Recent Developments of Electrical Drives - Part 12 docx

I-9. Torque Ripple in Inverter Driven Induction Motors 97 V C A B C H V A V B E/2 E/2 Figure 3. A basic inverter-induction motor drive. Drive system model An inverter driven inductionmotor canbe modeledby combiningthe spacephasor model of the machine with the supply model representing the non-sinusoidal voltage source inverter. Fig. 3 illustrates the ensuing model. All electromagnetic terms in (15) and (16) are expressed as space phasors by advancing from the actual three-phase machine model to an orthogonal model with alpha-beta and d-q windings, the voltages and currents of which are combined to give a deceptively simple representation of the drive system. Furthermore, it becomes possible to assess the effects of supply harmonics by simply including (or injecting) the significant supply harmonics into ˜ U S of (15). Simulation results The simulated alpha-beta terminal voltages containing the significant harmonics have been drawn into the simulation. With these voltages the ripple torque for no-load and load con- ditions shown in Figs. 4 and 5 are predicted, obtained by solving the system equations of (15) to (17). The simulation results show that the torque ripple is already of a considerable magnitude when the motor is not loaded. Under load, the ripple band is seen to widen. Electromagnetic quantities not shown here provide supportive evidence for the deterioration. The results are significant in that they indicate that the simulation method used is capable of estimating parasitic torque behavior in advance. Experimental verification The validity of the approach was tested experimentally for an inverter driven three-phase cage induction motor of 1.5 kW rating. Field orientation control with strong overmodula- tion was employed, resulting in supply harmonics both the order and magnitude of which resembled that of the six-step inverter at the motor terminals. Table 1 gives the relevant data for the test motor with which the foregoing simulations were conducted. Fig. 6 depicts the measured torque ripple band under steady state operating conditions with full load. As can be seen, both the behavior and the relative magnitude of the torque reflect strongly those of the simulation. Fig. 7 gives the torque ripple at no load. The ripple magnitude is seen to have increased with load when compared with the no-load regime: an observation which is also supported by simulation. 98 G¨ol et al. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 –1 –0.5 0 0.5 1 1.5 2 2.5 3 T el [Nm] t [s] Figure 4. Torque ripple at no load (simulated). 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 T el [Nm] t [s] Figure 5. Torque ripple at full load (simulated). I-9. Torque Ripple in Inverter Driven Induction Motors 99 Table 1. Data for the test motor Number of pole pairs (P)2 Rated voltage (U N ) 380/220 Y/ V Rated frequency (F)50Hz Rated power (P N ) 1.5 kW Rated speed (n N ) 1,405 rpm Rated current (I N ) 3.7 A Main inductance (L 1h ) 382 mH Stator inductance (L s ) 396 mH Rotor inductance (L r ) 393 mH Stator resistance (22 ◦ C) (R s ) 5.0  Rotor resistance (22 ◦ C) (R r ) 4.1  Rotor inertia (J r ) 0.008 kgm 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 –1 0 1 2 3 t [s] T el [Nm] Figure 6. Torque ripple at no load (measured). 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 8 9 10 11 12 13 t [s] T el [Nm] Figure 7. Torque ripple at full load (measured). 100 G¨ol et al. Evidently, the simple model representation does not allow the fine detail in the ripple band to be predicted in detail including the measured swings in the torque fluctuations. However the approximation achieved is satisfying. Conclusion The intuitive method of analysis based on the space phasor concept yields adequately ac- curate information about the nature of possible ripple torque generation in inverter driven induction motors. It is easy to assimilate and produces credible results with minimal com- putational effort. Although the approach has beendemonstrated for a six-step inverter drive, it is equally applicable to more sophisticated drive systems. References [1] T. Lipo, P.C. Krause, H.E. Jordan, Harmonic torque and speed pulsations in a rectifier-inverter induction motor drive, IEEE Trans., Vol. PAS-88, No. 5, pp. 579–587, 1969. [2] S.T.D. Robertson, K.M. Hebbar, Torque pulsations in induction motors with inverter drives, IEEE Trans. Ind. Gen. Appl., Vol. IGA-7, No. 2, pp. 318–323, 1971. [3] G.B. Klimann, A.B. Plunkett, Modulation strategy for a PWM inverter drive, IEEE Trans. Ind. Appl., Vol. IA-15, No.1, pp. 72–79, 1979. [4] K.P. Kovacs, J. Racz, Transiente Vorg¨ange in Wechselstrommaschinen, Budapest: Academiai Kiado, 1959. [5] R.J.W. Koopman, Direct simulation of AC machinery including third-harmonic effects, IEEE Trans. Power Apparatus Syst., Vol. PAS 88, No. 4, pp. 465–470, 1969. [6] R.H. Park, Two-reaction theory of synchronous machinery—Part I, AIEE Trans., Vol. 48, pp. 716–730, 1929. I-10. VIBRO-ACOUSTIC OPTIMIZATION OF A PERMANENT MAGNET SYNCHRONOUS MACHINE USING THE EXPERIMENTAL DESIGN METHOD S. Vivier 1 , A. Ait-Hammouda 1 , M. Hecquet 1 , B. Napame 1 , P. Brochet 1 and A. Randria 2 1 L2EP—Ecole Centrale de LILLE Ecole Centrale de Lille, Cit´e scientifique, B.P. 48, 59651 Villeneuve D’Ascq Cedex, France stephane.vivier@ec-lille.fr, michel.hecquet@ec-lille.fr, ait-hammouda.amine@ec-lille.fr, pascal.brochet@ec-lille.fr, Bellemain.Napame@ec-lille.fr 2 Alstom—2 Av de Lattre de Tassigny, 25290 Ornans, France andry.randria@transport.alstom.com Abstract. The aim of this paper is to use an analytical multi-physical model—electromagnetic, mechanic, and acoustic—in order to predict the electromagnetic noise of a permanent magnet syn- chronous machine (PMSM). Afterward, the experimental design method, with a particular design: “trellis design,” is used to build response surfaces of the noise with respect to the main factors. These surfaces can be used to find the optimal design or more simply, to avoid unacceptable designs of the machine, in term of noise for a variable speed application. Introduction The majority of the electric machines operate at variable speed. In most of cases, it involves a generation of noise and vibrations, for a given speed and frequency. For industries of manufacture, but also with the increasingly rigorous European stan- dards, it is necessary to take into account the noise and the vibrations from the design stage. Aclassicalmethod used to studyelectromagneticphenomena is the finiteelementmethod (FEM) in magneto-dynamics including the coupling with electrical circuits. However, in the case of strong coupling, taking into account the electromagnetic, vibro-acoustic, and thermic models in the same time would need a considerable computing effort. This would make the structure optimization practically impossible. In order to solve this problem, an analytical approach is considered instead. The aim of this work is to develop and use an analytical multi-physical model— electromagnetic, mechanic, and acoustic—of a synchronous machine with permanent mag- nets. The complete model is coded using the data-processing tool MATLAB R  , making possible the determination of fast and simple prediction models of the acoustic noise. S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 101–114. C  2006 Springer. 102 Vivier et al. –600000.0–800000.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 B [T] –400000.0 –200000.0 H [A m –1 ] Figure 1. 1/8 of synchronous machine with magnet characteristic. In order to reduce noises and vibrations, two main ways can be considered: by the control of the machine excitation [1], or by modifying the system structure. In this work, only the second solution is explored. Three models are presented: electromagnetic, mechanical of vibration, and acoustic. For each of them, comparisons with FEM and experiments have been made. Lastly, a study of sensitivity is presented in order to deduce the influential—or significant—factors on the noise. For that, the technique of the experimental designs is used. More particularly, the modeling of the noise will be achieved thanks to the new “trel- lis” designs. Several response surfaces are given; they represent the noise according to influential factors, with respect to different speeds of the machine. These surfaces are useful to deduce the parts of the design space to avoid. Presentation of the synchronous machine This machine is composed of eight rotor poles and 48stator slots. The powerof this machine is about 250 kW (Fig. 1). Analytical models The vibration analysis of electrical machines is a rather old problem. During the 40s and 50s, itwasdeeplystudiedbyvariousresearchers[2to5].Vibrationsofelectromechanicalsystems are due to excitation forces. Some of them have a magnetic origin. Other sources of vibra- tions, such as aerodynamic conditions, bearings, etc., will not be considered in this paper. An analytical model, considering electromagnetic phenomena, mechanical vibrations, and acoustic noises, was developed to take into account the overall noise produced by a variable induction in the air-gap [6 to 9] and by forces applied to the various structures. I-10. Permanent Magnet Synchronous Machine 103 0 10 20 30 40 50 60 70 80 90 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Amplitude [Tesla] Angle [°] 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Amplitude [tesla] Spectrum of Induction F.E.M Analytical Model F.E.M Analytical model Figure 2. Comparison of the form induction and FFT. Electromagnetic model It is assumed that forces in the air-gap of the machine are the main mechanical excitation. To characterize induction in the air-gap, the proposed method is based on the calculation of the air-gap permeance (P e ) and the magnetomotive force (mmf ) [6 to 8]. To establish the analytical expression of the permeance, some assumptions are made: r the magnetic circuit has a high permeability and a linear characteristic, r the tangential component of the air-gap flux density is negligible relative to the radial component. Results are given in Ref. [10] and just a comparison is recalled by Fig. 2. Using the finite element software OPERA-2D [11], the air-gap induction created by the magnet rotor as a function of space and time has been also calculated. In Fig. 2, a comparison on induction wave shapes vs. the angle is presented. The comparison results are very satisfactory, the induction distribution and the harmonic values determined analytically are validated numerically, as shown in Refs. [6,12]. The FFT of the radial forces vs. time (t) and angle (θ ) is presented below in Fig. 3. Vibratory model Vibrations are the consequence of the excitation of the mechanical system by electro- magnetic forces. Once the forces applied to the stator have been determined, the study of vibrations is possible. They correspond to the deformations whose amplitudes have to be calculated. For that purpose, some parameters have to be determined: r the damping, r the mode shapes and resonance frequencies for each mode. 104 Vivier et al. (2p,2fr) (4p,4fr) (8p,8fr) t θ Amplitude Figure 3. FFT 2D of radial force ( f r = p  N). For the damping coefficient, we have used the experimental measurements and the software PULSE [13] to determine the resonance frequencies, the mode shapes, and the damping. For example, some results are detailed in Fig. 4. The studied analytical model takes into account the yoke, the frame, the teeth, and the winding. The self vibration modes of the stator structure are determined, in various configurations: yoke only, yoke + teeth, yoke + teeth + carcass, and yoke + teeth + winding + carcass [14]. Mode 2 376 Hz (3.32%) Mode 3 1004 Hz (1.44%) Mode 4 1720 Hz (1.31%) Mode 5 2870 Hz (1.54%) Figure 4. Mode shape, resonance frequencies with (damping coefficient) obtained by measurements. I-10. Permanent Magnet Synchronous Machine 105 Table 1. Resonance frequencies (Hz) for each mode No. mode Analytical model FEM No. mode Analytical model Experimental 0 3,063 3,151 0 2,736 2,855 (1.46) 2 243 268 2 308 376 (3.32) 416 (3.22) 3 688 732 3 871 1,004 (1.44) 1,140 (1.74) 4 1,319 1,349 4 1,670 1,720 (1.31) 1,968 (2.44) 5 2,134 2,078 5 2,702 2,870 (1.54) 2,944 (1.46) (yoke and teeth only) (Complete stator: yoke + teeth + winding + carcass) Some results are presented in Table 1, with an experimental comparison. Resonance frequencies of the stator have been obtained by impact testing measurements, realized thanks to the impact test method. The comparison of the results with the analytical model are very satisfactory. Let us note that the damping coefficient ξ a cannot be given theoretically. However, Jordan [4] considers that for a synchronous machine, it stands between 0.01 and 0.04 (Table 2). The total vibratory spectrum obtained by our analytical model is presented in Fig. 5. The simulation results agree well with the theory. In addition, the proximity of the fre- quencyofexcitationmode0withthefrequencyoftheresonantmode0(at2,844Hz) explains the vibration peak located around 2,900 Hz. However let us point out that precautions must be taken when analyzing the results. The model giving the induction values is not perfect (the saturation phenomenon is neglected) and the formulae of Timar [3] giving the vibrations are also approximated. What is of interest is to determine the frequency of the main peaks and to be able to range their amplitudes. In order to study the vibrations generated by the operating conditions, an accelerometer is positioned on the frame of the machine. It measures the deformations of the frame. The vibratory spectrum gives lines identical to those obtained by the noise measurement; it displays a dominant line situated at 2,900 Hz, that corresponds to the theoretical excitation mode 0 predicted at 2,844 Hz (Fig. 6). Table 2. Main characteristics of the machine Speed 3,555 rpm Frequency of the supply f r 237 Rotational frequency f rot 237/p Frequencies of components of forces h ∗ 237(h = 2, 4, 6, ) (multiple of 2 f ) 106 Vivier et al. 0 1000 2000 3000 4000 5000 6000 0 20 40 60 80 100 120 474 Hz 945 Hz 2844 Hz 3318 Hz Figure 5. Total analytical vibratory spectrum with 3,555 rpm. In order to study the vibrations generated by the operating conditions, an accelerometer is positioned on the frame of the machine. It measures the deformations of the frame. The vibratory spectrum gives lines identical to those obtained by the noise measurement; it displays a dominant line situated at 2,900 Hz, that corresponds to the theoretical excitation mode 0 predicted at 2,844 Hz (Fig. 6). 486 244 58 23 5158 2900 0 20 40 60 80 100 120 140 Frequency Hz Amplitude dB Figure 6. Vibratory spectrum measured with 3,555 rpm with 1/12 of octave. . data-processing tool MATLAB R  , making possible the determination of fast and simple prediction models of the acoustic noise. S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical. induction in the air-gap [6 to 9] and by forces applied to the various structures. I-10. Permanent Magnet Synchronous Machine 103 0 10 20 30 40 50 60 70 80 90 -1 -0 .8 -0 .6 -0 .4 -0 .2 0 0.2 0.4 0.6 0.8 1 Amplitude. D’Ascq Cedex, France stephane.vivier@ec-lille.fr, michel.hecquet@ec-lille.fr, ait-hammouda.amine@ec-lille.fr, pascal.brochet@ec-lille.fr, Bellemain.Napame@ec-lille.fr 2 Alstom—2 Av de Lattre de