notes-CH4.ppt

8/21/2019 notes-CH4.ppt

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CH.4 Full-wave and Three-

phase 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

!ncontrolled and controlled single-phase and three- hase full-wave converters used as


2/55

4-2 Single-phase full-wave rectifiers

Fig. 4-1 Bridge rectifier ：

The lower peak diode voltage ake it ore

suita!le for high-voltage applications.


3/55


4/55

Fig. 4-2 center-tapped

transformer rectifier

"ith electrical isolation#

onl$

one

diode

voltage

drop

!etween the source and load#

suita!le for low-voltage# high-

current applications


5/55

Resistive load：

π≤≤π−π≤≤

="

##

wt ,wt sinVm

wt ,wt sinVm )wt ( v

∫ == π

π π #"$%$sin%

1 Vmwt d wt VmVo

$%"

RVm

RVo Io

π ==

"Im= Irms

power absorbed by the load resistor ：

rmsR I P R"=

power factor ： %f&'


6/55

R-L load： Fig.4-3


7/55

∑∞

=π++=

‧‧4"# , ,n )nwt cos( VnVo )wt ( v

π= VmVo "

+−−= 11

1

1"

nn

Vm

Vn π

RVo Io =

| jnwL R|

Vn

ZnVn In

+==

If & is relatively large, the load current is

essentially dc ( )

R>> L for

Io Irms

R

Vm

R

Vo

Io )wt ( i ω

≈π==≈ "

'ource harmonics are rich in the odd-numbered

harmonics

(ilters： reducing the harmonics

R>> Lω


8/55

R-L source load： Fig.4-


9/55

or con nuous curren opera on, e on y modification to the analysis that was done for

)-& load is in the dc term of the (ourier

series The dc component of current in this circuit

is

R

VdcVm

R

VdcVo Io

−=

−= π

"

The sinusoidal terms in the (ourier

analysis are unchanged by the dc

source, provided that the current is

continuous*iscontinuous current is analyzed li+e

section -


10/55

!apacitance output filter" Fig. 4-#


11/55

Assuming ideal

diodes

θ=

θ−− off diodes ,e sinVm

on pair diode one ,|wt sinVm| )wt ( v

)wRc /( )wt (

#

θ ： the angle where the diodes become

reverse biased, which is the same as for the half-wave rectifier and is

π+ω−=ω−=θ −− ) RC ( Tan ) RC ( Tan 11

α π +=wt

) sin( Vme sinVm ) RC /( )( α+π−=θ ωθ−α+π−

#=α−θ ωθ−α+π− sine )(sin ) RC /( )(

α .？ solved numerically for α /ea+-to-pea+

variation%ripple$：

$sin1%0$sin%0 α α π −=+−=∆ VmVmVmVo


12/55

In practical circuits where ω)C

,

" ,

"π α π θ ≈≈

minimal output voltage occurs at

α+π=wt

) RC /( ) RC /( )(

eVmeVm )( v ωπ−ω

π−

π+π−

==α+π ""#

[ ]

fRC Vm

RC Vm

RC Vm

eVmeVmVmVo ) RC /( ) RC /(

"

11

1

‧

‧

=ω π=

ωπ

−−=

−=−≈∆ ωπ−ωπ−

f w

e

π "

"1

1"

=

++++=！！！

is half that of the half-wave rectifier

π >>


13/55

Fig. 4-$ %a& 'oltage dou(ler

ig. 4-$ %(& )ual voltage rectifier

*full-wave rectifier%sw. open&+

voltage dou(ler%sw. closed&


14/55

L-! filtered output ： Fig.4-,

C holds the output voltage at a constant level# and

the soothes the current fro rectifier and

reduces the peak current in diodes.


15/55

Continuous

Current：

( & " π VmVoV == .# , full-wave rectified )

#, $%

" ==== Ic R

Vm R

Vo I I R L π

LiThe variation in can be estimate from the first

Ac term %n."$ in the (ourier series

The amplitude of the inductor current for n." is

L

Vm

L

/ Vm

L

V

Z

V I

πω=

ω

π=

ω==

"

"

4

"" "

"

"

where "1

1

1

1"=

+−

−π= n ,

nn

VmVn

(or Continuous current, L I I <

"

R

Vm

L

Vm

π<

πω"

"

ω>

R L 1

>

ω R

L


16/55

*iscontinuous current：

2hen is positive ( at ),

Vowt V v m L −= sin

[ ]

( )[ ]

! i

,wt for

wt Vowt Vm

L

wt d Vowt Vm L

wt i

L

wt

L

==


17/55

/rocedure for determining o：

%1$ 3stimate a alue for o slightly below m, and solve =α

%"$ 'olve numerically,β

$%$cos%cos#$% α β β α β −−−== VoVmi L

%$

'olve

[ ]∫

∫ β

α

β

α

α−−−αωπ=

π=

)wt ( d )wt ( Vo )wt cos(cosVm L

)wt ( d )wt ( i I L L

11

1

%4$ 'love o. R I L

%$ )epeat step %1$5%4$ until the computed o in

step%4$ e6uals the estimated o in step%1$

utput *oltage for discontinuous current is larger than

for continuous current.(see Fig4-,%d&)


18/55

4-3 controlled full-wave rectifiers

Resistive load： Fig.4-1


19/55

)cos( Vm

)wt ( d )wt sin( VmVo

α+

π

=

π= ∫

π

α

1

1

an"#e de#a$=α

$cos1% α π

+== R

Vm

R

Vo Io

π

α

π

α

π

π

α

4

$"sin%

""

1

$%$sin%1 "

+−=

= ∫

R

Vm

wt d wt R

Vm I rms

The power delivered to the load rmsR I P "=

The rms current in source is the same as the rms cur

the load


20/55

R-L load " Fig.4-11


21/55

Analysis of the controlled full-wave rectifier operating

in the discontinuous current mode is identical to that

of the controlled half-wave rectifier, e7cept that the

period for the output current is .π

[ ] ) /( )t ( o e ) sin( )t sin( Z

Vm )wt ( i ωτα−ω−θ−α−θ−ω= for β≤ω≤α t

R L , ) R

L( tan

) L( R Z

=τω=θ

ω+=

−1

""

(or discontinuous current

π α β +<

discontinuous current "


22/55

continuous current

#$% , ≥++= α π α π iwt

[ ]

c%rrent contin%o%s for

R

L

Tan

& )'(

& )' sin( e

e

'

$%

#1$sin%

#$sin%$sin%

$8%

$8%$%

ω

θ α

α θ

α θ α θ

θ α θ α π ωτ π

ωτ α α π

=≤

≥≥

≥+−

≥−−−+−

−+−

,9,4,"

1

$1sin%

1

$1sin%"

1

$1cos%

1

$1cos%"

cos"

$%sin1

$cos%$%

""

1

=

−−

−++

=

−−

−++

=

+=

==

++=

∫

∑

+

∞

=

n

n

n

n

nVm

n

n

n

nVm

a

aVn

Vmwt d wt VmVo

nnwt VnVowt v

n

n

nn

n

&

α α

π

α α

π

α π π

θ

α π

α

$%an

nTann '

−=θ


23/55

Fig 4-12


24/55

RVo Io

) In( Io Irms

| jnwL R|Vn

ZnVn In

*** ,n

=

+=

+==

∑∞

= 4"

""

"


25/55

R-L Source load ： Fig.4-14

The 'C)' may be turned on at any time that they are forward biased, which is at an angle

$%sin 1Vm

Vdc−≥α


26/55

(or continuous current case, the average bridge output volta

average load current is

The ac voltage terms are unchanged from the

controlled rectifier with an )-& load The ac current

terms are determined from circuit /ower absorbed by the dc voltage is

e# is Lif R IormsR I P arg "" ≈=

α

π

= cosVm

Vo"

RVdcVo Io −=

Vdc Io Pdc =

/ower absorbed by resistor in the load is


27/55

Controlled 'ingle-phase converter operating as an inverter ：

seeing Fig 4-14. 4-15


28/55

## :##

Vo

rectifier

operation

## 1;#:#


29/55

4-4 hree-phase rectifiers

Resistive load " Fig 4-1#


30/55


31/55

上、下半部 *iode，每次僅一個


32/55

3ach diode conducts one-third of the time, resulting in

av" oav" + I I ,,

1=

rmsorms + I I ,,

1=

rmsorms I I ,,

"=

Apparent power from the three-phase source is

rms , rms , L L I V −=


33/55

*** , , ,n , )n(

V V

V *

V

)wt ( wtd sinV / V

)t nwcos( V Vo )t ( v

L L ,m

n

L L ,m

L L ,m /

/ L L ,m

** , ,n

n

1;1"91

9

:-#

1

"

"

#

#

1;1"9

#

=−π

=

=π=π=

π++=

−

−

−π

π −

∞

=

∫

∑

,ince the output voltage is periodic with period '+

of the ac suppl$ voltage# the haronics in the output

are of order +k# k&'#/#0#1

Adevantage： output is inherentl$ like a dc voltage# and

the high-fre2uenc$ low-aplitude haronics ena!le filters

to !e effective.


34/55

For a dc load current %constant /& --- Fig4.1$


35/55

****t wcost wcost wcost wcost w(cos I i oa ##### 11

111

11

1?

?

1-

-

1"+−+−

π=

which consists of terms at fundamental fre6uency of

the ac system and harmonics of order 9+ ± 1, +.1,",,@

Filters(Fig.4-1,) are fre2uentl$ necessar$ to prevent

haronic currents to enter the ac s$ste.

3esonant filters for th and 5th haronics.

High-pass

filters

for

higher

order

haronics.


36/55

4- !ontrolled three-phase rectifiers


37/55

α π

π

α π

α π

cos$

%

$%sin

1

,

"

,

L Lm

L Lmo

V

wt wtd V V

−

+

+ −

=

= ∫

armonics for output voltage remain of order 9+, but

amplitude are functions of α

seeing Fig. 4-20


38/55

Twelve-pulse rectifiers： using two si6-pulse !ridges


39/55

p g p g

The purpose of the transformer connection is to ∆Υ


40/55

p p

introduce phase shift between the source and bridge

This results in inputs to two bridges which are

apart The two bridge outputs are similar, but also shifted

by

∆− Υ##

##

#

#

The delay angles for the bridge are typically the same

α π α π α π cos

9

cos

cos

,,,,,

L Lm L Lm L Lm

o- oo

V V V

V V V

−−−

∆ =+=+=

The pea+ output of the twelve-pulse converter occurs

midway between alternate pea+s of the si7-pulse converters

Adding the voltages at that point for gives °= #α

°==°= −− #:"1$1-cos%" ,,, α for V V V L Lm L Lm pea. o

'ince a transition between conducting 'C)s every°


41/55

'ince a transition between conducting 'C)s every

, there are a total of 1" such transitions for each

period of the ac source The output has harmonic

fre6uencies which are multiple of 1" times the source

fre %1"+ +.1,",@$

°#

,",111"

cos1

cos1

%cos

$%$%$%

$cos1

1cos

11

1cos

?

1cos

-

1%cos

"$%

$cos1

1cos

11

1cos

?

1cos

-

1%cos

"$%

###

#####

#####

=±=

−+=+=

++−+=

−+−+−=

∆

∆

. ,. order armonic ,i

***)t w00

t w

't w I 1

t it it i

t w0t wt w2 't w3t w I t i

t w0t wt w2 t w3t w I t i

ac

o- ac

o

o-

π

π

π

Cancellation of harmonics 9%"n-1$ 1 , n.1, ", @ has resulted

from this transformer and converter configuration

±

This principle can be e7panded to arrangements of hi h l b b i ti i d b f


42/55

higher pulse number by incorporating increased number of

si7-pulse converters with transformers which have the

appropriate phase shifts

The characteristic ac harmonics of a p-pulse converter

will be p+ ' # k&'#/#01± Bore e7pense for producing high-voltage transformers

with the appropriate phase shifts


43/55

ree-phase converter operating as a inverter ：

seeing 4-22.


44/55

The bridge output voltage o must be negative

operation Inverter '' &Vo ,

operation Rectifier '' &Vo ,

>


45/55

4-6 DC power transmission

․ >y using controlled twelve-pulse converter %generally$

․ !sed for very long distances of transmission lines

Advantages： %1$ # voltage drop7 in lines (/) # line loss

#= L

4

∞=C 4

% ↓c%rrent #ine %$ Two conductors re6uired rather than three%4$ Transmission towers are smaller% $ /ower flow in a dc transmission line is

controllable by adustment of delay angles

at the terminals

%9$ /ower flow can be modulated during disturbances on

one of the ac system 'ystem stability

increased

%?$ The two ac systems that are connected by

the dc line do not need to be in synchronization

)

Disadvantages： costl$ ac-dc converter# filter# and control

s$ste re2uired at each end of the

line to interface with the ac s$ste.

Fi 4 23 i i l t


46/55

Fig.4-23 using si0-pulse converter

° tifi:##


47/55

°


48/55

Fig.4-24 using twelve-pulse converter

(a !ipolar schee)


49/55


50/55

4-7 commutation ： effect of source inductance ( &

Single-phase (ridge rectifier" Fig.4-2 s 4

Assume that the load current is constant Io


51/55

Assume that the load current is constant Io

Commutation interval starts at ωt.π

$changed polarity% o%rce

om

t

o s

I )wt cos( Ls

V

I )wt ( wtd sinVm Ls

)wt ( i

++ω

−=

+ω

=

∫

ω

π

1

1

Commutation is completed at ωt.π Du

[ ] ## 1 I )%cos( Ls

V I )%( i m ++π+

ω

−=−=+π

)Vm

4 I ( cos )

Vm

Ls I ( cos% oo

"1

"1 11 −=

ω−= −−

&8 Coutation angle：

Ls 4 ω=

Average load voltage is


52/55

Average load voltage is

)V

4 I (

5V

)%cos( V

)wt ( d wt sinV V

m

som

mm

%o

−π=

+π

=π

= ∫ π

1

11

,ource inductance lowers the average output voltage

of full-wave rectifier.

hree-phase rectifier： Fig 4-2#


53/55

hree phase rectifier ： Fig.4 2#

*uring Commutation from The voltage1 +to+


54/55

*uring Commutation from # The voltage

across a is1 +to +

wt sinV v

v L L ,m 67

La""

−==

Current in starts at 9: and decreases ;ero in the

coutation interval

La

)

V

I 4 ( cos )V

I L( cos%

I )wt ( d wt sinV

La )%( i

L L ,m

s

L L ,m

a

% L L ,m

La

−

−

−

−

+π

π

−

−=ω−=

+ω

==+π ∫ #1#1

#

"1"1

"

1#

*uring the commutation interval from the+to+


55/55

*uring the commutation interval from # the

converter output voltage is

1 + to +

"

6C 7C

o

vvv

+=

""

"

#

7C 6C 7C 6C 6C

67 6C

*

*c La L 6C o

7C 6C 67C6 7C 67

vvvvv

vvvvvv

v'vv , vvv

+=

−−=

−=+−=

==++

Average output oltage： () 'ingle-phase rectifier

)V

I 4 (

V V

L L ,m

s L L ,m

o

−

− −π

= #1

,ource inductance lowers the average output voltage

of three-phase rectifiers.

Date post:	07-Aug-2018
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