TORQUE RIPPLE CALCULATION OF TWO-PHASE IM SUPPLIED BY THREE-LEG VSI INVERTER

Transcription

1 Maszyy Elektrycze - Zeszyty Probleowe Nr 3/7 (5) 3 Pavel Záskalický Techical Uiversity of Košice TORQUE RIPPE CACUATION OF TWO-PHASE IM SUPPIED BY THREE-EG VSI INVERTER Abstract: The proposed paper deals with steady state estiatio of a electroagetic torque ripples ad a curret wavefor of a two-phase iductio otor (IM), which is supplied by a three-leg voltage source IGBT bridge coected iverter (VSI). The coplex Fourier series aalysis of the iverter s output voltage was ade, to obtai a aalytical forula of supply voltage wavefor. The space phasor theory has bee applied to the two-phase iductio achie odel to obtai electroagetic torque ad supply curret wavefors for various operatio states. Keywords: two-phase otor, torque ripple, atheatical odel, space phasor. Itroductio The electrical low-power uites which are supplied by a sigle phase voltage are widely deployed by two-phase otors. They are settled i differet idustrial ad doestic devices. They are very ofte deployed as a pups drives i a washig achies ad dishwashers, but also i a circulatig pups for cetral doestic heatig. [],[] Fig. Two-phase IM drive topology A two-phase otor by their characteristics o differs fro the three-phase oes. Their advatage is sipler widig layout, which is of great iportace for autoated otor productio. Slovak Sciece Grat Agecy VEGA is ackowledged. The paper was ade uder project No: VEGA //5. The two-phase otors are at preset aufactured as either squirrel cage iductio or peraets agets sychroous otors. A peraet aget i this case is water ad lye resistat, which allows akig a pup with a absolute waterproof. The stator widig ca be cofigured i either a serial or parallel twophase syste. Norally, the widig are idetical. The widigs which for oe phase are coected to iduce opposite agetic polarity. Moder devices require drives eable speed variatio. For this reaso, desist fro auxiliary phase with capacitor ad two-phase otors are supplied by seicoductor coverters too. There are several ways to create a two-phase supply voltage. Oe of possibility is show i Fig..[5], The voltage source iverter (VSI) processes represet the fial stage i ters of geeratig the output voltage ad frequecy. This oe guaratees good operatig coditios throughout the whole speed rage by adaptig the output voltage to the load coditios. It is thus possible to aitai the agetizatio of the otor at the optial value. Fro the DC lik circuit, the iverter is supplyig variable DC voltage. The ai copoets are cotrolled seicoductors (IGBT trasistors), arraged i pairs i three-legs, as show i Fig.. A freewheelig diode is coected i parallel with each trasistor, because high iduced voltages ca occur across the iductive output load. Assue that variable voltages ad frequecies are geerated by a pulse aplitude

2 4 Maszyy Elektrycze - Zeszyty Probleowe Nr 3/7 (5) odulatio (PAM). PAM is used i coverters with variable iterediate DC-lik voltage. I a case where the rectifier is fully cotrolled, the aplitude of the output voltage is geerated directly. This eas that the output voltage for the otor is ade available i the DC lik circuit.[8]. Iverter atheatical odel To build a atheatical odel of iverter the coplex Fourier series is used. Fourier series are extreely useful as a way to break up a arbitrary periodic fuctio ito a set of siple ters that ca be solved idividually, ad the recobied to obtai the solutio to the origial proble. For iverter operatio study we assue that all seicoductor copoets are ideal. This eas that the switches ca hadle with uliited curret ad voltage. et s assue that the voltage drop ad curret leakage through switches are zero ad that they are able tured o ad off with o rise ad fall ties. We also assue that the iverter iput capacity is sufficietly high, so we ca suppose the coverter iput DC voltage to by costat for ay output currets. The trasistors i each leg are switched so that they for voltage ipulses over the half period of the desired output frequecy. The voltage ipulse of the first trasistors leg easured versus egative pole of the DC lik ca be expressed ],[3] jkt u U e Re cke () k While: U e is a DC lik output voltage ad ck is a Fourier coefficiet, defied as jk ck e jk valid for k The voltage ipulses i other legs a shifted by jkt u U e Rea cke k () j kt u U e Re a cke k jk a e is a phase shiftig factor. Motor phase voltages are give by a differece of the legs voltages as follow u u u u U Re - c e jkt e a k k u u u u U Re - c e jkt e a k k Fig. Phase otor voltages (3) Fig. shows the phase voltages u ad u wavefors calculated o the base of equatios (3). They are shifted by respectively. To siplify the calculatio of the AC otor quatities it is advatageously to eploy space phasors [3], [6]. Output voltage space phasor of VSI u is expressed as follow j uu a u (4) While: a e is a space shiftig factor I Fig. 3 is show the VSI output voltage trajectory. Calculatio was ade for DC-lik voltage U V ad frequecy f 5 Hz. e Fig.3 VSI output voltage trajectory

3 (A) Maszyy Elektrycze - Zeszyty Probleowe Nr 3/7 (5) 5 3. Curret calculatio For the calculatio stator ad rotor space phasor trajectories, there was used the equivalet circuit of IM depicted i Fig i i.5 i,i (A) Fig.4 Equivalet circuit of IM For further calculatio will used followig easured paraeters of two-phase IM P 5W ;U 6V / 5Hz; 3 rev / i; p ; R 3; R 5;. 8H;. 5 H; Referred to the equivalet circuit above, for the curret space phasor i the followig equatio is valid u i (5) k jk R sk jk Rjk R sk jk k sg th takig: sk is slip for k haroic k copoet where: sg for positive, sg for egative sequece Fig. 5 shows calculated the stator curret space phasor trajectory. Coputatio was ade for oial otor speed (A) Fig.5 Stator curret space phasor trajectory t (s) Fig.6 Stator currets wavefors By decopositio of curret space phasor to real ad iagiary part, we receive stator currets wavefors. They are show i Fig. 6. There a several possibilities to solve equivalet circuit for the curret i. The easiest oe is to deterie Thevei equivalet circuit. Thevei equivalet circuit presets a siple series cobiatio of eleets, as show i Fig. 7. Fig.7 Thevei equivalet circuit of IM To fid the Thevei voltage, we ust recalculate the DC-lik voltage for requested frequecy ad otor paraeters as follow Uth (6) R f is a agular frequecy The Thevei ipedace is give by j R j R j ad Z R j th th th (7) Because R, the Thevei resistace ad iductace are approxiately give by Rth R th (8)

4 (A) 6 Maszyy Elektrycze - Zeszyty Probleowe Nr 3/7 (5) O the base of Thevei equivalet circuit for the rotor curret space phasor followig equatio is valid i.5 uth R s R jk (9) k k th th (N) (A) Fig.8 Rotor curret space phasor trajectory Fig. 8 shows the rotor curret space phasor trajectory calculated o the base of the equatio (9). Correspodig two diesioal rotor curret copoets are show i Fig. 9. i,i (A) t (s) Fig.9 Rotor curret wavefors 4. Torque calculatio Electroagetic torque of two-phase iductio otor depeds o the stator ad rotor currets. Geerally followig equatio is valid e M p i i p i i () i i t (s) Fig. Electroagetic torque wavefor There is a possibility to express stator ad rotor liked fields i ters of product of iductace ad curret e M p i i i i () Usig space phasor relatioship, the equatio () coes ito the for M e p I i i * () Fig. shows the calculated tie plot of electroagetic torque of a two-phase IM supplied by a rectagular voltage. 5. Coclusio The proposed cotributio shows possibility of calculatio electroagetic torque of a twophase IM usig space phasor theory. The coputatio is siple ad gives sufficietly accurate values. The proposed atheatical ethod is easily applicable to ultiphase IM, too. The coditio is that we are able aalytically express course of supply iput voltages. The ethod of the atheatical represetatios of the voltages be doe ay be differet. This oe ca be doe i the for of ifiite series, but also by eas of trigooetric fuctios. 6. Refereces [] S.J. Chapa, Electric achiery fudaetals, Mc-Graw-Hill, 4 th editio, (5) [] J.Chatelei, Machies électriques, Vol. X, Editio Georgi, (983) [3] G. J.Machowski, J.W. Bialek, J.R. Buby, «Power syste Dyaics: Stability ad cotrol, Joh Wiley & Sos, ()

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