Chapter 3
AC to DC Converters
(Rectifiers)
Outline
3.1 Single-phase controlled rectifier
3.2 Three-phase controlled rectifier
3.3 Effect of transformer leakage inductance on
rectifier circuits
Power
3.4 Capacitor-filtered uncontrolled rectifier
3.5 Harmonics and power factor of rectifier circuits
3.6 High power controlled rectifier
3.7 Inverter mode operation of rectifier circuit
3.8 Realization of phase-control in rectifier circuits
2
3.1 Single-phase controlled
(controllable) rectifier
3.1.1 Single-phase half-wave controlled rectifier
Power
3.1.2 Single-phase bridge fully-controlled rectifier
3.1.3 Single-phase full-wave controlled rectifier
3.1.4 Single-phase bridge half-controlled rectifier
3
3.1.1 Single-phase half-wave
controlled rectifier
u2
Resistive load
VT
T
u1
Power
b)
uVT
u2
0
ug
t1

t
2
c)
ud
R
t
0
ud
id
d)
0 
t

uVT
a)
1
Ud 
2


2U 2 sin td (t ) 
e)
t
0
2U 2
1  cos
(1  cos )  0.45U 2
2
2
(3-1)
Half-wave, single-pulse
Triggering delay angle, delay angle, firing angle
4
3.1.1 Single-phase half-wave
controlled rectifier
Inductive (resistor-inductor) load
u2
VT
T
id
Power
u1
0

 t1
t
2
ug
uVT
a)
b)
u2
c)
t
0
ud
ud
+
+
d)
t
0 
id
e)
0

t
uVT
f)
0
t
5
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.
VT
VT
L
Power
L
u2
u2
R
R
b)
a)
did
L
 Rid 
dt
2U 2 sin t
(3-2)
ωt =  ，id= 0
R
 (t  )
2U 2
2U 2
id  
sin(   )e L

sin(t   )
Z
Z
(3-3)
6
Single-phase half-wave controlled
rectifier with freewheeling diode
u2
Inductive load (L is large enough)
VT
T
iVDR
u VT
a) u 1
u2
R
I VDR
Power
1

2
2 

t
Id
d)
O
iVT
(3-5)
I d2 d (t ) 
 
I d (3-7)
2
 
Id2 d (t ) 
Id
2
t
Id
e)
O
iVD
(3-6)
R

t
O
id
ud

t1
c)
 
I dVT 
Id
2
 
I dVD 
Id
2
1
2
O
ud
L
VDR
I VT 
b)
id
-
+
t
R
f)
t
O
uVT
g)
(3-8)
t
O
Maximum forward voltage, maximum reverse voltage
Disadvantages:
– Only single pulse in one line cycle
– DC component in the transformer current
7
3.1.2 Single-phase bridge
fully-controlled rectifier
Resistive load
ud
id
T
u1
VT 3
i2
VT 1
b)
a
 
t
uVT
1,4
ud
u2
R
c)
VT4
VT2
b
Power
0 
id
ud(id)
0
i2
t
0
t
d)
a)
For thyristor: maximum forward voltage, maximum
reverse voltage
Advantages:
– 2 pulses in one line cycle
– No DC component in the transformer current
8
3.1.2 Single-phase bridge fullycontrolled rectifier
Resistive load
Average output (rectified) voltage
1 
2 2U 2 1  cos
1  cos (3-9)
U d   2U 2 sin td(t ) 
 0.9U 2
 

2
2
Average output current
Power
Id 
U d 2 2U 2 1  cos
U 1  cos

 0.9 2
R
R
2
R
2
(3-10)
For thyristor
I dVT 
I VT
1
U 1  cos 
I d  0.45 2
2
R
2
1

2
2U 2
U2
1
 
2
(
sin

t
)
d
(

t
)

sin
2


 R

2 R 2

(3-11)
(3-12)
For transformer
I  I2 
1

 
(
2U 2
U
sint ) 2 d (t )  2
R
R
1
 
sin 2 
2

(3-13)
9
3.1.2 Single-phase bridge
fully-controlled rectifier
T
u2
t
O
id
ud
VT3
i2
t
O
Id
id
a
ud
u2
L
iVT O
t
Id
1,4
iVT O
Id
2,3
b
O
i2
R
VT4
VT2
Power
u1
VT1
Inductive load
(L is large enough)
uVT
O
Id
a)
1
 
 
2U 2 sintd(t ) 
t
t
Id
1,4
t
O
Ud 
t
b)
2 2

U 2 cos  0.9U 2 cos
Commutation
Thyristor voltages and currents
Transformer current
(3-15)
10
Electro-motive-force (EMF) load
With resistor
id
ud
E
R
Power
ud
E
O 


t
id
Id
t
O
a)
b)
Discontinuous current id
11
Electro-motive-force (EMF) load
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
Power
ud


0
=
E

t
id
O
2 2U 2
3 U 2
L
 2.8710
I dmin
I dmin
t
(3-17)
12
3.1.3 Single-phase full-wave
controlled rectifier
i1
Power
u1
T
ud
VT1
u2
u2 VT
2
ud
O 
R i
1
t
t
O
a)
b)
Transformer with center tap
Comparison with single-phase bridge fully-controlled
rectifier
13
3.1.4 Single-phase bridge
half-controlled rectifier
u2
a
u
ud
2
VD4
b
VD3
Power
id
VDR
T
VT 2
i2
VT 1
b)
t
O
ud

t
L
O
id
R
iVTO
iVD1
Id
iVTO
iVD
2

4
Id
t
3
iVDO
R
O
i2
O
Id
Id
t

t

Id
Half-control
I
Comparison with fully-controlled rectifier
Additional freewheeling diode
t
t
d
14
T
VD3
VT1
Another single-phase bridge
half-controlled rectifier
load
VD4
VT2
Power
u2
Comparison with previous circuit:
– No need for additional freewheeling diode
– Isolation is necessary between the drive circuits
of the two thyristors
15
Power
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
16
3.2 Three-phase controlled
(controllable) rectifier
3.2.1 Three-phase half-wave controlled rectifier
(the basic circuit among three-phase rectifiers)
Power
3.2.2 Three-phase bridge fully-controlled rectifier
(the most widely used circuit among three-phase
rectifiers)
17
3.2.1 Three-phase half-wave
controlled rectifier
Resistive load, = 0º
u2
O t1
T
VT1
Power
VT2
VT3
ud
R
id
ua
ub
t2
uc
t3
t
uG
O
ud
t
O
iVT
t
O
uVT
t
O
t
1
1
Common-cathode connection
Natural commutation point
uab
uac
18
Resistive load, = 30º
u2
ua
ub
uc
t
O
T
Power
VT1
VT2
VT3
ud
uG
t
O
ud
O
iVT
t1
t
1
R
id
O
uVT u
t
O
t
1 ac
uab
uac
19
Resistive load, = 60º
u2
O
T
ua
ub
uc
t
Power
VT1
ud
R
VT2
uG
VT3
O
ud
id
O
iVT
t
t
1
O
t
20
Resistive load, quantitative analysis
 When   30º, load current id is continuous.
5

6
1
Ud 
2
3

6

2U 2 sin td (t ) 
3 6
U 2 cos  1.17U 2 cos
2
(3-18)
1
Ud 
2
3

 
6

3 2 





2U 2 sin td (t ) 
U 2 1  cos(   )  0.6751  cos(   )
2
6
6




(3-19)
Average load current
U
Id  d
R
(3-20)
Thyristor voltages
1.2
1.17
Ud/U2
Power
 When >30º, load current id is discontinuous.
0.8
1
3
0.4
2
0
30
90
60
)
/(°
120
150
1- resistor load 2- inductor load
3- resistor-inductor load
21
Inductive load, L is large enough
ua
ud
O
T
u2
a
b
VT2
c
VT1
L
eL id
ud R
VT3
ub
uc
t

ia
O
ib
t
O
ic
t
O
id
t
O
uVT
t
Power
1
O
uac
Load current id is always continuous.
1
Ud 
2
3
5

6

6

uac
t
uab
(3-18)
3 6
2U 2 sin td (t ) 
U 2 cos  1.17U 2 cos
2
Thyristor voltage and currents, transformer current
I 2  I VT
1

I d  0.577I d (3-23)
3
U FM  U RM  2.45U 2
(3-25)
I VT(AV) 
I VT
 0.368 I d (3-24)
1.57
22
3.2.2 Three-phase bridge
fully-controlled rectifier
Circuit diagram
ia
T
VT1
id
a
n
Power
d1
VT 4
VT
6
VT3
VT 5
b
c
VT
load
ud
2
d2
Common-cathode group and common-anode group
of thyristors
Numbering of the 6 thyristors indicates the trigger
sequence.
23
Resistive load, = 0º
u2 = 0¡ãua
ud 1
ub
uc
O t 1
t
ud 2
u2 L
ud
Ⅰ Ⅱ
uab uac
Ⅲ Ⅳ Ⅴ Ⅵ
ub c ub a uca ucb uab uac
t
Power
O
iV T
1
O
uV T
uab uac
ub c ub a uca ucb uab uac
t
1
t
O
uab
uac
24
Resistive load, = 30º
ud 1 = 30¡ã
ua
O
ud 2
ud
ub
uc
t 1
t
Ⅰ Ⅱ
uab uac
Ⅲ Ⅳ Ⅴ Ⅵ
ub c ub a uca ucb uab uac
t
Power
O
uV T
1
uab uac
ub c ub a uca ucb
uab uac
t
O
ia
uab
uac
t
O
25
Resistive load, = 60º
ud 1
= 60¡ã
ua
ub
uc
t 1
t
O
ud 2
Power
ud
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
uab uac ub c ub a uca ucb uab uac
t
O
uV T
1
uac
uac
t
O
uab
26
Resistive load, = 90º
ud1
ua
ub
uc
ua
ub
t
O
ud2
ud
Power
O
uab uac
ubc uba uca ucb uab uac ubc uba
t
id
O
iVT
t
O
ia
t
O
t
1
27
Inductive load, = 0º
u2 = 0¡ãua
ud 1
ub
uc
O t1
Power
ud 2
u2 L
ud
t
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
uab uac ub c ub a uca ucb uab uac
t
O
id
O
iV T
t
O
t
1
28
Inductive load, = 30º
ud 1
= 30¡ãu
a
ub
uc
O t 1
ud 2
Power
ud
O
t
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
uab uac ub c ub a uca ucb uab uac
t
id
O
ia
t
O
t
29
Inductive load, = 90º
= 90¡ã
ud 1
Power
ud
uc
ua
t 1
O
ud 2
ub
t
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
uab uac ub c ub a uca ucb uab uac
t
O
uV T
1
uac
uac
t
O
uab
30
Quantitative analysis
Average output voltage
Ud 
1

2

3

3

6U 2 sin td (t )  2.34U 2 cos
(3-26)
3
For resistive load, When a > 60º, load current id is discontinuous.
3 



U d  
6U 2 sin td (t )  2.34U 2 1  cos(   )
(3-27)
 3 
3


Average output current (load current)
Power
Id 
Ud
R
(3-20)
Transformer current
1  2 2
2 
2
2
I2 
I d  0.816I d
 I d    ( I d )    
2 
3
3 
3
Thyristor voltage and current
(3-28)
– Same as three-phase half-wave rectifier
EMF load, L is large enough
– All the same as inductive load except the calculation of average output
current
Id 
Ud  E
R
(3-29)
31
3.3 Effect of transformer leakage
inductance on rectifier circuits
T
a
ik
b
c
R
Power
ud

ub
ua
LB
LB
LB
ud
ia
VT1
ib
VT2
ic
VT3
L
uc
t
O
id
O
ic
ia
g
ib
ic
ia
Id
t
In practical, the transformer leakage inductance has to be
taken into account.
Commutation between thyristors thus can not happen instantly,
but with a commutation process.
32
Commutation process analysis
Circulating current ik during commutation
ub-ua = 2·LB·dia/dt
Power
ik: 0
Id
ia = Id-ik : Id
0
ib = ik
Id
:
0
Commutation angle
Output voltage during commutation
dik
dik ua  ub
ud  ua  LB
 ub  LB

dt
dt
2
(3-30)
33
Quantitative calculation
Reduction of average output voltage due to the
commutation process
di k
1  g  56
3  g  56
U d 
(ub  ud )d(t )   5 [ub  (ub  LB )]d(t )
5



2 / 3 6
2   6
dt
Power
3

2
 g 


5
6
5
6
di
3
LB k d(t ) 
dt
2

Id
0
LBdik 
3
X BId
2
(3-31)
Calculation of commutation angle
cos  cos(  g ) 
– Id
– XB
,g
,g
2X BId
(3-36)
6U 2
– For   90o ,  , g
34
Summary of the effect on rectifier circuits
Circuits
U d
Power
cos  cos(  g )
Singlephase full
wave
XB

Id
Id X B
2U 2
Threephase halfwave
Singlephase
bridge
2X B

2I d X B
2U 2
Id
3X B
Id
2
2X BId
6U 2
Threephase
bridge
3X B

2X
B
Id
Id
6U 2
m-pulse recfifier
mX B
Id ①
2
Id X B
2U 2 sin
②

m
Conclusions
– Commutation process actually provides additional working
states of the circuit.
– di/dt of the thyristor current is reduced.
– The average output voltage is reduced.
– Positive du/dt
– Notching in the AC side voltage
35
3.4 Capacitor-filtered uncontrolled
(uncontrollable) rectifier
Emphasis of previous sections
– Controlled rectifier, inductive load
Power
Uncontrolled rectifier: diodes instead of thyristors
Wide applications of capacitor-filtered uncontrolled
rectifier
– AC-DC-AC frequency converter
– Uninterruptible power supply
– Switching power supply
3.4.1 Capacitor-filtered single-phase uncontrolled
rectifier
3.4.2 Capacitor-filtered three-phase uncontrolled
rectifier
36
3.4.1 Capacitor-filtered single-phase
uncontrolled rectifier
Single-phase bridge, RC load
id
Power
VD1
i2
u1
VD3
iC
ud +
u2
VD2
VD4
a)
i,ud
i
iR
C
ud
R
0


2
t

b)
37
3.4.1 Capacitor-filtered single-phase
uncontrolled rectifier
Single-phase bridge, RLC load
id
u2
L
+ uL VD3
iC
+
ud
VD2
VD4
Power
VD1
i2
u1
i2,u2,ud
iR
a)
ud
i2
R
C
u2

0


t
b)
38
3.4.2 Capacitor-filtered three-phase
uncontrolled rectifier
Three-phase bridge, RC load
ud u
ab u uac
d
VD1 VD3 VD5
T
ia
Power
a
b
c
VD4 VD6 VD2
a)
ia
id
iC
iR
ud +
C

0 

3

t
R
id
t
O
b)
39
3.4.2 Capacitor-filtered three-phase
uncontrolled rectifier
Power
Three-phase bridge, RC load
Waveform when RC1.732
ia
ia
O
t O
id
id
O
t O
a）RC= 3
t
t
b）RC< 3
40
3.4.2 Capacitor-filtered three-phase
uncontrolled rectifier
Three-phase bridge, RLC load
ia
VD1 VD3 VD5
T
iC iR
u d+ C
Power
a
b
c
t
O
id
ia
R
ia
b)
t
O
VD4 VD6 VD2
a)
c)
41
3.5 Harmonics and power factor of
rectifier circuits
Originating of harmonics and power factor issues in
rectifier circuits
– Harmonics: working in switching states—nonlinear
– Power factor: firing delay angle causes phase delay
Power
Harmful effects of harmonics and low power factor
Standards to limit harmonics and power factor
3.5.1 Basic concepts of harmonics and reactive power
3.5.2 AC side harmonics and power factor of
controlled rectifiers with inductive load
3.5.3 AC side harmonics and power factor of
capacitor-filtered uncontrolled rectifiers
3.5.4 Harmonic analysis of output voltage and current
42
3.5.1 Basic concepts of harmonics
and reactive power
For pure sinusoidal waveform
u (t )  2U sin(t   u )
(3-54)
For periodic non-sinusoidal waveform

Power
or
u (t )  ao   (an cos nt  bn sin nt )
(3-55)
n 1

u (t )  ao   cn sin(nt  n)
where
(3-56)
n 1
cn  an 2  bn 2
an  cn sin 
n  arctan(an / bn)
bn  cn cos
Fundamental component
Harmonic components (harmonics)
43
Harmonics-related specifications
Take current harmonics as examples
Content of nth harmonics
Power
In
HRIn  100%
I1
(3-57)
In is the effective (RMS) value of nth harmonics.
I1 is the effective (RMS) value of fundamental component.
Total harmonic distortion
Ih
THDi  100% (3-58)
I1
Ih is the total effective (RMS) value of all the harmonic components.
44
Definition of power and power factor
For sinusoidal circuits
Active power
1
P
2

2
0
uid (t )  UI cos 
(3-59)
Reactive power
Power
Q=U I sin
(3-61)
Apparent power
Power factor
S=UI
(3-60)
S 2  P2  Q2
(3-63)
l 
P
S
l=cos 
(3-62)
(3-64)
45
Definition of power and power factor
For non-sinusoidal circuits
Active power
Power
P=U I1 cos1
(3-65)
Power factor
P UI cos 1 I 1
(3-66)
l  1
 cos 1  n cos 1
S
UI
I
Distortion factor (fundamental-component factor)
n=I1 / I
Displacement factor (power factor of fundamental component)
l1 = cos1
Definition of reactive power is still in dispute.
46
Power
Review of the reactive power concept
The reactive power Q does not lead to net transmission of
energy between the source and load. When Q 0, the rms
current and apparent power are greater than the minimum
amount necessary to transmit the average power P.
Inductor: current lags voltage by 90°, hence displacement
factor is zero. The alternate storing and releasing of energy in
an inductor leads to current flow and nonzero apparent power,
but P = 0. Just as resistors consume real (average) power P,
inductors can be viewed as consumers of reactive power Q.
Capacitor: current leads voltage by 90°, hence displacement
factor is zero. Capacitors supply reactive power Q. They are
often placed in the utility power distribution system near
inductive loads. If Q supplied by capacitor is equal to Q
consumed by inductor, then the net current (flowing from the
source into the capacitor-inductive-load combination) is in
phase with the voltage, leading to unity power factor and
minimum rms current magnitude.
47
3.5.2 AC side harmonics and power factor of
controlled rectifiers with inductive load
Single-phase bridge fully-controlled rectifier
u2
VT3
ud
ud
u2
b
t
O
a
L
t
O
Id
id
R
VT4
u1
i2
VT2
Power
T
VT1
id
iVT O
t
Id
1,4
iVT O
Id
2,3
O
i2
a)
uVT
O
Id
t
t
t
Id
1,4
t
O
b)
48
AC side current harmonics of single-phase bridge
fully-controlled rectifier with inductive load
4
1
1
i2  I d (sin  t  sin 3 t  sin 5 t  )

3
5
4
1
 Id 
sin n t   2 I n sin n t
 n 1,3,5, n
n 1, 3, 5 ,
(3-72)
Power
where
2 2Id
In 
n
n=1,3,5,…
(3-73)
Conclusions
– Only odd order harmonics exist
– In  1/n
– In / I1 = 1/n
49
Power factor of single-phase bridge fullycontrolled rectifier with inductive load
Distortion factor
I1 2 2
n 
 0.9
I

(3-75)
Power
Displacement factor
l1  cos1  cos
(3-76)
I1
2 2
l  nl1  cos 1 
cos  0.9 cos
I

(3-77)
Power factor
50
Three-phase bridge fully-controlled rectifier
ud 1
= 30¡ãu
a
ub
uc
O t 1
ud 2
Power
ud
O
t
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
uab uac ub c ub a uca ucb uab uac
t
id
O
ia
t
O
t
51
AC side current harmonics of three-phase bridge
fully-controlled rectifier with inductive load
2 3
1
1
1
1
I d [sin t  sin 5t  sin 7t  sin 11t  sin 13t  ]

5
7
11
13
2 3
2 3
1

I d sin t 
I d  (1) k sin nt  2 I1 sin t   (1) k 2 I n sin nt


n
n 6 k 1
n 6 k 1
k 1, 2,3
k 1, 2, 3
(3-79)
where 
6
Power
ia 
Id
 I1 



 I  6 I , n  6k  1, k  1,2,3,
 n n d
(3-80)
Conclusions
– Only 6k1 order harmonics exist (k is positive integer)
– In  1/n
– In / I1 = 1/n
52
Power factor of three-phase bridge fullycontrolled rectifier with inductive load
Distortion factor
I1 3
n    0.955
I 
(3-81)
Power
Displacement factor
l1  cos1  cos
(3-82)
Power factor
I1
3
l  nl1  cos 1  cos   0.955 cos 
I

(3-83)
53
3.5.3 AC side harmonics and power factor of
capacitor-filtered uncontrolled rectifiers
Power
Situation is a little complicated than rectifiers with
inductive load.
Some conclusions that are easy to remember:
– Only odd order harmonics exist in single-phase
circuit, and only 6k1 (k is positive integer) order
harmonics exist in three-phase circuit.
– Magnitude of harmonics decreases as harmonic
order increases.
– Harmonics increases and power factor decreases as
capacitor increases.
– Harmonics decreases and power factor increases as
inductor increases.
54
3.5.4 Harmonic analysis of output voltage
and current
ud0  U d0 

b
n  mk
n
cos nt
ud

2 cos k


 U d0 1   2
cos nt 
 n  mk n  1

2 U2
Power
(3-85)
where
m

U d0 
2U 2
bn  
2 cos k
U d0 (3-87)
2
n 1

sin
m
(3-86)
O 
m
m
2
m
t
Output voltage of m-pulse
rectifier when 0º
55
Ripple factor in the output voltage
Output voltage ripple factor
UR
gu 
U d0
(3-88)
Power
where UR is the total RMS value of all the harmonic
components in the output voltage
UR 

2
2
2
U

U

U
 n
d0
(3-89)
n  mk
and U is the total RMS value of the output voltage
Ripple factors for rectifiers with different number of pulses
m
2
3
6
12
∞
gu（%）
48.2
18.27
4.18
0.994
0
56
Harmonics in the output current
id  I d 

d
n  mk
n
cos(nt   n )
(3-92)
Power
where
U d0  E
Id 
R
bn
dn 

zn
(3-93)
bn
R 2  (nL) 2
nL
 n  arctan
R
(3-94)
(3-95)
57
Conclusions for = 0º
Only mk (k is positive integer) order harmonics exist
in the output voltage and current of m-pulse rectifiers
Power
Magnitude of harmonics decreases as harmonic
order increases when m is constant.
The order number of the lowest harmonics increases
as m increases. The corresponding magnitude of the
lowest harmonics decreases accordingly.
58
For  0º
As an example,
for 3-phase bridge
fully-controlled rectifier
ud  U d 

c
n 6 k
n
0.3
2 U2L
Power
cn
Quantitative harmonic
analysis of output voltage
and current is very
complicated for  0º.
cos(nt  n)
n=6
0.2
n=12
0.1
n=18
0
30
60 90 120 150 180
/(¡ã)
(3-96)
59
3.6 High power controlled rectifier
3.6.1 Double-star controlled rectifier
Power
3.6.2 Connection of multiple rectifiers
60
3.6.1 Double-star controlled rectifier
Waveforms When  0º
Circuit
ud 1
ua
ub
uc
t
O
T
Power
a
b
ia
iP
n
n2
LP
n1
c' u
d
VT2
VT6
VT4
VT1
VT3
b'
id
uc'
ua'
uc'
ub'
ia'
t
t
O
L
VT5
1I
6 d
c
O
ud 2
a'
1I
2 d
1I
2 d
R
O
Difference from 6-phase half-wave rectifier
1I
6 d
t
61
Effect of interphase reactor
(inductor, transformer)
ud1,ud2 u '
b
ua
Power
VT1
uc'
ub
ua'
uc
ub'
1
u
2 P
n2-
iP
ua
ub'
ud 2 ud
n LP
+ - +n
1
L
ud 1
a)
O t1
t
。
up
60
R
VT6
b)
t
O
。
360
up  ud2  ud1
ud  ud2
(3-97)
1
1
1
 u p  ud1  U p  (ud1  ud2 ) (3-98)
2
2
2
62
Quantitative analysis when = 0º
ud1 
3 6U 2
1
2
1
[1  cos3t  cos6t  cos9t    ]
2
4
35
40
(3-99)
3 6U 2
1
2
1
[1  cos 3( t  60)  cos 6( t  60)  cos 9( t  60)  ]
2
4
35
40
3 6U 2
1
2
1

[1  cos 3 t  cos 6 t  cos 9 t  ]
(3-100)
2
4
35
40
Power
ud2 
3 6U 2 1
1
up 
[ cos3t  cos9t    ]
2
2
20
ud 
3 6U 2
2
[1  cos6t    ]
2
35
(3-101)
(3-102)
63
Waveforms
when > 0º
。
ud   30 u u '
a
c
O
Power
ud
Ud  1.17U 2 cos 
O
ud
O
  60。 u '
c
  90。 u '
c
ub
ua'
uc
ub'
t
ub
ua'
uc
ub'
t
ub
ua'
uc
ub'
t
64
Comparison with 3-phase half-wave
rectifier and 3-phase bridge rectifier
Voltage output capability
– Same as 3-phase half-wave rectifier
Power
– Half of 3-phase bridge rectifier
Current output capability
– Twice of 3-phase half-wave rectifier
– Twice of 3-phase bridge rectifier
Applications
Low voltage and high current situations
65
3.6.2 Connection of multiple rectifiers
Power
To increase the
output capacity
Connection
of multiple
rectifiers
Larger output voltage:
series connection
Larger output current:
parallel connection
To improve the AC side current waveform
and DC side voltage waveform
Phase-shift connection of multiple rectifiers
Parallel connection
1
LP
2
VT
T
c1
Power
b1
a1
L
c2
b2
a2
M
12-pulse rectifier realized by
paralleled 3-phase bridge rectifiers
67
Phase-shift connection of multiple rectifiers
Series connection
ia1
a1
I
id
0
u a1b1
A
▲
b1
Power
c1
▲
I
ia2
L
*
iab' 2
ud
II
30 °lagging
B
c2
▲
iab2
*
b2
iab 2
Id
2
I
3 d
t
1
I
3 d
3
I
3 d
c)
2 3
3 Id
t
0
a2
*
iA
3
(1+
R
u a2b2
II
360¡ãt
180¡ã
b)
0
1
C
Id
a)
1
0
iA
ia1
2 3
3
)Id
d)
0
3
3 Id
(1+
12-pulse rectifier realized by
series 3-phase bridge rectifiers
3
3
t
)Id
68
Quantitative analysis of 12-pulse rectifier
Voltage
– Average output voltage
Parallel connection:
Ud  2.34U2 cos
Series connection:
Ud  4.68U2 cos
Power
– Output voltage harmonics
Only 12m harmonics exist
Input (AC side) current harmonics
– Only 12k±1 harmonics exist
Connection of more 3-phase bridge rectifiers
– Three: 18-pulse rectifier (20º phase difference)
– Four: 24-pulse rectifier (15º phase difference)
69
Sequential control of multiple
series-connected rectifiers
u2
VT14 VT13
VT24 VT23
VT34 VT33
Power
u2
VT22 VT21
i
VT32 VT31
u2
VT12 VT11
Id
ud
Ⅰ
L
Ⅱ
O 
+
b)
ud
i
load
Id
2Id
Ⅲ
c)
a)
Circuit and waveforms of series-connected
three single-phase bridge rectifiers
70
3.7 Inverter mode operation of rectifiers
Power
Review of DC generator-motor system
EG  EM
Id 
R
EM  EG
Id 
R
should be avoided
71
Inverter mode operation of rectifiers
Rectifier and inverter mode operation of single-phase
full-wave converter
VT1
1
0
u10
iVT
1
u20 VT2
2
iVT
Power
2
ud

u10
1
L
u20
R
id
ud
L
iVT
1
0
VT2
Energy +
M EM
u10
VT1
id
ud
Energy
2
iVT
2
ud
u20
u10
R
M EM
+
u10
Ud>EM
t
O
t
O
Ud<EM
id
id=iVT +iVT
1
2
iVT
iVT
1
2
iVT
O
a)
Ud  EG
Id 
R
1
Id
t
id

id=iVT +iVT
iVT
iVT
2
1
2
O
iVT
1
2
b)
Id 
Id
t
EM  Ud
R
72
Necessary conditions for the inverter
mode operation of controlled rectifiers
Power
There must be DC EMF in the load and the direction
of the DC EMF must be enabling current flow in
thyristors. (In other word EM must be negative if
taking the ordinary output voltage direction as
positive.)
>90º so that the output voltage Ud is also negative.
EM > U d
73
Inverter mode operation of
3-phase bridge rectifier
u2
ua
ub
uc
ua
ub
uc
ua
ub
uc
ua
ub
t
O
= 
3
Power
ud uab uac ub c ub a uca
= 4
ucb uab uac ub c ub a uca
= 6
ucb uab uac ub c ub a uca
ucb uab uac ub c
 t1  t2  t3
t
O
= 
3
= 4
= 6
Inversion angle (extinction angle) 
 +  =180º
74
Inversion failure and minimum inversion angle
Power
Possible reasons of
inversion failures
– Malfunction of
triggering circuit
– Failure in thyristors
– Sudden dropout of AC
source voltage
– Insufficient margin for
commutation of
thyristors
Minimum inversion
angle (extinction angle)
min=+g+′ （3-109）
a
b
c
iV T
1
iV T
2
LB VT
1
LB
VT2
LB VT
L
id
3
iV T
ud
3
M EM
+
o
ua
ud
ub
uc
ua
ub
p
O
id
iV T
3
O

g  g

g > g

iV T
1
iV T
2
t
iV T
3
iV T
1
t
75
3.8 Realization of phase-control in
rectifier circuits
Power
Object
How to timely generate triggering pulses with
adjustable phase delay angle
Constitution
Synchronous circuit
Saw-tooth ramp generating and phase shifting
Pulse generating
Integrated gate triggering control circuits are very
widely used in practice.
76
A typical gate triggering control circuit
R15
VD11~VD14
220V
C7 + C 6
VD15
36V
VD7
+15V
RP2
Power
VS
R3
A
V1
R1
TS
R
R4
R7
C2
V2 R5
Q
C1
R2
R6
VD9
C5
R16
V7
V4
R17
R8
VD6
+15V
R18
R14
V5
R10
VD8
TP
R13
I1c
VD1 VD2
uts
C3
VD4
V3
R12
R11
R9
B
C3
VD10
V8
V6
VD5
up
RP1
uco -15V
Disable
XY
-15V
77
Power
Power
Waveforms of the typical
gate triggering control circuit
78
How to get synchronous voltage for the gate
triggering control circuit of each thyristor
uA
uB
uC
UAB
Ua
U
Usa - sc
TS
Power
TR
-Usb
Usb
Ub
D,y 5-11
D,y 11
Usc
ua
ub
uc - usa - usc usb
usc
- usb u sa
Uc
-Usa
For the typical circuit on page 20, the synchronous voltage of the gate
triggering control circuit for each thyristor should be lagging 180º to the
corresponding phase voltage of that thyristor.
79