constrants at the twotermnal port,.e. the voltagestff sources may never be shortcrcuted and the lnk current may never be opencrcuted. Hence, we may ca

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1 7th IEEE Power Electroncs Specalsts Conference / June 18 22, 26, Jeju, Korea ThE14 A Carrer Based PWM Algorthm for Indrect Matrx Converters Bngsen Wang Gr Venkataramanan Department of Electrcal and Computer Engneerng Unversty of WsconsnMadson 141 Engneerng Drve Madson, WI 76 USA bngsen@cae.wsc.edu; gr@engr.wsc.edu Abstract In ths paper, the ndrect matrx converter s systematcally studed wth the snglepolemultplepole representaton. A carrer based PWM algorthm s developed n two steps. Frst, the contnuous modulaton functons for all the throws are derved based on the desred snusodal nput currents. Then the swtchng functons are derved from the modulaton functons wth focus on the zero current commutaton. The proposed PWM algorthm s verfed by numercal smulaton and hardware expermentaton on a laboratory prototype matrx converter. I. INTRODUCTION Varous attractve features of the matrx converter, such as a potental for hgh power densty through elmnaton of bulky passve components, hgh qualty nput and output current/voltage waveforms, are leadng to contnuous research efforts to enable ther adopton wdely [16]. Whle the matrx converter was frst studed as a class of frequency changers realzed usng controlled swtches by Gyugy [7], sgnfcant progress n hgh frequency synthess was made by Venturn n 198 [81], wth voltage transfer rato lmted to one half. An mproved modulaton strategy wth a voltage transfer rato of.866 was publshed n 1989 [11, 12]. In addton to Venturn s modulaton methods, ndrect modulaton methods based on a fcttous dc lnk concept [118] and space vector technques have been developed further [17, 192]. Irrespectve of the modulaton methods used, robust commutaton s crtcal, as has been extensvely nvestgated and examned [24]. As an alternatve to the standard 9swtch topology, the rectfernverterwthoutdclnk structure and ts varatons have been caully examned [16, 17], leadng to a smple and robust commutaton on the bass of ts topologcal propertes [4]. A space vector modulaton scheme for the ndrect matrx converter (IMC was developed wth focus on the desred output voltage n [4]. In ths paper, a carrer based PWM algorthm s proposed and derved based on the desred snusodal nput currents and output voltages. The dervaton proceeds wth the calculaton of modulaton functons followed by the generaton of swtchng functons. Compared to the space vector modulaton scheme, the proposed carrer based PWM algorthm demands less computatonal complexty, whle retanng the features of smple and robust commutaton. Thus, resources of the dgtal sgnal processor commonly employed n the practcal realzaton wll be avalable to assume complex computatonal tasks for the controller rather than the modulator. Furthermore, wth more ncreasngly avalable feld programmable gate array (FPGA, the frmware mplementaton of carrer based PWM algorthm may feature the relablty of hardware and the flexblty of software. The paper s organzed as follows: In Secton II the IMC s represented usng snglepolemultplethrow (SPMT swtches. The modulaton functons are derved n Secton III followed by swtchng functons presented n Secton IV. The numercal smulaton and the expermental results are presented n Secton V and Secton VI, respectvely. A summary of the paper s presented n the concludng secton. II. IMC REPRESENTED BY SPMTS A famly of varous IMC realzatons has been developed from the topology shown n Fg. 1 by mposng addtonal operatng constrants such as undrectonal power flow and/or lmted power factor range. Some of the realzatons that ntroduce addtonal operatng constrants feature reduced number of controlledswtch count []. The focus of ths paper s on the IMC topology wthout reduced swtch count, snce bdrectonalpower capablty and the wde power factor range of ths converter are of prmary nterests to the authors. However, the PWM algorthm developed here can be readly appled to the reducedswtch topologes wth lttle or mnor modfcatons. The IMC topology shown n Fg. 1 conssts of a cascaded connecton of two brdge converters. One brdge s connected to a set of voltagestff sources v a, v b and v c, whch would be formed usng capactors n practcal mplementatons. The other brdge s connected to a set of currentstff sources u, v, w, whch would be realzed usng nductors n practcal mplementatons. The brdge connected to the voltagestff sources acts lke a current source converter n terms the voltage constrants at the threetermnal port and current /6/$2. 26 IEEE. 278

2 constrants at the twotermnal port,.e. the voltagestff sources may never be shortcrcuted and the lnk current may never be opencrcuted. Hence, we may call ths brdge as current source brdge or CSB. In a dual manner, the brdge connected the currentstff sources acts lke a voltage source converter n terms of the voltage constrants at the twotermnal port and the current constrants at the threetermnal port,.e. the stff currents may not be opencrcuted and the lnk voltage should never be shortcrcuted. Thus, we may call ths brdge as voltage source brdge or VSB. These nherent operatonal constrants to the IMC topology can be accommodated by the graphc representaton as shown n Fg. 2, where the CSB s represented by two snglepoletrplethrow (SPTT swtches S p and S n and the VSB s represented by three snglepoledoublethrow (SPDT swtches S u, S v, S w. Wth ths arrangement, the throws n each swtch can be modulated ndependently wthout volatng the termnal constrants. III. MODULATION FUNCTIONS For threephase ac power converson, the stff voltages and stff currents at the ac ports shown n Fg. 2 may be assumed to be balanced threephase quanttes gven by v a v b v c a b c CSB p VSB Fg. 1 Schematc of the IMC topology under study. u v w u v w ( α ( ( α ( π ( α ( π ( β ( ( β ( π ( β ( π va = Vcos t u = Iocos o t v cos 2 b = V t ; cos v = Io o t 2 v cos 2 c = V t cos w = Io o t 2 where V s source voltage ampltude; I o s the current source ampltude; α (t s the phase angle of the voltage source, gven by α( t = ωt α ; β o (t s the phase angle of the current source, gven by βo( t = ωot βo The desred fundamental components of the currents at the ac termnals of the CSB may be expressed as a_ = I_ cos ( β( t (2 b_ = I_ cos ( β( t 2 π / = I cos ( β ( t 2 π / b where β( = ω β Consequently, the power factor angle on the CSB becomes t t. φ α β (1 = ( A. CSB Modulaton Functons Each fundamental perod of the ac quanttes at the CSB port may be dvded nto sx sectors as shown n Fg.. If we assume the movng average of the dc lnk current to be constant, the modulaton of the CSB would be dentcal to that of a PWM current source acdc converter. If the desred erence nput phase current s maxmum durng one sector, then the correspondng throw of S p s closed durng the entre sector. The dc lnk current returns through one of the three throws of S n, through approprate commutaton. Smlarly, f the erence phase current s mnmum durng one sector, the correspondng throw of S n s closed durng the entre sector. The dc lnk current returns through one of the throws of S p through approprate commutaton. Notce, f the throws correspondng to the same phase of S p and S n are closed smultaneously (termed a zero state of the CSB, each of the ac lne currents are zero. Thus by varyng the nterval of the zero state, the current source converter may be modulated to regulate the ampltude of the ac fundamental component of the lne currents, wth a constant dc lnk current. S p p Sector 1 Sector 2 Sector Sector 4 Sector Sector 6 T ap T bp T cp a_ b_ c_ u n v a v c v b a b c S w S u S v T wp T wn T up T vp T un T vn v u v v v w v w T an T bn T cn S n Fg. Waveforms of typcal snusodal erence currents at the CSB ac port, dvded nto sx ntervals. Fg. 2 Schematc of the IMC represented usng SMPTs. 2781

3 Alternatvely, n the proposed modulaton strategy for the IMC, the ampltude of the ac fundamental component of the lne currents are regulated by modulatng the VSB, wthout usng zero states for the CSB and not compromsng any waveform qualty. For example, throw T ap s closed T an s open durng the entre nterval of Sector 1, and the correspondng modulaton functons m ap s 1 and m an s. The dc lnk current flows through T ap, and returns through T bn or T cn. In order for ths modulaton approach to be successful a key modulaton assumpton needs to be satsfed, namely, the averaged lnk current may be regulated (by approprately modulatng the VSB, as wll be llustrated n the followng subsecton to follow the erence current a_ durng Sector 1. Then, t follows that the modulaton functons m bn and m cn can be determned to be m m bn cn = = b_ a_ c_ a_ It may be notced that the ampltude of the erence currents I _ does not appear n (4, and thus the modulaton functons of the CSB only determne the phase angle of the syntheszed current relatve to the stff voltage source. In other words, whle the power factor angle of CSB currents s set by the modulaton functons, ampltude of the syntheszed currents are determned ndrectly by the VSB (naturally, through power balance consderatons. Extendng ths strategy to the other sectors of the CSB ac current erences, the modulaton functons for each throw of the CSB of can be calculated as tabulated n TABLE I. The correspondng waveforms of modulaton functons are also plotted n Fg. 4. B. VSB Modulaton Functons Havng determned the modulaton functons for the throws of CSB durng the entre perod, we proceed to calculate the modulaton functons of the VSB, to synthesze approprate output voltage, whle valdatng the key modulaton assumpton from the prevous subsecton. The modulaton functons for each of the three phaselegs for the VSB can be expressed as mu = Mocos ( αo( t ( mv = Mocos ( αo( t 2 π / m = M cos α ( t 2 π / w o o ( where M o s the modulaton ndex and < M o < 1 and the phase angle αo( t = ωot α. If njecton of trplen o harmoncs are adopted, M o can reach 2/ wthout overmodulaton. The power factor angle φ o at the VSB termnals s defned by φ = α β (6 o o o Valdaton of the key modulaton assumpton s made through makng the modulaton ndex M o to be a tme varyng functon, as descrbed further. Furthermore, n the conventonal snetrangle scheme, the modulaton functons (4 TABLE I. m ap 1 m bp MODULATION FUNCTIONS FOR THE THROWS OF CSB. Sector 1 Sector 2 Sector Sector 4 Sector Sector 6 a_ a_ c_ b_ b_ b_ 1 c_ a_ c_ c_ 1 a_ b_ a_ a_ 1 b_ c_ b_ b_ a_ c_ 1 c_ c_ 1 m cp m an m bn m cn 1 1 a_ b_ Sector 1 Sector 2 Sector Sector 4 Sector Sector 6 m ap m bp m cp m cn Fg. 4 Typcal waveforms of modulaton functons for the varous throws of CSB. for each phase leg s shfted and scaled to obtan the modulaton functons for throws of the swtches. For nstance, the modulaton functons for the throws T up, T vp and T wp are ( 1 mu mup = 2 ( 1 mv (7 mvp = 2 ( 1 mw mwp = 2 Now, the average of the lnk current may be determned to be p = mupu mvpv mwp (8 w Substtutng the expressons for the output currents u, v and w from (1 nto (8, the averaged lnk current may be determned to be p = MoIocosφ (9 o 4 The key modulaton assumpton requres that < p > should follow a_ durng Sector 1, and c durng Sector 2, etc. m an m bn 2782

4 correspondng to the unty modulaton ndex rows of TABLE I., under each sector. Thus durng Sector 1, a_ = MoIocosφ (1. o 4 In order to valdate (1, the ampltude M o of the VSB modulaton functon, beng the only free varable may be determned to be 4 I_ M o( t = cos ( β( t (11 Iocosφo usng (2 durng Sector 1. Through recprocty, M o (t durng Sector 1 may also be expressed as M o( t = cos ( β( t (12 Vcosφ The varaton of M o (t durng other sectors of the CSB current erence waveforms can be determned n a smlar manner, as tabulated n TABLE II. The resultant modulaton functon ampltude M o (t together wth the modulaton functons for each phaseleg of the VSB are plotted n Fg.. TABLE II. M o (t VSB MODULATION FUNCTION AMPLITUDE IN DIFFERENT SECTIONS. M o (t 4 V Vcosφ cos β ( 2 t π V cos φ cos β ( 2 t π V cos φ cos β( t Vcosφ cos β ( 2 t π V cos φ cos β ( 2 t π V cosφ o_ Sector 1 cos ( β( t Sector 2 ( Sector ( Sector 4 ( Sector ( Sector 6 ( Fundamental perod of the CSB ac quanttes dvded nto 6 sectors π 2π π t ω ω ω m u (t m v (t m w (t π ω o Fg. Waveforms of VSB modulaton ampltude M o (t and phase leg modulaton functons m u (t, m v (t, m w (t. 2π ωo t IV. SWITCHING FUNCTIONS Once the modulaton functons are derved, the swtchng functon h xy for each throw S xy can be determned wth specal care of the swtchng sequence. A. CSB Swtchng Functons The CSB swtchng functons are generated by comparng the modulaton functons shown n Fg. 4 wth a lnear carrer at the swtchng frequency. However, the algnment of the PWM pulses wth respect to the carrer has to be caully chosen to elmnate abrupt dscontnutes at sector boundares. It may be observed from Fg. 4, that durng each sector there are only two unclamped (or actve modulaton functons, one of them decreasng and the other ncreasng and all the other modulaton functons are clamped (or nactve at ether zero or unty. At each sector boundary one of the actve modulaton functons becomes nactve (clamped at zero or unty, and an nactve modulaton functon becomes actve. In order to preclude any dscontnutes n PWM pulses at sector boundares, the followng smple condtons needs to be satsfed: For postve throws, ncreasng pulse wdths have to be left algned and decreasng pulse wdths have to be rght algned; for negatve throws, ncreasng pulse wdths have to be rght algned and decreasng pulse wdths have to be left algned. Such a strategy usng a sawtooth carrer to generate the swtchng functon h bn and the h cn s obtaned by nvertng h bn as llustrated n Fg. 6. It can be observed that all PWM pulses are contnuous durng all sector boundares. B. VSB Swtchng Functons Swtchng events of VSB swtchng events are coordnated wth the CSB swtchng events, such that CSB commutaton takes place when the lnk current s zero [4]. To llustrate the coordnaton between the VSB and CSB, a sngle swtchng cycle of durng Sector 2 from Fg. 6 s zoomed n as llustrated n Fg. 7. It may be noted that the carrer waveform of the VSB carrer trangle s not symmetrcal as would be typcal. Instead, the rsng nterval d 1 T s and the fallng nterval d 2 T s of VSB carrer are determned by the CSB swtchng functons. In ths manner, the two CSB commutaton events n each swtchng cycle always take place durng the zero states of VSB,.e. when ether all the three top throws or all the three bottom throws of the VSB are connected to the lnk bus. Durng the zero states, the lnk current s zero. Theore CSB can commutate wth zero lnk current, whch elmnates the swtchng losses, and the need for overlap tme, elmnatng any possble short crcut between stff voltages feedng the CSB. V. NUMERICAL SIMULATION VERIFICATION In order to verfy the modulaton scheme descrbed n Secton III and Secton IV, a numercal smulaton has been carred out on a detaled model bult usng Matlab Smulnk. 278

5 Sector 1 Sector 2 Sector Sector 4 Sector Sector 6 CSB carrer h ap m bn m bp m cn m cp m an m ap h cn h bp Rght algned h an Left algned h cp h bn Rght algned Left algned Fg. 6 Waveforms llustratng the generaton of CSB swtchng functons. CSB modulaton functons CSB swtchng functons m bp h ap h bp CSB carrer Schematc of the power crcut used n ths smulaton s shown n Fg. 8 wth the parameters lsted n TABLE III. Selected waveforms from the smulaton results are plotted n Fg. 9 Fg. 11. The currents drawn from the source and load currents are shown n Fg. 9. Fg. 1 llustrates the source voltage of a phase v a, the flter capactor voltage of a phase v Cfa, the lnk voltage and the source current of a phase a. In ths smulaton, unty power factor on the nput sde s selected. Fg. 11 shows the output lnelne voltage v uv, the lneneutral voltage v u, the lnk voltage and the load current u. VSB modulaton functons m up m vp VSB carrer v a v b v c a L f b c v uv u v u v w m wp v Cfa C f L load R load VSB zero states VSB swtchng functons h up h vp Fg. 8 Schematc of the power crcutry used for smulaton and expermental test. h wp d 1 T s d 2 T s Fg. 7 Waveforms llustratng of the coordnaton between the VSB and the CSB commutaton processes TABLE III. LIST OF PARAMETERS FOR SIMULATION v a,b,c 1 V pk_l_n f swtchng 1 khz L f 2 μh R load 8 Ω C f μh L load 1 mh f nput 6 Hz f output 4 Hz 2784

6 a,b,c (A u, v, w (A v a t (s Fg. 9 Waveforms of source currents (top panel and load currents (bottom panel obtaned usng smulatons VI. EXEPRIMENTAL RESULTS In order to further valdate the proposed modulaton algorthms, a prototype of the converter has been constructed n the laboratory. A photograph of the prototype converter s shown n Fg. 12. The DSP board s used as the controller platform. The modulaton scheme descrbed n proceedng sectons s mplemented n the FPGA on the DSP board. The measured gatng sgnals for the sx throws n CSB and VSB are shown n Fg. 1. It can be observed at no nstant the top and bottom throws n the same phase leg are closed smultaneously,.e. no zero states n the CSB modulaton. Fg. 14 llustrates that the CSB only commutates when the lnk current s zero. The top three traces n Fg. 14 are swtchng sgnals for the three top throws n VSB. The swtchng events, as shown n the bottom trace, only happen when the top throws n VSB are all closed or all open, whch corresponds to zero lnk current for ether case. Selected waveforms (correspondng to the smulaton waveforms from Fg. 9 Fg. 11 are shown n Fg. 1 Fg. 16, llustratng excellent conformty. v Cfa a t (s Fg. 1 Waveforms of source voltage, flter capactor voltage, lnk voltage and source current, obtaned usng smulatons, from the top to bottom, respectvely. 2 v AB A t (s Fg. 11 Waveforms of load ll voltage, load ln voltage, lnk voltage and load current obtaned usng smulatons, from the top to bottom respectvely. T ap T bp T cp T an T bn T cn Fg. 12 A photograph of the prototype of the converter used for obtanng expermental tests t (s Fg. 1 Waveforms of measured of gatng sgnals for the CSB throws: from top to bottom T ap, T bp, T cp, T an, T bn, T cn, obtaned usng the laboratory prototype. 278

7 T up T vp T wp T ap t (s CSB Swtchng Events Fg. 14 Illustraton of the zero current commutaton of the CSB, from top to bottom are measured gatng sgnals for throws T up, T vp and T wp of the VSB and T ap of the CSB obtaned usng the laboratory prototype.. a, b, c (A u, v, w (A v a v acf t (s Fg. 1 Waveforms of source currents (top panel and load currents (bottom panel obtaned usng the laboratory prototype a (A t (s Fg. 16 Waveforms of source voltage, flter capactor voltage, lnk voltage and source current, obtaned usng the laboratory prototype, from the top to bottom respectvely. v uv v u u (A t (s Fg. 17 Waveforms of load ll voltage, load ln voltage, lnk voltage and load current obtaned usng the laboratory prototype, from the top to bottom respectvely. VII. CONCLUSIONS A carrer based modulaton scheme has been presented for the IMC, by dentfyng them to be a cascade connecton of a CSB and a VSB. The modulaton functons for the two brdges are developed from the modulaton functons derved from the desred CSB currents and VSB voltages. Then the swtchng functons are generated wth partcular focus on the coordnaton between the CSB and VSB swtchng events to realze a robust commutaton. The proposed algorthm has been verfed by both numercal smulaton and hardware experments, valdatng the effectveness of the modulaton scheme. The modulaton strategy s smple to mplement usng general purpose dgtal sgnal processors n conjuncton wth FPGAs as well as usng applcaton specfc dgtal sgnal processors wth bult n PWM operatonal modules. Although the 18actveswtch topology s used n the analytcal development and laboratory tests, the proposed modulaton algorthm can be appled the reducedswtchcount topologes wth no or mnmal modfcatons. Further the modulaton functons can be extended to the conventonal matrx converter (CMC by the mappng relatonshp between CMC and IMC, whch wll be explored n the future publcatons. ACKNOWLEDGMENT The authors would lke to acknowledge support from the Wsconsn Electrc Machne and Power Electroncs Consortum (WEMPEC at the Unversty of Wsconsn Madson. The work made use of ERC shared facltes supported by the Natonal Scence Foundaton (NSF under AWARD EEC REFERENCES [1] M. Aten, C. Whtley, G. Towers, P. Wheeler, J. Clare, and K. Bradley, "Dynamc performance of a matrx converter drven electromechancal actuator for an arcraft rudder," n Second IEE 2786

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