76 Rembowski and Pelikant Figure 3. Generated meshes for the shit larger than one third of the air gap width. Taking into account the stator node with number j neighboring with nodes i and i + 1 of the rotor mesh, f the linear interpolation the equation for this node can be described as follows (5): V J θ i − V i (γ i ) − V i+1 (θ i − γ i ) = 0 (5) Where θ i is the angle between nodes with the numbers i and i +1, and γ i is the angle between the stator mesh node j and the rotor mesh node i. j+n+1 i+k+2 j+n j i i+k j+n-i i+k-1 j+n-2 j-1 j-2 i-1 i-2 i+2 i+1 Figure 4. Part of the one level(z =const.) of the mesh with the overlappingregion(parts of electrodes dashed, dimensions enlarge)—symmetrical air gap. I-7. Analysis of Electrostatic Microactuators 77 j+n+1 i+k+2 j+n j i i+k j+n-i i+k-1 j+n-2 j-1 j-2 i-1 i-2 i+2 i+1 Figure 5. Part of the one level(z =const.) of the mesh with the overlappingregion(parts of electrodes dashed, dimensions enlarge)—nonsymmetrical air gap. In the nonsymmetrical model nodes of the stator and the rotor meshes for overlapping bounds have different angle θ and radius r (Fig. 5). The stator mesh node with number j neighbors with four nodes i, i + 1, i +k, and i + k + 1 of the rotor mesh. The equation describing the value of the potential in the j node can be written down using bilinear interpolation function in the following form (6): V j (γ i , r i ) = a 0 γ i r i + a 1 γ i + a 2 r i + a 3 (6) Using equation (6) for each node at both boundaries (outer for the rotor mesh and inner for the stator mesh) one obtains sub matrix of main matrix [M] containing five nonzero elements for each row. As a result one gets nonsymmetrical system of linear equations, which is solved using LDU decomposition method with permutation matrix (7). [M] = [P][L][D][U][ ˜ P] (7) Presented algorithm was implemented in a numerical program, which allows determining a field distribution for every angular position of the rotor and every possible movement of its rotation axis. Integral parameters The application allows calculating integral parameters for every position of the rotor—in particular the system energy that can be written down in general in form (8): W = wd = E 0 D dE d (8) 78 Rembowski and Pelikant According to Maxwell stress tensor, formula defining force components can be described as follows (9): F = S 1 ε (D × n)D − 1 2 | D | 2 n dS = ε S E 2 n − E 2 t 2 dS n + ε S E n E t dS t (9) Using explicit choose shape functions λ i in formula (8), one can calculate the total system co-energy as the sum of the energy accumulated in each of the mesh elements (10). W = e e ε e grad 2 V i λ i d e (10) Proceeding in the same way with the general expression (9) leads to formulas describing force components in relation to surface S, which consists of the sum of elementary surfaces S i in the single mesh elements. As the result one obtains the normal force component in form (11): F n = ε 0 2 i S i (grad 2 n V i λ i − grad 2 t V i λ i ) dS i (11) By analogy the tangent force components can be written down as follows (12): F t = i S i ε 0 (grad n V i λ i · grad t V i λ i ) dS i (12) Numerical verification The basis of the verification of the presented model was the numerical experiment. Air gap energy was calculated in the part common for both the rotor and the stator meshes and obtained results were compared. The quality of energy calculation was determined on the basis of numerical testing of the convergence of the solutions from both meshes (Figs. 6 and 8). Theinfluence of mesh density on the value of energy accumulated inthe air gap was analyzed for different positions of rotor (rotation and shift). It allows determining minimal mesh density for given accuracy of computations. A clear tendency of both curves to reach the same value was observed. It means a convergence of energy value and exact value. The convergence was observed ir respective of the rotor’s location. However, the slope of the curve changes, which results from different energy values for different locations of the rotor. At the same time the difference between the energy value calculated from stator mesh and the energy value calculated from rotor mesh was computed (Fig. 7). Convergence to zero of the above difference was observed. Like before the tendency appears irrespective of the rotor’s position. Convergence of solutions determined on the basis of the values of potentials in the nodes of both rotor and stator meshes confirm the thesis that the implemented model is correct. I-7. Analysis of Electrostatic Microactuators 79 Figure 6. Influence of the mesh density on the value of air gap energy for rotor position 0 ◦ . Calculating the changes of energy value for different rotor angular position (Fig. 9) allows determining static torque as follows (13). M = ∂W ∂γ (13) As a matter of fact, the approach based on Maxwell’s tensor is used (11, 12), whereas the above formula (13) is only a method of confirming the correctness of the results. Figure 7. Influence of the mesh density on the value of air gap energy for rotor position 30 ◦ . 80 Rembowski and Pelikant Figure 8. Ratio of air gap energy calculated from stator and rotor meshes. Figure 9. Dependence of air gap energy on the rotation angle of the rotor (about 25,000 mesh elements). Conclusions Another step in developing the model will be extending it to the analysis of microacutators with leant rotation axis. It will require interpolation by three variable function and not one variable function (symmetrical model)or twovariable function (model withshifted rotation axis) as so far. I-7. Analysis of Electrostatic Microactuators 81 The most important conclusion resulting from the carried out studies on the three- dimensional model forthe analysis of theelectrostatic micromotors isthat it allowseffective analysis for any position of the rotor—both rotation and rotation axis shift. Presented algorithm allows correct and exact representation of the changing width air gap in the model. Since significant part of the main matrix rows is calculated only once and it’s only recalculated fragment is the one representing the air gap, it is possible to reduce computation time. The results of the numerical tests confirm the thesis about the correctness of the model. Short computation times are obtained even with quite big number of mesh elements. References [1] G. Hainsworth, D. Rogger, P. Leonard, 3D finite element modeling of conduction supports in coilguns, IEEE Trans. Magn., Vol. 31, No. 3, pp. 2052–2055, 1995. [2] M. Mehergany, S.F. Bart, L.S. Tavrow, J.H.Lang, S.D.Senturia, M.F. Schelecht, A studyofthree microfabricated variable capacitance motors, Sens. Actuators, A21–A23, pp. 173–179, 1990. [3] A.Pelikant,Analizapolowo-obwodowasilnik´owelektrostatycznychielektromagnetycznychza- silanychimpulsowo,WydawnictwoPolitcchnikiL odikicy, ZesrytyNauKowenr 908, Rozprowy Naukowe, Z. 111 2002. [4] R. Perrin-Bit, J. Coulomb,A three dimensionalfine elements meshconnection for problem with movement, IEEE Trans. Magn., Vol. 31, No. 3, pp. 1920–1923, 1995. [5] I. Tsukerman, Overlapping finite elements for problems with movement, IEEE Trans. Magn., Vol. 28, No. 5, pp. 2247–2249, 1992. I-8. COUPLED FEM AND SYSTEM SIMULATOR IN THE SIMULATION OF ASYNCHRONOUS MACHINE DRIVE WITH DIRECT TORQUE CONTROL S. Kanerva 1 , C. Stulz 2 , B. Gerhard 3 , H. Burzanowska 2 ,J.J¨arvinen 3 and S. Seman 1 1 Laboratory of Electromechanics, Helsinki University of Technology, P.O. Box 3000, FI-02015 HUT, Finland sami.kanerva@hut.fi, slavomir.seman@hut.fi 2 ABB Switzerland Ltd, Large Drives, Austrasse, CH-5300 Turgi, Switzerland christian.stulz@ch.abb.com, halina.burzanowska@ch.abb.com 3 ABB Oy, Electrical Machines, P.O. Box 186, FI-00381 Helsinki, Finland bilal.gerhard@ch.abb.com, jukka.jarvinen@fi.abb.com Abstract. A compound drive simulator is presented, where a finite element method (FEM) model of the electric motor is coupled with a frequency converter model and a closed-loop control system. The method is implemented for SIMULINK and applied on a 2-MW asynchronous machine drive. The results are validated by measurements and the performance is compared with an analytical motor model. It is shown that simulation with the FEM model provides very good results and gives much better insight in the motor behavior than the analytical model. Introduction As the demands for performance of electric drive systems increase, also the simulation software must follow the requirements. Designers of frequency converters and electric motors rarely work in the same location, but they must be able to model both parts of the drive as accurately as possible. Naturally, different expertise is required to model electrical machines or power electronics, but the key issue is to couple these models together in a way that experts in both fields can profit from each other by using the most advanced simulation models in their design. Accurate modeling of digital control systems requires simulation in multiple timescales, because different sampling times are used for measurement, filtering, estimation, and mod- ulation. By including all detailed functions and sample times, it is possible to create very accurate simulation modelsof the converter control. In such a case, however, also a detailed electrical machine model is needed in order to get the maximum advantage of the drive model. Finite element method (FEM) is a widely known method to model electrical machines with high accuracy. For standard-type machines, two-dimensional field solution coupled S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives, 83–92. C 2006 Springer. 84 Kanerva et al. with simple circuit equations of the windings is usually accurate enough, when the cross- section geometry and material properties are known [1]. The most problematic in the drive simulation is to couple the FEM computation with the converter model. Most obvious method would be to couple the converter model in the FEM code and solve all the equations simultaneously with uniform time steps [2,3]. However, such an approachis hardly applicable to a detailed converter model with digitalclosed-loop control because of the amount of programming, and the demand for common time step length would make the simulation too heavy with respect to existing computing facilities. Reference [4] presents an indirect method for coupling time-stepping FEM simulation with SIMULINK using multiple sample times for different parts of the system model. The method was applied to a cage induction motor and a frequency converter with direct torque control (DTC). The model of the control system was developed in order to investigate control-related topics and verified for steady-state and transient operation of the drive. In its original state, it was using a motor model that was based on analytical equations. In this paper, thesame method isapplied to an asynchronous motor drive withDTC. The frequency converter modelis based ona realapplication, comprising adetailed model of the digital control system. The frequency converter model is implemented in SIMULINK and it is coupled with a two-dimensional FEM model of the asynchronous motor. The system is simulated in steady-state and transient operation, and the simulation results are validated by a comparison with the measured results. Compound model of inverter-fed electrical drive The general structure of the compound drive simulator is shown in Fig. 1. The model is implemented in SIMULINK but the execution of the model is controlled from within Script file: runA6ka.m Input data - Environment - Model - Operating - Starting conditions Calculation of initial conditions Setup of SIMULINK Run simulation Output results (Plots, ) Parameters Management Speed ControlTorque Control Motor Inverter Plot files DC circuit 3-Phase 3-Level Inverter Motor Process Measurements Inverter control Model Overall DC voltage Phase voltages Torque Speed Phase currents Half DC voltages DC currents Speed reference Torque reference Flux reference Inverter control Measurements Flux reference Speed control Torque reference Load model Speed reference conditions Figure 1. General structure of the drive simulator. I-8. Coupled FEM and System Simulator 85 + – neutral to motor Overall DC voltage Figure 2. Simulation model of the three-phase three-level inverter. MATLAB. This allows to specify the plant parameters, operating and starting conditions very easily. Based on the selected operating conditions, theinitial conditions for continuous and discrete time states are determined. This allows to start the simulation in a reasonably stable operating point. The machine model in this drive simulator can be selected to be the simple analytical or the precise FEM-based model. The main components of the simulated plant are DC circuit, inverter, motor, process, and control. Two basic control schemes can be selected: torque or speed control. Inverter and DC link The three-level inverter is modeled as a set of ideal switches, which can connect the phase voltages to either plus, neutral, or minus potential of the DC links. Fig. 2 gives a rough overview. The switching pattern is given by the drive control. The status of the switches together with the phase currents determines the currents in the DC bus bars of the DC link. The current in the neutral bus bar is used to calculate the potential of the neutral point of the DC link. The phase voltages transferred to the motor terminals are defined by DC link voltages and switching pattern. Analytical motor model and load The analytical motor model is used for simulations that will be compared to the FEM-based motor model.It isbased onthe well-known space vector representationof theasynchronous machine. Ituses boththe stator and the rotor fluxes as statevariables. The followingfeatures are present in the model: r constant air-gap and sinusoidal flux distribution along the air-gap r no iron losses r resistances and inductances are independent of frequency and temperature r the magnetizing inductance can saturate with increasing main flux Themodel needs phase voltagesandspeedasinputsandproduces phase currents and air-gap torque as outputs. The driven process is described by the differential equation of motion. A single inertia is used. The loadtorque mayfollow several functions of the speed (constant, linear, quadratic, or mixed). The mechanical mass is driven by the electromagnetic torque of the motor and gives the speed as output. 86 Kanerva et al. v3_vs v3_s A6ka VECTOR v3_s Vector -> Switching me t_load n Mechanical System n_rot t_load Load Model Inverter Induction Machine Model (analytical or FEM) Current Meas. Speed Meas. v3_s v3_is vdc1_t2 vdc2_t2 n_rot VECTOR Control_dtc6000_AD vdc_id Voltage Meas. Figure 3. Model of the drive system implemented in SIMULINK. Control The control model describes speed/torque control using a DTC algorithm. The main func- tions ofthe ACS6000drive areimplemented asdiscrete functions on different time levels to appropriately represent the behavior of the real drive. The detailed description of the DTC control cannot be in the scope of this paper. The top level of the SIMULINK environment is shown in Fig. 3. Model of the asynchronous motor Modeling by finite element method (FEM) The FEMmodel ofthe motor is based ontwo-dimensional finiteelement method and circuit equations of the windings [1]. The magnetic field in the core region is calculated using magnetic vector potential formulation, in which the vector potential and current density have only z-axis components. The phase windings in the stator or rotor are modeled as filamentary conductors with uniformly distributed current flowing through all the coils that belong to the same phase. The rotor bars aremodeled assolid conductors,inwhichthecurrentdensity variesaccording to eddy currents. The sources of the magnetic field are the phase currents, the voltage drop in the rotor bars and the magnetic force of the permanent magnets, depending on the type and construction of the machine. The relations between voltage and current are determined in the circuit equations of the stator and rotor windings, which also include the end-winding impedances and the short-circuit rings. As a result, only phase voltages are needed as an electrical input for the FEM model. The electromagnetic torque is calculated by virtual work principle, and the movement of the rotor is determined from the equation of motion. At each time step, new position is calculated for the rotor and the air-gap mesh is refined. FEM block for SIMULINK The FEM computation is implemented as a functional block in SIMULINK using dy- namically linked program code (S-function), as illustrated in Fig. 4. The stator voltage . node i. j+n+1 i+k+2 j+n j i i+k j+n-i i+k-1 j+n-2 j-1 j-2 i-1 i-2 i+2 i+1 Figure 4. Part of the one level(z =const.) of the mesh with the overlappingregion(parts of electrodes dashed, dimensions. enlarge)—symmetrical air gap. I-7. Analysis of Electrostatic Microactuators 77 j+n+1 i+k+2 j+n j i i+k j+n-i i+k-1 j+n-2 j-1 j-2 i-1 i-2 i+2 i+1 Figure 5. Part of the one level(z =const.) of the mesh with. model electrical machines with high accuracy. For standard-type machines, two-dimensional field solution coupled S. Wiak, M. Dems, K. Kom ˛ eza (eds.), Recent Developments of Electrical Drives,