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Investigations on Dodecagonal Space Vector Generation for Induction Motor Drives

Investigations on Dodecagonal Space Vector Generation for Induction Motor Drives . Presented by Anandarup Das CEDT, IISc, Bangalore. Motivation for the present research. Schemes to be presented

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Investigations on Dodecagonal Space Vector Generation for Induction Motor Drives

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  1. Investigations on Dodecagonal Space Vector Generation for Induction Motor Drives Presented by Anandarup Das CEDT, IISc, Bangalore

  2. Motivation for the present research. Schemes to be presented Hybrid space vector PWM strategy in linear and over-modulation region involving hexagonal and dodecagonal space vector diagrams. Development of two concentric dodecagons using conventional 3-level inverters with capacitor balancing. Further refinement of the above space vector structure into multiple 12-sided polygons with conventional 3-level inverters. Modulation strategies and PWM timing calculation of the above schemes. Discussion on experimental verification Steady state operation. Transient results with motor accelerated upto rated speed with open-loop V/f control Harmonic performance of phase voltage and phase current under these conditions Conclusion Flow of presentation CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  3. Motivation for the present research • Multilevel inverters are popular for high power drives because of low switching losses and low harmonic distortion in the output voltage. • In many multilevel inverter fed high power drives, the switching frequency of the inverter is limited because of large dv/dt stress on the devices and the motor and higher switching losses. • However, with low switching frequency, lower order harmonics e.g. 5th and 7th order can be a considerable percentage of the total current, in particular during over-modulation and 6-step operation. • So a trade-off is required to maintain the quality of the inverter output voltage without resorting to higher switching frequency. In this regard, a dodecagonal space vector diagram is very desirable that eliminates all the 5th and 7th order harmonics from the phase voltage, leaving the next set of harmonics at (12n±1), n=integer. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  4. 4 3 5 2 6 7 1 8 12 O 9 11 10 Evolution of space vector structures (Hexagonal and 12-sided) Hexagonal space vectors. 2-level 3-level 5-level 12-sided polygonal space vectors. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  5. Proposed research schemes • In the proposed work, a multilevel inverter topology is described which produces hexagonal space vector diagrams in lower-modulation region and a dodecagonal space vector structure in the higher modulation region. • In another scheme, a multilevel voltage space vector structure with vectors on the dodecagon is generated by feeding an open-end winding IM drive by two three level inverters. • In a third scheme, a high resolution PWM technique is proposed involving multiple dodecagonal space vector structures, that can generate highly sinusoidal voltages at a reduced switching frequency. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  6. Part-1A Combination of Hexagonal and Dodecagonal Voltage Space Vector Diagram for Induction Motor Drives

  7. Evolution of space vector structures (Hexagonal and 12-sided) Hexagonal space vectors. 2-level 3-level 5-level 12-sided polygonal space vectors. 4 3 5 6 2 7 1 O 8 12 9 11 10 CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  8. Voltage space vector diagram of the proposed scheme End of linear modulation • Consists of four concentric hexagonal diagrams with different radii (0.366kVdc , 0.634kVdc , 1kVdc and 1.366kVdc). • Operates in the inner hexagons at lower voltage to retain the advantages of multilevel inverter like low switching frequency. • At higher voltage, the outermost hexagon and the 12-sided polygonal space vector structure is used resulting in highly suppressed 5th and 7th order harmonics. • The leads to 12-step operation at rated voltage operation, leading to the complete elimination of 6n±1 harmonics. (n=odd) from the phase voltage. OE: 1.225kVdc CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  9. Inverter Topology • Consists of three cascaded 2-level inverters • Two inverters are supplied with a dc bus of 0.366kVdc while the third one is supplied with a dc bus of 0.634kVdc. C D A Switch status for different levels of pole voltage B O Pole voltage of overall inverter-vAO Pole voltage of INV3- vBO Pole voltage of INV2-vAB Pole voltage of INV1-vCD CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  10. Transformer connection for generation of 12-sided polygonal voltage space vector • Asymmetric DC-links are easily realized by a combination of star-delta transformers, since 0.634kVdc=√3 x 0.366kVdc. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  11. Comparison with hexagonal space vector structure • The modulation index (m), is defined as the ratio of the length of the reference vector to the length of the radius of the dodecagon. m = 0.966 in linear modulation and m =1 at 12-step operation. • Here, the radius of the dodecagon is 1.225kVdc.Thus the maximum fundamental phase voltage available from this space vector diagram is 0.806kVdc (in 12-step). • It is known that, the maximum fundamental voltage available from a conventional hexagonal space vector diagram in 6-step mode is 0.637Vdcand equal to 0.577Vdc at the end of linear modulation. • For comparison purpose, if the maximum fundamental voltage available in 6-step mode and 12-step mode are made equal to 0.637Vdc, then ‘k’ = 0.637/0.806=0.789. • For k = 0.789, the maximum phase voltage available here in linear modulation is 0.615Vdc and equal to 0.637Vdcin 12-step mode of operation. There is hence an increase in linear modulation range. Radius of dodecagon radius= 1.225kVdc CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  12. Modulating waveform • The modulating waveform for phase-A for 35Hz operation (linear modulation range) is shown. • The modulating waveform is synchronized with the start of the sector (sampling interval is always a multiple of twelve). • Because of asymmetric voltage levels, three asymmetric synchronized triangles are used; their amplitudes are in the ratio 1:√3:1. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  13. Switching sequence analysis • Three pole voltages are shown for a 60 degree interval at 35Hz operation. • Level shifted SVPWM is used here. • In ‘A’ phase the voltage level fluctuate between levels ‘3 ’ and ‘2 ’, and in ‘C’ phase the voltage level fluctuates between levels ‘1 ’ and ‘0 ’. • The sequence in which the switches are operated are as follows: (200), (210), (211), (311), (321), (311), (211), (210), (211), (311), (321), (211), (221), (321), (221), (210), (220), (221), (321), (331), (221), (220), where the numbers in brackets indicate the level of voltage. • This sequence corresponds to 2 samples per sector. There are altogether 12 sectors spanning from 00 to 300, 300 to 600 and so on. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  14. Experimental Setup • A digital signal processor (DSP), TMS320LF2812 is used for experimental verification. • For different levels of output in the pole voltage, three carriers are required. However, it is difficult to synthesize three carrier waves in the DSP, as such only one carrier is used and the modulating wave is appropriately scaled and level shifted. • A 3.7kW induction motor was fed by the proposed inverter operating under open loop constant V/f control at no load. The motor was made to run under no load in order to show the effect of changing PWM patterns of the generated voltage on the motor current, particularly during transient conditions. • In order to keep the overall switching frequency within 1 KHz, number of samples is decided as follow: • Upto 20 Hz operation: 4 samples per sector. • 20 Hz-40 Hz: 2 samples per sector. • Beyond 40 Hz: 1 sample per sector-extending up to final 12-step mode. • Individual inverters are switched less than half of the total cycle. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  15. Experimental results-Operation at 10 Hz Normalized harmonic spectrum of Phase current Phase voltage Phase voltage Phase current Phase voltage and current waveforms • Switching happens within the innermost hexagon space vectors. • As seen from the pole voltage waveforms, only the lower inverter is switched while the other two inverters are off, hence the switching loss is low. • Four samples are taken in each sector, so INV3 switching frequency is (12x4X10=480Hz). The first carrier band harmonics also reside around 48 times fundamental. [Space Vector] Overall inverter INV3 [Inverter Topology] INV2 INV1 Pole voltage waveforms CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  16. Experimental results-Operation at 30 Hz Normalized harmonic spectrum of Phase current Phase voltage Phase voltage Phase current Phase voltage and current waveforms • The space vectors that are switched lie on the boundaries of the second and third hexagon from the center. • Number of samples are reduced from four to two, thus switching frequency is (fs=12X2x30=720Hz). • INV3 and INV1 are switched about 1/3rd of the total cycle, while INV2 is switched about 20% of the cycle. Overall inverter [Space Vector] INV2 INV2 switches INV3 INV1 Pole voltage waveforms [Inverter Topology] CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  17. Operation at 47 Hz ( end of linear modulation range) Normalized harmonic spectrum of Phase current Phase voltage Phase voltage Phase current Phase voltage and current waveforms Overall inverter • One sample is taken at the start of a sector, so switching frequency is only around (12X47=564Hz). • The space vectors that are switched lie between the outer hexagon and the 12-sided polygon. INV2 INV3 INV1 [Space Vector] Pole voltage waveforms CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  18. Operation at 50 Hz ( 12-step operation) Normalized harmonic spectrum of Phase current Phase voltage Phase voltage Phase current Phase voltage and current waveforms • Complete elimination of 6n±1 harmonics (n=odd) from the phase voltage. • One sample is taken at the start of a sector (fs=12X1x50=600Hz). • Each inverter is switched only once in a cycle. Overall inverter INV2 INV3 INV1 Pole voltage waveforms Inverter Topology CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  19. Input current at 50 Hz ( 12-step operation) Phase voltage Phase current Input phase voltage Input line current • The input current to the inverter is not peaky in nature, because of the presence of the star-delta transformers. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  20. Motor acceleration with open loop V/f Control Phase voltage Phase current Transition of motor phase voltage and current from 24 samples to 12 samples per cycle at 40Hz Transition of motor phase voltage and current from outermost hexagon to 12-step operation. • Because of the suppression of the 5th and 7th order harmonics, the motor current changes smoothly during the transition when the number of samples per sector is reduced from two to one at 40Hz operation. • As the speed of the motor is further increased, the inverter switching states pass through the inner hexagons and ultimately the phase voltage becomes a 12-step waveform. • Under all operating conditions, the carrier is synchronized with the start of the sector. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  21. Total Harmonic Distortion upto 100th harmonic Harmonic performance of phase voltage and current 10Hz 30 Hz 48.25 Hz 50Hz Voltage THD 57.59% 27.51% 14.67% 17.54% Voltage WTHD 0.81% 0.7%0.97% 1.04% Current THD 12.31% 10.59% 15.6% 19.54% Current WTHD 0.28% 0.45% 1.2% 1.5% • It is seen that voltage WTHD is quite low for all the operating conditions, as such the torque pulsation and harmonic heating in the machine is minimized. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  22. Comparison with conventional structures • A simplified comparative study is made between the proposed topology and the existing multilevel inverter configurations viz. 3-level NPC and 4-level NPC inverters used for induction motor drives. • The conduction and switching losses incurred in the inverter, and motor phase voltage harmonic distortions are numerically calculated by computer simulation for comparison. • A linear turn-on and turn-off switching profile is used for loss calculation. Losses incurred in snubber circuits, protection circuits, gate drives and due to leakage currents are neglected. • A 2.3kV, 373kW induction motor is driven by a 3-level NPC, 4-level NPC and the proposed inverter.The inverter drives the induction motor under full load condition at around 0.85 p.f. lagging. Numbers of samples in a cycle are taken as 24. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  23. Loss comparison with conventional structures CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  24. Observations • The phase voltage WTHD for the proposed inverter shows considerable improvement, particularly at higher modulation indices and the 12-step mode of operation, because of the suppression or elimination of the 6n±1 (n=odd) harmonics. • Conduction losses are more dominant than switching losses for IGBT made inverters. As such, presence of the clamping diodes in NPC inverters increases the total losses of the inverter. The proposed inverter does not have any clamping diode and is devoid of any such losses. The switching losses also remain low for the proposed inverter. • It is seen that the conduction losses in the proposed inverter are always less than the conventional inverters. This is because in the proposed inverter, for any ‘level’ of pole voltage output, two current carrying switches remain in conduction. This is not always the case in NPC inverters; e.g. for a four level inverter, at higher modulation indices, three switches per phase carry the phase load current when the total dc bus voltage is obtained at the pole. Conduction losses in the proposed inverter are further less in over-modulation region because of the fact that the r.m.s. current in the inverter is less compared to conventional NPC inverters, due to the suppression or elimination of the 6n±1 (n=odd) harmonics. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  25. Synopsis • A multilevel inverter topology is described which produces hexagonal space vector structures in lower-modulation region and a dodecagonal space vector structure in the higher modulation region. • In the extreme modulation range, voltage vectors at the vertices of the outer dodecagon and the vertices from the outer most hexagon is used for PWM control, resulting in highly suppressed 5th and 7th order harmonics thereby improving the harmonic profile of the motor current. This leads to the 12-step operation at 50Hz where all the 5th and 7th order harmonics are completely eliminated. • At the same time, the linear range of modulation extends upto 96.6% of base speed. Because of this, and the high degree of suppression of lower order harmonics, smooth acceleration of the motor upto rated speed is possible. • Apart from this, the switching frequency of the multilevel inverter output is always limited within 1 kHz. The middle inverter ( high voltage inverter) devices are switched less than 25% of the output fundamental switching period. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  26. Part-IIGeneration of Multilevel Dodecagonal Space Vector Diagram

  27. Evolution of space vector structures (Hexagonal and 12-sided) Hexagonal space vectors. 2-level 3-level 5-level 12-sided polygonal space vectors. 4 3 5 6 2 7 1 O 8 12 9 11 10 CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  28. Multilevel dodecagonal space vector diagram • This is an extension of the single dodecagonal space vector structure into a multilevel dodecagonal structure. • Compared to conventional dodecagonal space vector structure, the device ratings and dv/dt stress on them are reduced to half. • The switching frequency is also reduced to maintain the same output voltage quality. • Here the added advantage is the complete elimination of 6n±1 harmonics, n=odd, from the phase voltage throughout the modulation index. • The linear modulation range is also extended compared to the hexagonal structure. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  29. Multilevel 12-sided polygonal space vector structure • Consists of two concentric dodecagonal space vector structures. • Unlike conventional hexagonal multilevel structure, here the sub-sectors are isosceles triangles rather than equilateral triangles. • Each sector is thus divided into four sub-sectors as shown. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  30. Inverter Structure • In order to realize the proposed space vector structure, two conventional three level NPC inverters are used to feed an open ended induction motor. • The two inverters are fed from asymmetrical dc voltage sources which can be obtained from the mains with the help of star-delta transformers and uncontrolled rectifiers. • Because of capacitor voltage balancing of the NPC inverters, only two dc sources are used. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  31. Switching state combination for realizing space vector 16 • INV1 produces vector X(220) while INV2 produces vector Y(0’2’2’). • When they are combined, the resultant vector Z(220, 0’2’2’) is produced. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  32. Timing calculation for dodecagonal space vector locations • Here, the timings for which adjacent vectors are switched are obtained as, • This is similar to conventional space vector PWM. • However, this requires calculation of sine values through a look-up table, which takes unnecessary memory and time in a DSP. • A better algorithm has been generated which can calculate the timings by sampling six reference rotating phasor. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  33. Timing relation among different space vectors V2, T2 • Instead of vectors on the vertices of the sector, three nearest enclosing vectors are now switched. This is done to achieve multilevel switching. • The time durations for which the original vectors need to be switched is modified. The new timing durations are achieved by volt-second balance. • The timing relation can be extended to other sub-sectors. V4, T’2 V’1, T’0 V1, T’1 V1, T1 • Note: • T’0 >= 0. • T1+T2+T0= T’1+T’2+ T’0=TS. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  34. Capacitor balancing scheme • The inner dodecagonal space vectors (points 1-12) have four multiplicities which are complementary in nature in terms of capacitor balancing. • The outer dodecagonal space vectors ( points 13-36) either do not cause any capacitor unbalancing, or have complementary states to maintain capacitor balancing. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  35. Inner 12-sided polygon-switching multiplicities for point-1 C2 is discharged, C4 is charged. C1 is discharged, C4 is charged. C1 is discharged, C3 is charged. C2 is discharged, C3 is charged. The four switching multiplicities are complementary in nature in terms of capacitor balancing. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  36. Outer 12-sided polygon-switching multiplicities Point-13, two multiplicities C3 is discharged, C1 & C2 are undisturbed. C4 is discharged, C1 & C2 are undisturbed. Point-14: no multiplicity, no capacitor disturbance Point-36: no multiplicity, no capacitor disturbance CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  37. Experimental results-15 Hz operation Normalized harmonic spectrum of Phase voltage Phase voltage Pole voltage- high voltage inverter Phase current Pole voltage-low voltage inverter Phase current • Four samples are taken in each sector and switching takes place entirely in the inner 12-sided polygon. • The phase voltage harmonics reside at 15x12x4=720 Hz, which is 48 times the fundamental. However, the switching frequency of the pole voltage of INV1 is (24x15=) 360Hz, while that of INV2 is (32x15=) 480Hz. • The higher voltage inverter switches about 50% of the cycle. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  38. Experimental results-23 Hz operation Normalized harmonic spectrum of Phase voltage Phase voltage Pole voltage- high voltage inverter Phase current Pole voltage-low voltage inverter Phase current • Three samples are taken in each sector and switching takes place at the boundary the inner 12-sided polygon. All the 6n±1 harmonics, n=odd, are absent from the phase voltage, while the rest are highly suppressed. • The switching frequencies of the pole voltage of INV1 and INV2 are respectively (18x23=) 414Hz and (24x23=) 552Hz, with output phase voltage switching frequency at 828Hz (=23x12x3). CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  39. Experimental results-40 Hz operation Normalized harmonic spectrum of Phase voltage Phase voltage Pole voltage- high voltage inverter Phase current Pole voltage-low voltage inverter Phase current • Two samples are taken in each sector and switching takes place between the inner and outer dodecagons. • This is also seen in the phase voltage waveform, since the outer envelope of the waveform at lower frequency becomes the inner envelope at higher frequency. • The harmonic spectrum of the phase voltage and current shows the absence of peaky harmonics throughout the range. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  40. Experimental results-48 Hz operation Normalized harmonic spectrum of Phase voltage Phase voltage Pole voltage- high voltage inverter Phase current Pole voltage-low voltage inverter Phase current • This is the end of the linear modulation of operation. • Here the number of samples per sector is two, as such the switching frequency sidebands reside around 24 times the fundamental. The switching frequency of the pole voltages of INV1 and INV2 is respectively (48x12=) 576Hz and (48x16=) 768Hz, with an output phase voltage switching frequency of 1152Hz (48x12x2). CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  41. Experimental results-49.9 Hz operation Normalized harmonic spectrum of Phase voltage Phase voltage Pole voltage- high voltage inverter Phase current Pole voltage-low voltage inverter Phase current • At the end of end over-modulation region, 24 samples are taken in a sector, corresponding to the vertices of the polygon. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  42. Experimental results-50 Hz operation Normalized harmonic spectrum of Phase voltage Phase voltage Pole voltage- high voltage inverter Phase current Pole voltage-low voltage inverter Phase current • This is the 12-step operation, where one sample is taken at the start of a sector. The phase voltage and current is completely devoid of any 5th and 7th order harmonics. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  43. Total Harmonic Distortion upto 100th harmonic Harmonic performance of phase voltage and current • It is seen that voltage WTHD is quite low for all the operating conditions, as such the torque pulsation and harmonic heating in the machine is minimized. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  44. Acceleration of the motor Phase voltage Phase current Transition of motor phase voltage and current from over-modulation to 12-step operation. Transition of motor phase voltage and current from inner to outer 12-sided polygon • In both the cases, the motor current changes smoothly as the motor accelerates. This happens because of the use synchronized PWM and total elimination of 6n±1 harmonics, n=odd, from the phase voltage throughout the modulation index. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  45. Experimental Results-capacitor unbalancing at 20 Hz • Capacitor unbalance is done at steady state with the motor running at 20 Hz speed. • Both side capacitors are deliberately unbalanced and after some time controller action is taken. Vc1, Vc2 Vc3, Vc4 Deliberate unbalancing Controller action taken C1,C2 : higher voltage side capacitors C3,C4 : lower voltage side capacitors CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  46. Experimental Results-capacitor unbalancing at 40Hz • Both the sides are made unbalanced at the same time and are seen to come back to the balanced state. • Compared to the 20 Hz case, it requires more time to restore voltage balance, since the number of multiplicities in the outer polygon is less. Vc1, Vc2 Vc3, Vc4 Controller action taken Deliberate unbalancing C1,C2 : higher voltage side capacitors C3,C4 : lower voltage side capacitors CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  47. Synopsis • An improved space vector diagram is proposed here that is composed of of two concentric dodecagons. It reduces the device rating and the dv/dt stress on the devices to half compared to existing 12-sided schemes. • The entire space vector diagram is divided into smaller sized isosceles triangles. PWM switching on these smaller triangles reduces the inverter switching frequency without compromising on the output voltage quality. • The proposed topology is realized by feeding an open-end induction motor with two conventional 3-level NPC inverters, where, the high voltage inverter always switches at nearly half the output phase voltage switching frequency. • Additionally, the mechanism for capacitor balancing, using switching state redundancies is also proposed for the full modulation range CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  48. Part-IIIA Voltage Space Vector Diagram Formed By Six Concentric Dodecagons

  49. Evolution of space vector structures (Hexagonal and 12-sided) Hexagonal space vectors. 2-level 3-level 5-level 12-sided polygonal space vectors. 4 3 5 6 2 7 1 O 8 12 9 11 10 CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

  50. Space Vector Structure • The space vector diagram consists of six concentric dodecagonal structures - A, B, C, D, E and F. • These are grouped as type-1 and type-2 dodecagons, where type-2 dodecagons (A, C and E) lead type-1 dodecagons (B, D and F) by 150. • The radii of these polygons are in the ratio r1: r2: r3: r4: r5: r6 = • 1: cos (π/12): cos (2π/12): • cos (3π/12) :cos (4π/12) :cos (5π/12). • The entire space vector diagram is divided into 12 sectors each of width 300. CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

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