Notes - Active Rectifiers and Dynamic Modelling of Three-phase Systems

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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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(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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(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


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Fig. 1 (e): Supply- and load-side voltage and current waveforms

frequency spectra for a three-phase diode rectifier with negligible


Fig. 1 (f): Supply- and load-side voltage and current waveforms

frequency spectra for a three-phase diode rectifier with negligible


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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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• 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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• 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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Fig. 3 (b): Supply- and load-side voltage and current waveforms

frequency spectra for a single-phase active rectifier with negligible


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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Fig. 3 (d): Supply- and load-side voltage and current waveforms

frequency spectra for a three-phase active rectifier with negligible


• 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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• 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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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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• 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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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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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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• 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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• 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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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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• 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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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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• 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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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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[ ][ ]

=

=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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• 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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[ ]

[ ]

+

−=

+

−=

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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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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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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• 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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• 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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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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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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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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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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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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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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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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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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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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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)

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39

• An expression for output voltage is obtained as

( ) ( ) ( ) ( ) ( ) ( ) ( )sisHsvsHsvsHsvo 132211 ++= (60)

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) q

osdss

sqss

Isdsi

VsdLIsVssvsv

LIsVssvsv

~

~~sin~

cos~

~cos~

sin~

1

2

1

=

−−−=

++=

ωφφφ

ωφφφ

(61)

• Substitute eqn. (61) into (60) to obtain

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( )[ ]( ) ( )[ ]q

osdss

sqsso

o

IsdsH

VsdLIsVssvsH

LIsVssvsHsv

sisHsvsHsvsHsv

~

~~sin~

cos~

~cos~

sin~

3

2

1

132211

+

−−−+

++=

++=

ωφφφ

ωφφφ (62)

• Further rearrange eqn. (62) to obtain

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )sdsGssGssGsvsGsv

sdVsHIsH

sLIsHLIsH

sVsHVsH

svsHsHsv

dsvo

oq

sdsq

ss

so

~~~~

~

~

~sincos

~cossin

23

21

21

21

+++=

−+

−

−+

+=

ωφ

ω

φφφ

φφ

ωφ

(63)

• The relevant transfer functions are defined as follows: The

input to output transfer function is given by

( )( )( )

( ) ( ) φφωφ

cossin~ 21

0~~~

sHsHsv

svsG

ds

ovv oin

+=====

(64)

• The transfer function relating the output voltage to the phase-

shift is obtained as

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40

( )( )( )

( ) ( )( )φφφ

ωφ sincos~ 21

0~~~

sHsHVs

svsG s

dvs

ov

in

o+==

===

(65)

• The transfer function relating output voltage to supply frequency

is obtained as

( )( )( )

( ) ( )( )dqs

dvs

ov IsHIsHL

s

svsG

in

o 21

0~~~

~ +−=====φ

ωω

(66)

• The transfer function relating the output voltage to duty ratio is

obtained as

( )( )( )

( ) ( )( )qo

vs

odv IsHVsH

sd

svsG

in

in 32

0~~~~ +−==

=== ωφ

(67)

• Additional transfer function that need to be defined are as

below:

( )( )( ) ( )

( )( )( )

( )( )

( )( )( )

( ) ( )( )( )

( ) ( ) ( ) ( ) ( )( )222

223

3

22

11

1 sssLssLo

o

sssL

o

o

ssL

o

o

Ls

o

LRsLCsRRsLRDsG

sG

LRsLR

sG

sGsH

sG

RsLDR

sG

sGsH

sG

DRL

sG

sGsH

ω

ω

ω

+++++=

++==

+==

==

(68)

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41

CURRENT CONTROLLER BASED ON THE SYNCHRONOUS

REFERENCE FRAME MODEL

• Figure 14 presents a model of a three-phase voltage source

converter in the abc-axes reference frames.

• It is evident from Fig. 14 that the system is highly coupled due

to the interaction between the phases. This makes designing of

the controller a very complex affair.

ss RsL +

1 dcvaiav

ss RsL +

1

+sC

1

3

1ss RsL +

1bv

cv

ad

bd

cd

dcv

+

+

++

+

+

+

++

+

−

−

−

−

−

−

−

++bi

ci

aadi

bbdi

ccdi

dci

∑=

−cban

nc dd,,3

1

∑=

−cban

na dd,,3

1

∑=

−cban

nb dd,,3

1

dcv

dcv

Fig. 14: Model of a three-phase VSC in the abc-axes reference frames

• The general voltage and current equations applicable to a three-

phase active rectifier are obtained as:

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42

0

3

1

3

1

3

1

,,,,

,,

,,

,

,,

,

,,

,

∑∑

∑

∑

∑

∑

==

=

=

=

=

==

−=

−−=

−=+

−−=

−=+

−−=

−=+

cbaksk

cbaksk

cbakdckk

c

cbanncdcsc

creflscscssc

s

cbannbdcsb

breflsbsbssb

s

cbannadcsa

areflsasassa

s

iv

ididt

dvC

ddVv

VviRdt

diL

ddVv

VviRdt

diL

ddVv

VviRdt

diL

(69)

• In the s-domain, the above equations become

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

3

1

3

1

3

1

,,,,

,,

,,

,,

,,

∑∑

∑

∑

∑

∑

==

=

=

=

=

==

−=

−−=+

−−=+

−−=+

cbaksk

cbaksk

cbakdckskc

cbanncdcscscsscs

cbannbdcsbsbssbs

cbannadcsasassas

sisv

sidsissCv

ddsvsvsiRsisL

ddsvsvsiRsisL

ddsvsvsiRsisL

(70)

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43

ss RsL +

1 dcvβiβv

ss RsL +

1

+sC

1

2

3

αv

βd

αd

αd

dcv

+

+

−

−

−

αi

dci

( ) dcddqqc ididi

dt

dvC −+=

2

3

βd

+ +

−

ββββ

dvviRdt

diL dcss −=+

αααα dvviR

dt

diL dcss −=+

Fig. 15: Model of the three-phase VSC in the αβ-axes reference

frames

• With reference to Fig. 15, the equations applicable to a three-

phase VSC in αβ-axes reference frames are obtained as

( )

αααα

ββββ

ααββ

dveiRdt

diL

dveiRdt

diL

idididt

dvC

dcss

dcss

dcc

−=+

−=+

−+=2

3

(71)

• using complex phasor notation , the above equations become

{ }( )

αβαβαβαβ

αβαβ

dveiRdt

diL

idiedt

dvC

dcss

dcc

vv

vv

−=+

−ℜ= ∗

2

3

(72)

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44

ssLR +

1eqi

+ssLR +

1

2

3

sC

1

sLω

sLω

edi

dci

dcv

ed

v

edd

eqd

eqv

+

+

++

+

−

−−

eqd

edd

ed

ed

id

eq

eqid

dcv

Fig. 16: Model of the three-phase VSC in the dq-axes reference

frames

• With reference to Fig. 16, the equations applicable to the three-

phase VSC converter in dq-axes reference frames are obtained

as

( )

eddc

ed

eds

eqs

ed

s

eqdc

eq

eqs

eds

eq

s

dced

ed

eq

eq

c

dvviRiLdt

diL

dvviRiLdt

diL

idididt

dvC

−=+−

−=++

−+=

ω

ω

2

3

(73)

• Comparing Figs. 14, 15 and 16, it is seen that controller design

in the abc-axes reference frames is very complicated whereas

the dq-axes reference frames model allows for a much simpler

controller design.

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45

• This makes the effort involved in the abc-to-dq reference frames

transformation worthwhile. In the abc-axes reference frames,

three separate controllers are required. Further, controller

variables are AC.

• This is unlike the dq-axes reference frames where only two

controllers are required and also the controller variables are DC.

• Figure 17 presents a current loop controller employing axes

decoupling through voltage feedforward.

+

+

+

+

+

+

-

-

+

s

KK ic

Pi +

s

KK ic

Pi +

refdI ,

refqI ,

dssq ILV ω+

qs ILω−

abcV abcI

4,1S

6,3S

2,5S

qI

dI dq

abc

Fig. 17: dq-axes reference frames current controller implementation

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46

COMPENSATION OF A FEEDBACK SYSTEM USING A DIRECT

DUTY RATIO PULSE-WIDTH MODULATOR

• The overall open-loop transfer function is given by

)()()()()(

)()()(

modmod1

1

sGsGsGsGsG

where

sGsGsG

plantconv

coL

==

=

(74)

( )sGc

refdcv ,′dcv

dcv′

+−

( )sGmod( )sGplant

( )sH

Fig. 18: Block diagram representation of overall open-loop transfer

function

• For a given transfer function G1(s), the transfer function of the

compensated error amplifier Gc(s) must be properly tailored so that

GoL(s) meets the performance requirements expected of the power

supply. Some of the desired characteristics of the open-loop

transfer function GoL(s) are as follows:

• The gain at low frequencies should be high to minimise the

steady state error in the power supply output

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47

• The crossover frequency is the frequency at which the gain of

GoL(s) falls to 1.0 (0dB).

• This crossover frequency ωcross should be as high as possible,

but approximately an order of magnitude below the switching

frequency to allow the power supply to respond quickly to the

transients, such as step changes of load.

• Phase margin (PM) is defined as

Phase margin=φoL+1800

• Where φoL is the phase angle of GoL at the crossover

frequency. Phase margin should be a positive quantity and

normally in the range 45-600.

• Figure 19 presents a current loop controller. Although a PI

controller is shown, other possibilities are available each with its

advantages and disadvantages. The type of controller adopted

should that which will meet the requirements of the system

without introducing unnecessary complexities.

• Current loop determines the dynamic response and hence it is

important to ensure adequate bandwidth.

swsT+1

1erefi

dcv

eqdi

+ +−

( )

i

cici

sT

sTK +1

21 sw

pwm

Ts

K

+

RL

RL

sT

K

+1

−

Fig. 19: Block diagram representation of current-loop controller

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48

• With reference to Fig. 19, the closed-loop current loop transfer

function is obtained as

( )( )

( )

( )

( )

( )

( )( )

( ) ( ) ( )

( )( )

( )cicipwmRL

RLsw

sw

swswRLRL

swsw

swsw

ci

RLcicipwm

cicipwmRLRLsw

swci

RLcicipwm

sw

pwm

ci

cici

swRL

RL

sw

pwm

ci

cici

sw

ci

RL

RL

sw

pwm

ci

cici

sw

sTKKK

TT

Ts

TTTssT

TTs

TTs

sT

sTsTKK

sTKKKsTT

ssTsT

sTsTKK

Ts

K

sT

sTK

sTsT

K

Ts

K

sT

sTK

sT

sH

sT

KsH

Ts

K

sT

sTK

sTsG

++

+

++++

++

++=

+++

++

++=

+

+

+++

+

+

+

=

+=

+

+

+=

1

2

2221

11

112

11

11

21

1

1

1

11

21

1

1

1

1

21

1

1

1

3

22

(75)

• Equation (75) can be simplified if TRL=Tci. Define the following

2

1

swpwm

s

sRL

sRL

TT

R

LT

RK

=

=

=

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49

• From definition of bandwidth, if damping is 0.7071, then,

ωbw=ωn, i.e.

bws

cipwmn

L

KKωω ==

2 (76)

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50

VOLTAGE LOOP COMPENSATION

• Figure 20 presents a block diagram representation of a voltage

loop controller.

dvsT+1

1refdcv ,

dci

dcv

+ +−

( )

v

cvcv

sT

sTK +1ciH

sC

1

−

edqi

erefdqi ,

Fig. 20: Block diagram representation of voltage loop controller

• With reference to Fig. 20, the closed-loop voltage loop transfer

function is obtained as

( )( )

( )( )

( )( )sCsTsTsT

sTKsH

sTsH

pwmfvcv

cvcvov

pwmci

211

1

21

1

++

+=

+=

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51

( )( )

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( ) ( ) ( )

( )( ) ( )

( )( ) ( )[ ]

( ) ( )

( ) ( )( )

( )

( ) ( )( )

( ) ( ) ( )sissCvsvsTsT

sHsTK

svsTsT

sHsTK

sissCv

svsvsTsT

sHsTKsi

sissCvsi

sCsisisv

svsvsHsT

sTK

sTsi

dcdcdcdvcv

cicvcv

refdcdvcv

cicvcv

dcdc

dcrefdcdvcv

cicvcvedq

dcdcedq

dcedqdc

dcrefdccicv

cvcv

dv

edq

+++

+=

+

+

+=

−+

+=

+=

−=

−+

+=

1

1

1

1

1

1

1

1

1

1

,

,

,

(77)

( ) ( )

( )( ) ( )

( )( ) ( )

( )

( )( )( ) ( )

( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )dvcvcicvcv

dvcvdc

refdc

dvcvcicvdvcvcv

cidvcvcvcvdc

dvcv

cicvcvdc

dcrefdcdvcv

cicvcv

sTCTssHsTK

sTsTsi

svsTCTssHsTsTKsT

sHsTsTKsTsv

sCsTsT

sHsTKsv

sisvsTsT

sHsTK

+++

+−

++++

++=

+

+

+=

−+

+

11

1

111

11

1

1

1

1

2

,2

,

(78)

( )( )

( )( ) ( )cvcvpwmcvcv

cvcvcv

sTKsTsTCTs

sTKsH

++++

+=

1211

12

(79)

Documents

Notes - Active Rectifiers and Dynamic Modelling of Three-phase Systems