Comparison of Different FE Calculation Methods forthe Electromagnetic Torque of PM Machines Dieter Gerling Institute for Electrical Drives, University of Federal Defense Munich Neubiberg, Germany Summary: Generally, the torque of electrical machines can be calculated applying different methods. In this paper, the Maxwell’s stress tensor method, the magnetic co-energy method, and the lumpedparameter method are investigated. As an example, a PM machine with surface mounted magnets in the rotor is analyzed by means of the FE-software package ANSYS. For the lumped-parameter method, the dq-parameters of the electric machine are derived with the Fixed Permeability Method (FPM). With this method the parameters of the PM machine can be calculated with high precision. The obtained results for the electromagnetic torque applying the different calculation methods are compared concerning accuracy, ease of use and computing time. Keywords: Torque Calculation, Finite Element Method, ANSYS, PM Machine, Fixed Permeability Method (FPM) NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 1 October 26 - 27, 2005 Wiesbaden, Germany The methods analysed here are Maxwell’s stress tensor-. the electromagnetic torque equals the derivative of the magnetic co-energy with respect to angular position at constant current. 1: Geometry of the studied PM machine Different methods based on finite element solutions have been used for the calculation of the electromagnetic torque of this machine.9 4 127.1. since it requires only the local flux density distribution along a specific contour around the air-gap of the machine. TABLE 1. The accuracy of torque calculation with this method relies on model discretization and on contour selection.] Stator outer radius [mm] Stack length [mm] 21.8 4 40 35 Fig. dqmodel of PM machine-. Figure 1 shows the geometry of the studied PM machine. This calculation method allows precise determination of machine parameters through the magnetic field solutions as it takes into account the actual distribution of windings. In this paper. The finite element method provides an accurate approach to torque evaluation from the derivation of electromagnetic field distribution. torque is a very important consideration for both analysis and design of electrical machines.27. Calculation of the electromagnetic torque with the third method is based on the dq-mathematical model of the PM machine. a PM machine with surface mounted magnets in the rotor is analysed with ANSYS.6 0. Germany . With this method at least two finite element solutions are required to obtain the co-energy change due to an incremental displacement. Introduction The basic task of any electric machine is to generate torque to accelerate and drive a load over a specific range of speeds. This method ensures that the saturation effect which occurs in the load model is not ignored during calculation of the • • dq-axes inductances (single current excitation of the stator windings). back emf (single magnet excitation).GEOMETRY DATA Rotor radius [mm] Air gap length [mm] Thickness of magnet [mm] Magnet pole arc [deg] Number of poles [ -. The dq-parameters of the machine are derived accurately using fixed permeability method (FPM) [2-4]. NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 2 October 26 . and magnetic co-energy method. The calculation algorithm for the dq-parameters of the PM machine with the fixed permeability method is presented in the fourth section of this paper. The main geometrical data are presented in the Table 1. and this inevitably increases the computing time. Thus. 2005 Wiesbaden. The use of Maxwell’s stress tensor is probably the simplest method. and non-linearity of magnetic materials of an electrical machine. details of geometry. According to the virtual work principle. the flux linkages depend on both current components. because with this method the field is solved twice to get an energy difference.2 Co-Energy Method The second most frequently used method is based on the stored magnetic co-energy change or the virtual work with a small displacement. The motor equations. which is well known and effective.2. 2. Te = ∆Wco-eng ∆θ (2) i = const In co-energy method the calculation time is doubled. on the model discretization and on the selection of the integration line or contour. on the other hand. can be calculated by integrating the Maxwell’s stress tensor. According to the virtual work principle. The electromagnetic torque based on the dq -formulation is Te = 3 p (ψ d ⋅ iq − ψ q ⋅ id ) 2 (3) while the voltage dynamic equations are d ψ d − ω ⋅ψ q dt d uq = R ⋅ iq + ψ q + ω ⋅ψ d dt ud = R ⋅ id + (4) where. the electromagnetic torque Te equals the derivative of the magnetic co-energy Wco-eng with respect to angular position at constant current. valid also when the iron is saturated. according to the following magnetic model ψ d = ψ d ( id . The electromagnetic torque Te in the motor air-gap. 2005 Wiesbaden. it is important to force at least three layers of elements in the small air-gap and to calculate the torque in the middle element layer.3 Electromagnetic Torque Based on the dq-Mathematical Model of PM Machine The PM motor is usually modeled and analyzed by means of the space-vector theory.27. and L is the active length of the machine. For two-dimensional electromagnetic field models: Te = 2π L µ0 ∫r 2 Br Bθ dθ (1) 0 where Br and Bθ are radial and tangential components of the flux density. iq ) (5) ψ q = ψ q ( id . in general. Taking advantage of the specific periodicity of electrical machines. are reported in the following. the integral can be performed from 0 to 2π electrical radians and the result multiplied by the number of pole pairs p. iq ) NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 3 October 26 .1 Maxwell’s Stress Tensor Method The Maxwell’s stress tensor method is probably the most commonly used since it is uncomplicated to apply and needs a relatively small calculation time. Electromagnetic Torque 2. 2. But. Germany . on a closed surface of radius r. If a Maxwell’s stress tensor is used. the accuracy of this method is obviously dependent on the type of the problem to be solved. iq ) = Lq ( id .and q-axis current components. a second calculation is carried out either with current or magnet single excitation. With this fixed permeability. When the motor operates with high torque. e. iq ) = Ld ( id . iq ) Compared with the equ. when the saturation occurs the ψ dm and ψ qm also depend on the d. (6). it is shown that the ψ dm increases with id .and q-axis flux linkages vary linearly with the corresponding d. 4].and q-axis current components. 3. ψ dm is the d-axis flux-linkage due to magnets. iq ) ⋅ id 2 (8) The above expression is used to calculate the electromagnetic torque with the third FE method. The flux line distribution and the flux density along the surface of a PM pole pair for this operation condition are shown in the figure 2. its variation reaches up to 24% of the ψ dm . while it decreases with id . The dq-parameters of the PM machine are calculated using the fixed permeability method (FPM) [2. the following expression for the electromagnetic torque is obtained Te = { } 3 p Ld (id . 2005 Wiesbaden. All following finite element calculations have been performed using the software package ANSYS. iq ) (7) ψ q ( id . This ensures that the saturation effect which occurs in the load model is not ignored during calculation of the dq-parameters of the PM machine. the magnitude of ψ qm (which is negative for the motor operation condition of the PM machine) increases with iq . For the non-saturation case the total PM fluxlinkage flows through the d-axis. NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 4 October 26 . Germany . Like Ld and Lq . Finite Element Analysis Results Based on the above calculation FE methods. saturation effects are not negligible and the dq flux linkages become involved functions of both d. when the saturation occurs. i.If the motor works in the magnetic linear region (low current). For each operation point the magnetic permeability of each element obtained from the previous FE simulation under the double excitation is fixed and stored for the further analysis. iq ) ⋅ id + ψ dm ( id . iq ) ⋅ iq + ψ qm ( id . Using the equ. (3) and (7). iq ) ⋅ id ⋅ iq + ψ dm (id .56 A (peak value) and δ = 0° operation condition. iq ) ⋅ iq −ψ qm (id . an additional q-axis PM flux-linkage component is present in the expression of the total flux-linkage. The main point of this method is that it transforms non-linear problems into linear ones by storing the permeabilities from the non-linear analysis. in the following the electromagnetic torque is calculated for I = 2. δ is the electrical angle between total flux linkage due to the stator currents and the rotor q-axis (in the literature this angle is known as torque angle or load angle). The complete magnetic model for a saturated motor can be thus expressed as ψ d ( id . 3. This is the main effect of the saturation. In an analysis made in [1] for a PM machine with inset magnets in the rotor (IPM machine). It is shown that the rotor surface is subjected to a relatively large flux density fluctuation due to the varying reluctance as teeth are passed.27. the d. while it decreases with iq components. In the same way. iq ) − Lq (id . hence high current. ψ d = Ld ⋅ id + ψ dm ψ q = Lq ⋅ iq (6) where.and q-axis current components. Germany . 2005 Wiesbaden.B [T] Fig.27. NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 5 October 26 . It is shown that the results obtained from the Maxwell’s stress tensor method are in good agreement with the results obtained with the magnetic co-energy method.6 T_Coen T [Nm] 0.4 0. 2: Flux line distribution and the flux density along the surface of a PM pole pair.2 0 -0.559 A.2 0 20 40 60 80 100 120 140 160 180 delta [elec. the electromagnetic torque is evaluated using the Maxwell’s stress tensor and the magnetic coenergy method for a fixed current excitation in the stator and for different incremental rotor positions. This torque component is known as cogging torque. Only if the teeth are saturated due to stator current. Torque-load angle 1.2 -0. Figure 3 compares the results obtained with these methods. This effect is normally not depending on the stator current. the load angle is equivalent with the rotor position). usually the cogging torque increases because of the wider effective slot openings. It is caused by the interaction between the magnets and the stator teeth.and the magnetic co-energy method. I=2.2 1 0. degree] Fig. I = I q ). As is known the PM machine with surface mounted magnets in the rotor gives the maximum torque for the case when the load angle is zero ( δ = 0° .4 -0. The cogging torque is often the largest component of torque pulsation in permanent magnet motors. Due to the slotting effect some harmonics (torque pulsations) are added in the toque curve of this type of the machine. 3: Electromagnetic torque versus load angle obtained with Maxwell’s stress tensor. The torque curves vary sinusoidally with the load angle (for a fixed current in the stator.6 -0.8 T_maxw 0. Firstly.8 -1 -1. Figure 4 shows the saturation condition of the iron parts of the PM machine for the given operation condition.27. Fig.2 T [Nm] 1 0. degree] Fig. ψ dm = 1.For the same operation condition as before in the following the electromagnetic torque versus rotor position is evaluated using Maxwell’s stress tensor method and the dq mathematical model of the PM machine. NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 6 October 26 . Using this method the saturation condition of the machine under the double excitation is fixed and stored for the further analysis.559 A. 4: The saturation condition (relative permeability) of the PM machine at I = 2.56 A and δ = 0° . The dq-parameters of the motor are: iq = 2. The fixed permeability method is used to derive the motor parameters.2 240 228 216 204 192 180 168 156 144 132 120 108 96 84 72 60 48 36 24 12 0 0 theta [elec. and the rotor yoke (in the main flux path).method and the dq-mathematical model of PM machine. I=2.8 T_maxw T_dq 0.4 0. id = 0 A. The dq-inductances and flux linkages due to magnets are derived under fixed permeability from the double excitation condition. 5: Electromagnetic torque versus rotor position obtained with Maxwell’s stress tensor. L q = 6.483 mH. The electromagnetic torque results versus rotor position obtained with the Maxwell’s stress tensor and from the dq mathematical model of the PM machine are presented and compared in the figure 5.663 ⋅10−4 Vs . the stator yoke (in the main flux path).6 0.559 A: Torque-rotor position 1. Germany . L d = 6.4 1.483 mH. delta=0°. Also here the obtained results show a good agreement between these methods. Three regions of the machine can be seen to have low relative permeability values: the stator teeth (in the main flux path).531 ⋅10−1 Vs. ψ qm = −7. 2005 Wiesbaden. For this case this method is very time consuming. ∗ψ qm . ∗ψ dm Double excitation condition Torque. ∗ψ qm . the dq model gives only the average components of the electromagnetic torque. Figure 6 shows the calculation procedure of the dq-parameters with fixed permeability method. T Total flux linkage: ψ T τ P − Pole pitch θn < τ P Fixed Permeability method Current flux-linkage: ψ Average values i Lq . The air-gap harmonics are ignored using this model. 2005 Wiesbaden. In co-energy method the calculation time is doubled.As the calculation of the electromagnetic torque with the Maxwell’s stress tensor method gives the total torque components in the air-gap. from which the permeability of each element is fixed and stored for the further analysis. Germany . because with this method the field is solved twice to get an energy difference. start Iˆ. Only one simulation for each operation point is needed with this method. The dq-theory is established based on the assumption that both the winding flux-linkage and the currents are sinusoidal. Also. a second calculation is carried out either with current or magnet single excitation. With this fixed permeability. the Maxwell’s stress tensor method is probably the most commonly used method for the torque calculation since it is uncomplicated to apply and needs a relatively small calculation time. The calculation time for the second simulation with the “frozen” permeability condition is faster because the electromagnetic problem is linearized. NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 7 October 26 . the torque calculation with dq-mathematical model of the PM machine parameters of which are derived with fixed permeability method needs two simulations for each operation point. 6: Calculation flow-chart of the dq-parameters of the PM machine. ∗ψ dm . 4.27. If the saturation effect occurs it is required that the calculation procedure to be repeated for different rotor positions. therefore to derive the average values for these parameters. Computing time As is discussed in the previous sections. The first simulation should be done under the double excitation. Ld . Ld . δ dq-transformation theory θ = 0 : ∆θ : τ P Lq . T Magnet flux-linkage: end ψ m = ψ T − ∗ψ i ∗ Fig. Weiyan L.. The finite element method provides an accurate approach to torque evaluation from the derivation of electromagnetic field distribution. Bolognani S. USA Jianhui H.: “The characteristics of the magnetic saturation in the interior permanent magnet synchronous motor”. Institute for Electrical Drives. J. St. IEEE IAS Annual Meeting. Cracow. 2004. Kim K. University of Federal Defense Munich. Jibin Z. Using these methods the electromagnetic torque versus load angle and rotor position for a given operation point is calculated. Louis. the Maxwell’s stress tensor.: “Finite element calculation of the saturation dq-axes inductance for a direct-drive PM synchronous motor considering cross-magnetization”. In this paper. Poland (ICEM 2004) Bianchi N. July 2005 Kwak S. Conclusions The torque is a very important parameter for both analysis and design of electrical machines.. Singapore (PEDS 2003) NAFEMS Seminar: „Numerical Simulation of Electromechanical Systems“ 8 October 26 .: “Magnetic models of saturated interior permanent magnet motors based on finite element analysis”.. International Conference on Electrical Machines.27. J. Jung H. 2005 Wiesbaden..5. References [1] [2] [3] [4] Gerling D. K.. 2003. International Conference on Power Electronics and Drive Systems. Dajaku G.: “Finite element analysis and torque calculation of the IPM2910 Machine”. The obtained results show a good agreement between these methods.. Germany . three calculation methods based on the results of electromagnetic field FE analysis are used to derive the electromagnetic torque of a PM machine. 1998. the co-energy method and the dq mathematical model for the PM machine.
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