13  

CHAPTER 2

MATRIX CONVERTER FUNDAMENTALS

In industrial applications there is a strong demand for power quality and energy

efficient power converters with reduced number of switches. MC has recently

received considerable interests, because it possesses the necessary features to fulfil

these current trends. The most desirable features of MC in power converters are

a. Generation of output voltage with the desired amplitude and frequency

b. Sinusoidal input current

c. Sinusoidal output voltage

d. Improved power factor

e. Regeneration Capability

These features of MC replace Voltage Source Inverters in IM drive applications. MC

uses nine bi-directional controlled switches in a 3 x 3 matrix form to produce variable

output voltage with variable frequency. The main advantage of MC is that it does not

have any dc link and energy storing elements which reduce the performance of the

converter.

2.1 Circuit Topologies

Various MC topologies have been studied since 1970 [11]. In 1980, Venturini and

Alesina presented the power circuit of the MC as a matrix of bi-directional power

switches and they introduced the name “MC” [12-13]. The circuit topology of Direct

MC shown in Fig. 2.1 and Indirect MC shown in Fig.2.2 has been studied since 1993.

[14-19].

14  

Fig. 2.1: Direct MC

Fig. 2.2: Indirect MC

In the beginning, power transistors were used as bi-directional switches in MC

topology [20-24]. The performance of Direct MC and Indirect MC based on SVM in

an IM drive is also compared [16]. It is shown that the output voltage of the

15  converters does not follow the input. This is due to the need of different commutation

methods to provide safe operation and also the power losses caused by different main

circuits. Also, the effect of non-linearity is more visible in Indirect MC than in Direct

MC. Hence the efficiency of Indirect MC is low compared to Direct MC [16].

In 2001, a new circuit topology of MC called Sparse Matrix Converter (SMC) shown

in Fig.2.3 was invented. The reduced number of components, less complexity in

modulation algorithm and no multistep commutation procedure of SMC attracts its

application in AC drives in Industrial operations. SMC is constructed with 15

transistors, 18 diodes and 7 isolated Driver Potentials [25-27]. The SMC provides

identical function like Direct MC with reduced number of switches. The improved

DC link current commutation scheme reduces the control complexity and increases

the safety and reliability with limitations in functionality. The modified SVM called

Very Sparse Matrix Converter (VSMC) topology shown in Fig. 2.4 consists of 12

transistors, 30 diodes and 10 isolated driver potentials. The limitations in functionality

of SMC are avoided in VSMC and also the number of transistors is reduced in VSMC

compared to SMC whereas the conduction loss is increased due to increase in number

of conducting diodes. The Ultra Sparse Matrix Converter (USMC) shown in Fig. 2.5

was developed with 9 transistors, 18 diodes and 7 isolated driver potentials to reduce

the conduction loss. But the maximum displacement angle between input voltage and

input current is limited to ±30°. In the year 2009, the Z source MC was proposed. It

comprises voltage fed structure and current fed structure. It overcomes the voltage

transfer limitations of MC [28].

16  

Fig. 2.3: Bidirectional Sparse MC

Fig. 2.4: Very Sparse MC

17  

Fig. 2.5: Ultra Sparse MC

2.2 Matrix Converter Protection Techniques

In order to prevent the semiconductor switches from over voltages and over currents,

an effective protection technique is required. In 2002, a protection circuit consisting

of input and output diode bridges and an electrolytic capacitor circuit was proposed

[17] & [19]. The major disadvantage of this protection circuit is that it increases the

size of the converter. In 2004, a protection circuit using small capacitor called

dynamic clamp circuit was proposed [29]. This clamp circuit consists of an IGBT and

a resistor and does not need a drive and control circuit for IGBT. In 2005, a shutdown

strategy under fault conditions was proposed for MC [30]. This method provides a

freewheeling path for load currents also. Hence, the motor currents can be reduced to

zero immediately without any overshoot voltages. In this method, a small capacity

clamp circuit using small capacitor is used to protect the circuit from unexpected over

voltage under a switch open circuit condition.

18  2.3 Current Commutation Methods in MC

The MC uses bidirectional controlled switches in matrix form. Due to the absence of

freewheeling path, the commutation of switches is the major problem in MC. In order

to transfer current safely from one bidirectional switch to the other, timing and

synchronization of the switch command signals must be chosen carefully.

In order to achieve safe commutation of switches, minimization of commutation time

was proposed in 2004 [26]. Another commutation method for reducing the losses

caused by the reverse leakage current was proposed in 2005 [31]. In this method the

output current direction and voltage polarities of bidirectional switches were used to

perform a two-step commutation.

2.4 Modulation Methods

The objective of MC modulation strategy is to obtain the target sinusoidal output

voltages and sinusoidal input currents with controllable input power factor. The

conditions of the switches are such that output circuit should never be open circuited

and any two input phases should not be short circuited.

Alesina and Venturini [32-33] introduced the “low frequency matrix” concept. They

developed a mathematical analysis which describes the low frequency behaviour of

the converter. The modulation method introduced is called direct transfer function

approach, in which the output voltages are obtained by the multiplication of the

modulation matrix with input voltages [34]. In this method, the output phase voltages

are formed separately and are between negative and positive envelopes of the input

voltage as shown in Fig.2.6. The maximum voltage transfer ratio achieved by this

19  method is 0.5 [35]. This method becomes inefficient for practical applications due to

limitations in voltage transfer ratio. This method is further improved by adding the

third order harmonic of the input frequency and the third order harmonic of the output

reference frequency to the fundamental output phase voltage reference. When the

magnitudes of those third order components are chosen correctly, the full voltage

transfer ratio of 0.866 can be achieved as shown in Fig.2.7. In this method,

independent control of each output phase is achieved and hence all the switching

states of MC presented in Table 2.1 and Table 2.2 are possible. However, in this

method, the switching combinations producing high common mode voltages cannot

be avoided.

Fig.2.6: Waveform with 50% Voltage Transfer Ratio

20  

Fig.2.7 Third Order Harmonic Injection to Achieve Voltage

Transfer Ratio of 86.6% [36]

In 1983, Rodrigues [37] introduced a switching method using PWM technique in

which each output line switches between the most positive and most negative input

lines. This method is called “fictitious DC link” method. This concept is also known

as “indirect transfer function approach”. In 1985, the use of Space Vectors in the

analysis and control of MCs was introduced by Kastner and Rodriguez [30]. In 1987,

G. Roy presented a scalar control algorithm for MC. In this method voltage transfer

ratio of 0.866 was achieved as in Venturini method. The only difference between the

Venturini method and Scalar Control method is that in Scalar control method, the

voltage transfer ratio is fixed at its maximum value and its effect on output voltage is

negligible in high frequency applications. In 2009, Bachir and Bendiabedellah [39]

presented a detailed comparative study between Venturini and Roy modulation

21  methods. The study result shows the performance of IM for various speed and torque

references. During 1989 - 1995, the principles of Space Vector Modulation (SVM)

and their applications in MC were published by Huber [31-36]. Many researchers

focus on SVM method [46]. SVM method uses simpler algorithms compared to

Venturini method. It is used to calculate the duty cycles of the active voltage vector in

each switching cycle period. This method is capable of controlling the input current,

output voltage and input power factor. The SVM Algorithm derived from indirect

transfer approach is proposed to increase the rms output voltage [47].

Table2.1 Switching Configuration

Group  State Phase Switching function values 

A  B  C SAa SAb SAc SBa SBb SBc SCa  SCb  SCc

I 

1  A  B  c 1 0 0 0 1 0 0  0  12  A  C  b 1 0 0 0 0 1 0  1  03  B  A  c 0 1 0 1 0 0 0  0  14  B  C  a 0 1 0 0 0 1 1  0  05  C  A  b 0 0 1 1 0 0 0  1  06  C  B  a 0 0 1 0 1 0 1  0  0

IIA 

1  A  C  c 1 0 0 0 0 1 0  0  12  B  C  c 0 1 0 0 0 1 0  0  13  B  A  a 0 1 0 1 0 0 1  0  04  C  A  a 0 0 1 1 0 0 1  0  05  C  B  b 0 0 1 0 1 0 0  1  06  A  B  b 1 0 0 0 1 0 0  1  0

IIB 

1  C  A  c 0 0 1 1 0 0 0  0  12  C  B  c 0 0 1 0 1 0 0  0  13  A  B  a 1 0 0 0 1 0 1  0  04  A  C  a 1 0 0 0 0 1 1  0  05  B  C  b 0 1 0 0 0 1 0  1  06  B  A  b 0 1 0 1 0 0 0  1  0

IIC 

1  C  C  a 0 0 1 0 0 1 1  0  02  C  C  b 0 0 1 0 0 1 0  1  03  A  A  b 1 0 0 1 0 0 0  1  04  A  A  c 1 0 0 1 0 0 0  0  15  B  B  c 0 1 0 0 1 0 0  0  16  B  B  a 0 1 0 0 1 0 1  0  0

III 1  A  A  a 1 0 0 1 0 0 1  0  02  B  B  b 0 1 0 0 1 0 0  1  03  C  C  c 0 0 1 0 0 1 0  0  1

22  

Table2.2 MC Switching State Selection 

S.No. Switching Configuration

Converter State

Output voltage

(v0)

Output Voltage Vector

Angle (α0)

Input Current

(ii)

Input Current Vector

Angle (βi)

1 +1 Sabb vab 0 2

√3 -

2 -1 Sbaa - vab 0 2

√3 -

3 +2 Sbcc vbc 0 2

√3

2

4 -2 Scbb - vbc 0 2

√3

2

5 +3 Scaa vca 0 2

√3 7

6

6 -3 Sacc - vca 0 2

√3 7

6

7 +4 Sbab vab 23

2√3

-

8 -4 Saba - vab 23

2√3

-

9 +5 Scbc vbc 23 2

√3

2

10 -5 Sbcb - vbc 23

2√3

2

11 +6 Saca vca 23

2√3

76

12 -6 Scac - vca 23

2√3

76

13 +7 Sbba vab 43 2

√3 -

14 -7 Saab - vab 43

2√3

-

15 +8 Sccb vbc 43 2

√3

2

16 -8 Sbbc - vbc 43 2

√3

2

17 +9 Saac vca 43 2

√3 7

6

18 -9 Scca - vca 43 2

√3 7

6

19 0a Saaa 0 - 0 0

20 0b Sbbb 0 - 0 0

21 0c Sccc 0 - 0 0

22 FRa Sabc - - - -

23 FRb Sacb - - - -

24 FRc Sbca - - - -

25 BRa Sbac - - - -

26 BRb Scab - - - -

27 BRc Scba - - - -

23  

2.4.1 Venturini Algorithm

Various modulation techniques can be applied to MC to achieve sinusoidal input

current and output voltage with minimised harmonic distortion and power loss. Also

the modulation technique should follow two important rules. The MCs are fed by

Voltage sources. Hence the input phases should not be short circuited. The load is of

inductive nature so the output phases should never be open-circuited. The modulation

control strategy satisfying the above conditions was first proposed by Alesina and

Venturini and is called Venturini Algorithm.

In MC each switch is used to connect or disconnect any phase of the input to any

phase of the load. This can be achieved only by selecting proper switching

configuration. In order to avoid the interruption of load current suddenly, at least one

switch in each column must be closed. Each switch can be defined with a

commutation function described as below:

 

0 1                                                      (2.1) 

 where , , is input phase and k , , is output phase. The two

conditions in equation (2.1) can be expressed as

 1                 (2.2) 

 The equation (2.2) shows that a 3 x 3 MC has got 27 possible switching states. In

order to select appropriate combinations of open and closed switches and to generate

the desired output voltages, modulation strategy for the MC must be developed for

which it is necessary to develop the mathematical model. It can be derived from

equation (2.1) as follows:

The load and source voltages with reference to supply neutral can be expressed as

24  

 and                (2.3) 

The load and source currents can expressed as

  and                (2.4) 

          (2.5) 

          (2.6) 

                   (2.7) 

where K represents input phases A,B,C and k represents output phases a, b, c and Ts

is the sequence time.

The modulation strategies are defined by using these continuous time functions.

         (2.8) 

         (2.9) 

Voltages , & and currents , & are average values of output voltage and

outupt current respectively.

The MC generates sinusoidal output voltage and sinusoidal input current to get

adjustable input power factor. Hence, the output voltage can be expressed as

cos cos

cos              (2.10) 

25  

cos cos

cos             (2.11) 

where , and are the average input voltage, input and output frequencies

respectively.

cos cos

cos             (2.12) 

 

cos cos

cos            (2.13) 

where and are the input and output displacement angles respectively. q is

defined as the ratio of output fundamental to input fundamental voltage value called

as voltage transfer ratio.

The modulation matrix M(t) can be expressed as,

            (2.14) 

where

1 2   

1 2 )) 

1 2   

1 2   

1 2   

1 2   

26  

1 2   

1 2   

1 2   

where is the modulation frequency.

Hence,        

               (2.15) 

                (2.16) 

The MC equation can be written as

∑ , , ∑ ,, , ∑ , , 1    (2.17) 

Venturini provided two solutions for the problem one with and

another with and are expressed as and respectively

as given below

1 2 cos 1 2 cos 1 2 cos

1 2 cos 1 2 cos 1 2 cos

1 2 cos 1 2 cos 1 2 cos

             

(2.18) 

 

 

1 2 cos 1 2 cos 1 2 cos

1 2 cos 1 2 cos 1 2 cos

1 2 cos 1 2 cos 1 2 cos

  

                    (2.19) 

27  In M1(t) the input phase displacement is equal to output phase displacement

. In M2 (t) the output phase displacement is reversal of input phase

displacement . The input displacement control can be obtained by

combining M1(t) and M2(t),

                        (2.20)     

where 1. If is equal to , then the input displacement factor of the

converter is equal to unity. Other choices of and will provide leading and

lagging displacement factor at the input and lagging and leading output displacement

factor at the output.

If = , then the modulation function can be expressed as

1               (2.21) 

If the target output voltage fits into the input voltage envelope for all operating

frequencies, then the average output voltage will be equal to the target output voltage

at each sampling sequence. Hence, the maximum value of q that can be obtained by

this method is only 50%. This makes the converter impractical because of the

maximum of 50% voltage transfer ratio.

The above method was improved by Venturini [48 - 49] by adding third harmonics of

input and output frequencies with the target output voltage matrix . This method

is called common mode addition technique. The maximum voltage transfer ratio

achieved by this method is 86.6%.  

cos 16

cos 3 13.46

cos 3

cos 23

16

cos 3 13.46

cos 3

cos 43

16

cos 3 13.46

cos 3

    (2.22) 

28  For unity displacement factor, the modulation factor becomes,

1.

sin sin 3       (2.23) 

where 0, , and K represents the input phases A, B, C.

If the displacement factor is other than unity, then the output voltage limit will be

reduced from 0.86Vim to smaller value.

2.4.2 Space Vector Modulation

In Space Vector approach, only one switch in each output phase of MC must be

conducting at a time. Hence, there are 27 possible switching combinations which can

be split into three groups as shown in Table 2.3. Out of the three groups only 21 space

vector groups are useful. Only 18 non-zero space vectors in group-I and 3 zero space

vectors in group-II are used. Fig.2.8a shows the output voltage vector and Fig.2.8b

shows the input current vectors. KV denotes the output voltage vector sector and KI

denotes the input current vector sector. The output phases are indicated as a, b and c

and input phases are indicated as A, B and C.

Table2.3 Selection of switching vector using SVM

29  

 

Fig.2.8a: Output Voltage Vector Fig.2.8b: Input Current Vector

 

The output voltage vector is given by

 

where 3imqV is the length of the output vector and oω is the frequency of the rotating

vector. At each sampling period, the output vector )(tvor can be generated from two

adjacent vectors whose direction matches the direction of the output vector. The

higher magnitude vectors, which maintain the input current vector )(tiir

in phase with

the input voltage vector )(tvor

, are considered to generate the output vector )(tvor . The

input current vector can be given as,

where  32πj

ea =             (2.25)        

The possible output voltages for the converter can be expressed as

)(32)( 2

ocoboao vaavvtv ++=r

                     (2.26) 

The required duty cycles for the switching configurations can be calculated as

 

KV = 1

KV = 2 

KV = 3 

KV = 4 

KV = 5 

KV = 6

±1, ±2, ±3

±4, ±5, ±6 

±7, ±8, ±9 

α0   

±1, ±4, ±7±3, ±6, ±9

βi 

±2, ±5, ±8 

KI = 1 

KI = 2 KI = 3 

KI = 4

KI = 5  KI = 6 

)cos(3

)( tqVtv oim

o ω=r

)(32)( 2

iCiBiAi iaaiiti ++=r

30  

                                      (2.27) 

 

i

i

II qφ

πβπασ

cos

)3

cos()3

cos(

32 0 −+

=

rr

                     (2.28) 

i

i

III qφ

πβπασ

cos

)3

cos()3

cos(

32 0 +−

=

rr

                     (2.29) 

i

i

IV qφ

πβπασ

cos

)3

cos()3

cos(

32 0 −−

=

rr

                     (2.30) 

where oαr , iβ

r

and iφ are the angles of output voltage vector, input current vector and

input phase displacement respectively.

The total modulation duty cycle for the complete modulation period is given as 

)(1 10 IVIIIII σσσσσ +++−=                       (2.31) 

For unity and 0.866 input power factor operation,

1≤+++ IVIIIIII σσσσ                        (2.32) 

 

2.5 Simulation Model

Implementation of MC is done using Matlab / Simulink tools. The Venturini

Algorithm with third harmonic injection and SVM techniques are used to provide the

switching pulses for MC. The MC consists of 9 IGBT switches connected between

each input phase and each output phase as shown in Fig.2.9.

i

i

I qφ

πβπασ

cos

)3

cos()3

cos(

32 0 ++

=

rr

31  

The Fig.2.10 shows the Matrix Venturini Modulation Unit with Input Filter.

Fig.

2.9:

IGB

T S

witc

hes A

rran

gem

ent

32  

Fig.

2.10

: Ven

turi

ni M

odul

atio

n M

etho

d

33  The timing pulses generated for phase A using Venturini Modulation are shown in

Fig.2.11. The pulses for Phase B and Phase C can be obtained similarly. Fig.2.12

shows the input current and its harmonic spectrum. Fig.2.13 shows the output voltage

and the harmonic analysis of the MC. Fig.2.14 shows the load voltage and load

current waveforms.

 

Fig.2.11: Timing Sequence for 3 Phases

 

 

Fig.2.12a: Input Current Waveform

34  

Fig.2.12b: Harmonic Analysis of Input Current

 

 

 

 

Fig.2.13a: Input Voltage

 

 

 

 

35  

 

Fig.2.13b: Harmonic Analysis of Input Voltage

Fig.2.14a: Load Voltage and Load Current

36  

Fig.2.14b: Harmonic Analysis of Output Current

The space vector algorithm is also tested with MC. The input voltage, current and

output voltages are analysed for the same load used in Venturini Algorithm method.

Fig.2.15 shows the SVM unit.

Fig.2.15: Space Vector Modulation Unit

37  The timing sequence of SVM method for phase A is shown in Fig.2.16. The input

current and input voltage and their harmonic spectrum are shown in Fig.2.17 and

Fig.2.18 respectively. Fig.2.19 shows the load waveforms and harmonic spectrum of

load current.

Fig.2.16: Timing Signals generated by Space Vector Modulation Unit

Fig.2.17: Input Voltage and Input Current using SVM method

38  

Fig.2.18a: Harmonic Spectrum of Input Voltage

 

 

 

 

Fig.2.18b: Harmonic Spectrum of Input Current without Filter

39  

Fig.2.18c: Harmonic Spectrum of Input Current with Input Filter

 

 

 

 

Fig.2.19a: Load Voltage and Load Current

40  

 

Fig.2.19b: Harmonic Spectrum of Output Current

It can be seen from the results that the harmonics present in the output and input

currents are less in Venturini Algorithm compared to SVM Algorithm because in

Venturini Algorithm the output voltage and current are calculated independently for

each phase whereas in SVM method the harmonics present in the switching signals

affect the output signals.

2.6 Conclusion

The MC is still not popular in industrial applications because of some practical issues.

The MC is expected to be the “All Silicon Converter”, because it does not need large

reactive elements to store energy. However a study shows that a MC of 4 kW needed

a larger volume of reactive components than a comparable DC link converter [50].

This shows that the MC is not an “All Silicon Converter” and that passive elements in

the form of input filter are needed. More work must be done in order to optimize the

size of the input filter. Another important drawback that is present in MC is the lack

of suitably packaged bi-directional switches. This limitation has recently been

41  overcome with the introduction of power modules which include the complete power

circuit of MC. The most important practical implementation problem in the MC

circuit is the commutation problem between two controlled bi-directional switches.

This has been solved with complicated multistep commutation strategies. The MC is

still facing strong competition from Voltage Source Inverter in terms of cost, size and

reliability. The major problem of MC for use with standard IMs in speed control

applications is the limitation in maximum voltage transfer ratio between the input and

output. In order to overcome this difficulty suitable modulation techniques have to be

derived.

The Venturini, SVM and other methods of modulation based switching techniques

produce output with high level of harmonics. This limits the use of MC in variable

speed operation especially in the application of IM drive circuits. To avoid this

difficulty, more research work has to be focussed on improving the modulation

technique to use MC in IM drive circuits replacing VSCs. The present work is

focussed on implementing the soft computing techniques in modulation strategies to

achieve speed control of IM using MC.