The Multi-Frequency Small-Signal Model for Buck and Multiphase Interleaving Buck Converters

Transcription

1 The Multi-Frequeny Small-Signal Model for Bu and Multihae Interleaing Bu Conerter Yang Qiu, Ming Xu, Kaiwei Yao, Juanjuan Sun, and Fred C Lee Center for Power Eletroni Sytem irginia Polytehni Intitute and State Unierity Blaburg, A 2461, USA Abtrat Thi aer introdue a multi-frequeny mall-ignal model for bu and multihae interleaing bu onerter Inluding the influene from the ideband-frequeny omonent generated by the ule-width modulation (PWM, thi model i aliable beyond half of the withing frequeny In oltage-mode-ontrolled bu onerter, the rooed model redit the meaured hae delay, while the onentional aerage model fail to exlain thi henomenon With thi new model, frequeny-domain harateriti are learly exlained for bu and multihae bu onerter Furthermore, the aliaing effet at half of the withing frequeny i examined Simulation and exerimental reult are reented to erify the rooed multi-frequeny model * Keyword mall-ignal model; multi-frequeny model; multihae bu I INTRODUCTION Future miroroeor will oerate at multi-ghz lo frequenie, and will onit of oer a billion integrated tranitor Conequently, more than 15-A urrent at the 8- oltage leel will be required [1] The ower-hungry roeor oe many tringent hallenge on it ower uly, the oltage regulator (R One eial iue i how to meet the requirement of fat tranient reone with fewer outut aaitor Beide the reaon related to ot, thi neeity alo exit beaue of the limited ae for R in the omuter ytem The multihae interleaing ynhronou bu onerter, a hown in Fig 1, i widely adoted in the R aliation [2~3] Many aer hae diued the tranient reone of multihae R and how to imroe it [4~9] It ha been hown that the feedba ontrol loo bandwidth lay a ery imortant role in the tranient reone With a higher bandwidth, fewer outut aaitor are needed for the required tranient erformane [7~9] Thu, it i adantageou to uh the bandwidth a high a oible with a limited withing frequeny * Thi wor wa uorted by Arteyn, Delta Eletroni, Hiro Eletroni, Infineon, Intel, International Retifier, Interil, Linear Tehnology, National Semiondutor, Renea, and Texa Intrument Thi wor alo made ue of Engineering Reearh Center Shared Failitie uorted by the National Siene Foundation under NSF Award Number EEC Any oinion, finding and onluion or reommendation exreed in thi material are thoe of the author( and do not neearily reflet thoe of the National Siene Foundation In the at, mot of the feedba ontroller deign hae been baed on the aerage model for bu onerter Howeer, beaue the tate-ae aeraging roe eliminate the inherent amling nature of the withing onerter, the auray of the aerage model i quetionable at frequenie aroahing half of the withing frequeny [1] A the reult, the relationhi between the ontrol-loo bandwidth and the withing frequeny ha not been learly undertood To redit the ub-harmoni oillation at half of the withing frequeny in ea-urrent-ontrolled onerter, amled-data aroahe, hybrid aroahe, and harmonibalane aroahe hae been rooed [1~14] For the oltage-mode ontrol, it i found that the aerage model hould alo be reexamined if a high bandwidth i deired A an examle, for a 1-MHz ingle-hae bu with oltage-mode ontrol, Fig 2 omare the loo gain alulated from the aerage model with that obtained by uing SIMPLIS oftware SIMPLIS erform ery fat mall-ignal AC analyi baed on a imilar heme a what ha been ued in the meaurement, exet that the withing rile are not onidered Hene the imulated tranfer funtion from SIMPLIS i almot the ame a that from meaurement In Fig 2, for the ae with a 1-Hz bandwidth of the oltage feedba ontrol loo, the aerage model agree with the imulation u to half of the withing frequeny Howeer, for a 4-Hz bandwidth deign, the aerage model i good only u to 1 Hz, ie, one-tenth of the withing frequeny Comared with the aerage model, the imulation reult ha a 3 o more hae delay at the rooer frequeny Thi exeie hae dro would reult in undeired tranient or tability roblem if a high-bandwidth onerter i deigned baed on the aerage model, whih annot redit the highfrequeny behaior I 1 I n L 1 L n Figure 1 An n-hae interleaing bu onerter I L /5/$2 25 IEEE 392

2 Gain (db Phae (degree Gain (db Phae (degree f =1Hz f =1MHz Frequeny (Hz (a o f =1MHz f =4Hz Frequeny (Hz (b Figure 2 Loo gain for a 1-MHz bu with oltage-mode ontrol: (a 1-Hz bandwidth, and (b 4-Hz bandwidth (olid line: SIMPLIS imulation reult; dotted line: aerage-model reult Although the harmoni-balane tehnique redit the hae delay at high frequenie [13~15], it i not traightforward to extrat hyial meaning out of thi omliated model In order to imlify the modeling, a well a to inetigate the ontrol-loo bandwidth limitation and to imroe the ontrol deign, it i eential to hae a lear iture of the onerter harateriti u to the withing frequeny Therefore, thi aer introdue the multi-frequeny model, whih i alid both below and aboe half of the withing frequeny Setion II reiew the tranfer funtion meaurement etu and the limitation of the aerage model After that, the onet of multi-frequeny modeling i deeloed in Setion III baed on a ingle-hae oltage-mode-ontrolled bu Fourier analye for the PWM omarator [16] are extended to inetigate the relationhi among different frequeny omonent In Setion I, thi model i alied to the multihae interleaing bu onerter A a ulementary to the rooed model, the aliaing effet at half of the withing frequeny i exlored in Setion Simulation and exerimental reult are roided to erify the analye II LIMITATIONS OF THE AERAGE SMALL-SIGNAL MODEL Fig 3 demontrate the hardware etu for meauring the tranfer funtion of a ingle-hae bu onerter uing oltage-mode ontrol With an inerted inuoidal oltage oure at the wet erturbation frequeny, f, Fourier analyi i onduted at meaured waeform After that, the tranfer funtion are alulated baed on the Fourier analyi reult Fig 4 how the inut and outut etra of the PWM omarator It i undertood that the PWM omarator i a non-linear funtion With a erturbation at frequeny f inerted at, and the withing frequeny, f, the outut of the PWM omarator, d, ha infinite frequeny omonent at f, -f, f -f, f f, -f f, -f -f, et Thee frequenie other than ±f are the ideband around f, -f, et If the oltage loo i loed, the generated ideband frequeny omonent are fed ba to The fed-ba omonent generate all thee frequeny omonent again Therefore, the ideband effet haen and the frequeny omonent are ouled Howeer, the traditional aerage model only inlude the f omonent If the other frequeny omonent an be ignored, the aerage model might be good enough Otherwie, the ideband frequenie hould be onidered in the model Beaue the feedba ontrol loo (inluding the ower tage and omenator funtion a a low-a filter, the highfrequeny omonent are attenuated Conidering f lower than the withing frequeny, the two mot dominant frequenie are f and f -f If f i ery low and f -f i muh higher than the bandwidth, the omonent at f -f an be well attenuated by the feedba loo Fig 5 how the imulated oltage waeform at for a 1-MHz ingle-hae bu with 1-m erturbation at 1 Hz The dominant omonent i the erturbation frequeny Figure 3 Tranfer funtion meaurement etu with oltage-mode ontrol (a (b Figure 4 Samling reult of PWM heme: (a inut etrum of the PWM omarator, and (b outut etrum of the PWM omarator 393

3 (m Time (m Figure 5 Simulated waeform with 1-Hz, 1-m erturbation for a 1-MHz ingle-hae bu onerter with oltage-mode ontrol (m (m Time (m (a Time (m (b Figure 6 Simulation with 99-Hz, 1-m erturbation for a 1-MHz inglehae bu onerter with oltage-mode ontrol: (a waeform at, and (b loe-u waeform at Howeer, when f i high, f -f beome relatiely low Thu, the low-a filter of the feedba loo annot gie enough attenuation to the omonent at f -f Fig 6 illutrate the imulated waeform at for the 1-MHz bu with a 1- m, 99-Hz erturbation Both the 99-Hz omonent and a 1-Hz omonent are dominant The 1-Hz omonent i generated from the PWM omarator and fed ba through both the ower tage and the omenator to Thi 1-Hz omonent generate 99-Hz omonent at d and o again A the reult, thee two frequeny omonent are ouled with eah other Thi i why the aerage model, in whih only the erturbation frequeny i onidered, i inalid at the highfrequeny region III THE MULTI-FREQUENCY SMALL-SIGNAL MODEL Fig 7 illutrate the model at the erturbation frequeny, f, for a ingle-hae oltage-mode-ontrolled bu onerter To analyze the mall-ignal reone in the entire frequeny region, all the ideband frequeny omonent are inluded Therefore, it i named the multi-frequeny model If f i lower than f, only one other omonent at f -f i imortant beaue the onerter funtion a a low-a filter Uing Fourier analyi, the PWM tranfer funtion ha been deried onidering the ame inut and outut frequeny [16] It i a ure gain a G PWM = =, (1 if n /2 Here, in i the inut oltage, and r i the eato-ea alue of the PWM ram In order to determine the relationhi between thee two frequeny omonent, a imilar aroah a in [16] i alied but with different inut and outut frequeny omonent A an examle, oltagemode ontrol with trailing-edge modulation i analyzed Howeer, the ame aroah i aliable to aerage-urrentmodel ontrol if the rile of the feedba ignal i mall enough It an alo be extended to analyze leading-edge and double-edge modulation The inut and outut waeform of the trailing-edge PWM omarator are hown in Fig 8 The ontrol oltage i ˆ = in t θ, (2 and the duty ratio for the -th yle i D T ˆ in [( 1 T DT ( D D T ] θ = = D, T (3 where T i the on-time of the -th yle, T i the withing eriod, and D = (4 Alying the mall-ignal aroximation, it i obtained that ˆ D = D in( ( 1 T φ θ, (5 where φ = DT (6 With the definition, the Fourier oeffiient for the eriodial ignal d i exreed a, T 1 jt = e dt (7 T (f =± 1, ± 2, A (f f 1/ r in (f H d(f o (f Figure 7 The multi-frequeny model at f with influene of all the omonent from the PWM heme r d T } } T Figure 8 Inut and the outut of the PWM omarator _ 394

4 In thi aer, it i aumed that N =, (8 M where N and M are oitie integer If the relationhi between and annot be exreed by (8, double integral are required for the deriation Howeer, the reult i the ame o thi aet will not be diued here for imliity Then, 2π ( N M 1 j( t d = e d[( t] 2π ( N M (9 Beaue d =, ( 1 T < t < ( 1 T T, (1 and d =, ( 1 T T < t < T, (11 it an be deried that = 2π ( N M ( 1( T T M j( t = 1 e ( 1( T d[( t] (12 If n /2, (12 i rewritten by alying the mall-ignal aroximation a jθ jd2π e e ˆ = (13 2 j The Fourier oeffiient for the ontrol ignal i jθ e ˆ = ; (14 2 j therefore, jd2 = π (15 With the ame aroah, it i obtained that at n /2, jd2π = (16 With (15 and (16, Fig 9 how the multi-frequeny model for a ingle-hae oltage-mode-ontrolled bu There are two feedba loo in the model, and eah loo rereent a ertain frequeny, but i influened by the other one Thi i the rooed multi-frequeny model, and it i alid at f <f and f f /2 The additional ignal added to d rereent the influene exreed in (15 and (16 Uing the deried model, it an be alulated that the loo gain at f i H( GLC( / T =, (17 1 H( GLC( / where i the tranfer funtion of the ower tage LC filter The numerator i atually the loo gain in the aerage model The denominator inlude the aerage-model loo gain at f -f, whih reflet the ideband effet, ie the influene of the other frequeny ( ( d( o( in / r ( in e -jd2π / r in e jd2π / r d(- o(- (- in / r (- -H (- -H ( Figure 9 The multi-frequeny model for a ingle-hae oltage-mode-ontrolled bu Outer loo: tranfer funtion at f frequeny; Inner loo: tranfer funtion at f -f frequeny U to the withing frequeny, Fig 1 omare the loo gain in the aerage model, in the SIMPLIS imulation and in the multi-frequeny model Fig 11 how the meaurement reult When f i loated in the low-frequeny region, f -f i higher than the oltage-loo bandwidth The denominator of (17 i aroximately equal to one Therefore, the aerage model i aurate When f beome high and i aroahing the withing frequeny, f -f goe into the oltage-loo bandwidth Under thi ondition, the denominator of (17 i aroximately the loo gain at f -f, whih i muh higher than one Thi high loo gain at f -f reult in a di around the withing frequeny in the loo gain of T Although the aerage model fail to redit thi di, it doe exit in both the exeriment and the withing model imulation For the ame reaon, with a low-bandwidth deign, the ideband effet ha inignifiant influene at the rooer frequeny, beaue it i muh le than f /2 Howeer, when the bandwidth i high, an exeie hae delay i exeted at the rooer frequeny a the reult of ideband effet A tated in (17, the higher the ontrol bandwidth, the larger imat on the hae margin from the ideband effet Therefore, the ideband effet limit the oibility of high-bandwidth deign To hel the deign at thee ae, it i neeary to ue the rooed multi-frequeny model intead of the onentional aerage model Gain (db Phae (degree f =1MHz Frequeny (Hz Figure 1 Loo gain for the ingle-hae bu with oltage-mode ontrol (dotted line: the aerage model; olid line: SIMPLIS imulation; dahed line: the multi-frequeny model 395

5 6 (f-f Gain (db (f (f f(f in / r in / r e -jd2π in / r d1(f e -j2π/n e -jd2π in / r d2(f o(f Phae (degree Frequeny (Hz Figure 11 Meaured loo gain for the ingle-hae bu with oltage-mode ontrol I THE MULTI-FREQUENCY MODEL FOR MULTIPHASE INTERLEAING BUCK CONERTERS To aly thi multi-frequeny model to the multihae interleaing bu onerter, Fig 9 need to be modified Not only do more aralleled hae aear in the diagram, but alo the relationhi between the frequeny omonent are hanged For the m-th hae in an n-hae interleaing bu, dm =, ( 1 T ( m 1 T / n< t < ( 1 T ( m 1 T T, m (18 and dm =, ( 1 T ( m 1 T Tm < t < T ( m 1 T (19 Through a roe imilar to that ued in the ingle-hae ae, (1 i till orret for the ame inut and outut frequeny For different inut and outut frequenie, it an be deried that if n /2, then dm Similarly, dm = = in r in r jd2π jd2π j( m 12π / n j( m 12π / n (2 (21 Baed on (1, (2 and (21, the mall-ignal model for an n-hae interleaing bu at f i illutrated in Fig 12 If r and the tranfer funtion are the ame for all the hae, the influene from the f -f omonent on f i anelled at o Therefore, the di that ha been obered in Fig 1 i no longer exeted to our around the withing frequeny In Fig 13, SIMPLIS imulation i ued to erify the multifrequeny model analyi baed on a two-hae interleaing bu with oltage-mode ontrol Comared with that of the ingle-hae bu, the lowet-frequeny di i around twie the withing frequeny Thi i beaue the f -2f omonent a the ideband of 2f annot be anelled e -j(n-12π/n e -jd2π in / r in / r H dn(f Figure 12 Multi-frequeny model at f for an n-hae interleaing bu Gain (db Phae (degree Frequeny (Hz f =1MHz Figure 13 Loo gain for a two-hae interleaing bu with oltage ontrol (olid line: SIMPLIS imulation reult; dotted line: aerage-model reult Generally, for an n-hae bu, the firt di aear around n-time the withing frequeny beaue of the ideband frequeny at f -nf For examle, the loo gain for a four-hae interleaing bu hown in Fig 14 ha the di around four time the withing frequeny Thi mean that theoretially, for oltage-mode-ontrolled bu onerter, the n-hae interleaing tehnique i aable of inreaing the bandwidth to n-time that of the ingle-hae bu Howeer, in the loo-gain meaurement reult hown in Fig 15 for a two-hae interleaing bu, a mall di exit around the withing frequeny although high bandwidth i ahieed with uffiient hae margin Thi i beaue in the imlementation, the aymmetry of the two hae reult in only a artial anellation of influene from the f -f omonent Aording to Fig 12, if the two hae r or are different, or een if the two hannel do not hae a hae hift of exatly 18 o, the influene from the f -f frequeny omonent annot be erfetly aneled Therefore, the bandwidth of oltage-mode-ontrolled multihae interleaing bu onerter i limited in the reality, although it an be inreaed theoretially There are ri to uh the oltage bandwidth higher when uing aymmetri hae Thi i why ome deigner laimed they had ahieed bandwidth higher than half the withing frequeny, while other laimed they ould not _ 396

6 Gain (db Phae (degree f =1MHz Frequeny (Hz Figure 14 Loo gain for a four-hae interleaing bu with oltage ontrol (olid line: SIMPLIS imulation reult; dotted line: aerage-model reult Gain (db Phae (degree f =1MHz Frequeny (Hz Figure 15 Meaured loo gain for a two-hae interleaing bu with oltage-mode ontrol ALIASING EFFECT AT HALF OF THE SWITCHING FREQUENCY A indiated by the analye in the etion aboe, it i oible to ahiee high bandwidth with multihae tehnique Howeer, in the reiou deriation of the multi-frequeny model, there i an aumtion of f f /2 Thu, it i neeary to exlore the erformane at thi frequeny At half of the withing frequeny, the ideband frequeny oerla with the erturbation frequeny, namely f i equal to f -f Therefore, inide the loed-loo ytem, only one frequeny omonent hould be onidered uoing the omonent higher than the withing frequeny ould be ignored With the ame method of Fourier analyi ued reiouly, the relationhi between inut and outut ignal of the PWM heme for a ingle-hae bu i deried a j2( θ Dπ / 2 = (1 e / 2 (22 In thi equation, the PWM gain i related to the erturbation hae θ Fig 16 demontrate the funtion of d /( * in / r in (22 when θ hange At ertain θ, the PWM ha no gain on the erturbation frequeny omonent While with other hae, the outut ha higher magnitude To erify the reult, a imulation with a 1-MHz inglehae bu i erformed and hown in Fig 17 A erturbation i added at the ontrol oint with frequeny of 5-Hz, ie, f /2 The outut oltage, o, i meaured in the frequeny domain With the relatie hae of 9 o, the outut ha the larget magnitude of the omonent at f /2 With 6 o hae, it beome maller and there i no f /2 omonent at o Beaue of thi aliaing effet, the loo gain ha no attenuation for the erturbation with ertain hae in the inglehae bu A aforementioned, the multihae tehnique anel the influene ome from the ideband frequenie, o it an imroe the ontrol-bandwidth Hene, it i imortant to analyze the aliaing effet for multihae bu ae to exlore whether there i any anellation a well Different from the ingle-hae bu, the PWM gain for multihae bu i related to whih hae it i in For the m-th hae in an n-hae interleaing bu, j( 12π / N j2( θ Dπ m / 2 = (1 e e / 2 (23 The um of eah hae d i alulated to tudy the aliaing effet for multihae bu, beaue the outut oltage inlude eery hae influene and ymmetrial hae hae the ame outut filter tranfer funtion The effetie d i n n _ eff / 2 = m( / 2 = / 2 (24 m= 1 Therefore, the total effet i rereented by a ontant gain, whih mean that the aliaing effet i anelled at f /2 for multihae bu The imulation reult with a 1-MHz two-hae bu i hown in Fig 18 Unlie the ingle-hae ae, the outut oltage ha the ame etrum for erturbation with different hae, whih mean the aliaing effet i aneled at f /2 Howeer, imilar to the ideband effet, the anellation i deended on the ymmetry among hae A the reult, when the bandwidth i higher than half of the withing frequeny, it i neeary to onider the arameter tolerane and the aliaing effet that annot be omletely anelled Gain of F(θ (db Phae of F(θ (degree θ-dπ Figure 16 Tranfer gain of d /( * in / r a a funtion of the erturbation relatie hae 397

7 bandwidth an be uhed higher than half the withing frequeny Howeer, the aymmetry among hae reult in deign ri to uh the ontrol-loo bandwidth in imlementation ACKNOWLEDGMENT Thi wor wa onduted with the ue of SIMPLIS oftware, donated in ind by Tranim Tehnology of the CPES Indutrial Conortium Figure 17 Single-hae bu imulated outut oltage etra with different erturbation relatie hae o θ-dπ= θ-dπ=3 o θ-dπ=9 o f /2=5Hz f =1MHz Figure 18 Two-hae interleaing bu imulated outut oltage etra with different erturbation relatie hae I CONCLUSION The onentional aerage model fail to redit the hae delay een inide the ontrol-loo bandwidth for highbandwidth deign in oltage-mode-ontrolled bu The influene from the ideband frequeny omonent, whih are generated by the PWM omarator, hould be onidered To redit the ytem behaior and imroe the ontrol deign, thi aer introdue the multi-frequeny model, whih i alid aboe half the withing frequeny Model are deeloed for both the ingle-hae bu and the multihae interleaing bu A a ulementary, the aliaing effet at half of the withing frequeny i alo diued Simulation and exerimental reult hae erified the rooed model and orreonding analye For oltage-mode ontrol, the multihae interleaing tehnique anel the ideband effet and the aliaing effet Therefore, theoretially the ontrol-loo REFERENCES [1] E Stanford, Intel Tehnology R Roadma, Intel Tehnology Symoium, 21 [2] X Zhou, P L Wong, P Xu, F C Lee and A Q Huang, Inetigation of Candidate RM Toologie for Future Miroroeor, IEEE Tran Power Eletron, No 2 [3] Y Pano and M M Joanoi, Deign Conideration for 12-/15-, 5-A oltage Regulator Module, IEEE Tran Power Eletron, Jan 22 [4] L D arga and N A Loi, Synthei of Zero-Imedane Conerter, IEEE Tran Power Eletron, Jan 1992 [5] R Redl, B P Eriman and Z Zany, Otimizing the Load Tranient Reone of the Bu Conerter, IEEE APEC, 1998 [6] S A Chiamenahalli, S Mahadean, E Stanford and K Merley, Effet of Target Imedane and Control Loo Deign on RM Stability, IEEE APEC 22 [7] P L Wong, F C Lee, P Xu and K Yao, Critial Indutane in oltage Regulator Module, IEEE Tran Power Eletron, July 22 [8] K Yao, M Xu, Y Meng and F C Lee, Deign Conideration for RM Tranient Reone Baed on the Outut Imedane, IEEE Tran Power Eletron, No 23 [9] P L Wong, Performane Imroement of Multi-Channel Interleaing oltage Regulator Module with Integrated Couling Indutor, Ph D Diertation, irginia Teh, 21 [1] A R Brwon and R D Middlebroo, Samled-Sata Modeling of Swithing Regulator, IEEE PESC, 1981 [11] G C erghee, C A Bruzo and K N Mahabir, Aeraged and Samled-Data Model for Current Mode Control: A Re-Examination, IEEE PESC, 1989 [12] R B Ridley, A New, Continuou-Time Model for Current-Mode Control, IEEE Tran Power Eletron, Aril 1991 [13] J Groe, Small-Signal Analyi Uing Harmoni Balane Method, IEEE PESC, 1991 [14] R Tymeri, Aliation of the Time arying Tranfer Funtion for Exat Small-Signal Analyi, IEEE Tran Power Eletron, Marh 1994 [15] R Tymeri, orerian, F C Lee and W T Baumann, Nonlinear Modeling of the PWM Swith, IEEE Tran Power Eletron, Aril 1989 [16] R D Middlebroo, Prediting Modulator Phae Lag in PWM Conerter Feedba Loo, POWERCON,