1

Chapter 7

DC-to-AC Converters

“Introduction to Modern Power Electronics”, 2nd Ed., John Wiley 2010by

Andrzej M. Trzynadlowski

Voltage-source inverter supplied form a diode rectifier

Chapter 7 2

Fig. 7.1

AC

LIN

E

RECTIFIERDC LINK

LOA

D

INVERTER

Single-phase voltage-source inverter

Chapter 7 3

Fig. 7.2

i i

V i

i

vo

o

D A

D A'

SA

SA'

SB

SB'

D B

D B'

Circuit diagram of a Z-source

Chapter 7 4

Fig. 7.3

FR

OM

RE

CT

IFIE

R

TO

INV

ER

TE

R

States, switching variables, and waveforms of output voltage and current in a single-phase VSI in the basic square-wave mode

Chapter 7 5

Fig. 7.4

0

0 t

t

Vi

Vi

43

STATE: 2 1 2 1

b

a

io

-

0

vo

2

t

States, switching variables, and waveforms of output voltage and current in a single-phase VSI in the optimal square-wave mode

Chapter 7 6

Fig. 7.5

0

0 t

t

Vi

Vi

43

0.81

b

a

io

STATE: 2 3 1 0 2 3 1 0

-

0

vo

2

t

Waveforms of output voltage and current in a single-phase VSI in the PWM mode, N = 10: (a) m = 1, (b) m = 0.5

Chapter 7 7

Fig. 7.6

Vi -

Vi

Vi -

Vi

io

vo

vo

io

t0

(a)

t0

(b)

Waveforms of output voltage and current in a single-phase VSI in the PWM mode, N = 20: (a) m = 1, (b) m = 0.5

Chapter 7 8

Fig. 7.7

Vi -

Vi

Vi -

Vi

io

vo

vo

io

t0

(a)

t0

(b)

Harmonic spectra of output voltage in a single-phase VSI: (a) basic square-wave mode, (b) optimal square-wave mode

Chapter 7 9

Fig. 7.8

(b)

(a)

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

Harmonic spectra of output voltage in a single-phase VSI in the PWM mode (m = 1): (a) N = 10, (b) N = 20

Chapter 7 10

Fig. 7.9

(b)

(a)

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

Harmonic spectra of output current in a single-phase VSI: (a) basic square-wave mode, (b) optimal square-wave mode

Chapter 7 11

Fig. 7.10

(b)

(a)

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

Harmonic spectra of output current in a single-phase VS in the PWM mode (m = 1): (a) N = 10, (b) N = 20

Chapter 7 12

Fig. 7.11

(b)

(a)

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.001

0.01

0.1

1

10

Waveforms of input current in a single-phase VSI: (a) optimal square-wave mode, (b) PWM mode (m = 1, N = 20)

Chapter 7 13

Fig. 7.12

ii

ii

t0

(a)

t0

(b)

Harmonic spectra of input current in a single phase VSI: (a) optimal square-wave mode, (b) PWM mode (m = 1, N = 20)

Chapter 7 14

Fig. 7.13

(b)

(a)

HARMONIC NUMBER

0 10 20 30 40 50

AM

PLI

TU

DE

(pu

)

0.0001

0.001

0.01

0.1

1

HARMONIC NUMBER

0 10 20 30 40 50

AM

PLI

TU

DE

(pu

)

0.0001

0.001

0.01

0.1

1

Three-phase voltage-source inverter

Chapter 7 15

Fig. 7.14

SA' DA' SB' DB' SC' DC '

A B CiA i iB C

v v

v

AB BC

CA

N

v AN

vBN

v CN

V i

SA DA SB DB SC DC

i i

Chapter 7 16

TABLE 7.1 States and Voltages of the Three-Phase Voltage-Source Inverter

______________________________________________________________________________

State abc 𝑣𝐴𝐵 𝑉𝑖Τ 𝑣𝐵𝐶 𝑉𝑖Τ 𝑣𝐶𝐴 𝑉𝑖Τ 𝑣𝐴𝑁 𝑉𝑖Τ 𝑣𝐵𝑁 𝑉𝑖Τ 𝑣𝐶𝑁 𝑉𝑖Τ

____________________________________________________________________________________________________________________

0 000 0 0 0 0 0 0

1 001 0 -1 1 -1/3 -1/3 2/3

2 010 -1 1 0 -1/3 2/3 -1/3

3 011 -1 0 1 -2/3 1/3 1/3

4 100 1 0 -1 2/3 -1/3 -1/3

5 101 1 -1 0 1/3 -2/3 1/3

6 110 0 1 -1 1/3 1/3 -2/3

7 111 0 0 0 0 0 0

______________________________________________________________________________

Switching variables and waveforms of output voltages in a three-phase VSI in the square-wave mode

Chapter 7 17

Fig. 7.15

c

b

a

t

t

t

22

t

t

t

t

t

541 _3

_3

_3

_3

vCN

vBN

vAN

vCA

vBC

Vi

STATE: 5 4 6 2 3 1

t

0

vAB

Waveforms of output voltage (line-to-neutral) and current in a three-phase VSI in the square-wave mode (RL load)

Chapter 7 18

Fig. 7.16

3_ Vi 2

3_2 Vi

43t

iA

vAN

-

0 2

Waveforms of input current in a three-phase VSI in the square-wave mode (RL load)

Chapter 7 19

Fig. 7.17

ii

02

t

Switching variables and waveforms of output voltages in a three--phase VSI in the PWM mode

Chapter 7 20

Fig. 7.18

c

b

a

t

t

t

2

t

t

t

t

t

vCN

vBN

vAN

vCA

vBC

t

0

vAB

Waveforms of the output voltage and current in an RL load of a three-phase VSI in the PWM mode (a) load angle of , (b) load angle of

Chapter 7 21

Fig. 7.19

iA

vAN

iA

vAN

3-

2_ Vi

3

2_ Vi

3_ Vi 2

3_ Vi 2

-

t0

(a)

t0

(b)

Waveforms of input current in a three-phase VSI in the PWM mode (a) load angle of , (b) load angle of

Chapter 7 22

Fig. 7.20

ii

ii

t0

(a)

t0

(b)

Carrier-comparison PWM technique (N = 12, m = 0.75)

Chapter 7 23

Fig. 7.21

rA rB

t

t

t

y

c

b

a

rC

0 2t

Third-harmonic modulating function and its components at m = 1

Chapter 7 24

Fig. 7.22

2

1

-1

0

3rd

st

tF(1, )1

t

Voltage space vector plane of a three-phase VSI (per-unit)

Chapter 7 25

Fig. 7.23

V6V2

V1

jq

d

VI

I

IV

II

V

III

V 5

V4V3

v *

m

1

-1

__V 3

2 __V 3

2

High-quality space sequence

Chapter 7 26

Fig. 7.24

b

c

Ts w

2_TY

2_TX

1

1

0

0

0

1

X Y(4) (6)

S TATE :

100 110 a

2_TY

2_TX

Y(6)

X(4)

110 100

2_TZ

2_TZ

(7)

111

Z 1 Z 2(0)

000

High-efficiency space sequence

Chapter 7 27

Fig. 7.25

b

c

Ts w

TZ2_TY

2_TX

2_TY

2_TX

1

1

0

0

0

1

X Y Z(4) (6) (7)

Y(6)

X(4)

S TATE :

100 110 111 110 100a

Switching pattern with half- and quarter-wave symmetries

Chapter 7 28

Fig. 7.26

1

0_ _

a

2

32

10 2t

Optimal primary switching angles as functions of the magnitude control ratio (K = 5)

Chapter 7 29

Fig. 7.27

MAGNITUDE CONTROL RATIO

0.0 0.2 0.4 0.6 0.8 1.0

OP

TIM

AL

PR

IMA

RY

SW

ITC

HIN

0

15

30

45

60

75

90

Harmonic spectrum of line-to-neutral voltage with the harmonic-elimination technique (K = 5, M = 1)

Chapter 7 30

Fig. 7.28

HARMONIC NUMBER

0 20 40 60 80 100

AM

PLI

TU

DE

(pu

)

0.0001

0.001

0.01

0.1

1

Switching patterns and output voltage and current waveforms; (1) carrier-comparison PWM with sinusoidal reference, (2) space vector PWM with high-efficiency state sequence, (3) programmed PWM with harmonic elimination

Chapter 7 31

Fig. 7.29

iA

iA

vAN

vAN

a

a

0

0

0

0

0

0

t

t

t

2

t

t

t

(3)

(2)

(1)

iAvAN

a

0

Waveforms of output current in a three-phase VSI: (a) regular PWM, (b) random PWM

Chapter 7 32

Fig. 7.30

iCiBiA

iCiBiA

t0

(a)

t0

(b)

Frequency spectra of the line-to-neutral output voltage in a three-phase VSI: (a) regular PWM, (b) random PWM

Chapter 7 33

Fig. 7.31

(a)

HARMONIC NUMBER

0 48 96 144 192

AM

PLI

TU

DE

(pu

)

0.0001

0.001

0.01

0.1

1

HARMONIC NUMBER

0 48 96 144 192

AM

PLI

TU

DE

(pu

)

0.0001

0.001

0.01

0.1

1

Comparison of random PWM techniques with the regular PWM

Chapter 7 34

Fig. 7.32

n n+1 n+2 n+3 n+4

n n+1 n+3 n+4

n n+3 n+4n-1

. . .

. . .

. . .

t

t

t

n n+3n+1 n+4 . . .t

SAMPLING/SWITCHING CYCLES

SAMPLING/SWITCHING CYCLES

n+2

n+2n+1

SAMPLING CYCLES

SWITCHING CYCLES

n+2

PWM

T smp

Tsw,ndelt

RPWM

VD-RPWM

Hysteresis current control scheme

Chapter 7 35

Fig. 7.33

A

N

i Ai A

i *A i

i *i

i *

B

B C

C

i B

i C

B

C

Characteristic of the hysteresis current controller

Chapter 7 36

Fig. 7.34

i Ah2

h2

0

1

a

Waveforms of output currents in a VSI with hysteresis current control: (a) 20% tolerance, (b) 10% tolerance

Chapter 7 37

Fig. 7.35

iCiBiA

iA iCiB

t0

(a)

t0

(b)

Waveform of output currents in a VSI with hysteresis current control at a rapid change in the magnitude, frequency, and phase of the reference current

Chapter 7 38

Fig. 7.36

iA

t 0

Space vector version of the hysteresis current control scheme

Chapter 7 39

Fig. 7.37

A

N

i *q

i

i d

q

z d

z q

a b c

i A

iC

S W ITC HIN G TA B L E

dq

ab c

i q

i d

i *d

B

C

Characteristic of a current controller for the space vector version of the hysteresis current control scheme

Chapter 7 40

Fig. 7.38

z d

h2

h4

0h4

h2

id

1

-1

Characteristic of a current controller for the space vector version of the hysteresis current control scheme

Chapter 7 41

Fig. 7.39

RAMP GENERATOR

CONTROLLER COMPARATOR

iA*

iA iA

y

a

zA

iA

Waveforms of the output current in a VSI with the ramp comparison current control: (a)(b)

Chapter 7 42

Fig. 7.40

iCiBiA

iA iCiB

t0

(a)

t0

(b)

Current-regulated delta modulation scheme for a current-controlled VSI

Chapter 7 43

Fig. 7.41

iA*

iA

aS& H

iA

Linear current control scheme for a VSI

Chapter 7 44

Fig. 7.42

A

B

C

N

a b c

i A

iCdq

abc

i *q

i d

i q

i d

i *d

PULSE W IDTHMODULATOR

i q

LINEAR CONTROLLERS

v*d

v*q

Current-source inverter supplied from a controlled rectifier

Chapter 7 45

Fig. 7.43

RECTIFIERDC LINK

INVERTERLOAD

AC

LIN

E

CURRENT FEEDBACK

Three-phase current-source inverter

Chapter 7 46

Fig. 7.44

B

N

i B

Ai A

i i

Ci C

v AN

v BN

v CN

I i

SCSBSA

SC 'SB'

SA'

v AB v

v

BC

CA

Switching variables in a three-phase CSI in the square-wave mode

Chapter 7 47

Fig. 7.45

c'

b'

a'

c

b

at

22

t

t

t

t

541 _3

_3

_3

_3

t

0

Idealized waveforms of output currents in a three-phase CSI in the square-wave mode

Chapter 7 48

Fig. 7.46

iC

iB

iA

iCA

iBC

iAB

t

22

t

t

t

t

541 _3

_3

_3

_3

Ii

t

0

Waveforms of output voltage and current in a three-phase CSI in the square-wave mode: (a) RL load, (b) LE load

Chapter 7 49

Fig. 7.47

iA

vAN

iA

vAN

t0

(a)

t0

(b)

Three-phase PWM current-source inverter

Chapter 7 50

Fig. 7.48

R E C T IF IE RD C L INK

INV E R T E R

AC

LIN

E

L O A D

i'A

i'

i'

B

C

Ai

i

i

B

C

i

i

C,AB

C,BC

iC,CA

A

B

C

N

C UR R E NT F E E D B A C K

Carrier-comparison method for the PWM CSI

Chapter 7 51

Fig. 7.49

Optimal switching pattern for the PWM CSI with two primary switching angles

Chapter 7 52

Fig. 7.50

Waveforms of the output current, capacitor current, and output voltage in a three-phase PWM CSI (wye-connected RL load, P = 9)

Chapter 7 53

Fig. 7.51

vAN

iC,AB

iA

0

0

0

0 t

2

t

t

t

iA'

0

Generic five-level inverter

Chapter 7 54

Fig. 7.52

vo

V1

C1

C2

C3

C4

V3

Vi

4

S4

S5

S3

S2

S1

Vi

4

Vi

4

V2

V4

V5

Vi

4

Vi

Half-bridge voltage-source inverter

Chapter 7 55

Fig. 7.53

io

V i

i i

vo

Three-level neutral-clamped inverter

Chapter 7 56

Fig. 7.54

V i

i i

C2

C1

G

D6

D5S1

S2

S3

S4

D1

D2

D3

D4

v

A

AN

i A

B

C

i iB C

v v

N

BN CN

vBN

v vAB BC

vCA

Voltage space vectors of a three-level neutral-clamped inverter

Chapter 7 57

Fig. 7.55

d

jq

V6 V15 V24

V12 = V25

V18V=V9 22V8

V7 V21

V2 V20V11

V19

-1 1

__V 32

__V 32

-j

V=V1 14 V = V10 23V5

V16=V3

V=V4 17

States, switching variables, and waveforms of output voltage in a three-level neutral-clamped inverter in the square-wave mode

Chapter 7 58

Fig. 7.56

c

b

a

t

t

t

2

t

t

t

t

t

vCN

vBN

vAN

vCA

vBC

Vi

STATE: 18 21 24 15 6 7 8 5 2 11 20 19

t

0

vAB

Waveforms of output voltage and current in a three-level neutral-clamped inverter in the square-wave mode

Chapter 7 59

Fig. 7.57

3_ Vi 2

3_2 Vi

43t

iA

vAN

-

02

One phase of a multilevel cascaded H-bridge inverter

Chapter 7 60

Fig. 7.58

Switched network for illustration of the operating principle of a resonant dc link

Chapter 7 61

Fig. 7.59

V i

R L

C

i i

vo I oDS

i S

i C

i D

Waveforms of voltage and current in the resonant dc link

Chapter 7 62

Fig. 7.60

t0

I2

t3t2t1

iC

g

iS

0

0

0

0

0

0ii

Io

vo

t

t

t

t

t

Vi

0

t iD

Three-phase resonant dc link inverter with an active clamp

Chapter 7 63

Fig. 7.61

V i

A B C

N

C c l S

L

D

C

Waveforms of line-to-line output voltages in a resonant dc-link inverter

Chapter 7 64

Fig. 7.62

t

t

t

vBC

vAB

vCA

0

0

0

Auxiliary resonant commutated pole inverter: (a) one phase with the auxiliary circuit, (b) the entire inverter

Chapter 7 65

Fig. 7.63

(a)

A C

A C

A C

L rV i

C i

C i

AC

D1

D2

S1

S2

C r

S

C r

A

L f

A B C

(b)

Idealized line-to-neutral voltage and line current waveforms in a VSI in the square-wave mode

Chapter 7 66

Fig. 7.64

a

a

iSA iDA'

iDA iSA'

0

iSA'

iSA

0

vAN

vAN

t0

(a)

t0

(b)

Block diagram of a photovoltaic utility interface

Chapter 7 67

Fig. 7.65

D IO D E R E C TIF IE R

T R ANS F O R ME R

INVE R T E R

D C LIN K

F ILTE R

C O NT R O LLE D R E C T IF IE R

GR

ID

P V AR R AY

Block diagram of an active power filter

Chapter 7 68

Fig. 7.66

Waveforms of voltage and current in an active power filter

Chapter 7 69

Fig. 7.67

UPS System

Chapter 7 70

Fig. 7.68

LINEFILTER

RECTIFIER DC LINK

GR

ID

LO

AD

LOADFILTER

BATTERY

STATIC SWITCH

INVERTER

Block diagram of an ac drive system with scalar speed control

Chapter 7 71

Fig. 7.69

LOAD

SPEEDSENSOR

MOTOR

INVERTERDC LINK

RECTIFIER

AC

LIN

E

VOLTAGE CONTROLLER

SLIP CONTROLLER

*

*

*sl

M

MM*

M

v *

p

*sy n

Use of the modular frequency changer of Figure 2.24 in an ac drive: (a) system with a braking resistor, (b) system with a step-up chopper

Chapter 7 72

Fig. 7.70

RECTIFIER

INVERTER BRAKINGRESISTOR

AC

LIN

EA

C L

INE

RECTIFIER INVERTER

BOOSTINDUCTOR

(a )

(b )TO MOTOR

TO MOTOR

PWM rectifier-inverter cascades for bidirectional power flow in ac motor drives: (a) current-type rectifier, inductive dc link, and current-source inverter, (b) voltage-type-rectifier, capacitive dc link, and voltage-source inverter

Chapter 7 73

Fig. 7.71

AC

LIN

E

LINEFILTER

DC LINK

AC

LIN

E

LINEFILTER

RECTIFIER DC LINK

(b )

(a)

INV ERTER

INV ERTERRECTIFIER

MOTOR

MOTOR

Switching pattern of the inverter in Example 7.2

Chapter 7 74

Fig. 7.72

1

0

1

S TATE : X Y(7)

111

Y X(1) (5)

Z(5) (1)

001 101 101 001

( )s

( )s

( )s

a

b

c

t

t

t

0 32 92 .5 15 7.5 21 8 25 0

1

0

0

Per-unit voltage vectors of the three-level inverter in Example 7.3

Chapter 7 75

Fig. 7.73

V 2 4

V=V 9 2 2

V1 2 = V2 5

jq

d

V2 1

V1 8

v *

V1 5

0 0.5 1

32j

3j 4