Unified Control Strategy Covering CCM and DCM for a Synchronous Buck Converter

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1 Unified Conrol Sraegy Covering CCM and DCM for a Synchronous Buck Converer Dirk Hirschmann, Sebasian Richer, Chrisian Dick, Rik W. De Doncker Insiue for Power Elecronics and Elecrical Drives RWTH Aachen Universiy Jägersr. 17/19, D Aachen, Germany Phone: , Fax: hi@isea.rwh-aachen.de Absrac In general, ransfer funcions of swich-mode converers are reaed differenly when operaing in coninuousconducion mode (CCM) or disconinuous-conducion mode (DCM). This complicaes he developmen of a conrol algorihm for a converer operaing in boh operaing modes. To solve his problem, a conrol algorihm is developed, which is applicable for CCM and DCM. Since he developed conrol algorihm is based on linear equaions insead of differenial equaions, i is easy o implemen and is calculaion effor is minor. I is based on equaing he vol-seconds across he inducor for boh operaing modes and adjusing he duy cycle d. All necessary equaions are derived and he funcionaliy of he developed algorihm is shown by means of simulaion and measuremens. v in S D S 1 D 1 v C i v R e C piezo I. INTRODUCTION The synchronous buck converer is a variaion of he well known and widely used buck converer, where he freewheeling diode is replaced by anoher power MOSFET, which in mos applicaions is always urned on when he op swich is urned off (complemenary swiching). There are hree advanages over he radiional buck converer. A firs, i allows bidirecional power flow. Secondly, in low volage applicaions he efficiency can be increased, because he onsae volage drop of he swich is less han he forward drop of he diode [?]. Finally, he conrol can be simplified. Conrolling he buck converer, one has o disinguish beween coninuous-conducion mode (CCM) and disconinuousconducion mode (DCM). This disincion is necessary because he ransfer funcion beween duy cycle d and he oupu volage or beween inpu volage v in and oupu volage becomes also dependen on load condiion in DCM. In he complemenary swiched synchronous buck converer he curren can reverse wihin he swiching period. Thus, i always operaes in CCM. The main disadvanage of his operaing mode is ha he swiches are always operaed even if no power is ransfered. Apar from he losses in he swiches and he inducor, his causes addiional ripple in he oupu volage. In he given applicaion, he synchronous buck converer is used o drive a piezo acuaor. The corresponding circui diagram is given in Fig. 1, where he effecive resisance R e accouns for various series resisances in he acual circui. The piezo acuaor is used o correc he cuing edge posiion of a high precision drilling ool wih high dynamics wihin Fig. 1. inpu sage low-pass filer Synchronous buck converer wih conneced piezo acuaor he micromeer range [?]. Hence, he piezo acuaor has o be driven over is full operaing range ( 8 V) up o is naural frequency f mech (approximaely 1 khz wih aached mechanics) wih a high degree of accuracy. This accuracy requires a low ripple in he oupu volage, which can be achieved by using a high swiching frequency f s and/or a large inducance. If f s f mech, he mechanical sysem canno follow he ripple of he oupu volage which are generaed by he swiching. Hence, in his case he accuracy is no reduced by he swiching [?]. Neverheless, hese volage ripples generae inernal forces in he piezo acuaor which have o be minimized because hey degrade he piezo acuaor and decrease is lifeime [?]. In addiion, if no resisance is conneced in parallel, he piezo acuaor, which elecrically behaves like a capacior, can keep is volage for a couple of minues. Therefore, if he deflecion of he piezo acuaor is no o be changed, swiching can be avoided. One can easily see ha in he given applicaion he synchronous buck converer should no always operae in CCM and a conrol sraegy which covers CCM and DCM is needed. In order o model he sysem, also he swiches have o be modeled. Already in [?] and[?] circui models were proposed ha are based on sae-space averaging and sricly disinguish beween CCM and DCM. In [?] i is saed ha for DCM he order of he sae-space model is reduced by one. This /7/$. 7 IEEE. 489

2 isrevisedin[?] and[?] and i is shown ha he ransfer funcion of a buck converer in DCM sill conains wo poles, where he second pole moves o higher frequencies when he sysem eners deeper ino DCM. Therefore, if he changes in he ransfer funcion are aken ino consideraion, i is possible o develop one single conrol algorihm for a sysem operaing in CCM as well as in DCM. Since in his applicaion he swiching frequency is much higher han he naural frequency of he low-pass filer ( f s f elec ), sae-space averaging can be applied o he synchronous buck converer [?]. In he following a sae-space model and a conrol algorihm for he synchronous buck converer will be derived. Subsequenly, he exension of his algorihm o DCM and is precondiions are described in deail. Finally, he approximaions will be validaed by means of simulaion and experimenal resuls. II. AVERAGED CONVERTER MODE To develop a sae-space model of he synchronous buck converer he model can be divided in wo ses of sae equaions, one for S being in on-sae or D conducing (v C = v in ) and one for S 1 being in on-sae or D 1 conducing (v C = V) as i was done in [?]. Here he circui was divided in wo pars: he inpu sage and a second order low-pass filer. I is assumed ha he naural frequency of he converer f elec is much smaller han he swiching frequency f s and ha he volage v C across he second order low-pass filer can be averaged over a swiching period. Thus, if he volage v C is averaged, only one se of sae equaions is needed. The basic equaions are given in (1) and (). where and R e A = 1 C x = A x + B u (1) y = C x + D u () [ ] i x =, u =[ v v C ] ou 1,B = C = [ 1 ],D =[]. 1, The ransfer funcion of he second order low-pass filer is: v ω s + σω s + ω (3) where v is he gain, σ is he damping raio and ω is he naural angular frequency. Using sae feedback, σ and ω can be chosen arbirarily. This was done in a way ha he damping raio σ = 1/, which provides a good rade-off beween seling ime and overshoo [?]. ω was increased unil he inpu variable, in his case he inpu volage v in, became he dynamic limiing elemen. To consider he dead ime, which is inroduced by he swiching period respecively sampling period, he sae-space model as well as he ransfer funcion have been discreized. Nex, he feedback gain marix K was compued o conver he sysem given in (1) and () o he sysem of (3). To provide uniy gain v, he seady-sae gain of he feedback loop was compued and is reciprocal value was used as prefiler R. Invered cener-aligned swiching mode was used as swiching paern. So far, sandard conrol heory was applied o a coninuous sysem of which he inpu variable is disconinuous and is calculaed once per swiching period. Here he inpu variable v C was averaged o v C. This can be done using he assumpion ha he dead ime of he inpu variable is very shor compared o reciprocal of he naural frequency of he sysem. In order o explain how he conrol algorihm was exended o DCM, firs he hardware will be described in deail. III. THE HARDWARE The sysem consiss of wo MOSFETs, one inducor, and he piezo acuaor. As menioned above, he ripple in he oupu volage is scaled by he inducance, which also scales he bandwidh of he second order low-pass filer. To provide a high bandwidh while smoohing he oupu volage sufficienly, he inducance was seleced in a manner ha he cu-off frequency of he low-pass filer f elec lies in he geomeric mean beween he naural frequency f mech of he acuaor and he swiching frequency f s. Due o hermal limiaions, he swiching frequency f s could no be increased above 5 khz. The dc-link volage was chosen 1 % above he maximum acuaor volage and a dc-link capaciance of 1 he acuaor capaciance C Piezo was insalled [?]. To conrol he sysem he rapid conrol prooyping sysem XCS was used [?]. This sysem provides up o 3 analog simulaneous sampling AD/DA channels. The PWM is generaed by a FPGA, where he duy cycle d, whichis he on-ime inerval of he upper swich S, is updaed once per swiching period. The AD-conversion is riggered a he beginning of each swiching period ( = ). Each ime he inpu volage v in (), he oupu volage (), andhe inducor curren i () are measured. Since he implemened conrol algorihm is closely relaed o he swiching paern, he used swiching paern will be explained briefly. Two basic swiching paerns can be disinguished: Cener-aligned swiching mode, where he on-sae inerval is aligned o he cener of he swiching period and edge-aligned swiching mode, where he on-sae inerval sars a he beginning of each period. Boh paerns can be invered, like i is depiced for cener-aligned swiching mode in Fig.. This swiching paern, invered cener-aligned swiching mode, was used in he synchronous buck converer. Thus, analyzing he curren wave shapes, he swiching period has o be divided ino hree inervals. The firs inerval where he swich is in on-sae, he second inerval where he 49

3 swich is in off-sae and he hird inerval where he swich is once more in on-sae. i 1 (a) on -4 off (b) on off (c) on 5 off on 3 4 off Fig. 3. Example curren waveforms of differen sysem saes Fig.. Differen swiching modes: (a) edge-aligned swiching mode (b) cener-aligned swiching mode and (c) invered cener aligned swiching mode IV. THE CONTRO AGORITHM The conrol algorihm is sraigh forward and easy o undersand. Firs he sae variables, i.e. inducor curren i and oupu volage, are calculaed. The conroller has o disinguish beween CCM and DCM. Then he required duy cycle d is calculaed. Finally, for DCM, he duy cycle has o be correced. The conrol algorihm can be divided ino five seps: A. Deecing DCM in he Curren Swiching Period B. Calculaing he Inducor Curren i C. Deermining he Desired Inpu Volage v C D. Deecing DCM in he Following Swiching Period E. Calculaing he Duy Cycle d new DCM for DCM Insead of solving he differenial equaions, he linear equaions are used and i is assumed ha he oupu volage does no change significanly during one swiching period. As will be shown in he simulaion resuls, his inroduces small variaions compared o heoreical resuls. However, he calculaion effor is reduced dramaically making his approach reasonable. In buck mode only he op swich S and in boos mode only he boom swich S 1 is operaed. Buck or boos mode can be deeced by comparing he reference volage v ref o he oupu volage.ifv ref >, he boos mode is seleced. When v ref <, he buck mode is seleced. If v ref = no swiching acion akes place. Since buck and boos mode are analogous, he conrol algorihm is discussed for buck mode only. A. Deecing DCM in he Curren Swiching Period The formulas o calculae he inducor curren depend on he sysem sae. There are 5 differen sysem saes ha can be disinguished depending on he measured inducor curren i (). The curren waveforms for cener-aligned swiching mode are depiced in Fig. 3. 1) i () > A and will no become A wihin his swiching period ) i () > A and will become A wihin his swiching period 3) i () < A and will become posiive, before i goes o A 4) i () < A and will become A wihin his swiching period 5) i () < A and will no be disconinuous wihin his swiching period Due o he high dynamics of he sysem, sae 5 can be excluded. Thus, he saes can easily be disinguished beween coninuous (sae 1) and disconinuous (sae -4) saes. If he measured curren i () is negaive, he curren of his period is disconinuous. If he measured curren i () is posiive, i has o be deermined if he sysem sae is coninuous or disconinuous. This is done by evaluaing (4). where i ( off ) (1 d) vou < A, (4) i ( off )=i ()+ d vin. (5) B. Calculaing he Inducor Curren i For he wo operaing modes, CCM and DCM, differen curren values are calculaed. Typically, in seady sae CCM he average inducor curren ī can be easily deermined by measuring he curren value in he cener of he swiching period. Due o he high dynamic operaion of he converer also for CCM his may lead o a high degree of inaccuracy. Hence, in CCM he inducor curren a he end of he swiching period i ( ) is aken as acual sae variable. This should be done because his curren sample conains he laes informaion available before he new duy cycle has o be deermined. The inducor curren i ( ) in CCM is calculaed by: i ( )=i ()+(d v in ) Ts (6) 491

4 This esimaor ries o eliminae he delay beween measuremen and calculaion of he nex duy cycle d. Since his value is prediced over a whole swiching period, i is imporan o exacly know he sysem quaniies. If no esimaor is used, an addiional sysem sae is creaed due o he delay and he order of he sysem increases o hird order. In simulaion his delay can be modeled by a firs-order hold elemen. Since his sae canno be measured, i canno be used for he feedback conrol and he sysem is less conrollable. For DCM, where he curren i ( )=A, he average curren ī of he curren swiching period is calculaed. This average curren ī is calculaed by deermining he ransfered charge of he hree inervals and dividing i by he duraion of he swiching period. Hence, he ransfered charge of he hree inervals is deermined. For he firs inerval his is (i ( off ) has already been calculaed in (5)): q firs inerval = d i()+i ( off ) To calculae he ransfered charge of he second inerval, firs i has o be deermined when he curren i becomes A. Thus, ε is calculaed, where ε is he fracion of he swiching period where he swich is in off-sae and he curren i remains A. For i ( off ) > A (sae & 3) his is done in (8). (7) ε = i ( off ) (8) If i ( off ) < A (sae 4) (8) has o be slighly changed o ε = i ( off ) (v in ). (9) For he saes - 4 he ransfered charge of he second inerval q second inerval can hen be deermined by q second inerval = i ( off ) ε For he hird inerval he ransfered charge is (1) q hird inerval = d i(i ( ), (11) where he curren i ( ) is calculaed by (1). Finally, he average curren i ( )= d vin (1) ī = q firs inerval + q second inerval + q hird inerval. (13) C. Deermining he Desired Inpu Volage v C The calculaed inducor curren ī or i ( ) respecively and he oupu volage ( ) are aken for he acual sae vecor x. For his reason, he oupu volage ( ) has o be calculaed from he measured oupu volage (). Since he oupu volage does no change significanly in one swiching period, his is done in a simplified manner by (14). ( )= ()+ i C piezo (14) Using he feedback gain marix K and he reference oupu volage v ref, he desired inpu volage v C can be deermined easily. Firs, CCM is assumed and he new duy cycle d new of he op swich S is calculaed o: d new = v C (15) V in This value is aken and i is checked if i leads o DCM in he following period. In case of DCM, he duy cycle d new has o be changed. D. Deecing DCM in he Following Swiching Period To deec DCM in he following period (4) can be applied once more, where i ( off ) of he following period has o be calculaed differenly. If in he curren period DCM was deeced, i ( off ) is calculaed by (16). Oherwise, he calculaion of he new i ( off ) becomes more complicaed and is accomplished by evaluaing (17). i ( off )= (d + d new) vin ( i ( off )=i ()+ d + d ) new vin (1 d) vou (16) (17) This predicion is over a long ime scale bu he resul is only binary (DCM or CCM). If DCM is deeced, he duy cycle d new has o be adaped o DCM. E. Calculaing he Duy Cycle d new DCM In DCM for a cerain ime inerval here is no volage applied o he inducor. Therefore, applying he same duy cycle d new, he average inducor volage is no wha i would be in CCM. Thus, he calculaed duy cycle d new will no change he sysem sae as inended. Using he fac ha in DCM he applied average inducor volage is larger han in CCM, he duy cycle d new can easily be correced wihou running he risk of geing back o CCM. To correc he calculaed duy cycle d new, he vol-seconds applied in DCM and CCM are equaed. d new DCM (v in ) ε new! = d new v in (18) 49

5 Insering (8) and (5) ino (18) and reducing he equaion o d new DCM leads o (19). If i ( off ) < A, hen (9) and (5) have o be insered ino (18). This also leads o (19). Therefore, his formula is valid for all sysem saes. This simple formula is needed o conver CCM quaniies ino DCM quaniies and makes sandard conrol heory applicable o a sysem operaing in CCM as well as in DCM. d new DCM = dnew v in + i ( ) (19) v in For boos mode (19) changes o (), where he duy cycle of he boom swich S 1 can be easily deermined by 1 d new DCM. d new DCM = dnew v in + i ( ) () Before implemening he given equaions hey should be simplified. To keep hem comprehensible in his work hey are wrien in heir original form. V. SIMUATION AND EXPERIMENTA RESUTS To es he conrol algorihm, a sep response from V o 7 V was simulaed. Here he developed conrol algorihm was compared o he same conrol using complemenary swiching. Hence, for he complemenary swiching conrol he sysem is always in CCM and all addiional calculaion effor o deec DCM is no needed. The wo volage and curren curves are given in Fig. 4. While he sysem is in CCM he developed conrol algorihm behaves exacly like he complemenary swiching conrol, as expeced. According o conrol heory for he seleced damping raio here should be a volage overshoo of approximaely 4 % [?], which would be 8 V for a 7 V sep response. Here he volage overshoo is less han 1 %. The reason for his difference is he linearizaion of he sysem equaions. In CCM he sysem saes a = are calculaed. Based on his informaion he new duy cycle d new is deermined. Therefore, here is no delay bu, as already menioned, linear equaions insead of he differenial equaions are used o calculae he sysem saes. This small sysemaic error causes he sysem behaviour o differ from heory. Since he differences are small and he reducion of calculaion ime is high, his is accepable. As he difference in volage beween and v ref decreases, he inducor curren i is reduced and he sysem eners DCM. As a resul, he second pole moves o higher frequencies and he sysem is addiionally damped. While he developed conrol algorihm reaches v ref, he complemenary swiching algorihm shows a consan volage ripple of 5V p p around he reference volage. This shows he advanages of he developed conrol algorihm. For large volage errors he sysem operaes wih a high degree of dynamics in CCM. When he volage error is reduced, i auomaically changes o DCM where ringing is eliminaed. Thus, swiching and conducion losses are reduced in areas where here is only lile power ransfered. As a resul, he overall efficiency is increased. Volage in V v complemenary swiching ou CCM and DCM conrol v ref Time in s x 1 4 Curren in A i complemenary swiching i CCM and DCM conrol Time in s x 1 4 Fig. 4. v ref Fig. 5. Comparison of differen swiching modes i Measuremen of sep response The conrol algorihm was finally verified by experimenal resuls. Fig. 5 shows he experimenal resuls for he same load condiions as he simulaion. Here CH1 is he reference volage v ref, CH is he oupu volage and CH3 is he inducor curren i. One can see ha he simulaion resuls mach he measuremens fairly well. There are only small differences due o disurbances and he limied bandwidh of he curren measuremen. Thus, he overshoo is increased minimally. Rise ime is approximaely he same compared o he simulaion. I should be menioned ha generaing V a he oupu of a synchronous buck converer is no a sable operaing area. Already small disurbances may lead o oscillaions. Therefore, his sep response should no be repeaed in a harsh environmen. VI. CONCUSIONS In his work a new conrol algorihm was inroduced ha can be applied for CCM and DCM of a synchronous buck 493

6 converer. I is easy o implemen and is based on sandard conrol heory. Hence, already available sofware ools can be uilized o develop he conrol parameers and o guaranee sabiliy. In he developed conrol algorihm he duy cycle d, which is calculaed for CCM, is changed for DCM o esablish equal vol-seconds across he inducor. The main advanages are he reduced swiching losses. Especially, when no power has o be ransfered, swiching can be avoided compleely. As a resul, he volage waveform is smoohed, which also improves he overall accuracy. Especially for piezo applicaions his is of paramoun imporance. 494