GENERALIZED FOURIER SERIES A USEFUL MATHEMATICAL TOOL IN POWER ELECTRONICS

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1 GEERALIZED FOURIER SERIES A USEFUL MATHEMATICAL TOOL I POWER ELECTROICS A. J Iwszkiewicz, B. Jcek Perz, C. Muel Perez Dosió. The Electrotechicl Istitute, Gdsk Brch, Pold. Electricl Egieerig Deprtmet, Vigo Uiversity, Spi The pper is relted to the prolem of cretig stepped wveforms y use the geerlized Fourier series. This method c e pplied especilly i power electroics s tool permittig to rech AC voltge wveforms pproximted to the sie wve shpe. This ovel pproch to the sythesis of AC voltge wveforms is sed o Fourier series of orthogol fuctios set. Applyig lyticl methods d expressios give i the pper it is possile to defie such optiml prmeters of the wveform i order to receive the lowest hrmoics cotet. The preseted mthemticl tool is very useful i desigig structures d cotrol lgorithms of multilevel coverters. Key words Fourier series, orthogol fuctios, pproximtio, hrmoics, multilevel coverters. ITRODUCTIO I power electroics there re my devices i.e. AC drives, uiterruptile power sources or compoets of distriuted power geertio systems, where the essetil demd is to geerte sie wve voltge (or curret) wveforms of reltively high qulity. The voltge or curret otied from DC sources like rectifiers, storge ccumultors, wid or photovoltic frms d fuel cells is coverted to AC usig power electroics coverters. The result of coversio my ssume shpe of rectgulr pulses or stepped wveforms. Their qulity should comply with the pproprite stdrds. I mthemtics the qulity of pproximtio is mesured y use of the verge squre error ut i power electroics pplictios the most populr d useful tool is the THD fctor (Totl Hrmoic Distortio). Although this fctor does ot crry y hdy iformtio out the shpe of the hrmoics spectrum. The commoly used device for the purpose of eergy coversio is two level VSI (Voltge Source Iverter), cotrolled usig PWM (Pulse Width Modultio) method. But i VSIs the dmissile method of coversio hs well kow disdvtges, relted to high frequecy switchig, like power losses i switchig elemets d ecessity of usig specil filters for high frequecy compoets i the output voltges. These disdvtges c e reduced usig multilevel coverters d mplitude modultio method. The pper presets mthemticl pproch to the cotrol strtegy of multilevel coverters. The relevt method for sythesis of the output wveforms pplyig geerlised Fourier series is discussed [, ]. Thks to otied useful d lytic descriptio it is esy to optimise the prmeters of the coverter output wveforms. The mthemticl wveform sythesis preseted i the pper is very helpful tool durig desigig of cotrol lgorithms d structures of multilevel iverters.. WAVEFORMS SYTHESIS BASED O FOURIER SERIES.. STADARD FOURIER APPROXIMATIO Give is sclig fuctio φ(x): for x <, ( x ) = () for other x The sclig fuctio φ (x) is defied s follows: ( x) = ( x ) for =..,,,,,,.. () The equtio defies set of rectgulr pulses of uitry mplitude d gle durtio equl to. The pulse positio o x xis depeds o prmeter so settig this prmeter permits to cotrol the time dely of pulse. A few exmples of the sclig fuctios re preseted i Fig.. Fig.. Exmples of the sclig fuctios: φ - (x), φ (x), φ (x)

2 I itervl x <, ), which legth is k (k ), fuctios φ (x) stisfy two coditios: = ( x) dx = (3) x x dx = if k m k ( ) ( ) m therefore the ottio () cretes the set of orthogol fuctios clled orthogol se. Sice ll fuctios φ (x) hve their orms equl to, this se is clled orthoorml se. The expsio of fuctio f (x) i geerlized Fourier series relted to the set of sclig fuctios (φ ) is s follows: ( ) x = = where ( f, ) c = f c ( x) (4) = f ( x) ( x) dx The expsio (4) is vlid for y fuctio f ( x) L <, > The Fourier series cotis ifiite umer of elemets d mkes it possile to pproximte fuctio f (x) y use of ifiite set ( sum) of dequtely scled fuctios φ (x). Prticulrly it is possile to expd fuctio f (x)=si(x) y summig ifiite set of rectgulr pulses. It is i cotrdistictio to typicl pplictio of the Fourier series where y fuctio f(x) is expded s set of hrmoics. Accordig to (4) d (5) the expsio of si(x) i the itervl x <,) is give s: si = ( ) = x = si ( x) ( x) dx ( x) x <, > (5) (6) The expressio (6) defies series of cosecutive rectgulr pulses represeted y fuctios φ (x). Amplitudes of pulses re differet d determied fter clcultio of the itegrl. This sttemet c e pplied for compositio of sie wveforms i power electroics. Rectgulr pulses represet essetil shpe of output voltge (curret) of iverter. turlly the compositio of stepped wveforms usig rectgulr pulses hs ee pplied i my pplictios ut it cocered verticl dditio of wveforms. The resultig phse voltge ws sythesized y the dditio of the voltges geerted y differet cells of cscded iverter. The preseted proposl reltes to the dditio of pulses log x xis or i time scle. Cosecutive pulses form the resultig output voltge or curret of the coverter. Prcticlly i power electroics pplictios, the pproximtio of sie wve c e relized usig fiite umer of the series memers d turl spirtio of desigers is to utilize the possily lowest umer. The ccurcy of pproximtio depeds o it. Its umericl vlue c e mesured i differet wys. I mthemtics the ccurcy of pproximtio is defied s the verge squre error δ, very useful criterio destied to tht purpose. If the pproximtio of the fuctio f (x)=si(x) hs ee doe i the itervl <,> y use of elemets, the verge squre error δ is determied y the followig expressio: δ = si( x) c ( x) dx = x <, >, > (7) I power electroics the most importt criterio of the ccurcy or rther the qulity of pproximted wveforms is THD fctor. For illustrtio of the prolem two exmples of pproximtio hve ee preseted i Fig. d Fig. 3. The hrmoic spectr hve ee locted er the wveforms i order to expose the evidet reltio etwee the umer of steps d the order of pperig hrmoics. Figures d 3 preset stepped wveforms otied fter pproximtio sed o the set defied ccordig to () d (). The results of Fourier pproximtio for selected re collected i Tle. The results of verge squre errors, clculted ccordig to the expressio (7), re preseted i the colum deoted y δ. The symol F deotes the umer of steps i the sced itervl herei it is the itervl <,π >. The prmeter F deotes the umer of demded supply DC voltge or curret sources. The correltio etwee these two prmeters is descried y the followig: sttemet: F if F W th = F 4 4 F else = E + F 4 (8) where W deotes the set of whole umers d E fuctio [x] kow s fuctio Etier {x}. The vlue F is very importt prmeter of multilevel coverters. Tle : The prmeters of Fourier pproximtio for differet F F δ THD F = π % F =6 π/ % F = π/ % F =6 π/ % F =4 π/ % Fig. presets the stepped wveform otied usig very low pproximtio level. The rtio of the

3 mesures of steps is equl to d it is the sme s i three phse iverter with coected lod d cotrolled y dequte set of rectgulr wves. I order to produce such wveform i oe phse pplictios the coverter eeds oly two DC sources esy to get e. g. y dividig oe DC supply voltge. Fig.. The pproximtio of the fuctio f (x)=si(x) for =6 (=π/3) d the spectrum lysis of the wveform Fig. 3. The pproximtio of the fuctio f (x)=si(x) for = (=π/6) d the spectrum lysis of the wveform Geerlly the pplictio of the Fourier series i the domi of pproximtio does ot demd the use of orthoorml se of sclig fuctios. The ecessry coditio is the orthogolity of the se. Therefore other set of orthogol fuctios c e pplied for pproximtio. It my e set cosistig of rectgulr pulses of differet legth k o the x xis: for x < k, k ( x) = k >, k =,,,... (9) for other x The sclig fuctio φ (x) is defied s follows: k ( ) = = x x k for =,,,... () A exmple of pproximtio usig two sclig fuctios: φ (x) φ (x) hs ee preseted i Fig. 4. Relevt prmeters re: =π/6 d =π/3. I this cse the hrmoic spectrum is ot regulr d cotis lower order hrmoics icludig the third oe. The totl hrmoic cotet is less th i cse preseted i Fig. : THD=.6 % i compriso to 3.9 % i cse of stdrd pproximtio. The pproximtio ccurcy reches the vlue δ=.... OPTIMIZED FOURIER APPROXIMATIO The proportio of step mesures preseted i Fig. does ot ssure the miiml hrmoic cotet of the wveform ville i coverter equipped with oly two DC sources. It will e show further tht chgig the rtio of the supply voltges d mtchig the durtio of pulses it is possile to decrese the THD fctor of the output wveform. Fig. 4. The pproximtio of the fuctio f (x)=si(x) for = ( =π/6, =π/3) d the spectrum lysis of the wveform I cse of the wveform preseted i Fig. ll eve hrmoic compoets re equl to zero d odd compoets k re give s: k 4 = [ V + ( V V ) cos ] k Ν () kπ

4 where V d V deote the mesures of steps d the gle of the first step. It ws ssumed here tht i every hlf cycle the wveform ws symmetricl with respect to the stright lie π = + π =..,,,,,,.... The reltio-ship () c e utilized to select the step levels i order to ccel selected hrmoic compoets. The prolem of the selective hrmoic elimitio is well kow s fudmetl of the stte of the rt. The exmple discussed here shows oly the results ville i the three level coverter. Assumig ccelig of third d fifth hrmoic compoets set of equtio c e writte: [ V + ( V V ) cos ] = π + ( V V ) cos( 3 ) = + ( V V ) cos( 5 ) = 4 V () V Solvig the set of equtios oe c receive: π V =.397, V =.948, =, =, 3 = 5 = 4 The results of clcultios re preseted i Fig. 5. The THD rtio is 3, %. ( ) f, θ = [ θ + ( θ ) cos[ ( k + ) ] = + (5) k k The three dimesiol picture of the lyzed fuctio f (,θ ) is preseted i the Fig. 6. f (lf, thet),5,,5,,5,9,8,7 thet,6,5,4,3,,,,349,698 lf,47,396,,,3,4,5,6,7 thet,-,5,5-,,-,5,5-, -,5,-,5,5-,,-,5,5-, -,5,8, lf [degrees] Fig. 6. Three dimesiol picture of the fuctio f (,θ ), d its projectio o (,θ ) ple showig the optiml re The prmeters of the optiml pproximtio for the miimum THD vlue re s followig: = π/9 (4 ), θ=.4, =, V =.3655, V =.936. The THD fctor for this wveform is.4 %. The optiml wveform d its hrmoic spectrum is preseted i the Fig. 7. Fig. 5. The pproximtio of the fuctio f (x)=si(x) for = ( =π/4, =π/) with third d fifth hrmoics ccelled d the spectrum lysis of the wveform Tkig s criterio the miiml vlue of the THD rtio it is possile to fid optiml rtio of step prmeters s miiml vlue of the expressio: THD = Cosiderig () k+ (3) 4V THD = f (, θ ) k =,,... (4) π d the miiml vlue of THD is relted to the miimum of the fuctio:

5 Fig. 7. The pproximtio of the fuctio f (x)=si(x) for = d miiml THD fctor with optiml prmeters ( =π/9, =5π/9, THD=.4 %) 3. APPLICATIO EXAMPLES The proposed coverter model c e used s the descriptio of multilevel coverter s output wveforms i the form of orthogol squre impulses. The output wveform is creted s sum of the cosecutive impulses shifted log the ωt xis. The ide of summig impulses is preseted i the Fig. 8. The preseted coverter cosists of two oe-phse ridge iverters (F d F) d summig circuit Σ. Fig. 9. Three-phse three-level voltge coverter uilt usig three oe-phse four-switches iverters 3. COCLUSIOS The ove descried method of sie wve pproximtio permits to defie mthemticl model of the coverter i which the sythesis of the lterte output wveform is sed o the geerlized Fourier series. The Fourier series cotis set of limited umer of orthogol fuctios represetig rectgulr pulses. The preseted pplictio exmples show the possiilities of the prcticl use of the proposed method. The full descriptio of the model d some more resultig implictios re iserted i the moogrph []. Fig. 8. The exmple of oe-phse voltge coverter geertig f =6 wveform The idividul cosecutive impulses re geerted i this wy tht the ed of the former impulse djois the egiig of the ext oe. Therefore it is possile to desig direct coupled solutios (without summig circuitry), similr to cscde coverters. The cotrol priciple for the idividul semicoductor switches withi the coverter is derived directly from the preseted ide of the output wveforms costructio. The three-phse three-level direct coupled voltge coverter uilt usig three oe-phse four-switches iverters supplied from commo voltge source is preseted i the Fig. 9. REFERECES [] IWASZKIEWICZ J: Mthemticl Models of Power Electroics Multilevel Coverters Alysis d Applictios, Proceedigs of Electrotechicl Istitute, vol. 7, pp. 4, Wrsw, 6, Pold. [] IWASZKIEWICZ J., Perz Jcek: Fourier Series d Wvelet Trsform Applied to Stepped Wveforms Sythesis i Multilevel Covertors Proceedigs of Electrotechicl Istitute, vol. 7, pp , Wrsw, 6, Pold.