that the resulting signthe scope trigger input has very steep slopes. One also has to take into account the calicoefficients of the power sensor that is used.

Correspondence between “nose-to-nose” and “swept-sine”

The correspondence between the “nose-to-nose” measurements and the “swepmeasurements was investigated for an HP54124 oscilloscope in its two operating modes,“HIGH BANDWIDTH” and “LOW BANDWIDTH”. In “LOW BANDWIDTH” mode thebandwidth of the oscilloscope is about 30GHz and in “HIGH BANDWIDTH” mode tbandwidth is about 50GHz. The “LOW BANDWIDTH” mode has the advantage of havinsignificantly better signal-to-noise ratio below 30GHz. The results of the comparisons are

Ta fTa

106


As cann one

tematice inputlower

is tohieveperfecteyde. Thee isopticalred by

ulatingnsfer

e

in Figure4.5-2.

On this graph the smoothest curves correspond to the “nose-to-nose” measurements.be seen the correspondence between the two methods is very good. Especially wheconsiders that it was assumed that both oscilloscopes were identical and that several syserrors have been neglected such as jitter, timebase drift and the presence of harmonics of thsinewaves. This experiment was later successfully repeated by the author, although only thebandwidth HP54121 could be used. These results are described in [11].

Comparison with electro-optic sampling techniques

Another approach to check the accuracy of the “nose-to-nose” calibration procedureuse very fast electro-optical sampling techniques [10]. Electro-optical samplers acbandwidths of several hundred GHz. What usually happens is that one assumes them to beuntil for example 50GHz. Hendersonet al.checked the “nose-to-nose” accuracy as follows. Thgenerate a fast electrical pulse by means of sending a short laser pulse into a fast photodiogenerated electrical pulse has a “full width half maximum” (FWHM) of about 10ps. This pulsthen measured by the sampling oscilloscope as well as by the (assumed perfect) electro-sampler. The oscilloscope impulse response is found by deconvolving the pulse as measuthe electro-optical sampler from the pulse as measured by the sampling oscilloscope. Calcthe Fourier transform of the impulse response results in the frequency domain tra

Figure4.5-2 Comparison of swept-sine response and frequency amplitude responsderived from “nose-to-nose” measurements.

0 10 20 30 40 50 60

0

-3

-6

-9

-12

Frequency (GHz)

Re

spo

nse

(d

B) Low BW

High BW

107


t all

doingoing aese

ffectdo theoscoperoomraturefew

uddenrouble.ents.mummalmicveryhat theigher

ctralnted introller.d bytimes)

ntaryabout

ing is

d asne by

ng the

characteristic. Hendersonet al.also did a swept-sine power measurement. They concluded thathree methods correspond very well.

4.5.4 Timebase errors

Introduction

As explained in Chapter 3 several timebase errors are present when one ismeasurements with a sampling oscilloscope. These errors will thus also be present while d“nose-to-nose” calibration procedures. In what follows will be explained how the effect of therrors can be minimized and or can be quantified.

Timebase drift

As mentioned in 3.6.0 timebase drifts due to temperature variations can significantly athe quality of the “nose-to-nose” measurements. The first thing one can do about this is tomeasurements in a room with a stable temperature. One should avoid to expose the oscillto fast temperature variations. Note that working in a thermostatic temperature controlledseems to be the ideal solution but can actually represent a dangerous pitfall. In the tempecontrolled lab at the university it is such that the temperature can drop very fast (just aminutes) from the thermostat upperbound to the thermostat lowerbound. When this stemperature drop manifests itself during the “nose-to-nose” measurements one can be in tThis is the reason why the air-conditioning is usually turned off while doing the measuremDuring one of the experiments in the lab the timebase drift was measured with a maxilikelihood estimator, described in [6]. A histogram of the time drift shows an almost norprobability density function with a standard deviation of 0.6ps. Without using the “logarithaverager” this would introduce an error of 75mdB at 35GHz. Although this does not looksignificant, one has to be aware that it is usually not easy to stabilize the temperature such tdrift standard deviation is lower than 1ps, and that the error rapidly increases for hfrequencies.

The next thing one can do about the timebase drift is to use the logarithmic speaveraging explained in 3.6.0. In order to do this one can not use the averaging as implemethe scope itself but one has to perform the logarithmic spectral averaging in an external conThe way it is implemented there is first one positive offset measurement immediately followeone negative offset measurement. This measurement is repeated many times (about 1000and the logarithmic spectral averaging is applied on the difference of the two complememeasurements. Doing this one assumes that there is no significant timebase drift during the10s it takes to perform the two complementary measurements. When the air-conditionturned off this is certainly a safe assumption.

Timebase jitter

The timing jitter present during the measurements is estimated using the methodescribed in 3.5 where the amount of timebase drift has been taking into account. This is doinverse delaying each measured pulse by the estimated timebase drift [6] before calculati

108


ectralrmales any ofg all

)s noated to

sent.4nsh theby theed onerrorpe ofpulse

nse bysion ofnd hasz thepe less

is theith a

ebasen be“4ns

all ine thehatnt of

averages that are used in order to estimate the amount of jitter. Note that “logarithmic spaveraging” can not compensate for the errors introduced by timing jitter. Assuming a nodistribution for the jitter, its standard deviation was estimated to be about 1.6ps. This causamplitude error of about 175mdB for a frequency of 20GHz and 538mdB for a frequenc35GHz. Using the method described in 3.5 this error can be compensated by multiplyinfrequency components with the frequency dependent function C(f) defined as follows:

, with f the frequency andτ the jitter standard deviation. (4.5-49It is important to note that the jitter only affects the amplitude characteristic and ha

influence on the phase characteristic, such that it is of no importance for measurements relcharacterizing the phase distortion of reference generators.

Timebase distortion

Next to the jitter and timebase drift a systematic timebase distortion will also be preThe effect of this error can easily be minimized by making sure that there is no “everydiscontinuity” (see 3.4) present in the 1ns window of the measurement. The delay at whicmeasured kick-out appears on the screen of the pulse receiving oscilloscope is determinedlength of the SMA-cable between the two oscilloscope test-sets and by the time delay specifioscilloscope A (see Figure4.4-1). If no timebase discontinuity error is present, the timebaseover a 1ns window can be approximated in first order by a linear curve. The worst case slothis error is about 2ps error per 1ns timebase range. This would imply that the measured imresponse is a slightly time compressed or expanded version of the actual impulse respoabout 0.2%. The same holds for the measured transfer characteristic, where the expancompression is seen on the frequency axis. Since this transfer characteristic is rather flat aonly little phase distortion this worst case 0.2% error can easily be neglected. At 20GHamplitude characteristic has a slope less than 0.2dB/GHz and the phase distortion has a slothan 0.5°/GHz, resulting in worst case errors of 8mdB for the amplitude and 0.02° for the phase.

Conclusion

The most important timebase error influencing the “nose-to-nose” measurementstimebase drift. Its effect can easily be eliminated by doing the measurements in a room wstable temperature and by applying the “logarithmic spectral” averaging technique. The timjitter distorts the amplitude of the measured transfer characteristic but this distortion cacompensated. The timebase systematic distortion will not have any measurable effect if thediscontinuity” is not occurring within the time window of the measurement.

4.5.5 Sampling head linearity

As mentioned in 4.2 one needs to be sure that the signal levels are small enough smorder for the sampler to behave linear. This means that the DC-voltage applied (in practicoscilloscope internal “OFFSET” function is used, such that we will call this value “offset” in wfollows) in order to create a kick-out pulse needs to be limited. To determine the amou

C f( ) e2πfτ( )2

2-------------------

=

109


entdwidthages:urierimentanlitudecurve

0mVand

hesecreatorhigher121Table

wing

nonlinear distortion the following experiment is done. Because of availability two differoscilloscope test-sets are used, one 54121T (bandwidth 20GHz) and one 54124T (ban50GHz). Several “nose-to-nose” measurements are performed, with the following offset volt50mV, 75mV, 100mV, 125mV, 150mV and 200mV. The amplitude and phase of the Fotransform of all experiments are calculated, each time normalized to the 50mV offset exper(with normalizing is meant that the kick-out pulse with e.g. 200mV offset is divided by 4). Inideal case, all curves should be identical. The differences between the normalized ampcharacteristics and the 50mV offset characteristic is shown in Figure4.5-3. Note that a

which is closer to 0 corresponds with a smaller offset voltage. For an offset voltage of 10there is a difference with the 50mV offset experiment smaller than 100mdB for the amplitudesmaller than 1° for the phase, and this for all frequencies smaller than 25GHz. Note that tvalues are achieved by arranging the set up such that the 54124T test set is the kick-outand the 54121T scope is the receiver. When the functionality of both scopes is switched, anonlinear distortion is present, probably because of the relatively lower bandwidth of the 54test-set. All this implies that an offset voltage lower than 100mV has to be used in order to beto achieve better than 100mdB accuracy at 25GHz (typically 75mV is used).

4.5.6 Repeatability and noise

To have an idea about the repeatability and the uncertainty due to noise, the follo

Figure4.5-3 Sampler linearity check: difference between normalized amplitudes ofmeasured kick-out spectra and the amplitude of kick-out spectrum with50mV offset applied (curves for 75mV, 100mV, 125mV, 150mV and 200mVoffset, curves with a lower offset voltage are closer to zero).

frequency (GHz)

ampl

itude

diff

eren

ces

(dB

)

110


rimentt pulsesed 24plitudets ofthirda 99%

for the, the

of theation is

erage

r thethepplying

nose”

experiment is done. Two oscilloscope test-sets (the same as used for the linearity expeexplained above) are connected and disconnected three times, and each time 1000 kick-ouare measured, with an offset voltage of 75mV. Note that the third measurement was performhours later than the first. For the three measurements the average is calculated of both amand phase (for the averaging “logarithmic spectral averaging” is applied to avoid the effecsmall timebase drifts). Next, the difference is calculated between the second andmeasurement sequence and the first. From the sample standard deviation of the amplitudeconfidence interval of these differences is also calculated. Note that the confidence intervalphase is derived from the amplitude confidence interval using the fact that, for sufficient SNRstandard deviation for the phase is equal to the standard deviation of the natural logarithmamplitude. The sample standard deviation of the phase is not used, since this standard deviinfluenced by the time base drifts.

The results are shown in Figure4.5-4 and Figure4.5-5. The figures show that, for an av

of 1000 pulses with 75mV offset, the 99% confidence interval is better than +/-100mdB foamplitude and +/-1° for all frequencies smaller than 35GHz. The figures also show thatrepeatability is better than these values. Note that the phase characteristics are aligned by aappropriate delays (to compensate for small timebase drifts).

4.5.7 Check on sampling aperture asymmetry

The worst case error on the phase characteristic estimated with a “nose-to-

Figure4.5-4 Differences between the amplitudes of the measured kick-out spectrawith indication of the 99% confidence interval.

: difference between the first and the second measurement: difference between the first and the third measurement

0 10 20 30 40 50

-0.2

-0.1

0

0.1

0.2

Frequency (GHz)

Am

plitu

de d

iff. (

dB)

111


he realplingsome

lid is thelmost

ing atequaltween

an be

argerwith a

beenatingfact

ch that

a

measurement procedure was derived in 4.2.0. This is a worst case and one expects tsampling aperture waveform to be pretty symmetric. Although the symmetry of this samaperture waveform p(t) can not be checked by doing “nose-to-nose” measurementsconsequences of this assumption can. One such consequence of the assumption to be vafact that the phase characteristic of the Fourier transform of the measured kick-out pulse is ainvariant towards changes in the sampling diodes bias voltages vB1 and vB2 (cf. 4.2.0),

nevertheless the amplitude characteristic will change drastically. As can be seen by look(4.2-42), (4.2-43) and (4.2-44) this phase invariance should perfectly be achieved with f(t)to zero. In practice, the bias voltages can be changed by switching both oscilloscopes be“LOW BANDWIDTH” and “HIGH BANDWIDTH” mode. The resulting amplitude difference ofthe Fourier transform of the measured kick-out is about 5dB for a frequency of 20GHz. As cseen on Figure4.5-5 the phase difference, however, is smaller than 2° for all frequencies smallerthan 20 GHz. Note the phase 99% confidence interval on the figure. The significant ldifference for higher frequencies is probably due to the fact that a 54121T scope is usedspecified bandwidth of only 20GHz (f(t) is not minimized until 35GHz).

4.5.8 Kick-out drop-outs due to pulse creator screen updates

One effect specific to the “nose-to-nose” measurement implementation has not yetmentioned. The effect is called kick-out drop-outs. What happens is that the pulse creoscilloscope A of Figure4.4-1 from time to time does not create a kick-out. This is due to thethat the oscilloscope is in an interrupt mode to update its screen during those moments, su

Figure4.5-5 Differences between the phase distortions of the measured kick-out spectrwith indication of the 99% confidence interval.

: difference between the first and the second measurement: difference between the first and the third measurement

0 10 20 30 40 50-1.5

-1

-0.5

0

0.5

1

1.5

Frequency (GHz)

Pha

se d

isto

rtio

n di

ff. (

deg)

112

4.6 Oscilloscope transfer function measurements

equal(thistheseeratorgive alusionate is0 pointte of

xternallue ofto say

121,aintyoint ofhased.

it is not sampling. At the side of the pulse receiving oscilloscope one notes sampled valuesto approximately zero where one clearly expects them to be significantly different from zeroappears as if some values dropped out of the waveform). The simplest way to avoiddrop-outs seems to be the use of a low actual sampling rate (determined by the TDR genrepetition rate). Practice shows that 200Hz is a safe value. Note that nobody could actuallygood explanation for this the drop-out dependency on the sampling rate, every concconcerning this is based on empirical investigation. The disadvantage of this low sampling rof course that the measurements take a long time. For every kick-out measurement two 100waveforms are acquired (one with positive and one with negative offset). At a sampling ra200Hz this takes about 12 s (10s of sampling and about 2s for transferring the data to the econtroller). This means that it takes about 3 hours and twenty minutes to acquire a typical va1000 kick-outs. Doing the three oscilloscope measurements takes three times as long, it is10 hours.

4.6 Oscilloscope transfer function measurements

Experimental results of the transfer characteristic of a sampling oscilloscope (HP54with a specified bandwidth of 20GHz), together with 99% confidence intervals for the uncertdue to noise (cf. 4.5.0), can be found in Figure4.6-1 to Figure4.6-4. As one sees the -3dB pthe oscilloscope is at 20GHz, which is conform to the specifications of the instrument. The pdistortion is smaller than 3° for all frequencies below 20GHz, which is considered as very goo

Figure4.5-6 Difference between phase characteristic in “LOW” and “HIGH”bandwidth mode, with indication of the 99% confidence interval due tonoise.

: difference between the two measurements.

5 10 15 20 25 30 35

0

1

2

3

4

Frequency (GHz)

Pha

se d

isto

rtio

n di

ff. (

deg)

113


n ordera solidbility

ce.

4.7 Conclusion

The “nose-to-nose” calibration procedure has been explained as an accurate method ito determine the impulse response of a broadband sampling oscilloscope. The method hastheoretical base and it was shown that it can be practically implemented with good repeataand accuracy. It is considered in this thesis as the consensus standard for a phase referen

Figure4.6-1 Measured amplitude characteristic.

Figure4.6-2 99% confidence interval for the amplitude uncertainty due to noise.

5 10 15 20 25 30-6

-5

-4

-3

-2

-1

0

Frequency (GHz)

Am

plitu

de (

dB)

5 10 15 20 25 300

10

20

30

40

50

60

70

Frequency (GHz)

Am

p. 9

9% c

onfid

ence

int.

(mdB

)

114

4.7 Conclusion

Figure4.6-3 Measured phase distortion characteristic.

Figure4.6-4 99% confidence interval for the phase distortion uncertainty due to noise.

5 10 15 20 25 30

0

2.5

5

7.5

10

12.5

15

Frequency (GHz)

Pha

se d

isto

rtio

n (d

eg)

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

Frequency (GHz)

Pha

se 9

9% c

onfid

ence

int.

(deg

)

115


ling

the

,

ing

The75.

encynt

ter,”

1,

n ofr

-22,

4.8 References

[1] Jan Verspecht and Ken Rush,”Individual Characterization of Broadband SampOscilloscopes with a “Nose-to-Nose” Calibration Procedure,”IEEE Transactions onInstrumentation and Measurement, Vol.IM-43, No.2, pp.347-354, April 1994.

[2] Jan Verspecht,”Broadband Sampling Oscilloscope Characterization with“Nose-to-Nose” Calibration Procedure: A Theoretical Analysis,”Conference Record of the1994 IEEE Instrumentation and Measurement Technology Conference, Hamamatsu, Japanpp.526-529, May 1994.

[3] Ken Rush, Steve Draving, John Kerley,”Characterizing high-speed oscilloscopes,”IEEESpectrum, pp. 38-39, September 1990.

[4] William L. Gans,”Dynamic Calibration of Waveform Recorders and Oscilloscopes UsPulse Standards,”IEEE Transactions on Instrumentation and Measurement, vol. IM-39,no. 6, pp. 952-957, December 1990.

[5] L.W. Nagel,”SPICE2: a Computer Program to Simulate Semiconductor Circuits,”Electronics Research Lab at the University of California, Berkeley, ERL-M520, May 19

[6] Istvan Kollar,”Signal Enhancement of Nonsynchronized Measurements for FrequDomain System Identification,”IEEE Transactions on Instrumentation and Measureme,vol. IM-41, no. 1, February 1992.

[7] K. Kurokawa,”Power Waves and the Scattering Matrix,”IEEE Transactions on MicrowaveTheory and Techniques, March 1965, pp. 194-202.

[8] Stephen Wolfram,”Mathematica - A system for doing mathematics by compuAddison-Wesley, Second Edition.

[9] Jan Van Tiel,”Convex analysis - An introductory text,” John Wiley & Sons Ltd., pp.41984.

[10] D. Henderson, A.G. Roddie and A.J.A. Smith,”Recent developments in the calibratiofast sampling oscilloscopes,”IEE Proceedings-A, Vol.139, No.5, pp.254-260, Septembe1992.

[11] J. Verspecht,”Annual report: Oct. 90 - July 91,” Hewlett-Packard internal report, pp. 16August 1991.

116

Chapter 5

l
The Absolute Calibration of a Vectoria“Nonlinear Network” Analyzer
Abstract - It is shown how the absolute calibration procedure is effectivelyimplemented for connectored and on wafer device measurements. The quality of thecalibration procedures is illustrated by repeatability and stability measurements. Theaccuracy of the calibration procedure is made traceable to national standards labsconcerning the relative part of the calibration and the power calibration, and to the“nose-to-nose” calibration procedure for the phase distortion calibration. Animportant device for this phase distortion calibration is the reference generator. It isshown how it is constructed and characterized, using a classical network analyzer(for the output match) and a “nose-to-nose” calibrated sampling oscilloscope (for thepulse waveform).

117

Chapter 5 The Absolute Calibration of a Vectorial “Nonlinear Network” Analyzer

ta asactualto twotainscausedthertionelow

f thecase,s areportsr RFatic

es theon, ins arebility

rementof theis new.ely theer 4.

solutes arerationividualascaderementding”

ationwaysolute

5.1 Introduction

The VNNA hardware is far from perfect. Besides random noise errors, the raw daacquired by the system described in 2.4 has significant systematic errors relative to thephysical voltage waves as present at the DUT ports. The systematic errors are divided ingroups: nonlinear distortion and linear distortion errors. Since the VNNA test set only conlinear passive devices like couplers, cables and probes, all nonlinear distortion errors areby the VNNA data acquisition part. This part contains solid state amplifiers anddownconvertor sampling diodes, which are typical nonlinear devices. Simple harmonic distotests performed at different RF and IF frequencies reveal that the harmonic distortion is b-60dBc for the system described in 2.4, provided that the input power at the level odownconvertor input is below -10dBm. During all experiments one takes care that this is thesuch that the nonlinear distortion errors can be neglected. The linear distortion errorintroduced by skin effect losses in the test port cables, unequal electrical lengths from DUTto data acquisition ports, linear distortions occurring in the couplers and the downconvertoand IF paths, non infinite directivity of the couplers,... . In order to eliminate these systemerrors a calibration procedure has been developed. Since this calibration procedure involvmeasurement of power, which is a dimensional quantity, one talks about absolute calibraticontrast to a classical linear network analyzer calibration where only dimensionless ratiodetermined and which is called a relative calibration. Special attention is given to the traceaaspects of the absolute calibration procedure. Since one is dealing with a new measutechnique, traceability towards national standards labs is not yet possible for all aspectscalibration. Especially the phase distortion calibration, based upon a reference generator,This implied that one had to build an in house phase reference consensus standard, nam“nose-to-nose” calibration procedure. This procedure has been explained in detail in Chapt

5.2 Consistency versus cascading independent of absolute calibration

5.2.1 Introduction

One of the questions that naturally arose when the research concerning the abcalibration started was: is there a way to prove that the absolute calibration coefficientwrong? If the answer to this question is yes, this could lead to a practical absolute calibmethod. One suggestion was to take two heavily nonlinear devices and to measure their indbehavior as well as the behavior of the cascade of the two. The measured behavior of the ccould then be confronted with the calculated behavior of the cascade based upon the measuof the individual behaviors. It turns out however that the so-called “consistency versus cascais always valid for a relatively calibrated system, no matter how wrong the absolute calibrcoefficients are. Although this might seem trivial by now, it certainly was not considered thatback in 1990. The proof that consistency versus cascading is independent from the abcalibration is given in what follows.

118

5.2 Consistency versus cascading independent of absolute calibration

tion

ave,alx of

eous

ectralted in

nd

5.2.2 Mathematical prove

Consider a VNNA which has been relatively calibrated, but whose absolute calibracoefficients are wrong. Mathematically expressed this means:

. (5.2-1)

In (5.2-1) an “a” stands for an incident voltage wave and a “b” for a scattered voltage wsubscript “M” stands for “as measured by the VNNA” and subscript “D” for “at the DUT signport”, subscripts “1” and “2” stand for the port number, and superscript “i” stands for the inde

the spectral component under consideration. The coefficient “ “ stands for the erron

absolute calibration coefficient.Consider now a first nonlinear device-under-test, called DUT1. This device will transform

the spectral components of the incident voltage waves at both ports into scattered spcomponents at both ports, note that this transformation is a nonlinear one. This is illustraFigure5.2-1. Due to the erroneous absolute calibration coefficients the model of DUT1 as

measured by the VNNA, called DUT1M is the one depicted in Figure5.2-2. Note that DUT1M

equals the physical DUT1 embedded in a structure which divides all incident waves by a

Figure5.2-1 The DUT as a nonlinear voltage wave transformer.

Figure5.2-2 The DUT as measured by the VNNA, called DUT1M.

aM1i

bM1i

aM2i

bM2i

αi

aD1i

bD1i

aD2i

bD2i

=

αi

DUT1

aD1i

aD2i

bD2i

bD1i

DUT1

aM1i

aM2i

bM2i

bM1i

αi

1 αi⁄ 1 αi⁄

αi

DUT1M

αi

119


at

and

l as

n

ersus

e pastNNA

ilar,

multiplies all scattered waves with . Also note that if DUT1 is a linear device, one can state th

DUT1 effectively equals DUT1M. Consider now a second nonlinear device-under-test

measure DUT2M. The model one gets by cascading the two erroneous models DUT2M and

DUT1M is called DUTCM and is depicted in Figure5.2-3. This can be simplified into the mode

depicted in Figure5.2-4. This reveals that DUTCM is actually the model that one would find whe

measuring the actual physical cascade of the two devices, called DUTC. This proves the

conjecture that an erroneous absolute calibration will not influence the consistency vcascading devices.

5.3 Classical absolute calibration approaches

A short description of the absolute calibration approaches used by other people in thhas already been given in 2.3, together with the hardware description of preexistent Vprototypes. It is clear that the relative part of the different calibration procedures are very sim

Figure5.2-3 The cascade of the two measured models DUT1M and DUT2M.

Figure5.2-4 Simplified model for DUTCM.

αi

DUT1

aM1i

aM2i

bM2i

bM1i αi

1 αi⁄ 1 αi⁄

αi

DUT1M

DUT2

αi

1 αi⁄ 1 αi⁄

αi

DUT2M

DUTCM

DUT1

αi

1 αi⁄DUT2

1 αi⁄

αi

DUTC

aM1i

bM1i

aM2i

bM2i

DUTCM

120

5.3 Classical absolute calibration approaches

ing thean RFphase

s notpes orthisthese

mall,theiry canracy ofrence

cticalvery

tion or[1].

f the

smallted inuringsincenciesof the

model

due to

also.2 thatith theture

tically

although some need to assume that some VNNA test ports are perfectly matched. Concernamplitude of the absolute calibration coefficients, all the measurements are traceable topower meter. Concerning the phase of the absolute calibration coefficients (modelling thedistortion of the VNNA), the situation is different. Two kinds of traceability paths exist.

All of the methods, except that of Urs Lott, assume that their data acquisition module iintroducing any phase distortion. This means that they assume broadband oscilloscodownconvertors without phase distortion. According to our knowledge the problem withapproach is that no specifications can be found concerning the phase distortion ofinstruments. Although the error they make with their assumption might actually be very sthey provide no way to check on this important issue. One way out of this would be to verifydata acquisition modules against a “nose-to-nose” calibrated oscilloscope. This way thetrace the measurement of the phase distortion of their data acquisition module to the accuthe “nose-to-nose” calibration procedure, which, by our knowledge, is the only phase refecalibration procedure presently available with known error bounds.

Urs Lott uses the approach of the “golden diode” as a phase reference. From a prastand point this seems a more interesting way to establish traceability. It would indeed beeasy to exchange reference diodes with national standards labs for comparison and verificacalibration. One problem with the applicability of this approach is due to the Page theoremThis theorem is valid for any nonlinear globally and locally passive resistance and one oconsequences is that:

, (5.3-1)

with P1 equal to the incident fundamental power and PN equal to the power generated at the Nth

harmonic. This theorem implies that the power generated at the higher harmonics will becompared to the fundamental power. Since the fundamental power itself needs to be limiorder to avoid nonlinear behavior of the data acquisition system, the signal-to-noise ratio dthe calibration will become smaller at the higher frequencies. This is even more annoyingthe test set and data acquisition will have a natural roll-off characteristic versus higher frequeby themselves typically caused by skin-effect losses. A second problem is that the accuracy“golden diode” approach is questionable. In his paper Lott estimates the error between theand the real device to be smaller than 10° at 15GHz and less than 16° at 20GHz (for afundamental frequency of 5GHz). These uncertainty bounds are rather large and aresignificant contributions of unknown parasitical effects.

It can be interesting at this point to note that the “nose-to-nose” calibration procedure isbased upon a “golden diode” approach. It is assumed for the mathematical description in 4the sampling diodes can be modelled by a nonlinear resistance. The big advantage w“nose-to-nose” is that all linear parasitical effects of the input circuitry (transmission strucfrom connector to sampling node and dynamic behavior of the termination load) are automataken into account.

PN

P1

N2------≤

121


deds ofrmonicliers

ling of

e

related(to ard tot the

d to theen theted int be

ed [7].

the’s areof the

nts

der toitrary

dto

tive

5.4 Absolute calibration procedure for connectored device measurements

5.4.1 Introduction

Originally the VNNA was build solely for the purpose of doing the measurements neefor the identification of so-called VIOMAP models, these are black box nonlinear modelnonlinear devices based upon the Volterra theory [2]. These models can then be used in habalance simulators to model the nonlinear behavior of amplifiers, mixers, frequency multipand cascades of these devices [3] [4]. Many applications of this are based on the model

connectored devices. For the VNNA of HP-NMDG APC-3.5connectors are typically used. Thabsolute calibration procedure for connectored devices [5] is explained in what follows.

5.4.2 Theory

The basic assumption for the absolute calibration is that the measured quantities areto the physical quantities by a linear relationship. As explained in 5.1 this will be the caselevel of at least -60dBc) if care is taken to limit the power going into the downconvertor boaa maximum value of -10dBm. A second assumption that simplifies the calculations is thameasured waves at port1 are only related to the physical waves at port1 and are not relatephysical waves at port2. In other words this means that there is no cross talk present betwetwo ports. This assumption is based on the physical construction of the test set as depicFigure2.4-1. The concept of the absolute calibration would still work if this cross talk can noassumed to be zero, but then more sophisticated relative calibration methods should be us

With these assumptions one can use the following error model:

. (5.4-1)

Definition of the wave variables is as given in Figure2.2-1, with superscript “i” beingfrequency index. Note that because of the assumption that cross talk is negligible 8 zeropresent. The goal of the calibration procedure will be the determination, for each frequency,

8 a priori unknown complex coefficients ( , , , , , , , ). Once these coefficie

are determined, it will be sufficient to use (5.4-1) together with the measured quantities in orknow the physical DUT quantities (which is our final goal). Note that the presence of an arb

linear phase shift in the vector Ki will have no other effect than time delaying both input anoutput signals, which is of no importance for our application. It will thus be sufficient

determine Ki on behalf of an arbitrary linear phase shift (in time domain an arbitrary delay).A classical linear vectorial network analyzer calibration [8] (this was called a rela

aD1i

bD1i

aD2i

bD2i

Ki

1 β1i

0 0

γ1i δ1

i0 0

0 0 α2i β2

i

0 0 γ2i δ2

i

aM1i

bM1i

aM2i

bM2i

=

Ki β1

i γ1i δ1

i α2i β2

i γ2i δ2

i

122


t for

this

ts arehen at1. Forer nownd the

d with

the

to bebe

can

tities.lts in:

calibration in 5.1) is then performed in order to determine all calibration coefficients excep

the ‘s. A short-open-load-through (also known as SOLT) calibration is performed for

purpose. What happens is the following. For each frequency seven raw measuremenperformed. For the first three measurements the power is incident to the port1 RF input. Tshort is connected to port1, an open is connected to port1, and a load is connected to porthe following three measurements the same procedure is repeated at port2, with the powincident to the port1 RF input. For the last measurement a through connection is made apower is incident to the port1 RF input. For the first three measurements one can write:

, (5.4-2)

, (5.4-3)

, (5.4-4)

with the numbers in the superscript referring to the index of the measurement performed, an

, and referring to the reflection coefficient of respectively the short, the open and

through for a frequency with index “i”. Note that these reflection coefficients are assumedknown a priori with a SOLT calibration. Using (5.4-1) the unknown DUT quantities cansubstituted for the known measured quantities:

, (5.4-5)

, (5.4-6)

. (5.4-7)

This represents a set of linear equations in the unknown coefficients , and , which

easily be solved:

. (5.4-8)

Note that all variables at the right hand side of (5.4-8) are measured or a priori known quanThe same calculations can also be done for the fourth to the sixth measurement. This resu

, with: (5.4-9)

Ki

Γ1iaD1

i1bD1

i1=

Γ2iaD1

i2bD1

i2=

Γ3iaD1

i3bD1

i3=

Γ1i Γ2

i Γ3i

Γ1i

aM1i1 β1

ibM1

i1+( ) γ1

iaM1

i1δ1

ibM1

i1+=

Γ2i

aM1i2 β1

ibM1

i2+( ) γ1

iaM1

i2δ1

ibM1

i2+=

Γ3i

aM1i3 β1

ibM1

i3+( ) γ1

iaM1

i3δ1

ibM1

i3+=

β1i γ1

i δ1i

β1i

γ1i

δ1i

Γ1ibM1

i1aM1

i1– bM1

i1–

Γ2ibM1

i2aM1

i2– bM1

i2–

Γ3ibM1

i3aM1

i3– bM1

i3–

1–

Γ1iaM1

i1–

Γ2iaM1

i2–

Γ3iaM1

i3–

=

β2Ni

γ2Ni

δ2Ni

Γ1ibM2

i4aM2

i4– bM2

i4–

Γ2ibM2

i5aM2

i5– bM2

i5–

Γ3ibM2

i6aM2

i6– bM2

i6–

1–

Γ1iaM2

i4–

Γ2iaM2

i5–

Γ3iaM2

i6–

=

123


to the

1)

)

ven

ed. It

omaino find

d with

odel

r the

e

that a

nsor.

e A

. (5.4-10)

With the seventh measurement a through connection is made and power is incidentport1 RF input, such that one can write:

, and using (5.4-1) this becomes (5.4-1

. (5.4-12)

This results in the knowledge of : . (5.4-13

Using (5.4-13), (5.4-10), (5.4-9) and (5.4-8) finally results in the knowledge of the se

relative calibration coefficients , , , , , and .

Next one has to determine . First the amplitude of this complex number is determin

is important to note here that the absolute value of a complex number describing a time dsinusoidal signal is by convention defined as the peak amplitude of this sinusoid. In order t

an eighth measurement is performed with a power meter sensor connected to port1 an

power incident to the port1 RF input. This measurement is done for each frequency. The m

that is used for the power meter sensor is given in Figure5.4-1. In this figure stands fo

wave from which the absolute amplitude is actually given by the power meter sensor, whil

stands for the sensor reflection coefficient and stands for the transmission factor. Note

table with the values of and is provided by the manufacturer of the power meter se

Usually the power meter read out is given in dBm. The relationship between peak-amplitudP

Figure5.4-1 Model of power meter sensor.

β2i

γ2i

δ2i

α2i

β2Ni

γ2Ni

δ2Ni

=

aD1i7

bD2i7

=

aM1i7 β1

ibM1

i7+ α2

i γ2Ni

aM2i7

δ2Ni

bM2i7

+( )=

α2i α2

i aM1i7 β1

ibM1

i7+

γ2Ni

aM2i7

δ2Ni

bM2i7

+--------------------------------------------=

β1i γ1

i δ1i α2

i β2i γ2

i δ2i

Ki

Ki

aSi

aD1i

bD1i ΓS

i

TSi

aSi

power sensor

ΓSi

TSi

TSi ΓS

i

124


)

d withel that

r the

ectral

ence

own on

-nose”

ing a

inth

)

and the power in dBm PdBm is given by:

. (5.4-14)

For this eighth measurement one can write:

, which can be written as (5.4-15

, such that (5.4-16)

. (5.4-17)

Note that all quantities at the right side of (5.4-17) are measured or known a priori.

A ninth measurement is finally performed in order to measure the phase of Ki. For thispurpose the reference generator is connected to port1, while the port1 RF input is terminatea load. More practical details concerning this reference generator are given in 5.6. The mod

is used for this reference generator is given in Figure5.4-2. In this model stands fo

reference generator output reflection coefficient at frequency “i” and stands for the sp

component generated at the frequency with index “i”. It is important to note that this refer

generator generates all spectral components at once and that the components are kn

behalf of a linear phase component. These components are characterized by a “nose-to

calibrated sampling oscilloscope. The reflection coefficients are measured by us

conventional vectorial linear network analyzer. More on this is described in 5.6. For this nmeasurement one can write:

, which can be written as (5.4-18

. (5.4-19)

It is then possible to calculate the phase of Ki:

Figure5.4-2 The model used for the reference generator.

AP 10

PdBm 10–

20---------------------------

=

aSi

TSiaD1

i8=

aSi

TSiK

iaM1

i8 β1ibM1

i8+( )=

Ki aS

i

TSi

aM1i8 β1

ibM1

i8+( )

----------------------------------------------=

ΓRi

aD1i

bD1i

ΓRi

aRi

reference generator

aRi

aRi

aRi

ΓRi

bD1i9

aRi ΓR

iaD1

i9+=

Ki γ1

iaM1

i9 δ1ibM1

i9+( ) aR

i ΓRi

Ki

aM1i9 β1

ibM1

i9+( )+=

125


f Kwingferenceer andlem of

aboutthatwould

and toeter.

ffects,ve tont of

ich isdes.

NAsent toNNAfor a

t can beenceowingr is

ed onentplitudet thesedent of

the

2. Thee aor thebeing

ope.The

. (5.4-20)

Note that this ninth measurement can in principle also be used to find the amplitude oi.Although this practice can be considered in the future, this is at present not done for the folloreasons. It is not easy to accurately determine the power of each spectral component of a regenerator. Two methods can be considered: measurement with a spectrum analyzmeasurement with a sampling oscilloscope. Concerning spectrum analyzers, there is a probaccuracy. A typical specification for state-of-the-art RF spectrum analyzers is an error of0.5dB, which is considered to be too high for our application. It is also important to knowspectrum analyzers are calibrated by means of power meters, such that the traceability pathend at a power meter anyway. It is then best to avoid the detour of the spectrum analyzerchoose the shortest path towards the primary standard by directly using a power mConcerning sampling oscilloscope measurements, one would have to take care of jitter enext to the fact that especially the gain of the oscilloscope samplers is slightly sensititemperature variations (this is due to the fact that a diode characteristic is dependetemperature). This sensitivity is much less for the phase distortion of the oscilloscope whprimarily determined by the transmission path from the input connector to the sampling dio

5.4.3 Practice

It is impossible to prove the accuracy of the absolute calibration procedure by doing VNmeasurements of a connectored device. This is due to the fact that it is impossible at preidentify accurate large signal models of connectored devices by any other means than Vmeasurements. This implies that cross-verification between the absolute VNNA calibrationconnectored measurement and another independent technique is simply not possible. Whadone is to check how well the calibration elements (including power meter and refergenerator) can be used as transfer standards. To investigate this in practice the follexperiment is done. A broadband (50GHz bandwidth) 6dB gain travelling wave amplifiechosen as the device-under-test (DUT). A harmonic distortion experiment is then performthis DUT with two significantly different measurement setups (using significantly differhardware). Both setups are absolutely calibrated and the results of both measurements (amand phase of fundamental and harmonics) are compared with each other. By showing tharesults are close to each other it is proven that the remaining systematic errors are indepenthe VNNA hardware used, such that the accuracy of an absolutely calibrated VNNA withprocedure as explained is traceable to the accuracy of the calibration element models.

Both setups used for this experiment are older versions of the setup described in 2.4.main difference is the use of the MTA internal analog-to-digital convertors, which havsignificantly lower overall performance compared to the HP-E1430 modules that are used flatest prototype. As reference generator a 1GHz step-recovery-diode comb generator [20] isused. This reference generator has been characterized by a broadband sampling oscillosc

ϕ Ki( ) ϕ

aRi

γ1i ΓR

i–( )aM1

i9 δ1i ΓR

i β1i

–( )bM1i9

+------------------------------------------------------------------------------

=

126


t setupstics ofg the

hethe

asured,ency

5.4-6.

es 4d twodBm)s the

ation ist toe onemainental

nents.

significant hardware difference between the two setups are the couplers used. For the firs14dB couplers are used and for the second setup 20dB couplers. Note that the characteriboth coupler types are quite different, not only concerning the amplitude but also concerninphase distortion characteristic (this was verified with classical VNA measurements).

The DUT is excited by an input signal with a fundamental frequency of 4GHz. Tamplitude of this signal is swept from -2.5dBm to 15dBm. For each input amplitudeamplitudes and phases of the fundamental component together with 3 harmonics are meboth for the incident and scattered waves at port1 and port2 of the DUT (the frequcomponents measured are 4GHz, 8GHz, 12GHz and 18GHz).

An example of the results of such an experiment can be seen in Figure5.4-3 to Figure

On the x-axis the incident fundamental power is indicated in dBm. On the y-axis one seindependent measurement curves (two measurements performed with setup 1 anmeasurements performed with setup 2). In Figure5.4-3 one finds on the y-axis the power (of the fundamental component (frequency 4GHz) at the DUT output, in Figure5.4-4 one findphase (degrees) of the same component. In Figure5.4-5 and Figure5.4-6 the same informgiven for the third harmonic (frequency 12GHz) at the output. It is important at this poinclarify how the phase of a measured spectral component is defined in the above. Each timmeasures the four voltage waves an unknown delay is present in the digitized time dowaveforms (one unknown delay for the four waveforms). One then uses the incident fundamas the “timing reference” to define the phases of all the measured spectral compoMathematically this is done by applying the delay in (5.4-1) such that (5.4-21) results.

Figure5.4-3 Power of the fundamental at the output versus incident fundamental powerfor the four measurements.

-2.5 0 2.5 5 7.5 10 12.5 150

2

4

6

8

10

12

14

Input power (dBm)

Out

put p

ower

(dB

m)

127


Figure5.4-4 Phase of the fundamental at the output versus incident fundamental powerfor the four measurements.

Figure5.4-5 Power of the third harmonic at the output versus incident fundamentalpower for the four measurements.

-2.5 0 2.5 5 7.5 10 12.5 15

-169

-168.5

-168

-167.5

-167

-166.5

-166

Input power (dBm)

Pha

se (

degr

ees)

0 2.5 5 7.5 10 12.5 15-30

-25

-20

-15

-10

-5

0

5

Input power (dBm)

Out

put p

ower

(dB

m)

128


for thechieved5 andr all 4

he two

1hat the

noise)hisan beThisre. Ifrming

. (5.4-21)

In Figure5.4-3 and Figure5.4-4 can be seen that there is a very close correspondencefundamental output component between the 4 measurements. This result can however be aby only using a power meter, without using a phase reference generator. In Figure5.4-Figure5.4-6 can be seen that the third harmonic component (frequency 12GHz) agrees fomeasurements within 300mdB for the amplitude and within 3° for the phase. Without applying anabsolute calibration by means of a reference generator the phase difference between tsetups would be about 16°.

Although a phase correspondence of 3° between the measured third harmonic with setupand with setup 2 (cf. Figure5.4-6) is considered as a good result, measurements show trepeatability for one setup is significantly higher than this 3° (repeatability typically better than0.5 degrees). This means that the repeatability error (due to connections as well as additivecan not explain the difference of 3° between the two different setups. Later it was shown that tdifference is influenced by inserting attenuators at the MTA inputs, such that the difference cattributed to small nonlinear effect caused by the MTA internal analog-to-digital convertors.small nonlinearity violates the basic linearity assumption of the absolute calibration proceduthese nonlinear effect would not be present, as is the case when using the more perfo

Figure5.4-6 Phase of the third harmonic at the output versus incident fundamentalpower for the four measurements.

0 2.5 5 7.5 10 12.5 15-30

-25

-20

-15

-10

-5

0

Input power (dBm)

Pha

se (

degr

ees)

aD1i

bD1i

aD2i

bD2i

Kie

iϕ aM11 β1

1bM11

+ –

1 β1i

0 0

γ1i δ1

i0 0

0 0 α2i β2

i

0 0 γ2i δ2

i

aM1i

bM1i

aM2i

bM2i

=

129


setups

evicesrated(this isntifieds) areted inlly then putuits

oneeviceilable one testused.lled

1] is

t thiseviceresentz the

ensors

HP-E1430 ADC’s, the author believes that the correspondence between the two differentwould only be limited by the repeatability (a correspondence better than 0.5 degrees).

5.5 Absolute calibration for on wafer measurements

5.5.1 Introduction

The extension of the measurement system from connectored devices to on wafer dwas induced by two things. First, it was necessary in order to confront the absolutely calibmeasurements with large signal models that are derived by independent measurementsdescribed in Chapter 6). As far as known, such large signal transistor models (which are ideby doing a lot of small-signal s-parameter measurements in a large number of bias pointonly available for on wafer devices. Secondly it was noted that many people that are interesVNNA measurements are actually dealing with on wafer measurements. These are typicapeople busy with the large signal modelling of on wafer transistors. These models are theinto simulators to help them in building complete microwave monolithic integrated circ(MMIC’s).

Concerning the VNNA absolute calibration for on wafer measurements [6], there isspecific problem which makes it more complex than the calibration for connectored dmeasurements. This is the fact that there are no power sensors or reference generators avawafer. The way to solve this is to use the principle of reciprocity between the probe tip and thset RF input connector. This implies that a more complex calibration procedure has to beAnother fact which makes the calibration more complex is the use of a so-caline-reflect-match method (LRM), in stead of the classical SOLT procedure. The LRM [9]-[1preferred since this method is known to be more accurate for on wafer measurements.

5.5.2 Theory

The use of reciprocity and outline of the calibration procedure

The basic error model that is used is the same as the one described by (5.4-1):

. (5.5-1)

This implies that the cross talk between the two wafer probes is neglected. Note thaassumption is much more questionable than it is the case for the connectored dmeasurements. It has to be recognized, however, that neglecting this cross talk is the pstate-of-the-art in linear small-signal on wafer measurements. For frequencies below 20GHcross talk is estimated to be at a level lower than -40dB [13]. Because there are no power s

aD1i

bD1i

aD2i

bD2i

Ki

1 β1i

0 0

γ1i δ1

i0 0

0 0 α2i β2

i

0 0 γ2i δ2

i

aM1i

bM1i

aM2i

bM2i

=

130


be tip

he port1were

alsouse

t1 RF

n theed in

then

1 and

is a

.

ionients

olute

or reference generators available on wafer the principle of reciprocity between the port1 pro

and the test set port1 RF input will have to be used to determine Ki. The absolute calibrationelements (power meter sensor and phase reference generator) are therefore connected to tRF input and the measurement results are theoretically transformed as if the elementsconnected to the probe tip. This idea was introduced by Ferreroet al. [12], but it was only appliedto absolute amplitude calibration. As will be explained later, new problems arise when onewants to use this reciprocity principle for phase distortion calibration. In order to practicallythis reciprocity principle one has to introduce the relationship between the waves at the porinput and the waves at corresponding probe tip (for clarity superscript “i” is omitted):

. (5.5-2)

This equation is the so-called t-parameter description of the electrical network betweeprobe tip and the test set RF input. Note that the definition of the waves is as depict

Figure2.2-1, such that is a voltage wave incident to the RF input. Reciprocity is

mathematically expressed as:

. (5.5-3)

This conjecture can be proven as follows.If one uses the scattering parameter description of the electrical network between port

RF-input 1 one can write:

. (5.5-4)

Note that is a wave incident to the network, reflected from the DUT, and that

wave scattered by the network, incident to the DUT. Reciprocity can be expressed as

This can be written under the form:

. (5.5-5)

Using (5.5-2) together with (5.5-5) one gets:

, (5.5-6)

which is equivalent to

. (5.5-7)

The outline of the whole calibration procedure is then the following. An LRM calibratprocedure is first executed in order to determine the seven relative calibration coeffic

, , , , , and . Next the RF input port1 is treated as a test port and an abs

aG1

bG1

t11 t12

t21 t22

aD1

bD1

=

aG1

t11t22 t12t21– 1=

aD1

bG1

s11 s12

s21 s22

bD1

aG1

=

bD1 aD1

s12 s21=

aD1

aG1--------

bD1 0=

bG1

bD1---------

aG1 0=

=

1t11------ t22

t12t21

t11-------------–=

1 t11t22 t12t21–=

β1i γ1

i δ1i α2

i β2i γ2

i δ2i

131


n theinput

)

)

)

RMport1e two

in a

ts are

1 RF

calibration procedure analog to the one described in 5.4 is carried out. This results iknowledge of the matrix which describes the relationship between the waves at the port1 RFand the measured waves at the same port:

. (5.5-8)

One already had (5.5-1):

, with Ki unknown and to be determined. (5.5-9

Eliminating the measured quantities in (5.5-8) and (5.5-9) results in:

, such that (looking at (5.5-2)): (5.5-10

. (5.5-11)

Expressing (5.5-3) results in:

, which can be written as: (5.5-12

, with (5.5-13)

. (5.5-14)

All quantities at the right hand side of (5.5-13) are known after execution of the Lcalibration at the level of the two probe tips and the absolute calibration at the level of theRF input connector. This implies that these measurements are sufficient in order to provid

solutions for Ki with an opposite sign. How one can know what sign to choose is discussedlater paragraph.

Detailed description of the measurements and the calculations

The same wave notations as defined in 5.4 will be used. A total of 10 measuremenperformed. These measurements are described as follows.1. Both probes are lifted from the calibration substrate and power is incident to the port

input.

aG1i

bG1i

Li 1 λi

µi νi

aM1i

bM1i

=

aD1i

bD1i

Ki 1 β1

i

γ1i δ1

i

aM1i

bM1i

=

aG1i

bG1i

Li 1 λi

µi νiK

i 1 β1i

γ1i δ1

i

1–

aD1i

bD1i

=

t11 t12

t21 t22

Li 1 λi

µi νiK

i 1 β1i

γ1i δ1

i

1–

=

det Li 1 λi

µi νiK

i 1 β1i

γ1i δ1

i

1–

1=

Ki

Qi±=

Qi

Li( )2 νi λiµi

–

δ1i β1

i γ1i

–----------------------

=

132


F

r is

put is

put is

put is

put is

ference

ticalt there

i

ision

quen-ctual

n line

the

sured

d ,

2. Probes untouched and power is incident to the port2 RF input.3. Both probes are connected to an on wafer 50Ω resistor and power is incident to the port1 R

input.4. Probes untouched and power is incident to the port2 RF input.5. Both probes are connected to a 50Ω characteristic impedance transmission line and powe

incident to the port1 RF input while the port2 RF input is connected to a matched load.6. Probes are untouched and power is incident to the port2 RF input, while the port1 RF in

connected to the connectored calibration load.7. Probes are untouched and power is incident to the port2 RF input, while the port1 RF in

connected to the connectored calibration open.8. Probes are untouched and power is incident to the port2 RF input, while the port1 RF in

connected to the connectored calibration short.9. Probes are untouched and power is incident to the port2 RF input, while the port1 RF in

connected to the connectored power meter sensor.10. Probes are untouched, the port1 RF input is connected to the connectored phase re

generator and the port2 RF input is connected to a matched termination.

The assumptions used for the LRM part of the calibration (measurements 1 to 6) are:1. The reflection coefficients from the two probes when lifted from the substrate are iden

(close to an open) and have an amplitude of one (this corresponds to the assumption thaare no losses, this is verified in [11]).

2. The on wafer loads can be modelled by a perfect 50Ω resistor in series with an a prior

unknown inductor, with inductance . The resistive parts are laser trimmed at a prec

of 0.3% and are so small that they can still be considered lumped constant resistors at frecies as high as 40GHz. The inductor models the transition from the probe tip to the aresistor. This means that the impedance of the load can be written as:

. (5.5-15)

3. The 50Ω characteristic impedance transmission line that is used is a perfect transmissiowith no losses and with a delay of 1ps. For ease of notation the s12parameter of this transmis-sion line at frequency “i” will be noted , it is defined as:

, (5.5-16)with equal to the delay of the transmission line (1ps in this case) and with equal tofundamental frequency of the used frequency grid.

The assumptions for the absolute calibration of the port1 RF input versus the meawaves (measurements 6 to 10) are the same as those used in 5.4.0:

1. The reflection coefficients of the load, open and short are perfectly known. They are note

and respectively.

Lload

Zloadi

50 jXloadi

+ 50 j2πLloadif 0+= =

Di

Di

ej2πτf 0i–

=τ f 0

Γ1i

Γ2i Γ3

i

133


nt) alle LRMents

sic

rticle.load

tain

own

oss in

g the

then

ith ais, in[11]:

2. The power meter sensor can be modelled as in Figure5.4-1.3. The reference generator can be modelled as illustrated in Figure5.4-2.

Except for the tenth measurement (the phase reference generator measurememeasurements are done one frequency component at the time. First the calculations of thcalibration will be carried out in order to determine the seven relative calibration coeffici

, , , , , and . This is explained in what follows. Note that only the ba

principles are explained in [11], no details concerning the implementation are given in the aThe main problem for the implementation is that one does not know the value of the

inductance a priori. It is however possible to calculate out of the measurements the value of

as a function of (the algorithm to do this is given later). With corresponds a cer

complex admittance which can be written as:

, (5.5-17)

with equal to the characteristic impedance, in our case 50Ω.

This implies that it is possible to write as a function of the measurements (kn

quantities) and of the unknown quantity . Because of the assumption that there is no l

the open probe tip, it will then be possible to estimate, at each frequency, , by solvin

equation:

. (5.5-18)

The secant method is used in order to find these values . A least-squares fit is

used in order to estimate the value of out of the calculated values :

, (5.5-19)

with N equal to the maximum frequency index. This corresponds to least-squares fit wuniform weighting of the residual. This is preferred since the quality of the measurementsfirst order, the same at all frequencies. Note that a different estimation algorithm is used in

β1i γ1

i δ1i α2

i β2i γ2

i δ2i

Γopeni

Xloadi Γopen

i

Yopeni

Yopeni

Gopeni

jX openi

+ Zc1–1 Γopen

i–

1 Γopeni

+----------------------= =

Zc

Gopeni

Xloadi

Xloadi

Yopeni

Xloadi( ) 0=

Xloadi

L load Xloadi

L loadest 1

2πf 0-----------

iX loadi

i 1=

N

∑

i 2

i 1=

N

∑------------------------=

134


o 0 ifwhyents.ation

ith

the

nown

t can

tities

sixththese

d. Thethe

ves by

, (5.5-20)

with W(i) equal to 1 if the corresponding frequency is above or equal to 10GHz, and equal tthe corresponding frequency is below 10GHz. It is unclear from the article, however,precisely this form is chosen by the authors and it was as such not used for our measurem

In what follows is explained how one can calculate the seven unknown relative calibr

coefficients and as a function of the measurements and of (w

). Note that this algorithm is used as well during the search for by

secant method (explained above) as afterwards for the final calculation of the seven unkcalibration coefficients. The derivation of the algorithm is as follows.

For the first two measurements one can write:

, and (5.5-21)

. (5.5-22)

The assumption is that this reflection coefficient is identical for both probes such that ibe eliminated, resulting in:

. (5.5-23)

One then eliminates the DUT quantities by substituting them with the measured quanusing (5.5-1):

. (5.5-24)

This represents one quadratic equation in the seven unknowns. The third to themeasurement can then be used in order to find six linear independent equations inunknowns, resulting in a total of seven unknowns and seven equations, which can be solvesix linear equations are found by writing down the equation describing the model ofcomponent present at the probe tip and by substituting the DUT waves by the measured wausing (5.5-1). The result, after some rewriting:

Lloadest 1

2πf 0-----------

W i( )Xloadi

i 1=

N

∑

W i( )ii 1=

N

∑-----------------------------------=

Γopeni

Xloadi

Xloadi

2πf 0iL load= Xloadi

bD1i1 Γopen

iaD1

i1=

bD2i2 Γopen

iaD2

i2=

aD1i1

bD2i2

aD2i2

bD1i1

=

aM1i1 β1

ibM1

i1+( ) γ2

iaM2

i2 δ2ibM2

i2+( ) α2

iaM2

i2 β2ibM2

i2+( ) γ1

iaM1

i1 δ1ibM1

i1+( )=

135


in

, (5.5-25)

with . (5.5-26)

The inverse of (5.5-25) can then be used in order to substitute , , , , and

(5.5-24) by 6 linear functions of :

, with (5.5-27)

and (5.5-28)

ΓLibM1

i3bM1

i3– 0 0 0 0

0 0 ΓLiaM2

i4 ΓLibM2

i4aM2

i4– bM2

i4–

D–ibM1

i50 0 0 aM2

i5bM2

i5

0 bM1i5

D–iaM2

i5D–

ibM2

i50 0

D–ibM1

i60 0 0 aM2

i6bM2

i6

0 bM1i6

D–iaM2

i6D–

ibM2

i60 0

β1i

δ1i

α2i

β2i

γ2i

δ2i

ΓLiaM1

i3–

0

DiaM1

i5

0

DiaM1

i6

0

γ1i

aM1i3

0

0

aM1i5

–

0

aM1i6

–

+=

ΓLi Xload

i

Xloadi

j2Zc–------------------------------=

β1i δ1

i α2i β2

i γ2i δ2

i

γ1i

β1i

δ1i

α2i

β2i

γ2i

δ2i

u1i

v1i

w1i

x1i

y1i

z1i

u2i

v2i

w2i

x2i

y2i

z2i

γ1i

+=

u1i

v1i

w1i

x1i

y1i

z1i

ΓLibM1

i3bM1

i3– 0 0 0 0

0 0 ΓLiaM2

i4 ΓLibM2

i4aM2

i4– bM2

i4–

D–ibM1

i50 0 0 aM2

i5bM2

i5

0 bM1i5

D–iaM2

i5D–

ibM2

i50 0

D–ibM1

i60 0 0 aM2

i6bM2

i6

0 bM1i6

D–iaM2

i6D–

ibM2

i60 0

1–

ΓLiaM1

i3–

0

DiaM1

i5

0

DiaM1

i6

0

=

136


ions.

nts

The

re the

on

idual

the

. (5.5-29)

The result is a quadratic equation in the variable , resulting in two possible solut

Applying (5.5-27) finally results in two solutions for the seven relative calibration coefficie

, , , , , and . That solution is chosen which corresponds to a value for

which is closest to the complex value 1 (reflection coefficient of an ideal open). Note that

. (5.5-30)

Next, measurements 6 to 10 will be used in order to determine , , and .

calculations are identical to those used for (5.4-8), (5.4-17) and (5.4-20). The results afollowing:

, (5.5-31)

and (5.5-32)

. (5.5-33)

Applying (5.5-13) will then result in the knowledge of the last unknown calibrati

coefficient Ki.

Determining the sign of the absolute calibration coefficient

As previously mentioned (5.5-13) reveals that there are 2 solutions for each indiv

component of the vector Ki. If there are N spectral components this represents a total of 2 to

u2i

v2i

w2i

x2i

y2i

z2i

ΓLibM1

i3bM1

i3– 0 0 0 0

0 0 ΓLiaM2

i4 ΓLibM2

i4aM2

i4– bM2

i4–

D–ibM1

i50 0 0 aM2

i5bM2

i5

0 bM1i5

D–iaM2

i5D–

ibM2

i50 0

D–ibM1

i60 0 0 aM2

i6bM2

i6

0 bM1i6

D–iaM2

i6D–

ibM2

i60 0

1–

aM1i3

0

0

aM1i5

–

0

aM1i6

–

=

γ1i

β1i γ1

i δ1i α2

i β2i γ2

i δ2i Γopen

i

Γopeni bD1

i1

aD1i1

---------γ1

iaM1

i1δ1

i1ibM1

i1+

aM1i1 β1

ibM1

i1+

---------------------------------------= =

Li λi µi νi

λ1i

µ1i

ν1i

bM1i6 Γ1

iaM1

i6– Γ1

ibM1

i6–

bM1i7 Γ2

iaM1

i7– Γ2

ibM1

i7–

bM1i8 Γ3

iaM1

i8– Γ3

ibM1

i8–

1–

aM1i6

–

aM1i7

–

aM1i8

–

=

Li aS

i

TSi µ1

iaM1

i9ν1

ibM1

i9+( )

----------------------------------------------------=

ϕ Li( ) ϕ

aRi

µ1i ΓR

i–( )aM1

i10 ν1i ΓR

i λ1i

–( )bM1i10

+-------------------------------------------------------------------------------

=

137


f the

er

lf of

irectplings lessdelay

r to

asewas

or the

nminedport1e timeneratorice for

tal

inplies

ar

e

r casepping

.

Nth power possibilities. To find the right solution, one has to look at the physical meaning o

complex vector Ki. As shown by (5.5-1) Ki equals the sampled (N points) complex transf

function describing the linear relation between and . In a first approximation, on beha

a linear phase shift, this complex transfer function is proportional to the inverse of the dcoupling characteristic of the coupler. We know that the phase distortion of this direct coucharacteristic (when delay has been removed!) is a smooth function, which typically variethan 90 degrees for 1GHz frequency difference. The first thing to do then is to remove the

from Qi, such that the phase of Qi can be unwrapped before the square root is taken in orde

find Ki’s which have a “consistent phase” relationship (with this is meant that the phcharacteristic is smooth). A special robust algorithm (explained later in this paragraph)developed in order to remove this delay. The final result are then only 2 possible values f

vector Ki (where we had 2 to the Nth power possibilities at the start): two solutions with aopposite sign for all components. Which of these 2 solutions to choose can easily be deterby looking at the time domain representation of the calibrated incident voltage wave at theprobe tip when the reference generator is connected to the port1 RF input. The polarity of thdomain pulse appearing at port 1 should be the same as the polarity of the reference gepulse as measured by the sampling oscilloscope. If the opposite is true, the wrong sign cho

Ki was made.

Experimental results

The algorithm to find Ki is described in what follows, together with some experimen

results. An example of the phase of Qi before adding any linear phase shift is shownFigure5.5-1. Note that the fundamental frequency for this experiment is 1GHz. First, one ap

a linear phase shift (corresponding to a delay in the time domain) to Qi such that the “length” of

the curve described by the points Qi in the complex plane is minimized. The amount of linephase shift to apply is then found by minimizing the function L(θ), which is defined as:

. (5.5-34)

For our experiment, this function is illustrated in Figure5.5-2. The value ofθ for which L(θ)is minimum is calledθmin. In the case of Figure5.5-2θmin equals about 0.602 radians. When th

corresponding linear phase shift is applied, the phase can easily be unwrapped (in ouunwrapping is even unneeded), the resulting phase is illustrated by Figure5.5-3.The unwra

is necessary in order to find the correct phase characteristic. The value of Ki is then given by:

. (5.5-35)

Measured values of the phase and amplitude of Ki are given in Figure5.5-4 and Figure5.5-5

aM1i

aD1i

L θ( ) Qiejθi Q

i 1–ejθ i 1–( )–

i 2=

N

∑=

Ki

Qie

jϕunwrap Q

ie

jθmini

2--------------------------------------------------------

=

138


at

gat the

aset the

ed

The physical interpretation of Ki is the following. Equation (5.5-1) together with (2.4-1) show th

the amplitude characteristic of Ki is primarily determined by the inverse of the coupler couplincharacteristic and the inverse of the downconvertor conversion loss (about 5dB), and th

phase characteristic of Ki is primarily determined by the phase distortion of the coupler (the phdistortion of the sampler being very small). Probably the most remarkable fact abou

amplitude characteristic of Ki is the ripple with a repetition rate of 2GHz. This can be explain

Figure5.5-1 Phase of Qi without applying any linear phase shift.

Figure5.5-2

Frequency (GHz)

Pha

se (

degr

ees)

3 6 9 12 15 18-180

-120

-60

0

60

120

180

θ (radians)

L (θ)

0 1 2 3 4 5 60

500

1000

1500

2000

2500

139


VNAwith ats in

Kept forrtion

by looking at the inverse of the coupler coupling characteristic as measured by a classical(cf. Figure5.5-6). The coupler that is used is based upon an equiripple design (1dB ripples)ripple repetition rate of 2GHz. All the maxima and minima occur at a 1GHz grid, which resul

the strange amplitude behavior of Ki (cf. Figure5.5-4). Looking at the phase characteristic ofi

(note that an arbitrary delay is always present) one notes a rather smooth characteristic, exca frequency of 1GHz. This can be explained by looking at the inverse of the phase disto

Figure5.5-3 Phase of Qi, with optimal linear phase shift applied.

Figure5.5-4 Amplitude of Ki.

3 6 9 12 15 18-90

-80

-70

-60

-50

-40

Pha

se (

degr

ees)

Frequency (GHz)

3 6 9 12 15 1816

17

18

19

20

21

Frequency (GHz)

Am

plitu

de (

dB)

140


elow

f Kand

eorem

characteristic of the coupler (compensated for delay), which is depicted in Figure5.5-7. B

2GHz this phase characteristic changes rapidly, explaining the strange phase behavior oi at1GHz. As one notes, the phase behavior is also “rippling” with a repetition rate of 2GHz,

with a ripple amplitude of about 7°. In the phase of Ki this ripple is not present because this timthe 1GHz grid corresponds to the ripple zero’s. Note that this is a good illustration of the the

Figure5.5-5 Phase of Ki.

Figure5.5-6 Inverse of the coupler amplitude characteristic.: amplitude sampled at multiples of 1GHz

3 6 9 12 15 18-45

-40

-35

-30

-25

-20

Pha

se (

deg)

Frequency (GHz)

3 6 9 12 15 18

13.5

14

14.5

15

15.5

16

Frequency (GHz)

Am

plitu

de (

dB)

141


ponseidaleristicpler

anosen,theblems,

f thecablerdwareable

Thiset-upors arethis

. Thisle. Thez to

stating that the phase and the natural logarithm of the amplitude of a complex frequency resfunction form a Hilbert transform pair [14]. This theorem implies that a hypothetical cosinusologarithmic amplitude characteristic (1dB ripple) corresponds to a sinusoidal phase charact(6.6° ripple) with the same repetition rate, which is very well approximated by our coucoupling characteristic.

5.5.3 Hardware and software implementation

Making the on wafer probing calibration theory work in practice was more difficult thmight have appeared on a first glance. As software environment Mathematica™ [15] was chboth to control all instruments as well as to do the calculations required. Duringimplementation many unforeseen problems were discovered and solved. Some of these protogether with their solutions, are mentioned in what follows.

One of the major problems that originally occurred was a serious degradation osignal-to-noise ratio for the higher frequencies. This problem was primarily caused bylosses increasing and available synthesizer power decreasing at higher frequencies. A haand software solution was then found such that, for all frequencies, the maximum allowpower was send to the DUT, where the maximum limit is set by the downconvertor linearity.results in a calibration with an optimal signal-to-noise ratio. For this purpose the source schosen for measurements 1 to 6 is the one depicted in Figure5.5-8. Note that the attenuatactually used as “switches”. With this is meant that for a port1 excitation the attenuator inpath is set to 0dB and the other attenuator to 110dB, and the opposite for a port2 excitationset-up is used in stead of using switches because the attenuators are more readily availabamplifiers used are of the HP 83006A type, they have a bandwidth ranging from 10MH

Figure5.5-7 Inverse of the coupler phase characteristic (compensated for delay).: phase sampled at multiples of 1GHz

0 3 6 9 12 15 18-50

-45

-40

-35

-30

-25

-20

-15

-10

Pha

se (

deg)

Frequency (GHz)

142


tualh theport1uired

powerThisoard,datar bothtable ismetry

powerirectly

26GHz, with a gain of 20dB and a maximum output power of 16dBm. Before the accalibration takes place a power check is performed with the probes lifted in the air and witsynthesizer generating a power of -10dBm. At all frequencies, the power is once send to theprobe tip and once to the port2 probe tip. The amplitude of the incident voltage wave as acqby the data acquisition (raw data) is each time measured, and it is calculated what thesetting of the source should be in order to have the maximum allowable value of 60mV.value corresponds to about -10dBm of power present at the input of the downconvertor bwhich corresponds to the maximum input power allowable for a linear operation of theacquisition. The calculated optimal synthesizer power settings for each frequency and foports are stored in a table and are used during the measurements 1 to 6. Part of such agiven in Table5.5-1. The difference between the two channels can be explained by the asym

of the cables, the amplifiers and the couplers for both ports. For the tenth measurement (themeasurement) the source set-up of Figure5.5-8 is not used and the synthesizer is d

Figure5.5-8 Source set-up for on wafer calibration.

Table5.5-1 Optimal power of calibration source at both ports.

Frequency(GHz)

Pport1(dBm)

Pport2(dBm)

1 -14.0 -12.6

5 -7.47 -6.00

10 -5.78 -3.18

15 -3.01 0.51

18 0.61 4.10

towards port 1 towards port 2

20dB 20dB

143


r they theeteraboutonics

d withvel is

nentlse likeisitionerence6.0) is

Whileade towhileLRM

of took theto doionss (thiswrong

twole pitfalln anddone aentsitors,

connected to the port2 RF input, without the use of an amplifier. This is the optimal set-up fopower experiment since it appears that the maximum power is now no longer limited bdownconvertor linearity but by the linearity of the signal source! Indeed, the power mmeasures the power of both fundamental and harmonics. If one wants to limit the error to35mdB, the harmonics need to be at a level as low as -55dBc. Because of the harmgenerated by the amplifiers, they are omitted for this measurement (check was performespectrum analyzer). Taking into account the harmonic distortion of the source, the signal leset to 7dBm at 1GHz, 9dBm at 2GHz and 15dBm at higher frequencies.

Another problem is the relatively low signal-to-noise ratio of each spectral compoduring the reference generator measurement. This is caused by the fact that one has a pusignal and one needs to limit the pulse amplitude in order to assure that the data acqubehaves linear. Because of the fact that the data acquisition is not synchronized with the refgenerator, regular averaging can not be used and logarithmic spectral averaging (cf. 3.applied (64 averages).

In general great care has to be taken and one needs to think about a lot of things.implementing and testing the calibration procedure several warning messages were mappear on the computer screen. One may not forget for example to turn off the biascalibrating. If bias is present at both port1 and port 2 while connecting both probes to theline standard one has the risk of damaging the probes and the calibration element becausemuch current. Another important thing not to forget before one starts measuring is to checplanarity of the wafer probes, which are of the ground-signal-ground (GSG) type. The waythis is explained in [16]. A bad planarity will make that only one of the two ground connectwill make contact to the calibration elements or the DUT, resulting in useless measurementis illustrated in Figure5.5-9). Nevertheless the fact that these measurements are completely

(one has 100Ω for the matched load calibration element for instance since only one of theparallel resistors is connected), they are as repeatable as regular measurements. A possibis that, after calibration, one measures the calibration elements a second time for verificatiothat one finds a perfect load and perfect transmission lines, such that one believes to havegood calibration. The only way to detect this kind of error is to measure verification elemwhich are significantly different from the calibration elements, such as coplanar capac

Figure5.5-9 Good and bad planarity of GSG wafer probes.

substrate substrate

wafer probe (front view) wafer probe (front view)

G S G G S G

GOOD PLANARITY BAD PLANARITY

144


f theused,ration

ed. Thephasenose”

is then thelity ofprobe

rm theablethree

ss thant abouts are

the

inductors or long transmission lines.

5.5.4 Quality of the calibration procedure

An important topic is the quality of the calibration procedure. Concerning accuracy orelative calibration one can say that existing state-of-the-art calibration procedures aretraceable to national standards labs. The accuracy of the amplitude of the absolute calibcoefficients is also traceable to national standards labs, since a calibrated power meter is uspower meter that is used has a specified accuracy of plus or minus 20mdB [17]. Thecharacteristic of the reference generator is traceable to the accuracy of a “nose-to-calibration procedure (Chapter 4), which has meaningful error bounds (4.5).

Next to the accuracy, the repeatability of the calibration procedure is as important, asquestion how long the calibration coefficients are valid. The repeatability of the procedure, oshort term, is determined by the data acquisition signal-to-noise ratio and by the repeatabithe contact between wafer probes and calibration elements. Things like dirt particles on thecontacts and wafer probe operator skills can have a significant influence. On the longer tevalidity of the calibration coefficients can be deteriorated by thermal effects and cmovements. In order to have an idea of the short term and longer term repeatabilitycalibration procedures were done: a second immediately after the first (it takes somewhat lehalf an hour to calibrate the system for 18 frequency components), and a third measuremen20 hours later. In what follows the relative calibration matrices of these three procedure

called , and respectively, and the absolute calibration coefficients , and

(with the superscript i still referring to the frequency index). The differences betweencalibration matrices and coefficients are defined as:

. (5.5-36)

One will also introduce the notations for the voltage waves at the device ports and

for the measured quantities:

and . (5.5-37)

M1i

M2i

M3i

K1i

K2i

K3i

∆M21i

M2i

M1i

–=

∆M31i

M3i

M1i

–=

∆K21i

K2i

K1i

–=

∆K31i

K3i

K1i

–=

vDi

vMi

vDi

aD1i

bD1i

aD2i

bD2i

= vMi

aM1i

bM1i

aM2i

bM2i

=

145


onefect.5-1)

nts

n in

ments

lows toerror

ion

In order to properly quantify and interpret the repeatability of the calibration procedure,will not merely look at the calibration coefficients themselves, but one will look for the total efthat these calibration coefficients have on the final data. In order to do this one will rewrite (5as:

, (5.5-38)

such that

, (5.5-39)

and finally

, (5.5-40)

with the “calibration error coefficients” (not to confuse with the calibration coefficiethemselves) defined as

and . (5.5-41)

An analog definition is used for and . Some experimental values are give

Table5.5-2 to Table5.5-4. Since one is only comparing between three different measure

some care is needed for the quantitative interpretation of this data. Nevertheless, the data aldraw some important conclusions. There is no significant difference between the calibrationcoefficients with subscript “21” and with subscript “31”. This implies that the calibrat

Table5.5-2 Experimental values for (dB).

1GHz 5GHz 10GHz 18GHz

-78 -67 -60 -52

-54 -45 -45 -45

-50 -50 -46 -36

-49 -60 -44 -39

-49 -53 -50 -46

-49 -54 -43 -49

-60 -49 -46 -47

-52 -51 -49 -41

vDi

K1iM1

ivM

i=

∆vDi( )21 ∆K21

i( )M1ivM

iK1

i ∆M21i( )vM

i+=

∆vDi( )21 ∆A21

i( )vDi

∆R21i( )vD

i+=

∆A21i ∆K21

i

K1i

-------------= ∆R21i ∆M21

i( ) M1i( ) 1–

=

∆A31i ∆R31

i

∆R21

∆R21 1 1,[ ]

∆R21 1 2,[ ]

∆R21 2 1,[ ]

∆R21 2 2,[ ]

∆R21 3 3,[ ]

∆R21 3 4,[ ]

∆R21 4 3,[ ]

∆R21 4 4,[ ]

146


that itoverallhis as

he

hase

due

coefficients can be assumed to remain stable for a period of at least 20 hours, which meansis save to do one calibration procedure a day. As a rough estimate one can say that thelevel is about -50dB for the lower frequencies, and -40dB for the higher frequencies, and t

well for the “absolute calibration error” ( ) as for the “relative calibration error” ( ). T

calibration errors in dB (noted “e”) can be converted into an amplitude error ( ) and a p

error ( ) by applying (5.5-42) and (5.5-43). Some values can be found in Table5.5-5:

(5.5-42)

(5.5-43)

To conclude one can give the following indicative numbers: amplitude error on the data

Table5.5-3 Experimental values for (dB).

1GHz 5GHz 10GHz 18GHz

-72 -69 -59 -54

-48 -47 -44 -47

-49 -46 -44 -37

-45 -46 -43 -63

-38 -54 -43 -37

-56 -48 -46 -41

-54 -57 -47 -46

-46 -58 -43 -43

Table5.5-4 Experimental values for and (dB).

Freq.

1GHz -44 -52

2GHz -49 -47

5GHz -58 -53

10GHz -67 -49

18GHz -43 -41

∆R31

∆R31 1 1,[ ]

∆R31 1 2,[ ]

∆R31 2 1,[ ]

∆R31 2 2,[ ]

∆R31 3 3,[ ]

∆R31 3 4,[ ]

∆R31 4 3,[ ]

∆R31 4 4,[ ]

∆A21 ∆A31

∆A21 ∆A31

∆A ∆R

Aerror

ϕerror

Aerror 20log10 1 10e 20/+( )=

ϕerror ArcTan 10e 20/( )=

147


ciesr the

rationd the

tant forshiped byed into

h by

iven

t the

nciesn of

hat the

to the repeatability of the calibration is typically less than 40mdB for the lower frequen(1GHz - 9GHz) and less than 120mdB for the higher frequencies (10GHz to 18GHz), fophase error one typically has an error less than 0.3° for the lower frequencies and 0.9° for thehigher frequencies.

5.5.5 Derivation of test-set characteristics

Some interesting features of the port1 side of the test-set can be derived from the calibcoefficients. These features are the insertion loss from the RF input to the probe tip anmismatch present at the probe tip and at the RF input. These characteristics can be imporexperiment design, or for verification of the test set. Using (5.5-10) one finds the relationbetween the port1 RF input voltage waves and the port1 probe tip voltage waves, describmeans of so-called t-parameters. For ease of interpretation this relationship can be convertan s-parameter description [18], resulting in:

, (5.5-44)

where and do contain an arbitrary delay (because of the presence of Ki in the t-parameter

description). The RF input mismatch will then be characterized by , the probe tip mismatc

and the insertion loss from RF input to probe tip by . Values for the mismatches are g

in Table5.5-6, while the insertion loss is illustrated by Figure5.5-10. One sees tha

mismatches of probe tip and RF input port are at a level of about -30dB for the lower frequeand -20dB for the higher frequencies. This mismatch is, however, not a smooth functiofrequency. This can be explained by complex standing wave patterns caused by the fact t

Table5.5-5 Some values of (dB) and (degrees).

e (dB) -60 -55 -50 -45 -40 -35

(mdB) 9 15 27 49 86 153

(deg) 0.06 0.10 0.18 0.32 0.57 1.02

Table5.5-6 RF input and probe tip mismatch (dB).

Freq. 1GHz 3GHz 6GHz 9GHz 12GHz 15GHz 18GHz

RF input -28 -37 -31 -29 -17 -23 -19

probe tip -27 -37 -27 -33 -21 -23 -18

Aerror ϕerror

Aerror

ϕerror

bG1i

aD1i

s11i

s12i

s21i

s22i

aG1i

bD1i

=

s12i

s21i

s11i

s22i

s12i

148

5.6 The reference generator

(about. Ascy by:

odel

tion

e skin

t is theombourcy ofbout

physical length of the test set (about 2m) is much longer than the wavelength of the signals20cm for 1GHz). The insertion loss is primarily due to skin effect losses in the cablesdescribed in [19], coaxial cable skin effect losses can be described as a function of frequen

. (5.5-45)

Some values of the difference between the actual insertion loss and the skin effect m

with α equal to 43.4 are given in Table5.5-7. The small values (maximum devia

about 200mdB) reveals that the actual insertion loss can be approximated very well by theffect model of (5.5-45).


5.6.1 Construction of the reference generator

The reference generator is constructed as depicted in Figure5.6-1. The key componen“step recovery diode” (SRD) module [20]. This module can be used as a so-called “cgenerator”. When fed at the input with a 0.5W sinusoidal signal of a certain frequency (incase 1GHz) a narrow repetitive pulse signal will appear at the output. The repetition frequenthe pulse will be equal to the frequency of the input signal. The amplitude of the pulse is a

Figure5.5-10 Insertion Loss from RF input to probe tip.

Table5.5-7 Difference between skin effect loss model and insertion loss (mdB).


Diff. 215 138 -110 -95 -24 -37 212

3 6 9 12 15 18-7

-6

-5

-4

-3

-2

-1

Frequency (GHz)

Inse

rtio

n Lo

ss (

dB)

lossdB f( ) α f=

µdB Hz⁄

149


ainental

zer isovidesgoest the

cableVNNArectlyork”,

cyncyll belevel,plingality

ffectr willown)SRD

d. Thesents a

ight

highly

at the

-12V, and it has a FWHM (full width half maximum) of about 150ps. In the frequency domthis signal represents a multiharmonic signal existing out of the superposition of a fundamfrequency component of 1GHz and significant harmonics up to about 20GHz. A synthesiused in order to generate the 1GHz signal. This signal goes into a powersplitter, one part prthe trigger signal that will be fed to the sampling oscilloscope trigger input, the other partinto a power amplifier which boosts the signal power to a level such that 0.5W is available aend of the long cable connecting the amplifier output and the input of the SRD module. Thisallows a flexible use of the reference generator, such that it can easily be connected to thetest set as well as to the sampling oscilloscope input. The output of the SRD module is diconnected to a broadband differentiating network, this is a so-called “impulse forming netw

model 5208 of PicoSecond Pulse Labs®. The purpose of this component is to flatten the frequenspectrum, which will result in a better signal-to-noise ratio (SNR) for the critical higher frequecomponents while maintaining an acceptable SNR for the lowest frequencies (this wiexplained in more detail in 5.6.0). A 20dB attenuator will reduce the signal to an acceptablesuch that the signal will not cause nonlinear effects when characterized by the samoscilloscope or when measured by the VNNA. An attenuator is chosen with high qu

APC-3.5® connectors in order to provide a good connector repeatability. An important side eof the attenuator is that it provides so-called “padding”. With this is meant that the attenuatoprovide the reference generator with a good output match and will prevent any (unknmismatches or nonlinear effects which might otherwise occur in the highly nonlinear stepdiode.

5.6.2 Characterization of the reference generator

For a good calibration the reference generator needs to be accurately characterizemodel that is used for the reference generator is depicted in Figure5.4-2. This model repre

perfectly linear generator, fully characterized by the parameters and . At first sight it m

look a bit strange to use this linear model since the actual reference generator contains anonlinear diode. The justification of the model is based on the value of the s21 parameter of the

circuit which is between the reference generator output port and the actual diode (note th

Figure5.6-1 Construction of the reference generator.

1GHz

trigger signal

amplifier(0.5W)

1.5m SMA cable

SRD module

differentiator

20dB att.

APC-3.5 conn. (m)

aRi ΓR

i

150


e

nsitionues forltage

neratorause a

theiodewill

nuatorationthe

erenceey cannents).

and

brated

ower

are

suring

use.

evealsference

30dB.ctual

mpling

n fact

circuit is reciprocal). A good estimate for this s21 parameter is found by measuring th

s-parameters of the differentiator cascaded with the 20dB attenuator, such that only the trabetween the diode terminals and the SRD module output connector is neglected. Some vals21 are given in Table5.6-1. Suppose now that the 18GHz frequency component of the vo

wave coming out of the attenuator has a value of -30dBm. Whenever used the reference geis connected to the test set RF input. Because of the small mismatch (Table5.5-6) this will creflected voltage wave of about -50dBm. This reflected voltage wave will travel throughattenuator and the differentiator and will hit the diode with a level of about -74dBm. The dwill behave as an harmonic mixer (the local oscillator signal being the SRD input signal) andgenerate a whole bunch of intermodulation products which are scattered towards the atteinput (all intermodulation products are situated at the 1GHz frequency grid). The intermodulproduct voltage waves will have a power smaller than -74dBm. They will travel towardsreference generator output after being multiplied with the appropriate s21 parameter, such that the

highest possible level of these parasitic intermodulation products as they appear at the refgenerator output is lower than -98dBm. This means that, if nonlinear effects are present, thbe neglected for our measurements (being at least 68dB down relative to the main compoThis justifies the use of a linear model.

The question then is how to identify the model parameters and . Both phase

amplitude of are determined by connecting the reference generator to a classical cali

vector network analyzer, while there is no power incident to the SRD module (if there is p

incident the network analyzer can not properly measure). The measured values of

illustrated in Figure5.6-2 and Figure5.6-3. One might ask what error one can make by mea

with no power incident on the SRD module, while power is incident during the actual

The same reasoning as above for justifying the linearity of the reference generator model rthat this error can be neglected. Figure5.6-2 even shows that one can almost assume the regenerator to be perfectly matched. Indeed, the worst output match is at a level of about -Since the RF input match for this frequency is at about -20dB, this would result in an ameasurement error at a level as low as -50dB.

The reference generator parameters are characterized by means of a broadband sa

oscilloscope which has been characterized by a “nose-to-nose” calibration procedure. I

Table5.6-1 Measured s21 of the differentiator and attenuator combination.


s21 -44dB -36dB -30dB -28dB -26dB -25dB -24dB

aRi ΓR

i

ΓRi

ΓRi

ΓRi

aRi

151


the

y haveby the5.6-5

that

(5.5-33) reveals that only the phase of needs to be determined. This facilitates

measurement since one does not have to take into account jitter effects (cf. 3.5), which onlan influence on the amplitude measurement. The time domain waveform as measuredsampling oscilloscope is depicted in Figure5.6-4. The spectral contents is given in Figure(amplitude) and Figure5.6-6 (phase with delay compensation). Figure5.6-5 reveals

Figure5.6-2 Amplitude of (dB).

Figure5.6-3 Phase of (deg).

3 6 9 12 15 18-42

-40

-38

-36

-34

-32

-30

-28

Frequency (GHz)

Am

plitu

de (

dB)

ΓRi

Frequency (GHz)

Pha

se (

deg)

3 6 9 12 15 18-180

-120

-60

0

60

120

180

ΓRi

aRi

152


ent isctralility ofm, 9about 3). Withphases

significant spectral energy is present from 1GHz to about 20GHz (note that no DC componpresent). In order to minimize the effects of the oscilloscope noise, a logarithmic speaveraging of 100 times is applied during the measurement. In order to estimate the repeatabthe measurement (with the 100 times logarithmic averaging applied) on the short termeasurements are done right after each other (the measurement of 100 waveforms takesminutes, so the total measurement time for the 9 averaged waveforms is about half an hourthe appliance of a delay compensation, the estimated standard deviation of the 9 measured

Figure5.6-4 Reference generator waveform.

Figure5.6-5 Spectral representation of reference generator (amplitude).

200 400 600 800 1000-240

-180

-120

-60

0

60

120

180

240

Time (ps)

Am

plitu

de (

mV

)

5 10 15 20 25-50

-45

-40

-35

-30

-25

-20

Frequency (GHz)

Am

plitu

de (

dBm

)

153


-7.

4mVremente of theference

is smaller than 0.1° for all frequencies smaller than 18GHz. This is illustrated in Figure5.6

This error can be explained by the oscilloscope noise standard deviation, which is aboutroot-mean-square for each sample. In order to also check the long term stability the measuwas repeated 24 hours later. The difference between that measurement and the averagphase measured the day before is illustrated in Figure5.6-8. One notes a larger phase difthan with the short term measurements, but still smaller than 0.3° for all frequencies smaller than

Figure5.6-6 Spectral representation of reference generator (phase).

Figure5.6-7 Standard deviation of the 9 phase measurements.

5 10 15 20 25-15

0

15

30

45

60

75

90

Frequency (GHz)

Pha

se (

degr

ees)

5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

Frequency (GHz)

Pha

se S

tand

ard

Dev

. (de

gree

s)

154


y thealso

for ther term

phaseis

eratorr to getboutase the

at allk andertorucingmaine thatch morers thealler

18GHz. Note however that the long term phase difference can no longer be explained boscilloscope noise. Next to the explanation of a small phase drift of the generator, it ispossible that the larger difference on the longer term is caused by connector repeatability (9 short term measurements, the connectors were never disconnected, for the longemeasurement this was the case).

In general one can conclude that it is possible to characterize the reference generatorwith a precision better than about 0.5° for all frequencies smaller than 18GHz. The accuracytraceable to the “nose-to-nose” calibration procedure.

5.6.3 On the optimization of the reference generator

As mentioned in 5.5.0 the SNR of the spectral components of the 1GHz reference genis rather bad, such that 64 times logarithmic spectral averaging needs to be applied in ordethe kind of calibration repeatability results described in 5.5.0. This averaging takes a5 minutes. One might wonder what reference generator signal can be used in order to decremeasurement time without decreasing the quality of the absolute calibration procedure.

The main problem with the reference generator signal is that a good SNR is wantedfrequency components of interest (in our case 18GHz) while the maximum peak-to-peapeak amplitude in the time domain needs to be limited in order to allow both the downconvboards as well as the sampling oscilloscope to characterize the signal without introdnonlinear distortions. For the sampling oscilloscope the maximum allowable time dopeak-to-peak amplitude is about 400mV (better than 50dB linearity in averaging mode, notpersistence mode, which needs to be used during the nose-to-nose measurements is musensitive to nonlinear distortions and should not be used here), and for the downconvertofinal digitized (raw data, no calibration applied) time domain peak amplitude needs to be sm

Figure5.6-8 Long term difference of the phase.

5 10 15 20 25-1.5

-1.2

-0.9

-0.6

-0.3

0

0.3

Frequency (GHz)

Pha

se d

iffer

ence

(de

gree

s)

155


litude,. Thiscope

t oftput of

peaksuredr this

nt. Inf thethis0dBpeakts (butgherat thef bothtwo inr withhas anciesf less

than about 60mV. Note that the oscilloscope linearity depends on the peak-to-peak ampwhile the downconvertor linearity depends on the peak amplitude of the time domain signalis due to the fact that an arbitrary hardware offset can be introduced with the oscillosmeasurements, which can not be done with the downconvertor board.

The first reference generator that was built did not use the differentiator circuiFigure5.6-1. With this reference generator set-up a 30dB attenuator was needed at the outhe SRD-module in order to bring the peak-to-peak amplitude down to 360mV (theamplitude of the downconverted signal equals 28mV). The time domain waveform as meaby the sampling oscilloscope is depicted in Figure5.6-9.The available power at 18GHz fo

reference generator is only -41.2dBm. At 1GHz, however, as much as -16dBm is presegeneral, one notes that a significant 1/f slope is present in the output spectrum oSRD-module. The signal can be improved by introducing the differentiator. Withdifferentiator a lot of energy is lost in the lower frequency components, such that a 2attenuator after the differentiator is sufficient in order to reduce the time domain peak-to-value to 415mV. The resulting spectrum has less energy at the lower frequency componenstill high enough) (e.g. -31dBm at 1GHz) but has significantly more energy at the hifrequencies (e.g. -34dBm at 18GHz). This signal is then preferred because the calibrationhigher frequencies is the most critical due to the significant cable losses. The spectra oreference generator signals are depicted in Figure5.6-10, the difference between theFigure5.6-11. Despite the fact that the peak-to-peak value of the reference generatodifferentiator is only 1.28dB higher than the reference generator without differentiator, onepower gain higher than 3dB for frequencies above 9GHz, and higher than 6dB for frequeabove 15GHz. This implies that, by introducing the differentiator (which enables the use o

Figure5.6-9 Reference generator waveform (without differentiator).

0 200 400 600 800 1000

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Time (ps)

Am

plitu

de (

V)

156


or 4!isinewhile

nly a

attenuation) one has reduced measurement time to get the same SNR at 18GHz by a factAn even much better solution is the use of so-called low-crest-factor microwave mult

sources [5]. These generate signals which optimize the SNR ratio in the frequency domainrespecting a maximum amplitude constraint in the time domain [22]. At this time, however, o

Figure5.6-10 Spectra of the two reference generators.: no differentiator used, 30dB attenuator: differentiator used, 20dB attenuator

Figure5.6-11 Difference between the two reference generator power spectra.

5 10 15 20 25

-60

-50

-40

-30

-20

Frequency (GHz)

Am

plitu

de (

dBm

)

5 10 15 20 25-15

-10

-5

0

5

10

15

Frequency (GHz)

Am

plitu

de d

iffer

ence

(dB

)

157


withof this

or thegain

ethere isnd in

beenpectrala phase

d forbrationratione forability

encerecoveryalyzer

lower frequency prototype is available [21], with a fundamental frequency of 100MHz and20 frequency components present (highest spectral component at 2GHz). The practical usemultisine source has been proven on a VNNA prototype based upon resistive bridges fsignal detection and a sampling oscilloscope for the data acquisition. An experimental SNRof 12.6dB was found compared with the use of a 100MHz repetition rate SRD module togwith a 46dB attenuator [5]. The time domain waveform of the microwave multisine sourcdepicted in Figure5.6-12. The amplitude and phase of the source spectrum can be fou

Figure5.6-13 and Figure5.6-14. Note that the phase characteristic that is shown hasunwrapped. That way one can clearly see that the unwrapped phase of the significant scomponents (until 2GHz) has a parabolic shape. It has been shown by Schroeder that suchcharacteristic results in a low-crest-factor signal [23].

5.7 Conclusion

In has been shown how the absolute calibration procedure is practically implementeconnectored and on wafer device measurements. It has been explained how the caliprocedure is traceable to national standards labs, concerning the relative part of the caliband the power calibration, and is traceable to the “nose-to-nose” calibration procedursampling oscilloscopes concerning the absolute part of the calibration. Repeatability and stmeasurements prove the quality of the calibration procedure.

The most important device for the phase distortion calibration is undoubtedly the refergenerator. Such a reference generator can easily be constructed based on so-called “stepdiodes”. The characterization of the reference generator is done with a classical network an

Figure5.6-12 Multisine Source Waveform.

2 4 6 8 10-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Time (ns)

Am

plitu

de (

V)

158

5.7 Conclusion

e (to
(to measure its output match) and a “nose-to-nose” calibrated sampling oscilloscopcharacterize the pulse shape).
Figure5.6-13 Multisine Source Spectrum: Amplitude.

Figure5.6-14 Multisine Source Spectrum: Unwrapped Phase.

0.5 1 1.5 2 2.5 3-70

-60

-50

-40

-30

-20

Frequency (GHz)

Am

plitu

de (

dBm

)

Frequency (GHz)

Pha

se (

deg)

0.5 1 1.5 2 2.5 3-900

-720

-540

-360

-180

0

180

159


for0.

aklys

a

rk”sium

t Ofltage

,

tion

es,”

forn

eter.O.

tic.O.

aferd

5.8 References

[1] M. Hasler and J. Neirynck,”Nonlinear Circuits,” Artech House Inc., pp.386-388, 1986.

[2] Mark Vanden Bossche,”Measuring Nonlinear Systems: A Black Box ApproachInstrument Implementation,” Doctoral Dissertation, Vrije Universiteit Brussel, May 199

[3] F. Verbeyst and M. Vanden Bossche,”VIOMAP, the s-parameter equivalent for wenonlinear RF and microwave devices,”IEEE Trans. Microwave Theory and Technique,vol.42, No.12, pp.2531-2535, December 1994.

[4] F. Verbeyst and Marc Vanden Bossche,”The Volterra Input-Output Map ofHigh-Frequency Amplifier as a Practical Alternative to Load-Pull Measurements,”IEEETransactions on Instrumentation and Measurement, Vol.44, No.3, pp.662-665, June 1995.

[5] Tom Van den Broeck and Jan Verspecht, “Calibrated Vectorial “Nonlinear NetwoAnalyzers,”Conference Record of the IEEE Microwave Theory and Techniques Sympo1994, San Diego, California, USA, pp.1069-1072, May 1994.

[6] J. Verspecht, Peter Debie, Alain Barel, Luc Martens,”Accurate On Wafer MeasuremenPhase And Amplitude Of The Spectral Components Of Incident And Scattered VoWaves At The Signal Ports Of A Nonlinear Microwave Device,”1995 IEEE MTT-SInternational Microwave Symposium Digest, Vol.3, pp.1029-1032, Orlando (Florida-USA)May 1995.

[7] H. Van hamme and M. Vanden Bossche,”Flexible Vector Network Analyzer CalibraWith Accuracy Bounds Using an 8-Term or a 16-Term Error Correction Model,”IEEETrans. Microwave Theory and Techniques, Vol.42, No.6, pp.976-987, June 1994.

[8] D. Rytting,”An Analysis of Vector Measurement Accuracy Enhancement TechniquProc. Hewlett-Packard RF & Microwave Symposium, pp.16-20, March 1982.

[9] H. J. Eul and B. Schiek,”Thru-match-reflect: One result of a rigorous theoryde-embedding and network analyzer calibration,”Proceedings of the 1988 EuropeaMicrowave Conference, September 1988.

[10] Andrew Davidson, Eric Strid and Keith Jones,”Achieving greater on-wafer S-paramaccuracy with LRM calibration technique,” Application Note, Cascade Microtech, Inc., PBox 1589, Beaverton, OR 97075-1589, USA.

[11] Andrew Davidson, Keith Jones and Eric Strid,”LRM and LRRM calibrations with automadetermination of load inductance,” Application Note, Cascade Microtech, Inc., PBox1589, Beaverton, OR 97075-1589, USA.

[12] Andrea Ferrero and Umberto Pisani,”An Improved Calibration Technique for On-WLarge-Signal Transistor Characterization,”IEEE Transactions on Instrumentation an

160

5.8 References

8510

ns,

ond

ut,”589,

.

4.

y &

for

rce

sing ad

low

Measurement, Vol. 42, No. 2, pp.360-364, April 1993.

[13] E. Strid, R. Gleason and K. Jones,”40GHz On-Wafer Measurements With The HP-Network Analyzer And Cascade Microtech Wafer Probes,”RF & Microwave MeasurementSymposium and Exhibition, March 1988.

[14] N. Balabanian, T. A. Bickart and S. Seshu,”Electrical Network Theory,” John Wiley & SoInc., Chapter6, 1969.

[15] Stephen Wolfram,”Mathematica - A System for Doing Mathematics by Computer”, SecEdition, Addison-Wesley Publishing Company, Inc., 1991.

[16] “Microwave Wafer Probe Calibration Constants - HP 8510 Network Analyzer InpInstruction Manual, Cascade Microtech, Inc., P.O. Box1589, Beaverton, OR 97075-1USA.

[17] “436A Power Meter - Operating and Service Manual,” Hewlett-Packard Co., April 1977

[18] G. Gonzalez,”Microwave Transistor Amplifiers - Analysis and Design,” Chapter_1, 198

[19] Carl T. A. Johnk,”Engineering Electromagnetic Fields & Waves,” Chapter 9, John WileSons, 1975.

[20] John L. Moll and Stephen A. Hamilton,”Physical Modeling of the Step Recovery DiodePulse and Harmonic Generation Circuits,”Proceedings of the IEEE, Vol. 57, No. 7,pp. 1250-1259, July 1969.

[21] T. Van den Broeck, R. Pintelon, and A. Barel,”Design of a Microwave Multisine SouUsing Allpass Functions Estimated in the Richards Domain,”IEEE Transactions onInstrumentation and Measurement, Vol. 43, No. 5, pp.753-757, October 1994.

[22] Van der Ouderaa E., J. Schoukens and J. Renneboog,”Peak Factor Minimization UTime-Frequency Domain Swapping Algorithm,”IEEE Transactions on Instrumentation anMeasurement, Vol. 37, No. 1, pp.145-147, 1988.

[23] M. R. Schroeder,”Synthesis of low-peak-factor signals and binary sequences withautocorrelation,”IEEE Transactions on Information Theory, Vol.16, pp.85-89, January1970.

161


162

Chapter 6

n
Consistency of the Absolute Calibratioversus Large Signal Models
f

e

Abstract - The validity of the absolute calibration procedure is checked versuslarge-signal models of field-effect transistors. These large-signal models, theso-called Root-model and a very similar model, are based upon a lot of small-signals-parameter measurements, executed in a large set of bias voltages. A shortexplanation concerning these models is presented. An harmonic distortion analysis ofthe transistors is then performed using the VNNA, and the same experiment issimulated based upon the large-signal model. The harmonic distortion measurementinvolves the determination of the phases and amplitudes of the spectral components oboth incident and reflected voltage waves. It is found that there is a goodcorrespondence between model and measurements. The results are shown both in thtime and the frequency domain.

163

Chapter 6 Consistency of the Absolute Calibration versus Large Signal Models

, iss noureble toent

e isith a

ever,As a

sk:notning

soundshowanyout anf the

thened largee samein a lotmodelmilar

some

ofs areers,

6.1 Introduction

A major problem with a new kind of calibration, such as the VNNA absolute calibrationto convince the scientific world of the validity of the whole procedure. Unfortunately there itrivial way to achieve this. The only scientifically sound way is to verify the calibration procedagainst all possible “consistency experiments”. These are scientific experiments which are ashow that the calibration is wrong. Unfortunately it is not possible to think of an experimwhich can prove the calibration to be right. A nice illustration of this scientific basic principlthe work of Hertz. He had doubts about the validity of the Maxwell equations and came up wlot of experiments that would show the Maxwell equations to be wrong. As turned out, howall experiments Hertz did failed: Hertz could not show the Maxwell equations to be wrong.result the Maxwell equations were accepted by the scientific world as being right.

The problem one is facing with the VNNA calibration is very similar. Many people aplease show me that the VNNA calibration procedure is right. This is, however, scientificallypossible. The only thing that is possible is to “challenge” the calibration procedure by desigconsistency experiments. Probably the only way to convince someone which has ascientific doubt about the whole issue is to ask him to design an experiment that is able tothe calibration to be wrong. There are then two possibilities: firstly, if he can not think ofexperiment, the person in question has no reason at all to doubt, secondly, if he can think abexperiment it has to be performed and if the experiment fails to show the wrongness ocalibration, the person in question again has no reason to doubt.

It should be clear by now that the process of convincing the whole scientific world ofvalidity of a new kind of calibration is very difficult and time consuming. In this thesis only ospecific kind of consistency experiments has been designed and performed: the calibratesignal measurements of an on wafer transistor are compared with large signal models of thdevice which parameters can be identified by performing many small signal measurementsof DC bias points and by integrating the results. Two such models are considered: thedeveloped by David Root of the Hewlett-Packard Company (the Root-model [2]) and a simodel developed by Philip Jansen, Dominique Schreurset al. [5] of the Flemish “InteruniversitairMicro Electronica Centrum” (IMEC) and the “Katholieke Universiteit Leuven” respectively.

In this chapter first a short description of the models themselves is given, thenexperimental results are shown. Finally some conclusions are drawn.

6.2 Large Signal Models

6.2.1 The Root-model

Introduction

The Root-model was originally designed for the large signal simulation of all kindsmicrowave field effect transistors on wafer in harmonic balance simulators [4]. The modelespecially useful for designers of MMICs who try to build microwave on chip power amplifi

164


he two

metry,r. The

etersuratelyysicalntials. Forre veryvery

d byurces,ertainmeterell asduced,that,

age isted inmodeled forsignalssibletionolationions.

e soes. It

inearitionsmely

t theles,

frequency multipliers and mixers. The Root-model has several advantages compared to tcompeting modeling approaches, called the empirical models and the physical models.

The physical models are based on the physical parameters of the device (like geodoping profiles,...) and use semi-conductor physics in order to predict the device behaviomain problem with this approach is that in most cases the uncertainty on the physical paramis too large, due to processing variations, and that several parameters are hard to accmeasure (like doping profiles). Even if the physical parameters are exactly known, the phmodel implies the solution by computer of a complicated set of nonlinear partial differeequations in two or even three dimensions. This are in general time consuming calculationthis reason physical models are seldom used by MMIC designers, although these models auseful for designers of individual transistors since they can simulate the influence of ephysical parameter variation.

A second kind of models are so called empirical models. This are models describeequivalent schemes containing all kinds of nonlinear elements (capacitors, current soresistors, inductors,...), where every nonlinearity has a parametric description of a cfunctional form. The main disadvantage of these models is that the process of paraextraction is seldom straight forward. Another disadvantage is that the functional form, as wthe equivalent scheme topology, needs to be adapted whenever a new kind of device is introlike for example the MESFET and the HEMT transistors. An advantage of these models isonce the parameters are properly extracted, simulations require only little time.

Finally there is the Root-model (and some very similar approaches). The main advantthat it is valid for a wide variety of components and that the model parameters can be extraca systematic and accurate way. Because of the availability of automation software, theseparameters can easily be measured by non-experts. Note that, although originally developFET-transistors, Root-models also exist for BJTs and diodes. The model is based on smallscattering parameter measurements at one frequency in a lot of different bias points. A poproblem is that, when not carefully implemented, the simulation of small intermodulaproducts can be erroneous due to interpolation errors between biasing steps. This interperror is not present with the empirical models since they only allow the use of smooth funct

The Root-model

The Root-model is depicted in Figure6.2-1 and Figure6.2-2 .The first figure shows thcalled extrinsic device model. This models the transistor as it is seen by the wafer probconsists out of the so called intrinsic device, which will be modelled by a quasi-static nonlmodel, and which is embedded in a linear parasitic network. The network models the transfrom wafer probe to the actual active region terminals by resistors in series with inductors, narg and lg for the gate terminal, rs and ls for the source terminal and finally rd and ld for the drain

terminal.The heart of the Root-model is actually the intrinsic device model. It is assumed tha

intrinsic device currents, Igs and Ids, are completely determined by the use of two state variab

165


s

e DC

filtershavethe

tionssitionthe RF

lator

ted as

namely the intrinsic terminal voltages VG and VD, and by the use of five scalar state function

defined on the two dimensional state variable space, namely the gate and drain charge QG and QD,

the RF drain current source and the DC gate and drain current sources and . Th

sources are always low-pass filtered and the RF sources high-pass filtered. Thesecharacterize some low-frequency anomalies which are typically present in GaAs, and whichtransition frequencies of only a few MHz [7]. Note that it are these effects which causenonlinear model not to be static, but only quasi-static. It is implicitly assumed that all excitahappen on a frequency grid with a fundamental frequency much higher than this tranfrequency, such that the DC current sources are just modelling the DC bias currents, andcurrent source is modelling all variations (even large signal) around the bias points.

With these assumptions the intrinsic device model enables the calculation of Igs(t) and Ids(t)

once Vg(t) and Vd(t) are known. Since the model is to be used in a frequency domain simu

one needs the Fourier transforms of the intrinsic device terminal currents. These are calcula

Figure6.2-1 The Root model: extrinsic device

Figure6.2-2 The Root-model: intrinsic device model

intrinsicdevice

VdsE

VgsE

Igs Ids

lg rg rd ld

rs

ls

source

gate drain

Igs Ids

VGVD

IDDC

VG VD,( )IGDC

VG VD,( )

QD VG VD,( )QG VG VD,( )

IDRF

VG VD,( )

IDRF

IGDC

IDDC

166


heon a

sitionuring

urcess of the

rce,urrentdel,

eters

Q

FETriorir bias.

FET

eddingminalrinsic

led inby

nts at aer toples inratio

olate in

follows:

(6.2-1)

, and

. (6.2-2)

In the above equations LP(ω) and HP(ω) stand for filter characteristics, needed since tmodel is quasi-static. Note that, because of the assumption that all excitations happenfrequency grid with a fundamental frequency much higher than the aforementioned tranfrequency, LP(ω) equals 1 at DC, and equals 0 for all other spectral components occurring dthe simulation, while the opposite is true for HP(ω).

Equation (6.2-1) and (6.2-2) state that the terminal currents originate from current soon one hand, and from charge sources on the other hand, where these sources are functionterminal voltages. The factor “jω” expresses the time domain derivative of the charge souneeded to convert a time varying charge into the corresponding current. Note that no RF csource is present in (6.2-1). Although it is theoretically not impossible to include it into the moits value is always neglected in practice.

Parameter extraction

The main problem is of course how to determine (read “measure”) the different paramof the Root-model: the six scalar constants modelling the embedding network, namely rg, lg, rs, ls,

rd and ld and the five two-dimensional state functions modelling the intrinsic device, namelyG,

QD, , and .

First the embedding network parameters are determined by the use of so called “coldmeasurements”. This technique, described in [8], allows the calculation of the six a punknown parameters by performing small signal s-parameter measurements with a particula

The FET is biased such that equals zero (that is why one talks about “cold

measurements”) and such that the gate-source junction is forward biased. Once the embnetwork has been characterized, it is possible to transform all measured extrinsic tervoltages into intrinsic terminal voltages (note that the intrinsic currents are equal to the extcurrents) by a simple linear transformation.

Next the five state functions need to be characterized. In order to do this they are sampthe two-dimensional (VG,VD) state variable space by changing the bias voltages and

performing accurate bias current measurements and small-signal s-parameter measuremefixed frequency (typically several GHz). Root [2] uses an adaptive sampling technique in ordacquire a lot of samples in areas where the state functions rapidly change and less samareas where the state functions are relatively flat. This way he tries to optimize theinterpolation accuracy versus number of measurements. Splines are used in order to interp

F Igs t( )( ) LP ω( )F IGDC

VG t( ) VD t( ),( )( ) jωF QG VG t( ) VD t( ),( )( )+=

F Ids t( )( ) LP ω( )F IDDC

VG t( ) VD t( ),( )( ) jωF QD VG t( ) VD t( ),( )( )+ +=

HP ω( )F IDRF

VG t( ) VD t( ),( )( )

IDRF

IGDC

IDDC

VdsE

167


is

to be

y using

s

v

and

te that

e biase ideasuredniques

state

between sampled points. The measurement of the DC current characteristics and

trivial and can be done by the use of accurate DC measurements. The measurement of QG,QD and

is, however, less trivial. The sampled values for these three state-functions need

calculated by means of the small-signal s-parameters (measurements which can be done bclassical linear network analyzers). This is done as follows.

Consider as intrinsic bias voltages (VG,VD) and apply small signal sinusoidal excitation

(with an angular frequencyω) at both gate and drain characterized by the complex numbersG

and vD. If i G and iD are the small signal sinusoidal terminal currents one can write (cf. (6.2-1)

(6.2-2)):

and (6.2-3)

. (6.2-4)

If one considers the definition of the small signal Y-parameter matrix one can then sta

, (6.2-5)

, (6.2-6)

, (6.2-7)

, (6.2-8)

, and (6.2-9)

. (6.2-10)

These equations reveal that the y-parameters of the intrinsic device at a certain voltagare directly correlated with the partial derivatives of the state functions at these voltages! This then to measure the y-parameters of the intrinsic device (by converting the meas-parameters into y-parameters) at a lot of bias points and to apply proper integration techto the measured y-parameters in order to calculate good estimates of the value of the

IGDC

IDDC

IDRF

iG jωVG∂

∂QG

vG jωVD∂

∂QG

vD+=

iD VG∂∂ID

RF

jωVG∂

∂QD+

vG VD∂∂ID

RF

jωVD∂

∂QD+

vD+=

VG∂∂QG

VG VD,

Im y11 VG VD,( )( )ω

-------------------------------------------=

VD∂∂QG

VG VD,


-------------------------------------------=

VG∂∂ID

RF

VG VD,

Re y21 VG VD,( )( )=

VG∂∂QD

VG VD,


-------------------------------------------=

VD∂∂ID

RF

VG VD,

Re y22 VG VD,( )( )=

VD∂∂QD

VG VD,


-------------------------------------------=

168


ns in

odelervestateo thet

argethe

These

iqueinsicified byrinsication,

drain

d. Inidistante one

thent gridsingnts atxtractticulareters

functions.

On the “conservation of charge”

If one takes a look at (6.2-5) to (6.2-10) one notes that the integration of these equatioorder to find the state functions only makes sense provided that

, (6.2-11)

and (6.2-12)

. (6.2-13)

These three relationships are called the “integrability conditions”. The intrinsic Root-mis only valid if these relationships hold for a certain DUT. Such a device is said to “conscharge”. This implies that the gate and drain charges can effectively be modelled byfunctions of the intrinsic terminal voltages, such that these charges will each time return tsame value whatever closed contour one follows in the (VD, VG) space. As shown by Root, mos

FET devices effectively conserve charge very well. The largest deviations from “chconserving” behavior [9]-[11] are typically present at small drain voltages or if one allowsdrain voltage to change sign (such that the functionality of drain and source is exchanged).conditions are, however, not present during the experiments described later in this chapter.

6.2.2 The model developed by Ph. Jansen et al.

The large signal HEMT transistor model that was developed by Philip Jansen, DominSchreurset al. is very similar to the Root-model. It is also based upon deembedding of an intrdevice, which is assumed to be charge conservative, such that state functions can be identintegrating the small-signal y-parameters of a device in the two-dimensional space of the intterminal voltages. At this moment, low-frequency anomalies are not taken into considersuch that only three state functions are used. Since these anomalies are typical for lowvoltages, which is not the case for our experiments [9], only little error results.

One of the main differences with the Root-model is that adaptive sampling is not usestead, a dense equidistant grid is used [7]. Note that care is taken in order to have an equintrinsic terminal voltage grid. This can only be achieved by an iterative measurement, sinccan only control the extrinsic voltages. Having an equidistant intrinsic voltage grid hasadvantages of easier file manipulation and integration. The advantage of a dense equidistais that there is no risk of inaccurate interpolation. This risk can not be avoided when uadaptive sampling. Another difference is that for the Root-model small signal measuremeone single frequency are used. If the modelling assumptions are correct this is sufficient to eall parameters of the model. If one takes a look at the small signal y-parameters at one parbias, one indeed notes (cf. (6.2-3) to (6.2-10)) that, theoretically, the real parts of the y-param

VD∂∂ Re y21( )

VG VD∂

2

∂∂ ID

RF

VD VG∂

2

∂∂ ID

RF

VG∂∂ Re y22( )= = =

VD∂∂ Im y21( )

VG∂∂ Im y22( )=

VD∂∂ Im y11( )

VG∂∂ Im y12( )=

169


to thertainrales theg timeless

itance

l datamonicr of thenics isationphasefirst

uding[1]. As

theatonic

rtionasuredrationr 5 do

gth ofis

ias ofport1ourceBmred ofoltagemplies

are constant versus the excitation frequency, and the imaginary parts are proportionalexcitation frequency. In practice this theoretical behavior will only be present within a cedegree of accuracy. The idea used by Ph. Jansenet al. is to measure the y-parameters at sevefrequencies and to do an averaging. It is believed that this averaging in general reducsystematic errors. The disadvantage is that, compared to the Root-model, it takes a very lonand a lot of computer memory in order to do all the measurements and calculations. Asignificant difference with the Root-model is the presence of a parasitic gate and drain capacin the embedding network.

6.3 Early consistency measurements

6.3.1 Introduction

An early report of a comparison between the Root-model and measured large-signadates back from 1991 and can be found in [2]. This article reports about a large-signal hardistortion analysis based upon spectrum analyzer measurements. As such, only the powegenerated harmonics could be compared since no information on the phase of the harmoavailable. In the article only the transmitted voltage wave are measured, no informconcerning the reflected voltage waves is available. The VNNA, however, also measures theof all generated harmonics (as well the reflected as the transmitted harmonics). Thecomparison between a Root-model of a device and calibrated VNNA measurements (inclthe phase of harmonics and the reflected voltage waves) was done in September 1994device an on wafer microwave transistor was provided by the department INTEC of“Universiteit Gent”, in cooperation with IMEC. A Root-model of this device was extractedtheir laboratory. The device was then transferred to our lab in Brussels, where an harmdistortion analysis took place based upon the calibrated VNNA. Finally the harmonic distoanalysis was simulated based upon the Root-model and the results compared with the medata. It should be noted that at the time of these first measurements the whole calibprocedure was still in its infancy, such that the good repeatability results described in Chaptenot apply.

6.3.2 Experiment description

The device used for the measurements is a GaAs MESFET transistor with a gate len0.7µm and a gate width of 300µm. For the harmonic distortion measurement, the deviceexcited at the gate terminal at a fundamental frequency of 3GHz, with a gate-drain voltage b-1V and a source-drain voltage bias of 3V. The transistor gate is connected to the VNNAterminal, and the transistor drain to port2. As such, the transistor is put into a common samplifier configuration. Port1 is excited with a variable input power, ranging from about -18dto 3dBm, with a step of 1dB. For each excitation value both amplitude and phase are measuthe fundamental and 5 harmonics, and this for both the reflected as well as the transmitted vwave. Note that all phases are referenced to the phase of the incident fundamental. This i

170


ponentphase

GHz,nt theo noteectralf theatedan beectionffect

the

of ther the

f thed fore thethiswers.o anneralors, isinsicl

in less

es ofts of

good

ingction

s intomodel.3-6. Ars. This

that the “phase of an harmonic” is defined as the phase of the associated Fourier series comwhen a delay has been applied to the waveform such that the incident fundamental has zero(cf. (5.4-21)).

6.3.3 Comparison in the frequency domain

The measurement and simulation results for the first three harmonics (at frequencies 36GHz and 9GHz) are shown in Figure6.3-1 to Figure6.3-8. The dots in the figures represemeasurements and the solid lines represent the Root-model simulations. It is important tthat the simulations were performed with as only source of excitation a fundamental spcomponent at the gate terminal. This will only be approximately valid in reality because omismatch of the wafer probes (cf. Table5.5-6). This mismatch will make that all generharmonics are partially reflected towards the DUT, such that these incident harmonics cconsidered as “unwanted” excitation signals. Considering the probe mismatch and the reflcoefficient of the DUT, the maximum error that can attributed to the neglecting of this eduring simulation is at a level of about -30dB, corresponding to an amplitude error forharmonics of 300mdB and a phase error of about 1.8°.

Compared with the experimental results as described by Root [2] the correspondenceharmonic amplitudes (Figure6.3-1 and Figure6.3-2), both for the reflected as well as fotransmitted voltage waves, is good, except for the amplitude of the third harmonic otransmitted voltage wave, for lower input powers. The fact that the correspondence is goohigh powers, but bad for low powers, can only be explained by errors in the Root-model, sincVNNA itself behaves linear (cf. 5.1). If there would be an error in the absolute calibration,error would show up in a systematic way in all measurements, as well at low as high input poWhen confronted with this information, Root replied that the model error is probably due tinterpolation error, which can be avoided when applying smaller biasing steps. As a gecomment Root explained that the accuracy of his large-signal model, due to interpolation erronly guaranteed for effective large-signal excitations. With such an excitation, the intrterminal voltages will sweep through an area in the (VG, VD) plane which is covered by severa

sampled biasing points, such that possible interpolation errors are smoothed, resultingmodeling errors.

No previous results of a “model versus measurement” comparison of the phasharmonics were found in literature. Looking at the fundamental frequency componenreflected and scattered voltage waves (Figure6.3-3 and Figure6.3-4) reveals a verycorrespondence. For the reflection, the error is everywhere smaller than 0.5°, for the transmissionthe error is systematically about 2°. This systematic error can be explained by a small error durthe relative calibration. Note that the phase significantly deviates from being a constant fun(which is the case for linear devices) if the input power gets larger and the transistor goecompression. The same kind of deviation is seen as well in the measurements as with theThe phases of the harmonics at 6GHz and 9GHz can be found in Figure6.3-5 to Figure6.general conclusion is that the measured phases become very noisy at the lower input powe

171


poweradraticcan

input

d the

ned by

at theomain.ltageanalogthismodel

ptable,is thes one

can easily be explained since the power of the harmonics decreases very rapidly when theof the input fundamental decreases (in first approximation the second harmonic has a qurelationship with the input amplitude and the third harmonic a third order relationship). Thisalso be verified by looking at Figure6.3-1and Figure6.3-2. If one takes a look at the higherpowers one has a correspondence of about 6° for the 2nd harmonic and about 15° for the 3rdharmonic. For the lower input powers, however, the difference between the model anmeasurements becomes exceedingly large for the 3rd harmonic (errors higher than 100°). Byusing the same reasoning as done with the amplitude, this inconsistency can only be explaimodel errors.

6.3.4 Comparison in the time domain

Since both phases and amplitudes of all spectral components of the voltage wavesDUT signal ports are measured the spectral data can also be interpreted in the time dFigure6.3-9 shows the incident voltage waveform at port1, together with the reflected vowaveforms as measured and as modelled by Root (gray line). Figure6.3-10 shows theinformation for transmitted voltage waveforms. Note that the DC information is not present inrepresentation. Looking at the figures one sees a very good correspondence between theand the measurements for the reflected voltage wave and a less, but certainly accecorrespondence for the transmitted voltage wave. A first thing to note about the waveformsspectral purity of the incident voltage wave, it appears to be a nice sinewave. This is a

Figure6.3-1

: Root-model : VNNA measurements

172


Figure6.3-2

Figure6.3-3



173


Figure6.3-4

Figure6.3-5



174


Figure6.3-6

Figure6.3-7



175


Figure6.3-8

Figure6.3-9 Reflected voltage waves in the time domain.


0 100 200 300 400 500 600

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Time (ps)

Am

plitu

de (

V)

: VNNA measurement : Root-model

: incident voltage wave

176

6.4 More advanced consistency measurements

eableof theior (onr canivativetrue

form.alf of aeing

e wascantnd bym theder tof thesemeter

expects since the generator is well matched (cf. Table5.5-6). This in contrast to the noticdistortion of the reflected and transmitted voltage waves. Indeed, a strong compressionhighest values of the transmitted wave can easily be distinguished. Note the inverting behavbehalf of a small delay) of the common source amplifier configuration. Nonlinear behavioalso be distinguished in the reflected wave. For positive slopes, the absolute value of the derof the reflected waveform is clearly less than that of the input waveform, while the opposite isfor negative slopes. This causes the reflected waveform to look a bit like a sawtooth waveNote that the reflected waveform is about the same amplitude and almost in phase (on behsmall delay) with the incident waveform. This is as expected, since the incident wave is breflected on the (small) gate capacitance, almost representing a perfect open at 3GHz.


6.4.1 Introduction

During the first on wafer measurements, described above, a lot of useful experiencacquired. This resulted in a significantly improved VNNA set-up. Probably the most signifiimprovement was achieved by using higher quality cables from couplers to wafer probes achanging the orientation of the couplers such that these cables were most remote frooperator. In the first version the operator had to reach over the (less quality) cables in orconnect the calibration elements to the RF generator input. As such, eventual movements ocables could not be avoided. The software was also adapted, allowing small-signal s-para

Figure6.3-10 Transmitted voltage waves in the time domain.

0 100 200 300 400 500 600

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

Time (ps)

Am

plitu

de (

V)

: VNNA measurement : Root-model


177


rimented by

oveted to

eterGHz.at, byetworkhis an

theeachts (fromiment

drainourcere

uencyower

d. NoteThe

del andignalnd of3dB

s

evel ofower

en as

measurements and easier debugging of the calibration procedure. A new consistency expewas then performed in June 1995. This time an on wafer HEMT transistor was used, providthe department ESAT of the “Katholieke Universiteit Leuven”, in cooperation with IMEC [6].

6.4.2 Short description of the experiments

The transistor used is an on wafer HEMT transistor, with a cut-off frequency ab100GHz. The transistor is in a common source configuration, with the gate terminal connecVNNA port1 and the drain terminal connected to port2.

In order to check the quality of the relative calibration, first a small signal s-parammeasurement is performed with the VNNA system at 18 frequency points from 1GHz to 18The result of this measurement is compared with the modeled s-parameters (note thconstruction, the model will be close to the s-parameters as measured by a classical nanalyzer since the model parameters are derived from these measurements). After tharmonic distortion analysis is performed, with an excitation frequency of 3GHz, wherepower at the gate terminal is swept from about -25dBm to -5dBm, with a step of 2dB. Forpower setting the calibrated amplitudes and phases are measured of 6 spectral componen3GHz until 18GHz) of the incident as well as the scattered voltage waves. During the experthe drain bias current is also registered (in order to detect self-biasing effects).

6.4.3 Small-signal s-parameter measurements

For doing the small-signal s-parameters the gate voltage is equal to 0.35V and thevoltage is equal to 1.5V. With these bias voltages the transistor will behave as a common-samplifier, with relatively good linearity and high gain. In order for the VNNA to measusmall-signal s-parameters, the frequency is swept from 1GHz to 18GHz, and for each freqtwo measurements are done, once with RF power incident to port1 and once with RF pincident to port2. Out of the two measurements the 4 s-parameters can easily be calculatethat the RF power is limited to about -15dBm in order to avoid nonlinear effects.experimental results can be found in Figure6.4-1 to Figure6.4-4.

As can be seen on these figures there is a very good correspondence between the mothe VNNA measurements. This is nontrivial, since the model is based on small-ss-parameters which were measured on a different wafer probe station, with a different kiwafer probes. For s21, for example, the maximum error occurs at 18GHz and is less than 0.

for the amplitude and less than 4° for the phase. The largest deviation can be seen in the22

parameter. At 18GHz the difference between the measurements and the model is at a l-23dB, which is still acceptable. Note that the differences become significantly smaller for lfrequencies. Out of this can be concluded that the relative calibration is of good quality.

6.4.4 The harmonic distortion measurement

Introduction

For doing the harmonic distortion measurements, a significantly different bias is chos

178


V and
the one used during the small-signal s-parameters. This time the gate voltage is equal to -0.1
Figure6.4-1 Smith chart representation of s11 and s22.

Figure6.4-2 Polar plot of s12 (note the expanded scale).

S22

S11: VNNA measurements: large signal model

0.05

0.1

0.15

: VNNA measurements: large signal model

179


Figure6.4-3 Amplitude of s21.

Figure6.4-4 Phase of s21.

2 4 6 8 10 12 14 16 1810

11

12

13

14

15

16

17

Frequency (GHz)

Am

plitu

de (

dB)


2 4 6 8 10 12 14 16 1890

105

120

135

150

165

180

Frequency (GHz)

Pha

se (

degr

ees)


180


onglylier or

thatrents as

biasmodel

(at allly ben bias

to betunedias aswer

withis canliketratednts and

ear to

the drain voltage equal to 1.25V. With these biasing conditions the transistor behaves strnonlinear. These biasing conditions are typical if the transistor is used as a frequency multipas a mixer and are thus very well suited for our consistency experiment. A small problemcame to the surface after all measurements were done was the fact that the drain bias curmeasured during the VNNA measurements did appear not to correspond with the draincurrents as measured during the classical s-parameter measurements used for theextraction. The difference between the two drain bias currents was approximately 0.3mAinput powers), which means an error of 100% for the lowest RF input power! This can onexplained by a DC measurement error. The drain current is rather insensitive to the draivoltage, but is very sensitive to the gate bias voltage. This indicates that the error needsassociated with the gate bias voltage. During the simulation, the gate bias voltage was thensuch that the drain bias current, as measured by the VNNA, corresponded with the drain bmodeled, and this for an excitation level of -10dBm. In order to do this it was sufficient to lothe gate bias voltage by only 20mV.

Self-biasing effect

During the power sweep, one notes that the drain current is systematically increasingincreasing input power present at the gate. This effect is sometimes called self-biasing. Theasily be explained by noting that the drain current waveform at large input powers will looka clipped sinewave, such that the transistor acts as a rectifier. The self-biasing effect is illusin Figure6.4-5. As one notes, there is a very good correspondence between the measuremethe model. Without the 20mV correction, explained above, the modeled current would appbe about 0.3mA higher.

Figure6.4-5 The self-biasing effect. ( : measured, : modeled)

-25 -20 -15 -10 -50

0.5

1

1.5

2

2.5

3

3.5

Input power (dBm)

Dra

in c

urre

nt (

mA

)

181


mittedare

lationasoningis may

an ismuchdBm.blingt theain atwavea first

still

s isements3-3 tont,

or they high

pointsdence

dinput

ith and in

d thedenceistencyith aittedV.d with

Harmonic distortion analysis: frequency domain

The amplitudes of the fundamental and three harmonics of the reflected and the transvoltage waves, both for the model (solid line) and for the VNNA measurements (dots)depicted in Figure6.4-6 and Figure6.4-7. It is important to note that, analog to 6.3, the simuuses as excitation one fundamental spectral component at the gate terminal. The same reas done in 6.3.0 concerning probe mismatch and the possible errors the neglecting of thintroduce is valid.

The correspondence between model and VNNA measurements is much better thachieved with the early consistency measurements (cf. Figure6.3-1 and Figure6.3-2), and isbetter than the results described in [3]. The best fit is achieved at input powers around -10Note that the transmitted voltage wave behaves strongly nonlinear, with a frequency douconversion loss of only 5dB at -10dBm input power. Another peculiar nonlinear effect is thacircuit has a fundamental frequency transmission loss at low powers, but has a small ghigher input powers. This can be explained by the self-biasing effect. The reflected voltageis a much softer nonlinearity. This can easily be explained because the gate acts inapproximation as a linear capacitor. At an input power of -6dBm the second harmonic islower than -26dBm.

The modeled and VNNA measured phase information of the first three harmonicillustrated in Figure6.4-8 to Figure6.4-13. The correspondence between model and measuris also much improved compared to the early consistency measurements (cf. Figure6.Figure6.3-8). Note the scales, with only 0.5° per division for the fundamental spectral compone2° per division for the second harmonic of the transmitted voltage wave, and 5° per division forthe third harmonics and the second harmonic of the reflected voltage wave. Note that fsmaller input powers the variance of the phase of the harmonics becomes excessivelbecause the corresponding harmonic powers become very low. This is why some measureddo not appear on the figures. Looking at the figures one can roughly say that the corresponbetween measurement and model is about 1° for the fundamental spectral component, 2° for thesecond harmonic of the transmitted voltage wave, and 5° for the third harmonics and the seconharmonic of the reflected voltage wave. The numeric information for the six harmonics at anpower of -10dBm is given in Table6.4-1 and Table6.4-2.

Harmonic distortion analysis: time domain

The time domain waveforms (based upon 6 spectral components) corresponding winput power of -10dBm, for both the reflection as well as the transmission, are depicteFigure6.4-14 and Figure6.4-16, respectively. The difference between the model anmeasurements is given in Figure6.4-15 and Figure6.4-17, respectively. The corresponbetween model and measurements is much better than was the case for the early consmeasurements (cf. Figure6.3-9 and Figure6.3-10). For the reflected voltage wave, wmaximum value of about 100mV, the maximum deviation is about 1mV and for the transmvoltage wave, with a maximum value of about 180mV, the maximum deviation is about 6m

Looking at the waveforms one notes the same quantitative characteristics as mentione

182

6.5 Conclusion

and an

onicistorson ofge set oftor. A

del [2].f thisNNAtor. A

AT ofarean the

the early consistency measurements (cf. 6.3.0): a steepening of the reflected waveforminversion and clipping of the transmitted waveform.

6.5 Conclusion

Two consistency checks for the absolute calibration are performed by doing an harmdistortion analysis of on wafer field effect transistors (a MESFET and a HEMT). These transare modeled by a large-signal model whose parameters are extracted by integratismall-signal s-parameter measurements. These s-parameters have to be measured in a largate and drain bias voltages. The first measurement is performed on a MESFET transisRoot-model of this transistor is extracted and the measurements are compared with the moConcerning the harmonic amplitudes of the transmitted voltage waves, the results oexperiment are comparable with the results as reported by David Root. After tuning of the Vtest-set, a second consistency measurement is performed, this time on a HEMT transismodel is used, similar to the Root-model, but developed by people of the department ES“Katholieke Universiteit Leuven”, in cooperation with IMEC. The results of this comparisonmuch better than the results achieved with the MESFET transistor, and are much better th

Table6.4-1 Fundamental and harmonic information for the reflected voltage wave.

freq. (GHz) b1m (dBm) b1s (dBm) b1m (°) b1s (°)

3 -9.71 -9.76 -13.2 -13.3

6 -34.4 -34.3 -131.8 -125.8

9 -46.2 -46.1 -153.7 -160.8

12 -57.9 -54.6 32.3 52.5

15 -62.4 -61.2 20.1 -9.19

18 -71.2 -67.4 -41.5 -141.5

Table6.4-2 Fundamental and harmonic information for the transmitted voltage wave.

freq. (GHz) b2m (dBm) b2s (dBm) b2m (°) b2s (°)

3 -8.83 -8.88 163.7 163.0

6 -15.1 -15.5 148.4 150.9

9 -27.9 -28.9 129.7 125.5

12 -47.2 -42.7 9.5 -17.4

15 -48.3 -47.9 -73.3 -69.3

18 -59.8 -58.6 -131.2 160.8

183


Figure6.4-6 Harmonic distortion analysis (reflected voltage wave, fundamental and 3

Input power (dBm)

Har

mon

ic o

utpu

t pow

er (

dBm

)

-25 -20 -15 -10 -5-60

-50

-40

-30

-20

-10

0

: large signal model : VNNA measurements

184

harmonics).

6.5 Conclusion

Figure6.4-7 Harmonic distortion analysis (transmitted voltage wave, fundamental and 3harmonics).

-25 -20 -15 -10 -5-60

-50

-40

-30

-20

-10

0

Har

mon

ic o

utpu

t pow

er (

dBm

)

Input power (dBm) : large signal model

: VNNA measurements

185


Figure6.4-8 Harmonic distortion measurement: Phase of the fundamental of thereflected voltage wave.

Figure6.4-9 Harmonic distortion measurement: Phase of the fundamental of thetransmitted voltage wave.

-25 -20 -15 -10

-14

-13.5

-13

-12.5

-12

Input power (dBm)

Pha

se (

deg)


-25 -20 -15 -10162

162.5

163

163.5

164

Input power (dBm)

Pha

se (

deg)


186

6.5 Conclusion

Figure6.4-10 Harmonic distortion measurement: Phase of the 2nd harmonic (@ 6GHz) ofthe reflected voltage wave.

Figure6.4-11 Harmonic distortion measurement: Phase of the 2nd harmonic (@ 6GHz) ofthe transmitted voltage wave.

Input power (dBm)

Pha

se (

deg)

-25 -20 -15 -10

-140

-135

-130

-125


-25 -20 -15 -10146

148

150

152

154

Input power (dBm)

Pha

se (

deg)


187


n then not be

andeableswer.

ors arefunction

hichand the

whichal with

prettyo usetooodel,

models, by

e is not

results as reported by David Root.An important question remains however: what causes the (small) deviations betwee

models and the measurements? One indeed notes that, although small, the differences caexplained by the effect of neglecting the probe mismatch during the simulations (cf. 6.3.06.4.0) or by the calibration repeatability (cf. 5.5.0). Since there is no standard available tracto a national standards lab (read “accepted by all scientists”), this is no trivial question to anOut of the measurements one can conclude that significant (read “measurable”) model errpresent, since the relative deviation between model and measurement can be seen to be aof the RF input power, and this can never be explained by an absolute calibration error, wdoes not change during the power sweep. On the other hand, the match between the modelmeasurements can be significantly improved by changing the calibration coefficients a bit,suggests that there is a systematic error on the calibration (this approach is actually identicthe “golden diode” calibration philosophy of Urs Lott, cf. 2.3.3).

By our knowledge, the accuracy aspects of the large signal models used are as yetmuch unknown territory. Next to the interpolation accuracy, it is even such that it is advised tthe Root-model for large-signal simulations with an excitation frequency which is notdifferent from the single frequency at which the small-signal s-parameters, needed for the mare measured. The fact that this is stated means that the validity conditions of the Root-assumptions are only partially met in practice. This can be caused, among other thingproblems encountered with the deembedding process and by the fact that the actual devic

Figure6.4-12 Harmonic distortion measurement: Phase of the 3rd harmonic (@ 9GHz) ofthe reflected voltage wave.

-25 -20 -15 -10

-165

-160

-155

-150

-145

Input power (dBm)

Pha

se (

deg)


188

6.5 Conclusion

Figure6.4-13 Harmonic distortion measurement: Phase of the 3rd harmonic (@ 9GHz) ofthe transmitted voltage wave.

Figure6.4-14 Incident and reflected voltage wave in the time domain; incident waveform.

-25 -20 -15 -10

115

120

125

130

135

140

145

Input power (dBm)

Pha

se (

deg)


0 100 200 300 400 500 600

-0.1

-0.05

0

0.05

0.1

Time (ps)

Am

plitu

de (

V)

: VNNA measurement : large signal model


189


e
Figure6.4-15 Difference between the modeled and measured reflected voltage wave in thtime domain.
Figure6.4-16 Incident and transmitted voltage wave in the time domain.

0 100 200 300 400 500 600

-0.001

-0.0005

0

0.0005

0.001

Time (ps)

Am

plitu

de (

V)

0 100 200 300 400 500 600

-0.15

-0.1

-0.05

0

0.05

0.1

Time (ps)

Am

plitu

de (

V)

: VNNA measurement : large signal model


190

6.5 Conclusion

meterlab, no

tionnd forlitudeator, tomakesusingtion

n

perfectly charge conservative. All this means that, although the small-signal s-parameasurements on which the model is based can be traced up to a national standardstraceability exists for the final model.

On the other hand, there exists a traceability path for the VNNA relative calibraprocedure which goes up to national standards labs (classical linear calibration elements), athe absolute calibration procedure which goes up to national standard labs for the amp(power sensor) and up to the nose-to-nose calibration for the phase (from reference genersampling oscilloscope, to the nose-to-nose calibration procedure). This fact suggests that itmore sense to use the calibrated VNNA in order to check the quality of a Root-model, thanthe Root-model as a “golden diode” in order to check the quality of the absolute calibraprocedure.

Figure6.4-17 Difference between the modeled and measured transmitted voltage wave ithe time domain.

0 100 200 300 400 500 600-0.006

-0.004

-0.002

0

0.002

0.004

0.006

Time (ps)

Am

plitu

de (

V)

191


t Ofltage

,

onvice

onsand

ium

e

istent

.

atednts,”ncy

cuit

thed

aAs

’s,”r

6.6 References

[1] J. Verspecht, Peter Debie, Alain Barel, Luc Martens,”Accurate On Wafer MeasuremenPhase And Amplitude Of The Spectral Components Of Incident And Scattered VoWaves At The Signal Ports Of A Nonlinear Microwave Device,”1995 IEEE MTT-SInternational Microwave Symposium Digest, Vol.3, pp.1029-1032, Orlando (Florida-USA)May 1995.

[2] D.E. Root, Siqi Fan and Jeff Meyer,”Technology Independent Large Signal NQuasi-Static FET Models by Direct Construction from Automatically Characterized DeData,”21st European Microwave Conference Proceedings, pp.927-932, 1991.

[3] D. E. Root and Siqi Fan,”Experimental Evaluation of Large-Signal Modeling AssumptiBased On Vector Analysis of Bias-Dependent S-Parameter Data from MESFETsHEMTs,” 1992 IEEE MTT-S International Microwave Theory and Techniques SymposDigest, pp.255-258, 1992.

[4] D. E. Root and K. J. Kerwin,”CAD for Microwave Integrated Circuits,” Appendix B of thbook by Y. Konishi,”Microwave Integrated Circuits,” Marcel Dekker, Inc., pp.584-586.

[5] Ph. Jansen, D. Schreurs, W. De Raedt, B. Nauwelaers and M. Van Rossum,”ConsSmall-Signal and Large Signal Extraction Techniques for Heterojunction FET’s,”IEEETransactions on Microwave Theory and Techniques, Vol.43, No.1, pp.87-93, January 1995

[6] D. Schreurs, Y. Baeyens, B. Nauwelaers, W. De Raedt and M. Van Rossum,”AutomGeneration of Intrinsic Large-Signal HEMT Models from S-parameter MeasuremeMIOP 95 - Microwaves and Optronics - 8th Exhibition and Conference on High FrequeEngineering Conference Proceedings, pp.106-110, Sindelfingen, Germany, May 1995

[7] N. Scheinberg, R. Bayruns and R. Goyal,”A Low-Frequency GaAs MESFET CirModel,” IEEE Journal of Solid-State Circuits, Vol.23, No.2, pp.605-608, April 1988.

[8] G. Dambrine, A. Cappy, F. Heliodore and E. Playez,”A New Method for DeterminingFET Small-Signal Equivalent Circuit,”IEEE Transactions on Microwave Theory anTechniques, Vol.36, No.7, pp.1151-1159, July 1988.

[9] Herman Statz, Paul Newman, Irl W. Smith, Robert A. Pucel and Hermann A. Haus,”GFET Device and Circuit Simulation in SPICE,”IEEE Transactions on Electron Devices,Vol.ED-34, No.2, pp.160-168, February 1987.

[10] Dileep Divekar,”Comments on “GaAs FET Device and Circuit Simulation in SPICE,”IEEETransactions on Electron Devices, Vol.ED-34, No.12, pp.2564, December 1987.

[11] I. W. Smith, H. Statz, H. A. Haus, and R. A. Pucel,”On Charge Nonconservation in FETIEEE Transactions on Electron Devices, Vol.ED-34, No.12, pp.2565-2568, Decembe1987.

192

Chapter 7

Conclusions and ideas for furtherresearch

Abstract - In this chapter some conclusions are drawn and ideas are given for furtherresearch.

193

Chapter 7 Conclusions and ideas for further research

d ofAtallyf thecident

or thes.

mostlly theistingre usedat and. Ason the

t larges how

of aned. Thet parts ofsolute

ortionNNAich is

-calledquencynd thezed bymplingis a

es. It isthe

eatedhere it

7.1 Past

When HP-NMDG started their research in 1990, they wanted to develop a new kinmicrowave instrumentation: the so-called vectorial “nonlinear network” analyzer (VNNA).that time only very few prototypes of such a VNNA were described in literature, and actunone of these prototypes corresponded to the demands of HP-NMDG, since none oprototypes could measure the amplitude and phase of the spectral components of both inand reflected voltage waves at both signal ports of the nonlinear microwave DUT. The goal fresearch described in this thesis was to develop a calibration procedure for these prototype

When looking at the VNNA calibration procedures as implemented by the others, theimportant questions probably concern the traceability of the calibration procedures, especiacalibration of the phases of the harmonics relative to the fundamental. Most of the excalibration procedures are based upon the assumption that the microwave samplers that afor the data acquisition introduce negligible phase distortion. The main problem is thspecification of a microwave sampler phase distortion can, as far as I know, nowhere be foua consequence traceability was not possible. Another approach that was used is basedso-called “golden diode”. This a nonlinear device from which one assumes to have a perfecsignal nonlinear model, including all parasitic effects. The remaining unanswered question ione can trace the accuracy of the “golden diode” approach.

7.2 Present

Five years later, the situation is much different. HP-NMDG has a flexible prototypeVNNA, with a bandwidth of 18GHz and a dynamic range of about 60dB. Although not planoriginally, the VNNA can measure both connectored devices as well as on wafer devicescalibration procedure for the instrument, developed in this thesis, has two major parts. A firsis the relative calibration. This procedure is identical to the linear calibration procedureclassical network analyzers. The second part of the calibration procedure is called the abcalibration. This procedure involves the determination of the power loss and phase distintroduced by the instrument. The power calibration is based upon comparison of the Vmeasurements with classical power meter measurements. The phase calibration, whconsidered as the most original contribution of this thesis, is based upon the use of a sophase reference generator. This is a signal generator which generates a fundamental freand harmonics, with an accurately known phase relationship between the harmonics afundamental. In order to know this phase relationship, the reference generator is characteria broadband sampling oscilloscope. The measurement accuracy of the broadband saoscilloscope is determined by the so-called “nose-to-nose” calibration procedure. Thismeasurement procedure using no other instrumentation besides three sampling oscilloscopbased upon the assumption that “the oscilloscope sampler kick-out is proportional tooscilloscope impulse response”. The oscilloscope sampler kick-out is a pulse which is crwhen the oscilloscope samples a DC voltage and which comes out of the input connector, w

194

7.3 Future

nt). Arder to

lativelabs

to theator. Itan beeantmpler

ssfulof thisse anor themeterpletelyn the

ration

many

ations labsration

t aresuch aslow tothat iterrors

lity,thenotes

can be detected by another oscilloscope (this is the actual “nose-to-nose” measuremetheoretical as well as practical study of this “nose-to-nose” procedure has been done, in obe able to determine the precision and accuracy of the method.

An important advantage of the calibration procedure is the traceability aspect. The repart of the calibration as well as the power calibration are traceable to national standardthrough the calibration elements and the power meter. The phase calibration is traceableaccuracy of the “nose-to-nose” calibration procedure through the use of the reference generhas been shown in this thesis how the accuracy of the “nose-to-nose” procedure cdetermined, starting from a topological model of the sampling head. With topological is mthat no model parameters need to be determined a priori, and that it is sufficient that the sahas a certain topology.

The quality of the calibration procedure has been shown by performing a succeconsistency check between calibrated measurements on one hand and simulationsexperiment based upon existing large-signal models on the other hand. For this purpoharmonic distortion analysis was performed of on wafer FET transistors. The parameters ftype of large-signal models used are extracted by means of small-signal s-parameasurements in a lot of bias points. The determination of these models is as such comrelying on classical linear measurement techniques. The correspondence betweemeasurements and the models is, as far as I know of, unprecedented in literature.

Finally, one can conclude that the initial research goal has been reached: a calibprocedure for VNNA’s is now available.

7.3 Future

7.3.1 Introduction

Fortunately for the researcher, the goal is never reached at a level of 100%, sinceanswers introduce new questions. How about the research work described in this thesis?

7.3.2 Full traceability towards national standards labs

By now there is traceability towards national standards labs for all aspects of the calibrprocedure except for the phase calibration. A future goal is to convince the national standardof the importance of such a phase standard. Their acceptance of the “nose-to-nose” calibprocedure as it exists today is only one possibility. Other possibilities worth while looking ahigh-speed electrical sampling techniques based upon short duration (< 1ps) laser pulses,the electro-optical samplers and the photo-conductive samplers. These techniques alconstruct very broadband samplers (with a bandwidth of several hundreds of GHz), suchmight be possible to assume that the sampling system has only negligible phase distortionfrom DC to about 50GHz (note that making this assumption is far from trivial). A possibiwhich I personally think is most promising towards the future, is the application of“nose-to-nose” calibration procedure on a photo-conductive sampling set-up. One indeed

195


n thebeinglitudeis is,

tionightrasiticnto thediodeever,itance.at thegoodn, it is

rdlye asibrationat herenceay aserator,se ins way

ts toin thision atrenceivial.this istion. kind ofThise data

that the sampler topology of this kind of samplers is actually the same as is present iHP54120 sampling oscilloscopes, with the only difference that a photo-conductive switch isused in stead of a diode switch. This offers the possibility of a calibrated sampler (both ampand phase characteristic) with a bandwidth of several hundred GHz! At this moment thhowever, pure science-fiction, comparable to the moon traveling in the age of Jules Verne.

7.3.3 Extending the sampler topological model

Looking at the sampler model which is used for proving the “nose-to-nose” calibraprocedure basic principle “the kick-out is proportional to the impulse response”, one mwonder whether this model indeed encompasses all significant effects. As far as linear paeffects are considered the answer is yes, since all these parasitical effects can be put igeneral s-parameter matrix description used to describe the network in which the samplingis embedded (this is referred to as “the input circuitry”). One nonlinear component is, hownot considered in the sampler model, namely the sampling diodes nonlinear junction capacA possible future research topic is to include this into the general model and to see whinfluence is on the accuracy of the “nose-to-nose” calibration procedure. Considering thecorrespondence between the swept-sine measurements and the “nose-to-nose” calibratioexpected that the influence, if any, is very small.

7.3.4 Adapting the calibration procedure towards commercial VNNA use

The whole calibration procedure of the VNNA, as described in this thesis, is haapplicable for commercial VNNA use. If one wants the VNNA measurements to becomcommon as s-parameter measurements are today, there is a need to adapt the calprocedures towards commercial every day use. One of the first things a user will say is thdoes not want to buy an expensive sampling oscilloscope in order to characterize his refegenerator. This can easily be solved by treating the reference generator in the same wclassical calibration elements are treated today: sell the user a fully specified reference genand provide the user with the possibility of verifying this reference generator on a regular baa “standard lab” which possesses a “nose-to-nose” calibrated sampling oscilloscope. Thione builds the hierarchical structure, typical for traceable calibrations.

Next there is the problem of a flexible frequency grid. It is very unlikely that a user wando measurements on a fixed frequency grid. Using the calibration procedure, as developedthesis, the user would have no problem to do the relative calibration and the power calibratany frequency grid, but he would need for each separate frequency grid a different refegenerator for doing the phase calibration. Finding a flexible solution to this problem is not trOne possibility would be the use of a tunable reference generator. Although not impossible,technologically difficult to realize. Another approach is based upon the idea of interpolaSuppose one knows the phase distortion at a 1GHz grid, it might then be possible to use ainterpolation procedure in order to find the phase distortion at other frequency grids.approach is as such based on the identification of a model for the microwave test set and th

196

7.3 Future

at theed byrs andhich

F phaseut aupon a

h this

ortanttion ofthat

e can do

NAon the

of thewafertudy of“error

acquisition. This is less trivial than might be apparent at first sight. One of the problems is thtotal phase distortion consists out of two independent parts: the microwave part, introduccables, connectors and couplers, and the intermediate frequency part, introduced by IF filteamplifiers. An interpolation only makes sense if both parts can be identified separately, wnecessitates extensive test procedures. A practical approach might be to characterize the Idistortion at manufacturing time (with verifications on a regular base) and to try to work oprocedure that enables the interpolation of the RF phase distortion characteristic basedfixed frequency grid reference generator.

Further research in the future will be needed in order to find out how one deals best witproblem of flexible calibration.

7.3.5 Putting error flags on VNNA measurements

Looking at calibrated data provided by a measurement instrument one can ask the impquestion: how close is the measured data to the physical reality? The quantitative determinathe accuracy of VNNA measurements is, however, far from trivial. First there is the problemone can not refer to a national standards lab primary phase reference standard, the best onis to refer to a consensus standard (cf. 7.3.0).

But even the availability of a primary phase reference standard would not solve all VNaccuracy aspects. There still is the problem that many things have a measurable influencefinal accuracy: additive noise, phasenoise of the downconvertor local oscillator, accuracyrelative and absolute calibration procedure, connection repeatability (especially for onmeasurements), sensitivity of the instrument versus temperature changes... . A detailed sall these aspects is needed before one will be able to put reliable (but not exceedingly large)flags” on the final data.

197


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