CH.4 Full-wave and Three- phase rectifiers (Converting AC to DC)
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Transcript CH.4 Full-wave and Three- phase rectifiers (Converting AC to DC)
CH.4
Full-wave and Threephase rectifiers
(Converting AC to DC)
4-1
Introduction
The average current in AC source is zero
in the
full-wave rectifier, thus avoiding
problems
associated with nonzero average
source currents,
particularly in transformers.
The output of the full-wave rectifier has inherently
less ripple than the half-wave rectifier.
Uncontrolled and controlled single-phase and threephase full-wave converters used as
rectifiers are
analyzed.
4-2
Single-phase full-wave rectifiers
Fig. 4-1 Bridge rectifier:
The lower peak diode voltage
for high-voltage applications.
make
it
more
suitable
Fig. 4-2 center-tapped
rectifier
transformer
With electrical isolation, only
one diode voltage drop between
the source and load, suitable for
low-voltage, high-current applications
Resistive
load:
Vm sin wt ,0 wt
v0 ( wt )
Vm sin wt , wt 2
Vo
1
0
Io Vo
Vm sin( wt )d ( wt ) 2 Vm
R
2Vm
Irms Im
(R)
2
power absorbed by the load resistor:
PR I 2 rmsR
power
factor :Pf=1
R-L load:
Fig.4-3
v ( wt ) Vo
Vn cos( nwt )
0
n2,4‧‧
,
Vo 2Vm
Io Vo
Vn
2Vm 1
1
n 1 n 1
In Vn
R
If L is relatively large, the
dc.
( L >> R )
i( wt ) Io
Vo 2Vm
R
R
Irms Io
Vn
Zn | R jnwL |
load current is essentially
for L >> R
Source harmonics are rich in the odd-numbered harmonics.
Filters:reducing the harmonics.
R-L source load:
Fig.4-5
For continuous current operation, the only modification to
the analysis that was done for R-L load is in the dc
term of the Fourier series .The dc component of current
in this circuit is.
2Vm
Vdc
Vo Vdc
Io
R
R
The sinusoidal terms in the Fourier analysis
are unchanged by the dc source, provided
that the current is continuous.
Discontinuous current is analyzed like
section 3-5.
Capacitance
output filter:
Fig.
4-6
Assuming ideal diodes
| Vm sin wt | , one diode pair on
v0 ( wt )
( wt ) /( wRc )
Vm
sin
e
, diodes off
:the angle where the diodes become reverse biased,
which is the same as for the half-wave rectifier
and is
Tan 1 ( RC ) Tan 1 ( RC )
wt
Vm sin e ( ) /( RC ) Vm sin( )
(sin )e ( ) /( RC ) sin 0
=?
solved numerically for
Peak-to-peak variation(ripple):
Vo Vm | Vm sin( ) | Vm(1 sin )
In practical circuits where
2
,
minimal
,
ωRC
2
wt
output voltage occurs at
v0 ( ) Vm e
( ) /( RC )
2 2
Vm e /( RC )
Vo Vm Vm e /( RC ) Vm 1 e /( RC )
Vm1 1
‧
RC
Vm
Vm
RC 2 fRC
‧
x x2 x3
e 1
...
1! 2! 3!
w 2f
x
is half that of the half-wave rectifier.
Fig. 4-7
(a) Voltage doubler
Fig. 4-7
(b) Dual voltage rectifier
=full-wave rectifier(sw. open)+
voltage doubler(sw. closed)
L-C filtered output: Fig.4-8
C holds the output voltage at a constant level, and the L
smoothes the current from rectifier and reduces the peak
current in diodes.
Continuous
Current:
Vx Vo 2Vm
I L I R Vo
R
2Vm
VL =0 , full-wave rectified
, Ic 0
(R)
i L can be estimate from the first
The variation in
Ac term (n=2) in the Fourier series.
The amplitude of the inductor current for n=2 is
I2
where
V2
V
4Vm / 3 2Vm
2
Z 2 2L
2L
3L
Vn
2Vm 1
1
, n2
n 1 n 1
For Continuous current,
2Vm 2Vm
3L
R
R
L
3
I2 IL
3L
1
R
Discontinuous current:
When
iL
,
is positive
(
Vm sin wt Vo
at
wt
Vo
sin
Vm
1
vL Vm sin wt Vo
1 wt
Vm sin wt Vod (wt )
iL ( wt )
L
1
Vm(cos cos wt ) Vowt
L
for wt , ,
iL ( ) 0 , ?
)
Procedure for determining Vo:
(1) Estimate a
(2) Solve
Value
for
Vo
slightly below Vm, and solve
?
iL ( ) 0 Vm(cos cos ) Vo( )
numerically,
(3) Solve
1
I L i L ( wt )d ( wt )
1 1
Vm(cos cos wt ) Vo( wt )d ( wt )
L
(4) Slove
Vo=
ILR
(5) Repeat step (1)~(4) until the computed
equals the estimated Vo in step(1)
Vo
Output Voltage for discontinuous current is
for continuous current.(see Fig4-8(d))
in step(4)
larger
than
4-3
controlled full-wave rectifiers
Resistive
load:
Fig.4-10
1
Vo Vm sin( wt )d ( wt )
Vm
( 1 cos )
Io
I rms
delay
angle
Vo Vm
(1 cos )
R R
1
Vm
2
(
sin
wt
)
d ( wt )
R
Vm 1 sin( 2 )
R 2 2
4
The power delivered to the load
P I 2 rmsR
The rms current in source is the same as the rms current in
the load.
R-L
load : Fig.4-11
discontinuous current :
Vm
io ( wt )
sin( t ) sin( )e ( t ) /( )
Z
for
t
Z R 2 ( L ) 2
tan 1 (
L
)
R
, L
R
For discontinuous current
Analysis of the controlled full-wave rectifier operating in the
discontinuous current mode is identical to that of the controlled
half-wave rectifier, except that the period for the output current
is .
continuous current
wt
, i ( ) 0
sin( ) sin( )e ( ) /( )
sin( ) 1 e /( )
0
sin( - ) 0
v0 ( wt ) Vo Vn cos( nwt n)
( - ) 0
Tan (
-1
0
n 1
L
Vo
)
R
for continuous current
1
Vm sin wt d ( wt )
Vn an bn
2
n Tan -1 (
bn
)
an
2Vm
cos
2
an
2Vm cos( n 1) cos( n 1)
n 1
n 1
bn
2Vm sin( n 1) sin( n 1)
n 1
n 1
n 2,4,6,....
Fig 4-12
In Vn
Zn
Vn
Irms Io
2
| R jnwL |
n 2 ,4...
Io Vo
R
(
In
2
)2
R-L Source load : Fig.4-14
The SCRS may be turned on at any time that they are
forward biased, which is at an angle
sin 1 (Vdc Vm)
For continuous current case, the average bridge output voltage is
Vo
2 Vm
cos
average load current is
Io
Vo Vdc
R
The ac voltage terms are unchanged from the controlled rectifier
with an R-L load. The ac current terms are determined from
circuit.
Power absorbed by the dc voltage is
Pdc Io Vdc
Power absorbed by resistor in the load is
P I 2 rmsR Io 2 R if
L is l arg e
Controlled Single-phase converter operating as an inverter:
seeing Fig 4-14. 4-15
.
For inverter operation, power is supplied by the dc source,
and power is absorbed by the bridge and is transferred to
the ac system.
Vdc and Vo must be negative
00 90 0
900 1800
Vo 0
rectifier operation
Vo 0
inverter operation
Pbridge Pac IoVo
4-4 Three-phase rectifiers
Resistive load :
Fig 4-16
上、下半部Diode,每次僅一個ON;同相上、下Diode不可同時ON;
Diode ON由瞬間最大線電壓決定。
A transition of the highest line-to-line voltage must take place
every 360 0 / 6 60 0
.
Because of the six transitions that occur for each period
of the source voltage, the circuit is called a six-pulse
rectifier.
vo(t)之基頻為3
電源頻率之6倍
Diode turn on in the sequence 1,2,3,4,5,6,1,..
ia i D1 i D 4
i i i
D3
D6
b
ic i D 5 i D 2
Each diode conducts one-third of the time, resulting in
I D ,avg
1
I o ,avg
3
I D ,rms
1
I S ,rms
2
I o,rms
3
3
I o,rms
Apparent power from the three-phase source is
S 3 VLL ,rms I S ,rms
v0 ( t ) Vo
V
n
n 6 ,12 ,18..
cos( nw0 t )
3Vm ,L L
1 2 / 3
V0
Vm ,L L sin wtd( wt )
/
3
/3
0.95Vm ,L L
Vn
6 Vm ,L L
( n 1 )
2
, n 6 , 12 , 18, ...
Since the output voltage is periodic with period 1/6 of the ac
supply voltage, the harmonics in the output are of order 6kω,
k=1,2,3,…
Adevantage:output is inherently like a dc voltage, and the highfrequency low-amplitude harmonics enable filters to be effective.
For a dc
load current (constant I0) --- Fig4.17
ia
2 3
1
1
1
1
I o (cos w0 t cos 5w0 t cos 7 w0 t cos 11w0 t cos 13w0 t ....
5
7
11
13
which consists of terms at fundamental frequency of the ac
system and harmonics of order 6k 1, k=1,2,3,…
Filters(Fig.4-18) are
currents to enter the
Resonant filters for
High-pass filters for
frequently necessary to prevent harmonic
ac system.
5th and 7th harmonics.
higher order harmonics.
4-5 Controlled three-phase rectifiers
Vo
1
(
2
3
3 3
3Vm , L L
Vm , L L sin wtd ( wt )
) cos
Harmonics for output voltage remain of order 6k, but amplitude are
functions of
. seeing
Fig. 4-20
Twelve-pulse rectifiers:using two six-pulse bridges
The purpose
phase 30 0 shift
This results in
apart. The two
of the transformer connection is to introduce
between the source and bridge.
30 0
inputs to two bridges which are
bridge outputs are similar, but also shifted by
30 0
.
The delay angles for the bridge are typically the same.
Vo Vo,Y Vo,
3Vm, L L
cos
3Vm, L L
cos
6Vm, L L
cos
The peak output of the twelve-pulse converter occurs midway
between alternate peaks of the six-pulse converters. Adding the
voltages at that point for 0 gives
Vo , peak 2Vm, L L cos(15) 1.932 Vm, L L
for 0
Since a transition between conducting SCRs every 30
, there are a total of 12 such transitions for each period of the
ac source. The output has harmonic frequencies which are multiple
of 12 times the source fre. (12k
k=1,2,…)
1
1
1
1
I o (cos w0t cos 5 w0t cos 7 w0t cos 11w0t cos 13 w0t ....)
5
7
11
13
2 3
1
1
1
1
i (t )
I o (cos w0t cos 5 w0t - cos 7 w0t cos 11w0t cos 13 w0t ....)
5
7
11
13
4 3
1
1
iac (t ) iY (t ) i (t )
I o (cos w0t - cos 11w0t cos 13 w0t ...)
11
13
iac , harmonic order 12k 1 , k 1,2,...
iY (t )
2 3
Cancellation of harmonics 6(2n-1) 1 , n=1, 2, … has resulted
from this transformer and converter configuration.
This principle can be expanded
number by incorporating increased
with transformers which have the
The characteristic ac harmonics
pk 1
, k=1,2,3…
to arrangements of higher pulse
number of six-pulse converters
appropriate phase shifts.
of a p-pulse converter will be
More expense for producing high-voltage transformers with the
appropriate phase shifts.
Three-phase converter operating as
seeing 4-22.
a inverter:
The bridge output voltage Vo must be negative.
0 90
, Vo 0 - - Rectifier
operation
90 180 , Vo 0 - - Inverter operation
4-6 DC power transmission
․ By using controlled twelve-pulse converter (generally).
․ Used for very long distances of transmission lines.
Advantages:(1)
(2)
X L 0 , voltage drop↓ in lines
( line current
X C , line loss
)
(3) Two conductors required rather than three
(4) Transmission towers are smaller.
(5 ) Power flow in a dc transmission line is controllable
by adjustment of delay angles at the terminals.
(6) Power flow can be modulated during disturbances on
one of the ac system. System stability increased.
(7) The two ac systems that are connected by the dc
line do not need to be in synchronization.
Disadvantages:costly ac-dc converter, filter, and control system
required at each end of the line to interface
with the ac system.
Fig.4-23
using six-pulse converter
rectifier
, 0 90
Vo1 ,Vo 2
,
90
180
inverter
For current being ripple free
Vo1 Vo 2
R
3Vm1, L L
Vo1
cos 1
Io
Vo 2
3Vm 2, L L
cos 2
Power supplied by the converter at terminal 1 is
P1 Vo1 I o
Power supplied by the converter at terminal 2 is
P2 Vo 2 I o
Fig.4-24
using twelve-pulse converter
(a bipolar scheme)
One of the lines is energized at Vdc and the other is energized
at - V dc . In emergency situations, one pole of the line can operate
without the other pole, with current returning through the ground path.
4-7 commutation :effect of source inductance (
Single-phase bridge rectifier: Fig.4-25
Xs)
Assume that the load current is constant Io.
Commutation interval starts at ωt= ( Source polarity changed )
1 t
i s ( wt )
Vm sin wtd( wt ) I o
Ls
Vm
( 1 cos wt ) I o
Ls
Commutation
is completed at ωt=
i( u ) I 0
=>
+u
Vm
1 cos( u ) I 0
Ls
Commutation angle:
2I o Ls
2I o X S
1
u cos ( 1
) cos ( 1
)
Vm
Vm
1
X S Ls
Average load voltage
is
Vm
1
V
sin
wt
d
(
wt
)
( 1 cos u )
m
u
2V
I X
m (1 o s )
Vm
Vo
Source inductance lowers the average output voltage of fullwave rectifier.
Three-phase rectifier:
Fig.4-26
During Commutation
is
v La
from
D1 to D3
, The voltage across La
v AB Vm ,L L
sin wt
2
2
Current in La starts at I0 and decreases zero in
commutation interval
1 u Vm ,L L
i La ( u ) 0
sin wt d ( wt ) I 0
La
2
2L a I 0
2X s I0
u cos 1 ( 1
) cos 1 ( 1
)
Vm ,L L
Vm ,L L
the
During the commutation
output voltage is
v BC v AC
vo
2
interval from
,
D1 to D3
v AB vBC vCA 0 ,
v AB v AC - vBC
.
vo v AC vL a vL c v AC v AB
.
v AC
Average output Voltage:
Vo
3Vm ,L L
(1
the converter
2
v AC vBC v AC vBC
2
2
類似 Single-phase rectifier
X s I0
)
Vm ,L L
Source inductance lowers the average output voltage of threephase rectifiers.