matrix converter principle

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Principle of operation of a Matrix Converter under Space Vector Pulse Width Modulation control Prasid Syam, Department of Electrical Engineering Bengal Engineering and Science University, Shibpur Matrix converter is a power level direct frequency changer Desirable characteristics of a power level direct frequency changer

1) Power conversion possible from any input frequency to any output frequency 2) Bidirectional power flow 3) Input current sinusoidal 4) Output current sinusoidal 5) Input displacement factor controllable 6) No intermediate energy storage element for power conversion

Matrix Converter can perform closely to an ideal frequency changer.

Fig.1 Matrix converter schematic

psyameebesus_nampet_besus [email protected]

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The switches Sjk are bidirectional switches.

The output load is considered to be inductive in nature, therefore, during switching, it is

assumed to act as constant current load, not changing with the step change in voltage.

Two basic constraints of operation :

1) Input phases never be short circuited through switches at any switching interval. This will protect the switches against flowing of input supply short-circuit current.

2) Any of the output phases never be open circuited through switching operations to break the inductive current in the load . Otherwise, an open circuit voltage can destroy the switching devices.

. The switching function of a switch, Sjk (t) is defined as,

Sjk(t) =1,when the switch is closed. Where, j∈ {A, B, C}, k∈{a, b, c} = 0, when the switch is open.

The constraints can be written as, Sja(t)+Sjb (t)+Sjc(t)=1,where j∈{A, B, C} The switching constraints result in twenty-seven allowable switching combinations as shown

in Table.1. Table-1 A B C vAB vBC vCA ia ib ic SAa SAb SAc SBa SBb SBc SCa SCb SCc Vom αo Iim βi

a b c vab vbc vca iA iB iC 1 0 0 0 1 0 0 0 1 Vim αi io βo

a c b -vca -vbc -vab iA iC iB 1 0 0 0 0 1 0 1 0 - Vim -αi+4π/3 io -βo

b a c -vab -vca -vbc iB iA iC 0 1 0 1 0 0 0 0 1 - Vim -αi io -βo+2π/3

b c a vbc vca vab iC iA iB 0 1 0 0 0 1 1 0 0 Vim αi+4π/3 io βo+2π/3

c a b vca vab vbc iB iC iA 0 0 1 1 0 0 0 1 0 Vim αi+4π/3 io βo+4π/3

1

c b a -vbc -vab -vca iC iB iA 0 0 1 0 1 0 1 0 0 - Vim -αi+4π/3 io -βo+4π/3

a c c -vca 0 vca iA 0 -iA 1 0 0 0 0 1 0 0 1 -2/√3vca π/6 -2/√3iA 7π/6

b c c vbc 0 -vbc 0 iA -iA 0 1 0 0 0 1 0 0 1 2/√3vbc π/6 2/√3iA π/2

b a a -vab 0 vab -iA iA 0 0 1 0 1 0 0 1 0 0 -2/√3vab π/6 -2/√3iA -π/6

c a a vca 0 -vca -iA 0 iA 0 0 1 1 0 0 1 0 0 2/√3vca π/6 2/√3iA 7π/6

c b b -vbc 0 vbc 0 -iA iA 0 0 1 0 1 0 0 1 0 -2/√3vbc π/6 -2/√3iA π/2

2

A

a b b vab 0 -vab iA -iA 0 1 0 0 0 1 0 0 1 0 2/√3vab π/6 2/√3iA -π/6

c a c vca -vca 0 iB 0 -iB 0 0 1 1 0 0 0 0 1 -2/√3vca 5π/6 -2/√3iB 7π/6

c b c -vbc vbc 0 0 iB -iB 0 0 1 0 1 0 0 0 1 2/√3vbc 5π/6 2/√3iB π/2


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a b a vab -vab 0 -iB iB 0 1 0 0 0 1 0 1 0 0 -2/√3vab 5π/6 -2/√3iB -π/6

a c a -vca vca 0 -iB 0 iB 1 0 0 0 0 1 1 0 0 2/√3vca 5π/6 2/√3iB 7π/6

b c b vbc -vbc 0 0 -iB iB 0 1 0 0 0 1 0 1 0 -2/√3vbc 5π/6 -2/√3iB π/2

2

B

b a b -vab vab 0 iB -iB 0 0 1 0 1 0 0 0 1 0 2/√3vab 5π/6 2/√3iB -π/6

c c a 0 vca -vca iC 0 -iC 0 0 1 0 0 1 1 0 0 -2/√3vca 3π/2 -2/√3iC 7π/6

c c b 0 -vbc vbc 0 iC -iC 0 0 1 0 0 1 0 1 0 2/√3vbc 3π/2 2/√3iC π/2

a a b 0 vab -vab -iC iC 0 1 0 0 1 0 0 0 1 0 -2/√3vab 3π/2 -2/√3iC -π/6

a a c 0 -vca vca -iC 0 iC 1 0 0 1 0 0 0 0 1 -2/√3vca 3π/2 2/√3iC 7π/6

b b c 0 vbc -vbc 0 -iC iC 0 1 0 0 1 0 0 0 1 -2/√3vbc 3π/2 -2/√3iC π/2

2

C

b b a 0 -vab vab iC -iC 0 0 1 0 0 1 0 1 0 0 2/√3vab 3π/2 2/√3iC -π/6

a a a 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 - 0 -

b b b 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 - 0 -

3

c c c 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 - 0 -

Each of these switching combinations relates output voltage to input voltage and input

current to output current. The switching combinations are divided into three groups. The each of the combinations can

be conveniently analysed by the generated space vectors of input current and output voltage. Assuming sinusoidal input voltage, input line to line voltage space vector is,

eVevevvv ijim

jjcabcabi

αππ .3/4.3/2. 3)(3/2 =++=

where, and input a-phase voltage oii t 30+= ωα

)cos( tVv iimia ω= {Note: The initial phase angle for t = 0 is 30o for the input line voltage space vector. The output line voltage space vector}

o

ooo t 30+−= φωα

oφ is an arbitrary angle and

)cos( ooomoph tVv φω −= In the same way, the input and output line current space vectors are expressed,


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eIeieiii ijim)3/4.j3/2.j(3/2

cbai

βππ =++=

iii t φωβ −=

)cos( iiima tIi φω −=

iφ input displacement angle

eIeieiii ojom

jjCBAo

βππ =++= )3/4.3/2.(3/2

looo t φφωβ −−=

)cos( loiomA tIi φφω −−=

lφ load displacement angle {Note: the input/output voltage vector and output/input current vector are rotating vector and can stay at any phase angle in space vector plane.}

Combination of group 1 The output voltage vector having a phase angle αo, which is dependent on phase angle, αi of the corresponding input voltage space vector and this angle is time varying ( space vector is not stationary). We cannot use this combination. In the same way the input current vector has a phase angle βi which is related to the phase angle βo of the output current vector and this angle is time varying ( space vector is not stationary). These combinations are not used in space vector modulation because the phase of output voltage and input current are not stationary. Combination of Group II a, b and c From the table1 we can get output line voltages and input phase currents in terms of input phase voltages and output phase currents respectively. They space vectors are stationary in the sense that the phase angles are not dependent on time.

The instantaneous expression for output line voltages ( for delta connected load these are also

the phase voltages) in terms of input phase voltages, derived from the Fig.1 is,

=

vvv

vCA

BC

AB

ol


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=

−−−−−−−−−

vvv

SSSSSSSSSSSSSSSSSS

co

bo

ao

AcCcAbCbAaCa

CcBcCbBbCaBa

BcAcBbAbBaAa

.

……………………………………… (1) vT iphphl .=

Where “TphL” is the instantaneous input-phase to output-line transfer function matrix of the 3-phase to 3-phase matrix converter. The local averaged value( the averaged value over a switching time period) of a switching function Sjk(t) is the duty cycle of a switch Sjk , denoted as djk. Similarly the instantaneous expression for input phase currents in terms of output load currents is,

=c

b

a

iph

iii

i

=

−−−−−−−−−

iii

SSSSSSSSSSSSSSSSSS

CA

BC

AB

AcCcAbCbAaCa

CcBcCbBbCaBa

BcAcBbAbBaAa

T

.

iT olphl.T= ……………………………………… (2)

Note that i is actually the current flowing in the phase if the load is connected in delta. oL

Now the low frequency equivalents ( taking the average value over a switching time period and rejecting high frequency harmonics) of the voltage and current equations and the transfer function matrices mentioned above can be written as,

10 d jk≤≤ , Where, j∈ {A, B, C}, k∈{a, b, c} ………………………………(3)

1ddd jcjbja =++ , Where j∈ {A, B, C} ………………………………… (4)

V.T 'V 'iphphlol

= ……………………………………… (5)

i.TT 'i '

olphliph = ……………………………………… (6) Where,


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−−−−−−−−−

=AcCcAbCbAaCa

CcBcCbBbCaBa

BcAcBbAbBaAa

phl

dddddddddddddddddd

T '

Let the input phase voltages are given as,

−=

+ )tos(ω

)tos(ωt)os(ω

i

i

i

iph

120

120oc

cc

Vv oim

…………… (7)

If desired local averaged value of the output line voltages are sinusoidal, i.e.

v

=

'v

'v

'v

CA

BC

AB

ol'

+−

+−

+−

=

+−

)12030cos()12030cos(

)30cos(3

ooo

ooo

ooo

O

O

om

ttt

Vϕωϕωϕω

…………………… (8)

Then with the low frequency input phase to output line transfer function chosen as,

+−

−−

−

++−−+−

+−=

)120cos()120cos(

)cos(

)12030cos()12030cos(

)30cos('

o

o

T

oooo

oooo

ooo

ii

ii

ii

phl

tt

t

tt

tmT

ϕω

ϕω

ϕω

ϕωϕω

ϕω

…………………………… (9)

Where iϕ is input displacement angle and m is the modulation index, 0≤m≤1 and

equation (7), (8), (9) satisfy equation (5) with,

)cos(...2/3 iimom mVV ϕ= ………………… (10) Here due to inductive nature of load, the output load currents( for a delta connected load these are the current flowing in phases of the delta connected load) can be assumed sinusoidal and it is given as,


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++−−

−+−−

+−−

=

)12030cos()12030cos(

)30cos(3/

oloo

oloo

oloo

O

Oomoph

ttt

Iiϕϕωϕϕωϕϕω

…. (11)

Where, lϕ is the load displacement angle and o

ω the output frequency. Then

substituting equation (9) and (11), into (6), the local averaged input phase currents( the line currents) are obtained as,

='i'i'i

'ic

b

a

iph

+−

−−=

−

)120cos()120cos(

)cos(

ot

oti

t

ii

ii

i

imIϕωϕω

ϕω

… (12) Where,

).cos(.2/3 . ϕl

omim mII = …………… (13)

From the expression of Vom, we can make trade off between the modulation index, m, the

voltage gain Vom/Vim, and the input displacement angle, iϕ , independently of the load

displacement angle.

Unity input power factor is obtained for iϕ =0. With i

ϕ =0 and m=1 the maximum

voltage gain of √3/2. Physically the only restriction is the equality of the input and output active powers, because, from the expressions of Vom and Iim ,

Pi=3/2.Vim.Iim.cos ( iϕ )

=3/2.Vom.Iom.cos ( lϕ )

=Po

Matrix Converter modeled as two-stage converter:

From the theory of matrix converter we get the low frequency input phase to output line transfer function as,


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+−

−−

−

++−

−+−

+−

=

)120cos()120cos(

)cos(

)12030cos()12030cos(

)30cos('

o

o

T

oooo

oooo

ooo

ii

ii

ii

phl

tt

t

tt

tmT

ϕω

ϕω

ϕω

ϕω

ϕω

ϕω

This transfer matrix can be thought of product of two matrices,

)(T)(T 'T 'iVSR

'.

T

OVSIphl ωω=

Now, T V)( iph

T

.iVSR

' ω

=

−

+

+−

−−

−

)tos(ω

)tos(ω

t)os(ω

i

i

i

120

120)120cos()120cos(

)cos(

oc

c

c

tt

t

V o

oii

oii

ii

T

im

ϕω

ϕω

ϕω

=3/2.Vim.cos ( iϕ ) =Constant (a voltage)

This is equivalent to the operation of a voltage source rectifier (VSR). Similarly by

multiplying the constant voltage obtained from VSR stage withT , averaged

output line voltage V can be obtained, provided equation (10) is satisfied. It is milar to a voltage source inverter (VSI) operation.

)(' ω OVSI

'ol

This approach, where matrix converter can be emulated as a back-to-back VSR-VSI converter is called indirect transfer function approach (ITF). Figure 3.3 shows an equivalent VSR-VSI form of a 3-phase AC-to-AC matrix converter. A virtual DC link is established by thus manner.


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Fig. 2 Equivalent VSR-VSI conversions

The indirect transfer function approach allows space vector PWM technique to be used for both VSR and VSI stage simultaneously for over all matrix converter control.

Matrix Converter control algorithm For Matrix Converter control, the most popular technique is indirect transfer function approach. Here the 9-switch matrix converter is emulated as 12 switch back-to-back VSR-VSI stages. Space vector PWM (SVM) technique is simultaneously applied to both VSR and VSI stage to control the matrix converter.

VSR – SVM A voltage source rectifier converts AC input voltages to DC output voltage and current. Figure 4.1 shows the VSR part of the equivalent VSR-VSI conversion of matrix converter. Here the VSR is loaded by the DC current, IDC = ip

Fig.3.VSR part of equivalent VSR-VSI model of matrix converter

Consider the switch matrix corresponding to VSR part as,

ncnbna

pcpbpa

T

SSSSSS

And the relation between input currents and DC link current is,

−=

+

DC

DC

ncnbna

pcpbpa

T

c

b

a

ii

SSSSSS

iii

.

At any instant two switches, one from upper row and another from lower row of the converter can be “ON” at rectifier side. The possible switch combinations for rectifier, corresponding switch matrix states, and expression of input phase current space vector in


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terms of DC link current are shown in table 2. The “ON” state of a switch is denoted by “1” and “OFF” state is denoted by a “0”. The input current space vector is defined as

)3/4.j3/2.j(3/2 eieii cbaiππ ++=i

From Table-2 it can be seen that the resulting stationary space vectors corresponding to the allowable switch combinations are of constant magnitude and phase. Hence these vectors are called the switching state vectors (SSVs). These SSVs are plotted and space vector hexagon is formed. Fig. 4a shows the input current hexagon.

Table 2 SSVs for input current hexagon

Number

ncnbna

pcpbpa

T

SSSSSS

ia

ib

ic

ii

ii∠

Switching state

vectors

VDC

1

100001

T +iDC

0

-iDC

(2/√3)iDC

π/6

I1

-vca

2

100010

T

0 +iDC

-iDC (2/√3)iDC π/2 I2

vbc

3

001010

T

-iDC +iDC 0 (2/√3)iDC 5π/6 I3 -vab

4

001100

T

-iDC 0 +iDC (2/√3)iDC -5π/6 I4 vca

5

010100

T

0 -iDC +iDC (2/√3)iDC -π/2 I5 -vbc

6

010001

T

+iDC -iDC 0 (2/√3)iDC -π/6 I6 vab


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7

001001

T

0 0 0 0 I0 0

8

010010

T

0 0 0 0 I0 0

9

100100

T

0 0 0 0 I0 0

Fig.4.a. Switching state vectors Fig. 4.b. Input current hexagon


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Fig. 4.c. Vector decomposition of input current

Let the expression of the desired input phase current to be generated in complex plane is,

eIi )it( ij

imiϕω −=

At any instant of time the rotating input current vectorii , can be approximated by two adjacent switching state vectors and zero switching state vector. Let at any instant the

input current vector ii lies in sector 2 and makes an angle θsc with vector I1 as shown in

Figure 4.b. Now from the theory of space vector PWM, III 0i ++= γµi From geometry(Fig. 4.c),

Iµ= ii .sin(60o-θsc)/Sin(60o), Iγ= ii .sin(θsc)/sin(60o)

But, I1.Tµ/Ts=Iµ and I2.Tγ/Ts=Iγ Hence, Tµ/Ts= dµ= mc. sin (60o-θsc) Tγ/Ts= dγ= mc. sin (θsc) And d0=T0/Ts=1- dµ- dγ Where, 0≤mc=Iim/IDC ≤1 and Ts is the total sampling or switching time.

So, ii =I1. dµ+I2. dγ+I0. d0

The switching for I1 makes, ia=IDC and ic=-IDC and switching for I2 makes, ib=IDC and ic= -IDC Hence the local averaged value of the input phase currents for a switching cycle within sector 2 of the VSR hexagon, are


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DCcDC

c

b

a

ImIdd

dd

iii

osc

sc

sco

...

)( )30cos()sin(

)60sin(

'''

=

+−

=

−−

−

θθ

θ

γµ

γ

µ

……………. (14)

By the substitution of, θsc =(ωi.t-φi)-30o, for 30o≤ (ωi.t-φi) ≤ 90o with in sector 2 the local averaged value of the input phase current follows the desired input current given by, (Note that initial phase at t=0 for the input current vector is iφ− .

='i'i'i

'ic

b

a

iph

+−

−−=

−

)120cos()120cos(

)cos(.

ot

ot

t

ii

ii

ii

imIϕωϕω

ϕω, Where mc=Iim/IDC

The VSR local averaged output voltage is determined as,

)cos(...23.

''

ϕi

imciph VmVT TVSRpn ==V

This is a constant. VSI – SVM A three –phase voltage source inverter converts a DC voltage to three-phase AC. Fig. 5 shows the VSI part of the equivalent VSR-VSI conversion of matrix converter. Here the VSI is supplied by the DC voltage VDC=Vpn , derived from VSR part.


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Fig. 5 VSI part of equivalent VSR-VSI model of matrix converter

Consider the switch matrix corresponding to VSI part as,

CnBnAn

CpBpAp

SSSSSS

T

And the relation between output voltages and input DC link voltage is,

=

−

+

DC

DC

T

Cn

Bn

An

VV

CnBnAn

CpBpAp

SSSSSS

VVV

'

'

'

where, V and V 'pnDC V=+'nnDC V=−

−+ −== DCDCpnDC VVVV

Here the switching constraint is that two switches of a same leg of the inverter can never be “ON” simultaneously (to prevent short circuit of the input dc voltage) and at any instant any three switches of the 6-switch inverter will be “ON” ( not to make any of the output phases open-circuited). The “ON” state and the “OFF” state of a switch is denoted by “1” and “0” respectively. If all upper switches or all lower switches are “ON” then output voltage become zero(Zero vectors). The three-phase system is transformed in to spatial (two-axis real and imaginary) co-ordinates by using the transform,

)(3/2 evevvv 3/4j3/2j

CABCABo

ππ ++=

From Table-3 it can be seen that the resulting vectors corresponding to the allowable switch combinations are of constant magnitude and phase. Hence like VSR stage, these vectors are called the switching state vectors (SSVs). These SSVs are plotted and space vector hexagon is formed. Figure 5a shows the output voltage hexagon.

Table 3 SSVs of output voltage hexagon


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N

o

CnBnAn

CpBpAp

SSSSSS

T

vAB

vBC

vCA

vo

vo∠

Output voltage

switching state

space vectors

+ IDC

1

110001

T

+VDC 0 -VDC (2/√3)VDC π/6

V1

iA

2

100011

T

0 +VDC -VDC (2/√3)VDC π/2

V2

-iC

3

101010

T

-VDC +VDC 0 (2/√3)VDC 5π/6 V3 iB

4

001110

T

-VDC 0 +VDC (2/√3)VDC -5π/6 V4 -iA

5

011100

T

0 -VDC +VDC (2/√3)VDC -π/2 V5 iC

6

010101

T

+VDC -VDC 0 (2/√3)VDC -π/6 V6 -iB

7

000111

T

0 0 0 0 - V0 0

8

111000

T

0 0 0 0 - V0 0


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Fig.5a. Switching state vectors Fig.5b. Output voltage hexagon

Let the expression of the desired output line voltage to be generated in complex plane is,

=vo √3.Vom.ej (ωo

t-φo

+π/6)

At any instant of time the rotating output voltage vectorv , can be approximated by two adjacent switching state vectors and zero switching state vector. Let at any instant the

input current vector

o

vo lies in sector 2 and makes an angle θsv with vector V1 as shown

in Fig. 5b. Now from the theory of space vector PWM,vo =Vα+Vβ+V0, then from geometry(Fig.6),

Fig.6. Vector decomposition of output voltage

Vα= vo .sin (60o-θsv)/sin (60o)


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Vβ= vo .sin (θsv)/sin (60o)

But, V1.Tα/Ts= Vα and V2.Tβ/Ts= Vβ Hence, Tα/Ts = dα= mv. sin (60o-θsv) Tβ/Ts = dβ= mv. sin (θsv) And d0=T0/Ts=1- dα - dβ Where, 0 ≤mv=(√3.Vom)/VDC ≤1 and Ts is the total sampling or switching time.

So, vo = V1. dα + V2. dβ +V0. d0

The switching for V1 generates, vAB=+VDC and vBC=0 and vCA=-VDC and the switching for V2 generates, vAB=0 and vBC=+VDC and vCA=-VDC. The local averaged value of the output line voltages for a switching cycle with in sector 2 of the VSI hexagon, are

DC

sv

sv

sv

vDC

CA

BC

AB

V)θ(

)θ()θ(

mV

d )(ddd

''' o

.

βα

β

α

v

v

v

−−

−

+−=

=

30cossin

60sin

0

………… (15) By substituting, θsv = (ωo.t-φo+30o)-30o, for 30o≤ (ωo.t-φo+30o) ≤ 90o with in sector 2 the local averaged value of the output line voltage follows the desired output line voltage given by,

=

'''

'CA

BC

AB

ol

v

v

v

v

+−

+−

+−

+−

)12030cos()12030cos(

)30cos(3

ooo

ooo

ooo

O

O

om

ttt

Vϕωϕωϕω

= Where, mv=√3 Vom/VDC

The VSI local-averaged input current is determined

as, )cos(...2/3.''

ϕ lvomol mIiT TVSIp ==i

Which is a constant current.

MC output voltage and input current SVM As the local averaged output voltage of the VSR and the local averaged input current of the VSI are constant, from the local averaged point of view the VSR and VSI stages can


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be directly connected. Since both the VSR and the VSI hexagons contain six sectors, there will be 6x6=36 combinations of sectors or operating sectors. If at any instant, the sector 2 of both output voltage and input current hexagon are active then, the low

frequency input phase to output line voltage transfer matrix T with in the switching

cycle becomes, phl

'

−−

−

−−

−

=

)30cos()sin(

)60sin(

)30cos()sin(

)60sin('

0

oo

o

sc

sc

sc

sv

sv

sv

phl

T

mT

θθθ

θθθ

Here, m=mv.mc and the local averaged output line voltages are,

=

+−+− co

bo

ao

CA

BC

AB

v

v

v

v

v

v

dddd T

dddd

.

)()('''

γµ

γ

µ

βα

β

α

Using vbc=vbo-vco and vac=vao-vco, it is finally obtained,

bcac

CA

BC

AB

vv

v

v

v

)dd(dd

)dd(dd

'''

+

=

+−+− βγαγ

βγ

αγ

βµαµ

βµ

αµ

………………………………………(16)

Where,

dαµ= dα. dµ=m.sin(60o-θsv).sin(60o-θsc)=Tαµ/Ts

dβµ= dβ. dµ=m.sin(θsv). sin (60o-θsc)=Tβµ/Ts

dαγ= dα. dγ=m.sin(60o-θsv).sin(θsc)=Tαγ/Ts


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dβγ= dβ. dγ=m.sin(θsv). sin (θsc)=Tβγ/Ts

and d0=1- dαµ-dβµ-dαγ-dβγ=T0/Ts

Here it can be seen that the output line voltages can be synthesized in each switching cycle from samples of two input line voltages, here in this case the input line voltages to be sampled are, Vac and Vbc. By inspecting the equations (15) and (16) it can be concluded that simultaneous output-voltage and input-current SVM can be obtained by applying the standard VSI-SVM sequentially in two VSI-sub topologies of the three-phase ac-to-ac matrix converter. For the first sub topology, the duty cycles of the adjacent voltage SSV’s are dαµ and dβµ and Vpn=Vac and for the second sub topology the duty cycles of the adjacent voltage vectors are dαγ and dβγ and Vpn=Vbc. For the remaining part of the switching cycle, do the output line voltages are made 0, by applying “0” SSV. The switching sequence within one switching cycle can be used is,

dαµ- dβµ- dβγ- dαγ- d0 or d1- d2- d3- d4- d0

Thus for a switching cycle a particular set of five switching combinations are selected. The selection is done on the basis of the position of the rotating input current vector and rotating output voltage vector at a particular sampling interval. Here it is assumed that the both rotating vectors are lying at sector 2. Here it is assumed that the both rotating vectors are lying at sector 2. For dαµ interval the active vectors are V1 and I1 ,for dβµ

interval the active SSVs are V2 and I1, for dαγ interval the active SSVs are V1 and I2, for dβγ interval the active SSVs are V2 and I2, d0 interval the active SSVs are V0 and I0. To each voltage-current SSV pair, there exists a particular switching combination for matrix converter. These switching combinations can be found out by comparing 12-switch VSR-VSI model and 9-switch matrix converter as shown in Fig. 7. From this figure it can be found out that for V1 - I1 combination SAa, SBc, SCc are on and for V2 - I1 combination SAa, SBa, SCc and for V1 – I2 combination SAb, SBc, SCc are on . Similarly for V2 – I2 combination SAb, SBb, SCc are on. Switching for V0 – I0 is chosen according to the minimum change of switches from the previous switching states. Here SAb, SBb, SCb will be on during d0 . This can be further observed from the Table-1, if SAa, SBc, SCc are ON, the first row of the group 2A will be active and that will generate same set of input current switching state vectors and output voltage switching state vectors. Similarly if SAa, SBa, SCc are ON, for V2 - I1 combination, the 4th row of group 2C will be active which will generate same set of input current switching state vectors.


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Fig.7. Switching combinations for matrix converter derived from equivalent VSR-VSI model

As harmonic contents of the output voltages and the input currents can be improved if the switching intervals are each halved and symmetrically distributed with in a switching period Ts, in the practical use, the following switching sequence is used :

dαµ/2- dβµ/2- dαγ/2- dβγ/2- d0- dβγ/2- dαγ/2- dβµ/2- dαµ/2. Relevant papers:

1.

2.

3.

L. Huber, and D. Borojevic, “Space vector modulated Three-Phase to Three-Phase Matrix Converter with Input Power Factor Correction”. IEEE Trans. Ind. Applicat., vol. 31, pp. 1234-1246, Nov./Dec. 1995. P. Nielsen, F. Blaabjerg and J. K. Pedersen,” Space vector modulated Matrix Converter with minimized number of switchings and a feedforward compensation of input voltage unbalance,” Proc. Power Electronics, Drives and Energy Systems for Industrial Growth, 8-11 Jan, 1996, pages 833-839 P. Wheeler, J. Rodriguez, J. Clare, L. Empringham, A. Weinstein, “Matrix Converters: A

Technology Review,” IEEE Trans. Ind. Electron., vol. 49, pp. 276-289, Apr. 2002

matrix converter principle

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