Energy and Power Engneerng, 22, 4, 7-6 http://dx.do.org/.4236/epe.22.435 Publshed Onlne May 22 (http://www.sp.org/journal/epe) 7 Modelng and urrent Programmed ontrol of a Bdretonal Full Brdge D-D onerter Shahab H. A. Moghaddam, Ahmad Ayatollah, Abdolreza ahmat Shool of Eletral Engneerng, Iran Unersty of Sene and Tehnology, Tehran, Iran Emal: shahab.h.moghaddam@gmal.om, {ayatollah, rahmat}@ust.a.r eeed February 7, 22; resed February 28, 22; aepted Marh 6, 22 ABSTAT Modellng of bdretonal full brdge D-D onerter as one of the most applable onerters has reeed sgnfant attenton. Mathematal modellng redues the smulaton tme n omparson wth detaled rut response; moreoer t s onenent for ontroller desgn purpose. Due to smple and effete methodology, aerage state spae s the most ommon method among the modellng methods. In ths paper a bdretonal full brdge onerter s modelled by aerage state spae and for eah mode of operatons a ontroller s desgned. Attaned mathematal model results are n a lose agreement wth detaled rut smulaton. Keywords: Aerage State Spae; Bdretonal; Detaled rut Smulaton; Full Brdge D-D onerter; Mathematal Modelng. Introduton Modelng of D-D onerter as one of the most applable ndustral onerters has aroused a lot of nterest. Sne modelng ges us nformaton about stat and dynam of the system, t s a rual fator n desgn and ontrol. Moreoer, attaned mathematal model an redue the smulaton tme n omparson wth the smulaton tme proded by yle by yle solng the dfferental equatons of the rut, as s the ase n matlab/ smulnk. Wth respet to renewable energy systems and optmum use of regenerated energy, nterfae onerters should be apable of transferrng power n both dretons. So bdretonal D-D onerters (BD) are one of the most mportant nterfaes that hae applatons suh as: hybrd or eletral ehles [], aerospae systems [2], teleommunatons, solar ells, battery hargers [3], D motor dre ruts [4], unnterruptable power supples [5-7], et. so far many BDs topologes hae been ntrodued and sureyed [8,9]. In applatons that transferred power s more than 75 watts, full brdge topology s a proper one []. Bdretonal full brdge (FB) onerters hae been studed n many papers lke [-3]. A general modelng method that deelops the dsrete tme aerage model s proposed n [2]. The operaton perod s dded to 3 nterals, the equalent rut and the dfferental equatons for eah nteral are wrtten n matrx form. After solng equatons and applyng approxmaton of Taylor expanson, the aeragng state etor n half yle ges us the fnal answer. Sne tme doman method employs numeral ntegraton to sole dfferental equatons the analyss s omplated and omputatonally ntense. Moreoer the nformaton about the dependene of the onerter s operatng ondtons on the rut parameters s not proded [4]. eferene [3] proposes a dsrete Small sgnal model wth the amount of onsderable alulaton, just to predt the peak response of state etors. There are also some dentfaton-based methods lke NAMAX [5, 6] and Hammersten [7,8] to model the D-D onerter. Hammersten model noles of a stat nonlnearty followed by a lnear dsrete-tme and tme-narant model, but dentfaton based methods onsder the system as a blak/gray box, therefore they do not prode any nsght nto rut detals. So many referenes use rut orented methods to model the onerter. Vorperan and Tymersk et al. [9,2] prode the rut swth model wth replang the PWM swth wth ts equalent rut n order to model the onerter. Ths method may pose some omplexty espeally n nonbas topologes. Another rut orented method that s ntrodued by Mddlebrook and uk [2-23] n 977 s the aerage state spae. An adantage of the state-spae aeragng method s ts effeny ompared to that of the swthed model beause there s not any swthng frequeny rpple and, onsequently, the smulaton tme requred by the aeraged model s muh lower than requred by the swthed model. Among all methods of
8 S. H. A. MOGHADDAM ET A. modelng, aerage state spae seems to be one of the most ommon, smplest, and effete methods, so n ths paper we model a bdretonal full brdge D-D onerter wth aerage state spae to gan the approprate transfer funtons for ontroller desgn purpose n both modes of operaton. 2. Prnple of Operaton Fgure shows the proposed bdretonal full brdge onerter where arrows represent the dreton of power flow. There are so many onfguratons for FB onerters [,4,24] wth the same bas topology whh dffers from one another n the ase of swthng sheme or employng the elements of onerter for the purpose of aheng ZVS or ZS. Sne modelng of ths onerter s the man purpose of ths study and hangng modes of operaton rely on ondtons of applaton; these ondtons are not taken nto onsderaton. Detaled rut desrpton an be reewed n the lterature [24,25]. There are some long and short nterals n eah mode, sne the short ones are not as sgnfant as long ones they an be left out. Meanwhle Fgure 2 shows the bas waeforms and pulse gatng of swthes n boost mode operaton, gnorng the short subnterals, Fgure 3 depts the buk mode waeforms. Small sgnal Fgure. Bas topology of the proposed FB bdretonal onerter. Fgure 2. Ideal steady-state oltage and urrent waeforms of the onerter n boost mode operaton durng one swthng yle. Fgure 3. Ideal steady-state oltage and urrent waeforms of the onerter n buk mode operaton durng one swthng yle. model of eah mode wll be extrated. For smplty of ontrol, gatng rut, and analyss, t s assume that n eery mode only swthes of one sde are gated meanwhle the opposte ones are operatng n ther dode modes. 2.. Boost Mode Operaton In boost mode wth respet to pulse gatng sgnals, there are two man nterals, onsstng of four swthes on and two dagonal swthes on. When four swthes turn on, nput ndutor oltage s equal to nput soure (low oltage sde) and the ndutor urrent nrease proportonal to the appled oltage. In ths nteral ndutor saes the energy to transfer t n the next nteral. Whle at the hgh oltage (HV) sde the load s fed by the energy that has been transferred to the output flter () durng perous nteral. Next nteral usually s known as the energy transfer nteral. For nstane, assume that S and S 4 are on and S 2 and S 3 are off. The nput oltage plus ndutor oltage s appled to transformer and s saled by n fator, rato of seondary to prmary oltage, then wll be retfed through the other sde of brdge. These proesses wll be repeated at next half swthng yle wth ths pont that the prmary appled oltage n next half swthng yle wll be negate but wll be retfed n the other sde of the brdge. 2.2. Buk Mode Operaton Aordng to Fgure 3 that shows the onentonal pulse gatng, hard swth, for buk mode operaton, agan there are two man nterals n a half swthng yle, frst when dagonal swthes turn on, and seond one when all swthes turn off. When dagonal swthes are on, for nstane S5 and S8, the power s transferred from hgh HV sde to V sde. In ths nteral the ndutor urrent nreases proportonal to the saled HV sde oltage mnus output (nomnal V) oltage.
S. H. A. MOGHADDAM ET A. 9 Wth turnng off the swthes, next nteral starts. Although the exstene of leakage ndutane preents the swthes go off mmedately after applyng gate turn off pulses and onduton of swthes wll ontnue through parast apators and dodes, but t s assumed that these subnterals are ery short and an be negleted. In off tme, the seondary sde s only fed by the ndutor stored energy, so the ndutor urrent dereases proportonal to output oltage. Next half swthng yle s the same, and only appled oltage of HV sde s negate that s retfed n V sde. 3. Small Sgnal Modelng Usng Aerage State Spae Employng aerage state spae method s dded nto three phases: ) Wth respet to swth ondtons, the rut s dded nto dfferent subnterals and state equatons are wrtten n the matrx form n eah nteral. State etors are defned as ndutors urrents and apators oltages. 2) Aeraged equatons are formed by takng weghted aerage of state equatons of eah nteral. 3) Aeraged equatons are wrtten n dfferental form then lnearzaton s done by perturbng arables. Employng aplae transform and omttng addtonal A and D terms (only frst order A terms), needed transfer funtons are aheed. For smplty of modelng the followng assumptons an be employed: Swthes are deal, there s no parast effet n swthes; Indutor has no resstane; Transformer s deal and there are no leakage and magnetzng ndutanes; Flter apators hae low ES (equalent seres resstane) that an be negleted; oad s onstant and for modelng of the load hange an addtonal urrent soure has been added at the output; Eah mode (buk or boost modes) starts wth zero ntal ondton. State equatons wll be wrtten n eah mode separately and wth some mathematal operatons one an dere needed transfer funtons. 3.. Boost Mode State Equatons As we saw n Seton 2. n ths mode two man nterals an be assumed. When all four swthes ondut the equalent rut wll be the same as shown n Fgure 4 and the dfferental equatons an be wrtten as follows: IN d d IN () d d z z (2) d IN d Z Ths state lasts for (d.5)t s where Ts fs s perod of swthng, d ton Ts s the effete duty rato and n s turn rato of seondary to prmary wndngs. In the next nteral when dagonal swthes ondut, the equalent rut an be skethed as the same n Fgure 5 and the state equatons an be wrtten as below: d d IN IN (4) n n (3) d d Z Z (5) n n d n IN d Z n IN Y o Y Z Ths state lasts for ( d)t s. The output state n both nterals s the same as Equaton (7). Now aeragng the state equatons durng half swthng yle wll result n the equaton that has the haraterst of two nterals: 2( d ) n At & Bt (8) 2( d) n (6) (7) Fgure 4. Equalent rut of boost mode operaton wth four V swthes on. Fgure 5. rut of boost mode operaton wth two V swthes on.
S. H. A. MOGHADDAM ET A. 2( d ) n X IN 2( d) Z n (9) IN Y X DU () Z Perturbng the state equatons around ther operatng ponts wll result n the equatons that an be used for derng the transfer funtons; to do so one just need to wrte oltages and urrents n the followng forms: I V V IN VIN n () Z z d Dd d di 2( Dd ) n d dv V Dd I 2 n V z V IN n Dd V d 2 V n IN n (2) (3) (4) d 2( Dd ) I V z (5) n For extratng transfer funtons addtonal A and D terms an be negleted. Wth applyng aplae transform to the equatons one an ahee the followng transfer funtons. n ns 2 2 2 2D n n S4 D 2 2nI S 8 D V 2nV d 2 2 2 ns n S4 IN D 2 D 2 (6) (7) 2nVS 8 D I (8) d 2 2 2 ns n S4 2 ns (9) 2 2 2 2 z ns n S 4 D 3.2. Buk Mode State Equatons The method s the same as mentoned n preous seton for the boost mode operaton. Two man operatng nterals are onsdered to llustrate rut ondtons. No matter what swthng sheme s employed, onentonal PWM, phase shft or PWM plus phase shft, there s always an effete duty yle, D eff. Fgures 6 and 7 dept the man equalent ruts meanwhle (2) and (2) desrbe state equatons. Note that n wrtng equatons, output oltage (VS) s mentoned as V and the nput oltage at HV sde s defned as V n. d n IN d Z (2) Ths nteral wll lasts for dt s seonds. Followng nteral wll last for the remaned half yle, (.5 d)t s. d IN d Z (2) Aeragng n half swthng yle and perturbng the oltage and urrents wll result the needed transfer funtons. 2D n n 2 S S 2VIN n d 2 S S (22) (23) Fgure 6. Equalent rut of buk mode operaton wth two HV swthes ondutng. Fgure 7. Equalent rut of buk mode operaton wth shortng HV sde.
S. H. A. MOGHADDAM ET A. 2VIN 2V S n n d 2 S S z S 2 S S IN (24) (25) be appled to duty yle n the both mathematal models and smulated rut. Fgure 9 shows the output oltage response to duty yle hange and Fgure represents the urrent waeforms whle Fgure represents the output oltage hange due to A step hange n load urrent that s modeled by a urrent soure, z. 4. Model Verfaton Hang aeraged state equaton and perturbaton one an ahee all transfer funtons needed for ontroller desgn. In ths seton erfaton of the attaned aeraged model s done by omparng the step responses of transfer funtons dered from aerage state spae and smulated rut n matlab/smulnk. 4.. Boost Mode Start up proess of open loop onerter system wth nput oltage of 24 V s smulated under these ondtons: D =.6, f s = 2 khz, = 2 μh, = 5 μf, V out = V = 3 V, P n = P out =.5 KW, = 6 Ω. The smulated start up proess of the predted mathematal model and detaled rut smulaton wth fxed duty yle s depted n Fgures 8 and, respetely. Steady state, peak response, and rse tme of output oltage n smulnk s the same as obtaned by the mathematal model but settlng tme dffers.7 seonds whh s aeptable. In order to hek ontrol to output and ontrol to ndutor urrent transfer funtons, a small step (.) an Fgure 9. Output oltage n the presene of. step hange n duty yle. Predted response by mathematal aeraged model; Detaled rut smulaton n matlab/smulnk. Fgure 8. Start-up proesses Smulated mathematal aeraged; Detaled rut smulaton n matlab/smulnk. Fgure. Indutor urrent n the presene of. step hange n duty yle. Predted response by mathematal aeraged model; Detaled rut smulaton n matlab/smulnk.
2 S. H. A. MOGHADDAM ET A. Fgure. Output oltage n the presene of A step hange n load urrent. Predted response by mathematal aeraged model; Detaled rut smulaton n matlab/ smulnk. 4.2. Buk Mode Start up proess of open loop onerter system wth nput oltage of 3V s smulated under these ondtons: D =.4, f s = 2 khz, = 2 μh, = 5 μf, V out = V = 24 V, P n = P out =.5 KW, =.384 Ω. From Fgure 2 t an be seen that the mathematal model has predted the response of the onerter ery well. Fgures 3 and 4 represent the behaor of output oltage and ndutor urrent n the presene of.2 step hange n duty yle. Fgure 5 shows onerter output oltage due to.5a step hange n load, t s obous that the mathematal model s n a lose agreement wth smulated rut. 5. ontroller Desgn Swth mode power supples (SMPS) ontrol methodology an be dded nto two man methods onsstng of oltage mode ontrol (VM) and urrent mode ontrol. urrent mode ontrol (M) s faster than oltage mode but t suffers from rngng, so mergng two methods an oer the shortage of both. urrent programmed ontrol (P) method that s presented n [,26,27] employs M and VM together. Fgure 6 shows the blok dagram of P whh s manly dded to peak, alley, and aerage [28]. Among all of those methods of P, the peak urrent programmed ontrol (PP) s one of the most ommon modes and easest one to understand. Fgure 2. Open loop start up output oltage wth fxed duty yle predted by mathematal model; Detaled rut smulaton wth matlab/smulnk. Fgure 3. Open loop response of output oltage due to.2 step hange n duty yle predted by mathematal model; Detaled rut smulaton wth matlab/smulnk.
S. H. A. MOGHADDAM ET A. 3 Fgure 6. Blok dagram of peak urrent mode ontrol. The method of PM modelng s the same as de- srbed n [], only up and down slopes and ther tmes hae been replaed to adapt the formulaton wth proposed topology. So the model of Fgure 6 an be aheed, where F m, F n and F n boost and buk modes are determned n (26). d F F F m n n Fgure 4. Open loop respons e of ndutor urrent due to. step hange n duty yle predted by mathematal model; Detaled rut smulaton wth matlab/smulnk. Fgure 5. Open loop response of output oltage due to A step hange load urrent predted by mathematal model; Detaled rut smulaton wth matlab/smulnk. Fm boost mode Fn F MT a s D 34Ts Ts n D 2 Fm MT a s 2 DTs buk mode Fn n 4DT s F (26) Not onsderng nput oltage and load aratons, the oerall blok dagram for both modes of operatons an be depted as Fgure 7 wth some algebra operatons; the loop gan transfer funton of (27) an be onluded. FG m V d loop gan HSG S (27) FG FFG m d m V d In order to examne the stablty and desgn the onerter s ontroller, boost and buk mode transfer funtons an be replaed to ahee loop gan of eah mode. Wth some algebra operatons one an get to (28) and (29) for boost and buk loop gan respetely. Fgure 8 shows the unompensated and ompensated loop gan of boost mode. ompensatng network for boost mode operaton s a
4 S. H. A. MOGHADDAM ET A. robustness. Other adaned ontrol algorthms an be appled to ths onerter easly wth the dered small sgnal model but that s beyond the sope of ths work. Fgure 9 represents unompensated and ompensated loop gan of buk mode. Fgure 7. Smplfed blok dagram of losed loop onerter wth negletng the effet of lne and load araton. 6. Smulaton esults In order to make sure that the desgned ontrollers are apable of ontrollng the onerter, they should be examned. In ontrol loop addtonal bloks lke duty yle lmter are employed beause at the start up there s no output oltage. Ths leads to % the duty yle, as a result after some yles; the rut s gone to ts steady state n whh ndutor and apator experene only a onstant soure and ths s when they go under short and open rut respetely. The other mportant blok s ompensatng ramp whh an oerome the unstable osllaton problem desrbed n [,22,3]. Fgures 2 and 2 show that the ontrollers are suessful to ontrol the onerter n the presene of A and A step load hange for boost and buk mode respetely. Fgure 8. Unompensated and ompensated loop gan of boost mode sold lne s unompensated and dashed lne s ompensated response. smple PID whh ts oeffents hae been seleted wth respet to the method presented by hen and Fruehauf [29]. GSFmHS 2nIS 8DV 2nVIN Tboost S den S I Fm F S 4 8 2nF FV 2 2 2 den S n S n 2n FmV 2n T buk S 2 D DF I FV G( S) FmH( S) 2nV IN den( S) 2 2nF F V 2 2 2 den S n S n n m IN m FmV IN S m IN (28) (29) A smple ntegral ontroller an meet the requrements of buk mode ompensatng network. Howeer, the desgned ontrollers are not the optmum ones n terms of Fgure 9. Unompensated and ompensated loop gan of buk mode, sld lne s unompensated and dashed lne s ompensated response. Fgure 2. esponse of ompensated losed loop rut n boost mode operaton n the presene of A step load hange.
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