ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005
MULTI-SLICE FINITE ELEMENT MODELLING OF INDUCTION MOTORS CONSIDERING BROKEN BARS AND INTER-BAR CURRENTS J. Gyselinck1 , J. Sprooten1 , L. Vandevelde2 and X.M. L´opez-Fern´andez3 1
Department of Electrical Engineering, Universit´e Libre de Bruxelles Franklin Roosevelt Avenue 50, B-1050 Brussels, Belgium phone: +32 2 650 26 69 – fax: +32 2 650 26 53 – e-mail:
[email protected] Department of Electrical Engineering, Ghent University Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium phone: +32 264 34 22 – fax: +32 9 264 35 82 – e-mail:
[email protected] Department of Electrical Engineering, University of Vigo Lagoas, Marcosende 9, 36200 Vigo, Spain phone: +34 986 812177 – fax: +34 986 812201 – e-mail:
[email protected] 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].
ISEF'2005-MTT-SA.19
ISBN 84-609-7057-4
(6-Pages)
ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005
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 lz (along the z-axis) consists of nsl unskewed (i) slices of axial length lz(i) (1 ≤ i ≤ nsl ), in which the rotor position, denoted by θrot , is shifted with respect to the average rotor position θrot . 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 (i) (i) rotor position of the i-th slice by lz and θrot respectively, the discretisation of the skew can be defined by means of the dimensionless coefficients η (i) and γ (i) : lz(i) = η (i) lz where θsk is the skew angle, and with commonly used [3]: η (i) =
(i)
and θrot = θrot + γ (i)
θsk , 2
(1)
P (i) η = 1 and −1 < γ (i) < 1. A uniform discretisation is
1 nsl
and γ (i) =
2i − nsl − 1 . nsl
(2)
Alternatively, a classical 1D Gauss integration scheme can be adopted [9]. Herein the position η (i) and the weight γ (i) of the nsl evaluation points in the reference interval [−1, 1] are such that the numerical integration is exact for all polynomials of degrees up to 2nsl − 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 nsl 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
ISEF'2005-MTT-SA.19
ISBN 84-609-7057-4
(6-Pages)
ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005
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 2nsl resistances 2RIB /η (i) distributed over the nsl slices and between each pair of adjacent bars, as is shown in Fig. 2. Directly connected in parallel, these 2nsl resistances amount to the IB resistance RIB . Note that each segment of the upper and the lower endring is shunted by a resistance RIB /η (1) = RIB /η (nsl ) . 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 (θsk = 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. 3. 5.2
reactance (Ω)
5.0
skew, no IB currents skew, IB currents
4.8 4.6 4.4
skew, measured
4.2 4.0 3.8 0.1
no skew, no IB currents 1
10
100
1000
10000
inter-bar resistance RIB (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 Ωmm2 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: RIB = 10 µΩ. By way of comparison, the endring segment resistance and inductance values (see Fig. 2) are Rers = 0.87 µΩ and Lers = 4.8 nH respectively; the rotor bars have a DC resistance of 107 µΩ.
ISEF'2005-MTT-SA.19
ISBN 84-609-7057-4
(6-Pages)
ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005
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 RIB are carried out. By comparison with the measured reactance, RIB 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 (RIB = ∞) or considered (RIB 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) =
X
¯ k exp[j(2πfk t − κk θ)]} , 0, these three values for i result in a direct, homopolar and inverse voltage and current component; if fk < 0 this is inverse, homopolar and direct. (Note that the parameter g having coefficient Np does not affect the fulfilment of this criterion.)
ISEF'2005-MTT-SA.19
ISBN 84-609-7057-4
(6-Pages)
ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005
For the motor and operation under study (Np = 2, Nr = 32, f0 = 50Hz, s = 0.1, η = 0, εs = 0), the fundamental frequency f0 = 50Hz (l = 0, εd = 0) is only slightly affected by skew and the broken bar, whereas the first rotor slotting and m.m.f. harmonics (l = ±1, εd = 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 εd that are multiples of 4. For instance, the lowest (direct) harmonic |1 − 2(1 − s)| · f0 = (1 − 2s)f0 = 40 Hz is produced with l = 0 and εd = −4; εd = 4 gives the homopolar 140 Hz. l = 0 and εd = ±8 produces 230 Hz (inverse) and 130 Hz (homopolar). l = 0 and εd = ±12 produces 320 Hz (direct) and 220 Hz (inverse). This way all the other frequencies in Fig. 5 can be traced as well. 1 no skew, healthy rotor no skew, 1 bar broken skew, 1 bar broken
relative current amplitude
relative current amplitude
1
0.1
0.01
0.001
50
140
230 310 400 490 frequency [Hz]
580
670
0.1
0.01
0.001
770
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)
RIB = ∞ RIB = 100µΩ RIB = 10µΩ
50
140
230 310 400 490 frequency [Hz]
580
670
770
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)f0 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 2sf0 torque harmonic, which may produce a speed variation, which in turn will create an additional harmonic (1 + 2s)f0 in the stator winding currents [12]. One may expect the latter harmonic to be attenuated by the IB currents as well.
amplitude of bar current [A]
700 600 500 400 slice 4 slice 3 slice 2 slice 1
300 200 100 0
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
bar number
Fig. 7. Locked-rotor motor reactance (with skew) as a function of IB resistance
ISEF'2005-MTT-SA.19
ISBN 84-609-7057-4
(6-Pages)
ISEF 2005 - XII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering Baiona, Spain, September 15-17, 2005
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. 1286-1291. [2] D. Dorrell and T. Miller, Inter-bar currents in induction machines, IEEE Trans. Ind. Appl., Vol. 39, No. 3, pp. 677-684, May/June 2003. [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. 4218-4225, Sept. 1999. [4] J. Gyselinck and X.M. L´opez-Fern´andez, 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. 621–622, extended paper (790) on CD-ROM, accepted for COMPEL 2005. [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. 152-158, March 1994. [6] G. M¨uller 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. 71-79, March 2003. [7] X.M. L´opez-Fern´andez and P. Marius, Magnetodynamic performance in cage induction motors with a broken bar, Proc. ISEF2003, Maribor, Slovenia, 18–20 Sept. 2003, pp. 261-266. [8] J. Sprooten and J.C. Maun, Induction machine fault detection and quantification by means of superposed analytical models, Proc. SPEEDAM, Capri, Italy, 16–18 June 2004, pp. 821-826. [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. 3233-3237, Sept. 2001. [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. 1291-1294, March 1992. [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. 2288-2295 [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. 484-491.
ISEF'2005-MTT-SA.19
ISBN 84-609-7057-4
(6-Pages)
