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Investigations on Dodecagonal Space Vector Generation for Induction Motor Drives

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### Investigations on Dodecagonal Space Vector Generation for Induction Motor Drives

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

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

Presented by

Anandarup Das

CEDT, IISc, Bangalore

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

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

3

5

2

6

7

1

8

12

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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

- 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

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

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

- 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

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

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

- 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

- 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

- 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

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

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

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

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

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

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

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

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

Loss comparison with conventional structures

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

- 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

- 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

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

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

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

- 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

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

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

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

- 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

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

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

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

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

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

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

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

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

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

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

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

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

- 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

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

- 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

- The power circuit of the inverter consists of 2 three level NPC inverters feeding an open end induction motor.
- These two inverters are fed from isolated dc voltage sources having voltage ratio of 1:0.366. This ratio of voltages is obtained from a combination of star delta transformers since 1:0.366= (√3+1):1.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Realization of space vector 27

- Point 27 can have two switching combinations- (110,002) or (221,002).

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

- PWM timing calculations are done separately for type-1 and type-2 dodecagons.
- Later they are used to find a uniform timing relation.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

[movie]

PWM timing calculations type-2 polygon

V2, T2

- Instead of switching points V1 and V2, the same reference vector VR can be realized by switching V3 and V4, but with different time durations.
- This will reduce the instantaneous error between the reference vector and the switching vectors, causing multilevel operation of the inverter.

V4, T’2

V3, T’1

- Note:
- k’ =V3/ V1=V2/ V4 is a fraction.
- T1+T2+T0= T’1+T’2+ T’0=TS.

V1, T1

T0’ ≥ 0

provided

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

PWM timing calculations type-2 polygon

V2, T2

V4, T’2

V1, T1

V3, T’1

- The same relation can be extended to the type-2 polygon.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

PWM timing calculations for both types of polygons

Points O,K and J are from type-1 polygon, while point F is on type-2 polygon.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

PWM timing calculations for both types of polygons

T2

r4

r5

T0

T1

T1+T2+T0= T’1+T’2+ T’0=TS.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

- A 3.7kW induction motor was fed by the proposed inverter under experimental condition, 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.
- In order to limit the switching frequency of the inverter, number of samples is decided as follow:
- Upto 10 Hz operation: 8 samples per sector.
- 10 Hz-30 Hz: 4 samples per sector.
- 30Hz-12step operation: 2 samples per sector leading to final 12-step mode.
- The samples are always taken synchronized with the start of the sector.
- A digital signal processor (DSP), TMS320LF2812 is used for experimental verification. The DSP is used for calculating the PWM timing durations. The actual gating signals to drive all the devices are generated using a SPARTAN XC3S200 FPGA. The FPGA stores the look-up table for the switching state combination of both the inverters for a particular space vector point.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Experimental results-10 Hz operation

Normalized harmonic spectrum of

Phase voltage

Phase voltage

Pole voltage- high voltage inverter

Phase current

Pole voltage-low voltage inverter

Phase current

- Switching happens within the innermost dodecagonal space vectors.
- Here, the number of samples per sector is taken as 8, as such the voltage and current harmonics reside around (12x8=) 96 times the fundamental.
- Individual devices of INV1 and INV2 are respectively switched at (10x16=) 160 Hz and (10x48=) 480 Hz.
- INV1 is switched only 1/3rd of a cycle, thereby the switching loss is less.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Experimental results-20 Hz operation

Normalized harmonic spectrum of

Phase voltage

Phase voltage

Pole voltage- high voltage inverter

Phase current

Pole voltage-low voltage inverter

Phase current

- Switching happens within the first and second dodecagonal space vectors.
- 4 samples are taken in a sector, so the first band of carrier harmonics reside around 48 times the fundamental.
- Because of the multilevel structure, all the harmonics are highly suppressed.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Experimental results-24.5 Hz operation

Normalized harmonic spectrum of

Phase voltage

Phase voltage

Pole voltage- high voltage inverter

Phase current

Pole voltage-low voltage inverter

Phase current

- Here the reference vector passes through the boundary of E dodecagon. As such, switching happens sometimes among points on the ‘D’ and ‘E’ dodecagons, while at other times among ‘E’ and ‘F’ dodecagons.
- The high voltage inverter switches about 20% of the cycle, thus the switching losses are minimized.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Experimental results-46 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 per sector are taken here.
- The phase voltage waveform of phase A distinctly shows the presence of 20 steps in a cycle, although 24 vectors are switched altogether.
- The phase voltage harmonics reside at (24x45=) 1080 Hz, while individual devices of INV1 and INV2 switch at (5x45=) 225 Hz and (15x 45=) 675 Hz.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

20-steps in phase voltage at Operation at 46 Hz

- Vectors numbered 37, 48, 50 and 60 switched at the positive peak of the phase-A waveform have same projection on A-phase axis.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

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

- The phase voltage waveform of phase A shows the 20 steps in a cycle.
- The switching vectors involved are located on the vertices of the A and B dodecagons, because of very small zero periods in a cycle.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

20-steps in phase voltage at 49.9 Hz

- Vectors numbered 49, 61 and 72 switched at the positive peak of the phase-A waveform have same projection on A-phase axis.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

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 of the inverter, when maximum fundamental voltage is available.
- Under this condition, INV1 and INV2 are switched 3 and 12 times respectively in a cycle.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Input current drawn from the grid

Motor phase voltage

Normalized harmonic spectrum of

Motor phase

current

input current

Grid side voltage

input current

- Because of the presence of the star-delta transformer at the input that forms the dc bus ratio, the input current is more sinusoidal than what is observed in a single transformer supplying an uncontrolled rectifier

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Motor acceleration with open loop V/f Control

Phase

voltage

Phase

current

Transition of motor phase voltage and current from 20 Hz to 30Hz

Transition of motor phase voltage and current from over-modulation to 12-step operation.

- In the first case, the reference vector starts from inside dodecagon E, crosses through the boundary of it and finally settles below the D dodecagon.
- In the second case, the number of samples per sector is changed from 2 to 1 at 12-step operation.
- Correct calculation of the PWM timings and complete elimination of the 5th and 7th order harmonics ensure that the motor current changes smoothly during the transition.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Comparison with a 5-level space vector diagram

- The harmonic distortion is less in the proposed scheme.
- This is more prominent in the higher modulation indices where the number of samples per sector is less, thus the switching frequency harmonics containing 5th and 7th order harmonics in the 5-level scheme come closer to the fundamental.
- Because of the total elimination of the these harmonics from the phase voltage in the present case, the ripple content in the phase voltage will be less.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

Harmonic performance in terms of flux ripple

- The rms value of the flux ripple is calculated and plotted above under constant V/f ratio and 24 samples in a fundamental period.
- It shows that for most of the operating conditions, the flux ripple is around 1% of the fundamental flux impressed on the machine, and thus restricts the torque ripple in the machine.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

- A new space vector diagram for induction motor drive is proposed, which divides the space vector plane into six concentric dodecagons.
- Here the space vector diagram is characterized by alternately placed type-1 and type-2 dodecagons which become closer to each other at higher radii. As such the harmonics in the phase voltage are reduced.
- Apart from this, the known benefits of dodecagonal space vector diagram like the complete elimination of all 6n±1 harmonics, (n=odd) from phase voltage and the extension of linear modulation range, are also retained here.
- The high voltage inverter having a voltage of about 3 times the lower one, is switched almost 1/3rd compared to the low voltage inverter.
- A comparison with 5-level inverter topology is also given which shows that the present scheme produces less harmonic distortion in the phase voltage.
- Because of the use of star-delta transformers for having the dc bus in the ratio 1:0.366, the input current has lesser harmonics compared to the case when a single transformer supplies the inverter.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

- Different dodecagonal space vector diagrams are proposed in this work.
- In one of the schemes, the space vector diagram consists of two concentric dodecagons, with the radius of the outer one twice the inner one. This has the benefit of reducing the device rating and dv/dt stress on the devices.
- This is then further refined to distribute six dodecagons in the space vector diagram. Switching on these closely spaced dodecagons will highly reduce the harmonic content in the phase voltage, apart from totally eliminating all the 5th and 7th order harmonics from the phase voltage.
- In another work, a 4-level inverter with asymmetric dc links is used to generate hexagonal space vector diagrams at lower modulation indices and a dodecagonal space vector structure at higher modulation index finally leading to 12-step operation of the inverter. This structure thus, incorporates the advantages of both hexagonal and dodecagonal space vector diagrams.
- With increased linear modulation range, less switching frequency and improved harmonic spectrum, the proposed concepts may be considered as an interesting addition to the field of multilevel inverters for high/medium voltage high power applications.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

- Anandarup Das, K. Sivakumar, Rijil Ramchand, Chintan Patel and K. Gopakumar, "A Combination of Hexagonal and 12-sided Polygonal Voltage Space Vector PWM control for IM Drives Using Cascaded Two Level Inverters", IEEE Trans. On Industrial Electronics, vol. 56, no. 5, May 2009, pp. 1657-1664.
- Anandarup Das, K. Sivakumar, Rijil Ramchand, Chintan Patel and K. Gopakumar, "A Pulse Width Modulated Control of Induction Motor Drive Using Multilevel 12- sided Polygonal Voltage Space Vectors", IEEE Trans. on Industrial Electronics, vol. 56, no. 7, July 2009, pp. 2441-2449.
- Anandarup Das, K. Sivakumar, Rijil Ramchand, Chintan Patel and K. Gopakumar, "A High Resolution Pulse Width Modulation Technique Using Concentric Multilevel Dodecagonal Voltage Space Vector Structures", Proc. of ISIE 2009, Jul. 2009. (Best paper award in the conference).
- Anandarup Das, K. Sivakumar, Rijil Ramchand, Chintan Patel and K. Gopakumar, "Multilevel Dodecagonal Space Vector Generation for Open-end Winding Induction Motor Drive Using Conventional Three Level Inverters", Proc. of EPE 2009, Sep 2009, pp 1-8.
- Anandarup Das, K. Sivakumar, Gopal Mondal and K. Gopakumar, "A Multilevel Inverter with Hexagonal and 12-sided Polygonal Space Vector Structure for Induction Motor Drive", Proc. of IECON 2008, Nov 2008, pp 1077-1082.
- Anandarup Das and K. Gopakumar "A Voltage Space Vector Diagram Formed By Six Concentric Dodecagons for Induction Motor Drives", communicated to IEEE Trans. on Power Electronics.

CEDT, INDIAN INSTITUTE OF SCIENCE, BANGALORE, INDIA

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