Calculation of Induction Machines Inductances under Outer Raceway Defect N. Bessous, S. E. Zouzou, N. Halem Abstract—The stator or rotor faults don’t have important percentage compared to bearings faults. For this reason, it’s necessary for us to worry about this kind of faults. The detection of this failure has been given by several methods. To progress this approach, we propose an analytical model for the diagnosis of induction motor. This paper presents a new precise model for the calculation of inductances in three-phase squirrel cage induction machine. This calculation is done under outer raceway defect. This paper determines a particular geometrical model under the bearing fault conditions and evaluates the inductances. We will investigate the effects of slots in rotor and stator. Rotor asymmetries lead to the change of winding function theory which will be transformed into the modified winding function that influences the inductances due to these conditions. Index Terms—Induction motors, bearing defect, outer raceway, non-symmetrical air-gap, eccentricity, inductance calculation, winding function approach, modified winding function approach.
I. INTRODUCTION
I
nduction motors have small air-gap as well as they are vulnerable to unimportant variations in the dimensions of motors. The precocious detection of abnormality in the induction motor helps us to avoid future serious damage. The air-gap between the stator and rotor the in the healthy machine is equal, but under other different conditions; eccentricity (static or dynamic), bearing defect, etc, the value of air-gap becomes unequal. In the last case of bearing fault, the axis of the rotor rotation does not coincide with the center of the stator. Bearings are defined as shared elements of electrical machine. They work so as to permit rotary motion of the shafts. Bearing fault accounts for over 41% of all motor failures, according to some statistical data [1]. Bearing consists of two rings called the outer (external) and the inner (interior) rings. A set of balls or bearing elements placed in raceways rotate inside these rings and its balls are surrounded by a cage. A continued pressure on the bearings causes fatigue failures on the outer or inner races of the bearings. The small pieces have the possibility N. BESSOUS is with the Laboratoire de Génie Electrique de Biskra (LGEB), Université Mohammed Khider, Biskra 07000, Algérie (e-mail:
[email protected]). S. E. ZOUZOU is with the Laboratoire de Génie Electrique de Biskra (LGEB), Université Mohammed Khider, Biskra 07000, Algérie (e-mail:
[email protected]). N. HALEM is with the Laboratoire de Génie Electrique de Biskra (LGEB), Université Mohammed Khider, Biskra 07000, Algérie (e-mail:
[email protected]).
978-1-4799-7743-7/15/$31.00 ©2015 IEEE
to make a break of the bearing called flaking or cracking. These failures may cause uneven running of the bearings that produces vibrations and increases noise levels. Other external sources can help this process, including infectivity, oxidization, dirty lubrication, bad installation, etc. The shaft voltages and currents could be considered as another source to bear failures. The flux disturbances such as rotor eccentricities produce these shaft voltages and currents [2]. High bearing temperature is another reason for bearing failure; the latter can be caused by degradation of the lubricant or the bearing. The following factors such as: winding temperature rise, motor operating speed, temperature distribution within motor, etc could cause the bearing temperature rise. Therefore, some useful information about the machine health and bearing health can be provided by the bearing temperature measurement [3]. The stator winding contains all harmonics of the stator windings distribution, except the fundamental one. In the machine, we can consider that the stator winding has a sinusoidal distribution. A model that is based on the actual geometry and winding form of stator windings and rotor loops machine was proposed [4] as suitable for time domain simulation of the induction motors [5]. We use the winding function and other equations within the theory of our calculations. By this approach, all space harmonics have a great importance without any restriction of symmetry of stators or rotors windings. So, this model analyses the electrical machines under asymmetrical conditions and faults, for example broken rotor bars [6, 7], fault condition in stator windings [8], eccentricity [9-12]. In [5, 12, and 31], the modified winding function approach (MWFA) for non-symmetrical air-gap in the induction motor was treated. The effects of skew on inductances are analyzed by [13, 14]. In [15, 16, 18], there is an evaluation of the different geometrical models to calculate inductances. A new inclusive method to calculate inductances of induction machine under good and faulty conditions, based on combined winding function approach (WFA). Another methods based on finite element model for calculation and approximation is studied [17, 19]. In all works, the air gap non-uniformity for outer raceway was not considered in the analysis of the electrical machines. In fact, few research efforts in the domain of diagnosis have been addressed for the computation of inductances under bearing defects. In this paper we apply a new method to analyze the bearing faults. This technique was used for computation of different inductances for our type of machine under the state’s presence of the defect outer race. The average radius of the machine is
150
considered constant. As the geometrical model of the machine is simplified; the calculation of inductances and length of the air-gap of the machine are approximated. Our objective is to put an approximation which is written by mathematical formulas of the all inductances in our machine. Therefore, we have contributed to the diagnosis of the electric machines as well as to make a simulation that follows the electromagnetic system under healthy or faulty conditions. The calculated inductances are based on Fourier series, which allow us a better use of the analysis or simulation with adequate time. II. BEARING DIAGNOSIS OF OUTER RACEWAY DEFECT An accurate and dependable technology for bearing harm detection is based on the vibration measurement on the bearing housing [20]. A spoiled bearing with movements of the mechanics leads to alter the air gap within the electrical machine [21]. These variations influence the current stator and can therefore serve as fault indicators for bearing damages. Changes in the current stator spectrum of the machine are related to the characteristic fault frequencies due to faulty bearings that can be detected. The faults to the bearing depend on the number of its elements; therefore, we can find them in: outer raceway, inner raceway, ball and cage. The bearing geometry (Fig. 1) and the mechanical rotor frequency fr have driven us to write the different formulas of the characteristic frequencies fc as [21, 22, 23]: Outer raceway: fc = f o =
Nb D f r (1 − b cos β ) Dc 2
(1)
the bearing, so we cannot consider it as an early fault, but a real fault. In other researches, one of the most studied faults is the hole in the outer race [3, 21, 24]. A fault in a bearing can be considered as a small hole, a pit or a missing piece of material on the corresponding elements (Fig. 2). Fatigue failure starts by small cracks under the normal operating conditions of a good alignment and balanced load, which located between the surface of the raceway and bearing elements, which increasingly spread to the surface generating vibrations and increasing noise levels [3, 24]. When it starts, the affected area expands fast contamination of the lubricant and causing localized overloading over the whole circumference of the raceway.
Rotor centered Situation at t≠k/fo
Rotor center displaced Situation at t=k/fo
Fig. 2. Radial rotor movement due to an outer bearing raceway defect.
The stress will continue periodically because of the rotation of the motor, and according to the position of the motor, we find a distribution of the force according to the center of gravity (Fig.3), this force occupies a contour of half circle,
Where:
Db , Dc : are the inner and the outer raceway diameters. β : the slip angle and N : Number of balls.
b It has been statistically shown in [3] that the vibration frequencies can be approximated for most bearings with balls between 6 and 12 by: (2) f c = f o = 0 .4 N b . f r Outer raceway
For a zone of load of 1800 Fig. 3. Extended of the distribution of load for a radial functional game downwards.
Db Inner raceway
Ball
Dc
β
Cage
Fig. 1. Ball bearing parameters.
In this work, one type of the bearing faults has been analyzed. We have considered a very serious fault which is a crack in the outer race, which a single point defect. This condition is considered as closer to the complete breaking of
In this case and with time, the outer race element is not permitted to rotate on the designed oil film that causes increased levels of heating. The excessive heating causes the grease to break down or appears like an overloading for the machine, which reduces its ability to grease the bearing elements and accelerates the failure course. This deformation has an influence on the air-gap, therefore on the winding function approach and finally on the inductances of the machine, all that drives us to find the real influence on the physical sizes current, torque, …etc. We consider this phenomenon like a periodic signal that carries the shape of a drop (Fig. 4), and we model is as an
151
oblong signal which has a height b and a width a (Fig. 5). The height and the width are very small, for this reason we can approach our signal to the impulse of Dirac. This function can or cannot be periodic (Fig. 6, Fig. 7).
sweep along the face of the stator and rotor of any position motor coil in the slots. The mutual inductance between any two windings i and j in any electric machine can be developed by [9, 10, 12]:
g(t)
2π
Lαβ (θ ) = μ0 r l ∫ nα (θ ,θr ) Nβ (θ ,θr ) g −1 (θ ,θr ) dθ
(3)
0
g0
t 1/fo
Fig. 4. The drop form of the air gap in the presence of an outer bearing raceway defect for θ =0. g(t)
a
where: μ0 is the permeability of free space, μr is the relative permeability, μ is the magnetic permeability, l is the machine length, r is the mean radius of the air-gap. θ is a particular angular position along the stator inner surface, θ ' is a particular angular position along the rotor, θ r is the angular position of the rotor with respect to the stator frame, g −1 (θ ,θ r ) it is the inverse air gap function which becomes 1 / g due to the assumption of uniform air gap. The
g0
terms nα (θ , θ r ) and N β (θ ,θ r ) are the distribution function
b
t 1/fo
Fig. 5. Rectangular form of air-gap length in the presence of an outer bearing raceway defect (θ =0).
The variation on the air-gap length for the outer race defect is approximated by a series of Dirac generalized functions. g(t)
g0
t
and the winding function in windings i and j respectively [10]. a relation joins all angles is defined like follows: (4) θ = θ ' + θr A.1. Stator Inductances It is known that the space harmonics of the MMF are related to air-gap formula, conductors in slots placement and to the number of pole pairs. The induction machine which is studied in this work is a three-phase machine, 3kW, 4 poles and 28 rotor bars. Figure 8.a represents a distribution function of an elementary coil in the stator, composed by Nc conductors and its expression is given by [10]: if 0 ≤ θ ≤ α s ⎧N c n(θ ) = ⎨ (5) elsewhere ⎩0
with:
1/fo
Fig. 6. Airgap length g and permeance Λ in the presence of an outer bearing raceway defect (θ =0).
In fact, the strokes (beats) are at random and linked to the position of the balls in every moment of time. Consequently, the figure of air-gap will be changed (Fig. 7). g(t)
2πQ (6) Ns The distribution function of the stator phase “q” comes from a sum of Ne elementary coils shifted one after the other by 2π/Ns and this is repeated for p poles (Fig. 8-b) [4, 13]. For the stator phase number 1, q=0; and q=1 or q=2 in favor of stator phase number 2 or 3 successively. This one becomes in a Fourier series:
αs =
nsq ( θ ) = C0 +
g0
2 Nt pπ
⎡ ⎛ K wh 2π ⎞ ⎤ cos ⎢h.p ⎜ θ − θ0 − q ⎟⎥ 3 p ⎠⎦ h =1 h ⎣ ⎝ ∞
∑
(7)
n(θ)
“a”
Nc
t T0=1/fo
θ
αs
3/fo nsq(θ)
Fig. 7. Airgap length form for random positions in the presence of an outer bearing raceway defect (θ =0).
3Nc
“b”
2Nc Nc
III. CALCULATION OF INDUCTANCES
θ
A. Calculation of Inductances Under Healthy Motor The winding functions theory is the basis to calculate all inductances for the induction machine. This method makes a
Fig. 8. a) Elementary coil of the first stator phase, b) distribution function of the first stator phase.
152
leakage inductance of the two rotor bars and the leakage inductance of the two end rings segments their expression is as follows :
where ⎧ ⎪ N = pN N , K wh = K prh .K dh c e ⎪ t ⎪ N sin (h.p.π e ) ⎪ Ns Q ⎪ ) , K ph = sin (h.p.πh ⎨ K dh = π Ns ⎪ N e sin (h.p. ) ⎪ Ns ⎪ ⎪ NtQ π = n q (θ ) , θ 0 = (Ne − 1 + Q) ⎪C 0 = N N s s ⎩
Lrii = Lmri + Lbi + Lb ( i ±1) + 2 Le
Lmri =
1
iαr
(10)
Lrirj =
(11)
(12)
μ 0 rl g
M sq − ri = L
The distribution function of the ith rotor loop is given by (Fig. 9) [10]: (15)
The rotor winding function is defined by: ⎡ ⎛ α 2 ∞ 1 1 ⎞ ⎤ ∑ sin(m 2r ) cos ⎢ m ⎜θr − (i − 2 )αr ⎟ ⎥ π m =1 m ⎝ ⎠ ⎦ ⎣
μ0 rl 2π ∫ nsq ( θ ,θr ) N ri (θ , θr )dθ g 0
x 10
(20)
-3
The derivative Lsar1 (H)
1
(13)
A.2. Rotor Inductances In order to get the inductances of the rotor loops, we precede in the same manner as for the stator case with the assumption that the rotor loops that are equal and divided one from another by a mechanical angle: 2π (14) αr = nb Where nb is the number of rotor bars.
N ri ( θ ' ) =
2
0
α α 2 ∞ 1 ⎡ ⎛ 1 ⎞⎤ nri ( θ r ) = r + ∑ sin(m r ) cos ⎢ m ⎜ θ ' − (i − )αr ⎟ ⎥ 2π π m =1 m 2 2 ⎠⎦ ⎣ ⎝
sq ri
=
The mutual inductance and its derivatives between the 1st rotor loop and the first stator phase are exposed in Fig. 10.
2π
∫ n sq (θ ) N s (q + 1 )(θ )d θ
(19)
and a rotor loop “ i ” depends on the angular position of the rotor loop. This is obtained from:
Lsar1
( )
sq s q + 1
=
μ0 rl 2π ' ' ' ∫ nri ( θ )N rj ( θ )dθ g 0
The mutual inductance Lsq ri between a stator phase “q”
We give the mutual inductance between any two stator phases by: L
2π
The mutual inductance between loop “ i ” and any loop “ j ” is not adjacent; we obtain it as :
We determine the self magnetizing inductance of any stator phase by the following equation:
μ rl 2π Lmsq = 0 ∫ n sq (θ ) N sq (θ )dθ g 0
(i+1)αr
Fig. 9. Distribution function of a i rotor Loop.
(9)
The summation of the self magnetizing inductance and the leakage inductance is considered as the definition of the inductance of a stator phase “q”, and is given by:
Lsq = Lmsq + Lsf
(18)
th
then 2 Nt ∞ K wh ⎡ 2π ⎤ cos ⎢ h.p(θ − θ0 − q N sq (θ ) = ) ∑ 3 p ⎥⎦ pπ h=1 h ⎣
μ 0 rl 2 π n ri ( θ ' )N ri ( θ ' )d θ ' g 0∫
nri(θ’)
where: Nc is the number of conductor per stator slot, Ne is the number of slots per pole and per phase, Nt is the number of stator turns in series, N s is the number of stator slots, Q is the number of slots per pitch turn. The winding function of phase “q” is defined by:
N sq(θ ) = nsq(θ ) − nsq(θ )
(17)
with:
(8)
0
-1
-2
0
1
2
3
4
Angle of the rotor (rad)
5
6
Fig. 10. The mutual inductance (blue) and its derivative (red).
B. Calculation of Inductances Under Faulty Motor The air gap of a machine in the case of a bearing damage will be changed, this modification will show new picks with large amplitude in spectral current (stator current) for every bearing defects. The function of air gap in the case of an eccentricity plus our faults, we also distinguish three types of eccentricity; the static air-gap eccentricity (SE), the dynamic air gap eccentricity (DE) and mixed eccentricity, the air-gap can take the following expression [21, 24]:
(16)
We define The inductance of a rotor loop “ i ” as the summation of the self magnetizing inductance, Lb , Le the
153
+∞ ⎡ k ⎤ g (θ , θ r ) = g 0 . ⎢ 1 − δ cos(θ + ψ (t )) ∑ δ (t − ) ⎥ fc ⎦ k =−∞ ⎣
(21)
where fc is the characteristic bearing fault frequency given by (1), (2) and (3), δ is the static or dynamic eccentricity degree and ψ (t ) is defined as follows : ⎧ 0 ⎪ ψ (t ) = ⎨ ωr t ⎪ω t ⎩ cage
L BA = μ 0 rl
∫
0
for an inner race defect for a ball defect
-1
n B (θ , θ r ) M A (θ , θ r ) g (θ , θ r ) d θ
B.1. Air-gap Permeance The first step in the theoretical analysis is the determination of the air-gap length g and the air-gap permeance Λ (the inverse of g) as a function of time t and angular position θ in the stator reference frame. The air-gap length is caused by the radial rotor movement with the presence of the defect, that we consider it always as a hole or a point of missing material in the matching bearing element [24]. The air-gap permeance Λ is proportional to the opposite of the air-gap length g and we define it as follows: Λ (θ , θ r ) =
μ g (θ , θ r )
(24)
where μ = μ μ , the permeance becomes: r
Λ(θ ,θr ) = Λ0 .
where
1 +∞ ⎡ k ⎤ ⎢ 1 − δ cos(θ +ψ (t )) ∑ δ (t − f ) ⎥ k =−∞ ⎣ ⎦ c
Λ0 =
(25)
k =−∞
by:
Lmsq = μ 0 rl
2π
n sq (θ ) M sq (θ ) g −1 (θ , θ r )d θ
∫
(30)
0
the modified winding function approach (MWFA) in this case is calculated by: 2π
M sq (θ ) = nsq (θ ,θ r ) −
−1 ∫ nsq (θ ,θ r ).g (θ ,θ r )dθ 0
(31)
−1
2π g (θ ,θ r )
g −1 (θ , θ r ) =
1 2π
2π
∫
g −1 (θ ,θ r )dθ
(32)
0
with g −1 (θ , θ r ) ≈
1 ⎧ ⎨ 1 + δ c0 cos(θ ) g0 ⎩
∞ ∞ ⎫ (33) + δ ∑ ck cos(θ + kωc t ) + δ ∑ ck cos(θ − kωc t ) ⎬ k =1 k =1 ⎭ where δ is the relative degree of eccentricity introduced by the outer race defect. We interpret this equation as a temporary static eccentricity of the rotor, that appears merely at t = k / f 0 for θ =0, the function g0(θ ,t) is figured in Fig. 8. we
simulate this eccentricity static by a degree δ s .
ω is the characteristic pulsation,
.
f c = 0, 4.Nb . f r
(26)
of eccentricity verifies δ ≺ 1 in order to keep away from the contact between the rotor and the stator. The series of Dirac generalized functions is expressed as a complex Fourier series development [26], i.e., +∞
The self-magnetizing inductance of any stator phase writes
c
μ
g0 The Fig. 7 illustrates the relationship between air-gap length g (θ ,θ r ) and the inverse of air-gap Λ (θ ,θ r ) at the position θ =0 for an outer raceway defect. The fraction 1 (1 − x) is approximated for small air-gap variations by the first-order term of its series development, i.e., 1 (27) = 1 + x + x 2 + ... ≈ 1 + x., for x ≺ 1 1− x The condition x ≺ 1 is always suitable because the degree
∑ δ (t −
B.2. Derivation of Stator Inductances for outer raceway B.2.1. Self-Magnetizing Inductance
(23)
and can be calculated with the same manner as above for different rotor positions.
(29)
+∞ +∞ ⎫ + δ ∑ ck cos(θ + ψ (t ) + k ωc t ) + δ ∑ ck cos(θ + ψ (t ) − k ωc t ) ⎬ k =1 k =1 ⎭
(22)
where M (θ ,θ r ) is the modified winding function of coil A A
0
⎧ Λ (θ , θ r ) ≈ Λ 0 . ⎨ 1 + δ c0 cos(θ + ψ (t )) ⎩
for an outer race defect
A characteristic fc frequency corresponds to the periodicity of occurrence of the anomalous physical phenomenon allied to the existence of the fault. Under bearing defects conditions, the inductance of a coil A due to current in coil B can be expressed as: 2π
with the Fourier series coefficients ck = f c ∀k . We combine equations (25)–(38) into an easy expression for the air-gap permeance wave, i.e.,
+∞ +∞ k ) = ∑ ck e − j 2π kf t = c0 + 2 ∑ ck cos(2π kf c t ) (28) fc k =−∞ k =1
and
θc = 2π f c t
(34)
We deduced:
θc = 0, 4.Nb .θ r = λcθ r
(35)
So the function is written: Kbh ⎡ 2π ⎤ cos ⎢ hp (θ − θ0 − q ) ⎥ 3p ⎦ h =1 h ⎣ N 2π − t .v.δ s c0 Kb1 cos(θ0 + q ) 3p πp
M sq (θ ) =
2 Nt πp
−
Nt
πp
∞
∑
.v.δ s .K b1 ∑ ck cos(θ0 + q
2π + k λcθ r ) 3p
Nt 2π (36) .v.δ s .K b1 ∑ ck cos(θ0 + q − k λcθ r ) 3p πp It indicates an important note regarding the number of pole pairs:
c
154
−
Thus if p = 1 , v = 1 and for p ≥ 2 , v = 0 For our study, the number of pair poles is 2, but this last value does not mean that there isn’t an influence of p in our coming study; the calculation of inductances are influenced the number of pole pairs.
Lmri (θr ) =
μ0rl ⎧ ⎛ αr ⎞ 1 ⎤ ⎛α ⎞ ⎡ .δsc0 sin⎜ r ⎟ cos ⎢θr + (i − )αr ⎥ ⎨ g0 ⎩ ⎜⎝ π ⎟⎠ 2 ⎦ ⎝2⎠ ⎣
1 ⎤ ⎛α ⎞ ⎛α ⎞ ∞ ⎡ + ⎜ r ⎟.δs sin ⎜ r ⎟ ∑ck .cos ⎢θr + kθc + (i − )αr ⎥ π 2 ⎦ ⎣ ⎝ ⎠ ⎝ 2 ⎠ k=1 1 ⎤ ⎛α ⎞ ⎛α ⎞ ∞ ⎡ + ⎜ r ⎟.δs sin ⎜ r ⎟ ∑ck .cos ⎢θr − kθc + (i − )αr ⎥ π 2 ⎦ ⎣ ⎝ ⎠ ⎝ 2 ⎠ k=1 2
We replace (7), (33) and (36) in (30), we get:
⎡ ⎛ αr ⎞ ⎤ ⎢sin ⎜ m ⎟ ⎥ ⎛ 4 ⎞ ⎢ ⎝ 2 ⎠⎥ +⎜ ⎟ ∑ (1+δsc0 + 2.δsck ) m ⎥ ⎝ π ⎠ m=1⎢ ⎢ ⎥ ⎣ ⎦ ∞
2
μ 0 rl ⎛ 2 N t ⎞ ∞ ⎛ K bh ⎞ ⎜ ⎟ .∑ g 0 π ⎝ p ⎠ h =1 ⎜⎝ h ⎟⎠
L m sq =
2
(37)
2
⎛ 1 ⎤⎞ ⎛α ⎞ ⎡ + 4.⎜δsc0 sin ⎜ r ⎟cos ⎢θr + (i − )αr ⎥ ⎟ 2 2 ⎦⎠ ⎣ ⎝ ⎠ ⎝
B.2.2. Mutual Inductance
2 ∞
⎛ ⎛ α ⎞⎞ + 4.⎜δsc0 sin ⎜ r ⎟⎟ ⎝ 2 ⎠⎠ ⎝
The mutual inductance between any two stator phases is 2
M sab
2 ∞
⎛ ⎛ α ⎞⎞ + 4.⎜δsc0 sin ⎜ r ⎟⎟ ⎝ 2 ⎠⎠ ⎝
2
μ rl ⎛ 2 N ⎞ ∞ ⎛ K ⎞ 2π 2π = Lmsq .cos(h ) = 0 ⎜ t ⎟ .∑ ⎜ bh ⎟ .cos(h ) (38) 3 g 0π ⎝ p ⎠ h =1 ⎝ h ⎠ 3
sin(m
+ 4δ s .sin(
αr
1 ⎡ ⎤ ∞ ).cos ⎢θ r + (i − )α r ⎥ .∑ ck cos(kθ c ) 2 2 ⎣ ⎦ k =1
∫
nri (θ ' , θ r ) M ri (θ ' , θ r ) g −1 (θ , θ r )d θ '
⎤ ⎦
⎡ ⎣
1
⎤ ⎦
⎡ ⎣
1
⎤ ⎦
∑ck cos ⎢θr − kθc + (i − 2)αr ⎥.cos ⎢θr + (i − 2)αr ⎥
k =1
∑ck cos[ kθc ]
k =1
2
2
⎛ 1 ⎤∞ 1 ⎤ ⎫ ⎛ α ⎞⎞ ⎡ ⎡ +8.⎜δs sin ⎜ r ⎟⎟ cos ⎢θr + (i − )αr ⎥ ∑ck cos[ kθc ] cos ⎢θr − kθc + (i − )αr ⎥ ⎬ 2 2 2 ⎦ ⎭ ⎣ ⎦ ⎣ ⎝ ⎠⎠ k =1 ⎝
2π
Lrjri = μ0 rl ∫ nrj (θ ' , θ r ) M ri (θ ' , θ r ) g −1 (θ , θ r )dθ '
(41)
(42)
0
(39)
Substituting (15), (25) and (39) in (42) leads to the expression of mutual inductances between rotor loops μ rl ⎧ ⎛ α ⎞ ⎡ 1 ⎤ ⎛α ⎞ Lrjri (θr ) = 0 ⎪⎨ ⎜ r ⎟.δsc0 sin ⎜ r ⎟ cos ⎢θr + (i − )αr ⎥ g0 ⎪⎩ ⎝ π ⎠ 2 ⎦ ⎝ 2 ⎠ ⎣ ∞ α α 1 ⎡ ⎤ + ⎛⎜ r ⎞⎟.δs sin ⎛⎜ r ⎞⎟ ∑ ck .cos ⎢θr + kθc + (i − )αr ⎥ 2 ⎦ ⎝π ⎠ ⎝ 2 ⎠ k=1 ⎣ α sin2 ⎛⎜ r ⎞⎟ ⎛ 4⎞ ∞ ⎝ 2 ⎠ cos ⎡(i − j)α ⎤ + ⎜ ⎟. ∑ r⎦ ⎣ m ⎝ π ⎠ m=1 2 ⎛ α ⎞ ⎡ ⎡ 1 ⎤ 1 ⎤ + 4.⎜δsc0 sin ⎛⎜ r ⎞⎟ ⎟ cos ⎢θr + (i − )αr ⎥.cos ⎢θr + ( j − )αr ⎥ 2 ⎦ 2 ⎦ ⎝ 2 ⎠⎠ ⎣ ⎣ ⎝ 2 ∞ ⎛ α ⎞ 1 ⎤ 1 ⎤ ⎡ ⎡ + 4.c0.⎜ δs sin ⎛⎜ r ⎞⎟ ⎟ cos ⎢θr + (i − )αr ⎥. ∑ ck cos ⎢θr + kθc + ( j − )αr ⎥ 2 ⎦ k =1 2 ⎦ ⎝ 2 ⎠⎠ ⎣ ⎣ ⎝ 2 ⎛ α ⎞ 1 ⎤ ∞ 1 ⎤ ⎡ ⎡ + 4.c0.⎜ δs sin ⎛⎜ r ⎞⎟ ⎟ cos ⎢θr + (i − )αr ⎥. ∑ ck cos ⎢θr − kθc + ( j − )αr ⎥ 2 ⎦ k=1 2 ⎦ ⎝ 2 ⎠⎠ ⎣ ⎣ ⎝ 2 ∞ ⎛ α ⎞ 1 ⎤ 1 ⎤ ⎡ ⎡ +8.c0.⎜δs sin ⎛⎜ r ⎞⎟ ⎟ cos ⎢θr + (i − )αr ⎥.cos ⎢θr + ( j − )αr ⎥ ∑ ck cos ⎣⎡kθc ⎦⎤ 2 ⎦ 2 ⎦ k=1 ⎝ 2 ⎠⎠ ⎣ ⎣ ⎝ 2 ⎛ α ⎞ 1 ⎤ ∞ 1 ⎤∞ ⎡ ⎡ +8.⎜δs sin ⎜⎛ r ⎟⎞ ⎟ cos ⎢θr + (i − )αr ⎥. ∑ ck cos ⎢θr + kθc + ( j − )αr ⎥ ∑ ck cos ⎡⎣kθc ⎤⎦ 2 ⎦ k=1 2 ⎦k=1 ⎝ 2 ⎠⎠ ⎣ ⎣ ⎝ 2 ⎛ α ⎞ ⎡ ⎡ 1 ⎤ ∞ 1 ⎤∞ +8.⎜δs sin ⎛⎜ r ⎞⎟ ⎟ cos ⎢θr + (i − )αr ⎥. ∑ ck cos ⎢θr − kθc + ( j − )αr ⎥ ∑ ck cos ⎣⎡kθc ⎦⎤ 2 ⎦ k=1 2 ⎦k=1 ⎝ 2 ⎠⎠ ⎣ ⎣ ⎝
with 2π
1
⎛ 1 ⎤∞ 1 ⎤ ⎛ α ⎞⎞ ⎡ ⎡ +8.⎜δs sin ⎜ r ⎟⎟ cos ⎢θr + (i − )αr ⎥ ∑ck cos[ kθc ] cos ⎢θr + kθc + (i − )αr ⎥ 2 2 2 ⎦ ⎣ ⎦ k=1 ⎣ ⎝ ⎠⎠ ⎝
The inductance of a rotor loop i is defined by (17)
Lmri = μ 0 rl
⎡ ⎣
The mutual inductance between loop “i” and any loop “j” not adjacent to it can be obtained by
αr
) 2 cos ⎡ m(θ ' − (i − 1 )α ) ⎤ M ri (θ r , θ ) = ∑ r ⎥ ⎢ π m =1 m 2 ⎣ ⎦ αr 1 ⎡ ⎤ + 2δ s c0 sin( ).cos ⎢θ r + (i − )α r ⎥ 2 2 ⎣ ⎦ ∞
⎤ ⎦
2 ∞
of the rotor can be expressed as, a modified winding function for the rotor written by: 2
1
⎛ 1 ⎤⎞ ⎛α ⎞ ⎡ +8.c0.⎜δs sin ⎜ r ⎟ cos ⎢θr + (i − )αr ⎥ ⎟ 2 ⎦⎠ ⎝2⎠ ⎣ ⎝
B.3. Derivation of Rotor Inductances for outer raceway Similarly to the stator and by means of equation (30) and by replacing nsq (θ ) by nri (θ ' , θ r ) , the modified winding function
'
⎡ ⎣
∑ck cos ⎢θr + kθc + (i − 2)αr ⎥.cos ⎢θr + (i − 2)αr ⎥
k =1
(40)
0
Substituting (15), (25) and (39) into (3), this yields to:
B.4. Derivation of Mutual Inductances Stator/Rotor for outer raceway
155
⎫⎪ ⎬ ⎪⎭
(43)
-3
Substituting (15), (25) and (36) in (23) leads to the expression of the mutual inductance M between =M sq − ri
sr
x 10
(c) 0.5
sar1
(H)
stator windings and rotor loops. After development we get μ0rl ⎧ ⎛4Nt ⎞ ∞ Kwh ⎛ αr ⎞ ⎡ 2π 1 ⎤ ⎨ ⎜ ⎟∑ sin hp. cos hp(θr −θ0 −q +(i− )αr)⎥ 3p 2 ⎦ g0 ⎩ ⎝ πp ⎠h=1h2.p ⎜⎝ 2 ⎟⎠ ⎢⎣
0
L
Msr(θr) =
1
-0.5
⎛2N ⎞ ∞ K ⎤ α⎞ ⎡ 2π 1 ⎛ +⎜ t ⎟.δsc0∑ wh sin⎜(hp+1). r ⎟cos⎢hp(θr −θ0 −q )+(hp+1)(i− )αr +θr)⎥ 2⎠ ⎣ 3p 2 ⎝ πp ⎠ h=1h.(hp+1) ⎝ ⎦ K α⎞ ⎡ ⎤ ⎛2N ⎞ 2π 1 ⎛ +⎜ t ⎟.δsc0∑ wh sin⎜(hp−1). r ⎟cos⎢hp(θr −θ0 −q )+(hp−1)(i− )αr −θr)⎥ h hp p .( 1 ) 2 3 2 p − π ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ h=1 ⎛2Nt ⎞ ∞ ∞ Kwh ⎡ ⎤ α⎞ 2π 1 ⎛ sin⎜(hp+1). r ⎟cos⎢hp(θr −θ0 −q )+(hp+1)(i− )αr +θr +kθc)⎥ +⎜ ⎟.δs ∑ck ∑ 2⎠ ⎣ 3p 2 ⎝ πp ⎠ k=1 h=1h.(hp+1) ⎝ ⎦
-1
0
1
2
3
4
Angle of the rotor (rad)
5
6
7
∞
∞
-3
k =9 . . . k =1
(H)
0.5
0
-0.5
(44)
-1
0
1
2
4 (rad) Angle 3of the rotor
5
6
7
Fig. 12. The mutual inductance between the first rotor loop of and phase a of the stator for different for different value of k and for δ s = 0.005 . -3
1
x 10
k =9 . . . k =1
sar1
(H)
0.5
L
As we said previously about the influence of the number of k stroke during the working of the motor, it is appeared in a clear manner and for a circumference of 2π , that there are several values which can take the k number. We say that a stroke of a ball has an influence less than two strokes and for two strokes less than three and so forth. The depth of the hole in the outer raceway which is represented by δ s , give an influence on the form of the mutual inductance, and the deformation increases by the increase of δ s under the supposition of constant k (Fig. 11). It is clear that the amplitude of the mutual inductance depends on the number of stroke of the ball by period; the figures 12 and 13 show a proportional relation with k.
x 10
1
sar1
K ⎤⎫ ⎛2N ⎞ α⎞ ⎡ 2π 1 ⎛ +⎜ t ⎟.δs ∑ck ∑ wh sin⎜(hp−1). r ⎟cos⎢hp(θr −θ0 −q )+(hp−1)(i− )αr −θr +kθc)⎥ ⎬ π .( 1 ) 2 3 2 p h hp p − ⎝ ⎠ ⎣ ⎦⎭ ⎝ ⎠ k=1 h=1 ∞
and (c) δ s = 0.01 ).
L
α⎞ ⎡ ⎤ ⎛2N ⎞ ∞ ∞ K 2π 1 ⎛ +⎜ t ⎟.δs ∑ck ∑ wh sin⎜(hp−1). r ⎟cos⎢hp(θr −θ0 −q )+(hp−1)(i− )αr −θr −kθc)⎥ − .( 1 ) 2 3 2 h hp p π p ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ k=1 h=1 ∞ ∞ ⎛2N ⎞ α⎞ ⎡ ⎤ K 2π 1 ⎛ +⎜ t ⎟.δs ∑ck ∑ wh sin⎜(hp+1). r ⎟cos⎢hp(θr −θ0 −q )+(hp+1)(i− )αr +θr −kθc)⎥ 2⎠ ⎣ 3p 2 ⎝ πp ⎠ k=1 h=1h.(hp+1) ⎝ ⎦
Fig. 11. The mutual inductance between the first rotor loop of and phase a of the stator for different eccentric cases (k=9, (a) δ s = 0.001 , (b) δ s = 0.005
0
-0.5
-1
0
1
2
4 (rad) Angle 3 of the rotor
5
6
7
Fig. 13. The mutual inductance between the first rotor loop of and phase a of the stator for different for different value of k and for δ s = 0.00005 .
-3
1
x 10
IV. CONCLUSION
(a)
0
L
sar1
(H)
0.5
-0.5
-1
0
1
2
3
4
Angle of the rotor (rad)
5
6
7
-3
1
x 10
(b)
0
L
sar1
(H)
0.5
-0.5
-1
0
1
2
3
4
Angle of the rotor (rad)
5
6
7
We have presented in this work a modified model to calculate the inductances of induction machines, and we consider non-uniformity of air-gap. Precisely, the geometrical models of induction machine under healthy, static eccentricity and outer raceway have been taken into account. The outer race defect can be assumed to be located at the angular position θ , the contact between a ball and the small hole in outer race (outer race defect) which gives a variable air-gap length. The functions are in relation to θ and θ r . So, it has no limit about non-uniformity in all directions. By means of the modified winding function and planned geometrical models, inductances of the induction machine that are calculated in these conditions. The theoretical analysis of the relationship between healthy and defected machine has been made, jointly
156
with a review of fault models that are used in the literature for the diagnosis of bearing faults in induction machines. The calculation of the inductances in the presence of bearing defect, allows us to put supervision on the factors of influence. Our calculation is also appeared according to the advantage in the time of calculation and even to make an analytic survey. We even see the importance of the number of pole pairs; the latter has an influence on the relations and the form of the inductances. Therefore, new inductances influence systematically the stator current and torque of the machine. Besides, our model allows us to develop an adequate model that treats all shortcomings that appear to the bearing. It has also been shown that by taking into account the harmonics, the influence of angle appears in the calculation of the inductances. This results an increase in the accuracy of calculating the machine inductances. Consequently, in the analysis of induction motors have a large number of poles. A precise geometrical model must be used or extra terms otherwise we take the real shape of the air-gap length. Our future work will be the simulation of this type of defect. It is an important step to make another study on the stator current (MCSA) under bearing defect.
[2] [3] [4] [5] [6]
[7]
[8] [9]
[10]
[11] [12]
[13]
[15]
[16] [17] [18]
[19]
[20] [21] [22]
V. REFERENCES [1]
[14]
P. F. Allbrecht, J. C. Appiarius and R. M. McCoy, “Assessment of the Reliability of Motors in Utility Applications-Updated,” IEEE Trans. Energy Conversion, vol. 1, pp. 39-46, 1986. S.Nandi and H.A. Toliyat, “Condition Monitoring And Fault Diagnosis of Electrical Machines- A Review,” in proc. 34th Annual Meeting of the IEEE Trans. Industry Applications, pp. 197-204, 1999. R. R. Schoen, T. G. Habetler, F. Kamran and R. G. Bartheld, “Motor bearing damage detection using stator current monitoring,” IEEE Trans. Industry Applications, vol. 31, pp. 1274-1279, 1995. H. A. Toliyat, T. A. Lipo, ''Transient analysis of cage induction machines under stator, rotor bar and end ring faults',” IEEE Trans. Energy Conversion, vol. 10, pp. 241 – 247, Jun. 1995. J. Faiz, I. Tabatabai, “Extension of winding function theory for nonuniform air-gap in electric machinery,” IEEE Trans. Magnetics, vol. 38, Nov. 2002. G. B. Kliman, R. A. Koegl, J. Stein, R. D. Endicott and M. W. Madden, “Noninvasive detection of broken rotor bars in operating induction motors,” IEEE Trans. Energy Conversion, vol. 3, pp. 873-879, Dec. 1988. G. Didier, E. Ternisien, O. Caspary, and H. Rzik, "Fault Fetection of Brokeb Rotor Bars in Induction Motor Using a Global Fault Index," IEEE Trans. Industry Application, vol. 42, Jan-Feb. 2006. G. M. Joksimovic, J. Penman “The Detection of Inter-Turn ShortCircuits in the Stator Windings of Operating Motors,” IEEE Trans. Industrial Electronics, vol. 47, pp. 1078 – 1084, Oct. 2000. J. R. Cameron, W. T. Thomson, and A. B. Dow, “Vibration and current monitoring for detecting air-gap eccentricity in large induction motors,” Processing Instrumentation and Electronics Energy, vol. 133, pp. 155163, 1986. A. Khezzar, M. Hadjami, N. Bessous, M. E. K. Oumaamar, and H. Razik, “Accurate modelling of cage induction machine with analytical evaluation of inductances,” in Proc. 2008 IEEE IECON Orlando, FL, USA Conf., pp. 1112–1117. K. N. Gyftakis, J. C. Kappatou, “A Novel and Effective Method of Static Eccentricity Diagnosis in Three-Phase PSH Induction Motors,” IEEE Trans. Energy Conversion, vol. 28, Jun. 2013. S. Nandi, S. Ahmed, H. A. Toliyat, “Detection of Rotor Slot and Other Eccentricity Related Harmonics in a Three Phase Induction Motor with Different Rotor Cages,” IEEE Trans. Energy Conversion, vol. 16, pp. 253–260, Sep. 2001. A. Ghoggal, S. E. Zouzou, H. Razik, M. Sahraoui and A. Khezzar, “An improved model of induction motors for diagnosis purposes – Slot
[23] [24] [25]
[26]
skewing effect and air–gap eccentricity faults,” IEEE Energy Conversion and Management, vol. 50, pp. 1336-1347, May 2009. H. R. Akbari, “Calculation Of Inductances Of Induction Machines Under Axial Non–Uniformity Conditions,” Journal of Electrical Engineering, vol. 60, pp. 149–154, 2009. N. Bessous, M. Hadjami, M. Y. Kaikaa and A. Khezzar, “Diagnosis of a mixed eccentricity fault in a squirrel cage Three-phase Induction Motor using Spectrum Analysis of Stator Current,” in 1st International Engineering Sciences Conference IESC’08, Syria, 2-4 Nov. 2008. S. Nandi, R. M. Bharadwaj and H. A. Toliyat, “Performance analysis of a three phase induction motor under mixed eccentricity condition,” IEEE Trans Energy Conversion, vol. 17, pp. 392–399, Sep. 2002. J. Faiz, B. M. Ebrahimi, B. Akin. H. A. Toliyat, “Comprehensive eccentricity fault diagnosis in induction motors using finite element method,” IEEE Trans, Magnetics, vol. 45, pp. 1764-1767, 2009. M. Y. Kaikaa, M. Hadjami, A. Khezzar, “Effects of the Simultaneous Presence of Static Eccentricity and Broken Rotor Bars on the Stator Current of Induction Machine,” IEEE Trans. Industrial Electronics, vol. 61, May. 2014. N. Halem, S. E. Zouzou, K. Srairi, S. Guedidi, F. A. Abbood, “Static eccentricity fault diagnosis using the signatures analysis of stator current and air gap magnetic flux by finite element method in saturated induction motors,” International Journal System Assurance Engineering Management, vol.4, pp. 118–128, Apr-June 2013. B. Trajin, “Détection automatique et diagnostic des défauts de roulements dans une machine asynchrone par analyse spectrale des courants statoriques,” JCGE'08 LYON, 16 and 17, Dec. 2008. H. Ocak, and K. A. Loparo, “Estimation of the running speed and bearing defect frequencies of an induction motor from vibration data,” Mechanical Systems and Signal Processing, vol. 18, pp. 515–533, 2004. W. Zhou; T. G. Habetler; R. G. Harley; “Stator Current-Based Bearing Fault Detection Techniques: A General Review,” IEEE Power Electronics and Drives, pp. 7–10, 6-8, Sept 2007. B. Li, M. Chow, Y. Tipsuwan, and J. Hung, “Neural-network-based motor rolling bearing fault diagnosis,” IEEE Trans. Industrial Electronics, vol. 47, pp. 1060–1069, Oct. 2000. M. Blödt, P. Granjon, B. Raison and G. Rostaing, “Models for Bearing Damage Detection in Induction Motors Using Stator Current Monitoring,” IEEE Trans. Industry Applications, vol. 55, Apr. 2008. N. A. Al-Nuaim and H. A. Toliyat, “A novel method for modeling dynamic air-gap eccentricity in synchronous machines based on modified winding function theory,” IEEE Trans. Energy Conversion, vol. 13, pp. 156–162, Jun. 1998. J. Max, J. L. Lacoume, “Méthodes et Techniques de Traitement du Signal,” 5th ed. Dunod, Paris, France, 2004.
VI. BIOGRAPHIES Noureddine Bessous was born in 1981. He received the B.Sc. degree in electrical engineering, from the University of Constantine, Constantine, Algeria, in 2005, and the M.Sc. degree from the University of Constantine, Constantine, Algeria, in 2007. Currently, he is working toward the Ph.D. degree on the diagnosis of faults of the inductions machines at the laboratory LGEB of Mohammed Khider University, Biskra, Algeria, now he is an assistant professor at the El-Oued University. Salah Eddine Zouzou was born in Biskra, (Algeria) in 1963. He received the B.Sc. degree from the Polytechnic National School of Algiers, Algiers, Algeria, in 1987 and the M.S and Ph.D degrees from the Polytechnic National School of Grenoble, Grenoble, France, in 1988 and 1991 respectively. His fields of research interests deal with the design and condition monitoring of electrical machines. He has authored or co-authored more than 50 scientific papers in national and international conferences and journals. Professor Zouzou is a Professor at the University of Biskra, Algeria. Noura Halem was born in 1984. She received the B.Sc. degree in electrical engineering, from the University of Biskra, Biskra, Biskra, Algeria, in 2007, and the M.Sc. degree from the University of El-Oued, El-Oued, Algeria, in 2010. Currently, she is working toward the Ph.D. degree on the diagnosis of faults of the inductions machines at the laboratory LGEB of Mohammed Khider University, Biskra, Algeria, now she is an assistant professor at the ElOued University.
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