Comparative Analysis of the Large and Small Signal Responses of "AC inductor" and "DC inductor" Based Chargers

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1 Comparaive Analysis of he arge and Small Signal Responses of "AC inducor" and "DC inducor" Based Chargers Ilya Zelser, Suden Member, IEEE and Sam Ben-Yaakov, Member, IEEE Absrac Two approaches of operaing inducors in swiched mode power converers are compared. The classic mehod, uilized in mos PWM converers, is o place he inducor in a pah ha has a non-zero average curren. Such inducors are referred o here as "DC inducors". In conras, inducors ha operae wih zero average curren are denoed "AC inducors". The heoreical analysis carried in his paper compares he small signal response of he curren in he DC and AC inducors as a funcion of various sysem parameers, such as inpu and pu volages and duy cycle, in he case of he DC inducor, and frequency in he case of he AC inducor. The analysis was suppored by large and small signal model developed in his sudy and verified agains analyically derived large and small signal equaions. The small signal curren response was derived by applying average models. The analyical derivaions and simulaions of his sudy were verified experimenally. Good agreemen was found beween he heoreical predicaions and he analyical resuls. Based on he resuls of his sudy one can conclude ha "AC inducor" based opologies are opimal for curren sourcing applicaions, such as baery chargers, where differences beween he average inpu and pu volages are expeced o exis and develop over ime. Index Terms average model, DC-DC conversion, baery chargers, small-signal response. I. INTRODUCTION The classic mehod, uilized in mos PWM chargers, is o place an inducor in a branch ha carries a non-zero average curren []. Such inducors are referred o here as "DC inducors". In conras, inducors ha operae wih zero average curren are denoed "AC inducors". This laer operaional mehod is similar o resonan load converers [, 3] bu in he opologies sudied in his paper, a resonan capacior is no applied [4] and he curren waveforms are non sinusoidal [3-5]. The reason for he ineres in AC inducor opologies is he fac ha hey achieve sof swiching [3, 6-8]. The objecive of his sudy was o compare he large and small signal responses of he currens in he DC and AC inducors and he pu volages, as a funcion of various sysem parameers, such as inpu volage, pu volage and duy cycle perurbaions in he case of he DC inducor, and as a funcion of frequency deviaions in he case of he AC inducor. II. ARGE SIGNA ANAYSIS OF AC&DC INDUCTOR To describe he behavior and delineae he differences beween DC and AC inducor charger opologies, we firs consider wo DC-DC baery chargers shown in Fig.. In boh circuis, an pu DC source approximaes he baery and i is assumed ha he inpu volages are larger han he baery volage. R represens he parasiic resisances of he inducor's branch. DC inducor. The volages applied o he main inducor of Fig. a are unipolar square wave signal D on he inducor's lef erminal and consan DC volage on is righ erminal (Fig. ). Neglecing he parasiic resisances of he inducor (R ) as well as a volage drop of he diode D he duy cycle of he main swich is deermined as: D = () in The average value of he inducor's curren and consequenly he average power ransferred canno be conrolled in his case solely by he duy cycle. Tha is, in order o regulae he power ransferred o he load, an exra curren conrol loop is mandaory. Wih his exra curren loop here is a risk of curren runaway, as a resul of a shif in he inpu or pu averages. AC inducor. I is assumed ha he exemplary fullbridge inverer of Fig. b operaes wih 50% duy cycle and hence generaes a symmerical bipolar square wave signal a is pu. (a) The Auhors are wih he Power Elecronics aboraory, Deparmen of Elecrical and Compuer Engineering, Ben-Gurion Universiy of he Negev, Beer-Sheva, 8405, Israel ( sby@ee.bgu.ac.il). This research was suppored by THE ISRAE SCIENCE FOUNDATION (gran No. 3/0) and by he Paul Ivanier Cener for Roboics and Producion managemen. (b) Fig.. Baery chargers. (a) Buck wih DC inducor. (b) Full bridge wih AC inducor.

2 Fig.. Key waveforms of an ideal Buck converer. In his case, he volages applied o boh erminals of he inducor in Fig. b are bipolar and he inducor's curren flows in boh direcions (Fig. 3). Due o he pu diode bridge (Fig. b), he polariy of he volage seen by he righ erminal ( r ) of he inducor reverses every zero crossing of he inducor's curren. During ime inerval - (Fig. 3) he volage a he inpu side of he inducor ( l ) is posiive ( ), he inducor's curren is posiive and diodes D and D 3 conduc. Consequenly, ne volage seen by he inducor is posiive and equal o ( - ). The inducor's curren is posiive and increases. A he polariy of he inpu volage swiches. The curren hrough he inducor keeps flowing in he same direcion so diodes D, D 3 keep conducing. The volage applied o he inducor will now be negaive and equal o (- - ) and he inducor's curren will sar decreasing. A ime insan 3 he inducor's curren crosses zero and is polariy reverses. As a resul, diodes D and D 4 urn on, he volage seen by he righ erminal of he inducor ( r ) is and he ne volage applied o he inducor is (- + ). Tha is, he magniude of he volage applied o he inducor during he ime inerval 3-4 is equal o ha during - bu opposie in polariy. Similarly, he inducor's volage during 4-5 is equal in magniude bu opposie in polariy o ha during - 3. Consequenly, he curren hrough he inducor reverses is polariy every inpu volage swiching cycle. In he ideal case, he volages seen by each inducor s erminal do no conain DC componens. In pracical AC inducor circuis, due o non-symmery of he inverer s pu volage, a DC componen may appear a he lef erminal of he AC inducor (Fig. b). This error componen can be removed by a series capacior, which of course canno be applied in he DC inducor case. During he ime inerval - (Fig. 3) he peak inducor curren Ipk is derived o be: Ipk = ( ) () where is he inducance of he inducor. Similarly, during he ime inerval - 3 Ipk is expressed as: + Ipk = ( 3 ) (3) From () and (3), aking ino accoun ha ( 3 - ) is half a swiching cycle, yields: ( ) + ( ) = I = (4) 3 pk F where F is he swiching frequency. Rearranging (4) back for Ipk we ge: Ipk = (5) 4F The pu curren is he recified inducor's curren as depiced in (Fig. 4). Since he pu curren is of riangular shape, is peak value is wice is average value and hence: I = Ipk = (6) 8F where I is he pu curren averaged over he swiching cycle. The average power delivered o he pu is hus: P = I (7) Subsiuing (7) ino (6) yields: = (8) 8F P III. SMA SIGNA ANAYSIS DC inducor. The average model approach described in [9] is used o analyze he small signal behavior of he sysem of Fig. a in open loop. Tha is, wih a fixed duy cycle D. The average model of his sysem is shown in Fig. 5. I denoes an inducor's curren averaged over he swiching cycle. Following he procedure described in [9], he small signal curren response o a small signal injecion in eiher inpu or pu volages, under consan duy cycle condiion is found o be: + Fig. 3. Basic waveforms of he full bridge AC inducor converer. Fig. 4. Oupu curren of an "AC inducor" converer.

3 Fig. 5. Average model of a buck charger. i K = (9) v s + R where i is he small signal average inducor curren; v is a disurbance in eiher inpu or pu volage and K is equal o D in case of he inpu volage or uniy in case of he pu volage. According o (9), he slower he disurbance in he volage he higher he raio i v is. I implies ha, for a consan D, even a small difference beween he average volages of he inducor's erminals may develop a very high curren which will be limied only by parasiic resisances in he inducor's curren pah. As menioned above, he power ransferred o he load is deermined in case of DC inducor by conrolling he average inducor's curren (pu curren). The inducor's curren can be conrolled by designing a curren feedback loop which will mainain he desired average curren by modulaing he duy cycle D. The design of a compensaing nework for his conrol loop requires knowledge of he small signal response of he inducor's curren o he modulaed (small signal changes) duy cycle D. This response was obained using he average model of Fig. 5 afer linearizaion, assuming ha inpu and pu volages ( in and respecively) are consan (Fig. 6). The small signal curren response o he modulaion in he duy cycle is found o be: i in = (0) d s + R where d is a small signal modulaion of duy cycle D and i is a small signal change of he inducors curren in response o his modulaion. Equaion (0) implies ha a small low frequency disurbance in he duy cycle D may invoke a large change in he inducor's curren, which will be limied mainly by he parasiic resisance of he inducor. AC inducor. Since he average curren hrough he main inducor is zero we will focus on he average value of he pu curren which is he recified inducor's curren. During he ime period - (Fig. 4) he volage applied o he inducor is ( - ) whereas during - 3 i is ( + ). Having in mind ha ( - )+( 3 - )=/F he average volage of he inducor will be given as: ( ) ( ) F ( + ) ( ) F 3 = () Equaion () can be represened by he schemaic diagram shown in Fig Equaing () and (3), we will find he raio beween he wo ime periods: 3 = () + Expressing ( 3 - ) as ( ) and subsiuing o F () he ime period ( - ) is found as follows: + ( ) = (3) 4 F Considering (3), he expression for he lef volage source of Fig. 7 is reduced o. Tha is, he magniude of he lef volage source is deermined by inpu and pu volages. The righ volage source can be se o be a funcion of he pu curren and he swiching frequency by applying (3). From (3) he ime period ( 3 - ) duraion is given as: Ipk ( ) = (4) 3 + Since he curren is of a riangular shape (Fig. 4), he peak is wice is average value and hence: I ( ) = (5) 3 + Applying (5) o he righ volage source of Fig. 7, he expression is rewrien as 4 I F. The resuled average model is shown in Fig. 8. A. Small signal curren response o a disurbance in he inpu volage. We assume ha he circui is operaed a some inpu volage, pu volage, and frequency F. The dependence of he inducor's curren changes i on a small signal disurbance in inpu volage in (v in ) was found by linearizing he wo depended sources of Fig. 8 and assuming ha all variables excep he inpu volage are consan (Fig. 9): v 4Fi in + i = (6) s Taking ino accoun he riangular shape of he inducor's curren i = / and rearranging: i i K = (7) v s + 4F in + where K =. ( )( ) F ( + )( 3 ) F Fig. 6. A linearized average model of a DC inducor charger for consan inpu and pu volages. Fig. 7. Schemaic diagram of equaion ().

4 4 Fig. 8. Average large signal model of "AC inducor". B. Small signal curren response o a disurbance a he pu volage. Following he same procedure as above bu for he disurbance in he pu volage v, we find: i K = (8) v s + 4F where K =. Boh (7) and (8) have finie response a low perurbaion frequencies. Tha is, in conras o he "DC inducor" case, he low frequency small signal curren in he inducor will be limied by he inducor's impedance a he swiching frequency (F) raher han by he low frequency impedance or he parasiic resisances. C. Inducor's curren response o a modulaing frequency. In an elecronic charger i is usually desirable o mainain a consan load curren. One possible way o achieve ha will be o conrol he swiching frequency. Tha is, he relevan conrol ransfer funcion required for he design of he compensaion nework, is he response of he inducor's curren o a modulaion f in he swiching frequency of he square-wave inpu volage (F). To find his ransfer funcion, he depended sources of Fig. 8 were linearized assuming ha he disurbance was in he swiching frequency (Fig. 0). All he volage sources were assumed o be consan. The small signal pu curren was found o be: ( 4 I f + 4i F) i = (9) s Taking ino accoun again ha i = i / and rearranging: i 8I = (0) f s + 4F Fig. shows he Bode diagram of (0) ploed for = 75µH, =00, =50, and F=6.5kHz. The las expression implies ha for low frequencies he inducor behaves as a resisor (fla response in he bode diagram of Fig. ). 4 I f + 4i Fig. 0. inearized average small signal model of AC inducor for small-signal injecion in F, (f). Tha is, even for a very slow disurbance in he swiching frequency (very slow modulaion of F) he changes in he inducor's curren will be negligibly small considering a pracical swiching frequency (ens of kiloherz). I follows ha he AC inducor charger, unlike he case of he DC inducor case, can be operaed in open loop. This is a corollary of he fac ha he AC inducor behaves as a curren source while he DC inducor behaves as a volage source. I. EXPERIMENTA The AC inducor case was esed experimenally using a half bridge opology wih a blocking capacior (Fig. ). The baery load was modeled in he experimenal seup by an RC nework wih very long ime consan so i can be considered pracically zero impedance a he esed small signal frequencies. The inducor was 75µH, load capacior 4700µF, and load resisor 5Ω. DC blocking capacior was (.5mF). The volage was 00DC. Oupu volage was ab 50DC. The swiching frequency was se o 6.5kHz (5kHz a he load side) and power ransferred o he load was ab 00W. Fig. 3 shows he response of he pu curren o he disurbance a he pu volage. The pu volage was modulaed by adding a 50Hz ac volage source in series wih he RC nework. Fig.. Bode diagram of pu curren o swiching frequency modulaion ransfer funcion (0). F + v in Fig. 9. Average small signal model of AC inducor charger linearized for a consan pu volage Fig.. Experimenal seup.

5 5 Fig. 3. Oupu curren response o pu volage modulaion. Upper race: Modulaing volage 0/div; ower race: Oupu curren A/div; Horizonal scale: 5ms/div. The injeced signal had an ampliude of 0 pk-pk. This resuled (Fig. 3) in an envelope ampliude of he inducor curren of ab 0.55A pk-pk. By subsiuing f of 50Hz, v of 0 ino (8) we ge he value of 0.54A which is in good agreemen wih he measured value. Oupu curren response o a modulaed swiching frequency is presened in Fig. 4. The frequency was modulaed by a khz sinusoidal modulaing signal. Maximum frequency deviaion was 0kHz (modulaion index of 0). The measured curren o frequency raio was ab 66[µA/Hz]. The raio calculaed from (0) is 64[µA/Hz]. An expanded view of he pu curren, modulaing signal as well as he volage across he inducor is shown in Fig. 5. The resuls of presen sudy show ha "AC inducors" are less sensiive o average volage disurbances a eiher inpu or pu side. As a resul, hey can be operaed as curren sources in open loop configuraion wih he need for curren feedback. On he oher hand, in he case of he "DC inducor" a small difference beween he average volages of he inducor's erminals may cause curren run-away and herefore, a igh curren feedback loop mus be applied. The analyical derivaions and simulaions of his sudy were verified experimenally. Good agreemen was found beween he heoreical predicaions and he analyical resuls.. DISCUSSION AND CONCUSIONS Based on he resuls presened in his sudy one can conclude ha "AC inducor" based opologies are opimal for curren sourcing applicaions, such as baery chargers, where differences beween he average inpu and pu volages are expeced o exis and develop over ime. Fig. 4. Oupu curren response o modulaion of swiching frequency. Upper race: Oupu curren A/div; ower race: Modulaing frequency 0kHz/div; Horizonal scale: 0.5ms/div. Fig. 5. Expanded view of AC inducor's key waveforms. Upper race: Inducor's volage: 00/div. Middle Trace: Inducor's curren 5A/div. ower Trace: Swiching frequency 0kHz/div (carrier frequency 6.5kHz). Horizonal scale: 50us/div The disadvanage (from he dynamics poin of view) is he higher pu impedance, similar o he case of a DC inducor converer wih an inner curren loop. REFERENCES: [] R. W. Erickson, "Converer circuis," chap. 6 in Fundamenals of Power Elecronics, Chapman&Hall, N.Y., 997. [] R. W. Erickson, "Resonan Conversion," chap. 9 in Fundamenals of Power Elecronics, Chapman&Hall, N.Y., 997. [3] S. Sao, S. Moisseev, and M. Nakaoka, "A novel synchronous recifiers based ZS-PWM DC-DC power converer wih exended sof-swiching operaional range," IEEE The 5h Inernaional Telecommunicaions Energy Conference, INTEEC'03, pp , 9-3 Ocober 003. [4] Y. Zhang, P. C. Sen, "A new ZS phase-shifed PWM DC-DC converer wih push-pull ype synchronous recifier," IEEE Canadian Conference on Elecrical and Compuer Engineering, CCECE 003, vol., pp , 4-7 May 003. [5] F. Krismer, J. Biela, and J. W. Kolar, "A Comparaive Evaluaion of Isolaed Bi-direcional DC/DC Converers wih Wide Inpu and Oupu olage Range," IEEE Indusry Applicaions Conference, Foureenh IAS Annual Meeing, vol., pp , -6 Ocober 005. [6] C. Hang-Seok, J.H. ee, B. H. Cho, J. W. Kim, "Analysis and design consideraions of zero-volage and zero-curren-swiching (ZZCS) full-bridge PWM converers," IEEE 33rd Annual Power Elecronics Specialiss Conference, PESC 00, vol. 4, pp , 3-7 June 00. [7] M. Xu, Y. Ren, J. Zhou, and F. C. ee, "-MHz Self- Driven ZS Full-Bridge Converer for 48 Power Pod and DC/DC Brick," IEEE Transacion on Power Elecronics, vol. 0, pp , 005. [8] S. TingTing, H. Nianci, "A novel zero-volage and zero-curren-swiching full-bridge PWM converer," Eigheenh Annual Applied Power Elecronics Conference, APEC 003, vol., pp , 003. [9] S. Ben-Yaakov, "Average simulaion of PWM converers by direc implemenaion of behavioral relaionships," Inernaional Journal of Elecronics, vol. 77, pp , 994.

Table of Contents. 3.0 SMPS Topologies. For Further Research. 3.1 Basic Components. 3.2 Buck (Step Down) 3.3 Boost (Step Up) 3.4 Inverter (Buck/Boost)

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