AC to DC converters

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Chapter 2 AC to DC Converters
Outline
2.1 Single-phase controlled rectifier
2.2 Three-phase controlled rectifier
2.3 Effect of transformer leakage inductance on rectifier circuits
2.4 Capacitor-filtered uncontrolled rectifier
2.5 Harmonics and power factor of rectifier circuits
2.6 High power controlled rectifier
2.7 Inverter mode operation of rectifier circuit
2.8 Thyristor-DC motor system
2.9 Realization of phase-control in rectifier
2.1 Single- phase controlled (controllable) rectifier

2.1.1 Single-phase half-wave controlled rectifier

Resistive load

T

VT

R

a

)

u

1

u

2

u

VT

u

d

i

d

0

ω

t

1

π

2

π

ω

t

u

2

u

g

u

d

u

VT

α

θ

0

b

)

c

)

d

)

e

)

0

0

ω

t

ω

t

ω

t

∫

+
=+==
π
α
α
α
π
ωω
π
2
cos1
45.0)cos1(
2
2
)(sin2
2
1
2
2
2d
U
U
ttdUU
（
2-1
）

Inductive (resistor-inductor) load

a

)

u

1

T

VT

R

L

u

2

u

VT

u

d

i

d

u

2

0

ω

t

1

π

2

π

ω

t

u

g

0

u

d

0

i

d

0

u

VT

0

θ

α

b

)

c

)

d

)

e

)

f

)

+

+

ω

t

ω

t

ω

t

ω

t

Basic thought process of time-domain analysis for power electronic
circuits
The time- domain behavior of a power electronic circuit is actually the
combination of consecutive transients of the different linear circuits
when the power semiconductor devices are in different states.

a

)

b

)

VT

R

L

VT

R

L

u

2

u

2

tURi
t
i
L
ω
sin2
d
d
2d
d
=+
ω
t
=
a
，
i
d
=0
)sin(
2
)sin(
2
2

)(
2
d
ϕωϕα
αω
ω
−+−−=
−−
t
Z
U
e
Z
U
i
t
L
R
（
2-2
）

（
2-3
）

Single- phase half- wave controlled rectifier with freewheeling diode
load (L is large enough) Inductive

a

)
L

T
VT
R

u

1
u

2
u

VT
u
d
VD
R
i
d

i

VD

u

2

u
d
i

d
u
VT
i

VT
I

d
I
d
ω

t

O
O
O
O
O
O
π

-
α

π

+
α

b
)
i

VD
R
ω

t
ω

t

ω

t

ω

t
ω

t
g

)
c
)
e
)
f
)
d
)
ω

T
1

Maximum forward voltage, maximum reverse voltage
Disadvantages:
–Only single pulse in one line cycle
–DC component in the transformer current

ddVT
2
II
π
απ
−
=
d
2
dVT
2

)(
2
1
ItdII
π
απ
ω
π
π
α
−
==
∫
d
2
2
dVD
2
)(
2
1
R
ItdII
π
απ
ω
π
απ
π
+

==
∫
+
ddVD
2
R
II
π
απ
+
=
（
2-5
）

（
2-6
）

（
2-7
）

（
2-8
）

2.1.2 Single- phase bridge fully-controlled rectifier

Resistive load

π

ω

t

0

0

0

i

2

u

d

i

d

b

)

c

)

d

)

u

d

(

i

d

)

α

α

R

T

u

1

u

2

a

)

i

2

a

b

VT
3

u

d

i

d

u

VT

1

,

4

ω

t

ω

t

VT
4

VT
1

VT
2

Average output (rectified) voltage:
Average output current:
For thyristor:
For transformer:

∫
+
=
+
==
π
α
αα
π
ωω
π
2
cos1
9.0
2
cos122
)(dsin2
1
2
2
2d
U
U
ttUU
（
2-9
）

（

2-10
）

2
cos1
9.0
2
cos122
22
d
d
αα
π
+
=
+
==
R
U
R
U
R
U
I

（
2-11
）

2

cos1
45.0
2
1
2
ddVT
α
+
==
R
U
II
π
απ
α
π
ωω
π
π
α
−
+==
∫
2sin
2
1
2
)(d)sin
2
(

2
1
2
2
2
VT
R
U
tt
R
U
I
（
2-12
）

π
απ
α
π
ωω
π
π
α
−
+===
∫
2sin
2

1
)()sin
2
(
1
2
2
2
2
R
U
tdt
R
U
II
(2-13)


Inductive load (L is large enough)

ω
t

ω
t

ω
t
ω
t

ω
t

ω
t

ω
t

ο

ο

ο

ο

ο

ο

ο

u
2

u
d

i
d

I
d

I
d

I
d

I
d

I
d

i
VT
1,4

i
VT
2,3

u
VT
1,4

i
2
,

b
)
R

T

u

1

u

2

a

)

i

2

a

b

VT
3

u

d

i

d

VT
4

VT
1

VT
2


Electro- motive-force (EMF) load With resistor

∫
+
===
απ
α
αα
π

ωω
π
cos9.0cos
22
)(dsin2
1
222d
UUttUU
（
2-15
）

a

)

b

)

R

E

i

d

u

d

i

d

O

E

u

d

ω

t

I

d

O

ω

t

α

θ

δ


With resistor and inductor
When L is large enough, the output voltage and current waveforms are
the same as ordinary inductive load.
When L is at a critical value

O

u

d

0

E

i

d

ω

t

ω

t

π

δ

α

θ

=

π

dmin
2
3
dmin
2
1087.2
22
I
U
I
U
L
−
×==

πω
（
2-17
）

2.1.3 Single- phase full- wave controlled rectifier

a

)

b

)

u

1

T

R

u

2

u

2

i

1

VT

1

VT

2

u

d

u

d

i

1

O

O

α

ω

t

ω

t

2.1.4 Single- phase bridge half-controlled rectifier
a)
T
a
b
R
L
O
b)
u
2
i
2
u
d
i
d
VT
1
VT
2

VD
3
VD
4
VD
R
u
2
O
u
d
i
d
I
d
O
O
O
O
O
i
2
I
d
I
d
I
d
I
d

I
d
α
ω
t
ω
t
ω
t
ω
t
ω
t
ω
t
ω
t
α
π−
α
π−
α
i
VT
1
i
VD
4
i
VT

2
i
VD
3
i
VD
R

Another single- phase bridge half-controlled rectifier
Comparison with previous circuit:
–No need for additional freewheeling diode
–Isolation is necessary between the drive circuits of the two thyristors

load

T

u

2

VT
2

VT
4

VT
1

VT
3

Summary of some important points in analysis
When analyzing a thyristor circuit, start from a diode circuit with the
same topology. The behavior of the diode circuit is exactly the same
as the thyristor circuit when firing angle is 0.
A power electronic circuit can be considered as different linear circuits
when the power semiconductor devices are in different states. The
time- domain behavior of the power electronic circuit is actually the
combination of consecutive transients of the different linear circuits.
Take different principle when dealing with different load
– For resistive load: current waveform of a resistor is the same as the
voltage waveform
–For inductive load with a large inductor: the inductor current can
be considered constant
2.2 Three- phase controlled (controllable) rectifier
2.2.1 Three- phase half- wave controlled rectifier
Resistive load, α= 0º
a

b

c

T

R

u

d

i

d

VT

2

VT

1

VT

3

u

2

O

O

O

u

ab

u

ac

O

i

VT

1
u

VT

1
ω

t

ω

t

ω

t

ω

t

ω

t

u

a

u

b

u

c

u

G

u

d

ω

t
1

ω

t
2

ω

t
3

Common-cathode connection
Natural commutation point

Resistive load, α= 30º

u

2

u

a

u

b

u

c

O

O

O

ω

t

O

ω

t

O

ω

t

u

G

u

d

u

ab

u

ac

ω

t

1

i

VT

1

u

VT

1

u

ac

ω

t

ω

t

a

b

c

T

R

u

d

i

d

VT

2

VT

1

VT

3

Resistive load, α= 60º

ω

t

u

2

u

a

u

b

u

c

O

O

O

O

u

G

u

d

i

VT

1
ω

t

ω

t

ω

t

a
b

c

T
R

u

d

i
d

VT

2

VT
1

VT
3

Resistive load, quantitative analysis
When α≤ 30º , load current id is continuous.

When α > 30º , load current id is discontinuous.
Average load current
Thyristor voltages
αα
π
ωω
π
α
π
α
π
cos17.1cos
2
63
)(sin2
3
2
1
22
6
5
6
2d

UUttdUU ===
∫
+
+
（
2-18
）







++=






++==
∫
+
)
6
cos(1675.0)
6
cos(1

2
23
)(sin2
3
2
1
2
6
2d
α
π
α
π
π
ωω
π
π
α
π
UttdUU
(2-19)

R
U
I
d
d
=
（

2-20
）

0

30

60

90

120

150

0

.

4

0

.

8

1

.

2

1

.

17

3

2

1

α

/(

°

)

U
d/
U
2



Inductive load, L is large enough

T

R

L

u

d

e

L

i

d

VT

3

u

d

i

a

u

a

u

b

u

c

i

b

i

c

i

d

u

ac

u

ab

u

ac

O
O
O
O
O
O
ω

t

α

u

VT

1
ω

t

ω

t

ω

t

ω

t

ω

t

a

b

c

VT

2

Thyristor voltage and currents, transformer current :

αα
π

ωω
π
α
π
α
π
cos17.1cos
2
63
)(sin2
3
2
1
22
6
5
6
2d
UUttdUU ===
∫
+
+
（
2-18
）

ddVT2
577.0
3

1
IIII ===
d
VT
VT(AV)
368.0
57.1
I
I
I ==
2RMFM
45.2 UUU ==
（
2-23
）

（
2-25
）

（
2-24
）

2.2.2 Three- phase bridge fully-controlled rectifier
Circuit diagram
Common- cathode group and common- anode group of thyristors