2

sLLLLLL

aveo VVVVV 3,235.1

223

., ===π

)3

2(2)(

)3

2(2)(

)(2)(

πω

πω

ω

+=

−=

=

tSinVtv

tSinVtv

tSinVtv

sC

sB

sA

Three-Phase, Half-Wave, Uncontrolled Rectifier

3

Three-Phase, Half-Wave, Controlled Rectifier

sLL

LLLL

aveo

VV

CosVCosVV

3235.1

223

.,

=

== ααπ

4

AC

AC

AC

Cd Rloadvd

Lsa

b

c

n

ia

id

D1 D3 D5

D4 D6 D2

Three-Phase, Full-Wave, Uncontrolled Rectifier

5

AC

AC

AC

D1

D4

D3 D5

D6 D2

Idvd

ia

n

a

b

c

AC

AC

AC

ia

n

a

b

c

Id

Id

n

D1

D3

D5

D2

D6

D4

vd

vPn

vNn

P

N

Full-Bridge Rectifier

NnPnd vvv −=

6

AC

AC

AC

ia

n

a

b

c

Id

Id

n

D1

D3

D5

D2

D6

D4

vd

vPn

vNn

P

N

7

⎪⎩

⎪⎨

⎧−=0

d

d

a II

iwhen D1 is conducting

when D4 is conductingwhen neither D1 or D4 is conducting

LLLLLLdo

LLabd

VVtdtCosVV

ttCosVvv

35.123)()(23/

166

),(2

6/

6/

===

<<−==

∫− π

ωωπ

πωπω

π

π

LLVA 2=

1

1

1

78.061

816.032

ssh

dds

dds

Ih

I

III

III

=

==

==

π

line current

8

π3&0.1 == PFDPF

18

Comparison of Single-Phase and Three-Phase Rectifiers

Line current in a 1-phase rectifier contains significantly more distortion.

Much poorer power factor in a 1-phase rectifier compared to a 3-phase rectifier.

Ripple in the DC current is smaller in a 3-phase rectifier.

The filter required may be much smaller in a 3-phase rectifier.

Lower output DC voltage ripple in a 3-phase rectifier.

Regulation in the DC voltage from a no-load to a full-load condition for a 3-phase rectifier is much smaller than a 1-phase rectifier.

19

AC

AC

AC

Loadvdn

LdLs

a

van

id

Three-Phase, Full-Wave, Controlled Rectifier

20

AC

AC

AC

Idn

a

b

c

1

3

5

4

6

2

vd

ia

ib

ic

P

N

AC

AC

AC

n

a

b

c

1 3 5

4 6 2

Idvd

ia

ib

ic

P

N

Full-Bridge Rectifier

NnPnd vvv −=

21

22

α

αααπ

απππ

π

αωω

ω

α

α

αα

α

α

CosIVIVP

CosVCosVCosVV

CosVVAVV

VVV

CosVtdtSinVA

tSinVv

dLLdd

doLLLLd

LLLLdod

LLLLdo

LLLL

LLac

35.1

35.123

)1(23233/

35.123

)1(2)()(2

)(2

0

==

===

−−=−=

==

−==

=

∫

23

Two-Quadrant Operation

24

Two-Quadrant Operation

38

12-Pulse AC/DC Rectifiers

39

⎟⎠⎞

⎜⎝⎛ +−=+=

⎟⎠⎞

⎜⎝⎛ −+−+=

⎟⎠⎞

⎜⎝⎛ +−+−=

L

L

L

θθθπ

θθθθθπ

θθθθθπ

1313111

11132

1313111

1117

715

51

232

1313111

1117

715

51

232

21

2

1

CosCosCosIN

iii

CosCosCosCosCosIN

i

CosCosCosCosCosIN

i

daaa

da

da

),3,2,1(112L=

±=m

mh

ds

LL

ddd

ILCosV

VVV

πωα

π323

221

−=

==

41

High Voltage DC (HVDC) Transmission Systems