Angular-based modeling of induction motors for monitoring

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Angular-based modeling of induction motors for monitoring Aroua Fourati a,b,n, Adeline Bourdon a, Nabih Feki b, Didier Rémond a, Fakher Chaari b, Mohamed Haddar b a LaMCoS, Contacts and Structural Mechanics Laboratory, University of Lyon, CNRS, INSA-Lyon, UMR 5259, 20 rue des Sciences, F-69621, Villeurbanne, France b LA2MP, Laboratory of Mechanics, Modelling and Manufacturing, National Engineering School of Sfax, Ministry of Higher Education and Research of Tunisia, BP 1173, 3038 Sfax, Tunisia

a r t i c l e i n f o

abstract

Article history: Received 20 June 2016 Received in revised form 28 November 2016 Accepted 19 December 2016

Understanding the occurrence of bearing defects in electrical current signals using Motor Current Signal Analysis (MCSA) requires the implementation of numerical models. In this paper, an electro-magnetic-mechanical model is proposed to describe the dynamic behavior of a squirrel cage induction motor coupled to a rotating shaft supported by elastic foundations. The aim of this research work is to gain understanding of the interaction between multiphysics subsystems, mainly in faulty cases, to decipher the transfer path from the defect to its manifestation in stator currents. A new method of writing dynamic equations for simplified simulations of an induction motor is developed using an angular approach. In addition to its capacity to extend the modeling to non-stationary operating conditions, the model proposed highlights the angular periodicity of the rotating motor’s geometry. The electromagnetic field of the motor is redistributed periodically when a geometric defect occurs on a rotating part of the global system. In this case, the electromagnetic torque of the induction motor may present angularly-periodic variations. After having presented the electromagnetic-mechanical coupling methodology, the influence of torque variations is investigated and the importance of the angle-time function is highlighted. & 2017 Elsevier Ltd All rights reserved.

Keywords: Angular approach rotating-machinery diagnosis Motor Current Signal Analysis (MCSA) electromagnetic-mechanical modeling Permeance Network Model (PNM)

1. Introduction Many industrial activities are faced with the challenge of ensuring effective monitoring to reduce machine downtime and improve operation under optimal conditions. In the case of rotating machinery, especially induction motors, the presence of a defect on a mobile part of the system generates disturbances associated with its characteristic frequencies (e.g. bearings). These disturbances may be caused by electric, magnetic or mechanical faults and are detectable on measured signals (acoustic signals, accelerations, rotation speed, electrical signals, etc.) [1]. Investigating these signals generally provides information about the defect. Thus numerous research works have focused on monitoring induction motors using techniques based on signal processing tools to identify the presence of a defect by detecting characteristic fault frequencies [2– 5]. In spite of many available techniques for monitoring motors, industrial sectors are still faced with unexpected faults in

n

Corresponding author at: INSA de Lyon – LaMCoS, Bâtiment Jean d'Alembert 20 rue des Sciences, 69621 Villeurbanne cedex, France. E-mail addresses: [email protected] (A. Fourati), [email protected] (A. Bourdon), [email protected] (N. Feki), [email protected] (D. Rémond), [email protected] (F. Chaari), [email protected] (M. Haddar). http://dx.doi.org/10.1016/j.jsv.2016.12.031 0022-460X/& 2017 Elsevier Ltd All rights reserved.

Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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Nomenclature

P ϕ Fm F ε nph ns nr θ θij

Id fs fsh fd

permeance magnetic flux magnetomotive force electromotive force Magnetomotive potential number of phases number of stator teeth number of rotor teeth rotor angular displacement relative to the stator jth rotor tooth angular displacement referring to the i th stator tooth identity matrix stator current fundamental frequency slot harmonic frequency defect characteristic frequency

Subscripts

( )s ( )r ( )t ( )sy ( )st ( )sl ( )rl ( )rt ( )ry ( )br ( )n ( )a ( )i ( )j

stator rotor tooth stator yoke stator tooth stator leakage rotor leakage rotor tooth rotor yoke branches nodal quantity active branch stator node where i ¼ 1 to ns stator node where j ¼ 1 to nr

electric motors and the resulting decrease in their predicted lifetime. Indeed, although these techniques provide useful information, they do not allow deciphering the occurrence of phenomena in the signals monitored. Several of these investigations proposed the use of the stator current spectrum to detect mechanical defects [6–8]. More intrusive models of motors are required to obtain a precise description of the electromagnetic behavior of the machine and decipher the transfer path from the mechanical defect to its manifestation in electrical signals. Several methods have been proposed in the literature to build models of motors. A simplified model based on a d–q representation of an asynchronous motor was proposed [9]. However, the d–q model is clearly inadequate when attempting to account for the effects of machine geometry, and more refined and realistic modeling appears necessary. Thus sophisticated models based on finite element modeling can be found [10]. Although these methods offer good precision on the motor dynamics, they require excessive computation time, especially when analyzing the couplings between realistic bearing failures at the early stage of development (bearing pitting) and induction motors. Relatively simplified models of induction motors can be found in the literature in which the machine is discretized as a finite number of nodes, such as in the multiple coupled circuit approach [11] and the Permeance Network Model (PNM) [12]. The PNM requires a limited number of nodes while offering sufficient precision for describing electromagnetic phenomena occurring during operation. An analytical model of a squirrel cage induction motor based on permeance network modeling was proposed in [13]. The model developed was coupled to a geared mechanical system and proved its efficiency for detecting gear defects using MCSA. This model also provided a satisfactory description of the motor’s electromagnetic behavior. Indeed, it was accurate enough to account for the effects of machine slotting and periodicities, rotor eccentricity, and air-gap variations which can be disturbed by the presence of mechanical faults. The model was well formulated but its performance was limited to stationary operating conditions and had to be extended to less energetic faults like bearing faults. With the ultimate goal of identifying the transfer path from bearing defects to electrical signals, it was demonstrated in [6,14] that a localized bearing defect induces torque variations. In fact, torque oscillations exist naturally in a healthy motor due to the angular variations of the air-gap field. However, bearing defects induce additional torque perturbations which are present at particular frequencies related to the geometry of the global rotating system and the rotational speed of the motor. Therefore mechanical-speed oscillations due to load-torque variations occur [15]. These variations are very weak and must be distinguished from macroscopic velocity variations representative of non-stationary operating conditions. Generally, rotating machinery presents periodic geometries in the angular domain. These geometries define characteristic frequencies and govern the kinematic relationships between rotation speeds of the machine’s technological elements. These frequencies are homogenous with respect to the number of events per revolution of the reference element, generally the shaft of the machine. Taking this into account, it therefore appears natural to express the model equations of the rotating machine in terms of the angular displacement of the shaft. In this context, the angular approach has been used increasingly for rotating machines, incorporating technological elements with discrete geometry such as bearings, gears, synchronous belts, etc. This approach involves two main features simultaneously: firstly, angular sampling, and, secondly, Instantaneous Angular Speed (IAS) measurement. The latter has recently emerged as a sensitive source of information for monitoring the mechanical parts of rotating machines [16]. Its sensitivity for detecting mechanical defects, such as bearing faults, was proven theoretically and with experimental measurements in [17,18]. Nevertheless, few papers have dealt with the application of angular approaches when writing classical equations of motion in the angular domain, to take into account two advantages: the angular periodicity of a rotating system, even under non-stationary conditions, and the cyclic characteristic fault frequencies of the rotating elements which are independent from the IAS [17]. In addition to the ability to represent Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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torque and IAS variations produced by a bearing defect, that fact that induction motors present discrete angularly-periodic geometries is a good reason for applying the angular approach to motor modeling. In this paper, an electromagnetic-mechanical model of a squirrel cage induction motor and a rotating shaft using an angular description is presented. It is important to mention that the notation ‘rotor’ used in this work is a purely electrical denomination used to designate the rotating part of the induction motor and that it differs from the theory of rotor dynamics. First of all, Section 2 provides a general representation of the multiphysics system. In Section 3, the theoretical aspects are presented and steps for calculating the magnetic flux derivative are introduced. Section 4 proposes an analytical development of the non-linear model of the induction motor associated with the angular approach and the time-angle relation used for switching from time to angle domains is highlighted in particular. Also, the non-linear distribution of the air-gap permeances as the main source of angular periodicity of the model is described and the advantages of angular description are underlined. Afterwards, the coupling methodology is presented in detail in Section 5. A perturbation function of an angularly-varying torque is introduced in Section 6, and it is chosen as the rational torque variation induced by a localized bearing defect, as given in [17]. Simulations are performed for a healthy machine and a torque-perturbed one, the torque perturbation being representative of a bearing defect. The main objective of these simulations is to determine the interaction between the main characteristic frequencies from electromagnetic behavior and those from bearing defects.

2. General electro-magnetic-mechanical model The multiphysics system of an induction motor and a rotating shaft is represented within three physical domains and two subsystems: an electromagnetic subsystem of the induction motor and a mechanical subsystem of the rotating shaft. The dynamic behavior of the coupled electromagnetic-mechanical model is expressed by a set of differential equations given by [13]

⎧ dQ( t ) ⎛ 0 ⎛ 0 ⎞ −I d ⎞ ⎪ +⎜ ⎟⋅Q( t ) = ⎜ −1⎟ Tem( t ) + Fext( t ) ⎪ dt ⎝M ⎠ ⎝ M−1K M−1C ⎠ ⎨ ⎪ dI( t ) dϕ( t ) + R⋅I( t ) + = V( t ) ⎪ L⋅ ⎩ dt dt

(

) (1)

The first equation represents the dynamic behavior of the mechanical subsystem. M , K and C are the constant matrices of mass, stiffness and damping, respectively. Fext( t ) and Tem( t ) represent the vectors of external forces and the electromagnetic torque, respectively, generated by the induction motor and applied on the shaft to ensure its rotation. ⎛ (t)⎞ The state variables are gathered in Q( t) = ⎜ ⎟ where (t) is the generalized displacement vector of the rotating shaft. ⎝ ̇ (t)⎠ As the shaft is generally modeled by a discrete representation, (t) represents the displacement of the degrees of freedom (DOF) on its different nodes. The second equation represents the dynamic behavior of the induction motor. A general representation of an induction motor is given and its dynamic behavior is described by stator-phases and rotor currents. The stator electrical circuit is characterized by stator resistances Rs and leakage inductances Ls . The electrical circuit of the rotor is defined by considering electrical parameters, where Rr , Rb , L r and Lb are ring resistance bar-to-bar, rotor bar resistance, ring leakage inductance barto-bar, and rotor bar leakage inductance, respectively. This model is resolved by considering basic electrical laws, essentially those of Kirchhoff and Lenz. This representation is described in detail in [13]. Vector I( t ) is the generalized vector of stator-phases and rotor currents, V( t ) is the stator-phases and rotor voltage vector d and dt ϕ( t ) is the derivative of the stator-phases and rotor magnetic flow vector versus time variable t . L and R are the matrices of inductances and resistances, respectively, defined as

⎛ Ls 0 ⎞ L=⎜ ⎟ ⎝ 0 L r⎠

⎛ Rs 0 ⎞ R=⎜ ⎟ ⎝ 0 Rr⎠ where L s and R s are the (nph , nph) diagonal matrices of the stator leakage inductances and stator resistances, respectively. L r and R r are the (nr , nr ) matrices of the rotor ring segment leakage inductances and bar and ring segment resistances, respectively. Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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⎛ 2 R r + 2 Rb −Rb ⎞ 0 … 0 −Rb ⎜ ⎟ 0 −Rb ⎜ ⎟ ⎜ ⎟ 0 ⋱ ⋮ ⎟ Rr = ⎜ ⎜ ⎟ 0 ⋮ ⎜ ⎟ 0 − R b ⎜ ⎟ ⎜ ⎟ 0 … 0 −Rb 2 R r + 2 Rb ⎠ −Rb ⎝ and

⎛ 2 L r + 2 Lb −Lb ⎞ 0 … 0 −Lb ⎜ ⎟ 0 −Lb ⎜ ⎟ ⎜ ⎟ 0 ⋱ ⋮ ⎟ Lr = ⎜ ⎜ ⎟ 0 ⋮ ⎜ ⎟ 0 −Lb ⎜ ⎟ ⎜ ⎟ 0 … 0 −Lb 2 L r + 2 Lb ⎠ −Lb ⎝ As this research work does not focus on electrical failures in the induction motor (such as rotor bar breaking, short circuiting of stator winding, etc.), it is important to note that the values of Rs Rr , Rb , Ls , L r and Lb are taken as constant and uniformly distributed for the different elementary stator and rotor components. However, the model can easily consider such non uniform phenomena. Further research work may be performed using this model of an induction motor to solve such problems.

3. Resolution requirements In an induction motor, the motion of the rotor is generated by forces produced between the stator and the rotor surfaces. These magnetic forces are the result of electromagnetic interactions between the air-gap magnetic flux produced by the multiphase stator winding and the induced rotor currents, and they are proportional to the magnetic flux density in the airgap. When faults related to the stator windings, air-gap variations and rotor occur, they alter the normal air-gap flux waveform. Thus quantities which are functions of the air-gap flux will be modified and information related to defects will be concentrated in these quantities [19]. Indeed, all monitoring strategies are related to flux wave variations, as the expression of the magnetic flux waveform contains frequency components characteristic of the motor’s electromagnetic interactions as well as the frequency related to the defects. This consideration means that the magnetic flux in the air-gap is a complex quantity and includes different multi-frequency components such as:

   

Stator and rotor MagnetoMotive Force (MMF) harmonics, Air-gap permeance harmonics, Fundamental electrical current harmonics, Defect harmonics.

The air-gap flux waveform can be represented by dummy air-gap magnetic tubes characterized by permeances. These air-gap permeances are angularly-periodic. As shown in Fig. 1, angular dependencies can be established between different combinations of each adjacent pair of stator and rotor teeth relative to the angular displacement of the rotor, enabling the definition of characteristic frequencies relative to the periodicity of the induction motor’s geometry. In this research work, we focus on the modulations of the harmonics produced by these angular periodicities, especially when combined with electrical current harmonics and defect periodicities. Hence an appropriate representation of the motor’s behavior requires a detailed description of the magnetic flux waveform expression. The intrusive development in this study is presented by the following steps:

 Step 1: Expressing the magnetic flow vector of the stator-phases relative to the flux in the stator teeth. This step is 



performed by defining a constant transfer matrix. Generating this matrix is essentially based on describing the flux path across the different teeth and coils. The specific characteristics of this matrix are described in Section 4.1. Step 2: Establishing a relation between the flux and MMF. The definition of this relation is based on the rotating wave expression whereby the magnetic flux waves in the air-gap are taken as the product of the air-gap permeances and MMF. Accordingly, the rotating parameters dependent on the angular position of the rotor during motor operation will be included in the computation. This development will be described in detail in Section 4.2. Step 3: Determining the link between the stator and rotor MMF and the stator and rotor current by defining a constant impedance matrix. This relation is obtained by considering electric-magnetic interaction. The type of stator winding is also taken into account to generate the stator-phase current from the currents in the stator teeth. The relation is defined in Section 4.3. Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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Fig. 1. Time and angular periodicities of adjacent stator and rotor teeth, stator per-phase current and defect perturbations for three different operating regimes.

The three steps presented previously will be combined to establish a relation between electric and magnetic quantities in the induction motor. This combination leads to the definition of the matrix J( θ ) linking the expression of the flux and current as follows

(

)

ϕ( t ) = J θ ( t ) ⋅I( t )

(2)

where matrix J( θ(t )) depends on the instantaneous angular position of the shaft. Its expression will be described later in the paper. We assume that the flux waveform depends on the angular displacement of the rotor. The expression of the flux is integrated in Eq. (1), and the resolution methodology leads us to derivate it versus the time variable. This situation imposes the generation of a time variable to derivate the flux wave. This definition can only be formulated by considering the condition in Eq. (3) which limits the investigations to a constant rotor rotation speed.

θ (t ) = ωt where ω is constant

(3)

As said previously, instantaneous torque fluctuations exist naturally in the electromagnetic torque produced by the induction motor. This reality induces IAS oscillations even for a machine operating under stationary conditions. It also involves estimating an approximate value of the flux derivative versus time for each elementary time interval in a time representation which imposes a constant rotation speed, in order to perform the resolution. Hence, in addition to the requirement of an accurate expression of the magnetic flux, the challenge is to express the flux derivative according to quantitative values. In this research work, we conserve the angular description of the stator vectors to avoid verifying the resolution restriction and thus contribute to the refinement of the PNM proposed, and at the same time, extend the model Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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for investigations under non-stationary conditions. Furthermore, Fig. 1 presents the time and the angle distributions of the air-gap permeances of an example of a single pair of teeth chosen arbitrarily from the stator and rotor teeth and its adjacent pairs of stator and rotor teeth. They are presented for three operating regimes illustrated by three IAS levels in order to emphasize the periodicity characteristic of this magnetic parameter. The angular periodicity of the air-gap permeances is kept only in the angular representation. Since the air-gap permeances depend only on the angular gap between the pair of teeth, in other terms on the angular position of the rotor during the movement of the motor, the angular periodicity of this parameter is kept for the different operating regimes, even under non-stationary conditions. However, while considering the time representation, the periodicity characteristic is hidden by non-stationary conditions. The same conclusions are shown for the perturbation induced by a bearing defect, which is not the case for the electrical fundamental frequency characterized by a constant in time frequency whatever the operating conditions. In this research work, we focus on this diversity of frequencies which represent time and angle periodicities simultaneously. The objective is to highlight the modulations induced by time and angle interactions. At the same time, we attempt to identify the interaction between those values representative of the multiphysics system in order to obtain a fluent description of the system’s dynamics and understand the occurrence of phenomena when the motor is operating. The main contribution of this study is the investigation of the flux distribution within the induction motor during shaft rotation. The objective is to understand the multiphysics interactions in order to dissociate angularly varying components from time dependent ones and distinguish periodicity modulations hidden by multiphysics interactions. This development improves understanding of the induction motor’s electromagnetic interactions. In particular, in the case of interaction with rotating mechanical components, these modulations interfere with the angular periodicities generated by the mechanical subsystem, mainly in the case of faults where the latter are expressed by an angularly-periodic impact. The development presented in this paper contributes to better understanding of the effect of varying excitations produced by a defective bearing on the resulting stator current, especially under non-stationary operating conditions. An angular approach in the induction motor description provides a complementary view of the nonlinear dynamic model.

4. Nonlinear model of the induction motor We distinguish three structural parts in the geometry of the induction motor: the stator, the rotor and the air-gap. Although we are dealing with a squirrel cage induction motor, we consider that the stator and the rotor geometries are formed by a yoke and a series of teeth and slots uniformly distributed, as presented schematically in Fig. 2. The model is built by considering the following assumptions: (a) the air-gap length variations around the stator and rotor circumferences are neglected, (b) the saturation of the magnetic circuit is neglected. 4.1. Step 1: Flux distribution in the stator and rotor We consider that nc conductors pass through each stator slot forming a single layer of diametrically distributed winding, as schematized in Fig. 2. The stator includes nph phases and ns /2 coils. For each coil, the flux field passes through a 2ns /nph stator slot. W is the transform matrix defined to enable the transition from stator per-phase fluxes to stator teeth fluxes. Referring to [20], the winding transform matrix depends on the topology of the stator winding, essentially from:

 the position of the coils in the stator slots,  the number of turns per coil,

Fig. 2. Axial view of the induction machine.

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 the method of connecting the coils to coil groups,  the method of connecting the coil groups to elementary phases,  coil orientation. The matrix W is defined in two steps by multiplying two matrices:

 The matrix of transition from stator-phase fluxes to stator-coils fluxes Wp → c  The matrix of transition from stator-coils fluxes to stator-teeth fluxes Wc → t Combining the two transfer matrices, W is given by

W = Wp → c

ns ph, 2

(n

)⋅Wc→ t( n2 ,n ) s

(4)

s

W verifies the following relation as described in [20]

ϕ nph (nph) = W(nph, ns)⋅ϕs(ns)

(5)

The fluxes in the rotor are considered to be the fluxes passing through the rotor bars. Stator and rotor fluxes are then expressed relative to the fluxes in the branches of the motor magnetic circuit via matrix G defined by

ϕ = G ( nph + nr, nb)⋅ϕ br

(6)

More explicitly,

⎛ 0 W 0( nph, nb −2ns)⎞ ( nph, ns) ⎟ G=⎜ ⎜0 ⎟ I 0 d ( n , n ) ( n , n ) n , n − 2 n r r r r ⎝ ( r b r) ⎠

4.2. Step 2: Flux distribution in stator, in rotor and MMF The induction motor is modeled using the PNM in order to simulate its magnetic behavior. The advantage of this model is its capacity to detect small magnetic disturbances and to provide a detailed representation of the magnetic state of the motor which is sensitive to possible faults.

4.2.1. Discretization of the induction motor The motor is discretized on 2ns stator nodes and 2nr rotor nodes. The stator is uniformly discretized on ns loops composed of two nodes in the yoke and two nodes at the extremity of the teeth, as shown in Fig. 3a. The circuit is linked by four branches that represent the flux circulation. Similarly, the rotor is divided into nr loops formed by two teeth and a part of the rotor yoke, as shown in Fig. 3b. The overall network contains nn nodes uniformly distributed where nn=2ns +2nr . In the complete PNM, there are ne branches in the gap. ne is relative to the stator teeth and the number of rotor bars; ne = ns × nr . The total number of branches in the overall network is nb=3ns +3nr + ne .

Fig. 3. Discretization. (a) Stator and (b) rotor.

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Fig. 4. Part of the permeance model of the induction motor.

4.2.2. The PNM The induction motor proposed is discretized into a finite number of dummy flux tubes, as shown in Fig. 4. We distinguish three main parts: stator magnetic circuit, air-gap dummy flux tubes and rotor magnetic circuit. Each tube is characterized by its permeance. Permeances representing the stator and rotor yoke, and teeth and leakage are constant. They are calculated as a function of the geometrical parameters of the stator and the rotor, while the air gap permeances are defined as a set of permeances connecting a pair of teeth, i.e. a stator tooth to a rotor tooth. During the rotor motion, the geometry of the flux tubes varies which induces variations in air-gap permeance values. The values of the stator and rotor permeances are estimated by neglecting the flux field variations across a tube. In this case, the analytical expressions of the permeances are defined as follows

Pk =

μS L

(7)

where μ , S and L are the permeability, cross section area and length of the tube, respectively. Pys , Pts, Pls, Plr , Ptr and Pyr are the constant stator and rotor permeances, respectively. 4.2.3. Angular periodic air-gap permeances The angular positions of rotor nodes vary during shaft rotation. As shown in Fig. 5, the angular gap between a pair of teeth is expressed as a function of the mechanical angle θ relative to the shaft rotation through the relation

θij(t ) = θ (t )+θij0

(8)

where θij0 is the initial angular gap between the pair of teeth. This variation induces flux tubes in the air-gap with geometries that are nonlinear functions of the rotor/stator displacement angle [20]. Since the flux tubes in the air-gap have variable dimensions, their permeances are parametrically nonlinear. A maximum flux passes whenever the pair of teeth is directly in front and the permeance value is equal to Pmax (case 1 in Fig. 6, point 1 in Fig. 7). The permeance value remains maximal until reaching the limit angle θt1(case 2, point 2). As the rotor moves, the angular gap between the two teeth increases, and the rate of flux moving via the air-gap flux tubes decreases, expressed by a decrease in the permeance value (case 3, point 3). Obviously no flux will flow from the stator tooth to the rotor tooth when reaching the limit value θt (case 4, point 4). Above this value, the air-gap flux is equal to zero and thus a zero permeance value (case 5, point 5). The limit angle values are calculated as a function of the geometrical characteristics of the induction motor. In the case of non-skewed rotor bars, these values are written as follows

θt =

L ts + L ss + L tr + L sr Dag

(9)

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Fig. 5. Evolution of the angular gap between a pair of teeth as a function of the angular position of the shaft. (a) Initial position of the shaft (b) During the rotation of the shaft.

and

θt1 =

L ts − L tr Dag

(10)

where Dag is the mean diameter of the induction motor. Lts , Lss , Ltr and Lsr are the stator tooth width, stator slot width, rotor tooth width and rotor slot width, respectively. In the case of skewed rotor bars, these angles are calculated relative to the geometrical relations described in [20]. Considering that all the parameters of the electromagnetic model, except air-gap permeances, are constants, and regarding Fig. 6 and Fig. 7, one major characteristic of the model can be emphasized: the dynamic behavior of the induction motor is controlled by the angular position of the shaft whatever the global speed of the system whether stationary or not. In this instance, we define the angular displacement of the shaft as the governing variable for angular modeling. For each pair of teeth, the value of the permeance is an analytical function of the mechanical angle. This value is updated for every rotor position relative to the formula [20]

⎧ ⎪ ⎪ Pmax ⎪ ⎪ ⎪ θ−θ ⎪ 1 + cos π θ − θt1 t t1 Pij( θ ) = ⎨ Pmax ⎪ 2 ⎪ θ − 2π + θ 1 + cos π θ − θ t1 ⎪ t t1 ⎪ Pmax 2 ⎪ ⎪ 0 ⎩

−θij0 ≤ θ ≤ θt1 − θij0 if

and 2π − θt1 − θij0 ≤ θ ≤ 2π − θij0

if

θt1 − θij0 ≤ θ ≤ θt − θij0

if 2π − θt − θij0 ≤ θ ≤ 2π − θt1 − θij0 if

θt − θij0 ≤ θ ≤ 2π − θt − θij0

(11)

The expression of Pmax is defined as

Pmax =

μ 0 L mL tr e

(12)

where μ0 is the air-gap permeability, L m is the machine length and e is the air-gap thickness. 4.2.4. Elementary cell: electric-magnetic interaction Induction motors present electric and magnetic behaviors simultaneously. The basis of any reliable model of induction motors relies on considering these characteristics simultaneously while ensuring interactions between them. An elementary cell of the stator and rotor is depicted in Fig. 8. The cell is composed of active and passive branches. The actives branches represent MMF. where k is the branch index. ∅k , Pk and εk are the flux, permeance and magnetic potential difference, respectively, characterizing the branch. Ft , k is the MMF source for the k th tooth, Ik is the phase current across the slot and nc, k Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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Case 1:

=0

Case 2:

Case 3:

Case 4:

Case 5: and

Fig. 6. Evolution of flow tubes as a function of the angular position of the shaft for θ ij0 ¼ 0.

is the number of conductors inside the corresponding slot. For each branch of the cell, the flux is calculated via the relation

∅k = Pk. ( εk − Fk )

(13)

Since the branches in the air gap are considered as passive, the fluxes in the air gap are given by the relation

∅ij = Pij( θ ). εij

(14)

Considering the relations developed in Eq. (13) and Eq. (14) for the different branches of the network, an equation in the compact matrix form can be formulated as follows

= P br( θ )(n

ϕ br (nb)

b, nb)

⋅( ε br (nb) − Fbr (nb))

(15)

The resolution of the PNM is based on the nodal representation of physical quantities. The orientation of the flux direction inside the network is organized by formulating a connection matrix. In this approach we consider that the matrix is constant in terms of size and coefficients even if there is no flux tube between a pair of teeth in the air gap. By defining a referential node, we define a (nn −1, nb) reduced connection matrix. τ transforms MMF in the branches to nodal values as follows

ε br (nb) = τt ⋅εn (nn −1)

(16)

Where the nodal difference of the magnetic potential matrix is defined by −1

εn = Pn( θ ) ⋅ϕ n (nn −1)

(17)

Following the flux directions in the network branches, the nodal permeance matrix is established by

Pn( θ )(n

n −1, nn −1)

= τ⋅P br( θ )(n

b, nb)

· τt

(18)

In other terms, the matrix can be written as

⎛ P11n P12n ⎞ 0 0 ⎜ ⎟ ⎜ P21n P22n( θ ) P23n( θ ) 0 ⎟ ⎟ Pn( θ ) = ⎜ ⎜ 0 P32n( θ ) P33n( θ ) P34n ⎟ ⎜⎜ ⎟⎟ P43n P44n ⎠ 0 ⎝ 0

(19)

The matrix combining the network permeances is a set of constant matrices and angularly varying matrices resulting from a dynamic model composed of constant and angularly varying permeances. Each component of the varying matrices is characterized by a non-linear 2π periodic function versus the angle. The elementary matrices are described in detail in Appendix A. In order to define the nodal magnetic flux vector, we consider the direction of the flux and the contribution of active branches in each node. The magnetic flux in the active branches is calculated as a function of the permeances in the active branches. Therefore we define the matrix of the active branch permeances Pan as

ϕn = Pan (nn −1, ns + nr)⋅Fn

(20)

By combining Eqs. (16), (17) and (20), we obtain Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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Fig. 7. Evolution of the non-linear air-gap permeance versus the rotation angle for a pair of teeth.

Fig. 8. Elementary cell of the stator and the rotor.

−1

ε br = τt ⋅Pn( θ ) ⋅Pan⋅Fn

(21)

The nodal MMF vector is calculated by setting a transform matrix τf that relates branch values to nodal values such that

Fbr = τf (nb, ns + nr)⋅Fn (ns + nr)

(22)

By substituting Eqs. (21) and (22) in Eq. (15), the expression relating the fluxes of the branches and nodal MMF is presented by defining the matrix X( θ ) as follows

ϕ br = X( θ )(n

b, ns + nr )

⋅Fn

(23)

where

(

−1

X( θ ) = P br( θ )⋅ τt ⋅Pn( θ ) ⋅Pan − τf

)

(24)

4.3. Step 3: MMF and currents in the stator and rotor The objective of this step is to establish matrix relations between MMF and currents in the stator and the rotor. This objective is obtained by combining two sub-steps in which the currents in the stator and rotor teeth are defined as an intermediate quantity. Matrices τs1 and τr1 are defined in order to link MMF to currents in the stator and rotor teeth. Then, matrices τs2 and τr2 link the stator teeth currents to the stator phase currents and the rotor teeth currents to the rotor bar currents, respectively. With reference to the electro-magnetic interaction law applied to the elementary cell defined in Fig. 8, the general relation setting the interaction between MMF and electrical currents is established as:

Ft, k − Ft, k +1 = −nc, k Ik

(25)

By generalizing Eq. (25) for the different cells of the stator and the rotor, MMF and currents are related in a compact matrix form by the following system:

⎧ τs1(ns, ns)⋅Fst(ns) =nc . Ist(ns) ⎨ ⎩ τr1(nr, nr)⋅Frt(nr) = Irt(nr) ⎪

⎪

(26)

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Matrix τs2 is defined to describe the current circulating through the stator teeth. The manner in which the stator winding is performed affects the resulting stator-phase current. Therefore, the matrix is defined by considering the type of winding, the slot number, the phase number, the teeth belonging to each phase and the direction of current circulation. By considering a diametrically distributed stator winding, the current distribution in the stator is defined as:

Ist = τs2(ns, nph)⋅Is(nph)

(27)

By applying the classical Kirchhoff current law to the different nodes of the rotor, as shown in Fig. 9, the relation between the currents of the rotor bars and the currents of the bar-to-bar rings is obtained by setting an orientation convention such that

Irb, k = Irr, k − Irr, k −1

(28)

The rotor structure is composed of nr bars and nr ring segments, as shown in Fig. 9. The rotor current distribution is deduced by applying Kirchhoff’s law. Relative to Eq. (28), we define matrix τr2 as

Irt = τr2(nr, nr)⋅Ir (nr)

(29)

By combining Eqs. (26), (27) and (29), we obtain

Fn = Z⋅I

(30)

⎛ nc τs1−1τs2 0 ⎞ ⎟⎟ Z (ns + nr, nph + nr) = ⎜⎜ 0 τ r1−1τ r2 ⎠ ⎝ ⎛ Is ⎞ I(nph + nr) = ⎜ ⎟ ⎝ Ir ⎠

(31)

where

4.4. Flux expression The relation between the fluxes and currents in the induction motor is obtained by combining Eqs. (6), (23) and (30). The expression is written by setting the matrix J( θ ) as

ϕ = J( θ )(n

ph + nr , nph + nr )

⋅I

(32)

where J( θ ) depends on the relative angular position of the rotor. It is expressed as follows: −1

J( θ ) = S1⋅Pn( θ ) ⋅S2 − S 3

(33)

It should be noted that S1, S2 and S3 are constant matrices obtained by the multiplication of angularly periodic matrices and constant matrices given by

S1 = G⋅P br( θ )⋅τt S2 = Pan⋅Z

(34) (35)

S 3 = G⋅P br( θ )⋅τf ⋅Z

(36)

4.5. Calculating the flux derivative: angular resolution of the electromagnetic equation As shown in Section 2, the resolution of the differential system involves computing the derivative of the flux expression developed in Eq. (32) versus the time variable. As matrix J( θ ) is explicitly defined relative to the angular variable, we can write the following relation mathematically:

d J( θ ) dϕ(t ) dI(t ) = ⋅I(t )+J( θ )⋅ dt dt dt

(37)

The equation shows that the resolution involves both angular and time variables simultaneously. Regarding the fundamental time–angle relation, two bijective functions are defined in order to switch from time to angle domains and vice versa:

θ = φ( t ) ⟺

t = ψ ( θ)

(38)

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Fig. 9. Rotor current orientation convention.

This condition is mathematically possible only if φ is strictly monotonic, so the rotation speed must be different from zero.

ω( t ) =

dφ ( t ) dθ = dt dt

(39)

Accordingly, the derivate can be simply deduced by considering the following relations:

d( ) d( ) d( ) d( ) = = ⋅ω( t ) = ⋅ω( θ ) dt dθ dθ dψ ( θ )

(40)

These relations allow writing the expression of the flux derivate either with time or angular representation. In a time description, the relation expressed in Eq. (37) becomes

dJ ( θ ) dϕ(t ) dI(t ) = ⋅ω( t )⋅I(t )+J( θ )⋅ dt dθ dt

(41)

In the angular description, the flux derivative written in Eq. (37) can be developed as follows:

d J( θ ) dϕ(t ) dI(ψ (θ )) = ⋅ω( θ )⋅I(ψ ( θ ))+J( θ )⋅ω( θ )⋅ dt dθ dθ

(42)

In this representation, it can be shown explicitly that the angularly periodic matrix has been dissociated from the time variable dependency. Also, in what follows, it can be deduced that the varying rotation speed function can be introduced in the model. The exact value of the derivative from the J( θ ) matrix can be calculated analytically by the derivation of the formulation in Eq. (33), as shown in the following formula:

d J( θ ) dθ where

−1

=−S 1⋅Pn( θ ) ⋅

dPn( θ ) dθ

dPn( θ ) dθ

−1

⋅Pn( θ ) ⋅S 2

(43)

is the derivative of the nodal permeance matrix versus the angle. The matrix is calculated by setting the

derivative of the air-gap permeance function versus the angle. Regarding the expression in Eq. (44), the derivative versus the angle of the air-gap permeances is an angularly periodic function given by:

⎧ ⎪ 0 ⎪ ⎪ ⎪ ⎪ θ−θ sin π θ − θt1 ⎛ ⎪ dPij( θ ) ⎪ 1 π ⎞ t t1 ⎟ = ⎨ − Pmax⎜ 2 2 ⎝ θt − θt1 ⎠ dθ ⎪ ⎪ θ − 2π + θ sin π θ − θ t1 ⎪ ⎛ π ⎞ t t1 ⎪ − 1 Pmax⎜ ⎟ 2 ⎪ 2 ⎝ θt − θt1 ⎠ ⎪ ⎪ 0 ⎩

(

(

−θij0 ≤ θ ≤ θt1 − θij0

)

)

if

and 2π − θt1 − θij0 ≤ θ ≤ 2π − θij0

if

θt1 − θij0 ≤ θ ≤ θt − θij0

if

2π − θt − θij0 ≤ θ ≤ 2π − θt1

if

θt − θij0 ≤ θ ≤ 2π − θt − θij0

(44)

As the nodal permeance matrix is defined in Eq. (18) as the product of three matrices, we can write its derivative as:

dPn( θ ) dθ

= −τ⋅P̃ br( θ )⋅τt

(45)

Where P̃ br( θ ) the derivate matrix of P br( θ ) versus the angle. Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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The substitution of the expression developed in Eq. (42) in the differential equation of the electromagnetic behavior of the induction motor represented in Eq. (1) results in a differential system that considers the angle-time relation:

⎧ ⎞ ⎛ dI t dJ θ ⎪ L + J( θ ) ⋅ω( θ )⋅ ( ) + ⎜⎜ R + ( ) ⋅ω( θ )⎟⎟⋅I( t ) = V( t ) ⎪ dθ dθ ⎠ ⎝ ⎨ ⎪ dt 1 = ⎪ ω( θ ) ⎩ dθ

(

)

(46)

5. Methodology of electromagnetic-mechanical coupling The mechanical part of the multiphysics coupling is represented by the rotating shaft and supports. The shaft is modeled while neglecting gyroscopic effects. It is discretized on three nodes. The dynamic behavior of the mechanical system is then accounted for by three nodes with 6 DOF per node. The shaft is modeled by beam elements that take into account the elastic effects in all directions and, potentially, the couplings between the DOF. The support is modeled by punctual stiffness and damping elements. The shaft is connected to the support by connecting forces in the end nodes, as shown in Fig. 10. Matrices M and K are composed of the assembly of the structural elements composed of the shaft and the support. The structural damping is introduced using a modal damping approach. The damping of the rigid modes is added as a damping elementary matrix between the shaft ending and support nodes. We denote the total number of DOF as nm . The mechanical state vector Q contains the displacements and velocity of all the DOF as follows: t

Q (2nm) = ( u1…uk …unu1̇ …u̇k …u̇n)

The electro-magnetic torque is the quantity that ensures the link between the electromagnetic and mechanical models. The torque produced by the motor is expressed by: ns

Tem( θ ) =

nr

∑∑ i

j

dPij( θ ) dθ

εij2

(47)

The rotation of the shaft is ensured by the electromagnetic torque provided by the induction motor applied on its central node. Indeed, the model of the induction motor presented previously is a 2D model with a rigid rotor. The model takes into account the skewing angle of the rotor bars in the axial direction. The effects due to electro-magnetic phenomena are merged at a single central node of the shaft. The objective of the electromagnetic-mechanical coupling is to obtain a single differential system combining electrical and mechanical state variables. It consists in resolving electromagnetic and mechanical differential equations simultaneously while conserving the angle-time relation. The general differential system is defined by combining the previous differential equations. The equations can be written in the state form like Eq. (48). The advantage of this integration methodology is to ensure direct convergence for all the state variables. In order to take into account the dynamic variations, the relative rotor angular position, the electromagnetic torque, the instantaneous angular speed, and the rotor eccentricity are updated at every step. The magnetic, electric and mechanic models are solved at each iteration, while ensuring input/ output relationships between subsystems, as shown in Fig. 11.

⎛ dI ⎞ ⎜ ⎟ ⎜ dθ ⎟ ⎛ I⎞ ⎜ dQ ⎟ ⎜ ⎟ A = θ ⋅ ( ) ⎜ ⎟ ⎜ Q ⎟ + B( θ , t ) d θ ⎜ ⎟ ⎝t⎠ ⎜ dt ⎟ ⎜ ⎟ ⎝ dθ ⎠ ⎡ ⎡ −1 ⎛ ⎢ − 1 ⋅⎢ L + J( θ ) ⋅⎜ R + ⎝ ⎢ ω( θ ) ⎣ ⎢ where A( θ ) = ⎢ 0 ⎢ ⎢ ⎣ 0

(

)

(48) d J( θ ) dθ

⎞⎤ ω( θ )⎟⎥ ⎠⎦

⎤ 0⎥ ⎥ ⎥ and ⎡ ⎤ 0 I 1 d − ω θ ⋅⎢ ⎥ 0⎥ ( ) ⎣ −M−1⋅K −M−1⋅C ⎦ ⎥ ⎥ 0 0⎦ 0

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Fig. 10. Schematic representation of the electro-magnetic-mechanical model.

⎛ −1 1 ⎜ ω( θ) ⋅ L + J( θ ) ⋅V( t ) ⎜ ⎡ 0 ⎤ ⎜ B( θ , t ) = ⎜ 1 ⋅⎢ ⎥⋅ Tr + Tem( θ ) ω( θ ) ⎣ M−1⎦ ⎜ 1 ⎜⎜ ω( θ ) ⎝

(

)

(

)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎠

The electromagnetic model of the induction motor is fed by the instantaneous angular speed, and the rotor eccentricity is obtained referring to the generalized displacement vector and the power supply as a function of time. The electromagnetic torque is computed by post-processing the current outputs for application to the shaft in addition to external forces that may depend on the time or/and angle variable. Time, rotation speed and the generalized displacement vector are obtained as outputs of the mechanical subsystem.

6. Influence of torque perturbations The disturbance considered in this work must be representative of a bearing defect. Based on the fact that a bearing defect induces an angularly-varying resistant torque, it seems interesting to summarize the dynamic behavior of a bearing defect by considering a perturbation function of the resistant torque [16]. The angularly periodic defect is characterized by its angular period θo , its angular length L θ and its maximal amplitude Aθ . The latter is assumed to be small and representative of a localized disturbance. Fig. 12a is used to display the disturbance modeling parameters. Each occurrence can be divided into three zones defined by angles θa , θb , θc , θd , as shown in Fig. 12b. In each zone, the expression of the disturbance is a function of the rotor angular position and of the characteristics of the disturbance [16]

( ( ) ( ( ))) ( ( ) ( ))

 Zone ab: Trp( θ ) = Ap ⋅sin  Zone bc: Trp( θ ) = 0  Zone cd: Trp( θ ) = Ap ⋅sin

π Lθ

⋅ θ−θa +

π Lθ

⋅ θ−θc

Lθ 2

The non-linear dynamic model introduced in the last section was implemented to simulate the electrical response of a 50 kW, 50 Hz, 400 V, 2-pole, 24-stator slot, 30-rotor slot, star-connected, standard squirrel cage induction motor. The parameters introduced for the simulation are detailed in Table B1 in Appendix B. Differential equations were implemented in a Matlab code using the ode15s function based on Backward Differentiation Formulas [21]. A standard PC (CPU 3 GHz and 16 Gb RAM) led to a computation time of approximately 25 min to simulate running of the motor-shaft system for 1 s. All the results presented in this paper are performed for 1480 revolutions. The simulation precision is about 800 points per revolution, a number capable of providing a good description of the perturbation form. Although it is not the main objective of this study to compare time and angle methods of integration, it remains interesting to highlight the importance of the angular representation of electrical machine behavior. This representation emphasizes two realities:

 the angular periodicity of the machine, even under non-stationary conditions, leads to an accurate description of the slot arrangement and thus better control of machine geometry during resolution,

 the considerable gain in computation time, ensured by the fact that angular permeances are explicitly described and Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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Fig. 11. The electro-magnetic-mechanical coupling methodology.

Fig. 12. Definition of the torque perturbation (a) Parameters and (b) Partition.

introduced in the numerical model. Regarding computation time, the simulations performed under the same conditions using the angular approach provide a gain of 72%. In this instance, it was possible to extend the simulations for longer durations. Figs. 13a, b and c illustrate the IAS, the electromagnetic torque and the stator first-phase current versus angular evolution, respectively. These curves are obtained as a result of the electric-magnetic-mechanical interactions in a healthy case. The current curve is governed by the 50 Hz imposed by the power supply. Small variations generated by the multiphysics periodicities exist in the current signal, but they are hidden by the current fundamental frequency. However, they appear clearly in the signals of the IAS and torque in the zoomed views. To illustrate the effects of torque variations under angular sampling conditions, numerical simulations were performed for angularly varying resistant torque. In this case, the torque applied on the rotating shaft is the algebraic sum of the Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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Fig. 13. Dynamic behavior of the multiphysic system in a healthy case. (a) IAS, (b) Electromagnetic torque and (c) First phase stator current.

Fig. 14. Angular FFT of the first phase stator current obtained in healthy case and in the case of resistant torque perturbations. (a) Global view and (b) Zoom view for angular frequencies between 2 and 10 event/revolution.

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Fig. 15. Time FFT of the first phase stator current obtained in healthy case and in the case of resistant torque perturbations. (a) Global view and (b) Zoom view for time frequencies between 100 and 500 Hz.

electromagnetic torque produced by the induction motor, the constant value of the resistant torque and the angularly varying resistant torque. The parameters of the resistant perturbation torque are defined by referring to the perturbation function introduced in Fig. 12. Considering that the bearing defect appears at 4.1 events/revolutions (value representative of the Ball Pass Frequency of Outer ring (BPFO)), the parameters of the disturbance are chosen as follows: θ0 = 42.π1 rad, L θ = 0. 21*θ0rad, Aθ = 1 Nm . The spectra of the first-phase stator current in Fig. 14a and b were obtained without windowing and in a logarithm scale to emphasize the contribution of small amplitude frequencies such as expected for torque perturbation. In addition to the peaks at the current frequency ( fs =1.007event/revolution) and at the principal slotting harmonics ( fsh =30events/revolutions) and its multiples modulated by fs (i.e. 28.993 events/revolutions, 31.007 events/revolutions, 58.993 events/revolutions), the spectra show peaks related to the perturbation frequency ( fd =4.1events/revolutions) and its harmonics modulated by fs : 3.093 events/revolutions, 5.107 events/revolutions, 9.207 events/revolutions for example. The frequencies relative to the slot harmonics can be expressed in analytical form as follows:

(

f = k1. fsh ± k2. fs

)

(49)

with integers k1 and k2. The frequencies relative to the defect can be expressed in a general form, as proposed in [8] as follows:

f = fs ± k3. fd

(50)

with an integer k3. Eq. (49) represents the frequency modulation of the stator current by the rotor slots harmonics. From the physical viewpoint, this can be justified by the fact that the stator and rotor slots present areas in the path of the magnetic flux. When a stator slot passes over a rotor slot, it causes a change in the resistance to flux in that area. This change in resistance to the flux leads to a change in the electric current producing the flux. The change in the current represents a distortion of the current waveform and the distorted waveform is described as being composed of a fundamental component plus a series of harmonic components. By adopting the same reasoning and considering that the defect produces localized variations in the magnetic field in the air-gap of the motor, the modulation between the stator current frequency and the characteristic defect frequency is justified as proposed in Eq. (50). Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i

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As recalled previously, the electromagnetic-mechanical model shown in reference to the angular approach provides with instantaneous information on the IAS and time for each angular step. Thus it seems interesting to represent the spectrum of the stator current versus time even when conserving the angular description of the global system. The objective of this representation is not to compare time and angular results; however, the two results seem to be complementary since the coupled model represent time and angle dependent characteristic frequencies simultaneously. In other terms ● The current frequencyis constant in time representation; it is equal to 50 Hz, while it varies in angular representation. ● The defect frequency and the slot harmonic frequency, which are angularly periodic remain located in constant frequencies in angular representation while they vary in time representation. In Fig. 15a, we represent the spectrum of the first-phase stator current versus time. It was obtained without windowing and on a logarithm scale. The spectrum shows peaks at the current frequency ( fs =50 Hz) and at the principal slotting harmonics ( fsh =1488 Hz) and its multiples modulated by fs . These frequencies are related to Eq. (49) where they are expressed in the time domain. In the zoomed view in Fig. 15b the peaks are related to the perturbation frequency ( fd =203. 5 Hz) and its harmonics modulated by fs : 153.5 Hz and 457 Hz, for example. These frequencies are related to Eq. (50) where the frequencies are expressed in the time domain.

7. Conclusions This paper describes the formulation of an electric-magnetic-mechanical model of a squirrel cage induction motor connected to a rotating shaft. In contrast to classical time approaches, the model proposed is sampled in the angular domain, by defining the time-angle function and a constant angle-step variable, making it possible to extend the model to non-stationary conditions. From the modeling standpoint, applying the angular approach to model the multiphysics system highlighted the angular periodicity of the geometry and led to investigating the frequency modulations produced by the electric-magnetic-mechanical interactions. The separation of angle-dependent parameters from time-dependent ones provided a detailed description of the electric-magnetic coupling and simplified the modeling, as the varying parameters were described with precision. Indeed, the simulations performed on the extended model led to a 72% gain in computation time, thereby allowing the exploration of much longer simulations and smaller perturbations in angular sampled signals. The Fourier analysis showed that the spectrum of the stator current exhibits modulations of frequencies characterizing multiphysics interactions, even without introducing external excitation to the system. The model was tested for torque variations induced by a small localized bearing defect. The results showed the modulation of the bearing defect frequency with characteristic frequencies of the multiphysics system and good detectability of the defect in both time and angular representations. Regarding the validation of the numerical results, an essential aspect is that the modulations exhibited in the dynamic response of the system were those predicted to result from multiphysics interactions. Indeed, the system reacts naturally to frequency modulations, even without introducing external efforts. Further work is necessary to introduce a bearing model that takes into account the bearing defect to investigate the relation between defect geometry and current detectability. Acknowledgments Authors gratefully acknowledge the support of the Rhone-Alpes Regional Council via mobility grant “Acceuil Doc” 13722. Appendix A. Analytical development of the elementary matrices of the nodal permeance matrix

P11n( θ )

(ns , ns )

⎛ 2Psy + Pst −Psy 0 … 0 −Psy ⎞ ⎜ ⎟ ⎜ −Psy ⎟ 0 ⎜ ⎟ 0 ⋱ ⋮ ⎟ = ⎜⎜ ⎟ 0 ⋮ ⎜ ⎟ 0 −Psy ⎟ ⎜ ⎜ ⎟ 0 … 0 −Psy 2Psy + Psy ⎠ ⎝ −Psy

(A.1)

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P22n( θ )(n , n ) s s

P33n( θ )(n , n ) r

r

nr ⎛ ⎜ Pst + 2Psl + ∑ P1, j( θ ) ⎜ j=1 ⎜ ⎜ −Psl ⎜ 0 ⋱ =⎜ ⋮ ⎜ ⎜ 0 ⎜ ⎜ ⎜⎜ −Psl 0 … 0 −Psl ⎝

ns ⎛ ⎜ Prt + 2Prl + ∑ Pi,1( θ ) ⎜ i=1 ⎜ −Prl ⎜ ⎜ 0 ⋱ =⎜ ⋮ ⎜ ⎜ 0 ⎜ ⎜ ⎜ −Prl 0 … 0 −Prl ⎝

P44n( θ )

(nr − 1, nr − 1)

⎞ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⋮ ⎟ 0 ⎟ ⎟ −Pfs ⎟ nr ⎟ Pst + 2Psl + ∑ Pns, j( θ )⎟⎟ j=1 ⎠

−Psl 0 … 0 −Psl

⎞ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⋮ ⎟ 0 ⎟ −Pfr ⎟ ⎟ ns ⎟ Prt + 2Prl + ∑ Pi, nr( θ )⎟ ⎠ i=1

(A.2)

−Prl 0 … 0 −Prl

⎛ 2Pry + Prt −Pry 0 … 0 ⎞ 0 ⎜ ⎟ ⎜ −Pry ⎟ 0 ⎜ ⎟ 0 ⋱ ⋮ ⎟ = ⎜⎜ ⎟ 0 ⋮ ⎜ ⎟ 0 −Pry ⎟ ⎜ ⎜ ⎟ 0 0 … 0 −Pry 2Pry + Prt ⎠ ⎝

(A.3)

(A.4)

⎛ −Pst 0 ⎞ ⎜ ⎟ ⋱ P12n( θ )(n , n ) = ⎜ ⎟ s s −Pst ⎠ ⎝ 0

(A.5)

⎛ −Pst 0 ⎞ ⎜ ⎟ ⋱ P21n( θ )(n , n ) = ⎜ ⎟ s s −Pst ⎠ ⎝ 0

(A.6)

P23n( θ )(n , n ) s r

P32n( θ )(n , n ) r

s

P34n( θ )(n , n r

⎛ −P ( θ ) … −P ( θ ) ⎞ 1, nr ⎜ 1,1 ⎟ =⎜ ⋮ ⋱ ⋮ ⎟ ⎜ ⎟ ⎝ −Pns,1( θ ) … −Pns, nr( θ )⎠

(A.7)

⎛ −P ( θ ) … −P ( θ ) ⎞ ns ,1 1,1 ⎜ ⎟ =⎜ ⋮ ⋱ ⋮ ⎟ ⎜ ⎟ P P − θ … − θ ( ) ( ) n n n 1, , ⎝ ⎠ r s r

(A.8)

r −1)

⎛ −Prt 0 ⎞ ⎜ ⎟ ⋱ ⎟ =⎜ −Prt ⎟ ⎜ 0 … ⎜ ⎟ ⎝ 0 0 ⎠

(A.9)

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P43n( θ )(n

r −1, nr )

⎛ −Prt ⎜ ⋱ =⎜ ⎜ ⎝ 0

21

0⎞ ⎟ ⋮ ⎟ ⎟ −Prt 0 ⎠ 0

(A.10)

Appendix B. Electro-magnetic-mechanical parameters See Appendix Tables B1 and B2

Table B1 Motor Parameters. Parameter

Signification

Value

ns nr Rs Ls Rr Rb Lr Lb Pst Psy

Number of stator slots Number of rotor slots Stator resistance (Ω) Stator leakage inductance (H) Ring resistance bar-to-bar (Ω) Rotor bar resistance (Ω) Ring leakage inductance bar-to-bar (H) Rotor bar leakage inductance (H) Stator tooth permeance (H) Stator yoke permeance (H)

24 30 0.051 3.02394e-3 6.41e-7 6.6397e-5 6.41e-8 5.9611e-7 2.984e-4 6.617e-4

Prt Pry

Rotor tooth permeance (H) Rotor yoke permeance (H)

3.174e-6 2.035e-5

Pmax

Maximal air-gap permeance (H)

3.0330e-06

Table B2 Characteristics of the mechanical model. Parameters

Values

Steel Young’s modulus (GPa) Steel density (kg/m3) Steel Poisson’s ratio Shaft length (m) Shaft diameter (m) Modal damper (N/m) Stiffness (N/(m/s))

210 7800 0.3 0.38 0.07 600 10e8

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Please cite this article as: A. Fourati, et al., Angular-based modeling of induction motors for monitoring, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2016.12.031i