1A Matrix Converter With Space Vector Control Enabling Overmodulation

A Matrix Converter with Space Vector Control Enabling Overmodulation J. Mahlein

A Matrix Converter with Space Vector Control Enabling Overmodulation

J. Mahlein, O. Simon, M. Braun

Universität Karlsruhe, Elektrotechnisches Institut

Kaiserstr. 12

D-76128 Karlsruhe, Germany

Tel. :++49-721-2783, Fax :++49-721-358854

[email protected]

Keywords : Matrix converter, Vector control, Modulation strategy

Abstract :

In this paper the design and testing of a matrix converter using a new method of vector overmodulation is described. The selection of the bi-directional switches as a main part of the converter is discussed and an overview of admissible commutation switching sequences is given. The theoretical output voltage limit which is 86.6% of the input voltage on sinusoidal operation can be increased up to 105%. Using overmodulation means to stress the grid and the load with non sinusoidal currents for higher voltage transfer ratio.

Introduction

The matrix converter is a simple 3 to 3 phase converter as shown in figure 1. By using 9 bi-directional switches the matrix converter is able to create a variable output voltage system of a desired frequency and magnitude [1], [2]. If an ordinary LC-filter is added the grid is loaded nearly by sinusoidal current.

This can only be done if the desired maximum output voltage is restricted to 86.6% of the input voltage which is seen as a disadvantage of this special type of converter.

The matrix converter is often compared with a DC voltage link converter. If the line side of the voltage link converter is realised by a simple 3 phase diode rectifier the maximum DC link voltage is 100% of the phase to phase input voltage. Stressing the converter with a load the DC link voltage will decrease to a minimum of 86.6% of the phase to phase voltage depending on the size of the link capacitor and the inductivity of the net and filter coils. The available sinusoidal output voltage ratio is between 100% and 86.6% of the input voltage. A higher voltage transfer ratio compared to the matrix converter has to be paid by costs for a large link capacitor.

M atrix converter

BDS11

BDS12

BDS13

LC -filter

ElectricBDS21

BDS22

BDS23

grid

BDS31

BDS32

BDS33

load

Figure 1: Schematic of the matrix converter

EPE '99 - Lausanne P. 1

Since all diode rectifiers do not enable sinusoidal load or power recovery to the grid, neither a variable power factor like the matrix converter does. Considering that the requirements for converters connected to the grid will increase with respect to power quality, the matrix converter has to compete with the self commutated line side voltage converter. This type of converter enables a greater output voltage than the line voltage by charging up the DC link. It has the need of 12 transistors and a large DC-link capacitor. The matrix converter consists of 9 bi-directional switches (BDS), 18 transistors if Insulated Gate Bipolar Transistors (IGBTs) and a proper commutation sequence is chosen. Both converter types have the need of filter elements and proper signal proceeding. Comparing this facts the matrix converter is a topology which will take place in several electrical drive applications.

Bi-directional switch configurations

To build up a matrix converter it is necessary to have a switch which is able to conduct current and to block voltage in both of its directions. For a small filter design and a low current ripple it is favourable to chose a high switching frequency of the converter. IGBTs offer these demands from small up to high power levels making them a perfect device for this application. Unfortunately, a bi-directional conducting IGBT is not available at this time and for proper commutation it will be necessary to have a bi-directional device which can be controlled in both current flow directions. This BDS device has to be built up from discrete components. There are three possible configurations to construct a BDS.

Rectifier bridge with IGBT

The configuration in figure 2 has the advantage that only one transistor is needed to build up a BDS but there are 5 semiconductors needed at least to build up a whole switch in which 3 semiconductors will lead a current. This will increase the conduction losses of the BDS. Neither is this configuration able to control the current direction making it useless for proper commutation which will be described later.

Figure 2: Rectifier bridge with IGBT

Emitter connected double configuration

The BDS is realised by two IGBTs with anti parallel diodes in which two semiconductors will lead a current at a certain time instant.

Figure 3: IGBTs emitter connected

The diodes have to be added to ensure a reverse blocking and a bi-directional conduction capability of the switch due to figure 3. With this configuration it is possible to build up a single output phase arrangement of the converter. For example a modular output phase arrangement can be built with the IGBTs of BDS 11, 21, 31 in figure 1 mounted together on a heat sink. The 3 phase converter can be simply built by adding two identical arrangements. Each BDS has to get its own galvanic isolated



gate unit. There are 9 isolated gate power supplies needed altogether. With this configuration the current direction can be controlled in both directions.

Collector connected double configuration

The collector connected double configuration (figure 4) has the same characteristic as the emitter configuration. Input phase arrangements (e.g. BDS 11, 12, 13 in figure 1) can be built which allows to save 3 galvanic isolated gate power supplies for the 3 phase converter because three IGBT emitters are connected to the same input phase and three belong to the same output phase. This circumstance makes the collector configuration more interesting for industrial applications. The disadvantage is a non modular signal structure of the converter.

Figure 4: IGBTs collector connected

Commutation sequence

The basic rule for matrix converter commutation is a switching sequence which does not interrupt the output currents to the inductive load or short-circuit the input voltage sources. Therefore, the demand of a BDS which can control the current in both directions becomes clear. If no clamp circuit is desired because of its extra power losses a step-by-step commutation strategy has to be chosen. The commutation strategy can be voltage controlled as described in [3] or current controlled as given in [4]. A mixture of both is conceivable but not necessary. All possible switching sequences have been investigated. In figure 6 a switching sequence is described by four numbers. Each number represents a switching instance of a transistor as numbered in figure 5 over the time of the commutation sequence. Starting situation is a current il with the transistor 1 and 2 switched on.

1

2

Vc

3

4

il

Figure 5: Single output phase

For example a switching sequence with a positive current il is 2314 which means that in a first step transistor 2 is switched off. Second step transistor 3 is switched on, third step transistor 1 is off and fourth transistor 4 on. After the sequence has passed the current il has commutated on transistor 3. For backward commutation the numbers of the transistors have to be exchanged and a sequence has to be chosen from figure 6. In order to compensate the delay times of the IGBTs a time interval has to be added after every switch command.

Theoretically, there are 24 combinations existing of which 12 are not suitable because they will short-circuit the voltage sources or interrupt the load current. Figure 6 shows the remaining sequences in their valid quadrants. Non marked sequences will lead to a proper commutation but the knowledge of



voltage polarity and current direction is needed making the signal processing more complicate than required.

i l

2 3 1 4

2 3 1 4

2 3 4 1

3 2 1 4

2 4 3 1

3 1 2 4

4 2 3 1

3 1 4 2

4 2 3 1

3 1 4 2

Vc

4 2 1 3

1 3 4 2

4 1 2 3

1 4 3 2

1 4 2 3

1 4 2 3

Figure 6: Legal commutation sequences

Current conditioned commutation

The two elliptically marked sequences 2314 and 1423 appear in two voltage quadrants which means that these sequences are independent from voltage polarity. The load current has to be measured in each phase and the current direction has to be evaluated for every switching cycle. A wrong detection of the direction which may appear near zero crossing of the current will interrupt the load current, which means that the blocking voltage at the transistor will rise rapidly with Lload ⋅ d il d t destroying the voltage sensitive semiconductors if no protection devices are added.

Voltage conditioned commutation

Two rectangularly marked sequences (4231, 3142) can be found which are free of current direction knowledge. Measuring the voltage polarity between the input phases is necessary for this kind of commutation. A polarity detection error which may occur at small voltage values will short circuit the input phases during the commutation process. Especially, the filter capacitors have to be connected with a low impedance commutation circuit to the switches which allows a short circuit current to rise fast. Fortunately, the commutation circuit has 4 semiconductors in series which means that a short circuit current will flow only if Vc is greater than the sum of the on-state voltages of the semiconductors. It has been found that the time interval between each switching step can be very short which means that short circuit currents do not last long. Another favourable effect is that the semiconductors are durable to high currents for a short time making it possible to build up a BDS which can withstand the stress of sequence errors without adding any protection device.

Vector modulation theory

The matrix converter is often described in its matrix equations like in [3] or [5]. The described methods are using transfer equations for each phase of the converter. To increase the output voltage to the maximum, a third harmonic system for the input system as well as for the output system has to be added to the transfer functions. These third harmonic systems have to be measured respectively calculated and synchronised making the modulation complicate to implement. The space vector theory offers a compact method to describe the whole set of the 9 BDS making the matrix converter theory as understandable as for a common DC-link converter. Space vector theory for matrix converters is described in [6], [7] for sinusoidal input and output current. If maximum power is desired e.g. high torque alternation at high voltage operation, space vector theory can be used


switching the matrix converter to an overmodulation mode. The disadvantage of overmodulation is that the grid and the load is stressed with harmonics.

Figure 7: Theoretical matrix converter

According to figure 7 the output voltages of a theoretical matrix converter can be written in a matrix T:

vA

v B

v

C

s11

s21

s12

s22

s

s

13

23

s31

va

s32

⋅

vb

s

v

33

c

va

T

⋅

v

(1)

PhPh

b

v

c

Switching matrix T consists of elements which are 1 if BDS is on or 0 if BDS is off. To observe the basic rule for matrix converter not to interrupt the currents to the outputs or short-circuit the input voltage sources there has to be one and only one 1 in each column of the matrix. The equation for the input currents is:

ia

ib

i

c

s11

s12

s21

s22

s

s

31

32

s

13 s ⋅

23 s33

i A

i B

i

C

iA

T T

⋅i

(2)

PhPh

B

i

C

Using the space vector theory like given in [6] it can be shown that the local averaged output line voltages can be written as:

AB

v

v BC

T VSI⋅ V

(3)

dc

vCA

The DC voltage in this term can be expressed by:



cos(ω

it)

3

T

2π

ˆ

ˆ

cos(ω

V

⋅

)

T

⋅

−t

)

v⋅ cos(φ

i

v⋅

3

dc

2

VSR

i

cos(ω

t −4π

)

i

3

In which

TVSR is set to:

(4)

cos (ω

i t −φ

VSR

cos (ω

i t− φ

T

cos (ω

i t −φ

i )

2π

−i

3

)

−

4π

)

i

3

(5)

As expressed in (3) and (4) the output voltage of the matrix converter are a product of two transfer

T

functions. The transfer vectors T VSI and T VSR have to be explained. Regarding figure 8 the matrix converter can be divided in two fictitious converter functions. Converter A at the input can be regarded as a common self commutated current source converter. A modulation strategy for the matrix converter can be found, enabling this type of operation.

C o nv erter A

ii

TV S R

v

ID C

i

VD C

TV S I

C o nv erter B v

o io

Figure 8: Decoupling of in- and output frequency

The matrix converter is offering 27 switch combinations which can be grouped in 3 families:

All output phases are switched to the same input phase (3 combinations)

Every output phase is switched to a different input phase (6 combinations)

Two output phases are switched to the same input phase, one output phase is switched to a different input phase (18 combinations)

Taking the combinations of family 3 and grouping them in 3 sets of 6 combinations as done in [6] it can be shown that these sets can be used as rectifying terms giving the input the desired behaviour of Converter A. Figure 9 shows a hexagon on which TVSR can be calculated.

The sets of family 1 will lead to a zero vector for the hexagon. By modulating the input current space vector this way that its locus diagram describes a inner circle in the hexagon having the same frequency as the grid the input line current will be sinusoidal with unity power factor. The transfer

vector doing this is named T VSR .


Im

I2

TV S R

i b

I3

III

II

I1

IV

I0

Ii a

R e

I4

V

V I

I6

i c

I5

Figure 9: VSR hexagon

T VSR

2

1 a a 2 T VSR , with a

j 2π

(6)

e 3

3

Considering that there is a fictitious DC voltage available converter B is able to create a sinusoidal output voltage by modulating an output voltage vector that describes a circle in the borders of the hexagon shown in figure 11. This hexagon is named VSI hexagon.

By modulating the output voltage with the VSI hexagon and the input current with the VSR hexagon the input frequency is decoupled from the output frequency.

Im V2

vO

vB C

III

II

V3

V1

V0

IvA B

IV

R e

V4

V

V6

V I

vC A V5

Figure 10: VSI hexagon

The coupling of input and output is described by the product of the two transfer vectors. If both locus diagrams describe the maximum inner circle to their hexagons it can be shown, that the voltage transfer ratio is 0.86.

Overmodulation

As discussed before the matrix converter can be treated as two different converters with their own modulation strategies. As known from common DC link converters it is possible to increase the output space vector by using the edges of the hexagon for 60° of ω it (table 1) instead of the circular locus diagram. To maximise the output voltage the space vector jumps from edge to edge in the diagram that way that a whole circulation of the 6 edges is done for one period of the desired fundamental frequency. The locus diagram will not longer describe a circle which will lead to harmonics.

In a first step only the output voltage VSI vector can be used to do this overmodulation mode. The value for voltage transfer ratio will be 95.4%.

In a second step if maximum voltage operation is desired the second hexagon VSR can be switched to overmodulation increasing the voltage transfer ratio for the fundamental component up to 105% of the input voltage.



0°≤ω ot<60°

vO = V1

60°≤ω

ot<120°

vO = V2

120°≤ω

ot<180°

vO = V3

180°≤ω

ot<240°

vO = V4

240°≤ω

ot<300°

vO = V5

300°≤ω

ot<360°

vO = V6

Table 1: Output voltage vectors at overmodulation


Simulation results

The consideration described above have been tested in a simulation. The result is seen in figure 11 and 12 for the input- side of the matrix converter.

ii ( t )

!

i

vi ( t )

v!

Figure 11: Input line current and voltage at overmodulation

Figure 12 shows the output voltage given at an RL-load. All magnitudes are standardised to the source magnitudes (fig. 7).

ia(t)

!

i

va(t)

v!

Figure 12: Output line current and voltage at overmodulation

The experimental setup

To verify the results of the space vector theory a matrix converter has been built with a digital signal processor system using a Texas Instruments TMS320C40 (figure 13).


P C

D S P ( T M S 3 2 0 C 4 0 )

P W M - m o d u l a t o r

commutation logic 1

cl2

cl3

double gateunit for BDS 1

gateunit 1

gateunit 2

input voltage

IGBT1

IGBT2

output

detection

bidirectional switch 1

current

detection

Figure 13: Signal structure of the matrix converter

The switching times of each BDS are calculated by the processor based on space vector theory and are given to the PWM-modulator. The modulator is realised with a FPGA ALTERA FLEX 8636 device and works with a switching frequency of 9.9 kHz. The commands for the BDS are given to the commutation logic, that decides the switching sequence for commutation. The commutation sequence chosen is voltage conditioned commutation as discussed previously. The BDS consists of two IGBTs 1200V/100A connected together by their emitters. The converter output power is about 25 kW at sinusoidal mode. The LC-filter used in the setup consists of three 30 µF MKP capacitors connected in triangle configuration and a line inductivity of 100 µH. The matrix converter is connected to an asynchronous machine 330 V/30 kW coupled to a DC-generator as a load.

Experimental results

Sinusoidal mode

Usually, the converter operates in a modulation mode that creates sinusoidal output voltages. At linear loads the input currents are sinusoidal, too. Figure 14 shows the input wave forms of the converter. The input power factor is nearly one.

A small phase shift is caused by the capacitors of the LC-filter and can be corrected by the converter with little loss in the voltage transfer ratio. The output frequency at the motor is about 50 Hz. The output currents are in fact a symmetrical three-phase system.

Input voltage (grey) and current without overmodulation

40

30

20

[A]

10

, I

U [V/10]

0

-10

-20

-30

-40

0

5

10

15

20

25

30

35

40

t [ms]

Output currents without overmodulation

30

20

10

[A]

0

I

-10

-20

-30

0

5

10

15

20

25

30

t [ms]

Figure 14: Input and output wave forms at harmonic mode

Overmodulation mode

Figure 12 shows the behaviour of the converter in overmodulation mode. The input current follows the input voltage but contains harmonics. The output frequency is about 90 Hz.



Input voltage (grey) and current with overmodulation

80

60

40

20

I[A]

,

U [V/10]

0

-20

-40

-60

-80

0

5

10

15

20

25

30

t [ms]

Output voltage with overmodulation

500

400

300

200

100

[V]

0

U

-100

-200

-300

-400

-500

0

5

10

15

20

25

30

t [ms]

Figure 15: Input and output wave forms at overmodulation mode

Compared with the simulation result, the output voltage and the input current contain an additional ripple. This ripple is caused by the LC-filter of the converter. If the converter works in overmodulation mode the filter design rules for sinusoidal mode which have been used in the experimental setup have to be changed. The used LC-filter has a resonant frequency of 1.6 kHz which is enough distance to the chosen switching frequency of the PWM-modulator which works under 9.9 kHz and the grid

frequency. As mentioned before the overmodulation mode causes additional harmonics to the grid. These harmonics stimulate the LC-filter to oscillations which can be seen in figure 15. These measurements show that overmodulation mode is not useful for common operation but available for a short time of operation if demanded.

Conclusion

In this paper an overview on circuit arrangements building up a bi-directional switch with IGBT devices is given. A summary of all possible commutation sequences is presented. An introduction to space vector modulation theory is done explaining the procedure to achieve overmodulation of the matrix converter increasing the voltage transfer ratio with disregard of harmonics. An experimental setup is presented on which measurements on vector modulation mode have been done. Simulation results and measurements on overmodulation mode are compared.

References

Späth H.; Söhner W.; Der selbstgeführte Direktumrichter als Stellglied für Drehstrommaschinen, Archiv für Elektrotechnik, Vol. 71, pp. 441-450, Springer 1988

Braun M.; Ein dreiphasiger Direktumrichter mit Pulsweitenmodulation zur getrennten Steuerung der Ausgangsspannung und der Eingangsblindleistung, Dissertation Technische Hochschule Darmstadt, 1983

Alesina A.; Venturini M; Analysis of optimum-amplitude nine-switch direct AC-AC converters, IEEE Transactions on Power Electronics, Vol 4, No 1., 1989, pp. 101-112

Wheeler P.W.; Grant D. A.; A low loss matrix converter for AC variable-speed drives; EPE Brighton 1993; pp. 27-32

Roy G., April G.-E.; Cycloconverter operation under a new scalar control algorhythm; IEEE Transactions on Power Electronics, Vol 6, No 1., 1991, pp. 100-107

Huber L., Borojevic D.; Space vector modulated three-phase to three-phase matrix converter with input power factor correction; IEEE Transactions on industry Applications, Vol. 31, No. 6, 1995; pp. 1234-1245

Christensson Å; Switch effective modulation strategy for matrix converters; EPE Trondheim 1997; pp. 4.193-4.198

Gyugyi L., Pelly B. R., Static power frequency changers, New York: Wiley; 1976

1A Matrix Converter With Space Vector Control Enabling Overmodulation

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