Recent Developments of Electrical Drives - Part 25 pot

Fixed structure Figure 13 presents the mode shapes and the natural frequencies of the statoric ring first in free condition and then when it is attached... Tang, Impact of stator winding

Time (s)

Vertical current J(1,M)

0

–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.45 0.05

Figure 9 Vertical current j M,Nresponse (impact excitation)

Fixed structure

Figure 13 presents the mode shapes and the natural frequencies of the statoric ring first in free condition and then when it is attached

1

0

1

2

3

4

5

6

7

1000 2000 3000 4000

Frequency (Hz)

FFT

×10 –4

5000 6000 7000 8000 9000 10000

Figure 10 FFT.

234 Lecointe et al.

Figure 11 Suspended stator and meshing.

Figure 12 Experimental results: modes 2-2D, 3-2D, 4-2D and modes 2-3D, 3-3D, 4-3D.

Table 4 Results for the sleek cylinders

Natural frequencies (Hz) Mode Analytical laws FE Software Analogies Hammer test

Cyl 1

Cyl 2

Free stator

Mode 2-2D : 912 Hz

Mode 2-3D : 1300 Hz

Mode 3-2D : 2480 Hz

Mode 3-3D : 3010 Hz Mode 4-3D : 3020 Hz

Mode 4-2D : 2480 Hz Mode 2-3D : 1330 Hz

Mode 3-2D : 944 Hz Fixed stator

Figure 13 Mode shapes and natural frequencies.

In free mode, the three first modes are perfectly identified However, some modes are not as perfectly defined as for the sleek cylinders For example, the shape of the mode 2 in two dimensions is undoubtedly deformed by the welded feet

When the stator yoke is fixed to the chassis, the natural frequencies are increased but the maximal relative variation is equal to 5.2% This evolution is attributed to the fixation of the structure If the natural frequencies are not modified a lot, the mode shapes differ from the shapes observed in free mode For example, the modes 3-2D and 3-3D in free mode become almost modes 4-2D and 4-3D when the structure is fixed The fixation prevents the structure from moving It is clearly visible at presented pictures

236 Lecointe et al.

Conclusion

In this paper, four different methods to determine the mechanical behavior of cylinders— sleek or equipped of feet—have been studied Analytical and analogy methods give accurate results but the FE software keeps the advantage of the 3D mode determination The extended Jordan’s laws allow to find very quickly the 2D radial frequencies with a simple computer whereas others methods requires specific software or modal analysis equipment In this way, the analytical method can be used for a first diagnosis

Thus, analytical laws can still be used to determine the natural frequencies of fixed electrical rotating machines Indeed, frequencies in both cases are very close However, it has been shown that the mode number is different Such a modification can influence the determination of noise emitted by electrical machines Indeed, analytical theories [9] of noise prediction consider non-fixed structure as a cylinder of infinite length or a vibrating sphere Consequently, the next step will study the influence of the machine fixation on noise emission At last, the analogy method will be extended to the third dimension

References

[1] P Fran¸cois, “La g´en´eration des bruits et la r´eponse des structures dans les moteurs asynchrones,

en particulier en ce qui concerne les ´ecoulements”, Revue G´en´erale de l’Electricit´e, Avril 1968,

pp 377–392

[2] B Cassoret, R Corton, D Roger, J.-F Brudny, Magnetic noise reduction of induction machines, IEEE Trans Power Electron., Vol 18, No 2, pp 570–579, 2003

[3] D.E Cameron, J.H Lang, S.D Umans, The origin and reduction of acoustic noise in doubly salient variable-motors, IEEE Trans Ind App., Vol 28, No 6, November/December, pp 1250–

1255, 1992

[4] C Couturier, B Cassoret, P Witczak, J.-F Brudny, “A Contribution To Study The Induction Machine Stator Resonance Frequencies”, ICEM 98, Vol 1, Istanbul, Turkey, September 2–4,

1998, pp 485–489

[5] W Cai, P Pillay, Z Tang, Impact of stator windings and end-bells on resonant frequencies and mode shapes of switched reluctance motors, IEEE Trans Ind App., Vol 38, No 4,

pp 1027–1036, 2002

[6] S.P Timoshenko, J.N Goodier, Theory of Elasticity, 3rd student edition, McGraw-Hill Compa-nies, New York, 1970

[7] H Jordan, Ger¨auscharme Electromotoren, Essen: W Girardet, 1950

[8] J.-Ph Lecointe, R Romary, T Czapla, J.-F Brudny, Five methods of stator natural frequencies determination Case of induction and switched reluctance machines, Mechanical Systems and Signal Processing, Elsevier, Vol 18, pp 1133–1159, 2004

[9] Ph.L Alger, The magnetic noise of polyphase induction motors, Trans Amer IEEE, Pt III A,

No 73, pp 118–125, 1954

II-9 DIAGNOSIS OF INDUCTION

MACHINES: DEFINITION

OF HEALTH MACHINE ELECTROMAGNETIC SIGNATURE

D Thailly, R Romary and J.F Brudny

CNRT Futurelec 2—R´eseaux et Machines Electriques du Futur, Laboratoire Syst`emes Electrotechniques et Environnement—Universit´e d’Artois, Facult´e des Sciences Appliqu´ees,

Technoparc Futura, 62400 B´ethune, France

delphine.thailly@fsa.univ-artois.fr, romary@fsa.univ-artois.fr, brudny@fsa.univ-artois.fr

Abstract This paper deals with the diagnosis of induction machines using data contained in the

radial external magnetic field This work presents a theoretical approach which permits to study the evolution of each flux density air-gap component through the stator The aim of this method is to find, by computation, the magnitude of measured spectral lines The study is made on the couples of toothing spectral lines and justifies why these couples do not have the same magnitude, what is not obvious in a first approach where the practical spectrum is directly compared with this one of the air-gap flux density

Introduction

From an economical point of view, it appears that, for a factory, predictive maintenance of electrical machines is essential Various procedures have been brought [1,2], and it should

be noted a recent orientation in the exploitation of data enclosed in magnetic field (MF) which surrounds the machine It was display that is advisable to dissociate two cases: the axial MF [3], which principally corresponds to the coil end leakage of armature winding, and the radial MF [4], which is the subject of our research

Previous studies [5] have shown that it is necessary to distinguish two frequency domains

of the radial MF according to the skin effect which appears in the magnetic sheets This paper

is restricted to low frequency components induced by slot effect in induction machines The aim is to investigate the flux density evolution through the stator frame to determine a correspondence between the flux density in the air-gap and the one which appears at the measured point outside the machine

This paper presents first, the theoretical approach, then, the principle of measurement of the radial MF, and finally, the comparison between results deduced from experimental tests and these ones obtained by analytical computations

2006 Springer.

238 Thailly et al.

Analytical expression of the air-gap flux density

The analytical expression of the normal air-gap flux density bgis achieved by multiplying the expression of the magneto-motive force (m.m.f.) which appears at the limit of the air-gap

by this one of the air-gap permeance per surface unit

To reduce the developments, it is assumed that the magnetic effects are only created by

the three phase p pole pair stator which windings are crossed by the currents i q (1≤ q ≤ 3).

Taking the stator phase 1 axis as spatial reference and locating any point of the air-gap by the variableα, the m.m.f expression ε(α), assuming first infinite the iron permeability, is

given by the following relation:

ε(α) =

3



q=1

i q

+∞



hs=1

hs une ven

A hscos



hs



p α − (s − 1)2π

3



(1)

where s is the slip.

Considering that the m.m.f evolutes in a linear way on the slot width leads to defines

A hsby the expression:

A hs= 2z

πhs(−1)

hs−1 2

sin



p N s hs



p N s hs

sin hs π

6

m sin hs π

6m

(2)

where rds is the ratio of the slot width to the slot step, N sis the number of stator slots per

pole pair, z is the total number of turns for one stator phase under a pole pair, and m is the

number of slots per pole per phase

The used expression of the permeance per surface unit, takes the stator and rotor slotting effects into account separately but, also the interaction between the both [6] So this quantity, function ofα but also of θ which represents the angular displacement between the stator

and rotor references, can be written as:

p( α, θ) =+∞

ks=0

+∞



kr=−∞

P kskr cos [(ks N s − kr N r

) p α + kr N r

N r represents the bars number per pole pair, P kskr is a term which depends on the machine

geometry, and ks and kr are two integers.

Let us assume that the stator is supplied by a balanced three phase sine currents of I

r.m.s value andω angular frequency It results that:

θ = θ0+ (1 − s) ω

Choosing a temporal origin such as, for t = 0, i1= I√2 andθ0= 0, leads to

b g=

hs

+∞



ks=−∞

+∞



kr=−∞

ˆb g

hs ,kr

cos{[1 + kr Nr(1 − s)]ωt − (hs + ksNs + kr Nr)pα} (5) with:

ˆb g

hs = 3I

√ 2

ext Rs

R OTOR

A IR -G AP

S TATOR

int Rs

r

bn bt

M

b = 0

b A

Figure 1 Section of the machine.

hs is an integer equal to 6 ν + 1 where ν is a positive, negative, or null integer Expression

(5) can also be expressed with the following form:

b g= +∞

h=−∞

+∞



k=−∞

ˆb g

where:

It has been shown [7] that this method defines precisely the frequencies and gives satis-factory results in the magnitudes

Evolution of the flux density through the stator

In order to establish the analytical expression of the flux density b ,in the surrounding of the

machine, at M point, it is necessary to study its evolution through the stator frame Fig 1 presents a section of a machine

Rs int and Rsext are the interior and exterior stator radii, and r is the distance of the sensor

from the motor shaft The relative magnetic permeability of the stator iron is denotedμ r

To determine the diffusion of the flux density through the stator [8], it is required to exploit the Maxwell equations:

Div b = 0, which describes the conservative aspect of the flux density,

Curl h = j which is the local expression of the Ampere theorem

240 Thailly et al.

where b , h, andjare, respectively, the flux density, the magnetic field, and the current density

vectors

The equations will be solved in the stator iron and in the external air So, as there is no current in these areas, it comes:

The flux density vector b can be expressed using the definition of the magnetic vector potential A , as well as:

b =

bn bt

0

so:

Curl h= Curl

 1

μ0μ r

Curl A



As A is only in the axial direction, equation (12) amounts, consequently, to resolve:

∂

∂r



r ∂ A

∂r



∂α

 1

r

∂ A

∂α



This equation admits one solution which has the following form [9]:

A=+∞

ς=0

λ1r ς + λ2r −ς sinς(α − α) (14)

Identification with equation (7) leads toςα= kωt and, as the sign of the flux density pole

number is just significant of the rotating sense of the component:ς = |h|p.

It is also possible to obtain the normal and tangential components of the flux density vector:

⎧

⎪

⎪

bn h ,k = 1

r

∂A

∂α = ˆbnh ,kcos (|h| pα − kωt)

bt h ,k = −∂A ∂r = ˆbth ,ksin (|h| pα − kωt)

(15)

So:

ˆbnh ,k =

|h| p r

λ1r |h|p + λ2r −|h|p

ˆbth ,k = − |h| pλ1r |h|p−1 − λ2r −|h|p−1 (16)

It can be noticed that these equations are valid whatever the area

In the following, the upper indexes “g,” “i ,” or “a” will be used to distinguish different

variables related to, respectively, air-gap, iron and external air areas

In the air

According to the flux density equals zero when r tends toward infinity, leads to:

Equation system (15) becomes:



bn a h ,k = |h| pλ a

bt h a ,k = |h| pλ a

It can be noticed that the normal component magnitude of the flux density is equal to the tangential one

In order to determine the second coefficientλ a , it is necessary to know the magnitude

of the flux density at the boundary r = Rsext At this point, if the value of the flux density components is noted

ˆb a

h ,k (r=Rsext )

it comes:

λ a

a

h ,k (r=Rsext )

Then, the equations of the flux density in the air are given by:



bn a h ,k = ˆb a

h ,k (r=Rsext ) (r)−|h|p−1cos (|h| pα − kωt)

bt a ,k = ˆb a ,k (r=Rsext ) (r)−|h|p−1sin (|h| pα − kωt) (20)

with:

r= r

In the stator

On the iron-air boundary (r= 1), the normal component of the flux density is preserved

as well as the tangential one of the magnetic field:

ˆbn i

h ,k (r=Rsext ) = ˆb a

,k (r=Rsext ) ˆbt i

h ,k (r=Rsext ) = μr ˆb a

h ,k (r=Rsext )

(22)

Going back to the developed Maxwell equations (15), it can be written, for r = 1:

⎧

⎪

⎪

ˆb a

h ,k (r=Rsext )= |h|p

Rs ext



λ i

ext + λ i

ext



μ r ˆb a

h ,k (r=Rsext ) = − |h| pλ i

ext − λ i

ext

The resolution of the system (23) leads to the following expressions ofλi

2

λ i

a

h ,k (r=Rsext )

2|h| pRs |h|p−1 ext

(24)

λ i

2 =(μ r + 1) ˆb

a

h ,k (r=Rsext )

242 Thailly et al.

Assuming thatμ r  1, the evolution equations of the flux density through the stator are finally express as following:

⎧

⎪

⎪

bn i h ,k = μ r

2 ˆb a

h ,k (r=Rsext )

bt h i ,k = μ r

2 ˆb a

h ,k (r=Rsext )

As, in this approach, it is the magnetic field radial component which is considered, it is just necessary to continue the study on the normal component of the flux density

In this case, as the normal flux density is preserved on the boundary between the air-gap and the stator and since this one can be obtained by the equation (7), it is consequently possible to establish a link between the flux density which exists in the air-gap and this which flows at the outside of the machine

If r = Rsint , then ˆbn i

h ,k (r =Rsint ) = ˆb g

h ,k

So, it can be deduced:

ˆb a

h ,k (r=Rsext ) = ˆb

g

h ,k

μ r

2



Rs i nt

Rs ext

−Rs i nt

Rs ext

From equation (20), it can be defined the Kh(r) coefficient which takes into account the decreasing of the magnitude of each component, regarding to its number of pole pair:

K h (r ) = ˆbn

a

h ,k

ˆb g

h ,k

(28)

whereK is a constant coefficient equals to:

μ r



Rsint

Rs ext

−Rsint

Rs ext

It can be noticed that the higher|h| is, the more rapidly Kh, and consequently, the corre-sponding component magnitude, decreases regarding its values in the air-gap

Measured flux

In order to measure the radial magnetic field in the surrounding of the machine, a coil, used

as a flux sensor, is set, in a plan parallel to the shaft, as it is shown on Fig 2

The 2β angle corresponds to the angular size of the sensor from the motor shaft (ls ≈

2βr) The sensor length Ls is sufficiently small with regard to this of the machine L, so that

the developed theory remains valid even the bars are skewed

The flux h ,k , linked by this sensor, is tied to the external normal flux density component

depending on its geometry

number is just significant of the rotating sense of the component:ς = |h|p.

It is also possible... stator

On the iron-air boundary (r= 1), the normal component of the flux density is preserved

as well as the tangential one of the magnetic field:

... Kh(r) coefficient which takes into account the decreasing of the magnitude of each component, regarding to its number of pole pair:

K h (r ) = ˆbn