Space vector modulation

1

Power Converter Systems

Graduate Course EE8407

Ryerson Campus

Bin Wu PhD, PEng

ProfessorELCE DepartmentRyerson University

Contact Info Office: ENG328Tel: (416) 979-5000 ext: 6484Email: [email protected] http://www.ee.ryerson.ca/~bwu/

2

VDM5000 Two-level VSI

Two-Level Voltage Source Inverter (VSI)

Source: Alstom

3

• Sinusoidal PWM• Space vector modulation

Lecture Topics

• To control inverter output frequency (fundamental)

• To control inverter output voltage (fundamental)

• To minimize harmonic distortion

Why Use PWM Techniques?

Two Level Voltage Source Inverter

4

• Inverter Configuration

Assumption:

dc capacitor very large dc voltage ripple free

dC

1S

2S

3S 5S

4S 6S

A

B

C

P

N

LOAD

Ai

OBi

CidV

Sinusoidal PWM

5

• Modulating and Carrier Waves

• vcr – Carrier wave (triangle) • Amplitude modulation index

cr

ma V

Vm

ˆ

ˆ

• Frequency modulation index

m

crf f

fm

0

v mAv Bmv Cmvcrv

crV̂ mV̂t

• vm – Modulating wave (sine)

Sinusoidal PWM

6

• Gate Signal Generation

1gv

4gv

dV

0

ANv

2

mAv crv

0

crmA vv 01 gv )0( 4 gv 1S on )off( 4S dAN Vv Phase A

crmA vv 04 gv )0( 1 gv 4S on )off( 1S 0ANv

Vg1 and Vg4 are complementary

Sinusoidal PWM

7

• Line-to-Line Voltage vAB

ABv

BNv

ANv

0

0

0

v mAv Bmv Cmvcrv

crV̂ mV̂

dV

dV

dV

2t

t

t

t

1ABv

1S

2S

3S 5S

4S 6S

B

C

P

N

dV

A

Sinusoidal PWM

8

• Waveforms and FFT

• ma = 0.8, mf = 15,

fm = 60Hz, fcr = 900Hz

• Switching frequency

fsw = fcr = 900Hz 0.1

0.2

0

0

0

THD = 92.07%

THD = 92.07%

THD = 7.73%

THD = 92.07%

dV

3/2 dV

ABv

AOv

Ai

2fm

12 fm

23 fm 14 fm

n

3

32

2

01 5 10 15 20 25 30 35 40 45 50 55 60

dVVAB 49.01

dn VVAB /

Sinusoidal PWM

9

• Harmonic Content

• Low order harmonics n < (mf -2) are eliminated

• VAB1 versus ma is linear

• VAB1,max = 0.612Vd

am

12 fm2fm

14 fm23 fm

43 fm

d

nAB

V

V

1n

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0

0

100

200

300

0

THD(%)

THD

Sinusoidal PWM

10

• Over-Modulation

• Fundamental voltage ↑

• Low-order harmonics ↑

0

0.1

0.2

0dV

2 3

0

1

2

-1

-2

mAv mCvmBv

crv

ABv

n

0

Ai

2 3

dVVAB 744.01

dn VVAB /

n1 5 10 15 20 25 30 35 40 45 50 55 60

Sinusoidal PWM

11

• Third Harmonic Injection PWM

crm VV ˆˆ1 • - Fundamental voltage increased

crmA VV ˆˆ • - No low order harmonics produced

• 3rd harmonic – zero sequence (to appear in vAN and vBN )

• No triplen harmonics in vAB (vAB = vAN - vBN)

3/ 3/2 2

3mv

1mv

31 mmmA vvv

0

v

t

1ˆmV mAV̂ crV̂

Sinusoidal PWM

12

• Switching States

Leg A Leg B Leg C Switching State

1S 4S ANV 3S 6S BNV 5S 2S CNV

P On Off dV On Off dV On Off dV

O Off On 0 Off On 0 Off On 0

1S

2S

3S 5S

4S 6S

B

C

P

N

dV

A

LOAD

Ai

OBi

Ci

Space Vector Modulation

13

• Switching States (Three-Phase)

• Eight switching states

1S

2S

3S 5S

4S 6S

B

C

P

N

dV

A


14

• Space Vector Diagram

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III

SECTOR IV SECTOR VISECTOR V

SECTORII

• Active vectors: to (stationary, not rotating)

• Zero vector:

1V

6V

0V

• Six sectors: I to VI


15

• Space Vectors

• Three-phase voltages

0)()()( tvtvtv COBOAO

• Two-phase voltages

)(

)(

)(

3

4sin

3

2sin0sin

3

4cos

3

2cos0cos

3

2)(

)(

tv

tv

tv

tv

tv

CO

BO

AO

• Space vector representation)()()( tvjtvtV

(2) (3)

3/43/20 )()()(3

2)( j

CO

j

BO

j

AO etvetvetvtV

where xjxe jx sincos

(3)

(1)

(2)

(4)


16

• Space Vectors (Example)

Switching state [POO] S1, S6 and S2 ON

dBOdAO VtvVtv3

1)(,

3

2)(

dCO Vtv3

1)( and

(5) (4)

(7)

(5)

(6)0

1 3

2 j

d eVV

Similarly,

3)1(

3

2

kj

dk eVV

.6...,,2,1k

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III


SECTORII


17

• Active and Zero Vectors

S p a c e V e c t o r S w it c h in g S t a t e ( T h r e e P h a s e s )

O n - s t a t e S w it c h V e c t o r

D e f in it io n

[ P P P ] 531 ,, SSS Z e r o V e c t o r 0V

[ O O O ] 264 ,, SSS 00 V

1V

[ P O O ] 261 ,, SSS 01 3

2 jd eVV

2V

[ P P O ] 231 ,, SSS 32 3

2

j

d eVV

3V

[ O P O ] 234 ,, SSS 3

2

3 3

2

j

d eVV

4V

[ O P P ] 534 ,, SSS 3

3

4 3

2

j

d eVV

5V

[ O O P ] 564 ,, SSS 3

4

5 3

2

j

d eVV

A c t iv e V e c t o r

6V

[ P O P ] 561 ,, SSS 3

5

6 3

2

j

d eVV

• Active Vector: 6

• Zero Vector: 1

• Redundant switching states: [PPP] and [OOO]

1S

2S

3S 5S

4S 6S

B

C

P

N

dV

A


18

(8)

• Reference Vector Vref

• Definition

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III


SECTORII

• Angular displacement

t

dtt0

)( (9)

jrefref eVV

• Rotating in space at ω


f 2

19

• Relationship Between Vref and VAB

• Vref is approximated by two active and a zero vectors

• Vref rotates one revolution, VAB completes one cycle

• Length of Vref corresponds to magnitude of VAB

1V

2V

refV

1VT

T

s

a

2VT

T

s

b

SECTOR I

Q


20

• Dwell Time Calculation• Volt-Second Balancing

0

0021

TTTT

TVTVTVTV

bas

basref

(10)

• Ta, Tb and T0 – dwell times for and , 21 VV

0V

• Ts – sampling period

• Space vectors

d

j

refref VVeVV3

2, 1

3

2 3

2 j

d eVV

00 V

, and

(11) (10)

bdsref

bdadsref

TVTV

TVTVTV

3

1)(sin

3

1

3

2)(cos

:Im

:Re

(11)

(12)

1V

2V

refV

1VT

T

s

a

2VT

T

s

b

SECTOR I

Q


21

• Dwell Times

Solve (12)

bas

d

refs

b

d

refs

a

TTTT

V

VTT

V

VTT

0

sin3

)3

(sin3

3/0 (13)


22

• Vref Location versus Dwell Times

refV Location 0

60

6

36

3

Dwell Times 0

0

b

a

T

T baTT baTT baTT 0

0

b

a

T

T

1V

2V

refV

1VT

T

s

a

2VT

T

s

b

SECTOR I

Q


23

• Modulation Index

cbs

asb

asa

TTTT

mTT

mTT

0

sin

)3

(sin

(15)

d

ref

a V

Vm

3 (16)


24

• Modulation Range

• Vref,max

32

3

3

2max,

ddref

VVV (17)

1V

0V

3V

2V

4V

5V

6V

j

POO

PPOOPO

OPP

OOP POP

refV

OOOPPP

SECTOR ISECTOR III


SECTORII

(17) (16)

• ma,max = 1

• Modulation range: 0 ma 1 (18)


25

• Switching Sequence Design

• Basic Requirement:

Minimize the number of switchings per

sampling period Ts

• Implementation:

Transition from one switching state to

the next involves only two switches in

the same inverter leg.


26

• Seven-segment Switching Sequence

dV

20T

2aT

2bT

2aT

BNv

ANv

CNv

0

1V

1V

2V

0V

2V

POOOOO PPO PPP PPO POO OOO

dV

dV

40T

40T

2bT

sT

0

0

0V

0V

• Total number of switchings: 6

• Selected vectors: V0, V1 and V2

• Dwell times: Ts = T0 + Ta + Tb


27

• Undesirable Switching Sequence

• Vectors V1 and V2 swapped

dV

20T

2aT

2bT

2aT

BNv

ANv

CNv

0

1V

1V

2V

2V

POOOOO PPO PPP PPOPOO OOO

dV

dV

40T

40T

2bT

sT

0

0

0V

0V

0V

• Total number of switchings: 10


28

• Switching Sequence Summary (7–segments)

Sector Switching Sequence

0V 1V

2V

0V

2V

1V

0V

I OOO POO PPO PPP PPO POO OOO

0V 3V

2V

0V

2V

3V

0V

II OOO OPO PPO PPP PPO OPO OOO

0V 3V

4V

0V

4V

3V

0V

III OOO OPO OPP PPP OPP OPO OOO

0V 5V

4V

0V

4V

5V

0V

IV OOO OOP OPP PPP OPP OOP OOO

0V 5V

6V

0V

6V

5V

0V

V OOO OOP POP PPP POP OOP OOO

0V 1V

6V

0V

6V

1V

0V

VI OOO POO POP PPP POP POO OOO

Note: The switching sequences for the odd and ever sectors are different.


29

• Simulated Waveforms

ABv

AOv

0

0

0

Ai

dV

3/2 dV

2 3

2 3

VIVISector

III

IIIIV

V

III

IIIIV

V

f1 = 60Hz, fsw = 900Hz, ma = 0.696, Ts = 1.1ms


30

• Waveforms and FFT

0

0.1

0.2

n

0

0

ABv

AOv

Ai

THD =80.2%

THD =80.2%

THD =8.37%

THD =80.2%

dV

3/2 dV

2

dVVAB 566.01

1 5 10 15 20 25 30 35 40 45 50 55 60

dn VVAB /

0 2 3


31

• Waveforms and FFT (Measured)

23

AOv

ABv d

n

V

VAB

0.2

0.1

0

(a) Waveforms 2ms/div (b) Spectrum (500Hz/div)

14

34

10 4758

THD = 80.3%

8

16 29 43


32

0 0.2 0.4 0.6 0.80

0.05

0.10

0.15

10 16 20

dnAB VV /

am

1n

2n 4 8

14

(a) Even order harmonics

57111317n = 19

0.05

0.10

0.15

dnAB VV /

0 0.2 0.4 0.6 0.8 am

1n

(b) Odd order harmonics

0

100

200

300

0

THD(%)

THD

• Waveforms and FFT (Measured)

Hz601 f sec720/1sT ( and )


33

• Even-Order Harmonic Elimination

BNv

ANv

CNv

0

5V

4V

0V

OOPOOO OPP PPP OPP OOP OOO

0

0

0V

0V

ABv0

dV

4V

5V

dV

dV

dV

Type-A sequence (starts and ends with [OOO])

BNv

ANv

CNv

0

0

0

ABv0

dV

5V

OOP4V

OPP

dV

dV

dV

4V

OPP5V

OOPPPP0V

0V

OOO PPP0V

Type-B sequence (starts and ends with [PPP])


34


1V

3V

2V

4V

5V

6V

SECTOR ISECTOR III

SECTOR IV SECTOR VI

SECTOR V

SECTOR II

a

ba

a

a

aa

b

b

bb

b

Type-A sequence

Type-B sequence

30

30

Space vector Diagram


35


(a) Waveforms 2ms/div

AOv

ABv

0.2

0.1

0

d

n

V

VAB

(b) Spectrum (500Hz/div)

23

13 47

35

7

17

65

THD = 80.5%

41

5

• Measured waveforms and FFT


36


17

1913

75

11

100

200

300

0

THD(%)

0 0.2 0.4 0.6 0.8 am0

0.1

0.2

0.3

dnAB VV /

1nTHD

Hz601 f sec720/1sT ( and )


37

• Five-segment SVM

dV

2aT

bT2aT

BNv

ANv

CNv

0

1V

1V

2V

0V

POOOOO PPO POO OOO

dV

sT

0

0

0V

20T

20T

dV

aT

1V

2V

0V

PPP PPO POO PPP

dV

sT

0V

20T

20T

(a) Sequence A

2V

PPO

dV

2bT

2bT

(b) Sequence B


38

• Switching Sequence ( 5-segment)

S ector S w itch in g S eq u en ce (A )

0V

1V

2V

1V

0V

I

O O O P O O P P O P O O O O O 0CNv

0V

3V

2V

3V

0V

II

O O O O P O P P O O P O O O O 0CNv

0V

3V

4V

3V

0V

III

O O O O P O O P P O P O O O O 0ANv

0V

5V

4V

5V

0V

IV

O O O O O P O P P O O P O O O 0ANv

0V

5V

6V

5V

0V

V

O O O O O P P O P O O P O O O 0BNv

0V

1V

6V

1V

0V

V I

O O O P O O P O P P O O O O O 0BNv


39

• Simulated Waveforms ( 5-segment)

dV

2 4

1gv

3gv

5gv

ABv

0

0

Ai

3/2

2 4

2 4

• No switching for a 120° period per cycle. • Low switching frequency but high harmonic distortion

• f1 = 60Hz, fsw = 600Hz, ma = 0.696, Ts = 1.1ms


40

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Space vector modulation