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Power ElectronicsPower Electronics
Chapter 3
AC to DC Converters
(Rectifiers)
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OutlineOutline3.1 Single-phase controlled rectifier3.1 Single-phase controlled rectifier
3.2 Three-phase controlled rectifier 3.2 Three-phase controlled rectifier
3.3 Effect of transformer leakage inductance on 3.3 Effect of transformer leakage inductance on
rectifier circuits rectifier circuits
3.4 Capacitor-filtered uncontrolled rectifier3.4 Capacitor-filtered uncontrolled rectifier
3.5 Harmonics and power factor of rectifier circuits3.5 Harmonics and power factor of rectifier circuits
3.6 High power controlled rectifier3.6 High power controlled rectifier
3.7 Inverter mode operation of rectifier circuit3.7 Inverter mode operation of rectifier circuit
3.8 Realization of phase-control in rectifier circuits3.8 Realization of phase-control in rectifier circuits
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3.1 Single-phase controlled 3.1 Single-phase controlled (controllable) rectifier (controllable) rectifier
3.1.1 Single-phase half-wave controlled rectifier3.1.1 Single-phase half-wave controlled rectifier
3.1.2 Single-phase bridge fully-controlled rectifier 3.1.2 Single-phase bridge fully-controlled rectifier
3.1.3 Single-phase full-wave controlled rectifier3.1.3 Single-phase full-wave controlled rectifier
3.1.4 Single-phase bridge half-controlled rectifier3.1.4 Single-phase bridge half-controlled rectifier
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3.1.1 Single-phase half-wave3.1.1 Single-phase half-wave controlled rectifier controlled rectifier
Half-wave, single-pulseHalf-wave, single-pulse
Triggering delay angle, delay angle, firing angleTriggering delay angle, delay angle, firing angle
Resistive loadResistive load
TV T
Ru 1 u 2
u V Tu d
id
a )
0 t1 2 t
t
t
t
u 2
u g
u d
u V T
0
b )
c )
d )
e )
0
0
2
cos145.0)cos1(
2
2)(sin2
2
12
22d U
UttdUU (3-1)
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3.1.1 Single-phase half-wave 3.1.1 Single-phase half-wave controlled rectifier controlled rectifier
Inductive (resistor-inductor) loadInductive (resistor-inductor) loadu 2
0 t1 2 t
t
t
t
t
u g
0
u d
0
id
0
u V T
0
b )
c )
d )
e )
f)
+ +
a ) u 1
TV T
u 2
u V T
u d
id
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Basic thought process of time-domain Basic thought process of time-domain analysis for power electronic circuitsanalysis for power electronic circuits
The time-domain behavior of a power electronic The time-domain behavior of a power electronic circuit is actually the combination of consecutive circuit is actually the combination of consecutive transients of the different linear circuits when the transients of the different linear circuits when the power semiconductor devices are in different states.power semiconductor devices are in different states.
a ) b )
V T
R
L
V T
R
L
u 2u 2
tURit
iL sin2
d
d2d
d
)sin(2
)sin(2 2
)(2
d
tZ
Ue
Z
Ui
tL
R
ωt = ， id= 0
(3-3)
(3-2)
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Single-phase half-wave controlled Single-phase half-wave controlled rectifier with freewheeling dioderectifier with freewheeling diode
Maximum forward voltage, maximum reverse voltageMaximum forward voltage, maximum reverse voltageDisadvantages:Disadvantages:– Only single pulse in one line cycleOnly single pulse in one line cycle– DC component in the transformer currentDC component in the transformer current
Inductive load (L is large enough) Inductive load (L is large enough) VT i
a)
T
u 1 u2
uVTL
R
d
ud
VD
i
R
VDR
u 2
u d
id
u V T
iV TId
Id
t1 t
t
t
t
t
tO
O
O
O
O
O
- +
b )
c )
d )
e )
f)
g )
iV D R
d2dVT 2
)(2
1ItdII
(3-5)
(3-6)
(3-7)
(3-8)
2I I
RdVD d
2 21( )
2 2I I d t I
RVD d d
2I I
dVT d
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3.1.2 Single-phase bridge 3.1.2 Single-phase bridge fully-controlled rectifier fully-controlled rectifier
For thyristor: maximum forward voltage, maximum reverse For thyristor: maximum forward voltage, maximum reverse voltagevoltageAdvantages: Advantages: – 2 pulses in one line cycle2 pulses in one line cycle– No DC component in the transformer currentNo DC component in the transformer current
Resistive loadResistive load
d
R
T
u1 u2
i2a
b
VT
1
VT
3
VT 2
VT
4
ud
i t
t
t0
0
0
i2
u did
b )
c )
d )
u d ( id )
u V T1 ,4
a)
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3.1.2 Single-phase bridge fully-3.1.2 Single-phase bridge fully- controlled rectifier controlled rectifierResistive loadResistive load
Average output (rectified) voltage Average output (rectified) voltage
(3-9) (3-9)
Average output currentAverage output current
(3-10)(3-10)
For thyristorFor thyristor
(3-11) (3-11) (3-12) (3-12)
For transformerFor transformer
(3-13)(3-13)
2
cos19.0
2
cos122)(dsin2
12
22d U
UttUU
2
cos19.0
2
cos122 22dd
R
U
R
U
R
UI
2
cos145.0
2
1 2ddVT
R
UII
2sin
2
1
2)(d)sin
2(
2
1 222VT
R
Utt
R
UI
2sin
2
1)()sin
2(
1 2222 R
Utdt
R
UII
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3.1.2 Single-phase bridge 3.1.2 Single-phase bridge fully-controlled rectifier fully-controlled rectifier
CommutationCommutationThyristor voltages and currentsThyristor voltages and currentsTransformer currentTransformer current
Inductive loadInductive load(L is large enough)(L is large enough)
T a
b R
L
a )
u 1 u 2
i2
VT
1
VT
3
VT
2
VT
4
u d
id
u 2
O t
O t
O t
u d
id
i2
b )
O t
O t
u V T1 ,4
O t
O t
Id
Id
Id
Id
Id
iV T 2 ,3
iV T 1 ,4
cos9.0cos
22)(dsin2
1222d UUttUU (3-15)
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Electro-motive-force (EMF) loadElectro-motive-force (EMF) load
Discontinuous current iDiscontinuous current idd
With resistorWith resistor
a ) b )
R
E
id
u d id
O
E
u d
t
Id
O t
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Electro-motive-force (EMF) loadElectro-motive-force (EMF) loadWith resistor and inductorWith resistor and inductor
When L is large enough, the output voltage and When L is large enough, the output voltage and current waveforms are the same as ordinary current waveforms are the same as ordinary inductive load.inductive load.When L is at a critical valueWhen L is at a critical value
O
ud
0
E
id
t
t
=
dmin
23
dmin
2 1087.222
I
U
I
UL
(3-17)
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3.1.3 Single-phase full-wave 3.1.3 Single-phase full-wave controlled rectifier controlled rectifier
Transformer with center tapTransformer with center tap
Comparison with single-phase bridge fully-controlled rectifierComparison with single-phase bridge fully-controlled rectifier
a) b)
u 1
T
R
u 2u 2
i1VT 1
VT 2 u d
u d
i1
O
O
t
t
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3.1.4 Single-phase bridge 3.1.4 Single-phase bridge half-controlled rectifier half-controlled rectifier
Half-controlHalf-control
Comparison with fully-controlled rectifierComparison with fully-controlled rectifier
Additional freewheeling diodeAdditional freewheeling diode
Ob )
u 2
O
u d
id Id
O
O
O
O
Oi2
Id
Id
Id
Id
Id
t
t
t
t
t
t
t
iV T1iV D4
iV T2iV D3
iV D R
a
b R
Lu
2
i2
ud
idV
T1
VT
2
VD
3
VD
4
VD
R
T
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Another single-phase bridge Another single-phase bridge half-controlled rectifierhalf-controlled rectifier
Comparison with previous circuit:Comparison with previous circuit:– No need for additional freewheeling diodeNo need for additional freewheeling diode– Isolation is necessary between the drive circuits of the two thIsolation is necessary between the drive circuits of the two th
yristorsyristors
T
u 2
VD
3V
D4
VT
1V
T2
lo ad
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Summary of some important Summary of some important points in analysispoints in analysis
When analyzing a thyristor circuit, start from a diode When analyzing a thyristor circuit, start from a diode circuit with the same topology. The behavior of the circuit with the same topology. The behavior of the diode circuit is exactly the same as the thyristor circuit diode circuit is exactly the same as the thyristor circuit when firing angle is 0. when firing angle is 0. A power electronic circuit can be considered as A power electronic circuit can be considered as different linear circuits when the power semiconductor different linear circuits when the power semiconductor devices are in different states. The time-domain devices are in different states. The time-domain behavior of the power electronic circuit is actually the behavior of the power electronic circuit is actually the combination of consecutive transients of the different combination of consecutive transients of the different linear circuits. linear circuits. Take different principle when dealing with different Take different principle when dealing with different loadload– For resistive load: current waveform of a resistor is the same For resistive load: current waveform of a resistor is the same
as the voltage waveformas the voltage waveform– For inductive load with a large inductor: the inductor current For inductive load with a large inductor: the inductor current
can be considered constantcan be considered constant
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3.2 Three-phase controlled 3.2 Three-phase controlled (controllable) rectifier (controllable) rectifier
3.2.1 Three-phase half-wave controlled rectifier3.2.1 Three-phase half-wave controlled rectifier
(the basic circuit among three-phase rectifiers)(the basic circuit among three-phase rectifiers)
3.2.2 Three-phase bridge fully-controlled rectifier 3.2.2 Three-phase bridge fully-controlled rectifier
(the most widely used circuit among three-phase (the most widely used circuit among three-phase
rectifiers) rectifiers)
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3.2.1 Three-phase half-wave 3.2.1 Three-phase half-wave controlled rectifier controlled rectifier
Common-cathode connectionCommon-cathode connection
Natural commutation pointNatural commutation point
Resistive load, Resistive load, = 0= 0ºº
T
R
u d
id
V T 2
V T 1
V T 3
u2ua ub uc
O t1 t2 t3
uG
Oud
O
O
uab uac
O
iVT1
uVT1
t
t
t
t
t
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Resistive load, Resistive load, = 30= 30ºº
T
R
u d
id
V T 2
V T 1
V T 3
u2 ua ub uc
O t
O t
O t
O t
O t
uG
ud
uab uac
t1iVT1
uVT1 uac
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Resistive load, Resistive load, = 60= 60ºº
T
R
u d
id
V T 2
V T 1
V T 3
t
t
t
t
u2 ua ub uc
O
O
O
O
uG
ud
iVT1
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Resistive load, quantitative analysisResistive load, quantitative analysis
When When 3030º, load current º, load current iidd is continuous. is continuous.
When When 3030º, load current º, load current iidd is discontinuous. is discontinuous.
cos17.1cos2
63)(sin2
3
21
226
5
62d UUttdUU
)
6cos(1675.0)
6cos(1
2
23)(sin2
321
26
2d
UttdUU
(3-18)(3-18)
(3-19)(3-19)
Average load current
Thyristor voltages
R
UI d
d (3-20)(3-20)
0 30 60 90 120 150
0.4
0.8
1.21.17
32
1
/(°)
Ud/
U2
1- resistor load 2- inductor load 3- resistor-inductor load
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Inductive load, L is large enoughInductive load, L is large enough
Load current Load current iidd is always continuous. is always continuous.
Thyristor voltage and currents, transformer currentThyristor voltage and currents, transformer current
a
b
c
T
R
Lu 2
u d
e Lid
V T 1
V T 2
V T 3
u d
ia
u a u b u c
ib
ic
id
u a c
u a b
u a cO t
O t
O t
O t
O t
O t
u V T1
cos17.1cos2
63)(sin2
321
226
5
6
2d UUttdUU
(3-18)(3-18)
ddVT2 577.03
1IIII (3-23)(3-23) d
VTVT(AV) 368.0
57.1I
II
2RMFM 45.2 UUU
(3-24)(3-24)
(3-25)(3-25)
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3.2.2 Three-phase bridge3.2.2 Three-phase bridge fully-controlled rectifier fully-controlled rectifier
Common-cathode group and common-anode group Common-cathode group and common-anode group of thyristorsof thyristorsNumbering of the 6 thyristors indicates the trigger sNumbering of the 6 thyristors indicates the trigger sequence.equence.
Circuit diagramCircuit diagram d
d
ba
c
id
ud
VT1
VT3VT 5
VT4
VT 6 VT 2
2
1
T
n
ia
load
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Resistive load, Resistive load, = 0= 0ººu 2
u d1
u d2
u 2Lu d
u ab u ac
u ab u ac u bc u ba u ca u cb u ab u ac
u ab u ac u bc u ba u ca u cb u ab u ac
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
u a u cu b
t1O t
O t
O t
O t
= 0¡ã
iVT 1
u VT 1
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Resistive load, Resistive load, = 30= 30ººu d1
u d2
= 30¡ã
ia
O t
O t
O t
O t
u d
u ab u ac
u a u b u c
t1
u ab u ac u bc u ba u ca u cb u ab u ac
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
u ab u ac u bc u ba u ca u cb u ab u acu VT 1
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Resistive load, Resistive load, = 60= 60ºº= 60¡ã
u d1
u d2
u d
u acu ac
u ab
u ab u ac u bc u ba u ca u cb u ab u ac
u a
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
u b u c
O t
t1
O t
O t
u VT 1
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Resistive load, Resistive load, = 90= 90ººud1
ud2
ud
ua ub uc ua ub
tO
tO
tO
tO
tO
ia
id
uab uac ubc uba uca ucb uab uacubc uba
iVT1
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Inductive load, Inductive load, = 0= 0ºº
u d1
u 2
u d2
u 2Lu d
id
tO
tO
tO
tO
u a= 0¡ãu b u c
t1
u ab u ac u bc u ba u ca u cb u ab u ac
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
iVT 1
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Inductive load, Inductive load, = 30= 30ººu d1
= 30¡ã
u d2
u du ab u ac u bc u ba u ca u cb u ab u ac
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
tO
tO
tO
tO
id
ia
t1
u a u b u c
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Inductive load, Inductive load, = 90= 90ºº= 90¡ã
u d1
u d2u ac u bc u ba u ca u cb u ab u ac
u ab
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵu d
u ac
u ab
u ac
tO
tO
tO
u b u c u a
t1
u VT 1
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Quantitative analysisQuantitative analysisAverage output voltageAverage output voltage
For resistive load, When For resistive load, When a a > 60> 60º, load current º, load current iidd is discontinuous. is discontinuous.
Average output current (load current)Average output current (load current)
Transformer currentTransformer current
Thyristor voltage and currentThyristor voltage and current– Same as three-phase half-wave rectifierSame as three-phase half-wave rectifier
EMF load, L is large enoughEMF load, L is large enough– All the same as inductive load except the calculation of average output All the same as inductive load except the calculation of average output
currentcurrent
cos34.2)(sin6
3
12
3
2
3
2d UttdUU
(3-26)(3-26)
)
3cos(134.2)(sin6
32
32d
UttdUU
(3-20)(3-20)
(3-27)(3-27)
dd2
d2d2 816.0
3
2
3
2)(
3
2
2
1IIIII
(3-28)(3-28)
R
EUI
d
d(3-29)(3-29)
R
UI d
d
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3.3 Effect of transformer leakage 3.3 Effect of transformer leakage inductance on rectifier circuits inductance on rectifier circuits
In practical, the transformer leakage inductance has to be takIn practical, the transformer leakage inductance has to be taken into account. en into account. Commutation between thyristors thus can not happen instantlCommutation between thyristors thus can not happen instantly, but with a commutation process. y, but with a commutation process.
R
a
b
c
T
Lu d
ic
ib
iaL B
L B
L B
ikV T 1
V T 2
V T 3
u d
id
tO
tO
ic ia ib ic ia Id
u au b u c
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Commutation process analysisCommutation process analysis
Circulating current Circulating current iikk during commutation during commutation
Commutation angle Commutation angle Output voltage during commutationOutput voltage during commutation
ub-ua = 2·LB·dia/dt
ik: 0 Id
ia = Id-ik : Id 0
ib = ik : 0 Id
2d
d
d
d bakBb
kBad
uu
t
iLu
t
iLuu
(3-30)(3-30)
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Reduction of average output voltage due to the Reduction of average output voltage due to the commutation processcommutation process
Calculation of commutation angleCalculation of commutation angle
– Id , ,
– XXB B ,,
– For 90 90o o , , ,
Quantitative calculationQuantitative calculation
dB0 B6
5
6
5 B
6
5
6
5 Bbb6
5
6
5 dbd
2
3d
2
3)(d
d
d
2
3
)(d)]d
d([
2
3)(d)(
3/2
1
IXiLtt
iL
tt
iLuutuuU
I
d
kk
k
(3-31)(3-31)
2
dB
6
2)cos(cos
U
IX (3-36)(3-36)
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Summary of the effect on rectifier circuitsSummary of the effect on rectifier circuits
ConclusionsConclusions– Commutation process actually provides additional working stateCommutation process actually provides additional working state
s of the circuit. s of the circuit. – di/dt of the thyristor current is reduced.di/dt of the thyristor current is reduced.– The average output voltage is reduced. The average output voltage is reduced. – Positive du/dtPositive du/dt– Notching in the AC side voltageNotching in the AC side voltage
dUd
B IX
dB2
IX
dB
2
3I
X
dB3
IX
d
B
2I
mX
)cos(cos 2
Bd
2U
XI
2
Bd
2
2
U
XI
2
dB
6
2
U
IX
2
dB
6
2
U
IX
mU
XI
sin2 2
Bd
Single-
phase full wave
Single-phase bridge
Three-phase
half-wave
Three-phase bridge
m-pulse recfifier
①
②
Circuits
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3.4 Capacitor-filtered uncontrolled3.4 Capacitor-filtered uncontrolled (uncontrollable) rectifier (uncontrollable) rectifier
Emphasis of previous sectionsEmphasis of previous sections– Controlled rectifier, inductive loadControlled rectifier, inductive load
Uncontrolled rectifier: diodes instead of thyristorsUncontrolled rectifier: diodes instead of thyristors
Wide applications of capacitor-filtered uncontrolled rWide applications of capacitor-filtered uncontrolled rectifierectifier– AC-DC-AC frequency converterAC-DC-AC frequency converter– Uninterruptible power supplyUninterruptible power supply– Switching power supplySwitching power supply
3.4.1 Capacitor-filtered single-phase uncontrolled3.4.1 Capacitor-filtered single-phase uncontrolled rectifier rectifier
3.4.2 Capacitor-filtered three-phase uncontrolled 3.4.2 Capacitor-filtered three-phase uncontrolled rectifier rectifier
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3.4.1 Capacitor-filtered single-phase 3.4.1 Capacitor-filtered single-phase uncontrolled rectifier uncontrolled rectifier
Single-phase bridge, Single-phase bridge, RCRC load load
a )
+RCu 1 u 2
i2
V D 1 V D 3
V D 2 V D 4
id
iC iR
u d
b)
0
i
u d
2 t
i,u d
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3.4.1 Capacitor-filtered single-phase 3.4.1 Capacitor-filtered single-phase uncontrolled rectifier uncontrolled rectifier
Single-phase bridge, Single-phase bridge, RLCRLC load load
a) b)
-
+R
C
L
+
u 1 u 2
i2
u d
u L
id
iC iR
VD 2 VD 4
VD 1 VD 3
u 2 u d
i2
0 t
i2 ,u 2 ,u d
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3.4.2 Capacitor-filtered three-phase 3.4.2 Capacitor-filtered three-phase uncontrolled rectifier uncontrolled rectifier
Three-phase bridge, Three-phase bridge, RCRC load load
a)
+a
b
c
T ia
RCu d
id
iCiR
VD 4 VD 6
VD 1 VD 3 VD 5
VD 2
b)
O
ia
u d
id
u du ab u ac
0 t3
t
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3.4.2 Capacitor-filtered three-phase 3.4.2 Capacitor-filtered three-phase uncontrolled rectifier uncontrolled rectifier
Three-phase bridge, Three-phase bridge, RCRC load load
Waveform when Waveform when RCRC1.7321.732
t t
tt
ia
id
ia
id
O
O
O
O
a RC= b ） RC<
3 3
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3.4.2 Capacitor-filtered three-phase 3.4.2 Capacitor-filtered three-phase uncontrolled rectifier uncontrolled rectifier
Three-phase bridge, Three-phase bridge, RLCRLC load load
a)
b)
c)
+a
bc
T ia
RCud
i di
Ci
R
VD 4 VD 6
VD1
VD3
VD5
VD 2
ia
ia
O
O t
t
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3.5 Harmonics and power factor of 3.5 Harmonics and power factor of rectifier circuitsrectifier circuits
Originating of harmonics and power factor issues in Originating of harmonics and power factor issues in rectifier circuitsrectifier circuits– Harmonics: working in switching states—nonlinearHarmonics: working in switching states—nonlinear– Power factor: firing delay angle causes phase delayPower factor: firing delay angle causes phase delay
Harmful effects of harmonics and low power factorHarmful effects of harmonics and low power factor
Standards to limit harmonics and power factorStandards to limit harmonics and power factor
3.5.1 Basic concepts of harmonics and reactive power3.5.1 Basic concepts of harmonics and reactive power
3.5.2 AC side harmonics and power factor of 3.5.2 AC side harmonics and power factor of controlled rectifiers with inductive load controlled rectifiers with inductive load
3.5.3 AC side harmonics and power factor of 3.5.3 AC side harmonics and power factor of capacitor-filtered uncontrolled rectifiers capacitor-filtered uncontrolled rectifiers
3.5.4 Harmonic analysis of output voltage and current3.5.4 Harmonic analysis of output voltage and current
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For pure sinusoidal waveformFor pure sinusoidal waveform
For periodic non-sinusoidal waveform For periodic non-sinusoidal waveform
oror
wherewhere
Fundamental componentFundamental component
Harmonic components (harmonics)Harmonic components (harmonics)
3.5.1 Basic concepts of harmonics 3.5.1 Basic concepts of harmonics and reactive power and reactive power
1
)sincos()(n
nno tnbtnaatu
(3-54)(3-54)( ) 2 sin( )uu t U t
(3-55)(3-55)
1
)sin()(n
nno tncatu (3-56)(3-56)
22nnn bac
)/arctan( nnn ba
sinnn ca
cosnn cb
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Harmonics-related specificationsHarmonics-related specificationsTake current harmonics as examplesTake current harmonics as examples
Content of Content of nnth harmonicsth harmonics
IInn is the effective (RMS) value of is the effective (RMS) value of nnth harmonics.th harmonics.
II11 is the effective (RMS) value of fundamental component is the effective (RMS) value of fundamental component..
Total harmonic distortionTotal harmonic distortion
IIhh is the total effective (RMS) value of all the harmonic components. is the total effective (RMS) value of all the harmonic components.
%1001
I
IHRI n
n
%1001
I
ITHD h
i
(3-57)(3-57)
(3-58)(3-58)
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Definition of power and power factorDefinition of power and power factor
For sinusoidal circuitsFor sinusoidal circuits
Active powerActive power
Reactive power Reactive power
Q=U IQ=U I sinsin
Apparent powerApparent power
S=UIS=UI
Power factorPower factor
=cos=cos
2
0cos)(
2
1UItuidP
222 QPS
S
P
(3-59)(3-59)
(3-60)(3-60)
(3-61)(3-61)
(3-63)(3-63)
(3-62)(3-62)
(3-64)(3-64)
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Definition of power and power factorDefinition of power and power factor
For non-sinusoidal circuitsFor non-sinusoidal circuits
Active powerActive power
P=U IP=U I11 coscos1 1
Power factorPower factor
Distortion factor (fundamental-component factor)Distortion factor (fundamental-component factor)
==II1 1 / I/ I
Displacement factor (power factor of fundamental component)Displacement factor (power factor of fundamental component)
11 = cos = cos11
Definition of reactive power is still in dispute. Definition of reactive power is still in dispute.
11111 coscos
cos I
I
UI
UI
S
P
(3-65)(3-65)
(3-66)(3-66)
Review of the reactive power conceptReview of the reactive power concept
The reactive power The reactive power Q Q does not lead to net transmission of enerdoes not lead to net transmission of energy between the source and load. When gy between the source and load. When Q Q 0, the rms current 0, the rms current and apparent power are greater than the minimum amount necand apparent power are greater than the minimum amount necessary to transmit the average power P.essary to transmit the average power P.
Inductor: current lags voltage by 90°, hence displacement factInductor: current lags voltage by 90°, hence displacement factor is zero. The alternate storing and releasing of energy in an ior is zero. The alternate storing and releasing of energy in an inductor leads to current flow and nonzero apparent power, but nductor leads to current flow and nonzero apparent power, but P P = = 0. Just as resistors consume real (average) power P, indu0. Just as resistors consume real (average) power P, inductors can be viewed as consumers of reactive power Q.ctors can be viewed as consumers of reactive power Q.
Capacitor: current leads voltage by 90°, hence displacement faCapacitor: current leads voltage by 90°, hence displacement factor is zero. Capacitors supply reactive power Q. They are oftctor is zero. Capacitors supply reactive power Q. They are often placed in the utility power distribution system near inductive en placed in the utility power distribution system near inductive loads. If loads. If Q Q supplied by capacitor is equal to supplied by capacitor is equal to Q Q consumed by indconsumed by inductor, then the net current (flowing from the source into the capuctor, then the net current (flowing from the source into the capacitor-inductive-load combination) is in phase with the voltage, acitor-inductive-load combination) is in phase with the voltage, leading to unity power factor and minimum rms current magnitleading to unity power factor and minimum rms current magnitude.ude.
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3.5.2 AC side harmonics and power factor of 3.5.2 AC side harmonics and power factor of controlled rectifiers with inductive load controlled rectifiers with inductive load
Single-phase bridge fully-controlled rectifierSingle-phase bridge fully-controlled rectifier
u2
O t
O t
O t
ud
id
i2
b)
O t
O t
uVT1,4
O t
O t
Id
Id
Id
Id
Id
iVT2,3
iVT1,4
T a
b R
L
a )
u 1 u 2
i2V
T1
VT
3
VT
2
VT
4u d
id
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AC side current harmonics of single-phase bridge AC side current harmonics of single-phase bridge fully-controlled rectifier with inductive loadfully-controlled rectifier with inductive load
ConclusionsConclusions– Only odd order harmonics existOnly odd order harmonics exist
– IInn 1/ 1/nn
– IInn / I / I11 = 1/ = 1/nn
where
,5,3,1,5,3,1d
d2
sin2sin14
)5sin5
13sin
3
1(sin
4
nn
n
tnItnn
I
tttIi
n=1,3,5,…n
IIn
d22
(3-72)(3-72)
(3-73)(3-73)
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Power factor of single-phase bridge fully-Power factor of single-phase bridge fully-controlled rectifier with inductive loadcontrolled rectifier with inductive load
Distortion factorDistortion factor
Displacement factorDisplacement factor
Power factorPower factor
I
I1 2 2
0 9.
coscos 11
cos9.0cos22
cos 11
1 I
I
(3-75)(3-75)
(3-76)(3-76)
(3-77)(3-77)
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Three-phase bridge fully-controlled rectifierThree-phase bridge fully-controlled rectifier
u d1
= 30¡ã
u d2
u du ab u ac u bc u ba u ca u cb u ab u ac
Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ
tO
tO
tO
tO
id
ia
t1
u a u b u c
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AC side current harmonics of three-phase bridge AC side current harmonics of three-phase bridge fully-controlled rectifier with inductive loadfully-controlled rectifier with inductive load
ConclusionsConclusions– Only 6Only 6kk1 order harmonics exist (1 order harmonics exist (kk is positive integer) is positive integer)
– IInn 1/ 1/nn
– IInn / I / I11 = 1/ = 1/nn
3,2,116
1
3,2,116
dd
da
sin2)1(sin2sin1
)1(32
sin32
]13sin13
111sin
11
17sin
7
15sin
5
1[sin
32
kkn
nk
kkn
k tnItItnn
ItI
tttttIi
(3-79)(3-79)
1
6
6, 6 1, 1,2,3,n
I I
I I n k kn
d
d
where
(3-80)(3-80)
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Power factor of three-phase bridge fully-Power factor of three-phase bridge fully-controlled rectifier with inductive loadcontrolled rectifier with inductive load
Distortion factorDistortion factor
Displacement factorDisplacement factor
Power factorPower factor
955.031
I
I
coscos 11
cos955.0cos3
cos 11
1 I
I
(3-81)(3-81)
(3-82)(3-82)
(3-83)(3-83)
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3.5.3 AC side harmonics and power factor of 3.5.3 AC side harmonics and power factor of capacitor-filtered uncontrolled rectifiers capacitor-filtered uncontrolled rectifiers
Situation is a little complicated than rectifiers with Situation is a little complicated than rectifiers with inductive load. inductive load. Some conclusions that are easy to remember:Some conclusions that are easy to remember:– Only odd order harmonics exist in single-phase circuit, and Only odd order harmonics exist in single-phase circuit, and
only 6only 6kk1 (1 (kk is positive integer) order harmonics exist in is positive integer) order harmonics exist in three-phase circuit. three-phase circuit.
– Magnitude of harmonics decreases as harmonic order Magnitude of harmonics decreases as harmonic order increases. increases.
– Harmonics increases and power factor decreases as Harmonics increases and power factor decreases as capacitor increases. capacitor increases.
– Harmonics decreases and power factor increases as Harmonics decreases and power factor increases as inductor increases. inductor increases.
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3.5.4 Harmonic analysis of output voltage 3.5.4 Harmonic analysis of output voltage and current and current
wherewhere
Output voltage of Output voltage of mm-pulse-pulse
rectifier when 0º
u d
tOmm
2m
U 22
tnn
kU
tnbUu
mkn
mknn
cos1
cos21
cos
2d0
d0d0
m
mUU
sin2 2d0
d02 1
cos2U
n
kbn
(3-85)(3-85)
(3-86)(3-86)
(3-87)(3-87)
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Ripple factor in the output voltageRipple factor in the output voltage
Output voltage ripple factorOutput voltage ripple factor
where Uwhere URR is the total RMS value of all the harmonic is the total RMS value of all the harmonic
components in the output voltagecomponents in the output voltage
and U is the total RMS value of the output voltageand U is the total RMS value of the output voltage
Ripple factors for rectifiers with different number of pulses Ripple factors for rectifiers with different number of pulses
d0
R
U
Uu
2d0
22R UUUU
mknn
(3-88)(3-88)
(3-89)(3-89)
m 2 3 6 12 ∞u
（%）48.2 18.27 4.18 0.994 0
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Harmonics in the output currentHarmonics in the output current
where where
)cos(dd nmkn
n tndIi
R
EUI
d0
d
22 )( LnR
b
z
bd n
n
nn
R
Lnn
arctan
(3-92)(3-92)
(3-93)(3-93)
(3-94)(3-94)
(3-95)(3-95)
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Conclusions for Conclusions for = 0= 0ºº
Only Only mkmk ( (kk is positive integer) order harmonics exist in is positive integer) order harmonics exist in
the output voltage and current of m-pulse rectifiersthe output voltage and current of m-pulse rectifiers
Magnitude of harmonics decreases as harmonic order iMagnitude of harmonics decreases as harmonic order i
ncreases when ncreases when mm is constant. is constant.
The order number of the lowest harmonics increases aThe order number of the lowest harmonics increases a
s s m m increases. The corresponding magnitude of the loincreases. The corresponding magnitude of the lo
west harmonics decreases accordingly.west harmonics decreases accordingly.
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For For 0 0ºº
Quantitative harmonic analysis Quantitative harmonic analysis
of output voltage and current is of output voltage and current is
very complicated for very complicated for 0 0ºº. .
As an example, As an example,
for 3-phase bridge for 3-phase bridge
fully-controlled rectifierfully-controlled rectifier
0 30 120 150 18060
0.1
0.2
0.3
90
n =6
n =12
n =18
/(¡ã)
U2L
c n
2
kn
nnd tncUu6
)cos( d
(3-96)(3-96)
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3.6 High power controlled rectifier3.6 High power controlled rectifier
3.6.1 Double-star controlled rectifier3.6.1 Double-star controlled rectifier
3.6.2 Connection of multiple rectifiers3.6.2 Connection of multiple rectifiers
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3.6.1 Double-star controlled rectifier3.6.1 Double-star controlled rectifier
Difference from 6-phase half-wave rectifierDifference from 6-phase half-wave rectifier
CircuitCircuit Waveforms When Waveforms When 00ºº
T
a b c
L
R
niP L P
u d
id
VT
2
VT
6
VT
4
VT
1
VT
3
VT
5
c 'a ' b '
n 1n 2
u d1u a u b u c
ia
u d2
ia'
u c' u a
' u b' u c
'
O t
O t
O t
O t
Id12
Id16
Id12
Id16
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Effect of interphase reactor (induEffect of interphase reactor (inductor, transformer)ctor, transformer)
n
L
R
+- +-
u d1
L P
u b'
u d2 u d
n 2 n 1
iP
u a
VT 1 VT 6
u P12
d1d2p uuu
)(2
1
2
1
2
1d2d1pd1pd2d uuUuuuu
(3-97)(3-97)
(3-98)(3-98)
up
ud1,ud2
O
O
60
360
t1 t
tb)
a)
ua ub ucuc' ua
'ub' ub
'
。
。
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Quantitative analysis when Quantitative analysis when = 0= 0ºº
]9cos40
16cos
35
23cos
4
11[
2
63 2d1 ttt
Uu
2
2
3 6 1 2 1[1 cos3( 60 ) cos 6( 60 ) cos9( 60 ) ]
2 4 35 40
3 6 1 2 1[1 cos3 cos6 cos9 ]
2 4 35 40
Uu t t t
Ut t t
d2
]9cos20
13cos
2
1[
2
63 2p tt
Uu
]6cos35
21[
2
63 2d t
Uu
(3-99)(3-99)
(3-100)(3-100)
(3-101)(3-101)
(3-102)(3-102)
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Waveforms Waveforms when when > 0> 0ºº
。90
。60
。30u d
u d
u d
tO
tO
tO
u a u b u cu c' u a
' u b'
u b u cu c' u a
' u b'
u b u cu c' u a
' u b'
cos17.1 2UUd
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Comparison with 3-phase half-waveComparison with 3-phase half-waverectifier and 3-phase bridge rectifierrectifier and 3-phase bridge rectifier
Voltage output capabilityVoltage output capability– Same as 3-phase half-wave rectifierSame as 3-phase half-wave rectifier
– Half of 3-phase bridge rectifierHalf of 3-phase bridge rectifier
Current output capabilityCurrent output capability– Twice of 3-phase half-wave rectifierTwice of 3-phase half-wave rectifier
– Twice of 3-phase bridge rectifierTwice of 3-phase bridge rectifier
ApplicationsApplications
Low voltage and high current situationsLow voltage and high current situations
3.6.2 Connection of multiple rectifiers3.6.2 Connection of multiple rectifiersE
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ower
Larger output current: Larger output current: parallel connectionparallel connectionConnection Connection
of multiple of multiple rectifiersrectifiers
To increase the To increase the output capacityoutput capacity
To improve the AC side current waveform To improve the AC side current waveform and DC side voltage waveformand DC side voltage waveform
Larger output voltage: Larger output voltage: series connectionseries connection
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Phase-shift connection of multiple rectifiersPhase-shift connection of multiple rectifiers
Parallel connectionParallel connection
12-pulse rectifier realized by 12-pulse rectifier realized by
paralleled 3-phase bridge rectifiersparalleled 3-phase bridge rectifiers
M
LTVT
1 2
c 1
b 1
a 1
c 2
b 2
a 2
L P
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Phase-shift connection of multiple rectifiersPhase-shift connection of multiple rectifiers
12-pulse rectifier realized by 12-pulse rectifier realized by
series 3-phase bridge rectifiersseries 3-phase bridge rectifiers
Series connectionSeries connection
0a)
b)
c)
d)
ia1
Id
180¡ã 360¡ã
ia2
iab2'
iA
Idiab2
t
t
t
t
0
0
0
Id23
33 Id
33 Id
Id32 3
(1+ ) Id32 3
(1+ )Id33
Id13
C▲
L
RB
A
1
*▲
▲
*
*
0
30° l aggi ng
3
iAc1
b1
a1
1
c2
b2
a2
iab2 ua2b2
ua1b1
ia1 id
ud
I
II
I
II
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VoltageVoltage– Average output voltageAverage output voltage
Parallel connection: Parallel connection:
Series connection:Series connection:– Output voltage harmonicsOutput voltage harmonics Only 12Only 12mm harmonics exist harmonics exist
Input (AC side) current harmonicsInput (AC side) current harmonics– Only 12kOnly 12k±±1 harmonics exist1 harmonics exist
Connection of more 3-phase bridge rectifiersConnection of more 3-phase bridge rectifiers– Three: 18-pulse rectifier (20Three: 18-pulse rectifier (20º phase differenceº phase difference))– Four: 24-pulse rectifier (15Four: 24-pulse rectifier (15º phase differenceº phase difference))
Quantitative analysis of 12-pulse rectifierQuantitative analysis of 12-pulse rectifier
22.34 cosU U d
24.68 cosU U d
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Sequential control of multiple Sequential control of multiple series-connected rectifiersseries-connected rectifiers
Circuit and waveforms of series-connectedCircuit and waveforms of series-connectedthree single-phase bridge rectifiersthree single-phase bridge rectifiers
L
i
a )
Ⅰ
Ⅱ
Ⅲ
u 2
u 2
u 2
Id
VT
11
VT
13V
T14
VT
12V
T 21
VT 23
VT 24
VT 22
VT
31
VT
33V
T34
VT
32
u db )
c )
i Id
2 Id
u d
O +
load
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3.7 Inverter mode operation of rectifiers3.7 Inverter mode operation of rectifiers
Review of DC generator-motor systemReview of DC generator-motor system
R
EEI
MG
d
R
EEI
GM
d should be avoided
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Inverter mode operation of rectifiersInverter mode operation of rectifiers
Rectifier and inverter mode operation of single-phaseRectifier and inverter mode operation of single-phasefull-wave converterfull-wave converter
R
EUI
Gd
d
R
UEI
dM
d
a) b)
R
+
-
EnergyM
1
0
2
u10
u20
udid
LVT1
VT2
u10udu20 u10
O
O t
t
Id
id
Ud>EM
EMM
R
+
-
1
0
2
udid
LVT1
VT2
u10udu20 u10
O
O t
t
Id
id
Ud<EM
EM
iVT1
iVT2
iVT1
iVT2
iVT1iVT2
iVT2
id=iVT +iVT1 2
id=iVT +iVT1 2
iVT1iVT2
iVT1
Energy
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Necessary conditions for the inverter Necessary conditions for the inverter mode operation of controlled rectifiersmode operation of controlled rectifiers
There must be DC EMF in the load and the direction There must be DC EMF in the load and the direction
of the DC EMF must be enabling current flow in thyriof the DC EMF must be enabling current flow in thyri
stors. (In other word stors. (In other word EEMM must be negative if taking th must be negative if taking th
e ordinary output voltage direction as positive.)e ordinary output voltage direction as positive.)
9090º º so that the output voltage so that the output voltage UUdd is also negativ is also negativ
e. e.
dM UE
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Inverter mode operation of Inverter mode operation of 3-phase bridge rectifier3-phase bridge rectifier
Inversion angle (extinction angle) Inversion angle (extinction angle) + + =180 =180º º
u ab u ac u bc u ba u ca u cb u ab u ac u bc u ba u ca u cb u ab u ac u bc u ba u ca u cb u ab u ac u bc
u a u b u c u a u b u c u a u b u c u a u bu 2
u d
tO
tO
= 4= 3 = 6
= 4= 3 = 6
t1 t3t2
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Inversion failure and minimum inversion angleInversion failure and minimum inversion angle
Possible reasons of invPossible reasons of inversion failuresersion failures– Malfunction of triggering Malfunction of triggering
circuitcircuit– Failure in thyristorsFailure in thyristors– Sudden dropout of AC sSudden dropout of AC s
ource voltageource voltage– Insufficient margin for cInsufficient margin for c
ommutation of thyristorsommutation of thyristors
Minimum inversion angMinimum inversion angle (extinction angle)le (extinction angle)
minmin==++++′′ （（ 3-103-10
99））
La
b
c
+
-Mu d
id
E M
L B
L B
L B
VT 1
VT 2
VT 3
o
u d
O
O
id
t
t
u a u b u c u a u b
p
iVT 1
iVT 2
iVT 3
iVT 1iVT 2
iVT 3iVT 1
iVT 3
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3.8 Realization of phase-control in 3.8 Realization of phase-control in rectifier circuitsrectifier circuits
ObjectObject
How to timely generate triggering pulses with How to timely generate triggering pulses with adjustable phase delay angleadjustable phase delay angle
ConstitutionConstitution
Synchronous circuitSynchronous circuit
Saw-tooth ramp generating and phase shiftingSaw-tooth ramp generating and phase shifting
Pulse generatingPulse generating
Integrated gate triggering control circuits are very Integrated gate triggering control circuits are very widely used in practice.widely used in practice.
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A typical gate triggering control circuitA typical gate triggering control circuit
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Waveforms of the typical Waveforms of the typical gate triggering control circuitgate triggering control circuit
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How to get synchronous voltage for the gate How to get synchronous voltage for the gate triggering control circuit of each thyristortriggering control circuit of each thyristor
For the typical circuit on page 20, the synchronous voltage of the gate For the typical circuit on page 20, the synchronous voltage of the gate triggering control circuit for each thyristor should be lagging 180º to the triggering control circuit for each thyristor should be lagging 180º to the corresponding phase voltage of that thyristor.corresponding phase voltage of that thyristor.
D,y 11D,y 5-11
TR TS
uA uB uC
ua ub uc - usa
- usb
- usc
usa
usb usc
Uc
Usc -Usa
Ub
Usb
-Usc
-Usb
Ua
Usa
UAB
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