Recent Developments of Electrical Drives - Part 13 ppsx

After this comparison, the permanent magnet synchronous machine was tested at various speeds, that allowed us to highlight a particularly dangerous speed.. An analytical relation linking

Acoustic model

Acoustic intensity I (x) can be written as a function of the frequency, the amplitude of

vibrations, the mode order and the stator surface [5,9]

I (x)= σ 8200 f r2Y2

md S e

4πx2(2m+ 1) The coefficientσ is called factor of radiation It represents the capacity of a machine to

be a good sound generator and can be calculated through two different ways according

to whether one assumes the machine to be a sphere or a cylinder σ is a factor which

varies withλ (wavelength) and the diameter of the machine It also depends on the mode

shape [3]:

σ = f



π D λ



, λ = c

f r

c: Traveling speed of sound (344 m/s); f r:Vibration frequency

It appears that I (x) is inversely proportional to the order of the mode, in addition the

acoustic intensity is proportional to the square of the vibration amplitude In general, we

define I and W in decibels We thus define the levels of acoustic pressure, acoustic intensity,

and sound power as follows:

L p = 20 log



P

P0



, L i= 10 log



I

I0



, L w= 10 log



W

W0



with:

P0 = 20 μPa, I0= 10−12W/m2, W0= 10−12W The spectrum of the total noise obtained by our analytical model is presented below (Fig 7)

0

20

40

60

80

100

3318 Hz

5688 Hz

Frequency Hz

m = 0

m=8

Figure 7 Spectrum of the noise of the simulated PMSM.

2900 459

58

4597

244 91

183

15

25

35

45

55

65

75

85

95

0,37 0,82 1,83 4,1 9,17 20,54 45,97 102,92 230,41 515,82 1154,78 2585,23

Frequency Hz

12 2f

f

Figure 8 Spectrum of the noise of PMSM measured with 3,555 rpm (1/12 octave).

Fig 8 presents the measured acoustic noise spectrum at the same speed (3,555 rpm) Lines are located at the same frequencies as in the vibration spectrum

The first line determined by measurements is located at 2,900 Hz (12 f ) In theory, the harmonic of teeth (12 f ) is located at 2,844 Hz.

The lines at low frequencies (between 24 and 459 Hz) are not found in theory, because they are mainly related to the background noise They are not generated by the PMSM, but

by the driving motor and the ventilator (Fig 9)

23

434 183 91 52

15

25

35

45

55

65

75

85

95

0,37 0,87 2,05 4,87 11,55 27,38 64,94

153,99 365,17 865,9

6 2053,5 4869,7

Frequency Hz

Figure 9 Spectrum of the noise of driving motor and ventilator measured with 3,555 rpm (1/12

octave)

1894 rpm

860 rpm

2469 rpm

3555 rpm

3165 rpm 3945 rpm

860 rpm

3555 rpm

3945 rpm

3165 rpm

1894 rpm 2469 rpm

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600

simulations measures

Experimental frequency of resonance F0 = 2855 Hz

Frequency Hz

Figure 10 Level of the 12th harmonic vs rotation speed Vibratory comparisons.

After this comparison, the permanent magnet synchronous machine was tested at various speeds, that allowed us to highlight a particularly dangerous speed Moreover, some results are overestimated but the quality of those is respected (Fig 10) In spite of the inaccuracies, major lines appear, which is of primary importance in view of noise reduction

To know which lines to reduce does not require to know its amplitude precisely Its frequency, on the other hand, must be well given Lastly, taking into account the complexity

of the studied phenomena and the many steps of calculations making it possible to lead to the results, the latter seem very satisfactory

Screening analysis

Once the different models finalized and assembled into a single “coupled model,” it becomes possible to study the variations of the main variables representing the vibration sources This

is achieved by the building of response surfaces, and by the launching of optimizations The privileged tool employed is the Experimental Design Method [15,16]

First of all, a sensibility analysis using the global coupled model is described

The overall audible noise produced by the synchronous machine stands as the studied variable (the response) An analytical relation linking the noise amplitude with five variation sources (the factors) has been established:

rthe stator slot opening (lse);

rthe height of the yoke (hyoke);

rthe opening of permanent magnet (alp);

rthe width of the air-gap (e);

rthe height of the permanent magnet (hmag)

A screening design [17] is calculated It gives the ability to determine the influent factors, with respect to the response, inside the design space This domain is implicitly defined by the intervals of variation, for the five factors (Table 3)

The Fig 11 gives a representation of the influence of each factor on the noise

Firstly, it shows that the opening of the permanent magnets (alp) is a very influential factor, since its variation from its middle value (31◦) to its upper limit (32◦) makes the noise increase by about 15 dB

Table 3 Intervals of variation—screening analysis

0 2 4 6 8 10 12 14

0.21478 95%

Figure 11 Factor influences on the noise amplitude.

The height of the yoke (hyoke), the width of the air-gap (e), and the stator slot opening (lse) are also significant factors according to this figure, since they all exceed the two 95% significance levels It means that the probability to declare these factor influential although they are not, is equal to 5% They all have a negative influence on the noise variations: their values have to be increased to reduce the noise amplitude

The height of the magnets (hmag) is not considered as an influent factor, if the same significance level is used

It is important to keep in mind that these conclusions only hold inside the design domain The previous results have been obtained for a fixed rotor speed (3,000 rpm) Imposing different speeds do not change the relative influence of the factors However, one can say that effect values increase for speeds around 3,000 rpm This aspect will be confirmed by the following study

Research of optimal conditions

In a second stage, our purpose was to “model” the part of the conception domain in which the global audible noise produced by the PMSM was smaller than a predefined limit: 80

Table 4 Intervals of variation—modeling stage

dB was considered as the maximal admissible noise intensity This frontier for the noise— the response—has been computed with respect the same factors except the height of the

permanent magnet (hmag) and in addition, the motor speed (N ).

These factors have been selected thanks to screening analyses realized with the complete coupled model Their intervals of variation are given by Table 4

Since we want to have a good description of the variations of the noise with respect to the five factors, it is necessary to increase the number of the different levels taken by each factor Considering five levels is in general enough Such a configuration leads to 55 = 3,125 experiments with the use of a grid design—that is a multi-level full factorial design This number of evaluations of the coupled model is relatively large, and it can be interesting

to take advantage of the new “trellis” deigns [17]

Trellis designs can be described as multi-level fractional grid design They are build from fractional two-level factorial designs judiciously superposed (Fig 12) For this reason, under important hypotheses, they keep their interesting mathematical characteristics, such as for instance the orthogonality property

When five factors are present, it is possible to use the two-level fractional factorial design defined as 25 −2, that is the quarter (22= 4) of the corresponding full factorial design 25 When this design is used to build the trellis design, it leads to definition of a five-level incomplete grid, with only 795 experiments—instead of 3,125 with a complete grid It takes approximately 16 h to compute this trellis design on a PC

Instead of using the 795 values of noise directly, we have exploited the interesting relative location of the experiments inside the design domain: an iterative procedure has been applied to estimate the noise values for each of the 3, 125 − 795 = 2,330 initially

nonevaluated experiments It has been shown that the overall error made for these 2,330 interpolations, realized thanks to the 795 initial experiments, is lower than 0.8%

Figure 12 Example of experiment sharing between two fractional designs (23 −1).

28

30

32

3000 3500 4000

4500 50

55

60

65

70

75

80

85

N (rpm)

Figure 13 Noise variations vs “alp” and “N ”

The different results that follow are deduced from the 795 first experimental points mixed with the estimated ones

It quickly appeared that the opening of the permanent magnets (alp) and the motor

speed (N ) were the two most influential variables over the noise production The following

response surface shows the corresponding variations, as shown in Fig 13

It is very clear that the noise is strongly reduced when the permanent magnet area—

in fact the corresponding angular opening—is equal to 30◦ This result is confirmed by practical considerations

The rotor speed has also a neat influence over the noise production A reso-nance phenomenon is visible near the speed value 3,500 rpm, whatever the factor alp values

The influence of the three other factors are relatively small in comparison However, we

can notice that the decrease of hyokeleads to move the resonance point toward lower rotation speed values

The 80 dB limit can be graphically represented with respect to alp, N , and hyoke, thanks

to iso-value surfaces (Fig 14)

Two iso-value surfaces are represented: one showing the noise equal to 80 dB, and the other to 84 dB The graphic is nearly symmetrical: 30◦standing as the central value for the magnet opening (alp) Then, the admissible subspace of the conception domain is modeled

by the zone delimited by the two central 80 dB surfaces This indicates that it is always possible to conceive a PMSM generating a noise lower than 80 dB, provided that the magnet

Figure 14 Noise iso-value surfaces (80 and 84 dB) vs “N ,” “hyoke”, and “alp.”

opening is chosen between 28.5◦ and 31.5◦ This interval can be extended for particular rotor speeds greater than 4,000 rpm or lower than 2,700 rpm

Conclusion

The purpose of this work is to present some results obtained from the exploitation of a complete coupled model of a permanent magnet synchronous machine Different multi-physical aspects are considered: electromagnetic, mechanic, and acoustic phenomena are taken into account thanks to a single analytical model

The Experimental Design Method is the privileged tool used to make the complex rela-tionships between the main variables appear

The first study—a screening analysis—shows that, whatever the rotor speed considered, the angular opening is a very influential factor: the particular value 30◦is certainly the best choice It is more difficult to set the other factors, since the rotor speed interacts with them However, the height of the permanent magnets is declared nonsignificant in term of acoustic noise

The second study is designed to work on more precise data For that purpose, a trellis design with five levels per factor and only 795 experiments, is computed The advantageous properties of this type of design allow the subsequent evaluations of 2,330 other points, with

an excellent accuracy, leading to practical design choices for lowering the limited noise

[1] M Gabsi, “Conception de machines sp´eciales et de leurs alimentations R´eduction du bruit d’origine ´electromagn´etique”, Habilitation `a diriger des recherches, Juillet 1999

[2] S Timochenko, Th´eorie des vibrations, Librairie Polytechnique CH Beranger, 1939 [3] P.L Timar, Noise and Vibration of Electrical Machines, Elsevier, 1989

[4] H Jordan, Electric Motor Silencer—Formation and Elimination of the Noises in the Electric Motors, W Giradet-Essen Editor, 1950

[5] S.J Jang, Low-Noise Electrical Motors, Oxford: Clarendon Press, 1981

[6] N Boules, Prediction of no-load flux density distribution in permanent magnet machines, IEEE Trans Ind Appl., Vol IA 21, No 4, pp 121–124, 1985

[7] J.D.L Ree, N Boules, Torque production in permanent magnet synchronous motors, IEEE Ind Appl Soc Conf Rec., Vol 87, pp 15–20, 1987

[8] Z.Q Zhu, D Howe, Instantaneous magnetic field distribution in brushless permanent mag-net DC motors Part III: Effect of stator slotting field, IEEE Trans Magn., Vol 29, No 1,

pp 143–151, 1993

[9] R Corton, Bruit magn´etique des machines asynchrones, proc´edure de r´eduction passive et active, th`ese, 2000, Universit´e d’Artois, France

[10] A Ait-hammouda, M Hecquet, M Goueygou, P Brochet, A Randria, “Analytical Approach

to Study Noise and Vibration of a Synchronous Permanent Magnet Machine”, ISEF’2003, Maribor, September 18, 2003, CD

[11] OPERA 2D, Reference Manual, VECTOR FIELDS, http://www.vector-field.co.uk

[12] R Breahna, P Viarouge, “Space and Time Harmonics Interactions in Synchronous Machines”, Proceedings of Electrimacs, 1999, pp 45–50

[13] Br¨uel and Kjaer, PULSE System: Modal Test Consultant, http://www.bksv.com

[14] S.P Verma, L Wen, “Experimental Procedures for Measurement of Vibration and Radiated Acoustic Noise of Electrical Machines”, Power System Research Group 2002, ICEM 2002,

p 432

[15] J.J Droesbeke, J Fine, G Saporta, Plans d’exp´eriences—Applications `a l’entreprise, Ed TECHNIP, 1997

[16] J Goupy, La M´ethode des plans d’Exp´eriences, Paris: Dunod, 1988

[17] S Vivier, “Strat´egies d’optimisation par plans d’exp´eriences et Application aux dispositifs

´electrotechniques mod´elis´es par ´el´ements finis”, Th`ese de doctorat, Universit´e des Sciences et Techniques de Lille, July 2002

I-11 ELECTROMAGNETIC FORCES AND MECHANICAL OSCILLATIONS OF THE STATOR END WINDING OF TURBO

GENERATORS

A Gr ¨uning and S Kulig

Institute of Electrical Drives and Mechatronics, University of Dortmund,

D-44227 Dortmund, Germany,

arne.gruening@uni-dortmund.de, stefan.kulig@uni-dortmund.de

Abstract Numerical methods of calculating the electromagnetic forces and of simulating the

oscilla-tion behavior of the stator end winding are introduced The end winding oscillaoscilla-tions of different turbo generators under forced vibrations are computed in a combined simulation Also eigenfrequencies and eigenmodes are determined The obtained results are surveyed by measurements Numerical sim-ulation of oscillation behavior is found a useful tool in end winding design although model parameter identification still offers improvement potential

Introduction

Due to the complex structure, the design of the stator end winding of large turbo generators and especially of the appendant support fixture still offers a huge potential for optimization Primarily the capability of the stator end winding to perform oscillations owing to the operant electromagnetic forces and the resultant eventuality of damages like fatigue or even cracks of the insulation gives reason to accomplish improvements [1]

When optimizing the end winding support fixture in order to reduce the occurring os-cillations and therewith the risk of damages, detailed knowledge of the vibration behavior under steady-state as well as under transient conditions is very beneficial

The most viable method to obtain this knowledge is the accomplishment of numerical computer simulations, whereas detailed measurements are difficult to perform and therefore only sometimes used to verify the simulation results As can be seen in Fig 1, the stator end winding of a turbo generator is a very complex entity with a huge number of components

of different mechanical properties But due to the support fixture it is also acting as a complex composite structure Therefore the appliance of simulation methods using a three-dimensional model of the complete end winding is precondition for obtaining reliable and useful results

In this connection numerical simulation methods based on a three-dimensional modeling

of the end region and facilitating investigations of the vibration behavior under various operation modes become increasingly the aim of development

S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 115–126.

Figure 1 Stator end winding.

In terms of the impact on the stator end winding a certain mode of operation can be characterized by the currents in the windings and, if the influence of the rotor winding is taken into account, the rotor movement, which determines the position of the rotor winding

On this account the procedure of investigating the vibration behavior of the stator end winding comprises the electromagnetic computation of the three-dimensional distribution

of the forces generated by the currents and afterward the three-dimensional simulation of the mechanical oscillation due to these forces The implementation of such a simulation method represents an interdisciplinary task combining electromagnetic and mechanical problems [2]

The present paper summarizes the results of an investigation, which was performed by

a team of mechanical and electrical engineers over a period of about five years, concerned with the development and coupling of three-dimensional numerical electromagnetic and mechanical simulation methods

Electromagnetic simulation

A numerical simulation method based on the application of Biot-Savart’s law to line circuit segments was used to compute the three-dimensional distribution of electromagnetic forces acting on the stator end winding

Computing the magnetic flux density B by Biot-Savart’s law, materials with non-linear

magnetization characteristics like iron cannot be considered directly A practicable way

to use this method nevertheless is to replace all iron parts by additional fictive current distributions emulating the influence of the iron parts on the magnetic field, according to