1. 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.
Uncontrolled　and　controlled　single-phase　and　three-phase　full-wave　converters　used　as　 rectifiers　are　analyzed.

2. 4-2　Single-phase　full-wave　rectifiers
Fig.　4-1 Bridge　rectifier：
The　lower　peak　diode　voltage　make　it　more　suitable　for　high-voltage　applications.

4. Fig.　4-2 center-tapped　transformer　rectifier　
　　With　electrical　isolation,　only　one　diode　voltage　drop　between　the　source　and　load, suitable　for　low-voltage, high-current　applications

5. Resistive load： power absorbed by the load resistor：
　power　factor ：Pf=1

6. R-L　load：　Fig.4-3

7. If　L　is　relatively large, the　load current　is　essentially　dc. ( )
Source　harmonics are　rich　in　the　odd-numbered　harmonics.
Filters：reducing　the　harmonics.

8. R-L source load：　Fig.4-5

9. Discontinuous current is analyzed like section 3-5.
　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.
　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.

10. 　Capacitance　output　filter: Fig. 　4-6

11. =？ solved numerically for
Assuming　ideal　diodes
：the　angle　where　the　diodes　become　reverse　biased, which is　the　same　as　for　the　half-wave　rectifier　 and　is
=？　　solved　numerically　for　
　Peak-to-peak　variation(ripple)：

12. minimal output voltage occurs at
　In　practical　circuits　where　ωRC ,
minimal　output　voltage occurs　at　
　is　half　that　of　the　half-wave　rectifier.

13. voltage doubler(sw. closed)
Fig. 4-7　(a) Voltage　doubler
Fig. 4-7　(b) Dual　voltage　rectifier
　　　　　　　=full-wave　rectifier(sw.　open)+
　　　　　　　　voltage　doubler(sw.　closed)

14. 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.

15. =0 , full-wave rectified Continuous Current：
　The　variation　in　 can　be　estimate　from　the　first　
Ac　term　(n=2) 　in　the　Fourier　series.
　The　amplitude　of　the　inductor　current　for　n=2　is
　where　
　For　Continuous　current, 　 

16. Discontinuous current： When is positive ( at )　,

17. 　Procedure　for　determining　Vo：
Estimate　a　Value　for　Vo　slightly　below　Vm, and　solve　　　
(2) Solve　 numerically,
(3) Solve　
(4) Slove　Vo=
(5) Repeat　step　(1)~(4) 　until　the　computed　Vo　in　step(4)
　 equals　the　estimated　Vo　in　step(1)
Output　Voltage　for　discontinuous　current　is　larger　than　
for　continuous　current.(see　Fig4-8(d))

18. 4-3 controlled full-wave rectifiers
Resistive　load： 　Fig.4-10

19. The rms current in source is the same as the rms current in the load.
　The　power　delivered　to　the　load
　The　rms　current　in　source　is　the　same　as　the　rms　current　in　
the　load.

20. R-L　load : Fig.4-11

21. 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　 .

22. continuous　current

23. 　Fig　4-12

25. 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

26. average load current is
　　For　continuous　current　case, the　average　bridge　output　voltage　is
average　load　current　is　
　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
Power　absorbed　by　resistor　in　the　load　is

27. Controlled Single-phase converter operating as an inverter：
seeing 　Fig　4-14. 　4-15 .

28.  Vdc and Vo must be negative
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
　　　rectifier　operation
　　　　
　　　　
　　　inverter　operation

29. 4-4　Three-phase　rectifiers
Resistive　load : Fig　4-16　　

31. A transition of the highest line-to-line voltage must take place every
　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　 .
　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 電源頻率之6倍
　Diode turn　on　in　the　sequence　1,2,3,4,5,6,1,..

32. Each diode conducts one-third of the time, resulting in
　Apparent　power　from　the　three-phase　source　is

33. 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　high-frequency　low-amplitude　harmonics　enable　filters　to　be　effective.

34. 　For　a　dc　load　current (constant I0) ---　Fig4.17

35. which　consists　of　terms　at　fundamental　frequency　of　the　ac　system　and　harmonics　of　order　6k
　Filters(Fig.4-18) 　are　frequently　necessary　to　prevent　harmonic　currents　to enter　the　ac　system.
　Resonant　filters　for　5th　and　7th　harmonics.
　High-pass　filters　for　higher　order　harmonics.

36. 4-5 Controlled three-phase rectifiers

37. 　Harmonics　for　output　voltage　remain　of　order　6k, but　amplitude　are　functions　of　
.　 seeing　 Fig. 4-20

39. Twelve-pulse rectifiers：using two six-pulse bridges

40. This results in inputs to two bridges which are
　The　purpose　of　the　 　transformer　connection　is　to　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 .
　The　delay　angles　for　the　bridge　are　typically　the　same.
　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 gives　

41. 　Since　a　transition　between　conducting　SCRs　every　
, 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,…)
Cancellation of harmonics 6(2n-1) , n=1, 2, … has resulted
from this transformer and converter configuration.

42. 　This　principle　can　be　expanded　to　arrangements　of　higher　pulse　number　by　incorporating　increased　number　of　six-pulse　converters　with　transformers　which　have　the　appropriate　phase　shifts.
　The　characteristic　ac　harmonics　of　a　p-pulse　converter　will　be
　pk 　 , 　k=1,2,3…
　More　expense　for　producing　high-voltage transformers　with　the　appropriate　phase　shifts.

43. Three-phase converter operating as a inverter：
　seeing　4-22.　

44. 　The　bridge　output　voltage　Vo　must　be　negative.

45. 4-6 DC power transmission
․ By　using　controlled　twelve-pulse　converter (generally).
․ Used　for　very　long　distances　of　transmission　lines.
Advantages：(1) 　 , 　voltage　drop↓　in　lines
(2) 　 , 　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.

46. Fig.4-23　　using　six-pulse　converter

47. For　current　being　ripple　free　
　Power　supplied　by　the　converter　at　terminal　1　is　
　Power　supplied　by　the　converter　at　terminal　2　is　

48. Fig.4-24 using twelve-pulse converter
　　　　　　(a　bipolar　scheme)

49. One of the lines is energized at and the other is energized at -
　One　of　the　lines　is　energized　at　　 and　the　other　is　energized　at In　emergency　situations, one　pole　of　the　line　can　operate　without　the　other　pole, with　current　returning　through　the　ground　path.

50. 4-7 commutation ：effect　of　source　inductance ( )
　 Single-phase　bridge　 rectifier: Fig.4-25

51. Commutation interval starts at ωt=
　Assume　that　the　load　current　is　constant　Io.
Commutation　interval　starts　at ωt=
Commutation　is　completed　at　ωt= +u
=> Commutation　angle：

52. Average load voltage is
　　Source　inductance　lowers　the　average　output　voltage　of　full-wave　rectifier.

53. Three-phase　rectifier：　　Fig.4-26

54. 　During　Commutation　from　　 , The　voltage　across　La is
　Current　in　 starts　at　I0　and　decreases　zero　in　the　
　commutation　interval

55. During the commutation interval from , the converter output voltage is
Average　output　Voltage：　　類似 Single-phase　rectifier
　Source　inductance　lowers　the　average　output　voltage　of　three-phase　rectifiers.