SIMULATION OF THE IMPACT OF HIGHER HARMONICS ON THE TRANSIENT PROCESS OF INDUCTION MACHINE FED FROM PWM INVERTERS

ISSN 1330-3651(Print), ISSN 1848-6339 (Online) DOI: 10.17559/TV-20150502231618 SIMULATION OF THE IMPACT OF HIGHER HARMONICS ON THE TRANSIENT PROCESS OF INDUCTION MACHINE FED FROM PWM INVERTERS Nena Marković, Sloboan Bjelić, Jeroslav Živanić, Uroš Jakšić Preliinary counication In this paper a funaental atheatical oel (algorith) for eterination of conition of inuction achine as well as the siulation proceure for easureent of certain values in progra MATLAB has been presente. A schee of easuring syste an results of siulation application are presente on output lists. Thus, easuring syste graually evelops an iproves as cobination of inforation an results of previous steps in oeling with new eleents essential for evelopent of researches when inuction achine is fe through PWM inverter. Keywors: algorith; inuction achine; inverter; easureent; siulation Siuliranje utjecaja viših haronika na prijelazni proces inukcionog stroja napajanog iz PWM invertera Prethono priopćenje U rau je prikazan osnovni ateatički oel (algorita) za oređivanje stanja inukcionog stroja kao i postupak siulacije jerenja pojeinih vrijenosti u prograu MATLAB. Na izlazni listaa prikazana je shea jernog sustava i rezultati prijene siulacije. Na taj način se jerni sustav postupno razvija i usavršava kao kobinacija inforacija i rezultata prethonih koraka u oeliranju s novi eleentia značajni za razvoj ijagnostičkog sustava kaa se inukcioni stroj napaja pooću PWMinvertera. Ključne riječi: algorita; ijagnostika; inukcioni stroj; jerenje; siulacija 1 Introuction In polyphase an ultiple-wining inuction achines in the air gap, besie funaental haronics also occur any haronics with frequency that is lower an higher than the funaental-network frequency, an have one nae- higher haronics. They are characterize by tie an space istribution in the inuction achine [1, 2]. Tie-epenent haronics reach the gap of stator an rotor of the inuction achine fro the achine erivatives (connections towar electric network, fro echanical connector such as shaft an fro theral erivatives) an space-epenent haronics occur ue to constructive particularities an non-linearity of the paraeters in the achine; this assuption is also vali for oel of ouble-pole inuction achine (p=1) [3, 4]. The first source of higher tie haronics in a steaystate regie is non-sinusoial voltage on erivatives of inuction achine. If syetric an non-sinusoial voltages also associate to iealize inuction achine the value of agnetic inuction in air gap shall have the for of associate voltage, which besies the funaental also contains the higher haronics. By knowing the given voltage for on erivatives of inuction achine it is possible to calculate an easure or siulate the spectral coposition of the fiels in the gap. The secon source of higher tie haronics in air gap is an engine shaft-echanical erivative of the inuction achine. During the changing of loaing oent M op or uring the changing of spee of rotation n (frequency) the higher haronics occur in the air gap. On the sie of echanical erivative higher haronics in air gap can also occur uring non-linear change of rotation n (frequency) an uring siultaneous non-linear change of oent M op an spee of rotation n (frequency) [5]. Tie-epenent higher haronics can also reach the air gap through theral erivatives-fro areas that surroun conucting parts of the inuction achine. In the case of non-linear changes of the teperature in the inuction achine environent higher tie-epenent haronics occur an shape of agnetic fiel becoes efore. The highest values of higher haronics aplitues occur at theral shocks [4, 6]. Space epenent haronics occur ue to specific construction an have big influence on energy transforation process in inuction achine. Iportant space haronics of an inuction achine are [5]: funaental haronic of agnetic excitation force (MEF) an MEF teeth an haronics occurre ue to non-linearity of electric achine paraeters, an technological haronics occurre ue to application of cobine frequencies (in oern inuction achines). Iportant influence on space haronics spectra has eccentricity of rotor an stator axes, conical or ellipsoial shape of rotor an eforities in wining positioning of stator an rotor [1, 5]. That eans that in air gap exists the spectru of fiel which can be ivie on sets of haronics, an ivision shoul be perfore accoring to tie an space origin. 2 Measureent of significant quantities of the inuction achine Previously entione ivision of higher haronics on tie-epenent an space-epenent can be only consiere conitionally because all haronics are connecte to the energy of a fiel in the gap an can be consiere only together in the space an tie, Fig. 1. Process equations in inuction achines present atheatical oel (algorith of theoretically iealize achine) for the analysis of agnetic fiel in the gap between rotor an stator an calculation (easureent/siulation) of spectru of voltages an currents haronics of inuction achine that gives the picture on its conition. Tehnički vjesnik 24, 1(2017), 265-271 265

Siulation of the ipact of higher haronics on the transient process of inuction achine fe fro PWM inverters N. Marković et al. Figure 1 Schee [7] an istribution of fiels of ouble-pole three-phase achine of inuction engine 3,4 MW. One flux line presents unit value 0,005 Wb/ [5] All oern nuerical ethos presently use by engineers for calculation of istribution of agnetic fiels in the gap start fro the etho of final ifferences or fro etho of finite eleents an efine initial conitions an coe to calculation of inuctivity an reactance as inuctive paraeters of the inuction achine [7]. The ai of this paper is the siulation proceure of easureent of certain values on the oel of threephase ieal ouble-pole inuction achine with sooth rotor an air gap; all paraeters are linear an voltage on erivatives is rectangular with PWM inverter [7, 8]. Inuction achine has 3 winings on stator with nuber of coils N A N B N C an three winings on rotor with nuber of coils N a N b N c. For three-phase inuctiоn achine in space syste of phase coorinates A, B, C, a, b, c ifferential equations for contours where phase voltages are present are as follows: ψ A VA = RAiA t ψ B VB = RBiB, ψ C VC = RCiC (1) ψ a Va = Raia t ψ b Vb = Rbib t. ψ c Vc = Rcic (2) The values of echanical quantities are: ωr p = J ωr J ( T Fω T ) e = J 2π 60 Δn = p Δt ( T Fω T ) e, (3) where V voltages A, B, C, a, b, c; i i rotor currents i r an stator currents i s ; R s, R r active resistances of stator an rotor; ψ i the resultant agnetic fluxes; ω r electrical angular velocity (ω r xp); ω angular velocity of the rotor; p nuber of poles inuction achines; T e electroagnetic torque inuction achine; T shaft echanical torque inuction achine; F cobine rotor an loa viscous friction coefficient; J cobine rotor an loa inertia coefficient. The resultant flux in phase A can be eterine fro the relation: ψ A = LAi A M ABiB M ACiC M Aaia, M Abib M Acic (4) ψ a = Laia M abib M acic M Aai A. M i M i (5) ab b ac c In the siilar way relations for resultant fluxes of other 5 contours which relates to phases B, C, a, b, c are written. In the real inuctivity coorinates an utual inuctivity in relations given for fluxes are coefficients with perioical values, which change as functions of haronic at rotation of inuction achine rotor [3, 7]. If expressions for fluxes in Eq. (4) an (5) are inclue in Eq. (1), (2) an (3), solutions with a few ozen eleents an perioically variable coefficients will be obtaine. In orer to avoi these perioical coefficients, atheatical transition-transforation to another appropriate syste of coorinates is necessary where in the air gap the fiel is rotating an can be presente by resultant vector of rotating fiel B ob. By projecting of resultant vector of rotor currents, voltage, an flux on certain axes (here on, q, 0 stator axes), expressions that correspon to atrices of transitional values can be eterine [7, 9]. 3 Matrix of transforation fro 0q coponents to uvw coponents an inverse atrix of reverse process of transforation fro uvw to 0q Syste atrix 0q (Park s transforation atrix) correspons best to inuction achines construction because it takes into account inequality of ipeances on longituinal an transverse axis an therefore is use in analysis of transitional processes in electric achines [2]. Transition atrix fro 0q coponents into uvw coponents an inverse atrix relate to inverse process of transition fro uvw into 0q are efine by Eq. (6) an (7) [10 13]: [ ] [ ][ ] 1 0q uvw = Tuvw T0q [ T ] [ T ] T T, (6) uvw q = 0 q uvw 0. (7) Measuring syste of iagnostics of technical conition of inuction achine base on the analysis of space an tie epenent haronics of agnetic fiel ust contain sensors for conversion of paraeters of space an tie haronics into unique electrical signal. With the assistance of AD converter obtaine inforation is entere into coputer for further processing [14, 15]. Toay, PC is ostly use for aitional processing an rarely special controller with appropriate software support. Possible schee with coputer support can be perfore in two ways, Fig. 2, through: 1. Siple analysis, easureent an recoring of higher haronics of voltage an stator currents. 2. Calculation of coefficients of atheatical oel (this proceure is applicable only for basic haronics [9, 16]). 266 Technical Gazette 24, 1(2017), 265-271

Figure 2Basic schee for iagnostic syste Accoring to the first way, the task coes own to progra writing or to usage of one of the progras for siulation of inuction achine in an open-loop spee control on a 3,6 kw, 230 V inustrial otor (MATHLAB/Siulink, progra Matheatica Wolfra etc.) [5]. The progra eory ust contain coposition of haronics at noral inuction achine conition an at engine running perioically to recor results an copare coposition of haronics regaring the noral engine. A coan for signalling at elevate vibrations or for isconnection of the engine fro the rive at higher isturbances ust be given to the progra. Realization of the first way for writing of a certain progra is not ifficult, but proceure itself has soe faults. Priarily, it is necessary to establish correlation between efects of echanical parts with paraeters that eterine haronics, an requires a huge aount of coputation when a large nuber of haronics shoul be retaine. On the other han, this proceure oes not allow the ientification of efects. The secon way shows that calculation of coefficients of atheatical oel requires the coplex progras, since coefficients are calculate in the real tie an ust be copare with values obtaine in the operating process of inuction engine. As iscusse the coplexity of the secon way is reflecte in writing of atheatical oel that contains ifferential equations for each haronic respectively, where assive syste of equations, very ifficult for solving, is obtaine. Regarless of applie way, 1 or 2, in both cases a progra which perfors analysis of higher haronics of stator voltages an currents is neee. Progra which refers to spectral analysis ust rely on signals obtaine fro sensors that contain paraeters of haronics coponents, an ata about the shall be use in aitional analysis of atheatical oel. 4 Siulation of ynaic operation regie of the inuction achine fe by rectangular voltage The inuction achine block operates in any state: as generator or engine. The operation oe is ictate by the sign of the echanical torque (positive for engines, negative for generators). The electrical part of the inuction achine is represente as the four-poles an the echanical part as a secon-orer syste. Electrical variables are brought to the stator. This is inicate by ( ) sign in the equations given below. All stator an rotor quantities are in the arbitrary two-axis reference frae (q frae). The subscripts use are efine as follows, Fig. 3: Subscript efinition: : axis quantity ; q axis quantity q; r rotor quantity; s: stator quantity; L: leakage inuctance; agnetizing inuctance. Applying Eq. (1) (7) the following is obtaine: Vqs = Rsiqs ϕ qs ωϕs, (8) ϕ = L i L i', (9) qs s qs qr Vs = Rsis ϕs ωϕs, (10) ϕ s = L sis Li' r, (11) V' qr = R' r i' qr ϕ' qr ( ω ωr ) ϕ' r, (12) ϕ ' = L' i' L i, (13) qr r qr qs V' r = R' r i' r ϕ' r ( ω ωr ) ϕ' qr, (14) ϕ ' = L' i' L i, (15) T e r r 3 = p 2 r s ( ϕ i ϕ i ) s qs qs s, (16) L s = Lls L, (17) L ' = L' L. (18) r lr Figure 3 Equivalent electric schees an evelopent on -axis an q-axis Mechanical syste of quantities is: 1 ω = ( Te Fω T ), 2H (19) θ = ω. (20) Paraeters of inuction achine block are efine as follows; all quantities are referre to the stator. Definition of paraeters is: R s, L is stator resistance an leakage inuctance; R' r, L' ir rotor resistance an leakage inuctance; L agnetizing inuctance; R s, L' r total stator an rotor inuctances; V qs, i qs q axis stator voltage an current; V' qr, i' qr q axis rotor voltage an current; V s, i s axis stator voltage an current; V' s, i' s axis rotor voltage an current; φ qs, φ s stator q an axis; φ' qr, φ' r rotor q an axis; θ nuber of pole pairs; θ r electrical rotor angular position(θ r p); H cobine rotor an loa inertia constant. Tehnički vjesnik 24, 1(2017), 265-271 267

Siulation of the ipact of higher haronics on the transient process of inuction achine fe fro PWM inverters N. Marković et al. 5 Siulation of atheatical oel of operation of inuction achine The software MATLAB Siulink Power Syste oel [7] is use for siulation of atheatical oel of operation of inuction achine, schee in Fig. 4, which uses value equations. Behaviour of three-phase inuction achine, as loaing in ynaic operation regie has been teste at three-phase inuction engine start-up, with noinal value of 3,6 kw fe by inverter that gives rectangular voltage for [17 20]. At the engine start-up fe fro inverter converter it is typical that spectru of haronics of engine current continuously changes uring the whole start-up interval [2, 21]. For obtaining of ore authentic responses at the output (stator an rotor currents, spee of rotation an oent) the following paraeters of inuction achine have been selecte: Paraeters inuction achine: Rotor type: Woun Reference frae: Stationary No. power, L-L, voltage an frequency: [P n =3 1200 (VA), V n =230 (Vrs), f n =50 (Hz)] Stator: [R s =0,435 (Ω), L is =2 2,0e 03 (H)] Rotor: [R' r =0,816 (Ω),L' ir =2,0e 03 (H)]-values referre to stator Mutual inuctance: L =69,31e 03 (H) Inertia, friction factor, pairs of poles [J=0,089 (kg 2 ), F=0 (N..s), p=2]. Figure 4 Inuction achine fe through PWM inverter. Inverter with ipulse wih oulation-ipulse range oulation. Low-voltage three-phase inuction engine 3,6 kw "Sever"-JUS ISO 9001 (source: www.sever.rs) n = 1410 in 1, Protection IP 54, Voltage 230 V, f=50 Hz 6 Siulation results Tie fors of voltage haronics V ab (1, 3, an 5 th ) (rectangular ipulses of sae aplitue, which, at the sae tie, present values of agnetic inuction in the gap of stator an rotor) are presente in Fig. 5. Fig. 6 represents appropriate spectru of current haronics for stator an rotor of siulate 3 r an 5 th haronic. Fig. 7 represents haronics spectrus of stator an rotor torque uring the starting of inuction achine an the ipacts of siulate 3 r an 5 th haronics, while Fig. 8 represents tie fors of spee of rotation of stator an rotor an the ipacts of the 3 r an 5 th haronic coing into inverter for feeing an operation of the achine. For 1 st basic an 3 r an 5 th haronic the sae tie fors of currents, rotation spee an oent are obtaine. By coparing tie fors of spectru of voltage haronics at the inuction achine input obtaine fro PWM inverter an currents of rotor an stator it can be seen that inuction achine anifests certain insensitivity to higher haronics eleents. Such insensitivity coul be ue to iperfection of the oel for siulation or atheatical proceure regaring ual transforation-transition process fro uvw (in this case phase a, b, c) into 0q coponents or ue to inverter operation. Deviation of current tie for fro sine for can be notice uring the run-up interval. Accoring to Fig. 5 it can be conclue that values of voltage aplitue are the sae for all haronics of voltage an the value of agnetic inuction V in air gap has the for of ipresse voltage. 268 Technical Gazette 24, 1(2017), 265-271

a) 1 st haronic b) 3 r - haronic siulate in PWM inverter c) 5 th - haronic siulate in PWM inverter Figure 5 Interphase rectangular voltagev ab fro PWM inverter at the inuction achine input a) 1 st haronic b) ipact of the 3 r haronic c) ipact of the 5 th haronic Figure 6 Haronic spectru of stator an rotor currents of inuction achine an ipacts of siulate 3 r an 5 th haronic coing to inverter for feeing an operation of the achine Tehnički vjesnik 24, 1(2017), 265-271 269

Siulation of the ipact of higher haronics on the transient process of inuction achine fe fro PWM inverters N. Marković et al. a) 1 st haronic b) ipact of the 3 r haronic c) ipact of the 5 th haronic Figure7 Haronic spectru of stator an rotor torque of the inuction achine an the ipacts of siulate 3 r an 5 th haronic coing to inverter for feeing an operation of the achine a) 1 st haronic b) ipact of the 3 r haronic c) ipact of the 5 th haronic Figure 8 The nuber of rotations of stator an rotor of inuction achine an the ipacts of siulate 3r an 5th haronic coing to inverter for feeing an operation of the achine 7 Conclusion For the purpose of presentation of testing results, a siulation oel of three-phase active power filter with referent current with ynaic oel of a three-phase inuction achine. Siulation of feeing of inuction achine by rectangular voltage haronics in all electrical an echanical tie variable quantities is create: stator an rotor currents, rotation spee an electroagnetic oent. Siulation an analysis of tie fors-iagras of voltages, stator an rotor currents, spee of rotation, electroagnetic oent (T) certainly oes not create a coplete spectral picture since siulation of space 270 Technical Gazette 24, 1(2017), 265-271

haronics of agnetic fiel along the scope of the stator is lacking. In siulation of engine run-up fro inverter converter spectru of haronics of engine current (rotor an stator) uring the run-up interval {0 0,1} has been continuously change an there was ipression that they are insensitive to haronics, that is, inuction achine oes not etect haronics at the input of the achine. However, iagnostics base only on consieration of istribution in tie oain at least creates initial conitions for correction of asyetries an higher haronics. On iagras at real proportion, which are obtaine by siulation, haronics that presents 0,1 part of the basic haronic have been analyze, since further reuction of values of haronic aplitues coul cause serious probles, an therefore the value 0,1 in this case presents sufficient sensitivity for analysis of spectru of haronics in the inuction achine. Parallel active evice for correction of haronics shoul etect unesirable coponents of current prouce in nonlinear loaing such as inuction achine, an inject the siilar current (for of current) into the point where the correction evice is. Corrector of asyetries is ieal resource whose currents correspon to unwante spectru of coponents in the network current that shoul be annulle with as less loss as possible. 8 References [1] Beaty, H. Motor require voltage liit-elec. // Worl. 5, (1978), pp. 52-63. [2] Vonina, D.; Nastran, J. Active Power Filter Copensating Haronic Distortion an Unbalance Non-Linear Loas. // Autoatika. 3-4, (1998), pp. 111-118. [3] Bjelić, S.; Marković, N.; Jakšić, U.; Živanić, J. Selection of linear filter eleents paraeters for easuring of voltage an currents coponents of irect an inverse orer. // Przeglą Elektrotechniczny. R. 89 NR 1a/2013, (2013), pp. 172-176. http://pe.org.pl/articles/2013/1a/44.pf [4] Montsinger, V. M.; Clea, J. E. 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Control Strategies for Active Power Filter in Three-Phase Four-Wire Systes. // Fifteenth Annual IEEE Applie Power Electronics Conference&Exposition, APEC 2000, 1, Piscataway NJ, USA, (2000), pp. 420-426. DOI: 10.1109/APEC.2000.826137 [14] Bjelić, S. Energetic converters in networks an installations. SVEN Niš, 2007. [15] Bjelić, S.; Jakšić, U.; Marković, N. Energetic converters. Kvark-Kraljevo, 2010. [16] Jeong, S. G.; Woo, M. H. DSP-Base Active Power Filter with Preictive Current Control. // IEEE Trans. In. Electron. 44, 3, (1997), pp. 329-336. http://serials.unibo.it/cgiser/start/it/spogli/f-s.tcl? prog_art=641122&language=italiano&view=articoli [17] Arees, M.; Hafner, J.; Heuann, K. A. Three-phase Fourwire Shunt Active Filter Using Six IGBTs. // EPE 95, 6th European Conference on Power Electronics an Applications, 1, Seville, Spain, (1995), pp. 874-879. [18] Akagi, H.; Nabae, N.; Ato, S. Control Strategy of Active Power Filters Using Multiple Voltage Source PWM Converters. // IEEE Trans. In. Appl. 22, (1986), pp. 460-465. DOI: 10.1109/TIA.1986.4504743 [19] Arees, M.; Watanabe, E. H. New Control Algoriths for Series an Shunt Three-phase Four-wire Active Power Filters. // IEEE Trans. Power Deliv. 10, 3, (1995), pp. 1649-1656. [20] Verelho, P.; Marques, G. D. An Active Power Filter an Unbalance Current Copensator. // IEEE Trans. In. Electron. 44, 3, (1997), pp. 321-328. [21] Dixon J.; Garcia, J. Control Syste for the Three--phase Active Power Filter which Siultaneously Copensates Power Factor an Unbalance Loas. // IEEE Trans. In. Electron. 42, (1995), pp. 636-641. DOI: 10.1109/41.475504 Authors aresses Nena Marković, MSc Avance Polytechnic School of Vocational Stuies Uroševac, 24. Novebar, 38218 Leposavić, Srbija E-ail: nen.ark74@yahoo.co Sloboan Bjelić, PhD Faculty of Technical Sciences Kosovska Mitrovica, University of Priština, KnezaMiloša 7, 38220 Kosovska Mitrovica, Srbija E-ail: sloboanbjelic49@yahoo.co Jeroslav Živanić, PhD Faculty of Technical Sciences Čačak, University of Kragujevac, Svetog Save 65, 32000 Čačak, Srbija E-ail: jeroslav.zivanic@ftn.kg.ac.rs Uroš Jakšić, PhD Avance Polytechnic School of Vocational Stuies Zvečan Nušićeva 6, 38227 Zvečan, Srbija E-ail: uros_jaksic@yahoo.co Tehnički vjesnik 24, 1(2017), 265-271 271