8.urmila bandaru 86-97
Upcoming SlideShare
Loading in...5
×
 

8.urmila bandaru 86-97

on

  • 517 views

 

Statistics

Views

Total Views
517
Views on SlideShare
517
Embed Views
0

Actions

Likes
0
Downloads
5
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

8.urmila bandaru 86-97 8.urmila bandaru 86-97 Document Transcript

  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 Design of Multilevel Inverter Driven Induction Machine Urmila Bandaru G.. Pulla Reddy Engineering College, Kurnool A.P., India E-mail: urmila913@gmail.com Subbarayudu Brindavan Institute of Technology, Kurnool A.P., India E-mail: dsr@gmail.comThe research is financed by Asian Development Bank. No. 2006-A171(Sponsoring information)AbstractMultilevel inverters have gained interest in recent years in high-power medium-voltage industry. Thispaper considered the most popular structure among the transformer-less voltage source multilevel inverters,the diode-clamped inverter based on the neutral point converter. This paper proposes a single carrier multi-modulation SVPWM technique with conventional space vector switching sequence. Simulation resultspresents comparison of single and multicarrier conventional space vector switching sequence with generalswitching sequence of nine-level diode-clamped inverter for stator currents, electromagnetic torque andspeed for constant modulation index and for constant V/f control method. Simulation is carried out inMATLAB-Simulink software.Keywords- Multilevel inverter, APODC, SVPWM, total harmonic distortion, Diode-clamped inverter,SCMMOS, MCMMOS, Induction machine, synchronously rotating reference frame1. IntroductionMultilevel inverters have drawn tremendous interest in high-power medium-voltage industry. In literature,inverters with voltage levels three or more referred as multilevel inverters. The inherent multilevel structureincrease the power rating in which device voltage stresses are controlled without requiring higher ratings onindividual devices. They present a new set of features that suits well for use in static reactive powercompensation, drives and active power filters. Multilevel voltage source inverter allows reaching highvoltages with low harmonics without use of series connected synchronized switching devices ortransformers. As the number of voltage levels increases, the harmonic content of output voltage waveformdecreases significantly. The advantages of multilevel inverter are good power quality, low switching losses,reduced output dv/dt and high voltage capability. Increasing the number of voltage levels in the inverterincreases the power rating. The three main topologies of multilevel inverters are the Diode clampedinverter, Flying capacitor inverter, and the Cascaded H-bridge inverter by Nabae et al. (1981).The PWM schemes of multilevel inverters are Multilevel Sine-Triangle Carrier Pulse Width Modulation-SPWM and Space Vector Pulse Width Modulation-SVPWM. Multilevel SPWM involves comparison ofreference signal with a number of level shifted carriers to generate the PWM signal. Due to its simplicityand its well defined harmonic spectrum which is concentrated at the carrier frequency, its sidebands, and itsmultiples with their sidebands, the SPWM method has been utilized in a wide range of AC driveapplications. However, the method has a poor voltage linearity range, which is at most 78.5 % of the six-step voltage fundamental component value, hence poor voltage utilization. Therefore, the zero sequencesignal injection techniques that extend the SPWM linearity range have been introduced for isolated neutral 86
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011load applications which comprise the large majority of AC loads. The carrier based PWM methods canoperate with high switching frequency and offer high waveform quality and implementation advantages.Carrier based PWM methods employ the per carrier cycle volt-second balancing principle to program adesirable inverter output voltage waveform. The programmed PWM technique, SVPWM involvessynthesizing the reference voltage space vector by switching among the nearest voltage space vectors.SVPWM is considered a better technique of PWM owing to its advantages (i) improved fundamentaloutput voltage (ii) reduced harmonic distortion (iii) easier implementation in microcontrollers and DigitalSignal Processor. This paper considered the most popular structure among the transformer-less voltagesource multilevel inverters, the diode-clamped converter based on the neutral point converter withSVPWM technique by Carrara et al. (1992). This paper proposes a single carrier multi-modulationtechnique for multilevel inverter driven induction machine, modeled in synchronously rotating referenceframe. Simulation results present comparison of single and multicarrier SVPWM of nine-level diode-clamped inverter. Improved fundamental component of voltage is observed with SCMM method. Reducedtotal harmonic distortion and current ripple can be observed with Alternate Phase Opposition DispositionCarrier (APODC) SVPWM technique by Leon et al. (1999). Simulation is carried out in MATLAB-Simulink software.2. Diode-Clamped Multilevel InverterA three-phase nine-level diode-clamped inverter is shown in Fig.1. Each phase is constituted by 16switches. The complementary switch pairs for phase ‘A’ are (Sa1, Sa1’), (Sa2, Sa2’), (Sa3, Sa3’), (Sa4, Sa4’), (Sa5,Sa5’), (Sa6, Sa6’), (Sa7, Sa7’), (Sa8, Sa8’) and similarly for B and C phases. Clamping diodes carry the full loadcurrent by Jose Rodriguez (2002).Table1 shows phase to fictitious midpoint ‘o’ of capacitor string voltage (V AO) and line-to-line voltage(VAB) for various switching’s. Table 1 Nine-Level Inverter Voltage States Sa1 Sa2 Sa3 Sa4 Sa5 Sa6 Sa7 Sa8 VAO VAB 1 1 1 1 1 1 1 1 +Vdc/2 Vdc 0 1 1 1 1 1 1 1 +3Vdc/8 7Vdc/8 0 0 1 1 1 1 1 1 +Vdc/4 6Vdc/8 0 0 0 1 1 1 1 1 +Vdc/8 5Vdc/8 0 0 0 0 1 1 1 1 0 5Vdc/8 0 0 0 0 0 1 1 1 -Vdc/8 3Vdc/8 0 0 0 0 0 0 1 1 -Vdc/4 2Vdc/8 0 0 0 0 0 0 0 1 -3Vdc/8 Vdc/8 0 0 0 0 0 0 0 0 -Vdc/2 0 87
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 Figure 1 Circuit Diagram of 3 Phase Nine Level Diode Clamped Inverter 88
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 818 718 618 518 018 418 318 218 118 828 728 628 528 028 428 328 288 817 128 717 617 517 017 417 317 217 117 827 838 738 638 538 038 438 338 238 816 327 138 716 727 627 527 027 427 227 127 616 516 016 416 316 216 116 826 837 848 748 648 548 048 448 348 248 148 815 715 726 637 537 037 437 337 237 137 737 615 626 526 026 426 326 226 126 515 015 415 315 215 115 808 708 608 508 008 825 836 847 747 647 547 047 447 408 308 208 108 810 710 725 736 636 536 036 436 336 347 247 157 610 625 525 025 425 325 225 236 136 510 010 410 310 210 110 125 846 807 858 758 658 558 058 820 835 735 746 707 607 507 007 407 458 358 258 814 158 714 720 620 635 646 546 046 446 346 307 207 107 614 514 205 535 435 435 335 235 246 146 014 020 868 420 768 320 220 120 135 806 758 668 568 068 414 757 314 657 214 114 824 830 845 745 607 557 057 457 468 368 268 813 606 506 168 713 724 730 630 546 006 406 306 357 257 157 545 045 613 624 524 035 445 345 245 206 106 030 878 430 778 513 013 420 867 330 230 130 145 424 767 324 667 678 840 805 658 314 756 224 124 578 078 313 656 213 556 567 823 834 734 740 507 113 067 467 478 378 278 812 605 505 056 178 712 723 623 634 046 005 456 356 367 267 167 540 040 405 612 523 435 440 305 205 256 156 512 034 877 888 340 866 434 334 788 688 240 140 105 012 855 320 423 766 777 234 844 800 755 323 223 677 577 134 588 088 488 700 214 312 655 666 123 822 833 733 744 600 212 112 566 066 077 477 377 388 288 644 500 555 711 722 622 633 544 055 455 466 366 266 277 177 188 811 533 044 000 611 511 522 033 400 300 355 255 155 166 022 433 444 011 865 422 876 887 787 344 687 244 200 100 850 411 322 333 587 750 311 765 776 676 233 576 133 144 087 487 832 704 211 222 076 843 804 604 650 665 565 122 065 476 376 387 287 721 643 111 050 465 276 732 743 543 504 550 450 365 265 176 187 821 532 404 350 165 621 632 032 043 004 304 150 021 421 875 432 886 443 786 343 686 586 204 521 243 250 854 860 760 321 775 332 675 232 575 075 143 086 486 842 803 132 286 703 754 654 660 221 560 121 060 460 475 375 386 731 742 354 104 260 275 175 831 642 603 503 554 054 454 360 186 631 531 542 042 003 403 303 203 254 154 160 031 431 885 442 785 685 242 585 142 103 342 870 770 670 570 131 070 085 485 331 231 802 853 864 764 664 564 064 464 470 370 385 285 741 702 753 653 553 053 453 353 364 264 270 841 170 185 641 602 502 002 402 302 202 253 153 164 541 041 441 341 241 141 102 880 780 680 580 080 480 852 863 874 774 674 074 474 374 380 280 574 701 752 763 663 563 363 263 274 174 063 463 801 601 652 163 180 552 052 452 352 252 152 501 001 401 201 101 884 301 584 784 684 084 862 873 484 284 773 673 573 073 473 384 751 762 373 173 662 562 062 462 362 273 851 651 262 184 551 051 451 351 251 162 151 872 283 761 883 783 683 583 083 483 383 861 172 772 672 572 072 472 372 272 183 661 561 061 461 361 261 161 882 782 682 582 082 482 382 282 871 771 671 071 271 171 182 571 471 371 881 681 581 081 481 381 281 181 781 Figure 2 Nine-Level Inverter State Space Vector Diagram3. SVPWM ImplementationImplementation of SVPWM involves (i) Sector identification-where the tip of the reference voltage vectorlies (ii) Nearest three voltage space vectors (NTV) identification (ii) Determination of the dwelling time ofeach of these NTVs (iv) Choosing an optimized switching sequence.Space vector diagram of nine-level inverter with its 729 (=39) vectors are shown in Fig 2. Each sector takes60 degrees. Sector1 diagram is shown in Figure 3. Each sector consists of 64 regions 177 vectors.3.1 Sector and Region identificationThree phase instantaneous reference voltages (1) are transformed to two phase (2). Every 60 degrees, fromzero to 360 degrees constitute a sector. Identification of the sector in which the tip of the reference vectorlies is obtained by (3). (1) 89
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 (2) ; (3)Where θ is the angle varies from 0 to 2π.Amplitude and angle of the reference vector are obtained from (3).From Park’s transformation the di-phase, α-β components are: (4) (5)Region is obtained by normalizing the di-phase components of the space vector (4)-(5) of an n-levelinverter through division by V dc/n-1, where Vdc is the dc link voltage.3.2 Determination of the duration of nearest three voltage space vectorsSwitch dwelling duration is obtained from (6)-(7). (6) (7)3.3 Determination of optimized switching sequenceConsider reference vector is lying in sector1 region 35 of Figure 3. The nearest three space vectors forswitching sequence are , , .The space redundant vectors for sector1-region 35, are, for 4 0 8, 3 5 7, 2 3 6, 1 2 5 (4 redundantvectors), for 4 4 8, 3 3 7, 2 2 6, 1 1 5 (4 redundant vectors), and for 3 4 8, 2 3 7, 1 2 6 (3 redundantvectors).An optimized switching sequence starts with virtual zero vector’s state. A virtual zero vector is withminimum offset from zero vector of two-level inverter. Based on the principles derived in literature fortwo-level inverter, for Region 35, the switching sequence is 4 0 8→4 4 8→ 3 4 8→ 3 4 7→ 3 3 7→ 2 3 7→2 3 6→ 2 2 6→ 1 2 6→ 1 2 5→ 1 1 5 during a sampling interval and 1 1 5→ 1 2 5→ 1 2 6→ 2 2 6→ 2 36→ 2 3 7 → 3 3 7→ 3 4 7→ 3 4 8 → 4 4 8 → 4 0 8 during the subsequent sampling interval. This sequenceuses all the space redundant vectors of each state by Anish Gopinath et al.(2007), (2009).4. Mathematical Modeling of Induction MachineThree-phase asynchronous or induction machines which contain a cage, are very popular motors in manyindustrial variable speed drive applications and all this is due to its simple construction, robustness,inexpensive, reliability, good efficiency and good self-starting capability and available at all power ratings.Progress in the field of power electronics and microelectronics enables the application of induction motorsfor high-performance drives, where traditionally only DC motors were applied. During starting up andother severe transient operations induction motor draws large currents, produces voltage dips, oscillatorytorques and can even generate harmonics in the power systems by Vivek Pahwa et al.(2009), Ogubuka etal.(2009). It is important to predict these phenomena. A transient free operation of the induction machine is 90
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011achieved only if the stator flux linkages are maintained constant in magnitude and its phase is stationarywith respect to the current by Krishnan (2003) and Leon et al. (1999). Various models have been developedand the qd0 or two axis model for the study of transient behaviors has been tested and proved to be veryreliable and accurate. 64 63 49 62 48 61 0 36 47 60 35 46 59 25 P 34 45 58 24 33 44 57 16 23 32 0 43 56 15 22 31 42 55 9 14 21 30 41 54 8 13 20 29 40 53 4 7 12 0 19 28 39 52 3 6 11 18 27 38 51 1 2 5 10 17 26 37 50 Figure 3 Sector1 Regions Space Vector Representation4.1 Synchronously Rotating Reference Frames ModelThe three phase balanced voltages are transformed to di-phase components which are in stationary rotatingreference frame. These are transformed to synchronously rotating reference frames using (11). The speed ofthe synchronously rotating reference frames is the stator supply angular frequency. e  vqds  T c vqds (8) e vqds  vqs e e vds , t vqds  vqs vds  t (9)   sin  T   cos c  sin cos   (11)   91
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 Vqs   Rs  Ls p e s Ls Lm p s Lm  iqs  e  e  e Vds     s Ls Rs  Ls p  s Lm Lm p  ids   (13) Vqr   e Lm p (s  r ) Lm Rr  Lr p (s  r ) Lr  iqr  e  e   e  Vds   (s  r ) Lm   Lm p  (s  r ) Lr Rr  Lr p  ids   The electromagnetic torque is 3PTe  Lm (iqsidr  iqrids ) e e e e (14) 224.2 V/f Control on Induction MotorMost of the industrial loads are operated based on constant Volts/Hz control method of speed because of itssimplicity. Neglecting the stator resistance drop, the ratio of supply voltage to frequency is maintainedconstant by varying these variables. When the frequency approaches to zero, the magnitude of the statorvoltage also tends to zero and the stator resistance absorbs this low voltage. The stator resistance drop iscompensated at low speed by injecting the boost voltage to maintain rated air gap flux thus full load torqueis available up to zero speed. At steady state operation, if load torque is increased, the slip increases withinstability limit and a balance will be maintained between the developed torque and the load torque. Thispaper considered the V/f control on induction machine by Krishnan (2003).5. Simulation Results and ConclusionsSimulation is carried out on nine-level diode-clamped inverter driven induction machine in synchronouslyrotating reference frame for two methods of Space Vector PWM technique at switching frequency 1.5 KHzwith APODC technique. (i) SCMM CSVPWM-Single Carrier Multi-Modulation for Conventional SVPWM (ii) MCMM CSVPWM-Multi-Carrier Multi-Modulation for Conventional SVPWMIn MCMM the carriers are in Alternative phase opposition disposition, where each carrier band is shiftedby 1800 from the adjacent bands. Multi-Carrier Multi-Modulation results reduced harmonic distortion with reduced fundamental component;however Single Carrier Multi-Modulation results reduced harmonic distortion with highly improvedfundamental component of voltage.Figure4 and Figure5 are responses of induction machine for stator currents, electromagnetic torque andspeed of MCMM and SCMM respectively.The response is observed for speed reversal from +314 rad/sec to -314 rad/sec at time of 0.86 seconds andfrom -314 rad/sec to +314 rad/sec at time of 0.85 seconds and 1.01 seconds respectively, in Figure6 andFigure7. Transients are more in MCMM compared to SCMM. Reduced oscillatory behavior is observed inMCMM. Settling time is less in SCMM compared to MCMM.Figure6 shows the toque response with speed reversal at 0.86 seconds. When compared to MCMM, SCMMresults more torque.Figure7 shows the speed response with change in toque from 0 to 50 N-m at 0.4 seconds. When comparedSCMM, dip in speed is more in MCMM.Figure8 and Figure9 are the pole, phase and line voltages for SCMM and MCMM methods respectively. 92
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011Figure12 and Figure13 show the stator currents, torque and speed response for constant V/f control. Theload torque requirement shifts from 0 to 50 N-m at 0.4 seconds. Modulation index is 0.906 (50 Hz) at 0seconds, 0.84 (46 Hz) at 0.8 seconds, 0.73 (40 Hz) at 1.2 seconds, 0.62 (35 Hz) at 1.6 seconds, 0.906 (-50Hz) at 2 seconds, 0.84 (-46 Hz) at 2.4 seconds, 0.73(-40 Hz) at 2.8 seconds, and 0.62 (-35 Hz), at 3.2seconds.Figure14 and Figure15 are the pole, phase and line voltages for SCMM and MCMM methods respectivelyfor constant V/f control.When frequency falls below 40 Hz the MCMM performance is poor compared to SCMM.Ripple content is high with V/f control in SCMM. 400 200 0 -200 -400 0 0.5 1 1.5 2 400 200 0 -200 -400 0 0.5 1 1.5 2 400 200 0 -200 -400 0 0.5 1 1.5 2 Figure 4 MCMM Stator Currents, Torque and Speed 400 200 0 -200 -400 0 0.5 1 1.5 2 400 200 0 -200 -400 0 0.5 1 1.5 2 400 200 0 -200 -400 0 0.5 1 1.5 2 Figure 5 SCMM Stator Currents, Torque and Speed 93
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 400 300 200 100 0 -100 -200 -300 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 300 200 100 0 -100 -200 -300 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Figure 6 SCMM, MCMM Stator Currents for speed reversal from +314rad/sec to -314 rad/sec 300 200 100 0 -100 -200 -300 -400 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7 300 200 100 0 -100 -200 -300 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7 Figure 7 SCMM, MCMM Stator Currents for speed reversal from -314rad/sec to +314 rad/sec 100 0 -100 -200 -300 -400 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 100 0 -100 -200 -300 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 Figure 8 SCMM, MCMM Electromagnetic torque for speed reversal at 0.86sec 94
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 320 300 280 260 240 0.35 0.4 0.45 0.5 0.55 0.6 350 300 250 200 0.35 0.4 0.45 0.5 0.55 0.6 Figure 9 SCMM, MCMM rotor speed when torque change from 0 to +50 N-m 100 50 0 -50 -100 0 0.01 0.02 0.03 0.04 0.05 0.06 200 100 0 -100 -200 0 0.01 0.02 0.03 0.04 0.05 0.06 200 100 0 -100 -200 0 0.01 0.02 0.03 0.04 0.05 0.06 Figure 10 SCMM Pole, Phase and Line Voltage 100 50 0 -50 -100 0 0.01 0.02 0.03 0.04 0.05 0.06 200 100 0 -100 -200 0 0.01 0.02 0.03 0.04 0.05 0.06 200 100 0 -100 -200 0 0.01 0.02 0.03 0.04 0.05 0.06 Figure 11 MCMM Pole, Phase and Line Voltage 95
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 500 0 -500 0 0.5 1 1.5 2 2.5 3 3.5 500 0 -500 0 0.5 1 1.5 2 2.5 3 3.5 2 0 -2 0 0.5 1 1.5 2 2.5 3 3.5 Figure 12 SCMM Stator Currents, Torque and Speed with V/f Control 500 0 -500 0 0.5 1 1.5 2 2.5 3 3.5 200 0 -200 0 0.5 1 1.5 2 2.5 3 3.5 2 0 -2 0 0.5 1 1.5 2 2.5 3 3.5 Figure 13 MCMM Stator Currents, Torque and Speed with V/f Control 100 0 -100 0 0.5 1 1.5 2 2.5 3 3.5 200 0 -200 0 0.5 1 1.5 2 2.5 3 3.5 200 0 -200 0 0.5 1 1.5 2 2.5 3 3.5 Figure 14 SCMM Pole, Phase and Line Voltages with V/f Control 96
  • Innovative Systems Design and Engineering www.iiste.orgISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)Vol 2, No 6, 2011 100 0 -100 0 0.5 1 1.5 2 2.5 3 3.5 200 0 -200 0 0.5 1 1.5 2 2.5 3 3.5 200 0 -200 0 0.5 1 1.5 2 2.5 3 3.5 Figure 15 MCMM Pole, Phase and Line Voltages with V/f Control4. References Nabae, Takahashi, Akagi.(1981) “A Neutral-Point Clamped PWM inverter”, IEEE Transactions on I.A., Vol. IA-17, No. 5, pp. 518-523. Carrara, Gardella, Marchesoni, Salutari, and Sciutto, (1992) “ A New Multilevel PWM Method: A theoretical Analysis”, IEEE Transactions on Power Electronics, Vol. 7, No. 3, July, pp.497-505. José Rodríguez, Jih-Sheng Lai, Fang Zheng Peng (2002) “Multilevel Inverters: A Survey of Topologies, Controls, and Applications”, IEEE Trans., VOL. 49, NO. 4, AUGUST Anish Gopinath, Baiju, (2007) “Space Vector PWM for Multilevel Inverters- A Fractal Approach” PEDS, Vol.56, No. 4, April, pp 1230-1237 Anish Gopinath, Aneesh Mohammed, and Baiju, (2009) “Fractal Based Space Vector PWM for Multilevel Inverters-A Novel Approach” IEEE Transactions on Industrial Electronics, Vol.56, No. 4, April, pp 1230-1237 Vivek pahwa, Sandhu (2009) “Transient Analysis of Three-phase Induction Machine using Different Reference Frames” ARPN Journal of Engineering and Applied Sciences, Vol. 4, No.8. Ogabuka, Eng (2009) “Dynamic Modelling and Simulation of a 3-HP Asynchronous Motor Driving a Mechanical Load” The Pacific Journal of Science and Technology, Vol. 10, No. 2. Krishnan (2003) “Electric Motor Drives” Pearson prentice Hall. Leon Tolbert, Fang Peng, ,Thomas Habetler (1999) “Multilevel PWM Methods at Low Modulation Indices” APEC ’99, Dallas, Texas, March 14-18, pp 1032-1039. 97