Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych Nr 62 Politechniki Wrocławskiej Nr 62 Studia i Materiały Nr 28 28 Janusz BIALIK*, Jan ZAWILAK* electrotechnology, synchronous motors, FEM modeling MAGNETIC FORCES CALCULATION IN TWO-SPEED, LARGE POWER, SALIENT POLE, SYNCHRONOUS MOTOR This paper deals with finite element calculation of magnetic forces inside two-speed, large power, synchronous motor. Prediction of such forces are very important in understanding vibration phenomena of electrical machines. In configuration for lower rotational speed pole numbers of magnetic field and numbers of excited poles of field-winding are not equal. Because of non symmetrical armature and field windings only one way of investigation is finite element (FE) modeling. The two-dimensional filedcircuit model is elaborated and the time-stepping approach in motor type GAe151/12p is used. Simulation are done for nominal load operation point, for two different rotational rotor speeds: n = 5 rpm (2p = 12) and n = 6 rpm (2p = 1), and corresponding nominal active powers: P = 6 kw and 15 kw. As a results of calculations are: results of magnetic field, magnetic pressure inside the machine and magnetic forces acting on stator yoke of investigated machine. 1. INTRODUCTION Two-speed synchronous, large power motors are nowadays used to drive main fans in polish coal and copper mains of the exhaust system. Positive aspects of using such motors are: two-step control of rotational speed, two-step start up of the motor, which limits maximum values of armature currents and the voltage drop, savings in energy consumption, less investments in comparison to alternatives solution, Negative aspects of presented motor are: magnetic asymmetry, distortion of the torque characteristic, different forces values acting on stator yoke. Two-speed synchronous motor are examples of no symmetrical machines (asymmetric armature and field winding). Therefore investigation can be done only with help of * Politechnika Wrocławska, Instytut Maszyn, Napędów i Pomiarów Elektrycznych, ul. Smoluchowskiego 19, 5-372 Wrocław, janusz.bialik@pwr.wroc.pl, jan.zawilak@pwr.wroc.pl
184 finite element modeling (FEM) methods [1, 2]. In this paper calculation results of the two speed synchronous motor type GAe151/12p are presented. This motor has two different speeds: n = 5 rpm (2p = 12) and n = 6 rpm (2p = 1) and corresponding nominal powers P = 6 kw and 15 kw. Determination of magnetic forces acting in mentioned motor is the goal of this paper. 2. MAGNETIC CALCULATION In all simulation, elaborated 2D field-circuit model is used. Details of the model can be found in [1, 2]. In Fig. 1 numeric model (field part) of investigated motor, together with cylindrical coordinate system (situated in air-gap,.3 mm below stator inner surface), and part of finite element mesh is shown. Second order approximation of the magnetic vector potential is used. To calculate the magnetic forces the magnetic flux density for next 1 (for higher rotational speed; 12 for lower rotational speed) time steps, for nominal load is determined. Fig. 1. Numeric model of investigated motor Magnetic field is sampled in 124 equidistance points in air-gap. Examples of a space magnetic flux density (radial component) are shown in Fig. 2. For the picture clarity only first three space distribution are show. Results are valid for both rotational speeds. Finally, collecting all calculation results (for all time steps), matrix of flux density B(m, n) is obtained, where M is a number of space samples, and N time
185 samples. By means of 2DFT, matrix B(m, n) can be converted into spectral domain B(µ, ν) [5]. In all further analysis the RMS value are taken on, both in space and time. Figure 3 and Figure 4. shows time/space distribution and modal/frequency spectrum of radial component of flux density. Result are presented for both rotational speeds and for field winding current I f = 2 A. 1,2 B n [T] 1,2 B n [T],8,4, -,4 -,8-1,2 6 12 τ [ o ] 18,8,4, -,4 -,8-1,2 6 12 τ [ o ] 18 Fig. 2. Instantaneous flux density distribution in air-gap nominal load, for 6 rpm (a) and for 5 rpm (b); distribution valid only for half machine circumference Fig. 3. Radial component of flux density in air-gap for 6 rpm: time/space distribution (a) modal/frequency spectrum (centered) (b) nominal load
186 Fig. 4. Radial component of flux density in air-gap for 5 rpm: time/space distribution (a) modal/frequency spectrum (centered) (b) nominal load 3. DETERMINATION OF MAGNETIC STRESSES Maxell stress tensor components in air-gap, in 2D calculations, in cylindrical coordinate system, can be calculated from the well know equations [3]. The 2DFT approach to magnetic stresses in air-gap can be used as well. In the Fig. 5 an example for time/space distributions of the magnetic stress (normal component), valid for both rotational speed are shown. In addition in Fig.6 the modal/frequency spectrum is presented. All results are valid for nominal load point of motor. According to [5] modal/frequency spectrum of magnetic pressure should be limited only to lowest harmonic in space, because from vibroacoustic point of view only longest waves are important. In case of large power, two-speed, silent pole, synchronous motors such approach can not be used. For all rotational speeds in modal/frequency spectrums of magnetic stresses (radial component), harmonics close to 1 khz are observed. These harmonics are connected with numbers of stator slots (18) and numbers of pole pairs (n = 5 rpm, p = 6 and n = 6 rpm, p = 5). For lower rotational rotor speed the harmonics number are 12 and 114, and for higher speed harmonics number 13 and 113. Magnitudes of these harmonics are similar to harmonic s amplitudes of small order. In addition these harmonics are very close to natural frequencies of a mechanical construction of two-speed synchronous motor [3]. Therefore vibration of such structure with big amplitudes can be expected.
187 Fig. 5. Radial component of magnetic pressure in air-gap (time/space distribution) for: 6 rpm (a) and 5 rpm (b) Fig. 6. Modal/frequency spectrum of radial component of magnetic pressure in air-gap for nominal load: a) for n = 6 rpm; b) n = 5 rpm 4. DETERMINATION OF MAGNETIC FORCES Currently are existing only two commercial programs, which allows to solve vibration problems with help of one finite element mesh Ansys [7] and J-Mag [4]. In case for lack of these programs user is forced to look for alternative solution, which allow to transfer magnetic load onto mechanical structure. Two methods exist to deal with the problem [5]:
188 calculation of magnetic forces which are applied to nodes of finite element mesh of structural model, structural load are applied as a magnetic pressure, applied to appropriate areas of structural model. Method of force calculation, which are applied to nodes of finite element structural mesh, is chosen. Reaction of magnetic field with mechanical structure of two-speed synchronous motor are represented with help of two force component: radial and tangential, which are applied to stator-iron teeth (actually to nodes of finite element structural mesh in stator teeth area). Magnetic force acting on one single stator tooth can be calculated in according to [6]. Method is explained in Fig. 7. Stator Iron 3 Armature winding 2 Rotor iron 1 4 Damper cage Fig. 7. Application of Maxwell stress tensor to calculation of magnetic forces acting on stator s yoke teeth. To every stator tooth a closed contour (consist 4 arcs) is assigned (in 3D model this will be 4 surfaces) where the appropriate magnetic stress components are determined. After integration of magnetic pressure components (radial or tangential) along defined contour, the magnetic force acting on one single stator tooth is obtained In Fig.8 the time distribution of magnetic force, acting on three chosen stator tooth are presented (only radial component). The Fourier analysis is used to obtain the harmonic spectrum of magnetic forces (Fig. 9). According to these results, more dense vibration spectrum with higher amplitudes for lower rotational speed (n = 5 rpm) are expected (harmonics are closer to the natural frequencies [3] of the investigated motor possible resonance phenomena can occur). In addition in Fig. 1 time distribution of tangential component of magnetic forces acting on teeth of stator yoke are presented.
189 4 35 3 25 Fn [N] Ząb 1 Ząb 2 Ząb 3 6 5 4 Fn [N] Ząb 1 Ząb 2 Ząb 3 2 15 1 5 Czas [s],1,2,3,4,5 3 2 1 Czas [s],1,2,3,4,5,6 Fig. 8.Time distribution of magnetic forces (radial component) acting on teeth of stator-yoke: a) for n=6 rpm, b) for n = 5 rpm 2 Fn[N] 2 Fn[N] 15 15 1 1 5 f [Hz] 2 4 6 8 1 5 f [Hz] 167 333 5 667 833 1 Fig. 9. Harmonic spectrum of magnetic forces (radial component) acting on teeth of stator-yoke: a) for n = 6 rpm, b) for n = 5 rpm 2 Fτ [N] Tooth 1 Tooth 2 Tooth 3 4 2 Fτ [N] Ząb 1 Ząb 2 Ząb 3-2 -4-6 t [s] -8,1,2,3,4,5-2 -4-6 Czas [s] -8,1,2,3,4,5,6 Fig. 1. Time distribution of magnetic forces (tangential component) acting on teeth of stator-yoke: a) for n = 6 rpm, b) for n = 5 rpm
19 5. CONCLUSIONS In this paper the detailed approach to calculate magnetic forces in two-speed, large power, salient pole, synchronous motors, using finite element model is presented. This approach in combination with time stepping analysis seems to be the only way to analyze vibration problems in unsymmetrical machines. According to calculation results more dense vibration spectrum on lower rotational speed n = 5 rpm can be expected. LITERATURE [1] BIALIK J., ZAWILAK J., ANTAL L., Polowo-obwodowy model dwubiegowego silnika synchronicznego, Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych Politechniki Wrocławskiej, Nr 56, Studia i Materiały, Nr 24, Wrocław, 24, pp. 43 54. [2] BIALIK J., ZAWILAK J., Drgania oraz siły pochodzenia elektromagnetycznego w dwubiegowych silnikach synchronicznych dużej mocy, Proceedings of XLI International Symposium on Electrical Machines SME 25, Jarnołtówek, 25, pp. 55 64. [3] BIALIK J., ZAWILAK J., Vibration modeling of the two-speed, large Power, synchronous motor, 6th IEEE International Symposium on Diagnostic for Electric Machines, Power Electronics and Drives, Cracow 6-8 September 27, pp. 173 177. [4] GIERAS J. F., WANG CH., CHO LAI J., Noise of polyphase electric machines, CRS Press Taylor & Francis Group USA 26. [5] WITCZAK P., WAWRZYNIAK B., Analiza magnetycznych sił wibracyjnych w maszynach z magnesami trwałymi w ujęciu modalno-częstotliwościowym, Proceedings of XLI International Symposium on Electrical Machines SME 25, Jarnołtówek, 25, pp. 214 219. [6] WITCZAK P., Magnetic force and stress analysis in electric machinery, International XIII Symposium on Micromachines and Servodrives Mis 22, Krasiczyn 22, pp. 169 176. [7] ANSYS Help, www.ansys.com OBLICZANIE SIŁ MAGNETYCZNYCH W DWUBIEGOWYCH SILNIKACH SYNCHRONICZNYCH DUŻEJ MOCY O BIEGUNACH WYDATNYCH W artykule zaprezentowano sposób obliczania sił magnetycznych działających w dwubiegowym silniku synchronicznym dużej mocy z zastosowaniem techniki MES. Wiedza o tych siłach jest pomocna w zrozumieniu zjawisk drgań w silnikach elektrycznych. W silnikach dwubiegowych w przypadku pracy na mniejszej prędkości obrotowej liczba biegunów magnetycznych i biegunów mechanicznych wirnika jest różna. Asymetria uzwojeń stojana oraz wirnika jest przesłanką do zastosowania metody elementów skończonych w analizach tego typu silników. Opracowany dwuwymiarowy polowo-obwodowy model obliczeń wraz z techniką kroków czasowych zostały zastosowane w obliczeniach sinika typu GAe151/12p. Symulacje przeprowadzono dla stanu obciążenia znamionowego dla dwóch różnych prędkości obrotowych: n =5 obr/min (2p = 12) i n = 6 obr/min (2p = 1), oraz mocach znamionowych: P = 6 kw i 15 kw. Jako wyniki obliczeń przedstawiono obliczenia pól magnetycznych w silniku, ciśnienia magnetycznego oraz sił magnetycznych działających na strukturę stojana badanego silnika.