ID PWM Inverters

PWM Inverters

Inverters

ClassificationsSingle phase & three phase

Voltage Source & Current source

Two-level & Multi-level

Voltage Source Inverter

Sinusoidal PWM

Space vector modulation

Topics

To control inverter output frequency (fundamental)

To control inverter output voltage (fundamental)

To minimize harmonic distortion

Why Use PWM Techniques?


Open loop voltage control

Closed loop current-control

VSIAC

motorPWMiref

if/back

VSI ACmotor

PWMvref


Inverter Configuration

4

Voltage Source Inverter (VSI) Six-Step VSI

Waveforms of gating signals, switching sequence, line to negative voltages for six-step voltage source inverter.

Gating signals, switching sequence and line to negative voltages

where, 561 means that S5, S6 and S1 are switched on

Six inverter voltage vectors for six-step voltage source inverter.

Switching Sequence:

561 (V1) 612 (V2) 123 (V3) 234 (V4) 345 (V5) 456 (V6) 561 (V1)

5


Waveforms of line to neutral (phase) voltages and line to line voltagesfor six-step voltage source inverter.

Line to line voltages (Vab, Vbc, Vca) and line to neutral voltages (Van, Vbn, Vcn)

Vab = VaN - VbN

Vbc = VbN - VcN

Vca = VcN - VaN

Line to line voltages

Van = 2/3VaN - 1/3VbN - 1/3VcN

Phase voltages

Vbn = -1/3VaN + 2/3VbN - 1/3VcN

Vcn = -1/3VaN - 1/3VbN + 2/3VcN

6


Harmonic spectrum of a square wave

Voltage Source Inverter (VSI)

Six-Step VSI

Amplitude of line to line voltages (Vab, Vbc, Vca)

Fundamental Frequency Component (Vab)1

Harmonic Frequency Components (Vab)h

: amplitudes of harmonics decrease inversely proportional to their harmonic order

dcdcdc V78.0V

6

2

V4

2

3

(rms))(V 1ab

3,.....)2,1,(n16nhwhere,

V78.0

dcab

h

(rms))(V h

7


AB DC4

V (1) V cos (4(b))n 2

8

Characteristics of Six-step VSI

It is called “six-step inverter” because of the presence of six “steps”

in the line to neutral (phase) voltage waveform

Harmonics of order three and multiples of three are

absent from both the line to line and the line to

neutral voltages and consequently absent from the

currents Output amplitude in a three-phase inverter can be controlled

by only change of DC-link voltage (Vdc)


Sinusoidal PWMModulating 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 mf should be an odd integer

if mf is not an integer, there may exist sub-hamonics at output voltage

if mf is not odd, DC component may exist and even harmonics are present at output voltage

mf should be a multiple of 3 for three-phase PWM inverter

An odd multiple of 3 and even harmonics are suppressed

Sinusoidal PWM

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 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

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 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

(a)

c c m1 AO DC

c m2 c c AO DC

c c m c c m DCAO c

c c c

DC m

c

T V v (t)S ON period = 2 V is V 2

2 2Vc

V v (t)S ON period = T T V is V 2

2Vc

T T v (t) T T v (t) V1V average for a period T + +

T 2 2V 2 2V 2

V v (t)..........(5)

2 V

Space Vector ModulationSwitching States (Three-Phase)

Eight switching states

1S

2S

3S 5S

4S 6S

B

C

P

N

dV

A

Space Vector ModulationSpace 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

Space Vector ModulationSpace 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)

Space Vector Modulation Space Vectors (Example)

Switching state [POO] S1, S6 and S2 ON

dCOdBOdAO VtvVtvVtv3

1)(,

3

1)(,

3

2)(

(5) (4)

0

1 3

2 j

d eVV

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

Space Vector Modulation 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

Space Vector Modulation

(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


Relationship Between Vref and VAB

Vref is approximated by two active and 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

Space Vector Modulation 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

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

d

j

refref VVeVV3

2, 1

32 3

2 j

d eVV


Dwell Times

Solve (12)

bas

d

refs

b

d

refs

a

TTTT

V

VTT

V

VTT

0

sin3

)3

(sin3

3/0 (13)

Space Vector Modulation 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


Modulation Index

cbs

asb

asa

TTTT

mTT

mTT

0

sin

)3

(sin

(15)

d

ref

a V

Vm

3 (16)


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)


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.

Space Vector Modulation 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

Space Vector Modulation 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

Space Vector Modulation 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.

Space Vector Modulation 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

Space Vector Modulation 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


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])


Even-Order Harmonic Elimination

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

ID PWM Inverters

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