‫كلية الهندسة‬

Faculty of Engineering

Dr. Shaimaa Abd El-Hamid Kandil

Contact details: shaimaa.kandil@h-eng.helwan.edu.eg
 Chapter (3)
Lecture (4)
Three Phase Rectifier

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

 Introduction
Rectifiers

Single Phase Rectifier Three Phase Rectifier

Controlled Rectifier Uncontrolled Rectifier

Half-wave Rectifier Full-wave Rectifier

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

 Power Electronics Converters
Today’s Lecture Outlines:
➢ Types of Three Phase Rectifiers.

➢ Principle of Operation.

➢ Analysis of Three Phase Half Wave Controlled Rectifier Circuits

✓ Resistive Load

✓ Battery Charger.

✓ Highly Inductive Load.

✓ Highly Inductive Load with FWD.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 4

 Three Phase Controlled Rectifier
Introduction
Three Phase Rectifiers

Half-Wave Rectifiers Full-Wave Rectifiers

(Three pulses) (Six Pulses)

The advantages of using three phase rectifiers are:

➢ The output power is high, and thus, they are used to supply power to large loads.

➢ The ripples in the output voltage are low, which means that the output voltage waveform is closer to

the DC waveform.

➢ The input power factor is higher as compared to single phase rectifiers.

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

 Three Phase Controlled Rectifier
Introduction
➢ Three phase rectifiers are divided into two main categories:
1- Uncontrolled three phase rectifiers.
2- Controlled three phase rectifiers.
Uncontrolled Three Phase Rectifiers

Uncontrolled 3-Phase Uncontrolled 3-Phase

Half-Wave Rectifier Full-Wave Rectifier

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 6

 Three Phase Controlled Rectifier
Introduction
➢ Three phase rectifiers are divided into two main categories:
1- Uncontrolled three phase rectifiers.
2- Controlled three phase rectifiers.
Controlled Three Phase Rectifiers

Controlled 3-Phase Controlled 3-Phase

Half-Wave Rectifier Full-Wave Rectifier

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 7

 Three Phase Controlled Rectifier
Introduction
Three Phase Voltage Source
✓ Three phase circuits can be balanced or unbalanced; however, we will
focus only on balanced three phase circuits.
✓ A balanced three phase voltage source is composed of three single
phase voltage supplies each having the same magnitude, but they are
phase shifted by 120 o from each other.

✓ The three phases are usually named a, b and c.

✓ There is a reference node from which the phase voltages are measured
and is called the neutral point n.

✓ The voltages that are measured from the neutral point are called the phase voltages.

✓ The voltages that are measured from phase to another phase are called the line voltages.

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

 Three Phase Controlled Rectifier
Introduction
Three Phase Voltage Source

✓ The balanced phase voltages can be expressed by:

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

 Three Phase Half -

Wave Controlled

Rectifier

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

 Three Phase Half-Wave Controlled Rectifier
Power Circuit: Principle of operation:

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 11

 Three Phase Half-Wave Controlled Rectifier
Power Circuit: Principle of operation:

➢ If Va> Vb & Va > Vc T1 conduct

Vo = Va

T1

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 12

 Three Phase Half-Wave Controlled Rectifier
Power Circuit: Principle of operation:

➢ If Va> Vb & Va > Vc T1 conduct

Vo = Va

➢ If Vb > Va & Vb > Vc T2 conduct

Vo = Vb

T1 T2

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 13

 Three Phase Half-Wave Controlled Rectifier
Power Circuit: Principle of operation:

➢ If Va> Vb & Va > Vc T1 conduct

Vo = Va

➢ If Vb > Va & Vb > Vc T2 conduct

Vo = Vb

➢ If Vc > Va & Vc > Vb T3 conduct

Vo = Vc

T1 T2 T3

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 14

 Three Phase Half-Wave Controlled Rectifier
Power Circuit: Principle of operation:
✓ The firing angles for the three thyristors are equal.
✓ The firing angle of each thyristor is measured from
the intersection between two phase voltages.

✓ Each thyristor can conduct for 360o/3 = 120o

✓ The output voltage waveform can be positive only or

have a negative portion depending on the firing
angle and the load.

T1 T2 T3

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 15

 Analysis of Three Phase Half Wave

Controlled Rectifier Circuits

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

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏
Vs
Va Vb Vc

wt

Vo
T3 T1 T2 T3 T1

wt
io
R- Load 𝜶 < 𝟑𝟎𝒐
The output voltage will be continuous iT1 wt
𝑣𝑜
𝑖𝑜 =
𝑅 iT2 wt
𝑖𝑎 = 𝑖 𝑇1
iT3 wt
𝑖𝑏 = 𝑖 𝑇2

𝑖𝑐 = 𝑖 𝑇3 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 17

 Three Phase Half-Wave Controlled Rectifier
𝒐
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 R- Load 𝜶 < 𝟑𝟎
Vs
Va Vb Vc
Circuit Performance parameters:
1- Average Output Voltage:
𝑜 +120𝑜
1 𝛼+30 wt
𝑉𝑜𝑎𝑣 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜

Vo
𝛼+5𝜋ൗ6
1 T1 T2 T3 T1
𝑉𝑜𝑎𝑣 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡 T3
2𝜋Τ3 𝛼+𝜋ൗ
6
wt
io
3 3
𝑉𝑜𝑎𝑣 = 𝑉 cos 𝛼
2𝜋 𝑝ℎ𝑚
iT1 wt

iT2 wt

iT3 wt

wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 18

 Three Phase Half-Wave Controlled Rectifier
𝒐
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 R- Load 𝜶 < 𝟑𝟎
Vs
Va Vb Vc
Circuit Performance parameters:
1- Average Output Voltage:
𝑜 +120𝑜
1 𝛼+30 wt
𝑉𝑜𝑎𝑣 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜

Vo
2- Average Output Current:
T3 T1 T2 T3 T1
𝑉𝑜𝑎𝑣
𝐼𝑜𝑎𝑣 =
𝑅 wt
io
3- RMS Output Voltage:

1 𝛼+30𝑜 +120𝑜
2
iT1 wt
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
iT2 wt

4- RMS Output Current:

iT3 wt
𝑉𝑜𝑟𝑚𝑠
𝐼𝑜𝑟𝑚𝑠 =
𝑅 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 19

 Three Phase Half-Wave Controlled Rectifier
𝒐
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 R- Load 𝜶 < 𝟑𝟎
Vs
Va Vb Vc
Circuit Performance parameters:
5- Average Thyristor Current:
𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = wt
3
6- RMS Thyristor Current:
Vo
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 = T1 T2 T3 T1
3 T3
5- Average Output Power: wt
2
𝑃𝑜 = 𝐼𝑜𝑟𝑚𝑠 ∗𝑅
io

6- Input Power Factor: iT1 wt

𝑃𝑜
𝑝. 𝑓𝑠 = iT2 wt
3𝑉𝑝ℎ𝑟𝑚𝑠 𝐼𝑝ℎ𝑟𝑚𝑠

7- RMS Phase Current: iT3 wt

𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 = 𝐼𝑝ℎ𝑟𝑚𝑠 =
3 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 20

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc

wt

Vo

T3 T1 T2 T3 T1
wt
io
R- Load 𝜶 > 𝟑𝟎𝒐
The output voltage will be discontinuous iT1 wt

𝑖𝑎 = 𝑖 𝑇1 iT2 wt

𝑖𝑏 = 𝑖 𝑇2 iT3 wt

𝑖𝑐 = 𝑖 𝑇3
wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 21

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc
Circuit Performance parameters:
1- Average Output Voltage:
𝜋
1 wt
𝑉𝑜𝑎𝑣 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜 𝑝ℎ𝑚

Vo
𝜋
1
𝑉𝑜𝑎𝑣 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+𝜋ൗ 𝑝ℎ𝑚
6
T3 T1 T2 T3 T1
wt

3 𝜋
io
𝑉𝑜𝑎𝑣 = 𝑉𝑝ℎ𝑚 1 + cos 𝛼 +
2𝜋 6
iT1 wt

iT2 wt

iT3 wt

wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 22

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc
Circuit Performance parameters:
1- Average Output Voltage:
𝜋
1 wt
𝑉𝑜𝑎𝑣 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜 𝑝ℎ𝑚

Vo
2- Average Output Current:
𝑉𝑜𝑎𝑣
T3 T1 T2 T3 T1
𝐼𝑜𝑎𝑣 =
𝑅 wt
io
3- RMS Output Voltage:

1 𝜋
2 iT1 wt
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
iT2 wt
4- RMS Output Current:
iT3 wt
𝑉𝑜𝑟𝑚𝑠
𝐼𝑜𝑟𝑚𝑠 =
𝑅 wt

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

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc
Circuit Performance parameters:
5- Average Thyristor Current:
𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = wt
3
6- RMS Thyristor Current:
Vo
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 =
3 T3 T1 T2 T3 T1
5- Average Output Power: wt
2
𝑃𝑜 = 𝐼𝑜𝑟𝑚𝑠 ∗𝑅
io

6- Input Power Factor: iT1 wt

𝑃𝑜
𝑝. 𝑓𝑠 = iT2 wt
3𝑉𝑝ℎ𝑟𝑚𝑠 𝐼𝑝ℎ𝑟𝑚𝑠
7- RMS Phase Current:
iT3 wt
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 = 𝐼𝑝ℎ𝑟𝑚𝑠 =
3 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 24

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏
Vs
Va Vb Vc

wt

Vo
T3 T1 T2 T3 T1
wt
io
R- Load 𝜶 = 𝟑𝟎𝒐
At a firing angle of 30o the output
voltage waveform touches the x-axis
iT1 wt

The output voltage will be critically iT2 wt

continuous
𝑖𝑎 = 𝑖 𝑇1 iT3 wt

𝑖𝑏 = 𝑖 𝑇2
wt
𝑖𝑐 = 𝑖 𝑇3 The same calculations for R- Load 𝜶 < 𝟑𝟎𝒐 𝒐𝒓 R− Load 𝜶 > 𝟑𝟎𝒐
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 25
 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc

wt

Vo

T3 T1 T2 T3 T1
wt
io
R- Load 𝜶 > 𝟑𝟎𝒐
The output voltage will be discontinuous iT1 wt

𝑖𝑎 = 𝑖 𝑇1 iT2 wt

𝑖𝑏 = 𝑖 𝑇2 iT3 wt

𝑖𝑐 = 𝑖 𝑇3
wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 26

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc
Circuit Performance parameters:
1- Average Output Voltage:
𝜋
1 wt
𝑉𝑜𝑎𝑣 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜 𝑝ℎ𝑚

Vo
2- Average Output Current:
𝑉𝑜𝑎𝑣
T3 T1 T2 T3 T1
𝐼𝑜𝑎𝑣 =
𝑅 wt
io
3- RMS Output Voltage:

1 𝜋
2 iT1 wt
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
iT2 wt
4- RMS Output Current:
iT3 wt
𝑉𝑜𝑟𝑚𝑠
𝐼𝑜𝑟𝑚𝑠 =
𝑅 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 27

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 α = 60o
Vs
Va Vb Vc
Circuit Performance parameters:
5- Average Thyristor Current:
𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = wt
3
6- RMS Thyristor Current:
Vo
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 =
3 T3 T1 T2 T3 T1
5- Average Output Power: wt
2
𝑃𝑜 = 𝐼𝑜𝑟𝑚𝑠 ∗𝑅
io

6- Input Power Factor: iT1 wt

𝑃𝑜
𝑝. 𝑓𝑠 = iT2 wt
3𝑉𝑝ℎ𝑟𝑚𝑠 𝐼𝑝ℎ𝑟𝑚𝑠
7- RMS Phase Current:
iT3 wt
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 = 𝐼𝑝ℎ𝑟𝑚𝑠 =
3 wt

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

 Three Phase Half-Wave Controlled Rectifier
Resistive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏
Vs
Va Vb Vc

wt

Vo
T3 T1 T2 T3 T1
wt
io
R- Load 𝜶 = 𝟑𝟎𝒐
At a firing angle of 30o the output
voltage waveform touches the x-axis
iT1 wt

The output voltage will be critically iT2 wt

continuous
𝑖𝑎 = 𝑖 𝑇1 iT3 wt

𝑖𝑏 = 𝑖 𝑇2
wt
𝒐 𝒐
𝑖𝑐 = 𝑖 𝑇3 The same calculations for R- Load 𝜶 < 𝟑𝟎 𝒐𝒓 R− Load 𝜶 > 𝟑𝟎
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 29
 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏
Vs
Va Vb Vc

wt

Vo
T3 T1 T2 T3 T1

wt
io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
𝜶 < 𝟑𝟎𝒐
iT1 wt
The output voltage will be continuous

𝑖𝑎 = 𝑖 𝑇1 iT2 wt

𝑖𝑏 = 𝑖 𝑇2 iT3 wt

𝑖𝑐 = 𝑖 𝑇3 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 30

 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 𝜶 < 𝟑𝟎𝒐
Vs
Va Vb Vc
Circuit Performance parameters:
1- Average Output Voltage:

1 𝛼+30 𝑜 +120𝑜
wt
𝑉𝑜𝑎𝑣 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
Vo
2- Average Output Current:
T3 T1 T2 T3 T1
𝑉𝑜𝑎𝑣
𝐼𝑜𝑎𝑣 =
𝑅 wt
𝑉𝑜𝑎𝑣 − 𝐸𝑏 io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
➢ with 𝐸𝑏 𝐼𝑜𝑎𝑣 =
𝑅
3- RMS Output Voltage: iT1 wt

1 𝛼+30 𝑜 +120𝑜
2 iT2 wt
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
iT3 wt
4- RMS Output Current:
wt
𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 31
 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏 𝜶 < 𝟑𝟎𝒐
Vs
Va Vb Vc
Circuit Performance parameters:
5- Average Thyristor Current:
𝐼𝑜𝑎𝑣
𝐼𝑇𝑎𝑣 = wt
3
6- RMS Thyristor Current:
Vo
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 = T1 T2 T3 T1
3 T3
5- Average Output Power: wt

𝑃𝑜 = 𝑉𝑜𝑎𝑣 ∗ 𝐼𝑜𝑎𝑣
io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠

6- Input Power Factor: iT1 wt

𝑃𝑜
𝑝. 𝑓𝑠 = iT2 wt
3𝑉𝑝ℎ𝑟𝑚𝑠 𝐼𝑝ℎ𝑟𝑚𝑠
7- RMS Phase Current:
iT3 wt
𝐼𝑜𝑟𝑚𝑠
𝐼𝑇𝑟𝑚𝑠 = 𝐼𝑝ℎ𝑟𝑚𝑠 =
3 wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 32

 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load : α = 60o
Vs
Va Vb Vc

wt

Vo
T3 T1 T2 T3 T1
wt
io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
𝜶 > 𝟑𝟎𝒐

The output voltage will be continuous iT1 wt

For : 𝜶 < 𝟑𝟎𝒐 iT2 wt

or 𝜶 ≥ 𝟑𝟎𝒐
𝑜 +120𝑜
𝛼+30
iT3 wt
1
𝑉𝑜𝑎𝑣 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
wt

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 33

 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load with FWD : α = 60o
Vs
Va Vb Vc

wt

Vo
T3 T1 T2 T3 T1
wt
io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
𝜶 > 𝟑𝟎𝒐

The output voltage will be discontinuous iT1 wt

iT2 wt

iT3 wt

iFWD wt

wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 34
 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load with FWD : α = 60o
Vs
Circuit Performance parameters: Va Vb Vc
1- Average Output Voltage:
𝜋
1
𝑉𝑜𝑎𝑣 = න 𝑉 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡 wt
2𝜋Τ3 𝛼+30𝑜 𝑝ℎ𝑚

2- Average Output Current:

𝑉𝑜𝑎𝑣
Vo
𝐼𝑜𝑎𝑣 =
𝑅 T3 T1 T2 T3 T1
3- RMS Output Voltage: wt

𝜋
io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
1 2
𝑉𝑜𝑟𝑚𝑠 = න 𝑉𝑝ℎ𝑚 𝑠𝑖𝑛(𝜔𝑡) 𝑑𝜔𝑡
2𝜋Τ3 𝛼+30𝑜
iT1 wt
4- RMS Output Current:
iT2 wt
𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
iT3 wt
5- Average Output Power:
iFWD wt
𝑃𝑜 = 𝑉𝑜𝑎𝑣 ∗ 𝐼𝑜𝑎𝑣
wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 35
 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load with FWD : α = 60o
Vs
Circuit Performance parameters: Va Vb Vc
6- Average Thyristor Current:
1 𝜋
𝐼𝑇𝑎𝑣 = න 𝐼 𝑑𝜔𝑡
2𝜋 𝛼+30𝑜 𝑜𝑎𝑣 wt

7- RMS Thyristor Current:

1 𝜋 Vo
𝐼𝑇𝑟𝑚𝑠 = න 𝐼𝑜𝑎𝑣 2 𝑑𝜔𝑡
2𝜋 𝛼+30𝑜
T3 T1 T2 T3 T1
8- Average Diode Current: wt
𝜋+𝛼−30𝑜
3 io 𝐼𝑜𝑎𝑣 ≈ 𝐼𝑜𝑟𝑚𝑠
𝐼𝐷𝑎𝑣 = න 𝐼𝑜𝑎𝑣 𝑑𝜔𝑡
2𝜋 𝜋
𝑜
3 𝛼+30 iT1 wt
or : 𝐼𝐷𝑎𝑣 = න
2𝜋 60𝑜
𝐼𝑜𝑎𝑣 𝑑𝜔𝑡
iT2 wt
9- Input Power Factor:
𝑃𝑜 iT3 wt
𝑝. 𝑓𝑠 =
3𝑉𝑝ℎ𝑟𝑚𝑠 𝐼𝑝ℎ𝑟𝑚𝑠
iFWD wt
10- RMS Phase Current:
𝐼𝑝ℎ𝑟𝑚𝑠 = 𝐼𝑇𝑟𝑚𝑠 wt
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 36
 Three Phase Half-Wave Controlled Rectifier
R-Eb Load : 𝜶𝟏 𝜶𝟐 𝜶𝟑 𝜶𝟏
Vs
Va Vb Vc
𝐸𝑏
𝛼𝑜 = 𝑠𝑖𝑛−1
𝑉𝑝ℎ𝑚
𝛼 ≥ 𝛼𝑜
wt

Vo

T3 T1 T2 T3 T1
wt
io
R- Load 𝜶 = 𝟑𝟎𝒐
At 𝜔𝑡 = 𝜋 − 𝛼𝑜 the output current
waveform touches the x-axis.
iT1 wt

𝑖𝑎 = 𝑖 𝑇1 iT2 wt
𝑖𝑏 = 𝑖 𝑇2
𝑖𝑐 = 𝑖 𝑇3 iT3 wt

Can you calculate the circuit

wt
performance parameters?

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

 Three Phase Half-Wave Controlled Rectifier
Highly Inductive Load with FWD :
To draw the thyristors anode cathode voltage:
𝑣𝑇1 = 𝑣𝑎 − 𝑣𝑜

𝑣𝑇2 = 𝑣𝑏 − 𝑣𝑜

𝑣𝑇3 = 𝑣𝑐 − 𝑣𝑜
The output voltage is a phase voltage so the voltage drop across the thyristors will be line voltages; so the
line voltages have to be drawn.
𝑣𝑎 − 𝑣𝑏 = 𝑣𝑎𝑏
𝑣𝑎 − 𝑣𝑐 = 𝑣𝑎𝑐

𝑣𝑏 − 𝑣𝑐 = 𝑣𝑏𝑐
𝑣𝑏 − 𝑣𝑎 = 𝑣𝑏𝑎

𝑣𝑐 − 𝑣𝑎 = 𝑣𝑐𝑎
𝑣𝑐 − 𝑣𝑏 = 𝑣𝑐𝑏

The line voltage lead the phase voltage by angle 30𝑜 , the phase shift between line voltages are 60𝑜
Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 38
 Three Phase Half-Wave Controlled Rectifier
For Highly Inductive Load with or without FWD with 𝛼 < 30𝑜 : α = 20o
The same for resistive load and
firing angle less than 30o.

T3 T1 T2 T3

𝒗𝑻𝟏 = 𝒗𝒂 − 𝒗𝒐
𝒗𝒂𝒄
𝒗𝑻𝟐 = 𝒗𝒃 − 𝒗𝒐

𝒗𝑻𝟑 = 𝒗𝒄 − 𝒗𝒐
𝒗𝒄𝒃

𝒗𝒄𝒂 𝒗𝒃𝒄
𝒗𝒃𝒄 𝒗𝒃𝒂 𝒗𝒂𝒃 𝒗𝒂𝒄

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

 Three Phase Half-Wave Controlled Rectifier
For Highly Inductive Load without FWD with 𝛼 > 30𝑜 : α = 60o

𝒗𝑻𝟏 = 𝒗𝒂 − 𝒗𝒐
T3 T1 T2 T3
𝒗𝑻𝟐 = 𝒗𝒃 − 𝒗𝒐

𝒗𝑻𝟑 = 𝒗𝒄 − 𝒗𝒐

𝒗𝒂𝒄

𝒗𝒃𝒂 𝒗𝒄𝒃 𝒗𝒂𝒄

𝒗𝒃𝒄
𝒗𝒄𝒂 𝒗𝒂𝒃 𝒗𝒃𝒄

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

 Three Phase Half-Wave Controlled Rectifier
For Highly Inductive Load with FWD with 𝛼 > 30𝑜 : α = 60o

𝒗𝑻𝟏 = 𝒗𝒂 − 𝒗𝒐
T3 T1 T2 T3
𝒗𝑻𝟐 = 𝒗𝒃 − 𝒗𝒐

𝒗𝑻𝟑 = 𝒗𝒄 − 𝒗𝒐

𝒗𝒂 𝒗𝒃 𝒗𝒄

𝒗𝒂𝒄

𝒗𝒄 𝒗𝒂 𝒗𝒃

𝒗𝒃𝒂 𝒗𝒄𝒃 𝒗𝒂𝒄

𝒗𝒃 𝒗𝒄 𝒗𝒂
𝒗𝒃𝒄
𝒗𝒄𝒂 𝒗𝒂𝒃 𝒗𝒃𝒄

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 41

 Three Phase Half-Wave Controlled Rectifier
Assignment (3)
➢ In the converter circuit shown, L is so large such that the current I is assumed constant,
𝒗𝒂 = 𝟐𝑽 𝐬𝐢𝐧 𝝎𝒕
𝒗𝒃 = 𝟐𝑽 𝐬𝐢𝐧 𝝎𝒕 − 𝟐𝝅Τ𝟑
𝒗𝒄 = 𝟐𝑽 𝐬𝐢𝐧 𝝎𝒕 + 𝟐𝝅Τ𝟑
𝑽 = 𝟏𝟐𝟎 𝑽
𝑹 = 𝟏𝟎 𝜴
𝜶 = 𝟑𝟎𝒐
𝑬𝒃 = 𝟐𝟎 𝑽

▪ Sketch waveforms of va, vb, vc, vo. iT1 , iT2 and VT3

▪ Calculate the average value of I0 and rms value of output voltage.

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 42

 Thank You

Study hard

See you the next lecture …….

Dr. Shaimaa A. Kandil………………………………………………………………………………………………………………………………………………………… 43