Walsh Functon Based Synthess Method of PWM Pattern for Full-Brdge Inverter Sej Kondo and Krt Choesa Nagaoka Unversty of Technology 63-, Kamtomoka-cho, Nagaoka 9-, JAPAN Fax: +8-58-7-95, Phone: +8-58-7-957 E-mal: kondo@voscc.nagaokaut.ac.jp Abstract-Ths paper apples the Walsh functon to smplfy the on-off pattern synthess of a sngle phase full-brdge PWM nverter wth low harmoncs. Owng to the dgtal nature of the Walsh functon, the synthess process becomes straghtforward. It s the remarkable feature of the proposed method over a conventonal method usng Fourer seres. Several examples are shown n the paper, whch verfes the valdty of the proposed synthess method. Ⅰ. INTRODUCTION The Fourer seres expanson s conventonally used to derve optmal PWM pattern wth low harmoncs []. The trgonometrc functon used n the Fourer seres causes a problem. It s that an equaton ncludng trgonometrc functon requres a numercal teraton process to fnd ts soluton. The teraton loop shown n Fg. s necessary for searchng optmal PWM pattern wth low harmoncs and t takes long calculaton tme. FFT order, where n s the number of zero axs crossng for one cycle. There are many way to create wal(n,t). In ths paper, the Walsh functon wal(n,t) s ndrectly generated by the recurrence relaton of H( ): H(, t) = t otherwse. [ n/]+ p H ( n+ p, t) = (-) {H( n, ( t+ ))} + n+ p (-) { H( n, ( t ))}, n=,,,3,..., p=([n/]%)= or, where, [n/] s maxmum nteger that s smaller than n/, and (m%) s the remnder of m dvded by. Then the functon H( ) gves the Walsh functon wal( ) as: () () wal(n,t) = H(n+p,t). (3) FFT - - wal(,t) wal(,t)=sal(,t) wal(,t)=cal(,t) Fg. Iteraton loop Outlne for searchng optmal PWM pattern The Walsh functon has only two level sgnal of ± and s sutable for the applcaton to swtchng process []. As t wll be descrbed n secton Ⅲ, the equaton relatng the on-off swtchng tmng wth the Walsh spectrum results n a lner algebrac equaton whch can be solved drectly wthout any teraton process. Hence, the PWM pattern synthess usng the Walsh functon becomes straght-forward and ts computaton s smpler and faster than that usng trgonometrc functon. Ⅱ. FUNDAMENTALS OF WALSH FUNCTION A. Defnton of Walsh Functon The Walsh functon forms an ordered set of rectangular waveforms takng only two ampltude values, + and -, over one normalzed frequency perod [, ] as llustrated n Fg.. The Walsh functon, wal(n,t), has Walsh s orgnal - - - - - - wal(3,t)=sal(,t) wal(,t)=cal(,t) wal(5,t)=sal(3,t) wal(6,t)=cal(3,t) wal(7,t)=sal(,t) 3 Fg. Walsh Functon Smlar to the trgonometrc functon, the Walsh functon can be classfed nto odd-functon and even-functon: sal(m, t) = wal(m-, t) [odd-functon], cal(m, t) = wal(m, t) [even-functon]. () The sal() and cal() are correspondng to sne-walsh and cosne-walsh, respectvely. Unlke the sne and cosne functons n the Fourer seres, the sal() and cal() have rectangular waveforms. Ths knd of bnary nature of the Walsh functon makes the calculaton of the seres expanson smpler than that of the Fourer seres expanson. t
B. Walsh Seres Expanson An approprately contnuous functon x(t) can be expressed by the Walsh seres n the smlar way as the Fourer seres expanson: xt () = Awal(,) t + { Acal(,) t + Bsal (,)} t, (5) where, A = x t t ()d A = x()cal(,)d t t t B = x t t t () sal(,)d C. Comparson between Fourer and Walsh Expansons of PWM pattern In order to compare calculaton tme between Fourer and Walsh expansons, t s assumed that the f(t)s a PWM swtchng pattern of whch frst quarter cycle s shown n Fg. 3(b) and whole one cycle s shown n Fg. 3(a). The Fourer and Walsh seres-expansons can be derved as follows. (6) Walsh seres expanson: f() t = B sal(,) t 35,,,... B = f ()sal(,)dt t t B = f ()sal(,) t t dt+ f ()sal(,) t t dt for, B + 3 3 = ( ) ( ). (8) For the Fourer expanson, the computaton of C n (7) requres the calculaton of cos(), whch takes relatvely long tme. On the other hand, for the Walsh expanson, the computaton of B n (8) requres only plus/mnus operatons. Ths beneft of the Walsh functon can make the calculaton faster. By the computer smulaton result, t was found that the computaton tme rato of (8)/(7) was.9. Ⅲ. PROPOSED SYNTHESIS METHOD OF PWM PATTERN / 3/ /T / - (a). One cycle of Full-brdge PWM nverter 3 /T (b). Frst quarter cycle Fg.3 Example of PWM pattern Fourer seres expanson: f() t = C sn( ωt) 35,,,... C = f ()sn( t ω t ) dt C for, C = f ()sn( t ωt) dt + f ()sn( t ωt) dt { = (cos( ω ) cos( ω)) ω (cos( ω ) cos( ω )) 3 3 } (7) A. Outlne of Proposed Method Fg. llustrates the outlne of the PWM pattern synthess process proposed n ths chapter. The goal of the synthess s to make the PWM pattern close to the commanded output waveform whch s llustrated as a sne-wave n Fg.. In the followng, t s assumed that the commanded output waveform s a sne-wave to make the explanaton smple. At the frst step, the values of B, whch are the Walsh seres coeffcents of the commanded waveform, are calculated. Ths wll be descrbed n the secton-b of ths chapter. Secondary, the equaton whch relates the PWM on-off tmng to the Walsh coeffcents B s derved. Ths wll be gven n the secton-c. Here at ths pont, f B =B for =,3,5,...,, the PWM wave should concde wth the commanded wave. Ths s true n theoretcal sense. But ths knd of coeffcents matchng can not be acheved actually, because the swtchng frequency of the PWM nverter can not be nfnty. Therefore, we lmt the maxmum value of for the coeffcent matchng takng account of the swtchng speed of the power devce. At the last step, the swtchng tmng of the PWM pattern can be determned drectly from the Walsh seres coeffcents B of the commanded waveform, whch s based on a equaton derved n the secton-d of ths chapter.
C. Walsh seres expanson of PWM pattern Walsh seres expanson Walsh seres expanson B B B = B Fg. 3 shows an example of whole one cycle of Full-brdge PWM nverter waveform whch has 3-level. By usng the symmetrcal property of the PWM pattern, t s enough to calculate only the frst quarter cycle. The PWM pattern s settng to be f(t) whch s the functon of. [ ] = N [ K ] [ B ] M [ T ] Fg. Outlne of propose PWM pattern synthess method B. Walsh Seres Expanson of Commanded Waveform In order to make the synthess smple, t s assumed that the commanded waveform of the PWM nverter output has the property of quarter-wave symmetry. As an example, the case when the output command equals to Msn( π t), where M s modulaton rato, wll be explaned n what follows. The Walsh seres expanson s gven as: Msn( π t) = { MB sal (, t)}, (9) = B = sn( πt) sal(, t) dt = 3,,,,... () Snce the output command s assumed to be the sne-wave, n the rght hand sde of (9), the coeffcent B - remans but the coeffcents A and A and B appeared n (5) and (6) become zero. The calculated values of B - for M= are shown n Fg. 5. B- B Ampltude B Ampltude B B 3 B 5 B 7 B 9 B B 3 B 5 6.37963e- -.67569e- -5.8833e- -.79e- -.565e- 6.8687577e-3 -.7638e- -6.76e- B 7 B 9 B B 3 B 5 B 7 B 9 B 3 -.539788e-3.833e-3 -.73538e-3.568e-3-7.88793e-3.3397e-3 -.5966e- -.9599e-.6.5..3... 3 order of harmonc - Fg. 5. Calculated values of coeffcents B to B 3 (M=) f () t = where t /. t = [, ],[ 3, ],... otherwse The Walsh seres expanson of f(t) s gven as: () f() t = { B sal(,)} t, () B = f ( t ) sal (, t ) dt = 3,,,,... D. Proposed Synthess Method of PWM pattern (3) In order to derve the synthess method of the on-off swtchng functon of the PWM nverter, the frst quarter cycle of the output command s dvded nto N subntervals, where N should be chosen to be an nteger of power of two. The sample quarter cycle of the PWM swtchng functon f(t) for N= s shown n Fg.6. - - - - K K K 3 K K K K3 K K 3 K3 K 33 K 3 K K K 3 K f(t) 3 T T T 3 T Fg.6 PWM desgn (N=) sal(,t) sal(3,t) sal(5,t) sal(7,t) PWM To make the swtchng functon f(t) close to the output command Msn( π t), the B n () should be set equal to the MB n (9), that s, MB = B. () As shown n Fg.6, the value of sal functon s + or - for N subntervals, whch makes the calculaton smple. Lettng the value of the sal functon for each subnterval be K j :
K j = j sal(, ). (5) N Accordng to the property of sal(), the followng relatons hold: Kj = Kj, (6) N ( Kj ) = N. (7) 3,,,... From (3) and (), the followng equaton holds: M B = f () t sal(,) t dt N d N N = K f ( t) t + K f ( t) dt+... N N ( N ) N N d NN d. (8) N + K f () t t + K f () t t Substtutng () nto (8), the followng equaton can be obtaned: M B K T K T where, T = ( ) + ( ) +... + K( N ) ( TN N ) + KN( N TN ) = K( ) + K( ) +... + K( N ) ( N ) + KN( N) + KT+ K( T ) +... + K( N ) ( TN ) + KN( TN ) (9) N =,,3,...,N/. () Rewrtng (9) n matrx form, B M B3... B N K K... K K K... K =... K K... K K K... K K K... K... KN KN... K N N N N NN N N N NN +... T T... TN. () The smple form of the above equaton s gven as: [ ] [ ] [ ] [ ] [ ] M B = K + K T. () By the property of the Walsh functon, the [K] matrx s a complete orthogonal matrx, and the relatons (6) and (7) hold. Takng these propertes nto account, the nverse matrx of [K] can be smply calculated as: [ K] = N [ K]. (3) Solvng () wth respect to the [] and usng (3), the swtchng tmng vector [] can be calculated as: [ ] = [ ] [ ] [ ] N K B M T. () The proposed synthess method s summarzed as follows: step-. calculate B - matrx by (), or use data lsted n Fg. 5 f N less than 6. step-. calculate T - by (), step-3. calculate [K] by (5), step-. then, determne the [] by (). Ⅳ. SIMULATION RESULTS The harmoncs of the proposed method and the conventonal trangular wave modulaton method wll be compared n the condton when the swtchng frequences of both methods are kept the same. A. The Case for Pure Sne-Wave Output In the frst quarter of one cycle the number of swtchng angle of both methods s set to N. For N= the calculaton of the Walsh s calculaton method and the conventonal trangular wave modulaton method are shown n Fg.7 and Fg.8 respectvely. For N=6 the calculaton of the Walsh s calculaton method and the conventonal trangular wave modulaton method are shown n Fg.9 and Fg.. As shown n Fg.7(c) and Fg.9(c), the relaton between modulaton rato and swtchng angle of the Walsh s calculaton method are lnear. Comparng Fg.7(b) and Fg.9(b), the calculaton wth larger N results n less harmoncs shfted to hgher frequency range. Comparng Fg.7(b) and Fg.8(b), the Walsh s calculaton method can decrease the harmoncs (3 rd,5 th,7 th ) more than the conventonal trangular wave modulaton method. For the calculaton by N=6, the harmoncs of both method are almost the same as shown n Fg.9(b) and Fg.(b). B. The case for the thrd harmonc ncludng In ths case, the fundamental wave ncludng the thrd harmonc s a command waveform for a sample smulaton. The ampltude of the thrd harmonc s settng to /6 on the fundamental component, then the command waveform s sn(πt)+/6sn(6πt). Both calculaton methods are compared n the same condton when N=6. The Walsh s calculaton method s shown n Fg. and the conventonal trangular wave modulaton method n Fg.. The harmoncs of both calculatons method are shown n TableⅠ.
..5..5..5 (a). PWM waveform (M=.). Ampltude Modulaton rato M.8.6... 5 5 5 3 35 5 5..9.8.7.6.5. Fourer s harmonc order (b). Fourer spectrum of waveform (a)...5..5..5 Quarter cycle of swtchng angle () (c). Relaton between modulaton rato M and relaton Fg. 7 Walsh s calculaton waveform when N=..5..5..5 (a). PWM waveform (M=.). Ampltude Modulaton rato M.8.6... 5 5 5 3 35 5 5..9.8.7.6.5. Fourer s harmonc order (b). Fourer spectrum of waveform (a)...5..5..5 Quarter cycle of swtchng angle ( ) (c).relaton between modulaton rato M and relaton Fg. 9 Walsh s calculaton waveform when N=6..5..5..5 Ampltude Modulaton rato M..8.6.. (a). PWM waveform (M=.). 5 5 5 3 35 5 5..9.8.7.6.5. Fourer's harmonc order (b). Fourer spectrum of waveform (a)...5..5..5 Quarter cycle of swtchng angle ( ) (c). Relaton between modulaton rato M and relaton Fg. 8 Trangular s modulaton waveform when swtchng angle=...5..5..5 Ampltude Modulaton rato M..8.6.. (a). PWM waveform (M=.). 5 5 5 3 35 5 5..9.8.7.6.5. Fourer's harmonc order (b). Fourer spectrum of waveform (a)...5..5..5 Quarter cycle of swtchng angle () (c). Relaton between modulaton rato M and relaton. Fg. Trangular s modulaton waveform when swtchng angle=6
..5..5..5 (a). PWM waveform. Ampltude.8.6... 5 5 5 3 35 5 5 Fourer's harmonc order (b). Fourer spectrum of waveform (a). Fg. The Walsh s calculaton for 3 rd harmonc ncludng when N=6..5..5..5 (a). PWM waveform. Ampltude.8.6... 5 5 5 3 35 5 5 Fourer's harmonc order (b). Fourer spectrum of waveform (a). harmoncs n lower length of the center (5 th to 3 st ) are smaller but the ampltude of harmoncs n the hgher length of the center (33 rd to 3 rd ) are larger than the ampltude of harmoncs calculated by trangular wave modulaton method. Harmoncs (th) 3 5 7 9 3 5 7 9 3 5 7 9 3 33 35 37 39 3 5 7 9 TABLE Ⅰ Fourer spectrum Walsh s calculaton.996.69.3....3.3.....6.9.9..8..5.35..3... Trangular s modulaton..67..........9.33.3.....3.33.9....3 Fg. Trangular waveform modulaton for 3 rd harmonc ncludng when N=6 From TableⅠ, by the calculaton of the trangular wave modulaton method we can verfy that the ampltude of fundamental and thrd harmonc matches to the command waveform. The ampltude of harmoncs n swtchng length are symmetry, when the 3 st and 33 rd harmoncs are suppose to be a center of swtchng. By the calculaton of the propose method, we found the ampltude of fundamental and thrd harmonc does not match to the command waveform. That s because we lmt the number N of Walsh seres coeffcents (B ) to desgned the PWM pattern. For matchng the ampltude of fundamental and/or the thrd harmonc, we should adjust nfntely large number of Walsh seres coeffcents. Also, the ampltude of harmoncs n swtchng length are not symmetry as the ampltude of harmoncs calculated by the trangular wave modulaton method. The ampltude of Ⅴ. CONCLUSION Ths paper proposes the algorthm to desgn full-brdge PWM pattern wth low harmoncs by usng the Walsh functon. The proposed algorthm process becomes straghtforward, n opposton to the algorthm process of the conventonal method usng Fourer seres, whch requres an teraton process for searchng the PWM pattern. The Walsh functon, whch has only two level sgnals of + and -, s sutable to desgn an algorthm process smpler and faster than that usng trgonometrc functon. REFERENCES [] I. Takahash, H. Mochkawa: A New Control of PWM Inverter Waveform for Mnmum Loss Operaton of an Inducton Motor Drve, IEEE Trans. on IA, Vol. IA-, No., p. 58 (985) [] T.J. Lang and R.G. Hoft Walsh Functon Method of Harmonc Elmnaton, pp.87-853, APEC conf., March 993 proceedngs.