‫كلية الهندسة‬

Faculty of Engineering

Dr. Shaimaa Abd El-Hamid Kandil

Contact details: shaimaa.kandil@h-eng.helwan.edu.eg
 Chapter (2)
Lecture (3)
Single Phase Rectifier……

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 2

 Single Phase Full-Wave Controlled Rectifier
Today’s Lecture Outlines:
Single Phase Full-Wave Controlled Rectifier

Center-Tap Transformer Full-wave

Full-Wave Rectifier. Bridge Rectifier.

➢ Resistive Load

➢ Battery Charger

➢ Highly Inductive Load

➢ Highly Inductive Load with FWD

➢ Single Phase Full-Wave Half-Controlled Rectifier.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 3

 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier.

The single-phase bridge uses 4 thyristors and produces

an output average voltage which is almost double that
produced from a half-wave rectifier operating at the
same conditions.

Operation:
Positive half-cycle: Negative half-cycle:

𝒗𝒐 = 𝒗𝒔 𝒗𝒐 = −𝒗𝒔
T1 & T2 are forward conduction, T3 & T4 are forward conduction,
T3 & T4 are reversed. T1 & T2 are reversed.
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 4
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier.

The single phase bridge uses 4 thyristors and produces

an output average voltage which is almost double that
produced from a half-wave rectifier operating at the
same conditions.
To ensure the proper operation:
▪ T1 and T2 are fired together at α, during the +ve half cycle, while
▪ T3 and T4 are fired together at π+α during the –ve half cycle.
▪ Thus, to analyze the circuit, we draw both vs and –vs

vs vs -vs

π 2π 3π w t

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 5

 Single Phase Full-Wave Controlled Rectifier

Center-Tap Transformer Full-Wave Rectifier.

Center-tapped Transformer

𝑉𝑠 𝑁𝑠
=
𝑉𝑃 2𝑁𝑝

𝐼𝐴 2𝑁𝑝
=
𝐼𝑃 𝑁𝑠
𝐼𝑠 = 𝐼𝐴 − 𝐼𝐵
𝐼𝐵 2𝑁𝑝
=
𝐼𝑃 𝑁𝑠
6
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 6
 Single Phase Full-Wave Controlled Rectifier

Center-Tap Transformer Full-Wave Rectifier.

Center-tapped Transformer
Operation:
Positive half-cycle: Negative half-cycle:

𝒗𝒐 = 𝒗𝒔 𝒗𝒐 = −𝒗𝒔
T1 is forward conduction, T2 is forward conduction,
T2 is reversed. T1 is reversed.
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 7
 Single Phase Full-Wave Controlled Rectifier
Center-Tap Rectifier. Resistive Load.

𝒗𝒔 = 𝑽𝒎 𝐬𝐢𝐧 𝝎𝒕

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 8

 Single Phase Full-Wave Controlled Rectifier
Center-Tap Rectifier. Resistive Load.
𝑵𝒑 : 2 𝑵𝒔

𝒗𝒔 = 𝑽𝒎 𝐬𝐢𝐧 𝝎𝒕

Operation:
During Positive Half Cycle:
T1 turns on , T2 off.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 9

 Single Phase Full-Wave Controlled Rectifier
i=0
Waveforms : 𝝎𝒕 = 𝝅
Center-Tap Rectifier. Resistive Load. vs Vm

𝑵𝒑 : 2 𝑵𝒔

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig2 T1 ON
π+a wt
𝒗𝒔 = 𝑽𝒎 𝐬𝐢𝐧 𝝎𝒕 T2 ON
vo

Operation: wt

During Positive Half Cycle:

T1 turns on , T2 off.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 10

 Single Phase Full-Wave Controlled Rectifier
i=0
Waveforms : 𝝎𝒕 = 𝝅
Center-Tap Rectifier. Resistive Load. vs Vm

𝑵𝒑 : 2 𝑵𝒔

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig2 T1 ON
π+a wt
𝒗𝒔 = 𝑽𝒎 𝐬𝐢𝐧 𝝎𝒕 T2 ON
vo

Operation: iT1 wt

During Positive Half Cycle:

iT2 wt
T1 turns on , T2 off.
io wt

is 𝐼𝑇1
𝑁𝑠
𝑁𝑝
wt

wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 11
 Single Phase Full-Wave Controlled Rectifier
i=0
Waveforms : 𝝎𝒕 = 𝝅
Center-Tap Rectifier. Resistive Load. vs Vm

𝑵𝒑 : 2 𝑵𝒔

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig2 T1 ON
π+a wt
𝒗𝒔 = 𝑽𝒎 𝐬𝐢𝐧 𝝎𝒕 T2 ON
vo

Operation: iT1 wt

During Positive Half Cycle:

iT2 wt
T1 turns on , T2 off.
During Negative Half Cycle: io wt

T1 turns off , T2 Conduct.

is 𝐼𝑇1
𝑁𝑠
𝑁𝑝
wt

wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 12
 Single Phase Full-Wave Controlled Rectifier
i=0
Waveforms : 𝝎𝒕 = 𝝅
Center-Tap Rectifier. Resistive Load. vs Vm

𝑵𝒑 : 2 𝑵𝒔

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig2 T1 ON
π+a wt
𝒗𝒔 = 𝑽𝒎 𝐬𝐢𝐧 𝝎𝒕 T2 ON
vo

Operation: iT1 wt

During Positive Half Cycle:

iT2 wt
T1 turns on , T2 off.
During Negative Half Cycle: io wt

T1 turns off , T2 Conduct.

is 𝐼𝑇1
𝑁𝑠
𝑁𝑝 𝑁
wt
-𝐼𝑇2 𝑁𝑠
𝑁𝑠 𝑝
𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇2 ∗ wt
𝑁𝑝
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 13
 Single Phase Full-Wave Controlled Rectifier
i=0
Waveforms : 𝝎𝒕 = 𝝅
Center-Tap Rectifier. Resistive Load. vs Vm
Circuit Performance parameters:
1- Average Output Voltage:
π 2π 3π w t
1 𝜋
𝑉𝑜𝑎𝑣 = න 𝑉𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡 ig1
𝜋 𝛼
-Vm
2- Average Output Current: a 2π+a wt
𝑉𝑜𝑎𝑣 ig2 T1 ON
𝐼𝑜𝑎𝑣 =
𝑅 π+a wt
3- RMS Output Voltage: T2 ON
vo
1 𝜋
𝑉𝑜𝑟𝑚𝑠 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 2 𝑑𝜔𝑡
𝜋 𝛼 𝑚
wt
4- RMS Output Current: iT1
𝑉𝑜𝑟𝑚𝑠 iT2
𝐼𝑜𝑟𝑚𝑠 = wt
𝑅
5- Average Output Power:
2
io wt
𝑃𝑜 = 𝐼𝑜𝑟𝑚𝑠 ∗𝑅

6- Input Power Factor: is 𝐼𝑇1

𝑁𝑠
𝑁𝑝 𝑁
wt
𝑃𝑜 -𝐼𝑇2 𝑁𝑠
𝑝
𝑝. 𝑓𝑠 =
𝑉𝑠𝑟𝑚𝑠 𝐼𝑠𝑟𝑚𝑠 wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 14
 Single Phase Full-Wave Controlled Rectifier
i=0
Waveforms : 𝝎𝒕 = 𝝅
Center-Tap Rectifier. Resistive Load. vs Vm
Circuit Performance parameters:
7- RMS Supply Current:
π 2π 3π w t
𝑁
𝐼𝑠𝑟𝑚𝑠 =𝐼𝑜𝑟𝑚𝑠 𝑁𝑠 ig1
𝑝 -Vm
a 2π+a wt
8- Average Thyristor Current: ig2 T1 ON
π+a wt
𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = T2 ON
2 vo
9- RMS Thyristor Current:
wt
𝐼𝑜𝑟𝑚𝑠 iT1
𝐼𝑇𝑟𝑚𝑠 = 2
iT2 wt

io wt

is 𝐼𝑇1
𝑁𝑠
𝑁𝑝 𝑁
wt
-𝐼𝑇2 𝑁𝑠
𝑝

wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 15
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier.

Battery Charger (R-E Load).

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 16

 Single Phase Full-Wave Controlled Rectifier
Waveforms : i=0
Full Bridge Rectifier. vs Vm 𝝎𝒕 = 𝝅 − 𝜶𝒐

Battery Charger (R-E Load). 𝛼𝑜 = 𝑠𝑖𝑛−1

𝐸𝑏
𝑉𝑚
π 2π 3π w t
𝛼 ≥ 𝛼𝑜
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
T3,4 ON
vo
Operation:
During Positive Half Cycle: wt
T1 & T2 turns on , T3 & T4 off.

𝜶𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 17
 Single Phase Full-Wave Controlled Rectifier
Waveforms : i=0
Full Bridge Rectifier. vs Vm 𝝎𝒕 = 𝝅 − 𝜶𝒐

Battery Charger (R-E Load). 𝛼𝑜 = 𝑠𝑖𝑛−1

𝐸𝑏
𝑉𝑚
π 2π 3π w t
𝛼 ≥ 𝛼𝑜
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
T3,4 ON
vo
Operation:
During Positive Half Cycle: iT1,2 wt
T1 & T2 turns on , T3 & T4 off.
iT3,4 wt

io wt

is wt

wt

𝜶𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 18
 Single Phase Full-Wave Controlled Rectifier
Waveforms : i=0
Full Bridge Rectifier. vs Vm 𝝎𝒕 = 𝝅 − 𝜶𝒐

Battery Charger (R-E Load). 𝛼𝑜 = 𝑠𝑖𝑛−1

𝐸𝑏
𝑉𝑚
π 2π 3π w t
𝛼 ≥ 𝛼𝑜
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
T3,4 ON
vo
Operation:
During Positive Half Cycle: iT1,2 wt
T1 & T2 turns on , T3 & T4 off.
iT3,4 wt

During Negative Half Cycle: wt

io
T1 & T2 turns off , T3 & T4 on.
is wt

wt

𝜶𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 19
 Single Phase Full-Wave Controlled Rectifier
Waveforms : i=0
Full Bridge Rectifier. vs Vm 𝝎𝒕 = 𝝅 − 𝜶𝒐

Battery Charger (R-E Load). 𝛼𝑜 = 𝑠𝑖𝑛−1

𝐸𝑏
𝑉𝑚
π 2π 3π w t
𝛼 ≥ 𝛼𝑜
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
T3,4 ON
vo
Operation:
During Positive Half Cycle: iT1,2 wt
T1 & T2 turns on , T3 & T4 off.
iT3,4 wt

During Negative Half Cycle: wt

io
T1 & T2 turns off , T3 & T4 on.
is wt

𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇4 wt

𝜶𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 20
 Single Phase Full-Wave Controlled Rectifier
Waveforms : i=0
Full Bridge Rectifier. vs Vm 𝝎𝒕 = 𝝅 − 𝜶𝒐

Battery Charger (R-E Load).

Circuit Performance parameters: π 2π 3π w t
1- Average Output Voltage: ig1,2
1 𝜋−𝛼𝑜 𝜋+𝛼 -Vm
𝑉𝑜𝑎𝑣 = න 𝑉𝑚 sin 𝜔𝑡 𝑑𝜔𝑡 + න 𝐸 𝑑𝜔𝑡 a 2π+a wt
𝜋 𝛼 𝜋−𝛼𝑜 ig3,4 T1,2 ON
=
𝑉𝑚
− cos 𝜋 − 𝛼𝑜 + cos 𝛼 +
𝐸
𝛼 + 𝛼𝑜 π+a wt
𝜋 𝜋 T3,4 ON
2- Average Output Current: vo
𝑉𝑜𝑎𝑣 − 𝐸
𝐼𝑜𝑎𝑣 =
𝑅 iT1,2 wt
3- RMS Output Voltage:
iT3,4 wt
𝜋−𝛼𝑜 𝜋+𝛼
1 2
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑚 sin 𝜔𝑡 𝑑𝜔𝑡 + න 𝐸 2 𝑑𝜔𝑡
𝜋 𝛼 𝜋−𝛼𝑜 io wt

4- RMS Output Current: is wt

𝑉𝑜𝑟𝑚𝑠 − 𝐸
𝐼𝑜𝑟𝑚𝑠 = wt
𝑅
𝜶𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 21
 Single Phase Full-Wave Controlled Rectifier
Waveforms : i=0
Full Bridge Rectifier. vs Vm 𝝎𝒕 = 𝝅 − 𝜶𝒐

Battery Charger (R-E Load).

Circuit Performance parameters: π 2π 3π w t

5- Output DC Power: ig1,2

-Vm
2
𝑃𝑜 = 𝐼𝑜𝑟𝑚𝑠 ∗ 𝑅 + 𝐸 ∗ 𝐼𝑜𝑎𝑣 a 2π+a wt
ig3,4 T1,2 ON
6- Input Power Factor: π+a wt
T3,4 ON
𝑃𝑜
𝑝. 𝑓𝑠 =
𝑉𝑠𝑟𝑚𝑠 𝐼𝑠𝑟𝑚𝑠
vo
7- RMS Supply Current: iT1,2 wt

𝐼𝑠𝑟𝑚𝑠 = 𝐼𝑜𝑟𝑚𝑠 iT3,4 wt

8- Average Thyristor Current: wt

io
𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = 2 is wt
9- RMS Thyristor Current:
𝐼𝑜𝑟𝑚𝑠 wt
𝐼𝑇𝑟𝑚𝑠 = 2
𝜶𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 22
 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Rectifier. vs Vm

Highly Inductive Load.

π 2π 3π w t
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
Io T3,4 ON

vo wt

wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 23

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Rectifier. vs Vm

Highly Inductive Load.

π 2π 3π w t
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
Io T3,4 ON

vo wt

wt
IT1,2
IT3,4 wt

wt
is
wt
𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇4
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 24
 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Rectifier. vs Vm

Highly Inductive Load.

Circuit Performance parameters: π 2π 3π w t

1- Average Output Voltage: ig1,2

-Vm
1 𝜋+𝛼
𝑉𝑜𝑎𝑣 = න 𝑉𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡 a 2π+a wt
𝜋 𝛼 ig3,4 T1,2 ON
2- RMS Output Voltage: π+a wt
Io T3,4 ON
1 𝜋+𝛼
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑚 𝑠𝑖𝑛(𝜔𝑡) 2 𝑑𝜔𝑡
𝜋 𝛼
vo wt
3- Average Output Current:
𝑉𝑜𝑎𝑣 − 𝐸
𝐼𝑜𝑎𝑣 = wt
𝑅 IT1,2
4- RMS Output Current:
𝐼𝑜𝑎𝑣 = 𝐼𝑜𝑟𝑚𝑠
IT3,4 wt
5- Output Power:
𝑃𝑜 = 𝑉𝑜𝑎𝑣 ∗ 𝐼𝑜𝑎𝑣 wt
is
wt
𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇4
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 25
 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Rectifier. vs Vm

Highly Inductive Load.

Circuit Performance parameters: π 2π 3π w t

6- Input Power Factor: ig1,2

𝑃𝑜 -Vm
𝑝. 𝑓𝑠 = 𝐼𝑠𝑟𝑚𝑠 = 𝐼𝑜𝑟𝑚𝑠 a
𝑉𝑠𝑟𝑚𝑠 𝐼𝑠𝑟𝑚𝑠 2π+a wt
ig3,4 T1,2 ON
7- Average SCR Current: π+a wt

𝜋+𝛼
Io T3,4 ON
1 𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = න 𝐼𝑜𝑎𝑣 𝑑𝜔𝑡 =
2𝜋 𝛼 2
vo wt
8- RMS SCR Current:
1 𝜋+𝛼 2
𝐼𝑜𝑟𝑚𝑠 wt
𝐼𝑇𝑟𝑚𝑠 = න 𝐼𝑜𝑎𝑣 𝑑𝜔𝑡 =
2𝜋 𝛼 2 IT1,2
IT3,4 wt

wt
is
wt
𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇4
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 26
 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Rectifier. vs Vm

Highly Inductive Load without Battery.

π 2π 3π w t
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
Io T3,4 ON

The same waveforms and the same analysis

except: vo wt

Average Output Current: wt

IT1,2
𝑉𝑜𝑎𝑣
𝐼𝑜𝑎𝑣 =
𝑅
IT3,4 wt

wt
is
wt
𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇4
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 27
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier.

Highly Inductive Load without Battery.

To draw The thyristors voltage:

𝒗𝑻𝟏
𝒗𝑻𝟑

𝒗𝑻𝟒 𝒗𝑻𝟐

T1 & T2 on , T3 & T4 off. T3 & T4 on , T1 & T2 off .

𝒗𝒐 = 𝒗𝒔 𝒗𝒐 = −𝒗𝒔

𝒗𝑻𝟏 = 𝒗𝑻𝟐 = 𝟎 𝒗𝑻𝟑 = 𝒗𝑻𝟒 = 𝟎

𝒗𝑻𝟑 = 𝒗𝑻𝟒 = −𝒗𝒐 = −𝒗𝒔 𝒗𝑻𝟏 = 𝒗𝑻𝟐 = −𝒗𝒐 = 𝒗𝒔

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 28

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Rectifier. vs Vm

Highly Inductive Load without Battery.

π 2π 3π w t
ig1,2
-Vm
a 2π+a wt
ig3,4 T1,2 ON
π+a wt
Io T3,4 ON
Period Thyristors Thyristors Off SCR
ON off voltage
wt
0 < 𝜔𝑡 < 𝛼 𝑇3 & 𝑇4 𝑇1 & 𝑇2 𝑣𝑠
𝒗𝑻𝟏,𝟐
𝛼 < 𝜔𝑡 < 𝜋 + 𝛼 𝑇1 & 𝑇2 𝑇3 & 𝑇4 −𝑣𝑠
𝜋 + 𝛼 < 𝜔𝑡 < 2𝜋 + 𝛼 𝑇3 & 𝑇4 𝑇1 & 𝑇2 𝑣𝑠 wt

𝒗𝑻𝟑,𝟒

wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 29

 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier. vs Vm

Highly Inductive Load with FWD

π 2π 3π
wt

io -Vm

vo wt

iT1 wt
iT4
iT2 wt
iT3
wt
iD
wt
is
𝑖𝑠 = 𝑖 𝑇1 − 𝑖 𝑇3 wt
a π+a 2π+a
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 30
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier. vs Vm

Highly Inductive Load with FWD

Circuit Performance parameters: π 2π 3π
1- Average Output Voltage: wt

1 𝜋 io -Vm
𝑉𝑜𝑎𝑣 = න 𝑉𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
𝜋 𝛼

vo wt
Voav with FWD is < or > Voav without FWD?
iT1 wt
iT4
iT2 wt
iT3
wt
iD
wt
Voav with FWD is > Voav without FWD is
wt
a π+a 2π+a
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 31
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier. vs Vm

Highly Inductive Load with FWD

Circuit Performance parameters: π 2π 3π
1- Average Output Voltage: wt

1 𝜋 io -Vm
𝑉𝑜𝑎𝑣 = න 𝑉𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
𝜋 𝛼
2- Average Output Current:
vo wt
𝑉𝑜𝑎𝑣
𝐼𝑜𝑎𝑣 =
𝑅
3- RMS Output Current: iT1 wt

𝐼𝑜𝑎𝑣 = 𝐼𝑜𝑟𝑚𝑠 iT4

4- RMS Output Voltage: iT2 wt
iT3
1 𝜋
𝑉𝑜𝑟𝑚𝑠 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 2 𝑑𝜔𝑡 wt
𝜋 𝛼 𝑚
iD
5- Average Output Power:
wt
𝑃𝑜 = 𝑉𝑜𝑎𝑣 ∗ 𝐼𝑜𝑎𝑣 is
6- Input Power Factor: 𝑃𝑜
𝑝. 𝑓𝑠 = wt
𝑉𝑠𝑟𝑚𝑠 𝐼𝑠𝑟𝑚𝑠
π+a
a 2π+a
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 32
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier. vs Vm

Highly Inductive Load with FWD

7- RMS Supply Current: π 2π 3π

wt
𝜋
1∗2
𝐼𝑆𝑟𝑚𝑠 = න 𝐼
2𝜋 𝛼 𝑜𝑎𝑣
2 𝑑𝜔𝑡
𝐼𝑠𝑟𝑚𝑠 ≠ 𝐼𝑜𝑟𝑚𝑠 io -Vm

8- Average Thyristor Current:

𝜋
vo wt
1
𝐼𝑇𝑎𝑣 = න 𝐼 𝑑𝜔𝑡
2𝜋 𝛼 𝑜𝑎𝑣
iT1 wt
9- RMS Thyristor Current:
iT4
1 𝜋 2 iT2 wt
𝐼𝑇𝑟𝑚𝑠 = න 𝐼 𝑑𝜔𝑡
2𝜋 𝛼 𝑜𝑎𝑣 iT3
wt
10- Average Diode Current:
1 𝛼
iD
𝐼𝐷𝑎𝑣 = න 𝐼𝑜𝑎𝑣 𝑑𝜔𝑡
𝜋 0 wt
is
11- RMS Diode Current:
1 𝛼 2 wt
𝐼𝐷𝑟𝑚𝑠 = න 𝐼 𝑑𝜔𝑡
𝜋 0 𝑜𝑎𝑣 π+a a2π+a
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 33
 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier.

Highly Inductive Load with FWD

To draw The thyristors voltage:

𝒗𝑻𝟐 𝒗𝑻𝟏 𝒗𝑻𝟏 𝒗𝑻𝟐

𝒗𝑻𝟑
𝒗𝑻𝟒 𝒗𝑻𝟑 𝒗𝑻𝟒

T1 & T4 on , T3 & T2 & FWD off. T3 & T2 on , T1 & T4 & FWD off. All SCRs off , FWD on.

𝒗𝒐 = 𝒗𝒔 𝒗𝒐 = −𝒗𝒔 𝒗𝒐 = 𝟎
𝒗𝒔
𝒗𝑻𝟏 = 𝒗𝑻𝟒 = 𝟎 𝒗𝑻𝟑 = 𝒗𝑻𝟐 = 𝟎 𝒗𝑻𝟏 = 𝒗𝑻𝟒 =
𝟐
𝒗𝑻𝟏 = 𝒗𝑻𝟒 = −𝒗𝒐 = 𝒗𝒔 𝒗𝒔
𝒗𝑻𝟑 = 𝒗𝑻𝟐 = −𝒗𝒐 = −𝒗𝒔 𝒗𝑻𝟑 = 𝒗𝑻𝟐 = −
𝟐

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 34

 Single Phase Full-Wave Controlled Rectifier
Full Bridge Rectifier. vs Vm

Highly Inductive Load with FWD

π 2π 3π
wt

io -Vm

vo wt

Period Thyristors 𝑣𝑎𝑘 𝑣𝑎𝑘

ON across across 𝒗𝑻𝟏,𝟒 wt
𝒗𝒔 𝒗𝒔
𝑇1 & 𝑇4 𝑇3 & 𝑇2
𝟐 𝟐
0 < 𝜔𝑡 < 𝛼 𝑛𝑜𝑛𝑒 𝑣𝑠 −𝑣𝑠
2 2 wt
𝒗𝒔 𝑣𝑠
𝛼 < 𝜔𝑡 < 𝜋 𝑇1 & 𝑇4 0 −𝑣𝑠 𝟐

𝜋 < 𝜔𝑡 < 𝜋 + 𝛼 𝑛𝑜𝑛𝑒 𝑣𝑠 −𝑣𝑠 𝒗𝑻𝟑,𝟐 −𝒗𝒔

𝟐
2 2
−𝑣𝑠 wt
𝜋 + 𝛼 < 𝜔𝑡 < 2𝜋 𝑇2 & 𝑇3 𝑣𝑠 0 −𝒗𝒔 −𝒗𝒔 −𝑣𝑠
𝑣𝑠 −𝑣𝑠 𝟐 𝟐
2𝜋 < 𝜔𝑡 < 2𝜋 + 𝛼 𝑛𝑜𝑛𝑒
2 2
π+a a 2π+a
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 35
 Single Phase Controlled Rectifier
➢ Single Phase Full-Wave Fully-Controlled Rectifier.

➢ Single Phase Full-Wave Half-Controlled (Semi-Controlled) Rectifier.

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig3
π+a wt
D2 T1 & D 2 D2 T1 & D 2

Operation:
During Positive Half Cycle:
𝟎 ≤ 𝝎𝒕 ≤ 𝜶 D2 conduct, T1 , T3 & D4 off.

𝜶 ≤ 𝝎𝒕 ≤ 𝝅 T1 & D2 turns on , T3 & D4 off.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 37

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig3
π+a wt
D2 T1 & D 2 D2 T1 & D 2
io
Operation:
During Positive Half Cycle:
𝟎 ≤ 𝝎𝒕 ≤ 𝜶 D2 conduct, T1 , T3 & D4 off.
vo wt

𝜶 ≤ 𝝎𝒕 ≤ 𝝅 T1 & D2 turns on , T3 & D4 off.

wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 38

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig3
π+a wt
D2 T1 & D 2 D2 T1 & D 2
D2
io D4
Operation:
During Positive Half Cycle:
𝟎 ≤ 𝝎𝒕 ≤ 𝜶 D2 conduct, T1 , T3 & D4 off.
vo wt

𝜶 ≤ 𝝎𝒕 ≤ 𝝅 T1 & D2 turns on , T3 & D4 off.

wt
During Positive Half Cycle:
𝝅 ≤ 𝝎𝒕 ≤ 𝝅 + 𝜶 D4 conduct with D2 as FWD, T1 , T3 off.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 39

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig3
π+a wt
D2 T1 & D 2 T1 & D 2
D2 T3 & D 4 D2
io D4
Operation:
During Positive Half Cycle:
𝟎 ≤ 𝝎𝒕 ≤ 𝜶 D2 conduct, T1 , T3 & D4 off.
vo wt

𝜶 ≤ 𝝎𝒕 ≤ 𝝅 T1 & D2 turns on , T3 & D4 off.

wt
During Positive Half Cycle:
𝝅 ≤ 𝝎𝒕 ≤ 𝝅 + 𝜶 D4 conduct with D2 as FWD, T1 , T3 off.
𝝅 + 𝜶 ≤ 𝝎𝒕 ≤ 𝟐𝝅 T3 & D4 turns on , T1 & D2 off.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 40

 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig3
π+a wt
D2 T1 & D 2 T1 & D 2
D2 T3 & D 4 D2
io D4 D4 D4
Operation:
During Positive Half Cycle:
𝟎 ≤ 𝝎𝒕 ≤ 𝜶 D2 conduct, T1 , T3 & D4 off.
vo wt

𝜶 ≤ 𝝎𝒕 ≤ 𝝅 T1 & D2 turns on , T3 & D4 off.

wt
During Positive Half Cycle:
𝝅 ≤ 𝝎𝒕 ≤ 𝝅 + 𝜶 D4 conduct with D2 as FWD, T1 , T3 off.
𝝅 + 𝜶 ≤ 𝝎𝒕 ≤ 𝟐𝝅 T3 & D4 turns on , T1 & D2 off.

During Positive Half Cycle:

𝟐𝝅 ≤ 𝝎𝒕 ≤ 𝟐𝝅 + 𝜶 D2 conduct with D4 as FWD, T1 , T3 off.
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 41
 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

π 2π 3π w t
ig1
-Vm
a 2π+a wt
ig3
π+a wt
D2 T1 & D 2 T1 & D 2
D2 T3 & D 4 D2
io D4 D4 D4

vo wt

iT1 wt

iT3 wt

iD2 wt

iD4 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… w

42t
 Single Phase Full-Wave Controlled Rectifier
Waveforms :
Full Bridge Semi-Controlled Rectifier. vs Vm

Highly Inductive Load

Circuit Performance parameters: π 2π 3π w t

1- Average Output Voltage: ig1
1 𝜋 -Vm
𝑉𝑜𝑎𝑣 = න 𝑉𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
𝜋 𝛼 a 2π+a wt
ig3
2- Average Output Current:
π+a wt
D2 T1 & D 2 T1 & D 2
𝑉𝑜𝑎𝑣 − 𝐸 D2 T3 & D 4 D2
𝐼𝑜𝑎𝑣 =
𝑅 io D4 D4 D4
3- Average Thyristor Current:

1 𝜋
vo wt
𝐼𝑇𝑎𝑣 = න 𝐼 𝑑𝜔𝑡
2𝜋 𝛼 𝑜𝑎𝑣
iT1 wt
4- Average Diode Current:
iT3 wt
1 𝜋+𝛼
𝐼𝐷𝑎𝑣 = න 𝐼𝑜𝑎𝑣 𝑑𝜔𝑡
2𝜋 0 iD2 wt

iD4 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… w

43t
 Single Phase Full-Wave Controlled Rectifier
Assignment (2)
▪ Study the following circuits, sketch the output waveforms then compare between both
circuits' performance parameters.

𝑣𝑠 = 120 2 sin 𝜔𝑡
𝑅 =5Ω

𝐸 = 20 𝑉
𝛼 = 60𝑜

L is so large to keep the output current constant.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 44
 Thank You

Study hard

See you the next lecture …….

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 45