Copyright reserved Prepared by Prof. MN Gitau Department of Electrical, Electronic and Computer Engineering, University of Pretoria 1 MODELLING ACTIVE RECTIFIERS • Conventional diode and phase controlled rectifiers draw currents that are rich in low order harmonics. • In particular, diode rectifiers with capacitor voltage filters operate with very low supply-side distortion power factor. • Diode rectifiers with capacitor voltage filter are widely used in off-line power supplies and drive applications as they operate with higher voltage gains compared with rectifiers employing current or LC-filters. • Proliferation of these rectifiers has led deterioration of power quality. This in turn has led to formulation of standards limiting harmonic injection into the grid in a bid to ensure power supply of acceptable quality. • Active rectifiers have been developed in an effort to reduce harmonic injection into the grid. • They are the practical realisations of near-ideal rectifiers. • Supply-side and load-side voltage and current waveforms as well as the frequency spectra of single- and three-phase rectifiers with capacitor voltage filters are as shown in Figs. 1 (a) and (b) below. • The highly distorted supply-side current waveforms are evident. Further, it is seen that the supply current frequency spectrum is very rich in low-order harmonics.
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Notes - Active Rectifiers and Dynamic Modelling of Three-phase Systems
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Copyright reserved
Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
1
MODELLING ACTIVE RECTIFIERS
• Conventional diode and phase controlled rectifiers draw currents
that are rich in low order harmonics.
• In particular, diode rectifiers with capacitor voltage filters
operate with very low supply-side distortion power factor.
• Diode rectifiers with capacitor voltage filter are widely used in
off-line power supplies and drive applications as they operate
with higher voltage gains compared with rectifiers employing
current or LC-filters.
• Proliferation of these rectifiers has led deterioration of power
quality. This in turn has led to formulation of standards limiting
harmonic injection into the grid in a bid to ensure power supply
of acceptable quality.
• Active rectifiers have been developed in an effort to reduce
harmonic injection into the grid.
• They are the practical realisations of near-ideal rectifiers.
• Supply-side and load-side voltage and current waveforms as
well as the frequency spectra of single- and three-phase
rectifiers with capacitor voltage filters are as shown in Figs. 1
(a) and (b) below.
• The highly distorted supply-side current waveforms are evident.
Further, it is seen that the supply current frequency spectrum is
very rich in low-order harmonics.
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
2
(a) Supply- and load-side voltage and current waveforms for a single-
phase diode rectifier with negligible supply-side inductance
Fig. 1(b): Supply- and load-side voltage and current waveforms
frequency spectra for a single-phase diode rectifier with negligible
supply-side inductance
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
3
(c) Supply- and load-side voltage and current waveforms for a single-
phase diode rectifier with substantial supply-side inductance
Fig. 1 (d): Supply- and load-side voltage and current waveforms
frequency spectra for a single-phase diode rectifier with substantial
supply-side inductance
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
4
Fig. 1 (e): Supply- and load-side voltage and current waveforms
frequency spectra for a three-phase diode rectifier with negligible
supply-side inductance
Fig. 1 (f): Supply- and load-side voltage and current waveforms
frequency spectra for a three-phase diode rectifier with negligible
supply-side inductance
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
5
PROPERTIES OF AN IDEAL RECTIFIER
• An ideal rectifier should draw a current that is in phase with the
supply voltage and also should not inject harmonics into the
grid.
• This suggests that an ideal rectifier appears as a resistive load to
the grid.
( )( )
( )cntrle
sav
e
ss
vR
VP
R
tvti
2
=
=
(1)
• Pav is the active power that is transferred to the output port of the
rectifier.
• Assuming lossless operation, the following relationships are
obtained:
es
rmsdc
s
rmsdc
oin
rmsdcrmsdcrmsdco
esssin
R
R
I
I
V
V
PP
RIIVP
RIIVP
==⇒
=
==
==
,,
2,,,
2
(2)
• A near-ideal rectifier can be realised in a number of ways.
• For example, a single-phase implementation could be realised
by connecting a full-bridge diode rectifier in cascade with a
boost DC-DC converter.
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
6
• The duty ratio of the boost DC-DC converter is then controlled
in such a manner that the supply current is in phase with the
voltage and also the input current ripple is very small.
• Another option entails using a full-bridge configuration where
each phase-arm comprises of diodes in anti-parallel with
controlled switches. Duty ratio control can be employed to
ensure that the rectifier draws a current that is in phase with the
voltage and also operate with very low current harmonic
injection.
• Figure 2 shows a circuit diagram of a single-phase single-switch
active or near-ideal rectifier.
oi( )tio
R
−
+
AC
( )tidc+
−
( )tvdc ( )tvo
( )tis
( )tvs C
Controller
( )td
( )( )tdM:1
converter
DCDC −
( )tvdc
( )tidc
Fig. 2: Active rectifier comprising of front-end diode rectifier in
cascade with a DC-DC converter
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
7
• Figure 3 presents supply-side and load-side voltage and current
waveforms and their respective frequency spectra for both
single-switch, single-phase active rectifier and full-bridge three-
phase active rectifier.
Fig. 3 (a): Supply- and load-side voltage and current waveforms for a
single-phase active rectifier with negligible supply-side inductance
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
8
Fig. 3 (b): Supply- and load-side voltage and current waveforms
frequency spectra for a single-phase active rectifier with negligible
supply-side inductance
Fig. 3 (c): Supply- and load-side voltage and current waveforms for a
three-phase active rectifier with negligible supply-side inductance
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
9
Fig. 3 (d): Supply- and load-side voltage and current waveforms
frequency spectra for a three-phase active rectifier with negligible
supply-side inductance
• With reference to Fig. 2, the input to the boost DC-DC converter
is a rectified single-phase AC voltage.
• For a sinusoidal supply voltage, the following expressions are
obtained
( )
( )( )
( )( )tdMtV
V
tv
tv
tVtv
tVv
m
o
dc
o
mdc
ms
==
=
=
ω
ω
ω
sin
sin
sin
(3)
• Consequently the conversion gain should be extremely high at
the zero crossing points and at its lowest when supply voltage is
at its peak.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
10
• If the conversion gain at the zero crossing points is not very
high, then current waveform is distorted in the neighbourhood of
the zero crossing points.
• Again, assuming lossless operation, the following expressions
for output current and power are obtained as follows
( )( ) ( ) ( ) ( )
( )( )
( )
e
maveoaveoaveo
eo
mToaveo
eo
m
eo
m
eo
dc
e
dc
o
dc
o
dcdco
R
VVIP
RV
VtiI
tRV
Vt
RV
V
RV
tv
R
tv
V
tv
V
titvti
Hz
2
2
2cos12
sin
2
,,,
2
,
22
22
50
==∴
==
−===
==
ωω
(4)
• Other converter topologies that can be considered for
implementing near-ideal rectifiers include:
o Buck-boost,
o Cuk,
o SEPIC
• Of all the converter topologies that are suitable for realising
near-ideal rectifiers, the boost has the most advantages to offer.
• In particular, it operates with the least switch stresses.
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
11
IMPLEMENTATION UTILISING BOOST CONVERTER
• Figure 4 shows a circuit diagram of a single-phase single-switch
active rectifier based on a boost DC-DC converter.
oi
( )tiC
R
−
+
( )tidc+
−
( )tvdc
( )tvo
( )tis
( )tvs
C
Controller
( )td
( )tvdc
( )tidc
( ) ( )titi Do =L
Fig. 4: Active rectifier comprising of front-end diode rectifier in
cascade with boost DC-DC converter
• Boost operation requires that the output voltage magnitude
should be greater that or equal to the peak AC input voltage.
• Converter controller has therefore to vary the duty ratio as
required to make input current proportional to input voltage.
• If the boost converter operates in the continuous mode, then, an
expression for the conversion gain is
( )( )( )td
tdM−
=1
1 (5)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
12
• An expression for duty ratio in continuous conduction mode is
then obtained as
( )( )( )
( )
o
dc
V
tv
tdMtd
−=
−=
1
11
(6)
• With reference to Fig. 4, an expression for inductor current
ripple at the boundary between continuous and discontinuous
conduction mode of operation is
( ) ( )L
Ttdtvi swdcL pkpk 2
=∆−
(7)
• Conditions for operation in the continuous conduction mode are
then obtained as
( )( )
( )
( )
−
<
<⇒
∆>=−
o
dcsw
e
esw
Le
dcTdc
V
tvT
LR
RT
Ltd
iR
tvti
pkpksw
1
2
2 (8)
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
13
TRANSIENT ANALYSIS OF THREE-PHASE SYSTEMS
• Steady-state analysis of three-phase systems is normally based on
conventional steady-state one phase equivalent circuit.
• This approach is inadequate for dealing with transient conditions in
both machine and power electronic converter systems.
• A more general mathematical model of a three-phase system is
required for control system design and dynamic studies in high
performance systems.
THREE-PHASE TO TWO-PHASE TRANSFORMS AND
COMPLEX SPACE VECTORS
• A three-phase system could be transformed into an equivalent two-
phase system by using Park’s Transforms.
• A single rotating space vector could also be used to represent
spatial variation of any of the three-phase quantities, e.g., voltage,
current, torque.
• Consider a balanced three-phase system and let the phase “a” be
the reference phase. Further, let the direct axis coincide with
phase-a axis. An anti-clockwise direction of rotation is assumed.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
14
a
b
c
β
α
Fig. 5: Three-phase voltage system representation
• A three-phase system can be transformed into a two-phase system
using the transform
[ ] [ ]
[ ]( )
( )
−
−−=
++
+−+
++
+−+
=
=
=
c
b
a
abc
c
b
a
c
b
aabc
S
S
S
ttt
ttt
C
S
S
S
C
S
S
S
CS
S
2
3
2
30
2
1
2
11
3
2
3
2sin
3
2sinsin
3
2cos
3
2coscos
3
2
32
φπ
ωφπ
ωφω
φπ
ωφπ
ωφω
αβ
φφαβ
β
α
(9)
• This is a power invariant transform and S represents voltage,
current, torque or any other three-phase system quantity.
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
15
• The two-phase system so obtained is still an AC system. Analysis
can be further simplified by transforming the AC quantities into
DC quantities.
• This is achieved using a transform that is usually referred to as the
two-phase to synchronous reference frame transform. It is effected
as follows:
d
q
β
αωt
Fig. 6: Two-phase to synchronous reference frame transformation
• With reference to Fig. 6, the following expressions are obtained:
tStSS
tStSS
q
d
ωω
ωω
βα
βα
cossin
sincos
−=
+= (10)
• In matrix form, eqn (2a) can be rearranged as
[ ] [ ]
−=
=
=
β
α
β
αφ
β
ααβ
ωω
ωω
S
S
tt
tt
S
SC
S
SC
S
S
dqdqq
d
cossin
sincos
2
(11)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
16
• It is possible to transform back to the two-phase system from the
synchronous reference frame transform using the following
transformation matrix:
−=
q
d
S
S
tt
tt
S
S
ωω
ωω
β
α
cossin
sincos (12)
• The reverse transformation from a two-phase to a three-phase
system is achieved using the following transformation matrix:
[ ]
−−
−=
=
β
α
β
ααβ
S
S
S
SC
S
S
S
abc
c
b
a
2
3
2
12
3
2
101
3
2
(13)
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Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria
17
TWO-PHASE MODEL OF AN ACTIVE AC-DC RECTIFIER
sL
sL
sL
+AT
−AT
+BT
−BT
+CT
−CT
+AD
−AD
+BD +CD
−BD −CD
fCZ
ci
oidci
Fig. 7: Circuit diagram of an active (synchronous) AC-DC converter
• Three-phase systems to be transformed in order to obtain a
model of a three-phase active rectifier include the following:
• supply voltage,
( )
+−=
+−=
+=
1
1
1
3
4sin
3
2sin
sin
φπ
ω
φπ
ω
φω
tVv
tVv
tVv
msc
msb
msa
(14)
• inductor voltage,
dt
diLv
dt
diLv
dt
diLv
scsL
sbsL
sasL
c
b
a
=
=
=
(15)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
18
• expression for supply current is,
( )
−−=
−−=
−=
3
3
3
3
4sin
3
2sin
sin
φπ
ω
φπ
ω
φω
tIi
tIi
tIi
msc
msb
msa
(16)
• voltage across the boost inductor resistor is,
( )
−−=
−−=
−=
3
3
3
3
4sin
3
2sin
sin
φπ
ω
φπ
ω
φω
tIRv
tIRv
tIRv
mssc
mssb
mssa
(17)
• reflected rectifier voltage is given by
( )
−−=
−−=
−=
2,
2,
2,
3
4sin
3
2sin
sin
φπ
ω
φπ
ω
φω
tVv
tVv
tVv
mreflreflc
mreflreflb
mreflrefla
(18)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
19
TRANSFORMING THE VOLTAGE QUANTITIES
Transformation of the voltage quantities to the stationary reference
frame (i.e. three-phase to two-phase) is achieved as follows
+
−
−
−−=
3
2sin(
)3
2sin(
sin
2
3
2
30
2
1
2
11
3
2
πω
πω
ω
β
α
tV
tV
tV
V
V
m
m
m
(19)
• In space vector form αβ-axes voltages can be expressed as
[ ] [ ]3/23/2,
3
2 ππβα
jcs
jbsas evevvv
−++= (20)
• The α-axis voltage is
tVtVV
tttVV
mm
m
ωω
πω
πωω
α
α
sin2
3sin
2
3
3
2
3
2sin(
2
1)
3
2sin(
2
1sin
3
2
==
+−−−=
(21)
• The β-axis voltage is obtained as
[ ] tVtV
V
tt
tt
VV
ttVV
mm
m
m
ωω
πω
πω
πω
πω
πω
πω
β
β
β
cos2
3cos
2
32
2
3
2sincos
3
2cossin
3
2sincos
3
2cossin
2
3
3
2
)3
2sin(
2
3)
3
2sin(
2
3
3
2
−=−=
−−
−
=
+−−=
(22)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
20
• In matrix form the α- and β-axis voltages can be expressed as
−
=
t
tV
V
V m
ω
ω
β
α
cos2
3
sin2
3
(23)
TWO-PHASE TO SYNCHRONOUS REFERENCE FRAME
TRANSFORMATION
• The transformation of voltage quantities from the two-phase to
synchronous reference frame is carried out as follows:
=
−
−=
mq
d
m
m
q
d
VV
V
tV
tV
tt
tt
V
V
2
30
cos2
3
sin2
3
cossin
sincos
ω
ω
ωω
ωω
(24)
• It is seen from eqn. (24) that there are only DC quantities and the
AC variations have been eliminated through this transformation.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
21
TRANSFORMING INDUCTOR VOLTAGE
Three-phase to two-phase transformation of inductor voltage is
achieved as follows:
[ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ]
=
=
=
=
=
=
dt
diCLC
dt
diCLCv
iCdt
dLC
iCdt
dLCv
dt
diLC
dt
diLCv
Lt
LtabcabcL
L
t
L
tabcabcL
abcabcabcL
βαφφ
φφ
βααβαββα
βαφφ
φφ
βααβαββα
φφαββα
,32
32
,,
,32
32
,,
32,
(25)
• Where the abc-to-αβ transformation matrix and its transpose are
[ ] [ ]
−
−−==
2
3
2
30
2
1
2
11
3
232
φφαβ CC
abc (26)
[ ] [ ]
−−
−==
2
3
2
12
3
2
101
3
232
ttabcCC
φφαβ (27)
• Thus, the product of the abc-to-αβ transformation matrix and its
transpose is obtained as
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
22
[ ][ ]
=
=10
01
2
30
02
3
3
232
32
tCC
φφ
φφ (28)
• With reference to eqns. (25) to (28), the inductor voltage in the αβ-
axes is obtained as follows
[ ]
=
=
dt
didt
di
L
dt
didt
di
LVL
L
L
L
Lβ
α
β
α
βα10
01, (29)
Two-phase to synchronous reference frame transformation of
inductor voltage
• In matrix form the inductors voltages in the αβ-axes can be
expressed as
[ ]
=
dt
diLVL
βαβα
,, (30)
• Inductors voltages in the dq-axes reference frames are then
obtained as
[ ] [ ][ ] [ ][ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
[ ] [ ][ ] [ ] [ ][ ]
+=
+=
=
==
•
•
dt
diCCiCCLV
dt
diCiCLCV
iCdt
dLCV
VCVCV
qLdt
dqdqqLd
t
dqdqqLd
qLdt
dqqLd
t
dqdqqLd
qLd
t
dqdqqLd
LsynchLdqqLd
,22,
22,
,2,
22,
,,
,2
,,
φφφφ
φφφ
αβαβ
βαφ
βααβ
(31)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
23
• The two-phase to dq-axes reference frames transformation
matrix is given by
[ ] [ ]tdqdq Ctt
ttC
φφ
ωω
ωω 22
cossin
sincos=
−= (32)
• The derivative of the transpose of the two-phase to synchronous
reference frame transformation matrix is obtained as
[ ]
−=
•
tt
ttC
t
dq ωωωω
ωωωωφ
sincos
cossin2 (33)
• From eqns. (32) and (33), the product of two-phase to synchronous
reference frame matrix and the derivative of its transpose as well as
the product of the two-phase to synchronous reference frame
matrix and its transpose are obtained as follows:
[ ][ ]
[ ][ ]
=
−
−=
−=
−
−
=
•
10
01
cossin
sincos
cossin
sincos
0
0
sincos
cossin
cossin
sincos
22
22
tt
tt
tt
ttCC
and
tt
tt
tt
tt
CC
t
dqdq
t
dqdq
ωω
ωω
ωω
ωω
ω
ω
ωωωω
ωωωω
ωω
ωω
φφ
φφ
(34)
• From eqns. (32) and (34), the inductor voltage in the synchronous
reference frame is obtained as
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
24
[ ]
[ ]
+
−=
+
−=
dt
didt
di
i
iLV
dt
didt
di
i
iLV
q
d
d
qqLd
q
d
q
dqLd
ω
ω
ω
ω
,
,10
01
0
0
(35)
• The inductor current and the rectifier reflected-voltage can be
transformed in manner similar to that employed in the case of the
supply voltage.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
25
SYNCHRONOUS REFERENCE FRAME MODEL FOR THREE-
PHASE ACTIVE AC-DC RECTIFIER
• The voltage equations describing a three-phase active rectifier in
the synchronous reference frame are obtained from the foregoing
analyses as follows:
mqreflq
sdsqsqreflLqsq
drefld
sqsdsdreflLdsd
VVdt
diLiLiRVVV
and
Vdt
diLiLiRVVV
2
3
0
,,
,,
=++−=+=
=+++=+=
ω
ω
(36)
• The power supplied to the load by the source can be expressed
as a function of d-axis and q-axis quantities as
ssmmmmin
sqsqsdsdin
IViViVP
iViVP
32
3
2
3
2
30 ==+=
+=
(37)
• In the preceding derivations, it was assumed that the rectifier
reflected voltage is sinusoidal. In a practical converter, the
reflected voltage will comprise of a fundamental component plus
harmonics with the harmonic order being dependent on the type of
switching scheme employed.
• Synchronous or active AC-DC rectifiers mostly employ PWM
switching schemes. It is therefore common to assume that the first
or lowest harmonic of significant magnitude is much higher than
the fundamental component. This pre-supposes that the switching
frequency is much higher than the fundamental frequency.
• The rectifier reflected voltage is in actual fact the inverted form of
the DC-bus voltage.
• To derive an expression for this reflected voltage in terms of the
DC-bus voltage and the switching functions applied to the gates of
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
26
the converter switches that are connected in anti-parallel to the
diodes.
• It is assumed that these functions are pure sinusoids. That is,
ignore the harmonics, as was the case before. Thus from the
following definition of the switching functions
( )[ ]( )( )( )
( )
++
+−
+
=
=
)3/2sin(
)3/2sin(
sin
2
2
2
φπω
φπω
φω
t
t
t
d
td
td
td
td
c
b
a
abc (38)
• Then, the reflected rectifier voltages in the abc-reference frames
are given by
[ ] ( )[ ]tdVV abcdcabcrefl =, (39)
• From which we can write the general expression for the reflected
rectifier voltages in the synchronous reference frame as
[ ] [ ][ ][ ]{ }[ ] [ ][ ] [ ]{ }[ ] [ ][ ]αβ
αβ
αβαβ
αβαβ
,,
,
,,
refldqdqrefl
abcdcabc
dqdqrefl
abcreflabc
dqdqrefl
VCV
dVCCV
VCCV
=
=
=
(40)
• If (ωt-ф1)=0, then the α- and β-axis terms of the reflected rectifier
voltages are given by
( )
( )
( )
+−
+=
++
+−
+
−
−−=
2
2
2
2
2
,
,
cos2
3
sin2
3
3
2
)3/2sin(
)3/2sin(
sin
2
3
2
30
2
1
2
11
3
2
φω
φω
φπω
φπω
φω
β
α
t
tdV
t
t
t
dVV
V
dc
dcrefl
refl
(41)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
27
• The dq-axes reference frame terms of the reflected rectifier
voltages are in turn obtained as
( )
( )
=
+−
+
−=
2
2
2
2
,
,
cos2
3
sin2
3
cos2
3
sin2
3
cossin
sincos
φ
φ
φω
φω
ωω
ωω
dc
dc
dc
dc
qrefl
drefl
dV
dV
tdV
tdV
tt
tt
V
V
(42)
• A general equivalent circuit for the synchronous rectifier
transformed into the dq-axes reference frame is as shown in Fig.
8 where the coupling between the d- and q-axis is represented
using a gyrator
sLω
1
dreflV ,
1:sinθd
( )
θ
φφ
cos
cos 21
s
ssq
V
Vv
=
−=
1:cosθd
di qi
sL sLsR sR
oi
ci dci
dco VV =
LR
( )
θ
φφ
sin
sin 21
s
ssd
V
Vv
=
−=
qreflV ,+ +−−
Fig. 8: A general equivalent circuit of an active rectifier in the dq-axes
reference frame
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
28
• Coupling of the axes has the effect of slowing down the
converter dynamic response. Some of the control strategies
address this shortcoming by employing decoupling techniques.
• If the phase-shift φ between the axes transformation matrix and
the supply voltage is zero, then the circuit reduces to that shown
in Fig. 9.
sLω
1
sL sLsR sR
rdVrqV
dcici
dcVoi
qidi
( )21sin φφ −= ssd Vv ( )21cos φφ −= ssq Vv
Fig. 9: Equivalent circuit of an active rectifier in the dq-axes reference
frame when the phase-shift is negligible
• With reference to Fig. 9, the voltage equations are obtained as
qrefldsqsssq
dreflqsdsssd
viLidt
dLRv
viLidt
dLRv
,
,
+−
+=
++
+=
ω
ω
(43)
• Using the converter model that was derived in the previous
sections simplifies current-loop controller design, as well as
allowing for more accurate and faster controllers to be designed
and implemented.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
29
• Additionally, using synchronous reference frame based
controllers enables us to control both the active and reactive
power independently; i.e. distortion and displacement power
factor control.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
30
DC ANALYSIS
• Figure 10 presents the DC equivalent circuit model of a three-
phase VSC in the dq-axes reference frames
d:1
qreflV ,
oV oi
qIdI
LR
dV
sLω
1
qV
+−
+−
Fig. 10: DC equivalent circuit of three-phase VSC in dq-axes
reference frames
• This can be carried out with reference to Fig. 10 as follows:
( )
( ) φφφ
φφφ
ω
φ
ω
coscos
sinsin
sin
21
21
ssqs
ssds
Ls
s
Ls
sdLqLoo
VVV
VVV
DRL
V
DRL
VDRIRIV
=−=
=−=
=
===
(44)
• The DC transfer function is obtained as
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
31
Ls
in
o
s
dcv
DRL
V
V
V
VG
ω
φsin=
==
(45)
• It is seen from eqn. (45) that the gain is controllable by
controlling the duty-ratio, D, and phase-shift, ф, and ranges
from zero to infinity. However, in a practical circuit, infinite
gain is impossible to achieve due to voltage and current
constraints.
• The above constraints limit the maximum value of DC gain to a
value close to unity, i.e.,
maxmax, DL
RG
s
Lv
ω= (46)
• Equation (46) shows that for a given Gv,max and RL, the source
impedance should be smaller than the value determined by eqn.
(46) in order to ensure a high enough DC output voltage.
• Equivalently, RL should be much larger than the source inductor
impedance to ensure that phase-shift, ф, and D are small enough
for power factor control.
IDEAL CURRENT SOURCE CHARACTERISTICS
• The rectified output current is given by
DL
VD
L
VDII
s
s
s
sdqo
ω
φ
ω
sin=== (47)
• Equation (47) is independent of the output DC voltage, Vo or
load resistance. It is purely dependent on circuit parameters and
switching function variables. Consequently, the converter
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
32
constitutes an ideal current source controlled by the switching
function characterised by D and phase-shift, ф.
• When the system response is very slow, the system can be
approximated using a first-order system given that the capacitor
is very large in practice.
• The maximum current is limited by the source impedance which
explains the large inrush current when the AC supply is first
connected or following large variations in capacitor voltage.
INPUT POWER P, Q, PF (RESISTIVE LOAD CASE)
• The input active and reactive power are obtained as follows
−=−=
==
+=
+=
+=+=
−=
=−
−=−
=
φφωω
φωω
ω
φφφ
ωφφ
φφ
ωω
ω
ω
ω
cossin1
sin3
sincossin
cossin
cossin
22
222
s
L
s
ssdsdsqsqin
s
L
s
sss
s
sssds
s
sdssds
sqssdssqsqsdsdin
s
sq
s
osd
ssdsqo
ssdosq
ssqsd
L
RD
L
VIVIVQ
L
RD
L
VIV
L
VVIV
L
VVIV
IVIVIVIVP
L
V
L
DVI
LIVDV
LIDVV
LIV
(48)
• From the definition of total power factor, and expression for
total input power factor is obtained as
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
33
+−
=
+==
φω
φω
φω
2
222
22
22
sin2sin1
sin
s
L
s
L
s
L
inin
in
in
inin
L
RD
L
RD
L
RD
QP
P
S
Ppf
(49)
• Optimum operation requires a high value of power factor. This
is achieved by proper selection of duty ratio, load power (i.e.
RL), boost inductance and phase-shift.
• Equation (49) suggests that power factor may not be unity at
very high output power.
• Unity power factor operation requires the following condition to
be met
2
..
2sin
2
1
02sin2
11
2
2
1
2
≥
=⇒
=−
−
s
L
s
L
s
L
L
RD
ei
L
RD
L
RD
ω
ω
φ
φω
(50)
• under unity power factor condition, output voltage is given by
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
34
D
VV
then
L
RD
if
RD
LD
L
RVV
so
s
L
L
s
s
Lso
≅
>>
= −
,2
2sin5.0sin
2
2
1
ω
ω
ω
(51)
• Equation (51) suggests that operation is similar to that for a
boost converter.
• When operating at very light load (i.e. when load resistance is
much larger than source impedance), the output voltage is not
sensitive to load resistance.
• Power factor is a maximum, even though it is not unity when
ratio of Q and P is a minimum, i.e.,
2
4
2sin
4
4
2
22
1
22
2
<
+
=
+
=
−
s
L
s
L
s
L
s
L
in
L
RD
L
RD
L
RD
L
RD
pf
ω
ω
φ
ω
ω
(52)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
35
INPUT POWER P, Q, PF (RESISTIVE LOAD CASE)
• When operating on no load, the system is capable of supplying
reactive power. The output DC voltage may be fixed to a
certain predetermined value by controlling the output current.
• In steady-state, active power and power factor are both set to
zero by adjusting the phase-shift to zero,
0
0
0
=
=
=
φ
in
o
pf
P
(53)
• The reactive power is then given by
−=
s
o
s
sin
V
DV
L
VQ 1
2
ω (54)
• It is seen from eqn. (54) that reactive power may be directly
controlled by controlling the duty ratio.
≥−
−
=↔
<−
−
=↔
=
11
1
11
1
2
2
,,
2
s
o
s
s
o
eqseq
s
o
s
o
seqs
eqs
s
in
V
DV
L
V
DV
CVC
V
DV
V
DV
LL
L
V
Q
ωω
ω
(55)
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
36
AC ANALYSIS
• With reference to Fig. 9, the following expressions are obtained
( ) ( )( )[ ]( )[ ]( ) ( )( )[ ]( )[ ]φφφ
φφφφ
φφ
φφφ
φφφφ
φφ
sin~
cos~
~sinsin
~coscos~~
~cos~~
cos~
sin~
~sincos
~cossin~~
~sin~~
++≅
++=+
++=+
++≅
++=+
++=+
ss
sssqsq
sssqsq
ss
sssdsd
sssdsd
vV
vVvV
vVvV
vV
vVvV
vVvV
(56)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )qreflqreflddsqqsssqsq
drefldreflqqsddsssdsd
vViILiIdt
dLRvV
vViILiIdt
dLRvV
,,
,,
~~~~~
~~~~~
++++−+
+=+
++++++
+=+
ωω
ωω
(57)
• From eqns. (56) and (57), the perturbed equivalent circuit of a
three-phase voltage source converter in the dq-axes is obtained.
• Figure 11 shows a perturbed equivalent circuit of a three-phase
VSC in the dq-axes reference frames.
• Figure 12 on the other hand shows a simplified version of the
small-signal equivalent circuit of a three-phase VSC in the dq-
axes reference frames.
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
37
ssL ssLsR sR
qVrqV
dci
ci
dcV oi
qidi
sC 1
LR
+−+− φφ cos
~sV
φsin~sv
sLω
1
+ −
+−
+−
φcos~sv
φφ sin~
sV
dVoVd
~
sd LIω~
sqLIω~
+−
qId~
Fig. 11: Perturbed equivalent circuit of a three-phase VSC in the dq-
axes reference frames
ssL ssLsR sR
qreflv ,
dci
ci
dcv oi
qidi
sC 1
LR
+− 1v
sLω
1
2v
+−
1iZ
( )sI1
+− ( )sVs
sLω
1
( )sI2
+ +
−−
( )sV2( )sV1
+−
1:d
Fig. 12: Small-signal representation of a three-phase VSC in the dq-
axes reference frames
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Department of Electrical, Electronic and Computer Engineering, University of Pretoria
38
• With reference to Figs. 11 and 12, the following expressions are
obtained
q
osdss
sqss
Idi
VdLIVvv
LIVvv
~
~~sin~
cos~
~cos~
sin~
1
2
1
=
−−−=
++=
ωφφφ
ωφφφ
(58)
Z( )sI1
+− ( )sVs
sLω
1
( )sI2
+ +
−−
( )sV2( )sV1 eqZ( )
s
s
L
sV
ω
( )sI2
+
−
( )sV2
eqZ
+− ( )sV
L
Zs
s
eq
ω
( )sI2+
−
( )sV2
Fig. 13: Using Norton’s Theorem to remove the gyrator
• Further, from Figs. 12 and 13, the following expressions are
obtained
( )( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( )Z
LZ
ZsisvL
Zsv
Z
sv
L
sv
L
Zsisv
L
svsi
seq
eqss
eq
eqs
s
s
s
s
2
22
2112
ω
ω
ωωω
=
−=
−=−
==
(59)
Copyright reserved
Prepared by Prof. MN Gitau
Department of Electrical, Electronic and Computer Engineering, University of Pretoria