III-3.4. ADVANCED MATERIALS FOR HIGH SPEED MOTOR DRIVES G. Kalokiris, A. Kladas and J. Tegopoulos Faculty of Electrical and Computer Engineering, National Technical University of Athens, 9, Iroon Polytechneiou Street, 15780 Athens, Greece tegopoul@power.ece.ntua.g Abstract. The paper presents electrical machine design considerations introduced by exploiting new magnetic material characteristics. The materials considered are amorphous alloy ribbons as well as neodymium alloy permanent magnets involving very low eddy current losses. Such advance materials enable electric machine operation at higher frequencies compared with the standard iron laminations used in the traditional magnetic circuit construction and provide better efficiently. Introduction The impact of innovative materials on the electrical machine design is very important. The advantages involved in machine efficiency and performance are important as mentioned in [1,2]. These materials enable electric machine operation at high frequencies when supplied by inverters, compared to the standard iron laminations used in the traditional magnetic circuit construction. The features and performance characteristics are analyzed by using field calculations and tested by measurements. In this paper, the study of asynchronous and permanent magnet machines based on such materials is undertaken. Low losses and high volumic power associated with high speed and converter machine operation are the main advantages of such applications [3–5]. Design procedure The proposed machine design procedure involves two steps. In a first step standard design methodology is used for preliminary design. In a second step the method of finite elements is implemented to calculate the machine efficiency and performance. Finally, a prototype is constructed in order to validate and compare the simulated machine characteristics to the corresponding experimental results [6,7]. The method of finite elements, is based on a discretization of the solution domain into small regions. In magnetostatic problems the unknown quantity is usually the magnetic vector potential A, and is approximated by means of polynomial shape functions. In two- dimensional cases triangular elements can easily be adapted to complex configurations and first order elements exhibit advantages in iron saturation representation [8,9]. The size of elements must be small enough to provide sufficient accuracy. In this way the differential S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 443–450. C 2006 Springer. 444 Kalokiris et al. equations of the continuous problem can be transformed into a system of algebraic equa- tions for the discrete problem. The practical problems necessitate usually several tenths of thousands of unknowns. However, appropriate numerical techniques have been developed, enabling to obtain the solution of such systems within reasonable time, even when personal computers are used. It should be mentioned that the 3D problems require considerably higher computational resources than the 2D ones. In the present paper the 2D finite element model adopted, involves vector potential formulation, while the magnetic flux m per pole can be calculated as follows: m = S 1 B · dS = C 1 A · dl ∼ = 2A gap L 0 (1) where L 0 is the length of the magnetic circuit in m, A is the magnetic vector potential, A gap is the vector potential value in the middle of the air-gap, B is the flux density in T, S 1 is the cross-sectional area normal to the direction of flux in m 2 , and C 1 is the contour surrounding the surface S 1 in m. The electromotive force at no load can be calculated as follows: E =− d m dt (2) The value of the voltage of the machine operated as generator under load conditions can be calculated by relation (3): V = E − RI − jL σ ωI (3) whereV is the voltage on stator windings in V, E is the electromotive force at no load in V, R is the stator resistance in , L σ is the stator leakage inductance in H, ω is the rotor angular velocity in rad/s, and I is the stator current in A. Then the magnetic flux and electromotive forces can be derived by using equations (1) and (2). Furthermore, the total resistance of stator winding can be calculated by the following relations: R = ρ l s (4) whereρ istheelectric resistivityof copper,l is thewindingtotal length,ands istheconductor cross-section. The winding length l can be estimated from relation (5): l = 2 × (l ax + l p ) × N w × P (5) where l ax is the machine’s axial length in meters, l p is the polar pitch in meters, and N w is the total number of series connected turns. Results and discussion The case of a permanent magnet machine has been considered. The machine designed has been checked through a 2.5 kW prototype, which has been connected to an appropriate power electronics converter. The air-gap width has been chosen 1 mm while a multipole “peripheral” machine structure has been adopted. The geometry of the permanent magnet machine is shown in Fig. 1 giving also the mesh employed for the two-dimensional finite element program of the machine involving, approximately 2,100 nodes 4,000 triangular elements. III-3.4. Advanced Materials for High Speed Motor Drives 445 Figure 1. Employed triangular mesh of the one pole part of the permanent magnet machine constructed. In a first step the no load operating conditions have been examined. The corresponding simulated voltage waveform is shown in Fig. 2 while the measured one is given in Fig. 3, respectively. In these figures a good agreement between the simulated and measured results can be observed. The simulation results concerning full load voltage of synchronous gen- erator are presented in Fig. 4. Fig. 5 gives the measured results under the same operating conditions. A good agreement can be observed in these figures between the simulated and measured results also in the case of full load. Moreover, measurements were realized for an asynchronous motor, which was supplied by an inverter with variable frequency. The motor is a three phase, four-pole, machine Figure 2. No load voltage waveform of permanent magnet machine (simulation). 446 Kalokiris et al. Figure 3. No load voltage waveform of permanent magnet machine (measurement). Figure 4. Full load voltage waveform of permanent magnet machine (simulation). Time (sec) 200 150 100 50 0 50 –100 –150 –200 0.00E+00 2.50E-00 5.00E-00 7.50E-00 1.00E-02 1.25E-02 Phase Voltage (V) Figure 5. Full load voltage waveform of permanent magnet machine (measurement). III-3.4. Advanced Materials for High Speed Motor Drives 447 a b Figure 6. Simulated field distribution in the machine under low load conditions. (a) Fundamental supply frequency of 300 Hz, (b) switching frequency of 10 kHz. supplied at a frequency of 400 Hz, at a voltage of 208 V while the nominal, speed is 10.800 rpm. The motor was tested under no load and low load operating conditions, for various frequencies. Fig. 6(a) shows the field distribution in the machine supplied at fundamental frequency of 300 Hz, while Fig. 6(b) gives the field distribution at the switching frequency of 10 kHz. Fig. 7 presents the respective measured phase voltage and current time variations. Fig. 8 shows the field distribution in the machine supplied at fundamental frequency of 100 Hz, while Fig. 9 presents the respective measured phase voltage and current time variations at the switching frequency of 10 kHz. Table 1 presents the measured and simulation results under no load conditions with a switching frequency of 1 kHz. Table 2 presents the same results under low load conditions. Table 3 presents the results related to a switching frequency of 10 kHz. The simulated torque T s is calculated by the relation: T s = F t · r g (6) where F t is the total circumferential tangential force in N and r g is the middle air-gap radius in m. The Maxwell’s stress tensor is calculated by relation (7): F t = 1 μ 0 C B n B t dl L 0 (7) 448 Kalokiris et al. a b Figure 7. Measured supply quantities in the machine for supply frequency of 300 Hz, under low load conditions. (a) Phase voltage time variation, (b) phase current time variation. Figure 8. Simulated field distribution in the machine, fundamental supply frequency of 100 Hz under low load conditions. III-3.4. Advanced Materials for High Speed Motor Drives 449 a b Figure 9. Measured supply quantities in the machine for supply frequency of 100 Hz under low load conditions. (a) Phase voltage time variation, (b) phase current time variation. Table 1. Measured and simulation results under no load conditions and f s = 1 kHz f 1 fundamental (Hz) I 1 measured (A) V 1 measured (V) T measured (Nm) V 1 simulated (V) T simulated (Nm) 20 1.587 4.717 0 2.041 0.055 50 2.267 14.350 0.1 13.279 0.088 75 2.417 20.962 0.15 21.096 0.11 Table 2. Measured and simulation results under low load conditions and f s = 1 kHz f 1 fundamental (Hz) I 1 measured (A) V 1 measured (V) T measured (Nm) V 1 simulated (V) T simulated (Nm) 50 2.754 13.769 0.4 14.540 0.23 Table 3. Measured and simulation results under low load conditions and f s = 10 kHz f 1 fundamental (Hz) I 1 measured (A) V 1 measured (V) T measured (Nm) V 1 simulated (V) T simulated (Nm) 50 2.368 13.584 0.22 13.353 0.18 100 2.874 22.646 0.38 25.790 0.3 200 5.179 53.533 1.05 75.970 1.03 450 Kalokiris et al. where B n and B t are the normal and tangential magnetic flux density components, respec- tively, to the integration surface of air-gap C in T, μ 0 is the permeability of air, and L 0 is the active part of the machine. In Tables 1–3 a good agreement between the measured and simulated results for both voltage and torque values can be observed. Conclusion The use of innovative materials, in electrical machines supplied by converters, can con- siderably affect the efficiency and performance in high speed operation. This has been investigated by using the finite element method for the machine analysis and verified by measurements. Moreover, the low cost involved makes such drives attractive rivals of the conventional ones. References [1] M.R. Dubois, H. Polinder, J.A. Ferreira, Contribution of permanent-magnet volume elements to no-load voltage in machines, IEEE Trans. Magn., Vol. 39, No. 3, pp. 1784–1792, 2003. [2] T. Higuchi, J. Oyama, E.Yamada, E.Chiricozzi, F. Parasiliti, M. Villani, Optimization procedure of surfacepermanent magnetsynchronous motors, IEEETrans.Magn., Vol.33, No.2, pp. 1943– 1946, 1997. [3] A. Toba, T. Lipo, Generic torque maximizing design methodology of surface permanent magnet Vernier machine, IEEE Trans. Ind. Appl., Vol. 36, No. 6, pp. 1539–1546, 2000. [4] G. Tsekouras, S. Kiartzis, A. Kladas, J. Tegopoulos, Neural network approach compared to sensitivity analysis based on finite element technique for optimization of permanent magnet generators, IEEE Trans. Magn., Vol. 37, No. 5/1, pp. 3618–3621, 2001. [5] D.C. Aliprantis, S.A. Papathanassiou, M.P. Papadopoulos, A.G. Kladas, “Modeling and Control of a Variable-Speed Wind Turbine Equipped with Permanent Magnet Synchronous Generator”, International Conference on Electrical Machines, Helsinki, Finland, 2000, pp. 558–562. [6] N.A. Demerdash, J.F. Bangura, A.A. Arkadan, A time-stepping coupled finite element-state space model for induction motor drives, IEEE Trans. Energy Convers., Vol. 14, No. 4, pp. 1465–1477, 1999. [7] T.M. Jahns, Motion control with permanent magnet AC machines, IEEE Proc., Vol. 82, No. 8, pp. 1241–1252, 1994. [8] C. Marchand, Z. Ren, A. Razek, “Torque Optimization of a Buried Permanent Magnet Syn- chronous Machine by Geometric Modification using FEM”, Proc. EMF’94, Leuven, Belgium, 1994, pp. 53–56. [9] H.C. Lovatt, P.A. Watterson, Energy stored in permanent magnets, IEEE Trans. Magn., Vol. 35, No. 1, pp. 505–507, 1999. III-3.5. IMPROVED MODELING OF THREE-PHASE TRANSFORMER ANALYSIS BASED ON NONLINEAR B-H CURVE AND TAKING INTO ACCOUNT ZERO-SEQUENCE FLUX B. Kawkabani and J J. Simond Laboratory for Electrical Machines, Swiss Federal Institute of Technology, EPFL-STI-ISE-LME, ELG Ecublens, CH-1015 Lausanne, Switzerland basile.kawkabani@epfl.ch Abstract. The present paper deals with a new approach for the study of the steady-state and transient behavior of three-phase transformers. This approach based on magnetic equivalent circuit-diagrams, takes into account the nonlinear B-H curve as well as zero-sequence flux. The nonlinear B-H curve is represented by a Fourier series, based on a set of measurement data. For the numerical simulations, two methods have been developed, by considering the total magnetic flux respectively the currents as state variables. Numerical results compared with test results and with FEM computations confirm the validity of the proposed approach. Introduction Traditionally in most of power system studies, the modeling of a three-phase transformer is reduced to its short-circuit impedance. The B-H curve introduced in some improved models and based on a set of measurement data, is approximated generally by several straight-line segmentsconnecting thepoints ofmeasurements. Butapparently, suchB-Hcurveobtainedis not smooth at the joints of the segments, and the slopes of the straight lines, representing the permeability, are discontinuous at these joints. Moreover, in the set of differential equations considering the currents as state variables, one needs the expressionsofthe derivatives of the inductances vs. the currents, which is impossible by using the above mentioned procedure. For that reason in the present study, the nonlinear B-H cur ve (or U-I curve) is represented by a Fourier series technique [1], based on the set of measurement data. An analytical ex- pression of a smooth B-H curve connecting the discrete measurement points can be defined. By using a magnetic equivalent circuit-diagram representing the three-phase transformer, all the self and mutual inductances can be expressed analytically in function of the magnetic reluctances of the cores. These inductances (and their derivatives) can be determined pre- ciselyusing the predetermined series Fourier representation, and adapted at each integration step in the numerical simulations. S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 451–460. C 2006 Springer. 452 Kawkabani and Simond Figure 1. Set of measured data U-I and its mirror image. Fourier series representation of B-H curve (U = f (I ) curve) A set of N +1 discrete measurement data U n and I n or B n -H n of a three-phase transfor mer (n = 0, 1, 2, 3, , N) is given. For the sake of making use of the Fourier series, a mirror image of this set of data is made about the U or B axis (Fig. 1). One has: f (H) = a 0 + ∞ k=1 a k · cos(ξ k · H) (1) with: a 0 = 1 H max · N n=1 B n · ( H n − H n−1 ) − 1 2 · α n · ( H n − H n−1 ) 2 (2) and a k = 2 H max · N n=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ α n ξ k · sin ( ξ k · H n−1 ) · ( H n − H n−1 ) + 1 ξ 2 k · α n · ( cos ( ξ k · H n ) − cos ( ξ k · H n−1 )) + B n ξ k · ( sin ( ξ k · H n ) − sin ( ξ k · H n−1 )) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (3) H max = H N (4) ξ k = k · π H max for the kth term (5) α n = B n − B n−1 H n − H n−1 (6) . (sec) 200 150 100 50 0 50 –100 –150 –200 0.00E+00 2.50E-00 5.00E-00 7.50E-00 1.00E-02 1.25E-02 Phase Voltage (V) Figure 5. Full load voltage waveform of permanent magnet machine (measurement). III-3.4. Advanced Materials for High Speed Motor Drives. Developments of Electrical Drives, 451 –460. C 2006 Springer. 452 Kawkabani and Simond Figure 1. Set of measured data U-I and its mirror image. Fourier series representation of B-H curve (U =. Institute of Technology, EPFL-STI-ISE-LME, ELG Ecublens, CH-1015 Lausanne, Switzerland basile.kawkabani@epfl.ch Abstract. The present paper deals with a new approach for the study of the steady-state