ComparativeStudiesofDifferentSwitchingPatternsforDirect ...downloads.hindawi.com/journals/ape/2012/852024.pdf · only one switch in each output phase must be conducting. This leads

Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2012, Article ID 852024, 8 pagesdoi:10.1155/2012/852024

Research Article

Comparative Studies of Different Switching Patterns for Directand Indirect Space Vector Modulated Matrix Converter

Amin Shabanpour, Sasan Gholami, and Ali Reza Seifi

School of Electrical and Computer Engineering, Shiraz University, Shiraz 71345, Iran

Correspondence should be addressed to Ali Reza Seifi, [email protected]

Received 17 May 2012; Accepted 17 November 2012

Academic Editor: Francesco Profumo

Copyright © 2012 Amin Shabanpour et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper presents a MATLAB/Simulink simulation of direct and indirect space vector modulation for matrix converter. Differentswitching patterns for both direct and indirect methods are simulated and compared. Three criteria are chosen to compare theperformance of switching patterns: (1) total harmonic distortion (THD); (2) harmonic spectrum analysis of output voltages; and(3) number of switching in each switching period. Switching strategies are completely implemented using the power library inMATLAB/Simulink environment.

1. Introduction

Three-phase matrix converter is an AC-to-AC converter withnine bidirectional switches. These switches are organizedin a 3 × 3 matrix and with this arrangement any outputphase can be connected to any input phase [1]. Someadvantages of using this converter are providing bidirectionalpower flow; control of output waveforms and input powerfactor; and absence of DC capacitor [2]. Different approachesfor switching of matrix converter have been proposed inliteratures [2–6]. Many aspects such as output harmonicspectrum, total harmonic distortion (THD), complexity ofimplementation, and number of switching play importantroles in determination of an appropriate modulation strat-egy. Space vector modulation has been successively improvedin recent years and is considered as a standard techniquein matrix converter modulations [2, 5]. Despite the factthat this concept is presented in various literatures, it isstill ambiguous for engineers to completely comprehend itsoperating principle.

Space vector switching methods for matrix converterare classified into two different strategies: (1) indirect spacevector modulation which takes the advantage of a virtual dclink [5] and ((2) direct space vector modulation that providesdirect conversion [2]. So far only one comparison betweenthe direct and indirect space vector modulation is reported

[7]. In [7] the direct and indirect control performances ofa matrix converter supplying an induction motor (IM) havebeen carried out; however, different switching patterns arenot investigated.

In this paper first the two methods of indirect and directspace vector modulation of matrix converter are reviewed.Then different switching patterns are introduced, and eachswitching pattern will be simulated in MATLAB/Simulinksoftware. Simulations and comparison are done under thesame conditions of the input power supply and the outputload. In order to compare the performances of switchingpatterns three criteria are considered: total harmonic dis-tortion, harmonic analysis of output voltage, and numberof switching. The rest of the paper is as follows: Section 2explores the matrix converter fundamental. Sections 3 and4 describe the indirect and direct space vector modulations,respectively. The switching patterns are given in Section 5.In Section 6 simulation results are discussed. Finally theconclusion is given.

2. Matrix Converter

The 3φ-3φ matrix converter scheme is shown in Figure 1.matrix converter comprises of nine bidirectional switchesso arranged that any of the three output phases can be

2 Advances in Power Electronics

a

b

c

A

B

C3-ph

ase

inpu

t

3-ph

ase

outp

ut

S11

S12

S13

S21

S22

S23

S31

S32

S33

Figure 1: Direct matrix converter.

connected to any input phase. Bidirectional switches makeit possible to connect any of input phases a, b, or c to anyof output phases A, B, or C at any moment. The inputphases of matrix converter must not be shorted due to theinput voltage sources, and the output phases must not beopened due to the inductive nature of the load. The switchfunction, Sαβ, can be defined as

Sαβ ={

0 Sαβ : open,

1 Sαβ : close,

α ∈ {a, b, c}, β ∈ {A,B,C}.(1)

The restriction is expressed as

Saβ + Sbβ + Scβ = 1. (2)

The output voltages and input currents of the matrixconverter can be represented by the switching function S andthe transposed ST , such as

[Vout] = [S]× [Vin],⎡⎢⎣VA

VB

VC

⎤⎥⎦ =

⎡⎢⎣S11 S21 S31

S12 S22 S32

S13 S23 S33

⎤⎥⎦.⎡⎢⎣Va

Vb

Vc

⎤⎥⎦,

[Iin] =[ST]× [Iout],

⎡⎢⎣IaIbIc

⎤⎥⎦ =

⎡⎢⎣S11 S12 S13

S21 S22 S23

S31 S32 S33

⎤⎥⎦.⎡⎢⎣IAIBIC

⎤⎥⎦,

(3)

where Va, Vb, and Vc are input phase voltages; VA, VB

and VC are output phase voltages; Ia, Ib, and Ic are inputcurrents; IA, IB, and IC are output currents.

3. Indirect Space Vector Modulation

A space vector is obtained from three phase quantitiesthrough the following transformation:

�X = 23

(χa + αχb + α2χc

),

α = e j(2π/3) = cos(

2π3

)+ j sin

(2π3

).

(4)

Many engineers are familiar with the space vectormodulation (SVM) for voltage source inverters (VSIs) [8];however, the modulation method for the matrix converter isunderstood to few engineers due to the high level of intricacyand limited materials to explain its fundamentals. Hence, itwould be easier and more conceivable to illustrate the switch-ing operation of matrix converter by adopting conventionalVSI topology and SVM concept. The indirect space vectormodulation (indirect SVM) was first proposed by Huber andBorojevic. [5], where matrix converter was described to anequivalent circuit consisting of current source rectifier andvoltage source inverter connected through virtual dc linkas shown in Figure 2. The idea of the indirect modulationtechnique is to separate the control of the input current andoutput voltage.

This is done by dividing the switching function S into theproduct of a rectifier and an inverter switching function:

⎡⎢⎣S11 S21 S31

S12 S22 S32

S13 S23 S33

⎤⎥⎦ =

⎡⎢⎣ S7 S8

S9 S10

S11 S12

⎤⎥⎦.[S1 S3 S5

S2 S4 S6

],

⎡⎢⎣VA

VB

VC

⎤⎥⎦ =

⎡⎢⎣ S7 S8

S9 S10

S11 S12

⎤⎥⎦.[S1 S3 S5

S2 S4 S6

].

⎡⎢⎣Va

Vb

Vc

⎤⎥⎦.

(5)

So the space vector for the two voltage source convertersshown in Figure 2 can be applied to the matrix convertershown in Figure 1. For example S1 · S7 + S8 · S2 in Figure 2 isequivalent to S11 in Figure 1. Two space vector modulationsfor current source rectifier and voltage source inverter stagesshould be implemented, and then the two modulation resultsshould be combined.

4. Space Vector Modulation forthe Rectifier Stage

The rectifier part of the equivalent circuit can be assumedas a current source rectifier (CSR) with the averaged valueof IDC and is derived as follows:

IDC =√

32Iout ·mv · cos(θout). (6)

Iout is the peak value of output current, θout is the outputload displacement angle, and mv = Vout/VDC. The inputcurrent space vector Iref is extracted as follows:

Iref = 23

(Ia + αIb + α2Ic

). (7)

Advances in Power Electronics 3

3-ph

ase

inpu

t

3-ph

ase

outp

ut

a

b

c

A

B

C

Rectification stage Inversion stage

S1 S3 S5 S7 S9 S11

S2 S4 S6 S8 S10 S12

IDC

VDC

Figure 2: Indirect matrix converter.

The nine rectifier switches have nine permitted com-binations to avoid an open circuit at the dc link. Thesecombinations include three zero and six nonzero inputcurrents given in Table 1.

The reference input current vector is synthesized byimpressing the adjoining switching vectors (Iγ) and (Iδ) withduty cycles (dγ) and (dδ), respectively. The reference vectorcan be expressed by the current-time product sum of theadjoining active vectors as illustrated in Figure 3:

Iref = dγ · Iγ + dδ · Iδ. (8)

The duty cycle of the active vectors are calculated by

dγ =Tγ

Ts= mc sin

(π

3− θi

),

dδ = Tδ

Ts= mc sin(θi),

doc = Toc

Ts= 1− dδ − dγ,

(9)

where θi indicates the angle of the reference current vector.The current modulation index, mc, defines the desiredcurrent transfer ratio such as

mc = Iref

IDC; 0 ≤ mc ≤ 1. (10)

5. Space Vector Modulation forthe Inverter Stage

The inverter can be assumed as a separate VSI. The switchingmethod is exactly similar to conventional VSI [8], but owingto its virtual DC link, VDC should be defined as follows:

VDC = 32Vin ·mc · cos(θin). (11)

Vin is the peak value of input voltage, and θin is the inputdisplacement angle.

ia

ic

ib

30◦

90◦

120◦

210◦

270◦

330◦

Iref

θi

dδIδ

dγ Iγd-axis

q-axis

Figure 3: Input current space vector in complex plane.

30◦

90◦

120◦

210◦

270◦

330◦

d-axis

q-axis

Vref

θv

dβIβ

dα Iα

VCA

VBC

VAB

Figure 4: Output voltage space vector in complex plane.

The output voltage space vector, Vref, is evaluated as.

Vref = 23

(Va + αVb + α2Vc

). (12)

The inverter switches have eight permitted combinationsto avoid a short circuit. These combinations include threezero and six nonzero input currents (see Table 2).

The reference output voltage vector is synthesized byimpressing the adjoining active vectors Vα and Vβ with theduty cycles dα and dβ, respectively. The reference vectorcan be expressed by the voltage-time product sum of theadjoining active vectors as illustrated in Figure 4:

Vref = dα ·Vα + dβ ·Vβ. (13)


Table 1: Current vectors for rectifier stage.

Type Vector Iref S1 S2 S3 S4 S5 S6

Active

I1 2/√

3IDC < −π/6 1 0 0 1 0 0

I2 2/√

3IDC < π/6 1 0 0 0 0 1

I3 2/√

3IDC < π/2 0 0 1 0 0 1

I4 2/√

3IDC < 5π/6 0 1 1 0 0 0

I5 2/√

3IDC < −5π/6 0 1 0 0 1 0

I6 2/√

3IDC < −π/2 0 0 0 1 1 0

Zero I0 01 1 0 0 0 0

0 0 1 1 0 0

0 0 0 0 1 1

Table 2: Voltage vectors for inverter stage.

Type Vector Vref S7 S8 S9 S10 S11 S12

Active

V1 2/3VDC < 0 1 0 0 1 0 1

V2 2/3VDC < π/3 1 0 1 0 0 1

V3 2/3VDC < 2π/3 0 1 1 0 0 1

V4 2/3VDC < π 0 1 1 0 1 0

V5 2/3VDC < −2π/3 0 1 0 1 1 0

V6 2/3VDC < −π/3 1 0 0 1 1 0

Zero V0 01 0 1 0 1 0

0 1 0 1 0 1

The duty cycles of the active vectors can be written as:

dα = Tα

Ts= mv sin

(π

3− θv

),

dβ =Tβ

Ts= mv sin(θv),

dov = Tov

Ts= 1− dα − dβ,

(14)

where θv indicates the angle of the reference voltagevector. mv is the voltage modulation index and defines thedesired voltage transfer ratio such as

mv =√

3Vref

VDC; 0 ≤ mv ≤ 1, (15)

6. Direct Space Vector Modulation

In direct space vector modulation the actual matrix convertercircuit is considered without any assumption of an equivalentcircuit. For operation of the matrix converter one andonly one switch in each output phase must be conducting.This leads to twenty-seven possible switching combinationsfor the matrix converter. Modulation is more complicatedbecause these vectors vary continuously and depend oninstantaneous magnitude of sources. The output voltagestates are usually classified in three groups:

(i) 18 combinations with fixed directions,

(ii) 3 zero vectors,

(iii) 6 rotating vectors.

The 6 combinations of rotating vectors in group 3 arenot used. Similar to indirect space vector modulation thereference output voltage vector is synthesized by impressingthe adjoining active vectors. The reference input current vec-tor is also synthesized by impressing the adjoining switchingcurrent vectors. Figure 5 shows the output voltage and inputcurrent reference space vectors. For any combination ofoutput voltage and input current sectors, four configurationscan be identified, which produce output voltage and inputcurrent vectors. Among the switching that can be selectedin each sector, ones that are shared input current andoutput voltage vectors are used. Table 3 shows the possibleconfigurations. Duty cycle calculations are given in [2]. Ifkv and ki are the sectors where Vref and Iref are placed andα and β are the phase angles within the sectors Vref and Iref,the duty cycles are calculated by

T1 = (−1)kv+ki 2√3q

cos(α− π/3) cos(β − π/3

)cos(ϕi) ,

T2 = (−1)kv+ki+1 2√3q

cos(α− π/3) cos(β + π/3

)cos(ϕi) ,

T3 = (−1)kv+ki+1 2√3q

cos(α + π/3) cos(β − π/3

)cos(ϕi) ,

T4 = (−1)kv+ki 2√3q

cos(α + π/3) cos(β + π/3

)cos(ϕi) ,

T0 = 1− T1 − T2 − T3 − T4.

(16)


Sector 1

αo

VoV o

+1, +3

+7,

+9

V o

(a)

Sector 1+1, +7

+3, +9

αi

ϕi βiii

Vi

(b)

Figure 5: The output voltage and input current reference space vectors.

T12

T22

T12

T22

T0

(a)

T14

T12

T24

T24

T02

T14

T24

T24

T02

(b)

Figure 6: Switching patterns for indirect modulation: (a) single sided; (b) double sided.

T1 T2 T0 T3 T4

(a)

T4 T12

T12

T22

T22

T02

T02

T32

T32

(b)

T1T01 T2 T02 T3 T03T4

(c)

T03T12

T22

T22

T32

T32

T42

T42

T012

T12

T012

T022

T022

(d)

Figure 7: Switching patterns for direct modualation: (a) asymmetrical single sided; (b) asymmetrical double sided; (c) symmetrical singlesided; (d) symmetrical double sided.

Powergui

A

B

C

a

b

c

A

B

C

a

b

c

Switching patterngenerator

Source

Matrix converter

A

B

C

a

b

c

Load

Discrete,Ts = 2e−006 s

Vabc

Vabc

Iabc

∼

∼

∼

+

+

+

Figure 8: Test case schematic in Matlab/Simulink.


Table 3: Switching configurations.

Switches on Vo Io

1, 5, 6 2/3VAB < 0 2/√

3ia < −π/64, 2, 3 −2/3VAB < 0 −2/

√3ia < −π/6

4, 8, 9 2/3VBC < 0 2/√

3ia < π/2

7, 5, 6 −2/3VBC < 0 −2/√

3ia < π/2

7, 2, 3 2/3VCA < 0 2/√

3ia < 7π/6

1, 8, 9 −2/3VCA < 0 −2/√

3ia < 7π/6

4, 2, 6 2/3VAB < 2π/3 2/√

3ib < −π/61, 5, 3 −2/3VAB < 2π/3 −2/

√3ib < −π/6

4, 5, 9 2/3VBC < 2π/3 2/√

3ib < π/2

4, 8, 6 −2/3VBC < 2π/3 −2/√

3ib < π/2

1, 8, 3 2/3VCA < 2π/3 2/√

3ib < 7π/6

7, 2, 9 −2/3VCA < 2π/3 −2/√

3ib < 7π/6

4, 5, 3 2/3VAB < 4π/3 2/√

3ib < −π/61, 2, 6 −2/3VAB < 4π/3 −2/

√3ic < −π/6

7, 8, 6 2/3VBC < 4π/3 2/√

3ic < π/2

4, 5, 9 −2/3VBC < 4π/3 −2/√

3ic < π/2

1, 2, 9 2/3VCA < 4π/3 2/√

3ic < 7π/6

7, 8, 3 −2/3VCA < 4π/3 −2/√

3ic < 7π/6

1, 2, 3 0 0

4, 5, 6 0 0

7, 8, 9 0 0

Table 4: Test case system parameters.

Parameter Value

Source voltage (peak) 100 V

System frequency 60 Hz

Load resistance 5Ω

Switching frequency 6 kHz

Modulation index 0.8

Sampling time 2 μs

In these equations ϕi is the displacement angle betweencurrent space vectors and input voltage space vectors,and q = Vo/Vi is the voltage ratio.

7. Switching Pattern

The order in which the vectors are placed along one period iscalled switching pattern.

A proper choice of the switching pattern should beapplied to the switches of the matrix converter in order toachieve the desired output. There are different combinationsfor ordering the time segments corresponding to duty ratios.In this paper two switching patterns are considered forindirect space vector modulation. A single and a doubledistributions of time periods during a switching period areselected. These patterns are shown in Figure 6.

Four switching patterns are simulated and analyzed fordirect space vector modulation: (1) asymmetrical singlesided, which uses only one of the three zero vectors; (2)

Table 5: Simulation results.

Pattern THD%Number of

Switching foreach switch

Single side indirect 69.59 88

Double side indirect 68.15 159

Asymmetrical single side direct 70.84 62

Asymmetrical double side direct 66.92 163

Symmetrical single side direct 67.21 130

Symmetrical double side direct 65.32 192

asymmetrical double sided switching pattern; (3) symmet-rical single sided, which utilizes all the three zero vectors and(4) symmetrical double-sided pattern. Figure 7 shows theseswitching patterns.

8. Simulation Results

In order to compare the performances of the direct andindirect space vector modulation techniques, these methodsare applied to an AC/AC matrix converter. This systemconsists of a simple source voltage that is connected to aresistive load through a matrix converter. The simulationswere performed with Matlab/Simulink software as shown inFigure 8. The main circuits were assumed to be ideal, andresults are evaluated under the same conditions for the inputpower supply and the output Load. The parameters of thesystem are shown in Table 4. Two patterns of Figure 6 andfour patterns of Figure 7 are simulated. Table 5 compares


2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Harmonic order

Mag

(%

of

fun

dam

enta

l)

(a)

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Harmonic order

Mag

(%

of

fun

dam

enta

l)

(b)

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Harmonic order

Mag

(%

of

fun

dam

enta

l)

(c)

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Harmonic order

Mag

(%

of

fun

dam

enta

l)

(d)

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Harmonic order

Mag

(%

of

fun

dam

enta

l)

(e)

2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

Harmonic order

Mag

(%

of

fun

dam

enta

l)

(f)

Figure 9: Harmonic spectrum of output voltage: (a) single side indirect; (b) double side indirect; (c) asymmetrical single side direct; (d)asymmetrical double side direct; (e) symmetrical single side direct; (f) symmetrical double side direct.

THD and the number of switching in each switchingperiod for these methods. Furthermore, Figure 9 illustratesharmonic spectrum of output voltage for the six patterns. Asthe higher harmonics could be removed easily by a low-passfilter, Figure 9 only shows the low-order harmonics, whichare harder to be eliminated. However, the THDs of Table 5all consist of the harmonics.

Based on these simulation results, the double-sidedpatterns show the following characteristics over the single-sided pattern:

(i) lower harmonic distortion,

(ii) greater number of switching.

In addition, the analysis for symmetrical and asymmet-rical patterns shows that the symmetrical pattern is moreappropriate choice for a lower harmonic distortion in spiteof greater switching losses.

9. Conclusion

This paper compares different switching patterns of directand indirect space vector modulations for three-phasematrix converter. Two methods of indirect and direct space

vector modulation of matrix converter were completelydescribed. Double-sided and single-sided patterns as wellas symmetrical and asymmetrical patterns were analyzedin Simulink. Comparison results were evaluated based onthree criteria: (1) total harmonic distortion (THD); (2)harmonic spectrum analysis of output voltages; and (3)number of switching in each switching period. As expectedthe double-sided as well as symmetrical patterns produceslower harmonic distortions. However, the number of switch-ing increases when using double-sided and symmetricalpatterns.

References

[1] P. W. Wheeler, J. Rodrı́guez, J. C. Clare, M. L. Empringham, andA. Weinstein, “Matrix converters: a technology review,” IEEETransactions on Industrial Electronics, vol. 49, no. 2, pp. 276–288, 2002.

[2] D. Casadei, G. Serra, A. Tani, and L. Zarri, “Matrix convertermodulation strategies: a new general approach based on space-vector representation of the switch state,” IEEE Transactions onIndustrial Electronics, vol. 49, no. 2, pp. 370–381, 2002.

[3] M. Venturini and A. Alesina, “Generalised transformer: a newbidirectional, sinusoidal waveform frequency converter with


continuously adjustable input power factor,” in Proceedings ofthe IEEE Power Electronics Specialists Conference (PESC ’80), pp.242–252, Atlanta, Ga, USA, June 1980.

[4] A. Alesina and M. G. B. Venturini, “Analysis and design ofoptimum-amplitude nine-switch direct AC-AC converters,”IEEE Transactions on Power Electronics, vol. 4, no. 1, pp. 101–112, 1989.

[5] L. Huber and D. Borojevic, “Space vector modulated three-phase to three-phase matrix converter with input power factorcorrection,” IEEE Transactions on Industry Applications, vol. 31,no. 6, pp. 1234–1246, 1995.

[6] J. Rodrı́guez, E. Silva, F. Blaabjerg, P. Wheeler, J. Clare, andJ. Pontt, “Matrix converter controlled with the direct transferfunction approach: analysis, modelling and simulation,” Inter-national Journal of Electronics, vol. 92, no. 2, pp. 63–85, 2005.

[7] M. Jussila and H. Tuusa, “Comparison of direct and indirectmatrix converters in induction motor drive,” in Proceedingsof the 32nd Annual Conference on IEEE Industrial Electronics(IECON ’06), pp. 1621–1626, Paris, France, November 2006.

[8] H. W. van der Broeck, H. Ch. Skudelny, and G. V. Stanke, “Anal-ysis and realization of a pulse width modulator based on voltagespace vectors,” IEEE Transactions on Industry Applications, vol.24, no. 1, pp. 142–150, 1988.

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ComparativeStudiesofDifferentSwitchingPatternsforDirect ...downloads.hindawi.com/journals/ape/2012/852024.pdf · only one switch in each output phase must be conducting. This leads