COMPLEX FOURIER SERIES MATHEMATICAL MODEL OF A UNIVERSAL MOTOR SUPPLIED BY A TRIAC

Transcription

1 Zeszyty Probleowe Maszyny Elektryczne Nr 2/23 (99) 59 Pavel Záskaliý, Ján Kaňuch Technical University of Košice, Slovakia COMPLEX FOURIER SERIES MATHEMATICAL MODEL OF A UNIVERSAL MOTOR SUPPLIED BY A TRIAC MODEL MATEMATYCZNY SILNIKA UNIWERSALNEGO ZASILANEGO POPRZEZ TRIAK PRZY WYKORZYSTANIU ZESPOLONYCH SZEREGÓW FOURIERA Abstract: The present contribution shows an analytical ethod of the calculus of the torque ripple and current wavefors of a universal otor supplied by a triac directly fro the network. The triac output voltage wavefor is forulated by the coplex Fourier series. The arature reaction of the otor is included in the calculus. The otor perforance is coputed using the circuit paraeters deterined by easureents. The calculated current wavefors are copared with the easured ones.. Introduction Despite their disadvantages, universal otor belong to the ost used electric achines in hoe appliances as well as workshop hand tools. Thanks to their excellent regulation properties they are eployed as drive otors of washing achines, professional ixers or ills. Their versatility is given by the fact that they can be supplied by both direct and alternative voltages. In both cases the otor speed is controlled by the value of the supply voltage. Fig.. Construction of universal otor used for washing achine drive Main advantages of the universal otor include: econoical and sooth speed regulation in wide range, easy start, large acceleration torque and sall acceleration current, good power factor. Sall universal otors are produced without copensating and coutation winding. The stator contains ostly two salient poles with exciting winding. The nuber of the coutator brushes is equal to the nuber of excitation winding poles. Since agnetic field of the achine is alternating, the agnetic circuit is lainated. The stator and the rotor are series connected. 2. Matheatical odel of the otor Matheatical analysis of a universal otor is based on the voltage equations, with certain siplifying assuptions []: echanical losses and also losses in iron are neglected, coutation influence is neglected. A perfect coutating arature is assued, utual inductance is supposed to be constant. Saturation effect is neglected. Fig.2 depicts the equivalent circuit of a twopole universal otor. The arature has two brushes on the diaeter of the coutator and it is shifted by an angle αa towards the axis of the exciting agnetic flux. The rotor rotates with echanical angular velocityω. Suppose that the otor is supplied by a variable voltage u. In such a case the following voltage equation can be written: di u = Ri + L + ui () dt For the electrootive force (EMF), the folowing general equation:

2 6 Zeszyty Probleowe Maszyny Elektryczne Nr 2/23 (99) di ui = M sinα a Mω cosα a (2) dt M is a utual inductance between stator and rotor. Fig. 2. Equivalent circuit of a universal otor The instantaneous value of the induced electroagnetic torque is given by the forula: 2 = Mi cosα a (3) 3. Matheatical odel of a triac convertor Fig. 3 shows the triac controlled universal otor drive [3], [4]. Suppose that the supply ains voltage is purely sinusoidal. jθ jθ e e u = U sinθ = U (4) 2 j and inductive character. For such a load current continues beyond the voltage zero, due to the inductance. Triac turns off after cessation of current. The turn off angle is arked as an angle β. Triac output voltage can be expressed in a for of the coplex Fourier series: jk e θ k = u = (5) Where is a Fourier coefficient, which is for the triac control defined: β π + β jkθ jkθ = ue ue 2π + 2π (6) α π + α After calculus we obtain for the Fourier coefficient following forula: U jα ( k ) jβ ( k ) = e e + 4π k jα ( + k) jβ ( + k ) e e + k For k ± U j α = j α β e e 4π + 2 For k = ± 2 j 2β ( ) ( ) In the Fig.4 is shown a typical triac output voltage wavefor. The wavefor was calculated on the basis a coplex Fourier series forula derived above. Fig. 3. Triac controlled universal otor drive Fig. 4. Triac output voltage wavefor Phase angle control of the triac can be used to control otor speed, as shown in Fig.3. The start of each half cycle is delayed by a control angleα. Suppose, otor as a load with oh

3 Zeszyty Probleowe Maszyny Elektryczne Nr 2/23 (99) 6 4. Motor current and electroagnetic torque calculation To deterine the otor current wavefor, we need to solve equation (). After substituting (2) and (4) into () we obtain. k = jkθ k = + a k = ( α ) + a ω ( ω α ) c e R M sin i L M cos j ki (7) The differential equation (7) has the following analytical solution: jkθ e i = (8) R + Mω sinα + jωk L M cosα ( ) ( ) k = a a Instantaneous value of intern electroagnetic torque is given by a forula (3). 5. Coputation results and their coparison with experiental data In order to copare the results obtained fro the atheatical odels of the achine and triac with easured values, the following paraeters of the otor were used: Machine power 8W / 23V 5Hz Stator resistance, 6Ω Stator inductance 7, 4 H Rotor resistance 3, 4 Ω Rotor inductance 2, 7 H Mutual inductance 63, H Using the equations of the previous sections, the wavefors of the supply voltage, otor current and electroagnetic torque were calculated. The otor worked at constant speed and was loaded by a constant torque. Based on the bibliography [2], was angle of arature reaction was a a assued to be5 o. Fig.5 and Fig.6 show the calculated and easured quantities at steady state at the speed 3 rp and control angleα = 5. Calculated effective current of the is I=2,976 A and easured value is 2,98 A. Calculated average value of otor oent is M AVI =,32 N and easured value is,3 N. To verify the correctness of the above equations were ade calculations for other speeds and otor load torque, at different triac switching angles. (V u (A i (N α=5 ; β=29 ; n=3rev/in ; I= A M AVi = N t (s) Fig. 5. Calculated otor quantities for 3 rp and α=5 Fig. 6. Measured otor quantities for 3 rp and α=5 So coputed behaviours preceded by quantities of otor were by easuring verified. Measureents of the otor confired the accuracy of calculated wavefors. Fig.7 and Fig.8 show the calculated and easured quantities at steady state at the speed 7 rp and control angleα = 3. Calculated effective current of the is I=2,84 A and easured value is 2,8 A. Calculated average value of otor oent is M AVI =, 29 N and easured value is,28 N.

4 62 Zeszyty Probleowe Maszyny Elektryczne Nr 2/23 (99) (V u (A i (N 2-2 α=3 ; β=29 ; n=7rev/in ; I=2.846 A M AVi =.2984 N t (s) Fig. 7. Calculated otor quantities for 7 rp and α =3 Fig. 8. Measured otor quantities for 7 rp and α =3 6. Conclusion The paper describes the analytical ethod for odelling the behaviour of a universal otor supplied by a triac directly fro the network. The ethod is based on the coplex Fourier series for the description of the otor supply voltage. Calculation of the supply voltage, current and torque by ethod shown in the introduction was cheed by easureent of the real otor. Calculated wavefors of the supply voltage and the otor current differ very little fro the oscilloscope easured wavefors different. The difference in easured and coputed otor torque is, N (i.e. 3%), which basically reaffirs the correctness of the analytical calculation. Anowledgent We support research activities in Slovakia. Project is co-financed fro EU funds. This paper was developed within the Project: "Centru excelentnosti integrovaného výskuu a využitia progresívnych ateriálov a technológií v oblasti autoobilovej elektroniky", ITMS (5%). The financial support of the Slovak Research and Developent Agency under the contract No. APVV -38- is anowledged. (5%). Bibliography []. Záskaliý P., Záskaliá M.: Analytical ethod of calculation torque ripple of a universal otor supplied by an IGBT chopper. Acta Technica, No.55, pp , 2, Prague, Czech republic. [2]. Ferková Ž., Fedor J., Dupej J.: Influence of agnetic field distribution on coutation of one phase coutator otor; Praxis of the Electrical Engineering, -2,pp.44-45, 24, Bratislava, Slovakia. [3]. Záskaliá M., Záskaliý P., Beňová M., Mahud A.R., Dobruý B.: Analysis of coplex tie function of converter output quantities using coplex Fourier transfor/series; Counications- Scientific letters of the University of Žilina, pp. 23-3, vol.2, No. 2, Žilina, Slovakia. [4]. Štepina J.: Syetrical Coponents in Rotating Electrical Machines Theory; Acadeia, Prague 968 (in Czech). [5]. Takeuchi T.J.: Theory of SCR Circuit and Application to Motor Control. Electrical Engineering College Press, Tokyo 968. [6]. Lander C.V.: Power electronics. McGraw-Hill Publishing, 3 rd edition, London New York, 993. Authors Prof. Ing. Pavel Záskaliý, PhD., is with Departent of Electrical Engineering and Mechatronic, Faculty of Electrical Engineering and Inforatics, Technical University of Košice, Letná 9, 4 Košice, Slovakia. Eail: pavel.zaskaliy@tuke.sk tel.: ; Ing. Ján Kaňuch, PhD., is with Departent of Electrical Engineering and Mechatronic, Faculty of Electrical Engineering and Inforatics, Technical University of Košice, Letná 9, 4 Košice, Slovakia. Eail: jan.kanuch@tuke.sk tel.:

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