High-Frequency Modeling and Analyses for Buck and Multiphase Buck Converters

Transcription

1 High-Frequency Modeling and Analye for Buck and Multihae Buck Converter Yang Qiu Diertation ubmitted to the Faculty of the Virginia Polytechnic Intitute and State Univerity in artial fulfillment of the requirement for the degree of Doctor of Philoohy in Electrical Engineering APPROVED Fred C. Lee, Chairman Daan van Wyk Ming Xu Yilu Liu Guo-Quan Lu November 30 th, 2005 Blackburg, Virginia Keyword: high frequency, ideband effect, multi-frequency model, buck converter, multihae 2005, Yang Qiu

2 High-Frequency Modeling and Analye for Buck and Multihae Buck Converter Yang Qiu Abtract Future microroceor oe many challenge to it dedicated ower ulie, the voltage regulator VR, uch a the low voltage, high current, fat load tranient, etc. For the VR deign uing multihae buck converter, one of the reult from thee tringent challenge i a large amount of outut caacitor, which i undeired from both a cot and a motherboard real etate erective. In order to ave the outut caacitor, the controlloo bandwidth mut be increaed. However, the bandwidth i limited in the ractical deign. The influence from the witching frequency on the control-loo bandwidth ha not been identified, and the influence from multihae i not clear, either. Since the widelyued average model eliminate the inherent witching function, it i not able to redict the converter high-frequency erformance. In thi diertation, the rimary objective are to develo the methodology of high-frequency modeling for the buck and multihae buck converter, and to analyze their high-frequency characteritic. Firt, the nonlinearity of the ule-width modulator PWM cheme i identified. Becaue of the amling characteritic, the ideband comonent are generated at the outut of the PWM comarator. Uing the aumtion that the ideband comonent are well attenuated by the low-a filter in the converter, the conventional average model only include the erturbation-frequency comonent. When tudying the high-frequency erformance, the ideband frequency i not ufficiently high a comared with the erturbation one; therefore, the aumtion for the average model i not good any more. Under thi condition, the converter reone cannot be reflected by the average model. Furthermore, with a cloed loo, the generated ideband comonent at the outut voltage aear at the inut of the PWM comarator, and then generate the erturbation-frequency comonent at the outut. Thi caue the ideband effect to haen. The erturbation-

3 frequency comonent and the ideband comonent are then couled through the comarator. To be able to redict the converter high-frequency erformance, it i neceary to have a model that reflect the amling characteritic of the PWM comarator. A the bai of further reearch, the exiting high-frequency modeling aroache are reviewed. Among them, the harmonic balance aroach redict the highfrequency erformance but it i too comlicated to utilize. However, it i romiing when imlified in the alication with buck and multihae buck converter. Once the nonlinearity of the PWM comarator i identified, a imle model can be obtained becaue the ret of the converter ytem i a linear function. With the Fourier analyi, the relationhi between the erturbation-frequency comonent and the ideband comonent are derived for the trailing-edge PWM comarator. The concet of multi-frequency modeling i develoed baed on a inglehae voltage-mode-controlled buck converter. The ytem tability and tranient erformance deend on the loo gain that i affected by the ideband comonent. Baed on the multi-frequency model, it i mathematically indicated that the reult from the ideband effect i the reduction of magnitude and hae characteritic of the loo gain. With a higher bandwidth, there are more magnitude and hae reduction, which, therefore, caue the ideband effect to oe limitation when uhing the bandwidth. The rooed model i then alied to the multihae buck converter. For voltagemode control, the multihae technique ha the otential to cancel the ideband effect around the witching frequency. Therefore, theoretically the control-loo bandwidth can be uhed higher than the ingle-hae deign. However, in ractical deign, there i till magnitude and hae reduction around the witching frequency in the meaured loo gain. Uing the multi-frequency model, it i clearly ointed out that the ideband effect cannot be fully cancelled with unymmetrical hae, which reult in additional reduction of the hae margin, eecially for the high-bandwidth deign. Therefore, one hould be extremely careful to uh the bandwidth when deending on the interleaving to cancel the ideband effect. The multihae buck converter with eak-current control i alo invetigated. Becaue of the current loo in each individual hae, there i the ideband effect that iii

4 cannot be canceled with the interleaving technique. For higher bandwidth and better tranient erformance, two cheme are reented to reduce the influence from the current loo: the external ram are inerted in the modulator, and the inductor current are couled, either through feedback control or by the couled-inductor tructure. A bandwidth around one-third of the witching frequency i achieved with the couled-inductor buck converter, which make it a romiing circuit for the VR alication. A a concluion, the feedback loo reult in the ideband effect, which limit the bandwidth and i not included in the average model. With the rooed multi-frequency model, the high-frequency erformance for the buck and multihae buck converter can be accurately redicted. iv

5 TO MY PARENTS FEIZHOU QIU AND CUIZHEN LIU AND TO MY WIFE JUANJUAN SUN v

6 Acknowledgment I would like to exre my incere areciation to my advior, Dr. Fred C. Lee, for hi continued guidance, encouragement and uort. It i an honor to be one of hi tudent here at the Center for Power Electronic Sytem CPES, one of the bet reearch center in ower electronic. In the at year, I am alway amued by hi great intuition, broad knowledge and accurate judgment. The mot reciou thing I learned from him are the ability of indeendent reearch and the attitude toward reearch, which can be alied to every aect of life and will benefit me for the ret of my life. I would alo like to thank Dr. Ming Xu for hi enthuiatic hel during my reearch at CPES. Hi elfle friendhi and leaderhi heled to make my time at CPES enjoyable and rewarding. From him, I learned o much not only in the knowledge of ower electronic but alo in the reearch methodologie. Hi valuable uggetion heled to encourage my uruing thi degree. I am grateful to the other member of my adviory committee, Dr. Daan van Wyk, Dr. Yilu Liu, Dr. Guo-Quan Lu, and Dr. Dan Y. Chen for their uort, comment, uggetion and encouragement. I am eecially indebted to my colleague in the VRM grou and the ARL grou. It ha been a great leaure to work with the talented, creative, helful and dedicated colleague. I would like to thank all the member of my team: Dr. Peng Xu, Dr. Pit-Leong Wong, Dr. Kaiwei Yao, Dr. Wei Dong, Dr. Francico Canale, Dr. Bo Yang, Dr. Jia Wei, Mr. Mao Ye, Dr. Jinghai Zhou, Dr. Yuancheng Ren, Mr. Bing Lu, Mr. Yu Meng, Mr. Ching-Shan Leu, Mr. Doug Sterk, Mr. Kiun Lee, Mr. Julu Sun, Dr. Shuo Wang, Dr. Xu Yang, Mr. Yonghan Kang, Mr. Chuanyun Wang, Mr. Dianbo Fu, Mr. Arthur Ball, Mr. Andrew Schmit, Mr. David Reuch, Mr. Yan Dong, Mr. Jian Li, Mr. Bin Huang, Mr. Ya Liu, Mr. Yucheng Ying, and Mr. Yi Sun. It wa a real honor working with you guy. I would like to thank my fellow tudent and viiting cholar for their hel and guidance: Dr. Peter Barboa, Mr. Dengming Peng, Dr. Jinjun Liu, Dr. Jae-Young Choi, Dr. Qun Zhao, Dr. Zhou Chen, Dr. Jinghong Guo, Dr. Linyin Zhao, Dr. Rengang Chen, Dr. vi

7 Zhenxue Xu, Dr. Bin Zhang, Dr. Xigen Zhou, M. Qian Liu, Mr. Xiangfei Ma, Mr. Wei Shen, Dr. Haifei Deng, M. Yan Jiang, M. Huiyu Zhu, Mr. Pengju Kong. Mr. Jian Yin, Mr. Wenduo Liu, Dr. Zhiye Zhang, M. Ning Zhu, M. Jing Xu, M. Yan Liang, M. Michele Lim, Mr. Chucheng Xiao, Mr. Hongfang Wang, Mr. Honggang Sheng, and Mr. Rixin Lai. I would alo like to thank the wonderful member of the CPES taff who were alway willing to hel me out, M. Terea Shaw, M. Linda Gallagher, M. Terea Roe, M. Ann Craig, M. Marianne Hawthorne, M. Elizabeth Tranter, M. Michelle Czamanke, M. Linda Long, Mr. Steve Chen, Mr. Robert Martin, Mr. Jamie Evan, Mr. Dan Huff, Mr. Callaway Ca, Mr. Gary Kerr, and Mr. David Fuller. My heartfelt areciation goe toward my arent, Feizhou Qiu and Cuizhen Liu, who have alway rovided uort and encouragement throughout my further education. Finally, with deeet love, I would like to thank my wife, Juanjuan Sun, who ha alway been there with her love, uort, undertanding and encouragement for all of my endeavor. vii

8 Thi work wa uorted by the VRM conortium Arteyn, Delta Electronic, Hiro Electronic, Infineon, Intel, International Rectifier, Interil, Linear Technology, National Semiconductor, Renea, and Texa Intrument, and the Engineering Reearch Center Shared Facilitie uorted by the National Science Foundation under NSF Award Number EEC Any oinion, finding and concluion or recommendation exreed in thi material are thoe of the author and do not necearily reflect thoe of the National Science Foundation. Thi work wa conducted with the ue of SIMPLIS oftware, donated in kind by Tranim Technology of the CPES Indutrial Conortium. viii

9 Table of Content Chater. Introduction.... Background: Voltage Regulator....2 Challenge to VR High-Frequency Modeling Diertation Outline...2 Chater 2. Characteritic of PWM Converter Introduction Characteritic of the Pule-Width Modulator Sideband Effect of PWM Converter with Feedback Loo Small-Signal Tranfer Function Meaurement and Simulation Previou Modeling Aroache Summary...37 Chater 3. Multi-Frequency Modeling for Buck Converter Modeling of the PWM Comarator The Multi-Frequency Model of Buck Converter Summary...52 Chater 4. Analye for Multihae Buck Converter Introduction The Multi-Frequency Model of Multihae Buck Converter Study for the Multihae Buck Converter with Unymmetrical Phae Summary...7 Chater 5. High-Bandwidth Deign of Multihae Buck VR with Current-Mode Control Introduction...73 ix

10 5.2 Bandwidth Imrovement with External Ram Bandwidth Imrovement with Inductor Current Couling Summary...96 Chater 6. Concluion Summary Future Work...99 Aendix A. Analye with Different PWM Scheme...00 Aendix B. Analye with Inut-Voltage Variation...09 Reference...7 Vita...22 x

11 Lit of Table Table 3.. Extended decribing function of the trailing-edge PWM comarator...44 Table 4.. Extended decribing function of the trailing-edge PWM comarator for the m- th hae in an n-hae buck converter...57 Table A.. Extended decribing function of the PWM comarator with different modulation Table B.. Extended decribing function from the inut voltage to the hae voltage... xi

12 Lit of Figure Figure.. Intel roadma of number of integrated tranitor in one microroceor.... Figure.2. Intel roadma of comuting erformance for the microroceor...2 Figure.3. The roadma of microroceor uly voltage and current...3 Figure.4. A ingle-hae ynchronou buck converter....4 Figure.5. A multihae buck converter...5 Figure.6. Current rile cancellation in multihae VR....5 Figure.7. Future microroceor demand more caacitor if today olution i till followed...6 Figure.8. The relationhi between VR bandwidth and the outut bulk caacitance for future microroceor baed on today ower delivery ath....7 Figure.9. VR efficiency uffer a lot a the witching frequency increae...8 Figure.0. Simulated loo gain of a -MHz buck converter with voltage-mode control..9 Figure.. A ingle-hae voltage-mode-controlled buck converter...9 Figure.2. Comarion of loo gain between SIMPLIS imulation and average model for a -MHz buck converter with voltage-mode control... Figure 2.. An oen-loo ingle-hae buck converter with V c erturbation...5 Figure 2.2. Inut and outut of the trailing-edge PWM comarator with V c erturbation.6 Figure 2.3. Samling reult of the PWM cheme....6 Figure 2.4. Aliaing effect haen at half of the witching frequency...8 Figure 2.5. Inut and outut waveform of the witche in buck converter with contant inut voltage....9 Figure 2.6. Inut and outut ectra of the witche in buck converter with contant inut voltage....9 xii

13 Figure 2.7. Simulated waveform with 0-kHz V c erturbation for a -MHz oen-loo buck converter Figure 2.8. Simulated waveform with 990-kHz V c erturbation for a -MHz oen-loo buck converter....2 Figure 2.9. The frequency-domain rereentation including only the erturbationfrequency comonent...22 Figure 2.0. The frequency-domain rereentation for the oen-loo buck converter with ideband comonent Figure 2.. Control voltage erturbation waveform at f / Figure 2.2. The frequency-domain rereentation for the oen-loo buck converter when f =f / Figure 2.3. The ideband effect in a voltage-mode-controlled buck converter Figure 2.4. Sace vector rereentation of outut voltage comonent at f with ideband effect...24 Figure 2.5. The frequency-domain rereentation for a voltage-mode-controlled buck converter including only the erturbation-frequency comonent...25 Figure 2.6. Simulated V o with a 0-kHz erturbation for a -MHz voltage-modecontrolled buck converter...26 Figure 2.7. Simulated V c with a 0-kHz erturbation for a -MHz voltage-modecontrolled buck converter...27 Figure 2.8. Simulated V o f and V o f for the -MHz voltage-mode-controlled buck converter...28 Figure 2.9. Simulated V o with a 990-kHz erturbation for a -MHz voltage-modecontrolled buck converter...29 Figure Simulated V c waveform with a 990-kHz erturbation for a -MHz voltagemode-controlled buck converter...30 xiii

14 Figure 2.2. Simulated V c ectra with a 990-kHz erturbation for a -MHz voltage-modecontrolled buck converter...30 Figure Simulated V o ectra a the reult of V c f and V c f -f for a -MHz voltagemode-controlled buck converter...3 Figure Network analyzer block diagram of meauring the control-to-outut tranfer function...33 Figure Partitioning of a converter ytem: linear ubytem and nonlinear ubytem Figure A tyical nonlinear ubytem...35 Figure Rereentation by extended decribing function for a tyical nonlinear ubytem Figure Nonlinearity in the ingle-hae voltage-mode-controlled buck converter with contant inut voltage...37 Figure 3.. Nonlinearity of the PWM comarator Figure 3.2. Inut and outut waveform of the trailing-edge PWM comarator...40 Figure 3.3. The model of the trailing-edge PWM comarator...45 Figure 3.4. Frequency-domain relationhi between the hae voltage and the duty cycle auming contant inut voltage...46 Figure 3.5. A voltage-mode-control buck converter with load-current erturbation Figure 3.6. The multi-frequency model of a ingle-hae voltage-mode-controlled buck converter...47 Figure 3.7. The average model of a ingle-hae voltage-mode-controlled buck converter Figure 3.8. Simlified multi-frequency model of a ingle-hae voltage-mode-controlled buck converter Figure 3.9. Loo gain of a -MHz buck converter with voltage-mode control...49 xiv

15 Figure 3.0. Meaured loo gain of a -MHz ingle-hae buck converter with voltagemode control...49 Figure 3.. Loo gain of a -MHz buck converter with voltage-mode control....5 Figure 4.. A multihae buck converter with V c erturbation Figure 4.2. Trailing-edge modulator waveform of a 2-hae buck converter with V c erturbation...55 Figure 4.3. The multi-frequency model of the m-th hae in an n-hae PWM comarator Figure 4.4. The multi-frequency model of the n-hae buck converter...58 Figure 4.5. Sace vector of the duty cycle for the multihae buck converter Figure 4.6. Simulated waveform with 990-kHz V c erturbation for -MHz oen-loo buck converter Figure 4.7. An n-hae buck converter with a load current erturbation....6 Figure 4.8. The multi-frequency model of the n-hae buck converter...6 Figure 4.9. Loo gain of a -MHz 2-hae buck converter with voltage-mode control...62 Figure 4.0. Sideband comonent at the outut of the PWM comarator Figure 4.. Simulated waveform with a.99-mhz V c erturbation for a -MHz 2-hae oen-loo buck converter...64 Figure 4.2. Simulated loo gain of a high-bandwidth -MHz 2-hae buck converter with voltage-mode control...65 Figure 4.3. Exerimental loo gain of a high-bandwidth -MHz 2-hae buck converter with voltage-mode control...65 Figure 4.4. Simulated V o waveform with 990-kHz V c erturbation for -MHz oen-loo buck converter...67 Figure 4.5. Sace vector rereentation in the 2-hae buck converter with inductor tolerance...69 xv

16 Figure 4.6. Simulated loo gain of -MHz 2-hae voltage-mode-controlled buck converter Figure 5.. A 2-hae buck converter with eak-current control Figure 5.2. A eak-current-controlled 2-hae buck converter with V c erturbation Figure 5.3. Simulated waveform for the -MHz eak-current-controlled 2-hae buck converter with 990-kHz V c erturbation...75 Figure 5.4. Simulated G vc of -MHz buck converter with eak-current control...76 Figure 5.5. Loo gain, T 2, of the -MHz 2-hae buck converter with eak-current control Figure 5.6. Loo gain, T v, of the -MHz 2-hae buck converter with voltage-mode control...77 Figure 5.7. Modulator in the voltage-mode control and current-mode control...78 Figure 5.8. Modulator in eak-current control with external ram...79 Figure 5.9. Loo gain, T 2, of the -MHz 2-hae buck converter with eak-current control, S e /S n = Figure 5.0. Q value of the f /2 double ole a a function of S e /S n for a 2-V-to-.2-V buck converter...8 Figure 5.. Phae-current-couling control for a 2-hae buck converter...82 Figure 5.2. A 2-hae couled-inductor buck converter...82 Figure 5.3. Waveform of the PWM comarator inut with inductor current information couling for 2-hae buck converter...83 Figure 5.4. Simulated G vc with inductor current information couling for 2-hae buck converter Figure 5.5. Inut waveform of the PWM comarator in a 2-hae couled-inductor buck converter Figure 5.6. Sytem block diagram of the couled-inductor buck converter with voltageloo oen xvi

17 Figure 5.7. Natural reone of the hae current, I L Figure 5.8. Forced reone of I L2 a a reult of I L variation...88 Figure 5.9. Forced reone of I L and I L2 a reult of V c variation Figure G vc tranfer function of a 2-hae couled-inductor buck converter Figure 5.2. Samle-hold effect in the couled-inductor buck converter...93 Figure Simulated T 2 loo gain of a 2-hae couled-inductor buck converter with α= Figure A 4-hae buck converter with 2-hae couled-inductor deign...94 Figure T 2 loo gain of a -MHz 4-hae buck converter with 2-hae couling Figure A.. Inut and outut waveform of the PWM comarator with different modulation cheme Figure A.2. Inut and outut waveform of the PWM comarator of contant-frequency control...0 Figure A.3. Magnitude of F m+ and F m- a a function of the duty cycle for the double-edge modulation...02 Figure A.4. The generalized multi-frequency model of a ingle-hae oen-loo buck converter...03 Figure A.5. Simulated V o waveform with 20% duty cycle for -MHz oen-loo buck converter with 990-kHz V c erturbation and different modulation cheme Figure A.6. Simulated V o waveform for -MHz oen-loo buck converter with 990-kHz V c erturbation and the double-edge modulation...05 Figure A.7. The generalized multi-frequency model of a voltage-mode-controlled buck converter...05 Figure A.8. Comarion of the magnitude of ideband effect, F m+ *F m-, auming V R =. 06 Figure A.9. Loo gain comarion among PWM method Figure B.. An oen-loo buck converter with the inut-voltage erturbation xvii

18 Figure B.2. The hae voltage waveform with the inut-voltage erturbation auming a contant duty cycle Figure B.3. Switche in the converter....0 Figure B.4. Decribing function of v d - /v in...2 Figure B.5. Multi-frequency model of the buck converter conidering the inut-voltage erturbation...2 Figure B.6. Comarion between the imulation and modeling with the inut-voltage erturbation...3 Figure B.7. Buck converter with erturbation at both the control voltage and inut voltage....3 Figure B.8. Function of the witche in the buck converter...3 Figure B.9. Multi-frequency model of the nonlinearitie of the buck converter....4 Figure B.0. Multi-frequency model of a voltage-mode-controlled buck converter with the inut-voltage erturbation....5 Figure B.. Comarion of the cloed-loo audio-ucetibility...6 xviii

19 Chater. Introduction. Background: Voltage Regulator In the at four decade, the Moore law, which tate tranitor denity double every eighteen month, ha uccefully redicted the evolution of microroceor, a hown in Figure. []. Currently, the latet roceor from Intel conit of hundred of million of tranitor [2]. It i redicted that in 205, there will be ten of billion of tranitor in a ingle chi [3]. Tranitor Figure.. Intel roadma of number of integrated tranitor in one microroceor. Year More integrated tranitor lead to better comuting erformance. A hown in Figure.2 [4], the comuting eed, a meaured in million of intruction er econd MIPS, increae dramatically in the at four decade. It i redicted that in around 205, the microroceor can deal with 0 trillion intruction er econd [4]. However, the more tranitor acked into maller ace, and the higher comuting erformance, the more ower the microroceor conume. Currently, a three-ercent increae in ower conumtion i required for a one-ercent imrovement in

20 Chater. Introduction microroceor erformance [3]. Since all the electric ower conumed by the microroceor i tranferred to heat eventually, tringent challenge have been oed on the thermal management. There i the reviion that if the develoment of the roceor till follow Moor law but without imrovement of the ower management, a ower lo denity of ten of thouand watt er quare centimeter i oible [3]. Figure.2. Intel roadma of comuting erformance for the microroceor. New ower management technologie for the tranitor in the microroceor have been introduced in the at decade. One of the olution i to decreae the microroceor uly voltage. Starting with the Intel Pentium roceor, microroceor began to ue a non-tandard ower uly of le than 5 V, and the uly voltage have been and will continuouly be decreaed. On the other hand, the increaing number of tranitor in the microroceor reult in continuou increae of the microroceor current demand, a hown in Figure.3 [5][6]. Although new technologie, uch a the multi-core tructure for the microroceor, may low down the trend, it i exected that the challenge to the ower uly i till tringent [6]. Moreover, due to the high comuting eed, the microroceor load tranition eed alo increae. In the mean time, the voltage deviation window during the tranient i becoming maller and maller, ince the outut 2

21 Chater. Introduction voltage kee decreaing. The low voltage, high current, fat load tranition eed, and tight voltage regulation imoe challenge on the ower ulie of the microroceor. Icc A Icc Vcc Year Vcc V Figure.3. The roadma of microroceor uly voltage and current. When uing the 5-V legacy voltage level, the microroceor wa owered by a centralized ilver box. Becaue the araitic reitor and inductor of the connection between it and the microroceor have a evere negative imact on the ower quality, it i no longer ractical for the bulky ilver box to rovide energy directly to the microroceor for the low-voltage high-current alication. Therefore, the voltage regulator VR i introduced a the dedicated ower uly. For the low-end microroceor VR, a ingle conventional buck or ynchronou buck toology, a hown in Figure.4, i utilized for ower converion [7][8][9]. A the microroceor ower conumtion increae continuouly, it i imoible to ue a ingle device a the to or bottom witche in the buck converter. To handle the required high current, more device in arallel are neceary. Meanwhile, the earlier VR oerated at low witching frequencie with high filter inductance. However, the large outut-filter inductance limit the energy tranfer eed. 3

22 Chater. Introduction In order to meet the microroceor requirement, huge outut-filter caacitor and decouling caacitor are needed to reduce the voltage ike during the load tranient. i o Q V in Q 2 C o R o Figure.4. A ingle-hae ynchronou buck converter. In order to reduce the VR outut caacitance to ave the total cot and to increae the ower denity, high inductor current lew rate are referred. With maller inductance, larger inductor current lew rate are obtained; therefore, a maller outut caacitance can be ued to meet the tranient requirement. In order to greatly increae the tranient inductor current lew rate, the inductance need to be reduced ignificantly, a comared with thoe in conventional deign. On the other hand, mall inductance reult in large current rile in the circuit oeration at the teady tate. The large current rile uually caue a large turn-off lo. In addition, it generate large teady-tate voltage rile at the VR outut caacitor. The teady-tate outut voltage rile can be o large that they are comarable to tranient voltage ike. It i imractical for the converter to work thi way. To olve the aforementioned iue, VPEC/CPES rooe to arallel hae intead of device, a hown in Figure.5 [0][][2][3][]. It conit of n identical converter with interconnected inut and outut. Baed on thi tructure, the interleaving technology i introduced by hae hifting the duty cycle of adjacent channel with a degree of 360 o /n. With the rooed multihae buck converter, the outut current rile are greatly decreaed, a hown in Figure.6. Therefore, the teady-tate outut voltage rile are ignificantly reduced, making it oible to ue very mall inductance in VR to imrove the tranient reone. 4

23 Chater. Introduction i o Q Q 2 V in i L C o R o Q 3 Q 4 i L2 Q 5 Q 6 i L3 Q 7 Q 8 i L4 Rile Cancellation Figure.5. A multihae buck converter. 2-Phae 3-Phae 4-Phae Duty Cycle Vo/Vin Figure.6. Current rile cancellation in multihae VR. 5

24 Chater. Introduction Beide the benefit of maller teady-tate voltage rile and tranient voltage ike, the multihae buck converter make the thermal diiation more evenly ditributed. Studie alo how that in high-current alication, the overall cot of the converter can be reduced uing thi technology. Therefore, many emiconductor comanie, uch a Interil, National Semiconductor, Texa Intrument, Analog Device, On Semiconductor, and Volterra, have roduced dedicated control IC for multihae VR. The concet of alying interleaving to VR i o ucceful that it ha become an indutry tandard ractice in the VR alication..2 Challenge to VR High-Frequency Modeling A the microroceor develo, the ower management related iue become much more critical for future microroceor and much more difficult to deal with. If today low-frequency olution i till emloyed to meet the future tranient requirement, more caacitor have to be aralleled. Baed on the microroceor ower delivery ath [4] and the ecification, it can be calculated [5][6] that the bulk caacitor number will increae by 40%, and the decouling caacitor number will double, a hown in Figure.7. A the reult, the cot of caacitor will increae 60%. Therefore, how to meet the requirement of fat tranient reone with fewer outut caacitor become one of the mot challenging iue to the VR deigner. Sening Point m 5 0.4m n m 560u*4 0.36m 0u*36 u*23 u i o Bulk Ca.4X Decouling Ca 2X Figure.7. Future microroceor demand more caacitor if today olution i till followed. 6

25 Chater. Introduction To reduce the outut caacitor, everal nonlinear method have been rooed [7][8]. Aroache uing hybrid filter have alo been introduced [9][20]. However, thee method are not yet ready for the ractical VR alication. From the tandoint of an indutry roduct, the linear control method i referred [8][2]. For the buck VR with linear control method, it ha been hown that there are two fundamental limitation to the inductor current lew rate [5][6][22][23][24][25]. Auming contant inut and outut voltage, the inductance value determine the lew rate when the duty cycle i aturated. Without the duty-cycle aturation, the feedback control loo bandwidth determine the lew rate. With a higher bandwidth, a fater inductor current lew rate i achievable. Conequently, fewer outut caacitor are needed for the deired tranient erformance, a hown in Figure.8 [5][6]. Therefore, to reduce the outut caacitance, high-bandwidth deign are mandatory. 0-2 Al-oly ca 560µF/7mΩ Caacitance F Ceramic ca 00µF/.4mΩ Bandwidth Hz Figure.8. The relationhi between VR bandwidth and the outut bulk caacitance for future microroceor baed on today ower delivery ath. In today ractice of multihae buck VR, the bandwidth can only be uhed to around /0~/6 of the witching frequency. Higher witching frequencie are required for higher bandwidth. For examle, to eliminate the bulk caacitor, a 390-kHz bandwidth i 7

26 Chater. Introduction neceary. Auming the bandwidth of one-ixth witching frequency i achievable, a witching frequency higher than 2.3 MHz i required. However, higher witching frequency mean more witching-related loe and lower efficiency. A an examle, Figure.9 comare the efficiency for a 4-hae ynchronou buck VR running at 300-kHz and -MHz witching frequencie. Thi 2-V inut, 0.8-V 70-A outut VR ue one HAT268 a the to witch and two HAT265 a the bottom witch for each hae. From 300 khz to MHz, the efficiency degrade around 5% [6]. Efficiency 89% 87% 300kHz 85% 83% 8% 79% MHz 77% 75% Outut Current A Figure.9. VR efficiency uffer a lot a the witching frequency increae. Becaue of the efficiency conideration, it i exected that the bandwidth can be uhed a high a oible with a certain witching frequency. Therefore, it i neceary to invetigate the bandwidth limitation for the buck and multihae buck converter. To tudy the iue of uhing the bandwidth, the witching-model imulation reult from SIMPLIS oftware are analyzed. Figure.0 comare the loo gain, T v, with different bandwidth for a -MHz ingle-hae voltage-mode-controlled buck converter, a in Figure.. With ame ole and zeroe but different DC gain in the comenator, there i more hae delay when the bandwidth i uhed from 00 khz to 400 khz. At 400 khz, the hae delay i 45 o for the 400-kHz-bandwidth deign, while it i 26 o for the 00- khz-bandwidth cae. Therefore, a higher bandwidth reult in additional hae delay in the loo gain. Beide, with a 3-ole 2-zero comenator, the hae delay i 270 o at the 8

27 Chater. Introduction witching frequency. Therefore, there exit a limitation for the control loo bandwidth, which i related to the witching frequency. Phae o Magnitude db f =MHz f c =00kHz 40 f c =400kHz Frequency Hz Figure.0. Simulated loo gain of a -MHz buck converter with voltage-mode control. Red olid line: 00-kHz-bandwidth deign; Blue dotted line: 400-kHz-bandwidth deign. V in v c V r PWM d v d L C v o R o f T v - H v + V ref Figure.. A ingle-hae voltage-mode-controlled buck converter. 9

28 Chater. Introduction In the at, mot of the feedback controller deign have been baed on the average model [26][27] for buck converter. The multihae buck ha alo been imlified to ingle-hae buck converter in the average model [23]. However, according to the obervation above, the witching frequency lay an imortant role in the loo gain at the high-frequency region. The highet achievable bandwidth i related to the witching frequency. Becaue the tate-ace averaging roce eliminate the inherent amling nature of the witching converter, the accuracy of the average model i quetionable at frequencie aroaching half of the witching frequency [28]. A an examle, for the -MHz ingle-hae buck converter with voltage-mode control, Figure.2 comare the loo gain calculated from the average model with that obtained in the witching-model imulation uing SIMPLIS. For the cae with a 00-kHz bandwidth of the voltage loo, the average model agree with the imulation u to around half of the witching frequency. However, for the 400-kHz bandwidth deign, the average model i only good u to 00 khz, i.e., one-tenth of the witching frequency. The imulation reult ha a 25 o more hae delay at the croover frequency a comared with the average model. Thi exceive hae dro would reult in undeired tranient or tability roblem if a highbandwidth converter i deigned baed on the average model, which cannot redict the high-frequency behavior. For a better undertanding of the characteritic of the control loo, the fundamental relationhi between the control-loo bandwidth and the witching frequency hould be clarified. To obtain an analytical inight, a imle model including the witching frequency information i deired. To addre thee iue, the rimary objective of thi diertation i to invetigate the influence from the witching frequency on the converter erformance. The methodology of high-frequency modeling for the buck and multihae buck converter i develoed and utilized to analyze their high-frequency characteritic. 0

29 Chater. Introduction Phae o Magnitude db SIMPLIS imulation Average model f =MHz f c =00kHz Frequency Hz a The 00-kHz bandwidth deign. Phae o Magnitude db SIMPLIS imulation Average model 25 o f =MHz f c =400kHz Frequency Hz b The 400-kHz bandwidth deign. Figure.2. Comarion of loo gain between SIMPLIS imulation and average model for a -MHz buck converter with voltage-mode control. Red olid line: SIMPLIS imulation reult; Blue dotted line: average-model reult.

30 Chater. Introduction.3 Diertation Outline Thi diertation conit of ix chater. They are organized a follow. Firt, the background information of VR and the need for VR high-frequency modeling are introduced. Then, the characteritic of the ule-width modular PWM converter are reviewed. The ideband effect a a reult of feedback control i identified. After that, baed on the harmonic balance aroach, the concet of multi-frequency modeling i develoed to addre the ideband effect for a ingle-hae voltage-mode-controlled buck converter. Next, following the ame aroach, thi model i alied to the multihae buck converter. At lat, the influence from the current feedback loo i invetigated. Several method to achieve high-bandwidth deign for VR alication are exlored. The detailed outline i elaborated a follow. Chater i the background review of exiting VR technologie and the need for high-frequency VR modeling. Multihae buck converter have become the tandard ractice for VR in the indutry. In order to imrove the tranient reone, the controlloo bandwidth mut be increaed. However, the bandwidth i limited in the ractical deign. The relationhi between the witching frequency and the control-loo bandwidth i not clear. Since the conventional average model eliminate the inherent witching function, it i not able to redict the high-frequency erformance. The rimary objective of thi diertation are to develo the methodology of high-frequency modeling for the buck and multihae buck converter, and to analyze their high-frequency characteritic. Chater 2 dicue the nonlinearity of the PWM cheme and review the exiting aroache to model thi nonlinearity. Becaue of the inherent amling function of the PWM comarator, ideband-frequency comonent are generated in the converter. With a feedback control loo, the ideband comonent aear at the inut of the comarator and generate the erturbation-frequency comonent again. Through the comarator, the ideband comonent and the erturbation-frequency comonent are couled. With the aumtion of low-a filter in the converter, the conventional average model only include the erturbation frequency and regard the PWM comarator a a imle gain. Therefore, it doe not reflect thee henomena. To be able to redict the converter highfrequency erformance, it i neceary to have a model that reflect the amling 2

31 Chater. Introduction characteritic of the PWM comarator. A the bai of further reearch, the exiting highfrequency modeling aroache are reviewed. The harmonic balance aroach i able to redict the high-frequency erformance but it i comlicated to utilize. However, for the alication with buck and multihae buck converter, once the nonlinearity of the PWM comarator i identified, a imlified model can be obtained. Chater 3 introduce the multi-frequency model to redict the ytem behavior. With the Fourier analyi, the relationhi between the ideband comonent and the erturbation-frequency comonent are derived for the PWM comarator. The concet of multi-frequency modeling i develoed baed on a ingle-hae voltage-mode-controlled buck converter. The influence of the ideband effect are invetigated quantitatively. In Chater 4, the rooed model i alied to the multihae buck converter. For voltage-mode control, the multihae technique ha the otential to cancel the ideband effect around the witching frequency. Therefore, it i theoretically oible to uh the control-loo bandwidth higher than the deign with ingle-hae buck converter. However, the aymmetry among hae reult in deign rik to uh the control-loo bandwidth in imlementation. Conidering the inductor with ractical tolerance a an examle, the limitation of bandwidth i dicued. Chater 5 analyze the multihae buck converter with eak-current control. In the current loo of each hae, there i a ideband effect that cannot be canceled with the interleaving technique. For higher bandwidth and better tranient erformance, two cheme are reented to reduce the influence from the current loo: the external ram are inerted to the modulator, and the inductor current are couled, either through feedback control or by the couled-inductor tructure. The amle-data model for the couledinductor buck converter i derived, which exlain the benefit of trong couling on bandwidth imrovement. Chater 6 i the ummary of thi diertation. 3

32 Chater 2. Characteritic of PWM Converter 2. Introduction A redicted by Moore law, future comuter microroceor will conit of billion of integrated tranitor. To reduce the ower conumtion, the oerating voltage will continue to dro. The allowed variation of the outut voltage will become maller for the VR. On the other hand, the higher eed of future roceor lead to more dynamic load. Conequently, one ecial iue exiting for the VR i how to meet the tringent voltage regulation requirement with le outut caacitor. Thi i a trict challenge becaue of cot related conideration, a well a limited ace for VR in the comuter ytem. To ave the outut caacitor, the VR inductor current lew rate mut be increaed. It ha been tudied that with linear control method, the feedback loo bandwidth lay a very imortant role in the tranient reone [5][6][22][23][24][25]. With a higher bandwidth, fewer outut caacitor are needed to meet the required tranient erformance ecification. On the other hand, high-bandwidth deign normally require high witching frequency, which i not referred from an efficiency aect becaue of the frequencyrelated loe. Puhing the control-loo bandwidth without increaing the witching frequency i more deirable. The relationhi between the control-loo bandwidth and the witching frequency i not clear. Conventionally, the control deign of the multihae buck VR utilize the average model, which doe not include the witching information. About the voltage loo gain of a ingle-hae buck converter hown in Figure.2, the average model fail to redict the erformance around or beyond half of the witching frequency, eecially with high-bandwidth deign. To exlain the dicreancie between the average model and witching-model imulation, and to clarify the limitation of the control-loo bandwidth, a model that can reflect the inherent witching characteritic of the converter i eential for further tudie. Once the model i obtained, it i oible to achieve guideline for the control deign and high-bandwidth olution. 4

33 Chater 2. Characteritic of PWM Converter To clearly undertand the witching feature of the converter, thi chater invetigate the nonlinear characteritic of the ule-width modulator PWM. Firt, the exitence of ideband comonent are oberved a a reult of amling. After that, the influence from the feedback loo i analyzed baed on a ingle-hae voltage-mode-controlled buck converter. The ideband effect i identified, i.e. the ideband comonent aear at the inut of the comarator and generate the erturbation-frequency comonent again. Then, the tranfer function meaurement and imulation are dicued, eecially on how they deal with the ideband effect. A the bai of further reearch, the exiting high-frequency modeling aroache are reviewed. 2.2 Characteritic of the Pule-Width Modulator Before a way can be found to identify the limitation of the average model and to redict the converter high-frequency erformance, it i eential to clarify the characteritic of the PWM converter. A an examle, Figure 2. illutrate the tructure when tudying the reone of an oen-loo ingle-hae buck converter with a erturbation at the control voltage, V c. For the mall-ignal analyi, it i aumed that the erturbation i mall enough that it doe not change the oerating oint of the converter. When tudying the erformance at a certain frequency, the erturbation i aumed to be inuoidal for imlicity. V in v d v c L PWM v r f d C Ro f v o Figure 2.. An oen-loo ingle-hae buck converter with V c erturbation. For the trailing-edge PWM comarator a hown in Figure 2.2, with a inuoidal erturbation frequency at f, the ectra of the comarator inut, V c, and that of the outut, 5

34 Chater 2. Characteritic of PWM Converter d, are illutrated in Figure 2.3 [29][30][3][32][33]. Becaue they are eriodical in the time domain, thee ignal have dicrete ectra in the frequency domain. V c V r d Figure 2.2. Inut and outut of the trailing-edge PWM comarator with V c erturbation. V c -f f a Inut ectrum of the PWM comarator. -f d f -f -f f -f -f f -f +f f +f b Outut ectrum of the PWM comarator. Figure 2.3. Samling reult of the PWM cheme. Clearly, the ectrum of d conit of the DC comonent, the comonent at the witching frequency, f, and it harmonic frequencie. The comonent at the erturbation frequency, f and -f, aear at the comarator outut a well. Meanwhile, becaue the 6

35 Chater 2. Characteritic of PWM Converter PWM comarator work like a amle-data function, it outut, d, ha infinite frequency comonent at f -f, -f +f, f +f, -f -f, etc [29][30][3][32][33]. Thee frequencie are called the ideband frequencie or the beat frequencie around f, -f, etc., which do not exit at the inut of the comarator. Hence, the PWM comarator i a tyical nonlinear function. For the PWM comarator, there are ecial cae exiting when f =kf /2, k=, 2, 3, A an examle, Figure 2.4 illutrate the cae when the erturbation frequency i exactly at half of the witching frequency. Under thi condition, the aliaing effect haen [30][3][34], which mean the ideband frequency overla with the erturbation frequency itelf, namely f i equal to f -f. Therefore, beide the DC and witching frequency comonent, the ytem contain comonent at f, -f, f +f, -f -f, etc. For examle, when f =f /2, there i only one frequency comonent below the witching frequency beide DC, which i different from the erturbation at other frequencie. From thi aect, again, the PWM comarator i a nonlinear function. In thi chater, only the erturbation at V c i conidered, a in Figure 2.. The inut voltage, V in, i aumed contant. Under thi condition, the buck converter hae voltage, V d, ha a imilar waveform a d, excet it magnitude i V in time high, a hown in Figure 2.5. Whether there i an aliaing effect or not, for the ectra in Figure 2.6, V d ha the ame number of frequency comonent a that of d and there i no additional frequencie generated. The only difference i that the magnitude of V d frequency comonent are V in time a high a thoe of d. Therefore, with the contant inut voltage aumtion, the function of the two witche in the buck converter i to magnify the duty cycle ignal, d, to be the hae voltage, V d, which i a tyical linear function. Meanwhile, the outut filter toology of buck converter doe not change during the witch on-time and off-time, o it i alo a linear function. Thu, all the comonent at V d aear at the outut voltage, V o, through the low-a filter formed by the outut inductor and caacitor. In ummary, the PWM comarator i the only nonlinearity for the oen-loo buck converter with erturbation on the control voltage. When analyzing the mall-ignal tability and the tranient erformance of a converter, it i not neceary to include the comonent at DC, the witching frequency, 7

36 Chater 2. Characteritic of PWM Converter and it harmonic. Only the conequence of the erturbation, i.e. the comonent at the erturbation frequency and the ideband frequencie, need to be included in the model. -f d f -f -f f -f -f f -f +f f +f a f <f /2. -f d f f -2f -f f -f -f +f f -f +2f b f /2<f <f. -f d f -f -f -f f f +f c f =f /2. Figure 2.4. Aliaing effect haen at half of the witching frequency. 8

37 Chater 2. Characteritic of PWM Converter d 0 V in V d 0 Figure 2.5. Inut and outut waveform of the witche in buck converter with contant inut voltage. -f d D f -f -f f -f -f f -f +f f +f -f V d D*V in f -f -f f -f -f f -f +f f +f Figure 2.6. Inut and outut ectra of the witche in buck converter with contant inut voltage. In the conventional average model, the decribing function of the PWM comarator only include the erturbation-frequency comonent, auming the other frequency can be well attenuated by the low-a filter in the converter. It i quetionable whether thi aumtion i till valid for the high-frequency erformance tudy. Therefore, it i eential to undertand the influence from the ideband frequencie on the ytem with multile frequencie. Firt, the henomena at a more general cae of f kf /2, k=, 2, 3,, i oberved and dicued. 9

38 Chater 2. Characteritic of PWM Converter A an examle, a -MHz ingle-hae buck converter i tudied with the etu a hown in Figure 2.. For thi converter, V in i 2 V, V o i.2 V, L i 200 nh, C i mf, R o i 80 mω, the eak-to-eak value of the awtooth ram, V r, i V. The inuoidal erturbation at the control voltage ha the magnitude of 5 mv. With thee arameter, the reone of the outut voltage i monitored with certain erturbation frequencie. Two cae with erturbation frequency at 0 khz and 990 khz are imulated. Vc / mv time/msec 00µSec/div a V c waveform. Vo / V time/msec 00µSec/div b V o waveform. Figure 2.7. Simulated waveform with 0-kHz V c erturbation for a -MHz oen-loo buck converter. With the erturbation frequency, f, at 0 khz, the imulated waveform at V c and V o are illutrated in Figure 2.7. According to Figure 2.6, V d contain comonent at 0 khz and the ideband frequencie of 990 khz,.0 MHz,.99 MHz, etc. Through the outut filter of the converter, thee comonent aear at V o a well. Since f i much lower than 20

39 Chater 2. Characteritic of PWM Converter the witching frequency, f, all of the ideband frequencie are much higher than f. Becaue of the low-a feature of the outut filter, the dominant comonent frequency at V o i f beide the DC and f comonent. In addition, Figure 2.8 how the waveform when f i 990 khz, which i beyond f /2. The ideband frequencie are 0 khz,.0 MHz,.99 MHz, etc. Becaue f -f =0kHz i much lower than f, the outut filter ha more attenuation at f than at f -f. Hence, the 0-kHz ideband frequency comonent i larger than the erturbation-frequency comonent. In the imulated V o waveform, the ideband comonent at 0 khz i the dominant one. Vc / mv time/msec 2µSec/div a V c waveform. Vo / V time/msec 00µSec/div b V o waveform. Figure 2.8. Simulated waveform with 990-kHz V c erturbation for a -MHz oen-loo buck converter. 2

40 Chater 2. Characteritic of PWM Converter In thee imulation cae, ignificant ideband comonent are oberved at V o when the erturbation frequency i aroaching f /2 or higher than f /2. Comared with that of the erturbation frequency, the magnitude of the ideband comonent cannot be ignored. Therefore, the ytem erformance cannot be reflected by only conidering the erturbation-frequency comonent, a hown in Figure 2.9. PWM V in LC Filter v c f df v d f vof Figure 2.9. The frequency-domain rereentation including only the erturbation-frequency comonent. Normally, the cae with the erturbation frequency, f, below the witching frequency, f, i conidered. Under thi condition, the lowet ideband frequency i f -f. Auming the other ideband comonent can be well attenuated by the low-a filter, Figure 2.0 rereent the ytem including the ideband comonent at f -f. For the highfrequency erturbation cae, the ideband comonent hould be addreed ince the lowa filter of the ower tage doe not have good attenuation for them. Figure 2.0. The frequency-domain rereentation for the oen-loo buck converter with ideband comonent. When the erturbation frequency i f /2, there i only the erturbation frequency aearing at the outut, a hown in Figure 2.4. It ha been demontrated [34] that the relative hae, θ, between V c and V r, a hown in Figure 2., influence the magnitude and hae reone. Therefore, the ytem i exreed in the frequency domain a hown in Figure 2.2, where θ i included in the PWM function. 22

41 Chater 2. Characteritic of PWM Converter V c V R θ T π d Figure 2.. Control voltage erturbation waveform at f /2. Figure 2.2. The frequency-domain rereentation for the oen-loo buck converter when f =f /2. In ummary, for the oen-loo buck converter, when the erturbation frequency i aroaching, equal to, or higher than half of the witching frequency, the ytem erformance cannot be rereented by the conventional average model. The ideband comonent and the aliaing effect mut be taken into conideration. The focu of thi diertation i not at half of the witching frequency; therefore, the remaining dicuion addree the ideband comonent only. Unle ecially mentioned, it i aumed that 0<f <f and f f / Sideband Effect of PWM Converter with Feedback Loo For the oen-loo buck converter in Figure 2., although there exit ideband comonent at the outut voltage, there i only one erturbation frequency at the PWM comarator inut. If the ideband comonent can be calculated baed on the erturbation, the ytem erformance i redictable. However, for buck converter with cloed-loo control, a in Figure., the ytem become much more comlicated. 23