The design of Low-Voltage Low-PowerAnalog Integrated Circuits and theirApplications in Hearing InstrumentsWouter Serdijn

The design of Low-Voltage Low-PowerAnalog Integrated Circuits and theirApplications in Hearing Instruments

PROEFSCHRIFTter verkrijging van de graad van doctoraan de Technische Universiteit Delft,op gezag van de Rector Magnicus,Prof.ir. K.F. Wakker,in het openbaar te verdedigen ten overstaan van een commissieaangewezen door het College van Dekanen,op maandag 28 februari 1994, te 16:00 uurdoorWOUTER ANTON SERDIJNgeboren te ZoetermeerElektrotechnisch Ingenieur

Delft University Press/1994

Dit proefschrift is goedgekeurd door de Promotor,Prof.dr.ir. J. Davidse.Overige leden van de promotiecommissie:Prof.dr.ir. C.A. GrimbergenProf.dr.ir. J.H. HuijsingProf.dr.ir. A.H.M. van RoermundProf.dr.ir. W.M.C. SansenProf.dr. H. WallingaDr.ir. A.C. van der Woerd

Published and distributed by:Delft University PressStevinweg 12628 CN DelftThe NetherlandsTelephone +31 15 783254Fax +31 15 781661ISBN 90-6275-955-6 / CIP

c 1994 by Wouter SerdijnCopyright All rights reservedNo part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical,including photocopying, recording or by any information storage and retrieval system, without permission from the publisher: Delft UniversityPress, Stevinweg 1, 2628 CN Delft, the Netherlands.Printed in the Netherlands

A change of speedA change of styleA change of sceneWith no regretsA chance to watchAdmire the distanceStill occupiedThough you forgetJoy Division: New dawn fades

to Marjo

Contents1 Introduction2 Ports

2.1 Introduction : : : : : : : : : : : : :2.2 Reducing errors : : : : : : : : : : :2.2.1 Compensation : : : : : : :2.2.2 Error feedforward : : : : : :2.2.3 Negative feedback : : : : :2.2.4 Indirect negative feedback :2.3 Operating in the current domain :2.3.1 Source and load : : : : : : :2.3.2 The desired topology : : : :2.3.3 The available technology : :2.3.4 The available power supply

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3 Modeling the bipolar transistor at low voltages and low currents 253.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : :3.2 Large-signal model of a bipolar transistor : : : : : : : : : :3.2.1 The transport current ICT : : : : : : : : : : : : : : :3.2.2 The resistances RB , RC and RE : : : : : : : : : : :3.2.3 The junction capacitances CJE and CJC : : : : : : :3.2.4 The diusion capacitances CDE and CDC : : : : : :3.2.5 Low-voltage low-power large-signal transistor model3.3 Small-signal model of a bipolar transistor : : : : : : : : : :3.4 Noise : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :3.4.1 Noise sources in the bipolar transistor : : : : : : : :3.4.2 Transformation of noise sources to the input : : : : :

4 Negative-Feedback Ampliers

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2526262828292929313131

35

4.1 Introduction (outline of the design method) : : : : : : : : : : : : : 354.2 The basic amplier conguration and the feedback network : : : : 384.2.1 Current ampliers : : : : : : : : : : : : : : : : : : : : : : : 38vii

4.34.44.54.64.7

4.8

4.2.2 Transconductance ampliers : : : : : : : : : : : : : : : : : :4.2.3 Transimpedance ampliers and voltage ampliers : : : : : :The input stage : : : : : : : : : : : : : : : : : : : : : : : : : : : : :The output stage : : : : : : : : : : : : : : : : : : : : : : : : : : : :Loop gain : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :4.5.1 Current ampliers : : : : : : : : : : : : : : : : : : : : : : :4.5.2 Transconductance ampliers : : : : : : : : : : : : : : : : : :High-frequency behavior : : : : : : : : : : : : : : : : : : : : : : : :Biasing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :4.7.1 Four fundamental ways of biasing : : : : : : : : : : : : : : :4.7.2 Biasing in the current domain : : : : : : : : : : : : : : : : :4.7.3 Biasing a symmetrical amplier with oating source andoating load : : : : : : : : : : : : : : : : : : : : : : : : : : :4.7.4 Biasing a symmetrical amplier with oating source andxed load : : : : : : : : : : : : : : : : : : : : : : : : : : : :4.7.5 Biasing a symmetrical amplier with xed source and oating load : : : : : : : : : : : : : : : : : : : : : : : : : : : : :4.7.6 Biasing a symmetrical amplier with xed source and xedload : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :An example: a microphone preamplier for use in hearing instruments4.8.1 The basic amplier conguration : : : : : : : : : : : : : : :4.8.2 The feedback network : : : : : : : : : : : : : : : : : : : : :4.8.3 The input stage : : : : : : : : : : : : : : : : : : : : : : : : :4.8.4 The output stage : : : : : : : : : : : : : : : : : : : : : : : :4.8.5 Loop gain : : : : : : : : : : : : : : : : : : : : : : : : : : : :4.8.6 High-frequency behavior : : : : : : : : : : : : : : : : : : : :4.8.7 Biasing : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

5 Automatic Gain Controls

5.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :5.2 AGCs with nite compression ratios : : : : : : : : : : : : : : : : :5.2.1 Controlled ampliers in cascade : : : : : : : : : : : : : : : :5.2.2 Dierently controlled ampliers : : : : : : : : : : : : : : : :5.2.3 Controlled knee level : : : : : : : : : : : : : : : : : : : : : :5.3 AGCs in the current domain : : : : : : : : : : : : : : : : : : : : :5.4 Controlled current ampliers : : : : : : : : : : : : : : : : : : : : :5.4.1 Four fundamental ways of controlling the gain : : : : : : : :5.4.2 The current-controlled type 1 symmetrical scaling currentamplier : : : : : : : : : : : : : : : : : : : : : : : : : : : : :5.4.3 The current-controlled type 2 symmetrical scaling currentamplier : : : : : : : : : : : : : : : : : : : : : : : : : : : : :viii

4142424444444647535455565758595960606162626263

6565666768697171727474

5.4.4 The voltage-controlled type 1 symmetrical scaling currentamplier : : : : : : : : : : : : : : : : : : : : : : : : : : : : :5.4.5 The voltage-controlled type 2 symmetrical scaling currentamplier : : : : : : : : : : : : : : : : : : : : : : : : : : : : :5.5 Comparators : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :5.5.1 Cascade of a non-linear one-port and a linear two-port : : :5.5.2 Ampliers with a saturated input-output relation : : : : : :5.6 Voltage followers : : : : : : : : : : : : : : : : : : : : : : : : : : : :5.7 An example: an automatic gain control for hearing instruments : :5.7.1 Design of the controlled amplier : : : : : : : : : : : : : : :5.7.2 Design of the comparator : : : : : : : : : : : : : : : : : : :5.7.3 Design of the voltage follower : : : : : : : : : : : : : : : : :5.7.4 Overall design : : : : : : : : : : : : : : : : : : : : : : : : :5.7.5 Experiment results : : : : : : : : : : : : : : : : : : : : : : :

6 Filters

757676777878798081828284

87

Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87Current integrators : : : : : : : : : : : : : : : : : : : : : : : : : : : 90Low-voltage low-power current integrators : : : : : : : : : : : : : : 91A capacitance-transconductance amplier : : : : : : : : : : : : : : 93A capacitance-transconductance amplier with enlarged voltage swing 956.5.1 Dynamic range : : : : : : : : : : : : : : : : : : : : : : : : : 956.5.2 Inuence of the output impedance of the voltage amplieron the transfer function : : : : : : : : : : : : : : : : : : : : 966.5.3 The voltage amplier : : : : : : : : : : : : : : : : : : : : : : 976.6 An example: a low-voltage low-power current-mode highpass leapfrog lter : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 986.6.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : 986.6.2 A rst approach : : : : : : : : : : : : : : : : : : : : : : : : 986.6.3 The integrator blocks : : : : : : : : : : : : : : : : : : : : : 996.6.4 The complete lter : : : : : : : : : : : : : : : : : : : : : : : 1016.6.5 Semicustom realization : : : : : : : : : : : : : : : : : : : : 1036.6.6 Measurements : : : : : : : : : : : : : : : : : : : : : : : : : 1036.16.26.36.46.5

7 A universally applicable analog integrated circuit for hearing instruments1097.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :7.2 A universally applicable analog integrated circuit for hearing instruments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :7.3 Specications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :7.3.1 General parameters : : : : : : : : : : : : : : : : : : : : : :ix

109

110111112

7.3.2 Audio parameters : : : : : : : : :7.4 A compression/expansion system : : : : :7.5 The lters : : : : : : : : : : : : : : : : : :7.5.1 The two rst-order highpass lters7.5.2 The second-order lowpass lter : :7.5.3 The oset lter : : : : : : : : : : :7.6 The controlled microphone preamplier :7.7 The envelope processor : : : : : : : : : : :7.8 The expander : : : : : : : : : : : : : : : :7.9 The pickup-coil preamplier : : : : : : : :7.10 Semicustom realization of the front-end :7.10.1 General parameters : : : : : : : :7.10.2 Audio parameters : : : : : : : : :7.11 Fullcustom realization of the front-end : :

8 Summary and conclusions

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Chapter 1Introduction"What do you like doing best in the world, Pooh ?""Well," said Pooh, "what I like best {" and then he had to stop and think.Because although Eating Honey was a very good thing to do,there was a moment just before you began to eat itwhich was better than when you were,but he didn't know what it was called.A.A. Milne: The house at Pooh cornerLow-voltage low-power circuit techniques are applied in the area of batteryoperated systems. In particular, they are of crucial importance for implantabledevices, such as pacemakers, blood owmeters and auditory stimulators 1, 2, 3, 4,5, 6, 7]. Also, as more and more complex systems are being integrated on the samechip, area minimization is becoming of primary importance. Typical examplesare portable radios, hand-carried radiotelephones, pagers and hearing instruments8, 9, 10, 11, 12, 13]. As the size of batteries is now becoming the limiting factor,it is not sucient to reduce the size of bulky components by integrating them thereduction of the power dissipation is also very important. As a consequence, thekey point is to develop, simultaneously, both low-voltage and low-power operatingintegrated circuits in order to reduce the battery size and chip area.There is, however, no general design theory available for the design of lowvoltage low-power integrated circuits. In most cases we have met in literature,the design is started by choosing a suitable circuit implementation from the largevariety of conventional electronic circuits. Then, this circuit is adapted in order tot within its new environment. As an example we mention an ordinary dierentialpair equipped with bipolar transistors.When used in practical circuits, the dierential pair needs an extra DC voltagereference. Especially in low-voltage circuits, this source must be quite accuratein order to fully exploit the possible output voltage swing of the dierential pair1

14]. Further, if the stage has to be biased individually, both base connectionsneed a resistive path to ground, to make the ow of base current possible. Thismeasure aects the small-signal behavior and noise properties. Apart from this, inlow-power circuits very large resistors are needed, which entails a large chip-areaconsumption. In 15] it has been shown that a dierential pair is only one of fourcongurations that exist for the implementation of a single dierential CE stageand that one of the other congurations, the `alternative dierential pair', is muchbetter suited for application in low-voltage low-power analog integrated circuits.Another example concerns the design of a class-AB output stage. In mostconventional output stages, e.g., in op amps or audio ampliers, this output stagecontains two complementary output transistors in CC conguration (i.e. withtheir emitters tied together) driving the load. This circuit, however, requires aminimum supply voltage that is larger than the sum of the base-emitter voltagesof both transistors. For circuits that use a 1-V power supply other solutions hadto be found 16, 17, 18].Various denitions of the term `low-voltage' can be found in literature. Designers of digital circuitry often think of 3.3 V as being low voltage. This is notsurprising, as today most of the digital circuits operate at 5 V. From this pointof view 3.3 V, which is about to become the new world standard, is indeed lowvoltage. From the application side, a circuit or system is often denoted as low voltage when it operates on a single battery. This, of course, depends on the type ofbattery one uses. A single zinc-air or mercury battery cell at the end of its lifetimedelivers only 1 V, whereas a single lithium cell delivers as much as 3 V. For thisreason we have chosen for a more design-driven denition: low-voltage analog

circuits do not have two or more junctions in series between two supplyrails. All the circuits described in this work comply with the above denition.

In dening the expression `low-power' the situation is more complicated. Preferably, a circuit should require as little power as possible: and every power reductionof a circuit, with respect to its earlier version while maintaining its functionality,makes it more `low-power'. Of course, this process cannot go on forever. A lowerbound is formed by the information capacity C of the electronic circuitry:!

S(1:1)C = B log 1 + N

B being the bandwidth of the circuit and S=N the ratio of the signal power tonoise power under the assumption of Gaussian noise 19].In most cases, the bandwidth properties of an electronic circuit depend onthose of the transistors used inside. A quantity serving as a gure of merit for thebandwidth properties of a transistor is the transit frequency fT . In Figure 1.1, thetransit frequency has been depicted as a function of the bias current for a typicalsmall integrated bipolar transistor in a high-frequency process. We can see that2

fT has a maximum at about 100 A, while it becomes proportional to the biascurrent for lower bias currents.10GfT1G

100M

10M

1M1n

100n

10

1m

IC

Figure 1.1: Transit frequency fT versus the bias current IC of a typical integratedbipolar transistorAn important source of noise in electronic circuits is related to the stochasticnature of the ow of charge carriers that pass a potential barrier. Because ofthe discrete nature of the emitted charge carriers, mostly electrons, the currentis quantized 19]. As a consequence, the maximal signal-to-noise ratio also isapproximately proportional to the bias current.Since both the bandwidth and the maximal signal-to-noise ratio are proportional to the bias current, we thus have to face the fact that low-power integratedcircuits have a smaller information capacity than conventional integrated circuitsthat do not have this `low-power' constraint.Since 1986, the Electronics Research Laboratory of the Delft University ofTechnology, Faculty of Electrical Engineering has had a project group `low-voltagelow-power electronics'. To date, the main research eld has been the developmentof circuits for hearing instruments and the underlying design theory. This work hasbeen carried out partly under the group's own control and partly in cooperationwith industry, and has resulted in two new generations of hearing instruments, bothnow in production. Moreover, these research eorts have increased both knowledgeand insight into the problems which specically concern this specialized branch ofelectronics. They, in turn, have resulted in specic design methodologies, devicemodeling and circuit architectures.This thesis deals with the design of low-voltage low-power analog integratedcircuits and their applications in hearing instruments. Its purpose is twofold: tooer a design procedure for low-voltage low-power circuits in general, and to give a3

new concept for a hearing instrument. It is assumed that the principal performancerequirements of the circuits are dictated by the overall system requirements, whichcannot be compromised in achieving low-voltage low-power operation. In view ofthe foregoing, it becomes clear that a low-voltage low-power design constraintentails special-purpose rather than general-purpose circuits. For this reason, thetypical general-purpose integrated circuit is not the focal point of this work. As,at the start of the research, low-threshold BiCMOS IC processes were not yetwell specied for our purpose, there is an emphasis on implementations in bipolarrealizations. However, many of the ideas expressed here are also valid for otherprocesses.In Chapter 2, it is shown that low-voltage low-power electronics operate bestin `the current domain'. In Chapter 3, transistor models for the large-signal, thesmall-signal, and the noise behavior, are discussed for bipolar transistors in a lowvoltage low-power environment. With these tools various important low-voltagelow-power functional blocks can be designed. These are: ampliers (Chapter 4),automatic gain controls (Chapter 5) and lters (Chapter 6). These three functionsare the main functions that can be found in a hearing instrument. Their applicationin a universally applicable IC for hearing instruments is presented in Chapter 7.

References1] W.H. Ko and M.R. Neuman: Implant biometry and microelectronics, Science, Vol. 156, pp. 351-360, April 1967.2] G. Weil, W.L. Engl and A. Renz: Integrated pacemakers, IEEE J. Solid-StateCircuits, Vol. SC-5, pp. 67-73, April 1970.3] R.W. Gill and J.D. Meindl: Low-power integrated circuits for an implantablepulsed Doppler ultrasonic blood owmeter, IEEE J. Solid-State Circuits, Vol.SC-10, pp. 464-471, December 1975.4] S. Gheewala, R.D. Melen and R.L. White: A CMOS implantable multielectrode auditory stimulator for the deaf, IEEE J. Solid-State Circuits, Vol.SC-10, pp. 472-479, December 1975.5] W.M.C. Sansen: On the integration of an internal human conditioning system, IEEE J. Solid-State Circuits, Vol. SC-17, pp. 513-521, June 1982.6] L.J. Scotts, K.R. Innger, J. Bobka and D. Genzer: An 8-bit microcomputer with analog subsystems for implantable biomedical applications, IEEEJ. Solid-State Circuits, Vol. 24, pp. 292-300, April 1989.4

7] H. McDermott: A custom-designed receiver-stimulator chip for an advancedmultiple-channel hearing prosthesis, IEEE J. Solid-State Circuits, Vol. 26,pp. 1161-1164, August 1991.8] T. Okanobu, H. Tomiyama and H. Arimoto: Advanced low-voltage single chipradio IC, IEEE Trans. Consumer Electronics, Vol. 38, pp. 465-475, August1992.9] D.W.H. Calder: Audio-frequency gyrator lters for an integrated radio pagingreceiver, Proc. IEE Conf. Mobile Radio Syst. Tech., pp. 21-24, 1984.10] M.J. Hellstrom: A family of integrated class-B hearing aid circuits, IEEETrans. Broadcast Telev. Receivers, Vol. BTR-11, pp. 73-78, December 1965.11] I.E. Getreu and I.M. McGregor: An integrated class-B hearing aid amplier,IEEE J. Solid-State Circuits, Vol. SC-6, pp. 376-384, 1971.12] F. Callias, F.H. Salchli and D. Girard: A set of four ICs in CMOS technologyfor a programmable hearing aid, IEEE J. Solid-State Circuits, Vol. 24, pp.301-312, 1989.13] A.C. van der Woerd: Analog circuits for a single-chip infrared controlledhearing aid, Analog Integrated Circuits and Signal Processing 3, 91-103(1993).14] H. Tanimoto, M. Koyama and Y. Yoshida: Realization of a 1-V active lterusing a linearization technique employing plurality of emitter-coupled pairs,IEEE J. Solid-State Circuits, Vol. 26, pp. 937-945, July 1991.15] A.C. van der Woerd and A.C. Pluygers: Biasing a dierential pair in lowvoltage analog circuits: a systematic approach, Analog Integrated Circuitsand Signal Processing 3, 119-125 (1993).16] J.H. Huijsing and D. Linebarger: Low-voltage operational amplier with railto-rail input and output ranges, IEEE J. Solid-State Circuits, Vol. SC-20, pp.1144-1150, December 1985.17] J. Fonderie, M.M. Maris, E.J. Schnitger and J.H. Huijsing: 1-V operationalamplier with rail-to-rail input and output ranges, IEEE J. Solid-State Circuits, Vol. SC-24, pp. 1551-1559, December 1989.18] F.J.M. Thus: A compact bipolar class-AB output stage using 1-V powersupply, IEEE J. Solid-State Circuits, Vol. 27, pp. 1718-1722, December 1992.19] J. Davidse: Analog electronic circuit design, Prentice Hall, London, 1991.5

6

Chapter 2PortsEt duodecim portae duodecim margaritae sunt,et singulae portae erant ex singulis margaritis.Et platea civitatis aurum mundumtamquam vitrum perlucidum.Apocalypsis Ioannis, 21:21

2.1 IntroductionIn this chapter we look at low-voltage low-power analog integrated circuits from amainly network-theoretical point of view.All electrical circuits or networks can be characterized by two fundamentalquantities. Although other quantities like energy and charge are equally well possible, they are usually taken to be voltages and currents. The properties of thenetwork are then completely specied by the relationships among these voltagesand currents and the network itself can be considered to be `a black box' withterminals connected to the external electrical world.The simplest networks are two-terminal networks or one-ports. Because ofthe charge conservation, the current owing into one terminal equals the currentowing out from the other terminal. This condition is called the port constraint.Examples of one-ports are:

ideal resistors ideal capacitors ideal inductors independent sources (e.g. voltage or current sources)7

series and parallel connections of one-portsIn Figure 2.1, a one-port and its sign conventions are given.+iv

one-port

-

Figure 2.1: A one-port together with its sign conventionsIn a linear system, for all these one-ports, the relationship between the currentand the voltage, usually given by a complex impedance, at the network terminalsspecies the behavior of the one-port completely.When we are interested in the transmission of information from one pointto another, the number of pairs of terminals becomes important. An importantexample of this class of network is the two-port. A two-port is a network withtwo pairs of terminals connected to the external electrical world, provided thatthese pairs of terminals behave as ports. There are some basic network elementsthat behave as two-ports, no matter how the terminal pairs are connected to theremainder of the circuit. These are:

controlled sources, i.e.,{{{{

current-controlled voltage sourcesvoltage-controlled voltage sourcescurrent-controlled current sourcesvoltage-controlled current sourcesideal transformers and gyrators

nullors connections of two-ports, i.e.,{{{{{

series-series connectionsparallel-parallel connectionsseries-parallel connectionsparallel-series connectionscascade connections8

ii

io

+vi

+two-port

vo

-

-

Figure 2.2: A two-port together with its sign conventionsIn Figure 2.2 a two-port and its sign conventions are given.The relationships between the two currents and the two voltages at the terminalpairs specify the two-port completely. Although there are many ways of writingthese equations, we use the transmission parameters in this work 1]. The two-portequations here express the input quantities as functions of the output quantitiesand take the form

vi = Avo + Bioii = Cvo + Dioor in matrix notation

!

!

(2.1)(2.2)!

vi = A BvoiiC Dioin which A, B , C and D are the transmission parameters or the chain parameters.The last name indicates that these parameters are natural ones to describe thecascade or chain connection of two-ports. The transfer parameters , , and are their reciprocal values and are dened as follows:voltage gain factor= = 1=A = vo=vi jio=0transconductance factor = = 1=B = io=vijvo=0transimpedance factor = = 1=C = vo=ii jio=0current gain factor= = 1=D = io=iijvo=0When the source impedance ZS and the load impedance ZL are known, thenthe input impedance Zi and the output impedance Zo of the two-port are givenbyAZ + BZi = CZL + DLB + DZSZo = A + CZS

(2.3)(2.4)

A special two-port which has proved to be very useful for modeling and designing circuits containing feedback is the nullor 2]. A nullor can be considered9

to be an ideal two-port of which the transmission parameters all equal zero, or alltransfer parameters are innite. The active part of a circuit with overall feedbackcan often be considered as an approximation of a nullor, thereby making it easierto estimate its behavior in a feedback conguration.

2.2 Reducing errorsAny transmission of information from the input of a network to the output is perturbed by both stochastic and systematic errors. By stochastic errors we mean inaccuracies in the input-output relation caused by noise or interference. Though impossible to eliminate, their inuence can be minimized by a proper design strategy.Systematic errors arise from network imperfections, such as oset, non-linearity,inaccuracy, drift and temperature dependence. Their inuence can be reduced bymeans of:

compensation error feedforward negative feedback (including indirect negative feedback)2.2.1 CompensationWhen the actual input-output relation of a network diers from the desired inputoutput relation, but in such a way that there is a unique relation between theinput and the output quantities, there is no irretrievable loss of information. It istherefore possible to pass the signal through a second network which compensatesfor the error in the original input-output relation.

Compensation with one-portsWhen two one-ports are connected in series, the currents in both one-ports areequal while the voltage across the connection is the sum of the voltages acrosseach one-port. Compensation of even-order terms in the voltage-current relationoccurs when two identical one-ports are connected in anti-series.When two one-ports are connected in parallel, the voltages across each of themare equal, while the total current equals the sum of the currents owing througheach one-port. Compensation of even-order terms in the current-voltage relationoccurs when two identical one-ports are connected in anti-parallel.10

Compensation with two-ports

Two two-ports can be combined to obtain a new two-port with dierent characteristics. Like one-ports, the ports of a two-port can be connected in series or inparallel. We thus have four possibilities: the input ports are connected in series and the output ports are connectedin series the input ports are connected in parallel and the output ports are connectedin parallel the input ports are connected in series, while the output ports are connectedin parallel the output ports are connected in parallel, while the input ports are connected in seriesIf we dene the transmission parameters of the rst and the second two-portas indicated by their indices, we are able to express the transmission parameters(without indices) of the connection as follows 3]:

A =B =C =D =

A =B =C =D =

series ; series connectionA1C2 + A2C1C1 + C2(A ; A )(D ; D )B1 + B2 + 1 C 2 + C2 112C1 C2C1 + C2D1C2 + D2C1C1 + C2parallel ; parallel connectionA1B2 + A2B1B1 + B2B1B2B1 + B2(D ; D )(A ; A )C1 + C2 + 1 B 2+ B2 112D1 B2 + D2B1B1 + B211

(2.5)(2.6)(2.7)(2.8)

(2.9)(2.10)(2.11)(2.12)

A =B =C =D =

A =B =C =D =

series ; parallel connection(B ; B )(C ; C )A1 + A2 + 1 D 2+ D2 112B1D2 + B2D1D1 + D2C1D2 + C2D1D1 + D2D1 D2D1 + D2parallel ; series connectionA1A2A1 + A2B1A2 + B2A1A1 + A2C1A2 + C2A1A1 + A2(C ; C )(B ; B )D1 + D2 + 1 A 2 + A2 11

2

(2.13)(2.14)(2.15)(2.16)

(2.17)(2.18)(2.19)(2.20)

When identical or complementary two-ports are used it is possible to obtain aproper compensation. These compensation techniques are commonly known asbalancing techniques.If a second two-port is connected in cascade with another two-port, the transmission parameters of the cascade connection can be expressed as follows:

ABCD

====

cascade connectionA1A2 + B1C2A1B2 + B1D2C1A2 + D1C2C1B2 + D1D2

(2.21)(2.22)(2.23)(2.24)

Now, in order to accomplish a proper compensation, the input-output relation ofthe second two-port should be the inverse function of the rst one.12

2.2.2 Error feedforward

Circuits that use the error-feedforward technique all have in common that theyrst obtain an error signal by subtracting an accurately known fraction of theoutput signal of a network (or a copy of the output signal) from the input signal(or a copy of the input signal), then pass this error signal through a network withcharacteristics similar to the rst network, and nally add the output signals ofboth networks to obtain a corrected output signal. A block diagram of a systemusing the error-feedforward technique is shown in Figure 2.3.+x

H1

y

+Kx

+

-

H2

Figure 2.3: Block diagram of a system using the error-feedforward techniqueThough this technique seems attractive, its use is restricted to some specialcases only, because its implementation is not without some diculties. We do notdeal with circuits using error feedforward in this work.

2.2.3 Negative feedback

A system is a feedback system if some variable, either the output variable or aninternal one, is used as an input to a part of the system in such a way that it is ableto aect its own value. A block diagram of a system using the negative-feedbacktechnique is shown in Figure 2.4.In the case of negative feedback, when the transmission around the loop Hfhas a negative sign, it is possible to nullify the error between the input signal x anda signal x which is obtained by passing the output signal through a subsystemwith well-known characteristics f . If Hf (which often is called the loop gain)approaches innity, the output y is related to the input x as the inverse inputoutput relation of that subsystem.When H and f are two-ports, there are four ways of applying (single-loop)feedback by means of two two-ports:0

series-series connection

13

+H

x

y

x

f

Figure 2.4: Block diagram of a system using the negative-feedback technique

parallel-parallel connection series-parallel connection parallel-series connectionThe transmission parameters of these four feedback congurations can be derivedfrom (2.5) to (2.20), with one major dierence: the input ports of the feedbacknetwork are connected to the output ports of the active two-port. Therefore, forthe parameters of the second two-port we use those of the feedback network, ofwhich the input and output ports have been exchanged 4]:

DA2 = ffBB2 = ; ffCfC2 = ; fAfD2 = f

(2.25)(2.26)(2.27)(2.28)

with f = Af Df ; Bf Cf (the determinant of the transmission matrix of thefeedback network).

Series-series connectionWhen used as a feedback conguration, this connection is designed to set the transconductance factor . See Figure 2.5. If the rst (active) network H approachesa nullor, we nd for the transmission parameters of the total network14

iL

ZS

H

vS

ZL

Tf

Figure 2.5: A transconductance amplier with direct negative feedback+iS

H

ZS

ZL vL

Tf

Figure 2.6: A transimpedance amplier with direct negative feedback

ABCD

= 0= ;1=Cf= 0= 0

(2.29)(2.30)(2.31)(2.32)

and for the transconductance factor

= iL=vS = ;Cf

(2:33)

Parallel-parallel connectionThis way of connecting an active two-port and a feedback network is designedto set the transimpedance factor . See Figure 2.6. Under the same assumptionthat the rst (active) network H approaches a nullor, we nd for the transmissionparameters of the total network

ABCD

= 0= 0= ;1=Bf= 015

(2.34)(2.35)(2.36)(2.37)

ZS

+

vS

H

ZL vL

Tf

Figure 2.7: A voltage amplier with direct negative feedbackand for the transimpedance factor

= vL=iS = ;Bf

(2:38)

Series-parallel connection

In order to set the voltage gain of a circuit with negative feedback, a seriesparallel connection is required. See Figure 2.7. When the rst two-port H againapproaches a nullor, we get for the transmission parameters of the total network

ABCD

====

1=Df000

(2.39)(2.40)(2.41)(2.42)

and for the voltage gain

= vL=vS = Df

(2:43)

Parallel-series connectionBy connecting the input of the active two-port in parallel with the output of thefeedback network, and the output of the rst in series with the input of the latter,we are able to set the current gain . See Figure 2.8. With the active two-port Hbeing a nullor we get

ABCD

====16

0001=Af

(2.44)(2.45)(2.46)(2.47)

iLiS

H

ZS

ZL

Tf

Figure 2.8: A current amplier with direct negative feedbackand for the current gain

= iL=iS = Af

(2:48)

2.2.4 Indirect negative feedback

In low-voltage circuits, due to the restricted voltage swing, it is often not possibleto sense the output current of a circuit, and/or to compare an input voltage directly. Only the transimpedance amplier does not have this problem. To realizevoltage, current and transconductance ampliers, a useful alternative then maybe a technique called indirect negative feedback. In an indirect-negative-feedbackcircuit the output and/or the input stage is copied, so that it has an equivalentinput-output relation, and the feedback signal is taken from and/or fed back tothat copy. Thus, it is possible to obtain a response of the circuit which is determined by the feedback network only, assuming that the copying does not introduceerrors.

Setting the voltage gain by means of indirect feedback

A voltage amplier with negative feedback and indirect voltage comparison isdepicted in Figure 2.9.T1 is the rst input stage which serves as the input for the input signal. T2is the second input stage, which is used to compare the voltage of the feedbacknetwork indirectly. Tf is the feedback network and Tr is the remainder of the activecircuitry. All networks are two-ports. When Tr approaches a nullor we obtain forthe transmission parameters of the total circuit

BA = ; A B +1B Df 2f 2B = 0DC = ; A B +1B Df 2f 217

(2.49)(2.50)(2.51)

ZSvS

+T1

Tr

ZL vL

T2Tf

Figure 2.9: A voltage amplier with negative feedback and indirect voltage comparison

D = 0(2.52)For the voltage gain of the total circuit we can write 2]1A B +B DvL=vS = A + B=Z + CZ + DZ =Z = ; Bf 2+ D Zf 2(2:53)LSS L11 SWe see that various other parameters have entered the expression of the voltagegain compared with the expression derived earlier for the `direct-feedback' voltagegain. When, for example, T1 and T2 are identical (e.g. two well-matched transistorsin the same operating point), the inuence of ZS can be counteracted by making ZSequal to Bf =Af , which is, in fact, the output impedance of the feedback network.This results invL=vS = ;Af

(2:54)

Setting the current gain by means of indirect feedback

A current amplier with negative feedback and indirect current sensing is depictedin Figure 2.10.T1 is the rst output stage which serves as the output of the total circuit. T2is the second output stage, which is used to sense the output current indirectly.Tf again is the feedback network and Tr is the remainder of the active circuitry.When Tr approaches a nullor, we get for the transmission parameters of the totalcircuit

A = 0B = 0AC = ; A B +1B D2 f2 f18

(2.55)(2.56)(2.57)

iLiS

Tr

ZS

T1

ZL

T2Tf

Figure 2.10: A current amplier with negative feedback and indirect current sensing

D =

; A2Bf B+1B2Df

(2.58)

For the current gain of the total circuit we can write 2]A B +B D1(2:59)iL=iS = AZ =Z + B=Z + CZ + D = ; A2 Zf + B2 fL SSL1 L1Once again various additional parameters have entered the expression of the current gain compared with the expression derived earlier for the `direct-feedback'current gain. When for example T1 and T2 are identical, the inuence of ZL canbe counteracted by making ZL equal to Bf =Df , which is, in fact, the input impedance of the feedback network. Hence

iL=iS = ;Df

(2:60)

Setting the transconductance factor by means of indirect feedbackIt is also possible to realize a voltage-current transfer by means of indirect feedback.This can be done in three dierent ways:

sensing the output current indirectly and comparing the input voltage directly

sensing the output current directly and comparing the input voltage indi

rectlysensing the output current and comparing the input voltage, both indirectly

These three possibilities are shown in Figures 2.11, 2.12 and 2.13, respectively.When Tr again approaches a nullor we get for the transmission parameters of19

iL

ZSvS

Tr

T1

ZL

T2Tf

Figure 2.11: A transconductance amplier with negative feedback and indirectcurrent sensing and direct voltage comparisoniL

ZSvS

T1

Tr

ZL

T2Tf

Figure 2.12: A transconductance amplier with negative feedback and direct current sensing and indirect voltage comparisoniL

ZSvS

T1

Tr

T2

T3

ZL

T4Tf

Figure 2.13: A transconductance amplier with negative feedback and indirectcurrent sensing and indirect voltage comparison20

the transconductance amplier with indirect current sensing and direct voltagecomparison

A =

; A2Af A+1 B2Cf; A2Af B+1 B2Cf

(2.61)

B =(2.62)C = 0(2.63)D = 0(2.64)And for the transconductance of the total circuit we can write 2]A A +B C1(2:65)iL=vS = AZ + B + CZ Z + DZ = ; A2 Zf + B2 fLS LS1 L1When for example T1 and T2 are identical, the inuence of ZL can be counteractedby making ZL equal to Af =Cf , which is, in fact, the input impedance of thefeedback network. HenceiL=vS = ;Cf(2:66)When Tr again approaches a nullor, we get for the transmission parametersof the transconductance amplier with direct current sensing and indirect voltagecomparisonABCD

= 0(2.67)B1(2.68)= ;C B + D Df 2f 2= 0(2.69)D1= ;(2.70)Cf B2 + Df D2And for the transconductance of the total circuit we can writeC B +D D(2:71)iL=vS = ; Bf 2+ D Zf 211 SWhen, for example, T1 and T2 are identical, the inuence of ZS can be counteractedby making ZS equal to Df =Cf , which is, in fact, the output impedance of thefeedback network. Hence

iL=vS = ;Cf21

(2:72)

Following the same procedure as above in order to nd the transmission parameters of the circuit of Figure 2.13, we see that none of these parameters is zero,which means that both the source impedance and the load impedance enters intothe expression for the transconductance factor. For this reason we do not dealwith this circuit any further.

2.3 Operating in the current domainGenerally, it is not possible to choose the input and output quantities of a circuitfreely. Their choice depends on:

the transducers at the input and/or output the desired topology the available technology the available power supply2.3.1 Source and load

When the input signal for a circuit comes from a transducer, the input quantity hasto be chosen to have the best reproducing relation to the physical input quantityof the transducer. When the output signal of a circuit has to drive a transducer,the output quantity has to be chosen to have the best reproducing relation to thephysical output quantity of the transducer 2].

2.3.2 The desired topology

Inside the circuit, when signals coming from several subcircuits with a commonterminal have to be added, current is a better choice for the information-carryingquantity than voltage. Currents can be added by simply connecting the outputterminals of the subcircuits in parallel. When a signal has to be distributed toseveral subcircuits, voltage is a better choice for the information-carrying quantitythan current. Voltages can be distributed by simply connecting the input terminalsof the subcircuits in parallel.

2.3.3 The available technology

When there is no adding or distributing inside a circuit, a preference for either voltage or current may depend on the available technology. Let us therefore considerthe inuence of parasitic immitances. The inuence of parasitic admittances in22

parallel with the signal path can be reduced by terminating the signal path witha low impedance. The parasitic admittances then have no voltages across theirterminals and thus no current ows in them. The inuence of parasitic impedancesin series with the signal path can be reduced by terminating the signal path witha high impedance. Then no current ows in the parasitic impedances and thusthere is no voltage across their terminals.In low-power integrated circuits, often the parasitic admittances, i.e., the nodecapacitances, have more inuence on the signal behavior than the parasitic impedances, i.e., the branch inductances and resistances. Therefore it is convenientto terminate the signal paths with low impedances as much as possible. In thissituation it is best to choose current as the information-carrying quantity. Circuitsthat have a current as the information-carrying quantity are from now on denotedas `operating in the current domain'.In literature, this technique is often recommended to improve the high-frequency performance of a system, resulting in socalled `current-mode' circuits 5, 6].However, it must be noted that the expression `current mode' has no rigorousmeaning: the behavior of electrical networks is always the product of an interplaybetween voltages and currents.

2.3.4 The available power supply

When we apply feedback to a circuit, the preference for either voltage or currentmay depend on the available power supply. Let us therefore consider the case ofseries feedback at the input or output. Generally, the active network must haveoating input or output ports. If the input voltages with respect to a commonreference are not zero, or the output currents are not equal, this may result inoset, inaccuracy or distortion.In order to overcome these imperfections, it may be attractive to use

compensation (by an anti-series connection of identical stages), or indirect feedback

A major disadvantage of the use of stages connected in anti-series at the input(voltage comparison) is that the power density spectrum of the equivalent noisevoltage is doubled. The anti-series connection of stages at the output results in adeterioration of the power eciency.The second possibility uses indirect feedback. Applied at the input (indirectvoltage comparison) this again produces a doubled noise spectrum, while whenapplied at the output (indirect current sensing) only the power eciency maydeteriorate slightly. Another major disadvantage of indirect voltage comparisonis that it requires two input stages with symmetrical or opposite non-linearities,23

in order to compensate for the non-linearities. In practice, this requires eithertwo balanced input stages or two complementary stages in a complementary ICprocess. The use of two balanced input stages again doubles the power densityspectrum of the equivalent noise voltage. A complementary IC process is oftennot available and, moreover, exact complementarity can never be accomplished.Indirect feedback at the output, however, calls for two identical output stages, tocompensate for the non-linearities. These can easily be made in any ordinary ICprocess. For this reason it is preferable that low-voltage analog integrated circuitsoperate in the current domain, i.e., have a current as the information-carryingquantity, as much as possible.

References1] F.D. Waldhauer: Anticausal analysis of feedback ampliers, The Bell Syst.Techn. Journal, Vol. 56, pp. 1337-1386, October 1977.2] E.H. Nordholt: Design of high-performance negative-feedback ampliers, Elsevier, Amsterdam, 1983.3] G.W. de Jong: Gentegreerde lters voor hoortoestellen, M.Sc. Thesis (inDutch), Delft University of Technology, Delft, the Netherlands, 1989.4] G. Zelinger: Basic matrix algebra and transistor circuits, Pergamon Press,Oxford, 1963.5] C. Toumazou, F.J. Lidgey and D.G. Haigh (editors): Analogue IC design:the current-mode approach, Peter Peregrinus, London, 1990.6] B. Wilson: Recent developments in current conveyors and current-mode circuits, IEE Proc., Vol. 137, Pt. G, pp. 63-77, April 1990.

24

Chapter 3Modeling the bipolar transistor atlow voltages and low currentsNo minority has a right to block a majorityfrom conducting the legal business of the organization.No majority has a right to prevent a minorityfrom peacefully attempting to become a majority.Robert M. Pirsig: Lila

3.1 IntroductionIn this chapter we look for mathematical models that describe the terminal behavior of bipolar transistors in low-voltage low-power circuits. These models can varyfrom coarse to rened, depending on the purpose for which they are needed. Forexample, when formulating an initial concept, a designer uses only those modelsthat describe the major function of the components. It is only at a later stage thatmodels are used that describe the components in more detail.For an active component, usually two kinds of models are given:

a large-signal, and a small-signal model

The choice depends on whether or not the signal quantities can be considered smallwith respect to the bias quantities. When the signal excursions are small, the nonlinear characteristics of the active devices can be regarded as being linear arounda certain operating point, which facilitates the calculation of the input-outputrelation.25

When designing a circuit in respect to its noise behavior a dierent model isimportant for the designer. We deal with this noise model in Section 3.4.

3.2 Large-signal model of a bipolar transistorOur starting point is the well-known Ebers-Moll model 1], partly because we inthis work do not deal with the physical processes, partly because this model hasproved to give a suciently accurate description of the transistor behavior under`normal' circumstances. The Ebers-Moll model for an NPN transistor is depictedin Figure 3.1.C

RC

RB

CDC

CJC

I3

I4ICT

BCJSLPNPonly

CDE

CJE

I1

CJSNPN andVPNP only

I2

RE

E

Figure 3.1: Ebers-Moll model for an NPN transistor

3.2.1 The transport current ICT

The model consists of a non-linear voltage-controlled current source ICT betweenthe intrinsic collector and emitter, controlled by the intrinsic base-emitter voltageVBE and the intrinsic base-collector voltage VBC :

ICT = ICC ; IECICC = IS (eVBE =VT ; 1)IEC = IS (eVBC =VT ; 1)26

(3.1)(3.2)(3.3)

with IS the saturation current and VT the thermal voltage kT=q, approximately26 mV at 300 K.Four leakage diodes model the base current IB :

IB = I1 + I2 + I3 + I4

(3.4)

I1I2I3I4

(3.5)(3.6)(3.7)(3.8)

====

ICC =BFISE (eVBE =E VT ; 1)IEC =BRISC (eVBC =C VT ; 1)

E , C , BF and BR being process parameters.I1 models the normal leakage base current caused by injection from the baseinto the emitter, while I3 models the normal leakage base current produced byinjection from the base into the collector. At low currents, however, three othercontributions to IB play an important role:

recombination of carriers at the surface the formation of emitter-base surface channels recombination in the emitter-base depletion layerWe do not here discuss the physics of these extra base currents, but simply statethat they also have an exponential dependence on VBE , but that the exponentialdiers by a factor E or C , which can have values between one and four. Theseadditional base currents can be included in the model by two non-ideal diodes.I2 models the non-ideal base current from base to emitter, and I4 models thenon-ideal base current from base to collector.At high currents, high-level injection occurs and the factor BF decreases withincreasing ICT . We do not deal with this eect in this work, because we areinterested in low-current behavior only.At high voltages, there are other eects that limit the operating area of atransistor. These eects are:

base-emitter Zener breakdown collector multiplication emitter crowding27

We do not here discuss these eects, because they do not occur in low-voltagelow-power applications.The transport current also depends on the voltages across the emitter-basejunction VEB and the collector-base junction VCB . This eect is called `basewidthmodulation' or `Early eect', and can be modeled by making the saturation currentIS dependent on VEB and VCB . In approximation:

IS = ISO (1 + VEB =VAR + VCB =VAF )(3:9)in which ISO , VAR and VAF depend on the process and the temperature only.

3.2.2 The resistances RB , RC and RE

Between the intrinsic transistor and the external connections, there are three resistances: the base resistance RB : This (parasitic) resistance can have a great eecton the small-signal and transient responses. It is a distributed element andtherefore strongly depends on the operating point. A typical value for anintegrated small geometry transistor is 1000 . the collector (bulk) resistance RC : This resistance depends on the collectorcurrent and voltage, and decreases the slope of the curves in the saturatedregion of the transistor. A typical value is 100 . the emitter (bulk) resistance RE : Because of the high doping level of theemitter, this resistance consists mainly of the contact resistance, which issmall (a typical value is 1 ). The emitter bulk resistance mainly aects theVBE -IC relation at higher operating currents.For these three resistances, it is valid that they do not have any inuence on thetransistor behavior when the currents used are small (e.g. 1 A) and the frequencyrange of interest is less than the transit frequency fT .

3.2.3 The junction capacitances CJE and CJC

The capacitances CJE and CJC model the incremental xed charges stored in thetransistor's depletion layers for incremental changes in the junction voltages. Eachcapacitance is a non-linear function of the voltage across the junction:

CandCJE = (1 + V JEOEB =VJE )MECCJC = (1 + V JCO=V )MCCB

28

JC

(3.10)(3.11)

in which CJEO is the small-signal depletion capacitance at zero base-emitter (bias)voltage and CJCO the equivalent capacitance at zero base-collector (bias) voltage.VJE and VJC are the built-in voltages of the base-emitter and base-collector junction and ME and MC are their grading coecients, which usually lie between 0.3and 0.5.In integrated circuits, there is also another junction capacitance present: a substrate capacitance, CJS , which for (vertical) NPN and PNP transistors is connectedbetween the intrinsic collector and substrate. For lateral IC transistor structures,this capacitance is connected between the intrinsic base and the substrate.CJS = (1 + VCJSO(3:12)SJ =VJS )MJwhere CJSO is the small-signal depletion capacitance at zero bias voltage, VSJ thevoltage across the junction and VJS is the built-in voltage of the junction. MJagain is a grading coecient.

3.2.4 The diusion capacitances CDE and CDC

The diusion capacitances model the charges QDE and QDC which are associatedwith the transport currents IEC and ICC respectively.

QDE = F ICCQDC = RIEC

(3.13)(3.14)

These relations indicate that the diusion capacitances are proportional tothe current. In practice, the diusion capacitances are only dominant when thecurrents exceed hundreds of micro-amps. Therefore they can be neglected in lowpower circuits.

3.2.5 Low-voltage low-power large-signal transistor model

Having now dealt with all the elements of the Ebers-Moll transistor model, we areable to state a simplied model which is valid for low-voltage low-power circuitswith a frequency range of interest which does not exceed the transit frequency fT .This model is depicted in Figure 3.2.

3.3 Small-signal model of a bipolar transistorAt this stage, it is possible to extract a linearized (small-signal) model from thelarge-signal model shown in Figure 3.2, which is valid when the transistor is biased29

C

CJC

I3

I4ICT

BCJE

CJSLPNPonly

I1

CJSNPN andVPNP only

I2

E

Figure 3.2: Simplied Ebers-Moll transistor model valid for low-voltage low-powerapplicationsin its forward active region. When the signal quantities are small compared tothe bias quantities, we are able to describe the signal behavior of any non-linearnetwork as a linear network. The circuit shown in Figure 3.2 then linearizes to thecircuit shown in Figure 3.3, with

gmIC

FIBrro

======

dIC =dVBE IC =VT the transconductancethe collector current in a certain operating pointdIC =dIB the current gainthe base current in a certain operating point

F =gmdVCE =dIC which accounts for the Early eectCCJC

B

CJSLPNPonly

r

CJE

gmvBEE

ro

CJSNPN andVPNP only

Figure 3.3: Small-signal transistor model derived from the simplied Ebers-Mollmodel30

3.4 Noise

3.4.1 Noise sources in the bipolar transistor

In a bipolar transistor biased in its forward active region, we are able to indicatethree noise sources among the three terminals whose power density spectra areproportional to the currents owing from one terminal to another. These sourcesproduce shot noise, and are uncorrelated:

the shot noise of the intrinsic collector-emitter current, between collector andemitter, has a power density spectrum S (iC ) = 2qICT 2qIC . the shot noise of the intrinsic base-emitter current, between base and emitter,has a power density spectrum S (iB ) = 2qIF 2qIB. the shot noise of the intrinsic base-collector current, between base and collector, has a power density spectrum S (iCO ) = 2qIR 0.

Further, the base resistance RB produces thermal noise, of which the powerdensity spectrum (in V2/Hz) equals 4kTRB .Finally, we are able to indicate a low-frequency (1/f) noise current source connected between the intrinsic base and emitter, which is the product of a processdependent noise mechanism. It has been found (experimentally) that the spectrumof this noise current generator equals

S (iBf ) = KIBa =f(3:15)(with a between 1 and 2) over a wide and useful range of collector bias currents2, 3, 4]. K is a process-dependent constant. Alternatively, it is possible to write4, 5]:S (iBf ) = 2qIBfl=f(3:16)in which fl is a representation of the noise corner frequency and is proportional toIBa 1. Because a lies between 1 and 2, the noise corner frequency decreases whenthe base current decreases. We therefore suppose that low-frequency noise makesonly an insignicant contribution to the total noise in most low-power circuits.;

3.4.2 Transformation of noise sources to the input

When the signal quantities are kept small with respect to the bias quantities,the noise sources can be considered stationary and therefore the noise is additiveto the signal. We are now able to replace the noisy transistor with a noise-freetransistor together with two external noise sources at the input 6], which usually31

are correlated. In a bipolar transistor, only the collector shot noise source iC needsto be transformed. This transformation results in two correlated noise sources BiCand DiC (B and D are the transistor transmission parameters), Figure 3.4.Note: the noise sources in this gure are represented by their Fourier transformsin order to account for correlations because of the transformations.CVRBB

+

BiCiB

-

noise-free transistorDiC

E

Figure 3.4: Noisy transistor replaced with a noise-free transistor and four noisesourcesWhen we consider that for a bipolar transistor

B = 1= = ;1=gmD = 1= = ;(1=F + jf=fT ) 5]IC = gmVTBF F

(3.17)(3.18)(3.19)(3.20)

we nd that the inuence of RB is negligible, if1RB < 2g

(3:21)

ff < pBT

(3:22)

m

In practice, this is true when the collector current does not exceed several tens ofmicroamps, so it can be stated that in most low-power circuits the noise causedby RB is negligible.Finally the inuence of Dic is negligible, ifF

Now we are able to state a simple noise model of a bipolar transistor, validfor plow-voltage low-power circuits and a frequency range of interest lower thanfT = BF . This model is depicted in Figure 3.5.32

CBiCB

noise-free transistoriB

E

Figure 3.5: Simplied noise model valid for low-voltage low-power applications

References1] I.E. Getreu: Modeling the bipolar transistor, Elsevier Science Publishers,Amsterdam, 1976.2] C.A. Bittmann, G.H. Wilson, R.J. Whittier and R.K. Waits: Technology forthe design of low-power Circuits, IEEE Journal of Solid-State Circuits, Vol.SC-5, pp. 29-37, February 1970.3] M.B. Das: On the current dependence of low-frequency noise in bipolar transistors, IEEE trans. Electron. Devices, Vol. ED-22, pp. 1092-1098, December1975.4] C.D. Motchenbacher and F.C. Fitchen: Low-noise electronic design, JohnWiley and Sons, New York, 1973.5] E.H. Nordholt: Design of high-performance negative-feedback ampliers, Elsevier, Amsterdam, 1983.6] Th. Komarek: Noise in electronic circuits, M.Sc. Thesis, Delft University ofTechnology, Delft, the Netherlands, 1973.

33

34

Chapter 4Negative-Feedback AmpliersThis example serves to illustrate how feedback{ plugging the system's output back into the system as input {ushers you to the xed points.Why should this be so?Why could the system not trash about randomly,somehow avoiding all xed points?Douglas R. Hofstadter: Metamagical themas

4.1 Introduction (outline of the design method)In almost any electronic system some kind of signal amplication can be identied. Amplication is obviously an indispensable function 1]. The aim of anamplier is to bring the information of the source signal to a higher energy level.This information must be transferred as accurately as possible, e.g., with minimal distortion, maximum speed and with minimal addition of interfering signalsand noise. In Chapter 2 we saw that these requirements can be met by applyingnegative feedback.Negative feedback is a principle that can be seen in many technical, biologicaland social processes. It implies that a driving quantity, based on the result of thedriven quantity, is corrected in such a way that the desired result is obtained 2].Negative feedback allows us to exchange the large gain provided by the (highlynon-linear) active devices for quality.According to the asymptotic-gain model 3] the transfer function Af of almostany negative-feedback amplier can be described in terms of an ideal amplicationfactor Af (the asymptotic gain), i.e., the transfer if the amplifying block hasnullor properties, and an error factor comprising the loop gain A , or1

35

;A

(4:1)Af = Af 1 ; A

The error factor ;A=(1 ; A ) describes the inuence of the non-ideal nature ofthe amplier. Ideally, this factor should assume unity value as closely as possible.For the designer, this means that the design of Af can be done in two successiveand independent design steps. The rst step is the determination of Af , whichcan be considered as the design aim, and the second step is the realization of anadequate loop transfer function A .For each design step, various quality aspects must be considered: noise. Being a random signal, noise already present in the incoming signalcannot be reduced by using negative feedback. However, the addition of extranoise can be kept to a minimum. Therefore, both the noise contribution ofthe feedback network and of the active part of the amplier have to beconsidered. distortion. The parameter values of active devices always vary with signalquantities and, therefore, the input-output relation of an amplier is nonlinear and distorted. The best method to reduce these imperfections is to makethe transfer independent of the parameters of the active devices as much aspossible by applying negative feedback with a linear feedback network. Atlarge values of the loop gain A , Af Af and the transfer will be almostindependent of the characteristics of the active devices. accuracy. The accuracy of an amplier depends on the spread in deviceparameters caused by fabrication tolerances and aging. Similar to distortion,the inaccuracy is reduced in a negative-feedback amplier when the feedbacknetwork is accurate and the loop gain is high. bandwidth. In low-power integrated circuits, the bandwidth is limitedbecause of the existence of parasitic capacitances. In Chapter 2 we foundthat their inuence can be minimized by operating in the current domain asmuch as possible. Expression (4.1) demonstrates an additional possibility:in negative-feedback ampliers the bandwidth can be enlarged by choosinga larger loop gain. output capability. In low-voltage low-power integrated circuits both thevoltage and current swing are limited. Special attention therefore has tobe paid to their maximum values. Often these appear at the output of anamplier. power eciency. Obviously, power eciency is of major interest in lowpower integrated circuits.1

1

1

36

integratability.

Since we are dealing with integrated circuits, we have tomake sure that all the required components are integratable. This meansthat we cannot make use of transformers or inductors and that the sum ofthe capacitor values and the sum of the resistor values are limited.An ideal design strategy would be one in which all quality aspects can betreated independently of each other (also called orthogonal). Unfortunately, thisis only seldom possible. An example: suppose we want to obtain an output currentof 100 nA (peak value). In view of the power eciency, the bias current of the(class-A) output stage is best chosen to be exactly 100 nA. Unfortunately, thisconicts with the designability of the amplier with respect to its high-frequencybehavior the poles of the output stage would sweep over a wide range, eventuallyleading to instability of the total amplier. However, following a hierarchic designstrategy (see for example 3]) keeps the interaction between the various qualityaspects small.This hierarchic design strategy comes down to the following steps: choose a proper basic amplier conguration. This choice is based onthe kind of electrical quantity we have at both input and output.

choose the character and numerical values of the (passive) feedbacknetwork. This choice is based on noise, accuracy, output capability andintegratability.

choose a proper input stage. This choice is based on noise. In low-power

circuits also the power eciency is important. choose a proper output stage. This choice is based on the output capability and power eciency. evaluate the loop gain. The other quality aspects, viz, distortion, accuracy and bandwidth are mainly determined by the loop gain as a functionof frequency. These aspects determine whether an amplier should have onestage (if the input stage can be the output stage as well), two stages (an input and an output stage) or more stages (the intermediate stages then haveto be chosen on the grounds of these quality aspects and power eciency). realize the desired high-frequency behavior. Especially oscillation hasto be avoided under all circumstances. choose a proper bias circuit. Because the design of the signal path iscompleted in the foregoing stages, this bias circuit is not permitted to havea major inuence on the signal transfer.These steps are the subjects of the following sections.37

4.2 The basic amplier conguration and the feedback networkFrom the large class of negative-feedback ampliers (using either direct, indirector active feedback 3]) we restrict ourselves to those that sense the output current indirectly. It was shown in Chapter 2 that this leads to congurations thatare especially suitable for low-voltage integrated circuits. Although multi-loopcongurations may be important in characteristic impedance systems, where theinterconnections are made by transmission lines, we deal only with single-loopcongurations in this chapter. An extensive description of multi-loop negativefeedback ampliers can be found in 3] and 4]. A practical example of a two-loopnegative-feedback amplier with indirect current sensing is discussed in Chapter 7.

4.2.1 Current ampliers

In Chapter 2, it was found that low-voltage low-power electronic circuits shouldpreferably operate in the current domain insofar as possible. As they have a currentat both input and output, current ampliers thus are the natural ampliers to usein such an environment.

Basic amplier congurationThe basic conguration of a current amplier with an indirect output is depictedin Figure 4.1. In this circuit, the output current is sensed indirectly by means ofQ1 and partially (by means of the impedances Z1 and Z2) fed back to the input.Q2 provides the output current. When all the transfer parameters of the op ampare innite, gm Z2 and gm ZL we can write for the ideal transfer Af ofthis two-impedance current amplier:1

Z +ZAf = ; 1 Z 22

(4:2)

1

Z1Q1 Q2iS

ZS

Z2

ZL

Figure 4.1: A current amplier with an indirect output38

Alternatively it is possible to choose Z1 and Z2 to be equal to zero and innite,respectively, and vary the current gain by means of varying the scaling factor nof transistor Q2. This amplier, which from now on is called a scaling currentamplier is shown in Figure 4.2. With Q2 gmQ2 ZL we are able to write forAf :g(4:3)Af = ;n = ; gmQ2mQ11

1

Q 1 Q2iS

ZS

ZL1

n

Figure 4.2: A scaling current amplierA well-known scaling current amplier is obtained when the op amp is replacedby a short circuit: a current mirror 5]. Its circuit diagram is depicted in Figure4.3. We deal with this amplier in a later section.Q1 Q2iS

ZS

ZL1

n

Figure 4.3: A current mirror

A special case: the two-resistance amplierWhen the impedances of the two-impedance amplier (Figure 4.1) are real, i.e.resistances, we obtain an amplier type that we can call a two-resistance amplier.A problem arises when we have to decide whether to choose this amplier or tochoose a scaling current amplier to obtain a real transfer function. We thereforetake a look at the noise production of both ampliers.

The feedback networkEven if the op amps are noise free, both amplier types add some noise to thesignal. This noise contribution originates from the two resistances R1 and R2together with the noise from transistors Q1 and Q2. These noise sources can be39

shifted towards the input and summed in a noise current source S (ineq) in parallelwith the signal source. We obtain:!

!

R2 2 R1 2S (inR2 ) + S (inQ1 ) + S (inQ2 )S (ineq) = R + R S (inR1 ) + R + R1212(4:4)With S (inR1 ) = 4kT=R1, S (inR2 ) = 4kT=R2 , S (inQ1 ) = 2qICQ1 , S (inQ2 ) =2qICQ2 and ICQ1 = ICQ2 = IC this can be rewritten as

!

!

!

R1 2 4kTR2 2 4kTS (ineq) = R + R R + R + R(4:5)R2 + 4qIC12112Note that only the collector shot noise of Q1 and Q2 is of importance. The baseshot noise is short-circuited by the op amp and does not make any contribution tothe output signal of the amplier. If for IC the minimal value (eciency !) whichequals (R1 + R2)=R2 times the peak value iSp of the signal current of the sourceis chosen, this results in

RS (ineq) = R +1 R12

!2

!4kTR2 2 4kTR2++R1R1 + R2 R2 R1 + R2 4qiSp

(4:6)

For the scaling current amplier, the input noise spectrum equals

n+1n+1S (ineq) = 2qICQ1 n = 2qiSp n(4:7)We are now able to compare both noise spectra. Substituting ;(R1 + R2)=R2 ,the gain of the two-resistance current amplier, for ;n, the gain of the scalingcurrent amplier, (4.7) results inR + 2RS (ineq) = 2qiSp R1 + R 21

2

(4:8)

Comparing (4.6) and (4.8) it follows that we do best to choose for a two-impedanceamplier only if R1 can be chosen larger than 2VT =iSp (VT = kT=q). In low-powerdesign this is only seldom possible. For example, if the signal current at theinput equals 25 nA (peak value) | this is a quite practical value in, e.g., ltercircuits (see Chapter 6) | R1 must be larger than 2 M. Even in full-custom ICdesign, because of the restricted chip area, this value is not easily realized. Otheradvantages of using scaling ampliers are that they have a better eciency andtheir current gain is not limited by the maximum voltage swing over R1.40

4.2.2 Transconductance ampliers

Another class of indirect-negative-feedback ampliers that is useful in low-voltagelow-power electronic circuits is the class of transconductance ampliers. In Chapter 6, it is shown that these are the natural building blocks when it comes todesigning low-voltage low-power controllable lters. For this reason we discussthem here.

Basic amplier congurationThe basic conguration of a transconductance amplier with an indirect outputis depicted in Figure 4.4. In this circuit, the output current is sensed indirectlyby means of Q1, transformed into a voltage (by means of impedance Z ) and fedback to the input. When all the transfer parameters of the op amp are innite,Q1 gmQ1 Z and Q2 gmQ2 ZL we can write for the ideal transfer Af :1

g 1Af = gmQ2 ZmQ1

(4:9)

1

Q1 Q2ZS

1

vS

Z

nZL

Figure 4.4: A transconductance amplier with an indirect outputFrom this expression it can be seen that the transconductance can be varied byvarying Z or the scaling factor n, which equals gmQ2 =gmQ1 .

The feedback networkEven if the op amp is noise free the amplier adds some noise to the signal. Thisnoise contribution originates from the impedance Z and transistors Q1 and Q2.These noise sources can be shifted toward the input and summed in a noise voltagesource S (vneq) in series with the signal source. We obtain:!

S(i)nQ(4:10)S (vneq) = S (vnZ ) + jZ j2 S (inQ1 ) + n2 241

The noise from Q1 and Q2 equals the collector shot noise 2qICQ1 and 2qICQ2 , respectively. If for ICQ1 the minimum value (eciency!) which equals the maximumof the ratio of the peak value vSp of the signal voltage of the source and Z , andICQ2 = nICQ1 is chosen, this results in:v n+1(4:11)S (vneq) = 4kT Re(Z ) + jZ j2 n 2q max ZSpFrom this expression, we can draw the conclusion that it is best to choose Z assmall as possible. However, for values of Z less than the reciprocal value of thedesired transfer function, it can be shown that the power eciency is seriouslydegraded. Z therefore has to be chosen in such a way that

the equivalent input noise voltage source S (vneq) is less than the maximum

acceptable noise contribution, andthe feedback impedance Z is larger than 1=Af , if possible, andZ is integratable1

In low-power integrated circuits, these demands often conict and an acceptablecompromise has to be found.

4.2.3 Transimpedance ampliers and voltage ampliers

The basic congurations of a transimpedance amplier and a voltage amplierare depicted in Figures 4.5 and 4.6, respectively. Because they are dealt withextensively in 3], and both have a voltage output, which makes them less suitablefor low-power applications, we do not discuss them here.Z

iS

ZS

ZL

Figure 4.5: A transimpedance amplier

4.3 The input stageTo here the op amp has been treated as if it were noise free. In practice, an op ampalways contributes some noise. However, this contribution can be kept suciently42

ZSvS

Z2

Z1

ZL

Figure 4.6: A voltage ampliersmall, as shown subsequently. All the noise sources in the op amp can be shiftedtoward the input of the op amp and modeled as a noise voltage source vn in serieswith the input of the op amp and a noise current source in in parallel with the opamp's input. This is depicted in Figure 4.7. These noise sources, in turn, can beshifted toward the input of the amplier and transformed into one noise source.vn+in

QA

-

Figure 4.7: Noisy op amp modeled by a noise-free op amp and two noise sourcesat the inputFor the current amplier depicted in Figure 4.1, this results in an equivalentnoise current source ineq with a power density spectrum S (ineq):

Y1Y2 2

(4:12)S (ineq) = S (in) + S (vn) YS + Y + Y 12in which Y = 1=Z .For the transconductance amplier shown in Figure 4.4, an equivalent noisevoltage source vneq is found.

S (vneq) = S (vn) + S (in)jZS + Z j2(4:13)When we integrate these two spectra over the frequency range of interest, weobtain an expression for the equivalent noise power at the input, Pneq, which isa function of the collector current of the rst stage of the op amp ICQA . As thespectrum of the noise current source in is proportional to this collector current(S (in) 2qICQA =BF ) and the noise voltage source vn is inversely proportional to43

this collector current (S (vn) 2qVT2=ICQA ) a minimum can be found for the noisepower 3].In practice, however, in many situations the source signal already contains somenoise and we are able to vary the collector current of the rst stage of the op ampover various decades around this optimum without seriously aecting the noisebehavior of the complete amplier. We discuss the use of this favorable propertyin a later section with relation to the amplier loop gain. From the eciencypoint of view, we should better not choose ICQA to be larger than the sum of thecollector currents of Q1 and Q2 or smaller than the base current of the next stage.

4.4 The output stageThe next step in our design of low-voltage low-power integrated negative-feedbackampliers is the choice of a proper output stage. This choice depends on theoutput capability and the power eciency. In both the current ampliers andthe transconductance amplier, this output stage is Q2. In view of the outputcapability, Q2 must be biased at a collector current, ICQ2 , which equals at leastthe peak value of the signal current coming from Q2. This signal current, in turn,equals Af times the source signal. Thus for the current ampliers1

ICQ2 Af iSpand for the transconductance amplier1

(4:14)

ICQ2 Af vSp(4:15)From the power eciency point of view, however, this collector bias currentshould better not be chosen to be much larger than the peak value of the signalcurrent. Thus, a compromise must be found: a practical value of 1.2 times as largewill do.1

4.5 Loop gainThe next aspects in the design of ampliers to be addressed are distortion, accuracyand bandwidth, mainly determined by one parameter, viz. the loop gain A . Weevaluate the loop gain for both amplier types.

4.5.1 Current ampliers

Now suppose the op amp can be modeled as a current amplier with a current gainG, an input impedance Zin and an output impedance Zout (Figure 4.8). This is a44

iinZin

Zout

G

Figure 4.8: The op amp modeled as a current amplier with a current gain G, aninput impedance Zin and an output impedance Zoutquite practical situation. For example, if the op amp is realized by two transistorsin cascade, G = F2 , Zin = F =gm and Zout = =gm . When Zout F =gm , we ndfor the loop gain A of the two-impedance amplier ( gm Z2):

ZZ1(4:16)A = ; 2 Z + Z +2 Z ==Z Z +SZ GF21Sin SinWith Af = ; Z2Z+2Z1 this can be rewritten asZ +ZZ1 1(4:17)A = 2 A Z + Z2 + Z1 ==Z Z +SZ GFf21Sin SinReplacing Z1 and Z2 by R1 and R2, respectively, this results in an expression forthe loop gain of the two-resistance amplier:1 1R +RZA = 2 A R + R2 + Z1 ==Z Z +SZ GF(4:18)f21Sin SinFor A of the scaling current amplier we nd (Q1 gmQ1 (ZS ==Zin )):g 1ZSA = ; g mQGF(4:19)mQ1 + gmQ2 ZS + ZinmQ2 this can be rewritten as:With Af = ; ggmQ11ZA = A ; 1 Z +S Z GF(4:20)fSinNow we are able to compare the two-resistance amplier and the scaling amplier with respect to their loop gains. For this purpose we calculate the ratio F ,dened as1

1

1

1

1

45

F =

Ascaling current amplier

Atwo

resistance current amplier

(4:21)

;

This results in

+ ZS ==Zin AF = 2 A Af ; 1 R2 + RR1 + > 2 fAf ; 1 > 1 8(Af < ;1)f2 R11

1

1

1

(4:22)

1

This expression indicates that F is larger than one as long as the absolute value ofthe asymptotic gain (the ideal transfer) is larger than one. However, the asymptotic gain of the two-resistance amplier equals (R1 + R2)=R2 and thus alwaysmeets this condition. Therefore F is always larger than one. For this reason, thescaling current amplier is always the best choice with regard to loop gain.From both (4.17) and (4.20) it also follows that the input impedance of theop amp, Zin, is best chosen to be as small as possible. For a bipolar input stage,this comes down to choosing the collector current of the rst stage, ICQA , to be aslarge as possible. Again, in view of the eciency, we had better not choose ICQAto be larger than the sum of the collector currents of Q1 and Q2 or smaller thanthe base current of the next stage.

4.5.2 Transconductance ampliersFollowing the same procedure as that given in the former subsection, an expressionis found for the loop gain A of the transconductance amplier:

ZgmQ1GFmQ1 + gmQ2 Z + ZS + Zinthis can be rewritten as:

A = ; gWith gmQ2 = gmQ1 ZAf

1

(4:23)

1ZA = ; 1 + ZA Z + Z + Z GF(4:24)fSinFrom this expression, it follows that also for a transconductance amplier, Zin isbest chosen to be as small as possible, and thus ICQA as large as possible. In theinterests of eciency we had better not choose ICQA to be larger than the sum ofthe collector currents of Q1 and Q2 or smaller than the base current of the nextstage.1

46

4.6 High-frequency behaviorIn this section, we deal with some aspects of the high-frequency behavior of lowvoltage low-power negative-feedback ampliers. As the common-emitter (CE)stage can be considered as a basic amplier stage, we take a closer look at thesmall-signal diagram of this stage connected between source (iS ZS ) and load(ZL ). See Figure 4.9. For the current gain Ai we nd

g ZZiAi = iL = ; Z + Z +mZf +S g Z ZSfLm S LSwith Zf = 1=j 2fC , ZS = ZS ==r ==(1=j 2fC ) and ZL = ZL==ro .0

0

0

0

(4:25)

0

C

iL

+iS

ZS

r

C

gmvr

-

ro

ZL

Figure 4.9: Simplied small-signal diagram of a CE stageFrom this expression, it can be concluded that there is a strong interaction betweensource and load due to the local (shunt) feedback by C , which degrades thedesignability of the high-frequency behavior. This interaction can be preventedby shunting the output with a low impedance. We thus need a current follower.Often, the demands made on this subcircuit can be much less than those made onthe total amplier, and it may be possible to use a single-stage solution. A wellknown example is the common-base (CB) stage. The combination of a CE and aCB stage is also known as a `cascode' stage. Another one-stage implementationof a current follower is the current mirror (CM), which can be regarded as the`low-voltage counterpart' of the CB stage. We call the latter solution a `cascade'stage. Both solutions are depicted in Figure 4.10.

A

B

Figure 4.10: Two ways of shunting the output of a CE stage with a low impedance:a cascode (CE + CB) stage (A) and a cascade (CE+CM) stage (B)47

Note that the cascode stage has an inverting transfer, while the cascade stageis non-inverting. Compared to a single CE stage, both stages also have a slightlydeteriorated noise behavior . However, this does not often pose a problem.If all the transistors are supposed to have the same parameters and to be inthe same operating point, if the inuence of ro and the substrate capacitancesis negligible and if gm is much larger than g (= 1=r ), then the transmissionparameters of the CE, CB and CM stage are given by

sCACE = sC ; g

m1BCE = sC ; g

msC(gm + sC )CCE = sC ; g

mg + s(C + C )DCE = sC ; g

and

m

sCACB = g m1BCB = gmsC (gm + sC )CCB =gmgm + sCDCB =gm

sCACM = sC ; g

m1BCM = sC ; g

msC (2g + 2sC )CCM = sC m; g

mDCM = gm +sCs(2C;g+ C)

m

(4.26)(4.27)(4.28)(4.29)(4.30)(4.31)(4.32)(4.33)

(4.34)(4.35)(4.36)(4.37)

Now we are able to nd the transmission parameters of the cascode (CE+CB)stage and the cascade (CE+CM) stage (Chapter 2). For the cascode stage weobtain48

gm + s(C + C)gm(sC ; gm ) sCg + s(C + C )= mg (sC ; g )m

m(g + sC )(g + s(C + 2C))sC= mgm(sC ; gm )(g + sC )(g + s(C + 2C))= mgm(sC ; gm )

ACE+CB =

(4.38)

BCE+CB

(4.39)

CCE+CBDCE+CB

(4.40)(4.41)

and for the cascade stage

sC(2gm + s(C + C))(4.42)(sC ; gm )2BCE+CM = gm +(sCs(2C;g+)22C )(4.43)

msC (g + sC )(2g + s(2C + 3C))(4.44)CCE+CM = m (sC ; g )2

m2C C +C2(g + s(C + 2C ))(gm + s 2C +4C +2C ))DCE+CM (4.45)(sC ; gm )2The reciprocals of these transmission parameters are the more commonly knowntransfer parameters ( and ) whose asymptotic Bode plots are plotted (upto !T ) in Figures 4.11, 4.12 and 4.13 for the CE, cascode and cascade stage, respectively.In low-power circuits C and C are both merely the result of xed chargesstored in the transistor's space-charge layers rather than charges associated withthe mobile carriers in the transistor (Chapter 3). They therefore lie in the sameorder of magnitude. We see that up to !T =2, the transfer parameters , and are the same for all stages. This means that in situations where the transfer isdetermined mainly by these parameters, i.e., when the source or load impedanceis small, there is no preference for either a CE, cascode or cascade stage.For the trans-impedance factor the situation is rather dierent. From (4.40)and (4.44) we see that CE+CB and CE+CM exceed CE by a factor between 1and F up to frequencies of almost !T . The interaction between source and loadis decreased and both the cascode and the cascade stage behave as single, nearlyunilateral, amplier stages. From the viewpoint of designability this is an enormousadvantage.ACE+CM =

49

CE

mag(H(f))

arg(H(f))

CE

/2

f(log)

f(log)

CE

mag(H(f))

arg(H(f))

CE

/2

f(log)

f(log)

CE

mag(H(f))

arg(H(f))

CE

/2

f(log)

f(log)

CE

mag(H(f))

arg(H(f))

CE

/2

fT/ F

f T/ F

f(log)

f(log)

Figure 4.11: Bode plots of the CE stage as function of the frequency up to !T

50

CE+CB

mag(H(f))

arg(H(f))

CE+CB

/2

f(log)

f(log)

CE+CB

mag(H(f))

arg(H(f))

CE+CB

/2

f(log)

f(log)

CE+CB

mag(H(f))

arg(H(f))

CE+CB

/2

fT/ F

f T/ F

f(log)

CE+CB

mag(H(f))

arg(H(f))

f(log)

CE+CB

/2

fT/ F

f T/ F

f(log)

f(log)

Figure 4.12: Bode plots of the cascode (CE+CB) stage as function of the frequencyup to !T

51

CE+CM

mag(H(f))

arg(H(f))

CE+CM

/2

f(log)

f(log)

CE+CM

mag(H(f))

arg(H(f))

CE+CM

/2

fT/2f(log)

fT/2f(log)

CE+CM

mag(H(f))

arg(H(f))

CE+CM

/2

fT/ F

f T/ Ff(log)

f(log)

CE+CM

mag(H(f))

arg(H(f))

CE+CM

/2

fT/ F

f T/ F

fT/2f(log)

fT/2f(log)

Figure 4.13: Bode plots of the cascade (CE+CM) stage as function of the frequencyup to !T

52

The cascode stage, however, is of less practical use in low-voltage circuits. Toclarify this we consider Figure 4.14. Two possible implementations of a cascodestage together with an appropriate biasing circuit are depicted.VCC

A

VCC

B

Ibias

2Ibias

Vbias

IbiasVbias

Figure 4.14: Biasing the cascode stageFrom Figure 4.14A, we observe that the output voltage swing is limited by thesupply voltage VCC , the voltage across the terminals of the current source Ibias andthe sum of the saturation voltages of both transistors, while in Figure 4.14B theoutput voltage swing is limited by the supply voltage, the saturation voltage ofthe CB stage and twice the voltage across the two current sources.Apart from that an extra voltage source Vbias is required, which must be ratheraccurate to prevent both transistors from saturating.The cascade stage does not have these major drawbacks (Figure 4.15A and B).In neither circuits is there a need for an extra voltage source, and the maximumoutput voltage swing is limited by the supply voltage, only one voltage across theterminals of a current source and only one saturation voltage. It is for this reasonthat this stage is supposed to be superior with respect to the designability of thehigh-frequency behavior in low-power low-voltage circuits.

4.7 BiasingIn this section, we deal with the various biasing schemes, and attempt to selectthe one that is most suited for low-power, low-voltage integrated negative-feedbackampliers.If we assume that the design of the signal path was completed in the foregoingstages, then the bias circuit is not permitted to have any inuence on the signaltransmission. This means that either the signal can be superposed on the biasquantities by connecting a voltage source in series with the signal source, or thesignal can be superposed on the bias quantities by connecting a current source53

VCC

A2Ibias

B

VCC

Ibias

Ibias

Figure 4.15: Biasing the cascade stagein parallel with the signal source. However, the rst solution implies that eitherthe signal source or the bias source must have oating output ports. Further, theconnection of two (voltage) sources in series restricts the possible voltage swingof the signal source, and therefore its dynamic range. Since the latter solutiondoes not have these disadvantages, it obviously is the better one, especially forlow-voltage circuits.

4.7.1 Four fundamental ways of biasing

Generally, there are four ways to accomplish the correct biasing of one or morestages in a negative-feedback amplier.First, there is the possibility of making no distinction between the bias quantities and the signal quantities. The circuit already contains a negative feedbackloop which can assure proper biasing. Very often, this introduces unpredictableand undesired signal components which cannot be distinguished from the actualinformation (temperature drift, oset). As an example, we consider a negativefeedback transimpedance amplier (the simplest one being a shunt stage). Here,the base-emitter voltage and the voltage drop over the feedback resistor determinethe output voltage. It is clear that this kind of biasing is suited for restrictedapplications only.Another possibility is the separation of the bias and signal quantities in thefrequency domain. In this case, a bias circuit can have an additional feedbackloop, which is active only at very low frequencies. We therefore need a network ofwhich the transfer parameters depend on the frequency (e.g. a lowpass lter). Ofcourse, this technique can be applied only when the signal quantities do not havea DC componentThe third possibility is the separation of the bias and signal quantities in thetime domain. As an example we mention auto-zero techniques. In general, thesetechniques can be applied only when the input signal shows a time-discrete char54

acter. During the time when there is no relevant information in the input signal,a feedback loop is made active (by means of a switch) and restores the outputquantity at a reference level.The nal possibility is the separation of the bias and signal quantities incommon-mode (CM) and dierential-mode (DM) signals. The DM quantities referto the ideally isolated situation, where the circuit can be considered as a twoport.The CM quantities refer to ground or one of the other supply rails. These quantities are called orthogonal when they do not inuence each other, as is the casewith balanced circuits. Although, in practice, this is never completely true, themain advantage of this biasing technique is that there is no need for a lowpasslter