Matrix Converter Based AC/DC Rectifier

IEEE Region 8 SIBIRCON-2010, Irkutsk Listvyanka, Russia, July 11 — 15, 2010

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Matrix Converter Based AC/DC Rectifier Senad Huseinbegovic* and Omer Tanovic† *

Faculty of Electrical Engineering, Sarajevo, B&H, [email protected] † Faculty of Electrical Engineering, Sarajevo, B&H, [email protected]

Abstract— In this paper we present a driver for the DC machine based on a matrix converter. 3 – Phase matrix converter with one and two output DC voltages is considered. Mathematical 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 increasingly 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 configuration [2], [3].

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 matrix 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. 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 matrix converter will be presented. Common AC/DC converters 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 relations working range of AC/DC converters will be derived. All simulations presented in this paper were obtained in software package PSIM. Fig. 4. AC/DC converter scheme with one output DC voltage

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.

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:

( )

Vi1 = Vm sin ωt Vi 2

Vi 3

∆T =

2π   = Vm sin  ωt −  3   2π   = V m sin  ωt +  3  

(1)

Output voltage should be of the form: V12 = Vo1 − Vo 2

(2)

The values of the output voltage signals Vo1 i Vo2 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: 1

Vo1 =

Ts

(vi1t1 + vi 2t 2 + vi3t3 )

1

Vo 2 =

Ts

(3)

(vi 2t1 + vi3t 2 + vi1t3 )

4 3Ts 9Vm

 5π   − sin  ωt +  6    π     − sin ωt + 6      − sin  ωt − π   2  

π   2  5π   sin  ωt +  6   π  sin ωt +  6  sin  ωt −

3

  3  Vo 4  3 4  4

(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 0 < t1 < Ts ), with some calculations, we obtain the functional dependency between voltages Vo1 and Vo2: 3V 2 2 2 V + 3V V + V ≤ m o1 o1 o 2 o2 16

(9)

The area of possible output voltages V12 determined with (9) is given in Fig. 5 (for Vm = 220 2 V).

where: Ts = t1 + t 2 + t 3

(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 periods when switches are on in phases 1, 2, and 3 respectively. 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:

Vo1  V  = 1  o2  T s V  m  Vo =

1 Ts

 vi1 vi 2 vi3   t1  vi 2 vi3 vi1  t 2  V V V  t   m m m  3 

Vi ∆T

Fig. 5. Area of possible output voltages V12

B. Two output DC voltages The scheme of the AC/DC converter based on the matrix converter, with two output DC voltages, is shown in Fig. 6. Again the input to the matrix converter is a symmetric three phase voltage signal.

(5)

(6)

with V

 Vo1  = Vo 2  o   Vm 



 v i1 V = v i 2 i V  m

vi2 vi3 Vm

We have to find the values of which satisfy (7): −1 ∆T = TsVi Vo

v i3  v i1 

Vm  

 t1  ∆T =  t 2  t   3

t i (i=1,2,3) from (6) (7) Fig. 6. AC/DC converter scheme with two output DC voltages

under the condition that det V ≠ 0 stands, because of the i existence of inverse matrix Vi −1 . From (5) and (6) fol-

9Vm ≠ 0 , so the inverse matrix Vi −1 exists 4 and the following equality is valid: lows det Vi = −

Nine switches in the converter structure are divided into three groups of three switches. For the calculation of control 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:


V12 = Vo1 − Vo 2

(10)

V23 = Vo 2 − Vo3

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: Vo1 = Vo 2 = Vo 3 =

(Vi1t1 + Vi 2 t 2 + Vi 3t 3 )

1 Ts

(Vi 2 t1 + Vi3 t 2 + Vi1t 3 )

1 Ts

(11)

(Vi3 t1 + Vi1t 2 + Vi 2 t 3 )

1 Ts

where:

Ts = t1 + t 2 + t3

Vo =

1 Ts

1 Ts

Vi1 V  i2 Vi 3  Vm

Vi 2

Vi 3 

Vi 3

Vi1

Vi1 Vm

  t1   t 2  Vi 2    t  3  Vm 

Ts

Vi ∆T

(14)

V − V Vi1 − Vi 2  i3 i 2  Vm

V −V V − V   t1  i2 i3 i 3 i1  V −V V − V t 2  i1 i 3 i2 i1    t V V m m   3 

 sin (ωt )   2π     sin  ωt −  3   

V ′ ∆T i

9Vm2 16

(18)

The area of possible output voltages determined with (18) is given in Fig. 7, for Vm = 220 2 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.

(15)

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

( )

Ts

(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:

1

G jω =

1

3  4 3 ⋅V 4 o  3 4 

Fig 8. LC output filter

with: Vo′ =

2π   sin  ωt +  3  

(13)

V  Vi1 Vi 2 Vi3   o1     t1  V  V = Vi 2 Vi 3 Vi1  ∆T = t  V = o0 2 o  i V Vi1 Vi 2  t  V  i3  3  o2     Vm  Vm Vm Vm  Equation (14) doesn’t have unique solution, and therefore times ti (i=1,2,3) are unobtainable. In order to find these times, we will use (10), and therefore we form following equation:

1

  sin (ωt )  4Ts   2π ∆T = sin  ωt − 9Vm   3  2  sin  ωt + π   3

V122 + V12V23 + V232 ≤

with:

V  V12  =  23   Vm 

27 V . Therefore inverse 4 m matrix Vi′ exists, and we can write that:

has been met because det Vi′ =

(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:

Vo1    Vo 2  = Vo3     Vm 

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(16)

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

( )2 + RL ( jω ) + 1

(20)

LC jω

We assume that:

U i = U iDC + U iAC

(21)


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

U oDC = R ⋅ I DC

Bode Diagram 50

Magnitude (dB)

656

(22)

0

-50

-100

0 R=1 Ohm R=10 Ohm R=100 Ohm

U oAC =

R ⋅ I AC 1 + jωRC

(23)

Phase (deg)

-45 -90 -135 -180 1 10

2

10

3

4

10

10

I DC

U = iDC R

I AC

U iAC = jωL + 1+ jRωRC

Quality coefficient K of the filter will be defined as ratio of load’s AC voltages in the cases without and with the filter:

jωL + 1+ jRωRC U iAC K= = R U oAC 1+ jωRC

(24)

IV. SIMULATION RESULTS All simulations have been performed on a PC with Pentium IV processor at 2.6 GHz and 2048 Mbytes of RAM, in software package PSIM. Two examples will be presented. Both examples show the response of the matrix converter with resistor load and output LC filter. Resistance value of the load is R=15Ω. Values of filter parameters are L=25mH and C=0.5mF, with the sampling frequency 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.

If we adopt ,

(25)

it follows that:

K=

jωL + 1+ jωR RC R 1+ jωRC

= ω 2 LC − 1

(26)

and:

Fig. 11. Phase one input current: Iil

LC =

1+ K

(27)

ω2

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. Bode Diagram 20

Magnitude (dB)

0 -20 -40 -60

Fig. 12. Frequency spectra of a current: Iil

-80

0 K=1 K=10 K=100

-45 Phase (deg)

6

10

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

with:

-90 -135 -180 2 10

5

10

Frequency (rad/sec)

10

3

10

4

10

5

Frequency (rad/sec)

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

Fig. 10 shows Bode-plots of the filter for different values 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. Fig. 13. Output voltage Vo1


Fig. 14. Output voltage Vo2

Fig. 15. Voltage V12

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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. 21. Output voltage V12

Fig. 16. Phase one input current: Iil

Fig. 22. Output voltage V23

V. CONCLUSION Fig 17. Frequency spectra of a current: Iil


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 working 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 paper. To obtain voltages with better characteristics, at the

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output, one must apply control loops with adequate algorithms 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 Optimum-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 Venturini Based Matrix Converter”, IEEE ISIE, pp. Vol.4, pp. 32143219, July 2006. [6] M.Milovanovič, “A Novel Unity Power Factor Correction Principle 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”, Advances in Induction Motor Control (Ref. No. 2000/072), IEE Seminar, 2000, pp 7/1-7/6. [8] Angkititrakul and R.W.Erickson, “Control and Implementation of a New Modular Matrix Converter”, Nineteenth Annual IEEE Applied 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.