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ISSN 1843-6188 Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13) THREE-PHASE INDUCTION MOTOR’S MAGNETIZING CHARACTERISTIC BASED ON FINITE ELEMENT METHOD Aurel. I. CHIRILĂ, Constantin GHIŢĂ, Valentin NĂVRĂPESCU, I. Dragoş DEACONU Electrical Engineering Department, University Politehnica of Bucharest Splaiul Independenţei, 313, Sector 6, Bucharest, Romania E-mail: [email protected] where Lm0 represents the unsaturated value of the magnetizing inductance, im0 is the current threshold value of the magnetizing current over which the machine can be considered to become saturated, and is a coefficient specific to each electrical machine. Expression (1) is nonlinear and its determination supposes some approximations. These approximations are more or less precise depending on the currents flowing through the induction machine’s windings. Relation (1) does not offer entirely a sufficient approximation especially in the cases where the machines are working in hard loading conditions such as start-ups or electrical breaks. In these cases it is necessary to know more precisely the dependency between the magnetizing inductance and the currents flowing through the machine’s windings. The approximations errors for the dependency described by (1) can be diminished if the magnetic field inside the machine for different values of the windings currents is known. Starting from this point, the magnetic energy stored in the statoric and rotoric ferromagnetic armatures can be computed and so the magnetizing inductance or the magnetizing reactance (by multiplying the inductance with the statoric current pulsation). The magnetic field computation inside the machine can be performed with various methods, the most precise and used in the last period being the finite element method [7], … , [10]. In the paper a numeric method to obtain the magnetizing reactance of an induction motor starting from the magnetic field computation found within the armatures and the magnetic energy stored inside these armatures is presented. The experimental measurements validate the obtained numerical results based on the finite element method. Abstract: The paper presents a 2D numeric method for computing the magnetizing reactance Xm of a three phase induction machine, for various values of the magnetizing current im. In other words, the magnetizing characteristic is determined Xm = f(im). The magnetizing reactance is computed starting from the magnetic energy found within the ferromagnetic armatures of the machine and its windings. It is concluded that a high saturation state of the machine leads to a decreasing magnetizing reactance. The results are validated by experiments, based on no-load and short-circuit trials. It is noted that the measured values and the calculated ones are different by a few percents. Keywords: induction machine, magnetizing reactance, finite element method 1. INTRODUCTION The magnetizing reactance of a machine is an important parameter and corresponds to a main magnetic flux that links the machines stator and rotor, passing two times over the air-gap. This reactance is present within the equivalent circuit and can be determined experimentally by a no-load trial at synchronous speed. The magnetizing reactance is influenced by the saturation state of the machine [1], i.e. a saturated machine has a lower magnetizing reactance. In technical literature various papers on the parameters saturation influence can be a found [2], …, [5]. In order to take into account the saturation effect of the ferromagnetic armatures of the induction machine, the variation law of the magnetizing reactance Xm with the magnetizing current im is needed, i.e. Xm = f(im), within the mathematical model expressed with space vectors [6]. This dependency is input into expression relating the flux linkages and machine’s currents. If the magnetizing reactance Xm is divided by the stator’s currents pulsation ω then it is found the magnetizing inductance Lm computed with the following expression Lm = Xm / ω. An empirical analytical expression for the magnetizing inductance Lm variation with the induction machine magnetizing current im is pointed out in [1]: Lm0 , im im0 Lm0 , im im 0 Lm (im ) 2 1 L i 1 1 m0 m im 0 im 2. MOTOR DESCRIPTION The induction motor used for numeric simulations and experimental measurements is low power and low voltage, squirrel-cage type. Its main characteristics are shown in Table 1. The magnetizing characteristic of the statoric and rotoric ferromagnetic armatures is given in Fig. 1. In Fig. 2 is presented a cross-section of the induction machine. The machine has sheets, with parallel wall teeth both for stator and rotor, and the slots are trapezoidal type with rounded bottoms. (1) 65 Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13) field strength. After the magnetic field strength H is computed then the magnetic flux density B and the current density J are obtained with the following expressions: Table 1. Rated data Quantity Value 3 kW 50 Hz 220 / 380 V 11 / 6.3 A 1427 rpm 0.8 Power Frequency Voltage Current Speed Power factor B H J curl H (3) In non-conducting mediums the magnetic field is computed as follows: from successive iterations the magnetic field strength Hb generated by the currents flowing through the motor windings (that cause the current density J b ), is computed using the equation: 2 1.8 curl Hb J b 1.6 1.4 (4) Then, using Hb the scalar potential is obtained as follows: 1.2 [T] B [T] ISSN 1843-6188 1 div grad Hb 0 0.8 0.6 (5) while the magnetic field strength H in non-conducting mediums is given by the equation: 0.4 0.2 0 0 2000 4000 6000 8000 10000 H grad Hb 12000 H [A/m] For non-conducting mediums, based on already known magnetic field strength H, the magnetic flux density B and current density J are obtained with (3) in a similar manner as for conducting mediums. The finite element methods as a numerical analyzing method is applicable for any engineering field giving an approximate solution. The computation error is imposed by the set tolerances within the iterative solving algorithm of the equations system. Taking into account that the electrical machines have quite long ferromagnetic armatures, the magnetic field computation is precise enough when supposing a 2D numerical simulation case. In order to perform the computation of the magnetic field based on the finite element method, 2D CAD techniques have been applied in order to draw the cross-section of the motor. Then, the physical material properties of the model’s mediums have been allocated for each one (electrical conductivities, magnetic permeabilities). Further, the sources of the magnetic field are imposed (motor’s windings currents) along with specific boundary conditions. The final step before solving is the mesh generation obtained by a programmable operation. Once the values of the magnetic field are obtained the magnetic energy is obtained and with it the magnetizing reactance corresponding to the imposed windings’ current values. Figure 1. The magnetizing characteristic of the statoric and rotoric ferromagnetic armatures. Figure 2. Induction machine cross-section. 3. MAGNETIC FIELD COMPUTATION The magnetic field for the considered motor can be computed in 2D or 3D using numeric methods. The most used numerical method is the finite element method that is also applied in this paper using specialized software described in [8]. In conducting mediums, the equation that has to be solved with the finite element method by neglecting the displacement current is: curl curl H H t (6) 4. MAGNETIZING REACTANCE COMPUTATION It is supposed that the induction machine’s rotor is driven at synchronous speed and so the machine’s rotor is synchronous with synchronous magnetic field generated by the statoric three-phase winding. This hypothesis reflects in the T-equivalent circuit by the elimination of the rotoric branch. At synchronous speed the magnetic field seen by the rotor is constant with time, and so the back-emf of the rotoric windings is zero. (2) where and represent the electrical conductivity and the magnetic permeability respectively, and H is the magnetic 66 Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13) ISSN 1843-6188 In consequence, the rotoric windings have no currents and can be considered as air (infinite resistivity or zero electrical conductivity). From magnetic point of view the spinning of the rotor has no importance whatsoever. All the hypotheses previously mentioned justify the use of a harmonic regime for solving the magnetic field problem with the finite element method. The harmonic regime supposes that the mediums are linear. However it is possible to use the same equations in case of nonlinear mediums by the piecewise linearization of the domain. Thus, each mesh element has its own magnetic permeability, but globally the problem is nonlinear. The software manufacturers do not provide the computation algorithms for such problems but some information can be found in [11], [12]. In this case the solving tolerances have to be set to tight enough values. The source of the magnetic field is determined by the currents flowing through the stator’s windings, i.e. their current densities. The three currents generate a symmetrical three-phase system. The next step is to numerically solve the problem and to obtain the magnetic flux density B. Based on this quantity the magnetic vector potential A is deducted, with the following expression: B curl A current Im is expressed in [A], and the magnetizing reactance Xm is expressed in [Ω]. 5. NUMERICAL AND EXPERIMENTAL RESULTS The simulations and experiments are performed on the motor described in paragraph 2. The obtained mesh, after some adaptations, is shown in Fig. 3. It is constructed so that it is more dense in the air-gap where it is stored an increased amount of magnetic energy and less dense in the rest where the magnetic energy is not so dense. (7) Figure 3. The cross-section mesh. The magnetizing reactance relates to the magnetic energy stored within the computation domain D , where Σ represents a closed surface that contains the entire machine, its armatures and windings. The surface Σ can be extended towards the infinite, without any change in the total magnetic energy. The magnetic energy found within the domain D is the volume integral of the magnetic energy density, given by the expression [10]: BH* dv D D 4 1 1 A H* n d AJ * dv 4 4 D Wm wm dv The magnetic flux density map for 1.9A, obtained using the 2D finite element method is presented in Fig. 4, obtained using the 2D finite element method. From Fig. 4 it can be seen that an increased magnetic field is found near the magnetic poles at the yoke level. The teeth corresponding to the magnetic poles are also supplementary magnetically stressed. Due the symmetry of the machine the magnetic field is also periodic. The equations of the magnetic field have been integrated for 9 values of the magnetizing current (ranging between 1.1 and 4.3A), the magnetic potentials vector have been computed along with the magnetic energy using (9). Then, the values of the magnetizing reactance have been obtained using (10) for each value of the stator current. The results of the numerical simulations are shown in Fig. 5, the curve with full line. Up to 2.2A the magnetizing reactance is still constant, that is, the machine is not saturated. Over this value the reactance is decreasing almost linearly with about 10.5 Ω/A. (8) where B and H* are the amplitudes of the complex magnetic flux density, the conjugated magnetic field strength, and J* represents the complex conjugate of the current density. The first term in (8) cancels when the surface is infinitely extended [10], [13], [14]. Thus, in harmonic regime, the time-averaged value of the magnetic energy is: 1 Wm AJ * dv (9) 4 D Based on the obtained magnetic energy Wm, the magnetizing reactance for one-phase of the induction motor is given by: Xm 2 Wm 2 3 Im (10) where the magnetic energy is expressed in [J], the pulsation is expressed in [rad/s], the magnetizing Figure 4. The magnetic flux density map for 1.9A. 67 Scientific Bulletin of the Electrical Engineering Faculty – Year 10 No. 2 (13) ISSN 1843-6188 machine’s electrical currents is also non-linear, and in consequence the mathematical model of the machine based on space vectors becomes non-linear. If the magnetizing reactance of the induction machine is computed using a 2D numerical method and measured experimentally it can be concluded that there is a good agreement, but the computed values are less with a few percents than the measured values. This happens due to the fact that the finite element method is 2D type and does not take into account the magnetic effects of the end-windings, which contribute to the magnetic energy and the magnetizing reactance. It is preferred to obtain the magnetizing characteristic of the induction motor by 2D numerical method, which is simple and fast to apply, eventually followed in the end by a small correction. For the validation of the numerical results an experimental trial under no-load conditions has been conducted. The measurements have been performed with advanced equipment: Fluke 434 power analyzer, HP 34401A digital multi-meter, DeLorenzo DL1059 threephase AC voltage regulator. 7. REFERENCES [1] Figure 5. Magnetizing reactance variation with the per phase magnetizing current. [2] [3] The variation of the magnetizing current has been achieved by the variation of the stator’s winding supplying voltage. For each value of the supplying voltage (and so the magnetizing current) the leakage reactance has been obtained (using a graphical-analytic method described in [15]), then the no-load factor has been computed and in the end the back-emf of the stator winding. Based on these quantities the per-phase magnetizing reactance Xm is deducted. The results of the experimental measurements are also shown in Fig. 5 (dots line). It can be concluded that the measured values are just few percents higher than the computed ones with the finite element method, because the employed numerical method is only a 2D a can not take into account the endwinding magnetizing reactance (that is indeed very low). It can be stated that the errors obtained between the computed and the measured ones are systematic errors and can be corrected. Thus, in order to get the values of the magnetizing reactance with enough precision it is necessary to increase the values with about 1 – 2 percents by energetic means, using 2D numerical method. It has to be said that the 2D approach requires a less computing effort, a less programming effort and a not so high-performance computational system. This is why it is preferred to compute the magnetizing reactance of the induction motors using 2D finite element method, and then a little positive correction. [4] [5] [6] [7] [8] [9] [10] [11] [12] 6. CONCLUSIONS [13] The magnetizing reactance of a three-phase induction machine is an important parameter. This characteristic is non-linear, that is the magnetizing reactance decreases when the machine’s saturation increases. If this nonlinear variation is considered then the dependency between the machine’s magnetic fluxes and the [14] [15] 68 Ursem R.K., Vadstrup P., Parameter identification of induction motors using stochastic optimization algorithms, Applied. Soft Computation, vol. 4, no. 1, pp. 49–64, 2004. Vas P., Sensorless Vector and Direct Torque Control, Oxford University Press, 1998. Boldea I., Parametrii maşinilor electrice, Editura Academiei, Bucureşti, 1991. Ilina D., Theoretical and experimental contributions concerning the determination of the AC electrical machines parameters, Phd. thesis, Politehnica University of Bucharest, 2010. Chirilă A. I., Deaconu I. D., Ghiţă C., Ilina D., Induction motors, dynamic regimes analysis methods, The 6th International Conference on Electromechanical and Power Systems (SIELMEN 2007), Chişinău, Rep.Moldova, October 4-6, 2007. Ghiţă C., Maşini electrice, Editura Matrixrom, Bucureşti, 2005. Udrişte C., Iftode V., Postolache M., Metode numerice de calcul, Editura Tehnică, Bucureşti, 1996. Infolytica Corporation - Electromagnetic Field Simulation Software [online] [2009]; Available from: URL:http://www.infolytica.com/ Champion E.R., Ensminger Jr. J.M., Finite Element Analysis with Personal Computers, Editura CRC Press, 1988. Mocanu C.I., Teoria câmpului electromagnetic, Editura Didactică şi Pedagogică, Bucureşti, 1981. Jack A.G., Mecrow B.C., Methods for magnetically nonlinear problems involving significant hysteresis and eddy currents, IEEE Transactions on Magnetics, 26(2):424-429, March 1990. Williamson S., Ralph J. S., Finite-element analysis for non-linear magnetic field problems with complex current sources, Proceedings IEE, Vol. 129, Part A, No. 6, pag. 391–395, 1982. Stratton J.A., Electromagnetic Theory, Editura Mc Graw-Hill, New York and London, 1941. Simonyi K., Electrotehnică teoretică, Editura Tehnică, 1974. Chirilă A. I., Theoretical and experimental approaches regarding the operation and optimization of induction machines within electrical drive systems, Phd. thesis, Politehnica University of Bucharest, 2010, Romania.