THREE-level neutral-point clamped inverter is being used

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1 Harmonc Analyss of DC-Lnk Capacor Curren n Snusodally Modulaed Neural-Pon-Clamped Inverer K.S. Gopalakrshnan and G. Narayanan Absrac The volage rpple and power loss n he DCcapacor of a volage source nverer depend on he harmonc currens flowng hrough he capacor. Ths paper presens double Fourer seres based harmonc analyss of DC capacor curren n a hree-level neural pon clamped nverer, modulaed wh sne-rangle PWM. The analycal resuls are valdaed expermenally on a 5-kVA hree-level nverer prooype. The resuls of he analyss are used for predcng he power loss n he DC capacor. Index Terms Capacor curren, capacor loss, dode-clamped nverer, double Fourer seres, harmonc analyss, neural-ponclamped nverer, hree-level nverer. C O P C z C SR SR SR R R R SY SY SY Y Y SB SB Y SB B B B I. INTRODUCTION THREE-level neural-pon clamped nverer s beng used wdely hese days [] []. The harmonc specrum of he oupu volage has been suded exensvely for boh wo-level [][4] and hree-level nverers [][5]. Harmonc analyss of he DC-capacor curren n a wo-level nverer based on double Fourer seres and geomerc wall model [] s presened n [6]. A bref descrpon on he ulsaon of hs echnque for deermnaon of harmonc specrum of DC-lnk curren for hree-level dode-clamped nverer s gven n [7]. Ths paper presens a dealed dscusson on he applcaon of he above echnques for he harmonc analyss of DC-lnk curren n a hree-level nverer. The analycal resuls are valdaed exensvely hrough expermenal resuls. Furher, he resuls of he harmonc analyss are used o evaluae he power loss n DC capacors. The DC-lnk curren n a dode-clamped nverer s expressed mahemacally n erms of load curren and devce swchng sae n secon II. Harmonc analyss of he DClnk curren based on double Fourer seres and geomerc wall model echnques s presened n secon III. The resuls of he analyss are valdaed expermenally on a 5-kVA NPC nverer. The analycal and expermenal resuls are compared n secon IV. The power loss n he DC capacor s esmaed usng he above harmonc analyss and he equvalen seres ressance (ESR) of he capacor a dfferen operang condons n secon V. The conclusons are presened n secon VI. K.S.Gopalakrshnan s wh he Deparmen of Elecrcal Engneerng, Indan Insue of Scence, Bangalore - 56, (ashwnkrshnan@gmal.com) G.Narayanan s wh he Deparmen of Elecrcal Engneerng, Indan Insue of Scence, Bangalore - 56, (gnar@ee.sc.erne.n) Fg.. C SR4 N SY4 SB4 R4 Y4 B4 Three-level neural-pon-clamped nverer II. DC-LINK CURRENT IN AN NPC INVERTER FOR IN-PHASE SPWM SCHEME The power crcu of a hree-level neural-pon-clamped nverer s shown n Fg.. The hree legs are swched by comparng hree-phase snusodal modulang sgnals agans wo level-shfed, n-phase, rangular carrers as llusraed n Fg.. The R-phase modulang wave f(ω), he op carrer wave g ( ), and he boom carrer wave g ( ) are descrbed mahemacally n (), where M s he modulaon ndex; s he carrer angular frequency; ω s he fundamenal angular frequency. f(ω) = Msn(ω) < ω < g ( ) = / < < g ( ) = / < < g ( ) = / < < g ( ) = + / < < The harmonc conen of he lne curren n a snusodally modulaed NPC nverer s que low [], [5]. Snce hese lne curren harmoncs are nsgnfcan o nfluence he DC-lnk curren [6], he hree-phase load currens R, Y and B can be expressed as shown n (), where I M s he peak fundamenal ()

2 M f(ω) ω -M - -/ / / Fg.. Modulang sgnal f(ω) curren, and φ s he load power facor angle. R = I M sn(ω φ) Y = I M sn(ω φ) (a).77 -ω g ( ) c / / g ( ) -- / -+ / - -.-/ /. R R - g ( ) - / / -- / -. g ( ) -+ / -.-/ /. R B = I M sn(ω + φ) () The load curren (say R ) flows hrough he op swch (say SR), whenever he swch s ON,.e. when he modulaon sgnal s greaer han boh he carrers. Ths s llusraed n Fg.a, whch consders a swchng cycle correspondng o M=.9, ω=5, and f(ω) =.77. The op-swch curren ( R ) s zero oherwse as shown by Fg.a and Fg.b; Fg.b consders a swchng cycle n he negave half cycle of f(ω),.e. M=.9, ω = and f(ω) =.. These condons are expressed mahemacally as shown n Table I. R When When < R Msn(ω) > ωc Msn(ω) > ωc > Msn(ω) > Msn(ω) TABLE I CONDITIONS FOR DETERMINING TOP-SWITCH CURRENT R IN A SWITCHING CYCLE The sum of he hree op-swch currens ( R, Y and B ) gves he DC-lnk curren as shown by (). = R + Y + B () The DC lnk curren consss of DC and AC componens. The DC componen ( ) flows from he DC source or he recfer. The enre ac componen (.e. ) s assumed o flow hrough he DC capacor. Hence he harmonc componens of he DC capacor curren ( C ) are he same as hose of he DC-lnk curren ( ). Furher, he harmonc analyss of he DC-lnk curren can be broken up no ha of he op-swch currens R, Y and B. Snce hese are symmerc, he harmonc analyss of one of hem (say R ) would suffce. (b) Fg.. Top swch curren R n a swchng cycle when (a) M=.9, ω=5 and (b) M=.9, ω= III. HARMONIC ANALYSIS OF DC-CAPACITOR CURRENT Harmonc analyss of he DC capacor curren C reduces o ha of he op-swch curren R as dscussed above. The curren R depends on f(ω), g ( ) and g ( ) whch, n urn, depend on ω and. Hence R s a funcon of wo ndependen varables ω and. Furher, s perodc n boh hese varables. Hence R s analysed based on double Fourer seres [6] [] n hs paper. A. Double Fourer Seres Analyss Fourer seres s used for expressng perodc funcons n one varable as summaon of he DC, fundamenal and harmonc componens. Smlarly, double Fourer seres s used o express perodc funcons n wo varables, as summaon of snusodal componens whose frequences can be represened as m + nω, where m s a non-negave neger, and n s an neger. The negers m and n are called carrer ndex and fundamenal ndex, respecvely []. Any funcon h(ω, ), whch s ndependenly perodc n boh ω and, s expressed as gven below []. + m= h(, ω) = A + + [A ncos(nω) + B nsn(nω)] n= [A mcos(m) + B msn(m)] m= n= n [A mncos(m + nω) + B mnsn(m + nω)] (4)

3 Where, A mn = / / B mn = / / h(, ω)cos(m + nω)d dω h(, ω)sn(m + nω)d dω The complex form of A mn and B mn can be wren as follows: C mn = A mn + jb mn = / h(, ω)e j(mωc+nω) d dω (6) / In equaon (4), he frs summaon erm, A, corresponds o he DC componen of he waveform. The second summaon erm, ( n= [A ncos(nω) + B n sn(nω)]), corresponds o he fundamenal and base-band harmoncs. The hrd summaon erm, ( m= [A mcos(m ) + B m sn(m )]), s he sum of carrer harmoncs. The fnal summaon erm, ( m= n= n [A mncos(m +nω)+b mn sn(m + nω)]), corresponds o he sde-band harmoncs []. (5) duraon for whch R = R whn hs swchng cycle. Furher, he segmens BQ and AP represen he nervals when R = a he sar and end of he swchng cycle, respecvely. The swchng cycle n Fg.b, can be represened by a vercal lne AB (.e. ω=-. rad) n he un cell shown n (4b). Snce R = n he enre duraon of he sub cycle, he lne segmen PQ s of zero lengh as shown n Fg.4b. Boh hese pons P and Q concde wh each oher, and le on he horzonal axs as shown n Fg.4b. (a). A P R = R = R ω o -. Q = - B R -/.9 / / B. Un Cell The op-swch curren R s a funcon n and ω as menoned earler. Hence can be defned n a wodmensonal space wh and ω as he orhogonal axes. Furher, R s perodc n boh and ω. wh a perodcy of. Hence one needs o consder only an nerval of along each of he axes. Such a regon bounded by one perod of boh varables s ermed as un cell. In hs dscusson, he un cell s defned as he regon formed by < < and / < ω < / (see Fg.4a and Fg.4b). C. Conour Plo Double Fourer seres and geomerc wall modelng echnques have been wdely used for harmonc analyss of PWM oupu waveforms of wo-level and hree-level nverers []. The PWM waveform has only dscree values (say hgh or low). Hence he un cell can be dvded no wo (ses of) regons dependng on he value of he waveform []. The boundary separang he wo regons s ermed as conour plo[]. (More generally, he un cell s dvded no a number of regons, each correspondng o a parcular value of he funcon). Such a mehod of analyss s called he geomerc wall mehod. Ths mehod s used for harmonc analyss of DC-lnk curren n hs work. The op-swch curren R akes he values of R or zero. Hence he un cell ges dvded no wo regons as dscussed below. Consder he swchng cycle n Fg.a. I can be represened by a vercal lne AB (.e. ω=5 =.9 rad) n he un cell as shown n Fg.4a. The lne segmen PQ represens he (b) / A R = = P, Q R ω o = B R -. / / Fg. 4. (a) Represenaon of he swchng cycle, F g.a n a un cell (b) Represenaon of he swchng cycle, F g.b n a un cell The loc of he pons P and Q represen he boundary separang he regons where R = and R = R. Therefore, he loc of he pons P and Q yeld he conour plo of R shown n Fg.5. From Fg.5, can be seen ha he curren R = R n he regon beween Msn(ω) < < Msn(ω). In all he oher regons, R = D. Evaluaon of Harmonc Componens The harmonc componens of he curren R can be found by applyng double Fourer negral on he funcon (.e. R ) n he un cell. Ths negral reurns a zero value n all he regons excep he regon Msn(ω) < < Msn(ω), where he funcon assumes he value R. Ths reduced negral s gven n (7). The harmonc componens of currens Y and B are obaned from (7), by mulplyng he same by e jn/ and e jn/, respecvely, as shown n (8) and (9).

4 4 M -M --/ R = R = =Msn(ω o ) R = R =-Msn(ω o ) R = R = ω o / / Capacor curren (A) Tme Fg. 5. Conour plo of R Fg. 6. Capacor curren waveform a I M =4 A, modulaon ndex of.9 a power facor of R = Y = B = Msnω Msnω Msnω Msnω Msnω Msnω R e j(mωc+nω) d dω (7) R e j(mωc+nω) d dω e j(n/) (8) R e j(mωc+nω) d dω e j(n/) = R + Y + B () The harmonc componens of he DC-lnk curren are obaned by summng he respecve componens of R, Y and B as ndcaed by (). The harmonc componens are calculaed usng (7) o (). A MATLAB code s wren for he purpose. IV. ANALYTICAL AND EXPERIMENTAL RESULTS The expermenal seup consss of an IGBT based 5 kva nverer. The conroller plaform s TMSLF47A DSP processor. The swchng frequency s.5 khz. The load s a hree-phase R-L load. The per-phase nducance of he load s mh. The power facor of he load s vared by changng he ressance of a rheosa n all he hree phases of he load. The load curren s kep consan a 4 A, by varyng he DC bus volage appropraely. The capacor curren s measured usng a fluke-4s probe. The measured capacor curren waveforms a modulaon ndces M=.9 and M=.5 are presened n Fg.6, Fg.7, Fg.8 and Fg.9. The waveforms are shown a power facors of.7 and.844 a boh modulaon ndces. The analycal and expermenal harmonc specra of he capacor curren waveforms n Fg.6 o Fg.9 are presened n Fg. o Fg.7. The analycally deermned specra peranng o Fg.6, Fg.7, Fg.8 and Fg.9 are shown n Fg., Fg., Fg.4 and Fg.6 respecvely. The respecve expermenal specra are gven n Fg., Fg., Fg.5 and Fg.7. The analycal specra are obaned as dealed n secon III. The expermenal specra are obaned by (9) Capacor Curren (A) Tme Fg. 7. Capacor curren waveform a I M =4 A, modulaon ndex of.9 a power facor of.7 applyng Fas Fourer Transform (FFT)on he measured curren waveform usng MATLAB. As seen from Fg. o Fg.7, he analycal and expermenal specra mach reasonably well wh each oher. Also, he capacor curren conans hrd harmonc and swchng frequency componens. The amplude of he hrd harmonc componen decreases wh ncrease n power facor. On he conrary, he magnude of he swchng-frequency componen ncreases wh load power facor. The hrd harmonc s he only sgnfcan low frequency componen presen n he curren specrum. Hence he volage rpple n he DC capacor s deermned manly by he hrd harmonc componen. V. EVALUATION OF POWER LOSS IN THE DC-LINK CAPACITOR The power loss n a DC-lnk capacor s due o he rpple curren flowng hrough and ESR of he capacor. The ESR of he capacor s dependen on frequency [8]. Equaon () gves he power loss n a DC-elecrolyc capacor. P = IC(n)ESR(n) n= () Here I C (n) s he RMS value of n h harmonc of he capacor curren; ESR(n) s he ESR a n h harmonc frequency (.e. n mes he fundamenal frequency). The capacor used s ALCON PG6DI, 45V, 47uF elecrolyc capacor [9]. The ESR values of he capacor a dfferen

5 X: Y:.64 5 Capacor curren(a) Tme(s) Fg. 8. Capacor curren waveform a I M =4 A, modulaon ndex of.5 a power facor of.844 Fg.. Analycal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.9 and power facor of Capacor curren(a) Tme(s) Fg. 9. Capacor curren waveform a I M =4 A, modulaon ndex of.5 a power facor of.7 Fg.. Expermenal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.9 and power facor of.844 frequences, as suppled by he manufacurer, are shown n Table II. A curve f on hese daa pons s done usng MATLAB. The resulng equaon for he capacor ESR a a base emperaure of 5 C s gven n (). A plo of he ESR versus frequency s shown n Fg.8..5 ESR(n) = ( + (.5n +.8 () )) Usng he harmonc specra evaluaed n secon III and ESR as gven by Fg.8, he power loss n he DC capacor s evaluaed. The power loss s evaluaed a he operang condons correspondng o Fg. o Fg.7. These are shown n Table III. Frequency(Hz) ESR(ohm) TABLE II ESR VALUE AT VARIOUS FREQUENCIES [9] VI. CONCLUSION Double Fourer negral and geomerc wall model are used o oban he DC-lnk capacor curren specra n a snusodally modulaed neural-pon clamped nverer. The analycally obaned specra are valdaed expermenally. I s shown ha he domnan componens n he frequency specra are he hrd harmonc and swchng-frequency componens. Wh a knowledge of he DC-capacor curren harmonc specrum, he power loss n he DC capacor s esmaed a dfferen operang condons. REFERENCES [] J. Rodrguez, S. Berne, P. K. Semer, and I. E. Lzama, A survey on neural-pon-clamped nverers, IEEE Trans. Ind. Elecron., vol. 57, no. 7, pp. 9-, Jul.. [] Kouro, Samr, Marusz Malnowsk, K. Gopakumar, Josep Pou, L. G. Franquelo, Bn Wu, Jose Rodrguez, M. A. Prez, and J. I. Leon, Recen advances and ndusral applcaons of mullevel converers, IEEE Trans. Ind. Elecron., vol. 57, no. 8, pp , Jul.. [] D. G. Holmes and T. A. Lpo, Pulse Wdh Modulaon for Power Converers, IEEE press seres on power engneerng. Pscaaway, NJ: IEEE Press,. [4] J. F. Moynhan, M. G. Egan, and J. M. D. Murphy, Theorecal specra of space-vecor-modulaed waveforms, IEE Trans. Elec. Power Appl., vol. 45, no., pp January 998. [5] B. P. McGrah, and D. G. Holmes, An analycal echnque for he deermnaon of specral componens of mullevel carrer-based PWM mehods, IEEE Trans. Ind. Elecron. vol. 49, no. 4, pp ,. [6] M. H. Berhoff, F. W. Fuchs, DC-Lnk Harmoncs of Three-Phase Volage-Source Converers Influenced by he Pulsewdh-Modulaon SraegyAn Analyss, IEEE Trans. Ind. Elecron., vol. 55, no. 5, pp.85-9, May 8 [7] G. I. Orfanoudaks, M. A. Yurach, S. M. Sharkh, Analyss of dclnk capacor curren n hree-level neural pon clamped and cascaded H-brdge nverers, IET Trans. Power. Elecron. vol. 6, no. 7, pp.76-89, Augus [8] F. D. Keferndorf, M. Forser, T. A. Lpo, Reducon of DC-bus capacor rpple curren wh PAM/PWM converer, IEEE Trans. Ind. Appl., vol. 4, no., pp.67-64, March/Aprl 4 [9] Alumnum Elecrolyc Capacors - caalogue no. PG-6DI--. ALCON elecroncs Pv Ld, Jan-. URL:

6 X: Y:.598 X: 7 Y:.79 X: 4 Y:.4 Y: X: Fg.. Analycal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.9 and power facor of Fg. 6. Analycal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.5 and power facor of Fg.. Expermenal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.9 and power facor of Fg. 7. Expermenal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.5 and power facor of Fg. 4. Analycal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.5 and power facor of.844 ESR(ohm) Fg. 8. Plo of ESR versus harmonc number Fg. 5. Expermenal harmonc specrum of DC-capacor curren for I M =4 A, modulaon ndex of.5 and power facor of.844 Modulaon ndex Load curren(a) Power facor Power loss(w) TABLE III POWER LOSS ESTIMATED AT VARIOUS OPERATING CONDITIONS