MONOLITHIC ACTIVE RESONATOR FILTERS FOR HIGH FREQUENCIES

Transcription

1 HELSINKI UNIVERSITY OF TEHNOLOGY Departent of Electrical and ounications Enineerin Electronic ircuit Desin Laboratory MONOLITHI ATIVE RESONATOR FILTERS FOR HIGH FREQUENIES Risto Kaunisto Noveber Dissertation for the deree of Doctor of Science in Technoloy to be presented with due perission of the Departent of Electrical and ounications Enineerin for public exaination and debate in Auditoriu S4 at Helsinki University of Technoloy Espoo, Finland on the 7th of Noveber,, at o clock noon. ISBN ISSN i

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3 ABSTRAT This doctoral thesis deals with onolithic active resonators and their use in hih-frequency filters. The ephasis has been put on noise and distortion properties of active resonators, as these are crucial in potential applications. Two active resonator types are considered: passive L resonators with active neative resistance copensation, and yrator-based active inductor resonators. An introduction to the theory of passive resonators is iven, and the basic quality factor and noise characteristics are discussed in detail. Filter structures based on parallel resonators are studied and techniques for frequency tunin briefly introduced. Based on a three-port equivalent, different neative resistor structures suitable for interation are cateorized, and their fundaental sall-sinal and tunin properties derived. The noise properties of the topoloies are analyzed and copared. The Volterra-series ethod is applied in the distortion estiations for each neative resistor type. Practical exaples of interated neative resistor are iven with realistic easured data. Hih-Q active inductors based on interated hih-frequency yrators are analyzed usin the total loop phase shift as an essential paraeter. Theoretical liitations of hih-frequency perforance and tunin are found. Noise and distortion properties are assessed in the sae anner as with neative resistors to ive rounds for direct coparisons. Practical issues of onolithic active inductor resonators are tackled and realized topoloies with easured results are presented. Active resonator filters eployin either of the resonator types are discussed. Their noise and distortion perforance derived fro the respective resonator results is calculated. Autoated tunin techniques are briefly discussed. Exeplary desins are presented with easured data. The two realized active resonator filters with neative resistance resonators operate in the 4 GHz reion with.% and % relative bandwidths, 4-MHz tunin ranes, and 9-dB and -db noise fiures respectively. The D power consuption is a low 5 W per resonator. The active inductor filter has a center frequency of.4 GHz with alost -GHz tunin rane. The noise fiure is a hih db as estiated by the theory. Syste considerations show that active filters cannot directly replace passive filters in traditional radio architectures due to their relatively poor perforance, but as a new potential application, an LO sinal eneration syste for direct-conversion transitters with a onolithic band-pass filter is presented. Both GaAs and Si-BiMOS realizations show the feasibility of the concept. With the coparable quality factors of 45 and and approxiately the sae -db output copression points of db, the BiMOS topoloy consues only a fraction of D power but still ives ore than 8 dbc irror reection thanks to its dual-ixer topoloy. Keywords: analo interated circuits, active resonators, neative resistors, active inductors, onolithic radio-frequency filters i

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5 PREFAE I couned with ine own heart, sayin, Lo, I a coe to reat estate, and have otten ore wisdo than all they that have been before e in Jerusale: yea, y heart had reat experience of wisdo and knowlede. And I ave y heart to know wisdo, and to know adness and folly: I perceived that this also is vexation of spirit. For in uch wisdo is uch rief: and he that increaseth knowlede increaseth sorrow. Ecclesiastes :6 :8 This doctoral thesis is a result of research work at the Electronic ircuit Desin Laboratory, Institute of Radio ounications IR, in 995. The research has been part of the Acadey of Finland s proects New Radio Systes and Their RF Technoloy SARF , and Interated ircuit Solutions for Adaptive and Wideband Radio ounication ircuits startin fro 999. In addition, soe financin was provided by the National Technoloy Aency TEKES in the proect Proraable Radio Receivers ORAVAT. Durin , I had the privilee to attend the Graduate School on Electronics, Telecounications and Autoation GETA, which enabled e to concentrate fully on postraduate studies and to ake university visits abroad. In the course of post-raduate studies, I have been fortunate to receive additional fundin fro several foundations acknowleded here: the Finnish ultural Foundation, the Nokia Foundation, the Eil Aaltonen s Foundation, the Foundation of Technoloy and the Electronics Enineers Foundation. I want to express y ratitude to y supervisor, Prof. Veikko Porra for his uidance and contribution in the raduatin process even when his schedule was tiht. I would also like to thank Dr. Petteri Alinikula of Nokia Research enter who acted as y instructor when I started y career in our laboratory in 99. He oriinally provided e with a subect that was broad enouh for doctoral studies. Lic.Tech. Kari Stadius and e were the first RF students in the laboratory. We have worked toether ever after and shared a workin roo alost all the tie. Kari and I have had countless aruents on professional issues as well as others, which, I believe, have been beneficial for both of us. I warly thank hi for havin borne with e all these years! Another colleaue of ine, Lic.Tech. diss. Aarno Pärssinen, whose developent fro an apprentice to a aster I have witnessed durin these years, has always been an enthusiastic and supportive discussion partner, for which I a rateful. Besides, these two uncles have been excellent lunch copanions! Lic.Tech. Jan Riska has co-operated with e on the applications of the circuits; I thank hi for that. Finally, the rest of the staff at our laboratory, both forer and present, also deserves thanks for a pleasant and relaxed workin environent. In 995 and 996 I spent a total of seven onths as a visitin researcher at University ollee London in Enland. The visit ave new perspectives and resulted in several papers in conferences and ournals. I would like to thank doctors David Haih and Danny Webster for their contributions and initiatives for a visit I ade to Macquarie University in Sydney Australia in 997. I want to thank the pre-exainers Prof. Joseph Tauritz of University of Twente and Prof. Ian Robertson of University of Surrey for their contribution to the copletion of this thesis. iii

6 Finally, I a deeply rateful to y parents Raia and Sakari and to y sister Raisa s faily for their love and irreplaceable support in the oents of oy and success, as well as despair and disappointent. It has not always been easy to keep up the otivation but thanks to the I did not ive up. To reach the final oal without the would have been very difficult and alost eaninless. The support of y dear relatives and friends will also be reebered. In Tapiola, Espoo, October Risto Kaunisto iv

7 SYMBOLS AND ABBREVIATIONS Sybols A, A excitation voltae aplitudes in Volterra kernels A i current ain A v voltae ain A filter filter pass-band loss B & derivative of susceptance at center anular frequency B in input susceptance b,, b diital loic levels of a binary word zero-bias capacitance,, n,n couplin capacitances in coupled-resonator filters be base-eitter capacitance c couplin capacitance in second-order filters yrator capacitance d ate-drain capacitance s ate-source capacitance in input capacitance p parallel capacitance p, p parallel capacitances in two-ports r resonator capacitance r,, rn resonator capacitances in coupled-resonator filters r capacitance associated with the actual resonance in coupledresonator filters s series capacitance v unction capacitance in varactors c, c, c coefficients in the power series approxiation of non-linear capacitance c i sall-sinal input capacitance F noise fiure F,, F n block noise fiures in a syste F filter filter noise fiure F R receiver noise fiure F tot total chain noise fiure f center frequency f,in, f,ax iniu and axiu center frequencies f in input frequency f LO local oscillator frequency f ax axiu oscillation frequency f sr self-resonance frequency f T transition frequency G characteristic conductance G,, G n block ains in a syste G yrator conductance G FB feedback conductance G in input conductance G l load conductance G s source conductance tunin conductance G tun v

8 sall-sinal output conductance, coefficients in the power series approxiation of non-linear transconductance c, c coefficients in the power series approxiation of non-linear copound transconductance b sall-sinal base conductance ce sall-sinal collector-eitter conductance ds sall-sinal drain-source conductance sall-sinal transconductance n relative noise conductance tot total sall-sinal conductance H, H, H Volterra kernels of order, and H iin Volterra kernel for input current I -R current throuh neative resistance I bias bias current I p current throuh parallel capacitance I s current throuh series capacitance I c collector current I dc D operatin current I dss zero-bias drain current I in input current I L current throuh inductance I Ls current throuh series inductance I Lp current throuh parallel inductance i current throuh yrator capacitance A coponent i in input current A coponent i in,actind active inductor input current A coponent i in,neres neative resistor input current A coponent i NL, i NL second- and third-order non-linear current sources i o, i out output current A coponent k Boltzann constant.87 - J/K L,, L n resonator inductances in coupled-resonator filters L cop inductance due to copression L fund fundaental-frequency inductance coponent L p parallel inductance L s series inductance L super super-inductance n pn-unction radin factor P IM power of third-order interodulation product P in input power Q quality factor Q, Q res unloaded quality factor at resonance Q,actind active inductor unloaded quality factor Q,noise effective unloaded noise quality factor Q, Q cap capacitor quality factor Q process-defined capacitor quality factor Q diff quality factor in differential resonators Q e external quality factor Q ax,,q ax,cap axiu capacitor quality factor Q ax,l,q ax,ind axiu inductor quality factor Q L, Q ind inductor quality factor vi

9 Q L Q l q R R cap R cop R ind R l R leff R p R peff R Qen R s R sens R sall-sinal R so r i r n r o S, S, S, S Sf T t pd V -R V bias,f V bias,q, V qb V V c V c, V c- V cc, V dd V p V s V in V V Lp V Ls V out V pp V res V ss V T V t v v in v in,cop v in,cop,actind v in,cop,neres v in,cop,reson Y diff Y in Y p process-defined inductor quality factor loaded quality factor eleentary chare.67-9 characteristic resistance resistance associated with capacitance loss resistance due to copression resistance associated with inductance load resistance effective load resistance after transforation parallel resistance effective parallel resistance after transforation Q-enhancin resistance series resistance sensin resistance sall-sinal resistance source resistance sall-sinal input resistance relative noise resistance sall-sinal output resistance s-paraeters noise spectral density standard teperature 9 K ean propaation delay tie voltae across neative resistance frequency control voltae quality factor control voltae voltae across capacitance control voltae positive and neative control voltaes positive supply voltae voltae across parallel capacitance voltae across series capacitance input voltae built-in unction potential voltae across parallel inductance voltae across series inductance output voltae peak-to-peak voltae voltae across a resonator neative supply voltae theral voltae 4.99 V threshold voltae voltae across yrator capacitance A coponent input voltae A coponent input voltae at copression A coponent active inductor input voltae at copression A coponent neative resistor input voltae at copression A coponent resonator input voltae at copression A coponent differential adittance input adittance parallel adittance vii

10 y, y, y, y y p, y p Z in Z nor Z out Z res Z s W diss W stored W stored,ind W stored,cap w α n β γ a ϕ, φ & τ ϕ, φ φ i φ o ϕ & & & & & L & LO & in & Qax & Qax, & Qax,L & self & sr iˆ, iˆ n iˆc iˆd iˆ, ˆ in i n, in iˆ n, A i i ˆ n, ne i ˆ n, res i ˆ n, Z v vˆ, v ˆ n vˆ y-paraeters parasitic yrator port adittances input ipedance noralized ipedance output ipedance resonator ipedance series ipedance dissipated enery stored peak electroanetic enery stored peak anetic enery stored peak electric enery relative anular frequency n th low-pass prototype filter coponent current ain coefficient in bipolar transistors noise aa coefficient aplitude error phase shift, phase error --db anular bandwidth in second-order filters transit tie phase anle input phase output phase derivative of phase anle at center anular frequency center anular frequency excitation anular frequencies in Volterra kernels anular frequency at inductance zero local oscillator anular frequency iniu usable anular frequency anular frequency for axiu quality factor anular frequency for axiu capacitor quality factor anular frequency for axiu inductor quality factor self-resonance anular frequency noise spectral density current collector shot noise spectral density current channel theral noise spectral density current input-referred noise spectral density current noise spectral density current associated with current aplifier neative resistor noise spectral density current resonator noise spectral density current noise spectral density current associated with ipedance total root-ean-square noise voltae noise spectral density voltae noise spectral density voltae associated with transconductance viii

11 vˆ vˆ ˆ in, v n, in n, A v vˆ n, Z vˆr input-referred noise spectral density voltae noise spectral density voltae associated with voltae aplifier noise spectral density voltae associated with ipedance noise spectral density voltae associated with resistance Abbreviations A ATRES Al APLA BDR BFL BiMOS BJT BW BW -db bal -V AD II D G LK OMP D D DS DR EL FDD FET GaAs G- GMMT-F GSGSG GSM HBT HD I-V I IP IF IIP IM IM ISM alternatin current active resonator aluiniu siulation software packae by Aplac Solutions blockin dynaic rane buffered FET loic bipolar and copleentary-etal-oxide-seiconductor bipolar unction transistor bandwidth --db bandwidth balanced capacitance-voltae coputer-aided desin second-eneration current conveyor coon drain coon ate clock copression depletion-ode direct current diital cellular syste dynaic rane eitter-coupled loic frequency division duplex field-effect transistor alliu arsenide transconductance-capacitance process technoloy provided by GE-Marconi round-sinal-round-sinal-round lobal syste obile heterounction bipolar transistor third-order haronic distortion current-voltae interated circuit input copression point interediate frequency third-order interodulation distortion intercept point interodulation third-order interodulation distortion industrial-scientific-edical ix

12 I/Re IQ IR L LNA LO MDS MESFET MIM MMI MOS MRR NF NOR OP OTA- PIN PLL pn R RF RFI RL rs RX SFDR Si SiGe TDD TX unbal WDMA ratio of iainary and real coponents in-phase quadrature-phase iae reection inductor-capacitor low-noise aplifier local oscillator Microwave Desin Syste; siulation software packae by Hewlett-Packard etal-extrinsic-seiconductor field-effect transistor etal-insulator-etal onolithic icrowave interated circuit etal-oxide-seiconductor irror-reection ratio noise fiure not-or output copression point operational transconductance aplifier capacitor positive-intrinsic-neative phase-locked loop positive-neative resistor-capacitor radio frequency radio-frequency interated circuit resistor-inductor-capacitor root ean square reception spurious-free dynaic rane silicon silicon-eraniu tie division duplex transission unbalanced wide-band code-division ultiple access x

13 ONTENTS ABSTRAT... I PREFAE... III SYMBOLS AND ABBREVIATIONS...V Sybols...v Abbreviations...ix ONTENTS... XI. INTRODUTION.... Motivation for the Thesis.... Research ontribution.... Oranization of the Thesis.... PASSIVE RESONATOR THEORY...5. General Definitions Series and Parallel Resonators Series-Mode and Parallel-Mode Transforations Lare-Sinal Behavior...6. oncept of Quality Factor General Rearks Resonator Q Factor...8 Theory...8 Practice Q in Two-Port ircuits and Differential Resonators.... Passive Resonator Noise....4 Resonators in Filters oupled Resonator Filters Resonator Q versus Filter Q Effective Terination Ipedances Noise in Resonator Filters Lare-Sinal Effects in Resonator Filters Frequency ontrol of Resonators Varactor diodes...9 Noise... Distortion Ipedance ultiplication... Noise and Distortion urrent Steerin apacitance Matrices... References...4. NEGATIVE RESISTOR RESONATORS...5. Historical Perspective...5. General Definitions...5 xi

14 .. Series and Parallel opensation...6. Sinle-Transistor Neative Resistors Series-Mode Neative Resistances...7 Tunin Parallel-Mode Neative Resistances...9 Tunin....4 Twin-Transistor Neative Resistances... Tunin....5 Noise in Neative Resistors Relative Noise Resistance and onductance Transistor Noise Model Noise in Twin-Transistor Transconductors Noise in Series-Mode Neative Resistors Noise in Parallel-Mode Neative Resistors Noise in Inverted- Parallel-Mode Neative Resistors Measurin Relative Noise Resistances and onductances Distortion in Neative Resistors Introduction Volterra Technique Non-Linear Behavior of the Series-Mode Neative Resistor...4 First-order kernels...4 Second-order kernels...4 Third-order kernels...4 Distortion coponents...4 Measured results Non-Linear Behavior of the Parallel-Mode Neative Resistors...45 Measured results Non-Linear Behavior of the Positive- Parallel-Mode Neative Resistors Effect of the Differential Transconductor Dynaic Rane of Neative Resistance Resonators Definition Neative Resistance Resonator Noise Dynaic Rane of Neative Resistor Resonators oparison Practical Neative Resistor Resonators Feasibility for Interation Realized Sinle-Ended MESFET Neative Resistor Resonator...5 References GYRATOR RESONATORS Historical Perspective Passive Manetic Gyrators Electronic Gyrators MMI Gyrators Non-Ideal Gyrators Non-Ideal Transconductor Effects of Finite Transconductor Bandwidth and Phase La Effects of Non-ideal Port Ipedances Non-Ideal Active Inductor Non-Ideal Active Inductor Bandwidth Active Inductor Hih-Q Operation...65 xii

15 4.. ontrollin Resonance Frequency and Q...68 Effect of...68 Effect of r o Active Inductor Noise Active Inductor Distortion Introduction Volterra Kernels Distortion oponents Dynaic Rane Practical Active Inductors Hara s ircuits and Its Derivatives Bipolar Active Inductors Q-Enhanceent Realized MESFET Active Inductors Realized Bipolar Active Inductors...8 GaAs-HBT active resonator...8 Silicon-BJT active resonator...84 References ATIVE RESONATOR FILTERS Introduction Noise in Active Resonator Filters Noise in Neative Resistor Resonator Filters Noise in Active Inductor Resonator Filters Dynaic Rane of Active Resonator Filters Practical Feasibility in Systes G GSM...94 FDD heterodyne receiver filters...94 TDD heterodyne receiver filters...94 Iae-reection filters...94 TDD direct-conversion receiver filters...95 Direct-odulation transitter filters...95 Frequency synthesizer G WDMA Bluetooth Autoated Tunin Techniques Master-Slave Tunin ouplin Factor Tunin Adaptive Transconductor Biasin Realized Active Resonator Filters Active Neative Resonator Filters...98 Active resonator...98 Band-pass filters...99 Realized circuit and results Active Inductor Filters... GaAs-HBT filter... Si-BJT Filter Application ase I: Local Oscillator Generation ircuit for Direct onversion Transitters in GaAs-MESFET Technoloy Introduction LO Sinal Generation ircuit...5 xiii

16 5.7. Desined MESFET ircuit...7 Frequency divider...7 Mixer...9 Active band-pass filter Experiental Results onclusions Application ase II: Local Oscillator Generation ircuit for Direct onversion Transitters in BiMOS Technoloy Introduction Divider... Siulated results...4 Measured results Polyphase Filter Mixers Band-Pass Filter...8 Measured results Output Buffers Entire Syste...9 Siulated results...9 Measured results oparisons and onclusions... References ONLUSIONS...5 xiv

17 . INTRODUTION. Motivation for the Thesis The breakthrouh of wireless personal telecounication in recent years has created a deand for saller and cheaper portable handsets. Much effort has been ade to fulfil these requireents, often with success. The nuber of discrete coponents in a cellular telephone has shrunk into few interated circuits in the base-band and IF-sections, the sae applyin to any parts in the RF section. Filters, however, have not been aon those coponents. RF filters are definitely the ost difficult RF parts to be interated. This is a serious disadvantae, as they appear in several locations in an RF front end. New radio architectures, such as direct conversion receivers, can possibly reduce the nuber of RF filters in the receiver chain but by no eans dispense with the. Moreover, the hih-quality passive filters are the ost expensive and bulky individual coponents in the RF section, and they are cubersoe in autoated anufacturin processes. Althouh interated filters would rectify all these drawbacks, no coercially sinificant proress has been ade in realizin the yet. The reason is clear: the current syste specifications are too tiht for active filters. Unlike their passive counterparts, active filters have noise and distortion; they consue power and need constant tunin for aintainin accuracy. Hih frequencies brin ore probles, as traditional well-known desin ethods are not applicable: no hih ain eleents are available, and the clock frequency for discrete-tie filters, such as switched-capacitor or switched-current filters, becoes ipracticably hih. On the other hand, distributed eleents applicable at illieterwave interated circuits, such as icrostrip structures, are too lare for interation in the frequency rane concerned < 5 GHz. Active resonator filters see to offer the best possibilities for onolithic hih-frequency realizations. The obective of this study is to investiate the possibilities offered by interated active resonators in icrowave filter desin, reconize their liitations and find potential applications in the field of obile telecounications. Different interated circuit technoloies have been probed in order to find the optial perforance for each technique. Active resonators theselves are not by any eans a new approach for filter synthesis. They have been reported throuhout the short history of electronics. Due to their nuerosity, it would be a tedious task to refer to the coprehensively, and they would have little relevance to this study. Soe of the historic references will be iven in the correspondin chapters, thouh. In the context of radio-frequency interated circuits, active resonator topoloies presented in the recent decade have ore sinificance to this work, and the ost sinificant scientific contributions to the subect will be discussed in further chapters. However, studies with practicable results and proper easured data are relatively few.. Research ontribution Active resonators can be divided into two roups: passive L resonators with active neative resistance copensation, and active inductor resonators. The theoretical and practical desin issues of these approaches will be tackled in this thesis. Practical exaples in the for of realized circuits will be iven. In the present study, all the different icrowave neative resistor topoloies are shown to derive fro a sinle three-port. Thus, they can be cateorized, and the coon fundaental properties reconized. This has also iven the otivation for coparison in ters of perforance and feasibility. The ephasis is put on noise and distortion properties, as they are the ain concern in applications. onsequently, yrator-based active inductors are also

18 analyzed in detail. A ethod for understandin loss eneration and its eventual cancellation with appropriate loop phase shift is presented. A very typical isconception found in any papers is to reard active resonators as direct alternatives to passive resonators. This is the result of totally nelectin noise and distortion studies. By careful analyses, the author ais to ive a realistic, althouh soewhat pessiistic, view of this issue. The author shows that siplified odels for noise and non-linearity of transistors can be used as a basis for in-depth analyses. Althouh the absolute theoretical results ive an optiistic view, their relative accuracy is ood and useful for practical diensionin, if peritted by the chosen process and practical issues. The Volterra-series ethod for calculatin distortion responses is applied to the topoloies, ivin ore understandin on the effect of each non-linear ter. Several active resonators and filters have been desined by the author, with the ephasis on low noise, low power consuption and sall size. Different process technoloies are experiented, and their suitability for active resonator desin is assessed. ellular phones are coonly rearded as potential applications for onolithic active filters. It is shown, however, that the noise and dynaic rane perforance of active filters cannot be adequate for direct replaceent of passive filters in current cellular architectures. However, a new application for wireless systes, an LO sinal eneration circuit for direct-conversion transitters, is presented. Two such circuits with proisin easured perforance have been desined and the results presented in the thesis. The thesis concentrates on the analysis and the realization of neative resistors and active inductors in resonators only. It does not deal with other filter techniques, even if they were presently applicable to the frequency rane in interest. Althouh iportant, the detailed analysis and the desin of frequency-tunin devices, e.. varactors, are also out of the scope of this work. The focus is on the core resonators and filters, and external tunin circuits are only briefly discussed. Based on several proects at the Electronic ircuit Desin Laboratory, the results have been published in nuerous scientific ournals and conference proceedins. References to these will be iven in the appropriate locations in the text. The contribution of the author in these papers has been the theoretical backround and all the practical desins of active resonator filters. Mr. Kari Stadius has participated in the practical desin issues, and Prof. Veikko Porra and Dr. Petteri Alinikula have acted as proect supervisors. In the course of doctoral studies, the author has visited University ollee London, where he has participated in the research tea led by Dr. David Haih. Durin the visit, the author has experiented with different resonator desins, which have been published in oint papers with Dr. Haih s tea. The syste considerations in hapter 5 have been published in a oint paper of Dr. Alinikula and the author, where the forer has contributed to the syste aspects and the latter to the filter desin issues. The GaAs version of the LO sinal eneration circuit has been developed in a laboratory proect, where Mr. Jan Riska has been responsible for practical desin of the circuits except for the band-pass filter. The author has been the proect leader and in chare of the syste desin and the ipleentation of the band-pass filter. Prof. Kari Halonen and Dr. Alinikula have been the supervisors of the proect. In the Si-BiMOS version, Mr. Riska has desined the ixer with its auxiliary circuitry, whereas the authors contribution is in the frequency divider desin, the filter and output buffer realizations, and the overall syste desin.. Oranization of the Thesis In hapter, an introduction to passive resonator theory is iven. It is essential to understand the definitions and the terinoloy of passive resonators prior to considerin their active counterparts. The definition of quality factor is discussed in detail, as inconsistent definitions can be found in the literature. Noise properties and lare-sinal effects in passive

19 resonators and filters are discussed, as they for a reference for active desins. Also, a review of frequency tunin techniques for resonators is iven. hapter deals with the analysis and desin of neative resistor resonators. The different topoloies are cateorized and their fundaental sall-sinal and tunin properties are derived. The ephasis is put on noise and distortion analyses and coparison of different neative resistor types, as they are the ost liitin issues in practical filter desin. The Volterra-series ethod is applied in the distortion estiations. At the end, realized neative resistor resonators are presented and their easured perforance presented. The subect of yrator-based active inductor resonators is elaborated in hapter 4. By utilizin the concept of loop phase error, the operation conditions for hih-q active inductors are derived, and the effects of tunin paraeters studied. The noise and distortion perforance is assessed in the sae anner as with neative resistors, so that direct coparisons are possible. Finally, practical issues of yrator inductor desin and realized topoloies with easured data are discussed. In hapter 5, the topic of active resonator filter desin is tackled. Derived fro the respective resonator results, the noise and dynaic rane perforance of the filters theselves are presented. Autoated tunin techniques are introduced althouh their ipleentation is outside the scope of the thesis. Utilization of active filters in odern cellular systes is shown to be difficult but as a new potential application, an LO sinal eneration syste eployin a onolithic band-pass filter is presented for direct-conversion transitters. The easureent results fro two test circuits show that the concept is feasible. Finally, hapter 6 contains a suary of the work carried out in this thesis.

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21 . PASSIVE RESONATOR THEORY. General Definitions For each active resonator, one can identify a passive equivalent at a iven narrow frequency band. As the passive counterpart is without exception superior in ters of essential perforance factors, such as noise and distortion, it fors a self-evident reference. Therefore, it is essential to first understand the behavior and perforance liits of resonators enerally before lookin into active circuits. In view of active resonator desin, this chapter ives a review of essential aspects of passive resonators, their quality factors, noise and tunin issues, and passive resonator filter desin... Series and Parallel Resonators An electrical resonator is fored when devices capable of storin electrical and anetic enery interact with each other. In electronics, such devices are capacitors and inductors. The two types of L resonators are series and parallel resonators. Fiure. Series and parallel L-resonators If an L resonator were ideal, i.e. lossless R s, R p, the electric and anetic enery in its coponents would transfor into each other in an oscillatory anner at the rate of called the resonance frequency, when excited by a voltae series resonators or current parallel resonators ipulse. In practice, passive resonators are never ideal, however, and possible initial oscillations are daped by the loss resistors R s or R p. Z s Rs Y, p, s R p Lp L s, p s, p. When resonators are fed fro an external source, they for a frequency-dependent network the ipedance of which reaches its iniu series resonators or axiu parallel resonators at resonance. This feature is useful in desinin frequency-selective circuits, such as filters. If used as a part of a passive filter circuit, a resonator is always loaded by surroundin coponents and source resistances, and even if the resonator is lossless, the circuit reains stable. In this case, an ideal resonator would ive zero pass-band attenuation and lowest noise, which are naturally desirable properties for any passive filter. Thus, axiu-q resonators are needed for ood-quality filters... Series-Mode and Parallel-Mode Transforations As inductor losses are usually doinatin in a onolithic resonator, all possible internal loss resistances are identified as inductor losses here. A lossy inductor can be defined with its loss resistance either in series or in parallel or both with the inductance. One can always transfor a series-ode circuit into a parallel-ode circuit and vice versa: 5

22 Fiure. Series-ode and parallel-ode inductors R L p p L Rs R s R Ls s Ls R s Q Ls R Q, s Ls Q Q >>. The quality factor of an inductor is a easure of its ideality; for a series-ode and a parallel-ode inductor it is defined as L Q R s s Rp L p. As the usae of rounded parallel resonators in onolithic band-pass filters is ore feasible, and ost active resonators are inherently parallel, I shall concentrate only on parallelode circuits fro now on. Nevertheless, ost calculations would apply directly to series resonators with little odifications... Lare-Sinal Behavior Within a resonator, very hih voltae/current levels across the reactive coponents can be observed. In a series and parallel resonator, the voltae/current anitudes over the inductor and the capacitor at resonance are V I Ls Lp V I s p Q V, Q I in in, I V Ls Lp I s V p V in R p R I s in.4 when the resonator is excited by the voltae V in series resonators or the current I in parallel resonators. This iplies that series resonators suffer fro internal voltae swins Q ties hiher than the input voltae, the sae applyin to current swins in parallel resonators. If the voltae/current handlin capability of the reactive coponents is liited, this leads to linearity probles. This will becoe an issue in hih-q filters if the resonators are active and thus ore or less non-linear. For instance, a quality factor of corresponds to -db voltae/current peakin at the inductor terinals. If the resonator is loaded, the external loss can be included in R s or R p, and Equation.4 still applies.. oncept of Quality Factor.. General Rearks The concept of quality factor Q has often several interpretations, dependin on the context. This will lead to confusion if the particular ethod of calculation is not revealed. There is, however, only one definition that is physically correct and applicable in ost cases: the quality 6

23 factor of a syste is the ratio of the stored enery and the dissipated enery per one frequency cycle in the syste. W Q π W stored diss cycle.5 This definition ives the two fundaental conditions where Q exists: the syste ust be able to store enery, i.e. it ust contain reactive or reactance-siulatin eleents, and it ust also be dissipative. For a syste that produces enery, quality factor is not defined, as it would becoe neative W diss <. Therefore, in ters of the network theory, Q cannot be defined for an active circuit. The syste can contain active coponents, i.e. transistors, but when lookin inside throuh its terinals, it ust act as a passive syste. W stored in Eq..5 describes the syste s capability for storin enery, i.e. it is the peak value of either anetic or electric enery, whichever is reater I L and V are the peak voltae and current applied to the coponent: W stored, ind L I L, Wstored, cap V.6 The peak eneries becoe equal at resonance, and that is the point where the definition of W stored chanes the anetic enery becoes larer than the electric enery, or vice versa. W diss is always the total dissipated enery durin one cycle. For a pure inductor in a parallel connection, Eq..5 ives the failiar expression [.]: V π V R Q π L R L W stored W diss.7 Yue and Won [.] suest that for a realistic inductor with parasitic capacitances, a different definition of Q should be used. The parasitic electric enery would be counterproductive and finally cancel out the anetic enery of the inductance, resultin in a quality factor of zero at self-resonance. Matheatically this corresponds to the widely used forula Q I{Z}/Re{Z}, where Z is the ipedance of the inductor with parasitics. It can be conteplated that this ideal violates the fundaental principle in Eq..5, thouh. There would be no enery stored at the self-resonance frequency, and it would be ipossible to use the circuit as a resonator at that point if the inductor Q sank to zero at resonance. Therefore, in order to preserve consistency with the fundaental quality factor definition, all practical inductors or capacitors for that atter should be identified as L tanks. At low frequencies well under self-resonance, this is soewhat awkward, and Eq..7 is quite an accurate approxiation. It is iportant to understand the distinction between the losses of an inductor and those of an L tank. The inductor loss is attributed to the anetic enery storae only, but the dissipation in a resonator consists of both anetic and electric enery losses. Hence, if we talk about the inductor Q value in conunction with a real-world inductor, the exaination ust be liited to low frequencies, and we ust not take any eventual losses in parasitic capacitors into account. In other words, any chane in the resistive behavior that is caused by parasitic capacitors does not affect the inductor Q but will chane the resonator Q. 7

24 .. Resonator Q Factor Theory We can easily calculate the stored and dissipated eneries with the aid of Eq..6, and after VRPH DSSUR[LPDWLRQV & L ½ et the followin expression for the resonator Q, irrespective of the resonator type: Q Q L Q Q Q QL,,.8 Exactly the sae result can be obtained by reducin the circuit to one of the basic resonator types via the series-parallel or parallel-series transforations, and then deterinin Q in the conventional way Eq... The results iven by Eq..8 are illustrated in Fiure., where the quality factor of a typical parallel resonator with a series-loss inductor and capacitor is plotted. Q ind Qcap 8 Q res 6 Q ax,cap Q 4 Q ax,ind inductive reion I / Re resonance reion capacitive reion / self Fiure. Quality factor of a parallel resonator with a series-loss inductor and capacitor. Pure inductor and capacitor Qs and the I/Re ratio of the resonator are also plotted. When the circuit is used in the inductive or capacitive reion, its Q is directly the inductor Q or the capacitor Q itself, ust like explained in the previous section. It is noteworthy to reark that the definition I{Z}/Re{Z} ives a false approxiation at the upper end of the inductive reion. In the resonance reion Eq..8 is the only valid definition of Q. It is interestin to reark that there is a axiu in both inductive and capacitive quality factors, and that this axiu does not necessarily stand at resonance. In the case of series-loss resonatin coponents, the followin relations apply: 8

25 Q ax, L Q ax, R R ind R R cap ind cap,, Q Q ax, L ax, 4 4 Q ax, L R ind Q ax, L L R cap.9 %\ DGGLQJ D VXLWDEO\ VL]HG KLJK4 FDSDFLWRU WR WKH FLUFXLW LW LV SRVVLEOH WR PDNH & Qax,L DQG & coincide and thus have the axiu available quality factor at resonance. Practice The previous studies are not necessarily easily applicable to a practical case. They assue that the loss resistance is constant over the frequency band in interest. In practice, however, it is dispersive, due to current crowdin effects in the conductive aterials. Moreover, in onolithic resonators the conductance of the seiconductor is frequency-dependent. The capacitive reion can be difficult to characterize, as parasitic inductances ay affect the easureents at hih frequencies. At resonance, it is possible to define the resonator Q value via easureents, thouh. It can be defined as the inverse of the resonator ipedance --db bandwidth Fiure.4: Q R p Lp. If the resonator is loaded by a resistance R l, the quality factor sinks to R prl Ql L p l e R R Q Q Lp p l e, Q R. where Q l and Q e are the loaded and external quality factors. For instance, if also the resonatin capacitor is lossy, the total Q value of the resonator can be calculated fro Eq.. where Q e is the capacitor Q value. If y of a resonator is known, e.. fro easureents, the unloaded quality factor can also be defined as Q B { y }, B { y } B, G Re I G. because the slope of B at resonance equals p Fiure.5a. Perhaps a ore eleant way of deterinin the resonator unloaded Q is to exaine the derivative of phase at resonance: ϕ Q, ϕ ar y. 9

26 Z res nor...8 z nor.6.4. BW w / w Fiure.4 Manitude of parallel resonator ipedance;, Q π/ 5 π/4 I{y } slope B' ϕ y π/4 slope ϕ' -π/.5..5 / / Fiure.5 a Resonator susceptance at resonance b Phase of y at resonance.. Q in Two-Port ircuits and Differential Resonators Very often active resonators ust be realized as differential circuits; any of the active resonator topoloies are by nature differential. A differential resonator is fed by sinals in 8 phase shift. In realizable circuits, they usually for a two-port; a typical exaple is shown in Fiure.6. Actually, practical interated inductors are siilar two-port resonators. Definin Q for such a circuit is probleatic, as it ay vary dependin on fro which port it is easured if the rounded capacitors are of different sizes. If the circuit is rounded fro one end, it reduces to a one-port, and Q is easily attained, but the inforation on the capacitor at the rounded end is lost. The solution to this proble is to reard the circuit as a differential one-port. The rest of the differential circuit reards the resonator as a noral parallel RL resonator, as the round level is floatin in ters of the differential sinal. For characterization a differential resonator is easured like an ordinary two-port in a sinle-ended environent, but the results ust be anipulated to ive the actual circuit paraeters:

27 Fiure.6 Differential resonator y y y Y diff, y y y y y.4 Phase ibalance at the input results in chanes in the functional quality factor of the differential resonator, althouh the actual Q, set by the coponent values, reains the sae. If φ desinates the phase deviation fro the ideal 8, the detectable Q value of the circuit becoes R φ p Lp p tan, Q.5 Q L Qdiff In the case of a pure differential inductor p, the chanes in Q are serious, as shown in Fiure.7 RUPDO UHVRQDWRUV ZKHUH & /L p p ½ are not affected by this phenoenon, thouh. As the phase error conceptually creates an extra reactive coponent, the resonance frequency is also chaned, but at a sall φ this is neliible. p Q diff Q Q Q f / de Fiure.7 Differential Q. Passive Resonator Noise The two quantities used in noise calculations are noise spectral density Sf and total rs noise voltae v. The forer ives the noise voltae or current density at a certain bandwidth, usually Hz. The latter ives the total noise voltae of the circuit over all frequencies liited by the transfer function. These two are related:

28 v S f df.6 The noise spectral density Sf enerally ives ore inforation about the resonator noise perforance itself than the rs noise voltae or current v. It is used for definin spot noise values near resonance, and consequently the filter noise fiure. When definin the filter perforance, the pass-band noise is ost sinificant, but the total rs noise value does not ive specific inforation on that. For instance, when the Q of a passive resonator approaches infinity its spot noise at resonance oes to infinity at the sae tie, resultin in instability, but the total rs noise still ives a fixed value of kt/, as illustrated in Fiure.8. Norally, the rs noise voltae is used in the syste desin, where the noise sources are wide-band but the output noise liited by filter transfer functions. It is assued that the filters have only a band-shapin effect but not any noise contribution. This is not the case with active filters, where the filterin function itself is noisy. The total rs noise voltae is still needed in the dynaic rane definitions, where the absolute axiu and iniu voltae levels across the resonator are studied. Fiure.8 Noise spectral densities and total rs noise voltaes of an unloaded resonator I shall denote noise spectral density voltaes currents as ˆv î hereafter for siplicity. A passive unloaded parallel resonator shown in Fiure.9 contains only one noise source, naely the series resistance of the non-ideal inductor. Fiure.9 Passive unloaded resonator with noise sources

29 The noise voltae at the output node is the product of the noise source voltae and the noise transfer function: vˆ vˆ R [ L R ] At the resonance frequency this yields 4kTR L R.7 ˆ 4kT v R 4kTRQ.8 Thus, the hiher the quality factor of the resonator the hiher the spot noise at resonance. Of course, if Q were infinite the resonator would be in fact noiseless, since there would be no noise sources left. However, it would be unstable and bound to oscillate, which in turn can be iained as infinite noise at a sinle frequency. The total rs output noise voltae of an unloaded resonator is v ktr vˆ df π ktr π kt π R d L R.9 which is the well-known forula for the noise of passive resonators, and in fact sinle-pole Rfilters as well. If a resonator is to be used as a buildin block of a filter, it is always loaded by the source and load ipedances and possibly other resonators. If the source and load ipedances are capacitively coupled, as usual, the couplin capacitances can be ebedded into the resonatin capacitance after ipedance transforation. The terinatin ipedances becoe transfored as well, and their values increase substantially. The transforation is frequency-dependent but fairly constant at a narrow bandwidth around the center frequency. The resultin circuit is shown in Fiure.. Fiure. Passive resonator with loadin In order to facilitate further noise derivations, I shall use inected output noise current sources instead of output noise voltaes, as the forer are independent on loadin. The inected output noise current of a passive resonator is iˆ 4kT R peff 4kTR, R L R peff L R R.

30 where the effective parallel resistance R peff is transfored fro the actual series resistance of the inductor. Now, the noise spectral density at the output becoes vˆ ˆ i Z n 4kTR R L R L Rl Rl. At resonance this yields vˆ 4kTR R R l 4kTR QQ e L R R l Q l 4kTRQ l 4kTR Q Q e Q Q e. where Q e is the external quality factor fored by loadin, Qe Rl L, and Q l is the loaded quality factor Ql Q Qe. It can be seen that the noise is directly proportional to the load, i.e. a hih Q e iply hih noise. As narrow-band filters have hih effective source and load ipedances, they inherently enerate ore noise. If the loadin is zero Q e, Q l Q the result is the sae as in Eq..8. For copleteness, the total rs noise voltae is calculated below. Without loadin, this yields kt/, as previously. v ktr π R Rl d L R L Rl kt R L Rl Rl R..4 Resonators in Filters.4. oupled Resonator Filters Resonator band-pass filters are ost straihtforward to construct fro capacitively coupled parallel resonators. The siplest possible second-order filter of this kind is shown in Fiure.. Hiher order filters are fored by cascadin ultiple second-order blocks with altered capacitor values, correspondin to the desired prototype function. Fiure. oupled-resonator filters; second-order and n th -order 4

31 Diensionin of coupled-resonator filters has been derived by ohn [.]. The low-pass prototype filter paraeters and types can be freely chosen and then translated into the band-pass coupled-resonator topoloy. For different filter functions Butterworth, hebyshev, Bessel etc. and orders, prototype coponent values are listed in filter reference books, such as n desinates the n th low-pass prototype coponent, the actual filter coponent values can be calculated fro the set of equations.4: rk L r k, k rk w w r R w R α α k, k r r, k k r R rk k rk k so α α so, so, k, k, k [, n], k n, n rn [, n ] w rn Rlαn w R α n, n rn rn l n, n R n l n, n.4 w is the anular --db bandwidth of the filter if the prototype filter is one, as usual, and R so is the source resistance. For a narrow-band second-order sinle-resonator filter with equal terinations, the equations can be approxiated as r r,, r r L Q R l so.5 ZKHUH. %XWWHUZRUWK UHVSRQVH n and Q l & ± & in second-order filters. The circuit looks like a loaded resonator with a loaded quality factor of Q l..4. Resonator Q versus Filter Q The loaded quality factor of a resonator in a second-order band-pass filter is often called the filter Q. The filter Q is actually defined as the inverse of the relative bandwidth, and it is the sae as the actual loaded Q of a sinle parallel resonator. It has no relation with the actual resonator Q in hiher-order band-pass filters, thouh. Therefore, I will refrain fro usin this ter hereafter..4. Effective Terination Ipedances When the source or load ipedance is raised, the correspondin couplin capacitor or diinishes, and eventually it can be oitted altoether. Generally, if the desired bandwidth can be achieved via the couplin capacitors without ipedance transforation, their values becoe zero. Then the filter becoes siply a loaded parallel resonator the loaded Q of which is deterined by the source and the load. This is the case when the resonator is coupled with buffer circuits instead of capacitors Fiure.. The advantae of this topoloy is that no ipracticably sall couplin capacitors are needed, even if Q is very hih. However, noise and linearity properties are ipaired due to liitations in the active buffer circuits. 5

32 Fiure. Buffered resonator.4.4 Noise in Resonator Filters Since the calculation of noise perforance becoes increasinly difficult when the nuber of resonators rises, I shall confine to sinle-resonator second-order filters only. The fundaental noise characteristics observed are nevertheless universal. The ost convenient and ost widely used quantity for describin noise perforance of a icrowave filter is noise fiure F. By definition, it is the ratio of the sinal-to-noise ratios at input and output, or the total input referred noise level copared to the source noise level at the standard teperature T 9 K. Usually, noise fiure is expressed as spot noise fiure at a defined bandwidth, usually Hz. Both input noise voltaes and currents can be used: 4kTRso vˆ F 4kTR so in vˆ in 4kTR so iˆ in 4kT R so.6 The input referred noise current î in is attained by reducin the resonator inected noise current to the input. Usin the forulation of c iven in Eq..4, it becoes at resonance iˆ in Rso c iˆ iˆ α R r so.7 Now, the spot noise fiure of a sinle-resonator filter at the center frequency is iˆ in F 4kT R so iˆ α 4kT r.8 Notably, the contribution of the source ipedance disappears. However, if R so alone is chaned will also chane and alter the noise level. Fiure. Second-order coupled resonator band-pass filter By Eq.. the resonator noise current is 6

33 iˆ 4kTR 4kT, R L Q Q r >>.9 Thus, the spot noise fiure of a hih-q passive resonator filter becoes F α Q or F Ql α Q. where Q l corresponds to the terination-loaded quality factor of the resonator. The result clearly proves the iportance of hih quality inductors in passive resonator filters. Fiure.4 shows how the filter noise fiure behaves, when the inductor Q is varied in the rane of typical interated spiral inductors and the --db bandwidth if kept constant as %. 5 NF / db Q Fiure.4 Passive resonator Q vs. noise fiure The relative bandwidth of the filter is an equally iportant factor for noise perforance, but usually it is fixed by the specifications and cannot be freely enlared. In a realistic resonator, both the inductor and the capacitor are non-ideal, and the expression for noise becoes respectively F αq l QL Q. where Q L and Q are the inductor and capacitor varactor quality factors. For instance, with typical interated ood-quality inductor/varactor Q values of 5, the absolute iniu noise fiure with noiseless copensation of Q would be as hih as 9.6 db for a second-order filter, when tareted to the Bluetooth syste as a potential exeplary application f.44 GHz, BW -db 8 MHz. Thus, for low-noise operation at narrow bandwidths, it would be iperative to have passive resonatin coponents of virtually unrealistically hih quality in the first place. Since the inductor and capacitor quality factors are proportional to frequency, filters with hiher center frequencies have lower noise fiures. Therefore, coupled-resonator filters are especially suitable for icrowave frequencies, provided that the self-resonance of the individual coponents will not yet affect their quality factors at the filter center frequency. In a hiher-order filter the individual noise sources each contribute to lower noise levels than in a second-order sinle-resonator filter, since the resonators load each other. ouplin 7

34 between the resonators is, however, so weak that the total noise of the filter is always hiher than that of a sinle-resonator filter with the sae bandwidth. In the previous studies, the resonators are assued noisy but lossless, which is naturally ipossible as far as passive resonators are concerned. Any loss resistance in the resonators will chane the transfer characteristics of the filter and increase its pass-band loss, as well as induce noise to the syste. However, in active resonator filters the situation is quite like discussed here: the active resonator is desined for zero-loss, but active coponents will introduce soe excess noise. As far as syste noise properties are concerned, also the loss of the filter is sinificant. Even if the filter were practically noiseless, its loss would still derade the total syste noise fiure, especially if it is located first in the chain. Accordin to Friis s forula, the syste noise fiure is in this case A filter is the filter loss F tot F filter A filter F F F4 Fn... G G G G G... G n. A hih filter pass-band loss will set reater noise perforance deands on the followin syste blocks. The strenth of active filters is that their losses can easily be cancelled, or even better, they can have ain. In a syste, an active resonator filter with a -db noise fiure but zero loss as ood as a passive filter with a 5.9-dB noise fiure and loss, if F 5 db and the followin staes are inored..4.5 Lare-Sinal Effects in Resonator Filters A passive resonator is practically linear. Only the hysteresis phenoenon in the inductor iht attribute to non-linearity at very hih sinal levels. However, in the context of active resonator filters, it is necessary to understand how the filterin function affects the lare-sinal perforance of the syste. In a second-order coupled-resonator filter Fiure.a the voltae level across the resonator is V res V in Ql R r so. usin the sae arkins as in Eq..5. The expression shows clearly how lare an effect the quality factor has on the voltae peakin at the resonator terinals. If the loaded Q is doubled the bandwidth halved and the rest of the paraeters reain unchaned the resonator voltae level is increased db. Or correspondinly, the hihest allowed input level is decreased db if the resonator lare-sinal properties are kept constant. It is also worth noticin that lare source and load resistances ive better lare-sinal perforance. opression in active resonators shows itself in diinishin unloaded quality factors copared to the sall-sinal values. This has effect on both the filter bandwidth and pass-band loss. The latter directly attributes to the copression point of the whole filter. The Q deradation can be understood as an increase of loss a decrease of current at the fundaental frequency when part of the response current is transferred to hiher haronic frequencies. If the total parallel loss resistance due to copression is arked as R cop, the output voltae of the second-order filter becoes 8

35 V out V in Ql r R cop.4 --db output voltae copression takes place when Ql R r cop %.5 If Q l is low a relatively sall R cop can be tolerated. Or, on the other hand, voltae peakin at the resonator input is low, resultin in a hih R cop at that particular input voltae. In band-pass and low-pass filters, haronic distortion has practically no effect, as the haronic distortion coponents are located clearly out of the band. However, third-order interodulation distortion can be detected, when the two excitation frequencies and the interodulation products are very close to each other at the pass-band. In very narrow-band filters, the two frequencies are practically equal, and the IM response is identical with the copression response. Fiure.5 Second-order resonator filter with copression resistance.5 Frequency ontrol of Resonators Precise control of the onolithic filter center frequency necessitates tunable resonators, since absolute coponent values are inaccurate in the presence of process variations. The resonance frequency of a passive resonator is defined by its L product, and the only way to chane it is to alter either the inductance or the capacitance. A passive tunable inductor is ipossible to realize as an interated circuit, whereas the varactor diode, althouh often rearded as an active eleent, functions as an electrically tunable passive capacitor. With the help of transistors, techniques like ipedance ultiplication or current steerin can be adopted for transforin a constant passive capacitor or inductor into a variable one. This section introduces the ost used frequency control techniques..5. Varactor diodes When a pn-unction diode is reverse-biased, its depletion capacitance can be controlled with the bias voltae. Especially, if the diode is optially desined for this purpose, i.e. to ive axial capacitance per device area and lare tunin rane, it is called a varactor. For RF frequencies, a varactor looks like a series connection of a capacitor and a resistor Fiure.6. The resistor corresponds to the contact and intrinsic resistances in the seiconductor. It is responsible for the quality factor deradation and the noise of the varactor. A onolithic varactor also suffers fro parasitic capacitances to the substrate, which soewhat cuts down 9

36 the tunin rane. Extrinsic inductances set the upper frequency liit after which self-resonance will occur. In conventional interation processes, the base-eitter/collector unction of a bipolar transistor ay be used as a varactor althouh not especially optiized for that purpose. Fiure.6 Varactor and its equivalent circuit Typical tunin curves of a pn-varactor are shown in Fiure.7 [.5]. The increental resistance of the increasinly conductin diode accounts for the fall of the quality factor curve towards positive bias voltaes. The best Q value is attained at hih reverse-bias conditions..4. apacitance Q-value 5 / pf..8 Q V c / Fiure.7 Measured capacitance and Q-value fro an 8-ruple BJT base-eitter-diode YDUDFWRU DW *] P %L&6 SURFHVV Noise The noise eneration echanis in a reverse-biased varactor diode is very uch siilar to that in a noral lossy capacitor or inductor. As the D current throuh the device is neliible, only theral noise is present, and it can be directly attributed to the series resistor of the equivalent circuit. Therefore, in a resonator filter a varactor behaves like stated in Eq... Fiure.8 Noise equivalent circuit of a pn-varactor Distortion The depletion capacitance of a pn-varactor is usually forulated as v V V n.6

37 where the radin factor n can vary fro conventional.4 to hyper-abrupt.6. is the zerobias capacitance and V the built-in potential of the unction. The expression in clearly nonlinear, and hence pn-varactors are distortion-eneratin coponents. Meyer and Stephens [.4] have analyzed distortion in varactor diodes utilizin the Volterra series ethod. Accordin to WKHLU VWXG\ WKH WKLUGRUGHU LQWHUPRGXODWLRQ DQG WKH FRPSUHVVLRQ & & & coponents of both series and parallel resonators with varactors can be expressed as Qc n P IM P, v c cvin c v c.5 n in.7 where P is the averae power in the loss resistor. Equation.7 suests that it would be possible to null IM with the particular value of n. The zero is quite sharp, thouh, and ipractical to realize. Multiple-varactor connections can be used for enhancin reatly distortion perforance. They are particularly suitable for differential resonators owin to their antisyetry and cancellation of even haronic coponents. The two types of connections, back-to-back and antiparallel, are shown in Fiure.9: Fiure.9 Back-to-back and antiparallel varactors It is shown in [.4] that the antiparallel connection is inferior, as its IM is hiher than that of a sinle varactor. It is also ore difficult to arrane its biasin in practice. The back-to-back connection halves the voltae swin over individual diodes, and therefore distortion is also reduced. Moreover, it can be shown that if n.5 for atched diodes, which is often the case, the third-order distortion ter approaches zero. The drawback is the halved total capacitance value necessitatin double-sized devices, when copared to sinle varactors of the sae net capacitance..5. Ipedance ultiplication An ipedance ultiplicator consists of a passive coponent, usually a capacitor, whose voltae or current is sensed and ultiplicated by an appropriately selected aplifier. By chanin the ain of the aplifier, the net ipedance of the circuit becoes variable. Dependin on the type of the aplifyin eleent, ipedance ultiplicators are either series- or parallel-ode circuits shown in Fiure.. The circuits can also transfor a positive ipedance into neative if the ain sin invertin non-invertin is chaned. Fiure. a Series-ode ipedance ultiplicator with voltae aplifier; b Parallel-ode ipedance ultiplicator with current aplifier

38 Fiure. a Miller circuit for capacitance ultiplication; b Parallel transconductor capacitance ultiplier Usually the circuits are applied to capacitance synthesis because of the easy practical ipleentation and the fact that the resultin capacitance values becoe larer than the oriinal. The series-ode circuit with a capacitor is also known as the Miller capacitance circuit [.5]. Parallel-ode ultiplication circuits have been reported e.. in [.6]. In practical ipleentations, the non-idealities of the controlled sources, especially nonzero output ipedances, liit the perforance of the ipedance ultiplication circuits. The input and output resistances of the voltae aplifier directly diinish the quality factor in the Miller capacitance circuit, and the sae applies to the parallel ultiplicators. In addition, the current aplifier in the parallel circuits ust be realized as a transconductor with a current-tovoltae transforer, which, in the siplest for, iplies a current-sensin resistor R sens in series with the actual ultiplicated ipedance. This resistor has usually a draatic reductive effect on the circuit Q. Hih-frequency properties of ipedance ultiplicators are very uch dependent on the bandwidth of the voltae or current aplifier. Near-ideal perforance is usually obtained only at relatively low frequencies. With the sae process paraeters, aplifier-based ipedance synthesizers have always ore liited operatin bandwidths than varactor diodes. Noise and Distortion In ipedance ultiplication circuits noise is enerated by the ultiplicated ipedance itself and by the active aplifier circuit. The noise sources of the passive ipedance if existent are also ultiplicated by the sae factor, and the resultin noise contribution corresponds to that of the passive ipedance of the ultiplicated value. The difference between the circuit types can be found in the active circuit noise that is also transfored in the series topoloies, but left unchaned in the parallel circuits. This is a result of the circuits not bein strictly dual, as the control quantities are taken fro different locations due to practical issues. This has a aor effect on the equivalent input noise that is lower in the parallel-ode ipedance ultiplicators. However, the need of current sensin resistance in practical parallel realizations causes severe noise perforance deradation. The lare-sinal perforance of an ipedance ultiplication circuit is deterined by the linearity of the active device/circuit. Where a current sensin resistor is required, its value has also effect on distortion: a saller resistor ives lower distortion, as the aplifier input voltae is diinished.

39 Fiure. Noise echaniss in ipedance ultiplication circuits.5. urrent Steerin The differential Gilbert analo ultiplier confiured as a variable-ain aplifier can be used for synthesizin variable ipedances [.7]. Multiplication is arraned by suin copleentary sinal currents via controllable current splitters. The ultiplicated ipedance is easily inserted as a series-feedback coponent Fiure.. The fundaental drawback with this approach is the inherent loss resistor fored by the transconductance of the aplifyin transistor. Toether with parasitics and losses in the switchin transistors, it will clearly reduce the achievable Q factor. Noise and distortion properties can be directly derived fro those of the analo ultiplier core. opared to the ipedance ultiplicators, the presence of the series steerin transistors will unavoidably increase noise and soewhat worsen the lare-sinal properties. Fiure. Variable ipedance based on Gilbert analo ultiplier; the principle and the differential ipleentation [.7].5.4 apacitance Matrices A capacitance atrix is a convenient ethod of controllin diitally the resonance frequency of an L-resonator. It consists of suitably sized passive capacitors that are connected in parallel via solid-state switches. For instance, by usin binary-weihted capacitor values a four-bit controllin word can tune the total capacitance in 6 discrete steps Fiure.5.

40 G / S / F M.G.5G.G 8.f 4.f. -4.f -8.f. 5.M.G.5G.G f / Hz G / S / F -.µ -4.µ p 5.f. -5.f -.p Fiure.4 Siulated parallel conductance and capacitance of a current-steerin capacitor as functions of frequency V c P9 DQG FRQWUROOLQJ YROWDJH I *] P %L&6 SURFHVV The quality factor, as well as the noise perforance of a capacitance atrix, is liited by the losses in the switch transistors. Lare-sinal properties are also inferior to a passive capacitor alone due to the solid-state switches, but still better than in active capacitance synthesizers. 4.µ.µ. V c / V Fiure.5 apacitance atrix References [.]. Yue and S. Won, On-hip Spiral Inductors with Patterned Ground Shields for Si- Based RF I s, IEEE Journal of Solid-State ircuits, Vol., pp , May 998. [.] S. ohn, Direct-oupled-Resonator Filters, Proceedins of the IRE, Vol. 45, pp , February 957. [.] A. Zverev, Handbook of Filter Synthesis, John Wiley, New York, 576 p., 967. [.4] R. Meyer and M. Stevens, Distortion in Variable-apacitance Diodes, IEEE Journal of Solid-State ircuits, Vol., pp , February 975. [.5] K. Stadius, R. Kaunisto and V. Porra, A Hih Frequency Haronic VO with an Artificial Varactor, Proceedins of IEEE International onference on Electronic, ircuits and Systes, Lisbon Portual, Vol., pp. 6-64, Septeber 998. [.6] S. Pipilos, Y. Tsividis, J. Fenk and Y. Papananos, A Si.8 GHz Filter with Tunable enter Frequency and Quality Factor, IEEE Journal of Solid-State ircuits, Vol., pp , October 996. [.7] W. hen and J. Wu, A -V -GHz BJT Variable Frequency Oscillator, IEEE Journal of Solid-State ircuits, Vol., pp. 46-4, Septeber

41 . NEGATIVE RESISTOR RESONATORS. Historical Perspective The concept of usin positive feedback to increase the quality factor of a resonator actually dates fro the early years of radio electronics itself. It was introduced in the reenerative receiver which was independently discovered by de Forest, Arstron, Lanuir and Meissner in 9, but only Arstron was iven the full credit [.][.]. Only in 94 the case was settled in de Forest s favor, but by that tie the reenerative receiver had been ade obsolete by iproved receiver techniques, such as the superheterodyne technique. In reenerative receivers, the aplified received voltae is fed back to the input circuit in such a phase that causes voltae increase at the input. This decreases the effective resistance of the input circuit and thereby provides reater aplification and selectivity for the input sinal [.][.4]. Reenerative receivers were, however, subect to oscillation if the feedback was increased beyond a certain point, causin serious interference to nearby receivers. Alon with difficult controllability, this ade reenerative receivers radually loose their popularity. The reenerative principle has lived throuh to our days in odern interated resonators. They still rely on the sae principle, positive feedback, in one for or another, althouh it always brins up the risk of instability. In fact, a hih-q active resonator operates on the brink of oscillation, which naturally sets hih deands on the controllin circuit accuracy. Fiure. Arstron s reenerative receiver [.]. General Definitions By definition, a neative resistor is a circuit eleent where voltae applied across its terinals creates in-phase current of the opposite direction. Apart fro few exceptions e.. tunnelin diodes, this functionality requires controlled sources, i.e. transistors, in the realizations. Hih-frequency neative resistors are typically one- or twin-transistor circuits with suitable feedback coponents. Their response is, however, never purely resistive due to built-in parasitic reactive coponents. These can be ebedded into the actual resonatin eleents, but due to series-parallel transforations, the resistive part often becoes frequency-dependent. In addition, bandwidths of the active devices are liited resultin in siilar effects. Different neative resistor topoloies have inherently different noise and distortion properties, and soe are practically ore suitable for interated circuits than others. An active neative resistor resonator is fored by a passive inductor-capacitor L resonator and an active neative resistor circuit that copensates the resistive losses. It is ost straihtforward to connect the neative resistor directly to the loss resistor in such a way that it entirely cancels the loss at all frequencies. In other words, by connectin a neative resistor of suitable anitude in series with the series loss resistance or in parallel with the parallel loss 5

42 resistance. Unfortunately, this is not always possible in practice, as there is usually no sinle definite loss resistance in an L resonator, and it would be ipractical to copensate the each with separate neative resistors. It is still possible to cancel the losses with a sinle neative resistor, but only at a narrower bandwidth... Series and Parallel opensation Reardless of the resonator type, there are two approaches for copensatin the resonator internal losses: series copensation and parallel copensation. Obviously, in the series copensation ethod the neative resistor is connected in series with the resonator, and respectively, in parallel copensation it is in parallel with the resonator. Intuitively, it is clear that series copensation is ore suitable for series L resonators and parallel copensation for parallel resonators, but there are also other issues to be considered. Fiure. Series copensation ethod Fiure. Parallel copensation ethod Fiure. shows the principle of series copensation for both series and parallel L resonators. The net loss of the resonator is described by a sinle frequency-dependent resistor R which is then cancelled by a neative resistor of equal size. If we calculate the voltae and the current across the neative resistor at resonance, when external current I in is applied across the whole circuit, we et V I R RIin, I R in. Respectively, in Fiure. the parallel copensation schee is illustrated. Now, at external voltae V in, the voltae and the current of the neative resistor at resonance are V Vin I R, R V in R. Thus, series resonators are current driven and parallel resonators voltae driven. It is reasonable to say that active neative resistors stand up with lare voltae swins better than lare current swins, when current non-linearities are doinant in the transistors. For practical perforance issues, it is often ore preferable to connect both the resonator and the neative resistor to the round. Therefore, parallel copensation is extensively used in interated circuits. 6

43 . Sinle-Transistor Neative Resistors All sinle transistor neative resistors can be described with a siple three-port shown in Fiure.4. When two of the ports are terinated with suitable ipedances, a neative resistor is seen at the reainin port. The type of the transistor is of little consequence to the sallsinal fundaentals. Due to liitations in the realizations of passive port ipedances and in the active device biasin, not all possible variations are practically applicable, thouh. Fiure.4 Basic structure of sinle-transistor neative resistors.. Series-Mode Neative Resistances For an RF sinal, a series-ode sinle-transistor neative resistor looks like a series connection of reactance and neative resistance. This type is fored when Port in Fiure.4 is the input port. If the transistor is siply rearded as a voltae-controlled current source the sall-sinal equivalent circuit can be fored as shown in Fiure.5: Fiure.5 Series-ode neative resistance a b c d Fiure.6 Possible transistor confiurations for rounded neative resistors: a oon-drain with capacitors [.5] [.8]; b oon-drain with inductors[.9][.]; c oon-ate with capacitors; d oon-ate with inductors [.] 7

44 With one terinal rounded four different topoloies can be found Fiure.6. The input ipedance of the circuit becoes Z in Z Z ZZ. The real part of the input ipedance is neative only if both Z and Z are of the sae type, i.e. either inductive or capacitive. The correspondin passive equivalent circuits are shown in Fiure.7. In practice, it is often ore convenient to use capacitors as Z and Z since the atesource or base-eitter capacitance of the transistor can be used as Z, or better, ebedded into Z. On the other hand, biasin can be ore easily arraned with an inductive Z. a b Fiure.7 Equivalent circuits of the series-ode neative resistance: a Z and Z capacitive; b Z and Z inductive Practical transistors have non-zero output conductances and other parasitic lossy eleents that add positive resistance to the circuit. If inductors are used as feedback ipedances, their liited quality factors have a siilar effect. Toether with the losses of the biasin coponents for the transistor, these factors cause deviation fro the ideal case, and ore neative resistance will be needed to copensate the. Fiure.8. shows soe neative resistance values easured fro a discrete BFR9A bipolar transistor in a capacitive coon-collector connection Fiure.6a. -5 R / Ω I c A I c 6 A I c A -5.M 5.M.M 5.M freq / Hz Fiure.8 Measured neative resistance values fro a coon-collector BFR9A, pf, pf Tunin By definition, the value of neative resistance can be adusted by chanin the transistor bias or by tunin either of the capacitors or, e.. by usin varactors. If is oitted and s is used as Z, tunin by becoes ineffective since and s have a link throuh the constant transit frequency f T of the transistor: 8

45 f T R f T πs s πf.4 In this case, varactors reain the only option for tunin [.5] [.7]. However, chanin the varactor capacitance directly affects the equivalent series capacitance value and thus the center frequency of the resonator. This, toether with the fact that hih-quality varactors are difficult to realize in interated circuits, akes this approach soewhat unattractive. In the presence of as in Fiure.8, transconductance of the active device can be used for resistance tunin but aain due to the - s link, the reactive part is chaned to a lesser extent. The neative resistance value can also be tuned by addin variable positive resistance e.. in parallel with Z [.9][.]. While bein a convenient ethod of tunin, the additional resistor inects excess noise into the circuit. In interated circuits, a tunable resistor can be realized as a transistor operatin in the triode reion, but it ay be difficult to achieve adequate resistance tunin rane... Parallel-Mode Neative Resistances The other two sinle-transistor neative resistors are inherently parallel-ode circuits, i.e. their sall-sinal equivalent circuits are parallel connections. Fiure.9 Parallel-ode neative resistor, types I and II a b c d Fiure. Transistor confiurations for rounded neative resistors, type I: a oon-source; b oon-source, swapped; c oon-drain; d oon-drain, swapped [.][.] 9

46 a b c d Fiure. Transistor confiurations for rounded neative resistors, type II: a oon-source; b oon-source, swapped; c oon-ate; d oon-ate, swapped The input adittances becoe Y in Z [ Z Z ] Z.5 Now the real part of the input adittance is neative only if Z is capacitive and Z inductive, or vice versa. The equivalent passive circuits are shown in Fiure.. a b Fiure. Equivalent circuits of the parallel-ode neative resistors a Z inductive, Z capacitive, hih-pass ;b Z capacitive, Z inductive, low-pass The value of effective resistance is a cobination of two ters one of which is always positive. The su becoes neative only when the operatin frequency is hiher hih-pass or lower low-pass than the self-resonance frequency of the inductor-capacitor pair, i.e. /L ½, dependin on the utual position of the reactive eleents. Aain, it is ore practical to have the capacitor in parallel with the ate-source capacitance of the transistor e.. Fiure.b, where it can even be oitted. In order to set the self-resonance frequency of L and sufficiently uch lower than the operatin frequency, a relatively lare inductor is required, especially if is replaced with the sall s. This is a serious practical drawback of the hihpass confiuration, when lookin at onolithic realizations. A lare spiral inductor has a low self-resonance frequency, which sets the upper frequency liit of the neative resistor function. Dependin on its self-resonance frequency, the neative resistor looks either inductive or capacitive at the operatin frequency. In a coplete resonator the reactive eleent external to the neative resistor ust be chosen copliantly: the external resonator coponent ust always be of the sae type as Z. The topoloy in Fiure.b and Fiure.b were experiented with a BFR9A, and the easureent results are shown in Fiure. and Fiure.4. Tunin Like the series-ode circuits, a parallel-ode neative resistor can be tuned by chanin the value of either or. Since the positive part is dependent only on the transconductance of the transistor and therefore variable, it is ore functional to chane the transistor bias for tunin

47 than in the series-ode circuits. This has also the benefit of leavin the self-resonance frequency untouched. I c A I c A - I c 5 A -4 R / Ω M 5.M.M 5.M freq / Hz Fiure. Measured neative resistance values fro a coon-eitter BFR9A, pf, L nh, inductor self-resonance at 8 MHz I c A I c A - I c 5 A - R / Ω M.M 4.M 6.M 8.M freq / Hz Fiure.4 Measured neative resistance values fro a coon-base BFR9A, pf, L nh, inductor self-resonance at 8 MHz.4 Twin-Transistor Neative Resistances The fundaental difference between sinle- and twin-transistor neative resistances is the orientation of the transconductance eleent they involve. With two transistors it is possible to for a positive transconductance and, with suitable positive feedback, a neative resistor. As illustrated in Fiure.5, a twin-transistor neative resistor is essentially a variation of the parallel-ode circuit Fiure.9 the only difference bein the inversion of. Accordinly, the input adittance is Y in Z [ Z Z ] Z.6

48 Now the real part is neative if Z and Z are of the sae type: capacitive, inductive or, althouh unfavorable in ters of noise, even purely resistive. The aor benefit of positive transconductor neative resistances is the independence of the resistance value on frequency. Therefore, they have autoatically wide operational bandwidths. Fiure.5 Twin-transistor neative resistor, types I and II a b c Fiure.6 Transistor confiurations for rounded neative resistors a type I, inductive; b type I capacitive [.4][.5]; c type I differential Fiure.7 Equivalent circuits of the twin-transistor neative resistors; capacitive and inductive The D-G transistor cobination in Fiure.6a,b functions as a positive transconductor with a copound of /. If Z is shorted Z and Z is reoved Z, the circuit reduces to a sinle neative resistor / a positive transconductor with unity

49 feedback, and the well-known cross-coupled differential pair shown in Fiure.6c is fored [.4] [.]. Tunin Tunin of twin-transistors neative resistors is easily arraned by adustin the current throuh the devices. When the transconductances of the transistors are chaned, the input capacitances are also affected. This results in a sliht frequency dependence of the circuit..5 Noise in Neative Resistors.5. Relative Noise Resistance and onductance In ters of noise, active neative resistors can be rearded as passive resistors whose noise voltae and current spectral densities are vˆ iˆ n n 4kTr n 4kT n R G.7 where R and G are the neative resistance and conductance values. The coefficients r n and n are so-called relative noise resistance and conductance, respectively. They ive a fiure of how uch ore or less the active neative resistor produces noise than a passive positive resistor of the sae size. This concept provides a convenient way of coparin noise properties of different neative resistor topoloies. In series-ode circuits, i.e. where the resistor is in series with the rest of the resonator, lare series resistances produce hih noise voltae levels across the unloaded input nodes. On the other hand, in parallel-ode circuits lare parallel conductances produce hih short-circuit noise currents and consequently hih noise voltae levels across the parallel circuit. Therefore, it is practical to use relative noise resistances and noise voltae analyses for describin seriesode neative resistances, whereas relative noise conductances and noise currents should be used for parallel-ode neative resistors. In neative resistors, the noise source usually appears to be associated with a reactive eleent, atheatically rearded as a noisy capacitance or inductance. For this reason, r n and n are frequency-dependent in contrast to passive resistors..5. Transistor Noise Model Essentially three noise echaniss can be observed in seiconductor coponents [.]. In each seiconductor ohic resistivity ives rise to theral noise eneration. When current flows throuh a seiconductor unction, shot noise always occurs due to the fluctuation in the nuber of chare carriers in any tie interval. Inversely proportional to frequency, flicker or /f noise, caused by slow tie-constant electron trappin, has a doinant effect at low frequencies. The collector shot noise can be rearded as the ain noise source in bipolar transistors in hand calculations. It is forulated as a current source between the collector and the eitter: iˆ c qi c.8 Althouh oitted here, other noise sources in a bipolar transistor would be the theral noise in collector, eitter and base resistances, and the base shot noise. When operatin at hih frequencies the /f noise can be nelected as lon as circuits are considered linear.

50 Reeberin that I c VT kt q.9 The collector shot noise can be written in a for that resebles theral noise: iˆ c kt. In field-effect transistors unction FETs, MOSFETs, MESFETs and HEMTs the doinatin noise source is the theral noise in the channel which is represented by a noise current source: 8 iˆ d kt. Now, it is easy to see that in a siplified case noise in all transistor types can be accounted for with a sinle current source in parallel with the transconductance. Its value becoes i 4kTγ ˆ n. Fiure.8 Siplified transistor noise odel 7KH IDFWRU LV FDOOHG WKH QRLVH JDPPD FRHIILFLHQW DQG LWV YDOXH LV IRU ELSRODU WUDQVLVWRUV DQG / for FETs. However, in short-channel MOSFET devices it can be uch larer. Althouh uch siplified, Equation. ives a convenient way of describin noise of a sinle transistor transconductor, and it will be used hereafter in calculations..5. Noise in Twin-Transistor Transconductors A twin-transistor copound connection is needed for realizin the positive transconductor in inverted- parallel-ode neative resistors. Both transistors induce utually uncorrelated noise, but if they are identical the output noise corresponds to that of a sinle half-sized transistor only. The individual transistors in a copound transconductor are sized as in order to et a cobined transconductance of, and therefore the noise properties are identical to sinle-transistor transconductors. 4

51 Fiure.9 Noise in twin-transistor, positive- transconductors, and the sinle-transistor equivalent.5.4 Noise in Series-Mode Neative Resistors The equivalent circuit of the series-ode neative resistor with the noise source is shown in Fiure.. Fiure. Equivalent circuit of a series-ode neative resistor with the transistor noise source The relative noise resistance r n can easily be defined for the circuit: r n Z γ Z. Obviously, noise can be iniized by iniizin the reactance Z and axiizin Z, but keepin their product constant for unchaned neative resistance. A very iportant reark can be ade fro Eq..: an active neative resistor can be with a proper selection of the reactance ratio less noisy than its passive counterpart r n <! This is one of the fundaental reasons why neative resistance resonators are superior to other active resonator types in ters of noise. This also iplies that noise characteristics, i.e. losses, of the actual passive L resonator coponents ay have reater effect on the overall noise perforance than the losscopensatin active circuit. When a neative resistor is used as a part of a resonance circuit, its reactive coponents for conveniently either the capacitor or the inductor of the resonator. However, in reality these coponents are not ideal. Their losses directly contribute to the total noise of the resonator, as stated previously, and slihtly chane the transistor noise transfer characteristics. Those reactive coponents that have the lowest Q values should be chosen for the feedback coponents of the neative resistor. As the resonator capacitors are often realized as low-q varactors, and interated spiral inductors also have Q values of the sae anitude, the difference is not necessarily lare, thouh..5.5 Noise in Parallel-Mode Neative Resistors The noise equivalent circuits of both parallel-ode neative resistors are shown in Fiure.. 5

52 Fiure. Equivalent circuits of a parallel-ode neative resistors with the transistor noise source In parallel circuits, the relative noise conductance is used as a easure of noise perforance. In the case of the type I, the noise source corresponds directly to the output noise current, and we et n w Z γ γ, Z w sr.4 for a hih-pass neative resistor where Z is inductive and Z FDSDFLWLYH DQG & sr is their utual resonatin frequency. The relative noise conductance rises with increasin frequency and the best results are obtained near the self-resonance of the L pair. The type II of parallel-ode circuits is ore suitable for low-noise applications, as its relative noise conductance becoes n γ Z Z w γ.5 Noise diinishes rapidly with increasin frequency, and potentially very low-noise parallelode neative resistors can be synthesized in this way. To further enhance noise perforance in parallel-ode circuits, it is iportant to iniize the nuber of noise-inducin coponents in the whole desin. As one always needs three reactive coponents for a coplete resonator, of which at least one is a noisy inductor and one is a noisy external tunin varactor, the third coponent should be noiseless: a passive capacitor. This iplies that the neative resistor itself should look inductive at the operatin frequency, i.e. the self-resonance frequency of L and should be lower than the operatin frequency hih-pass types. In this case, the neative resistor would not induce any additional noise except for the transistor noise..5.6 Noise in Inverted- Parallel-Mode Neative Resistors Noise inducin echaniss are the sae in sinle- and twin-transistor parallel-ode neative resistances, but since no resonance is present the relative noise conductances are frequency-independent: I : II : γ n γ n or or γ L n L γ n L L.6 6

53 Accordin to Section.5., when a twin-transistor transconductor fors the non-invertin transconductor, the noise perforance reains unchaned..5.7 Measurin Relative Noise Resistances and onductances The equivalent noise resistance can be defined when the neative resistor is realized as a two-port and its noise fiure F and y-paraeters have been easured, usin the followin relation R is the characteristic resistance of the easurin syste: r n R F R.7 Results fro a BFR9A with two different capacitance ratios are plotted in Fiure.. With the larer / ratio the circuit is indeed less noisy than its passive counterpart at ost frequencies, even thouh the behavior of the real circuit is ore coplicated and the device noise larer than in theoretical estiations. The theoretical ratio of the relative noise resistance curves is 5 and the easured ratio is about. Hence, the relative accuracy of the theoretical odel is fairly ood. r n / 5 /. M 5M M 5M M freq / Hz Fiure. Measured relative noise resistance fro a coon-collector BFR9A, I c 5 A Fro the easured paraeters the relative noise conductance of a parallel-ode neative resistor can be de-ebedded as n G G F y y y G s.8 The relative noise conductances of the easured parallel-ode circuits are plotted in Fiure.. The results confir the estiated behavior with the exception of the risin slope of the coon-base type II circuit fro 65 MHz on. This is due to the self-resonance of the nonideal inductor at 5 MHz, which brins up its own peakin noise. At low frequencies the type I has lower noise as predicted by the theoretical odel. The crossin frequency should be ½ f sr, but in the easured circuit f sr ] LW LV PXFK KLJKHU HYHUWKHOHVV WKH W\SH,, RI WKH parallel-ode neative resistors is clearly best in ters of noise perforance at hiher frequencies where also the neative resistance value is in a practicable rane. 7

54 parallel-ode I parallel-ode II n.m 5.M.M freq / Hz Fiure. Measured relative noise conductance fro a coon-eitter and coon-base BFR9A, I c 5 A, L 56 nh, 9 pf.6 Distortion in Neative Resistors.6. Introduction The neative resistors discussed in this thesis are active circuits includin one or ore transistors. As transistors are always non-linear in the real world, the neative resistors are also unavoidably non-linear and distort the incoin sinal. This will have effect on the perforance of the whole resonator. In this chapter, distortion in neative resistors is analyzed on a detailed level. A non-linear neative resistor can be described with a conductance function v where the A voltae across the resistor is the stiulus and the A current throuh the resistor is the response: i in v in. The choice of usin voltae as the stiulus can be ustified, if the resonator coponents are to be connected in parallel with the neative resistor, therefore sharin the sae voltae excitation. The contribution of each resonator coponent can then be exained separately as current responses. The type of the non-linear function v depends on the source of non-linearity, i.e. the transistors. Bipolar transistors create exponential resistance functions and field-effect transistors typically polynoial functions. The echanis of how the non-linear behavior of the transistor affects the whole circuit is defined by the surroundin passive circuitry. Therefore, with a proper selection of ipedances within a neative resistor, overall distortion can potentially be iniized. In non-linear resistors, the concept of A resistance is soewhat abiuous. At very low input currents the conductance function is practically linear and the conductance value is iven by the derivative JY in / Y in at zero or siply by the division v in /v in. When the input voltae swin is increased, the current throuh the resistor radually ets copressed because a part of the response current is shifted into hiher haronic frequencies. The fundaental frequency response cannot directly be seen fro the oriinal non-linear function any ore, and we ust perfor a haronic analysis in order to obtain it. Havin done that we can define the lare-sinal conductance 8

55 Re{ fundaental response current} stiulus voltae v in.9 which is a polynoial function of the input voltae. The lare-sinal input conductance is also called the real part of the describin function of the circuit. Fiure.4 shows an exeplary non-linear fundaental response resistance curve: db copression i in / A -6 / S - sall-sinal value v in / V -5 Fiure.4 Voltae-current and conductance curves of a non-linear neative resistor A practical easure of non-linearity is the --db copression point of the neative conductance. It is the input voltae aplitude akin the fundaental frequency response drop db or.9%. The drop of the neative conductance will show directly in quality factor deradation, when the neative resistor is a part of a parallel resonator. It also provides a ood ethod of coparin different neative resistor topoloies in ters of distortion perforance..6. Volterra Technique The Volterra series technique is coonly used for hand calculations of haronic responses. Due to its rapidly increasin coplexity in lare circuits with several non-linearities, the Volterra technique is practical only for very siplified cases. Therefore, the result is always a rouh approxiation of the real situation, and for ore accurate analysis nuerical siulation ethods, such as the haronic balance technique, ust be used. Nevertheless, the Volterra ethod can ive an insiht into how the diensionin of the electronic circuit should be done for optial perforance. v in / V v cop Fiure.5 Siple non-linear odel of a transistor The prerequisite for the Volterra analysis is that the circuit under exaination is weakly non-linear around the quiescent point. The circuit is weakly non-linear, or quasi-linear, if its non-linear behavior can be accurately described by the first three ters of its Volterra series 9

56 Order Frequency of Response response A H H A A A H, H, A A A H, A H, A H, A H, A A H,, 4 A A H,, 4 A A H,, 4 4 A A H,, A A H,, A A H,, A H,, 4 A H,, 4 A H,, 4 A H,, 4 Type of response linear second-order interodulation second haronics D shift third-order interodulation third-order desensitization third-order copression or expansion third haronics Table. Different responses at the output of a non-linear syste described by Volterra kernels of order, and, excited by two sinusoids A FRV& t and A FRV& t [.] [.]. This is also the point beyond which the calculations becoe overly coplicated. Thus, we ust assue that the non-linearities in the circuit, i.e. in the transistors, can be expressed accurately enouh with a three-ter power series. In a eneric field-effect transistor, the two doinant non-linear coponents are the input capacitance and the transconductance Fiure.5. The strenth of the Volterra series lies in its capability to handle both static and dynaic non-linearities. The ordinary Taylor series can be used only for describin static eoryless non-linearities resistances or transconductances, but for non-linear reactive coponents capacitors or inductors it is useless. However, the non-linear transistor input capacitor brins excessive coplexity to the expressions, and therefore only the static transconductor will represent the transistor non-linearity hereafter. Even after this siplification, the effect of auxiliary circuitry will be clearly visible. Strictly speakin, without dynaic non-linearities, the Volterra and Taylor expansions becoe the sae, and the Volterra analysis techniques would not necessarily have to be utilized. It is, however, convenient to use the straihtforward echanical calculation ethods developed for the Volterra analysis even in this case. In the followin calculations the ethod of non-linear current sources will be utilized [.][.]. It is a fairly siple but elaborate way of deterinin the haronic coponents of a non-linear circuit. The results of the calculations are the Volterra kernels of order one, two and three: H M& M& M&, H M& M& M&. Fro the Volterra kernels, the correspondin haronic coponents can be defined as shown in Table...6. Non-Linear Behavior of the Series-Mode Neative Resistor First we shall look into the series-ode neative resistor described in Section.. and calculate its Volterra kernels. The non-linear odel for the circuit is shown in Fiure.6. 4

57 4 Fiure.6 Siplified non-linear odel for the series-ode neative resistor First-order kernels The first-order Volterra kernels are nothin but the transfer functions of the linearized circuit when the excitation voltae equals one. For the circuit in Fiure.6 we can write the Kirchoff current equation for the node :,, Y H Y H Solvin Eq.. ives the first-order Volterra kernel for node :, Y Y Y H orrespondinly, the first-order input current kernel becoes iin Z Z Z Z Y Y Y Y H Y H,, Second-order kernels For calculatin the second-order Volterra kernels, a non-linear current source ust be placed in parallel with each non-linearity. The expressions of the second-order non-linear current sources are tabulated in Table., and the neative resistance circuit with the additional non-linear current sources is illustrated in Fiure.7. For hiher-order kernel calculations, the input voltae excitation is zeroed. The second-order Volterra kernels can now be solved in the siilar way than the first-order kernels:,,,,,,, Y Y Y i H Y Y i H i Y H Y H NL iin NL NL Fro Table. we et the expression for the non-linear current source:....

58 4,, Y Y Y Y Y Y H H i NL It is easier to use the actual adittance expressions for Y and Y fro now on. In the followin calculations, they are treated as capacitors and. The dual expressions can be used for inductors, respectively. Now, the second-order Volterra kernels becoe,,,, H H iin Fiure.7 Second-order non-linear current source in the series-ode neative resistor Third-order kernels The third-order Volterra kernels can be calculated in a siilar anner as the second-order ones, and the expressions for the correspondin non-linear current sources can be found in Table.., Y Y Y i H NL iin Type of basic nonlinearity Expression for second-order non-linear current source transconductance,, H H k k capacitance,, H H c k k Table. Non-linear second-order current sources for the different basic non-linearities [.] Type of basic nonlinearity Expression for third-order non-linear current source transconductance,,,,,, H H H H H k k k k k capacitance [ ],,,,,, H H c H H H c k k k k k,,,,,,,,,,,, H H H H H H H H k k k k k k k k Table. Non-linear third-order current sources for the different basic non-linearities [.].4.5.6

59 4 To facilitate the calculations the non-linear current source expressions can be derived separately for each haronic coponent of interest. By usin Table. and Table. as references, they are tabulated in Table.4. Thus, for the series-ode neative resistor we et OMP NL HD NL i i,, and further OMP iin HD iin H H 4, 4, The third-order interodulation products can be defined in the sae way. When neative resistors are used in narrow-band filters, the two excitation frequencies ust be very near each other in order to ensure that the IM products are at the pass-band as well. It is therefore reasonable to approxiate that the third-order IM response anitude is identical with the FRPSUHVVLRQ UHVSRQVH & & if the two excitation voltaes have the sae anitude. Distortion coponents After havin calculated the appropriate Volterra kernels we can finally identify each haronic coponent in question. With the aid of Table., they can be written as shown in Table.5. The coplexity of the expressions arises fro the fact that series-ode neative resistances should be excited by current, not voltae, and connected in series with the actual series resonator. The practical filters discussed here eploy, however, parallel resonators with voltae excitation, and hence also series-ode neative resistances ust be treated as parallel neative conductances. It is also very difficult to realize sufficiently sall neative resistance values for series copensation in practice. The theory shows that by diensionin the circuit paraeters suitably, the third-order copression kernel will cross zero and becoe neative. Instead of copression, expansion will take place in that case, and probles ay occur in the for of instability at hih sinal levels. By ensurin that << this proble is circuvented. The results in Table.5 show that the shunt capacitance at the transistor output has the reatest ipact on distortion, and so it should be iniized. However, this is in contradiction with the requireent for lowest noise, if the value of is respectively increased for.7.8 Type of response Expression for third-order non-linear current source third haronic,,,, H H H third-order interodulation _& & [ ],,,,,,,, H H H H H H copression or expansion [ ],,,,,,,, H H H H H H Table.4 Third-order non-linear current sources for the different types of responses

60 44 aintainin the desired neative resistance value [Section.5.4]. A trade-off between lowest noise and lowest distortion exists. The --db conductance copression for the series-ode neative resistor takes place, as explained in Section.6., when { } { }.9% fundaental Re copression Re in in i i Usin the expressions fro Table.5, we can write cop in K K v Re.9% 4, The theoretical --db conductance copression voltae is then,.45 K v cop in Measured results The theoretical estiations were verified by easurin the s-paraeters of a BFR9A-based series-ode neative resistor as a function of the input power. The circuit is the sae as in Fiure. ZLWK WKH VDPH ELDV SRLQW 7R FRQYHUW WKH UHIHUHQFHG RXWSXW SRZHU IURP WKH analyzer into the peak voltae across the circuit the followin transforation was used: 4 in in in in Z Z P v As seen fro Fiure.8 the circuit with a hih and a low has a clearly lower copression point than the other, which is in ood areeent with the theory in qualitative ters. Due to the liited power sweepin capability of the s-paraeter analyzer, the --db or.9% resistance copression points could not be reached in easureents. The %- copression points are 4 V and 9 V, respectively. Type of response Response i in, capacitive Z Z fundaental v in second haronic v in third haronic 4 4 v in copression < in v 4 4 Table.5 Haronic distortion coponents for the series-ode neative resistor with capacitors.9...

61 R / R s a ll-sina l..95 / 5 /. v in / V Fiure.8 Measured resistance copression fro a coon-collector BFR9A, I c 5 A, f 5 MHz.6.4 Non-Linear Behavior of the Parallel-Mode Neative Resistors By usin the sae calculation procedure as in the previous chapter, we can define the Volterra kernels and consequently the haronic coponents of the parallel-ode neative resistors [Section..]. The expressions are uch sipler as voltae excitation is well suited for parallel-ode circuits. The results are suarized in Table.6 for the types I and II. Only the hih-pass confiurations, i.e. inductive Z and capacitive Z, are discussed; the dual lowpass expressions can be obtained fro these easily. It sees that the type I suffers fro reater distortion than the type II at frequencies near the VHOIUHVRQDQFH IUHTXHQF\ & sr, but vice versa if the operatin frequency is sufficiently lare. For the type I, there is a contradiction between low-noise and low-distortion requireents, but interestinly this does not apply to the type II. [Section.5.5]. The --db conductance copression points can be defined in the sae anner as previously: v v in, cop in, cop w, w sr Type I Type II. At frequencies hiher than twice the self-resonance frequency, the type I has better copression perforance. Measured results The easured conductance copression curves of both parallel-ode neative resistance types are shown in Fiure.9. The circuits are aain the sae as in the noise easureents. 45

62 Type of response Response, type I inductive Z capacitive Z fundaental vin w second haronic v in w v third haronic in 4 w Response, type II inductive Z capacitive Z vin w v in 4w v in 4 9w v copression in v in 4 w 4 w Table.6 Haronic distortion coponents for the parallel-ode neative resistors. G / G sall-sinal.95.9 parallel-ode I parallel-ode II.85 v in / V Fiure.9 Measured conductance copression fro a coon-base and coon-eitter BFR9A, I c 5 A, f 5 MHz, f sr 5 MHz The self-resonance frequency of both easured circuits was about 5 MHz, and as the easurin frequency was three ties hiher 5 MHz, the type I ouht to have approxiately eiht ties hiher a copression point than the type II by the theory. By exainin Fiure.9, an excellent atch between the theory and the practice can be observed % copression voltaes 5 V and 7.6 V, iven the fact that even the slihtest variation in the resonance frequency chanes the distance between the copression points considerably..6.5 Non-Linear Behavior of the Positive- Parallel-Mode Neative Resistors The haronic coponents of the parallel-ode neative resistors with non-invertin transconductors follow basically the sae forat as those of noral parallel-ode neative resistors. They are listed in Table.7. The responses with inductive ipedances Z and Z can be easily obtained by a substitution &&, : /&/, in the expressions. Distortion perforance cannot be enhanced without alterin the fundaental neative conductance value, unless is chaned siultaneously. It is obvious that if the voltae division ratio in the transistor input is lowered / increased, the distortion perforance of the whole circuit ets better at the expense of a larer need of transconductance. This requires ore bias current throuh the device, which is often undesirable. 46

63 Type of response Response, type I capacitive Z and Z fundaental v second haronic third haronic copression in v in v in 4 4 v in v Response, type II capacitive Z and Z in v in v in 4 v in 4 Table.7 Haronic distortion coponents for the positive- parallel-ode neative resistors The -db copression points for the positive- neative resistors becoe v v in, cop in, cop Type I Type II Effect of the Differential Transconductor The non-invertin transconductor required by positive- neative resistors cannot be realized with a sinle transistor, and a twin-transistor D-G copound transconductor is needed. The sae transistor confiuration is also the basis of the differential circuits. It is coonly known that a differential transconductor suppresses even-order distortion coponents due to its anti-syetrical nature. This also applies to the sinle-ended copound circuit in Fiure., providin that the two coponent transconductors are of the sae size. Fiure. Siple non-linear odel for the copound transconductor The circuit in Fiure. corresponds to a sinle non-invertin transconductor of the value i out vin vin cvin 4 c v in.5 47

64 The cobined transconductance value has been halved, and twice the bias current of an invertin sinle-transistor transconductor will be needed. The second-order coponent has disappeared, but the relative value of the third-order coponent has rown <. The absence of the second-order coefficient c has an effect on the distortion perforance in the series-ode circuits where c contributes directly to the third-order haronic distortion and copression responses. On the other hand, in the parallel-ode neative resistors the second-order coefficient has no effect on the third-order responses, and the deraded c even worsens the situation. Therefore, twin-transistor parallel-ode neative resistors do not benefit fro the copound transconductances..7 Dynaic Rane of Neative Resistance Resonators.7. Definition The definition of dynaic rane is futile if the frequency band is not liited. Therefore, it is necessary to exaine the whole neative resistance resonator with a well-specified liited frequency band. When a neative resistor is connected in parallel with a lossy L resonator, its sall-sinal value ouht to copensate the unloaded resonator losses copletely at resonance. The current throuh the neative resistor is the su of the fundaental response current and the copression current, as described earlier. Now, the fundaental response is totally cancelled by the passive resonator dissipation resistance, and the only first-order ter left is the current throuh the load resistor G l. We can write the copression condition: v in, cop G Re l v { i copression } Re{ i copression } in in, cop G l 89.% Q l in v in, cop.9%.6 This equation ives the input voltae at which the total resonator input current, or the resonator input power, is copressed db fro its sall-sinal value. We can cobine Equations.9 and.6, and as the fundaental response in Eq..9 PXVW QRZ EH HTXDO WR & /Q, we et Re { i cop v } Re{ i cop v } in, neres in, cop, neres in, neres in, cop, reson Q vin, cop, neres vin, cop, reson Since the third-order copression ters are proportional to v, the expression becoes Q l.7 Q v in, cop, reson in, cop, neres Ql v.8 This iportant equation shows the relation between the --db copression voltaes of the neative resistor and the correspondin copensated L resonator at resonance. The sae result has been obtained in [.] and [.4] after soewhat inaccurate analyses, thouh. The dynaic rane of an active resonator can now be defined as the ratio: Q DR Q l v in, cop, neres rs noise voltae.9 48

65 .7. Neative Resistance Resonator Noise The total noise of a neative-resistance copensated resonator consists of the contributions of the passive resonator noise and the neative resistance noise. At a narrow band, the parallel conductance of the neative resistor is equal to that of the lossy resonator, so that the unloaded net conductance is zero. The inected noise current becoes ˆ in ˆ i n ˆ in, ˆ i res n, res ˆ in, ˆ i ne n, ne 4kTG 4kTG p p r n n series ode parallel ode.4 provided that the loaded Q of the resonator is hih enouh. Thus, near resonance the noise voltae spectral density of the neative resistance resonator is vˆ n G l iˆ n L 4kTG p r, n n L Gl L L.4 At resonance this yields vˆ n 4kT G p r, 4, n n ktr rn n Ql G l.4 onsequently, the total rs noise voltae of the neative resistance resonator becoes v n π vˆ d kt Q l n Q rn, n.4 It is assued that the relative noise resistances and conductances are constant over the narrow resonator bandwidth, when Q l is hih. The noise perforance of the different neative resistor types can be easily copared by lookin into their relative noise resistances or conductances. Nevertheless, the reatest effect has the ratio of the loaded Q of the resonator and the unloaded Q of the inductor/capacitor. With low-q coponents, noise is excessive in narrow-band filters reardless of the neative resistor type. Fiure. Neative resistance resonator noise.7. Dynaic Rane of Neative Resistor Resonators By cobinin the results fro Equations.9 and.4, we can define the dynaic rane of any neative resistance resonator: 49

66 vin, DR lo v cop n Q lo Ql kt v in, cop rn, n.44 The ratio Q /Q l is crucial when the dynaic rane is to be axiized. For instance, alost a -db enhanceent in the dynaic rane is achieved if the passive inductor quality factor is increased fro 5 to 5. For a series-ode neative resistor we et by Eq.. v DR lo in, cop vn.45 Q lo K Q l kt γ.45 Respectively, the dynaic ranes of the parallel-ode neative resistors can be derived fro the appropriate equations. For the types I and II, they becoe: Q DR lo Ql Q DR lo Ql And finally for the positive- circuits.45 kt w γ w.45 kt γ w Type I Type II.46 Q DR lo Q l Q DR lo Q l.45 kt kt γ γ.45 Type I Type II.47 The resonatin capacitance includes any eventual stray capacitance coin fro the neative resistance, and therefore there is a connection between the internal capacitance of the neative resistor and the total resonatin capacitance. Preferably, the contribution fro the neative resistor capacitances is kept sall, as they tend to vary alon with resistance tunin..7.4 oparison To et a hands-on view of the theoretical dynaic ranes of each neative resistor type it is necessary to copare the topoloies with realistic nuerical values of coponents. The exeplary paraeters chosen for the coparison are: & 4 GHz, Q l resonatin L nh, Q 5 S, A/V, $9 7UL4XLQW 67 [P c S, c, c -. A/V for copound transconductors with double-size transistors For siplification, the neative resistor coponents are chosen in such a way that the resonance condition is fulfilled, i.e. no external capacitance is connected. This can be rearded as the 5

67 worst case scenario but ives a ood basis for coparison. The nuerical results are athered in Table.8: Type of neative resistor rs noise voltae copression voltae dynaic rane / µv / dbv / db series-ode > < parallel-ode type I type II twin-transistor type I parallel-ode type II Table.8 oputational noise voltaes, copression points and dynaic ranes for the different neative resistor topoloies There is practically little difference between the dynaic ranes of the neative resistor topoloies considered here, althouh the parallel-ode circuits see to be the best. Due to the differential pair, the twin-transistor circuits suffer fro slihtly hiher third-order distortion coponents than the others. It ust be noted, however, that if invertin-transconductor circuits were to be realized in quasi-differential for, their distortion perforance would be inferior to that of the circuits with non-invertin transconductors. In actual circuits, it is ipossible to achieve the fiures shown in Table.8, as the distortion echaniss are uch ore coplex in reality. For instance, the non-linear input capacitances of the transistors have non-neliible effect on perforance, but it is reasonable to assue that all the circuits suffer fro the in a siilar way. The distortion definitions here only ive soe uidelines for diensionin neative resistors, and the absolute nuerical fiures are only valid for theoretical coparisons between different neative resistance structures. In the calculations it is assued that the resonatin capacitance is noiseless and linear. This is not true in realistic resonators, which has a considerable effect on noise and distortion, and thus dynaic rane in the resonator. The properties of the neative resistors theselves are not affected, but in a direct coparison with yrator-based resonators, the non-ideal resonatin capacitor, often a varactor, ust be taken into account..8 Practical Neative Resistor Resonators Practical aspects of neative resistor resonator desin are discussed in this section. Realized circuit exaples are iven on the basis of published articles. Measured data is presented when available..8. Feasibility for Interation The previous sections deal with the underlyin theoretical backround of different neative resistor types. However, any of the seeinly ood topoloies suffer fro considerable perforance deterioration when realized as interated circuits. The ain causes are process tolerances, especially in passive coponents, biasin arraneents and supply voltae sensitivities. Spiral inductors are the lowest-quality passive coponents in interated circuits, and therefore they should be avoided whenever possible. In neative resistors that eploy inductors, i.e. series-ode circuits with inductors and sinle-transistor parallel-ode circuits, positive loss resistance is enerated by the loss resistances of the inductors. This can be copensated by increasin the neative resistance value at the expense of power consuption, which is naturally undesirable. The inductive series-ode resistors also include two spiral inductors increasin the required die area. 5

68 All the neative resistor topoloies, except the twin-transistor type with unity feedback, rely on passive auxiliary coponents in synthesizin neative resistance. Thus, the values of these coponents directly affect the neative resistance value. As absolute variations of interated passive coponents are lare, the uncertainty of the resultin neative resistance is hih. Topoloies where the ratio of passive coponent values rather than their absolute values is deterinin are far better suited for interation, as the relative coponent variations are sall. The parallel-ode inverted- neative resistors eet this requireent, and for their robustness, they are best suited for realization as interated circuits. Naturally, all neative resistor types can be tuned and adapted to variations, but if their tunin ranes are narrow, this ay not suffice. Biasin of the active devices introduces ore non-idealities to the practical circuits. D bias voltaes and currents ust be provided for the transistor inputs and outputs. In any cases soe of these can be brouht into the circuit via D-coupled resonator inductors, and since the inductors are inseparable parts of the total resonator circuits, no perforance reduction follows. For instance, the outputs of the differential neative resistor transistors [Section.4] et their D bias voltaes and currents throuh the actual resonatin inductors connected to the positive supply. Topoloies where an inductor is connected between the base and the collector of the transistor Fiure.b,d and Fiure.b,c do not need further base bias in bipolar realizations, if zero base-collector voltae is peritted. Followin the sae pattern, an inductor between the ate and the source of the transistor enables easy biasin to I dss in FET circuits Fiure.6b and Fiure.a,c. The freedo of choosin bias points is lost in these cases, thouh, and the transconductances of the FETs becoe fixed akin tunin difficult. For totally independent biasin, the inductors ust be A-coupled via capacitors, and the bias voltaes and current ust be provided via as lare resistors as possible. The bias resistors have also a neative effect on noise perforance. The resistance tunin of neative resistance circuits is usually best achieved by adustin the device transconductances. This necessitates the utilization of a controllable current source that feeds the core transistors. The current source is realized as an additional transistor with nonzero output conductance. The current source transistor loads the neative resistor, increasin the aount of neative resistance needed for copensation. In the best topoloies, the current source connects to a low-ipedance point where its non-idealities are least visible, like in differential parallel-ode neative resistors. Supply voltae sensitivity is an issue in interated circuits where its stability cannot always be ensured. This is a proble in sinle-transistor topoloies, as their perforance is very uch affected by the transistor parasitics which in turn are supply-dependent. The extent of supply sensitivity is so reat in these circuits that the operatin voltaes can be rearded as tunin voltaes. The twin-transistor positive- neative resistors do not suffer fro this phenoenon. onsiderin all the practical probles discussed here, it becoes clear that the positive- neative resistors, such as the differential cross-coupled pairs, are ost feasible in ters of interation, althouh sinle-transistor topoloies iht show better theoretical perforance in soe cases..8. Realized Sinle-Ended MESFET Neative Resistor Resonator A neative resistance can be realized with a positive transconductance and unity feedback usually found in VO circuits Fiure.. If the identical transistors are odeled with the transconductance, the output conductance ds and the ate-source capacitance s, the sinleended input adittance becoes ds s Yin.48 5

69 correspondin to a neative resistor in parallel with a capacitor. When connected in parallel with a lossy L resonator, the iniu Q value of the inductor that ust can be copensated with a iven is Q L ds.49 where L and are the passive resonator coponent values. In practice, Q-values of less than five can easily be copensated. Since the basic operation relies only on the transconductance, sensitivity to parasitics and to the supply voltae is inherently low, and tunin is trivial by chanin the drain currents. The input nodes are the sae for the sinal and for the drain bias, which akes it possible to use a sinle passive inductor as both the resonatin and biasin eleent. The nodal ipedances are low, and the axiu possible voltae swin can be obtained at the input. Fiure. Differential neative resistance An interated sinle-ended resonator with active neative resistance loss copensation was desined for the GE-Marconi F GaAs-MESFET process [.4]. The tareted center frequency was.4 GHz, which is tunable within /- MHz via a varactor-connected MESFET. The scheatic of the circuit is shown in Fiure., and a photoraph of the layout in Fiure.4. The chip size without the RF and bias pads is.6.6. The sizes of the transistors are µ. For optiu noise perforance, they are biased to approxiately % of I dss. The external bias adusts the current and of the two MESFETs, and thus the Q value of the resonator. In a practical filter, this adustent is essential for optiu response and stability. A copensation capacitance can be added in parallel with the current source MESFET to lower the effective ain at hih frequencies. This brins the absolute conductance iniu within the frequency tunin rane and further desensitizes the circuit aainst variations of the transistor s. The varactor size is 4 8µ ivin a capacitance rane of. -.6 pf. Hewlett-Packard s MDS and Helsinki University of Technoloy s APLA siulation software, and the Parker-Skellern MESFET odel with enhanced capacitance descriptions were used throuhout the desin [.5][.6][.7]. The odel ives an excellent atch between siulations and easureents in the noral operatin reion of a device but unfortunately, it does not odel resistive losses in a varactor-connected MESFET properly. The difficulty of obtainin a eaninful equivalent circuit in the cold FET reion into which the I-V and -V odels fit, and the hih sensitivity of the circuit to the bias dependence of the ate resistance, are the ain probles in the odel extraction. This led to far too hih a varactor Q value in the siulations, resultin in a downward shift of the conductance iniu frequency, and 5

70 naturally a requireent for larer currents to copensate the extra loss. These were not taken into account in the oriinal siulations. onsequently, the frequencies of the conductance iniu and the resonance do not coincide as expected, and the power consuption is soewhat hiher. The effective series resistance of a varactor is a non-linear function of its bias, ivin lower values alon with decreasin V. A very ood areeent between the easured perforance of the resonator and the AD odel was obtained by increasin the series resistance of the varactor with a 5-Ω resistor in the device odel. Vdd n p Out 4x75 k8 5k x x 5p 5k 5k 5p Bias,f x Bias,Q 5k 5p Fiure. Scheatic diara of the desined MESFET neative resistor resonator Fiure.4 Microphotoraph of the resonator circuit The easured input adittance is shown in Fiure.5. The operatin point is Idc 9 A, Vbias,f.8 V, and Vbias,Q -.65 V. The unaccounted varactor loss has shifted the conductance iniu down to.9 GHz. This does not cause any stability probles as lon as no resonance occurs in the reion where Re{Yin} is neative. Both the conductance and susceptance curves cross zero at alost the sae frequency point resultin in a very hih unloaded Q. Fiure.6a shows the resonance frequency as a function of the varactor tunin voltae Vbias,f for different Q-tunin voltaes Vbias,Q. The chane in frequency, when Vbias,Q is varied, is due to the bias-dependent ate-source capacitances in the MESFETs even with a constant frequency tunin voltae. The Q tunin, or the conductance tunin characteristics are shown in Fiure.6b. Hih-Q operation and, on the other hand, stability can be assured with the wide tunin rane around zero conductance. 54

71 ,, 5,µ 5, ReY in /S, G, IY in /S -5,µ -5, -, B -,,6,8,,,4,6,8, f/ghz Fiure.5 Measured Y in of the MESFET neative resistor resonator f res /GHz,6,5,4,,,4,6,8,, V bias,f /V -.85 V -.75 V -.65 V -.55 V G/S,,5, 5,µ, -5,µ -, -, V -.75 V -.65 V -.55 V,6,8,,,4,6,8, f/ghz Fiure.6 a Resonance frequency tunin V bias,f ; b onductance tunin V bias,q The insensitivity to the supply voltae is deonstrated in Fiure.7. The increase of S is only about.5 db, when the operatin voltae is raised fro.5 V to. V, and.5 db fro. V to 4. V. These inor deviations can easily be copensated by adustin the biases accordinly., -, S /db -, -, -,4 -,5 V d 4.V V d.5v V d.v V d.5v,6,8,,,4,6,8, f/ghz Fiure.7 Sensitivity to the supply voltae 55

72 The -db power copression point was easured by feedin a variable-power sinal at the resonance frequency into the circuit and observin the reflected power level. The difference between the two power levels i.e. the ain of the circuit is shown in Fiure.8. The result is 9.5 db correspondin to an input voltae swin of approxiately. V pp. This is not the conductance copression point discussed in Section.6, thouh, as power copression is very uch affected by the varactor. The ain factor liitin the lare sinal perforance is not the active neative resistance itself but the capacitance non-linearity of the diode connected MESFET. This can be iproved by addin another varactor in the back-to-back connection [Section.5.]. This cuts the resonatin capacitance value in half, and in order to aintain the sae resonance frequency the inductor value or preferably the varactor size ust be doubled. The noise perforance of the resonator could not be easured by standard eans, since the circuit was only realized as a one-port. More inforation can be obtained fro the filter-ode easureents discussed in hapter 5., -,5 P out -P in /db -, -,5 db copression: P in 9.5 db -, P in /db Fiure.8 Lare sinal perforance of the MESFET resonator References [.] E. H. Arstron, The Reenerative ircuit, Proceedins Radio lub of Aerica, April 95. [.] E. H. Arstron, Soe Recent Developents of Reenerative ircuits, Proceedins of the IRE, Vol., pp. 44-6, Auust 9. [.] W. O. Swinyard, The Developent of the Art of Radio Receivin fro the Early 9 s to the Present, Proceedins of the IRE, Vol. 5, pp , May 96. [.4]. Buff, Radio Recevers Past and Present, Proceedins of the IRE, Vol. 5, pp , May 96. [.5] A. Presser, Varactor-Tunable, Hih-Q Microwave Filter, RA Review, Vol. 4, pp , Deceber 98. [.6] S. handler, I. Hunter and J. Gardiner, Active Varactor Tunable Bandpass Filter, IEEE Microwave and Guided Wave Letters, Vol., pp. 7-7, March 99. [.7] U. Karacaolu and I. Robertson, MMI Active Bandpass Filters Usin Varactor-Tuned Neative Resistance Eleents, IEEE Transactions on Microwave Theory and Techniques, Vol. 4, pp. 96-9, Deceber 995. [.8] J. Macedo and M. opeland, A.9-GHz Silicon Receiver with Monolithic Iae Filterin, IEEE Journal of Solid-State ircuits, Vol., pp , March

73 [.9] W. Aparin and P. Katzin, Active, Self-Adustin L-S Band MMI Filters, IEEE GaAs I Syposiu Diest, pp. 4-44, 994. [.] W. Aparin and P. Katzin, Active GaAs MMI Band-Pass Filters with Autoatic Frequency Tunin and Insertion Loss ontrol, IEEE Journal of Solid-State ircuits, Vol., pp. 68-7, October 995. [.]. han and T. Itoh, Microwave Active Filters Based on oupled Neative Resistance Method, IEEE Transactions on Microwave Theory and Techniques, Vol. 8, pp , Deceber 99. [.] D. Adas and R. Ho, Active Filters for UHF and Microwave Frequencies, IEEE Transactions on Microwave Theory and Techniques, Vol. 7, pp , Septeber 969. [.] B. Hopf, I. Wolff and M. Guliei, oplanar MMI Active Bandpass Filters Usin Neative Resistance ircuits, IEEE Transactions on Microwave Theory and Techniques, Vol. 4, pp , Deceber 994. [.4] R. Kaunisto, D. Webster and D. Haih, Iproved MMI Active Filters Based on Passive L Resonators with Active Neative Resistance ircuits, Proceedins of IEE olloquiu on Advanced Sinal Processin for Microwave Applications, Stevenae UK, pp. /-/6, Noveber 996. [.5] R. Kaunisto, K. Stadius and V. Porra, Active MMI Filters with Neative resistance opensation, Electronic Letters, Vol. 4, pp. 6-7, June 998. [.6] W. Kuhn, F. Stephenson and A. Elshabini-Riad, A MHz MOS Q-Enhanced L Bandpass Filter, IEEE Journal of Solid-State ircuits, Vol., pp. -, Auust 996. [.7] W. Kuhn, N. Yanduru and A. Wyszynski, Q-Enhanced L Bandpass Filters for Interated Wireless Applications, IEEE Transactions on Microwave Theory and Techniques, Vol. 46, pp , Deceber 998. [.8] R. Duncan, K. Martin and A. Sedra, A Q-Enhanced Active RL Bandpass Filter, IEEE Transactions on ircuits and Systes II, Vol. 44, pp. 4-47, May 997. [.9] S. Pipilos and Y. Tsividis, Desin of Active RL Interated Filters with Application in the GHz Rane, 994 IEEE International Syposiu ircuits Syst. Di., pp [.] S. Pipilos, Y. Tsividis, J. Fenk and Y. Papananos, An Si.8 GHz RL Filter with Tunable enter Frequency and Quality Factor, IEEE Journal of Solid-State ircuits, Vol., pp , October 996. [.] P. Gray and R. Meyer, Analysis and Desin of Analo Interated ircuits Second Edition, John Wiley, New York, 77 p., 984. [.] P. Wabacq and W. Sansen, Distortion Analysis of Analo Interated ircuits, Kluwer Acadeic Publishers, Dordrecht, 5 p., 998. [.] S. Maas, Non-linear Microwave ircuits, Artech House, Norwood, 478 p., 988. [.4] W. Kuhn, F. Stephenson and A. Elshabini-Riad, Dynaic Rane of Hih-Q OTA- and Enhanced-Q L RF Bandpass Filters, Proceedins of IEEE Midwest Syposiu on ircuits and Systes, pp , 994. [.5] A. Parker, Ipleentin SPIE Models with Hih-Order ontinuity and Rate Dependence, IEE Proceedins of ircuit, Devices and Systes, Vol. 4, pp April 994. [.6] A. Parker, J. Scott, Modellin and haracterisation of GaAs Devices, hapter 6 of Low Power HF Microelectronics, Ed. G. Marchado, IEE Books, London, 8 p., 996. [.7] D. Webster, M. Darvishzadeh and D. Haih, Iproved Total hare apacitor Model for Short hannel MESFETs, IEEE Microwave and Guided Wave Letters, Vol. 6, pp. 5-5, October

74

75 4. GYRATOR RESONATORS 4. Historical Perspective 4.. Passive Manetic Gyrators In 948, a new two-port network eleent was proposed by B. D. H. Telleen of Philips Research Laboratory [4.]. Telleen realized that all two-ports containin only resistors, capacitors, inductors and transforers are linear, constant, passive and reciprocal. In order to find a new fundaental eleent, he had to rule out one of these properties. He considered the last one, reciprocity, to be of least iportance in the network theory, and suested that a new non-reciprocal eleent, the yrator, can be rearded as the fifth fundaental circuit eleent. The ideal passive yrator is described by i i v v 4. Fiure 4. Sybol for the ideal yrator as proposed by Telleen The quantity is called yration conductance. Telleen noticed that a capacitance connected to the secondary terinals will look like an inductor at the priary terinals L /, or vice versa L. This is the ost iportant property of the yrator, as it enables the synthesis of inductors. More enerally, any adittance Y connected to the secondary terinals is converted to its dual /Y. This phenoenon is called iittance conversion. The nae yrator oriins fro the yroscopic ters that occur in the state equations of coupled rotatin asses. In fact, a echanical yrator a yroscopic coupler can be deonstrated. Telleen presented the electric equivalents of yroscopic state equations and suested that a construction in Fiure 4. could function as a yrator. Fiure 4. onstruction of a passive yrator The ediu between the electrodes of the priary terinal ust consist of particles carryin both peranent electric and peranent anetic dipoles. By Telleen, these conditions can be et with sall ferroanetic filins in an appropriate fluid. Alternatively, 59

76 edia with yroanetic properties can be used. Manetic yrators have also been realized by usin the Hall effect in a seiconductor [4.] or the Faraday effect in a ferrite [4.]. Yet another ethod of constructin a passive yrator is to exploit the non-reciprocal properties of coupled electric and anetic transducers the piezoelectric-piezoanetic yrator [4.4]. It is clear that these kind of passive yrators have little practical iportance in filter desin, and they should be rearded only as deonstrations of the yrator effect. However, ferroanetic circulators are widely used as yrators in icrowave circuits. 4.. Electronic Gyrators When the two yrator conductances are equal, the yrator is passive and non-dissipative. In eneral, this need not be the case, and the yrator conductances can be unequal but still opposite in sin. Then, dependin on their ratios, the yrator is either active or dissipative. The conductance atrix can be split into two parts: y 4. Now the atrices correspond to ideal voltae-controlled current sources, i.e. transconductors, with opposite sins. This brins up the principle of the electronic yrator: y Fiure 4. Electronic yrator To for an active inductor, a capacitor is connected to one port. The ipedance seen fro the other port is then, accordin to the iittance conversion theore, L 4. where is the port capacitance, the yrator capacitance. One of the first electronic yrators was presented by Boert in 955 [4.5], but the concept really ade advances durin the followin years, when transistors becae readily available. Several electronic yrators based on antiparallel transconductors were presented in the 96 s [4.6][4.7][4.8]. The ain obective at that tie, as well as nowadays, was to realize inductorless filters. As transistors were becoin cheaper, it was predicted that they would replace the costly wound inductors in soe sense or another in the future. Orchard [4.9] suested that the best way of desinin inductorless filters is to replace each inductor in a conventional L ladder filter by a synthesized inductor, i.e. a capacitively terinated yrator. This is because of the inherent insensitivity of L ladder filters, althouh rather any active coponents are needed. 6

77 Fiure 4.4 Typical yrator topoloy fro 965 [4.6] The first partially interated yrators bean to show in the early 97 s alon with the developent of interated operational aplifiers [4.][4.]. However, the rapid evolution of other active continuous-tie and especially discrete-tie filterin techniques, like switchedcapacitor filters, ade yrator filters obsolete very soon in ters of interation. Only after hihfrequency interated circuits and MMIs becae a hot topic in the late 98 s, interated icrowave yrators becae potential circuit eleents for active filterin. 4.. MMI Gyrators The need for iniization of obile radio RF front ends led to rowin interest in hihfrequency interated circuits in the 98 s. Many of the functional blocks of a radio transceiver could be realized onolithically, but filters reained a aor proble. Althouh interated spiral inductors were now available, their quality factors were unacceptably low for filterin. They also occupied a lot of expensive chip area. Traditional active filters could not be used, as hih ain aplifiers or sufficiently fast clock sinals were unrealizable. In this context, S. Hara revived the old idea of yrator-based active inductors in 988 [4.]. Hara s active inductors are shown in Fiure.5. They are nothin but ordinary yrators where the ate-source capacitance of the invertin MESFET acts as the terination capacitor. The invertin stae is cascode-connected for increased bandwidth. In the first version [4.], the non-invertin transconductor is siply realized as a resistor, whereas the second version [4.] eploys a coon-ate MESFET. Later, the probles concernin the Q value of Hara s inductors were elaborated by the author, which lead to studies on phase copensation techniques [4.4] [4.7]. Fiure 4.5 Hara s two types of active inductors [4.][4.] 6

78 4. Non-Ideal Gyrators Ideal electronic yrators do not exist in practice; therefore, it is iperative to consider the echaniss causin iperfections and their effects on the yrator perforance. The y-atrix in Equation 4.4 describes a non-ideal yrator: y y p e φ φ y e p 4.4 The coplex yration adittances e -φ and e -φ describe the phase shift in the transconductors. The non-zero diaonal eleents y p and y p represent the parasitic adittances at the correspondin ports. They occur as a result of the finite output conductances in the transconductors. Iperfections due to an external biasin network will alter the port ipedances and thus affect on the yrator perforance, too. Fiure 4.6 Non-ideal yrator 4.. Non-Ideal Transconductor The siplified non-ideal transconductor used in the theoretical studies is shown below. It consists of an input resistance r i and a capacitance c i, which can be identified as the base ate resistor and the base-eitter ate-source capacitor of a bipolar field-effect transistor in a sinle-transistor invertin transconductor. The output resistor r o corresponds to the output conductance of the transistor, respectively. In addition, the transconductance eleent itself has liited bandwidth and phase shift due to the transit-tie effect. If copared to the previous factors, it is of little iportance, thouh. Other phase-shiftin eleents not shown here, such as output and Miller capacitances, can be ebedded into the yrator or the resonatin capacitor and the input capacitor c i. Fiure 4.7 Non-ideal transconductor 6

79 4.. Effects of Finite Transconductor Bandwidth and Phase La Nelectin the transit-tie effect, the input network of the transconductor can be described also with a coplex transconductor eleent accordin to the followin relation: v i v e,, φ arctan r c in in φ ri c i ri ci i i i 4.5 Fiure 4.8 Non-ideal transconductor with a coplex The pole at input results in a liited transconductor bandwidth and chanes the inductance value at hih frequencies. The phase la -φ i corresponds to neative resistance and leads to instability in the absence of Q-deradin losses. 4.. Effects of Non-ideal Port Ipedances Fiure 4.9 Active inductor with non-zero output conductances The output ipedance of one transconductor and the input ipedance of the followin one connect directly in parallel with the correspondin port ipedance and cause deviations in the inductance value and the quality factor. The parasitic capacitive coponents increase the value of the yrator capacitance and hence the yrator inductance, whereas the resistive coponents decrease the quality factor. The voltae across the yrator capacitor and the input voltae of the other can be expressed as v i io r o r o r, φ o o e φ o o v arctan r in,, 4.6 if the loadin due to the input of the followin transconductor is ebedded into r o and r o. The phase lead φ o corresponds to positive resistance and lowers the Q value. The output resistance of the invertin transconductor r o connects directly in parallel with the rest of the circuit. 6

80 64 4. Non-Ideal Active Inductor obinin the results fro Section 4., we can write the equation for the input adittance of a non-ideal active inductor: arctan arctan arctan,, i i i i o i i o i i o i i o in c r c r r c r r c r r e Y φ φ φ φ φ Fiure 4. Series and parallel equivalent circuits of a non-ideal active inductor The quality factor of the inductor becoes sec tan φ φ o r Q When used as an active resonator the circuit is connected in parallel with the resonatin capacitor r. Then the resonance frequency and Q at resonance becoe tan cos φ φ r o r r r Q L If we disreard the phase shift caused by the input connections of the transconductors valid e.. in MESFETs, and set, r o r o r o / o, r, the resonance frequency can be written as cos φ o o o o r r and the quality factor at reduces to if, >> o o o o r Q

81 The sae result is also obtained in [4.9] and [4.8]. If the / o ratio is low, as in MESFETs, Q will becoe ipractically low. In eneral, /tan φ is a very steep function at sall aruents, and the quality factor of a yrator-based active inductor collapses rapidly even if the net phase shift is only a few derees. Ideally, it approaches infinity when φ arctan ro r 4. By Eq. 4.7, this requires that a phase-lain R-network exist at the input of either/both transconductors. 4.. Non-Ideal Active Inductor Bandwidth Accordin to Eq. 4.7 and Fiure 4., the series-ode inductance value of an active inductor is cos φ L ri ci ri ci r iciri ci ro 4. This iplies that the positive inductance bandwidth of the circuit is r c i i L, ri ci ro ri ci r o >> r c i i 4.4 if r i r i r i and c i c i c i. Thus, the input poles of the transconductors define the inductor bandwidth. Usually, however, the self-resonance frequency of the active inductor is lower than the inductance bandwidth, and it sets the liit of usability. The self-resonance circuit is fored by parasitic capacitance at the output, i.e. the output capacitance of one transconductor and rouhly the input capacitance of the other. cos φ sr, c c, o i >> i o c c Active Inductor Hih-Q Operation The third and the ost severe echanis liitin the active inductor frequency rane is its hih-q bandwidth. The parallel conductance of an active inductor is Fiure 4. tot o sin φ For siplicity, we assue that phase la is present only at one transconductor input. Now we et after substitutions in Eq. 4.7 tot o ro ri ro ci r c r i i o This function has a lobal iniu at Qax : 65

82 tot Q ax ri ro ci ri ci ro ri ciro 4.8 The other requireent is that parallel conductance ust becoe zero at this frequency. Accordin to the condition iven in Eq. 4. Q ax ri ci ro ro r r r c i o o i r r 4.9 With Eq. 4.8 and 4.9 cobined, we finally et ri ciro ro r c r r i i o o r r r Q ax o r 4. and further ro r r ici Q ax ro r # ro r ro Q ax ro r ± r Q ax o 4 r 4 Q ax o r r 4. Equation 4. fixes the transconductor input and output poles for infinite Q at Qax. The transconductor values ust be appropriately chosen to set the resonance to this frequency Eq.4.9. It is noteworthy to reark that it is not sufficient to adust only the transconductor input R-network for the axiu-q frequency; the loss in conunction with the yrator capacitance ust be adustable as well. The iniu attainable hih-q frequency is square root of three ties hiher than the output pole frequency: in r c r r r i i o o o r r 4. Fiure 4. is a raphical representation of Eq. 4. when r o r is set to a typical value of ns. µ n n R / s n p r o r i c i p p G G G 4G 5G f / Hz Fiure 4. R-products within a hih-q active inductor 66

83 Since the R-products are hihly non-linear functions of frequency, their linear adustin does not ove the hih-q reion linearly. Therefore, it is enerally unfeasible to et the track with a varyin resonance frequency by e.. adustable resistors. As an exaple, paraeters for GHz and 4 GHz iven by the previous equations are tabulated in Table 4.. The correspondin output conductances and Q-values are plotted in Fiure 4.. r ic i r o r r o GHz 78, ps 4 pf 5 Ω pf Ω 9,5 S 4 GHz 9,5 ps 4 pf 8 Ω,5 pf Ω 9, S Table 4. Theoretical paraeter values for active inductors G tot / S.. GHz resonator 4 GHz resonator Q GHz resonator 4 GHz resonator. -. G G G 4G 5G 6G f / Hz G G G 4G 5G 6G f / Hz Fiure 4. a Output conductance and b Q of GHz and 4 GHz active inductors As shown in Fiure 4., the hih-q bandwidth of an active inductor is stronly dependent on frequency bein uch wider at hih frequencies. The nearer the resonance frequency is to the iniu hih-q frequency the narrower and ore sensitive the hih-q reion is. Fiure 4. shows how the hih-q bandwidth depends on the ratio k Qax / in. As a rouh approxiation, the active inductor bandwidth of Q > when k is three ties wider than the iniu k, or five ties wider when k. Since the iniu hih-q frequency cannot be lowered liitlessly in practical interated circuits, the operatin frequency of a hih-q active inductor ust be sufficiently hih. k Qax / in k Q k k / Qax Fiure 4. Hih-Q bandwidth of an active inductor 67

84 4.. ontrollin Resonance Frequency and Q Effect of The resonance frequency of an active inductor resonator can easily be tuned by varyin the transconductor values. As approxiately applies ~ 4. the resonance frequency is directly proportional to the transconductor values if they are equal. Usually the transconductors are on the sae bias line, which akes this requireent easy to fulfill reeberin that transconductance is proportional to the D current throuh the device. Accordin to Eq. 4.8, the position of the Q axiu is independent on the transconductance product. The hih-q frequency does not track with the shiftin resonance frequency when the transconductors are tuned, which is naturally a serious drawback. The parallel conductance value is proportional to the transconductor product Eq. 4.7, and althouh the location of the conductance iniu Q axiu does not ove, its value, toether with the Q factor, will chane the transconductors are tuned. This behavior is illustrated in Fiure 4.4, where the upper curves correspond to lower transconductances and vice versa. 4.. G / S G 4.G 6.G 8.G.G f / Hz Fiure 4.4 onductance with chanin One can see fro Fiure 4.4 that althouh the conductance iniu position reains unchaned, the zero-crossin frequency does chane. It is indeed possible to desin a hih-q resonator not for the conductance iniu but for the zero-crossin frequency. This frequency is defined by the anitude of the parallel conductance which can be tuned e.. with r o as well as with. Since the resonance condition fixes inductance and the transconductors at each, tunin of r o reains the only practical way to adust the Q value. The fact that conductance becoes neative on either side of the resonance frequency will ake the definition of Q abiuous, but will not result in instability provided that no parasitic L-resonance is present within the reion of neative conductance. Effect of r o As stated previously, also loss resistance r o in conunction with the yrator capacitance ust be variable in addition to the transconductors, so that both inductance and loss can be optiized siultaneously. As seen fro Eq. 4.8, the position of the Q axiu can be adusted with r o. An equally effective alternative would be to tune the input resistance r i, but it is ore difficult to arrane in practice. The resistance r o also affects the anitude of the parallel conductance Eq. 4.7 and unfortunately also the yrator inductance, and hence the 68

85 resonance frequency Eq. 4.. This brins up the proble of concurrent tunin: resonance frequency tunin and Q tunin cannot be perfored separately. This sets reat deands on the eventual autoated tunin circuitry. The effect of chanin r o on the net conductance of the yrator is shown in Fiure 4.5: 4... G / S G 4.G 6.G 8.G f / Hz Fiure 4.5 onductance with chanin r o 4.4 Active Inductor Noise Noise is a very iportant issue in yrator-based active inductors. The very nature of the yrator with the yration conductances iplies that even in an ideal case the noise contribution fro these resistive coponents cannot be avoided. It can be said that an electronic yrator has a iniu noise production independent of the electronic desin [4.9][4.]. The additional QRLVH FRQWULEXWLRQ RI WKH DFWLYH GHYLFHV FDQ WKHQ EH H[SUHVVHG ZLWK WKH FRHIILFLHQW DV previously. When capacitors are connected at each port of a yrator to for an active resonator, the yrator noise sources reain the sae, and the circuit can be represented as in Fiure 4.6. Alternatively, the noise voltae sources at the inputs of the transconductors could be transferred into current sources at the outputs. When lookin at the passive equivalent circuit of this confiuration, we see that the noise source fro can be expressed as a series voltae source in conunction with the siulated inductance L. The other noise source fors a parallel current source for the resonatin capacitor, respectively. We can iaine that the active resonator susceptances becoe noisy in contrast to passive resonators where resistive coponents enerate noise. As each susceptance at resonance is Q ties larer than the loss conductance in a passive resonator, it can be concluded that the total noise of an active inductor resonator ust be Q ties that of a passive resonator with identical paraeters [4.]. Fiure 4.6 Active inductor resonator with noise sources and its passive equivalent circuit 69

86 The two noise coponents, the voltae source and the current source, can be forulated as v ˆ ˆ ˆ ˆ n, v, in, v 4.4 where v and v are the equivalent input noise voltaes of the transconductors. The total noise voltae spectral density over the resonator becoes vˆ vˆ L Gl L vˆ L 4.5 At the center frequency this yields vˆ vˆ L v Q ˆ l 4.6 In hih-frequency circuits the transconductors are usually realized as sinle transistors, and usin the transistor input noise approxiation 4N7J, we et the final for: vˆ Q 4 γ l kt 4kTγQl 4.7 This shows aain the relation between the hih loaded quality factor and hih noise. Obviously, axiized and iniized result in lowest noise, but due to practical liitations, they cannot usually be chosen optially. The total rs noise of an active inductor resonator can be obtained fro Eq. 4.5 by interation: v vˆ π L G l d vˆ L d π L L G L l 4.8 With the previous assuptions, this becoes v ktγ Q ktγ Ql l 4.9 In a siplified case where and, this yields v ktγ Q l 4. which is in ood areeent with the calculations in [4.9] and [4.]. At this point it is interestin to see how the active inductor resonator noise copares to that of neative resistor resonators. Fro Eq. 4.7 we can identify the effective noise resistance R which can be associated with that of the passive inductor R R s r n, n in a neative resistance resonator [Section.5.]. Thus, it is possible to define the effective unloaded noise quality factor for a yrator resonator, which shows what would be the anitude of Q in the passive inductor inducin the sae aount of noise as an active inductor. It becoes 7

87 Q, noise rn, n γ 4. With realistic paraeters Q,noise is very low, typically less than one. Therefore, it is reasonable to draw the conclusion that active inductor resonators are always clearly inferior to neative resistor resonators in ters of noise perforance. The noise calculations here are very uch idealized, and they do not account for additional noise fro Q-enhancin phase-shift circuits or their frequency-dependence. Nevertheless, they do show the effect of the ain paraeters to be diensioned and ive a suitable basis for coparisons. These results can even be extended to - filters constructed fro interators instead of yrators, since the sinal paths of the - biquadratic filters reduce to those of yrator filters [4.]. Hence, the inherent noise behavior is siilar. 4.5 Active Inductor Distortion 4.5. Introduction The non-linearities in the transconductors ive rise to distortion effects in yrator-based active inductors. Siilarly as in neative resistors, copression will cause increasin loss at the fundaental frequency and thus deradation of Q. As active inductors are priarily supposed to affect the phase characteristics of the resonator, the phase behavior at copression ust also be exained. The Volterra-series ethod is also applicable for yrators with the sae conditions as discussed in Section.6. Althouh very uch siplified, the non-linear transconductor or transistor odel used here is the sae as in neative resistor calculations. In spite of the fact that realistic active inductors suffer fro uch ore coplicated distortion echaniss than presented here, the calculations ive insiht on how the diensionin of the essential circuit paraeters contribute to the distortion perforance, at least in qualitative ters. The lare-sinal inductance of an active inductor can be expressed in the siilar way as the lare-sinal conductance of a neative resistor. Its reactance can be referred as the iainary part of the describin function: stiulus voltae vin L I{ fundaental response current} 4. The chane in the response current, and thus in the inductance, at the fundaental frequency with increasin input voltae can be described with the --db inductance copression of the circuit. The -db or.9% inductance chane will shift the resonance frequency approxiately 6% upwards, which is in ost cases unacceptable and ust soehow be copensated in potential applications. Even if the yrator-based active inductor is lossless in the sall-sinal approxiation, the non-linear effects will introduce loss conductance at hiher input voltae levels, resultin in a reduction of the circuit quality factor. Just like in neative resistors, this can be accounted for by exainin the real part of the fundaental response current. 7

88 4.5. Volterra Kernels Fiure 4.7 depicts an active inductor with non-linear transconductors. The transconductor i becoes non-invertin, when the polarity of its controllin voltae is inverted. This affects only odd-order ters in the current expression. For atheatical reasons, a conductance G ust be added in parallel with the yrator capacitor, otherwise soe distortion coponents becoe infinite in theory. Fiure 4.7 Gyrator resonator with non-linear transconductors Followin the sae procedure as previously with neative resistors, we can calculate the first-order Volterra kernels for the yrator inductor. When the excitation voltae is set to unity, we et H,, H iin, G G 4. The second-order Volterra kernels becoe respectively: H H, iin,, G, G i i NL, NL, i NL, 4.4 where the non-linear currents are i i NL, NL,, G G, 4.5 Finally, the expressions for the third-order Volterra kernels are H H, iin,,, G,, G i NL, i NL, i NL, 4.6 7

89 The third-order non-linear current sources for each type of non-linear response are as follows: i i NL, NL, i NL,, G G G, G G,,,,, G G G G HD IM, OMP 4.7 In hih-q inductors G &, and therefore the kernel expressions can be considerably siplified. For instance, Eq. 4.7 is approxiated as i i NL, NL, i NL,,,,,,,, G HD IM, OMP Distortion oponents Utilizin Table., we can now identify each distortion coponent in interest. The expressions are athered in Table.5 in which G is oitted where possible. We can see that when G is very sall the expression for the copression response becoes doinated by the last lare iainary ter. Hence, the doinant copression echanis in hih-q active inductors is the inductance copression. Obviously, if the second-order non-linearities in the active devices were iniized the proble would be alleviated. Differential transconductors with suppressed even-order non-linearities can be used for this purpose. In addition, the real part of the copression response, which is responsible for Q deradation, would be iniized in differential confiurations. opressin inductance and the resultin shift of resonance have a reductive effect on Q in practical active inductors, since the phase shiftin circuitry necessary for hih-q operation is narrow-banded without exception. The Q axiu is therefore sharp and ore or less constant in frequency. As the resonance frequency is deviated fro this point, the circuit Q will drop quickly, althouh the input sinal level is not yet hih enouh for the actual conductance copression to show up. The --db inductance copression point can ark the upper liit of the active inductor dynaic rane. As the total iainary input current at the fundaental frequency is i in v in v in,, G 4.9 the circuit can be represented by two parallel inductors, and the condition for --db/.9% copression is 7

90 Type of response Response i in fundaental vin second haronic third haronic v,, in v,,,, in 4 copression v < in,,,, 4 G Table 4. Haronic distortion coponents for the yrator-based active inductor L L cop fund v G in, cop,, v in, cop,,q, actind.9% 4.4 where Q,actind & /G is the unloaded quality factor of the active inductor. The copression point is vin, cop 9. 7,, Q, actind 4.4 At very hih Q values the theoretical copression point approaches zero, or in other words, the input current oes to infinity. This is naturally ipossible in practical circuit realizations, where the biasin and other auxiliary circuits liit the current swin. Therefore, even when the active inductor is tried to be copletely lossless, its copression voltae is non-zero, thouh sall Dynaic Rane Now that both the noise and copression characteristics are known we can deterine the theoretical dynaic rane of a yrator-based active resonator. It is assued that the unloaded Q of the active inductor is hih enouh not to increase the noise level substantially. When an active inductor fors a parallel resonator with a capacitor, the fundaental response current of the resonator becoes zero at resonance. Followin the sae procedure as with neative resistors, the condition for the total resonator current copression is G v l in, cop i in G v copression l in, cop 89.% i copression in G v l in, cop.9% 4.4 obinin Equations 4.4 and 4.4, we can write: { i cop v } { i cop v } in, actind in, cop, actind in, actind in, cop, reson L vin, cop, actind vin, cop, reson Q l 4.4 The third-order copression ters are proportional to v, and after cancellation we finally et 74

91 v in, cop, reson v in, cop, actind Q l 4.44 The result connects the copression voltaes of active inductor resonators with those of bare active inductors. It is shown in [4.]-[4.4] that the sae expression applies to any OTA- resonator/filter, reardless of the used topoloy. In the context of this thesis, it is very iportant to note that the copression point of an active inductor resonator is Q ties lower than that of a neative resistor resonator, if the axiu voltaes for the active circuits are equal. Here, Q refers to the unloaded quality factor of the copensated passive L resonator. By Eqs 4.9 and 4.44, we can be directly write the equation for the dynaic rane of an active inductor resonator: 9.7 DR lo Q ktγ l,,q, actind 4.45 opared to neative resistor resonators, the dynaic rane is Q ties lower, provided aain that the active circuit copression points are equal. Thus, even with L quality factors as low as, the deradation in the dynaic rane is substantial db. To et a coparable view on the anitude of the active inductor dynaic rane, we substitute the nuerical paraeter values used in Section.7.4 in the equation: & 4 GHz, Q l siulated L nh, Q. pf,.79 pf, S,,, A/V,,, $9 TriQuint MESFET 4 P With these paraeter values Eq ives DR G% ZKLFK LV HYHQ LQ WKLV LGHDOL]HG FDVH clearly lower than those of neative resistor resonators [Section.7.4]. rs noise voltae / µv copression voltae / dbv dynaic rane / db Active inductor resonator 97 Table 4. oputational noise voltae, copression point and dynaic rane for the active inductor resonator In practical active inductor resonators, Q enhanceent in the for of phase shifters is always needed for an hih unloaded Q. As these are stronly frequency-dependent and usually narrow-banded, they have effect on the noise transfer functions. Moreover, reardless of the construction, they will inect excess noise into the circuit. These factors unaccounted here will chane the noise characteristics of active inductors and so diinish the available dynaic rane. The phase shiftin network often includes active coponents that introduce additional distortion sources. The yrator port where the yrator capacitor is connected is subect to hiher voltae swins than the input port, as the yrator capacitor is usually saller than the resonatin capacitor. This intensifies the effect of the phase shifter non-linearity. Additional reduction of dynaic rane due to these eleents is substantial in realistic circuits. The input capacitances of the transconductors are included in both and, often doinatin the forer. As these are non-linear, they will anifest theselves as increased distortion in the circuit. If the circuit is desined in such a way that ost of capacitance coes fro passive capacitors, not fro the transconductors, this distortion contribution is saller. If 75

92 the non-linear capacitances had been taken into account in the Volterra kernel calculations, they would have ade the expressions too coplicated and unillustrative for the scope of this study. There is reason to underline aain that the fiures presented are results of very uch siplified analyses, and therefore not attainable in practice. However, they ive ood rounds for udin the perforance of each topoloy and diensionin the essential circuit paraeters, and show the order of anitude of the non-ideal effects. In spite of their favorable properties in ters of practical interation no need for area-consuin and low-quality passive inductors, the yrator-based active resonators are less attractive alternatives for hih-frequency active filters than the neative resistor resonators. 4.6 Practical Active Inductors Several realized active inductors and resonators for icrowave frequencies are presented in this section. They all share the sae basic structure based on a yrator, i.e. they include an invertin and a non-invertin transconductor without exception. The ost iportant properties of the circuits are explained and techniques for increasin their Q value are discussed Hara s ircuits and Its Derivatives The new possibilities of MMI interation brouht up the idea of active inductance siulation with MESFETs in the late 98's, when extensive work for developin hihperforance interated active inductors was ade by Hara et al. [4.][4.]. Their first realization eploys siply a resistor as the non-invertin transconductor. The yrator capacitance is fored solely by the ate-source capacitance of the transistor. Fiure 4.8 a Hara s first eneration active inductor realized with the GMMT-F process; b Measured inductance and series resistance [4.5] The circuit approxiates a series-connection of an inductor and a resistor: L s s, Rs GFB 4.46 As the feedback resistor siulates badly a transconductor, Hara s first-eneration inductor is inherently lossy and therefore not a prospective candidate for filter desin. Hara s second eneration topoloy shown in Fiure 4.9a has better perforance as the non-invertin transconductor is realized with a coon-ate transistor. A soewhat different approach is presented by Zhan et al. [4.6]. By eployin the theory of second-eneration 76

93 current conveyors II, they have anaed to develop a floatin active inductor. The theoretical perforance is, however, actually siilar to the inductor of Hara et al. Unfortunately, the properties of these circuits have been calculated by approxiatin the transistors with the ate-to-source capacitance s and the transconductance only. However, the drain-to-source conductance ds of the real transistor is a crucial paraeter when evaluatin active inductor circuits. When this eleent is added to the transistor odels, the perforance derades rapidly, since it is connected directly across the output as parallel conductance. Rerettably, the theoretical calculations tend to becoe ore coplicated and siplifications are necessary. The liited ds of a GaAs MESFET was first taken into account in [4.5]. Two new active inductor circuit confiurations were presented, and the crucial paraeter ds was held in theoretical exainations all the tie. The topoloies were found by a systeatic search fro the roup of potential feedback confiurations. The circuits cobined in a sinle scheatic drawin are shown in Fiure 4.9b. Fiure 4.9 a Hara s second-eneration active inductor; b Alinikula s active inductor The circuit in Fiure 4.9a is practically lossless at low frequencies only in the ideal case. The addition of ds clearly worsens the characteristics by lowerin the equivalent parallel resistance Fiure n 5 5.n 4.n 4 4.n.n.n 8 L / H.n Rs / Ω L / H.n 6 4 Rs / Ω.n L R s.n L R s. M G f / Hz f / Hz Fiure 4. a L and R vs. frequency Hara; b L and R vs. ds, f GHz Hara;,, 5 S, ds, ds, ds 6 DQG& s, s, s. pf The calculations and the siulations were ade on the assuption that all the active eleents in the circuits considered are of equal size. This is by no eans the optial diensionin for the best achievable perforance, but considerably siplifies the practical ipleentation. It allows the transistors to be placed in series with the sae drain current and akes siple ate biasin possible. These are all favorable aspects in interated circuit desin. In noral FET processes the ratio / ds can be as low as. Thus, this unpleasant property alost solely dictates the perforance, while the benefits of the possibly hih f T are few. The only way to iprove the results is to use another technoloy or to reduce the effect of ds. The latter can be done with a circuit topoloy shown in Fiure 4.a [4.4] [4.7]. When 77

94 approxiatin the perforance without ds, the proposed circuit is found to be ore lossy at soe frequencies than the circuit suested by Hara et al. However, the perforance of this circuit is less dependent on ds. Fiure 4. a Principle of the active inductor circuit by the author; b Equivalent circuit of the active inductor, ds inored The theoretical perforance of the proposed circuit in the ideal case ds can be easily calculated: its inductance, series resistance and parallel capacitance becoe L s s, R s s, s 4.47 These calculated quantities correspond to the equivalent circuit coponents in Fiure 4.b. 5.n 5 5.n 4.n 4 4.n.n.n 8 L / H.n Rs / Ω L / H.n 6 4 Rs / Ω.n L R s.n L R s. M G f / Hz f / Hz Fiure 4. a L and R s vs. frequency author; b L and R s vs. ds, f GHz The siulated results are illustrated raphically in Fiure 4.. As the frequency rows the losses rise ore rapidly than in the previous case, while the inductance is rather constant. The paraeter values reain the sae as in Fiure 4.. If the results in Fiure 4.b are copared with those in Fiure 4.b, one can reark that now ds has obviously less effect on the perforance, especially when it is lare. When ds is added to the transistor odels the equations becoe L s ds - -, Rs ds - ds - ds, R p -, s 4.48 The equations 4.48 and the correspondin equivalent circuit in Fiure 4. include a parallel and a series resistor in order to show the effect of the / ds ratio. 78

95 Fiure 4. Equivalent circuit of the active inductor with ds In addition to the previous one, the suested topoloy has a few other benefits: above all its siplicity and copatibility with processes. However, it should be noted that the absolute value of inductance is rather low, ainly due to the non-scaled transistors. In test structures, uch hiher values can be achieved with scaled transistors and separate biasin. Another way to increase inductance is to add a yrator capacitor to the ate of the lower transistor at the cost of diinished bandwidth. The reason why the proposed active inductor topoloy ives better results when ds is taken into account, lies in the fact that the non-invertin coon-ate transistor has a low input ipedance copared to the output ipedance of the invertin coon-source transistor. Therefore, the forer loads the latter considerably causin perforance deradation. The cascode connection for the invertin transconductor used by Hara et al. has approxiately twice the output ipedance of a sinle transistor, and hence the loadin effect is ephasized in Hara s circuits. Practical ipleentations of active inductors suffer fro deraded perforance coparin to the ideal case. The existence of the Miller capacitance d in the transistors affects the frequency characteristics of the circuit. The couplin capacitors liit the lowest usable frequency and increase the series resistance, not to ention the space on the chip they require. The RF decouplin of the bias line by eans of active loads clearly raises the losses, too. The active inductor circuit was first tested as a part of a siple L-filter processed by the GE-Marconi foundry [4.6][4.7@ 7KH SURFHVV ZDV *&DUFRQL ZLWK D P channel lenth and a -GHz f T. The low perforance of the filter could easily be noticed. This was due to the low Q-value of the inductors, which is unavoidable in noral hih- ds GaAs active inductor structures Bipolar Active Inductors The / ds ratio is the ost iportant factor contributin to the sall-sinal perforance of an active inductor. In FET processes, this paraeter is unavoidably sall, while in bipolar processes it can be several thousands. This ives a stron otivation to use bipolar transistors LQVWHDG RI 7V 7KH OLPLWHG RI WKH ELSRODU WUDQVLVWRU FDXVHV VRPH GHYLDWLRQ EXW EHLQJ norally in the rane of it has alost no effect whatsoever. A ore sinificant effect results fro the base-spreadin resistance of the bipolar transistor. The series resistance of the bipolar active inductor tends to ake a dip at a certain frequency, as shown in Fiure 4.4b. If required, this feature can partly be eliinated by keepin the base resistance as low as possible, that is, by choosin an appropriate transistor structure. Proper structures coonly have ultiple base contacts and a lon stripe-shaped eitter reion. The base resistance also affects the inductance value, worsenin the flatness of the frequency response but also increasin the hihest achievable value. In fact, the inductance in the low-loss frequency reion can be called super-inductance with the reactance value of X & L super. This dispersive effect can be advantaeous in filter desin, as it virtually increases the nuber of poles and thus akes the response steeper. 79

96 4.n.n L R s 5 L / H.n.n Rs / Ω. M f / Hz G -5 Fiure 4.4 a Sall-sinal representation of a bipolar active inductor; b L and R s vs. frequency,, 75 S, ce, ce 6 J b, b S and be, be 8 pf The resistance dip is well predicted by the theoretical studies in Section 4.. In fact, the base resistance of the bipolar transistor toether with its base-eitter capacitance fors a phaseshiftin network described earlier. Hence, this Q-enhancin quality is inherent in bipolar active inductors, even to such an extent that instability ay becoe a serious proble. It ust be noted that in real circuits there are ore loss-eneratin echaniss, such as biasin, which affect the depth of the resistance dip. The lare of the BJT reduces the inductance value, but at the sae tie the bier be copensates this chane. The larer also facilitates the A-couplin: the couplin capacitors need not be as lare as in MESFET circuits with the sae aount of phase copensation. A aor advantae of the bipolar active inductor is its low power consuption. opared to the MESFET inductors, only a fraction of the operatin current is needed to achieve adequate perforance. This should be a welcoed feature in low-power applications Q-Enhanceent The fall of the inductor series resistance in bipolar circuits is priarily due to the base resistance of the upper feedback transistor. If this resistance can be kept low, the iniu value and the correspondin frequency of the series resistance dip can be controlled with an additional resistor connected to the base of the feedback transistor. Thus, it is possible to increase the Q value at a certain narrow frequency band. Applied to FET-based circuits, this technique is particularly useful, since FETs lack sufficiently lare ate resistances. The inherently hih losses of MESFET-based inductors can be cancelled within a certain frequency band in this way. Basically, Q enhancin is based on a phase-lain network in the transistor inputs [Section 4.]. Two different ethods for phase copensation have been used: phase shift with an ordinary R-network [4.7][4.8], or phase shift within the positive transconductor as a for of ate resistance in the coon-ate FET, as described earlier [4.6][4.7][4.8]. R-networks are uncoplicated to desin and realize, but they tend to becoe quite narrow-banded and difficult to adust externally. A sinle R-stae ives a phase copensation of φ arctan R 4.49 Since the phase error of a practical active inductor varies alon with process variations and tunin, the phase copensation circuit ust be adaptive. This can be ipleented with variable resistors e.. triode-connected FETs or variable capacitors varactor diodes. If a resistor R Qen is connected at the ate of the coon-ate transistor, the additional phase shift across the transistor becoes 8

97 φ c arctan R Qen s 4.5 This ives an alternative way of phase copensation, althouh the basic operation is the sae as in R-networks. This schee eploys only one additional resistor, fixed or adustable, without bulky capacitors, and is therefore copact Realized MESFET Active Inductors A GaAs-MESFET active inductor circuit utilizin the Q-enhancin resistor technique is shown in Fiure 4.5. The controllin resistance is realized with a MESFET operatin in the triode reion. The equivalent series resistance value, and thus the location of the Q-axiu, can be controlled by the external D voltae V bias,q applied to the ate of this device. As shown in the theoretical discussions in Section 4., the loss resistance in conunction with the yrator port ust be variable, too. This adustent can be ade by controllin the topost RFdecouplin MESFET with another external voltae V bias,f. In this circuit, the transconductances of the active devices were chosen to be constant, and the voltae V bias,f also controls the inductance value. The easured inductance and Q-value of the circuit at the Q-axiu are plotted in Fiure 4.6. Vdd vdd Vbias,f 5k x5u current source p Q-enhancin resistor 5k xu p 5k x5u in Vbias,Q p x5u p5 5k port Fiure 4.5 a Principle of the Q-enhancin resistance; b MESFET active inductor with Q-enhancin resistance, processed with GE-Marconi F Fiure 4.6 a Measured inductance and resistance of the MESFET active inductor; b Measured unloaded Q of the MESFET active inductor 8

98 Siulated MESFET active inductors utilizin the other type of phase shifter, i.e. a noral R network, has been reported in [4.7] and [4.8] for deonstration purposes only. Two different resonator topoloies have been taken into inspection. The first one in Fiure 4.7a eploys a odified differential pair as the non-invertin transconductor. The benefit of this structure is that now the non-invertin stae is a coon-drain coon-ate cobination with hih input ipedance. Therefore, loadin at the invertin transconductor output, and the sensitivity to / ds, is saller. The transconductance values in this topoloy are always saller than in sinle-device transconductors, resultin in hiher inductance values. This akes its utilization in resonators rather difficult, as the resonatin capacitor becoes very sall. The second active resonator topoloy in Fiure 4.7b is a variation of the previous circuit in Fiure 4.5. The phase copensation for hih-q operation is realized usin both ethods: a Q- enhancin resistor at the ate of the coon-ate MESFET and an additional R network at the invertin stae input. Two phase shifters were needed, as the transconductances of the devices were sall in the used process the low-power, low-threshold voltae GE-Marconi L. Vdd Vdd u u Q tune 5k 5u p 5k 5p u u IN 5u 5k.5k.5p 47.5p u u p u u 5p.p 5p 5p 5k 5u IN Vss u 5k u.5p 45 5k 5u.8p Q tune F tune F tune Fiure 4.7 Active inductor resonators with R phase shifters The siulated S of both resonators are shown in Fiure 4.8. The total power consuptions becae 49 W and 7 W, respectively. Less transconductance is needed for hih-q operation in the differential-based active inductor resonator, and thus its power consuption is lower Fiure 4.8 Siulated S of the two active resonators 8

99 4.6.5 Realized Bipolar Active Inductors As stated before, bipolar technoloies with hih / ce ratios and in-built phase-shiftin capabilities are best suited for low-power hih-q active inductors. At icrowave frequencies, there are two alternatives for interation of bipolar devices: hih-speed silicon processes BJT and heterounction GaAs or SiGe processes HBT. In ters of basic theory or practical desin, no essential differences can be found between the technoloies, except for the fact that HBT processes have considerably hiher transition frequencies and better-quality passive coponents. In fact, HBT technoloies with sei-insulatin substrates cobine the benefits of GaAs-MESFET and Si-BJT processes. Both Si-BJT and GaAs-HBT technoloies have been applied to the desin of active resonators in this section. GaAs-HBT active resonator The underlyin topoloy of the HBT active inductor [4.9] is aain based on the Q-enhanced version in Fiure 4.5a. The inductance is fored in a yrator realized with a coon-eitter and a coon-base transistor connections Fiure 4.9a. The hih losses in a standard yrator confiuration are copensated by the extra Q-enhancin resistor in the base of the noninvertin transistor, ivin additional phase shift and ensurin hih-q operation at the desired frequency band. The scheatic diara of the realized circuit is shown Fiure 4.9b. The current of the transistor chain is controlled via the current irror Q,Q5. This adustent controls the yrator transconductances in Q and Q, and thus the inductance value enablin the center frequency tunin of the resonator. The PIN-diode connected HBT Q4 fors a voltae-controlled resistor which shunts part of the sinal to the round and adusts the Q value of the resonator. Q isolates the sinal fro the supply, and Q6 is for biasin. Only a little additional phase copensation in the for of R Qen is needed, thanks to the very hih / ce ratio of the HBTs. Vcc k 4 Q 4x4u in k4 Vfbias Q6 xu 5p 5p RQen 8 Q xu Q4 xu Vqbias r k 5 p input RQen Q5 4x4u k k Q xu p port Fiure 4.9 a HBT realization of a yrator with a Q-enhancin resistor in the base of the non-invertin transistor; b Scheatic diara of the realized HBT active resonator A GaAs/Al. Ga.7 As HBT technoloy with SPIE copatible transistor odels was used in the desin. The f T and f ax of the devices were GHz and 55 GHz, respectively. The easured inductance and series resistance curves toether with the resultin unloaded Q are shown in Fiure 4.. The operatin current of the resonator, drawn fro a -V supply, varies fro. A to. A, dependin on the frequency adustent. At the optiu, it is 8

100 L / H Q.8 A leadin to a power consuption of only 8. W. This is a fraction of that of GaAs MESFET active inductors. 5.n 5 4.n.n.n L R 4 R / Ω.n..5G.6G.7G.8G.9G.G f / GHz.5G.6G.7G.8G.9G.G f / GHz Fiure 4. a Measured inductance and resistance of the HBT active inductor; b Measured unloaded Q of the HBT active inductor Silicon-BJT active resonator In this section, a differential Si-BJT icrowave active inductor with a inial nuber of passive coponents is presented [4.]. The core of the resonator is a conventional active inductance siulatin circuit. Two of these are connected appropriately to for a differential structure. Thanks to the existence of virtual round nodes, any of the cubersoe passive couplin coponents can be oitted in the differential realization. The Q-enhanceent is realized with a tunable active phase shiftin stae between the two transconductors. Furtherore, the base resistance of a bipolar transistor in a non-invertin coon-base stae contributes to phase correction. Accordin to Section 4., tunin of the phase shift copensation affects the Q value but unavoidably also the inductance. On the other hand, when the inductance is tuned by chancin the values of the transconductances with bias current, the output conductance values, and thus the phase error, are chanin accordinly. Therefore, both quantities ust always be tuned siultaneously. Fiure 4. shows the scheatic diara of the differential active resonator. It attenuates the even-order haronic coponents and potentially ives better distortion perforance than a sinle-ended topoloy. It ust be noted, however, that the dynaic rane enhanceent described in Section 4.5 does not apply here, as the transconductors theselves are not differential. The transistors Qp,n are the invertin and Qp,n the non-invertin staes for the positive and neative sinals. Qp4,n4 and the resistor-connected Q5 for the active feedback/phase-shiftin network that can be tuned with the voltae V qb to produce an exact phase shift for hih-q operation. The effective phase shift can be approxiated with φ corr arctan Gtun s4 G tun s4 s4 4.5 where corresponds to the base-eitter capacitances of the devices Qp,n actin as the yrator capacitors, and G tun is the RF conductance of Q5. The coon-collector stae also isolates the hih-ipedance output of the non-invertin transconductor toether with the active loads Qp,n, and provides D level shiftin. The wide tunin rane for quality factor ensures stability in all conditions. The transconductances of the devices, and thus the inductance and the resonance frequency, are controlled by their I c, i.e. the bias current I bias. The effective hih-q inductance tunin rane is fro.9 nh to.7 nh, correspondin to resonance frequencies fro 84

101 .5 GHz to GHz. In contrast to earlier active inductors, this topoloy does not eploy passive coponents that have effect on the perforance or sensitivity of the circuit. Especially, no hih quality floatin capacitors difficult to realize on silicon are required. The only capacitors in the resonator are ra and rb, the rounded resonatin capacitors. Their losses are easily tuned out. The circuit operates at a -V supply voltae and draws 5 W D power dependin on the current frequency bias settin. Fiure 4. Scheatic diara of the differential BJT active resonator and its sall-sinal equivalent circuit The siulated conductance and susceptance at GHz and the resultin Q-value are plotted in Fiure 4., respectively. onductances at different tunin voltaes and the frequency tunin characteristics are shown in Fiure G in 5 B in /S -5 Gin /S Q B in - -5.G.5G.G.5G 4.G f/hz.g.5g.g.5g 4.G f/hz Fiure 4. a Siulated adittance of the BJT active resonator at GHz; b Siulated unloaded Q of the BJT active resonator The siulated output noise voltae is 7 nv/hz / which is ore than twice the theoretical absolute iniu with the sae Q l. Additional noise sources in the transconductors as well as in the auxiliary devices are not included in hand calculations, and this accounts for the difference. It should be noted that the theoretical values ive the absolute perforance liit only. A test chip includin a differential resonator and a differential band-pass filter was processed with a.8-µ Bi-MOS technoloy. A icrophotoraph of the resonator circuit is shown in Fiure 4.4. The active circuit area is µ. 85

102 5 Vqb.8 V.G.G.8G G in /S f res /Hz.6G.4G Vqb. V -5.G.5G.G.5G 4.G.G.G f/hz Ibias/A Fiure 4. a onductance tunin vs. control voltae V qb ; b Resonance frequency tunin vs. bias current I bias Fiure 4.4 hip layout of the active inductor The easured sall-sinal characteristic curves are shown in Fiure 4.5. The operatin frequency rane in wider than predicted by siulations, but the hihest achievable hih-q resonance frequency is MHz lower than anticipated G / S.. B / S f in.9 GHz f ax.88 GHz -4..G.5G.G f / Hz Fiure 4.5 Measured conductance and susceptance of the BJT active resonator Since differential resonators are two-ports, the y-paraeters, correspondin to a differential one-port, ust be calculated fro the actual easured values by usin the forula in Section 86

103 ... As far as easureent techniques are concerned, one of the reatest advantaes of twoport circuits is that their noise fiures can be easured, and the correspondin one-port noise source values de-ebedded [Section.5.7]. This is why no easured noise data could be obtained fro the previous sinle-ended active inductor resonators. Only by constructin a filter with well-known coponent values, and then easurin its noise properties, one can deebed inforation on the actual sinle-ended resonator. The de-ebedded noise current of the differential BJT resonator in the vicinity of resonance is shown below Q l. For coparison, respective easured and de-ebedded data fro a neative resistor resonator [5.4] is plotted in the sae fiure alon with the passive equivalent. The active inductor resonator suffers fro about four ties hiher a noise level than the neative resistor resonator, which is well in line with the theoretical estiations. The easured output noise voltae at resonance is 8. nv/hz ½, while the theoretical iniu calculated fro Eq. 4.7 would be.9 nv/hz ½. 5.p 4.p i noise / A/Hz /.p.p Active inductor resonator Active neative resistance resonator Passive resonator, Q.p. -M -M M M f / Hz Fiure 4.6 Noise currents of different resonator types, de-ebedded fro easured data 5.4n 5.n 5.n L / H 5.n 5.n 4.9n V in / V Fiure 4.7 Measured inductance expansion of the BJT active resonator Fiure 4.7 depicts the easured inductance copression, or in this case expansion, of the circuit at.4 GHz. The extrapolated -db expansion point is approxiately 8 V. Applyin the definition for dynaic rane iven in Section 4.5.4, we can conclude that the dynaic rane of this active inductor is rouhly 87

104 8V DR lo 49dB 8nV/ Hz.4GHz 4.5 when the loaded quality factor is BW 4 MHz. References [4.] B. Telleen, The Gyrator, a New Network Eleent, Philips Research Report, Vol., pp. 8-, April 948. [4.] W. Mason, W. Hewitt and R. Wick, Hall Effect Modulators and Gyrators Eployin Manetic Field Independent Orientations in Geraniu, Journal of Applied Physics, Vol. 4, pp , February 95. [4.]. Hoan, The Ferroanetic Effect at Microwave Frequencies and Its Applications; the Microwave Gyrator, Bell Systes Technical Journal, Vol., pp. -, January 95. [4.4] M. Onoe and M. Sawabe, A Piezoelectric-Piezoanetic Gyrator, Proceedins of the IRE, Vol. 5, pp , Septeber 96. [4.5] B. Boert, Soe Gyrator and Ipedance Inverter ircuits, Proceedins of the IRE, Vol. 4, pp , July 955. [4.6] B. Shenoi, Practical Realization of a Gyrator ircuit and R-Gyrator Filters, IEEE Transactions on ircuit Theory, Vol., pp. 74-8, Septeber 965. [4.7] W. Holes, S. Gruetzann and W. Heinlein, Hih-Perforance Direct-oupled Gyrators, Electronics Letters, Vol., pp , February 967. [4.8] R. Riordan, Siulated Inductors Usin Differential Aplifier, Electronics Letters, Vol., pp. 5-5, February 967. [4.9] H. Orchard, Inductorless Filters, Electronics Letters, Vol., pp. 4-5, June 966. [4.] H. Orchard, Inductorless Bandpass Filters, IEEE Journal of Solid-State ircuits, Vol. 5, pp. 8-8, June 97. [4.] H. Vooran and A. Biesheuvel, An Electronic Gyrator, IEEE Journal of Solid-State ircuits, Vol. 7, pp , Deceber 97. [4.] S. Hara, T. Tokuitsu, T. Tanaka and M. Aikawa, Broad-Band Monolithic Active Inductor and Its Application to Miniaturized Wide-Band Aplifiers, IEEE Transactions on Microwave Theory and Techniques, Vol. 6, pp. 9-94, Deceber 988. [4.] S. Hara, T. Tokuitsu and M. Aikawa, Lossless Broad-Band Monolithic Active Inductors, IEEE Transactions on Microwave Theory and Techniques, Vol. 7, pp , Deceber 989. [4.4] R. Kaunisto, P. Alinikula and K. Stadius, Active Inductors for GaAs and Bipolar Techniques, Proceedins of Norchip Seinar, Trondhei Norway, pp. 7-76, Noveber 99. [4.5] P. Alinikula, R. Kaunisto and K. Stadius, Monolithic Active Resonators for Wireless Applications, Proceedins of IEEE Microwave Theory and Techniques Syposiu, San Dieo USA, pp. 5-54, May 994. [4.6] R. Kaunisto, P. Alinikula and K. Stadius, Q-Enhancin Technique for Hih Speed Active Inductors, Proceedins of IEEE International Syposiu on ircuits and Systes, London UK, pp , May 994. [4.7] R. Kaunisto, P. Alinikula and K. Stadius, Active Inductors for GaAs and Bipolar Technoloies, Analo Interated ircuits and Sinal Processin, Vol. 7, pp. 5-48, January

105 [4.8] Y. Wan and A. Abidi, MOS Active Filter Desin at Very Hih Frequencies, IEEE Journal of Solid-State ircuits, Vol. 5, pp , Deceber 99. [4.9] D. Blo and J. Vooran, Noise and Dissipation of Electronic Gyrators, Philips Research Report, Vol. 6, pp. -, 97. [4.] J. Vooran and D. Blo, Noise in Gyrator-apacitor Filters, Philips Research Report, Vol. 6, pp. 4-, 97. [4.] A. Abidi, Noise in Active Resonators and the Available Dynaic Rane, IEEE Transactions on ircuits and Systes I, Vol. 9, pp , April 99. [4.] G. Groenewold, The Desin of Hih Dynaic Rane ontinuous-tie Interatable Bandpass Filters, IEEE Transactions on ircuits and Systes, Vol. 8, pp , Auust 99. [4.] W. Kuhn, F. Stephenson and A. Elshabini-Riad, Dynaic Rane of Hih-Q OTA- and Enhanced-Q L RF Bandpass Filters, Proceedins of IEEE Midwest Syposiu on ircuits and Systes, pp , 994. [4.4] W. Kuhn, F. Stephenson and A. Elshabini-Riad, A MHz MOS Q-Enhanced L Bandpass Filter, IEEE Journal of Solid-State ircuits, Vol., pp. -, Auust 996. [4.5] P. Alinikula and R. Kaunisto, New Active Inductor onfiurations for RF and Microwave Applications, Proceedins of European onference on ircuit Theory and Desin, Davos Switzerland, pp. 9-4, 99. [4.6] G. Zhan, M. Villeas and. Ripoll, Microwave Active Filter Usin GaAs Monolithic Floatin Active Inductor, Microwave and Optical Technoloy Letters, Vol. 8, pp. 8-88, 99. [4.7] D. Haih, GaAs MESFET Active Resonant ircuit for Microwave Filter Applications, IEEE Transactions on Microwave Theory and Techniques, Vol. 4, pp. 49-4, July 994. [4.8] D. Haih, D. Webster, R. Kaunisto,. Nduiuba, A. Khanifar, M. Darvishzadeh, T. Parker, J. Scott and I. Thayne, Developents in RF Desin, Proceedins of IEE olloquiu on RF Desin, London UK, February. [4.9] R. Kaunisto, P. Alinikula, K. Stadius and V. Porra, A Low-Power HBT MMI Filter Based on Tunable Active Inductors, IEEE Microwave and Guided Wave Letters, Vol. 7, pp. 9-, Auust 997. [4.] R. Kaunisto, K. Stadius and V. Porra, A -GHz Silicon-BJT Active Resonator and Filter, Proceedins of IEEE International onference on Electronics, ircuits and Systes, Lisbon Portual, Vol., pp. 97-, Septeber

106

107 5. ATIVE RESONATOR FILTERS 5. Introduction Second-order band-pass filters are developents of siple loaded resonators. ouplin to the surroundin circuitry is a way of turnin one-port loaded resonators into two-port band-pass filters without chanin their loadin and quality factors. Therefore, any of the active resonator properties discussed in the previous chapters are directly applicable to the correspondin filters. This ives a convenient way of exainin the properties of the equivalent stand-alone resonators: by easurin the filter perforance, the correspondin one-port resonator perforance can be obtained fro the filter data, if necessary. Active resonators ust be tuned very accurately in order to ensure proper filter function. The band-pass loss ust be iniized by nullin the resonator losses, while still aintainin the syste stability. The iportance of the tunin accuracy is even reater if ore than one resonator is eployed in the filter. onsiderin realizations as interated circuits, twinresonator fourth-order filters can be rearded as the upper liit, thouh, as the siultaneous absolute and relative tunin of the resonators becoes quickly overly difficult. If the coupledresonator approach [Section.4] is used, the two resonators are identical akin utual atchin easier and decreasin the nuber of separate tunin voltaes or currents required. The couplin between the resonators and the terination resistances is ore coplex in twinresonator filters, and the specific atheatical studies shown in this work are not valid per se. The ore fundaental properties, such as the relation between noise and filter bandwidth, are applicable whatsoever. Bein the ost essential phenoena in active resonator filters, noise and distortion properties are discussed in this chapter. Feasibility issues, particularly in view of relevant syste applications, and autoated tunin techniques required for coercial applications are studied. Realized onolithic filter test topoloies by the author are shown, and easureent results presented. Finally, the concept and two realizations of a local oscillator eneration circuit for direct-conversion transitters are presented as possible applications for active resonator filters. Aparin [5.] Karacaolu [5.] Pipilos [5.] Kaunisto [5.4] Kaunisto [5.5] Kuhn [5.6] Type neative neative neative neative neative active inductor resistor resistor resistor resistor resistor Tech. GaAs GaAs BiMOS GaAs BiMOS MOS f rel. BW NF P BDR.5 GHz.6 GHz 4.9% 5.5% 7 db 8 db db 78 db 76 db. GHz 4.7 GHz 5.% 8.5% 7.5 db db 8 db no. P D W 5 W 6 W.6 4 GHz.6 GHz..6 GHz.%.9% % 4 db * 9 db db -4 db * - db 6 db 59 db 4 db 7 db 5 W W.9.88 GHz MHz.7%.% 9 db db * - db -8 db db 6 db ** 5 W W Table 5. Perforance coparison of soe recent hih-frequency active resonator filter desins; * calculated fro the iven data; ** for the actual filter bandwidth 9

108 5. Noise in Active Resonator Filters The expressions of noise fiures are derived for filters with both active resonator in this section. As shown in Section.4.4, the noise fiure of a second-order resonator filter is iˆ F 4kT r 5. where î LV WKH QHW QRLVH FXUUHQW VSHFWUDO GHQVLW\ LQMHFWHG LQWR WKH UHVRQDWRU LQSXW QRGH û& LV the --db bandwidth of the filter, and r is the effective resonatin capacitance. Usin this equation, we can now calculate the noise fiures for different noise currents. 5.. Noise in Neative Resistor Resonator Filters The noise current of a neative resistor is r n is the relative noise resistance for series-ode neative resistors, and n is the relative noise conductance for parallel-ode neative resistors [Section.5]: iˆ n 4kTrn, n G 5. When a lossy passive resonator is copensated with a parallel-connected neative resistor whose resistance is equal in anitude but opposite in sin, the total noise current becoes iˆ n 4kT rn, n G 4kT rn, n r, Q Q >> 5. Hence, the noise fiure of a neative resistor resonator filter is Q r l F n, Q n 5.4 where Q accounts for all losses in the passive resonator, includin the tunable capacitor losses. In ters of noise, neative resistor resonators behave ust like ordinary passive resonators. With sall values of r n, n reasonable noise perforance can be attained, provided that the passive resonator quality factor is hih enouh. As the cobined Q of an interated spiral inductor and a pn-varactor is axially in the order of, it has a doinatin effect on the iniu noise fiure theoretically possible, even if r n, n for the particular neative resistor were sall. 5.. Noise in Active Inductor Resonator Filters Accordin to Section 4.4, the noise current spectral density of a hih-q active inductor resonator at resonance is vˆ n r iˆ, γ n vˆ n 4kT, L 5.5 Applyin Eq..8, we et the noise fiure of an active inductor band-pass filter: 9

109 F γ r γq l r r 5.6 With an obvious choice of the paraeters and, this yields F 4γQ l 5.7 In the order of anitude, the noise fiure of a second-order active inductor band-pass filter is typically Q ties hiher than that of a neative resistor with a passive-part quality factor of Q [5.][5.7] [5.]. This is a fundaental stateent in favor of neative resistance filter. 5. Dynaic Rane of Active Resonator Filters The dynaic rane of an active resonator filter is directly associated with the resonator dynaic rane involved, since a second-order band-pass filter is nothin but a loaded parallel resonator with current excitation. The ipedance transforation at the filter input and output is taken into account in the loaded quality factor of the resonator. It increases voltae peakin proportional to Q l over the resonator [Section.4.5], but the noise voltae at the resonator output experiences the sae aount of attenuation at the filter terinals. The dynaic ranes of transconductor-capacitor filters and Q-copensated L filters have been copared by Kuhn at al. [5.9][5.]. They have shown that the dynaic rane of a G- filter is Q ties lower copared to that of a Q-enhanced L filter with a passive unloaded Q of Q. The sae result throuh a ore coplete analysis has been attained in the previous chapters of this thesis. DR neres Q ~ Q, DRactind ~ l Ql 5.8 The increase in the dynaic rane level can be explained by exainin the sinal currents that ust be delivered by the active devices in the two cases. In the yrator-capacitor circuit, the sinal current is deterined by the sinal voltae divided by capacitive reactance. In the neative resistance filter, the sinal current is deterined by the sinal voltae divided by neative resistance. Since the anitude of the neative resistance is a factor of Q larer than the capacitive reactance, a factor of Q less current is required in the neative resistance filter for a specified sinal voltae. 5.4 Practical Feasibility in Systes Interated RF filters seeinly offer several benefits for applications in the RF front ends of cellular phones. The currently used passive filters cannot be adusted to cover ultiple systes and frequency bands, and the only feasible solution is to use selectable filters for each desinated frequency band. Passive filters are bulky and expensive, althouh uch proress has been ade durin the last decade. On the other hand, onolithic RF filters are easily tunable, sall and inexpensive in both price and anufacturin costs. The drive towards increasin iniaturization would ake the unparalleled eleents for future handheld systes. This section deals with syste requireents for RF filters and ives an insiht into why the realizable interated RF filters, with a few exceptions, are not feasible in ost telecounication systes after all. 9

110 5.4. G GSM The relative bandwidth of a sinal channel in GSM systes is.% khz/9 MHz. This akes channel selection filterin ipossible. Instead, the ain function of the RF filters is to relax the dynaic rane requireents for the followin staes, i.e. by reovin the unwanted sinal power fro other wireless systes as well as possible. Obviously, the dynaic rane of the filter itself becoes a critical paraeter. In [5.] five possible applications for active RF filters are considered:. Frequency-Division-Duplex FDD heterodyne receiver filters,. Tie-Division-Duplex TDD heterodyne receiver filters,. Iae-reection filters, 4. TDD direct-conversion receiver filters, and 5. direct-odulation transitter filters. The syste architecture is shown in Fiure 5.. In the followin text, excerpts fro [5.] are referred. Fiure 5. Receiver and transitter RF-chains in a cellular phone architecture FDD heterodyne receiver filters In pure FDD systes, the receiver and the transitter are both on at the sae tie. The first filter of the receiver chain, the duplexer, ust be able to attenuate the transitter sinal whose power level is typically db. The transission and reception bands are close to each other GSM 9: MHz and MHz, resultin in unacceptably hih stop-band sinal levels for any active filter. TDD heterodyne receiver filters In TDD systes, the transitter is turned off durin reception. Now, the axiu sinal strenth experienced by the first receiver filter is caused by out-of-band blockin sinals that can have a axiu power level of approxiately db in diital cellular systes. The outof-band sinals need to be attenuated about db down to the level of the in-band blockin sinals. When applyin active filters to TDD duplexers, the ain concern is noise. The specifications are deandin: the iniu detectable sinal is db in GSM systes with a carrier-to-noise ratio requireent of about db. Hence, the axiu syste noise fiure for the receiver becoes: F R < 74 lo khz 9 db 5.9 As the duplexer is the first eleent in the chain, its noise fiure ust definitely be uch saller than F R Friis s forula, Eq... This is ipossible for active filters. Iae-reection filters The second RF filter in the receiver chain in used ainly for iae reection. The iae band reection requireents vary between 6 9 db in different systes. The filterin is distributed to the duplexer and the second filter. If the IR-ixer has an iae reection ratio of 4 db and the duplexer provides additional 5 db, up to 4-dB reection is expected 94

111 fro the second filter, which is anaeable. The overall power rane of the received sinal would be too lare for active RF filters without prior ain control, typically -9-5 db in diital cellular phones. However, the LNA has stepped ain control in ost cases for this purpose. The syste noise floor of a filter in a frequency-division receiver architecture is defined by the channel bandwidth, as shown in Eq Therefore, the baseband-referred syste dynaic rane of the filter ets sinificantly larer than its own RF-band dynaic rane. Kuhn [5.6] suests that as far as the syste dynaic rane is concerned, an active filter with soe ain as a byproduct of Q enhanceent can have coparable perforance with a passive-filter LNA cobination. However, this does not alleviate the requireent for RF noise fiure, which is typically 8 db for the second filter if the LNA has a ain of db. The other ore proisin alternative is to use active band-stop filters for iae reection. The filter notch is tuned to attenuate the iae band, while the reception band reains unchaned. With this technique, ore than 65-dB iae reection has been achieved fro the second filter itself [5.]. A series L resonator can be used in conunction with a series-ode neative resistor with potentially low-noise capabilities [Section.5.4]. The ain advantae of this approach is that the reception band lies far fro the resonance frequency, and thus the voltae swin over the neative resistor is low, naely V w R V in Q w 5. where w && and Q is the resonator unloaded Q. For instance, in [5.] w.9ghz/.5ghz.76 and with a realistic Q, a 5-dB reduction in voltae swin across the neative resistor occurs, copared to the voltae swin over the series resonator, which itself is low off resonance. Therefore, the dynaic rane of the filter is acceptable for a cellular receiver. TDD direct-conversion receiver filters In a direct-conversion transceiver, the odulation and deodulation are carried out directly at the carrier frequency. This arraneent reduces the nuber of required filters, as neither IF nor iae-band filters are needed. The dynaic rane requireents for the first filter in the receiver chain are essentially the sae as for the heterodyne receiver. Direct-odulation transitter filters In the transitter chain, the last filter after the power aplifier is used for cleanin up the transission spectru. Naturally, it cannot be active, as the power level is very hih. The filter between the IQ-odulator and the power aplifier is used for filterin the odulator output sinal fro the leakin spurious frequencies and for attenuatin the noise floor outside the transission band. The ost iportant specification for the filter is its noise contribution at the stop-band. Unfortunately, the stop-band noise of an interated RF filter easily exceeds the stopband reection by db. Frequency synthesizer The only RF block in a cellular phone, where the required dynaic rane is sall, is the frequency synthesizer. In eneratin the LO sinal in a direct odulator, a proble will occur if the hih-power output sinal couples back to the oscillator chain. The couplin can be avoided if the LO sinal is enerated inside the odulator fro two reference sinals. Then, however, the enerated LO sinal ust be filtered prior to the ixers. This is a potential application for onolithic RF filters, and it will be elaborated in the followin sections 5.7 and

112 5.4. G WDMA The transceiver architecture for WDMA cellular phones is siilar to Fiure 5., and basically the sae considerations for active filter usae apply. The G syste applies FDD, and thus the duplexer requireents are strinent for adequate isolation between the receiver and the transitter chains. Accordin to [5.], the receiver noise fiure ust not exceed 9 db includin the loss of the duplexer. As it is practically the sae as in GSM systes, siilar receiver chain noise requireents apply to filters. The pass-band shape and selectivity specifications are ore deandin in WDMA receivers, which akes the application of active filters even ore unfeasible. The dynaic rane specifications are not alleviated either. The iae reection needs to be >84 db for the entire receiver, and the notch filter concept can be used accordinly Bluetooth The short-rane wireless protocol Bluetooth could offer soe possibilities for active RF filter application. Since it has been desired that utilization of siple, sall and cheap transceiver circuits would be possible, the specifications are relatively loose. The Bluetooth syste is operatin at the ISM band MHz with -MHz channels and frequency hoppin. The band is wide enouh Q l for active resonator filters with reasonable dynaic ranes, but still too narrow for purely passive onolithic resonators. Bluetooth is a TDD architecture, where the noise perforance of the first filter is the liitin specification. Usin Eq. 5.9 and referrin to the Bluetooth specifications [5.4], the receiver chain noise fiure can be as hih as db, since the sensitivity requireent is 7 db. This is within the capabilities of active resonator filters, especially if cobined with a low-noise aplifier. The specification for SFDR is 5 db, which is soewhat ore challenin in ters of active filter desin. 5.5 Autoated Tunin Techniques For full control over an interated active resonator, both the center frequency and the unloaded quality factor, or loss copensation, ust be externally variable. The absolute process tolerances are too wide for the paraeter accuracy required by filters, and the strenth of active resonators lies in their adaptivity. It is acceptable to tune each resonator anually in test circuits like in this thesis. However, if the application has a real-life taret, the center frequencies of individual resonators have to be tuned via a coon reference, and the pass-band loss ust be autoatically iniized without riskin stability. When ultiple resonators are included in the filter, the couplin between the becoes critical, if the desired prototype function Butterworth, hebyshev etc. and the pass-band shape are to be aintained over the entire operatin frequency band. The task becoes tedious if the nuber of resonators exceeds two or three, for it is practically ipossible to develop an autoatic tunin syste for this purpose Master-Slave Tunin The aster-slave control schee is an effective ethod of realizin resonance frequency and loss control reardless of the resonator type [5.][5.5]. It is based on an unloaded oscillatin slave resonator in a dual-loop confiuration. The slave resonator is identical to that/those in the actual filter. The block level diara of the aster-slave-tunin concept is shown in Fiure 5.. The frequency control loop is essentially a phase-locked loop where the slave resonator acts as the voltae-controlled oscillator. It locks the resonance frequency to an external subharonic reference sinal that is defined by the divider-by-n. It is iportant that the reference 96

113 input does not fall upon the filter band to prevent interference. The low frequency also enables the use of traditional diital phase detectors in the PLL. The oscillation anitude should be as sall as possible for iniizin couplin between the aster oscillator and the filter. The Q control loop autoatically liits the oscillatin aplitude to a certain sall value via a rectifier and an interator. Thus, the sall-sinal behavior is ensured in the aster resonator, and the characteristics reain well atched to the slave filter resonators. The output aplitude of the aster oscillator/resonator is detected in the rectifier, and the resultin D voltae is copared with a reference voltae in the interatin operational aplifier. This voltae is chosen appropriately to aintain low oscillation aplitude. The aster and slave resonators ust naturally be well atched on the die, otherwise instability or excess pass-band loss ay result. The proble is ore severe in very narrow-band filters, where the arin between the unloaded and loaded Q of the resonators is saller. Another issue is the stability of the loops theselves. As the frequency and Q controls of an active resonator are never independent, adustin one will chane the other, too. To avoid any instability resultin fro such interactions, the bandwidth of the Q control loop should be larer than that of the PLL [5.]. Fiure 5. Master-slave autoatic control schee 5.5. ouplin Factor Tunin In a ulti-resonator filter, the couplin between the resonators ust be accurate and tunable if the resonator paraeters chane. Aparin [5.] has solved this proble by usin atched varactor diodes as couplin capacitors between the resonator staes. Their control voltae is coon with that of the resonator varactors. This approach ties the couplin to the center frequency tunin, but ood pass-band shapes have still been attained with this technique. Perhaps a ore sophisticated schee has been presented by Kuhn [5.6][5.]. Here, the two resonators are anetically coupled throuh the suitable placeent of the inductors on the die, and a couplin neutralization circuit is desined for accurate control over the couplin coefficient. The neutralization circuit ensures that the phase relationship between the inductors is correct for a flat pass-band response. The control is, however, not autoatic. The sae principle for active inductor resonators have been used in [5.5]. 97

114 5.5. Adaptive Transconductor Biasin The loss control in a neative resistor resonator ensures axiu-q characteristics. When the input sinal level is raised and the neative resistor starts to copress, a constant tunin current is not able to aintain the hihest available Q any ore. If the transconductor bias adapts to the sinal level across the resonator, this effect can be circuvented in soe extent, and the upper liit of the dynaic rane is enhanced. Notably, the bias current control ust be accurate not to inflict instability. 5.6 Realized Active Resonator Filters 5.6. Active Neative Resonator Filters Both GaAs-MESFET and Si-BJT technoloies have been utilized in deonstratin the desin of neative resonator filters. The MESFET realization is based on the copensated resonator fro Section.8. with a slihtly odified topoloy and coponent values. The varactor diode odel has been enhanced in order to et a ood atch between siulations and easureents. I shall concentrate on only the MESFET filter here, as the BJT versions will be treated in Section The filters in question have been published by the author in [5.4]. Preliinary studies on the issue can also be found in [5.6]. Active resonator Fiure 5. shows the scheatic diara of the eployed active resonator. It consists of a passive spiral inductor, a pair of varactor-connected back-to-back MESFETs M4, M5, and an active tunable neative resistance. The neative resistance is a sinle-ended version of the cross-coupled differential pair well suited for interation. The equal transconductances of the differential transistors can be varied, and thus the aount of neative resistance and loss copensation, via the current source M. A -nh spiral inductor has been added to provide ate bias for the coon-drain MESFET M. It reduces noise copared to the previous resonator, as no hih-value hih-noise bias resistors need to be connected to the hih-ipedance output node. The inductor has a sall effect on the resonance frequency of the resonator, which ust be taken into account. Fiure 5. Tunable active resonator with neative resistance copensation The varactors enable frequency tunin for the resonator. As their Q values chane durin tunin, the neative resistance copensation ust be adusted accordinly for zero loss at each center frequency. The back-to-back connection ensures the axial lare-sinal perforance of the varactors. 98

115 The neative supply voltae is beneficial in two reasons: the depletion-ode MESFETS are biased with neative ate-source voltaes, and no lare shunt capacitor is required for providin the RF round for the resonatin inductor. Band-pass filters Two coupled-resonator band-pass filters have been constructed fro the resonator: a second-order sinle-resonator filter and a fourth-order filter with two identical resonators Fiure 5.4. The active resonators are connected to each other and to the 5-Ω terinations with sall couplin capacitors. These are realized as interdiital capacitors or series-connected MIM Metal-Insulator-Metal capacitors. The center frequencies are.8 GHz for the secondorder band-pass filter and.4 GHz for the fourth-order band-pass filter. Fiure 5.4 onfiurations of the second-order filter and the fourth-order filter Realized circuit and results The filters were siulated with the advanced Parker-Skellern MESFET device odels [5.7] ivin excellent atches between siulations and easureents. The GE-Marconi F -GHz D-MESFET process was used for the realization. Fiure 5.5a shows the easured transfer functions for the second-order filter at the noinal and the extree center frequencies. In Fiure 5.5b the easured and siulated responses of the fourth-order filter are presented. In both filters, the tunin rane is rouhly the sae 4 MHz. The siulated and easured responses for the second-order filter also atch extreely well. The siulated curves are not shown, as they coincide with the easured ones. - - H / db - V bi as,f. V V bias,f -.5 V V bias,f -. V H / db G.6G.8G 4.G 4.G f / Hz -4.G.5G.G.5G 4.G 4.5G 5.G f / Hz Fiure 5.5 a Measured responses of the second-order filter over the frequency tunin rane; b Measured and siulated dashed noinal frequency response of the fourth-order filter The noise perforance of the filters is of reat interest. By usin the theoretical expression for the noise fiure of the neative resistor resonator filter [Section 5..] 99

116 F Ql QL Q, γ n n 5. with the process-defined values Q L 5 and Q, we can calculate the theoretical noise fiure for the sinle-resonator filter to be 6.9 db. The doinant effect of the filter bandwidth and the resonator coponent Q values should be noted aain; the noise contribution of the neative resistance is ideally only. db. The easured noise fiures are soewhat hiher than the calculated values, ainly due to the very siplified and optiistic MESFET noise odel in Eq. 5.. The easured noise perforances of both filters are plotted in Fiure 5.6. The value of neative resistance is dependent on the input power, resultin in noteworthy Q deradation at hih input levels. Therefore, it is essential to retune the neative resistance circuit when the input power is raised. By this action, the dynaic rane of the filter can be extended. The liit is set by stability, and the copensation cannot be increased beyond a certain point without causin oscillation. The input power level, still with a zero pass-band loss at this point, can be defined as the axiu input power for the filter. Practically, this is equivalent to the copression point as the pass-band attenuation of the filter starts to row after this power level. The easured axiu power levels and the third-order interodulation intercept points of the filters are illustrated in Fiure 5.7. Soe essential fiures of erit are tabulated in Table 5.. The spurious-free and blockin dynaic ranes SFDR/BDR are db hiher than in active inductor filters. The wide-band fourth-order filter has clearly better power handlin capabilities due to the reasons discussed in Section 5.. The spurious-free dynaic rane SFDR is defined as the distance of the input power at which the third-order IM product rises off the noise floor, to the noise floor. More widely used is the blockin dynaic rane BDR that is the distance of the input power at --db copression to the noise floor. They can be calculated by usin the followin relations: Noise floor -74 db/hz NF lo BW SFDR IIP Noise floor BDR IP Noise floor 5. 5 nd-order filter 4th-order filter NF / db 5.G.G.4G.6G.8G 4.G f / Hz Fiure 5.6 Measured noise fiures of the filters

117 - - IP - db. IP -6.4 db P out / db -5-6 IIP - db. I D /A P out / db - -4 IIP 8.5 db -7-8 Operatin current P in / db P in / db Fiure 5.7 Measured power responses of a the second-order filter and b the fourth-order filter Second-order filter Fourth-order filter enter frequency.8 GHz.4 GHz Freq. tunin rane ± MHz ± MHz - db bandwidth no. 4 MHz 4 MHz NF 9 db db Max. input power - db -6.4 db IIP - db 8.5 db SFDR / BDR 44 db / 59 db 57 db / 7 db Operatin current no. 5. A A Table 5. Measured perforance fiures of the two filters 5.6. Active Inductor Filters Active inductor resonators can be used in filters in the sae anner as neative resistor resonators, and the frequency responses are fully copatible. The only difference can be observed in the tunin ranes that are larer in active inductor filters. This is due to the inductance tunin bein able to vary the resonance frequency within wider liits than the capacitance tunin in varactors. The GaAs-HBT and Si-BJT resonators, treated in Section 4.6.5, are applied to filter desin by the author in [5.5] and [5.8]. More contribution to the issue can be found [5.9], [5.]. GaAs-HBT filter A sixth-order tri-resonator band-pass filter was desined for experiental purposes, althouh it was understood that the tunin of the individual resonators would be tedious. The filter is based on the sixth-order hebyshev prototype filter, and it was constructed by chainin three active resonators with capacitive couplin Fiure 5.8. The drawback of this approach is the very sall couplin capacitances required for narrow-band operation that are difficult to interate. This proble can be alleviated if the overall ipedance level is raised, but since the circuit was desined for on-chip easureents, the ipedance level reained 5 Ω. To facilitate the easureents, all the resonators have coon bias voltaes. This prevents precise adustent of individual resonators and full control over the response. In a practical circuit, real-tie tunin of several independent biases would be very difficult, and therefore

118 coon biasin was explored. The chip size is... The icrophotoraph of the chip is shown in Fiure IN Res Res Res OUT Fiure 5.8 Top-level scheatic of a sixth-order filter with rounded resonators The easured response is shown in Fiure 5.. The tunin rane is.7 GHz.9 GHz but due to the coon biasin, the pass-band is clearly distorted at both extrees. The frequency tunin voltae is swept fro. V to.6 V, and the Q tunin voltae is set within.5 V.95 V to ive zero loss at the pass-band. The optiu is at. GHz with MHz --db bandwidth. The operatin current of the whole filter varies fro 7 A to A dependin on the frequency adustent. At the optiu, it is 8. A, resultin in only 5-W power consuption with a V supply voltae. This is only a fraction of that of the GaAs-MESFET active inductor filters. Fiure 5.9 Microphotoraph of the chip. The physical size is... The chip contains a tripleresonator filter and a slave resonator oscillator with an output buffer aplifier Non-linearities cause severe pass-band shape deradation at hiher sinal levels. The shape reained satisfactory up to --db input level that can be rearded as the upper liit of the dynaic rane. The axiu power level can be increased by raisin the operatin current, naturally at the cost of hiher power consuption. A noise fiure estiate of 5 db was obtained with the Y-factor ethod, resultin in a dynaic rane of 4 db.

119 5 Vfbias.6 V Vfbias. V Vfbias. V -5 - S /db G.G.G.G.4G.5G Fiure 5. Measured S of the HBT filter. The tunin rane extends fro.7 GHz up to.9 GHz with V fbias settins of. V -.6 V. The optiu is at. GHz V fbias. V Si-BJT Filter A differential second-order pass-band filter with one bipolar active resonator [Section 4.6.5] was desined for GHz. The --db bandwidth is 5 MHz.7% at GHz. The topoloy of the filter is siilar to that of Fiure 5.4a with.5 pf couplin capacitors on both differential sinal lines. The intended tunin rane is.5 GHz within which the pass-band loss can be cancelled out with the Q tunin of the resonator. The siulated transfer function at both tunin extrees is plotted in Fiure 5.a. The noise fiure for the filter was siulated to be 7 db, which is aain sinificantly ore than the theoretical estiation. The siulated --db copression point is as low as - db resultin in a blockin dynaic rane of only 5 db. The realized filter suffered fro a --db transfer attenuation, what sees an apparent processin error. Therefore, the easured data is actually fro the resonator-only easureents back-annotated into the siulator. As the couplin capacitors were properly odeled, the procedure ives realistic results. The transfer functions obtained in this anner are shown in Fiure 5.b. The tunin rane is wider, but the axiu frequency is MHz lower than predicted. f/hz 5 Ibias A Ibias 5 A S / db - S / db -5 f,in.9 GHz f,ax.88 GHz G.6G.8G.G.G f / Hz -.8G.G.G.4G.6G.8G.G f / Hz Fiure 5. Filter transfer functions at both ends of the tunin rane: a siulated, and b de-ebedded easured

120 5 Active inductor filter Active neative resistance filter NF / db 5 5 -M -M M M f / Hz Fiure 5. De-ebedded easured noise fiure of the filter with the correspondin result fro Fiure 5.6 The de-ebedded easured noise fiure of the filter is 9 db Fiure 5.. With the siulated --db copression point and the loaded Q of 6, this yields BDR 6 db. alculated fro the easured resonator-only value in Section 4.6.5, it is 4 db, which is in ood areeent with the siulations. It is noteworthy that the dynaic ranes fro this differential active inductor filter are coparable in anitude with the nubers fro the sinle-ended HBT filter. This was expected keepin in ind the theory in Section Althouh the circuit is differential, the transconductors theselves are sinle-ended, and the second-order non-linearities are not cancelled in the transconductors. The power consuption of the filter is 6.6 W, dependin on the center frequency settin. Fiure 5. KRWRJUDSK RI WKH ILOWHU WKH DFWLYH GLH DUHD LV [ P 4

121 5.7 Application ase I: Local Oscillator Generation ircuit for Direct onversion Transitters in GaAs-MESFET Technoloy 5.7. Introduction The continuous drive towards hiher iniaturization and reduction of the overall cost of the handheld terinals has ade the direct conversion concept attractive aon different receivertransitter architectures. The key advantae of the direct conversion topoloy is the reduced need of RF circuitry, and that no expensive and bulky IF filters are needed. onsequently, the interation level is increased. The direct conversion architecture suffers fro soe iportant drawbacks that the desiner has to consider when selectin the appropriate architecture: the oscillator feedthrouh and oscillator backward transission in the receiver and the disturbance of the local oscillator by the power aplifier in the transitter. In a direct conversion transitter, couplin between bond wires and packae pins, and therefore leakae of the power aplifier output, corrupts the in-band local oscillator sinal spectru Fiure 5.4. In the followin, a circuit is deonstrated which, if interated on the sae die with the transitter, circuvents the effect of couplin of the power aplifier output to the local oscillator sinal. Fiure 5.4 Leakae of power aplifier output to the local oscillator sinal input 5.7. LO Sinal Generation ircuit In a direct conversion transitter, the transitted carrier frequency is equal to the local oscillator frequency. The direct conversion IQ odulator perfors both odulation and upconversion of the baseband sinal. In the odulator, the quadrature sinals, I and Q, are upconverted in the quadrature ixers. The odulator is followed by a power aplifier which aplifies the transitted sinal and provides the required output power. When eneratin an LO sinal in a direct conversion transitter, probles will occur if the odulated hih-power output sinal is coupled back to the oscillator chain. Illustrated in Fiure 5.4, the power aplifier output has a odulated hih-level wavefor and a spectru centered around the LO frequency. ouplin between bond-wires and packae pins of the noisy power aplifier output sinal corrupts the local oscillator sinal spectru and odulates this sinal. The ipure local oscillator sinal is then used for upconvertin the baseband sinal, resultin in a distorted RF output sinal. In order to save power, the power aplifier is in any cases switched on and off periodically. The switchin of the power aplifier creates sharp transitions in the sinal wavefor, and as a consequence, undesired haronics and spurious frequency 5

122 coponents worsen the proble with the leakae of the power aplifier output to the local oscillator sinal input. Distortion of the LO sinal spectru in a direct conversion transitter is alleviated if the coupled RF output sinal does not create any in-band frequency coponents. This can be accoplished by eneratin the desired local oscillator sinal to be used in the up-conversion ixer of the direct conversion transitter fro two local oscillator sinals [5.][5.]. The local oscillator sinal is enerated as shown in Fiure 5.5, where one of the input sinals is divided by two and then ixed with the other input sinal. The wanted LO sinal is then iven as one of the ixin products. The frequencies of the input sinals should be selected in such a way that they do not create any in-band ixin, interodulation or haronic sinals that derade the purity of the wanted LO sinal. If the desired local oscillator sinal bandwidth is f LO,ax f LO,in the requireent for the input sinal frequencies is: nf [ f f ] n, [ 5, 4,...,4,5] ± f LO LO LO,in, LO, ax 5. f LO f [ f f ] LO f LO, f LO LO,in, LO,ax 5.4 The power associated with hiher haronics than five of f LO and f LO is very sall and can therefore be nelected. If one of the input sinals is divided by two Eq. 5.4, the enerated sinal is in practice haronically uncorrelated with the input sinals. In this case, the coupled RF output sinal of the direct conversion transitter will ix with either f LO, f LO or with itself and enerate therefore only out-of-band coponents. The up-conversion ixers of the direct conversion transitter convert the baseband sinal around every frequency coponent in the local oscillator sinal spectru. Direct odulation IQ odulator I On chip local oscillator eneration circuit f/ w /9 w w w Q Fiure 5.5 Direct odulator with on-chip LO eneration Because there is no filterin between the upconversion ixers and the power aplifier, all the unwanted frequency coponents will start to saturate the power aplifier at lower power levels. Therefore, all ixin products but the wanted one, i.e. hiher order ixin ters at Q& LO òp& LO, n,,,..., are filtered out with an active band-pass filter, as well as the other ixin product at the irror frequency. However, dependin on the frequency plan, the irror frequency can be very close to the wanted sinal, and the filter iht not be able to filter out this coponent. Alternatively, a ixer topoloy, which reoves the irror frequency, could be used as shown in Fiure 5.6. In this confiuration, the irror frequency is reoved by 9Û SKDVH VSOLWWLQJ DQG FRPELQLQJ DQG WKH EDQGSDVV ILOWHU RQO\ DWWHQXDWHV DOO WKH XQZDQWHG ixin products. In practice, due to parasitic coponents and layout considerations the irror 6

123 frequency attenuation can be as low as dbc [5.]. The fact that the input power for the filter is constant akes this circuit topoloy very attractive for applyin fully onolithic active resonator filters. The inherently poor noise properties of these filters do not liit their usability in the application, and their liited lare-sinal handlin can be copensated with aplifyin output buffers. For the proposed circuit to function as explained, it is essential that the circuit is interated on the sae die with the transitter. If the sinal is taken off-chip at soe point, for exaple to perfor filterin, the RF output can couple to this pin and distort the enerated local oscillator sinal as explained Desined MESFET ircuit A test circuit of the proposed on-chip local oscillator sinal eneration circuit was desined. The oal was to test the practical feasibility of the presented idea. The topoloy of the realized circuit is shown in Fiure 5.5. The required buildin blocks were fabricated usin the standard *&DUFRQL '67 WHFKQRORJ\ 7KH JDWH ZLGWKV RI WKH DFWLYH GHYLFHV DUH P and the cut-off frequency f T GHz. The threshold voltae of the active devices is low, i.e. V t -.8 V, which, toether with the lack of enhanceent ode devices, akes the process suitable for analo desin only. onsequently, the divider, which is a diital circuit, is not very well optiized with this technoloy. The GaAs technoloy was selected because of easy access to the foundry, and because hih quality and well-odeled inductors with Q values over and varactors that facilitate the ipleentation of the active band-pass filter. However, the circuit can be realized with any technoloy, providin that inductors and varactors are available. f f /9 out Fiure 5.6 Mirror reection topoloy of the local oscillator eneration The taret syste for the desined circuit is a DS 8 direct conversion transitter. The desired output frequency of the proposed circuit is the sae as the TX band of the syste, i.e MHz. The input frequency f LO can be selected accordin to Fiure 5.7, which is a raphical equivalent of Eq. 5.. The wanted output frequency is then produced by selectin the other input frequency f LO appropriately. Tunin of the output sinal is carried out by varyin f LO accordinly. In our case, the input frequency f LO was fixed as 464 MHz in the specifications, and the frequency f LO is therefore 94 7 MHz. The desined circuit blocks are all differential; therefore input and output buffers are needed to convert the sinle-ended inputs to differential and the differential output to sinle-ended, as ZHOO DV WR PDWFK WKH FLUFXLW WR IRU IDFLOLWDWLQJ PHDVXUHPHQWV $ PLFURSKRWRSUDSK RI WKH desined circuit is presented in Fiure 5.8. Frequency divider GaAs flip-flop circuits as 6-NOR ate ede triered flip-flops and EL aster-slave flipflops have been adapted fro silicon MOS or bipolar technoloies. With a sliht odification of the conventional 6-NOR ate D-flip-flop [5.], we et a fully syetrical D-flip-flop circuit that can be used as a frequency divider Fiure 5.9a. The axiu tolin frequency 7

124 of the flip-flop is f ¼t pd where t pd is the ean propaation delay tie in seconds per ate of the flip-flop when used as a divide-by-two circuit. The advantae of the fully syetrical topoloy is that a sinle-phase clock sinal can be used as the input sinal and that the output is differential. Fiure 5.7 Accordin to Eq. 5. possible selections of frequency f LO ûi LV WKH GLVWDQFH RI WKH UHVSHFWLYH KDUPRQLF RU PL[LQJ SURGXFW WR WKH FHQWHU IUHTXHQF\ ] DQG PLQ ûi PDUNV WKH iniu allowed distance for out-of-band sinals. If f LO is selected within the allowed shaded areas no in-band frequency coponents are created Fiure 5.8 Microphotoraph of desined circuit. The inductor in the upper riht corner is not a part of the circuit 8

125 vdd w w / w / out A B Fiure 5.9 a Ipleented fully syetrical frequency divider; b BFL NOR-port The ost straihtforward way of ipleentin a NOR ate is to connect transistors in parallel with an active load, i.e. a D-MESFET with ate connected to source. In this way, up to five-input NOR ates are possible. Because only D-MESFETs are provided by the foundry, a level shifter stae for V out -to-v in copatibility is needed. The loic block is biased with the positive supply voltae, and the level shifter stae that is ipleented with two MESFETs connected as diodes in series is biased by a current source connected to the neative supply voltae. The level shifter stae is connected to the loic block output throuh a coon source D-MESFET, and hence the level-shifter circuit acts as a buffer stae hence the nae, buffered FET loic, BFL, Fiure 5.9b. The nuber of diodes in the level shifter circuit is related to the low threshold voltae V t of the active device used as the loic eleent. If the ate to source voltae of the coon source transistor in the level shifter is approxiately. V, and the voltae over a diode is.8 V, we et approxiately n is the nuber of diodes vss V t..8n V 5.5 Fro Eq. 5.5 it is easily seen that two diodes are needed in the level shifter stae. In order to iniize the capacitive load of the NOR ate and the current throuh the ate, the input transistors are of iniu width. The output wavefor of the divider is a square wave with V pp.5 V. The power consuption of the BFL ate, and therefore the divider, is directly proportional to the needed supply voltaes and to the threshold voltae of the active device. The level shifter stae is biased between the neative and positive supply voltaes of V and -.5 V. This, toether with the low V t, akes the power consuption of the GaAs MESFET divider hih, naely 8 W. With a ore suitable choice of process or technoloy, it is possible to desin divide-by-two circuits that have a power consuption less than W. The used technoloy is by no eans suitable for diital I desin. Mixer Fiure 5. introduces the realized ixer, a doubly-balanced Gilbert analo ultiplier. The doubly-balanced connection of this ixer will cancel even-order spurious coponents at the IF output, which is iportant in onolithic ixer ipleentations, where spurious sinals can interfere with other circuits interated on the sae I throuh parasitic couplin paths. The ixer consists of a linear voltae-to-current converter coprisin coon-source FETs in saturation. The output current of the RF transconductance stae is coutated by the local oscillator. The coutation process conserves the total current and therefore a fraction of the RF current is downconverted, the reainin RF current is upconverted around one or ore 9

126 haronics of the LO. The voltae conversion ain of the Gilbert cell ultiplier is set by the choice of the transconductance and the load resistance. The lare voltae swin of the frequency divider output, used as the LO to the ixin core, quickly switches the FETs fro their saturation reion to their cutoff reion, and vice versa. Thus, the switchin FETs that are biased at ½I dss operate like ideal switches. The transconductance FETs are biased in I dss, and all the FETs of the ixer core are of equal size. Due to the poor filter lare sinal handlin capabilities, the axiu output power of the ixer is liited to -5 db. The conversion of the sinle-ended input sinal to differential is SHUIRUPHG E\ D PDWFKHG GLIIHUHQWLDO SDLU,QSXW PDWFKLQJ WR LV UHDOL]HG ZLWK D FRPPRQ ate transistor ivin an input return loss better than db. OUT OUT LO RF LO Fiure 5. Gilbert cell ixer with input buffer Active band-pass filter A fully interated band-pass filter which attenuates all unwanted ixin results and spurious frequencies was desined for the circuit. The ipleented filter is a varactor-tuned parallel L resonance circuit Fiure 5., where the cross-coupled differential active neative resistance circuit was used for loss copensation of the resonance tank [5.][5.4][5.][5.6]. As shown previously, if the transistor is odeled with the transconductance, the output conductance ds and the ate-source capacitance c s, the input adittance is siply Y in ds cs 5.6 This corresponds to a neative resistance in parallel with a capacitance > ds. The aount of neative resistance can be altered by adustin the current and accordinly the transconductance of the source coupled MESFETs, and thus the Q value of the tank is chaned. If too uch neative resistance is put in parallel with the resonance tank and the resultin tank resistance becoes neative, the circuit will be unstable and starts actin like an oscillator. Therefore, extra care has to be taken when adustin the tank Q. The noinal center frequency of the filter is.75 GHz. The center frequency was ade tunable by varyin the voltae of varactor-connected MESFETs. The sizes of the varactors are 6 P JLYLQJ D FDSDFLWDQFH UDQJH RI ± S HDFK 7KH IUHTXHQF\ WXQLQJ UDQJH LV MHz. When adustin the capacitance of the varactor pair the series resistance is also chaned, which affects the tank Q-value. Therefore, the tank Q-value has to be tuned as the frequency is tuned. In a production circuit, however, a control circuit would have to be used for eneratin control bias voltaes for the filter. The control circuitry could be realized with a dual-loop

127 aster slave schee as in [5.][Section 5.5.], which autoatically adust the bias voltaes to aintain -db pass-band insertion loss and stable center frequency in the presence of process tolerances and variations with operatin conditions. The dynaic rane of the filter has less sinificance in this application than usually, as the input power level reains constant. Thus, excess in-band noise enerated by the copensation circuit is of little interest. However, bein the lowest in the whole circuit, the copression point of the filter dictates the axiu attainable output power. The copression point of a narrowband filter is easily db lower than that of the copensation transconductor itself [Section 5.] if the passive inductor Q is low. Thus, hiher output power levels are possible only with aplification after the filter. In out In V bias,f Vbias,Q Fiure 5. Fully onolithic band-pass filter with output buffer Experiental Results The perforance of the circuit was verified with on-chip easureents. In Table 5., soe key fiures of the easured circuit are listed. Siulations were perfored with Hewlett Packard Microwave Desin Syste. Very ood areeent between easured results and siulations of the AD odel was achieved. P f lo 464 MHz - db Q-value 45 P f lo 94-7 MHz - db OP - db P f lo MHz -5 db Spurious frequencies -4 dbc filter f tunin GHz Noise MHz -47 db/hz Table 5. Soe perforance results of the presented circuit Fiure 5.a shows the easured output spectru of the fabricated circuit. The output power of the wanted sinal at.7 GHz is -5 db. The input power levels are kept at their noinal values, - db. The axiu output power is liited by the filter lare-sinal characteristics. The easured --db output copression point is - db. If the Q value is tuned siultaneously as the input power is increased, the filter power handlin capabilities are iproved. This is due to the deradation of the neative resistance when the input sinal level is raised. The frequency tunin rane of the filter is.56.9 GHz. In Fiure 5.b the frequency responses of the band-pass filter at both extrees and at the noinal frequency are shown. As

128 the input frequency is tuned, the spectral coponents ove accordinly alon the frequency axis. The loaded Q value was easured as the ratio of the center frequency to the two-sided -- db band-with and was found 45. Fiure 5. a Power spectru at output; b Frequency response of band-pass filter at both extrees and at the noinal frequency The ixer irror frequency at 74 MHz is attenuated dbc. In principle, the irror coponent attenuation can be iproved with a filter of hiher order. As a consequence of addin ore staes to the filter, the bias arraneents becoe coplicated and sensitiveness towards process variations rise. Alternatively, the ixer topoloy with reection of the irror frequency could be used. The phase noise of the output sinal in an issue in LO sinals. The ixer and the band-pass filter do not add hardly any phase noise to the sinal in theory, and the phase noise of the realized divider is very sall accordin to a coparison ade in [5.4]. Hence, the phase noise of the output sinal is practically the su of the phase noises of the two input sinals. The noise floor of -47 db/hz was easured at MHz offset fro the filter center frequency. To et a clear picture of how the noise floor of the proposed circuit would affect the total noise of a direct conversion odulator, the local oscillator sinal eneration circuit should be interated on the sae chip with the odulator. In soe applications the noise floor of the direct conversion odulator should be around -4 db/hz. If it were assued that the odulator ixer LO leakae were 5 db and the ixer were passive, which also attenuates the LO-based noise floor, the noise floor at the output of the local oscillator sinal eneration circuit should be -65 db/hz onclusions A local oscillator eneration circuit for a direct conversion transitter has been desined. The circuit deonstrates that a local oscillator sinal can be enerated fro two reference sinals as proposed, which stronly reduces the effect of couplin between the transitter antenna path and the local oscillator, provided that the circuit is interated on the sae die with the odulator. Thus, on-chip LO filterin could provide iproved perforance, and due to loose specifications, be the first application for active icrowave filters in cellular systes, even thouh the hih noise floor caused by the active device can couple to the sinal path and increase the noise of the odulator. The results obtained fro this circuit ive valuable inforation for further developent of the on-chip LO eneration circuit. With a ore suitable process a lare iproveent considerin especially the power dissipation can be achieved. The studies and results presented in this section have been published in [5.5] and [5.6].

129 5.8 Application ase II: Local Oscillator Generation ircuit for Direct onversion Transitters in BiMOS Technoloy 5.8. Introduction Referrin to the previous section, BiMOS technoloies offer several benefits for LO eneration circuit desin. The diital part, i.e. the divider with its auxiliary circuits, can be desined with copleentary MOS transistors, which sinificantly reduces the overall power consuption and the die area in spite of the reater coplexity. Proven MOS topoloies for the divider can be easily applied. In addition, bipolar transistors with hih-frequency capabilities are available for RF sections. The irror-reection topoloy shown in Fiure 5. was experiented in this circuit. The WHVW FKLS ZDV IDEULFDWHG ZLWK D P %L&6 VLOLFRQ SURFHVV HVSHFLDOO\ WDUJHWHG IRU 5,& desin. The process provided spiral inductors and varactors for uncoplicated ipleentation of the band-pass filter, and the neative resistor filter topoloy could be eployed. The circuit was realized as differential with -Ω terinations so that the power specifications are coparable to the sinle-ended version. The LO low frequency input port has a very hih input ipedance, and therefore the input quantity cannot be power but voltae. A iniu input voltae of 5 V sinle-ended was used as a desin taret. Fiure 5. Mirror-reection topoloy for LO eneration circuits 5.8. Divider The divide-by-two frequency divider provides the low-frequency sinal to the ixers. The irror-reection operation requires output sinals in all four quadratures, which is easily realizable with the conventional latch-based topoloy. The divider block consists of an input coparator, a divider core and output buffers. The low-frequency sinal LO is aplified and potentially clipped in the twin-stae input coparator shown in Fiure 5.4a. The coparator staes are noral MOS-inverters with coon-ode feedback for D-offset stabilization. The siulated ain at 464 MHz is 4 db, which enables functionin even at low input aplitudes. The conventional MOS-latch Fiure 5.4b is used in the divider core. The cross-coupled connection of two latches fors a -divider with direct I and Q outputs Fiure 5.5. The axiu division frequency is partly deterined by the bias current of the latches. In this case, D $ ELDV FXUUHQW HQVXUHV RSHUDWLRQ XS WR *] Output buffers were needed for preventin the ixers fro loadin the divider and for liitin the sinal swin to its optial value for the ixers, 5 V.

130 Fiure 5.4 a Input coparator; b Latch circuit Fiure 5.5 Divide-by-two frequency divider based on latch circuits Siulated results The siulated input and all four output wavefors of the divider core are shown in Fiure 5.6. The coparator input sinal aplitude is 5 V. The siulated power consuption for the whole diital part is 4 W. Measured results The diital section of the circuit the input coparator, the divider core and the output buffers was processed also as a stand-alone unit for testin purposes. The easured output wavefors are plotted in Fiure 5.7,W PXVW EH QRWHG WKDW WKH FLUFXLW ZDV ORDGHG E\ durin the easureents instead of the actual hih input ipedance of the ixer. Therefore, the easured output swin is very low. In addition to the actual input frequency 464 MHz, a lower frequency of MHz was also used. At 464 MHz the phase error due to the slihtly different sinal leads becoes visible, as well as the liited bandwidth of the used 5-MHz saplin oscilloscope. 4

131 The unloaded power consuption of the diital part was easured to be.7 W bein slihtly saller than predicted. The axiu operatin frequency is.9 GHz with an input sinal aplitude of 75 V. At hih frequencies, the input coparator ain becoes insufficient for sall input sinals. The iniu operational input aplitude is 5 V at the noinal frequency of 464 MHz. The lower the frequency the ore the input sinal is attenuated due to the -pf couplin capacitors in the coparator, and the iniu functional input level is raised. Fiure 5.6 Siulated divider core wavefors: A input, B I-output I, Q-output 5.. I Q. I Q V 5Ω / V. -5. V 5Ω / V n -.n..n.n t / s -.n -5.n. 5.n.n t / s Fiure 5.7 a Measured output of the stand-alone divider at f in MHz, v in V unbal, 75 V bal; b Measured output of the stand-alone divider at f in 464 MHz, v in V unbal, 75 V bal 5

132 5.8. Polyphase Filter The hih-frequency input sinal LO is fed to a two-stae balanced polyphase filter, shown in Fiure 5.8, which creates the quadrature sinals needed. The I and Q branches are then buffered and aplified with differential pairs for copensatin the loss in the polyphase filter. Fiure 5.8 Polyphase filter 4. typ in ax.5 typ in ax a error / db phase error / de G.8G.G.G.4G -..6G.8G.G.G.4G f / Hz f / Hz Fiure 5.9 a Manitude error of the polyphase filter; typical and the siulation corners; b Phase error of the polyphase filter; typical and the siulation corners Mixers Accordin to Fiure 5., the cobined output of the ixers is v o Asin t sin t Acos t cos t Acos t 5.7 6

133 and the irror frequency AFRV& & t is suppressed, provided that the phase shift between the branches is exactly 9º. If a phase error û DQG D UHODWLYH DPSOLWXGH ûa error between the branches occur, as inevitably happens in practice, the irror-reection ratio MRR becoes P MRR P a a a cos ϕ a cos ϕ 5.8 Theoretically one deree of total phase error without anitude error corresponds to an MRR RI G%F ZKHUHDV D VROH G% DPSOLWXGH HUURU ûa.89 will lead to a hih 5-dBc MRR. To avoid these deradations, one ust take care of ood atchin and uncoproisin syetry between the I and Q branches in the layout desin. The ixer cores are conventional balanced Gilbert cells optiized for the iven input sinal levels. Their siulated power consuption is W each. The scheatic diara of the whole ixer without the bias arraneents can be seen in Fiure 5.. Fiure 5. Gilbert-cell ixer onversion ain / db ain typ in ax IR typ in ax IR / dbc P / db f -f 7 MHz f f 74 MHz typ in ax P Ω / db -9.G.5G.G.5G.G f / Hz Fiure 5. a Siulated conversion ain and irror reection ratio; b Siulated output spectru of the ixers 7

134 5.8.5 Band-Pass Filter The band-pass filter shown in Fiure 5. is a second-order capacitively coupled L resonator with a controllable neative resistance for Q enhancin. The neative resistance circuit is a cross-coupled transistor pair that copensates the low quality factors < of the interated spiral inductors and enables arbitrarily hih unloaded Qs for the L resonator. The aount of neative resistance is adusted by chanin the bias current. The filter center frequency is controlled by a pair of varactor diodes in the back-to-back confiuration. The siulated tunin rane of the filter is GHz when the frequency tunin voltae is varied fro.v to.v. The Q tunin current is siultaneously set for iniu filter insertion loss A 4 A, resultin in the power consuption startin fro W. The axiu siulated stable loaded Q is around. For testin purposes, the band-pass filter was also processed as a stand-alone circuit. The power handlin requireents in the actual LO eneration circuit P out - db required an HPLWWHUGHJHQHUDWLRQ UHVLVWRU RI LQ WKH QHJDWLYH UHVLVWRU FLUFXLW EXW WKLV ZDV RPLWWHG LQ the stand-alone filter. The tunin rane is practically identical, but the Q control currents are lower than A due to the lack of the ain-decreasin eitter resistor the power consuption is less than 5 W. Fiure 5. Neative resistance resonator filter Measured results The stand-alone filter responses iniu, axiu and noinal are shown in Fiure 5.a. The easured tunin rane is.48.8 GHz Fiure 5.b, and dependin on the tunin current settin, the power consuption ranes fro W to.5 W. The easured loaded Q value at id-band is. The easured tunin rane is shifted down 5 MHz fro the expected value. This is caused by the inaccurate varactor foundry odel that predicted too low varactor capacitances. In addition, only one varactor size was odeled, which ade the selection of noinal resonance frequencies very coarse. The lare-sinal perforance of the stand-alone filter is depicted in Fiure 5.4. Due to the hih Q l, the copression point is very low -67 db. As the easured noise fiure is approxiately db at the whole frequency band, the dynaic rane becoes 8 db Output Buffers The output buffer is an ordinary eitter-coupled pair followed by an eitter follower. Its function is to prevent the terination ipedances fro loadin the hih-ipedance output of the band-pass filter, aplify the sinal for the axiu output power, and atch the output to 8

135 7KH PDWFKLQJ GUDZV D ODUJH DPRXQW RI '& FXUUHQW DQG WKHUHIRUH WKH VLPXODWHG SRZHU consuption of the output buffer is 65 W representin 65% of the whole circuit value. If used as a part of a bier syste with hiher interconnection ipedances, the power consuption can be reatly reduced.. V f, V V f,8 V V f,6 V f control voltae Q control current S / db V f / V.4 I q / A G.4G.5G.6G.7G.8G.9G.G f / Hz f / GHz.5 Fiure 5. a Frequency response of the stand-alone filter; b enter frequency and Q control -5-6 IP -67dB -7-8 IIP -58dB P out / db P in / db Fiure 5.4 opression and IM perforance of the stand-alone filter, f.65 GHz Entire Syste Siulated results The siulated total ain fro the f input to the output is 6 db OP db. The noinal power consuption of the total circuit is predicted to be 86 W 4 W W A, ost of which 65 W is consued in the output buffer. Due to the perfect balance in the siulator, the siulated noinal irror-reection values are infinitesial. 9

136 Measured results In Fiure 5.5a, the syste frequency response is plotted. alculated fro the plot, the loaded quality factor is as hih as, coplyin well with the siulations. The increase of Q in the full circuit can be explained with hiher source and load ipedances seen by the filter in the syste. The tunin rane is shifted further downwards, which iplies that the syste filter is loaded by extra capacitance not accounted for by the siulator. By redesinin the L network, this proble should be solved. The power response is shown in Fiure 5.5b. As seen, the specified output power of - db is reached in stron copression. The output copression point is - db, and the total ain is db. The output spectru of the whole circuit is shown in Fiure 5.6. The irror reection with the filter on is >7 dbc, and with the filter off 9 dbc. The LO frequency is attenuated 6 dbc and dbc, respectively. The low filter-off reection ratios are due to the iperfect balance of the input and output baluns in the easureents, thouh tried, and the process-inflicted isatch errors on the chip. 5 - P ou t -P in / db 5 f no.6 GHz P GHz / db - - OP - db f in.5 GHz f ax.7 GHz -4-5.G.4G.5G.6G.7G.8G f out / Hz P GHz / db Fiure 5.5 a Output frequency response; b Power-sweep response, P LO - db MHz: -6 dbc 64 MHz: < -7 dbc P out / db -6 P out / db M.G.5G.G.5G.G f / Hz -9.G.4G.6G.8G.G f / Hz Fiure 5.6 a Output spectru, P LO - db; b Spectru close-up

137 The easured power consuption is W, which coplies with the siulated value extreely well. The noise floor easured at -MHz offset fro the carrier is -9 db/hz. For curiosity, the phase noise at the output was also easured. When copared to the source phase noises the additional noise enerated by the circuit is sall but nevertheless detectable. The ost obvious chane can be detected in the raised noise floor. Fiure 5.7 Photoraph of BiMOS LO eneration circuit The icrophotoraph of the processed chip is shown in Fiure 5.7. As the desin is obviously sensitive to ibalance, the layout was drawn as syetrically as possible, and all the differential sinal leads are exactly of the sae lenth. All the differential blocks are exact irror copies of each other. The active chip area is.9 x.7 without the pads and the pad rin. As seen fro the photoraph, the actual functional blocks occupy rouhly only a half of this area, since the layout is pad-restricted the pads are set for five-finer GSGSG probe KHDGV ZLWK P SLWFKHV 7KH UHVW RI WKH DUHD LV ILOOHG ZLWK 5 JURXQGLQJ FDSDFLWRUV IRU WKH D voltae lines oparisons and onclusions Finally, it is interestin to copare soe key perforance fiures of both realized LO eneration circuits. They have been athered in Table 5.4. The results are very uch coparable, since the ost crucial specification, the output power copression, is approxiately the sae in both realizations. The differences in other properties ste fro the characteristics of the GaAs-MESFET and BiMOS technoloies and the circuit topoloies chosen. It is very uch obvious that the MESFET technoloy is not suitable for the application, except for deonstration purposes.