Recent Developments of Electrical Drives - Part 41 potx

III-2.4. Comparison Between Sinusoidal and Trapezoidal Waveforms 401 Figure 4. E.m.f., currents, electromagnetic power, and torque for a three-phase trapezoidal waveform system. For the e.m.f., the width of the interval corresponding to the peak value depends on the width of the magnet. It could vary between 100 ◦ and 150 ◦ . Deviation from the ideal forms of e.m.f. and current implies torque ripple. Finite element models Due to the fact that the ideal wavefor ms are not achieved, numerical models are elaborated to calculate the harmonic content of the current and e.m.f. If the shape of the magnet is defined it is possible to use a finite element model to calculate the air-gap magnetic field. High power disk PM synchronous generators have large diameters, and a flat representation of the machine is used to represent a symmetry period. Finite elements models are elaborated, for the sine-wave and trapezoidal square- wave machines. The models are built for an average value of the diameter. The magnetic characteristics are introduced for each region of the machine: the stator, the rotor, the magnets, and the air-gap. Due to the symmetry of the machine’s geometry, only half- stator and one rotor are designed. The magnetic periodicity is 9. For each model, boundary conditions are imposed: the tangential component of the magnetic induction is equal to zero for the line delimiting the two stators and the normal component of the magnetic induction is equal to zero for the inferior limit of the rotor. Sinusoidal waveforms For the sinus machine, the slot pitch is 3.6 slots/pole, the width of the magnet reported to the pole pitch is 0.6, and a 5/6 short-pitch winding is used. The flux for each coil is determined, 402 Vizireanu et al. Figure 5. Flux lines for nominal charge conditions. and consequently the phase e.m.f. Fig. 5 presents a zoom of the geometry over three poles and the flux lines for nominal charge conditions. For the sinusoidal waveforms machine, the e.m.f. obtained for no-load conditions are presented in Fig. 6. A FFT analysis (Fig. 7) shows the time harmonics spectrum: 4.3% of third harmonic, 0.7% of fifth harmonic, and 0.1% of seventh harmonic. For a constant rotational speed, and imposing sinusoidal currents in phase with the e.m.f, the electromagnetic torque computed for a 750 kW generator has the waveform presented in the Fig. 8. The blue line is the result done by the finite elements software, while the red Figure 6. The e.m.f. for no-load conditions. 050 100 150 200 250 300 Frequency (Hz) Mag (% of Fundamental) 4 3 2 1 0 H3 H5 H7 Figure 7. E.m.f. time harmonics. III-2.4. Comparison Between Sinusoidal and Trapezoidal Waveforms 403 Figure 8. Electromagnetic torque. line represents the torque calculated as the sum of products between phase e.m.f. and phase currents divided by the rotational speed. The curves are almost identically, and it confirms that the armature reaction has a low influence due to the large air-gap. Due to the fifth and seventh harmonics of the e.m.f. the torque presents a 0.5% 6th harmonic pulsation. A comparison is done between the simulation model and a real system (Table 1). The compared parameters are: the air-gap flux density, the per-unit RMS and harmonics values of the e.m.f., the synchronous reactance, and the per-unit electromagnetic torque. Trapezoidal waveforms For the trapezoidalmachine,thedimensionsof theslotsheight, the innerandouterdiameters are maintained. The goal is to impose the same core and copper volumes for both machines. This allows to impose the same level of losses and to compare the performances for the same level of heating. To obtain a trapezoidal e.m.f. waveform, the slot pitch in this case is 3 slots/pole and the winding is a full-pitch one. Table 1. Comparison between simulation and experimental results Compared parameters Simulation Measure Air-gap flux density (T) 0.77 0.77 FEM (pu) 0.98 1 Third Harmonic (%) 4.3 3.7 Fifth Harmonic (%) 0.7 0.7 Seventh Harmonic (%) 0.1 0.1 Phase current (u.r.) 1 1 Current THD (%) 1 1 Synchronous reactance (%) 0.52 0.52 Electromagnetic torque (pu) 0.98 1 404 Vizireanu et al. Figure 9. Trapezoidal e.m.f. for different magnet width. The width of flat zone of the e.m.f. depends on the magnet’s width. An e.m.f. close to the ideal trapezoidal waveform is obtained for a width of the magnet equal to 90% of the pole pitch (Fig. 9). Larger magnets induce higher leakage flux between magnets. The number of coils decreases by 20%, but the width is 20% larger, compared to the sinusoidal machine. Larger slots are used to obtained same volume of copper for trapezoidal waveform machine as for the sinusoidal waveform machine. For both systems, the same DC bus voltage is imposed. To avoid oscillations of the current during commutation of phases, it can be analytically proved that the peak value of the line voltage at the output of the generator should be half of the DC bus voltage. Therefore, the peak value of the e.m.f. for trapezoidal waveform is approximately half of the peak value for sinusoidal waveform. The number of turns/coil is decreased to obtain the proper peak value of the phase e.m.f. The three-phase no-load e.m.f. calculated for nominal speed and a magnet width equal to 90% of the pole pitch are presented in Fig. 10. The width of the e.m.f.’s flat zone is 120 electrical degrees and significant third harmonic due do the large coils and magnets can be noticed. Figure 10. Trapezoidal e.m.f. for no-load conditions. III-2.4. Comparison Between Sinusoidal and Trapezoidal Waveforms 405 EMF Rph Loyo DC Voltage V1 V3 V5 V4 V8 V2 Figure 11. Matlab-Simulink model of the machine and converter. Simulation model A simulation model is used to study the behavior of the wind generator system, from the generator’s shaft until the DC bus. The model uses the e.m.f. waveforms computed with the finite element model. Other parameters of the machine are also introduced to simulate the system:statorresistance,stator reactance,rotorposition. The air-gaplinkageinductance,the slots leakage inductance, and the mutual inductances are computed with the finite element model. Due to large air-gap and large slots, these inductances are linear, practically. The command strategy of the rectifier depends on e.m.f. waveform, imposing rectangular or sinusoidal currents, with or without harmonics injection. For the rectifier, the model allows also to introduce the parameters of each component (diode, transistor, snubber) which permits to estimate the converter losses. The DC bus allows the decoupling of the converters, and the simulation model will be simplified. In this way the group inverter-transformer-grid is replaced by a DC voltage source connected in the DC bus circuit (Fig. 11). The control system imposes the generator speed depending on the wind speed, and using a speed-torque characteristic, a reference torque is generated, which consists in imposing a reference for the quadrature component of the current, Iq. The direct component, Id, is imposed zero. Finally, using an inverse Park transformation, the command voltages are generated (Fig. 12). Therefore, for both types of machine, the currents are imposed in phase with the e.m.f. (Fig. 13). For the trapezoidal waveform machine, the currents are controlled using hysteresis regulators. Figure 12. Control block for sinusoidal waveform machine. 406 Vizireanu et al. Figure 13. E.m.f. and currents for sinusoidal machine. Sinusoidal waveforms In the case of sinusoidal waveforms, the e.m.f. contains fifth and seventh order harmonics, which induce torque oscillations if the current is purely sinusoidal. It is possible to inject current harmonics, as suggested by [2], in order to minimize the torque ripple, but these current harmonics influence also the harmonic content of the DC bus current. An FFT analysis of the electromagnetic torque (Fig. 14) shows a small 6th and 12th harmonics, which confirms the results obtained with the finite element model. Compared to the results presented in Fig. 8, high frequency harmonics are associated with the PWM frequency. For sinusoidal waveforms, the DC bus current obtained has the shape with an envelope done by the three-phase currents (Fig. 15). The harmonic content has very small low- frequency harmonics, and taking into consideration that the DC bus voltage is keptconstant, the power transferred through the DC bus has low fluctuations. Figure 14. Electromagnetic torque for sinusoidal currents. III-2.4. Comparison Between Sinusoidal and Trapezoidal Waveforms 407 1000 800 600 400 200 0 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 times (s) Signal (Input 1) Mag (% of DC component) 30 20 10 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz) Figure 15. DC bus current for sinusoidal phase currents. Trapezoidal waveforms For trapezoidal e.m.f. wavefor ms, the imposed currents have square-waveforms, in phase with the e.m.f. (Fig. 16). In this case problems are encountered during the commutation between two phases. The current in the phase not under commutation has a distortion which depends on the DC bus voltage, the rotational speed, the phase resistance, and the phase inductance. Proper values of these parameters lead to a minimum torque ripple (Fig. 17). In this case the DC bus current presents intervals where it drops to zero (Fig. 18), which means that energy transfer to the grid will have high oscillations. These oscillations have low frequencies (6k order) and depend on the rotational speed. Therefore, an additional filter on the DC bus is required to reduce them. Figure 16. Currents for trapezoidal waveform machine. 408 Vizireanu et al. Figure 17. Electromagnetic torque for trapezoidal phase currents. The number of turns/coil and the slots area decrease significantly for a trapezoidal waveform machine,comparedtoa sinusoidalwaveform machine. Then,thephase resistance and inductance decrease too. The phase resistance for the trapezoidal waveform machine will be: R (tr) = R (sin) N (tr) N (sin) · S (tr) S (sin) (8) Figure 18. DC bus current for trapezoidal phase currents. III-2.4. Comparison Between Sinusoidal and Trapezoidal Waveforms 409 Figure 19. Comparison between electromagnetic torque of the sinusoidal and trapezoidal waveforms machines. where N is the number of turns/coil and S is the number of slots of the studied machines. In order to have the same warming of stator winding with the same stator core volume for both machines, equal copper losses are imposed for both machines: P J(sin) = 3 · R (sin) I 2 rms (9) P J(tr) = 2 · R (tr) I 2 (tr) (10) Comparing with the sinusoidal waveform machine, a comparable level of torque ripple is obtained for a magnet width equal to 90% of the pole pitch (Fig. 19). The trapezoidal waveform machine has 150% of the magnets volume but only 48% of the copper volume of the sinusoidal waveform machine. In this case, the copper losses of trapezoidal waveform machine are lower. The electromagnetic torque fluctuation obtained for the trapezoidal waveforms machine depends on the width of the magnets (Fig. 20). Larger magnets give an electromagnetic torque with minimum oscillation. Decreasing the width of the magnets generates higher torque oscillations and lower average torque value (Fig. 20). In Table 2, the electromagnetic torque obtained by a trapezoidal waveform machine and different magnets width is compared to the torque obtained by a sinusoidal waveform Figure 20. Electromagnetic torque for different magnet’s width. 410 Vizireanu et al. Table 2. Comparison of electromagnetic torque characteristics for different magnet width Magnet Torque (pu) H6 (%) H12 (%) H18 (%) H24 (%) 0.6 1.10 16.7 7.00 3.10 1.32 0.7 1.20 6.15 2.98 1.80 1.15 0.8 1.25 2.27 1.03 0.63 0.34 0.9 1.28 0.985 0.52 0.36 0.10 machine with the same copper volume. If the copper volume is the same for both machines, the rated cur rent of the trapezoidal waveform machine can be increased to obtain the same amount of losses as the sinusoidal machine. As the stator core volume is not changed, the stator cooling area is the same such as the warming of coils. The power-to-weight ratio of trapezoidal waveform machines is 28% higher than sinu- soidal machine but the magnets weight is 50% higher (Table 2). Conclusions A study concerning the influence of the e.m.f. and currents waveform over the torque ripple and over the DC bus current harmonics has been done. Detailed analytical and numerical approaches were elaborated for sinusoidal and trapezoidal waveforms. Ma- chines models have been elaborated using finite element models, and the system machine- converter has been simulated. The interest is to study the possibility of increasing the power density of permanent magnet synchronous machine without altering the quality cri- teria imposed for wind generator systems: low torque oscillation and low DC bus current harmonics. Simulations show that increasing the magnets width, higher power density is obtained for trapezoidal waveforms machine. But problems are also encountered due to the fact that the DC bus voltage should be adapted to the rotational speed to avoid phase current distortions and torque ripple. Another disadvantage is that the DC bus current has high low-frequency harmonics, compared to the sinusoidal waveform machine. That means that additional active and/or passive components should be added, increasing the price of the converter. A solution to decrease the harmonics of the DC bus current is to increase the phase num- ber. Even for sinusoidal waveform machine, poly-phased machine seems a good solution to reduce the torque ripple, to minimize the DC bus current oscillation, and to overcome the technological limits of the power electronic components. Acknowledgment The work presented in this paper was done within ARCHIMED project of “Centre National de Recherche Technologique en G´enie Electrique”, with the support of ERDF, French government, and R´egion Nord—Pas de Calais. . avoid oscillations of the current during commutation of phases, it can be analytically proved that the peak value of the line voltage at the output of the generator should be half of the DC bus voltage. Therefore,. e.m.f. The three-phase no-load e.m.f. calculated for nominal speed and a magnet width equal to 90% of the pole pitch are presented in Fig. 10. The width of the e.m.f.’s flat zone is 120 electrical. level of torque ripple is obtained for a magnet width equal to 90% of the pole pitch (Fig. 19). The trapezoidal waveform machine has 150% of the magnets volume but only 48% of the copper volume of the