Drive system model An inverter driven induction motor can be modeled by combining the space phasor model of the machine with the supply model representing the non-sinusoidal voltage sour
Trang 1A B C H
VA
VB E/2
E/2
Figure 3 A basic inverter-induction motor drive.
Drive system model
An inverter driven induction motor can be modeled by combining the space phasor 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
Trang 298 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
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
t [s]
Figure 5 Torque ripple at full load (simulated).
Trang 3Table 1 Data for the test motor
Number of pole pairs (P) 2
Rated voltage (U N) 380/220 Y/ V Rated frequency (F ) 50 Hz
Rated power (P N) 1.5 kW
Rated speed (n N) 1,405 rpm
Rated current (I N) 3.7 A
Main inductance (L1h ) 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
–1
0
1
2
3
t [s]
T el
Figure 6 Torque ripple at no load (measured).
8
9
10
11
12
13
T el
Figure 7 Torque ripple at full load (measured).
Trang 4100 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 been demonstrated 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
Trang 5I-10 VIBRO-ACOUSTIC OPTIMIZATION
OF A PERMANENT MAGNET
SYNCHRONOUS MACHINE USING THE EXPERIMENTAL DESIGN METHOD
S Vivier1, A Ait-Hammouda1, M Hecquet1, B Napame1,
P Brochet1and A Randria2
1L2EP—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
2Alstom—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
A classical method used to study electromagnetic phenomena is the finite element method (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 MATLABR, 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.
2006 Springer.
Trang 6102 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 48 stator slots The power of 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,
it was deeply studied by various researchers [2 to 5] Vibrations of electromechanical systems are due to excitation forces Some of them have a magnetic origin Other sources of vibra-tions, such as aerodynamic condivibra-tions, 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
Trang 70 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
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
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:
rthe magnetic circuit has a high permeability and a linear characteristic,
rthe 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:
rthe damping,
rthe mode shapes and resonance frequencies for each mode
Trang 8104 Vivier et al.
t
θ Amplitude
Figure 3 FFT 2D of radial force ( fr = pN ).
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.
Trang 9Table 1 Resonance frequencies (Hz) for each mode
No mode Analytical model FEM No mode Analytical model Experimental
416 (3.22)
1,140 (1.74)
1,968 (2.44)
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-quency of excitation mode 0 with the frefre-quency of the resonant mode 0 (at 2,844 Hz) 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
Frequency of the supply f r 237
Rotational frequency frot 237/p Frequencies of components of forces h ∗ 237(h = 2, 4, 6, ) (multiple of 2 f )
Trang 10106 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
Figure 6 Vibratory spectrum measured with 3,555 rpm with 1/12 of octave.