One-Stage and Two-Stage Schemes of High Performance Synchronous PWM with Smooth Pulse-Ratio Changing

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1 One-Stage and Two-Stage Scheme of Hgh Performance Synchronou PWM wth Smooth Pule-Rato Changng V. Olechu Power Engneerng Inttute Academy of Scence of Moldova hnau, Republc of Moldova F. Blaaberg Inttute of Energy Technology Aalborg Unverty Aalborg Eat, Denmar B.. Boe Department of Electrcal Engneerng The Unverty of Tenneee noxvlle, USA Abtract Th paper preent detaled decrpton of onetage and two-tage cheme of novel method of ynchronou pulewdth modulaton (PWM) for voltage ource nverter for ac drve applcaton. The propoed control functon provde accurate realzaton of dfferent veron of voltage pace vector modulaton wth ynchronzaton of the voltage waveform of the nverter and wth mooth pule-rato changng. Voltage pectra do not contan even harmonc and ub-harmonc (combned harmonc) durng the whole control range ncludng the zone of overmodulaton. Example of determnaton of the bac control parameter of the nverter wth low wtchng frequency, a well a reult of t mulaton, are preented. I. INTRODUCTION Control of power converter feedng adutable peed ac drve baed on the prncple of modulaton of pule gnal. Parameter and charactertc of power converon ytem are n dependence of the PWM method and technque ued. A fat development of method of dgtal PWM ha occurred n the recent year []-[3]. Voltage ource nverter are nowaday the mot wdepread converter for dfferent applcaton, epecally for nducton motor drve. Control of the output gnal of three-phae varable-voltage varable-frequency nverter manly baed on the correpondng cheme of voltage pace vector modulaton, whch the mot utable for adutable peed drve []. Typcal vector PWM of the output voltage of the nverter baed on aynchronou prncple. It accompaned by the non-ymmetrcal voltage waveform, the pectra of whch contan even harmonc and ub-harmonc (combned harmonc) whch are very underable n mot applcaton ncludng adutable peed ac drve [],[]-[5]. So ynchronzaton of the output voltage of the nverter an mportant practcal problem, epecally for the power ytem wth low wtchng frequency. Some technque provdng ymmetry of the voltage waveform durng PWM control have been decrbed n [5],[6], but they were baed on underable abrupt pule-rato changng durng tranton between ubzone durng the pule droppng proce. In order to provde ynchronou ymmetrcal voltage control wth mooth pule-rato changng, a novel method of feedforward PWM for drve nverter ha recently been propoed wth applcaton to the one-tage cheme of algebrac dcontnuou PWM wth 30 0 non-wtchng nterval [7],[8]. Alo mplfed algebrac algorthm for contnuou cheme of PWM have been propoed and analyed [9]. Th paper preent a theoretcal decrpton and ome mulaton reult of both one-tage and more flexble twotage cheme of ynchronou pulewdth modulaton. Control algorthm are baed here on accurate trgonometrc control functon and appled to both conventonal contnuou and bac dcontnuou veron of voltage pace vector modulaton. II. ONE-STAGE SCHEME OF SYNCHRONOUS PWM A. Space Vector of the Output Voltage of the Inverter Three-phae voltage ource nverter are the bac worhore of adutable peed electrc drve. Fg. a how a mplfed tructure of three-phae voltage ource nverter upplyng the nducton motor (IM) a a load. It the bac converter topology for the maorty of low-power and medum-power applcaton. Fg. b preent the correpondng vector (wtchng tate equence) of the output voltage of the nverter (x actve vector -6, and two zero vector 0 and 7 wthn x ector I- VI). The conventonal defnton for the tate equence (voltage vector) of the nverter ued: 00; 2 0; 3 00; 0; 5 00; 6 0; 7 ; ( - wtchon tate, 0 wtch-off tate). Th vector form of repreentaton very convenent for analy and ynthe of the voltage waveform for three-phae converter wth dfferent veron of pulewdth modulaton. a) /02/$ IEEE 97

2 b) Fg.. Bac tructure of three-phae voltage ource nverter wth an nducton motor (IM) a a load (a), and t output voltage vector (b). B. One-Stage Contnuou Synchronou PWM Fg. 2 how the wtchng tate equence, gate gnal of the wtche of three phae, and lne-to-lne output voltage of the nverter for the mot popular (conventonal) contnuou veron of voltage pace vector modulaton (CPWM) [],[2]. It a typcal cheme of contnuou pulewdth modulaton becaue the modulaton prncple baed on contnuou operaton of all x wtche of the nverter durng every wtchng perod (ub-cycle). The actve wtchng tate are tuated n the centre of the ub-cycle. Zero vector tate (0 and 7) are changng tep-by-tep after every ub-cycle. The equence of wtchng here nde the frt preented 60 0 cloc-nterval. equence. For further analy of the PWM proce t convenent to ue ndex equal to the number of notche nde a half of 60 0 cloc-nterval (ncludng the notch on the border of the clocnterval). For the bac waveform preented n Fg. 2 th number equal to the number of wtchng perod (ubcycle), ncludng the central, nde a half of the clocnterval. In partcular, for the curve preented n Fg. 2, 5. Number of pule N n the half-wave of the lne output voltage can be determned n th cae a: N 6-3, () and N27 for the waveform preented n Fg. 2. The fundamental frequence whch correpond to the control mode wth ymmetrcal PWM for both conventonal and ynchronou contnuou PWM, can be calculated n a general form a: F 2Nτ 6(2 )τ (2) Durng the ncreae of the fundamental frequency of the drve ytem, the followng value of the frequency correpondng to the next control mode wth ymmetrcal PWM (ndex reduced by one here), can be determned: (3) F 6(2 3) τ Fg. 3 how the waveform of wtchng tate equence and the output voltage of the nverter wth ymmetrcal PWM, whch correpond to the next control ub-zone, where and N2. The fundamental frequency ncreaed here to 27/2 wtchng equence γ γ2 γ3 γ λ5 λ λ3 λ2 λ phae a phae b phae c Vab Fg. 2. Control and output gnal for the quarter-perod of the nverter wth contnuou PWM (CPWM) at the boundary frequency F 5. In Fg. 2 the gnal repreent the total wtch-on duraton durng the wtchng perod (ub-cycle) τ, gnal γ (mnor component of two part of the -gnal) are generated on the boundare of the correpondng. The wtchng frequency F n th cae equal to F / 2τ. The wdth of notche λ repreent the duraton of zero tate Fg. 3. Control and output gnal for the quarter-perod of the nverter wth contnuou PWM (CPWM) at the boundary frequency F. compared wth the prevou frequency (Fg. 2) n the cae the wtchng frequency contant. The fundamental frequence (2)-(3), whch are the bac n ynthe of ymmetrcal voltage waveform, can be called a the boundary (threhold) frequence tranent between control ub-zone of the tepby-tep proce of ynchronou contnuou PWM (CPWM). The control proce for conventonal vector PWM baed on contnuou rotatng of the voltage pace vector and t charactered by aynchronou varaton of the output voltage of the nverter durng control of the fundamental frequency of the drve ytem. It lead to aymmetrcal voltage waveform 975

3 between the ynchroned frequence (2)-(3), the pectra of whch contan addtonal even harmonc, ub-harmonc and combned harmonc, whch are epecally underable for hgh power ytem wth low wtchng frequence [],[]. To provde ymmetrcal varaton of the voltage waveform wth mooth pule-rato changng, pecalzed control of the gnal tuated near the boundare of the 60 0 cloc-nterval hould be executed n accordance wth the propoed method of ynchronou contnuou PWM. So durng the movement between two frame fundamental frequence F and F - (ee Fg. 2 and Fg. 3) correpondng to ynchronou PWM wth nteger odd number of the wtchng perod nde the 60 0 cloc-nterval, pecal control functon are ued for varaton of thee boundary gnal ( λ λ, and γ ). In partcular, for one-tage cheme of ynchronou CPWM for the waveform preented n Fg. 2, where 5: λ 5 λ λ, 5 ; for the curve preented n Fg. 3 ( here): λ λ λ,. wtchng equence 3 2 λ phae a phae b phae c Vab γ γ2 γ3 λ3 3 λ2 2 λ 2 3 Fg.. Control and output gnal for quarter-perod of the nverter wth dcontnuou PWM (DPWM). C. One-Stage Dcontnuou Synchronou PWM In contrat wth contnuou PWM the cheme of dcontnuou pulewdth modulaton are charactered by the or non-wtchng nterval for every wtch of the nverter module durng the perod of the output voltage. The cheme of dcontnuou PWM wth the non-wtchng nterval can be ymmetrcal or aymmetrcal. The mot famou cheme of dcontnuou PWM ha ymmetrcal output voltage and t charactered by the 60 0 non-wtchng nterval [],[3]. At the ame tme t nown, that a better qualty of the output voltage of the nverter provded by the cheme of dcontnuou ymmetrcal PWM wth the 30 0 non-wtchng nterval [3],[9]. Fg. how the wtchng tate equence, control gnal for three wtche and lne output voltage of the nverter for one-tage cheme of drect ynchronou PWM, appled to dcontnuou ymmetrcal pulewdth modulaton wth thee 30 0 non-wtchng nterval (DPWM, Method n [3]). The general wtchng tate equence n th cae. Here are alo cloc-pont gnal λ ( λ ) (wth the neghbourng ( )), t wdth are reduced moothly tll cloe to zero value at the boundary frequence F, calculated n accordance wth (), whch are tranent between control ub-zone: F. () 6(2.5)τ Specfc properte of dcontnuou PWM allow provdng a better qualty of the output voltage and current of the nverter n comparon wth contnuou PWM at hgher value of the fundamental frequency of drve ytem [],[3],[9]. D. Bac Control Functon for One-Stage Synchronou Pulewdth Modulaton Contnuou ymmetrcal varaton of the voltage waveform of the nverter wth ynchronou PWM provded by a pecal algorthm. It baed on a tep-by-tep control of pecaled parameter of the PWM cheme between the boundary frequence F, tuated on the ax of the fundamental frequency F of the drve ytem [7]-[9]. It provde contnuou adutment of the voltage waveform wth mooth pule-rato changng untl the maxmum fundamental frequency F m of the rated control mode of ac drve. F calculated n a general form n accordance wth (2) (for contnuou PWM) and () (for dcontnuou PWM). The modulaton ndex m F / Fm n th cae. Index equal to the number of notche nde a half of the 60 0 clocnterval, and t determned n a general form from (5), where fracton rounded off to the nearet hgher nteger: / 6F + τ, (5) 2τ where for CPWM and.5 for DPWM. The relatve duraton of the actve wtchng tate and pule of the output voltage of the nverter for thee ymmetrcal cheme of PWM can be wrtten n accordance wth the prncple of voltage pace vector modulaton [9]: n(60 0 α ) + n( α ) (6) γ n( n( α ), (7) α ) + n( α ) 976

4 where: α - angle poton of the centre of the correpondng -gnal from the begnnng of the 60 0 cloc-nterval; - the gnal, whch formed n the centre of the 60 0 clocnterval, where α 30 ; 2, -; + 0. Realaton of (6)-(7) can be baed on ether mplfed pure algebrac control functon [8]-[9], or prece trgonometrc one. Eq. (8)-(3) preent a et of accurate trgonometrc control functon, baed on ome tranformaton of (6)-(7), for determnaton of parameter for control and output gnal of the nverter n abolute value (econd) for tandard calar control mode of the drve ytem durng the whole range ncludng the zone of overmodulaton: for 2,...-: co[( 3 ) τ ov] (8) γ + { tan[( 3 ) τ ]} ov2 (9) τ ] (0) co[( 3 ) ov γ { tan[( ( λ ) / 2]} ov2 2) τ + () λ τ ( + ) / 2 (2) λ + ( τ ) ov λ, (3) where:. τ m untl F ov F, and τ m after F ov ; [ ( F F ) /( F F )] - coeffcent of ynchronaton; coeffcent of overmodulaton ov untl F ov, and ov [ ( F Fov ) /( Fov2 Fov )] between F ov and Fov Fm ; coeffcent of overmodulaton ov2 untl F ov2, and ov2 [ ( F Fov2 ) /( Fm Fov2 )] n the zone between F ov2 and F m ; 3 0 for CPWM and for DPWM. It neceary to note that real emconductor wtche of the nverter are not deal devce and they are charactered by ther mnmum on-tme and off-tme tate. So wdth of the boundary gnal (, λ, γ ) can not be le than the requred wtchng mnmum, and commutaton of every wtch of the nverter ha to be organed by the control ytem n accordance wth th condton []. Fg. 5 llutrate the varaton of the cloc-pont gnal for calar control mode of the drve ytem wth ynchronou dcontnuou PWM n the lnear control zone untl the zone of overmodulaton near the maxmum fundamental frequency of the ytem F m 60 Hz. For a phycal clearne the control proce charactered by low wtchng frequency. The curve n Fg. 5 correpond to the control mode wth nonlnear dependence () on the duraton of wtchng perod τ from the fundamental and maxmum frequence F and F m (the peed rato (dapaon between the ntal and maxmum Fgure 5. Varaton of the cloc-pont gnal (a), and number of pule n half-wave of lne voltage (b) for dcontnuou PWM wth varable τ. frequency of calar control mode) equal D 2 for th cae): Fm τ () 6F( DFm + Fm DF) After olvng (), the boundary frequence are calculated n accordance wth (5) for th PWM veron: F D F m( ) (5) D Fg. 5a how the wdth varaton of τ,, λ,. There a ynchronou qua-lnear varaton of λ and here. They are decreaed multaneouly tll cloe to zero wdth at the boundary frequence. Fg. 5b preent the number of pule n half-wave of the output voltage of the nverter, whch changng by eght pule after every boundary frequency for dcontnuou one-tage veron of ynchronou PWM. III. TWO-STAGE SCHEMES OF SYNCHRONOUS PWM In order to provde more mooth varaton of the number of pule n half-wave of the output voltage of the nverter, a two-tage cheme of ynchronou PWM can be ued. In addton to the nown one-tage cheme of PWM t characterzed by the pecfc econd control tage durng mooth tep-by-tep proce of pule-rato changng. Fg. 6 llutrate th econd tage for the mentoned above veron of dcontnuou PWM, whch nclude two novel parameter of the cheme of modulaton: cloc-pont gnal and the notche λ neghbourng wth the cloc-pont gnal. 977

5 boundary frequency. Here alo mooth qua-lnear varaton of the notche λ (2), neghborng wth the - gnal. The coeffcent of ynchronzaton 2 determned from (22) n thee control ub-zone: λ + (20) /2F (.25) τ / 2 / 2 (2) λ + [ ( F F ) /( F F )]. (22) Fg. 6. Control and output gnal for quarter-perod of the econd part of twotage cheme of dcontnuou PWM. The algorthm of two-tage ynchronou PWM charactered by two control ub-zone, whch are changng each other tep-by-tep durng adutment of the fundamental frequency. It baed on two threhold (boundary) frequence F and F, determned for dcontnuou PWM veron a a functon of the number of notche (6) nde a half of 60 0 cloc-nterval correpondngly n accordance wth () and (7): / 6F + 0.5τ (6) 2τ Fg. 7 how the varaton of the bac control gnal for a two-tage cheme n dcontnuou PWM untl the zone of overmodulaton. The parameter of the control proce are the ame a n the example preented n the prevou part. Fg. 7a llutrate the varaton of the cloc-pont gnal λ and whch are changng each other tep-by-tep on the boundary frequence. Fg. 7b preent a varaton of the gnal λ and whch are the next wth the correpondng cloc-pont gnal. Fg. 7c how the pule number varaton N n the halfwave of the nverter output voltage. It characterzed by more flexble mooth pule rato changng compared wth the rato changng of the one-tage cheme, preented n Fg. 5b. Here non-unform tep-by-tep changng the number of pule (by 6 pule, by 2, by 6, and o on). Fg. 8 how more n detal lne output voltage of the nverter whch correpond to the boundary part of the 60 0 F. (7) 6(2 2.5)τ In the frt of two control ub-zone, when F > F F, the voltage waveform for dcontnuou PWM are le the preented n Fg., and the bac et of control functon the ame too, but here are another value of the coeffcent of ynchronzaton, whch equal to for determnaton of [ ( F F ) /( F F )] (8) λ and γ, and [ 0.75( F F )/( F F )] (9) for determnaton of the gnal. So here a lnear decreae of the duraton of the gnal, and then the uncton of two equal halve nto one gnal, tuated on the edge of the 60 0 cloc-nterval, at the F boundary frequence. Control n the econd ub-zone, when F > F F +, baed on the frame et of functon, decrbed n the II.D. It alo characterzed by the lnear varaton of the -gnal n accordance wth (20), t wdth cloe to zero at the next Fg. 7. Varaton of the cloc-pont gnal (a), the gnal neghbourng wth the cloc-pont gnal (b), and number of pule n half-wave of the lne-tolne output voltage of the nverter (c). 978

6 Fg, 8. Part of the lne output voltage of the nverter wth two-tage cheme of ynchronou DPWM durng control of the fundamental frequency. cloc-nterval wth the cloc-pont gnal and λ. Control mode correpond here to the non-lnear varaton of ub-crcle τ durng control of the fundamental frequency F from 5 Hz untl 60 Hz n accordance wth () where peed rato D2. In partcular, for the left curve n Fg. 8, whch correpond to control between the boundary frequence F 35 Hz and F 3 0 Hz, here a mooth decreae of the clocpont gnal untl zero at the F 3 frequency, wth reducton of the number N pule n voltage half-wave from 2 to 9. If the fundamental frequency le between F 3 0 Hz and F 3 5 Hz (rght curve n Fg. 8), mooth multaneou decreae the wdth of the gnal λ and γ untl zero n th ub-zone. Number of pule n voltage half-wave reduced after the F 3 frequency from 9 to 3. There can be alo dfferent modfcaton of the bac twotage cheme of ynchronou PWM. A an example, Fg. 9 llutrate the varaton of ome control parameter for modfed two-tage DPWM wth more unform dtrbuton of the boundary frequence on the ax of the fundamental frequency of the ytem ( , F /[6(2 3) τ ] and ( / 6F + τ ) / 2τ for th cae). IV. OUTPUT VOLTAGE AND CURRENT OF INVERTERS WITH SYNCHRONOUS PWM Fg. 0 preent quarter-perod of the lne output voltage of the nverter wth bac two-tage cheme of dcontnuou PWM durng control proce from the ntal fundamental frequency F 5 Hz untl the maxmum frequency F m 60 Hz. It correpond to calar V/F control of the drve ytem wth low wtchng frequency, wth control of τ n accordance wth () (D2). Left ncrpton how here runnng value of the fundamental frequency F, ndex, number N of pule n lne voltage half-wave, and alo whch cloc-pont gnal, or λ, formed n th ub-zone. Fg. preent the pectral charactertc for every of the preented voltage waveform. Fg. 9. Varaton of the cloc-pont gnal (a), the gnal neghbourng wth the cloc-pont gnal (b), and number of pule n half-wave of the lne-tolne output voltage (c) of the nverter wth modfed two-tage PWM. Fg. 0. Quarter-perod of lne voltage of the nverter durng adutment. 979

7 Fg.. Ampltude of low order harmonc of the-lne-to-lne output voltage of the nverter wth two-tage cheme of ynchronou dcontnuou PWM. 980

8 The bac pecularty of the voltage waveform for nverter wth both one-tage and two tage cheme of ynchronou PWM, both contnuou and dcontnuou, ther quarterwave ymmetry durng the whole control range ncludng the zone of overmodulaton. So, a t hown n Fg., ther pectra nclude only odd voltage harmonc (wthout trple harmonc) for any non-nteger rato F /F between the wtchng and fundamental frequence and for any value of modulaton ndex m. So even harmonc and combned harmonc (of the fundamental frequency) are elmnated from the voltage pectrum n the nverter wth ynchronou PWM, and t epecally mportant for power drve ytem wth low wtchng frequency. Comparon of ntegral (averaged) pectral charactertc of the output voltage of nverter wth one-tage and two-tage cheme of ynchronou PWM how t practcal dentty for the both cheme. Fg. 2 preent reult of mulaton of adutable peed drve ytem wth 5 hp -pole 60 Hz nducton motor fed by the nverter wth two-tage cheme of ynchronou dcontnuou PWM. Here are pectrum of the lne voltage (ampltude of t harmonc dvded to dc-ln voltage), phae current a of the nducton motor and t pectrum (n ampere, wthout the frt harmonc). Fundamental frequency of the ytem 50 Hz, modulaton ndex m Left curve correpond to PWM wth average wtchng frequency 2.25 Hz, rght curve to wtchng frequency equal to.5 Hz. Fg. 3 how the averaged weghted total harmonc dtorton factor of lne voltage of the nverter (WTHD, [9]) for ynchronou one-tage cheme of contnuou PWM (old lne) and tandard aynchronou contnuou PWM ([2], dotted lne). WTHD ha been determned a a functon of the rato F /F between the wtchng and fundamental frequence, for F30 Hz, m0.6, n accordance wth the averagng approach [9]. Fg. 3 llutrate an advantage of ynchronou PWM for the nverter wth low wtchng frequence, untl frequency rato equal to about 00. Fg. 2. Spectrum of the lne voltage, phae current and t pectrum for the ytem wth ynchronou dcontnuou two-tage PWM. Fg. 3. WTHD of voltage for aynchronou () and ynchronou (2) PWM. V. CONCLUSION Both one-tage and two-tage cheme of drect ynchronou PWM for three-phae nverter have been propoed and analyed n th paper. It allow provdng ynchronaton of the output voltage of the nverter durng the whole control range ncludng overmodulaton, wth quarterwave ymmetry of the voltage waveform, the pectra of whch do not contan even harmonc and ub-harmonc (combned harmonc), wth flexble hoc-le pule-rato changng. Two-tage cheme of ynchronou PWM provde moother pule-rato changng compared wth the one-tage one. Analy and comparon of the average WTHD factor for the one-tage and two-tage cheme of ynchronou PWM how t practcal dentty for the both cheme. The method and cheme propoed can alo be demnated to other topologe of nverter and converter. ACNOWLEDGMENT Th reearch ha been upported partly by the Danh Techncal Reearch Councl n the range of the Proect of the NATO Fellowhp Program n Denmar (upervor Dr. F. Blaaberg), and by Award No. ME2-305 of the Moldovan R&D Aocaton (MRDA) and the U.S. Cvlan R&D Foundaton for the Independent State of the Former Sovet Unon (CRDF). REFERENCES [] J. Holtz, Pulewdth modulaton for electronc power converon, IEEE Proc., Vol. 82, No. 8, pp. 9-23, Augut 99. [2] H.W. van der Broe., H.C. Sudelny and G. Stane, Analy and realzaton of a pule wdth modulator baed on voltage pace vector, Proc. of IAS 86, pp [3] J.W. olar, H. Ertl and F.C. Zach, Influence of the modulaton method on the conducton and wtchng loe of a PWM converter ytem, IEEE Tran. on Ind. Appl., Vol. IA-27, No. 6, pp , December 99. [] N. Mohan, T.M. Undeland and W.P. Robbn, Power Electronc, 2 nd ed., John Wlley & Son, 995. [5] G. Narayanan and V.T. Ranganathan, Synchroned PWM tratege baed on pace vector approach. Part : Prncple of waveform generaton, IEE Proc. Electr. Power Appl., vol. 6, no. 3, pp , June 999. [6] G. Narayanan and V.T. Ranganathan, Synchroned PWM tratege baed on pace vector approach. Part 2: Performance aement and applcaton to V/F drve, IEE Proc. Electr. Power Appl., vol. 6, no. 3, pp , June 999. [7] V. Olechu and B.. Boe, Fat algorthm of modulaton of nverter output voltage for nducton motor drve, Proc. of ISIE 99, pp [8] V. Olechu, Novel feedbac and feedforward tratege of modulaton for converter for adutable electrc drve, Techncal Electrodynamc, no. 2, pp. 2-28, June [9] V. Olechu and F. Blaaberg, Drect ynchronzed contnuou and dcontnuou PWM baed on algebrac approach, Proc. of IECON 0, pp