# Mathematical Modeling and Dynamic Simulation of a Class of Drive Systems with Permanent Magnet Synchronous Motors

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2 2. Mathematical modeling of the drive system 2.1. Features of the drive system A simplified block diagram of the drive system under consideration is represented in fig. 1, where the notations are as follows: PC position controller; SC1, SC2 speed controllers; RC1, RC2 current reference blocks; CC1, CC2 three-phase current controllers; IC1, IC2 inverter control blocks; VI1, VI2 voltage source inverters; UR uncontrollable rectifier; C filter capacitor; PS1, PS2 position sensors; PF1, PF2 position feedback blocks; SF1, SF2 speed feedback blocks; M1, M2 motors; L1, L2 loads at the respective coordinate axes; V sr1, V sr2 speed reference signals; V cr1, V cr2 current reference signals; V pf1, V pf2 position feedback signals; V sf1, V sf2 speed feedback signals; V cf1, V cf2 current feedback signals; V d DC link voltage; ω 1, ω 2 motor speeds; θ 1, θ 2 angular positions; T l1, T l2 load torques applied to the respective motor shafts. Fig. 1. Block diagram of the drive system under consideration The motors used are characterized by having permanent magnet-produced fields on the rotors and armature windings on the stators. Both subsystems have identical cascade structures with subordinate regulation. The motors have been controlled in brushless DC motor mode in accordance with the respective rotor positions. Control loops optimization and tuning have been done sequentially, starting from the innermost ones Mathematical description In order to obtain a suitable simulation model some assumptions have been made, such as: motors are unsaturated; eddy-currents and hysteresis effects have negligible influence on the stator currents; motors are symmetrical three-phase machines; self and mutual inductances are constant and independent of the rotor position; devices in the power-electronic circuits are ideal. 332

3 Fig. 2 shows the equivalent circuit diagram of PMSM and the voltage source inverter for one coordinate axis. The corresponding notations are as follows: S1 S6 power electronic switches; D1 D6 freewheeling diodes; v a, v b, v c voltages applied on stator phases a, b and c respectively; i a, i b, i c phase currents; e a, e b, e c back EMF voltages; R a, R b, R c stator phase resistances; L a, L b, L c phase self inductances; L ba, L ca, L cb mutual inductances. Fig. 2. Equivalent circuit diagram of PMSM and the voltage source inverter The vector-matrix model of such a set is described as follows: v a R s 0 0 i a L a L ba L ca v b = 0 R s 0 i b + L ba L b L cb d dt v c 0 0 R s i c L ca L cb L c i a i b i c + e a e b e c, (1) where R s = R a = R b = R c is the stator phase resistance. The back EMF voltage waveforms are expressed by the equation: e a e b e c = ω d dθ Φ a Φ b Φ c, (2) where Φ a, Φ b, Φ c are the stator magnetic fluxes of the motor phases a, b and c respectively. Since the motor windings are star connected (fig. 2), the next relation is valid: i a + i b + i c =0. (3) Assuming there is no change in the rotor reluctance with angular position, then: L a = L b = L c = L; (4) L ba = L ca = L cb = M. (5) Taking into consideration Eqs. (4) and (5), the vector-matrix Eq. (1) is arranged as follows: v a i a i v b = R s i b d a e a + L s i b + e b, (6) dt v c i c i c e c where L s = L M is the stator phase inductance. 333

4 The motor electromagnetic torque can be expressed as: T = e ai a + e b i b + e c i c, (7) ω and the mechanical dynamics equations are as follows: J dω dt = T T l Dω; (8) dθ dt = ω, (9) where J is the total inertia referred to the respective motor shaft; D the viscous damping coefficient. 3. Computer simulation and performance analysis Every specific working regime of the two-coordinate drive system requires an appropriate control algorithm. For this reason, using the MATLAB/SIMULINK software package some computer simulation models have been developed to analyze the system under consideration. Detailed investigation has been carried out by means of mathematical modeling and computer simulation for the respective transient and steady-state regimes at various working conditions Phase currents formation The current controllers have a programmable hysteresis band which determines the respective modulation frequency. Fig. 3 illustrates the principle of current pulses formation. The used notations are as follows: i r reference phase current waveform; 2 i r = i max i min reference hysteresis band; i max and i min maximum and minimum current values, respectively; T m = t on + t off modulation period. Fig. 3. Principle of current pulses formation The modulation frequency is expressed by: a) b) Fig. 4. Influence of the hysteresis band f m = 1 T m. (10) The hysteresis band influence of the first current controller is shown in fig. 4. It represents the phase current waveforms for two different hysteresis bands (2 i r1 and 2 i r2 respectively). As evident, narrowing the zone, i.e. increasing the chopping frequency will result in reduction of the current pulsations. The analysis carried out shows that this frequency is not of a constant value and depends on the following factors: 334

5 hysteresis band; M. Mikhov / Applied and Computational Mechanics 3 (2009) reference motor speed; power circuit electromagnetic time-constant; initial current value. This analysis allows selecting of the optimal hysteresis band in accordance with the respective power switches. In the research carried out the maximum modulation frequency has been limited to a value of f m max < Hz. Fig. 5 illustrates the dynamic maintenance of zero phase current in the respective interval by means of pulse width modulation. Fig. 5. Dynamic maintenance of zero phase current Fig. 6 shows the phase current waveform i a1 and the respective trapezoidal back EMF voltage e a1. In this case the simulation results have been obtained for a quasi-steady-state regime at rated load applied to the motor shaft. The three-phase current waveforms i a1, i b1,andi c1 for the same load torque are represented in fig. 7. Fig. 6. Phase current and back EMF voltage Fig. 7. Three-phase current waveforms 335

6 3.2. Breaking regime M. Mikhov / Applied and Computational Mechanics 3 (2009) Various breaking modes have been investigated aiming at fast and efficient stopping and reversing. Some time-diagrams obtained at reverse speed control with electrical braking are shown in fig. 8, where T b1 is the braking torque. The reference speed values in this case are ±ω 1r = ±157 rad/s. As evident, the braking mode applied ensures good dynamics with maximum reverse time less than 0.1 s Current limitation and speed stabilization The phase currents limitation is provided through the current reference signals, formed by the respective speed controllers. These controllers have been optimized in such a way, that the static errors possibly caused by disturbances are eliminated. Fig. 9 illustrates motor speed stabilization when the load torque changes. The respective transient and steady-state regimes are represented, as well as the drive system reaction to a disturbance expressed as load change. Fig. 8. Reversing with electrical breaking Fig. 9. Current limitation and speed stabilization In this case the load torque applied is equal to the rated value T l1rat and the respective disturbance is T l1 =0.5T l1 rat. The load change is applied during the time interval of t = (0.2s 0.3s). As evident, at an appropriate tuning of the speed controller, static speed error does not appear when T l1 changes. The reference speed is ω 1r = 157 rad/s and the starting current is limited to the maximum admissible value of I 1max, which provides for a maximum starting motor torque. 336

7 3.4. Two-coordinate position control Two algorithms for trajectory formation have been investigated, namely simultaneous and consecutive movements along both coordinate axes. Fig. 10 represents some simulation results obtained for two-coordinate position control with consecutive motion along the coordinate axes. The reference distances are indicated with θ 1r and θ 2r respectively. a) b) Fig. 10. Time-diagrams for consecutive position control along both coordinate axes Fig. 10a illustrates the realization of a reference motion along the first coordinate axis. The respective motor works at maximum speed of ω 1 = ω 1rat, which ensures fast operation of the drive system. Fig. 10b shows the subsequent positioning along the second coordinate axis. In this case the movement is carried out with a lower speed of ω 2 <ω 2rat. Fig. 11. Trajectory of simultaneous movement along both coordinate axes 337

8 A simultaneous movement along both coordinate axes is illustrated in fig. 11, when the two motor speeds are equal. The motors used are identical, both with rated power P rat =0.6 kw and rated angular speed ω rat = 157 rad/s. 4. Conclusion The performance of a two-coordinate drive system with permanent magnet synchronous motors has been discussed in this paper. Both motors have been controlled in brushless DC motor mode in accordance with the respective rotor positions. Using the MATLAB/SIMULINK software package a number of computer simulation models have been developed to analyze this class of drive systems. Detailed study has been carried out for the dynamic and static regimes at various loads, disturbances and work conditions. The analysis shows that the control approach applied ensures good performance, which makes it suitable for a variety of applications in the mechanical industry. The research carried out as well as the results obtained can be used in the design, optimization and tuning of such types of drive systems. They could be also applied in the teaching process. Acknowledgements This work has been supported by the Technical University of Sofia under the research project No. 091NI142-08/2009. References [1] Boldea, I., Nasar, S. A., Electric drives, CRC, Boca Raton, [2] Chen, C., Cheng, M., A new cost effective sensorless commutation method for brushless DC motors without phase shift circuit and neutral voltage, IEEE Trans. on Power Electronics, 2 (22) (2007) [3] Mikhov, M. R., Investigation of a permanent magnet synchronous motor control system, Technical Ideas, 3/4 (38) (2001) [4] Mikhov, M. R., Analysis and simulation of a servo drive system with hysteresis current control, Proceedings of the International Conference on Automatics and Informatics, Sofia, Bulgaria, 2002, pp [5] Ong, C., Dynamic simulation of electric machinery using MATLAB/SIMULINK, Prentice Hall, New Jersey, [6] Safi, S. K., Acarnley, P. P., Jack, A. G., Analysis and simulation of the high-speed torque performance of brushless DC motor drives, IEE Proceedings Electric Power Applications, 3 (142) (1995) [7] Sozer, Y., Torrey, D. A., Adaptive torque ripple control of permanent magnet brushless DC motors, Proceedings on the Applied Power Electronics Conference, Anaheim, USA, 1998, pp [8] Yoon, Y., Kim, D., Lee, T., Choe, Y., Won, C., A low cost speed control system of PM brushless DC motor using 2 Hall-Ics, Proceedings of the International Conference on Mechatronics and Information Technology, Jecheon, Korea, 2003, pp

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