Comparison between Back-to-Back and Matrix Converters Based on Thermal Stress of the Switches

Transcription

1 Cmparsn between Back-t-Back and Matrx Cnverters Based n Thermal Stress f the Swtches D. Casade, Member, EEE,G. Grand, Member, EEE,C. Rss, A. Trentn, L. Zarr Dpartment d ngegnera Elettrca, Unverstà d Blgna, Vale Rsrgment, 416 Blgna, taly e-mal:(dmenc.casade, gabrele.grand, claud.rss, andrew.trentn, luca.zarr@mal.ng.unb.t Abstract A cmparsn between a matrx cnverter and a back-t-back cnverter feedng a passve lad s presented n ths paper, wth the am f determnng the cnverter tplgy whch yelds the hghest utput pwer per swtches number. The cmparsn has been perfrmed fr dfferent values f the utput frequency. Fr each utput frequency the lad pwer has been ncreased untl ne f the swtchng devces reaches the maxmum thermal stress, s defnng the maxmum utput pwer f the cnverter. Fr ths purpse, a smplfed thermal mdel has been used t evaluate the junctn temperature f the swtches n the bass f the swtch lsses. An accurate cmputer mdel f bth cnverters has been mplemented takng nt accunt the mdulatn laws and the real characterstcs f the swtchng devces. Smulatn results are presented shwng the dfferent behavur f the tw cnverters as a functn f the utput frequency. t has been verfed that matrx cnverters perfrm better than back-t-back cnverters at lw utput frequences. ndex Terms Matrx cnverter, Back-t-Back cnverter, Swtch thermal stress, Swtch lsses.. NTRODUCTON Three-phase matrx cnverters prvde b-drectnal pwer flw, snusdal nput/utput wavefrms, and cntrllable nput pwer factr. Fr these reasns matrx cnverters have receved cnsderable attentn n the last years, and they may becme a gd alternatve t back-tback cnverters. Furthermre, the matrx cnverter allws a cmpact desgn, due t the lack f dc-lnk capactrs [1]- []. The matrx cnverter has been already cmpared wth the back-t-back cnverter btanng sme mprtant but nt cnclusve results. The cmparsn s extremely dffcult due t the hgh number f system parameters (.e. nput flter and lad parameters, swtchng frequency, utput frequency, mdulatn strateges, etc. and t the nherent dfferences between the tw cnverter tplges, such as the maxmum vltage transfer rat. By means f prper cntrl algrthms, the matrx cnverter s able t generate balanced and snusdal utput vltages, whse ampltude can be regulated frm zer t apprxmately 87% f the nput vltage ampltude [4]. The utput vltage f the back-t-back cnverter s related t the DC-lnk vltage, and can be equal r greater than the nput vltage [5]. The swtchng frequences f the tw cnverters are related t the adpted mdulatn strateges and shuld be chsen wth care n rder t make a far cmparsn. Furthermre, bth cnverters need an nput flter fr reducng the nput current harmncs, and the flter parameters are strctly related t the swtchng frequency. n [5]-[8] the cmparsn s perfrmed n terms f ttal swtch lsses, evaluatng the cnverter effcency fr gven peratng cndtns. On the ther hand, n [9] t has been clearly emphaszed that n matrx cnverters the swtch lsses are nt equally shared amng the swtches, beng the dstrbutn related t the utput frequency. Thus, cnsderng nly the ttal swtch lsses as the key-parameter fr the cmparsn may be msleadng. n ths paper the cmparsn between matrx and backt-back cnverters s perfrmed by evaluatng the maxmum utput pwer that each cnverter s able t delver t the lad fr dfferent utput frequency. The cmparsn s carred ut assumng the same types f GBTs and ddes. The maxmum utput pwer s determned takng nt accunt the thermal lmts fr each swtch n the bass f thermal mdel prpsed n [1]. Then, the utput pwer f bth cnverters has been ncreased step by step, evaluatng the ttal lsses f each swtch n the bass f the current and vltage wavefrms acheved n steady-state cndtns. Mntrng the thermal behavur f all swtches, the maxmum utput pwer has been determned as ne f the swtchng devces reaches the maxmum thermal stress. The maxmum utput pwer has been dvded by the number f swtches n rder t defne a quantty representng the utlsatn degree f the swtches, partcularly useful fr the cmparsn. An accurate cmputer mdel f bth cnverters, takng nt accunt the mdulatn laws and the real characterstcs f the swtchng devces, has been mplemented n rder t emphasze hw the behavur f the tw cnverters changes as a functn f the utput frequency. Wth reference t the matrx cnverter, an analytcal apprach s als presented n Appendx, t gve a qualtatve explanatn f sme phenmena ccurrng at partcular values f the utput frequency. t has been shwn that matrx cnverters perfrm better than back-t-back cnverters n the lw frequency range and that matrx cnverters are n general mre sutable fr drve systems whch requre hgh start-up currents.. MATRX CONVERTER SCHEME The schematc crcut f a matrx cnverter feedng a passve lad s shwn n Fg. 1. The system s cmpsed by a vltage supply system, a lne mpedance, an nput flter, a

2 R lne L lne L f R lne Llne Lf Cdc VN C f nput vltage Pwer Crcuts Cmmutatn Cntrl Cntrl System matrx cnverter and a lad mpedance. The nput flter s generally needed t smth the nput currents and t satsfy the EM requrements. Several cntrl technques fr matrx cnverters have been prpsed n lterature [1]-[4]. Amng these, the mst smple s the ne based n detectng the zer-crssng f ne nput vltage fr synchrnsng the nput current. Ths cntrl technque perfrms crrectly assumng an deal pwer supply, but n presence f nput vltage dsturbances these are reflected n the utput sde determnng lw rder vltage harmncs. t s pssble t cmpensate these effects calculatng the duty-cycles necessary t generate balanced and snusdal utput vltages n the bass f the nstantaneus values f the nput vltages, as shwn n Fg. 1. n ths paper, the utput vltage s syntheszed by means f the symmetrcal Space Vectr Mdulatn (SVM technque, that uses nly ne zer vectr fr swtchng perd, thus determnng 8 cmmutatns wthn a cycle perd. The cmmutatn strategy s perfrmed n steps, accrdng t the methd prpsed n [11], whch requres the sgn measurement f bth utput current and nput vltage.. BACK-TO-BACK CONVERTER SCHEME The back-t-back cnverter ncludes tw nverters: the frst perates as an actve rectfer, the secnd as a tradtnal vltage surce nverter (VS, as represented n Fg.. t s assumed that bth cnverters are cntrlled wth a symmetrcal SVM technque wth ne zer vectr fr cycle perd (tw-phase mdulatn, and wth cmmutatn dead-tme cmpensatn [1]. Ths mdulatn strategy requres the mnmum number f swtch cmmutatns per cycle perd, and t s wdely used n practcal applcatns. The system ncludes a vltage supply system wth a lne mpedance, an nput flter, a back-t-back cnverter and a lad mpedance. V. COMPARSON BASS Output current Fg. - Schematc drawng f the matrx cnverter. The matrx cnverter requres 18 GBTs and 18 ddes, whereas the back-t-back cnverter requres 1 GBTs and 1 ddes. The cmparsn s carred ut assumng fr the tw cnverters the same GBTs and ddes. The am f the cmparsn s t evaluate the utlsatn degree f the semcnductr devces, determnng the maxmum utput pwer per swtch crrespndng t the maxmum thermal stress that the swtches can wthstand. The thermal stress f the swtches has been cntnuusly mntred durng the peratn, whle ncreasng the utput pwer. As ne swtch f the cnverter reaches the thermal V N nput vltage Cf nput current Cntrl System AC/DC SVM Cntrl DC/AC SVM Cntrl Fg. 1 - Schematc drawng f the back-t-back cnverter. stress lmt, the crrespndng pwer delvered t the lad has been cnsdered as the maxmum utput pwer f the cnverter. As far as the nput flter s cncerned, the matrx cnverter uses a L-C flter tplgy, dfferently frm the backt-back cnverter, that uses a C-L flter tplgy. n fact, the matrx cnverter can be cnsdered at nput sde as a current surce, whereas the back-t-back cnverter appears as a vltage surce. A reactve current flws thrugh the nput flter capactr f the matrx cnverter, whch results n a reduced pwer factr especally at lw utput pwer. As a cnsequence, the capactr has been chsen n rder t assure at least a pwer factr f.8 wth 1% f the rated utput pwer. After the selectn f the capactr, the nput flter nductance f the matrx cnverter has been chsen n rder t satsfy the EEE Recmmended Practces and Requrements fr Harmnc Cntrl n Electrcal Pwer Systems (EEE Std The nductance f the back-t-back nput flter has been desgned such that the crrespndng vltage drp at the lne frequency s abut.5 p.u.. After the chce f the nductr, the nput flter capactr has been selected n rder t satsfy the EEE recmmended standards. t s als knwn that the desgn f the nput flters s strctly related t the swtchng frequency. Wth reference t matrx cnverter a swtchng frequency f 8 khz has been chsen. Ths frequency crrespnds t a cycle perd f 15 µs that s suffcent fr the dgtal mplementatn f the cntrl algrthm. The back-t-back cnverter needs the defntn f tw swtchng frequences, fr the nput and the utput stage respectvely. The swtchng frequency f the utput stage has been ncreased up t 16 khz n rder t acheve the same number f swtch cmmutatns per secnd, and then a smlar THD f the utput vltage. The swtchng frequency f the nput stage, nstead, has been decreased t 6.6 khz wth the am t reduce the swtchng lsses. n fact, hgher swtchng frequences fr the nput stage are nt necessary t cmply wth the nput current harmnc standards, usng cmmn values f nductance and capactance. Althugh the desgn f the nput flter s an mprtant ssue, the nput flter parameters are nt crucal fr the determnatn f the maxmum utput pwer f the tw cnverters cmpatble wth the swtches thermal stress. Therefre, a mre detaled analyss cncernng the nput flter s nt necessary fr the am f the paper. As an example, n Tab. the nput current harmncs f the tw cnverters peratng at rated pwer are cmpared wth the recmmended lmts, when the rat between the

3 TABLE CURRENT DSTORTON LMTS FOR GENERAL DSTRBUTON SYSTEMS N PERCENT OF THE MAXMUM DEMAND LOAD CURRENT ndvdual harmnc rder (dd harmncs Order h<11 11 h<17 17 h< h<5 5 h TDD Lmts (% Matrx (% B-t-B (% maxmum shrt-crcut current n the pnt f cmmn cuplng and the maxmum demand lad current s between 1 and 1. All the system parameters are shwn n Tab.. t s pssble t verfy that usng the flter parameters gven n Tab., bth cnverters cmply wth the EEE recmmended standards. t s knwn that n matrx cnverters the ttal lsses f the swtchng devces are nt equally shared amng the swtches, and the lsses dstrbutn s mre r less unfrm dependng n the utput frequency. As a cnsequence, the cmparsn between the tw cnverters has been carred ut assumng the utput frequency as a parameter. The perfrmance f the tw cnverter tplges has been tested fr dfferent values f the utput frequency n the range -15 Hz. The vltage has been changed wth the frequency accrdng t the well-knwn cnstant V/Hz law, as n nductn mtr drves. Thus, the utput vltage s vared prprtnally t the frequency untl 5Hz. Fr hgher frequences the phase t phase utput vltage s kept cnstant,.e. V and 8V fr the matrx cnverter and the back-t-back cnverter respectvely. At lw frequences, the utput vltage has been changed n rder t cmpensate the vltage drp n the statr wndng resstance. V. THERMAL MODEL OF THE SWTCHES Under the assumptns made, the maxmum utput pwer delvered by the cnverters t the lad has been determned takng nt accunt the maxmum thermal stress f the swtches. The junctn temperature f the swtchng devces has been mntred by means f the dynamc thermal mdel f a sngle GBT shwn n Fg [1]. The mdel represents the junctn temperature θ j n terms f the statc junctn-t-case thermal resstance R and the thermal tme cnstant τ = R C j. Althugh ths mdel s smplfed and t des nt take nt accunt the nn-unfrm nternal devce temperature, t pnts ut that the devce thermal transfer functn H ( s s a lw-pass flter wth a cut-ff frequency f θ = 1/ πτ, whch s cmparable wth the cnverter utput fundamental frequency ( f ut, but much lwer than the swtchng frequency ( f sw. As a cnsequence, the junctn temperature s strctly dependent n the utput frequency. P lss C j θ j R θ case Fg. Smplfed thermal mdel fr the GBT. TABLE SYSTEM PARAMETERS Smulatn Parameters Back t Back Matrx Cnverter V N 8 V(RMS, 5 Hz 8 V(RMS, 5 Hz R lne.11ω.11ω L lne.167mh.167mh V DC 6V - C DC µf - C f 5µF (Y 4 µf (Y L f 1. mh.5 mh R.64 C/W.64 C/W C j 1. mj/ C 1. mj/ C θ case 7 C 7 C f sw 6.6kHz (ac-dc,16khz (dc-ac 8kHz cs ϕ lad.8.8 V ut f ut < 5 Hz, cnst. V/Hz f ut > 5 Hz, V ut = 8 V Fr bth cnverters the maxmum junctn temperature and the case temperature have been assumed 15 C and 7 C respectvely. Then, the maxmum junctn-t-case temperature s θ,max = 8 C. V. SMULATON RESULTS f ut < 5 Hz, cnst. V/Hz f ut > 5 Hz, V ut = V Ddes HFA16PB1 HFA16PB1 GBTs RG4PH5U RG4PH5U The behavur f the tw cnverters has been analysed usng the electrnc crcut analyss prgram MCROCAP 7., whch prvdes accurate mdels f the pwer swtches, mprvng the smulatn relablty. Frstly, the behavur f bth cnverters has been verfed wth reference t the rated utput frequency f 5 Hz. Fg. 4 shws the junctn-t-case temperature behavur fr all the 18 matrx GBTs crrespndng t the maxmum utput pwer. As can be seen, the thermal stress f the swtches s nt equal, snce 6 GBTs are mre stressed than the thers. n fact, when the utput frequency s equal t the nput frequency, the utput current s nt shared unfrmly amng all the swtches. The prtn f the utput current that flws thrugh each swtch depends n the dsplacement angle between the nput and utput vltage vectrs, and n the lad pwer factr [7]. The current wavefrm and the junctn-t-case temperature behavur f ne the mst stressed swtches are shwn n detals n Fg. 5. The junctn-t-case GBTs temperature behavur f the back-t-back cnverter s represented n Fg. 6. The upper traces are related t the utput stage, whereas the lwer traces crrespnd t the nput ne. t s evdent that the thermal stress fr nput stage s lwer than that f the utput ne. The cnverse ccurs fr regeneratve peratn f the back-t-back cnverter. Fg.7 shws n detals the current wavefrm and the junctn-t-case temperature f ne swtch. t can be seen n Fg. 6 that the thermal stress f the back-t-back cnverter utput stage swtches s mre unfrm n cmparsn wth the thermal stress f the matrx cnverter swtches. As a cnsequence, the maxmum utput pwer per swtch that can be acheved wth a matrx cnverter at 5 Hz (P matrx /18 s expected t be lwer than that f the back-t-back cnverter (P b-t-b /1. Fg. 8 shws the junctn-t-case temperature wavefrms

4 C θ ms/dv Fg. 4 - Junctn-t-case temperature wavefrms f the matrx cnverter swtches fr the maxmum utput pwer crrespndng t an utput frequency f 5 Hz. C 1 Output stage ms/dv Fg. 5 Junctn-t-case temperature and current wavefrms f the ne f the mst stressed matrx cnverter swtches, fr the same peratng cndtns f Fg θ nput stage 5ms/dv Fg. 6 - Junctn-t-case temperature wavefrms f the back-t-back cnverter swtches fr the maxmum utput pwer crrespndng t an utput frequency f 5 Hz. f the matrx cnverter GBTs crrespndng t the maxmum utput pwer and wth an utput frequency f 6 Hz. The current wavefrm and the junctn-t-case temperature behavur f ne f the mst stressed swtches are shwn n Fg. 9. The junctn-t-case GBTs temperature behavur f the back-t-back utput nverter, crrespndng t the maxmum utput pwer at 6 Hz, s represented n Fg. 1. Fg. 11 shws n detals the current and the junctn-t-case temperature wavefrms f ne swtch. Cmparng Fg. 6 and Fg. 1, bth referrng t back-tback cnverter, t s evdent that the thermal stress at 6 Hz s qute smlar t that at 5 Hz, and all swtches are equally stressed. On the cntrary, wth reference t matrx cnverter the cmparsn between Fg. 4 and Fg. 8 emphasses that the utput frequency has a relevant effect n the thermal stress C ms/dv Fg. 7 Junctn-t-case temperature and current wavefrms f ne f the mst stressed back-t-back cnverter swtch, fr the same peratng cndtns f Fg. 6. f the swtches. At 5Hz the GBTs can be dvded nt three grups, havng a qute dfferent behavur n terms f thermal stress, whereas at 6Hz the 18 GBTs are n turn equally stressed. The maxmum utput pwer achevable by the tw cnverters as a functn f the utput frequency s summarsed n Fg. 1. t s evdent that the utput pwer f matrx cnverter s always hgher than that f back-t-back cnverter, shwng a decrease arund the frequency values f 5 and 1 Hz. Fg. 1 shws the lad current crrespndng t the maxmum utput pwer as a functn f the utput frequency, fr the matrx and the back-t-back cnverters. n ths fgure t s evdent the better perfrmance f the matrx cnverter n terms f maxmum utput current, especally n the lw utput frequency range. The reasn f s lw values f the utput current fr the back-t back cnverter 1 θ ms/dv Fg. 8 Junctn-t-case temperature wavefrms f the matrx cnverter swtches fr the maxmum utput pwer crrespndng t an utput frequency f 6 Hz. 1 C ms/dv Fg. 9 - Junctn-t-case temperature and current wavefrms f the mst stressed matrx cnverter swtches, fr the same peratng cndtns f Fg θ 4 4ms/dv Fg. 1 - Junctn-t-case temperature wavefrms f swtches f the back-t-back cnverter utput stage fr the maxmum utput pwer crrespndng t an utput frequency f 6 Hz. 1 ms/dv Fg. 11 Junctn-t-case temperature and current wavefrm f ne swtch f the back-t-back cnverter utput stage, fr the same peratng cndtns f Fg.1.

5 P ut [kw] f ut [Hz] Fg. 1 Maxmum utput pwer as a functn f the utput frequency. 8 ut [A] f ut [Hz] Fg. 1 Maxmum utput current as a functn f the utput frequency. 7 A V Fg. 14 -V peratng area fr matrx cnverter and back-t-back cnverter. P n ut [kw] 1,6 swtches 1,,8, s that, at lw frequency, the utput current s nt equally shared amng the sx swtches. Ths stuatn s smlar t the ne ccurrng when the matrx cnverter perates at 5 Hz. On the cntrary, the matrx cnverter s able t delver hgh currents at lw frequency because these currents are equally shared amng the 18 swtches. Furthermre, t can be nted that n Fg. 1, as n Fg. 1, the utput current f the matrx cnverter shws mre r less evdent decreases at the frequency values f 5, 5 and 1 Hz. Mre detals abut these aspects, that are strctly related t the quas-perdc peratn f matrx cnverters, are gven n Appendx. A further cmparsn between the tw types f cnverter s shwn n Fg. 14, where the peratng area n the -V plane s represented. Ths fgure clearly emphasses that matrx cnverters are mre sutable fr drve systems that n general requre hgh start-up currents. n rder t make a far cmparsn, ne shuld take nt accunt that the tw cnverter tplges are realzed wth a dfferent number f swtches (.e. 18 fr the matrx cnverter and 1 fr the back-t-back cnverter. Fr ths purpse tw mre sgnfcant quanttes have been ntrduced, whch represent the maxmum utput pwer per swtch and the crrespndng utput current per swtch. These new quanttes are represented n Fgs. 15 and 16 respectvely. Fg. 15 shws that the utput pwer per swtch f the matrx cnverter s always lwer than that f back-t-back cnverter, except fr frequency values rangng frm t abut Hz. On the ther hand, t can be seen frm Fg. 16 that the lad current per swtch f the matrx cnverter s always hgher than that f the back-t-back cnverter, except fr frequency values arund 5 Hz. Frm Fgs. 15 and 16 t can be cncluded that n terms f maxmum utput pwer per swtch the tw cnverter tplges shw practcally a smlar perfrmance, whereas n terms f utput current per swtch the matrx cnverter shuld be preferred t back-t-back cnverter partcularly n the lw utput frequency range. These results culd be usefully emplyed n the chce f the cnverter tplgy fr drve systems, nce the peratng cndtns and the verlad capablty were specfed n detals. V. CONCLUSON n ths paper a cmparsn between matrx and back-tback cnverters has been carred ut takng nt accunt the thermal lmts f the swtchng devces and the dfferent number f swtches requred n the tw tplges. The same GBTs and ddes have been used fr the cmparsn. Therefre, the maxmum utput pwer per swtch and the crrespndng utput current per swtch have been regarded as the mst representatve quanttes fr a far cmparsn. The analyss made t pssble t emphasse the dfferent behavur f the tw cnverters as the utput frequency changes. The mst mprtant dfference between the tw tplges has been fund n the lw frequency range, where the matrx cnverter swtches are equally stressed, dfferently frm the back-t-back cnverter. Ths results n the pssblty t better explt the current capablty f the swtches fr drve systems, whch usually requre hgh verlad capablty n the lw frequency range. V. APPENDX An analytcal apprach s presented, demnstratng that the hghest temperature f the swtches f a matrx cnverter crrespnd t the frequency values f 5, 5 and 1 Hz. n general, the wrst peratng cndtn ccurs when the utput frequency s equal t the nput frequency. n the fllwng, the phase angles f the nput vltage f ut [kw] Fg. 15 Maxmum utput pwer per swtch as a functn f the utput frequency. n ut swtches [A] fut [Hz] Fg. 16 Maxmum utput current per swtch as a functn f the utput frequency.

6 and utput vltage vectr are dented wth α and α, respectvely. n steady state peratn, fr gven lad and vltage transfer rat, the mean pwer lss P f a sngle swtch n a cycle perd T c depends n α and α. Ths dependence can be expressed as fllws P = P (α, α (1 The functn P s perdc wth respect α and α,.e. P(α, α = P(α + π, α = P(α, α + π ( As shwn n (, P s a quas-perdc functn and t can be expressed by means f a duble Furer seres as fllws [1]: P ( α, α = Re{ P ( k= h= J ( kα + hα k, he } where P k, h s the Furer ceffcent f ndex h and k. n steady-state cndtn α and α can be expressed as a functn f tme as fllws α = ω t (4 α = ω t (5 where ω and ω are the nput and utput angular frequency, respectvely. Substtutng (4 and (5 n ( leads t the fllwng frm f the pwer lsses: P ( t = Re{ P e (6 k= h= J ( kω + hω t k, h } The junctn-t-case temperature f the swtch can be fund thrugh the thermal mdel f the GBT represented n Fg.. t can be derved frm the thermal mdel that the transfer functn H (ω between the junctn-t-case temperature and the pwer lsses s a lw pass flter, expressed as fllws: R H ( ω = (7 1+ jωτ Applyng (7 t (6 leads t the fllwng expressn f the junctn-t-case temperature f the swtch: ϑ j( kω + hω t ( t = Re{ Pkh H ( kω hω e } (8 k= h= t s wrth ntng that the ampltude f the harmncs f P(t decreases prprtnally t the prduct f the harmnc ndexes k and h. n addtn, the harmncs wth frequency much hgher that the cut-ff frequency f the thermal flter gve a neglgble cntrbute t the temperature. As a cnsequence, (8 can be apprxmated by a sum wth a fnte number f terms, as fllws ϑ t = P H ( + P H ( ω cs( ω + ϕ + (,1 t, 1 P1, 1H ( ω ω cs(( ω ω t + ϕ1, 1 + P1, H ( ω ω cs(( ω ω t + ϕ1, + P, 1H ( ω ω cs((ω ω t + ϕ, 1 (9 where ϕ k,h s the phase angle f P k, h. An apprxmate estmate f the maxmum junctn-tcase temperature can be derved frm (9, as fllws: ϑ,max ( ω = P H ( + P,1 H ( ω + P1, 1H ( ω ω (1 + P1, H ( ω ω + P, 1H (ω ω As can be seen frm (1 the maxmum junctn-t-case temperature depends n the utput frequency. n rder t determnate the values f ω that make (1 t assume a relatve maxmum, the dervatve f (1 s needed: ϑ,max τ = [ P,1 H ( ω ( ω + P1, 1H ( ω ω ( ω ω + ω R P 1, H ( ω ω ( ω ω + P H (ω ω ( ω ω ], 1 (11 t shuld be nted that fr ω ω the dmnant term n (11 s P H ( ω ω ( ω ω 1, 1, whereas the ther terms, multpled by the factr H ( ω, are neglgble. As a cnsequence, ω ω can be cnsdered an apprxmate slutn f (11, crrespndng t a relatve maxmum f (1. Wth smlar prcedure, the slutns ω ω, ω ω / can be fund, whch crrespnd t lcal maxma f lwer mprtance than the prevus ne. X. REFERENCES [1] A. Alesna, M. G. B. Venturn, Sld-state pwer cnversn: a Furer analyss apprach t generalzed transfrmer synthess, EEE Transactns. n Crcut and Systems, vl. CAS-8, n. 4, Aprl, 1981, pp [] A. Alesna, M. Venturn, Analyss and Desgn f Optmum- Ampltude Nne-Swtch Drect AC-AC Cnverters, EEE Transactns n Pwer Electrncs, vl. 4, n.1, January, 1989, pp [] P.W. Wheeler, J. Rdrguez, J. C. Clare, L. Emprngham, Matrx Cnverter. A technlgy Revew, EEE Transactns f ndustral Electrncs, vl.49, n., Aprl,, pp [4] D. Casade, G. Serra, A. Tan, L. Zarr, Matrx Cnverter Mdulatn Strateges: A New General Apprach Based n Space-Vectr Representatn f the Swtch State, EEE Transactns f ndustral Electrnc, vl. 49, n., Aprl,, pp [5] L. Malesan, L. Rssett ; P. Tent, P. Tmasn, "AC/DC/AC PWM cnverter wth reduced energy strage n the DC lnk," EEE Transactns n ndustry Applcatns, vl. 1, n., March-Aprl 1995, pp [6] S. Bernet, S. Pnnalur, R. Techmann, Desgn and Lss Cmparsn f Matrx Cnverters and Vltage-Surce Cnverters fr Mdern AC Drves, EEE Transactn n ndustral Electrncs, vl. 49, n., Aprl,, pp [7] M. Apap, J.C. Clare, P.W. Wheeler, M.Bland, K. Bradley, An Apprach t the Analyss n Effcency and Devce Cntrl Pwer Lss Dstrbutn fr Matrx Cnverters, n Prceedngs f EPE Cnference,, SBN: , n. 59, pp [8] C. Klumpner, F. Blaabjerg, P. Thgersen, "Evaluatn f the Cnverter Tplges Suted fr ntegrated Mtr Drves", EEE ndustry Applcatns Cnference, 1-16 Octber, vl., pp [9] J. K. Kang, H. Hara, E.Yamamt, E. Watanabe, Analyss and Evaluatn f B-Drectnal Pwer Swtch Lsses fr Matrx Cnverter Drves, EEE ndustral Applcatn Cnference, vl.1, n. 1, Octber,, pp [1] F. Flcr, C. G. L Banc, A Smplfed Thermal Analyss Apprach fr Pwer Transstr Ratng n PWM-Cntrlled DC/AC Cnverters, EEE Transactns n Crcuts and System, vl. 45, n.5, May, 1998, pp [11] D. Casade, A. Trentn, M. Matten, M. Calvv, Matrx Cnverter Cmmutatn Strategy Usng bth Output Current and nput Vltage Sgn Measurement, n Prceedngs f EPE Cnference,, SBN: , n. 111, pp.1-1. [1] D. Casade, G. Serra, A. Tan, L. Zarr, F. Prfum, Perfrmance Analyss f a Speed Sensrless nductn Mtr Drve Based n a Cnstant Swtchng Frequency DTC Scheme, EEE Transactns n ndustry Applcatns, vl.9, n., March-Aprl,, pp [1] R. C. Hlbrn, Chas and Nnlnear Dnamcs. Oxfrd, U.K: Oxfrd Unvesty Press