Walsh Function Based Reconstruction Method of. Command Waveform in PWM Inverter

Transcription

1 Walsh Fucio Based Recosrucio Mehod of Commad Waveform i PWM Iverer Kri Choeisai ad Seiji Kodo Nagaoka Uiversiy of echology 63- Kamiomioka, Nagaoka , Japa Fa: , Phoe: kri@s.agaokau.ac.jp Swichig harmoics icludig Absrac his paper proposes a Walsh fucio based recosrucio-mehod of commad volage waveform of a PWM iverer. he Fourier harmoics of PWM iverer oupu volage show ha he PWM swichig harmoics spread oward lower frequecy rage ad i makes difficul o recosruc he PWM commad volage waveform especially i he case whe he commad volage waveform has sharp edge. O he oher had, he Walsh harmoics show very good separaio bewee origial commad volage compoe ad PWM swichig compoe. herefore, he Walsh fucio based recosrucio-mehod ca recosruc he PWM commad volage waveform correcly. hese remarkable feaures of he proposed mehod are clarified by several simulaio resuls. Key Words: Walsh fucio, rigoomeric fucio riagular-wave PWM mehod Com PWM Modulaio Mehod PWM ON-OFF Paer Fig. he proposed recosrucio-mehod Fig. shows he proposed recosrucio-mehod of commad volage waveform of a PWM iverer. he proposed recosrucio-mehod operaes as a Demodulaor, which recosrucs he commad waveform (Com rec close o he origial commad waveform (Com. I order o desig he recosrucio-mehod, i is ecessary o horoughly ivesigae he PWM operaio. his paper shows he aalysis of he PWM operaio ad proposes a demodulaio-mehod of PWM ON-OFF paer by usig Walsh fucio. 2 Walsh Fucio Recosrucio Mehod Demodulaor Com rec Iroducio A moder DSP-based PWM iverer geeraes he PWM ON-OFF swichig paer usig a digial imer. he DSP oupus he PWM ON-OFF paer bu i does o oupu he PWM commad waveform. I his case, i order o fid he malfucio of a PWM iverer, i is ecessary o recosruc he PWM commad waveform from he PWM ON-OFF paer. However, he swichig harmoics i he PWM ON- OFF paer make i difficul o recosruc he PWM commad waveform by usig Fourier aalysis. Walsh fucio cosiss of recagular waveforms akig oly wo values, +, -, over oe period [,] as illusraed i fig.2. Walsh fucio, wal(,, has Walsh s origial order, where is he sequecy which is he umber of zero-ais crossig for oe period. Similar o rigoomeric fucio, Walsh fucio ca be classified io eve-fucio ad odd-fucio as shows i able.. Walsh fucio ca epress a arbirary ime fucio f( i erms of Walsh series as shows i ( ad (2. f ( C f Cwal(, ( wal(, d ( (2

2 he muliplicaio heorem of Walsh fucio is give i (3. wal( c, wal( where m c o, wal( modulo - 2 addiio o m, (3 Eve Fucio Odd Fucio able. Eve/Odd fucio of rigoomeric ad Walsh fucio rigoomeric Walsh fucio fucio cos(2π si(2π cal( j, wal(2 j, sal( k, wal(2k -, I case whe o < c ad c 2 k, he ew geeraed sequecy m is shifed o higher sequecy rage as shows i fig.3. 3 Ivesigaio of PWM Operaio 3. Model of Comparaor he riagular-wave PWM mehod is show i fig.4 (where ri is he riagular-wave. he PWM mehod geeraes PWM ON-OFF paer by usig comparaor, which makes difficul o ivesigae he PWM operaio because comparaor is oliear compoe. he proposed mehod uses a square-wave isead of comparaor as shows i fig., which makes i possible o ivesigae he PWM operaio by epadig square( wih he liear fucio. By usig Fourier series ad aylor epasio, he square( may also be wrie as (4 ad (. herefore, i is possible o make he model of comparaor by he muliplicaio of sigal as show i fig.6. y square( where 2 π a + a ( π ( π 3π a + a a + a + a (2 +! (4 ( 3.2 Operaio i PWM Mehod Sice Com is a ipu sigal of PWM mehod as shows i fig.4, he harmoics of sigal Com-ri should be he addiio of Com harmoics ad -ri harmoics. Fig. shows he sigal 2+ ad heir Fourier/Walsh harmoics i he model of comparaor fig.6 (whe Com is he si-wave. + a 64π 2+ Fig.2 Walsh fucio wal( o, wal( m, o c m Sequecy wal( c, m c o Fig.3 Walsh muliplicaio-mehod Com ri + + PWM y Fig.4 riagular-wave PWM mehod Fig. Relaio bewee sigal ad y - Fig.6 Model of comparaor - 2+ a 2+ a 3 a 3 a ysquare( - Σ y

3 a Sigal 2+ b Fourier harmoics c Walsh harmoics Fig. Operaio i riagular-wave PWM mehod Whe lookig a Fourier harmoics of sigal i fig., is have may harmoics of ri ha spread oward low harmoics rage (whe he low harmoics rage is o swichig ad is added o he Com harmoics. herefore, i ca o separae he Com harmoics ou from he harmoics of PWM ON-OFF paer by usig Fourier aalysis. O he oher had, Walsh harmoics of Com ha eis i low harmoics rage did o have ri harmoics because ri harmoics are oly a swichig harmoic ad closed o 2 h harmoic. Walsh harmoics of sigal 3 ad i fig. show ha, ri harmoics did o spread oward low harmoics rage. I because of, sigal 3 ad are he muliplicaio of sigal ha makes harmoics shifed o he higher harmoics rage as showed i fig.3. I addiio, (4 ad ( old ha, he summaio of he low harmoics of sigal 2+ are he harmoics of PWM ON-OFF paer ad is equal o low harmoics of sigal, which are he Com harmoics. 4 Proposed Recosrucio Mehod he above resuls lead o he observaio of PWM operaio ha, he low harmoics rage of PWM ON- OFF paer does o iclude ri harmoics ad equals o Com harmoics as illusraed i fig.8. I he case whe Com is he quarer-wave symmerical waveform as si-wave, i s harmoics are oly odd-harmoics of odd-fucio. herefore, i ca use his propery o decrease umber of harmoics i ( ad (2 i order o simplify he epressio of he commad waveform Com( as showed i (6 ad (. Com( C 4i 3 i C 4i 3 wal(4i 3, f ( wal(4i 3, d (6 (

4 he proposed recosrucio-mehod of he PWM commad waveform is summarized as below. Calculae Walsh harmoics of he PWM ON-OFF paer C 4i-3 by (. 2 Se C 4i-3, where C 4i-3 swichig sequecy. 3 he PWM commad waveform is he recosrucio of C 4i-3 by (6. Recosrucio Resuls. ype of Carrier ad Eve Fucio Fig.9 shows he variaio of PWM ON-OFF paer i case whe he duy-raio of riagular-wave (Duy has bee chaged. From fig.9, whe Duy[%], he PWM ON-OFF paer is he quarer-wave symmerical waveform, bu whe Duy2[%] or [%], he phase is shifed lead o he PWM ON-OFF paer is o he quarer-wave symmerical waveform ad i cosiss of he harmoics of odd-fucio ad eve-fucio. herefore, whe usig carrier ype2 or ype3 as showed i fig., he harmoics of evefucio of he PWM ON-OFF paer have bee geeraed ad is reproduce he oises icludig i he recosrucio waveform. However, whe he commad waveform (Com is he quarer-wave symmerical waveform, harmoics of Com are oly he odd-harmoics of odd-fucio. herefore, i ca separae Com harmoics ou from he harmoics of PWM ON-OFF paer by usig ( eve if he harmoics of eve-fucio have bee geeraed..2 ype of Carrier ad Harmoics Fig. shows he aalysis resuls whe ype of carrier ad Com have bee chaged. ypes of carrier are showed i fig.. he Com i fig.(a- is siusoidal ampliude.8, fig.(b- is peak of siusoidal flaed waveform ad fig.(c- is siusoidal compesaed waveform from square-wave ampliude.636. he sequecy of carrier is seig o be 64. I order o see oly he harmoics of Com, he harmoics i fig. is seig o be he odd-harmoics of odd-fucio. From fig., he swichig sequecy of ype,3 PWM are 64 h ad ype2 PWM is 28 h ha seems o be 2 imes of ype,3 PWM. I fac, ype2 PWM has he 64 h harmoic as ype,3 PWM bu his harmoic is he harmoic of eve-fucio ha does o show i fig.. Fig.2 shows he recosrucio waveforms i Fig.8 Walsh harmoics of PWM ad commad waveform (Com Duy raio2[%] Duy raio[%] Duy raio[%] - - Phase shifed riagular-wave ad Commad PWM ON-OFF paer Fig.9 Variaio of PWM ON-OFF paer (a ype (b ype2 (c ype3 ype: riagular-wave ha reverses ampliude a half period of commad waveform. ype2: riagular-wave ype3: saw-ooh waveform Fig. ypes of carrier for he modulaio case whe he ype of carrier has bee chaged. I used lower 64 h harmoics for he recosrucio ad he origial commad waveform showed i fig.(a-. Whe lookig a Fourier harmoics i fig., ype PWM has may swichig oises as showed i fig.(a-4 ad is reduce oises icludig i he recosrucio waveform as showed i fig.2(a. ype2 PWM has he oise harmoics as showed i fig.(a-6 ha higher ha he recosrucio harmoics lead o he recosrucio waveform does o have he oises icludig as showed i fig.2(c. ype3 PWM has he oise harmoics ha colleced a swichig harmoic (64 h as showed i fig.(a-8 ad is reproduce he oises o he recosrucio waveform fig.2(e.

5 Fig. Fourier/Walsh harmoics of PWM waveform O he oher had, Walsh harmoics did o have swichig compoe (i has oo few of swichig compoe whe compared wih Fourier harmoics lead o he recosrucio waveform i fig.2(b,(d,(f did o have swichig oises icludig ad is almos he same waveform. From he resuls above, eve if he commad waveform is he siusoidal, he swichig oises ca spread oward low harmoics rage of Fourier harmoics whe he ype of carrier has bee chaged. O he oher had, he swichig oises did o spread oward low harmoics rage of Walsh harmoics whe he ype of carrier has bee chaged. herefore, Walsh harmoics ca separae he Com compoe ou from he PWM ON-OFF paer correcly..3 Commad Waveform ad Harmoics Fig.3 shows he recosrucio waveform i case whe Com has bee chaged. he Com of fig.3(a,(b is i fig.(b- ad he Com of fig.3(c,(d is i fig.(c-. I used ype2 carrier for he modulaio ad recosruced waveform by usig lower 64 h harmoics. Fourier recosrucio waveform fig.3(a did o have oises icludig because oise harmoics i fig.(b-6 did o spread oward low harmoics rage. However, a he half period of waveform i fig.3(c had oises icludig because oise harmoics spread oward low harmoics rage as showed i fig.(c-6. From he observaio of fig. ad fig.3 hold ha, swichig oises i fig.(b-6 did o spread oward low harmoics rage because he Com (fig.(b- had jus a few of harmoics as showed i fig.(b-2. However, swichig oises i fig.(c-6 spread oward low harmoics rage because he Com (fig.(c- had much of harmoics as showed i fig.(c-2. O he oher had, he swichig oises did o spread oward low harmoics rage of Walsh harmoics whe he Com has bee chaged or he

6 Com has much of harmoics. Fig.3(b,(d showed ha, he Walsh recosrucio waveforms did o have swichig oises icludig ad is close o he origial commad waveform. 6 Coclusios Walsh harmoics of PWM ON-OFF paer old ha, he PWM commad compoe eiss i he lef-had side of swichig sequecy ad swichig oise compoe eiss i he righ-had side as showed i fig.8. From secio.2 ad.3 hold ha, he variaio of carrier ype ad he commad waveform did o have effecs o he low harmoics rage of he Walsh harmoics. herefore, he proposed Walsh fucio based recosrucio-mehod ca recosruc he commad waveform close o he origial commad waveform as showed i fig.2 ad fig.3. (a Fourier recosruced from ype PWM (c Fourier recosruced from ype2 PWM (b Walsh recosruced from ype PWM (d Walsh recosruced from ype2 PWM Referece [] K. Choeisai, S. Kodo: Syhesis Mehod of PWM Paer Based o Walsh Fucio, pp , ras.iee of Japa, Vol.9-D, No.4, (999-4 [2] S. Kodo, K. Choeisai: Walsh Fucio Based Syhesis Mehod of PWM Paer for Full- Bridge Iverer, pp. 2-26, PCC-Nagaoka cof., O9-3, (99-8 [3].J. Liag, R.G. Hof: Walsh Fucio Mehod of Harmoic Elimiaio, pp , proceedigs APEC cof., (993-3 (e Fourier recosruced (f Walsh recosruced from ype3 PWM from ype3 PWM Fig.2 Recosruced waveform i case whe ype of carrier is chaged (Comfig.(a- (a Fourier recosruced of Fig.(b- waveform (b Walsh recosruced of Fig.(b- waveform (c Fourier recosruced (d Walsh recosruced of Fig.(c- waveform of Fig.(c- waveform Fig.3 Recosruced waveform i case whe commad waveform is chaged (ype2 PWM