A NEW LINEARIZATION METHOD FOR CANCELLATION OF THIRD ORDER DISTORTION

Transcription

1 A NEW LINEARIZATION METHOD FOR CANCELLATION OF THIRD ORDER DISTORTION by KONRAD MIEHLE A hesis submied o he fauly of The Universiy of Norh Carolina a Charloe in parial fulfillmen of he requiremens for he degree of Maser of Siene in he Deparmen of Elerial and Compuer Engineering Charloe 00 Approved by: Dr. Thomas P. Weldon Dr. Farid Tranjan Dr. H. Cherukuri

2 00 Konrad Miehle ALL RIGHTS RESERVED ii

3 ABSTRACT iii KONRAD MIEHLE. A New Linearizaion Mehod for Canellaion of Third Order Disorion (Under direion of DR. THOMAS P. WELDON This hesis presens a new linearizaion mehod ha offers he poenial for reduing size, power, and os of wireless devies suh as ellular phones. In he new mehod, a less linear amplifier is ombined wih a more linear amplifier in suh a manner ha he ombined performane grealy exeeds he performane of he wo original amplifiers. This unexpeed resul is ahieved in a simple arhieure ha lends iself o sraighforward inegraed irui implemenaion wihou he need for bulky or expensive exernal omponens. To gain a beer undersanding of prior linearizaion ehniques, an overview of funion, advanages and disadvanages of prior linearizaion ehniques is presened. The simpliiy of he new linearizaion mehod is shown wih a simple mahemaial relaionship for anellaion of hird order disorion in erms of gain and hird order oupu inerep poins of he wo amplifiers. In order o prove he simple mahemaial relaionship, a ross-oupled differenial pair was implemened as a fully inegraed irui in a 0.5 m omplemenary meal oxide semionduor (CMOS proess. Measured resuls show very good improvemen for low inpu power levels. The prooype was esed for differen frequenies and shows an improvemen of db of hird order disorion for a frequeny of 0 MHz. Harmoni balane simulaions wih Agilen Advaned Design Sysem (ADS were performed and show good agreemen wih he measured resuls. The new linearizaion mehod also works for devies oher han amplifiers suh as mixers.

4 ACKNOWLEDGEMENTS iv I would like o hank he people who made i possible for me o sudy and work a he Universiy of Norh Carolina a Charloe. I would like o hank Dr. Weldon for his guidane. He gave me he opporuniy o work independenly on always hallenging projes. I am hankful o Dr. Tranjan and Dr. Cherukuri for supporing me and being members of my ommiee. Konrad Miehle July 00

5 TABLE OF CONTENTS v LIST OF FIGURES viii LIST OF TABLES xiii LIST OF ABBREVIATIONS xv CHAPTER : INTRODUCTION. Thesis Saemen. Organizaion 4 CHAPTER : PRIOR LINEARIZATION METHODS 5. Feedbak Linearizaion 5.. RF Feedbak 6.. Envelope Feedbak 7.. Polar Loop Feedbak 8..4 Caresian Loop Feedbak 0. Feedforward. Predisorion 4.. RF/IF Predisorion 5.. Adapive Baseband Predisorion 6.4 Envelope Eliminaion and Resoraion (EER 7.5 Linear Amplifiaion wih Nonlinear Componens (LINC and Combined Analog Loked Loop Universal Modulaor (CALLUM 9 CHAPTER : THEORETICAL BACKGROUND 5. Modeling of Nonlinear Behavior 5. Effes of Nonlineariy in Radio Sysems

6 vi.. Two-Tone Tes.. Desensiizaion 4.. Cross Modulaion 5..4 Inermodulaion 5. Measures of Nonlineariy 8.. db Compression Poin 9.. Inerep Poin 4.4 Phase Disorion 48 CHAPTER 4: DESCRIPTION OF NEW LINEARIZATION METHODS Basi Priniple of New Mehods Canellaion Condiions Using Power Series Expansion 5 4. Canellaion Condiion in Terms of Oupu Inerep Poin Theoreial Evaluaion of Canellaion Condiion Preliminary Mahemaial Analysis of Canellaion Condiion 57 CHAPTER 5: INTEGRATED CIRCUIT REALIZATION OF NEW LINEARIZATION METHOD Cross-oupled Differenial Pair Basi Tes Seup Measured Resuls Measured Resuls for 0 MHz Tesed wih Wafer Probe Saion Measured Resuls for Addiional Frequenies Power Effiieny Consideraions 0

7 5.5 Summary of Measured Resuls 0 vii CHAPTER 6: SIMULATION OF CROSS-COUPLED DIFFERENTIAL PAIR 04 CHAPTER 7: CONCLUSION 6 7. Summary of Work 6 7. Fuure Researh 8 REFERENCES APPENDIX A: POWER SERIES EXPANSION 6 A. Seond Order Nonlineariy 6 A. Third Order Nonlineariy 6 A. Seond and Third Order Nonlineariies 6 A.4 Seond and Third Order Nonlineariies wih Two-Tone Tes Signal 7 APPENDIX B: MATHCAD PROGRAM FOR TWO-STAGE AMPLIFIER 8 APPENDIX C: SMALL SIGNAL GAIN OF CROSS-COUPLED DIFFERENTIAL PAIR 40 APPENDIX D: TEST PRINTED CIRCUIT BOARD LAYOUT 44 APPENDIX E: BSIMV TRANSISTOR MODEL 45 APPENDIX F: ADS SCHEMATIC USED FOR SIMULATION OF TWO-STAGE AMPLIFIER 47 APPENDIX G: MATHCAD PROGRAM FOR FOUR-STAGE AMPLIFIER LINEARIZATION 48 APPENDIX H: RESULTS OF PROTOTYPE USING DISCRETE DEVICES 5

8 viii LIST OF FIGURES Figure. RF feedbak linearizaion mehod...6 Figure. Envelope feedbak linearizaion...7 Figure. Polar Loop feedbak linearizaion...9 Figure.4 Caresian Loop feedbak linearizaion...0 Figure.5 Basi feedforward linearizaion... Figure.6 Predisorion linearizaion...5 Figure.7 Adapive baseband predisorion linearizaion...7 Figure.8 Envelope eliminaion and resoraion...9 Figure.9 Linear amplifiaion using nonlinear omponens...0 Figure. Inpu and oupu signals of linear amplifier...7 Figure. Transfer haraerisi of a linear amplifier...7 Figure. Seond order nonlineariy...8 Figure.4 Third order nonlineariy...9 Figure.5 Seond and hird order nonlineariies... Figure.6 Two-Tone signal... Figure.7 Inermodulaion disorion...8 Figure.8 db-ompression poin...4 Figure.9 Third order inerep poin IP...45 Figure.0 Derivaion of hird order inerep poin...46 Figure. Seond order inerep poin IP...47 Figure 4. Blok diagram of new linearizaion mehod...5

9 ix Figure 4. Ideal limier volage ransfer haraerisi...57 Figure 4. Topology of new linearizaion mehod...58 Figure 4.4 Calulaed (Mahad oupu power levels...60 Figure 4.5 Calulaed (Mahad hird order disorion levels...6 Figure 4.6 Calulaed (Mahad gain...6 Figure 5. Cross-oupled differenial pair...65 Figure 5. Basi es seup for wo-one measuremen...7 Figure 5. Measured oupu frequeny sperum of amplifier A a 0 MHz...74 Figure 5.4 Measured oupu frequeny sperum of amplifier A a 0 MHz...74 Figure 5.5 Measured oupu frequeny sperum of omposie amplifier a 0 MHz...75 Figure 5.6 Measured signal leakage beween inpu and oupu of irui a 0 MHz...75 Figure 5.7 Measured linear signals and hird order produs of A, A and omposie amplifier a 0 MHz...8 Figure 5.8 Measured hird order oupu inerep poin OIP of A, A and omposie amplifier a 0 MHz...8 Figure 5.9 Measured gain of A, A and omposie amplifier a 0 MHz...8 Figure 5.0 Modified es seup for wo-one measuremens...8 Figure 5. Measured oupu frequeny sperum of amplifier A a MHz...85 Figure 5. Measured oupu frequeny sperum of amplifier A a MHz...86 Figure 5. Measured oupu frequeny sperum of omposie amplifier a MHz...86 Figure 5.4 Measured signal leakage beween inpu and oupu of irui a MHz...87 Figure 5.5 Measured oupu frequeny sperum of amplifier A a MHz...90 Figure 5.6 Measured oupu frequeny sperum of amplifier A a MHz...90

10 x Figure 5.7 Measured oupu frequeny sperum of omposie amplifier a MHz...9 Figure 5.8 Measured signal leakage beween inpu and oupu of irui a MHz...9 Figure 5.9 Measured oupu frequeny sperum of amplifier A a 0 MHz...94 Figure 5.0 Measured oupu frequeny sperum of amplifier A a 0 MHz...94 Figure 5. Measured oupu frequeny sperum of omposie amplifier a 0 MHz...95 Figure 5. Measured signal leakage beween inpu and oupu of irui a 0 MHz...95 Figure 5. Measured oupu frequeny sperum of amplifier A a 0 MHz...98 Figure 5.4 Measured oupu frequeny sperum of amplifier A a 0 MHz...98 Figure 5.5 Measured oupu frequeny sperum of omposie amplifier a 0 MHz...99 Figure 5.6 Measured signal leakage beween inpu and oupu of irui a 0 MHz...99 Figure 6. Simulaed (ADS oupu frequeny sperum of amplifier A a 0 MHz..07 Figure 6. Simulaed (ADS oupu frequeny sperum of amplifier A a 0 MHz..07 Figure 6. Simulaed (ADS oupu frequeny sperum of omposie amplifier a 0 MHz...08 Figure 6.4 Simulaed, measured and alulaed linear oupu power of amplifier A a 0 MHz... Figure 6.5 Simulaed, measured and alulaed linear oupu power of amplifier A a 0 MHz... Figure 6.6 Simulaed, measured and alulaed linear oupu power of omposie amplifier a 0 MHz... Figure 6.7 Simulaed, measured and alulaed hird order disorion of amplifier A a 0 MHz...4

11 xi Figure 6.8 Simulaed, measured and alulaed hird order disorion of amplifier A a 0 MHz...4 Figure 6.9 Simulaed, measured and alulaed hird order disorion of omposie amplifier a 0 MHz...5 Figure 7. Four-sage amplifier linearizaion...9 Figure 7. Simulaed (Mahad hird order disorion suppression of four-sage amplifier linearizaion... Figure 7. Simulaed (Mahad hird order disorion anellaion of four-sage amplifier linearizaion... Figure 7.4 Topology for four-sage amplifier linearizaion using differenial pairs...5 Figure 7.5 Simulaed (ADS linear and hird order disorion oupu power of amplifier A using four-sage linearizaion...8 Figure 7.6 Simulaed (ADS linear power and hird order disorion of error signal showing reduion of linear omponen...9 Figure 7.7 Simulaed (ADS linear oupu power and hird order disorion of final oupu from Fig. 7.4 showing hird order suppression...0 Figure C. Two half-iruis of ross-oupled differenial pair...4 Figure C. Linearized small signal irui for firs half-irui...4 Figure C. Linearized small signal irui for seond half-irui...4 Figure D. Fron side of prined irui board used in es seup in Fig. 5. and Fig Figure D. Bak side of prined irui board used for es seup in Fig. 5. and Fig

12 xii Figure F. ADS shemai of ross-oupled differenial pair used for simulaions wih ADS...47 Figure H. Blok diagram of new linearizaion mehod using disree devies...5 Figure H. Top layou for es board for disree devies used in es seup in Fig. H..5 Figure H. Tes seup for irui in Fig. H. using disree devies...55 Figure H.4 Measured oupu frequeny sperum of amplifier A (ERA5 a 5 MHz using es seup from Fig. H...56 Figure H.5 Measured oupu frequeny sperum of amplifier A (MAR a 5 MHz using es irui of Fig. H...56 Figure H.6 Measured oupu frequeny sperum a 5 MHz of amplifiers A+A using es seup of Fig. H....57

13 xiii LIST OF TABLES Table. Prior linearizaion ehniques... Table. Summary of hird order disorion suppression of prior linearizaion ehniques....4 Table. Summary of linear and higher order erms of power series expansion... Table 4. Parameer of amplifiers A and A for Mahad alulaion...59 Table 5. Proess parameers for NMOS ransisor for a 0.5m CMOS proess...70 Table 5. Summary of measured resuls for 0 MHz using seup of Fig Table 5. Summary of bias ondiions for anellaion of hird order disorion...78 Table 5.4 Summary of measured resuls for MHz using seup from Fig Table 5.5 Summary of measured resuls for MHz using seup from Fig Table 5.6 Summary of measured resuls for 0 MHz using seup from Fig Table 5.7 Summary of measured resuls for 0 MHz using seup from Fig Table 5.8 Summary of hird order disorion suppression and improvemen in OIP wih variaions in frequeny...0 Table 6. Comparison of bias ondiion of prooype and simulaion...05 Table 6. Summary of simulaion resuls for 0MHz using shemai in Appendix F.09 Table 7. Independen parameers for four-sage amplifier linearizaion... Table 7. Summary of gain and hird order inerep poin of four-sage amplifier linearizaion from Fig Table 7. Summary of bias urrens and aspe raios for four-sage amplifier linearizaion...7

14 xiv Table H. Summary of supply volages and urrens of amplifiers A and A from Fig. H...54 Table H. Summary of measured resuls for 5 MHz using irui in Fig. H. wih es seup in Fig. H....58

15 xv LIST OF ABBREVIATIONS ACPR ADS AM/AM AM/PM CALLUM CDMA CMOS DSP EER FET IF IIP IIP IMD IP IP IP IPdB LINC LNA LO Adjaen Channel Power Raio Advaned Design Sysem Ampliude Modulaion o Ampliude Modulaion onversion Ampliude Modulaion o Phase Modulaion onversion Combined Analog Loked Loop Universal Modulaor Code Division Muliple Aess Complemenary Meal Oxide Semionduor Digial Signal Proessing Envelope Eliminaion and Resoraion Field Effe Transisor Inermediae Frequeny Seond order Inpu Inerep Poin Third order Inpu Inerep Poin Third order Inermodulaion Produ Inerep Poin Seond order Inerep Poin Third order Inerep Poin Inpu db Compression Poin Linear amplifiaion using Nonlinear Componens Low Noise Amplifier Loal Osillaor

16 MOS OPdB OIP OIP PdB PA PLL QAM QPSK RF SSB VCO Meal Oxide Semionduor Oupu db Compression Poin Seond order Oupu Inerep Poin Third order Oupu Inerep Poin db Compression Poin Power Amplifier Phase Lok Loop Quadraure Ampliude Modulaion Quadraure Phase Shif Keying Radio Frequeny Single-Sideband Volage Conrolled Osillaor xvi

17 CHAPTER : INTRODUCTION Despie onsiderable advanes in ehnology, nonlineariy of devies oninues o limi he performane of wireless sysems and devies. In he design of radio reeivers, nonlineariy resris he abiliy of a radio o reeive weak signals in he presene of nearby sronger signals []. In radio ransmiers, nonlineariy an ause he ransmied signal o spill over ino adjaen frequeny hannels, inerfering wih oher users []. Therefore, he presen hesis presens new approahes o redue nonlineariy in devies and sysems. In many appliaions, a speifi nonlineariy known as hird order nonlineariy auses pariular problems. When a signal enouners suh a hird order nonlineariy, he resuling signal disorion is referred o as hird order inermodulaion disorion. I is his hird order inermodulaion disorion ha ofen auses greaes diffiuly in radio ransmiers and radio reeivers. To overome his problem, ransmiers and reeivers are ofen designed wih higher levels of power onsumpion. However, suh inreased power onsumpion is obviously undesirable in porable devies suh as ellular phones. In addiion, larger and more expensive devies usually aompany he higher power level. The approah presened in his hesis also seeks o miigae suh limiaions. In he ase of radio reeivers, hird order nonlineariy auses weak signals o be bloked in he presene of srong signals. In pariular, wo srong signals a frequenies adjaen o a desired weak signal an inerfere wih he reepion of he weak signal

18 beause of he hird order nonlineariy. For example, a pair of srong signals a 00 and 0 MHz an inerfere wih he reepion of a weak signal a 0 MHz, despie he fa ha all hree signals are on differen frequeny hannels. This siuaion is ommon in ellular phone sysems where nearby ellular phones may presen srong signals while a user is rying o reeive a weak signal from a disan radio or base saion. The linearizaion approahes in his hesis also address hese problems. In he ase of radio ransmiers, hird order nonlineariy auses ransmied signals o splaer ino adjaen hannels. Suh splaering auses inerferene wih adjaen radio hannels ha may be alloaed o oher users. In pariular, a srong ransmier signal an be disored suh ha i spreads ino an adjaen hannel. For example, a elevision broadas on hannel nine may inadverenly spill undesired ransmier power ino hannel eigh or en. Again, he presen hesis addresses suh radio ransmier problems. For boh ransmiers and reeivers, ommon approahes o improving lineariy ofen require bulky omponens suh as delay lines. Alhough suh soluions may be aepable for large insallaions suh as ellular base saions and owers, hese large omponens have obvious disadvanages for small porable appliaions suh as ellular phones. Therefore, anoher inenion of he hesis is o develop approahes and mehods ha an be implemened on inegraed iruis wihou he need for bulky exernal omponens. Suh inegraed irui implemenaions an be pariularly imporan in reduing os of high volume produs suh as ellular phones. In addiion o he ehnial problems addressed by his hesis, a simple analyi approah o linearizaion problems is presened. The simple approah is based on radio

19 frequeny oneps of hird order inerep poin and resuls in simple algebrai equaions o anel nonlineariies. Earlier approahes ommonly use umbersome polynomial expansions and ypially provide lile insigh for he purposes of designing new approahes o linearizaion. Imporanly, he simple analysis mehods give greaer insigh for he developmen of new linearizaion arhieures. Furhermore, he analysis is no dependen on underlying assumpions suh as square-law meal oxide semionduor field effe ransisor (MOSFET behavior. Finally, an unexpeed resul of he presen hesis is ha a bad devie an be added o a good devie o form a omposie devie ha is muh beer han eiher of he wo original devies. In more ehnial erms, a very nonlinear devie an be added o a slighly nonlinear devie o form a omposie devie ha is muh more linear han eiher of he wo original devies. Alhough earlier researhers used a similar ross-oupled differenial pair mehod, he presen work provides he foundaion for new four-sage approahes wih advanages over he ross-oupled differenial pair [], [7].. Thesis Saemen This hesis fouses on new linearizaion mehods for anellaion of hird order inermodulaion disorion. A simple analyi approah is presened whih promises o be a basis for various new linearizaion ehniques. The analyi approah has a simple algebrai relaionship, leading o a wide variey of new ideas for inegraed irui implemenaions. The new mehod is appliable no only o radio ransmiers, bu also o radio reeivers. Furhermore, i an be fully inegraed and does no require addiional exernal devies, suh as delay lines.

20 . Organizaion 4 Chaper desribes prior linearizaion mehods inluding feedbak, feedforward, and predisorion mehods. Chaper presens heory and desripions of nonlinear iruis as well as imporan oneps suh as hird order inerep and ompression poin. The new mehod of anellaion of hird order nonlineariy is desribed in Chaper 4. Chaper 5 presens he ross-oupled differenial pair implemened as inegraed irui and disusses measured resuls. Measured and simulaed resuls are ompared in Chaper 6. Finally, Chaper 7 summarizes his work and presens preliminary resuls for fuure researh. Par of his fuure researh is a fully inegrable four-sage implemenaion, whih promises o anel hird order disorion up o he db ompression poin.

21 CHAPTER : PRIOR LINEARIZATION METHODS In his haper, prior researh on linearizaion mehods is firs reviewed before desribing heoreial bakground maerial in Chaper. These prior approahes provide he onex for developmen of he new mehods of linearizaion in Chaper 4. In he following, prior linearizaion mehods are roughly spli ino hree aegories: Feedbak, Feedforward, and Predisorion [].. Feedbak Linearizaion The firs general aegory of linearizaion is Feedbak linearizaion. Feedbak linearizaion linearizes ransmiers or single PA s by foring he oupu o follow he inpu. The Feedbak ehnique an be eiher direly applied o he RF signal or indirely o he modulaion, i.e. envelope, phase or I (in-phase and Q (quadraure omponens []. In he following seion, RF feedbak using dire feedbak is firs disussed. Then, Envelope feedbak is desribed where he envelopes of inpu and oupu signals are exraed and ompared. Finally, Polar and Caresian loop feedbak are disussed, where he linearizaion sheme inludes he omplee ransmier.

22 6 Nonlinear Amplifier Inpu + Subraor - Oupu Feedbak Nework Figure. RF feedbak linearizaion mehod: oupu of nonlinear amplifier is fed bak and subraed from he inpu signal. Feedbak nework is ypially a onsan aenuaion nework... RF Feedbak In radio frequeny (RF feedbak he oupu signal is fed bak wihou deeion or down-onversion. RF feedbak is illusraed in Fig.. The RF signal is inpu o a subraor on he lef side in Fig... The oupu signal of he amplifier, on op of Fig.., is fed bak o he subraor and subraed from he RF inpu signal. The feedbak nework, a he boom of Fig.. an eiher be passive or aive. An amplifier an be used for an aive feedbak nework or resisors or ransformers an be deployed as passive feedbak neworks. The feedbak nework an redue disorion appearing a he oupu of he nonlinear amplifier in Fig... Volage-onrolled urren feedbak and urren-onrolled volage feedbak are ommonly used for his mehod beause hey are simple and heir disorion performane is prediable []. However, due o he ime delays in he feedbak nework, loop sabiliy is a problem in his design. As a resul, RF feedbak is limied o narrowband sysems. A

23 7 Inpu Direional Coupler Modulaor Direional Coupler Oupu Nonlinear Amplifier + - Differenial Amplifier Inpu Envelope Deeor Oupu Envelope Deeor Figure. Envelope feedbak linearizaion: on he lef side an inpu oupler splis he inpu signal. The envelope deeor a he lef dees he hanges in envelope of he inpu signal. On he righ side, he oupu of he amplifier is sampled by a oupler. The sample is fed ino an envelope deeor and he hanges in oupu envelope are deeed. A differenial amplifier ompares he wo envelope signals and onrols a modulaor. The modulaor adjuss he inpu signal o he amplifier. loss of gain is also an issue in his ehnique, pariularly in ransmiers, where high oupu power is desired [], [4]... Envelope Feedbak A seond feedbak linearizaion mehod is envelope feedbak. Fig.. shows an envelope feedbak sheme. In Fig.., a porion of inpu and oupu signals is sampled by a oupler a he inpu and oupu and he envelopes of he wo sampled signals are deeed by means of an envelope deeor a he boom of Fig... Boh envelope signals are subraed using a differenial amplifier. The resuling error-signal onrols a modulaor, whih modifies he envelope of he RF inpu signal. The oupu signal of he modulaor is amplified by he nonlinear power amplifier. In envelope feedbak, boh modulaor and PA are inluded in he linearizaion proess. Furhermore, envelope feedbak an be applied o eiher a ransmier or a single

24 PA. However, a disadvanage of he envelope feedbak is ha i only aouns for disorion in signal-ampliude and no in signal-phase [], [4]. 8.. Polar Loop Feedbak Polar Loop is an improved version of envelope feedbak. The Polar Loop sheme ompensaes ampliude and phase disorion. In mos ases, envelope and phase omparison akes plae a he inermediae frequeny (IF. Typially, he polar loop ehnique is deployed o a omplee ransmier raher han a single PA [], [], [4], [5], [6]. A Polar Loop linearizaion ehnique is illusraed in Fig... The operaion of a polar loop is similar o envelope feedbak. A sample of he oupu signal is downonvered o a onvenien IF by he loal osillaor (LO a he righ boom orner of Fig... The IF signal is hen separaed ino phase and ampliude by a limier and demodulaor, respeively. This proedure is illusraed a he boom ener in Fig... On he lef side in Fig.., a single sideband (SSB modulaed signal, as inpu signal, is spli and separaed ino phase and ampliude by a limier and demodulaor, respeively, similar o he sampled oupu signal. In he ener of Fig.., boh oupu and inpu ampliudes are ompared wih an error amplifier and he resuling error signal onrols a modulaion amplifier. The phase signals of inpu and oupu are muliplied uilizing a mixer shown a he boom of Fig... The resuling signal onrols a volage-onrolled osillaor (VCO afer passing a loop filer and being amplified. The new-formed phase and ampliude error signals are ombined wih a modulaing amplifier a he op of Fig... Finally, he ombined signals are amplified wih a power amplifier. In Fig.., he loop, whih deermines he phase error-signal, is essenially a phase lok loop (PLL wih

25 9 VCO Modulaion Amplifier Power Amplifier Oupu SSB Generaor Inpu Demodulaion Loop Amplifier - + Error Amplifier Aenuaor Loop Filer Demodulaion Downonversion Limier Limier LO Figure. Polar loop feedbak linearizaion: on he lef side, he inpu is a single sideband modulaed (SSB signal whih is spli and inpu o a limier and demodulaor. On he righ side, he oupu signal is down-onvered wih a loal osillaor (LO, spli, and also inpu o a limier and demodulaor. The ampliude informaion is oupu of he demodulaors and is fed o an error amplifier. The errorsignal onrols a modulaion amplifier. The oupu signals of he limiers are spli and fed bak o he demodulaor. The seond oupus of he spliers are ompared using a mixer. The oupu signal of he mixer is filered and amplified wih a loop amplifier and onrols a volage onrolled osillaor (VCO. The ampliude modulaed oupu signal of he VCO is inpu o he power amplifier []. a mixer as phase deeor, a VCO and an amplifier in he loop [5]. A disadvanage of he PLL is he loking problem a low ampliudes [5]. The PLL is also an example of he omplexiy of he polar loop ehnique. Anoher disadvanage of he polar loop linearizaion is ha he exraed inpu and oupu phase bandwidh mus be a leas en imes he RF bandwidh a he oupu [].

26 ..4 Caresian Loop Feedbak 0 Anoher feedbak linearizaion ehnique is he Caresian Loop feedbak whih is depied in Fig..4. In his ehnique, he inpu and oupu baseband signals shown on he lef in Fig..4 are proessed in Caresian raher han polar form. The inpu baseband signals are available as quadraure omponens, I and Q, on he lef side in Fig..4. Boh signals are fed ino differenial amplifiers and are up-onvered o RF using a quadraure ampliude modulaor (QAM. The signals a he oupu of he QAM up-onverers are Power Amplifier Direional Coupler Differenial Amplifier Carrier Frequeny Osillaor Aenuaor Baseband Amplifier Figure.4 Caresian loop feedbak linearizaion: on he lef side he inpu signals are available as I and Q omponens and are inpu o differenial amplifiers. On he righ side he oupu signal is sampled. The sampled signal is, afer aenuaion, fed o a quadraure ampliude modulaor (QAM where he sample is down-onvered ino I and Q omponens. Baseband amplifiers amplify he I and Q omponens and lose he loop o he differenial amplifiers. The oupu signals of he differenial amplifiers are up-onvered wih a QAM and amplified wih a power amplifier. The QAM modulaor onsiss of wo mixer and 90 degree phase shif nework o reae in-phase and quadraure omponens [].

27 ombined forming a omplex RF signal, whih is amplified by a nonlinear PA. To lose he loop o he differenial amplifiers, he oupu signal on he righ side in Fig..4 is sampled, down-onvered and resolved ino omponens I and Q [], [], [4], [5]. The QAM modulaors ypially onsis of wo mixers and a 90 phase shif nework. The Caresian Loop feedbak forms a omplee linear ransmier. Nonlineariies of all bloks, suh as he QAM up-onverer in he loop, are removed. However, as in all feedbak sysems, he degree of lineariy improvemen depends on he delay around he loop []. Hene, is performane depends on he bandwidh of he feedbak sysem. Higher frequeny omponens experiene more ime delay in he feedbak sysem han low frequeny omponens. Thus, he feedbak sysem an beome unsable a signals wih high frequeny omponens and ause unwaned osillaions. To preven suh unsable behavior, feedbak sysems are limied in bandwidh. Faulkner [7] repors a Caresian feedbak, whih employs a phase orreion irui o sop insabiliy and limi he level of he ou-of-band noise floor. This sheme uses he supply urren o he final sage amplifier as a measure of oupu power. The auhor also shows ha he phase orreion seings orrespond o he oupu power ondiion of he amplifier. However, he orreion performane is limied by hyseresis in he supply urren feedbak signal, whih adjuss he phase orreor in he feedbak pah. Furhermore, Lee e al [8] uilizes a regulaor for aurae osillaor phase adjusmen in Caresian feedbak. However, boh shemes in [7] and [8] need addiional iruiry whih inreases overhead, omplexiy and os.. Feedforward The seond major linearizaion mehod is Feedforward. In onras o feedbak

28 sysems, feedforward linearizaion sysems orreion is no dependen on pas evens. Feedforward linearizaion is based on presen evens in he sysem and linearizaion is independen of amplifier delays. Addiionally, basi feedforward sysems are unondiionally sable and ideally do no redue amplifier gain [], [9], [0]. Fig..5 presens a shemai of a basi feedforward sysem. The linearized PA wih feedforward orreion onsiss of a main and error amplifier, direional ouplers, and delay lines. The inoming signal is spli ino wo pahs wih one pah going o he main amplifier, and he oher pah going o a delay elemen, shown in Fig..5 on he righ side. The signal a he oupu of he nonlinear amplifier onains he desired and disorion omponens. The delayed inpu signal is subraed from a sample of he oupu signal of he nonlinear amplifier. The sample of he oupu signal of he nonlinear amplifier is aken by means of a oupler a he op of Fig..5. Ideally, he arrier is aneled and he signal a he oupu of he subraor onains only he disorion omponen. The error signal is amplified by an amplifier depied a he boom of Fig..5 and ombined wih a delayed version of he oupu signal of he main amplifier. This seond ombinaion ideally anels he disorion omponens of he main amplifier and leaves he desired signal unhanged. The firs loop anels he arrier and he seond loop redues he disorion omponen.

29 Figure.5 Basi feedforward linearizaion: on he lef side he inpu signal is spli by a splier whih is usually a passive nework. The spli signals are inpu o he main amplifier and a subraor afer being delayed. The ime delay ompensaes he ime delay of he main amplifier. A sample of he oupu of he amplifier is exraed wih a oupler and fed o he seond inpu of he subraor. The oupu of he subraor is inpu o an error amplifier on he righ side. A direional oupler ombines he oupu of he error amplifier and he delayed oupu of he main amplifier. The ime delay on he righ side is ompensaing he ime delay of he error amplifier. The firs loop eliminaes he arrier frequeny; he seond loop anels disorion produs in he main pah [9]. Feedforward an be used over a wide bandwidh of abou 0-00 MHz []. However, inaurae mahing of devies in ampliude and phase an impair he performane of he feedbak sysem. Sine his sysem is feedforward in naure, aleraions over aging and emperaure degrade he orreion of lineariy. A way o redue hese effes is o use muliple feedforward loops []. A single feedforward loop an be onsidered as he main amplifier. However, by using his onfiguraion, he omplexiy of a feedforward sysem inreases fas. In addiion, exernal devies are neessary suh as delay lines. Woo e al [] propose a new adapaion sheme for he feedforward linearizaion ehnique. The proposed sheme uses imperfe anellaion a he error amplifier o

30 4 aoun for disorion reaed in he error amplifier. However, he proposed new adapaion sheme inrodues feedbak ino he feedforward linearizaion ehnique, whih in urn an ause sabiliy and bandwidh problems.. Predisorion The hird major linearizaion ehnique is predisorion. Predisorion is based on he idea of insering a nonlinear elemen prior o a RF PA suh ha he ombined ransfer haraerisi of he wo devies is linear. Fig..6 shows he underlying priniple. The ransfer-haraerisi of he predisorer, shown on he righ side in Fig..6, mus be omplemenary o he ransfer-haraerisi of he PA, depied in he ener in Fig..6. The ombined ransfer haraerisi of he linearized amplifier is illusraed on he righ in Fig..6. Predisorer and amplifier ransfer haraerisi ideally ompensae eah oher reaing a linear inpu-oupu haraerisi. Predisorion an be aomplished a eiher RF, IF or baseband and has he abiliy o linearize he enire bandwidh of an amplifier or sysem [], [4]. There is also he possibiliy o asade he disorion elemen a he oupu of he PA. This ase is alled posdisorion [], [5]. RF/IF predisorion, whih is desribed firs, does no use feedbak. Nex, adapive baseband predisorion using feedbak is disussed.

31 5 Inpu PinP PouP PinA PouA Oupu Predisorer Nonlinear Amplifier PouP PouA PouA + = PinP PinA PinP Predisorer Amplifier Linearized Amplifier Figure.6 Predisorion linearizaion: on he op, he amplifier is asaded wih a predisorer. The predisorer an be a passive diode nework or a digial signal proessor. On he boom, he ransfer haraerisis of he predisorer, (on he lef is added o he ransfer haraerisi of an amplifier (in he ener. The righ side shows he ombined ransfer haraerisi, whih is ideally linear... RF/IF Predisorion RF and IF predisorion are similar in operaion and do no have a feedbak pah. Therefore, he nonlineariy o be aneled mus be known in advane. In RF predisorion, he predisorer operaes a he final arrier frequeny. IF predisorers work a an inermediae frequeny, hereby making i possible o use he predisorion nework for differen arrier frequenies by adjusing he loal osillaor (LO frequeny [9]. The problem in his sheme is o design and fabriae a predisorion irui, whih losely resembles he required funion of Fig..6. The ransfer haraerisi of a predisorion nework mus be uned o every single PA, even PA s wih idenial design [].

32 6 Typially, he predisorer aemps o disor only he hird-order haraerisi. This ype of nonlinear nework is named ubi predisorer. These iruis make use of he nonlinear haraerisis of devies suh as diodes and ransisors []. However, main disadvanages of RF/IF predisorion are relaively modes lineariy improvemen, and an inabiliy o deal wih many orders of disorion []... Adapive Baseband Predisorion A seond ype of predisorion is alled adapive baseband predisorion (also alled digial predisorion. Digial predisorion is basially a Caresian feedbak wih digial signal proessing (DSP added. This ehnique linearizes omplee ransmiers. Fig..7 displays an adapive baseband predisorion sheme. Predisorion ours a he baseband level and manipulaes usually I and Q omponens. On he righ side of Fig..7, he oupu signal is sampled, down-onvered by a QAM down-onverer (ino omponens I and Q, and inpu o a digial signal proessing (DSP uni. Similarly, inpu I and Q omponens are direly fed ino he DSP blok. In order o reae predisorion, weighing oeffiiens are sored in lookup ables in he DSP seion. These oeffiiens an be updaed by new oeffiiens derived from he fed bak in-phase and quadraure omponens. The oupu of he DSP is he predisored signal. This predisored signal is hen up-onvered by a QAM and amplified wih a PA illusraed a he op of Fig..7. The primary disadvanages of digial predisorion are is relaive omplexiy and bandwidh limiaions ied o he auray and ompuaional rae of he speifi DSP []. Furhermore, power onsumpion is inreased due o he digial signal proessor. In addiion, digial predisorion exhibis sorage and proessing overhead for he lookup ables [], [].

33 7 Power Amplifier Direional Coupler Digial Signal Proessing Carrier Frequeny Osillaor Aenuaor Baseband Amplifier Figure.7 Adapive baseband predisorion linearizaion: on he lef side he inpu signals, as I and Q omponens are inpu o a digial signal proessor. On he righ side a sample of he oupu signal is aken by a oupler and afer aenuaion downonvered ino I and Q omponens. The I and Q omponens are amplified wih baseband amplifiers. The oupu of he amplifiers lose he loop o he digial proessor. The predisored oupu of he digial proessor is up-onvered and amplified wih a power amplifier. Down and uponversion is performed wih a QAM onverer. The QAM onsiss of wo mixers and a 90 degree phase shif nework [5]..4 Envelope Eliminaion and Resoraion (EER This seion disusses envelope eliminaion and resoraion (EER. EER is a ehnique whih ries o inrease lineariy and power effiieny simulaneously. The basi mehod of EER is o disassemble he RF inpu signal ino phase and envelope omponens and ombine boh afer amplifiaion. EER is based upon he priniple ha

34 8 any narrow-band signal an be produed by simulaneous ampliude (envelope and phase modulaion []. The shemai in Fig..8 shows an EER ehnique wih limier and envelope deeor o exra he phase and envelope informaion, respeively. The limier, shown a he boom lef of Fig..8, eliminaes he envelope and hus makes i possible for a high-effiien nonlinear PA o amplify he onsan-envelope signal. Finally, he envelope amplifier (lass-s modulaor, depied on he op of Fig..8, modulaes he final RF power amplifier and reaes an amplified replia of he inpu signal a he oupu. EER an be employed o a omplee ransmier or a single PA. If EER is used o design a ransmier, a DSP is ypially uilized o generae he envelope and phase informaion. EER reahes good lineariy wih high effiieny. However, diffiulies in he design lie in he limied bandwidh of he lass-s modulaor and orre alignmen of he envelope and phase signals. Furhermore, large envelope variaions an drive he RF power amplifier ino uoff ausing signifian disorion []. Mehods of implemening a feedbak sheme as a par of he EER ehnique are inrodued in [4] and [5]. In he proposed feedbak proedure, envelopes of inpu and oupu signals are deeed and ompared wih a differenial amplifier. The oupu of he differenial amplifier hen modulaes he final sage amplifier. However, he resuling feedbak poses a problem onerning bandwidh limiaion and loop sabiliy. Furhermore, omplexiy of he irui opology is inreased.

35 9 Inpu Splier Envelope Deeor Modulaor Oupu Limier Nonlinear Amplifier Figure.8 Envelope eliminaion and resoraion: on he lef, he inpu signal is spli. The envelope of he inpu signal is deeed wih an envelope deeor in he op signal pah. The envelope is inpu o a modulaor. The oupu signal of he modulaor hanges he supply volage of an amplifier. On he boom he inpu signal is fed o a limier eliminaing he envelope informaion. Essenially, he inpu signal is disassembled ino envelope and phase by envelope deeor and limier, respeively. The modulaor is usually a lass-s amplifier..5 Linear Amplifiaion wih Nonlinear Componens (LINC and Combined Analog Loked Loop Universal Modulaor (CALLUM Lasly, linear amplifiaion wih nonlinear omponens (LINC and ombined analog loked loop universal modulaor (CALLUM are presened. LINC and CALLUM boh linearize a omplee ransmier. Fig..9 depis a basi LINC ehnique [], [5]. The inpu signal is separaed ino wo onsan envelope and phase-modulaed omponens by a DSP blok, illusraed on he lef side in Fig..9. Eah signal is uponvered and drives a nonlinear power amplifier. Boh PA inrease he signals by he same amoun. Finally, he oupus of boh power amplifiers are summed on he righ side in Fig..9. The final oupu signal is (ideally an amplified, disorionless version of he inpu signal [6]. Nonlinear, high effiien power amplifiers are uilized in he LINC

36 0 Nonlinear Amplifier Inpu Signal separaion/ generaion Nonlinear Amplifier + Oupu Figure.9 Linear amplifiaion using nonlinear omponens (LINC: he inpu signal on he lef side is inpu o a signal separaion irui. The oupu signals are of onsan ampliude wih 90 degree phase differene and amplified wih nonlinear amplifiers. The oupus of he wo nonlinear amplifiers are added on he righ side, rereaing an amplified and linearized inpu signal. The signal separaion nework is ypially a digial nework. linearizaion mehod. The nonlinear power amplifiers reae unwaned disorion omponens wih opposie phase a heir oupu. As resul, he disorion omponens anel eah oher afer summaion. CALLUM [7] is a ehnique, whih is derived from LINC. I uses Caresian feedbak. The oupu signal is fed bak, down-onvered o I and Q omponens by a QAM, and ompared wih he baseband (I and Q omponens. High effiieny an be ahieved wih LINC and CALLUM beause of he use of effiien power amplifiers. However, LINC has diffiulies wih gain and phase mahing in boh pahs, hus resuling in imperfe anellaion of disorion. CALLUM suffers from he limiaions of a feedbak sysem. I is ypially used for narrow-band appliaions. Finally, power and effiieny is los beause he summaion akes plae a

37 he oupu of he power amplifiers []. Oher linearizaion ehniques are proposed in [8], [9] and [0]. Yang e al. [8] presens a mehod where a seond order inermodulaion erm is generaed a he inpu of a main amplifier (gae of a ransisor. The seond order inermodulaion omponen is hen fed forward hrough an auxiliary feedforward amplifier o he oupu of he main amplifier (drain of a ransisor, resuling in reduion of hird order disorion a he final oupu. However, he proposed mehod allows only moderae hanges in inpu power. Ding e al. [9] propose a linearized low noise amplifier (LNA. The irui is arranged as a differenial LNA. Two idenial LNA, one LNA is he main amplifier, he seond amplifier is onsidered as he auxiliary amplifier, are onneed in parallel. A fraion of he urren in he auxiliary amplifier is subraed from he urren in he main amplifier. Inermodulaion produs are redued if hey are opposie in phase. The inpu o he auxiliary amplifier is muliplied wih a faor, in order o ahieve he desired reduion in hird order disorion. However, i is no learly saed in [9] how is reaed. The faor indiaes he presene of an addiional ideal amplifier wih gain, a he inpu of he auxiliary LNA. More disadvanages are, for insane, inreased power onsumpion and gain loss due o subraion of he linear omponens. Webser e al. [0] proposes a mehod for reduion of hird order inermodulaion produs pariularly for use in monolihi mirowave inegraed iruis. This new mehod is alled derivaive superposiion ehnique. In his ehnique wo or more MOSFETs are onneed parallel. The nonlinear drain urren of a MOS ransisor an be expressed as a funion of gae-soure volage wih a Taylor series expansion. The hird oeffiien of he series is he hird derivaive of he ransonduane of a MOSFET.

38 Furhermore, he hird derivaive of he ransonduane is dependen on he gae widh and is sign (posiive or negaive varies wih bias urren. If he wo ransisors are biased, suh ha he hird derivaives of he ransonduane have differen signs, hen a reduion of hird order disorion is ahieved by subraion. However, variaions in ransisors due o manufauring and emperaure limi he linearizaion performane. Finally, a mehod o linearize he DC ransfer haraerisi of differenial amplifiers is presened in [] This mehod uses wo differenial pairs wih ross-oupled drains o ahieve subraion a he oupu of he ombined differenial pairs, hus he name ross-oupled differenial pairs. The approah for reduion of disorion used in [] employs he fa ha he ransfer haraerisi of differenial pairs an be desribed wih a Taylor series expansion. Wih he proper adjusmen of bias urrens and aspe raios of he ransisors, anellaion of he hird order omponen of he Taylor series an be reahed and hird order disorion redued. However, he approah presened is based on he square-law behavior of MOS ransisors. The derived expression for anellaion in [] is no valid if higher order effes suh as veloiy sauraion in ransisors appear. Furhermore, Taylor series expansion is very umbersome for omplex irui opologies, and does no give informaive insigh ino he design approah of iruis. Conversely, he analyi mehod presened in he presen hesis is general and does no depend on underlying assumpions suh as he square-law haraerisi of MOS ransisors. Finally, he groundwork presened in his hesis is appliable o a wider variey of linearizaion implemenaions beyond he simple ross-oupled differenial pair. Neverheless, he ross-oupled differenial pair provides simple baseline validaion of a speifi ase of a wider range of approahes disussed in his hesis.

39 Table. summarizes he linearizaion ehniques. The linearizaion ehniques proposed in [8], [9] and [0] bes fi ino he feedforward aegory. In addiion, a omparison of suppression of hird order disorion for prior linearizaion mehods is lised in Table.. All daa is he resul of a wo-one es inpu signal exep where oherwise noed. Adjaen hannel power raio (ACPR is defined as disorion level in he adjaen hannel o he desired hannel relaive o he power level in he desired hannel in db. Table. Prior linearizaion ehniques [] Feedbak Feedforward Predisorion RF feedbak Basi feedforward RF/IF predisorion Caresian Loop Polar Loop Envelope Eliminaion and Resoraion Linear amplifiaion using nonlinear omponens (LINC/Combined analog loked loop universal modulaor (CALLUM Adapive baseband predisorion New linearizaion mehod Adapive baseband predisorion

40 Table. Summary of hird order disorion suppression of prior linearizaion ehniques 4 Linearizaion ehnique Third order disorion suppression Referene RF Feedbak db [] Envelope Feedbak 0 db [] Polar Loop Feedbak 0 db [] Caresian Loop Feedbak 0 db [] Feedforward RF/IF Predisorion 0 40 db db 0 db in ACPR 0 db [9] [] [] [4] Adapive Baseband Predisorion 0 db [5] Envelope Eliminaion & Resoraion (EER 5 db [4] LINC 5 db [5] Seond order inermodulaion feedforwarding 8 db [8] Bea feedforwarding 5 db [9] Derivaive superposiion 5 dbm oupu power ompared o lass AB amplifier [0]

41 CHAPTER : THEORETICAL BACKGROUND Before proeeding o desribe he new linearizaion mehod in Chaper 4, he presen haper desribes he oneps of nonlineariy, underlying mahemaial models, and he effes of nonlineariy on radio sysems. Depending on he operaing region (bias and signal level and he degree of nonlineariy in he inpu-oupu ransfer haraerisi, nonlinear iruis an roughly be haraerized as weakly nonlinear or srongly nonlinear [5]. The ransfer haraerisi of weakly nonlinear and memoryless iruis an be desribed by a power series expansion suh as a Taylor series expansion, where he series is runaed afer he firs few erms [5]. For srongly nonlinear iruis, he power series beomes unwieldy beause he series needs many erms wih large oeffiiens o auraely model he nonlinear behavior. Conversely, weakly nonlinear iruis are more easily modeled wih a few erms. In praie, an amplifier has o operae well below he gain ompression (sauraion o be in he weakly nonlinear region. An amplifier is onsidered srongly nonlinear, if he amplifier operaes lose o he db ompression poin.. Modeling of Nonlinear Behavior The inpu-oupu haraerisi of weakly nonlinear iruis is ommonly modeled wih a power series expansion [] as shown in (..

42 ( = nx( = + n= 0 6 y 0 x( + x ( + x ( +... (. In equaion (., x( orresponds o a ime varying inpu signal and y( o he resuling ime varying oupu signal. Coeffiien 0 orresponds o a DC offse and o he linear omponen. Coeffiiens and represen nonlineariies of seond and hird order, respeively. For a linear irui, he series (. would redue o y(= x(, where 0 = = =0 and =0, for example. For a volage amplifier, where x( and y( are volages, oeffiien would be equal o he volage gain. Fig.. illusraes he linear behavior of an amplifier in ime and frequeny domain. In Fig..(a, he inpu signal x( on he lef side is equal o a signal of he form Asin(ω 0 where A is he maximum ampliude and equal o one and ω 0 is he frequeny in radians per seond. If =0, hen he oupu signal y(, depied on he righ side in Fig..(a equals 0 Asin(ω 0. Fig..(b and ( depi he frequeny spera of inpu and oupu signals, respeively. The inpu and oupu spera only have one frequeny, he same frequeny a inpu and oupu. The ampliude of he oupu signal is en imes greaer han he ampliude of he inpu signal. The frequeny spera of he inpu and oupu signals an be obained by applying he Fourier ransform. The Fourier ransform of a sinusoid, os(ω 0 is equals o (. [7]. jω 0} os( ω 0 e d = δ ( ω ω 0 + δ ( ω + ω F {os( ω = 0. (.

43 7 0 x( y( X( ime Amplifier (a Y( 0 0 ime (b 0 ( Figure. Inpu and oupu signals of linear amplifier: he lef side in (a shows he inpu signal x( in ime domain wih ampliude. The righ side in (a depis he oupu signal y( in he ime domain. (b displays he frequeny sperum of he inpu signal X( wih a speral line a 0 and ( shows he sperum of he oupu signal Y( wih a speral line a 0. Coeffiien =0. The frequeny is denoed ω in (. and is he dira impulse funion. Only he posiive frequeny is shown in Fig..(b and ( for he example of a linear irui. For he above ase, he oupu signal y( over inpu x( is depied in Fig... 0 y 0 x 0 Figure. Transfer haraerisi of linear amplifier; he onsan slope equals =0.

44 The sraigh-line wih a slope of =0 displays a linear ransfer haraerisi. 8 In ases where he amplifier is no linear and exhibis a seond order nonlineariy, he ransfer relaionship would be equaion (. [] y ( = x( + x (, (. where 0 = =0, =0 and =. Equaion (. is presened graphially in Fig... The oupu signal y( is shown in Fig..(a, where he inpu is x(=asin(ω 0 and A=. The oupu sperum Y(ω in Fig..(b shows a DC omponen a ω=0 and a seond harmoni a ω 0. Boh omponens have absolue ampliudes of A / =. The Y( 0 y( (a (b y Figure. Seond order nonlineariy in amplifier: (a Oupu signal y( in ime domain, signal exhibis posiive DC offse; (b Oupu signal Y( in frequeny domain, DC omponen a =0 and seond harmoni a = 0 wih fundamenal a 0 ; ( inpu-oupu ransfer haraerisi has nonlinear behavior; =0, =, 0 = =0 8 ( x

45 9 fundamenal signal appears a ω 0 wih an absolue ampliude of A =0. Fig..( depis he ransfer haraerisi. Due o he seond order nonlineariy, he ransfer relaionship deviaes from a linear sraigh-line inpu-oupu relaionship. The full expansion of equaion (. is presened in Appendix A. If he amplifier exhibis a hird order nonlineariy, (. an be simplified o (.4 y ( = x( + x (, (.4 where 0 = =0, =0 and =-. Fig..4 demonsraes he resulan nonlinear behavior in 7.5 Y( 7.75 y( (a (b 7 y Figure.4 Third order nonlineariy in amplifier: (a Oupu signal y( in ime domain; signal exhibis ompression; (b oupu signal Y( in frequeny domain, hird order harmoni a = 0 and fundamenal a 0 ; ( inpuoupu ransfer haraerisi of amplifier wih hird order nonlineariy deviaes from linear haraerisi; =0, 0 = =0, =-.5 7 ( x

46 0 ime and frequeny domain, where x(=asin(ω 0 and A=. The oupu signal y( is displayed in he ime domain in Fig..4(a. The lipping of oupu waveform y( in he ime domain is an effe of hird order nonlineariies. The oupu sperum Y(ω in Fig..4(b exhibis a hird harmoni a ω 0, wih an absolue ampliude of A /4 =0.75. The sperum Y(ω does no show a DC omponen. Furhermore, he fundamenal signal a ω 0 has an absolue ampliude of ( A+ A /4 =7.75. Finally, he ransfer haraerisi of equaion (.4 is depied in Fig..4( where he lipping effe is also visible for higher inpu signal levels. Again, a full expansion of equaion (.4 is presened in Appendix A. The fourh ase ombines seond and hird order nonlineariies, hus reaing seond and hird harmonis as shown in Fig..5. The desribing equaion (. is alered o equaion (.5. y ( = x( + x ( + x ( (.5 In equaion (.5, 0 =0, =0, = and =-. The oupu signal y( in he ime domain is illusraed in Fig..5(a, wih an inpu waveform x(=asin(ω 0, where A=. Oupu y( exhibis a posiive DC offse. Oupu sperum Y(ω, depied in Fig..5(b onains a DC omponen a ω=0 as well. Furhermore, seond and hird order omponens in equaion (.5 reae seond harmoni a ω 0, and hird harmoni a ω 0 in he oupu sperum Y(ω in Fig.5(b. The inpu-oupu ransfer haraerisi, whih deviaes from a linear ransfer haraerisi, is displayed in Fig..5(. Appendix A presens a full expansion of equaion (.5 wih x(=asin( 0. The linear, seond and hird order erms, resuling from he full expansion of equaion (.5, are summarized in Table..

47 0 6 Y( 7.75 y( (a (b 0 y 0 x 6 ( Figure.5 Seond and hird order nonlineariies in amplifier: (a oupu signal y( in ime domain showing DC offse and ompression; (b frequeny sperum of oupu signal Y( ; fundamenal 0, seond harmoni 0 and hird harmoni 0 as well as a DC omponen a =0 appear in he oupu sperum; ( depis he nonlinear inpu-oupu ransfer haraerisi. 0 =0, =0, =, =- In Table., A is he inpu signal ampliude and Coeffiiens, and are linear, seond and hird order oeffiiens of equaion (.5, respeively. The previous disussion showed various erms of he power series expansion. All of hese erms are srongly dependen on he oeffiiens, and. The oeffiiens are usually no easily available. Furhermore, he auray of he oeffiiens is dependen on biasing and ehnology parameers, and deermined by devie models suh as ransisor models. Pariularly for ransisor models, he auray of he oeffiiens

48 Table. Summary of linear and higher order erms of power series expansion DC erm Firs order erm a fundamenal frequeny Seond order erm a seond harmoni Third order erm a hird harmoni A A A A A depends on he inlusion of seond order effes in ransisors, for insane veloiy sauraion and mobiliy reduion.. Effes of Nonlineariy in Radio Sysems In he following seions, a variey of nonlinear effes in radio sysems are disussed. Firs, wo-one es as a means of haraerizing nonlineariy is presened. Seondly, he nonlinear effes desensiizaion, ross modulaion and inermodulaion are disussed... Two-Tone Tes In many radio sysems, an ampliude modulaion sheme is used o ransmi informaion. To exerise signal ampliude, a wo-one es provides a simple ime varying signal envelope in order o es a irui suh as an amplifier. In he wo-one es, he inpu es signal onsiss of wo sinusoids wih frequenies losely spaed in he band of ineres as shown in (.6. x = A os( ω + A os( ω (.6 (

49 In equaion (.6, A and A are he ampliudes of he sinusoids and ω and ω are he frequenies in radians per seond. Usually, boh sinusoids have equal ampliude A =A. Fig..6(a illusraes a wo-one signal in he ime domain wih A =A =. The peak ampliude is A and equals wie he ampliude of eah sinusoid. The envelope hanges wih he differene frequeny ω -ω. The orresponding frequeny sperum in Fig..6(b onsiss of he wo frequenies ω and ω wih equal ampliudes. X( x( (a (b Figure.6 Two-Tone signal: (a ime domain wih varying envelope from 0 o A ; (b frequeny domain showing wo sinusoids a frequenies and. If an amplifier exhibis a nonlinear ransfer haraerisi suh as (.5 and is exied wih a wo-one signal (.6, he oupu signal will onain seond and hird order harmonis and inermodulaion produs. The oupu signal y( hen is y( = x( + x ( + x ( = ( A os( ω + A os( ω + ( A os( ω + A os( ω + ( A os( ω + A os( ω (.7 Wih he use of ommon rigonomeri ideniies, (.7 an be expanded as shown in [7] and Appendix A. A few imporan erms are used o obain key resuls in he following

50 disussions. 4.. Desensiizaion In radio reeivers, one effe of nonlineariy is when weak signals are being bloked by srong signals a differen frequenies. This siuaion an our when a radio reeiver has o proess a weak signal radiaed from a remoe radio sender in he presene of a srong signal emied from a lose radio ransmier. When his srong signal is proessed along wih he weak desired signal, a radio sysem an exhibi an effe alled desensiizaion [], [8]. This effe an be explained wih he fundamenal erm of he full expansion of equaion (.7, whih is desribed wih (.8. + A + A A os( ω (.8 4 ( A where (.8 represens he oupu sinusoid a he same frequeny as he inpu, i.e. (.8 represens he linear omponen of he oupu signal. Coeffiiens and represen he linear omponen and hird order nonlineariy of he nonlinear amplifier, respeively. If A is he ampliude of he weak desired signal in (.8 and A he ampliude of a srong inerferer and A <<A, (.8 an be approximaed as shown in (.9, where he erm + A / represens he linear gain of he nonlinear amplifier. + A A os( ω. (.9 ( For <0, he gain as given by + A /, is a dereasing funion of A []. This derease in gain dereases he desired signal srengh a he oupu of he amplifier.

51 .. Cross Modulaion 5 A seond effe of nonlineariy in radio sysems is ross modulaion. Cross modulaion is he nonlinear effe where modulaion from one arrier is ransferred o anoher [], [8]. To illusrae ross modulaion, onsider an inpu es signal where he es signal is an ampliude-modulaed arrier a frequeny ω added o a sinusoidal waveform a frequeny ω. The resuling signal an be desribed wih equaion (.0: x = A os( ω + A ( + m os( ω m os( ω. (.0 ( In equaion (.0, A and A are he peak ampliudes of he arriers wih frequenies ω and ω in radians per seond, m is he modulaion index (0 m, and ω m is he modulaing frequeny in radians per seond. If ampliude A in he fundamenal omponen of he full expansion of equaion (.7 is replaed by A (+mos(ω m, hen he fundamenal omponen of he full expansion of (.7 beomes (. A + A + A A ( + mos( ωm os( ω. (. 4 ( As an be seen from (., he ampliude of arrier now inludes a ime-varying ampliude modulaion erm (+mos(ω m ha indues modulaion on he unmodulaed arrier a frequeny ω. Cross modulaion is pariularly riial in muli arrier radio sysems...4 Inermodulaion A hird effe of nonlineariy on radio iruis is inermodulaion. In inermodulaion, a nonlinear irui exied wih wo inpu sinusoids resuls in an oupu,

52 onaining omponens a new frequenies. In he general ase, hese new frequenies will be of he form as desribed wih equaion (. [9]. 6 ω = m ω ± nω. (. In equaion (., ω is he new frequeny; ω and ω are he original frequenies in radians per seond. m and n are posiive inegers inluding zero. m+n is equal o he order of disorion. No all new reaed frequenies belong o he group of inermodulaion produs. Inermodulaion produs are he resul of he ineraion of fundamenals and harmonis of differen order. Harmonis are ypially no referred o as inermodulaion produs. Inermodulaion produs arise from he x (, x ( and higher order erms of he power series in (.. Produs reaed by he x ( erm are seond order produs and produs reaed by he erm x ( are hird order produs. Reurning o he wo-one inpu signal of equaion (.6, i an be seen from he full expansion of (.7 ha he following inermodulaion produs appear: Seond order inermodulaion produs: A A os(( ω + ω + A A os(( ω ω (. For A =A, erm (. hanges o erm (.4: A os(( ω + ω + A os(( ω ω (.4

53 7 In erms (. and (.4, A and A are he ampliudes of he wo-one inpu signal and is he seond order oeffiien. ω and ω are he fundamenal frequenies in radians per seond. Third order inermodulaion produs: os(( 4 os(( 4 os(( 4 os(( 4 A A A A A A A A ω ω ω ω ω ω ω ω (.5 Again, for A =A erms(.5 hange o erms (.6: os(( 4 os(( 4 os(( 4 os(( 4 A A A A ω ω ω ω ω ω ω ω (.6 In erms (.6, A and A are he ampliudes, is he hird order oeffiien of he power series, and ω and ω are he wo inpu frequenies in radians per seond. In erms (. o (.6 harmonis are no onsidered. Term (.6 indiaes ha as inpu ampliude A inreases by a faor of wo, he hird order inermodulaion produ goes up by a faor of eigh due o he exponen of value hree in he erms A /4. A faor of wo in volage equals 6 db and a faor of eigh equals 8 db.

54 8 Fundamenals s IMD5 IMD IMD IMD Figure.7 Frequeny sperum of inermodulaion disorion: fundamenal ones a frequenies and, hird order inermodulaion disorion (IMD produs a frequenies - and -, and fifh order inermodulaion disorion (IMD5 produs a frequenies - and - Of grea ineres are inermodulaion produs appearing in he viiniy of he arrier frequeny. These are hird order inermodulaion produs of he form - and -. Fig..7 depis he wo ones wih he inermodulaion disorion (IMD of ineres. The wo fundamenal ones of a wo-one signal are he wo innermos lines in Fig..7. Third order inermodulaion produs appear a frequenies ω -ω and ω -ω. The ouermos ones a frequenies - and - are fifh order inermodulaion produs.. Measures of Nonlineariy In order o ge a beer undersanding of he level of nonlinear disorion of a irui, measures are defined whih haraerize he iruis nonlinear behavior. Two ommonly used measures o desribe nonlineariies of a irui are db ompression poin (PdB and hird order inerep poin (IP. Simple es signals suh as hose desribed in he previous seion are ommonly used o es radio iruis for lineariy and find hese measures. Firs, he db ompression poin as a measure for gain

55 ompression will be disussed. The derivaion of he inerep poin will be shown nex. 9.. db Compression Poin The db ompression poin is defined as he power level where he small signal gain is degraded by db. PdB an be inpu (IPdB or oupu referred (OPdB and is usually given in unis of dbm. Typially, he oupu ompression poin is used, and will be desribed in he following seion. To measure he db ompression poin, a single sinusoidal es signal y(=aos( is usually supplied o a nonlinear irui wih A as he peak ampliude and as he frequeny in radians per seond. If he irui under es has he nonlinear ransfer haraerisi of (. up o hird order, he resuling oupu signal is of he form presened wih equaion (.7. y( = A + ( A + A os( ω + A os(ω + A os(ω ( Coeffiiens, and represen he firs, seond and hird order nonlineariies of he irui under es, respeively. The erm A+ A /4 represens he small signal gain a he fundamenal frequeny. Assuming ha A is greaer han he oher erms onaining A, he small signal gain equals for small inpu signals. If <0, he gain approahes zero for suffiienly high inpu ampliude levels []. The db ompression poin is measured by inreasing he inpu ampliude A from zero unil he measured gain dereases by db. A ompression of db (derease in gain by db orresponds o an ideal volage gain degradaion by a faor of = []. Sine he linear unompressed gain is, db ompression hen ours as desribed wih equaion (.8.

56 ( A 4 40 =. (.8 In equaion (.8, A is he inpu signal ampliude, is he linear oeffiien and is he hird order oeffiien. Solving for he inpu signal ampliude A where db ompression ours for <0 resuls in equaion (.9, where A dbin is he inpu signal level when db ompressed. A dbin = 0. ( The oupu signal ampliude A dbou, when db ompressed, an be found by seing A dbin =A dbou. The gain equals when ompressed by db. Subsiuing A dbin wih (.9 resuls in A dbou =0.47 ( / 0.5. Conversely, one an solve for, given A dbin. Equaion (.0 is desribing oeffiien. = (.0 A dbin Again, and are he firs and hird order oeffiiens of he power series desribing he ransfer haraerisi of a nonlinear devie. Fig..8 illusraes he db ompression poin by displaying oupu power over inpu power in logarihmi sale. The dashed line in Fig..8 depis he linear gain wihou ompression and he solid urve is he ompressed gain. The inpu power level where he wo gain urves differ by db is

57 4 0 linear Oupu Power in dbm 5 0 OPdB db ompressed IPdB Inpu Power in dbm Figure.8 db-ompression poin: linear ransfer (dashed line and ompressed ransfer haraerisi (solid line; inpu referred db ompression poin, IPdB is he inpu power level where he gain is ompressed by db, he ompressed oupu power level a his inpu power level is alled oupu referred db ompression poin OPdB alled db inpu ompression poin. Conversely, he orresponding oupu power level of he ompressed gain urve is denoed db oupu ompression poin... Inerep Poin Inermodulaion produs, pariularly hird order disorion produs, are of major onern in ommuniaion sysems. As disussed above, he frequenies of hese produs appear lose o he desired band and are diffiul o filer. The level of inermodulaion produs is dependen on inpu power level. The inerep poin (IP is a single parameer ha haraerizes he behavior of inermodulaion produs independen

58 4 of inpu power level []. The inerep poin an be inpu or oupu referred. Of mos ineres are seond order inerep poin (IP and hird order inerep poin (IP. The inerep poin is defined as he poin where he fundamenal linear omponen and inermodulaion produs have equal ampliude a he oupu of a nonlinear irui. In mos praial iruis, inermodulaion produs will never be equal o he fundamenal linear erm beause boh ampliudes will ompress before reahing his poin. Neverheless, he inerep poin is useful o haraerize iruis. In he following seion he hird order inerep poin will be derived, using a wo-one es signal wih equal ampliudes A=A =A as inpu o a nonlinear irui. A he heoreial inerep poin, he unompressed linear omponen A would equal he hird order inermodulaion produ from equaion (.6, as follows from equaion (.7 and resuls in equaion (.. 4 A = A (. Ampliude A in equaion (. is he inpu signal ampliude of he wo-one es, is he small signal gain and represens he hird order oeffiien. Solving for A=A IIP gives 4 A IIP = (. Ampliude A IIP in equaion (. is he inpu signal level where he linear omponen is equal o he hird order inermodulaion produ. Coeffiiens and desribe linear and hird order nonlineariies. Typially, power levels are of ineres in radio sysems. The inpu power level

59 4 wih uni Was is found by dividing (A IIP by wo imes he inpu resisane R in, if A IIP is a volage level [9]. The resuling inpu power is desribed wih expression (., where P IIP is he inpu power level in Was, is he small signal gain and is he hird order oeffiien. P IIP = (. R in The hird order inpu inerep poin in uni dbm is found using he following equaion. PIIP IIP = 0 log0 dbm ( W In (.4, IIP is he inpu referred IP (IIP [9]. In equaion (.4, IIP is alulaed in respe o mw. The oupu ampliude A OIP is equal o A IIP as shown in equaion (.5, where is he linear and is he hird order oeffiien. A 4 =, (.5 OIP AIIP = Similar o he inpu power level, he oupu power level is alulaed by dividing (A OIP by R L where R L is he load resisane and A OIP is a volage ampliude. This is shown wih expression (.6. P OIP = R. (.6 L

60 44 As saed above, and are linear and hird order oeffiiens, respeively. The oupu referred IP (OIP in dbm is given by he following equaion (.7, where P OIP is he oupu power level in Was. POIP OIP = 0 log0 dbm, ( W Equaion (.6 indiaes ha OIP depends on he linear oeffiien, he oeffiien of hird order nonlineariy, and R L. If approahes zero hen P OIP approahes infiniy as expeed sine he ubi erm in (. is muliplied by. Inpu and oupu inerep poins are relaed by he gain G in db: OIP=IIP+G. Fig..9 illusraes he irumsanes of he hird order inerep poin graphially. The plo in Fig..9 displays he oupu power versus he inpu power in dbm. The upper solid line desribes he linear omponen and he lower solid line desribes he hird order disorion produ dependen on inpu power levels. Boh solid lines ompress and do no inrease furher due o ompression and he limi in supply volage. The dashed lines in Fig..9, indiae exrapolaion of he solid lines. The poin where he dashed lines inerse is alled hird order inerep poin. The horizonal oordinae of he inerep poin is usually referred o as hird order inpu inerep poin IIP and he verial oordinae is he hird order oupu inerep poin OIP. As saed before, as he inpu power inreases by db, he linear oupu power also inreases by db, bu he hird order disorion inreases by db due o a slope of hree of he hird order disorion omponen on a logarihmi sale. In Fig..9, he inpu IP equals 5 dbm and he oupu IP is 6 dbm.

61 45 Slope= Slope= Figure.9 Third order inerep poin IP: he linear omponen has a slope of and he hird order disorion has a slope of. Solid lines indiae he behavior of a praial irui, where he oupu power levels lip. The dashed lines indiae exrapolaion of he ompressed urves and inerep a he inerep poin. IP an be inpu (IIP=5 dbm or oupu (OIP=6 dbm referred. From measuremen of he oupu sperum using power levels, a more praial equaion for alulaion of OIP an be found wih expression (.8 [9]. P P ou rd OIP = + P ou, (.8

62 46 In he expression (.8 for he OIP, P ou is he fundamenal (linear power level in dbm, P rd is he hird order disorion produ power level in dbm and OIP is he hird order oupu inerep poin in dbm. Fig..0 illusraes he derivaion of equaion (.8. Fig..0 shows he oupu frequeny sperum of a nonlinear irui wih a wo-one es signal as inpu. The innermos speral lines are he fundamenal ones of he wo-one es a frequenies and and he ouermos speral lines are hird order disorion produs. The linear power levels a he fundamenal frequenies and have equal P P ou P rd - - Figure.0 Derivaion of hird order inerep poin: fundamenal ones a and have a linear oupu power level of P ou and hird order disorion have a power level of P rd ; hird order disorion P rd rises hree imes faser wih inrease in inpu power level han fundamenals, P equals P ou P rd =(OIP P ou oupu power levels of P ou and he hird order inermodulaion produs a frequenies - and - have an oupu power level of P rd in dbm. The differene P equals P ou P rd. P ou is he oupu power level of he fundamenal frequeny, and P rd is equal o he power level of he hird order inermodulaion produ. OIP is he power level of he hird order oupu inerep poin. A similar inerep poin an be defined for he seond order inermodulaion

63 47 Slope= Slope= Figure. Seond order inerep poin IP: linear omponen has slope of one, seond order disorion produ has a slope of ; solid lines indiae he behavior of a praial irui where he oupu power levels lip. The dashed lines indiae exrapolaion of he ompressed urves and inerep a he inerep poin. IP an be inpu (IIP=0 dbm and oupu (OIP=0 dbm referred. produ [8], desribed wih expression (.9. OIP + P (.9 = Pou P nd ou OIP is he seond order inerep poin in dbm, P ou is he linear oupu power in dbm and P nd is he seond order inermodulaion produ power level in dbm. Fig..

64 48 illusraes seond order inerep poin IP. Similar o Fig..9, he solid lines in Fig.. are linear and seond order disorion omponens. Boh solid urves are exrapolaed as shown wih he dashed lines. The poin where he dashed lines inerse is alled seond order inerep poin. As demonsraed before, he fundamenal omponen inreases linearly unil lipping ours. The seond order inermodulaion produ level rises wih a slope of wo due o he seond order erm in equaion (.. IP an be referred o inpu (IIP or oupu (OIP..4 Phase Disorion Seion. onsiders only disorion resuling from hanges in inpu ampliude, ommonly alled ampliude-modulaion-o-ampliude-modulaion (AM/AM onversion. Mos praial iruis exhibi memory effes beause of sorage elemens suh as apaiors and induors. In his ase, phase disorion an also arise from apaiive or induive nonlineariies. This is ommonly referred o as ampliude-modulaion-o-phasemodulaion (AM/PM onversion. I is espeially riial in sysems where signals wih phase modulaion, suh as quadraure phase shif keying (QPSK, are proessed. The mehanism leading o AM/PM onversion an be explained using he fundamenal linear omponen from (.7 as shown in (.0 [6]. ( A + A os( ω (.0 4 The inpu signal ampliude is denoed A. and are he firs and hird order oeffiiens, respeively. The ampliude of equaion (.0 onsiss of a firs order, A, and hird order, A /4, omponen. Beause of a nonlinear apaiane, a phase

65 differene an exis beween hese wo omponens. The response is hen he veor sum of wo phasors depied as expression (. [6]. 49 V 4 jθ = A + A e (. The sum of he phasors is denoed V and he differene in angle of linear and hird order erm in (. is denoed. and, again, are firs and hird order oeffiiens, respeively. In (., he hird order omponen A /4 exhibis a phase angle differen from he linear omponen. Addiion of he wo phasors, linear and hird order, leads o a phasor V wih a new angle. This new angle is dependen on variaions of he inpu signal ampliude A, even when remains onsan [6]. As a resul, hanges in inpu ampliude give rise o hanges in angle of he linear omponen of he oupu signal. This mehanism is alled AM/PM onversion. Sine he hird order omponen rises hree imes faser han he firs order oupu omponen on a logarihmi sale, AM/PM onversion will be mos serious as he irui is driven ino sauraion.

66 CHAPTER 4: DESCRIPTION OF NEW LINEARIZATION METHODS Given he heoreial bakground of he previous haper, he new linearizaion mehods are desribed in he presen haper. The presen haper also presens a simple algebrai expression, useful in praie, whih desribes he ondiion for anellaion in erms of oupu inerep poin and gain. The simpliiy of he underlying heoreial approah is pariularly useful for he design of a wide variey of new linearizaion opologies, beyond he simple opology buil in his hesis. In he following, basi priniples of he new linearizaion approah are presened. Then, ondiions for aneling hird order disorion are given for a simple embodimen. Finally, an evaluaion of he ondiion of anellaion is performed wih Mahad. Tes daa for a simple inegraed irui implemenaion is given laer in Chaper 5. More sophisiaed linearizaion embodimens are presened in Chaper Basi Priniple of New Mehods The underlying priniple of he new linearizaion mehods is ha a highly disored signal an be subraed from a less disored signal o eliminae he disorion in he final oupu signal [0]. This overall disorion improvemen is ahieved by ombining a more linear amplifier wih a less linear amplifier in suh a way ha he ombined performane is signifianly improved and exeeds he performane of he wo original amplifiers. This unexpeed resul an be illusraed wih a simple arhieure, as shown in Fig. 4.. In Fig. 4., he inpu signal x is applied o an inpu oupler whih splis

67 5 (a Inpu x Coupler k (b (d A x y ( kx=x G, OIP + _ (f Oupu y A y G, OIP (e Figure 4. Blok diagram of new linearizaion mehod: oupler splis and sales inoming signal x and x, x is inpu o more linear amplifier A, x is inpu o less linear amplifier A; oupus are subraed y -y ; frequeny sperum in every pah is shown in (a o (f; gain G and hird order oupu inerep poin deermine haraerisi of amplifier [0]. he signal ino x and x shown on he lef side in Fig. 4.. The oupler has a oupling faor k and allows signal x, o be saled relaive o x. The definiion of oupling oeffiien k is: k = x /x. Signal x is hen inpu o more linear amplifier A depied in he upper half of Fig. 4.. Signal x is inpu o less linear amplifier A illusraed in he lower half of Fig. 4.. Oupu y of amplifier A is subraed from he oupu signal y of amplifier A by means of a subraor as shown on he righ side in Fig. 4., resuling in final oupu y. The lineariy of he wo amplifiers is adjused by parameers gain G (db and oupu inerep poin OIP (dbm for amplifier A and by parameers gain G (db and oupu inerep poin OIP (dbm for amplifier A. The proper relaion of he parameers for anellaion of hird order disorion will be derived in Seion 4.. In Fig. 4., he inpu es signal is a wo-one es signal of he form of equaion (.6. The frequeny spera of all signal pahs are shown in Fig. 4. labeled from (a o

68 5 (f. Firs, frequeny sperum (a is ha of a wo-one es signal a he inpu of he oupler. Frequeny spera (b and ( a he inpus of amplifiers A and A, respeively, show signals a he same frequeny ones as sperum (a, when he oupler does no add any disorion. In Fig. 4., he sperum a he oupu y of he more linear amplifier A is shown in (d. The wo ouermos lines of sperum (d are hird order disorion produs, whereas he wo innermos speral lines orrespond o he linear omponen. Third order inermodulaion produs (ouermos lines appear due o nonlineariies in amplifier A. Similarly, hird order disorion is reaed a he oupu of amplifier A, shown in sperum (e. In sperum (e, he fundamenal ones have lower ampliude relaive o he hird order disorion level han he sperum (d shown a he oupu of amplifier A. If he parameers of he amplifiers in Fig. 4. are adjused properly, he hird order disorion produs of (d and (e (ouermos speral lines have he same ampliude. Under suh ondiions, he final oupu sperum shown in (f only onains he fundamenal ones, afer subraion of signal y from y. The hird order disorion is ideally eliminaed. The fundamenal ones in (f are no aneled beause fundamenal ones in (d and (e are differen in ampliude. Furhermore, he gains of he wo amplifiers are no equal. Finally, he oupu signal y only onsiss of he fundamenal speral lines orresponding o an amplified version of he inpu frequeny sperum [0]. In summary, he unexpeed resul is ahieved by adding a bad (less linear amplifier o a good (more linear amplifier resuling in a beer omposie amplifier. 4. Canellaion Condiions Using Power Series Expansion The ondiion for anellaion of hird order disorion in Fig. 4. an be derived

69 using power series polynomial expansions (.. Amplifier A an be modeled wih a power series as shown in (4.. 5 y = + (4. a0 + ax + a x a x Here, a 0, a, a and a desribe he nonlinear behavior of amplifier A. In he same manner, nonlineariies of amplifier A an be modeled wih power series (4.: y = + (4. b0 + b ( kx + b ( kx b ( kx Again, b 0, b, b and b desribe he nonlinear behavior of amplifier A. The inpu signal is x and y and y are he oupu signals of amplifiers A and A, respeively. Oupu y is subraed from oupu y resuling in final oupu y. Final oupu y an be desribed wih (4.. y = y y = a0 b0 + ax b ( kx + a x b ( kx + ax b ( kx. (4. As an be seen from (4., hird order disorion is aneled and eliminaed when a -b k =0 or k=(a /b /. In order o preven he anellaion of he linear omponens (a -b k mus no equal zero. The same mehod an be used o eliminae oher nonlineariies of order n suh as seond order or fifh order by seing k=(a n /b n /n [0]. 4. Canellaion Condiion in Terms of Oupu Inerep Poin I is no sraigh-forward o obain oeffiiens of power series (4. and (4.. A more praial way o express he ondiion of anellaion is ahieved using measured oupu inerep poins OIP and gains G. To derive he ondiion for anellaion,

70 54 onsider Fig. 4. and le he oupling oeffiien k be one, resuling in an equal disribuion of he inpu signal beween he inpus of he wo amplifiers. Then, le amplifier A have gain G in db and hird order oupu inerep poin OIP in dbm, similarly amplifier A has a gain of G and hird order inerep poin OIP. The hird order oupu inermodulaion disorion levels are denoed P rd and P rd for amplifier A and A respeively, and hey are found by solving equaion (.8 for P rd = P ou -OIP, where P ou is he oupu power level in dbm of he linear omponen. P ou an be subsiued wih P ou =G+P in, where P in is he power level a he inpu of he wo amplifiers. Third order inermodulaion disorion levels of he wo amplifiers an hen be deermined by equaion (4.4 for amplifier A and equaion (4.5 for amplifier A. P rd ( G + Pin OIP = (4.4 P rd ( G + Pin OIP = (4.5 In (4.4 and (4.5, OIP and OIP are he hird order oupu inerep poins of amplifier A and amplifier A, respeively. G is he gain of amplifier A and G is he gain of amplifier A. The ondiion for equal hird order inermodulaion disorion levels is reahed by seing P rd =P rd resuling in (4.6. ( G + OIP (4.6 Pin OIP = ( G + Pin (4.6 an be simplified, resuling in he simple expression (4.7 desribing he anellaion ondiion.

71 55 ( OIP OIP = ( G G (4.7 In (4.7, G -G is he gain differene and mus no be zero o preven annihilaion of he desired signal []. OIP OIP is he differene of hird order oupu inerep poins. Expression (4.7 desribes he ondiion for anellaion of hird order disorion and as one an see, (4.7 is independen of inpu power levels. A derivaion of he heoreial gain of he omposie amplifier, G C =G -G is shown as follows. The subraion a he oupu of he wo amplifiers in Fig. 4. akes plae a he volage or urren level. If volages are subraed, hen he final oupu volage would be v =v o -v o. The volages v o and v o would be oupu volages of amplifier A and amplifier A, respeively, and v would be he oupu volage of he omposie amplifier. The oupu volages an be expressed wih he linear volage gain g of he amplifiers, as follows g g v in = g = g v in g g v in (4.8 In above equaion (4.8, g is he linear volage gain of he omposie amplifier, g and g are he linear volage gains of amplifier A and amplifier A. To find gain G C in dbm, he logarihm wih base 0 is applied o equaion (4.8, resuling in expression (4.9, where G C is he heoreial gain in db of he omposie amplifier and G and G are gains in db of amplifier A and amplifier A, respeively. GC 0 0 = 0log (0 0 (4.9 0 G G Similarly, noise figure an be ompued for he omposie amplifier.

72 4.4 Theoreial Evaluaion of Canellaion Condiion 56 A mahemaial evaluaion of he ondiion of anellaion desribed wih equaion (4.7 is performed in Seion 4.4. wih varying inpu power levels using Mahad []. Before proeeding o his evaluaion, a basi relaionship beween OIP and db ompression poin is esablished. This relaionship is used in Seion 4.4. for he mahemaial evaluaion wih Mahad. Using he resuls from Chaper, a heoreial relaionship an be esablished as shown in (4.0 A A OIP dbou = =.4 (4.0 or wih expression (4. in erms of db ompression poin and hird order oupu inerep poin OIP OPdB = 0 log0.4 = 0. 6dB. (4. In equaions (4.0 and (4., A OIP and A dbou are he ampliudes for hird order inerep poin and db oupu ompression poin, respeively. Coeffiiens and are linear and hird order oeffiiens. Furhermore, OPdB and OIP are db oupu ompression poin and hird order oupu inerep poin. In mos praial RF sysems, he differene in OPdB and OIP is abou 0 o 5 db [9], ypially db.

73 57 To also illusrae he heoreial value of db, a simulaion of an ideal limier was performed using Agilen Advaned Design Sysem (ADS []. The limier has he ransfer funion depied in Fig. 4.. The inpu-oupu haraerisi has an ideal linear region wih volage gain and slope of one. The oupu signal is limied o + V and V. A wo-one es signal was applied o he inpu of he limier and he inpu signal ampliudes were inreased from zero o greaer han he ompression poin. As a resul, he db ompression poin was found o be approximaely dbm wih a hird order inermodulaion disorion level of 9.6 dbm. The alulaed OIP equals 4. dbm, hus a differene beween OPdB and OIP (4. dbm dbm is. db. In many iruis, OIP is approximaely db greaer han he db ompression poin [9]. +V V ou V in -V Volage gain A v = Figure 4. Ideal limier volage ransfer haraerisi: wih linear gain of and limis ± V a he oupu 4.4. Preliminary Mahemaial Analysis of Canellaion Condiion In his seion, a preliminary mahemaial analysis of he anellaion ondiion

74 58 desribed wih equaion (4.7 is performed using Mahad. The purpose of he evaluaion is o gain insigh ino he performane of he new linearizaion ehnique. The Mahad program ould also be used for design purposes if he alulaed resuls show good agreemen wih he measured resuls. For he following alulaion wih Mahad, i is assumed ha he db ompression poin of he amplifiers ours a an oupu power level of (OIP- dbm. Oherwise, he oupu power level of he amplifiers is defined as gain added o he inpu power in dbm, G+P in. Fig. 4. illusraes he irumsanes. As shown on he lef side in Fig. 4., he inpu is equally disribued beween more linear amplifier A and less linear amplifier A. Oupu of amplifier A is subraed from oupu of amplifier A as depied on he righ side in Fig. 4.. Boh amplifiers are haraerized by parameers gain G and hird order oupu inerep poin OIP. The subraion of he oupu signals akes plae a he A P o + P in G, OIP P ou _ A P o G, OIP Figure 4. Topology of new linearizaion mehod: inpus of he wo amplifiers are onneed ogeher as primary inpu, amplifier A is more linear amplifier, amplifier A is less linear amplifier; oupu signals are subraed; gain G and hird order oupu inerep poin desribe he amplifiers nonlinear haraerisi [0]

75 volage or urren level and no he power level. The final oupu power P ou is alulaed wih equaion (4. 59 Pou Po Po 0 0 = 0log(0 0. (4. The oupu power levels in dbm of amplifiers A and A are denoed P o and P o, respeively. The final oupu in dbm is P ou. Independen parameers are gain, G, hird order inerep, OIP of amplifier A and he gain differene G G. The gain of amplifier A is alulaed using he gain differene and OIP is alulaed wih he use of equaion (4.7. Gain and hird order oupu inerep poins of amplifier A and amplifier A from Fig. 4. are summarized in Table 4.. The omplee Mahad program is aahed in Appendix B. Table 4. Parameer of amplifiers A and A for Mahad alulaion Amplifier A Amplifier A Gain -9 db -40 db OIP -6 dbm -.5 dbm Calulaed resuls are disussed in he following paragraphs, using Fig. 4.4 hrough Fig Firs, Fig. 4.4 shows alulaed linear oupu power level over inpu power level. The solid line haraerizes he linear oupu power of he omposie amplifier (A+A. The dashed urve desribes he oupu power level of amplifier A and he dash-doed rae depis he oupu power level of amplifier A.As referene, he hird order disorion suppression is shown as doed urve in he upper righ orner of Fig Suppression is defined as he amoun of reduion of hird order inermodulaion

76 60 0 Linear Power (dbm, Suppression (db Amplifier A IMD suppression Composie amplifier Amplifier A Inpu Power in dbm Figure 4.4 Calulaed (Mahad oupu power levels: linear oupu power of amplifier A (dashed, amplifier A (dash-do and omposie (solid amplifier; suppression of hird order disorion is doed line. The inpu power is varied. Verial axis displays linear oupu power in dbm and hird order disorion suppression in db. produ level a he final oupu relaive o he hird order inermodulaion produ level of amplifier A. A hird order disorion suppression of greaer han 80 db an be reahed before diminishing o a level of abou.9 db as shown in Fig Furhermore, in Fig. 4.4, hird order disorion suppression begins o derease a an inpu power level of 5.4 dbm. As one an see from Fig. 4.4, amplifier A ompresses a he same inpu power level of 5.4 dbm whereas he oupu power level of amplifier A oninues o inrease up o dbm inpu power. Due o ompression of amplifier A, he gain differene deviaes from he value neessary o uphold he anellaion ondiion of equaion (4.7. The oupu power level of he omposie amplifier, in Fig. 4.4, shown as solid line, lips also a an inpu power level of dbm and remains a a maximum oupu level of 9. dbm.

77 6 In addiion, he final oupu power is redued ompared o he oupu power of A. In he region beween ompression poins of amplifier A and A, he final oupu power inreases wih greaer slope beause less power is subraed from he oupu of amplifier A. Seondly, o gain more insigh in he mehanism of hird order anellaion, hird order disorion produs of he wo amplifiers in Fig. 4. and he omposie amplifier are ploed in Fig Fig 4.5 shows he hird order disorion power level versus he inpu power level. The solid line displays he hird order disorion of amplifier A only, hird order disorion of amplifier A is depied as dashed line in Fig. 4.5 and he hird order disorion a he final oupu is shown as dash-doed line. As a referene, he hird order disorion suppression is displayed as doed line on he upper righ orner in Fig As one an see in Fig. 4.5, he hird order inermodulaion produ of amplifier A also ompresses a an inpu power of 5.4 dbm whereas hird order disorion of A oninues o inrease. As a resul of he differen hird order power levels, hird order produs no longer anel and hird order disorion raises rapidly a he final oupu of he omposie amplifier. This mehanism resuls in imperfe suppression. IMD of he omposie amplifier assumes approximaely he same value as IMD of amplifier A when A ompresses.

78 6 0 Third Order Disorion (dbm, Supp (db Amplifier A Composie amplifier IMD suppression Amplifier A Inpu Power in dbm Figure 4.5 Calulaed (Mahad hird order disorion levels: hird order Inermodulaion disorion of A (solid, A (dashed and omposie (dashdo amplifier over inpu power; suppression of hird order disorion is inluded as doed line. Gain, as one adjusable parameer in his heoreial examinaion, is anoher imporan parameer. Fig. 4.6 shows gain urves over he inpu power of omposie amplifier and amplifier A of Fig. 4.. The gain of amplifier A is desribed wih he op dashed urve. The solid urve in Fig. 4.6 haraerizes he gain of he omposie amplifier. Gain of amplifier A remains, as defined in he Mahad program, onsan unil ompression begins. The omposie amplifier exhibis diminished gain due o subraion of oupu power levels of he wo amplifiers A and A. The inrease in gain of he omposie amplifier is a resul of subraion of redued oupu power of amplifier A. Amplifier A ompresses a an inpu power level of dbm.

79 6 8 0 Amplifier A Gain in db 4 6 Composie amplifier Inpu Power in dbm Figure 4.6 Calulaed (Mahad gain: gain of amplifier A (doed and gain of omposie amplifier (solid, G =-9 db, G C =-.8 db, gain of omposie amplifier rises o 0.4dB. To summarize he heoreial onsideraions: he new linearizaion ehnique promises an improvemen in hird order disorion for he opology in Fig. 4. for appliaions wih low inpu power signals, suh as reeivers. Some improvemen in lineariy is ahieved above he ompression poin of he less linear amplifier A.

80 CHAPTER 5: INTEGRATED CIRCUIT REALIZATION OF NEW LINEARIZATION METHOD This haper presens daa for he ommonly known ross-oupled differenial pair in he onex of he preeding analysis. The opology of he ross-oupled differenial pair is used o show he validiy of equaion (4.7 for inegraed iruis and demonsraes he overall approah. The ross-oupled differenial pair is also a basi building blok for more sophisiaed opologies disussed laer in Chaper 7. Measured resuls, disussed wihin his haper, onfirm equaion (4.7 for fully inegraed iruis. This measured daa forms a baseline for fuure researh. 5. Cross-oupled Differenial Pair The new linearizaion ehnique lends iself o inegraed irui realizaion beause no exernal pars are neessary. For a firs prooype, a ross-oupled differenial pair [] was hosen as he es irui, beause of is simple subraing mehanism. The irui of he ross-oupled differenial pair is depied in Fig. 5.. Compared wih Fig. 4., he subraion of he signals a he oupu of he amplifiers is aomplished by rossoupling he drains of ransisors M and M wih ransisors M and M4 in Fig. 5., hene he name ross-oupled differenial pairs. Beause drain urrens are subraed, his mehod of subraion is someimes alled urren differening []. Curren differening is defined in [] as he proess of subraing wo nonlinear urrens wih equal nonlinear erms bu unequal linear erms, resuling in a linear inpu-oupu

81 65 Ibias Ibias IRD IRD V DD=5V Ibias Ibias R D=50Ω R D=50Ω M M M M4 Vodiff Vidiff M5 M6 Vbias Vbias GND ISS ISS Amplifier A Amplifier A Figure 5. Cross-oupled differenial pair: amplifier A onsiss of differenial pair M and M, and urren soure M5, amplifier A onsiss of differenial pair M and M4 wih ransisor M6 as urren soure; drain resisors R D =50, he supply volage is denoed V DD =5V; drains of M and M are ross-oupled wih drains of ransisors M and M4. Tail urrens I SS and I SS an be adjused wih V bias and V bias. Inpu V idiff and oupu V odiff volages are differenial volages. dependene. In Fig. 5., amplifier A onsiss of ransisors M and M, omprising a differenial pair, and ransisor M5 represening a urren soure. Similarly, amplifier A onsiss of a differenial pair, omprised of M and M4, and a urren soure M6. All ransisors are of n ype hannel. Boh amplifiers share drain resisanes R D =50. The supply volage equals 5 V. Drain urrens I SS and I SS of M5 and M6, respeively, an be varied wih bias volages V bias and V bias, respeively. The differenial inpu volage is denoed V idiff and he differenial oupu volage is denoed V odiff. The inpu erminals

82 66 are onneed parallel. The number raio nex o he ransisors in Fig. 5. is he aspe raio of he ransisors, W/L in mirons. In order o preven apaiive nonlineariies from ineraion wih he nonlinear behavior of ransisors, n-diffusion resisanes R D were kep low in value. The n-diffusion resisors exhibi nonlinear apaiane due o a pn-junion beween n-diffusion and p- doped subsrae. As a resul, he larger he n-diffusion layer is, he larger he nonlinear parasii apaiane [4]. The small signal differenial volage gain of he ross-oupled differenial pair in Fig. 5. is denoed A vdiff and an be alulaed wih equaion (5.. vodiff A = vdiff ( g ma g ma RD v = (5. idiff In (5., differenial inpu volage and oupu volage are denoed v idiff and v odiff, respeively. Transonduane g ma is ransonduane of ransisors M and M, and g ma is ransonduane of ransisors M and M4. R D is he drain resisor. Equaion (5. is valid if all ransisors in Fig. 5. have square-law behavior. Furhermore, hannel lengh modulaion is negleed, hus drain-soure resisane is infinie. A full derivaion of he small signal low frequeny gain is shown in Appendix C. The large signal DC ransfer haraerisi of a single differenial pair an be found in he lieraure [4] as W k W I = I I = I k V L, (5. d d d SS n L idff n Vidiff 4I SS

83 67 assuming square law behavior of he ransisors omprising he differenial pair. However, in he ase of he presened prooype, he ransisors do no exhibi square-law behavior due o veloiy sauraion. The differenial drain urren of a single differenial pair is denoed I d. If only amplifier A is aivaed, hen I d and I d are drain urrens of ransisors M and M. Lengh L and widh W are gae lengh and widh of ransisors M and M and I SS is he bias urren and is denoed I SS for amplifier A. In ase only amplifier A is aivaed, hen I d and I d are drain urrens of ransisors M and M4. Similarly, lengh L and widh W are gae lengh and widh of ransisors M and M4 and I SS is denoed I SS for amplifier A. For equaion (5., he differenial inpu volage V idiff, is limied o V idiff =(I SS /(k n W/L 0.5 for he ransisors o remain in sauraion region [4] and ransonduane faor k n equals 0 C ox of an n-hannel meal oxide semionduor (MOS ransisor. Equaion (5. depis a nonlinear funion. In order o find he onribuing harmoni disorion, (5. an be expanded ino a MaLaurin series. A MaLaurin series is a Taylor series expanded around zero (V idiff = 0. A single differenial pair an be desribed by he following MaLaurin series []: W kn W 0 L I = + d I SS k n Vidiff Vidiff (5. L 8 I SS The parameers and variables in (5. are he same as in (5.. As one an see from equaion (5., even order erms are zero, resuling in eliminaion of even order disorion, however, due o he hird order oeffiien, hird order inermodulaion

84 68 produs are sill presen. The ransfer haraerisis of amplifier A and amplifier A in Fig. 5. an be desribed wih equaion (5.. For proper adjusmen of ransisor parameers, hird order disorion an also be redued or eliminaed. By seing he hird order oeffiien, =-0.5[(k n W/L /I SS ] 0.5 of he wo amplifiers, A and A equal, hird order nonlineariies are eliminaed beause of subraion of drain urrens. This proedure was disussed in Chaper 4. The resuling ondiion for anellaion of hird order nonlineariies in erms of aspe raios and bias urrens is shown in equaion (5.4 []. ( W ( W L L A A = I I SS SS (5.4 In equaion (5.4, (W/L A is he aspe raio of ransisors M and M, and (W/L A is he aspe raio of ransisors M and M4. Index A and A in equaion (5.4 refer o amplifier A and amplifier A, respeively. The aspe raios mus no be equal o preven anellaion of he linear erm in equaion (5.. For he irui in Fig. 5. he raio an be alulaed using aspe raios of ransisors M and M o (/9 / =.56. In Chaper, a dependene of OIP on firs and hird order oeffiiens was derived. Equaion (.5 desribes he linear oupu ampliude of he hird order inerep poin in erms of and. Firs and hird order oeffiiens are known for he presen ase of differenial pairs (5. as shown in (5.5 and (5.6: W = g m = I SS kn (5.5 L

85 69 W kn L = (5.6 8 I SS The ransonduane faor k n is equal o 0 C ox and W/L is he aspe raio. I SS is he bias urren of he differenial pair. The ransonduane is denoed g m. Subsiuing and in equaion (.5 wih (5.5 and (5.6 leads o A W I SS k n L = I SS (5.7 W kn L I OIP = SS In equaion (5.7, A OIP is he urren level a he oupu hird order inerep poin. The imporan resul of equaion (5.7 is, ha he OIP is only dependen on he bias urren. The oupu power a he oupu hird order inerep poin would be alulaed by muliplying (A OIP wih R L /: P OIP =5.I SS R L. As one an see, he hird order oupu inerep poin is proporional o I SS. The inegraed irui was fabriaed in a MOSIS 0.5 m omplemenary mealoxide semionduor (CMOS sandard proess and pakaged in a 40-pin dual-in-line erami hip pakage [5]. Table 5. summarizes proess parameers of he 0.5 m CMOS proess for he bah wih number TAK-AZ.

86 70 Table 5. Proess parameers for NMOS ransisor for a 0.5m CMOS proess (bah: TAK-AZ [5] Threshold volage: V T0 Eleron mobiliy: 0 Transonduane faor: 0 C ox Body-Effe faor: Fermi poenial: F 0.75 V 475 m /V/s 6 A/V 0.48 V V 5. Basi Tes Seup Before presening he measured resuls, he basi es seup wih he uilized measuremen equipmen is desribed. The inegraed iruis nonlinear behavior was haraerized wih a wo-one es. Fig. 5. illusraes he seup. The wo sinusoidal ones are reaed wih wo signal generaors of ype HP E4400B, uned o wo differen onsan frequenies losely spaed. The signals are hen ombined wih he help of wo 0 db aenuaors in order o provide suffiien isolaion beween he signal generaors. The ombined sinusoidal ones are inpued o a hird aenuaor wih db aenuaion. The oupu of he aenuaor is hen inpu o he amplifier es prined irui board (PCB. The firs resuls presened in he following seion were measured a a frequeny of 0 MHz on he wafer probe saion RF- from Casade Miroeh. For his reason, a bias ee was neessary o bias he MOS ransisor a he inpu of he es irui in Fig. 5.. The bias volage was V gae =4.5 V. The pads of he hip were direly onaed wih wafer probes of ype ACP40 (air oplanar probe for 40 GHz bandwidh. The oupu signal of he amplifier under es was amplified using a Miniiruis ERA-SM amplifier [6] mouned on a PCB, and oupu signal sperum reorded wih a sperum analyzer of ype HP E440B. A apaiane beween es irui and Miniiruis amplifier makes sure

87 7 Signal Generaor HP E4400B Aenuaor 0dB Aenuaor db Bias Tee C L Tes irui C Sperum Analyzer Vgae Amplifier: MiniCiruis ERA-SM HP E440B Figure 5. Basi es seup for wo-one measuremen: on he lef side, wo sinusoidal signals are generaed by wo signal generaors, he wo 0dB aenuaors are passive devies, he bias ee in he middle onsiss of a apaiane and induane and serves as means of supplying bias volage V gae o he inpu of he es irui. The es irui is mouned on a PCB bu was measured using a wafer probe saion, he following devie C is a oupling apaior; he oupu sperum of he es irui is displayed wih a sperum analyzer. The inpu power a he es irui is approximaely 7 db below he reading of he signal generaor ha he DC operaing poins of he wo devies are no affeed. The Miniiruis amplifier wih a gain of 0 db, was added o inrease he power level and overome he resriions of he sperum analyzers noise floor. All measuremens were performed single-ended, wih he gaes of ransisors M and M4 in Fig. 5. biased wih 4.5 V, bu onneed o small-signal ground via a 00 nf apaior. 5. Measured Resuls In he following seion, measured resuls are presened and disussed. In order o prove he heory of he new linearizaion mehod, in pariular equaion (4.7, hird order inermodulaion produ and oupu power levels of he oupu signal were measured and reorded. Third order oupu inerep poins were alulaed from he measured values. The hird order anellaion performane was measured a frequenies,, 0, 0 and 0 MHz. The hird order anellaion performane a higher frequenies was redued,

88 7 due o bandwidh limiaions of hip pakage and signal leakage beween inpu and oupu of he es irui. The displayed power level on he signal generaors was aken as inpu power level, whih is wih respe o a mahing 50 load. Due o aenuaion, he inpu power a he es irui is approximaely 7 db below he reading of he signal generaor. During he measuremen of amplifier A, amplifier A was urned off and during he measuremen of amplifier A, amplifier A was urned off. For he omposie amplifier, boh amplifiers A and A were urned on and adjused for anellaion of hird order inermodulaion produs. 5.. Measured Resuls for 0 MHz Tesed wih Wafer Probe Saion Firs, oupu frequeny spera for a es frequeny of 0 MHz are shown in Fig. 5., 5.4, 5.5 and 5.6. This se of plos depis frequeny spera of more linear amplifier A, less linear amplifier A, omposie amplifier, and signal leakage. Signal leakage was measured beween inpu and oupu when boh amplifiers were deaivaed. The inpu power level for he se of figures for he presen ase was 0 dbm a he signal generaor and 7 dbm a he inpu of he es irui. Similarly, he reading a he sperum analyzer is approximaely 0 db higher han he oupu power of he es irui. In Fig. 5., he wo speral lines in he ener are he fundamenal frequenies a 0 MHz and 0. MHz and wih oupu power levels of 9. dbm. Third order inermodulaion produs of 75.7 dbm power level appear a frequenies 09.9 MHz and 0. MHz, depied as ouermos speral lines in Fig. 5.. Fig. 5.4 piures he oupu frequeny sperum of amplifier A. Fundamenal ones have a power level of 40.6 dbm and hird order disorion produs have a power level of 75.9 dbm. Due o proper adjusmen of drain urrens of he wo amplifiers, A and A, hird order

89 7 disorion power levels are almos equal. By subraing he wo oupu signals, reduion in hird order disorion was ahieved. Fig. 5.5 shows he resul in he oupu sperum of he omposie amplifier. In Fig. 5.5, he hird order inermodulaion produs were redued below he noise floor of he sperum analyzer o approximaely 98.4 dbm. Comparing hird order disorion of Fig. 5. and Fig. 5.5, a hird order suppression of approximaely db was aained. The fundamenal speral lines, in Fig. 5.5 have a measured power level of. dbm. Again, omparing Fig. 5.5 wih he sperum of amplifier A in Fig. 5., a loss in gain of approximaely db was measured. This resul is an effe of subraing he wo fundamenal signals of Fig. 5. and Fig The signal leakage beween inpu and oupu of he es irui is shown in Fig If boh amplifiers are urned off, a power level a he fundamenal frequenies of approximaely -56 dbm is sill available a he oupu. The leakage is mos likely aused by parasii apaianes in he pad frame of he inegraed irui. Parasii apaianes an appear beween signal line and power and ground rings in he pad frame. The leakage is signifianly below he oupu power level of he less linear amplifier A.

90 74 Figure 5. Measured oupu frequeny sperum of amplifier A a 0MHz: amplifier A is urned off. Fundamenal signals a 0MHz and 0.MHz and hird order disorion a frequenies 09.9MHz and 0.MHz. Oupu power is measured a oupu of Fig. 5., rue oupu power is 0dB lower, irui of Fig. 5. a urren 0.87mA. Figure 5.4 Measured oupu frequeny sperum of amplifier A a 0MHz: amplifier A is urned off. Fundamenal signals a 0MHz and 0.MHz and hird order disorion a 09.9MHz and 0.MHz. Oupu power is measured a oupu of Fig. 5., rue oupu power is 0dB lower, irui of Fig. 5. a urren 0.06 ma.

91 75 Figure 5.5 Measured oupu sperum of omposie amplifier a 0MHz showing anellaion of hird order disorion a frequenies 09.9MHz and 0.MHz. Oupu power is measured a oupu of Fig. 5., rue oupu power is 0dB lower, irui of Fig. 5. a urrens 0.86 and 0.06 ma Figure 5.6 Measured signal leakage beween inpu and oupu of es irui a 0MHz: boh amplifiers are urned off. Fundamenal frequenies a 0 and 0.MHz; Resuls measured using es seup of Fig. 5..

92 76 Table 5. summarizes he resuls for he se of frequeny spera for a es frequeny of 0 MHz using he es seup from Fig. 5.. The firs olumn liss he inpu power P in, displaying he reading of he signal generaor. The inpu power a he es irui is 7 db lower. The fundamenal oupu power, denoed P ou, measured a he Table 5. Summary of measured resuls for 0 MHz using seup of Fig. 5. Amplifier A (Fig. 5. Amplifier A (Fig. 5.4 Composie amplifier A+A (Fig. 5.5 P in (dbm (a generaor P ou (dbm (a sperum analyzer G (db P rd (dbm OIP (dbm Comparison: A wih omposie amp Gain loss: db Third order disorion suppression:.6 db Improvemen of OIP: 8.5 db sperum analyzer is shown in he seond olumn. The linear power a he oupu of he es irui is approximaely 0 db lower. Column number hree from he lef, shows he gain, denoed G. The gain is alulaed as P ou -P in using measured value from Table 5.. Furhermore, hird order disorion level, denoed P rd, as displayed a he sperum analyzer is olleed in olumn four from he righ side. The hird order oupu inerep poin OIP is alulaed wih equaion (.8 using measured resuls and is displayed in he olumn on he righ in Table 5.. Furhermore, he firs row on he op shows

93 77 measured resuls of amplifier A olleed from Fig. 5., he seond row from he op in Table 5. liss resuls of amplifier A from Fig Finally, he hird row liss resuls of he omposie amplifier aken from Fig The row a he boom of Table 5. shows omparisons of he resuls of amplifier A and omposie amplifier. The measured resuls show an improvemen in hird order disorion of.6 db and an improvemen of OIP of 8.5 db. A gain loss of db of he omposie amplifier relaive o amplifier A was measured. In addiion, he measured resuls in Table 5. an be used o prove expression (4.7 whih is saed here again for review: (OIP -OIP =(G -G. Subsiuing he measured resuls ino he lef side of equaion (4.7 leads o: (OIP -OIP =( = 4.0dB and subsiuing resuls ino he righ side of equaion (4.7 gives: (G -G = ( = 4.dB The resuls of boh sides are almos equal, showing proof of he validiy of equaion (4.7. The bias ondiions of he es irui in Fig. 5. for anellaion of hird order disorion are summarized in Table 5.. A ombinaion of ail urrens I SS =87 A and I SS =6 A, wih he proper aspe raios, leads o reduion of hird order disorion whih an be proven by employing equaion (5.4: (W/L A =, (W/L A =9, hene (/9 / =.56 and (I SS /I SS 0.5 =(87/6 0.5 =.. The disrepany beween he wo values.56 and. ould be explained wih he low bias volage of V of ransisor M6. Transisor M6 is mos likely a he low end of srong inversion approahing moderae inversion, hus no longer follows he ransisor square-law haraerisi.

94 78 Table 5. Summary of bias ondiions for anellaion of hird order disorion V bias I SS Amplifier A (M5 in Fig V 87 A Amplifier A (M6 in Fig V 6 A Composie amplifier (M5+M6 in Fig. 5. Toal bias-urren: 0 A Bias volage V Gae of M hrough M4 4.5 V In Chaper 4, heoreial onsideraions indiae ha suppression of hird order disorion is redued a higher inpu power levels. In order o verify a derease in hird order suppression, he inpu power level was varied from 0 dbm o +0 dbm. Again, all hree amplifiers, A, A and omposie amplifier, were esed. The es frequeny was mainained a 0 MHz. Fig. 5.7 depis linear oupu power levels and hird order inermodulaion produs versus he inpu power level aken from he signal generaor. The inpu power a he es irui is 7 db lower. The oupu power levels are reorded a he sperum analyzer and are 0 db higher han he oupu power of he es irui. The op hree urves in Fig. 5.7 desribe he haraerisi of he linear oupu power. The oupu power level of amplifier A is denoed P ou ; oupu power of amplifier A is denoed P ou and oupu power of he omposie amplifier is denoed P ouc. The linear oupu of omposie amplifier, P ouc, is approximaely db lower han P ou up o approximaely dbm inpu power. As previously disussed, he loss in oupu power is a resul of subraion of he linear fundamenal omponens.

95 79 Due o ompression of A, beginning a an inpu power level of dbm, less signal is subraed from he more linear amplifiers (A oupu signal and he final oupu power of he es irui, P ouc, inreases faser and approahes power level P ou. The db inpu ompression poins of amplifier A and A were measured and found o be 9 dbm and 4 dbm, respeively, using es seup in Fig. 5.. The db ompression poins were aken from he signal generaor display. The db ompression poin of he omposie amplifier is approximaely equal o ha of amplifier A. The remaining hree urves, a he boom of Fig. 5.7, illusrae hird order inermodulaion disorion, IMD, of A, A and omposie amplifier. For an inrease in inpu power of dbm, IMD inreases by dbm for A and A. Good anellaion of hird order disorion an be ahieved for low inpu power levels up o abou dbm. As an be seen from Fig. 5.7, as oupu power of A ompresses, hird order disorion anellaion is diminished beause hird order inermodulaion produ of A also ompresses. As a resul, hird order disorion levels are no longer equal for higher inpu power levels. Suppression is almos zero as amplifier A reahes is db inpu ompression poin of 9 dbm, however, here is sill an improvemen in suppression a he db ompression poin of amplifier A of abou 8 db. These measured resuls agree wih he alulaed behavior. As an imporan measure for nonlineariy, he haraerisi of he hird order oupu inerep poin, OIP, over inpu power is shown in Fig The upper rae is he hird order oupu inerep poin of he omposie amplifier depied as solid line in Fig The doed urve displays he haraerisi of he hird order oupu inerep poin of amplifier A and he dashed urve a he boom of Fig. 5.8 shows OIP of amplifier

96 80 A. Again, he inpu power in Fig. 5.8 is aken from he signal generaor and is 7 db lower a he es irui inpu. As a resul of suppression of hird order disorion, OIP is improved by more han db for low inpu power levels. An improvemen of abou 7 db is measured a he db ompression poin of A (inpu referred 4 dbm. Informaion abou he behavior of he gain of he es irui is given in Fig Fig. 5.9 shows he gain of amplifiers A, A and omposie amplifier versus he inpu power. In Fig. 5.9, he gain is alulaed using inpu power levels aken from he display of he signal generaor and he oupu power reorded a he sperum analyzer. The upper doed urve in Fig. 5.9 displays he gain of amplifier A, he solid rae desribes he gain behavior of he omposie amplifier. The gain haraerisi of amplifier A is shown a he boom of Fig. 5.9 as dashed urve. As expeed, he gain of he omposie amplifier is lowered by db beause of subraion of he wo fundamenal oupu signals of A and A. As previously explained, he omposie gain inreases as A ompresses due o he fa ha less oupu signal of A is subraed from oupu signal of A. In Fig. 5.9, he overall gain finally dereases as amplifier A ompresses. To summarize, he linearizaion ehnique implemened in his opology anels hird order disorion for inpu power levels below he db ompression poin of he less linear amplifier. Some improvemen in hird order disorion is ahieved a inpu power levels beween db ompression poins of more linear and less linear amplifiers.

97 Pou Linear signals -40 PouC IMD_ Oupu Power in dbm Pou Third order disorion IMD_ -90 IMD_C Inpu Power in dbm Figure 5.7 Measured linear signals and hird order produs of A, A and omposie amplifier for 0MHz over inpu power: Pou is oupu power of A, Pou is oupu power of A and PouC is oupu power of omposie amplifier; hird order disorion is denoed IMD, index, and C refer o amplifier A, amplifier A and omposie amplifier. Oupu power is measured a oupu of Fig. 5., rue oupu power is 0 db lower. Inpu power is a signal generaors of Fig. 5., rue inpu power is 7 db less, irui of Fig. 5. a urrens 0.87 and 0.06 ma.

98 8 0 5 OIP in dbm Composie amplifier Main amplifier A Auxiliary amplifier A Inpu Power in dbm Figure 5.8 Measured hird order oupu inerep poin OIP of amplifier A, A and omposie amplifier a 0MHz over inpu power. Inpu power is a signal generaors of Fig. 5., rue inpu power is 7 db less Main amplifier A Gain in db Composie amplifier Auxiliary amplifier A Inpu Power in dbm Figure 5.9 Measured gain of A, A and omposie amplifier a 0MHz over inpu power; inpu db ompression poins, PdB =9dBm, PdB =dbm, PdB C =9dBm. Oupu power is a oupu of Fig. 5., rue oupu power is 0 db lower. Inpu power is a signal generaors of Fig. 5., rue inpu power is 7 db less.

99 8 5.. Measured Resuls for Addiional Frequenies Given he foregoing behavior of he performane of anellaion of hird order disorion wih various inpu power levels, he quesion arises how he new mehod of anellaion performs over a range of frequeny. To answer his quesion, es resuls for differen frequenies, inluding,, 0 and 0 MHz are disussed below. The following measuremen resuls are based on a slighly differen es seup from Fig. 5.. The measuremens were performed wih he es irui on a PCB; no wafer probe saion was used. Fig. 5.0 illusraes he new es seup. A wo-one es was employed in he es irui. On he lef side in Fig. 5.0, wo es signal generaors generae wo sinusoidal ones. Boh ones are ombined using wo 0 db aenuaors o ahieve good isolaion beween he generaors. The es signal, afer passing a db aenuaor, is inpu o he a-oupled es irui. The RF oupu signal is also a-oupled Signal Generaors HP E4400B Aenuaor 0dB Aenuaor db C 00nF C 00nF R 00kΩ Tes Cirui Vgae R 00kΩ C 00nF VDD=5V + GND C4 00nF Sperum Analyzer HP E440B Figure 5.0 Modified es seup for wo-one measuremens: on he lef side, wo signal generaors provide wo sinusoidal signals, hen he wo signals are ombined using wo passive 0dB aenuaors, he following devie is a db passive aenuaor; he inpu signal is a-oupled by a apaior C and inpu o he es irui. The es irui is mouned on a PCB. Resisors R and R are used for biasing he es irui; he oupu signal is a-oupled and inpu o a sperum analyzer of ype HP E440B.

100 84 and inpu o a sperum analyzer. The differenial inpu of he es irui is biased wih volage V gae =4.5 V. The negaive inpu is se o a-ground by uilizing a apaior C=00 nf. Resisors R and R in Fig. 5.0 provide isolaion for he RF signal. Measuremens are performed single-ended. The inpu power level a he es irui was kep onsan a 0. dbm for all following presened measured resuls. The bias urrens for he omposie amplifier for he presen se and for all following ses of frequeny spera for differen frequenies were adjused o I SS =00 A for amplifier A and I SS =0 A for amplifier A. The oal DC urren equals 0 A. Firs, a se of frequeny spera wih es resuls for a frequeny of MHz is presened similar o he se of frequeny spera for a es frequeny of 0 MHz. Fig. 5. shows a plo of he oupu signal of amplifier A. Fundamenal signals appear a MHz and.05 MHz wih a power level of 45. dbm, shown as he wo speral lines in he ener of Fig. 5.. Third order inermodulaion produs appear a frequenies 0.95 MHz and. MHz wih a power level of 85 dbm. Anoher plo for he less linear amplifier, A, is given in Fig. 5.. The oupu power level of he fundamenal ones is 6. dbm and hird order disorion is 84.6 dbm. Afer subraion of he oupu signals, he oupu signal sperum of he omposie amplifier resuls in a sperum, shown in Fig. 5.. The hird order disorion is redued below he noise level of he sperum analyzer and a hird order suppression of a leas 5 db was ahieved. The fundamenal oupu power was diminished by abou.5 db. The inpu power level a he es irui was kep onsan a 0. dbm. Finally, Fig. 5.4 shows he signal leakage of he fundamenals wih a value of -85 dbm. For he measuremen of he leakage, boh amplifiers were

101 urned off. As expeed, he leakage a MHz is signifianly lower han he leakage a a frequeny of 0 MHz, dbm. 85 Figure 5. Measured oupu frequeny sperum of amplifier A a MHz: amplifier A was urned off; fundamenal ones a MHz and.05mhz, hird order disorion a 0.95MHz and.mhz. Resuls measured using es seup of Fig. 5.0.

102 86 Figure 5. Measured oupu frequeny sperum of amplifier A a MHz: fundamenal ones a MHz and.05mhz, hird order disorion a frequenies 0.95MHz and.mhz. Resuls measured using es seup of Fig Figure 5. Measured oupu frequeny sperum of omposie amplifier a MHz showing anellaion of hird order disorion. Resuls measured using es seup of Fig. 5.0.

103 Figure 5.4 Measured signal leakage beween inpu and oupu of es irui a MHz: fundamenal ones appear a MHz and.05mhz wih power level of 85.dBm. Resuls measured using es seup of Fig

104 88 Table 5.4 Summary of measured resuls for MHz using seup from Fig. 5.0 P in (dbm P ou (dbm G (db P rd (dbm OIP (dbm Amplifier A (Fig. 5. Amplifier A (Fig. 5. Composie amplifier A+A (Fig Comparison: A wih omp. amp. Gain loss:.4 db Third order suppression: 5. db Improvemen of OIP: 0.9 db Table 5.4 summarizes he measured resuls for a frequeny of MHz similar o Table 5..For a frequeny of MHz, hird order disorion is suppressed by 5 db resuling in an improvemen of approximaely db in OIP as shown in Table 5.4. Subsiuing he measured resul from Table 5.4 ino equaion (4.7, similar o he ase of a es frequeny of 0 MHz leads o he following alulaions: (OIP-OIP=(G-G (OIP-OIP=( =48.6dB (G-G=( =48.dB The resuls show ha expression (4.7 is also valid for a es frequeny of MHz. Nex, a se of measured resuls for a frequeny of MHz is presened. The se also onsiss of 4 plos wih frequeny spera of oupu signals of A, A, omposie amplifier and signal leakage. As before, a wo-one es is applied where eah one has a power level of 0. dbm a he inpu of he es irui. Firs, Fig. 5.5 piures he

105 89 frequeny sperum of he oupu signal of amplifier A when A is urned off. The fundamenal ones a MHz and.05 MHz have a power level of 44.8 dbm, whereas he hird order disorion produs have a power level of 84. dbm a frequenies.95 MHz and. MHz. Seondly, he oupu frequeny sperum of amplifier A, when A is urned off, is shown in Fig Fundamenal linear omponens have power levels of abou 60.7 dbm and hird order disorion levels are 84. dbm. The hird frequeny sperum, in his presen se of frequeny spera, Fig. 5.7, depis he oupu frequeny sperum of he omposie amplifier adjused for anellaion of hird order disorion. The fundamenal omponens show a power level of 46. dbm. Compared o Fig. 5.5, fundamenal power levels are redued by abou.4 db. Third order omponens are no visible anymore; heir power level was redued below he noise floor of he sperum analyzer. An improvemen in hird order disorion of a leas 9 db was ahieved. The signal leakage for MHz is shown in Fig Wih boh amplifiers urned off, a signal leakage of he fundamenal ones of 75.6 dbm is sill presen. The measured resuls are olleed in Table 5.5 similar o Table 5..

106 90 Figure 5.5 Measured oupu frequeny sperum of amplifier A a MHz: amplifier A is urned off; fundamenal ones a MHz and.05mhz, hird order disorion a frequenies.95mhz and.mhz. Resuls measured using es seup of Fig Figure 5.6 Measured oupu frequeny sperum of amplifier A a MHz: amplifier A is urned off; fundamenal ones a fundamenal ones a MHz and.05mhz, hird order disorion a frequenies.95mhz and.mhz. Resuls measured using es seup of Fig. 5.0.

107 9 Figure 5.7 Measured oupu frequeny sperum of omposie amplifier a MHz showing anellaion of hird order disorion. Resuls measured using es seup of Fig Figure 5.8 Measured signal leakage beween inpu and oupu of es irui a MHz; fundamenal ones a frequenies MHz and.05mhz. Resuls measured using es seup of Fig. 5.0.

108 9 Table 5.5 Summary of measured resuls for MHz using seup from Fig. 5.0 P in (dbm P ou (dbm G (db P rd (dbm OIP (dbm Amplifier A (Fig. 5.5 Amplifier A (Fig. 5.6 Composie amplifier A+A (Fig Comparison: A wih omp. Amp. Gain loss:.4 db Third order suppression: 8.9 db Improvemen of OIP:. db The measured resuls of Table 5.5 show a reduion of hird order disorion of approximaely 9 db leading o an improvemen of. db in OIP. Furhermore, a loss of.4 db in gain is measured. Furhermore, equaion (4.7 is valid for he inegraed irui esed wih a frequeny of MHz as shown in he following alulaions using measured resuls from Table 5.5: (OIP -OIP = (G -G (OIP -OIP = ( =47.8dB (G -G =( =47.7dB For he hird se of frequeny spera, a wo-one es wih frequenies 0 MHz and 0.05 MHz and inpu power level of 0. dbm was deployed. The oupu frequeny sperum of amplifier A is presened firs in Fig In Fig. 5.9, 44. dbm power levels were measured for he fundamenal signals, and 8.8 dbm was measured for hird order disorion produs. Third order disorion produs appear a 9.95 MHz and 0.

109 9 MHz. Nex, he frequeny sperum a he oupu of amplifier A is illusraed in Fig Fundamenal ones show a power level of 59. dbm and hird order disorion levels have a value of 8. dbm. Third order disorion levels of A are almos equal o hird order disorion levels of amplifier A. Aivaing boh amplifiers A and A resuls in he omposie amplifier wih oupu frequeny sperum depied in Fig. 5.. Again, hird order disorion is redued by abou 0 db. Fundamenal ones have a power level of 45.6 dbm. The overall gain suffers a reduion of approximaely.4 db. Finally, he amoun of signal leakage was measured as 64.7 dbm and is shown in Fig. 5.. Third order oupu inerep poin and gain are summarized ogeher wih measured resuls in Table 5.6.

110 94 Figure 5.9 Measured oupu frequeny sperum of amplifier A a 0MHz: amplifier A is urned off; fundamenal ones a 0MHz and 0.05MHz, hird order disorion a frequenies of 9.95MHz and 0.MHz. Resuls measured using es seup of Fig Figure 5.0 Measured oupu frequeny sperum of amplifier A a 0MHz: amplifier A is deaivaed; fundamenal ones a 0MHz and 0.05MHz, hird order disorion a frequenies of 9.95MHz and 0.MHz. Resuls measured using es seup of Fig. 5.0.

111 95 Figure 5. Measured oupu frequeny sperum of omposie amplifier a 0MHz showing anellaion of hird order disorion. Resuls measured using es seup of Fig Figure 5. Measured signal leakage beween inpu and oupu of es irui a 0MHz; fundamenal ones a frequenies 0MHz and 0.05MHz. Resuls measured using es seup of Fig. 5.0.

112 96 Table 5.6 Summary of measured resuls for 0 MHz using seup from Fig. 5.0 P in (dbm P ou (dbm G (db P rd (dbm OIP (dbm Amplifier A (Fig. 5.9 Amplifier A (Fig. 5.0 Composie amplifier A+A (Fig Comparison: A wih omp. Amp Gain loss:.4 db Third order suppression: 9.9 db Improvemen of OIP:.8 db Table 5.6 liss measured resuls for a es frequeny of 0 MHz similar o Table 5.. A suppression of hird order disorion of approximaely 0 db is ahieved resuling in an amelioraion of.8 db in OIP. The omposie amplifier suffers a gain loss of.4 db. Again, he following alulaions illusrae he validiy of equaion (4.7 using measured resuls from Table 5.6: (OIP -OIP =(G -G (OIP -OIP =( =44.4dB (G -G =( =44.7dB In he following presenaion, a fourh and las se of frequeny spera for a es frequeny of 0 MHz is disussed. Two sinusoidal signals of frequenies 0 MHz and 0.05 MHz are applied o he es irui. The four frequeny spera of he se are shown in Figs. 5. o 5.6. The inpu power level a he inpu of he es irui was kep onsan a 0. dbm. Firs, he oupu sperum of amplifier A is shown in Fig. 5..

113 97 The fundamenal ones have a power level of 44. dbm. Third order inermodulaion produs a frequenies 9.95 MHz and 0. MHz show a power level of 8. dbm. Seondly, Fig. 5.4 depis he oupu power level of amplifier A. Third order disorion of A, -8.7 dbm, has approximaely he same power level as hird order disorion of A. Furhermore, he fundamenal power levels have a value of abou 5.6 dbm. The oupu signal of he omposie amplifier exhibis he frequeny sperum as illusraed in Fig Third order inermodulaion produs are sill visible bu redued o 05.6 dbm. The hird order disorion suppression an be alulaed as 4.5 db. The fundamenal ones show a power level of 45. dbm. Compared wih he sperum of A, a reduion of gain of approximaely db is noed. Finally, as previously saed, he leakage for a frequeny of 0 MHz is shown in Fig The fundamenal ones are shown wih a leakage power level of 55.0 dbm. The resuls for a es frequeny of 0 MHz are summarized in Table 5.7.

114 98 Figure 5. Measured oupu frequeny sperum of amplifier A a 0MHz: amplifier A is deaivaed; fundamenal ones a 0MHz and 0.05MHz, hird order disorion a frequenies of 9.95MHz and 0.MHz. Resuls measured using es seup of Fig Figure 5.4 Measured oupu frequeny sperum of amplifier A a 0MHz: amplifier A is deaivaed; fundamenal ones a 0MHz and 0.05MHz, hird order disorion a frequenies of 9.95MHz and 0.MHz. Resuls measured using es seup of Fig. 5.0.

115 99 Figure 5.5 Measured oupu frequeny sperum of omposie amplifier showing anellaion of hird order disorion; hird order disorion redued o 05.6dBm a marker. Resuls measured using es seup of Fig Figure 5.6 Measured signal leakage beween inpu and oupu of es irui a 0MHz; fundamenal ones a frequenies 0MHz and 0.05MHz. Resuls measured using es seup of Fig. 5.0.

116 00 Table 5.7 Summary of measured resuls for 0 MHz using seup from Fig. 5.0 P in (dbm P ou (dbm G (db P rd (dbm OIP (dbm Amplifier A (Fig. 5. Amplifier A (Fig. 5.4 Composie amplifier A+A (Fig Comparison: A wih omp amp. Gain loss:. db Third order suppression: 4.5 db Improvemen of OIP: 0. db The measured resuls are summarized in Table 5.7 for a frequeny of 0 MHz, similar o Table 5., indiae a suppression of hird order disorion of 4.5 db. As a resul OIP is improved by 0. db. A gain loss of. db was measured. Equaion (4.7 is proved rue for a frequeny of 0 MHz, shown in he following alulaions using measured resuls and equaion (4.7. (OIP -OIP =(G -G (OIP -OIP =( =8.0dB (G -G =( =8.5dB The resuls of he evaluaion of equaion (4.7 are almos equal, hus onfirm he validiy of (4.7 also for a es frequeny of 0 MHz. The variaions in hird order disorion suppression in dependene of frequeny are lised in Table 5.8. Saring a a low frequeny, suppression inreases and reahes is highes value of approximaely 0 db for 0 MHz. For inreasing frequenies,

117 0 Table 5.8 Summary of hird order disorion suppression and improvemen in OIP wih variaions in frequeny Frequeny (MHz Third order suppression (db Improvemen in OIP (db Reduion in Gain (db suppression dereases and arrives a he lowes value of.6 db a 0 MHz. One explanaion for variaions in suppression ould be variaions in gain of he omposie amplifier wihin he measured bandwidh. Variaions in gain an be reaed if he wo amplifiers, A and A, have ripples in heir gain haraerisi. Higher signal leakage a higher frequenies onribues mos likely o he lower suppression a higher frequenies. Signal leakage ould mask he oupu signal of he less linear amplifier A. 5.4 Power Effiieny Consideraions For handheld devies, power onsumpion plays a major role sine heir baery lifeime depends on power onsumpion. The following seion briefly disusses a few onsideraions onerning power effiieny. As alulaed in Table 5., he hird order oupu inerep poin was improved by approximaely 8.5 db (OIP -OIP wih an inpu power level of 7 dbm a he inpu of he es irui. Transferring he improvemen level ino a linear power raio leads o he raio of he possible new RF power o old RF power (P RFnew /P RFold = = 7.08 wih a value of In essene, he omposie amplifier performs like an amplifier wih 7 imes he oupu power apabiliy.

118 0 On he oher hand, he auxiliary amplifier onribues o he DC power onsumpion wih approximaely 6 A. The oal DC urren in he irui adds up o 0 A. Power onsumpion is defined as P DC =V DD *I SS. Consequenly, sine he supply volage remains onsan, he raio of new DC power o old DC power an be inrodued. I is P DCnew /P DCold =0A/87A=.056. This leads o one definiion of power effiieny whih is RF power (ypially a he drain of a ransisor divided by DC power, P RF /P DC []. Now he raio of he faors of RF power and DC power an be used o find he effiieny improvemen, 7.08/ , where faor 6.7 is he power effiieny improvemen of he omposie amplifier. In essene, a -Wa amplifier would have he hird order performane of a 7.08 Wa amplifier a he os of.056 inrease in DC power. One way o view his siuaion is, he omposie amplifier an deliver more RF power wih higher power onsumpion (I SS = 6 A, or ake an amplifier wih unalered RF power wih redued DC power onsumpion; for insane, redued bias urren. The seond way of viewing he resul is very favorable for baery-powered handheld devies suh as ellular phones beause less power onsumpion exends baery life. 5.5 Summary of Measured Resuls The presened measured resuls show ha he simple algebrai expression (4.7 is valid over a wide frequeny range from MHz o more han 00 MHz. A reduion of hird order inermodulaion produs of up o 9 db an be ahieved resuling in an improvemen of.8 db in hird order oupu inerep poin. Some hird order suppression ours above he db ompression poin of he less linear amplifier. Canellaion of hird order disorion holds up o he db ompression poin of he less linear amplifier. However, anellaion of hird order disorion degrades a he db

119 0 ompression poin of he more linear amplifier beause he wo amplifiers ompress. A maximum gain loss beween more linear amplifier and omposie amplifier of db was measured a a frequeny of 0 MHz. Gain loss ours due o subraion of fundamenal signals a he final oupu in Fig. 5.. Equaion (5.4 as ondiion of eliminaion of hird order disorion proposed in [] does no hold for he prooype, possibly beause of veloiy sauraion. Conversely, equaion (4.7 is proved o be valid and does no depend on operaing ondiions of ransisors. The measured resuls indiae ha he presened opology, as only one possible implemenaion of he new linearizaion mehod, is useful for appliaions wih low inpu power levels suh as radio reeivers. To eliminae hird order disorion up o he db ompression poin of a power amplifier, a new opology is being invesigaed. This newer opology is presened in Chaper 7 and promises hird order disorion anellaion up o he db ompression poin of he higher-power amplifier.

120 CHAPTER 6: SIMULATION OF CROSS-COUPLED DIFFERENTIAL PAIR Wihin Chaper 6, simulaion resuls for he ross-oupled differenial pair are disussed for a frequeny of 0 MHz and onsan inpu power. Furhermore, simulaed, measured and alulaed resuls are ompared for various inpu power levels. To gain informaion abou he auray of simulaion resuls regarding nonlinear behavior of inegraed iruis, he ross-oupled differenial pair in Fig. 5. was simulaed using Agilen s Advaned Design Sysem (ADS [] (The ADS shemai is in Appendix F. I would be advanageous o predi he nonlinear behavior of inegraed iruis during he design phase using simulaion ools. In addiion, he auray of simulaion resuls depends on aurae devie models. Comparison of measured and simulaed resuls an give informaion abou he auray of ransisor models. ADS is apable of performing a harmoni balane simulaion inluding nonlinear effes suh as inermodulaion and gain ompression. The simulaion was se-up similar o he es seup in Fig. 5.. Two aenuaors a he inpu are no neessary, sine here is only one signal soure. An amplifier and an addiional aenuaor are added o he oupu of he es irui, o obain he proper oupu power levels. The bias ee was no used in he simulaion. The simulaion was also performed single-ended. To ahieve he mos aurae resuls, ransisor models for his pariular proess wih lo number TAK-AZ, were obained from he fabriaion servie MOSIS [5]. The ransisor models used are of ype BSIM version. The BSIMv ransisor model ard an be found in Appendix D.

121 05 Harmoni balane simulaion allows he user o inpu wo or more sinusoidal frequeny ones ino a irui and alulae inermodulaion produs and harmonis. A wo-one es an be simulaed using harmoni balane simulaion. The es irui was simulaed wih a wo-one es a frequenies of 0 MHz and 0. MHz. Firs, he bias ondiions of he ross-oupled differenial pairs were varied in order o find he opimal ondiion for anellaion of hird order disorion. The bias volages a he gaes of ransisors M hrough M4 in Fig. 5. were kep onsan a 4.5 V, whereas he bias urrens were hanged by modifying he gae volages (V bias, V bias in Fig. 5. of ransisors M5 and M6. The inpu power level of 0 dbm a he signal soure was kep onsan. Table 6. liss bias ondiions for he measured prooype and simulaion resuls under he ondiion of anellaion. The inreased bias urren of amplifier A in he simulaion migh be a resul of inaurae modeling of ransisor M6 in Fig. 5.. Transisor M6 is biased a he low end of srong inversion where he ransisor does no longer exhibi square-law haraerisi. The lower bias urren ould also be an effe of veloiy sauraion. Afer he prerequisie for anellaion was esablished, he harmoni balane Table 6. Comparison of bias ondiions of prooype and simulaion Amplifier A Amplifier A Prooype Simulaion Prooype Simulaion V bias.06 V 0.9 V V V I SS 87 A 87 A 6 A 86 A

122 06 simulaion provided he resuls shown in Figs. 6. hrough Fig. 6.. The oupu power was aken a he final oupu of he simulaion seup in order o be able o ompare measured and simulaed resuls. Fig. 6. depis he oupu frequeny sperum of amplifier A when amplifier A was urned off. The fundamenal speral lines appear a 0 MHz and 0. MHz wih a power level of 9.7 dbm and are shown as innermos speral lines. Third order inermodulaion produs are reaed a frequenies 09.9 MHz and 0. MHz wih power levels of 75.0 dbm, depied in Fig. 6. as ouermos speral lines. The hird order oupu inerep poin an be alulaed o 7.05 dbm. Nex, he oupu frequeny sperum of amplifier A is illusraed in Fig. 6., wih fundamenal ones a 0 MHz and 0. MHz and power level of 8.0 dbm. Third order disorion produs have a power level of 74. dbm a frequenies 09.9 MHz and 0. MHz. The hird order inerep poin of he less linear amplifier, A, equals abou 9.7 dbm. A hird sperum in Fig. 6., shows he frequeny onen of he oupu signal of he omposie amplifier. A reduion of hird order disorion is learly visible. Third order disorion was redued o 5 dbm. The fundamenal ones have a power level of 4 dbm. Comparing he power levels of he fundamenal ones in Fig. 6. and Fig. 6., a derease in gain of approximaely 4 db an be noed.

123 07 Fundamenals IMD IMD Figure 6. Simulaed (ADS oupu frequeny sperum of amplifier A a 0MHz: Amplifier A urned off; fundamenal omponens are a 0MHz and 0.MHz wih power level of 9.7dBm, hird order disorion a frequenies 09.9MHz and 0.MHz and 75dBm power level. Resuls using irui in Fig 5.. Fundamenals IMD IMD Figure 6. Simulaed (ADS oupu frequeny sperum of amplifier A a 0MHz: Amplifier A disabled; fundamenal omponens are a 0MHz and 0.MHz wih power level of 8dBm, hird order disorion a frequenies 09.9MHz and 0.MHz and 74.dBm power level. Resuls using irui in Fig. 5..

124 08 Fundamenals IMD IMD Figure 6. Simulaed (ADS oupu frequeny sperum of omposie amplifier a 0MHz showing anellaion of hird order disorion: hird order produs are redued o 5 dbm. Resuls using irui in Fig. 5.. The simulaion resuls of Fig. 6., 6. and 6. are summarized in Table 6.. Table 6. liss simulaion resuls in he same manner as Table 5.. Inpu power P in, fundamenal oupu power P ou and hird order disorion levels P rd are lised and OIP and gain are alulaed for amplifier A, A and omposie amplifier. An improvemen in hird order suppression of 40 db was simulaed, resuling in an inrease in OIP of 4 db. Due o higher bias urren of amplifier A, omposie amplifier suffers a gain loss of 4 db. Measured resuls for 0 MHz indiae only a loss of db in fundamenal oupu power. The mos likely reason is he inreased ransonduane of he less linear amplifier g ma in he ase of simulaion due o he higher bias urren of 86 A. The volage gain of he ross-oupled differenial pair is desribed wih (g ma -g ma R D (5.. An inrease in ransonduane g ma resuls in lower gain of he omposie amplifier. Applying alulaed OIP and gain o equaion (4.7 proves is validiy.

125 Table 6. Summary of simulaion resuls for 0 MHz using shemai in Appendix F Amplifier A (Fig. 6. P in (dbm (a signal soure P ou (dbm (a final oupu Gain (db P rd (dbm OIP (dbm Amplifier A (Fig. 6. Composie amplifier A+A (Fig. 6. Comparison: A wih omposie amp Gain loss: 4. db Third order suppression: 40 db Improvemen of OIP: 4 db The following alulaions show prove of equaion (4.7 using simulaion resuls from Table 6.. Subsiuing simulaed resuls ino (4.7 leads o (OIP -OIP =(G -G (OIP -OIP =( =5.4dBm (G -G =( =4.9dB As one an see, equaion (4.7 also proves o be a useful equaion desribing he anellaion ondiion very well. In addiion o he simulaions wih onsan inpu power, harmoni balane simulaions were performed wih varying inpu power. The main ineres lies in he behavior of hird order anellaion beween he db ompression poins of less and more linear amplifiers. One an expe ha for low inpu power levels he ondiion for anellaion desribed wih equaion (4.7 holds and ha hird order suppression

126 0 gradually begins o derease for inpu power levels greaer han he db ompression poin of he less linear amplifier. The inpu power level was varied from 0 dbm o +0 dbm. Fig. 6.4 hrough Fig. 6.6 display he simulaed linear fundamenal oupu power of amplifier A, amplifier A and omposie amplifier ogeher wih measured daa from Chaper 5 and alulaed resuls from Chaper 4. The inpu power was aken from he signal soure and no he inpu of he es irui. Similar, he final oupu power of he ADS shemai shown in Appendix F was reorded. The plo in Fig. 6.4 depis linear oupu power over inpu power of amplifier A wih amplifier A deaivaed. The solid urve depis alulaed daa, he urve marked wih riangles shows he measured daa in Fig. 6.4, and simulaed daa is marked wih squares. One an see ha for low inpu power levels up o approximaely 4 dbm inpu power, alulaed, measured and simulaed daa, agree very well. For inpu power levels above 4 dbm, simulaed daa deviaes from measured and alulaed daa. Calulaed daa does no have a gradual lipping behavior due o he definiion of oupu power in Mahad. The simulaed daa has a db inpu ompression poin of abou 6 dbm, approximaely db lower han he PdB (9 dbm of measured daa. Nex, a omparison of simulaed, measured and alulaed oupu power of amplifier A is presened in Fig.6.5 similar o Fig Again, he solid line presens he alulaed daa. Measured daa is marked wih riangles in Fig For lower inpu power, measured and alulaed daa agree very well. Conversely, simulaed oupu power level, marked wih squares in Fig. 6.5, is abou db higher. The reason for he inreased gain is higher bias urren, hene higher gain for amplifier A. A db inpu

127 ompression poin of abou 4 dbm was found. Finally, Fig. 6.6 depis he linear oupu power of he omposie amplifiers for simulaed, measured and alulaed daa. The solid line haraerizes he alulaed oupu power. As before, measured and alulaed daa agree well. Simulaed daa has a db inpu ompression poin of abou 6 dbm. As expeed in Fig. 6.6, simulaed oupu power levels are approximaely db lower han measured and alulaed daa beause of he greaer bias urren in amplifier A, hus greaer ransonduane g ma. All simulaed daa in Fig. 6.4 hrough Fig 6.6 show less linear behavior ompared o measured daa for high inpu power levels. A possible reason ould be ha he es irui beomes srongly nonlinear for high inpu power levels. As a resul, ransisor models and simulaion ool lose auray in alulaing he power levels. -5 Calulaed Linear 0upu Power in dbm Measured ADS Simulaed Inpu Power in dbm Figure 6.4 Simulaed, measured and alulaed linear oupu power of amplifier A a 0MHz: Amplifier A is urned off. Simulaed resuls using ADS shemai.

128 - -4 ADS Simulaed Linear Oupu Power in dbm Measured Calulaed Inpu Power in dbm Figure 6.5 Simulaed, measured and alulaed linear oupu power of amplifier A a 0MHz: Amplifier A is urned off. Simulaed resuls using ADS shemai. -5 Linear Oupu Power in dbm Measured Calulaed ADS Simulaed Inpu Power in dbm Figure 6.6 Simulaed, measured and alulaed linear oupu power of omposie amplifier a 0MHz: Amplifiers A and A are urned on for anellaion of hird order disorion. Simulaed resuls using ADS shemai.

129 A omparison of hird order inermodulaion produs of simulaed, measured and alulaed daa is illusraed in Fig. 6.7 hrough Fig The plos are separaed ino resuls of amplifier A, A and omposie amplifier similar o Figs 6.4 o 6.6 and he raes are marked in he same manner. Firs, Fig. 6.7 displays hird order disorion of amplifier A when amplifier A is urned off. Simulaed daa, illusraed wih squares in Fig. 6.6, deviaes from measured daa and exhibis higher disorion levels a abou 4 dbm a higher inpu power levels. Measured daa is shown in Fig. 6.6 as riangles. The solid line in Fig. 6.7 shows he alulaed daa from Chaper 4. Seondly, hird order disorion of amplifier A only is depied in Fig. 6.8, where amplifier A is deaivaed. Again, simulaion of amplifier A reveals higher hird order disorion for inpu power levels greaer han dbm ompared o measured daa. Calulaed daa from Chaper 4 is inluded in Fig. 6.8 as a solid line. Finally, Fig. 6.9 depis hird order disorion versus inpu power levels wih simulaed, measured and alulaed daa. Fig. 6.9 is also showing anellaion of hird order inermodulaion produs for low inpu power levels. The haraerisi of alulaed daa in Fig. 6.9 is shown as he solid urve. For higher inpu power levels, hird order inermodulaion produs rise rapidly beause hird order disorion of amplifier A and amplifier A are no equal and no longer anel. As before, simulaed daa has higher hird order inermodulaion produs for inpu power levels greaer han 4 dbm ompared o measured daa. The alulaed daa in Fig. 6.9 exhibis ideal anellaion of hird order disorion for inpu power levels below he db ompression poin of amplifier A.

130 4-0 Calulaed Third Order Disorion in dbm ADS Simulaed Measured Inpu Power in dbm Figure 6.7 Simulaed, measured and alulaed hird order disorion of amplifier A a 0MHz: Amplifier A is urned off. Simulaed resuls using ADS shemai. Third Order Disorion in dbm ADS Simulaed Measured Calulaed Inpu Power in dbm Figure 6.8 Simulaed, measured and alulaed hird order disorion of amplifier A a 0MHz: Amplifier A deaivaed. Simulaed resuls using ADS shemai.

131 5-40 ADS Simulaed -50 Third Order Disorion in dbm Measured Calulaed Inpu Power in dbm Figure 6.9 Simulaed, measured and alulaed hird order disorion of omposie amplifier a 0MHz: Amplifiers A and A are aivaed for anellaion of hird order disorion. Simulaed resuls using ADS shemai. In onlusion, simulaion resuls of fundamenal oupu power in Fig. 6.4 hrough Fig. 6.6 exhibi more ompression han he measured resuls do, whih indiaes ha he simulaed irui is more nonlinear han he measured prooype. As a resul of he inreased nonlineariy of simulaed daa, hird order disorion is also inreased for he simulaion resuls in Fig. 6.7 hrough Fig The deviaion of simulaed and measured resuls is mos likely a resul of he inabiliy of he simulaion ool o auraely model he ompressed inegraed irui.

132 CHAPTER 7: CONCLUSION This haper summarizes and onludes he work presened in his hesis. Firs, a summary of he work from previous hapers is given. The seond seion suggess fuure work. The seond seion also presens a differen opology wih four amplifier sages applying he new linearizaion mehod. This opology promises good linearizaion up o he sauraion poin of he main amplifier. 7. Summary of Work This work inrodued new mehods of linearizaion. The new linearizaion ehniques are based on an unusual onep: a good amplifier ombined wih a bad amplifier resuls in a beer amplifier. In his onex, a good amplifier is a more linear amplifier and a bad amplifier is a less linear amplifier. A basi opology, Fig. 4., was presened using wo amplifiers onneed in parallel where he oupu signals are subraed. Wih proper adjusmen of amplifier parameers, suh as gain and hird order oupu inerep poin OIP, nonlinear omponens of he final oupu signal an be eliminaed and linear omponens ideally are no affeed. Theoreial ondiions for anellaion of hird order disorion in erms of power series oeffiiens and in erms of gain and hird order oupu inerep poin were derived. The derived equaion (4.7 for anellaion of hird order disorion, (OIP -OIP =(G -G is espeially useful in praie where gain and OIP of amplifiers is available. Due o his simple onep, he

133 7 new linearizaion ehniques lend hemselves o inegraed irui implemenaion. Some of he advanages of he new linearizaion mehods are no need for exernal expensive pars, low omplexiy and small size. Mahemaial onsideraions in Chaper 4 showed ha his pariular opology ahieved good resuls in anellaion of hird order inermodulaion produs for low inpu power levels up o he db ompression poin of he less linear amplifier. Chaper showed an overview of prior linearizaion ehniques suh as feedbak linearizaion, feedforward linearizaion and predisorion. Mos of hese linearizaion mehods are fairly omplex or suffer from devie mahing resriions and feedbak limiaions. Many of he prior linearizaion ehniques are only appliable o ransmier and no reeiver designs. Conversely, he new linearizaion ehnique an also be used for reeivers. In Chaper, effes of nonlineariy were presened. Desensiizaion ours when a srong inerferer and a weak desired signal are proessed simulaneously by a radio. The radio reeiver beomes less sensiive o he desired signal. Cross modulaion was disussed as a seond effe of nonlineariy. In ross modulaion, he modulaion of a arrier is imposed on he adjaen arrier. Finally, hird order inermodulaion was shown o be problemai beause hird order inermodulaion produs appear lose o he desired band of ineres and are diffiul o filer. Inermodulaion ours when fundamenal and harmonis inera and reae new, so alled, inermodulaion produs. Furhermore, wo ommon measures of nonlineariy were presened in Chaper. Oupu inerep poin and db ompression poin desribe a devie s nonlinear behavior. The ross-oupled differenial pair was presened in Chaper 5. This pariular

134 8 irui is fully inegrable and was uilized as a firs prooype. Previous researh showed ha hird order disorion anellaion an be expressed in erms of bias urrens and aspe raios of ransisors (5.4. The ross-oupled differenial pair was implemened in a 0.5 m CMOS proess. Measuremen and simulaion resuls revealed ha good eliminaion of disorion ours a low inpu power levels (up o 0 db improvemen when no ompression ours in he amplifiers, however, some improvemen of lineariy an sill be seen beween ompression poins of less linear and more linear amplifiers. Addiional measuremens for differen frequenies showed ha hird order disorion anellaion holds for a frequeny range of MHz o 0 MHz for he prooype. Furhermore, an inrease in power effiieny by a faor of 6 due o improvemen in lineariy ould be ahieved. Finally, simulaion resuls presened in Chaper 6 agree very well wih he measured resuls for low inpu power levels. Simulaed daa shows less linear behavior of linear oupu power and higher hird order disorion levels for high inpu power levels ompared o measured resuls. This haraerisi of simulaed daa suggess ha he simulaion ool, ADS, is no able o simulae he sauraed amplifier auraely. 7. Fuure Researh In he following disussion, Fuure Work is presened. Fuure researh is neessary o bring he new linearizaion mehod o produ level. To apply he new linearizaion ehnique o power amplifiers, new opologies mus be designed o aoun for linearizaion up o he db ompression poin of he PA. Suh a opology is illusraed in Fig. 7. using four amplifier sages. The irui in Fig. 7. onsiss of four amplifiers, one aenuaor and wo subraors. On he lef side, he inpu signal is equally disribued

135 9 Inpu A G, OIP (b Aenuaor A G, OIP + (f Oupu (a A (d + _ Error Signal A4 _ (h G, OIP ( (e G 4, OIP 4 (g Figure 7. Four-sage amplifier linearizaion: A, A, A and A4 are amplifiers wih parameers G and OIP. Afer aenuaion by an aenuaor, he oupu signal of A is subraed from oupu signal of A. The error signal is amplified by amplifier A4 and subraed from oupu signal of amplifier A. A is he power amplifier. The aenuaor is ypially a passive nework. The frequeny sperum of every pah is shown in (a o (h. beween amplifier A and amplifier A. The oupu signal of amplifier A is aenuaed and inpu o a subraor. The seond inpu o he subraor is onneed o he oupu of amplifier A. The par desribed so far is alled he firs loop, whih anels (ideally he arrier. In Fig. 7., he oupu of amplifier A is also onneed o he inpu of amplifier A. The oupu signal of he firs subraor is alled error signal and is inpued o amplifier A4. The error signal ideally only onains hird order disorion. Oupu signals of amplifiers A and A4 are subraed wih a seond subraor a he oupu of he irui, hus eliminaing hird order disorion. The las desribed mehanism in he seond loop ideally anels hird order disorion. Fig. 7.(a o Fig. 7.(h show frequeny spera of every signal pah in he irui. A wo-one es signal is applied o he inpu of amplifiers A and A shown in Fig. 7.(a. Fig. 7.(b depis he oupu sperum of amplifier A onsising of fundamenal ones and hird order disorion.

136 0 Similarly, frequeny sperum ( in Fig. 7. shows he frequeny onen of he oupu of amplifier A wih higher hird order disorion produs han amplifier A. The sperum in Fig. 7.(d is an aenuaed version of sperum (b. Afer subraion, he linear signals are eliminaed (ideally, as shown in Fig.7.(e. The error signal is amplified, resuling in frequeny sperum Fig. 7.(g. The oupu of amplifier A has a sperum, depied in Fig. 7.(f, wih inreased ampliude of fundamenals and hird order disorion. If hird order disorion levels in (f and (g have equal ampliude, he oupu signal has a frequeny sperum as illusraed in Fig. 7.(h, showing eliminaion of hird order disorion. Amplifier A needs o be fairly linear. Conversely, amplifier A should be designed wih higher disorion levels. Amplifier A is he power amplifier o be linearized and should be he firs amplifier o lip. Error amplifier A4 mus be designed for high lineariy o preven reaion of more disorion. Nex, heoreial onsideraions are performed o gain insigh ino he performane of he opology in Fig. 7.. Eah of he amplifiers in Fig. 7. an be desribed wih gain and hird order oupu inerep poin OIP. In order o gain informaion abou he haraerisi and he level of hird order disorion suppression, Mahad was used o perform preliminary alulaions using gain, G, and OIP. For he following alulaions, some assumpions have o be made. Firs, he linear oupu power for all amplifiers was defined as inpu power plus gain, P in +G. When he amplifier reahes ompression, he oupu power level is defined as OIP-dBm. The db ompression poin is ypially abou db lower han he hird order oupu inerep poin, as was disussed in Chaper 4. Sine eah single amplifier has wo independen parameers, i seems ha here are 8 independen parameers wih four

137 amplifiers. The opposie is he ase. Some of he parameers are dependen on eah oher. To anel he arrier in he firs loop, wo equal ampliudes have o be subraed a he firs subraor in Fig. 7.. In order o have equal ampliudes a he inpus of he subraor, he following dependene exiss: G =G -A. The oupu signal of amplifier A has o be aenuaed o reah he neessary power level. Amplifier A is required o be less linear, hus has lower gain han amplifier A. As a rule of humb, he gain differene beween amplifier A and amplifier A is 0 db. The parameers shown in Table 7. are independen and are hosen as lised. All parameers of all amplifiers an be alulaed using independen parameers in Table 7.. A summary of amplifier parameers is given in Table 7.. The aenuaion is 0 db. The omplee Mahad program is aahed in Appendix E. Table 7. Independen parameers for four-sage amplifier linearizaion Independen Parameers Desripion (using Fig. 7. G =0 db OIP =0 dbm G =0 db Gain of amplifier A Third order oupu inerep poin of amplifier A Gain of amplifier A Delaoip =0 db Differene in oupu inerep poin of amplifiers A and A, deraed o inpus of firs subraor Oipoverdrive=0 db oipoverdrive=(oip +G OIP

138 Table 7. Summary of gain and hird order inerep poin of four-sage amplifier linearizaion from Fig. 7. A A A A4 G OIP G OIP G OIP G 4 OIP 4 0 db 0 dbm 0 db -0 dbm 0 db 0 dbm 4.7 dbm 0 dbm Of ineres is he behavior of he linearizaion of amplifier A ha is operaed around he db ompression poin. To exerise he amplifier in he riial operaing region, he inpu power was swep from 40 o 0 dbm. Fig. 7. depis hird order disorion suppression and linear oupu power of amplifier A as a funion of inpu power. The suppression is defined as he differene beween hird order disorion of he final oupu minus hird order disorion of amplifier A. The linear oupu power of A reahes he ompression poin a an inpu power level of dbm. The solid line in Fig. 7. illusraes he haraerisi of hird order suppression. Suppression is greaer han 40 db up o he ompression poin of he power amplifier A. The opology in Fig. 7. promises eliminaion of hird order disorion a he db ompression poin of he power amplifier. The hird order disorion haraerisis a he final oupu are shown in Fig. 7.. For low inpu power levels up o dbm, hird order inermodulaion produs of A and A4 are equal and anel a he final oupu. Above he db ompression poin of amplifier A, all disorion levels reah a final maximum and lip.

139 5 Linear Oupu Power (dbm, Suppr (db IMD Suppression Linear oupu power of A Inpu Power in dbm Figure 7. Simulaed (Mahad hird order disorion suppression of four-sage amplifier linearizaion: linear oupu power of amplifier A (doed line and hird order disorion suppression a he final oupu (solid line. Suppression up o db ompression poin of A. Linear power ompresses a dbm 0 Amplifier A 6 Third Order Disorion in dbm 8 54 Final oupu Amplifier A Inpu Power in dbm Figure 7. Simulaed (Mahad hird order disorion anellaion of four-sage amplifier linearizaion: doed line shows IMD of amplifier A and dashed line illusraes IMD of error amplifier A4, solid line depis anellaion of hird order disorion a final oupu up o a inpu power level of dbm.

140 4 One possible irui opology for he four-sage approah is shown in Fig The irui in Fig. 7.4 is based on differenial pairs and has a differenial inpu volage and differenial oupu volage. Amplifier A is omprised of MOS ransisors M and M and amplifier A onsiss of ransisors M and M4. Transisors M o M5 buil a ross-oupled differenial pair in Fig.7.4 o ahieve subraion and reae an error signal. Addiional resisors in he drains of ransisors M and M orrespond o he aenuaor in Fig. 7.. The main signal an be aken from he drains of ransisors M and M whereas he error signal is available a he ross-oupled drains. Furhermore, amplifier A is omprised of MOS ransisors M5 and M6 forming a differenial pair and amplifier A4 is buil using ransisors M7 and M8. The subraion a he oupu is also ahieved by ross-oupling he drains of ransisors M5 o M8. For a prooype, he urren soures of he differenial pairs need o be implemened using ransisors. So far only few preliminary harmoni balane simulaions have been performed using ADS for he four-sage linearizaion. The bias urrens of he ransisors were adjused o ahieve reduion in fundamenals in he error signal and reduion of hird order disorion a he final oupu. Preliminary alulaions sugges ha he ransonduane g m of amplifiers A and A have o be equal o anel he linear fundamenal omponens. Transonduane of amplifiers A and A4 mus no be equal o preven anellaion of he linear omponens. Table 7. liss he bias urrens and aspe raios of he ransisors in Fig. 7.4 used for he simulaion wih ADS. These bias urrens and aspe raios ahieve good resuls for he simulaion, bu furher simulaions have o be performed o find an opimum in bias ondiions for a desired reduion of hird order inermodulaion disorion.

141 5 R D=kΩ V DD=5V R D=kΩ M M4 R=kΩ R=kΩ Vidiff M M Amplifier A Amplifier A I I V DD=5V R D=kΩ R D=kΩ M5 M6 M7 M8 Vodiff Amplifier A I Amplifier A4 I4 Figure 7.4 Topology for four-sage amplifier linearizaion using differenial pairs. Amplifier A and A have ross-oupled drains and Amplifier A and A4 have ross-oupled drains. Amplifier A4 is he error amplifier. (The shemai is onepual and omis deoupling iruis and biasing deail.

142 6 The resuls of a harmoni balane simulaion wih varying inpu power levels form 40 o 0 dbm are shown in Fig. 7.5 o 7.7. Firs, Fig. 7.5 depis he linear oupu power and he hird order disorion of amplifier A in Fig. 7.4 over he inpu power. The upper urve displays he linear oupu power as solid rae. The rae below shows he hird order disorion produ as dash-doed urve. As one an see, he linear oupu power of amplifier A does no ompress, whereas IMD ompresses. This phenomenon ould no be explained ye. Seondly, Fig. 7.6 illusraes he haraerisi of he error signal over he inpu power as oupu of he subraor. Again, he upper rae shows he linear omponen as solid urve and he dash-doed urve displays he hird order disorion. The linear omponen rises slowly wih inreasing inpu power. The reason for his behavior is he anellaion mehanism of he fundamenal omponens. Furhermore, Fig. 7.6 indiaes ha he hird order disorion produ is no affeed by he anellaion mehanism. Lasly, Fig. 7.7 depis he final oupu signal of Fig. 7.4 over inpu power. The upper rae displayed as solid urve, is he linear oupu power when amplifiers A and A4 are deaivaed, hen amplifiers A and A are asaded and no subraion ours. This resul is reorded as referene for he alulaion of improvemen of hird order disorion. Equally, he hird urve from he op is he hird order disorion of asaded amplifiers A and A when amplifier A and A4 are urned off. If anellaion is aivaed, hen he four sages show a linear oupu power depied as doed urve a he op of Fig The fourh rae from he op shows hird order inermodulaion disorion when anellaion is urned on. In addiion, he verial lef line marks he db ompression poin of he asaded amplifier A and A when A and A4 are urned off.

143 7 PdB has a value of 0 dbm a an inpu power of 8 dbm. The db ompression poin of he omplee linearized amplifier also has a value of approximaely 0 dbm bu a an inpu power level of abou dbm. Furhermore, hird order disorion is suppressed by approximaely 0 db up o he db ompression poin of asaded amplifiers A and A when A and A4 are deaivaed. This resul was suggesed from he mahemaial resuls of Mahad. However, a gain loss of abou 7 db was simulaed. This problem should be easily solvable by adjusing he proper parameers of he irui in Fig In summary, a hird order disorion suppression of 0 db was ahieved resuling in an improvemen in OIP of 7.75 db. This improvemen was reahed a he db ompression poin when amplifiers A and A4 were deaivaed. Table 7. Summary of bias urrens and aspe raios for foursage amplifier linearizaion Bias urren I Aspe raio W/L Amplifier A ma 5 Amplifier A 9 A 60 Amplifier A 60 A 00 Amplifier A4 99 A 000 For fuure work i is suggesed o oninue opimizing he irui in Fig In addiion, oher possible irui opologies for he four-sage amplifier linearizaion should be onsidered differenially and single-ended.

144 8 linear hird order disorion Figure 7.5 Simulaed (ADS linear and hird order disorion oupu power of amplifier A using four-sage linearizaion from Fig.7.4 a 00 MHz.

145 9 linear hird order disorion -8dBm Figure 7.6 Simulaed (ADS linear power and hird order disorion of error signal showing reduion of linear omponen: solid rae is linear omponen and dash-doed urve is hird order disorion. Resuls using Fig. 7.4 a 00 MHz.

146 0 PdB=-0dBm no linearizaion IMD no linearizaion PdB=-0dBm wih linearizaion IMD wih linearizaion -8dBm -dbm Figure 7.7 Simulaed (ADS linear oupu power and hird order disorion of final oupu from Fig. 7.4 showing hird order suppression: upper wo urves depi linear oupu power, lower raes show hird order inermodulaion disorion; PdB of irui from Fig. 7.4 wihou linearizaion is 0dBm a 8 dbm inpu power, a 00 MHz.

147 So far, only differenial iruis were implemened uilizing he new linearizaion mehod beause of he simple subraion mehanism. Mos appliaions in radio sysems, suh as reeiver and ransmier, require single-ended designs. Consequenly, he subraion mehanism has o be designed in a single-ended fashion. Anoher reommendaion is ha researh work should be pu ino invesigaing he effe of phase disorion. Many modern modulaion shemes inorporae phase modulaion, hene hey need aurae phase reonsruion of he amplified signal. In he ase a hand, boh amplifiers exhibi AM/PM onversion perhaps beause of differen ransisor sizes, hus differen parasii apaianes. Pariularly pn-junions are presen in almos every devie in inegraed irui design. pn-junions have nonlinear volagedependen parasii apaianes. The downsaling of ransisor sizes o submiron regions oninues and modifies he square-law I-V haraerisi of MOS devies. Shor hannel effes, suh as veloiy sauraion and mobiliy reduion, arise and hange he square-law relaion o a more linear haraerisi. This will affe he new linearizaion mehod, hus i is reommended ha researh should be done in his area. The presened irui opology was implemened in a 0.5 m CMOS proess. I is also of grea ineres o invesigae he nonlinear behavior in differen proess ehnologies, suh as 0.5 m CMOS proess, silion-germanium BiCMOS proess, gallium-arsenide or a proess using silion-on-insulaor ehnology.

148 REFERENCES [] P. B. Keningon, High-Lineariy RF Amplifier Design, Areh House, 000 [] B. Razavi, RF Miroeleronis, Prenie Hall, 998 [] F. H. Raab e al, Power Amplifiers and Transmiers for RF and Mirowave, IEEE Transaions on Mirowave Theory and Tehniques, vol. 50, no., Marh 00 [4] A. Kaz, Linearizaion: Reduing Disorion in Power Amplifiers, IEEE Mirowave Magazine, vol., no. 4, Deember 00 [5] P. B. Keningon, Mehods Linearize RF Transmiers And Power Amps (Par, Mirowaves & RF, vol. 7, no., Deember 998, pp. 0-6 [6] V. Perovi, W. Gosling, Polar Loop Transmier, IEE Eleronis Leer, vol. 5, no.0, May 979, pp [7] M. Faulkner, An auomai phase adjusmen sheme for RF and Caresian feedbak linearizers, IEEE Transaion on Vehiular Tehnology, vol.49, May 000, pp [8] J. L. Dawson, T. H. Lee, Auomai phase alignmen for high bandwidh Caresian feedbak power amplifiers, IEEE Radio and Wireless Conferene, 000, RAWCON, 000, pp [9] P. B. Keningon, Mehods Linearize RF Transmiers and Power Amps (Par, Mirowaves & RF, vol. 8, no., January 999, pp [0] H. S. Blak, U. S. Paen,0,67, Issued Deember 97 [] Y. Y. Woo, Y. Yang, J. Yi, J. Nam, J. Cha, B. Kim, An Adapive Feedforward Amplifier For WCDMA Base Saions Using Imperfe Signal Canellaion, Mirowave Journal, April 00

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152 6 APPENDIX A: POWER SERIES EXPANSION For all following power series expansions ommon rigonomeri ideniies are used. The inpu signal is denoed x(, he oupu signal is denoed y(. A is he ampliude of he inpu signal., and are linear, seond and hird order power series oeffiiens, respeively. The frequeny in radians per seond is denoed 0. A. Seond Order Nonlineariy os( sin( ( sin sin( ( ( ( sin( ( A A A A A x x y A x ω ω ω ω ω + = + = + = = A. Third Order Nonlineariy sin( 4 sin( 4 ( ( sin sin( ( ( ( sin( ( A A A A A x x y A x ω ω ω ω ω + = + = + = = A. Seond and Third Order Nonlineariies sin( 4 os( sin( 4 ( ( sin ( sin sin( ( ( ( ( sin( ( A A A A A A A A x x x y A x ω ω ω ω ω ω ω + + = + + = + + = =

153 7 A.4 Seond and Third Order Nonlineariies wih Two-Tone Tes Signal aion rdorder inermodul os(( 4 os(( 4 aion rdorder inermodul os(( 4 os(( 4 aion ndorder inermodul os(( os(( rdharmonis os( 4 os( 4 ndharmonis os( os( linear os( 4 ( os( 4 ( DC omponen os( os( ( os( os( ( os( os( ( ( ( ( ( os( os( ( A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A x x x y A A x ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω = = + + = + = For A =A =A, y( simplifies o: rd order inermod produ os(( 4 os(( 4 rd order inermod produ os(( 4 os(( 4 nd order inermod produ os(( os(( rd Harmoni os( 4 os( 4 nd Harmoni os( os( linear os( 4 9 ( os( 4 9 ( omponen DC ( A A A A A A A A A A A A A A A y ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω =

154 APPENDIX B: MATHCAD PROGRAM FOR TWO-STAGE AMPLIFIER 8 ip_ampkm.mp Mahad program o alulae hird order suppression and gain ompression using wo amplifier sages. Forrnulae p=linear siganl p=rd order level G=gain db, OIP=rd order oupu inerep dbm General ompressed gain oupu power level formula: p( Pin, G, OIP := min( Pin + G, OIP p=oupu power inluding ompression General hird order produ oupu power level formula: p( Pin, G, OIP := OIP ( OIP p( Pin, G, OIP p=third order IMD oupu Gain differene dg( g, g := g g g( g, dg := g dg Calulaion of OIP oip oip, dg ( := oip dg Calulaion of final linear oupu omponen p finalpl( pin, g, oip, g, oip := 0log 0 p( pin, g, oip 0 p( pin, g, oip 0 0 Calulaion of final hird order omponen p finalpou( pin, g, oip, g, oip := 0 log 0 p( pin, g, oip 0 p( pin, g, oip Due o anellaion, argumen in log funion is 0 for equal hird order omponens, in order o preven undefined add 0-40 means -400dB

155 9 Calulaion of suppression a final oupu suppression ( pin, g, oip, g, oip := p( pin, g, oip finalpou( pin, g, oip, g, oip Final oupu hird order omponen is subraed from hird order oupu of main amplifier A Calulaion of final oupu hird order inereppoin ( finalpl( pin, g, oip, g, oip finalpou( pin, g, oip, g, oip finaloip( pin, g, oip, g, oip := + finalpl( pin, g, oip, g, oip... Example alulaion Independen Parameer: G, OIP and gain differene Pin := 0 Inpu power G:= 9 Gain of amplifier A dg :=.4 Gain differene OIP := 6 OIP of amplifier A Calulaion of dependen parameer: G and OIP G:= g( G, dg OIP := oip( OIP, dg Gain of amplifier A OIP of amplifier A Summary of amplifier parameer: G = 9 OIP = 6 G = 40.4 OIP =. Calulaion of power levels a oupu of amplifiers and final oupu p( Pin, G, OIP = 9 p( Pin, G, OIP = 75 p( Pin, G, OIP = 40.4 p( Pin, G, OIP = 75 finalpl( Pin, G, OIP, G, OIP =.7 finalpou( Pin, G, OIP, G, OIP = 400 suppression ( Pin, G, OIP, G, OIP = 5

156 APPENDIX C: SMALL SIGNAL GAIN OF CROSS-COUPLED DIFFERENTIAL PAIR 40 The small signal gain of he ross-oupled differenial pair is derived in he following par. The irui is depied in Fig. 5.. The half-irui mehod is used for he derivaion [4]. The irui in Fig. 5. an be separaed ino wo half iruis, shown in Fig. C.. The firs half-irui on he lef side in Fig. C., onsiss of ransisors M and M4 wih drain resisane R D. The inpu volages are v idiff / and v idiff /. The seond half-irui on he righ side of Fig. C. is omprised of ransisors M and M wih drain resisor R D. Inpu volages are denoed v idiff / and v idiff /. The oupu volages are v o and v o. The differenial inpu volage equals v idiff =v idiff / (-v idiff /. The body-effe is no in effe beause body and soure of he ransisors have he same volage poenial. For he derivaion of he small signal gain, he half-irui on he lef an be ransferred o he small signal irui depied in Fig. C.. The MOS ransisor model is well known and an be found in he lieraure [8], [4]. Transonduane of ransisor M is denoed g m and ransonduane of ransisor M4 is denoed g m4. g ds is he onduane of he ransisors in ase hannel lengh modulaion is presen and R D =/G D is he drain resisane. The following par shows he derivaion of oupu volage v o as a funion of v idiff.

157 4 VDD VDD RD RD vo vo v idiff/ M M4 -v idiff/ M -v idiff/ v idiff/ M Figure C. Two half-iruis of ross-oupled differenial pair: soure erminals of all ransisors are onneed o virual ground. i i4 gmv idiff/ -gm4v idiff/ vo v idiff/ gds io RD gds4 -v idiff/ Figure C. Linearized small signal irui for firs half-irui: on he lef and righ side are volage onrolled urren soures, he urrens of he urren soures are g m v idiff / and g m4 v idiff /. Wih he use of Kirhhoff s urren law, he following equaions an be derived: i i i 4 v o = v = -v o = i = o o g g + i ds ds4 4 - ( g + g v = R m g m m4 o D g v v = -v m4 idiff idiff g o g ds ds - v + g ds4 o g ds4 + G D - g v m idiff v idiff + g m4 v idiff (7. (7. (7. (7.4

158 4 The same derivaions an be performed wih he seond half-irui. Fig. C. illusraes he small signal irui for he seond half-irui. Again, wih he use of i i -gmv idiff/ gmv idiff/ vo -v idiff/ gds io RD gds v idiff/ Figure C. Linearized small signal irui for seond half-irui: on he lef and righ side are volage onrolled urren soures, he urrens of he urren soures are g m v idiff / and g m v idiff /. Kirhhoff s urren law, he following equaions an be found: i i i o v o = v = v = i = o o ( g g + i g ds ds m + g g v = R o D g m m m v v idiff g idiff = v o ds g ds + g v ds o g + G ds D v + g idiff m v idiff g m v idiff (7.5 (7.6 (7.7 (7.8 The differenial oupu volage is v odiff =v o -v o. The following derivaion shows he differenial oupu volage as a funion of differenial inpu volage. v odiff = v o v o = ( g g + ( g g m m m m4 gds + gds + GD gds + gds4 + G D v idiff (7.9

159 for g m =g m =g ma and g m =g m4 =g ma, he differenial volage gain an be found using ( vodiff A = = + vdiff ( g ma g ma (7.0 vidiff g g where g =g ds +g ds4 +G D and g =g ds +g ds +G D. If hannel lengh modulaion is negleed, hen g ds =g ds =g ds =g ds4 =0 and (7.0 an be modified o equaion (7. vodiff A = vdiff ( g ma g ma RD v = (7. idiff where A vdiff is he differenial volage gain of he ross-oupled pair and v odiff and v idiff are he differenial oupu and inpu volages, respeively.

160 44 APPENDIX D: TEST PRINTED CIRCUIT BOARD LAYOUT Figure D. Fron side of prined irui board used in es seup in Fig. 5. and Fig Figure D. Bak side of prined irui board used for es seup in Fig. 5. and Fig. 5.0

161 45 APPENDIX E: BSIMV TRANSISTOR MODEL TAK SPICE BSIM VERSION. PARAMETERS SPICE f5 Level 8, Sar-HSPICE Level 49, UTMOST Level 8 * DATE: De 6/0 * LOT: TAK WAF: 0 * Temperaure_parameers=Defaul.MODEL CMOSN NMOS ( LEVEL = 49 +VERSION =. TNOM = 7 TOX =.4E-8 +XJ =.5E-7 NCH =.7E7 VTH0 = K = K = K = KB = W0 = E-8 NLX = E-9 +DVT0W = 0 DVTW = 0 DVTW = 0 +DVT0 = DVT = DVT = U0 = UA = E- UB =.8676E-8 +UC =.75854E- VSAT =.55909E5 A0 = AGS = B0 =.59099E-6 B = 5E-6 +KETA = E- A = E-5 A = RDSW =.5845E PRWG = E- PRWB = WR = WINT =.9856E-7 LINT =.607E-8 +XL = 0 XW = 0 DWG = E-8 +DWB = E-8 VOFF = NFACTOR =.064 +CIT = 0 CDSC =.4E-4 CDSCD = 0 +CDSCB = 0 ETA0 = ETAB = -.808E- +DSUB = PCLM =.508 PDIBLC=.6066E- +PDIBLC =.6609E- PDIBLCB = DROUT = PSCBE = 6.77E8 PSCBE =.65457E-4 PVAG =.6876E- +DELTA = 0.0 RSH = 8.8 MOBMOD = +PRT = 0 UTE = -.5 KT = -0. +KTL = 0 KT = 0.0 UA = 4.E-9 +UB = -7.6E-8 UC = -5.6E- AT =.E4 +WL = 0 WLN = WW = 0 +WWN = WWL = 0 LL = 0 +LLN = LW = 0 LWN = +LWL = 0 CAPMOD = XPART = 0.5 +CGDO =.E-0 CGSO =.E-0 CGBO = E-9 +CJ = 4.966E-4 PB = MJ = CJSW =.94E-0 PBSW = 0. MJSW = CJSWG =.64E-0 PBSWG = 0. MJSWG = CF = 0 PVTH0 = PRDSW = PK = WKETA = LKETA=5.669E-4 * MODEL CMOSP PMOS (LEVEL = 49 +VERSION =. TNOM = 7 TOX =.4E-8 +XJ =.5E-7 NCH =.7E7 VTH0 = K = K = K =

162 +KB = W0 = E-8 NLX = 4.775E-8 +DVT0W = 0 DVTW = 0 DVTW = 0 +DVT0 =.8745 DVT = DVT = U0 = UA =.69805E-9 UB =.06068E- +UC = E- VSAT = E5 A0 = AGS = B0 =.7508E-6 B = 5E-6 +KETA = E- A = 0 A = 0. +RDSW = E PRWG = PRWB = E- +WR = WINT =.0584E-7 LINT =.88055E-8 +XL = 0 XW = 0 DWG = E-8 +DWB =.00866E-8 VOFF = NFACTOR= CIT = 0 CDSC =.4E-4 CDSCD = 0 +CDSCB = 0 ETA0 = ETAB = DSUB = PCLM = PDIBLC = PDIBLC =.907E- PDIBLCB = DROUT = PSCBE = E9 PSCBE = E-0 PVAG = DELTA = 0.0 RSH = 0 MOBMOD = +PRT = 0 UTE = -.5 KT = -0. +KTL = 0 KT = 0.0 UA = 4.E-9 +UB = -7.6E-8 UC = -5.6E- AT =.E4 +WL = 0 WLN = WW = 0 +WWN = WWL = 0 LL = 0 +LLN = LW = 0 LWN = +LWL = 0 CAPMOD = XPART = 0.5 +CGDO =.88E-0 CGSO =.88E-0 CGBO = E-9 +CJ = E-4 PB = MJ = CJSW =.7065E-0 PBSW = 0.99 MJSW = CJSWG = 6.4E- PBSWG = 0.99 MJSWG = CF = 0 PVTH0 = E- PRDSW = PK =.798E- WKETA = 5.660E- LKETA=-6.878E- * 46

163 APPENDIX F: ADS SCHEMATIC USED FOR SIMULATION OF TWO- STAGE AMPLIFIER 47 Figure F. ADS shemai of ross-oupled differenial pair used for simulaions wih ADS

164 APPENDIX G: MATHCAD PROGRAM FOR FOUR-STAGE AMPLIFIER LINEARIZATION 48 ip -4amp7.md Mahad program o alulae suppression of hird order disorion and gain ompression urve for a four-sage amplifier linearizaion Forrnulae p=linear siganl p=rd order level G=gain db, OIP=rd order oupu inerep pon in dbm General ompressed gain oupu power level formula: ( p( Pin, G, OIP := min Pin + G, OIP p=oupu power inluding ompression General hird order inermodulaion produ oupu power level formula: p( Pin, G, OIP := OIP ( OIP p( Pin, G, OIP p=third order oupu disorion Derived / alulaed parameers alg( g := g 0 ala( g, g := g g Rule of humb for gain G of amplifier A Aenuaor loss aloip( OIP, a, delaoip := OIP a delaoip delaoip=differene in oupu inerep poin, deraed o inpus of firs subraor Derived value of OIP o assure anellaon of linear signal a oupu of firs subraor Calulaion of hird order disorion level a oupu of amplifier A alp_ou = Third order disorion level a oupu of amplifier A p( p( pin, g, o, g, o 0 alp_ou( pin, g, o, g, o := 0log0 + 0 p( p( pin, g, o, g, o 0

165 49 Calulaion of hird order disorion level a oupu of firs subraor (error signal alp_errou = Third order disorion level a oupu of firs subraor (error signal ( 0 log 0 alp_errou pin, g, o, g, o, a := p( pin, g, o 0 ( p( pin, g, o a 0 0 Subraion akes plae a volage level Calulaion of hird order disorion level a oupu of amplifier A4 alp_4ou = Third order disorion level a oupu of amplifier A4 alp_4ou( pin, g, o, g, o, a, g4, o4 := p( alp_errou( pin, g, o, g, o, a, g4, o4 Calulaion of hird order disorion level of final oupu alfinalp_ou = Third order disorion level a final oupu ( := 0 log alfinalp_ou pin, g, o, g, o, a, g, o, g4, o4 Subraon akes plae a he volage level alp_ou( pin, g, o, g, o 0... alp_4ou( pin, g, o, g, o, a, g4, o4 0 Calulaion of suppression of hird order disorion a final oupu suppression = Third order disorion level suppression a final oupu suppression( pin, g, o, g, o, a, g, o, g4, o4 := alp_ou( pin, g, o, g, o... + ( alfinalp_ou( pin, g, o, g, o, a, g, o, g4, o4 rd order disorion of amplifier A minus rd order IMD a final oupu Calulaion of linear power a final oupu alpou = linear oupu power a final oupu alpou( pin, g, o, g, o := p( p( pin, g, o, g, o asaded amplifier A and amplifier A

166 50 Example alulaion =============================== Free parameers: P := 40 Inpu Power G := 0 OIP := 0 Amplifier A Parameers G := 0 oipoverdrive:= 0 Amplifier A Parameers oipoverdrive = (OIP+G - OIP delaoip := 0 delaoip=differene in oupu inerep poin deraed o inpus of firs subraor Derived Parameers: G := alg( G Gain of amplifier A A := ala( G, G Value of neessary aenuaion OIP := aloip( OIP, A, delaoip OIP of amplifier A OIP OIP + G oipoverdrive := OIP of amplifier A OIP4 := OIP 0 OIP 4 of amplifier A4 G4 := alp_ou( P, G, OIP, G, OIP alp_errou( P, G, OIP, G, OIP, A Gain of amplifier A4 Summary of parameers of amplifiers A o A4 G = 0 G = 0 G = 0 G4 = 4.74 A = 0 OIP = 0 OIP = 0 OIP = 0 OIP4 = 0 ( = 9.7 Third order inermodulaion disorion IMD a oupu of A alp_ou P, G, OIP, G, OIP alp_errou P, G, OIP, G, OIP, A ( = IMD of oupu firs subraor alp_4ou P, G, OIP, G, OIP, A, G4, OIP4 ( = 9.7 IMD of oupu of amplifier A4 alfinalp_ou P, G, OIP, G, OIP, A, G, OIP, G4, OIP4 ( = 7.54 IMD of final oupu suppression P, G, OIP, G, OIP, A, G, OIP, G4, OIP4 ( = Third order disorion suppression alpou( P, G, OIP, G, OIP 0 = Linear oupu power a final oupu

167 5 Calulaion for inpu power sweep Veor Arrays for plos: N := Pin := N 0 N Inpu power swep from -40 o 0 dbm Third order suppression as a funion of inpu power ( Sup := suppression Pin, G, OIP, G, OIP, A, G, OIP, G4, OIP4 N N Linear oupu power of amplifier A as a funion of inpu power ( Pou := alpou Pin, G, OIP, G, OIP N N Third order disorion of oupu of amplifier A as a funion of inpu power ( po := alp_ou Pin, G, OIP, G, OIP N N Third order disorion of oupu of amplifier A4 as a funion of inpu power ( p4o := alp_4ou Pin, G, OIP, G, OIP, A, G4, OIP4 N N Third order disorion of final oupu as a funion of inpu power ( po := alfinalp_ou Pin, G, OIP, G, OIP, A, G, OIP, G4, OIP4 N N Error signal as a funion of inpu power ( perror := alp_errou Pin, G, OIP, G, OIP, A N N Final linear oupu power as a funion of inpu power ( finalp := alpou Pin, G, OIP, G, OIP N N

168 APPENDIX H: RESULTS OF PROTOTYPE USING DISCRETE DEVICES 5 The new onep for linearizaion was applied o a seond prooype and was buil wih disree devies in order o gain informaion abou he feasibiliy of he ondiion for anellaion desribed wih expression (4.7. A shemai of he prooype is shown in Fig. H.. The inpu signal is inpu o a 80 degree hybrid of ype AMT- on he lef side in Fig. H.. The hybrid disribues he inpu signal equally beween more linear amplifier A of ype ERA5-SM a he op of Fig. H. and less linear amplifier A of ype MAR- SM a he boom of Fig. H.. The oupu signal of A is fed ino a seond 80 degree hybrid of ype AMT- on he righ side of Fig. H. and oupu signal of A is aenuaed by a db pi-aenuaor and hen inpu o he 80 degree phase shif por of he seond hybrid. Oupu signal of A experienes a phase shif of 80 degree passing hrough he seond hybrid. As a resul, he final oupu signal is he differene beween oupu signals of A and A. The seond hybrid a he oupu as as subraor. All devies used are produs of Miniiruis [6] and mouned on a prined irui board. A layou of he op view of he PCB is depied in Fig. H.. The ransmission lines in he layou were designed as oplanar wave guides wih 50 impedane. The bak side of he PCB is ompleely mealized and as as ground plane. The maerial of he PCB is FR4 wih relaive dieleri onsan of 4 o 5. The PCB is equipped wih SMA onneors for ineronneion wih oher devies.

169 5 A (ERA5 Inpu Terminaions A (MAR Aenuaor db Oupu Figure H. Blok diagram of new linearizaion mehod using disree devies: inpu signal is spli equally in-phase wih a 80 degree hybrid and fed o amplifiers A and A and oupu signals are subraed wih hybrid. The fourh por of hybrids (AMT- is erminaed wih 50 resisors. The db aenuaor has pi form and is passive. Amplifiers A and A are ERA5-SM and MAR-SM. Volage Supply for ERA5 (A Oupu Volage Supply for MAR (A Inpu Figure H. Top layou for es board for disree devies used in es seup in Fig. H.; raes are oplanar 50 ransmission lines.