Information Technology, System Engineering
TALLINNA TEHNIKAÜLIKOOLI
VÄITEKIRJAD
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THESES
OF TALLINN TECHNICAL UNIVERSITY
Olev Märtens
AC Measurement Converters:
Analog and Digital Solutions
Institute of Electronics, TTU • Tallinn 2000
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Current work has been done at the chair of Measurement Electronics, Institute of
Electronics, Tallinn Technical University
Supervisor of this work has been Prof. Mart Min.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
3
Abstracts
The current work is about precise AC measurement solutions.
In the first chapter the unique solutions of AC/DC measurement converters with switched
capacitors in the negative feedback path are proposed and analyzed. Methodology for analyzing
of the AC/DC measurement converters with several capacitors has been proposed and used. The
benefits of such solutions are found out and explained theoretically.
In the second chapter the similar phase-sensitive detector solutions are proposed, explained
and analyzed. Also one practical implementation for the PLL (phase-lock-loop) has been
described.
In the third chapter the DSP- (digital signal processor) based solutions are proposed and
described, one for biomedical bio-impedance measurements (for digital post-processing of
signals, and for totally DSP-based multi-frequency measurements), and the other for AC
measurements purposes. Practical implementations are described also.
Current work includes mathematical analyses of some solutions known before, but also several
new (patent-level) solutions with improved performance have been proposed. The proposed and
analyzed solutions have been implemented in real prototypes. So the solutions have been
practically realized, verified, and tuned in details.
Main issues of the work have been described in the 16 international and local publications and
presented at several scientific conferences, and most of the results have found practical usage in
the industry and applied science.
Keywords
AC measurement converter, instrumentation, AC/DC converter, RMS value, peak value, average
value, phase-sensitive detector, synchronous detector, PLL, frequency domain, time domain,
accuracy, inaccuracy, state space equations, mathematical analysis, analog converter, digital
signal processing, sampling.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Kokkuvõte
Käesoleva töö pealkiri on “Vahelduvpinge mõõtemuundurid: analoog- ja digitaallahendused”.
Töö käsitleb lahendusi vahelduvpinge täppismõõtmisteks.
Esimeses peatükis on kirjeldatud ja analüüsitud unikaalseid vahelduvpinge mõõtemuundureid,
mis sisaldavad ümberlülitatavaid kondensaatoreid negatiivse tagasiside ahelas. On välja
arendatud taoliste vahelduvpinge muundurite analüüsi metoodika. On kirjeldatud taoliste
lahenduste eeliseid.
Teises peatükis on kirjeldatud ja analüüsitud sarnaseid faasitundlikke mõõtemuundureid,
samuti ühte konkreetset PLLi lahendust.
Kolmandas peatükis on kirjeldatud mitmeid DSP-põhjalisi lahendusi- biomeditsiinilisteks
mõõtmisteks (digitaalse signaali-järeltöötlusega, ja paljusagedusliku täieliku digitaalse
töötlusega variandid), samuti vahelduvpinge väärtuse mõõtmisteks. Ka praktilised tulemused on
kirjeldatud.
Käesolev töö sisaldab nii tuntud lahenduste matemaatilist analüüsi, aga ka uusi (patentseid)
parendatud parameetritega lahendusi. Välja pakutud ja analüüsitud lahendused on realiseeritud
prototüüpidena. Nii on lahendused verifitseeritud ja häälestatud.
Töö põhitulemused on ära toodud 16-s rahvusvahelises ja kohalikus publikatsioonis ja esitletud
mitmetel teaduslikel konverentsidel, ja leidnud kasutamist tööstuses ja teadusuuringutes.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
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Declaration
I declare that this thesis is my original, unaided work. It is submitted for the degree of Doctor
of Philosophy in Electronics and Biomedical Engineering at Tallinn Technical University,
Tallinn, Estonia. It has not been submitted before for any degree or examination in any other
university.
Olev Märtens
Tallinn - Mustamäe , 22-nd January 2000
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Contents
Acknowledgments........................................................................................……............
8
List of Publications ……………………………………………………………….........
9
Industrial (research) applications …………………………………………..................
11
Introduction.................................................................…..............................……..........
Introduction to the topic.............................................................................….....….....
Analog versus digital solutions ……………………………………….……...……........…
Tasks of the work.......................................................................................................
Methods and tools ………………………………………………………………….…......…
Contents of the current work .................................................................…...…...........
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12
13
13
13
14
1. Precise AC/DC converters with switched capacitors in the feedback path ..........
1.1. Basic functioning and circuit diagrams...............................................................
1.2. Low-frequency error analysis of the asymmetrical converter ..........................
1.3. Low-frequency error analysis of the symmetrical converter...................…..........
1.4. Analysis of the proposed circuits with state-space equations ..............................
1.4.1. Introduction to analysis ................................................................…......…
1.4.2. Analysis of the circuit by Fig. 1.1. ...........................................…… …......
1.4.3. Analysis of the circuit by Fig. 1.2. ................................................…......…
1.4.4. Analysis of the circuit by Fig. 1.3. ..........................................….….…......
1.4.5. Analysis of the circuit by Fig. 1.4. ................................................…......…
1.4.6. Analysis of the circuit by Fig. 1.5. ..........................................................…
1.4.7. Analysis of the circuit by Fig. 1.7. ..............................................…..…......
1.4.8. Time-domain analysis of the converters .........................................….......
1.5. Practical realizations............................................................................….........
1.5.1. Precise AC/DC measurement converter ...................................................
1.5.2. Precise AC/DC measurement converter with improved performance .......
1.5.3. AC/DC measurement converter with input conditioning .............…..........
1.6. Conclusions and results …………………………………………..……………..........
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15
22
26
30
31
35
39
43
46
51
53
55
57
57
58
61
62
2. Precise synchronous detectors with switched capacitors in the feedback path .....
2.1. Introduction to the topic ......................................................................................
2.2. Basic ideas and advantages of the proposed solutions .......................................
2.3. Solutions and functioning .....................................................................................
2.4. About practical using of proposed solutions .......................................................
2.5. Conclusions and results ........................................................................................
63
63
63
64
69
70
3. DSP-based AC measurement converters.....................................................................
3.1. Apology of digital solutions of AC measurement solutions....................................
3.2. Digital post-processing of the bio-impedance signal ............................................
3.2.1. About the analogue part of the Bio-impedance Measurement Interface ..
3.2.2. Analogue measurement interface circuit for bio-impedance measurement...
3.2.4. Digital (post)processing of the bio-impedance measurement signal .............
71
71
71
72
73
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
3.2.5. Proposed solution...............................................................................…....… 74
3.2.6. Results................................................................................................……… 76
3.3. DSP-Based Device for Multi-Frequency Bio-Impedance Measurement ..........… 79
3.3.1. Introduction .......................................................................................…….... 79
3.3.2. DSP in bio-impedance measurements...............................................………. 79
3.3.3. Description of the hardware ...........................................................…….….. 79
3.3.4. Description of the algorithm .........................................................…….....… 80
3.3.5. Results ...........................................................................................………… 82
3.4. DSP-based AC Measurement Converters ...............................................…….….... 83
3.4.1. Introduction ...............................................................................……........… 83
3.4.2. DSP-based peak-to-peak and average rectified value measurements ……… 83
3.4.3. Proposed algorithm for numerical RMS-measuring implementation ......… 84
3.4.4. DSP-realization of the RMS measurement converter ................…..……..… 86
3.4.5. Estimation of the principal inaccuracy (medium frequencies) ........…….... 86
3.4.6. Analysis of the low frequency error and settling time of the converter .….... 87
3.4.7. Practical realizations of the prototypes ...........................................……....... 90
3.4.8. Conclusion …………………………………………………………….......... 90
4. General conclusions and results ..................................................................……........... 91
5. Claims ………………………………………………………………………..........…… 92
References ……………………………………………………………………………….. 93
Appendixes .................................................…..................................................……........... 97
Appendix A. Low frequency error calculations for the AC/DC converter according to
Fig.1.2(asymmetrical) and Fig1.4 (symmetrical scheme)- programs for HP85....................97
A1. Low frequency error calculations for the AC/DC converter according to Fig.1.2
(asymmetrical scheme........................................................................................................... 97
A2. Low frequency error calculations for the AC/DC converter according to Fig. 1.4
(symmetrical scheme)............................................................................................................ 98
Appendix B. Practical circuit diagrams and component lists……………………......... 100
Appendix B1. Practical realization of the AC/DC converter by p.1.5.1 ……….. .. 100
Appendix B2. Practical realization of the AC/DC converter by p.1.5.2…...........…102
Appendix B3. Practical realization of the AC/DC converter by p.1.5.3 ................ 104
Appendix B4. Realization of the PLL for the stereo-decoder………...................... 106
Appendix C. Numerical calculation of the settling process in the time domain- programs
for HP85. ………………………………………………………………………………….. 107
Appendix C1. Analysis of the AC/DC converter according to p.1.5.1. …….…… 107
Appendix C2. Analysis of the AC/DC converter according to p.1.5.2. …….…… 108
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Acknowledgments
I am very grateful to my family about understanding and support during this work.
From latest years (results described in chapter 3) special thanks belong to the supervisor of
the current work, Prof. Mart Min- Head of the Department of Electronics of the Tallinn
Technical University (TTU), for encouragement and practical help of finishing the post-graduate
studies, and also on dealing with challenging digital signal processing (DSP) solutions. Thanks
belong also to all nice people at the TTU, department of electronics.
One part of this work- regarding creating, practical implementing, and analyzing of solutions of
precise AC/DC measurement converters (chapters 1 and 2 of the current work) was mostly done
in 1983-1990 at the Tallinn Design Office of Radio-Electronics (TREKB), belonging to the
Manufacturing Association of Radio-Electronics “RET”, where I worked as a research-engineer.
Special thanks from this time belong in the first order to Mr. Toom Pungas, who encouraged very
much to continue post-graduate studies, to find innovative and patentable solutions, and to be an
engineer who can “read and write” technically. Special thanks belong also to Mr. Rein Kipper,
to the patent specialists in the company- Mr.Toomas Lumi and Mr.Riho Pikkor, to Mr. Tiit
Põldver and Mrs. Tiina Vasserman for practical help at preparing of manuscripts this time, to
Mr. Mati Kembo for preparing conferences and collections of research papers, and to my all
other friendly colleagues of this time. Thanks also to Prof. Leonid Volgin about supervising of
scientific work at this time.
Thanks also to the TTU (Tallinn Polytechnical Institute at this time) and Tallinn Secondary
School No. 21 before that - for my basic and engineering education.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
List of Publications
[1] M.Min, O.Märtens, T.Parve. “Multisine Dynamic EBI spectrum Analyzer for in Vivo
Experiments,” Medical & Biological Engineering & Computing, Vol. 37, 1999,
Supplement 2, Part 1: Proc. of the European Medical and Biological Engineering
Conference EMBEC'99, November 4-7, 1999, Wien, Austria, pp.144-145.
[2] M.Min, O.Märtens, T.Parve. “Lock-in Measurement of Bio- Impedance Variations, ”
Measurement (Journal of IMEKO, Elsevier Science Publishers Inc.), vol.27, No. 1,
(Jan. 2000), pp. 21-28.
[3] O. Märtens, “DSP-based AC Measurement Unit”, in the Proceedings of the 1999 Finnish
Signal Processing Symposium FINNSIG ‘99 (University of Oulu, Finland, May 31,1999),
pp. 247-251.
[4] O.Märtens, “AC measurements with digital signal processing”, In: Proc. of the 6th
Baltic Electronics Conference BEC'98 ( 7-9 October 1998, Tallinn), Tallinn, Estonia,
1998, pp.121-122.
[5] O. Märtens, “DSP- Based RMS Measurement Converter”, Rahvusvahelise telekommunikatsioonipäeva konverentsi ettekannete materjalid TELEKOMMUNIKATSIOON
'98,15.mai 1998.(Papers of the International Telecommunication Day)- Tallinn Technical
University, 1998. - pp.66-72. (in English)
[6] O.Märtens, “ DSP-Based Device for Multifrequency Bio-Impedance Measurement,“ Medical & Biological Engineering & Computing, Vol.37, 1999, Supplement 1: Proc. of
the 11th Nordic-Baltic Conf. on Biomedical Engineering NBCBME'99, Tallinn, 6-10 June
1999, pp.165-166.
[7] O.Märtens, H.Märtin, M.Min, T.Parve and A.Ronk, “Digital post-processing of the bio
-impedance signal”, In Proc. X. Int. Conf. on Electrical Bio-Impedance.-ICEBI'98 (5-9
April 1998, Barcelona) Barcelona, Spain, 1998, pp.445-448.
[8] M.Min, H.Haldre, H.Härm, O.Märtens, H.Märtin, T.Parve, and A.Ronk., “Lock-in
measurement of the hand-hand bio-impedance,” In: Proc. of the 8th Int. IMEKO TC-13
Conf. on Measurement in Clinical Medicine and the 12th Int. Symp. on Biomedical
Engineering BMI'98 (September 16-19, 1998, Dubrovnik, Croatia), Zagreb, Croatia, 1998,
pp.2-49 - 2-52.
[9] O. Märtens, “Precise Synchronous Detectors”, Rahvusvahelise telekommunikatsiooni
päeva konverentsi ettekannete materjalid TELEKOMMUNIKATSIOON '99, 14.mai
1998. (Papers of the International Telecommunication Day)-Tallinn Technical University,
1999. - pp.74 -79 (in English).
9
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
[10] O. Märtens, T.Pungas, “Precision Average-Sensing AC/DC Measurement Converters,”
IEEE Trans. Instrum. Meas., vol.42, No1, (Feb. 1993), p. 71-73.
[11] O.I.Myartens and T.A.Pungas. “ Comparative Analysis of Analog Converters of MeanRectified AC Voltages”, Measurement Techniques. - Consultants Bureau, Plenum
Publishing Corp. - New York. ISSN 0543- 1972. Vol.33, No.1, January 1990, pp.70-73.
(Translation of Izmeritel'naya Tekhnika (in Russian), No.1, pp.45-47, January, 1990.)
[12] O.Märtens, “AC/DC Converter”, Author's certificate of the U.S.S.R. No. 1,431,020.
-1988, bullet. No.38.
[13] O.Märtens, “AC/DC Converter”, Author's certificate of the U.S.S.R. No. 1,676,038.
-1991, bullet. No.33.
[14] O.Märtens, K.Märtens, T.Pungas. “AC/DC Converter”, Author's certificate of the U.S.S.R.
No. 1,317,365.-1987, bullet. No.22.
[15] Pungas T. A. , Myartens O.I. ( T. Pungas, O. Märtens). “ Vysokotochnoe vypryamlenie
peremennogo napryazhenia metodom kommutacii kondensatora” (“Precise AC conversion
by method of the switched capacitors”)- Collection of the papers " Opyt, rezul'taty,
problemy: povyshenie konkurentosposobnosti radioelektronnoi apparatury" (Experience,
results, problems: increasing of the competition level of the radioelectronics equipment).
- Tallinn: "Valgus" Publishing House, 1986.- No.4, pp.140-146.
[16] Myartens O.I., Pungas T.A. (O.Märtens, T.Pungas) “Analiz vysokotochnogo izmeritel'nogo
vypriamitel'nogo preobrazovatel'ya s kommutiruemymi kondensatorami” (“Analysis of the
precise measurement rectifying converter with switched capacitors”)- Collection of
the papers " Opyt, rezul'taty, problemy: povyshenie konkurentosposobnosti radioelektronnoi apparatury" (Experience, results, problems: increasing of the competition
level of the radioelectronics equipment). - Tallinn: "Valgus" Publishing House, 1986.No.4, pp.147 -159.
Currently in the process of publishing:
[17] O.Märtens, “Precise Synchronous Detectors With Improved Dynamic Reserve,”
( submitted for publication at IEEE Transactions on Instrumentation and
Measurement in March 1999, revised in December 1999).
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
11
Industrial (research) applications
The results of the current research work concerning AC/DC measurement converters,
developed in 1983-1990, which are described in chapters 1 and 2, have been used in the
following industrial and scientific equipment:
•
AC/DC converters described in the p.1.5.1 and 1.5.2 in the voltmeter V3-60A and
measurement converter TV9-2 at the Production Association “RET” (Tallinn)
• AC/DC converter described in the p.1.5.3 in the multimeter Q1518 (Association “Vibrator”
in Leningrad)
• Synchronous detector according to proposed solutions in the spectrometric measurement
system “Faza” for the Soviet space lab “Mir”.
Latest DSP-based measurement solutions (1998-1999) according to p.3.2 and 3.3 have been
done in the collaboration and for the company “Pacesetter AB” (Sweden), a subsidiary of the
“St.Jude Medical Inc.” (USA).
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Introduction
Introduction to the topic
Current work is related to the design of improved Test &Measurement equipment, more
precisely to the development of precise measurement converters of parameters of the AC signals,
including both- analog and digital solutions.
The term “AC converters” is used here as general name for AC measurement devices, as often
(but not always) the AC value is firstly converted to DC value, and then further, the DC value is
measured.
Today mostly all of the precise measurements turn to measurements of electrical quantities.
Further more, most of the non-electrical values are first converted to electrical quantities, and
then measured. And large part of all measurements of electrical quantities are actually
measurements of some AC characteristics. And special electronical equipment is needed for such
purpose with the best performance obtainable.
Nowadays products become more and more sophisticated, giving more and more performance
in quality, more functionality etc. Measuring for testing and verification of such products need
measurement devices with improved characteristics- mainly in quality (accuracy, input
dynamical and frequency ranges), but also in functionality and flexibility, and also in the
productivity, that is the measurement speed in other words
Rapid development of the digital and computer and communication technologies, has two main
influences to the design of the precise measurement devices. First- as all the digital and
communication devices have more and more “bandwidth” (processing power and speed), so
more accuracy (resolution in “effective” bits) and speed (measurements per time-unit, e.g.
second) can be used. So there is a permanent pressure towards design of the measurement
devices with improved performance and speed. Second - as today much of the functionality of
the measurement units can be often easily implemented on programmable devices- e.g. digital
signal processors (DSP-s), we can flexibly change and add different functions of very
sophisticated signal processing and/or communication (networking of measurement devices etc.).
Practical question for this case is developing of the practical algorithms/software modules for
such measurement tasks, and also interfacing of these algorithms to existing hardware and
software platforms.
And still, in spite of the quick development of the digital/programmable electronics, a lot of the
signal (pre-)processing for measurement purposes can be most efficiently done with
analog/mixed circuits.
So the main directions of development of the measurement devices are the following:
1. Improvement of the basic metrological characteristics (accuracy, input frequency and
dynamical range) and the speed of measurements.
2. Efficient interfacing to the digital, computer and communication equipment, or even merging
of measurement devices together with this equipment.
3. Extended flexibility of functioning (programmability, universal and efficient hardware and
software platforms and architectures).
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
13
Analog versus digital solutions
One important question nowadays is - to choose analog or digital solution. Both approaches
has their own benefits, and limitations. Part of these limitations are fundamental, part of them
move quickly away. The last is especially true about digital signal processing solutions, where
more and more computational power at lower and lower price, and energy consumption is
available.
As we can see in this work, proposed analog solutions can have very good accuracy, and can
operate over the relatively wide frequency range with reasonable cost. As we can see for
synchronous (phase-sensitive) implementations of the proposed solutions, such analog converters
can have large dynamic reserve, not available yet in digital systems.
And the digital systems have much functionality and flexibility, as all depends on the actual
software. And mathematically implemented digital conversion algorithms are of course more
precise compared to analog "mathematical" units.
Probably there are a lot of cases, where the best solution is to combine analog and digital
methods, having an analog pre-processing with large dynamic reserve, and the digital postprocessing with effective "custom" presentation of the results.
Tasks of the work
Tasks of the current work have been:
1. To develop a new class of AC measurement converters having switched capacitors in the
negative feedback path, and having a very good combination of basic metrological
characteristics (accuracy, input frequency and dynamical range).
2. To develop the methodology for analyzing of such converters.
3. To do analysis of such converters in the frequency and time domains.
4. To develop synchronous detectors belonging to the same class, again to achieve a very good
combination of basic metrological characteristics (accuracy, input frequency and dynamical
range).
5. To propose, analyze and implement digital solutions for the AC measurements, based on
DSPs - for general purpose AC measurements, and for bio-impedance measurement
applications, with the aim to get improved performance in these fields, together with having
benefits of the digital processing (flexibility, programmability, functionality , etc.)
6. To test and verify the developed solutions and to get results in developed prototypes.
Methods and tools
The following tools and methods have been used in the current work:
• modelling/analyzing of the electronic circuits by using of the state space equations
• patent studies
• matrice algebra
• theory of the digital filters
• MATLAB software simulation
• practical prototyping and testing
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Contents of the current work
Chapter 1 is about the proposed class of precise AC /DC analog measurement converters,
utilizing the idea of switched capacitors in the negative feedback path of the op-amps for getting
maximum technical performance. Benefits of such solutions are explained. Formulas for nominal
values of transfer coefficients are given. Analysis of such circuits by state-space equations
(averaged over the half-periods) is proposed and implemented. Analytical formulas for
inaccuracies are found. Also the time-domain analysis is applied for such converters. Practical
implementations of prototypes are described.
Chapter 2 describes the solutions of the similar converters for phase-sensitive (synchronous)
implementations. Main formulas for such circuits are given. Practical application in the PLL
(phase-lock-loop) is shown.
Chapter 3 is about DSP-solutions of measurement converters, covering both hardware and
software parts of the solutions. Specially has been investigated solutions for the bio-impedance
measurements. Also the AC measurement device has been implemented, with emphasis to getting
of the true RMS (Root-Mean-Square) value measurement.
Conclusion and results of the work are given in the final part.
The dissertation includes the theoretical part and description of the practical implementation of
the proposed and analyzed solutions, done in parallel through-out all the work. .
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
15
1. Precise AC/DC converters with switched capacitors in the
feedback path
1.1. Basic functioning and circuit diagrams
Conventional AC/DC measurement converters include some kind of multiplier, multiplying the
input signal by certain reference signal, and is followed by some low-pass-filter to filter out the
useful signal from the ripple and noise. Often this multiplier is realized as a circuit, the gain of
which is discretely switched between +1 and -1 by the half-waves of the input signal [ 1, 2].
Historically the important idea has been using of the switched diodes in the negative feedback
path of the (op-)amp, proposed already by Ballantine in 1938 [3].
A special class of AC/DC measurement converters with improved performance has been
developed, and is described below. Initially first of such converters, described lower, were
proposed by Mr. Toom Pungas [4, 5], and implemented in the industrial measurement devices
V9-10 and V3-60.
Using of some capacitors in the AC conversion and filtering circuits has been known earlier [68], and these ideas has been further developed to get better performance.
The rectifying diodes switching capacitor(s) in the feedback path of op-amps, is the most
important feature of proposed solutions. The circuit diagrams are proposed, described and
discussed in the current chapter.
All the proposed circuits utilize the same basic ideas, allowing to achieve outstanding
specifications:
1. The op-amp is used as an AC amplifier, and is separated from the signal path by capacitors in
most of the proposed circuits. Therefore these detectors have a small or no DC offset and drift
voltages, allowing to detect small signals precisely.
2. Rectifying diodes are operating in the negative feedback path of the op-amp, so their “on”resistances do not generate inaccuracy in the first approximation.
3. These measurement converters suppress also the ripple, as the detector output is connected for
AC component (via a capacitor) to the inverting input of the op-amp.
4. The negative feed-back path of the op-amp is active in the both half-periods of the input
signal, closed via switched capacitor. So the instant values of the input signal are averaged
(integrated) on these capacitors, and as the rectified and averaged signal depends only on
useful signal component, all other components are suppressed, and the converter is not
sensitive to instant values of the input signal.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
5. There are proposed the circuit diagrams, in which the gain is unity and almost not depending
on the resistor ratios, and also other circuit diagrams, for which the gain directly depends on
resistor ratios, and practically any reasonable gain can be so achieved for useful signal
component.
One way, how this has been achieved, is usage of the circuit diagrams, where, for input AC
voltage the two (equal) resistors are switched in parallel, and for output DC signal the same
resistors are switched in series, and where the transfer coefficient is determined so by the ratio
of the serial and parallel resistances of the same resistors, what is not depending directly on the
resistor ratio (e.g. [ 9]).
The converters described here actually measure an “average” value, or more correctly, the
“average-rectified” value of the input signal. Often measuring of the RMS (Root-Mean-Square)
value ,or peak (or peak-to-peak) value is needed. For fixed ‘wave-form’ (e.g. sine-wave) the
ratio between different values (‘average’, ‘RMS’ ‘peak’) is constant, and can be easily recalculated [10, 11].
The half-wave converter according to Fig. 1.1 [5, 12, 16] is functioning in the following way.
Input AC voltage Uin causes an AC current with the value of Uin/R1 through the resistor R1, as
the other end of the R1 is connected for AC to the inverting input of the op-amp A1 (through
capacitor C2). As an input resistance of the op-amp is high, practically the same current Uin/R1
flows (via C1, V1 or V2, and C3) to the output of the op-amp. The diodes V1 and V2, controlled
by the input signal, rectify this AC current by half periods of the input signal. A DC component of
one half-periods of the rectified signal flows through V1 and R2, and during other half-periods
via V2 and R1. The DC voltage drops are generated accordingly on resistors R1 and R2. The DC
voltage drop on R1 serves as the output voltage of the detector. So, as the AC voltage to current
conversion, and conversion of the rectified DC current to the output DC voltage are performed on
the same resistor R1, the gain of the converter does not depend on any resistor ratio. The detector
suppresses AC ripple, as for AC the inverting input of the op-amp is tied to the output of the
detector. And this circuit has also no output voltage offset, as the op-amp is DC isolated from the
output of the detector by the capacitor.
Resistor R3 is for determining the DC voltage at the output of the op-amp, and must have
relatively high value not to cause any significant AC inaccuracy, as it is in parallel with “off”resistances of V1 and V2 for AC.
Because this detector is a half-wave converter, the transfer coefficient of the converter is for
input sine-wave with equal half-periods
Uout= (√2/π)* Uin
,
(1.1)
where Uin is the root-mean-square (RMS)-value of the input AC voltage , and Uout is the output
DC voltage.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
17
Fig. 1.1 Half-wave AC/DC converter
AC/DC converter in Fig. 1.2 operates in the similar way. Only the input AC current is
determined by the parallel resistance of R1 and R2. So the gain for this full-wave AC/DC
converter is
Uout= (√2/π)* R2/(R1||R2) * Uin
,
(1.2)
,
(1.3)
In the case R1=R2 the gain of the converter is
Uout= (2*√2/π)* Uin
Fig. 1.2 Full-wave AC/DC converter A.
The AC/DC converter by Fig. 1.3 operates also in the similar way. The input AC current is
determined here again by the parallel resistance of R1 and R2. Only the output DC voltage is
18
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
generated on the serial connection (for DC) of R1 and R2. So the gain for this full-wave
converter is
Uout= (√2/π)* (R1+R2)/(R1||R2) * Uin
,
(1.4)
,
(1.5)
In the case R1=R2 the gain of the converter is
Uout= (4*√2/π)* Uin
Fig. 1.3 Full-wave AC/DC converter B.
For AC/DC measurement converters by Fig.1.1...1.3 it is shown lower, that due to the
asymmetrical circuit diagram (for one half-periods the negative feedback path of the op-amp is
closed directly, for second half-period via “switched capacitor”), these structures can have
significant low-frequency error. The analysis of this error is given in the p. 1.2.
Proposed circuit diagrams with improved low-frequency performance have been proposed and
described just here lower [13].
The converter by Fig. 1.4 functions again in the similar way, as previous converters. The input
AC current is determined here again by the parallel resistance of the resistors R1 and R2. One
DC output voltage is generated on the resistor R1 and other output voltage on the resistor R2:
Uout1= (√2/π)* (R1/R1||R2) * Uin
,
(1.6)
Uout2= - (√2/π)* (R2/R1||R2) * Uin
,
(1.7)
,
(1.8).
In the case R1=R2=R
Uout1= -Uout2= (2√2/π)* Uin
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
19
An advantage of this converter is an improved low-frequency accuracy, as this circuit is
symmetrical for both half-waves. The analysis of this inaccuracy is given in the p.1.3.
Fig. 1.4 Full-wave AC/DC converter C.
The converter by Fig. 1.5 operates again in the similar way. The input AC current is determined
here again by the parallel resistance of R1 and R2. But these resistors are blocked for DC by
capacitors C1 and C2. So the rectified DC current component will generate first and second
output DC voltages on resistors R3 and R4 accordingly:
Uout1 = (√2/π)* (R3/R1||R2) * Uin
,
(1.9)
Uout2 = - (√2/π)* (R4/R1||R2) * Uin
,
(1.10)
The required gain of the converter can easily be achieved, as it is directly determined by the
resistor ratio. At the same time, due to the integrating capacitors C3 and C4 in the feedback path
the circuits are tolerant to high instant values of the total input signal.
In this circuit the resistor R5 is for determining the DC output voltage of the op-amp, and must
have again relatively high value for avoiding the AC inaccuracy.
20
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig. 1.5. Full-wave AC/DC converter D.
The converter by Fig. 1.6 operates also in the similar way. The input AC current is determined
here again by the parallel resistance of R1 and R2. But these resistors are blocked for DC by
capacitors C1 and C2. So the rectified DC current component will generate the first and second
output DC voltages on resistors R1+R3 and R2+R4 accordingly:
Uout1 = (√2/π)* ((R1+R3) /R1||R2) * Uin
,
(1.11)
Uout2 = - (√2/π)* ((R2+R4)/R1||R2) * Uin
,
(1.12)
Fig. 1.6. Full-wave AC/DC converter E.
The converter by Fig. 1.7 [14] operates in the following way. At the common point of C2 and
C3 the voltage Ux= ~Uin * (R2/R1) arise, and this AC voltage generates the total AC current
Ux/ (R2|| R3||R4) from the output of the op-amp. This AC current is rectified by diodes V1 and
V2, and the rectified DC current component will generate the first and second output DC
voltages on resistors R3 and R4 accordingly:
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Uout1 = (√2/π)* (R2/R1) * (R3 / (R2|| R3||R4)) * Uin
,
(1.13)
Uout2 = - (√2/π)* (R2/R1) * (R4/(R2|| R3||R4)) * Uin
,
(1.14)
21
Fig. 1.7 Full-wave AC/DC converter F.
Measurement converter according to Fig. 1.8 works in the similar way, as the previous one. Only
additionally to parallel connection of R2, R3, and R4 there is the resistor R6 in the AC path:
Uout1 = (√2/π)* (R2/R1) * (R3 / (R2|| R3||R4||R6)) * Uin
,
(1.15)
Uout2 = - (√2/π)* (R2/R1) * (R4/(R2|| R3||R4||R6)) * Uin
,
(1.16)
Fig. 1.8. Full-wave AC/DC converter G.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
1.2. Low-frequency error analysis of the asymmetrical converter
One special feature of the described AC/DC converters with switched capacitors in the
negative feedback path is the presence of the special component of the low-frequency conversion
inaccuracy. This appears due to the final frequency-depending impedance of these switched
capacitors. First we shall do this analysis for the AC/DC converter according to the fig. 1.2.
Equivalent circuit diagrams for the “positive” and “negative” half-waves of the input signal are
given in the fig. 1.9, a) and b) respectively.
Fig. 1.9. Equivalent circuit diagrams for AC/DC converter by fig.1.2
Let us assume, that input resistors are equal,
R1=R2=R ,
(1.17)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
23
and the input signal is a pure sine-wave
Uin(t) = Um ⋅sin( ωt)
,
(1.18)
where Um is the amplitude value of the input sine-wave, and ω=2πf- input frequency, t-time.
We assume also that the current through capacitor C2 is close to zero, and that the output voltage
does not have any ripple (inaccuracy from the ripple can be estimated by known methods
separately).
During the “positive” input half-waves the current through the capacitor C1 is defined by the
following equation
iC’(t)= [(Uin’ (t) - Uout ] / R
,
(1.19)
where iC’(t) is the current through C1 during this half-period as a function of the input Uin’ (t)
and output Uout voltages.
So a voltage on the capacitor C1 is defined by the equation
t
UC’(t) = UC,0 + (1/C) ⋅∫ iC’(t) dt ,
(1.20)
0
where UC, 0 is the initial value of the voltage on the C1, C is the capacitance of the C1.
Combining (1.17)....(1.20) together we shall get
1- cos( ωt)
UC’(t) = UC0 +  Um ωRC
t
 Uout
RC
,
(1.21)
So the voltage on the C1 at the end of the “positive”/ beginning of the “negative” half-wave has
the value:
2
π
UC,1 = UC’ ( T/2) = UC,0+  Um +  Uout
ωRC
ωRC
,
(1.22)
Average value of this voltage has the value of the integral of (1.21) at 0...T/2:

T/2
1
UC’ = (2/T) ∫ UC’ (t) dt = UC,0 +  Um 0
ωRC
π

Uout ,
2ωRC
(1.23)
For “negative” half-waves the current through C1 is defined as
iC’’ (t) = [ Um sin ( ωt) - UC’’ (t) + Uout ] / R ,
(1.24)
24
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
where the time is counted from the beginning of the “negative” half-wave.
Voltage on the C1 during the “negative” half-wave is the following:
t
UC’’(t) = UC,1+ (2/T)∫ iC’’ (t) dt ,
0
This equation can be changed to
UC’’(t) = UC,1 + (Uout - UC,1) ⋅ (1- e
-t /(RC)
(1.25)
)+
sin [ ωt - arctan (ωRC) ]
ωRC ⋅ e - t/(RC)
+{  +  } ⋅ Um .
√ [ 1+ (ωRC) 2 ]
1+ (ωRC) 2
(1.26)
From this equation we shall get the instant value of this voltage at the end of the “negative” halfwave (and beginning of the “positive” half-wave):
UC,0=UC’’(T/2)=UC,1+ (Uout - UC,1)⋅( 1-e
-π/(ωRC)
ωRC⋅ e -π/(ωRC)
)+ 
,
1+ (ωRC) 2
(1.27)
and the average value of this voltage over the “negative” half-wave will be

T/2
ωRC
UC’’= (2/T) ∫ UC’’(t)dt =[ 1 +  (e
0
π
ωRC
+  (1-e
π
-π/(ωRC)
) - 1)⋅Uout +
2 1+0.5⋅(ωRC)2) ⋅( 1-e -π/(ωRC) )
-π/(ωRC)
)⋅UC,1+   ⋅ Um .
π
1+ (ωRC) 2
(1.28)
Average value of the current through capacitor C3, and so the DC component of the input current
of the converter, are both zero. From this fact we can get, that DC components of currents through
R1 and R2 are equal (with opposite sign), and

(UC - Uout) /R= Uout /R
.
(1.29)
From that expression we shall get




Uout = 0.5⋅ UC = 0.25 ⋅Uc’ + 0.25⋅UC’’
,
(1.30)
So, the transfer coefficient of the converter is defined by the system of equations (1.22) , (1.23),
(1.27), (1.27) and (1.30).
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
25
For more compact way of writing of these equations we shall use the symbols x and y.
x= ωRC
y= e - π/x
,
,
(1.31)
(1.32)
Finally the system of equations (1.22) , (1.23), (1.27), (1.28) and (1.30) will have the following
form:
-
UC,0- UC,1 - (π/x) ⋅ Uout= - (π/x) ⋅ Uin
,
(1.33)

UC,0- y⋅UC,1-(1-y) Uout= (π/2) ⋅x/(1+x2)⋅(1+y)⋅ Uin ,
(1.34)
UC,0+ (x/ π)⋅ (1-y) ⋅ UC,1+ [ - (π/2x) -3 - x/π⋅(1-y)] ⋅ Uout=
1+ 0.5⋅x2 ⋅ (1-y)

= [- (π/2x) -   ] ⋅ Uin
1+ x2
,
(1.35)
As the average rectified value is defined by amplitude value:

Uin= 2⋅ Um/π
,
(1.36)
So, for medium and high frequencies (x>>1, y --> 0) , the system of the equations has the
solution Uout= Uin_aver. The low-frequency error (as relative difference of the transfer
coefficient from the nominal value) has the form
δ (x)= [ Uout (x) - Uout 0 ]/Uout 0 ⋅100%
.
This relationship δ (x), calculated on PC, is shown graphically in Fig. 1.10.
(1.37)
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig. 1. 10. Low-frequency error for non-symmetrical circuit
Results of this analysis show the presence of significant low-frequency inaccuracy, and the need
of the using capacitors with large capacitance. For example, if R1=R2= 1 kΩ, and frequency
range begins from 20 Hz, then for achieving the inaccuracy less than 0.03%, the C1 must have
the value more than 100 µF. At the same time using of such capacitors does not allow to achieve
the settling times better than 10 s (for inaccuracies in order of 0.03% or better). And further
more, capacitors of such value have often dielectrical absorption effects, and real settling times
will be in tens of seconds.
Such kind of the low-frequency inaccuracy can be explained by the non-symmetrical circuit
diagram of the converter, where for one half-periods of the input signal the feedback path of the
op-amp is closed via switched capacitor (C1), and for other half-periods is closed directly.
For achieving the improved low-frequency performance there has been proposed a new
(symmetrical) circuit diagram, analyzed in the p. 1.3..
1.3. Low-frequency error analysis of the symmetrical converter
The converter with improved low-frequency performance has been proposed in Fig.1.4 [12,
13]. In the circuit diagram of this converter the negative feedback path is closed via one switched
capacitor (C1) for one half-periods, and via an other switched capacitor (C2) for other halfperiods. Additional benefit of the proposed circuit is the presence of the second output (having
opposite polarity).
Lower we shall analyse the low-frequency inaccuracy for this circuit.
Equivalent circuit diagrams for the “positive” and “negative” half-waves of the input signals are
given in the Fig. 1.11, a) and b) respectively.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
27
Fig. 1. 11. Equivalent circuit diagrams for AC/DC converter by Fig.1.4
For “positive” half-waves of the input signal the currents through the capacitors C1 and C2
have the value:
iC1’ (t)= iC2’ (t)= [ Uin’(t) -U C1’(t)] / R
,
(1.38)
For this equation we shall get the voltage of C1 for the sine-wave input signal:
sin [ ωt - arctan (ωRC1) ]
UC1’(t)= [ + sin (arctan(ωRC1)) ⋅ e - t/(RC1) ] ⋅Um+
√ [ 1+ (ωRC1) 2 ]
+ e - t/(RC1) ⋅UC1,0
.
.
(1.39)
Here and henceforth we use the marking with UC1,0 and UC2,0 the voltages on C1 and C2 at the
beginning of the ”positive” half-period, and by UC1,1 and UC2,1 the voltages on C1 and C2 at the
beginning of the ”negative” half-period.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
28
From the last equation we shall get:
UC1,1= UC1’ (T/2) = e
-π/(ωRC )
1
2
ωRC1⋅ (1+ e -π/(ωRC1))
⋅UC1,0 +  ⋅  ⋅ Um ,
π
1+ (ωRC1) 2
(1.40)
So, the average value of the voltage on C1 during the “positive” half-period has the following
value:

T/2
ωRC1
UC1’ = (2/T) ∫ UC1’ (t) dt =  ⋅ (1- e -π/(ωRC1))) ⋅ UC1,0 +
0
π
+
2 1+ 0.5 ⋅(ωRC1) 2
 ⋅  ⋅ (1- e
π
1+(ωRC1) 2
-π/(ωRC )
1 )⋅
Um
,
(1.41)
As the currents through capacitors C1 and C2 have the same value, so the charges of these
capacitors change in the same way:
UC1,1 - UC1,0 = (C2/C1) ⋅ (UC2,1 - UC2,0)
,
(1.42)


UC1’ - UC1,0 = (C2/C1) ⋅ (UC2’ - UC2,0)
.
(1.43)
In the same way we can get the equations for the “negative” half-waves of the signal:
UC2,0= e
-π/(ωRC1)
⋅UC2,1 +

ωRC2
UC2‘’ =  ⋅ (1- e
π
+
ωRC2⋅ (1+ e -π/(ωRC1))
 ⋅ Um
1+ (ωRC1) 2
-π/(ωRC )
2 ))
,
(1.44)
⋅ UC2,1 +
2 1+ 0.5 ⋅(ωRC2) 2
 ⋅  ⋅ (1- e
π 1+(ωRC2) 2
-π/(ωRC )
2 )⋅
Um
.
(1.45)
As the average current through C3 is equal to zero, the average currents (DC components)
through R1 and R2 are equal, and output voltages of the converter are also equal:
Uout 1= Uout 2
where
,
(1.46)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Uout 1= 0.5⋅ (
and
Uout 2= 0.5⋅ (


UC1’ + UC1’’)


UC2’ + UC2’’)
29
(1.47)
(1.48)
After conversions of the equations (1.40)...(1.48) we shall get the system of equations

- y1⋅ UC1,0 + UC1,1 = (π/2) ⋅ [x1/( 1+ x1 2)] ⋅(1+y1) ⋅ Uin
,
(1.49)
- y1⋅ UC2,1 + UC2,0

= (π/2) ⋅ [x2/( 1+ x2 )] ⋅(1+y2) ⋅ Uin
2
,
(1.50)
UC2,0 + (x2/ π) ⋅ (1-y2) ⋅ UC2,1 + (x1/x2) ⋅ [(x1/π) ⋅(1-y1)-1] ⋅ UC1,0 -- 2⋅Uout=
x1 1+0.5 ⋅ x12 ⋅ (1-y1) 1+0.5 ⋅ x22 ⋅ (1-y2)

=[  ⋅  +  ] ⋅ Uin ,
x2 1+ x12
1+x22
(1.51)
UC1,1 + (x1/ π) ⋅ (1-y1) ⋅ UC1,0 + (x2/x1) ⋅ [(x2/π) ⋅(1-y2)-1] ⋅ UC2,1 -2⋅Uout=
x1 1+0.5 ⋅ x22 ⋅ (1-y2) 1+0.5 ⋅ x12 ⋅ (1-y1)

=[  ⋅  +  ] ⋅ Uin ,
x2 1+ x22
1+x12
(1.52)
UC1,0 - UC1,1 - (x1/x2) ⋅ UC2,0 + (x1/x2) ⋅ UC2,1 = 0
(1.53)
,
where we have used the following notation:
x1= ω⋅R⋅C1
,
(1.54)
x2= ω⋅R⋅C2
,
(1.55)
y1= e-π/x1
,
(1.56)
y2= e-π/x2

Uin= 2⋅ Um/π
,
(1.57)
,
(1.58)
For medium and higher frequencies (x1>>1, x2>>1, y1-->0, y2-->0), according to
(1.49)...(1.53), the output voltages of the converter are equal to the average rectified value of the
input signal:

Uout1 = Uout2= Uin  .
(1.59)
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
The system of the equations (1.49)...(1.53) has been solved numerically, expressing a lowfrequency inaccuracy as a function of the average normalized frequency x
x= 0.5⋅x1+0.5⋅x2=ω⋅R ⋅ (0.5⋅ C1 + 0.5⋅ C2)
(1.60)
for different differences of x1 and x2 (actually of capacitors C1 and C2)
δC = (C1 - C2) / (0.5⋅ C1 + 0.5⋅ C2) = (x1-x2) /x
.
(1.61).
The calculation of this inaccuracy of this low-frequency error (for the AC/DC converter by
Fig.1.4) has been done on PC.
The results of the calculations are graphically given in the Fig, 1.12, for “matching”
inaccuracies of C1 and C2 of values 5%, 10% and 20%. The results show the significant
improvement of the low-frequency accuracy, compared with asymmetrical circuits, e.g.
according to Fig.1.2. As a result, the capacitors with much reduced values can be used, and the
conversion speed will be much higher.
Brief medium-frequency inaccuracy analysis has been done in the previous chapter. More
detailed analysis will be done in the next chapter.
Fig. 1. 12. Low-frequency error for symmetrical circuits
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
31
1.4. Analysis of the proposed circuits with state-space equations
1.4.1. Introduction to analysis
AC/DC converters with switched capacitors in the negative feedback path can give very good
performance, as shown theoretically in the previous parts of this work, and also by practical
implementations. Anyhow, such converters include often many capacitors, and so the analysis of
such high-order nonlinear circuits is quite complex. And the optimization of parameters is very
important to get good combination of the conversion speed/settling time from one side, and
accuracy on the lower frequencies from the other side.
An universal method of the analysis of such complex circuits is the method of the state-space
equations [17].
To simplify the analysis, it is assumed, that the diodes are working as electronic switches,
having some maximal forward drop voltage value, and some minimal reverse resistance. This
simplification can be done, as the diodes are in the negative feedback path, and their “nonidealities” are therefore suppressed. And using of maximum/minimum values of the mentioned
non-idealities will give maximal estimation of the inaccuracy. Real performance of the circuits
will be better of course. At the same time, using of the (constant) limit values for conversion inaccuracy sources allow to use linear equations for analysis. For non-ideality of diodes we
introduce voltage generators for forward voltage drop, and reverse resistances to the equivalent
circuit diagrams. Non-zero phase shift of switching time moments can be analyzed later on.
Generally such AC/DC converters can be described by a system of differential equations
during one half-waves of the input signal, and by another system of the differential equations for
other half-waves. These equations can be derived from the equivalent circuit diagrams of the
corresponding converters.
Such kind of analysis was preliminarily proposed by Mr. Rein Kipper (together with Mr.
Toom Pungas) [18]. Some of such analysis has been done in [ 19, 20].
Let us assume that the (differential) equations (state space equations) for first (“positive”) halfperiods have the form
iC1’= C1 ⋅ (dUC1’/dt)= -A1,0‘ ⋅ Uin (t) + A1,1’⋅ UC1 +A1,2’⋅ UC2 +
+ A1,3’⋅ UC3 + .......+ A1,N’⋅ UCN - A1,D1’⋅ UD1 + A1,D2’⋅ UD2
iC2’= C2 ⋅ (dUC2’/dt)= -A2,0‘ ⋅ Uin (t) + A2,1’⋅ UC1 +A2,2’⋅ UC2 +
+ A2,3’⋅ UC3 + .......+ A2,N’⋅ UCN - A2,D1’⋅ UD1 + A2,D2’⋅ UD2
...................................................................
iCN’= CN ⋅ (dUCN’/dt)= -AN,0‘ ⋅ Uin (t) + AN,1’⋅ UC1 +AN,2’⋅ UC2 +
+ AN,3’⋅ UC3 + .......+ AN,N’⋅ UCN - AN,D1’⋅ UD1 + AN,D2’⋅ UD2
,
(1.62)
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Here, and in the coming discussion, we use the following notations:
iCi - current through i-th capacitor
Ci - capacitance value of the i-th capacitor
Uin(t) - input voltage
Uci - voltage on i-th capacitor
UD1 and UD2 - voltage drop on diodes D1 and D2,
dUCi/dt - is the derivative of the voltage on the i-th capacitor
‘-stands for the variables for the first (“positive”) half-period, and
‘’-for the second (“negative”) half-period.
AN,i - is marking corresponding coefficients of the equations, defined by the actual circuitry
behind these formal equations.
Here, as well as in the following discussions, the voltage drops on the switching diodes are
handled as independent variables, similarly with input voltages.
In the same manner we can write the formal set of equations for the other (“negative” halfperiods):
iC1’’= C1 ⋅ (dUC1’’/dt)= -A1,0‘’ ⋅ Uin (t) + A1,1’’⋅ UC1 +A1,2’’⋅ UC2 +
+ A1,3’’⋅ UC3 + .......+ A1,N’’⋅ UCN - A1,D1’’⋅ UD1 + A1,D2’’⋅ UD2
iC2’’= C2 ⋅ (dUC2’’/dt)= -A2,0‘’ ⋅ Uin (t) + A2,1’’⋅ UC1 +A2,2’’⋅ UC2 +
+ A2,3’’⋅ UC3 + .......+ A2,N’’⋅ UCN - A2,D1’’⋅ UD1 + A2,D2’’⋅ UD2
. ..................................................................
iCN’’= CN ⋅ (dUCN’’/dt)= -AN,0‘’ ⋅ Uin (t) + AN,1’’⋅ UC1 +AN,2’’⋅ UC2 +
+ AN,3’’⋅ UC3 + .......+ AN,N’’⋅ UCN - AN,D1’’⋅ UD1 + AN,D2’’⋅ UD2
,
(1.63)
For compact presentation of the sets of equation we shall use the matrice presentation. For that
purpose we shall introduce the following matrices and vectors:
C1
C2
....
CN




,
(1.64)

A0’ = 


A1,0’
A2,0’
.....
AN,0’




,
(1.65)

A0’’ = 


A1,0’’
A2,0’’
.....
AN,0’’




,
(1.66)
C=




O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)

A’ = 


A1,1’ ;
A2,1’ ;
.....
AN,1’ ;
A1,N’
A2,N’
......
AN,N’





A’’ = 


A1,1’’ ;
A1,2’’ ; ...... ; A1,N’’
A2,1’’ ;
A2,2’’ ; ...... ; A2,N’’
..... ....... ....... ......
AN,1’’ ;
AN,2’’ ; ...... ; AN,N’’




A1,2’ ;
A2,2’ ;
.......
AN,2’ ;
•
 dUC1 / dt 
UC =  dUC2 / dt 

........ 
 dUCN / dt 
•
 dUC1‘/ dt
UC’ =  dUC2 ‘/ dt

........
 dUCN ‘/ dt
...... ;
...... ;
.......
...... ;
,
(1.67)
,
(1.68)
,
(1.69)




,
(1.70)
•
 dUC1 ‘’/ dt 
UC’’ =  dUC2 ‘’/dt 

........ 
 dUCN ‘’/ dt 
,
(1.71)

AD1’= 


A1,D1’ 
A2,D1’ 
..... 
AN,D1’ 
,
(1.72)
 A1,D1’’ 
AD1’’=  A2,D1’’ 
 ..... 
 AN,D1’’
,
(1.73)

AD2’= 


,
(1.74)
A1,D2’ 
A2,D2’ 
..... 
AN,D2’ 
33
34
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)

AD2’’= 


A1,D2’’ 
A2,D2’’ 
..... 
AN,D2’’
,
(1.75)
Using of the presentation of variables as matrices/vectors, given above, the sets of equations
(1.62) and (1.63) will take the following form:
•
C ⋅ UC’ = -A0’⋅ Uin + A’ ⋅ UC - AD1’⋅ UD1 - AD2’⋅ UD2
,
(1.76)
•
C ⋅ UC’’ = -A0’’⋅ Uin + A’’ ⋅ UC - AD1’’⋅ UD1 - AD2’’⋅ UD2
,
(1.77)
Further we assume that the analysis is done for medium frequencies, for which the instant
values of voltages on capacitors are not changing during one period of the input signal. And from
other side, no high-frequency side effects are present (parasitic capacitances etc.). In other
words, for medium frequencies the transfer coefficient of the converters is frequency
independent. Further more, the currents through capacitors can be handled as “slowly” changing,
and for this “slowly changing” value we can take the average values of the real currents for
different half-periods. And instead of the instant values of the input voltage Uin(t) we can use the
averaged value over one half-period of the input signal- for positive half-period:

Uin(t) = |Uin|
,
( 1.78)
and for negative half-period:

Uin(t) = - |Uin|
.
( 1.79)
Now we can write the equation set for the whole period in the following way:
iC = h ⋅ iC‘ + (1- h) iC ‘’
,
(1.80)
where h is the relative duration of the first (“positive”) and (1-h) of the second (“negative”) halfperiod.
For signals with equal duration of the half-periods (e.g. sine-wave) h= 1/2, the set of statespace equations will have the following form:
•

C ⋅ UC’’ = - [h⋅ A0’- (1-h) ⋅ A0’’ ] ⋅ |Uin| + [h⋅ A’+ (1-h)⋅ A’’] ⋅ UC - [h⋅ AD1’+ (1-h) ⋅AD1’’] ⋅ UD1 - [h⋅ AD2’+ (1-h) ⋅AD2’’] ⋅ UD2
,
(1.81)
For more compact presentation of the equations we shall further use the following notations:
A0 = h⋅ A0’- (1-h) ⋅ A0’’
,
(1.82)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
A = h⋅ A’- (1-h) ⋅ A’’
,
(1.83)
AD1 = h⋅ AD1’+ (1-h) ⋅AD1’’
,
(1.84)
AD2 = h⋅ AD2’+ (1-h) ⋅AD2’’
,
(1.85)
So the circuits can be described by general set of equations:
•
C ⋅ UC = -A0⋅ |Uin| + A ⋅ UC - AD1⋅ UD1 - AD2⋅ UD2
,
35
(1.86)
Now, further we shall consider the two important special cases of analysis of the AC/DC
converters, described in the current work.
The first important case is finding of the static transfer coefficient, without taking into account
the settling process. For this case the voltages on the capacitors are not changing, or in other
words, the derivative of the voltages on capacitors are equal to zero:
•
UC’ =
0
0
...
0




For that case the matrice equation will have the following form:

UC = A-1⋅A0⋅ |Uin| + A-1⋅ AD1⋅ UD1 + A-1⋅AD2⋅ UD2
,
,
(1.87)
(1.88)
The second important case is the time domain analysis of the settling process of the output signal,
e.g. for getting the settling time of the converter. For that purpose we can simply solve the set of
state-space equations by numerical methods. In [17] it has been shown that correct solution for
the difference equation (1.88) has the following form:

A⋅∆t
A⋅∆t
-1
UC ( t+ ∆t) = e ⋅ UC (t) + (e
- 1) ⋅ A ⋅ A0⋅ Uin 
,
(1.89)
The most convenient way of solving of the equations set above, is to use the difference equation
of Euler, what is the piece-wise-constant approximation of the vector UC.

UC ( t+ ∆t) = (1+A⋅∆t) ⋅ UC (t) + ∆t⋅ A0⋅
Uin 
.
(1.90)
In the starting point the vector UC must be initialized, typically with zero values. Following
values we shall get from this matrice equation. The time step ∆t for numerical solving of the
equation (integration) must be much smaller than the time constants of the AC/DC converter.
Results of analysis of the practical circuits according to Fig.1.1 - 1.6 are given in the next
points.
1.4.2. Analysis of the circuit by Fig. 1.1.
The equivalent circuit diagrams for the half-period converter according to Fig. 1.1. for
“positive” and “negative” half-waves of the input signal are given in Fig.1.13 a) and b). The
diagram by Fig.1.13 a) can be described by the following equations:
36
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig 1.13 Equivalent circuit diagram for the half-wave converter
e1 = (-UC1 + UC2 +UC3- UD2) /K1
,
(1.91)
iR2= (Uin-UC2+e1) /R2
,
(1.92)
iC2= (UC1-UC2-UC3+UD2) / R3
,
(1.93)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
iR1= (-UC1+ UC2 -e1) / R1
,
(1.94)
iC1 = iR2 - iREV1- iC2
,
(1.95)
iC3 = iR1 - iC1 - iREV1
,
(1.96)
iREV1= (UC1+ UD2) / RREV1
,
(1.97)
37
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “positive” half-periods:
iC1 = (1/R2) ⋅ Uin + [-1/ (K1⋅ R2) - 1/ RREV1 - 1/ R3]⋅ UC1 +
+ [ -1/R2+1/R3 +1/(K1⋅R2)] ⋅ UC2+(1/R3 +1/ (K1⋅ R2) ⋅ UC3 +
+ [-1/R3-1/RREV1 - 1/(K1⋅R2)] ⋅ UD2
,
(1.98)
iC2 = (1/R3) ⋅ UC1 - (1/R3) ⋅ UC2 - (1/R3) ⋅ UC3 + (1/R3) ⋅ UD2 ,
(1.99)
iC3 = -(1/R2) ⋅Uin +[-1/R1+1/ (K1⋅ R1)+1/ (K1⋅ R2) +1/R3]⋅ UC1 +
+ [1/R1+1/R2-1/ (K1⋅ R1) -1/ (K1⋅ R2) -1/R3] ⋅ UC2 +
+ [- 1/R3 -1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UC3 +
+ [1/R3 +1/(K1⋅ R1) +1/(K1⋅ R2) ]⋅ UD2
.
(1.100)
For “negative” half-periods (equivalent circuit diagram by Fig. 1.13 b) ) the circuit can be
described by the following equations:
iC1= iR1 - iREV2
,
(1.101)
e1 = (UC2+UC3+UD1) /K1
,
(1.102)
iR2 = (Uin - UC2 + e1) / R2
,
(1.103)
iC2 = (-UC2-UC3 - UD1) / R3
,
(1.104)
iR1 = (-UC1 + UC2 - e1) / R1
,
(1.105)
iD1 = iC1 + iC2 - iR2
,
(1.106)
iC3 = iD1 + iREV2
,
(1.107)
iREV2 = (UC1 + UD1) / RREV2
,
(1.108)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
38
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “negative” half-periods:
iC1 = (-1/ R1 - 1/ RREV2 )⋅ UC1 + [ 1/R1 - 1/(K1⋅R1)] ⋅ UC2 - (1/ (K1⋅ R1) ⋅ UC3 + [-1/(K1⋅R1) -1/ RREV2] ⋅ UD2
iC2 = - (1/R3) ⋅ UC2 - (1/R3) ⋅ UC3 - (1/R3) ⋅ UD1
,
(1.109)
,
(1.110)
iC3 = -(1/R2) ⋅Uin -(1/R1)⋅ UC1 +[1/R1+1/R2-1/ (K1⋅R1) -1/(K1⋅R2)-1/R3] ⋅
⋅ UC2 + [- 1/R3 -1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UC3 +
+ [-1/R3 -1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UD2
.
(1.111)
In the case h=1/2 (equations of the both half periods have the same weight 1/2), and using the
notation
R1= R ⋅ (1+ δR /2)
,
(1.112)
R2= R ⋅ (1- δR /2)
,
(1.113)
,
(1.114)
we shall get the averaged matrice coefficients for the circuit
A0=
A1=
 -1- δR /2 
 0

 0

-1+δR /2-(R/RREV1) -(R/RREV2)-1/K1-R/R3

1
 -2+ δR + 2/K1+R/R3
 1/K1+ R/RREV2
AD1=  1
 2/K1 + R/R3



 1/K1+ R/R/RREV1+ R/R3
AD2=  -1
 - 2/K1 - R/R3



; -δR /2+R/R3 ; R/R3

; -2
; -2

;4-4/K1-2⋅R/R3; -4/K1-2⋅R/R 
,
(1.115)
,
(1.116)
,
(1.117)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
39
Further we shall find the determinant of the system, using only the the first order errors. The
determinant of the system has the value:
D = -8 ⋅ [1+ 1/K1 + R/(2⋅R3) + R/RREV1+ R/RREV2 ]
,
(1.118)
the determinant of the output voltage (voltage on the capacitor C2) has the value:
D2 = - 4
,
(1.119)
and determinants for the output voltage from the voltage drops on the diodes:
DD1= 4/K1 + (2⋅R)/R3 + (4⋅R)/RREV2
,
(1.120)
DD2= 4/K1 + (2⋅R)/R3 + (4⋅R)/RREV1
,
(1.121)
So, for the static analysis, the output voltage of the converter by Fig.1.1 can be defined as :
Uout = 0.5 ⋅ [1- 1/K1 - R/(2⋅R3) - R/RREV1- R/RREV2 ] ⋅Uin_aver - [1/(2⋅K1) + R/ (4⋅R3) + R/(2⋅RREV2) ⋅ UD1 - [1/(2⋅K1) + R/ (4⋅R3) + R/(2⋅RREV1) ⋅ UD2
,
(1.122)
From this equation we see, that the transfer coefficient of the converter has the nominal value of
1/2, and the conversion inaccuracy for the medium frequencies can be estimated, if UD1=UD2=UD ,
RREV1 = RREV2 = RREV :
δ = - 1/K1- R/ (2⋅R3) - (2⋅ R)/RREV - [1/K1+R/R3+(2⋅ R)/RREV ] ⋅ (UD/Uin_aver)
.
(1.123)
1.4.3. Analysis of the circuit by Fig. 1.2.
The equivalent circuit diagrams for the full-period converter according to Fig. 1.2. for “positive”
and “negative” half-waves of the input signal are given in Fig. 1.14 a) and b). The diagram by
Fig.1.14 a) can be described by the following equations:
40
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig 1.14 Equivalent circuit diagram for the full-wave converter
e1 = (-UC1 + UC2 +UC3- UD2) /K1
,
(1.124)
iR2= (Uin-UC2+e1) /R2
,
(1.125)
iC2= (UC1-UC2-UC3+UD2) / R3
,
(1.126)
iR1= (-Uin - UC1+ UC2 -e1) / R1
,
(1.127)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
iC1 = iR2 - iREV1- iC2
,
(1.128)
iC3 = iR1 - iC1 - iREV1
,
(1.129)
iREV1= (UC1+ UD2) / RREV1
,
(1.130)
41
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “positive” half-periods:
iC1 = (1/R2) ⋅ Uin + [-1/ (K1⋅ R2) - 1/ RREV1 - 1/ R3]⋅ UC1 +
+ [ -1/R2+1/R3 +1/(K1⋅R2)] ⋅ UC2+(1/R3 +1/ (K1⋅ R2) ⋅ UC3 +
+ [-1/R3-1/RREV1 - 1/(K1⋅R2)] ⋅ UD2
,
(1.131)
iC2 = (1/R3) ⋅ UC1 - (1/R3) ⋅ UC2 - (1/R3) ⋅ UC3 + (1/R3) ⋅ UD2
,
(1.132)
iC3 = [(-1/R1)-(1/R2) ] ⋅Uin +[-1/R1+1/ (K1⋅ R1)+1/ (K1⋅ R2) +1/R3]⋅ UC1 +
+ [1/R1+1/R2-1/ (K1⋅ R1) -1/ (K1⋅ R2) -1/R3] ⋅ UC2 +
+ [- 1/R3 -1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UC3 +
+ [1/R3 +1/(K1⋅ R1) +1/(K1⋅ R2) ]⋅ UD2
,
(1.133)
For “negative” half-periods (diagram by Fig. 1.14 b) ) the circuit can be described by the
following equations:
iC1= iR1 - iREV2
,
(1.134)
e1 = (UC2+UC3+UD1) /K1
,
(1.135)
iR2 = (Uin - UC2 + e1) / R2
,
(1.136)
iC2 = (-UC2-UC3 - UD1) / R3
,
(1.137)
iR1 = (-Uin -UC1 + UC2 - e1) / R1
,
(1.138)
iD1 = iC1 + iC2 - iR2
,
(1.139)
iC3 = iD1 + iREV2
,
(1.140)
iREV2 = (UC1 + UD1) / RREV2
,
(1.141)
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “negative” half-periods:
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
42
iC1 = (-1/ R1 - 1/ RREV2 )⋅ UC1 + [ 1/R1 - 1/(K1⋅R1)] ⋅ UC2 - (1/ (K1⋅ R1) ⋅ UC3 + [-1/(K1⋅R1) -1/ RREV2] ⋅ UD2
iC2 = - (1/R3) ⋅ UC2 - (1/R3) ⋅ UC3 - (1/R3) ⋅ UD1
,
(1.142)
,
(1.143)
iC3 =[-(1/R1) -(1/R2) ] ⋅Uin -(1/R1)⋅ UC1 +[1/R1+1/R2-1/ (K1⋅R1) -1/(K1⋅R2)-1/R3] ⋅ UC2 + [- 1/R3 -1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UC3 +
+ [-1/R3 -1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UD2
,
(1.144)
Again, in the case h=1/2 we shall get the averaged matrice coefficients for the circuit
A0=
-2
 0
 0



A1=
-1+δR /2-(R/RREV1) -(R/RREV2)-1/K1-R/R3; -δR /2+R/R3 ; R/R3

1
; -2
; -2
 -2+ δR + 2/K1+R/R3
;4-4/K1-2⋅R/R3; -4/K1-2⋅R/R
 1/K1+ R/RREV2
AD1=  1
 2/K1 + R/R3
,



 1/K1+ R/R/RREV1+ R/R3
AD2=  -1
 - 2/K1 - R/R3



(1.145)
,
(1.146)
,
(1.147)
,
(1.148)
Further we shall find the determinant of the system, using only the first order errors. The
determinant of the system has the value:
D = -8 ⋅ [1+ 1/K1 + R/(2⋅R3) + R/RREV1+ R/RREV2 ]
,
(1.149)
and the determinant of the output voltage (voltage on the capacitor C2) has the value:
D2 = - 8 ⋅ (1-δR/2)
,
(1.150)
and determinants for the output voltage from the voltage drops on the diodes:
DD1= 4/K1 + (2⋅R)/R3 + (4⋅R)/RREV2
,
(1.151)
DD2= 4/K1 + (2⋅R)/R3 + (4⋅R)/RREV1
,
(1.152)



O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
43
So, for the static analysis the output voltage of the converter by Fig.1.2 can be defined as :
Uout = [1- δR/2-1/K1 - R/(2⋅R3) - R/RREV1- R/RREV2 ] ⋅Uin - [1/(2⋅K1) + R/ (4⋅R3) + R/(2⋅RREV2) ⋅ UD1 - [1/(2⋅K1) + R/ (4⋅R3) + R/(2⋅RREV1) ⋅ UD2
,
(1.153)
From this equation we can see, that the transfer coefficient of the converter has the nominal value
of 1, and the conversion inaccuracy for the medium frequencies can be estimated, assuming
UD1=UD2=UD , RREV1 = RREV2 = RREV :
δ = -δR/2 - 1/K1- R/ (2⋅R3) - (2⋅ R)/RREV - [1/K1+R/R3+(2⋅ R)/RREV ] ⋅ (UD/Uin)
,
(1.154)
1.4.4. Analysis of the circuit by Fig. 1.3.
The equivalent circuit diagrams of the full-period converter according to Fig. 1.3. for “positive”
and “negative” half-waves of the input signal are given on Fig. 1.15 a) and b). The diagram by
Fig. 1.15 a) can be described by the following equations:
e1 = -UD1 /K1
,
(1.155)
iR2= (Uin+UC2+e1) /R2
,
(1.156)
iC2= iR1-iREV2
,
(1.157)
iR1= (-Uin + UC1- UC2 -e1) / R1
,
(1.158)
iC1 = iR1 - iREV2
,
(1.159)
iREV2= (UC1+ UD1) / RREV2
,
(1.160)
44
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig 1.15 Equivalent circuit diagram for the full-wave converter
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “positive” half-periods:
iC1 = (1/R1) ⋅ Uin + [-1/R1 - 1/ RREV2 ]⋅ UC1 + (1/R1) ⋅ UC2+
+ [-1/RREV2 - 1/(K1⋅R1)] ⋅ UD1
,
(1.161)
iC2 = [(-1/R1)-(1/R2) ] ⋅Uin +[-1/R1+(1/R1)⋅ UC1 +
+ (-1/R1-1/R2) ⋅ UC2 +
+ [1/(K1⋅ R1) +1/(K1⋅ R2) ]⋅ UD1
,
(1.162)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
45
For “negative” half-periods (diagram by Fig.1.15 b) ) the circuit can be described by the
following equations:
iC1= -iR2 - iREV1
,
(1.163)
e1 = (UC1+ UD2) /K1
,
(1.164)
iR2 = (Uin + UC2 + e1) / R2
,
(1.165)
iC2 = iR1-iR2
,
(1.166)
iR1 = (-Uin +UC1 - UC2 - e1) / R1
,
(1.167)
iREV1 = (UC1 + UD2) / RREV1
,
(1.168)
After arithmetical conversions we shall get the following values for the currents of the capacitors
for the “negative” half-periods:
iC1 = -(1/R2) ⋅ Uin + [-1/ (K1⋅R1) - 1/ RREV2)]⋅ UC1 -(1/R2) ⋅ UC2 +
+ (-1/ (K1⋅ R2) -1/ RREV1] ⋅ UD2
,
(1.169)
iC2 = - (1/R1-1/R2) ⋅Uin +[1/R1-1/ (K1⋅R1) -1/(K1⋅R2)]⋅UC1 +
+ (-1/R1 - 1/R2) UC + [-1/(K1⋅ R1) -1/(K1⋅ R2) ]⋅ UD2
,
(1.170)
Again, in the case h=1/2 we shall get the averaged matrice coefficients for the circuit
A0=
-2

 0



A=
-1+δR /2-(R/RREV1) -(R/RREV2)-1/K1 ; -δR ;

 2- δR - 2/K1
; -4



,
(1.171)
,
(1.172)
 1/K1+ R/RREV2
AD1= 
 -2/K1



,
(1.173)
 1/K1+R/RREV1
AD2= 
 2/K1



,
(1.174)
Further we shall find the determinant of the system, using only the first order errors. The
determinant of the system has the value:
46
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
D = 4 ⋅ [1+ 1/K1 + R/RREV1+ R/RREV2 ]
,
(1.175)
the determinant of the output voltage (voltage on the capacitor C1) has the value:
D1 = 8
,
(1.176)
and determinants for the output voltage from the voltage drops on the diodes:
DD1= -4/K1 - (4⋅R)/RREV2
,
(1.177)
DD2= -4/K1 - (4⋅R)/RREV1
,
(1.178)
So, for the static analysis, the output voltage of the converter by Fig.1.1 can be defined as :
Uout = 2 ⋅ [1- 1/K1 - R/RREV1- R/RREV2 ] ⋅Uin_aver - [1/⋅K1 + R/(2⋅RREV2) ]⋅ UD1 - [1/(2⋅K1) + R/ (4⋅R3) + R/(2⋅RREV1)] ⋅ UD2
,
(1.179)
From this equation we see, that the transfer coefficient of the converter has the nominal value of
2, and the conversion inaccuracy for the medium frequencies can be estimated, assuming
UD1=UD2=UD , RREV1 = RREV2 = RREV :
δ = - 1/K1 - (2⋅ R)/RREV - (1/K⋅+ R/RREV ) ⋅ (UD/Uin_aver)
.
(1.180)
1.4.5. Analysis of the circuit by Fig. 1.4.
The equivalent circuit diagrams for the full-period converter according to Fig. 1.4. for “positive”
and “negative” half-waves of the input signal are given on the Fig.1.16 a) and b). The diagram by
Fig. 1.16 a) can be described by the following equations:
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig 1.16 Equivalent circuit diagram for the full-wave converter
e1 = (-UC2 +UC3- UD2) /K1
,
(1.181)
iR1= (Uin + UC1 - e1) /R1
,
(1.182)
iR2= (Uin +UC2 + e1) / R2
,
(1.183)
iREV1= (UC1+UC2+ UD2) / RREV1
,
(1.184)
iR3= (-UC2+ UC3 - UD2) / R3
,
(1.185)
47
48
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
iC1 = -iR1 - iREV1
,
(1.186)
iC2 = -iC1 - i3
,
(1.187)
iC3 = - iR2 - iC2 - iREV1
,
(1.188)
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “positive” half-periods:
iC1 = (1/R1) ⋅ Uin + (-1/ R1 - 1/ RREV1 )⋅ UC1 +
+ [ - 1/(K1⋅R1) -1/ RREV1] ⋅ UC2+ [1/ (K1⋅ R1) ] ⋅ UC3 +
+ [- 1/(K1⋅R1) - 1/RREV1 ] ⋅ UD2
,
(1.189)
iC2 = (1/R1) ⋅ Uin + (-1/R1 - 1/ RREV1) ⋅ UC1 +
+[- 1/(K1⋅R1) -1/ RREV1 -1/R3] ⋅ UC2 + [1/ (K1⋅ R1)+ 1/R3) ] ⋅ UC3 +
+[- 1/(K1⋅R1) - 1/RREV1 -1/R3] ⋅ UD2
,
(1.190)
iC3 =[ -1/R1-1/R2 ] ⋅Uin +[ 1/R1]⋅ UC1 +
+ [-1/R2 + 1/ (K1⋅ R1) +1/ (K1⋅ R2) +1/R3] ⋅ UC2 +
+ [ -1/(K1⋅ R1) -1/(K1⋅ R2) - 1/R3 ]⋅ UC3 +
+ [ 1/(K1⋅ R1) +1/(K1⋅ R2) +1/R3 ]⋅ UD2
.
(1.191)
For “negative” half-periods (diagram by Fig. 1.16 b) ) the circuit can be described by the
following equations:
e1 = (UC1+UC3+UD1) /K1
,
(1.192)
iR1 = (-Uin + UC1 - e1) / R1
,
(1.193)
iR2 = (Uin + UC2 + e1) / R2
,
(1.194)
iREV2 = (UC1 + UC2+ UD1) / RREV2
,
(1.195)
iR3 = (UC1 + UC3+ UD1) / R3
,
(1.196)
iC2 = - iR2 - iREV2
,
(1.197)
iC1= iC2 - I3
,
(1.198)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
iC3 = iR1 + iC1 + iREV2
,
49
(1.199)
After arithmetical conversions we shall get the following values for the currents through the
capacitors during the “negative” half-periods:
iC1 = - (1/R2) ⋅ Uin + [ - 1/(K1⋅R2) -1/ RREV2] ⋅ UC1 +
+ (-1/ R2 - 1/ RREV2 )⋅ UC2 + [- 1/ (K1⋅ R2)- 1/RREV2 -1/R3] ⋅ UC3 +
+ [- 1/(K1⋅R2) - 1/RREV2 -1/R3] ⋅ UD1
,
(1.200)
,
(1.201)
iC2 = - (1/R2) ⋅ Uin +[- 1/(K1⋅R2) -1/ RREV2 ] ⋅ UC1 +
+ (-1/R2 - 1/ RREV2) ⋅ UC2 - [1/ (K1⋅ R2)] ⋅ UC3 +
+[- 1/(K1⋅R2) - 1/RREV2 ] ⋅ UD1
iC3 =[ -1/R1-1/R2 ] ⋅Uin + [1/R1 - 1/(K1⋅R1) -1/ (K1⋅R2) -1/R3]⋅UC1- [ 1/R2]⋅ UC2 + [ -1/(K1⋅ R1) -1/(K1⋅ R2) - 1/R3 ]⋅ UC3 +
+ [- 1/(K1⋅ R1) -1/(K1⋅ R2) -1/R3 ]⋅ UD1
,
(1.202)
In the case h=1/2 (equations of the both half periods have the same weight 1/2), and using again
the notations
R1= R ⋅ (1+ δR /2)
,
(1.203)
R2= R ⋅ (1- δR /2)
,
(1.204)
,
(1.205)
we shall get the averaged matrice coefficients for the circuit
A0=
 -2
 -2
 0



A1=
-1+δR/2-(R/RREV1)-(R/RREV2) -1/K1-R/R3
; -1-δR /2-(R/RREV1) -(R/RREV2)
- 1/K1-R/R3
;
- R/R3

-1+δR/2-(R/RREV1)-(R/RREV2) -1/K1-R/R3
; -1-δR /2-(R/RREV1) -(R/RREV2)
- 1/K1-R/R3
;
R/R3

 2 - 2/K1-R/R3- δR
; -2+ δR +2/K1+R/R3
; -4/K1-2⋅R/R3
,
(1.206)









50
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
 1/K1+ R/R3+ R/RREV2 
AD1=  1/K1 + R/RREV2

 2/K1 + R/R3

,
(1.207)
 1/K1+ R/RREV1

AD2=  1/K1 +R/R3+ R/RREV1 
 - 2/K1 - R/R3

,
(1.208)
Further we shall find the determinant of the system, using only the first order errors. The
determinant of the system has the value:
D = -4 ⋅ [1+ 1/K1 + R/(2⋅R3) + R/RREV1+ R/RREV2 ]
,
(1.209)
the determinant of the first and second output voltages (voltage on the capacitors C1 and C2)
have the values:
D1 = ( - 4+ 2⋅δR) ⋅ (R/R3)
,
(1.210)
D2 = ( - 4- 2⋅δR) ⋅ (R/R3)
,
(1.211)
and determinants for the output voltage from the voltage drops on the diodes:
DD1 = (2/K1+ R/R3 + 2⋅ R/RREV2 )
,
(1.212)
DD2 = (2/K1+ R/R3 + 2⋅ R/RREV1 )
,
(1.213)
For assumption RREV1 = RREV2 = RREV and UD1 = UD2= UD
Uout1= (1- 0.5⋅δR -1/K1- 2⋅R/RREV - R/(2⋅R3) ) Uin_aver - (1/K1+ R/(2⋅R3) + R/RREV ) ⋅ UD
,
(1.214)
- Uout2= (1+ 0.5⋅δR -1/K1- 2 ⋅R/RREV - R/(2⋅R3) ) Uin_aver - (1/K1+ R/(2⋅R3) + R/RREV ) ⋅ UD
,
(1.215)
So, for both outputs the nominal transfer coefficient is 1 (actually -1 for the second output), and
relative inaccuracy for medium frequencies can be estimated for first and second outputs:
δ1 = -0.5⋅δR -(2 ⋅R)/ RREV - R/(2⋅R3) - [ 1/K1+R/(2⋅R3) + R/ RREV ] ⋅ (UD/Uin_aver)
δ2 = 0.5⋅δR -(2 ⋅R)/ RREV - R/(2⋅R3) -
,
(1.216)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
- [ 1/K1+R/(2⋅R3) + R/ RREV ] ⋅ (UD/Uin_aver)
,
51
(1.217)
If the output voltage will be taken differentially between the first and second outputs, the transfer
coefficient will be 2, with inaccuracy
δ = -(2 ⋅R)/ RREV - R/(2⋅R3) - [ 1/K1+R/(2⋅R3) + R/ RREV ] ⋅ (UD/Uin_aver)
,
(1.218)
not depending on the inaccuracy δR of the resistor ratio (R1/R2).
1.4.6. Analysis of the circuit by Fig. 1.5.
The equivalent circuit diagrams for the full-period converter according to Fig. 1.5. for “positive”
and “negative” half-waves of the input signal are given in the Fig.1.17 a) and b). The diagram by
Fig.1.17 a) can be described by the following equations:
iC1 = (-Uin-UC1+UC3 - e1) /R1
,
(1.219)
iC2 = ( Uin - UC2+UC4 + e1) /R2
,
(1.220)
iR3= (UC3 - e1) / R3
,
(1.221)
iR4= (UC4 + e1) / R4
,
(1.222)
iR5= (-UC4 + UC5 - UD2) / R5
,
(1.223)
iC3 = -iC1 - iR3- iREV1
,
(1.224)
iC4 = iC3 + iR5
,
(1.225)
iC5 = - iC2 - iC4- iR4 - iREV1
,
(1.226)
iREV1 = ( UC3 + UC4+UD2 ) /RREV1
,
(1.227)
,
(1.228)
e1= (- UC4 + UC5- UD2 ) /K1
52
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig 1.17. Equivalent circuit diagram for the full-wave converter
The diagram by Fig. 1.17 b) for “negative” half-periods can be described by the following
equations:
iC1 = (-Uin-UC1+UC3 - e1) /R1
,
(1.229)
iC2 = ( Uin - UC2+UC4 + e1) /R2
,
(1.230)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
iR3= (UC3 - e1) / R3
,
(1.231)
iR4= (UC4 + e1) / R4
,
(1.232)
iC3 = iC4 - iR5
,
(1.233)
iC4 = -iR2 - iR4 - iREV1
,
(1.234)
iR5= ( UC3 + UC5 + UD1) / R5
,
(1.235)
iC5 = iC1 + iC3 + iR3 + iREV2
,
(1.236)
iREV2 = ( UC3 + UC4+UD2 ) /RREV2
,
(1.237)
e1= ( UC3 + UC5 + UD1 ) /K1
,
(1.238)
53
After deriving of the averaged space equations (equations for currents of the capacitors) we shall
get (in the case R1=R2=R, R3=R4= R’ , UD1 = UD2= UD , RREV1 = RREV2 = RREV) , that the
nominal value of the transfer coefficient is R’/R and the relative inaccuracy can be estimated as
δ = -0.5⋅ R’/R5 - 2⋅R’/RREV -1/K1⋅(1+R’/R) +
+ [1/K1⋅(1+R/R’)+R/RREV + R/(2⋅R5)] ⋅ (UD /Uin_aver)
,
(1.239)
1.4.7. Analysis of the circuit by Fig. 1.7.
To simplify the analysis we shall assume the circuit being ideal symmetrical - R3=R4= R,
UD1= UD2= UD , RREV1 = RREV2 = RREV. It can be shown, for that case, that UC1 = UC2 = 0, UC3 =
UC4 = Uc= Uout.
The equivalent circuit diagrams for the full-period converter according to Fig. 1.7. for
“positive” and “negative” half-waves of the input signal are given in the Fig. 1.18 a) and b). The
diagram by Fig.1.18 a) can be described by the following equations for the “positive” halfperiods:
e1= [ -(R2/R1) ⋅ Uin - UC - UD ) ]/K1
,
(1.240)
iR2= (Uin + e1) / R2
,
(1.241)
iREV1 = ( 2⋅UC +UD ) /RREV1
,
(1.242)
iR5= ( - UC + UD) / R5
,
(1.243)
UA = -(R2/R1) ⋅ Uin - (R1+R2)/R1⋅ e1
,
(1.244)
54
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
iR3= (UC + UA) / R
,
(1.245)
iR4= (UC - UA ) / R4
,
(1.246)
iC2 = -iR3 - iREV1
,
(1.247)
iC3 = iC2 + iR2 + iR5
,
(1.248)
Fig 1.18. Equivalent circuit diagram for the full-wave converter
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
55
For “negative” half-periods the circuit (by Fig.1.18. b) ) can be described by the following
equations:
e1= [ -(R2/R1) ⋅ Uin + UC + UD ) ]/K1
,
(1.249)
iR2= (Uin + e1) / R2
,
(1.250)
iREV2 = ( 2⋅UC +UD ) /RREV1
,
(1.251)
iR5= ( UC + UD) / R5
,
(1.252)
UA = -(R2/R1) ⋅ Uin - (R1+R2)/R1⋅ e1
,
(1.253)
iR3= (UC + UA) / R
,
(1.254)
iR4= (UC - UA ) / R4
,
(1.255)
iC3 = -iR4 - iREV2
,
(1.256)
iC2 = iC3 - iR2 - iR5
,
(1.257)
After deriving of the averaged space equations (equations for currents through the capacitors) we
shall get that the nominal value of the transfer coefficient for both outputs is:
K= (R2+ R/2) / R1
,
(1.258)
and the relative inaccuracy can be estimated as
δ = -2⋅(R1+R2)/(K1⋅R1) - R/(2⋅R5) - 2⋅R/RREV - [R/RREV + R/(2⋅R5)+ (R1+R2)/(K1⋅R1) ] ⋅ (UD /Uin_aver)
,
(1.259)
1.4.8. Time-domain analysis of the converters
Analysis of the AC/DC converters can be generally done by state space equations, averaged
over the periods of the input AC signal, as it was explained in 1.4.1. Such kind of state space
equations can be easily solved by numerical methods. Actually numerical integration is needed.
Still we shall further derive the analytical representations of the time-domain analysis for the
basic circuit diagrams.
Let us assume that we have three capacitors in the circuit (three state space variables). We shall
use the operator (‘s’) representation of the set of state-space equations:
-A1,0⋅ Uin_aver + (A1,1 - s⋅C1) ⋅UC1 +A1,2 ⋅UC2 + A1,3 ⋅UC3 - AD1,1 ⋅UD1 - AD2,1 ⋅UD2= 0
,
-A2,0⋅Uin_aver + A2,1 ⋅UC1 +(A2,2- s⋅C2) ⋅UC2 + A2,3 ⋅UC3 - AD2,3 ⋅UD1 - AD2,2 ⋅UD2= 0
,
-A3,0⋅Uin_aver + A3,1⋅UC1 +A3,2 ⋅UC2 + (A3,3 - s⋅C3) ⋅UC3 - AD1,3 ⋅UD1 - AD2,3 ⋅UD2= 0
56
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
,
(1.260)
Here, and further during this discussion we shall use, instead of the actual instant values of the
nput voltage Uin(t) the “average-rectified” value of the input signal, Uin_aver, with plus sign for
the “positive” half-period, and with minus sign, for the “negative” half-period. This is valid, as
converters under discussion are sensing “average” value, and the actual form’(shape) of the input
signal is not important at all, in the first approximation. Now the general matrice of the system
will have the form:
A=
 A1,1- s⋅C1
 A2,1
 A3,1
; A1,2
; A1,3

; A2,2- s⋅C2; A2,3

; A3,2
; A3,3 -s⋅C3 
,
(1.261)
So we can use the formerly found matrices for finding coefficients Ai,j with adding the ‘s⋅Ci’
component where needed.
So, after doing these modification, and founding the ‘operator’-transfer coefficients, while
taking into account only main members (ignoring the ‘high-order’ small values) the following
coefficients for the AC/DC converters are valid,
- for half-wave converter according to Fig. 1.1
K= 0.5 {1-1/K1- R/(2⋅R3) -(2⋅R)/RREV [2/K1+ R/R3 + (2⋅R)/RREV ]⋅(UD/Uin_aver)}
,
(1.262)
,
(1.263)
K= 2 {1 -(2⋅R)/RREV - [1/K1 + ⋅R/RREV ]⋅( UD /Uin_aver)} ,
(1.264)
-for full-wave converter according to Fig. 1.2
K= 1 {1 - δR/2 - 1/K1- R/(2⋅R3) -(2⋅R)/RREV [2/K1+ R/R3 + (2⋅R)/RREV ]⋅( UD /Uin_aver)}
-for full-wave converter according to Fig. 1.3
-for full-wave converter according to Fig. 1.4, for the first output (Uout1)
K= 1 {1 - δR/2 - 1/K1- R/(2⋅R3) -(2⋅R)/RREV [1/K1+ R/(2⋅R3) + R/RREV ]⋅( UD /Uin_aver)}
,
(1.265)
,
(1.266)
and for the second output (Uout2)
K= 1 {1 + δR/2 - 1/K1- R/(2⋅R3) -(2⋅R)/RREV [1/K1+ R/(2⋅R3) + R/RREV ]⋅(Ud/Uin_aver)}
Examples of the numerical analysis of the ‘real’ and more sophisticated converter models in the
time domain are given in the Appendix C.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
57
1.5. Practical realizations
1.5.1. Precise AC/DC measurement converter
This converter was designed after critical analysis of the earlier converters, used in the AC/DC
converter V9-10 and AC voltmeter V3-60, for modification of the V3-60(A). All the instruments
were designed and manufactured at the plant ‘RET’ (Tallinn).
Principal circuit diagram and component list of the designed converter are given in the
Appendix B1.
This solution is based on the proposed circuit diagram by Fig.1.4 [13].
The converter has the following main features:
- frequency range of the input signal- from 20 Hz to 100 kHz
- “basic” inaccuracy less than 0.035% (for frequency range from 100 Hz to 20 kHz)
- additional inaccuracy in the frequency range from 20Hz to 100 kHz is less than 0.08%, and up
to 100kHz less than 0.06%
- settling time of the output voltage is less than 10 seconds
The converter was tested with AC calibrating system V1-26/V1-9 (20Hz...100kHz) and
calibrator V1-16 (lower and higher frequencies). Results of the measurements are given in table
1.1.
Table 1.1. Conversion inaccuracies of the precise AC/DC converter
Frequency
Absolute inaccuracy, relative to 1 V full scale, in %
Uin= 1V
10 Hz
20 Hz
40 Hz
100 Hz
400 Hz
1 kHz
4 kHz
10 kHz
20 kHz
40 kHz
70 kHz
100 kHz
200 kHz
300 kHz
-0.088%
-0.065%
-0.044%
-0.019%
+0.002%
0
-0.013%
-0.018%
-0.012%
-0.002%
-0.022%
-0.045%
+0.051%
+0.131%
Uin= 0.1V
Uin= 0.01V
+0.038%
+0.001%
-0.001%
-0.001%
+0.001%
0
-0.002%
-0.003%
-0.004%
-0.004%
-0.009%
-0.014%
-0.012%
-0.012%
+0.042%
+0.008%
+0.007%
+0.004%
-0.006%
0
-0.001%
-0.001%
-0.006%
-0.002%
-0.006%
-0.009%
-0.018%
-0.028%
58
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
400 kHz
500 kHz
700 kHz
1 MHz
+0.209%
+0.306%
+0.423%
+0.423%
-0.011%
-0.012%
+0.003%
-0.204%
-0.036%
-0.048%
-0.173%
+0.047%
1.5.2. Precise AC/DC measurement converter with improved performance
This converter has been developed for the precise digital multi-meter/voltmeter. This converter
has the ‘basic’ inaccuracy of 0.01%, covers the frequency range from 10 Hz to 1 MHz, and has
significantly improved speed- conversion time is less than 1 second.
Functional diagram of the converter is given in Fig. 1.19. Main part of the converter is
implemented by the circuit diagram by Fig.1.4, where differential output is used [13]. Output
differential DC amplifier is based on use of the op-amp D3. It is important, that the difference of
output voltages on C1 and C2 settle much quicker than these voltages separately. For decreasing
of the low-frequency (nonlinear) inaccuracy, special compensation circuitry (D2, V3, V4 ) is
introduced [15].
Principal circuit diagram and component list of the designed converter are given in the
Appendix B2.
The converter was tested with the AC calibrating system V1-26/V1-9 (20Hz...100kHz) and
calibrator V1-16 (lower and higher frequencies). Results of the measurements are given in table
1.2.
Photo of the implemented prototype of the converter is given in the Fig. 1.20.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Fig 1.19. Functional diagram for the precise AC/DC converter
59
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Table 1.2. Conversion inaccuracies of the improved AC/DC converter
Frequency
Absolute inaccuracy, relative to 1 V full scale, in %
Uin= 1V
10 Hz
20 Hz
40 Hz
100 Hz
400 Hz
1 kHz
4 kHz
10 kHz
20 kHz
40 kHz
70 kHz
100 kHz
200 kHz
500 kHz
1 MHz
+0.001%
-0.015%
-0.014%
-0.010%
.0.002%
0
-0.001%
-0.006%
-0.010%
-0.025%
-0.050%
-0.063%
-0.055%
-0.098%
+0.090%
Uin= 0.1V
Uin= 0.01V
-0.011%
-0.007%
-0.003%
-0.002%
0
0
0
-0.001%
-0.003%
-0.05%
-0.007%
-0.006%
-0.012%
-0.046%
-0.058%
-0.003%
-0.002%
-0.001%
0
0
0
0
0
0
-0.002%
-0.004%
-0.004%
-0.011%
-0.036%
-0.069%
Fig 1.20 Photo of the improved AC/DC converter
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61
1.5.3. AC/DC measurement converter with input conditioning
A simple solution for AC measurement part of the desktop digital multimeter has been
designed. A special feature of the solution is application of an input conditioning (buffering, and
scaling of the input AC signal) in the same unit, having only a small number of components. The
solution is based on the circuit diagram Fig.1.7.
Principal circuit diagram and component list of the designed converter are given in the
Appendix B3.
The converter has the following features:
- input AC ranges- 1, 10, 100, 300V
- nominal output voltage 1V (0.3V for 300 V range)
- input impedance - 1 MΩ
- ‘basic’ inaccuracy (ranges 10, 100 V , frequencies from 100 Hz to 4 kHz)- less than 0.1%
- frequency range from 20 Hz to 100 kHz
- maximum inaccuracy- 0.1% of the full scale
- resolution is better than 0.001% of the full scale
- conversion time is less than 1 second
Photo of the implemented prototype of the converter is given on the Fig. 1.21.
Fig 1.21. Photo of the AC/DC converter with input conditioning
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
1.6. Conclusions and results
A special class of the AC/DC measurement converters with switched capacitors in the negative
feedback path has been developed.
It has been shown that in such circuits the significant reduction of sensitivity of the transfer
coefficient of the converter from the resistor ratios can be achieved.
The converters with basic inaccuracy 0.01% and even better can be easily designed.
Analysis of the low-frequency inaccuracy has been done for different structures.
Circuits with improved low-frequency performance have been developed.
Also circuits with input scale conditioning have been developed for designing of simple AC
measurement devices with small number of components.
The methodology of analysis of the circuits under discussion has been developed. This method
is based on the averaging of matrice coefficients over positive and negative half-periods of the
input signal.
Mathematical models in the form of averaged matrices of the state-space equations has been
found for described circuits.
Further, static transfer coefficients for the converters have been found from these models, and
analysis in the time domain has been performed and explained.
Several prototypes of AC/DC measurement converters with improved metrological
characteristics has been realized, and tested. So the main results are verified practically.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
63
2. Precise synchronous detectors with switched capacitors in the
feedback path
2.1. Introduction to the topic
Electronic measurements today are often directly or indirectly related to AC measurements,
and often to synchronous (phase-sensitive) measurements. Phase sensitive measurements are
efficient, if some excitation AC signal is available as reference signal directly, or can be regenerated at the measurement side [21]. Often modulation and demodulation are used for
improvement of the accuracy and noise suppression in the measurement systems, and the
synchronous signal processing is efficient also in this case [22].
Synchronous detectors (SD) are generally used to extract small useful signals as precisely as
possible in the presence of significant noise. Often instant amplitudes of the noise can many, even
hundreds and thousands times exceed the useful signal. So it is essential to have SDs, being as
precise as possible for small useful signals, and being not sensitive to large instant values of the
total input signal.
Further more, if there is no reference signal available directly, it can be regenerated by phaselocked-loops (PLL). Typical PLLs include a SD in the input, drived by the quadrature component
of the output signal and operating as a phase detector. The output of the SD is kept as close to
zero as possible via the feedback path. The high gain is needed to keep small error in the closed
PLL loop. But the conventional SDs have large 1/f noise and DC drift, and especially the large
“in-phase” component at the input can overload the small quadrature component detector,
limiting so the gain value, and also the preciseness of the PLL loop operation.
So, maybe even the most important specification of the synchronous detector is the input
dynamic reserve, usually named simply as dynamic reserve [21, pp. 45 -54], characterizing
the overloading capability of the SD input.
A class of SDs being precise and having significantly increased dynamic reserve compared
with conventional solutions, are described in this paper.
2.2. Basic ideas and advantages of the proposed solutions
Conventional SDs include some kind of multiplier, multiplying the input signal by certain
reference signal, and is followed by some low-pass-filter to filter out the useful signal from
ripple and noise signals [21, 23-25]. Often this multiplier is realized as a circuit, the gain of
which is discretely switched between +1 and -1 by the reference signal. An input AC and an
output DC amplifiers can be added to such circuit optionally, to work with smaller input signals
and get more output signal. Still, adding of these amplifiers can change the nominal input and
output voltage ranges, where the detector works, but it cannot improve the dynamic reserve of
the detector. All the signal components, as the useful one carrying information, as well as all
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
other components (noise, disturbances, internal offsets and drifts at the DC output) are amplified
in parallel by the same value, and also the overloading capability remains the same.
A special class of synchronous detectors with improved dynamic reserve, has been developed,
and is described below.
Proposed synchronous detectors were firstly developed as the precise asynchronous AC/DC
measurement converters [4, 5, 12, 16] for high accuracy AC voltmeters. But the same circuit
ideas can be used successfully also in SDs [26].
In the case of SDs instead of rectifying diodes the electronic switches switched synchronously
by reference signal, are used in the feedback path of op-amps. The circuit diagrams are
proposed, described, and discussed in the current chapter.
All the proposed circuits utilize the same basic ideas, allowing to achieve outstanding
specifications:
1. The op-amp is used as an AC amplifier, which is separated from the signal path by capacitors
in most of the proposed circuits. Therefore these detectors have a small or no DC offset voltage,
allowing to detect small signals precisely.
2. Rectifying switches (e.g. FET -switches) are operating in the negative feedback path of the opamp, so their on-resistances do not generate inaccuracy in the first approximation.
3. These synchronous detectors suppress also the ripple, as the detector output is connected (via
a capacitor) to the inverting input of the op-amp for AC component.
4. The negative feed-back path of the op-amp is active for both half-periods of the reference
signal, closed via switched capacitor. So the instant values of the input signal are averaged
(integrated) on these capacitors, and as the synchronously rectified and averaged signal depends
only on useful signal component, all other components are suppressed, and detector is not
sensitive to instant values of the total input signal.
5. There are proposed schemes, for which the gain is unity and almost not depending on the
resistor ratios, and also other circuit diagrams, for which the gain directly depends on resistor
ratios, and practically any reasonable gain can be achieved for useful signal component.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
65
2.3. Solutions and functioning
Figure 2.1. Half-wave synchronous detector
The half-wave SD according to Fig.2.1 is functioning in the following way. Input AC voltage
Uin causes an AC current with value of Uin/R1 through the resistor R1, as the other end of the
R1 is connected to the inverting input of the op-amp A1 for AC (through capacitor C2). As an
input resistance of the op-amp is high, practically the same current Uin/R1 flows (via C1, K1 or
K2, and C3) to the output of the op-amp. The switches K1 and K2 (for example, FET-switches),
controlled by the reference signal, rectify synchronously this AC current by half periods of the
reference signal. A DC component of one half-periods of the rectified signal flows through K1
and R2, and other half-periods via K2 and R1. According DC voltage drops are generated on
resistors R1 and R2. The DC voltage drop on R1 serves as the output voltage of the detector. So,
as AC voltage to AC current conversion, and conversion of the synchronously rectified DC
current to the output DC voltage are performed by the same resistor R1, the gain of the detector
does not depend on any resistor ratio. The detector suppresses AC ripple, as for AC the inverting
input of the op-amp is tied to the output of the detector. And this circuit has also no output voltage
offset, as the op-amp is isolated from the output of the detector by capacitor for DC.
Resistor R3 is for determining the DC output voltage of the op-amp, and must have relatively
high value not to cause any significant AC inaccuracy, as it is in parallel with off-resistances of
K1 and K2 for AC.
So, as this detector is an half-wave one, the transfer coefficient of the detector is 1/2 relatively
to the synchronously-rectified-mean-value, and for input sine-wave and reference signal with
equal half-periods
Uout= (√2/π)* Uin* cos ϕ
,
(2.1)
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
where Uin is root-mean-square (RMS)-value of the input AC voltage, Uout is the output DC
voltage, and ϕ is a phase angle between Uin and Uref.
Figure 2.2. Full-wave synchronous detector A.
The synchronous detector by Fig. 2.2 operates in the similar way. Only the input AC current is
determined by the parallel resistance of R1 and R2. So the gain for this full-wave synchronous
detector is
Uout= (√2/π)* R2/(R1||R2) * Uin* cos ϕ
,
(2.2)
,
(2.3)
In the case R1=R2 the gain of the detector is
Uout= (2*√2/π)* Uin* cos ϕ
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67
Figure 2.3. Full-wave synchronous detector B
The SD by Fig. 2.3 operates also in the similar way. The input AC current is determined here
again by the parallel resistance of R1 and R2. Only the output DC voltage is generated on the
serial connection (for DC) of R1 and R2. So the gain for this full-wave synchronous detector is
Uout= (√2/π)* (R1+R2)/(R1||R2) * Uin* cos ϕ
,
(2.4)
,
(2.5)
In the case R1=R2 the gain of the detector is
Uout= (4*√2/π)* Uin* cos ϕ
Figure 2.4. Full-wave synchronous detector C.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
The SD by Fig. 2.4 functions again in the similar way. The input AC current is determined here
again by the parallel resistance of the resistors R1 and R2. One DC output voltage is generated
on the resistor R1, and another one on the resistor R2:
Uout1= (√2/π)* (R1/R1||R2) * Uin* cos ϕ
,
(2.6)
Uout2= - (√2/π)* (R2/R1||R2) * Uin* cos ϕ
,
(2.7)
,
(2.8).
In the case R1=R2=R
Uou1= -Uout2= (2√2/π)* Uin* cos ϕ
An advantage of this detector is improved low-frequency accuracy, as this circuit is symmetrical
for both half-waves.
Figure 2.5. Full-wave synchronous detector D
The SD by Fig. 2.5 operates again in the similar way. The input AC current is determined here
again by the parallel resistance of R1 and R2. But for DC these resistors are blocked by
capacitors C1 and C2. So the rectified DC current component will generate first and second
output DC voltages on resistors R3 and R4 accordingly:
Uout1 = (√2/π)* (R3/R1||R2) * Uin* cos ϕ
, (2.9)
Uout2 = - (√2/π)* (R4/R1||R2) * Uin* cos ϕ
, (2.10)
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69
The required gain of the synchronous detector can easily be achieved, as it is directly
determined by the resistors ratio. At the same time, due to the integrating capacitors C3 and C4 in
the feedback path, the circuits are tolerant to high instant values of the total input signal.
In this circuit the resistor R5 is for determining the DC output voltage of the op-amp, and must
have again relatively high value not to cause the AC inaccuracy.
2.4. About practical using of proposed solutions
Main sources of the gain inaccuracy are tolerances of the scaling resistors (according to
equations (2.1)......(2.10) ), finite resistance of the electronic switches, and finite gain of the opamp, especially on high frequencies.
At lower frequencies an additional error occurs from the final impedance of capacitors. This
error has systematic and random (ripple) components.
Analysis and simulation of the circuits described can be easily done, using the state-space
equations for both periods of the reference signal, and averaging these equations over the total
periods of the reference signal.
Theoretical and practical investigation of the circuits described here show that accuracy of
0.01% with resolution of 0.0001% (1 ppm) can be achieved for medium frequencies (up to
some tens of kHz), and circuits can have acceptable accuracy and linearity up to 1 MHz.
These circuits has been used in practical precise stereo-decoder solutions including the PLL
part. Structural diagram of the PLL-loop with proposed SD for this application is given in
Fig.2.6. Circuit diagram of the practical realization with component values is given in the
Appendix B4.
Such kind of SD has been used in the equipment in the research space radiometric
instruments. The usage of proposed solutions for complex bio-impedance measurements is just
under the investigation.
Figure 2.6. Structural diagram for the PLL-loop with proposed SD
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2.5. Conclusions and results
A special class of synchronous detectors with significantly improved metrological
characteristics (accuracy, frequency and dynamical ranges) has been developed.
These novel detectors have switched capacitors in the negative feedback path. Various circuit
diagrams for such converters are given.
It has been shown that such the converter can have extremely wide dynamic range of the input
signal. One application field of such converters are the PLL (phase-locked-loop) circuits, where
wide dynamic range can be effectively exploited.
The performance analysis of such detectors is given.
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3. DSP-based AC measurement converters
3.1. Apology of digital solutions of AC measurement solutions
Nowadays the digital signal processors (DSP) offer the universal technology, being
technically and costly effective, giving precise and flexible solutions for various problems,
earlier realized in hardware, mainly using analogue circuits.
One application of DSPs can be AC measurement converters (for AC voltmeters or digital
multimeters).
The signal processing, used for AC measurement, can be easily implemented in the software of
the DSP.
Even more complicated non-linear and adaptive algorithms of the signal processing can be
implemented in the DSP-s, giving more accuracy and functionality.
Using of the DSP-solutions can also minimise the using of the precise analogue components,
seen in the earlier measurement converters. These components are often expensive, parameters
are changing under temperature and time etc.
Further more, having DSP 'onboard', a lot of 'pre-', 'post-' and 'secondary-' (additionally to
main task) processings can be implemented with no extra cost of the hardware. Such kinds of
tasks can be filtering, custom-representation of the measurement results, communication over
different fixed and wireless media etc. etc.
One example of such kind of approach is to use adaptive filter for correction of the frequencyamplitude characteristics of the input circuits (amplifiers, dividers), as proposed in [ 27].
3.2. Digital post-processing of the bio-impedance signal
Current application is realized using TMS320C50-based DSP Starter Kit (DSK) [28, 29]. This
board co-operates together with a specially designed analogue pre-processing board, being
realized as extension board to DSK.
In this application there was realized a digital post-processing of analogue signals for
extraction of the heart beat and respiratory components from the bio-impedance measurement
signal.
First, the bio-impedance measurement is carried out using special analogue measurement
converter, including the AC current excitation and the phase-sensitive synchronous measurement
of the bio-impedance, as the complex voltage response to the excitation current.
Further, synchronously measured response signal is feed (through the rough analogue prefilters) via Analogue Interface Chip (AIC) to the DSP, where adaptive digital filtering (FIR filters) of heart-beat and respiratory signals was implemented. Also the IIR-filtering was
implemented in the DSP for compensation of distortions caused by the analogue-filters.
The processed digital signals were feed via the serial RS232 link to the PC for presentation of
the processed signals graphically. Special high-speed communication software over RS232 was
designed between DSK and PC. This software allows speed up to 115 kbits/s, and is working in
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
72
the “background” (using timer periodic interrupt), allowing the DSP to be loaded nearly 100%
with signal processing (filtering etc.).
Much interesting information can be obtained from Electrical Bio-Impedance measurement. The
bio-impedance can be defined as a ratio of the voltage response to the forced current. The
inverse ratio of the impedance gives conductivity.
General equivalent electrical circuit diagram for measurement of the electrical bio-impedance
is given in Fig.3.1. Practical aspects of the measurement are discussed below.
Figure 3.1. The equivalent electrical circuit diagram for bio-impedance measurement
3.2.1. About the analogue part of the Bio-impedance Measurement Interface
In comparison with the technical impedance measurement devices the electrical bio-impedance
(EBI) measurement devices must be designed to meet the specific needs determined mostly by
the nature of the bio-object. The main specifics considerations are the following ones [31, 32].
The value of measurement/excitation signal allowed is strongly limited (10 µA). Accordingly
the level of electromagnetic interference is relatively high and varies greatly with the
environment.
Basic value of the bio-impedance at the higher frequencies, where it commonly is of the
interest, is somewhere around 1 kΩ, but can vary significantly (even 10 times) during
measurement session.
Only small variations of the impedance in time, corresponding to certain biological processes
(heart beating, breathing), are of the interest. Commonly (in the case of noninvasive
measurements) these are in the limits of 0.1 %.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
73
Excitation signal frequency over 100 kHz can be needed.
Accuracy of the measurement unit can be modest, but must be guaranteed (for repeatability of
the measurements).
For portable and implantable devices restrictions from the side of power supply can be
essential. The minimal mean value of current consumption needed can be as low as < 10 A for
impantable units and supply voltage can be unipolar and as low as 3.3 V or even 2.5 V (Libattery).
Some not very certain and well investigated things can be observed in EBI measurement:
• object can be nonlinear, and
• effects from electrodes are not well defined.
In addition, a lot of things of straightly biophysical nature can affect the measurements, like the
artefacts caused by muscular contractions (movements). In the case of long time monitoring (in
hospitals, in everyday life) different other artefacts can be met.
3.2.2. Analogue measurement interface circuit for bio-impedance measurement
Though there exists a strong tendency to digitize the measurement circuitry nearly completely,
the limitations characterizing digital techniques do not enable to minimize the analogue part to
consist of a good enough A/D converter only. So some more analogue interfacing parts are still
needed to convert the signal that is got from the object under test into the form that is suitable for
the digital (post)processing part.
To illustrate the situation, one can calculate that the voltage drop due to the excitation current
of 10 microamperes at the bio-impedance of typically about 1 kΩ is of 10 mV. The variation or
the component of interest is approximately 0.1% of it or only 10 microvolts.
This is just nearly a LSB value for a 100mV FS 14-bit (including sign bit) ADC. To measure
quadrature components at 50 kHz excitation (carrier) frequency, the sample rate must be at least
200 kHz. Not impossible, one can say, but also not just very suitable for low power biomedical
devices.
The excitation current source can be accomplished on the basis of digital signal synthesis with
a digital-to-analog converter (DAC) operating as the analogue output device. In this case
practically sine-wave output can be obtained at low frequencies, but at higher frequencies this
techniques does not give good enough results. Therefore some simpler wave-forms are often
used (mainly the rectangular wave-form). If a nearly sine-wave excitation current source is used,
there will not appear very significant problems with the higher harmonics, though the rectifying
type synchronous detectors commonly used, are sensitive to the higher odd harmonics of the
signal. As the reaction of the object at different frequencies differs, the result of measurement at
the frequency of the main harmonic can be affected by the results of detection at the frequencies
of other harmonics.
To prevent significant errors from the higher harmonics, the excitation signal source must not
have high content of them, or the synchronous detector must be insensitive to the most significant
higher harmonics (the lowest ones).
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
3.2.4. Digital (post)processing of the bio-impedance measurement signal
Fortunately, in most conditions, the signals caused by breathing and heart beating differ in their
frequency (rate), so it is possible to separate them using filtering. Each period of the heart beat
signal consists of a shorter part that corresponds to the contraction of the heart, and of the
slightly longer part that corresponds to the refraction of the heart. To investigate this process
through the impedance measurement, the wave-form of the impedance variation is of interest. So
not only the main, but also some lower frequency higher harmonics could be filtered out, and the
linear phase frequency response of the filter (constant delay) prevents big distortions of the
wave-form.
The heart beating frequency can vary approximately in the range from 1 to 3 Hz (60 to 180
beats per minute). The frequency of heart beating (the heart rate) is not very stable, but its mean
value does not change quickly.
The frequency of breathing can vary in the range from 0.1 to 1 Hz (or 6 to 60 breaths per
minute) approximately. The frequency of breathing is also not stable, and can even stop for a
while, but in general its mean value does not change very quickly.
If the wave-form of the changing of impedance with breathing is of interest, the main and some
lower frequency higher harmonics could be filtered out with linear phase shift (constant delay)
filter.
The main problem is, that the artefact signals are much stronger than the signals of interest.
But some problems can cause also the fact that the main frequency component of the heart beating
can be passed as a higher harmonic of the breathing signal. The spectra of these signals are
overlapping, especially at higher breathing rates.
As it can be noticed, the properties of filters needed for separating of the breathing and heart
beat signals, must be quite specific. It is evident that using analogue filters is practically
impossible to fulfill this complicated post-processing task.
Using digital filtering extremely high selectivity, needed to separate different components of the
signal from each other, can be easily achieved. In addition, the linear phase-frequency response
can be achieved needed for avoiding the wave-form distortion in time domain.
Further more, the DSP can be used to compensate the distortions caused by analogue filtering
of the signals, e.g. for suppression of the excitation signal "carrier", anti-aliasing etc.
In the current work the digital signal processing part is considered.
3.2.5. Proposed solution
General solution of the Bio-Impedance analyzer unit is given on the block-diagram in Fig. 3.2
[33].
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75
Figure 3.2. Block diagram of the Bio-Impedance Analyzer Unit
The DSP part of the system was designed using a TMS320C50-based DSP-Starter Kit, what
was connected to the analog measurement part. The DSP fulfils the following tasks (Fig. 3.3),
described below.
Breathing
Filters
0.1... 0.5 Hz
0.2... 1.0 Hz
Raw
data
compensated
Raw
data
averaged
Raw
input
data
Compensating
filter
Averaging
(IIR-Filter)
200
50
samples/s
samples/s
0.4... 2.0 Hz
M
0.8... 4.0 Hz
U
X
RS-232
115 kbit/s
1.0... 5.0 Hz
1 Hz
HPF
1.2... 6.0 Hz
1.4... 7.0 Hz
Heart
Beating
Filters
1.8... 9.0 Hz
Figure 3.3. Functional diagram of the used DSP-solution.
1. Averaging of the input samples for suppression of the 50-Hz disturbances and noise - “oversampled” 200 samples/s input data stream is converted to 50 samples/s stream by averaging
input values by groups of 4 samples.
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2. As the analog part includes a first order low-pass filter (cut-off frequency is 1 Hz), a digital
IIR filter was designed to compensate the introduced error, using the reverse transfer function
(1+jω), implemented by the software.
The resulting filter was designed by using Bilinear Z-Transform method [30].
3. Then the compensated signal is further FIR-filtered in 8 parallel bandpass channels. Four of
them are intended for extracting of the breathing signal:
•
•
•
•
from 0.1 to 0.5 Hz
from 0.2 to 1.0 Hz
from 0.4 to 2.0 Hz
from 0.8 to 4.0 Hz
and four of them are for extracting of the heart signals:
•
•
•
•
from 1.0 to 5.0 Hz
from 1.2 to 6.0 Hz
from 1.4 to 7.0 Hz
from 1.8 to 9.0 Hz
All these FIR- filters were designed in the same way, using the Windowing method [30]. In the
current case the Hamming window was used. The number of chosen coefficients was 301, giving
for the 50 samples/s rate the absolute transition width of ca 0.5 Hz, and delay time of 3.0
seconds.
A special auxiliary C-language software was developed for automatic generation of the FIRfilter coefficients, directly into the DSP-assembler language tables.
4. Serial communication to PC for presentation and logging of the data. A special serial
communication routine (compatible to standard RS-232) was developed for current application,
working:
• at high speed - up to 115 kbits/s
• in the background mode, using the interrupts of timer, needing processing power only for
every bit switching time for a very short time. Therefore almost all the processing power
remains free for other processing purposes (filtering etc.)
• with buffered sending out of data. So data can be prepared for transmitting continuously.
3.2.6. Results
The DSP-unit for processing of the bio-impedance signal to extract breathing and heart signals
has been developed.
An example image, captured from the PC-screen, is shown in Fig. 3.4. There can be seen
“raw” (input) data for DSP, and output signals from different band-pass filters. Parts of the plot
are shown separately (“raw” data in Fig. 3.5 a, “Breathing signal Filter No.2” in Fig. 3.5 b, and
“Heart Signal Filter No.2 “ in Fig. 3.5 c).
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77
Further improvement of processing can be achieved by automatic selection of the “bestmatching” filter. Different algorithms based on Least-Mean-Squares-Method, Spectral-Density
methods etc. have been investigated for this purpose Also using of non-linear and/or adaptive
filters can improve the performance.
Figure 3.4. PC screen: Input (‘Raw data 1’) data on the highest line, and breathing (‘B1’...’B4’)
and heart (‘H1’....’H4’) signal filters outputs
a)
78
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
b)
c)
Figure 3.5. Separate pictures of “Raw Data” (a) and outputs of breathing filter “B2” (b), and
heart signal filter “H2” (c)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
79
3.3. DSP-Based Device for Multi-Frequency Bio-Impedance Measurement
3.3.1. Introduction
Measurement of the bio-impedance, including the invasive animal experiments with the
implantable cardiac pacemakers, is the important part of bio-medical investigations [ 34, 35].
As the electrodes can be implanted in the heart of an animal, it is possible to get valuable
information about the intracardiac processes by different electrical measurements. And just bioimpedance measurement is very informative, as the intracardiac bio-impedance varies together
with the processes taking place during each heart cycle. This time-variation of the impedance
carries different information at different frequencies, needed for research purposes and also for
control of the pacing rate.
The goal of this work is to design the DSP (digital signal processing) based equipment for
studying the possibility to measure the complex bio-impedance for getting directly comparable
time variation curves at different frequencies simultaneously.
3.3.2. DSP in bio-impedance measurements
Digital Signal Processing (DSP) in nowadays a powerful technology for precise, efficient,
cost-effective and flexible signal processing in many fields, including bio-impedance
measurements.
One way of using the DSP at bio-impedance measurements is to realize sophisticated digital
filtering of the signals, e.g. as proposed and realized in [ 33].
More efficient can be an implementation of the whole bio-impedance measurement processing
in the digital domain. Generic structure of the universal DSP-system is given in Fig.3.6. Such a
system includes an analogue output for the excitation signal, and an analogue input for the signal
got from electrodes, A/D and D/A converters, anti-aliasing low-pass filters (LPF), and
programmable gain amplifier (PGA) in the input signal path. So, as this circuitry is generic
DSP-system, actual bio-impedance measurement and processing is realized in the software (SW)
of the DSP-system.
3.3.3. Description of the hardware
A hardware prototype was realized using 16-bit fixed-point DSP-processor from Texas
Instruments, as the A/D and D/A converters were used DSP101/102 and DSP201/202 from
Burr-Brown (sample-rate up to 200ks/s, 16-18 bits resolutions), and the LPFs (tuned to 100 kHz
cut-off frequency) was realized on LTC1562 from Linear Technology.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Figure 3.6. Block diagram of the DSP system
3.3.4. Description of the algorithm
Idea of the current solution is to use multi-frequency excitation, and parallel processing of all
the frequency components of the input signal. Much more information can be obtained so
simultaneously.
Block diagram of the SW-algorithm is given in Fig. 3.7. The SW consists of the set of cosine
and sine-wave generators (totally 8 for the current implementation). These generators are based
on look-up tables [36], where samples are tabulated during the initialization stage for every sineand cosine-wave. In the first implementation the length of every table was taken N= 200, and as
we have sampling frequency fs=200 ks/s, and in every period of the generated sine/cosine wave
there must be at least 2 samples (Nyquist criteria), the maximum frequency achieved can be 100
kHz, minimum frequency, and also frequency step will both be 1 kHz. More memory-effective
solution for cosine and sine-wave generators could be using of IIR filter-like generator
algorithms.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Re1
Channel 1
X
LPF
Im1
X
LPF
Channel 2
X
Digital
input
Di
Re2
LPF
Im2
X
LPF
Channel 8
X
LPF
X
Gencos1
81
Re8
Im8
LPF
f1
Gensin1
Gencos2
f2
Gensin 2
Gencos8
f8
Gensin 8
f1 to f8
Digital output Do
GenÓcos
Figure 3.7. Proposed digital signal processing algorithm
Efficient algorithms for multi-frequency signal processing are proposed, e.g. in [37]. In the first
implementation this approach was not used, as arbitrary frequencies may be needed. When there
will be more information about reasonable frequency selection, this approach can be discussed.
Digital output stream (for excitation, via A/D converter) is got as a sum of all the cosinewaves. Then digital input stream (including all frequency components) is multiplied (correlated)
by all the cosine- (for real components) and sine-waves for (imaginary components), and filtered
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
by digital LPF filters, suppressing the ripple and taking the average value. In the first
implementation these LPFs are realized as the first order IIR filters.
In the first implementation the frequency set can be initialized for any frequencies from 1 to 100
kHz, multiples of 1kHz.
These LPFs also limit the bandwidth of the demodulated impedance components to about 30 Hz
to avoid the power line interference and to obtain frequency selectivity. This is presumed to be
wide enough for the relatively smooth bio-impedance signals having correspondingly a relatively
narrow bandwidth in comparison with ECG signals contending narrow peaks.
3.3.5. Results
The DSP-based multi-frequency bio-impedance measuring and processing system was
proposed, analyzed and realized [38]. The instrument has been used in hand-to-hand thoracic
bio-impedance measurement, and will be implemented in intracardiac animal experiments at
Pacesetter AB (Sweden).
Experimental study of the equipment show that using the DSP-based multi-frequency device
enables to measure the bio-impedance reliably with the 10 mV excitation voltage, or 10 µA
current, only.
Both the real and the imaginary parts of the impedance were measured using the bio-impedance
simulator containing physiological solution.
The new method and technology, including specially designed hardware and software, has
made possible to perform intracardiac time-domain measurement of bio-impedance variation
simultaneously at different frequencies. As a result, the measurement will be significantly
shortened, and what is even more important, the results will be obtained at the same timemoments and in exactly the same conditions.
Photo of the designed unit (“EBI-BOX”) is given in Fig. 3.8
Figure 3.8. Prototype of the DSP-Based Multi-frequency Bio-impedance measurement unit
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
83
3.4. DSP-based AC Measurement Converters
3.4.1. Introduction
As it was mentioned already, the AC measurements are an important part of electrical
measurements.
Intensity of AC signals can be defined by different ways. One important electrical value to be
measured is the root-mean-square (RMS) value of the voltage (or current), related closely to
power measurements.
Up to the latest time such converters were realized using thermal heating sensors
(thermocouples e.g.), or analog signal conversion techniques. A good overview of analog
converters is given in [39].
But also measurements of other parameters, like the peak-to-peak value and average rectified
value can be under interest.
Nowadays the digital signal processors (DSP) offer the universal technology, being
technically and costly effective, and giving precise and flexible solutions for various problems,
earlier realized in hardware, mainly using of analog circuits [40]. Benefits of the using of the
DSPs has been given in p.3.1.
3.4.2. DSP-based peak-to-peak and average rectified value measurements
Peak-to-peak measurement converters are used for measurement of signal parameters, related to
amplitude (peak) value.
Analog realization of the peak-to-peak measurement converter can be done with the solution
according to Fig.3.9. Maximum (positive) value of the input AC voltage is fixed via diode D1
on the capacitor C1 and minimum (most negative value) via D2 on the capacitor C2. The output
DC voltage will be so Uout= max(Uin(t) ) -min (Uin (t)) . Resistors R1 and R2 provide
discharging time-constants for capacitors C1 and C2. Practical analog measurement converters
can be more complex, for example for compensating of the forward voltage drops on diodes.
Fig.3.9. A simple analog peak-to-peak value converter
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
The set of equatios for numerical approximation of the analog circuit can be the following:
UC1 = Uin (t) ,
UC1 = UC1* (1-∆T/T0) ,
for case (Uin (t) > UC1 ) ,
for case (Uin (t) < UC1 ) ,
(3.1)
(3.2)
UC2 = Uin (t) ,
UC2 = UC2* (1-∆T/T0) ,
for case (Uin (t) < UC2 ) ,
for case (Uin (t) > UC2 ) ,
(3.3)
(3.4)
,
(3.5)
Uout = UC1 - UC2
Uin(t) is there the current value for input sample, ∆T is the sampling period (must be much less
than T0 for this approximation), and T0 is equivalent time constant T0= R1*C1= R2*C2.
The average-rectified-value analog converters has been widely used in multimeters for AC
measurements, as they are easy to implement, and are precise for measurement of pure sinewaves Block diagram of a such converter is given in Fig. 3.10.
Uin (AC)
y= |x |
(Absolute value module)
Uout (DC)
n-order LPF
Fig. 3.10. Block diagram of the average-rectified value converter
Digital approximation of this algorithm can be done by the following equation set, where the
low-pass filter (LPF) is the first-order RC-filter with time-constant T0.
Uout= Uout + (abs (Uin(t)) - Uout) * ∆T/T0
.
(3.6)
There Uin(t) is the current value of an input sample, ∆T is the sampling period (must be much
less than T0 for this approximation), T0 is the equivalent time constant T0= R*C for LPF.
Practical realizations of the measurements of the average-rectified, peak-to-peak and RMSvalues are described in p. 3.4.7.
3.4.3. Proposed algorithm for numerical RMS-measuring implementation
As known, the RMS value of the signal is defined for continuos analog signals as
T
Urms= √ [ (1/ T) ∫ Uin(t)2 dt]
,
(3.7)
0
where T is the period over which RMS value is defined and Uin(t) is input voltage as
function of a time.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
85
Using for the RMS calculations the discrete time numerical samples, the formula will be:
Urms=
N-1
√ [ (1/ N) ∑ Uin (i) 2 ]
0
,
(3.8)
where N is the number of input samples, over what the RMS-value is calculated
Uin(i) is for input sampled values
Direct using of the formula (3.8) is possible, but needs calling of the square-root function. This
function is typically not supported by the hardware of the DSP, and takes a large number of
machine cycles. Further more, for AC-voltmeter applications it can be more desirable to have the
converted result estimation after every input sample processing, not only after all N samples are
processed.
For analog converters the implicit way of solution of formula (3.7) for RMS-measurement has
been proposed in [39] (Fig. 3.11). This scheme uses multiplier/divider with transfer function
Y= X 2 /Z, where the input value is squared and divided to converted RMS value, and averaged
by low-pass filter (LPF) (e.g. the first-order RC circuit).
Uin (t)
X
Y
LPF
Uout
Y= X 2 /Z
Z
Fig.3.11. The implicit RMS-converter with analog processing
For the first order LPF, this solution can be described by the following integral equation:
T
Urms (T) = Urms (0)+ (1/ RC) ∫ ( Uin(t)2 / Urms(t) -Urms(t) ) dt ,
(3.9)
0
which can be approximated for time-discrete numerical case by the following iteration formula :
Urms (i) = Urms(i-1) + (1/ M) ( Uin (i) 2 /Urms(i-1)
-Urms(i-1)) ,
(3.10)
This algorithm realizes quasi-continuos root-mean averaging of the input signal, and needs only
multiplication and division. Approximation is true as sampling period is coming close to zero
Ts = 1/fs --> 0, and coefficient M depends on the ratio of the sampling period to equivalent LPF
equivalent time constant 1/ M= Ts / (R*C) , or in other words, equivalent time period for the
calculation scheme ( 3.10) is M in units of Ts or (M * Ts ) .
So, in the current work, time domain methods are investigated. In some case frequency domain
analysis can be also interesting [ 41].
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
3.4.4. DSP-realization of the RMS measurement converter
Generally the DSP-based RMS-measurement converter (or converter with any other algorithm
in the software) can be implemented according to the following block-diagram (Fig.3.12). The
converter includes the input A/D converter, for converting input analog signal into digital
samples, and a DSP, doing the software calculation of the RMS-value over the digital samples.
Calculated value can be used more-or-less directly or converted to analog signal. Nowadays, the
digital presentation is mostly used, for indication, and further processing using computers etc.
Analog Input
Input
conditioning
A/D
Converter
Anti-aliasing
LPF
DSP
with special
(RMS-)
Software
S/H
circuit
D/A
Converter
Analog output
Digital output
Fig. 3.12. Block-diagram of DSP-based RMS-measurement converter
3.4.5. Estimation of the principal inaccuracy (medium frequencies)
For medium frequencies (where specific low- and high-frequency errors are not present) the
accuracy is limited only by the inaccuracy of the used analog front-end (A/D converter mainly) ,
and the limited resolution of arithmetics, limited with number of bits.
Let us assume that our system has M-bits of effective accuracy for DC, or in other words,
inaccuracy is ± 1/2 of the LSB-value, or δ= 1/2M+1 of the full scale (f.s.) of the A/D converter,
or of the peak-value of the AC signal .
Let us have N samples, over which we calculate the RMS-value of the signal, and all samples
are measured with the same inaccuracy ± 1/2M+1 of the peak-value of the signal:
That means that equation (3.8) will have the following form
N-1
Urms = √ [(1/ N) ∑ ( Uin (i) 2 ±2*Uin(i)* δ + δ 2)]
0
,
(3.11)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
87
For extracting the small relative error we shall arrange the error-part separately:
N-1
N-1
N-1
Urms= √ [(1/N)*∑ Uin (i) 2 *( 1 ± ∑ ( 2* Uin (i)* δ ) / ∑ Uin (i) 2 ] .
0
0
0
(3.12)
Using the mean-rectified value of the signal:
N-1
Umean = (1/ N)* ∑ Uin (i)
,
0
and the sampling error value
(3.13)
δ= 1/2M+1 *Upeak
(3.14)
,
the final equation for RMS value will have the following form:
Urms= Urms0 ( 1 ± δ* Umean /Urms2) ,
or using (3.14)
Urms = Urms0 ( 1 ± (1/2M+1) *Upeak *Umean /Urms 2)
.
(3.15)
For sine wave, as we know Upeak/Urms= √ 2 = ca 1.414, Umean/Urms= 2√ 2/ π= ca
0.909, and Upeak*Umean/Urms 2= 4/ π= ca 1.2732. So, for the RMS measurement converter,
compared with direct current voltage (DC) measurement, we have loss in the accuracy for the
sine wave of 1.27 times (or decrease of the number of bits of log2 1,27= 0.34 bits).
3.4.6. Analysis of the low frequency error and settling time of the converter
Settling time of the converter is depending on the time constant (M*Ts), as described earlier.
For normalized time constant M*Ts= 1 the settling curves were simulated (using equation (4)
with MATLAB for some combinations of input parameters. The results of the simulation are
given in Fig. 3.13.
The low-frequency error is resulted by the pulsation at the output of the converted signal (Fig.
3.14), and occurring of the difference in the RMS and mean values of the pulsating signal. Lower
we give some estimation for this pulsation, and for the systematical error from this pulsation.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
Figure 3.13. Settling of the conversion in the time domain
Curve 1- Uout(0) = 0.10, Ts = T /100
Curve 2- Uout(0) = 0.02, Ts = T /100
Curve 3- Uout(0) = 0.10, Ts = T /20
Curve 4- Uout(0) = 0.10, Ts = T /10
Let us have the input signal
√ 2* Uin_rms *sin (ω
ω t). After squaring we shall get
2
2*Uin_rms [0.5+0.5*sin (2ω
ω t)]. After dividing the squared value to the output signal, and after
the first order low-pass filtering with T=RC (transfer coefficient 1/(√
√ (1+(ω
ω T)2), the signal has
DC component ca Uin_rms and AC component Uin_rms* 1/√
√ (1+(2ω
ω T)2), and for ω>>1 the
AC component or pulsation is
Upuls= Uin_rms/(2ωT) ,
(3.16)
For finding the estimation of the systematical error we can think that the converter by Fig. 3.15
equals the RMS-values of the input and output signals or for ω>>1 the following equation must
be valid:
Uin_rms2 = Uout02 + (Uin_rms /(2ωT))
2
,
(3.17)
from what we can find the estimation of the systematical low-frequency error:
Uout= Uin_rms √ (1- 1/ (2ωT)2 ) ≈ Urms (1 - 1/ 8* (ωT)
2
,
(3.18)
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
89
Functions for pulsation and systematic low-frequency error formulas (3.16) and (3.18) are
given in Fig. 3.14.
Fig. 3.14. Low-frequency pulsation and systematical error functions
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
3.4.7. Practical realizations of the prototypes
One practical realization of the prototype for measurement of the RMS-value, the peak-to-peak
value, and the average rectified values, was realized by using the TMS320C50 -based DSP kit.
The board has additionally to 16-bit fixed-point DSP-chip 14-bit analog-interface-chip (AIC)
with A/D and D/A converters for audio frequencies, including input anti-aliasing filter and
Sample/Hold circuit [42].
The software was written in mixed C and assembler languages.
The data is sent to DSK standard RS-232 serial interface to PC. Users interface was designed
on PC under Windows.
Practical investigation show the accuracy (after calibration the board for DC input voltage
against the inaccuracy f the on-board reference voltage ) over the frequency range 20 Hz-10 kHz
being below 0.1% of full scale for input AC voltages from 0.01...1 V.
Such solution can be useful for medium-performance audio-band measurements, in some
combined multi-purpose instrument.
Second solution for RMS-measurement was done by using the same hardware, as described at
p.3.3.3. This solution has improved bandwidth and accuracy. This solution includes DSP101/102 A/D converter from Burr-Brown - with what resolution of 16 bits and bandwidth of
measurement up to 100 kHz can be easily achieved [43].
3.4.8. Conclusion and results
So, theoretical and practical investigations show that the proposed hardware solution is
suitable for precise measurement of AC voltage. Resolution and potential accuracy of 4 1/2 to 5
1/2 digits (up to 0.001% of the full scale) can be achieved. Frequency range up to 100 kHz can
be covered. At lower frequencies the performance is limited only by measurement time. As
actually measurement algorithm is implemented in the software, this can be easily modified.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
91
4. General conclusions and results
1. Special class of the AC/DC measurement converters with switched capacitors in the negative
feedback path has been developed: It has been shown that significant reduction of the
sensitivity of the transfer coefficient of the converter from the resistor ratios can be achieved
in such circuits. Circuits with improved low-frequency performance has been developed
inside this class. Methodology of analysis of the circuits under discussion has been
developed. Mathematical models for the described circuits has been found, and the circuit
analysis has been done. The prototypes of AC/DC measurement converters with improved
metrological characteristics has been realized and tested.
2. A special class of synchronous detectors with significantly improved metrological
characteristics (accuracy, frequency and dynamical ranges) has been developed. It has been
shown that such converter can have an extremely wide dynamic range of the input signal. One
application field of such converters are PLL (phase-locked-loop) circuits, where the wide
dynamic range can be effectively exploited.
3. Digital Signal Processor (DSP) based solutions for AC measurements has been proposed,
developed, and realized, focusing to the RMS (root-mean-square) value measurements. The
benefits of the 'digital conversion' has been analyzed (flexibility, programmability, extended
functionality etc.). Some extended measurement functions can be easily added to the main
ones. One such an example is multi-frequency vector-measurement of the bio-impedance,
what can be easily and cost-effectively realized in DSP technique. In an other application the
DSP does effectively very extensive post-processing (conditioning, band-pass filtering in
many sub-bands, communication) of the bio-impedance signal. The performance of the
measurements has been analyzed. Theoretical analysis and practical realization of the 'digital'
solutions show, that their performance can be comparable with analog solutions, but
functionality and flexibility are much better.
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
5. Claims
1. Developed and analyzed class of the AC/DC measurement converters (with switched
capacitors in the negative feedback path) has the best performance available today in the
world for some class of applications- resolution of 1 ppm, accuracy for medium frequencies
(from 400 Hz to 4 kHz) better than 0.01%, and frequency range from 10Hz to 1MHz. This is
valid for measurement of the signals with known wave-form.
2. The solutions, where capacitors in the negative feedback path are switched by external
reference synchronous signal, have, additionally to excellent metrological characteristics,
also a very wide dynamical range of the input signal, as such converters due to "synchronous
(integrating) capacitors" in the negative feedback path have inherent suppression of the
disturbances and noise.
3. As a rule, the best performance of the measurement systems today is achieved by combining
analog pre- and digital post-processing.
4. As digital signal processing methods can today already support extremely wide functionality
and flexibility, and extended classical measuring functions and completely new measurement
functions can be effectively implemented. One example of such an extended functionality is
processing of the sophisticated wave-forms, e.g. vector processing of multi-frequency signals
in frequency and time domains.
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
93
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O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
97
Appendixes
Appendix A. Low frequency error calculations for the AC/DC converter
according to Fig.1.2(asymmetrical) and Fig1.4 (symmetrical scheme)- programs
for HP85.
A1. Low frequency error calculations for the AC/DC converter according to Fig.1.2
(asymmetrical scheme.
98
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
A2. Low frequency error calculations for the AC/DC converter according to Fig. 1.4
(symmetrical scheme).
O.Märtens. AC Measurement Converters: Analog and Digital Solutions (Theses....)
99
Appendix B. Practical circuit diagrams and component lists
Appendix B1. Practical realization of the AC/DC converter by p.1.5.1
Fig B1.1 Principal circuit diagram for the precise AC/DC converter
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
Table B1.1. Component list for the precise AC/DC converter
101
102
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Appendix B2. Practical realization of the AC/DC converter by p.1.5.2
Fig B2.1. Principal circuit diagram for the improved AC/DC converter
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
Table B2.1. Component list for improved AC/DC converter
103
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Appendix B3. Practical realization of the AC/DC converter by p.1.5.3
Fig B3.1. Principal circuit digram for the AC/DC converter with input conditioning
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
Table B3.1. Component list for the AC/DC converter with input conditioning
105
106
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Appendix B4. Realization of the PLL for the stereo-decoder
Figure B4.1. Circuit diagram for the PLL-loop with proposed SD
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
Appendix C. Numerical calculation of the settling process in the time domainprograms for HP85.
Appendix C1. Analysis of the AC/DC converter according to p.1.5.1.
Figure C.1 First equivalent circuit diagram (converter described in p.1.5.1)
107
108
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
Appendix C2. Analysis of the AC/DC converter according to p.1.5.2.
Figure C.2 Second equivalent circuit diagram (converter described in p.1.5.2)
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
109
110
O.Märtens AC Measurement Converters: Analog and Digital Solutions (Theses....)
Comment First graph is for one output signal (Uout1). Second graph is for differential output
case (Uout=Uot1-Uout2)