CHAPTER 3 QUASI-RESONANT BUCK CONVERTER

27 CHAPTER 3 QUASI-RESONANT BUCK CONVERTER Hstrcally, prr t the avalablty f cntrllable swtch wth apprecable vltage and current-handlng capablty, the swtch-mde DC-DC cnverter cnssts f thyrstrs whch pertans t the famly f pwer semcnductr devces havng cntrllable fur layer structures and nclude devces lke Slcn Cntrlled Rectfer (SCR), ght Actvated Slcn Cntrlled Rectfer (ASCR), Slcn Blateral Swtch (SBS) and s n (Sngh 2002). Each thyrstr requres a current cmmutatn crcut that n turn cnssts f a resnant crcut and ther auxlary thyrstrs and ddes. Because f the cmplexty and substantal lsses n the cmmutatn crcut, the thyrstrs are replaced by cntrllable swtches t mprve ther pwer handlng capablty. The cntrllable swtches, fr nstance, n Pulse-Wdth Mdulated Cnverter are perated n swtch mde where they are requred t turn n and ff the entre lad current durng swtchng and hence are subjected t hgh swtchng stress and lss whch ncreases lnearly wth the swtchng frequency f the cnverter; als, the vltage and current trajectres f the swtchng devce durng turn-n and ff peratn play a vtal rle n the swtchng lss and stress and an addtnal dsadvantage s the electrmagnetc nterference (EMI) prduced due t hgh d/dt and dv/dt caused by swtch-mde peratn (Jaml Asghar 2004). A need t rase swtchng frequency and t reduce electrmagnetc nterference led t augmentng the cntrllable swtches n basc swtch-mde cnverter whch cnssts f a smple resnant crcut t shape the swtch

28 vltage and current wavefrm nt quas-snusdal and t yeld zer-vltage and zer-current swtchng cndtns (ander 1993). Such cnverter s termed as Quas-Resnant Cnverter (Sngh and Khanchandan 2007). It s vewed as a hybrd cnverter between Pulse-Wdth Mdulated Cnverter and Resnant Cnverter (Kwang-Hwa u et al, 1987) and t utlzes the prncple f nductve r capactve energy strage and transfers n a smlar manner as Pulse-Wdth Mdulated Cnverter (Kwang-Hwa u and ee 1990). 3.1 GENERAIZED STATE-SPACE AVERAGING METHOD State-Space Averagng (SSA) methd (ehman and Bass 1996) s cmmnly used t analyze Pulse-Wdth Mdulated Cnverter and s valuable because t prvdes a methd f analyzng bth AC and DC behavur f a cnverter n a systematc manner. Its result s that the dfferent state matrx crrespndng t each swtchng nterval f cnverter s replaced by a sngle equvalent matrx whch s the weghted average f ndvdual swtchng nterval matrx (Wtulsk and Ercksn 1990) and s manly based n tw assumptns namely () the swtchng frequency s much hgher than the natural frequency f the cnverter n each mde f peratn and () the nput t cnverter n each mde f peratn must be slw tme-varyng varables cmpared wth the swtchng frequency (Janpng Xu et al 1994). In Quas-Resnant Buck Cnverter, the natural frequency n the quas-resnant tank s n the same rder as the swtchng frequency and the state varables asscated wth the resnant tank are fast tme-varyng n each swtchng cycle f the cnverter cmpared wth thse n the lw-pass flter frmulate the State-Space Averagng methd cannt be appled t ts mdelng. Therefre, Generalzed State-Space Averagng s needed fr the mdelng and analyss f Quas-Resnant Buck Cnverter t establsh mre accurate results; here, a tme average nstead f weghted average s prpsed and nly the basc assumptn () f State-Space Averagng methdlgy s used whle the nput

29 varables can be fast tme-varyng. The State equatn f a perdcally swtched netwrk wth k dfferent swtched mdes n each swtchng cycle s X(t) = A x(t)+b (t), =1,2...k (3.1) The th Equatn f (3.1) s defned n the tme nterval (t -1, t ) where t -1 = 0 + 1 j and t = t -1 + j1 It s assumed that j > 0 and j = 1,2...k are fxed. defnng = d T, If the nput cntrl varable B s bunded and f s >> f 0, then, The Equatn (3.1) fr the perdcally swtched netwrk can be characterzed by the Generalzed State Space Averagng Equatn as mentned n (3.2) k x = { d A }x+1/t B ()d t =1 =1 t -1 k (3.2) T s the swtchng perd and s defned as T = k j j=1 f s = 1 T s the swtchng frequency, f 0 s the hghest natural frequency f state matrx A. Zer Current Swtchng (ZCS) (Ned Mhan et al 2003) Quas-Resnant Buck Cnverter s cted as an example n rder t demnstrate the applcatn f Generalzed State-Space Averagng n a straght frward manner.

30 3.2 MODEING OF QUASI-RESONANT BUCK CONVERTER Mdelng a pwer-electrnc cnverter necesstates dervng many mathematcal expressns that descrbe the cnverter peratn and s a tme-cnsumng prcess (Jan Sun and Hrst Grtstllen 1997). In mdelng Zer-Current Swtchng Quas- Resnant Buck Cnverter, n the resnant tank state, the natural frequency (497.61 khz) s f the same rder as the swtchng frequency f Cnverter (200 khz) whereas n the flter state the natural frequency f Cnverter (2.5165 khz) s much lwer than the swtchng frequency (200 khz). It s bvus that nce the state varables asscated wth lw-pass flter are determned, the state varables n resnant tank shall be determned n each mde f peratn. The state varables n the resnant tank reaches zer perdcally n each swtchng cycle and hence can be cnsdered t be lst as s the case fr Pulse-Wdth Mdulated DC-DC cnverter n the dscntnuus cnductn mde. Thus, the key varables f Quas- Resnant Buck Cnverter are the state varables asscated wth the lw-pass flter. Therefre, fr the mdellng and analyss f Quas-Resnant Buck Cnverter, the reduced-rder state-space equatn s frmulated by cnsderng the varables asscated wth the flter state as the state varables whle the varables f resnant tank are cnsdered as nput cntrl varables nly. Hence, by Equatn (3.2), the Generalzed State-Space Averagng methdlgy s appled t the analyss f Quas-Resnant Buck Cnverter wth accuracy whch s smlar t the State-Space Averagng methd appled fr the analyss f PWM Cnverter. 3.3 ANAYSIS OF QUASI-RESONANT BUCK CONVERTER Quas-Resnant Buck Cnverter perated n dscntnuus mde f cnductn mde s depcted n Fgure 3.1. It uses a undrectnal swtch S (Hart 1997) and s unable t return the excessve tank energy t the surce. Cnsequently, ts cnversn frequency s t be vared ver a wde range t

31 mantan vltage regulatn fr a varable lad (Jvanvc et al 1989); the cntrl f varable frequency s undesrable because ptmal utlzatn f magnetc cmpnents s nt pssble and s dffcult t handle the generated nse. Hwever, t ffers many dstnct advantages such as self-cmmutatn, lw swtchng stress and lss, hgh effcency and pwer densty, reduced nse and electrmagnetc nterference and faster transent respnse t bth lne and lad varatns and t was successfully mplemented n dc-dc cnverters capable f peratng at 5MHz (Jvanvc et al 1989). Hwever, t results n severe cmprmse n the desgn f flter cmpnents and dynamcs f the cnverter. The nductr r and the capactr C r cnsttute the seres resnant crcut wth ts scllatn ntated by the turn-n f the swtch S at tme t = 0. The nductr 0 and capactr C 0 near the lad cnsttutes the flter crcut f the cnverter (Bmbhra 2008). An analyss under steady state s perfrmed by analyzng the behavur f the state varables f the flter state usng the afresad methdlgy wth the assumptns mentned hereunder. Fgure 3.1 Zer-Current Swtchng Quas Resnant Buck Cnverter () All the elements ncludng the swtch are deal whch smplfes the generatn f basc equatns and relatnshps. () The swtchng frequency (f s ) s much hgher than the natural frequency f the lw-pass flter ( 0 -C 0 ) and hence the state

32 varables asscated wth the flter state shall be regarded as cnstant n each swtchng cycle f peratn and () The ratng f flter cmpnents must be much hgher than the ratng f the resnatng cmpnents. The reduced-rder statespace equatn s frmulated by analyzng the crcut n ts fur mdes f peratn (Gupta and Sngh 2002) as mentned hereunder. 3.3.1 Inductr Chargng Mde The swtch S n the Quas-Resnant Buck Cnverter s turned n at t = 0. The current n the nductr r ( r ) rses lnearly and the dde D s n. Fgure 3.2 Inductr chargng mde Because f the lad current (I 0 ) free wheelng thrugh the dde t appears as a shrt crcut and the nput vltage appears acrss the nductr r ; the vltage acrss capactr C r s zer between t 0 and t 1. The reduced-rder state equatn f the Zer Current Swtchng Quas-Resnant Buck Cnverter depcted n Fgure 3.2 s represented n Equatn (3.3). dv C -1 1 O RC v dt O C O C O = d -1 O 0 O dt O (3.3)

33 and the duratn f ths peratn mde s r = 1 v g (3.4) 3.3.2 Resnant Mde At t = t 1, r = I 0 and the dde D s turned ff; nductr current and the capactr vltage vary snusdally wth the resnant frequency untl t 2. Fgure 3.3 Resnant mde Fgure 3.4 Wavefrm f nductr current and capactr vltage f Zer Current Swtchng Quas-Resnant Buck Cnverter The current eventually drps t zer at tme t 2 and the swtch s turned ff resultng n zer current swtchng. The vltage acrss the capactr

34 C r (V Cr ) reaches twce the nput vltage V g as shwn n Fgure 3.4. The reduced-rder state equatn f the Zer Current Swtchng Quas-Resnant Buck Cnverter depcted n Fgure 3.3 s represented n Equatn (3.5). dv C -1 1 0 O dt RCO C V O C O = + v d Cr -1 O O 0 O dt (3.5) and the duratn f ths peratn mde s = (3.6) 2 where -Zn -1 = sn ( ) v g = 2f = r 1 s the resnant angular frequency n radans/s 2 r C r Z = n r C r s the characterstc mpedance n hms and V (t) = V g(1-cst) Cr (3.7) 3.3.3 Capactr Dschargng Mde The swtch S s pened wthut pwer lss after the nductr current reaches zer at t 2. Beynd the tme t 2, the pstve capactr vltage keeps the dde D reverse based and the capactr dscharges nt the lad. The capactr vltage lnearly decreases and drps t zer at tme t 3 ; at t = t 3, the dde D turns n. The reduced-rder state equatn f the Zer Current Swtchng (ZCS) Quas-Resnant Buck Cnverter depcted n Fgure 3.5 s represented n Equatn (3.8).

35 Fgure 3.5 Capactr dschargng mde dv C -1 1 0 RC v dt C C = + v d -1 C r dt (3.8) and the duratn f ths peratn mde s CrV g(1- cs ) = (3.9) 3 t = t 3 and Equatn (3.9) s vald untl the capactr vltage reaches zer at V (t) = - t + V g(1-cs ) Cr Cr (3.10) 3.3.4 Free Wheelng Mde The lad current (I 0 ) just freewheels thrugh the dde D untl a tme T where the swtch S s turned n and the cycle s repeated. The reduced-rder state equatn f the Zer Current Swtchng Quas-Resnant Buck Cnverter depcted n Fgure 3.6 s represented n Equatn (3.11).

36 Fgure 3.6 Free wheelng mde dv C -1 1 RC v dt C C = d -1 0 dt (3.11) and the duratn f ths peratn mde s = T- (3.12) 4 1 2 3 Rewrtng Equatns namely (3.3), (3.5), (3.8), (3.11) n the same way as fr (3.1) gves -1 1 RC C A =A =A =A = 1 2 3 4-1 0 0 0 B = B =, B = B = V 1 4 0 2 3 Cr (3.13)

37 The Generalzed State-Space Averagng (GSSA) methd (Janpng Xu and ee 1998) shall be nw appled t the mdelng f (3.13) and s btaned as dv C -1 1 RC C v V C dt g f = +. s.h (V g, ) d -1 2f r 0 dt (3.14) where zn vg H (v, ) = + + (1-cs ) g 2Vg zn (3.15) The Equatns (3.14) and (3.15) f the Zer Current Swtchng Quas-Resnant Buck Cnverter s vald nt nly fr characterzng ts steady-state but als fr characterzng ts tme-dman behavur. T perfrm ts small-sgnal characterstc analyss (Slbdan Cuk 1976) n rder t nvestgate ts dynamc behavur, perturbatn s ntrduced t the fur varables namely v g = V g+vˆ g (3.16) v = V +vˆ C C C = I + ˆ (3.17) (3.18) ˆ f = F +f s s s (3.19) and t s further assumed that

38 V g >> vˆ g (3.20) V C >> vˆ C (3.21) I >> ˆ (3.22) ˆ F >> f s s (3.23) where the captal letters stand fr DC steady state cmpnents and the letters wth a hat symbl stands fr the small-sgnal perturbatns. Substtutng the Equatns (3.16) t (3.23) n the Generalzed State-Space Averagng Equatns (3.14) and (3.15) and neglectng all the secnd and hgher rder terms f the small-sgnal perturbatns, the AC small-sgnal state equatn s btaned as n Equatn (3.24). dvˆ C 1 1 - ˆv C RC C dt = 1 d ˆ 1 ˆ + J J V - 0 ˆ H H F s dt 0 M(1- )v + R ˆ + fˆ g s (3.24) Usng the aplace transfrmatn t the ac small-sgnal state equatn, the nput-t-utput vltage transfer functn f Quas-Resnant Buck Cnverter s btaned as depcted n Equatn (3.25). J M 1- ˆv H = ˆv g s C +s -RC +1- R H H 2 J J (3.25) where the cnversn rat f the cnverter M s dependent n the swtchng frequency, nput and utput vltage and s depcted n Equatn (3.26)

39 V F M = = s H V g,i V 2F g n (3.26) and Zn vg J v, = - 1-cs g 2vg Zn (3.27) The small-sgnal mdel f Quas-Resnant Buck Cnverter btans a gd apprxmatn f the behavur f cnverter arund the peratng cndtn. It s clear that J H vares between -1.0 t 0.0 fr the half-wave mde f peratn where the swtch permts undrectnal flw f current and J H remans near zer fr the full-wave mde f peratn n whch swtch current flws bdrectnally n the crcut. 3.4 DESIGN OF QUASI-RESONANT BUCK CONVERTER A Quas-Resnant Buck Cnverter s desgned wth tw assumptns f Generalsed State-Space Averagng methdlgy mentned under; () Swtchng frequency f Quas-Resnant Buck Cnverter s much hgher than the natural frequency f the lw pass flter and hence the state-varables v C and are regarded as cnstant n each swtchng cycle f peratn; () Ratng f the flter cmpnents and C s much hgher than the ratng f the resnatng cmpnents r and C r f Quas-Resnant Buck Cnverter. Desgn parameters asscated wth Quas-Resnant Buck Cnverter s mentned n Table 3.1 and the bjectve s t accurately regulate the utput

40 vltage f Quas-Resnant Buck Cnverter at 54V n spte f dsturbances ether n supply vltage r lad current (Ercksn and Dragan Maksmvc, 2006) wth ether PI cntrller r nn-lnear cntrller such as Fuzzy r Neur cntrller r else, the bjectve s t regulate the utput f Quas- Resnant Buck Cnverter wth a cntrller at 54V rrespectve f fve dfferent peratng cndtns chsen hereunder; () Maxmum lne and lght lad cndtn; () Mnmum lne and maxmum lad cndtn; () Mnmum lne and lght lad cndtn; (v) Mdrange lne and lad cndtn; and (v) Maxmum lne and maxmum lad cndtn. In addtn, the tw vtal tme-dman specfcatns namely the peak versht and settlng tme are used t measure the dynamc perfrmance f cnverter n whch the settlng tme s assumed as the tme at whch the utput f the cnverter reach and stay wthn 2% f the desred utput. Table 3.1 Desgn parameters N. Parameter Symbl Value 1. Input Vltage V g 120-100V 2. Output Vltage V 54 V 3. Resnant Inductr r 1.6 µh 4. Resnant Capactr C r 0.064 µf 5. Flter Inductr 0.2 mh 6. Flter Capactr C 20 µf 7. ad Resstance R 100-10 8. Swtchng Frequency f s 200 khz 9. Natural Frequency f 2.5165 khz 10. ad Current I 0.54-5.4 A 11. Peak resnant Current I M 20A 12. Output Pwer (max) P 0 2.916kW