Electrcal Engneerng esearch (EE), Vlume, 5 A New Mdelng Methd and ntrller esgn fr a Zeta nerter Elshan aaran Hagh, Ebrahm Babae *, ela Mhammadan Faculty f Electrcal and mputer Engneerng, Unersty f Tabrz, Tabrz, Iran *rrespndng Authr: e babae@tabrzu.ac.r Abstract In ths paper, a new methd based n sgnal flw graph technque and Masn s gan frmula are appled t mdel a dc dc Zeta cnerter. The mdel s the transfer functns frm cntrl t utput and frm nput t utput. Zeta cnerter s a rather cmplex system, whch has fur passe elements and may perfrm n ether buck r bst peratng mdes. mpared wth ther cnerters n the same grup, such as uk and SEPI cnerters, the dynamc mdellng and analyss f the Zeta cnerter hae been nt reprted n the lterature. Fr mdellng target and t get rd f cmplex equatns n mdellng, a new methd based n sgnal flw graph apprach s appled t manpulate the equatns. After mdellng and dng dynamc stablty analyss n cntnuus cnductn mde, and pre the necessty f a cntrller desgn fr t. A mdel based PI cntrller s desgned fr the cnerter and the smulatn results are presented t erfy the crrectness f the acheed mdel and mdel based cntrller and the sgnal flw graph perfrmance. Keywrds Masn Gan Frmula; Mdel Based ntrller; Mdellng; Stablty Analyss; Sgnal Flw Graph; Zeta nerter Intrductn Nwadays, the dc dc cnerters hae mstly used n many applcatns such as a pwer supply n electrnc systems. In the prtable deces wth the battery, the cnerter cnerts the nput ltage t the prper utput ltage t use t by the electrnc lad. The battery ltage can ary n a wde range dependng n a charge leel. At the lw charge leel, t may drp belw the lad ltage. Thus, t cntnue supplyng the cnstant lad ltage, the cnerter must be able t wrk n bth step up and step dwn mdes. In ths paper, the Zeta cnerter s chsen t nestgate. Zeta cnerter s a frth rder dc dc cnerter whch s capable f buckng r bstng the nput ltage. In rder t prde an nsde ew f the cnerter s actn, and btan the needed nfrmatn fr cntrller desgn, mdellng plays an mprtant rle []. Mdellng dc dc cnerters s an effecte way t get pen lp frequency respnse and nestgate the stablty f the clsed lp systems. Mdellng s the frst stage fr desgnng cmpensatr r cntrller crcuts. Preus wrks present dfferent mdels fr dc dc cnerters []. nsderng that the pwer electrnc cnerters nclude nnlnear elements lke swtches and ddes, then mdellng f them needs lnearzatn. S, applyng aeragng and lnearzatn technques has a specal mprtance. After lnearzatn, the nsde mdel f the system s extractable and therefre nestgatng the frequency respnse wll be pssble [ 5]. Mdellng dc dc cnerters by usng small sgnal lnearzatn and aeragng technques cause cmplexty n the btaned equatns. Slng these equatns fr basc cnerters lke; buck, bst and buck bst s nt a prblem, but fr hgh rder cnerters, dealng wth them wll be mre dffcult [6]. In [7 8], n rder t dere a mathematcal mdel f a dcdc cnerter and study the transent states f t, a cmbnatn f aplace and Z transfrms has been emplyed. A cnerter ncludng swtches s a nnlnear system, but t can be decmpsed t tw lnear crcuts; ne fr n state and the ther fr ff state f the swtch. These tw lnear crcuts are llustrated by means f tw sgnal flw graphs. mpstn f tw sub graphs by usng swtchng branches reaches t the graph f the whle cnerter. Swtchng branches are the nly nnlnear parts f the cnerters. Thus mdellng prcess s lmted t the swtchng branches. In [9 ], a sgnal flw graph methd has been presented fr mdellng a dc dc cnerter, then by usng Masn s gan frmula any desred transfer functn f the system mght be extracted and used t desgn any prper cntrller, but the presented graph methd gets a sphstcated prcess t achee the small 8
Electrcal Engneerng esearch (EE), Vlume, 5 sgnal mdel. The prpsed technque n ths paper, ges greatly smplfed mathematcal and graphcal representatn f the system and s based n sgnal flw graph f the cnerter. Ths methd has great adantage because f smplcty and beng capable f gng any desred transfer functn f the system t desgn a cntrller fr a determned arable. Graphcal mdellng technque s used fr sureyng nnlnear and dynamc behar f swtchng cnerters. Ths type f cnerter has dfferent cnfguratns and all hae lnear subsystems []. The am f ths paper s t dere the mdel f the Zeta cnerter by usng the prpsed sgnal flw graph based methd. The pen lp system has sme dsadantages. Fr example, the utput cannt be cmpensated r cntrlled f there s a aratn r dsturbance at the nput. Fr the case f Zeta cnerter, the changes n the nput ltage r the lad current wll cause the cnerter s utput ltage t deate frm the desred alue. T desgn a prper cntrller r cmpensatr a gd mdel f the system s needed t nestgate the stablty by usng dfferent tls. One f these tls s frequency dman analyss [ 5]. In rder t get frequency respnse f the cnerters, after mdellng the rt lcus plts and the frequency dman respnse n frm f Bde magntude and phase dagrams are acheed and analyzed. Many cntrl strateges hae been presented t btan the desred utput ltage n dc dc cnerters. In past, the cntrller desgn was based n usng small sgnal lnearzatn. near PI and PI cntrllers f the cnerters are usually desgned by usng standard frequency respnse technques based n small sgnal mdel f the cnerter by usng lnear cntrl theres such as Zegler Nchl s methd, rt lcus technque, hysteress methd, Bde plt and etc. The Zegler Nchls methd s an expermental ne that s wdely used. The man cntrbutn f ths paper s t dere the mdel f the cnerter usng the presented sgnal flw graph methd and then desgnng a mdel based cntrller fr the Zeta cnerter. In ths prcess, by applyng the mentned sgnal flw graph technque and Masn s gan frmula, the nput t utput, cntrl t utput transfer functns are btaned. Then, the frequency dman respnses n frm f Bde magntude and phase dagrams are acheed and the prper tuned PI cntrller s desgned. At last, the smulatn results based n the presented mdelng and cntrllng methds are brught t shw the gd perfrmance and behar f the mdelng and cntrl methds. S S (a) The pwer crcut (b) Equalent crcut n n state. S (c) Equalent crcut n ff state. FIG.. ZETA ONVETE. eew n - Zeta nerter The pwer crcut f the Zeta cnerter shwn n Fg. (a) has ne swtch, ne dde, tw nductrs and tw capactrs. In ths fgure, and are the surce ltage and the utput r lad ltage, respectely., and are the ltages n capactrs, and, respectely. and are nductrs currents. The lad s 9
Electrcal Engneerng esearch (EE), Vlume, 5 suppsed as a resstr,. Zeta cnerter can perfrm t generate a pste ltage f the nput ltage whch can be less r hgher than nput. The cnerter acts n tw peratng mdes, ne f them s related t n state and the ther s the ff state f the swtch. Fgs. (b) and (c) llustrate the equalent crcuts f the n state and ff state peratng mdes, respectely. Extractn f the State Equatns Here, the pen lp perfrmance characterstcs f the cnerter are tested. The startng pnt s the extractng a mathematcal and analytcal mdel f the system. The mdel s a mathematcal descrptn f the behaur f the real system that s enugh fr perfrmng stablty test. The state equatns are extracted by usng krchhff s ltage and current laws fr the tw cases f n and ff states f the swtch. These equatns are calculated n lps and ndes wth just ne passe element, s all btaned equatns are frst rder nes. One f the adantages f ths methd s n hgh rder crcuts, wth ncreasng the rder f the crcut and energy sang elements, the all extracted state equatns wll be f the frst rder and reduce the cmplexty f the equatns and ease the analyss. The state equatns are extracted fr tw peratng mdes. The frst mde s when the swtch s n and the dde s ff, and the secnd mde refers t the case that the swtch s n ff state and the dde cnducts. The state equatns fr the crcut shwn n Fg. (b) are as fllws: d d () () d () () d (5) Substtutng Eq. () n Eq. (5), the fllwng equatn s acheed: d (6) The state equatns fr the case that the swtch s ff and the dde cnducts (Fg. (c)) are as belw: d (7) d d (8) (9) () d () Substtutng Eq. () n Eq. (), the fllwng equatn s acheed: d ()
Electrcal Engneerng esearch (EE), Vlume, 5 Extractn f the Aeraged State Equatns The state equatns at the n tme nteral f the swtch s multpled by d and at the ff tme nteral s multpled by d, then they are added t each ther. d s cnsdered as the swtch cnductn ceffcent and d as the swtch nn cnductn ceffcent. d can ary n and nteral. In the fllwng equatns, the arables n frm f x are aerage alues whch are dfferent frm the nstantaneus arables n frm f x. As mentned abe, Eq. () t Eq. () and Eq. (6) are multpled by d, and Eq. (7) t Eq. (9) and Eq. () are multpled by d. Then sum f the cuples wll be calculated and the result s the aeraged state equatns. Hence, the aeraged equatns are related t bth n and ff tme nterals f the swtch as fllws: d d ( d) () d d d () d (5) d d ( d) (6) Small Sgnal nearzatn In aeraged state equatns, dc and ac arables are substtuted and rewrtten wth X and x, respectely. nsderng all arables n frm f x X x, Eq. () t Eq. (6) are rewrtten as belw: s ( I ) ( d)( V) ( d)( V ) (7) s( I ) ( d)( V ) ( d)( V ) V (8) V s ( V) I (9) s( V ) ( d)( I ) ( d)( I ) () The prduct f tw dc arables s a dc ne and the prduct f ne dc and ne ac arable s an ac arable, als the prduct f tw ac arables s zer. Thus, Eq. (7) t Eq. () are btaned as belw: s ( V V) d( ) () s ( V V) d () () s s ( I I ) d ( ) () Usng dc equatns, V, I and I alues ersus V and are calculated as fllws: V V (5)
Electrcal Engneerng esearch (EE), Vlume, 5 V V V V (6) I V V ( ) (7) I ( ) I (8) I V I ( ) (9) nsderngv, I and I alues and replacng them n Eq. () t Eq. () and smplfyng them, the small sgnal ac equatns are calculated as belw: V s d () ( ) V s d () ( ) s () V s d () ( ) Prpsed Sgnal Flw Graph The ac small sgnal equatns gen n Eq. () t Eq. () are used t plt the sgnal flw graph as shwn n Fg.. In these equatns, there are fur arables;,, and, and tw nputs; d and. The graph s depcted fr all ndependent arables ( sx, x ) by usng ndes and paths between tw ndes. In rder t extract the transfer functns frm sgnal flw graph, the Masn gan frmula wll be used. d s s V ( ) s V ( ) s s s V ( ) s s FIG.. SIGNA FOW GAPH FO ZETA ONVETE. Extractng Transfer Functns Based n the Prpsed Methd Fr calculatng the transfer functns, frst the gan f dstnct lps, nn tuchng lps and frward paths frm each nput t utput are extracted as Table. Transfer functns and d n are calculated by usng Masn s gan frmula yut yn P kk. In ths k frmula, n s the number f frward paths frm nput t the utput and s calculated as fllws:
Electrcal Engneerng esearch (EE), Vlume, 5 ( dstnct lps gan) ( duble nntuchng lps gan) ( trpple nntuchng lps gan) Omttng the path P, s btaned, and mttng the path P, p s btaned. eplacng the btaned alues, the transfer functn wll be as shwn n Table. Omttng the path P d, s btaned, and mttng the path P d, s btaned. Als, remng P d, p s acheed. eplacng the btaned alues, the transfer functn d wll be btaned as gen n Table. TABE. TANSFE FUNTION EXTATION POESS FOM SIGNA FOW GAPH OF ZETA ONVETE stnct lps stnct lps gan Nn tuchng lps gan ( ) p s ( ) p p: s,, s,, s, s p: s,, s,, s p s ( ) p, p: s,, s,, s s p : s,, p s s p, s p s Paths frm t P :, s,, s,, s,, s, P :, s,, s, Paths frm d t P : d, s,, s,, s,, s, d P : d, s,, s,, s, d P : d, s,, s, d Frward paths Frward paths Gan ( ) s P P s Gan V Pd s V d ( ) s P V d ( ) s Transfer functn frm t P P P( p) p p p p p p p,,, ( ) s ( ) ( ) ( ) s s s s Transfer functn frm d t P d d d PdPd Pd( p) p p p p p p p,,, V V V s s ( ) ( ) ( ) ( ) ( ) s s s s P
Electrcal Engneerng esearch (EE), Vlume, 5 TABE. SIMUATION PAAMETES 8.9 H 8.9 H F. F 6.5 F V.57 d TABE. INFOMATION OBTAINE FOM POES AN ZEOS OF THE TANSFE FUNTION OF Ples Zers Value.5 6.5.5 9.69. ampng.7.67 Oersht (%) 86. 8.9 Frequency (rad/s).6 9.7. x 5 t cus - - -5 - -5 5 (a) OOT OUS Magntude (db) Phase (deg) - 8 FIG. ESPONSE OF TANSFE FUNTION OF -8 5 6 (b) BOE IAGAMS Smulatn esults In smulatn, the parameters f Zeta cnerter are cnsdered as gen n Table. Frequency espnse Analyss fr Transfer Functn The rt lcus, magntude and phase Bde dagrams fr extracted transfer functn f are shwn n Fg.. The nfrmatn btaned frm these dagrams s gen n Table. nsderng the rt lcus shwn n Fg. (a), t s understd that has fur cmplex ples n left hand sde f the magnary axs and tw cmplex zers n magnary axs. nsderng Bde dagrams (Fg. (b)), the magntude cure f the functn begns wth a cnstant alue f.6 db( gan ) and at the frequency f.6khz du t the ples.5.5, an undersht ccurs n magntude cure and ts slpe becmes db / dec and the phase cure begns frm and a phase reductn f 8 takes place. At the frequency f.khz du t the zers. n magnary axs, an ersht ccurs n magntude cure and db / dec s added t ts slpe. S, the slpe becmes zer and 8 s added t phase cure du t the mentned zers. Fnally at frequency 97.kHz because f the ples 6.5 9.69, an undersht ccurs n magntude cure and ts slpe becmes db / dec and at the phase cure a phase reductn f 8 takes place and the phase becmes 8. nsderng the pltted respnses fr all alues f the gan, the transfer functn s stable and gan margn becmes. nsderng Bde dagram, when the magntude s db, the phase equals t 7, s the phase margn whch equals t the dfference f 8 and ths phase wll be 9. ) Frequency espnse Analyss fr d Transfer Functn The rt lcus and Bde dagrams fr the extracted transfer functn f d are shwn n Fg.. The nfrmatn btaned frm these dagrams s gen n Table. nsderng the rt lcus shwn n Fg. (a), t s understd that d has fur cmplex ples n left hand sde
Electrcal Engneerng esearch (EE), Vlume, 5 f the magnary axs and tw cmplex zers rght hand plane. Frm Bde dagrams (Fg. (b)), gan margn f the system s 7.5dB and the phase margn s.5. S, the transfer functn s unstable and needs a cmpensatr n nrmal cndtns. Frm rt lcus dagrams, the mentned system has tw cmplex zers n rght hand sde f the magnary axs. In rder t hae a mre stable system the prper cntrller shuld be desgned fr the mentned transfer functn. In next sectn desgnng the prpsed mdel based PI cntrller wll be explaned. x 5 t cus - - - -.5 - -.5.5 (a) OOT OUS Magntude (db) Phase (deg) - 6 FIG. ESPONSE OF TANSFE FUNTION OF d -6 5 6 (b) BOE IAGAMS TABE. INFOMATION OBTAINE FOM POES AN ZEOS OF THE TANSFE FUNTION OF d Ples Zers Value.5 6.5.7.5 9.69. ampng.7.67.669 Oersht (%) 86. 8.9 Frequency (rad/s).6 9.7. Ampltude Step espnse 8 6.5.5.5.5 (msec) FIG. 5. STEP ESPONSE OF THE OPEN OOP SYSTEM refrence errr cntrller s () plant Ts () utput FIG. 6. THE WHOE SYSTEM ONFIGUATION WITH THE PI ONTOE The Prpsed Mdel-Based ntrller Fg. 5 shws the step respnse f the system. Ths shws that the pen lp system des nt reject dsturbances n the nput ltage and cannt regulate the utput ltage. At the fllwng, a mdel based PI cntrller by usng Zegler Nchl s tunng methd wll be explaned. The whle cnfguratn f the cntrller besde the cntrlled system s pltted n Fg. 6. In rder t cntrl the cnerter based n the preusly extracted mdel, the PI cntrller s added t the cntrlled transfer functn f the cnerter. The PI cntrllers are wdely used because they hae nly three parameters that has t be tuned fr the prcess cntrl. The nput f a PI cntrller s an errr sgnal whch s the dfference between the measured alue and a desred reference sgnal fr a prcess arable. The cntrller mnmzes the errr by tunng three cnstant parameters namely the prprtnal term ( K P ), the ntegral term ( K I ) and the dfferental term ( K ). Because f ts smple cnfguratn, the PI cntrller s the mst cmmnly used cntrl methd n ndustry. The mprtant 5
Electrcal Engneerng esearch (EE), Vlume, 5 ams f the PI cntrllers are: elmnatng steady state errr f the step respnse (due t the ntegral actn) reducng the peak ersht (prdng dampng due t derate actn) There are many methds fr tunng the PI parameters r gans. The mst famus methd fr acheng the mentned gal s Zegler Nchls tunng methd. Table 6 shws the parameters btaned fr the cntrller. Fg. 7 shws the step respnse f the system after applyng the desgned PI cntrller. Fg. 8 shws the Bde dagrams f the system befre and after applyng the desgned PI cntrller. Fg. 9 shws the Nyqust dagram f the clsed lp system. The results pre the stablty f the btaned system wth the prpsed cntrller. Stablty f the btaned cntrlled system s well llustrated n Fg. 9. Frm the smulatn results, t s bus that the dsturbance s rejected and ths cntrller behaes ery desrably n tme dman. heckng the bde dagrams depcts that the system has becme a stable ne. Als, calculatns brng ut the gan margn as 6. db and a phase margn f 89.7deg whch are farte frequency dman respnse characterstcs. Then, applyng the prpsed cntrller, the cnerter system becmes stable and regulated. TABE 6. PI PAAMETES OBTAINE K P K I 8.6 K se tme f clsed lp respnse. Settlng tme f clsed lp respnse.78 Oersht f clsed lp respnse Gan Margn f clsed lp respnse 6. Phase Margn f clsed lp respnse 89.7 lsed lp stable? Yes Magntude (db) Phase (deg) - Bde agram 6 8 pen lp -8 clsed lp -6 5 6 FIG. 8. BOE IAGAMS OF THE OPEN OOP AN OSE OOP ONVETE SYSTEM Step espnse Nyqust agram Ampltude.8.6.. Imagnary Axs.5 -.5...6.8.. FIG. 7. STEP ESPONSE OF THE OSE OOP ONVETE SYSTEM - - -.5.5 FIG. 9. NYQUIST IAGAM OF THE OSE OOP ONVETE SYSTEM nclusns In ths paper, mdellng and analyzng f the Zeta cnerter peratng n cntnuus cnductn mde were presented. Applcatn f the prpsed based n sgnal flw graph methd and Masn s gan frmula fr extractng asscated transfer functns and mdellng a dc dc Zeta cnerter was nestgated. In ths prcess, fr mdellng the aerage and small sgnal lnearzatn technques were emplyed. Extractng the fnal transfer functns frm ntal state space equatns were descrbed step by step. Applcatn f sgnal flw graph methd and Masn s rule helps t sle the hgh rder cmplex equatns smply. The presented methd deals wth the equatns wth mre state arables as well. In ths paper, the transfer functns frm nput t utput and cntrl t utput were btaned. Then, by usng the prpsed methd the ples, zers, rt lcus plt and Bde dagrams were extracted and the acheed frequency respnse was analyzed. Fnally, the effcency f the prpsed tuned 6
Electrcal Engneerng esearch (EE), Vlume, 5 PI cntrller as a mdel based technque t desgn a prper cntrl system s shwn. The prpsed cntrl system emplys the PI cntrller wth a specal adjustment fr prper settng f the cntrl respnses. EFEENES [] Kapat, S. ntrl methds fr mprng the perfrmance f dc dc cnerters. Ph.. Thess, Kharagpur, Inda, 9. [] Gatt, G.; Marngu, I.; Mcc, A.; Serp, A. An mpred aeraged mdel fr bst dc dc cnerters n Prc. IEON, 7, Venna,. [] Babae, E.; Mashnch Mahery, H. Inestgatn f buck bst dc dc cnerter peratn n M and the effect f cnerter elements n utput respnse usng mathematcal mdel based n aplace and Z transfrms. Electrc Pwer mpnents and Systems Jurnal, n press.. [] Prewasser,. Mdellng, cntrl and dgtal mplementatn f dc dc cnerters under arable swtchng frequency peratn. Ph. Thess, Klagenfurt Unersty,. [5] Prewasser,.; Agstnell, M.; Unterreder, h.; Marsl, S.; Huemer, M. Mdellng, ntrl, and Implementatn f dc dc nerters fr Varable Frequency Operatn. IEEE Trans. Pwer Electrn. l. 9, n., 87,. [6] Wng,.K.; Man, T.K. Small sgnal mdellng f pen lp SEPI cnerters. IET Pwer Electrn. l., n. 6, 858 868,. [7] Babae, E.; Mashnch Maher, H. Analytcal slutn fr steady and transent states f buck dc dc cnerter n M. Arab. Jurnal Sc. Eng., l. 8, n., 8 97,. [8] Mashnch Mahery, H.; Babae, E. Mathematcal mdellng f buck bst dc dc cnerter and nestgatn f cnerter elements n transent and steady state respnses. Electrcal Pwer and Energy Systems, l., 99 96,. [9] K, W.H. Sgnal flw graph n lp gan analyss f dc dc PWM M swtchng cnerter. IEEE Trans. rcut Sys., l. 5, n. 6, 6 655, 998. [] Smedley, K.; uk, S. Swtchng flw graph nnlnear mdellng technque. IEEE Trans. Pwer Electrn. l. 9, n., 5, 99. [] Veerachary, M. General rules fr sgnal flw graph mdellng and analyss f dc dc cnerters. IEEE Trans. Aerspace Electrn. Sys., l., n., 59 7,. [] Veerachary, M. Analyss f Furth Order nerters: A Flw Graph Apprach. IEEE Trans. Ind. Electrn. l 55, n.,, 8. [] Palm,..; Saldan, J.A.M.; ams, J.. Sgnal Flw Graphs fr Mdellng f Swtchng nerters wth educed edundant Pwer Prcessng. IET Pwer Electrn. l. 5, n. 7, 8 6,. [] Veerachary, M.; Sngamanen, B.S. Stablty Analyss f ascaded Pwer Electrnc System. IEEJ Trans. Elec. Electrn. Eng. l., n. 6, 76 77, 9. [5] enaudneau, H.; Martn, J.; Mbarakeh, B.N; Perfederc, S. nerters ynamc Mdellng wth State Obserer Based Parameter Estmatn. IEEE Trans. Pwer Electrn. l. 99, n., 9,. 7