MULTI-SLICE FINITE ELEMENT MODELLING OF INDUCTION MOTORS CONSIDERING BROKEN BARS AND INTER-BAR CURRENTS
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1 MULTI-SLICE FINITE ELEMENT MODELLING OF INDUCTION MOTORS CONSIDERING BROKEN BARS AND INTER-BAR CURRENTS J. Gyselinck 1, J. Sprooten 1, L. Vandevelde 2 and X.M. López-Fernández 3 1 Department of Electrical Engineering, Université Libre de Bruxelles Franklin Roosevelt Avenue 50, B-1050 Brussels, Belgium phone: fax: Johan.Gyselinck@ulb.ac.be Department of Electrical Engineering, Ghent University Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium phone: fax: Lieven.Vandevelde@ugent.be Department of Electrical Engineering, University of Vigo Lagoas, Marcosende 9, Vigo, Spain phone: fax: xmlopez@uvigo.es Abstract This paper deals with the finite element (FE) modelling of squirrel-cage induction motors having one or more broken bars. A two-dimensional (2D) multi-slice FE model allows to consider the position of the bar breakage (at one of the endrings rather than in the middle), the skew of the rotor bars and the interbar (IB) currents. The latter are effected by resistances distributed in the electrical circuit that connects the different slices of the cage. The multi-slice model is applied to a 3kW induction motor. An order-ofmagnitude estimate of the IB resistance follows from a short-circuit test (with healthy rotor). Next, the effect of a broken bar and of the IB currents on the stator current spectrum is studied. Introduction The existence of inter-bar (IB) currents in cast cage rotors of small induction motors is due to the absence of perfect insulation between the cage and the core [1]. These parasitic currents, which flow from bar to bar through the iron rotor core, are limited by the high but finite bar-core contact resistance rather than by the resistivity of the steel laminations. The accurate measurement of the IB resistance is by no means trivial. Some measuring methods and results of extensive experimental work are discussed in [1, 2]. Nominally identical rotors can have significantly different IB resistances, even when manufactured at the same plant, using the same equipment, and on the same day. Therefore, order-of-magnitude estimation of the bar-to-bar resistance may be thought to be sufficient [1]. Enhanced per-phase equivalent circuits of (healthy) induction motors show that IB currents are strongly promoted by rotor skew and may have a significant effect on their starting performance [2]. At load their influence is usually much less pronounced. Skew and IB currents can be taken into account more precisely when using a multi-slice FE-model. In such a model, the IB currents are easily effected by inserting lumped resistances in the electrical circuit of the cage [3, 4]. When one or more rotor bars are broken, the IB currents are locally promoted, attenuating the magnetic disturbance due to the broken bars [7] and thus rendering their detection more difficult [5, 6, 7, 8]. Indeed, depending on the finite IB resistance, current continues to flow into the broken bar from its healthy side (if the breakage occurs at an endring), through the laminations and toward the adjacent bars. A simple analytical expression for the axial current distribution in the broken bar(s) can be derived [5, 6].
2 This paper is concerned with a more detailed multi-slice FE analysis of this effect. After a brief discussion of the multi-slice FE model, its application to a 3kW induction is discussed. Particular attention is paid to the frequency spectrum of the stator phase currents in presence of skew and a broken rotor bar. Multi-Slice FE Model with IB currents A multi-slice FE model of a machine of total axial length l z (along the z-axis) consists of n sl unskewed slices of axial length l z (i) (1 i n sl ), in which the rotor position, denoted by θ (i) rot, is shifted with respect to the average rotor position θ rot. An approximation with three slices is shown in Fig. 1. endring FE slice 1 FE slice 2 FE slice 3 endring Fig. 1. Multi-slice model: approximation of skew by means of three slices of equal length Fig. 2. Electrical circuit of rotor cage with three FE slices and distributed IB resistance In each slice a 2D magnetic field is assumed and the same FE discretisation is commonly adopted in stator and rotor [3, 4, 9]. The layer of finite elements connecting stator and rotor mesh, the so-called moving band, is rotor position dependent, and varies from slice to slice. Denoting the length and the rotor position of the i-th slice by l z (i) and θ (i) rot respectively, the discretisation of the skew can be defined by means of the dimensionless coefficients η (i) and γ (i) : l (i) z = η (i) l z and θ (i) rot = θ rot + γ (i) θ sk 2, (1) where θ sk is the skew angle, and with η (i) = 1 and 1 < γ (i) < 1. A uniform discretisation is commonly used [3]: η (i) = 1 and γ (i) = 2i n sl 1. (2) n sl n sl Alternatively, a classical 1D Gauss integration scheme can be adopted [9]. Herein the position η (i) and the weight γ (i) of the n sl evaluation points in the reference interval [ 1, 1] are such that the numerical integration is exact for all polynomials of degrees up to 2n sl 1. The Gauss scheme allows significant savings as for a given accuracy less slices are required [9]. In a single-slice FE model, the stator windings are each modelled as a so-called stranded conductor whereas, the rotor bars are each modelled as a so-called massive conductor (thus ignoring and allowing for skin effect respectively) [10]. The series connection of the corresponding conductors in the n sl of a multi-slice model, their connection to the endrings, and the voltage supply of the stator windings are considered through electrical circuit equations. The endwindings and endrings are taken into account by means of lumped resistances and inductances. One thus obtains a large system of differential equations
3 in terms of the nodal vector potential values and bar voltages of each slice, and a number of loop currents [10, 3, 4]. The electrical circuit of the rotor cage is easily extended with a view to the approximate inclusion of the IB currents. The latter are indeed allowed for by 2n sl resistances 2R IB /η (i) distributed over the n sl slices and between each pair of adjacent bars, as is shown in Fig. 2. Directly connected in parallel, these 2n sl resistances amount to the IB resistance R IB. Note that each segment of the upper and the lower endring is shunted by a resistance R IB /η (1) = R IB /η (n sl ). The generic FE and electrical circuit equations remain valid, but the number of independent current loops in the cage circuit now depends on both the number of rotor bars and the number of slices. Application to a 3kW Induction Motor We consider a 4-pole 220 V 50 Hz 3 kw induction motor. The commercial version of the rotor has 32 closed and skewed slots (θ sk = 12.4 ). Three other rotors having open and/or unskewed slots have been constructed for research purposes. Using a single-slice and a multi-slice FE model, the stator current waveforms at noload and at full load, and with either of the four different rotors, can be calculated with a satisfactory precision. The effect of the IB currents on the short-circuit and load operation has been studied in [4]; this study was limited to the healthy rotors with open slots. It has been observed that the IB currents have a minor effect at load (e.g. on torque output and losses), whereas the skew effectively reduces slotting harmonics and results in a decrease and increase of copper and iron losses respectively. In this paper we focus on the combined effects of the IB currents and a broken rotor bar. A FE discretisation having 6000 first order triangular elements per slice is used. The stator and the rotor iron are separated by three layers of elements, the middle one being the moving band. A typical flux pattern at load is shown in Fig skew, no IB currents reactance (Ω) skew, measured skew, IB currents 4.0 no skew, no IB currents inter-bar resistance R IB (µω) Fig. 3. Cross-section of 3kW induction motor flux pattern at full load Fig. 4. Locked-rotor motor reactance (with skew) as a function of IB resistance IB resistance estimation from locked rotor test From the contact resistance value of 0.04 Ωmm 2 found in [2] and the dimensions of the 3kW motor at hand (core length 127 mm and rotor bar periphery 31 mm), a rough order-of-magnitude estimation follows: R IB = 10 µω. By way of comparison, the endring segment resistance and inductance values (see Fig. 2) are R ers = 0.87 µω and L ers = 4.8 nh respectively; the rotor bars have a DC resistance of 107 µω.
4 An estimate of the IB resistance estimate may also be obtained through short-circuit measurements and calculations with both the unskewed and skewed rotor (at reduced 50 Hz voltage supply) [4]. Measurements learn that the skew brings about an increase of the short-circuit motor reactance of 0.4Ω. Ignoring completely the IB currents, the increase can be estimated from the magnetising reactance and the skew factor of the fundamental 4-pole field [2], which produces a short-circuit reactance increase of 1Ω. This value agrees well with the increase predicted by means of the FE model if the IB currents are not taken into account (see Fig. 4). Assuming that the difference with these two values can be attributed to the IB currents, multi-slice FE simulations with different values of the IB resistance R IB are carried out. By comparison with the measured reactance, R IB should be in the range from 10 µω to 100 µω, as can be concluded from Fig. 4. Load operation with healthy and faulty rotor Time-stepping simulations at load under rated sinusoidal voltage supply are carried out. The stator phases are delta connected. For taking into account the skew a multi-slice model with 4 slices and Gauss distribution is used. The skewed cage is either healthy or has a broken bar (breakage in the first slice, i.e. near one of the endrings). The IB currents are either ignored (R IB = ) or considered (R IB equal to 10 µω or 100 µω). In order to reduce the computation time, magnetic saturation is ignored and a 10% slip (1350 rpm) is imposed. By simulating 25 fundamental time periods (with 150 steps per period) and by performing the Fourier analysis on the 20 last periods, i.e. [0.1s,0.5s], neat frequency spectra are obtained. Fig. 5 shows the frequency spectrum of the stator phase currents for 3 cases without IB currents. The origin of the frequencies found can be explained by means of the rotating field theory [11]. Indeed, the airgap flux density can be written as a series of travelling waves: B(θ, t) = k R{ B k exp[j(2πf k t κ k θ)]}, (3) where B, f k and κ k are the complex representation, the frequency and the spatial order respectively of the k-th induction component; θ is the angular coordinate in mechanical radians with respect to a stator reference frame, and j = 1 is the imaginary unit. Assuming an integral-slot stator winding with normal phase belt width (60 ) and in series connected coil groups, the frequencies f k and spatial orders κ k of the induction components that may occur in a three-phase induction motor having N p pole pairs and N r rotor bars, under sinusoidal voltage supply of frequency f0, and at slip s are given in terms of the integer parameters η, l, ɛ s, ɛ d and g: f k = ( 1 + 2η + (ln r + ε d ) 1 s N p ) f 0, (4) κ k = (1 + 6g + 2η)N p + ln r + ε s + ɛ d, (5) where g is related to the stator m.m.f. harmonics and slotting, l is related to rotor m.m.f. harmonics and slotting, ε s to the static rotor eccentricity if any, ε d to the dynamic rotor eccentricity or one or more broken bars if any, and η to iron saturation [11]. An induction component (f k, κ k ) induces a sinusoidal voltage of frequency f k in the stator windings only if the order κ k can be written as κ k = (i + 6m)N p, with m integer and i equal to 1, 3 or -1; if f k > 0, these three values for i result in a direct, homopolar and inverse voltage and current component; if f k < 0 this is inverse, homopolar and direct. (Note that the parameter g having coefficient N p does not affect the fulfilment of this criterion.)
5 For the motor and operation under study (N p = 2, N r = 32, f 0 = 50Hz, s = 0.1, η = 0, ε s = 0), the fundamental frequency f 0 = 50Hz (l = 0, ε d = 0) is only slightly affected by skew and the broken bar, whereas the first rotor slotting and m.m.f. harmonics (l = ±1, ε d = 0), at 670 Hz (inverse) and 770 Hz (direct), are significantly reduced thanks to the skew; the latter harmonic practically disappears. The harmonics due to the broken bar correspond to values of ε d that are multiples of 4. For instance, the lowest (direct) harmonic 1 2(1 s) f 0 = (1 2s)f 0 = 40 Hz is produced with l = 0 and ε d = 4; ε d = 4 gives the homopolar 140 Hz. l = 0 and ε d = ±8 produces 230 Hz (inverse) and 130 Hz (homopolar). l = 0 and ε d = ±12 produces 320 Hz (direct) and 220 Hz (inverse). This way all the other frequencies in Fig. 5 can be traced as well. relative current amplitude no skew, healthy rotor no skew, 1 bar broken skew, 1 bar broken relative current amplitude R IB = R IB = 100µΩ R IB = 10µΩ frequency [Hz] frequency [Hz] Fig. 5. Frequency spectrum of stator phase current at 10% slip in absence of IB currents (without and with broken bar, without and with skew) Fig. 6. Frequency spectrum of stator phase current at 10% slip with 1 broken bar influence of IB resistance Fig. 6 evidences the attenuating effect of the IB currents on the magnetic disturbance due to a broken bar: the associated harmonics diminish as the IB resistance is decreased. The significant reduction of the (1 2s)f 0 frequency is of particular interest as this may hamper broken-bar detection relying on the appearance of this harmonic [5, 12]. Note that a broken bar results in an additional 2sf 0 torque harmonic, which may produce a speed variation, which in turn will create an additional harmonic (1 + 2s)f 0 in the stator winding currents [12]. One may expect the latter harmonic to be attenuated by the IB currents as well. 700 amplitude of bar current [A] slice 4 slice 3 slice 2 slice bar number Fig. 7. Locked-rotor motor reactance (with skew) as a function of IB resistance
6 The distribution of the current in the rotor bars when one bar (bar number 16) is broken is depicted in Fig. 7. Thanks to the finite IB resistance, current continues to flow into the broken bar from its healthy side. The current in the adjacent bars on either side increases, especially in slice 1, in which the bar breakage is situated. Note that the current in bar 15 is greater than the one in bar 17. This asymmetry is also reported in [7]. Note also the slight oscillation of the current profile further away from the broken bar. The simple analytical expression for the bar current profile given in [5, 6] does not feature this oscillation as the coupling of the rotor cage with the stator windings is not taken into account. Conclusions This paper has dealt with multi-slice FE modelling of squirrel-cage induction motors in the presence of IB currents and a broken bar. The multi-slice model has been applied to a 3kW induction motor for which different rotors were available. An order-of-magnitude estimate of the IB resistance was obtained by means of a short-circuit test with healthy rotor (unskewed and skewed version). Next, the effect of a broken bar and of the IB currents on the stator current spectrum is studied. The origin of the harmonics has been explained and the attenuating effect of the IB currents on the magnetic disturbance due to a broken bar has been evidenced. References [1] S. Williamson, C. Poh and A. Smith, Estimation of the inter-bar resistance of a cast cage rotor, Proc. Electric Machines and Drives Conference (IEMDC), 1 4 June 2003, Vol. 2, pp [2] D. Dorrell and T. Miller, Inter-bar currents in induction machines, IEEE Trans. Ind. Appl., Vol. 39, No. 3, pp , May/June [3] S. Ho, H. Li and W. Fu, Inclusion of interbar currents in a network-field coupled time-stepping finite-element model of skewed-rotor induction motors, IEEE Trans. Magn., Vol. 35, No. 5, pp , Sept [4] J. Gyselinck and X.M. López-Fernández, Inclusion of inter-bar currents in multi-slice FE modelling of induction motors influence of inter-bar resistance and skew discretisation, Proc. ICEM2004, Cracow, Poland, 5 8 Sept. 2004, pp , extended paper (790) on CD-ROM, accepted for COMPEL [5] R. Walliser and C. Landy, Determination of interbar current effects in the detection of broken rotor bars in squirrel cage induction motors, IEEE Trans. Energy Conv., Vol. 9, No. 1, pp , March [6] G. Müller and C. Landy, A novel method to detect broken rotor bars in squirrel cage induction motors when interbar currents are present, IEEE Trans. Energy Conv., Vol. 18, No. 1, pp , March [7] X.M. López-Fernández and P. Marius, Magnetodynamic performance in cage induction motors with a broken bar, Proc. ISEF2003, Maribor, Slovenia, Sept. 2003, pp [8] J. Sprooten and J.C. Maun, Induction machine fault detection and quantification by means of superposed analytical models, Proc. SPEEDAM, Capri, Italy, June 2004, pp [9] J. Gyselinck, L. Vandevelde and J. Melkebeek, Multi-slice FE modelling of electrical machines with skewed slots - the skew discretisation error, IEEE Trans. Magn., Vol. 37, No. 5, pp , Sept [10] P. Lombard and G. Meunier, A general method for electric and magnetic combined problems in 2D and magnetodynamic domain, IEEE Trans. Magn., Vol. 28, No. 2, pp , March [11] L. Vandevelde and J. Melkebeek, Numerical analysis of vibrations of squirrel-cage induction motors based on magnetic equivalent circuits and structural finite element models, Conference Record of the 2001 IEEE Industry Applications Conference / 36th IAS Annual Meeting, Chicago, Illinois, USA, September 30 October 4, 2001, Vol. 4, pp [12] A. Bellini, F. Filippetti, G. Franceschini, C. Tassoni and G.B. Kliman, Quantitative evaluation of induction motor broken bars by means of electrical signature analysis, Conference Record of the IEEE Industry Applications Conference, 8 12 Oct. 2000, Vol. 1, pp
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