Matrix Converter Based AC/DC Rectifier

Senad Huseinbegovic* and Omer Tanovic† * Faculty of Electrical Engineering, Sarajevo, B&H, shuseinbegovic@etf.unsa.ba

† Faculty of Electrical Engineering, Sarajevo, B&H, otanovic@etf.unsa.ba

Abstract— In this paper we present a driver for the DC ma-chine based on a matrix converter. 3 – Phase matrix conver-ter with one and two output DC voltages is considered. Ma-thematical relations for the Venturini control algorithm of the matrix converter have been derived. On the basis of mathematical relations working range for both AC/DC converters have been defined. Also relations for the design of the converter output filter have been derived. Simulations have been performed in software package PSIM.

I. INTRODUCTION

Matrix converters have been considered for the first time in [1], and from that time on have become increa-singly attractive circuits in power electronics. The basic structure of a 3-phase-to-3-phase matrix converter is shown in Fig. 1. Each of the nine switches can either block or conduct current in both directions, thus allowing any of the output phases to be connected to any of the input phases. In a matrix converter structure in Fig. 1. each of the nine switches could be of the common confi-guration [2], [3].

Fig. 1. Structure of a three phase matrix converter

Authors in [3]-[8] propose different concepts for AC

motor drive. Common DC motor driver can be found in [9]. In this paper a AC/DC converter on the basis of a ma-trix converter will be presented. Common AC/DC conver-ters can be found in [2], [10]. Mathematical analysis for the driver will be derived based on the Venturini control method [2],[3]. On the basis of those mathematical rela-tions working range of AC/DC converters will be derived. All simulations presented in this paper were obtained in software package PSIM.

II. AC/DC RECTIFIER BASED ON THREE PHASE MATRIX

CONVERTER

Derived from schemes of AC/DC converters from [2], [10] and Fig.1, the general structure of AC/DC converter based on a matrix converter is given in Fig.2.

Fig. 2. General structure of AC/DC converter based on matrix converter

Desired outputs of the matrix converter are DC voltage

signals, e.g. Vo1. The values of the output voltage signals in particular moments of time will be equal to input AC voltages (Fig. 3). Filtering the output voltage we get a desired DC voltage.

Fig. 3. Output voltage Vo1 waveforms composed of segments of the input

3-phase voltage waves

III. MATRIX CONVERTER CONTROL STRATEGIES

A. One output DC voltage

The scheme of the AC/DC converter based on the ma-trix converter, with one output DC voltage, is shown in Fig. 4. The input to the matrix converter is a symmetric three phase voltage signal.

Fig. 4. AC/DC converter scheme with one output DC voltage

Six switches in the converter structure are divided into

two groups of three switches. The switch structure is the same as the one taken in [2] and [3]. For the calculation of

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control signals the Venturini method, [4], is used. The input symmetric three phase voltage signal is given as:

( )

+=

−=

=

3

2sin3

3

2sin2

sin1

πω

πω

ω

tmViV

tmViV

tmViV

(1)

Output voltage should be of the form:

2112 oVoVV −= (2)

The values of the output voltage signals Vo1 i Vo2 in particular moments of time will be equal to input AC vol-tages, so we can assume that the output voltages will be the functions of the input voltages as:

( )

( )3123121

2

3322111

1

tivtivtivsT

oV

tivtivtivsT

oV

++=

++=

(3)

where:

321 tttsT ++= (4)

In the last two equations we assumed symmetrical

mode. It would be possible to assume antisymmetrical mode, but we will not consider it. Time Ts represents the period of a PWM signal, while t1, t2, and t3 are time pe-riods when switches are on in phases 1, 2, and 3 respec-tively. Parameter T is the period of the input voltages. Although the output voltage is not exactly a DC signal it can be filtered to get such a signal.

From (3) and (4) we get the matrix equation:

=

3

2

1

132

32112

1

t

t

t

mVmVmViviviviviviv

sTmVoVoV

(5)

TiVsT

oV ∆=1

(6)

with

=∆==

3

2

1

132

321

2

1

t

t

t

T

mVmVmViviviviviviv

iV

mVoVoV

oV

We have to find the values of it (i=1,2,3) from (6)

which satisfy (7):

oViVsTT1−=∆ (7)

under the condition that 0det ≠i

V stands, because of the

existence of inverse matrix 1−iV . From (5) and (6) fol-

lows 04

9det ≠−= m

iV

V , so the inverse matrix 1−iV exists

and the following equality is valid:

oV

tt

tt

tt

mV

sTT

+−−

++−

−+−

=∆

4

3

6sin

2sin

4

3

6

5sin

6sin

4

3

2sin

6

5sin

9

34

πω

πω

πω

πω

πω

πω

(8)

From the previous follows that times t1, t2 and t3 could

be calculated so that output voltage mean values Vo1 and Vo2 have the same values as the assigned.

From the previous (assuming the inequality s

Tt << 10 ),

with some calculations, we obtain the functional depen-dency between voltages Vo1 and Vo2:

16

232221

321

mV

oV

oV

oV

oV ≤++ (9)

The area of possible output voltages V12 determined

with (9) is given in Fig. 5 (for 2220=mV V).

Fig. 5. Area of possible output voltages V12

B. Two output DC voltages

The scheme of the AC/DC converter based on the ma-trix converter, with two output DC voltages, is shown in Fig. 6. Again the input to the matrix converter is a symme-tric three phase voltage signal.

Fig. 6. AC/DC converter scheme with two output DC voltages

Nine switches in the converter structure are divided into

three groups of three switches. For the calculation of con-trol signals the Venturini method, [3], is used again. The input symmetric three phase voltage signal is given as in (1). Output voltage should be of the form:

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32

2112

23 oVoVV

oVoVV

−=

−= (10)

The values of the output voltage signals Vo1, Vo2 i Vo3

in particular moments of time will be equal to input AC voltages, so we can assume that the output voltages will be the functions of the input voltages as:

( )

( )

( )3221131

3

3123121

2

3322111

1

tiVtiVtiVsT

oV

tiVtiVtiVsT

oV

tiVtiVtiVsT

oV

++=

++=

++=

(11)

where:

321 tttTs ++= (12)

In last equations symmetrical mode has been assumed,

as in [9]. Time Ts represents a period of the carrier. Times t1, t2 and t3 represent switch-on times in phases 1, 2 and 3, respectively.

Based on (11) and (12) we form matrix equation:

=

3

2

1

1

2

321

1

3

2

1

23

13

t

t

t

mVmVmV

ViVV

VViV

iViViV

sT

mV

oV

oV

oV

ii

ii (13)

TiV

sToV ∆=

1 (14)

with:

=∆==

3

2

1

23

13

1

2

321

2

0

1

t

t

t

T

mVmVmV

ViVV

VViV

iViViV

iV

mV

oV

oV

oV

oV

ii

ii

Equation (14) doesn’t have unique solution, and there-fore times it (i=1,2,3) are unobtainable. In order to find these times, we will use (10), and therefore we form fol-lowing equation:

−−−

−−−

=

3

2

1

123123

133221123

12

t

t

t

mVmVmViViViViViViViViViViViViV

sTmV

V

V

(15)

with:

Ti

V

sToV ∆′=′

1 (16)

The last matrix equation has a form like (5). (16) has a solution only under condition 0det ≠′iV . This condition

has been met because mViV4

27det =′ . Therefore inverse

matrix iV ′ exists, and we can write that:

( )

( ) oV

tt

tt

tt

mVsT

T ⋅

−

+

−

+

=∆

4

3

3

2sin

3

2sin

4

3sin

3

2sin

4

3

3

2sinsin

9

4

πωπω

ωπω

πωω

(17)

From the last equation we can find times t1, t2 and t3 so

that mean values of voltages at the output V12 and V23 follow desired values of these voltages.

From the previous, with some calculations, we obtain the functional dependency between voltages V12 and V23:

16

9 22

2323122

12mV

VVVV ≤++ (18)

The area of possible output voltages determined with

(18) is given in Fig. 7, for 2220=mV V.

Fig. 7. Area of possible output voltages

C. LC Filter Design

The scheme of the filter from Fig. 2 is shown in Fig. 8. It is an LC filter, i.e. a filter composed of capacitor C and inductance L.

Fig 8. LC output filter

Transfer function of the filter shown on fig. 8 is given by (19), where R represents R-load at filters output.

( )( ) ( ) 1

2

1

++

=

ωωω

jR

LjLC

jG (20)

We assume that:

iACiDCi UUU += (21)

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where UiDC and UiAC represent DC and AC components of input voltage. Similar equation can be written for out-put voltage:

DCoDC IRU ⋅= (22)

ACoAC IRCj

RU ⋅

+=

ω1 (23)

with:

R

UI iDC

DC = RCj

R

iACAC Lj

UI

ωω ++=

1

Quality coefficient K of the filter will be defined as ra-

tio of load’s AC voltages in the cases without and with the filter:

RCjR

RCjR

oAC

iACLj

U

UK

ω

ωω

+

++==

1

1 (24)

If we adopt , (25) it follows that:

12

1

1 −=+

=+

+LC

LjK

RCjR

RCjR

ωω

ω

ω (26)

and:

2

1

ωK

LC+= (27)

Bode-plots have been shown on fig. 9 for K=1, K=10

and K=100. Other values are: R=10Ω, f=2kHz, and C=47µF.

-80

-60

-40

-20

0

20

Mag

nitu

de (

dB)

102

103

104

105

-180

-135

-90

-45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

K=1K=10K=100

Fig. 9. Impact of quality coefficient K on Bode-plots of the filter

Fig. 10 shows Bode-plots of the filter for different val-

ues of the loada R=1 Ω, R=10 Ω i R=100 Ω. Other values are K=10, f=2kHz, C=47µF, and L=1.5mH.

-100

-50

0

50

Ma

gnitu

de (

dB

)

101

102

103

104

105

106

-180

-135

-90

-45

0

Pha

se

(deg

)

Bode Diagram

Frequency (rad/sec)

R=1 OhmR=10 OhmR=100 Ohm

Fig. 10. Impact of load value on Bode-plots of the filter

IV. SIMULATION RESULTS

All simulations have been performed on a PC with Pen-tium IV processor at 2.6 GHz and 2048 Mbytes of RAM, in software package PSIM. Two examples will be pre-sented. Both examples show the response of the matrix converter with resistor load and output LC filter. Resis-tance value of the load is R=15Ω. Values of filter parame-ters are L=25mH and C=0.5mF, with the sampling fre-quency of f=1kHz.

In Figs. 11.-15. we can see the transient response of a system depicted in Fig. 4. Assigned value of the output voltage is V12=70V.

Fig. 11. Phase one input current: Iil

Fig. 12. Frequency spectra of a current: Iil

Fig. 13. Output voltage Vo1

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Fig. 14. Output voltage Vo2

Fig. 15. Voltage V12

In Figs. 16.-22. we can see the transient response of a

system depicted in Fig. 6. Assigned values of the output voltages are V12=48V and V23=110V.

Fig. 16. Phase one input current: Iil

Fig 17. Frequency spectra of a current: Iil

Fig. 18. Output voltage Vo1

Fig. 19. Output voltage Vo2

Fig. 20. Output voltage Vo3

Fig. 21. Output voltage V12

Fig. 22. Output voltage V23

V. CONCLUSION

In this paper we have shown that a matrix converter can be also used as an AC/DC rectifier. The particular rectifier that was considered in the example was the three phase rectifier. For the control of a two way switch system in the structure of a matrix converter the Venturini method was used. Relations for the periods of conduct of a switch were derived, and those relations were used for finding a work-ing range of the output voltages. Also relations for the design of the output filter were derived.

At the end we can say that examples presented in this paper justify application of MC as a AC/DC rectifier. This has also been mathematically proven throughout this pa-per. To obtain voltages with better characteristics, at the

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output, one must apply control loops with adequate algo-rithms on analyzed system.

REFERENCES [1] L.Gyugyi and B.R. Pelly, Static Power Frequency Changers, New

York: John Wiley&Sons, Inc. 1976

[2] W.Shepherd and L.Zhang, Power Converter Circuits, Marcel Dekker, Inc.2004.

[3] J.Mahlein, O.Simon and M.Braun, “A Matrix with Space Vector Control Enabling Overmodulation”, Proceedings of EPE’99, paper 394, pp. 1-11, 1999.

[4] A. Alesina and M.G.B. Venturini, “Analysis and Design of Opti-mum-Amplitude Nine-Switch Direct AC-AC Converters”, IEEE Transactions on Power Electronics, Vol. 4, No 1, pp. 101 -112, January 1989.

[5] S.F. Pinto, J.F. Silva and P. Gamboa, “Current Control of a Ventu-rini Based Matrix Converter”, IEEE ISIE, pp. Vol.4, pp. 3214-3219, July 2006.

[6] M.Milovanovič, “A Novel Unity Power Factor Correction Prin-ciple in Direct AC to AC Matrix Converters”, Proceedings of IEEE/PESC’98, Vol. 1, pp. 746-752., May 1998.

[7] E.Watanabe, S.Ishii, E.Yamamoto, H.Hara, J.-K.Kang, A.M.Hava, “High Performance Motor Drive Using Matrix Converter”, Ad-vances in Induction Motor Control (Ref. No. 2000/072), IEE Se-minar, 2000, pp 7/1-7/6.

[8] Angkititrakul and R.W.Erickson, “Control and Implementation of a New Modular Matrix Converter”, Nineteenth Annual IEEE Ap-plied Power Electronics Conference and Exposition APEC 2004, Anaheim California, February 2004.

[9] S.Huseinbegović, N.Hadžimejlić, O.Tanović: “3-Phase Matrix Converter Driving a DC Machine”, IX BHK CIGRÈ, Neum, 2009.

[10] N.Mohan, T.Undeland, W.Robins, Power electronics, Converters, Applications and Design, John Wiley&Sons, Inc.1989.

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