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International Journal of Electronics

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Matrix converter controlled with the

direct transfer function approach:
analysis, modelling and simulation
a a b c c
J. Rodríguez , E. Silva , F. Blaabjerg , P. Wheeler , J. Clare
a
& J. Pontt
a
Department of Electronic Engineering, Universidad Técnica
Federico Santa María, Avda Blaceres 401, Valparaíso, Chile
b
Institute of Energy Technology, Aalborg University,
Pontoppidanstraede 101, DK-9220, Aalborg East, Denmark
c
School of Electrical and Electronic Engineering, University of
Nottingham, University Park, Nottingham, NG7 2RD, England
d
Department of Electronic Engineering, Universidad Técnica
Federico Santa María, Avda Blaceres 401, Valparaíso, Chile E-mail:
Version of record first published: 21 Aug 2006.

To cite this article: J. Rodríguez , E. Silva , F. Blaabjerg , P. Wheeler , J. Clare & J. Pontt (2005):
Matrix converter controlled with the direct transfer function approach: analysis, modelling and
simulation, International Journal of Electronics, 92:2, 63-85

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 International Journal of Electronics
Vol. 92, No. 2, February 2005, 63–85

Matrix converter controlled with the direct transfer function

approach: analysis, modelling and simulation

J. RODRÍGUEZy, E. SILVA*y, F. BLAABJERGz, P. WHEELERx,

J. CLAREx and J. PONTTy
yDepartment of Electronic Engineering, Universidad Técnica Federico Santa Marı́a,
Avda Blaceres 401, Valparaı́so, Chile
zInstitute of Energy Technology, Aalborg University, Pontoppidanstraede 101,
DK-9220, Aalborg East, Denmark
Downloaded by [University of New Hampshire] at 06:00 25 February 2013

xSchool of Electrical and Electronic Engineering, University of Nottingham, University Park,

Nottingham, NG7 2RD, England

(Received 17 April 2004; in final form 9 December 2004)

Power electronics is an emerging technology. New power circuits are invented

and have to be introduced into the power electronics curriculum. One of the
interesting new circuits is the matrix converter (MC), and this paper analyses
its working principles. A simple model is proposed to represent the power
circuit, including the input filter. The power semiconductors are modelled
as ideal bidirectional switches and the MC is controlled using a direct transfer
function approach. The modulation strategy of the converter is explained in a
complete and clear form. The commutation problem of two switches and the
generation of overvoltages are clarified. The paper also includes a soft-switching
commutation method that allows for a safe commutation of the switches.
Finally a complete simulation scheme, using MatlabÕ –SimulinkÕ , is discussed.
Keywords: Matrix converter; Modelling; Simulation; Teaching power conversion;
Circuit analysis

1. Introduction

The transformation and control of energy is one of the most important processes
in electrical engineering. In recent years, this work has been done with the use of
power semiconductors and energy storage elements such as capacitors and induc-
tances. Several converter families have been developed: rectifiers, inverters, choppers,
cycloconverters, etc. Each of these families has its own advantages and limitations.
The main advantage of all static converters over other energy processors is the high
efficiency that can be achieved. One of the most interesting families of converters is
that of the so-called matrix converters (MCs).
The matrix converter is an array of bidirectional switches functioning as the main
power elements. It interconnects directly the three-phase power supply to a three-
phase load, without using any DC link or large energy storage elements, and there-
fore it is called the all-silicon solution.
The most important characteristics of the matrix converter are: (1) simple and
compact power circuit; (2) generation of load voltage with arbitrary amplitude and

*Corresponding author. Email: eduardo.silva@elo.utfsm.cl

International Journal of Electronics

ISSN 0020–7217 print/ISSN 1362–3060 online # 2005 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/00207210512331337686
 64 J. Rodrı´guez et al.

frequency; (3) sinusoidal input and output currents; (4) operation with unity power
factor; and (5) regeneration capability. These highly attractive characteristics are the
reason for the present tremendous interest in this topology.
The development of this converter starts with the early work of Venturini
and Alesina (Venturini 1980, Venturini and Alesina 1980). They presented the
power circuit of the converter as a matrix of bidirectional power switches and
they introduced the name ‘matrix converter’. Another major contribution of these
authors is the development of a rigorous mathematical analysis to describe the low-
frequency behaviour of the converter. In their modulation method, also known
as the direct transfer function approach, the output voltages are obtained by
the multiplication of the modulation matrix with the input voltages. A conceptually
different control technique using the ‘fictitious DC link’ idea, was introduced
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later (Rodrı́guez 1983, Huber et al. 1989).

The simultaneous commutation of the controlled bidirectional switches used
in matrix converters is very difficult to achieve without generating overcurrent or
overvoltage spikes that can destroy the power semiconductors. This fact affected
negatively the interest in matrix converters for several years until advanced multi-
step commutation strategies appeared, that allowed safe operation of the switches
(Empringham et al. 1998, Youm and Kuon 1999).
Another important limitation of matrix converters was the large number of power
semiconductors required to implement the bidirectional switches. This problem has
now been overcome with the introduction of power modules in the market with the
complete power circuit of the converter in a single chip.
The solutions of these important drawbacks are the reason for a new interest in
this topology. In effect, an important journal dedicated in year 2002 a special section
to this technology as a recognition of the increasing interest in this area (Rodrı́guez
2002, Wheeler et al. 2002b).
Today, research activity is mainly dedicated to studying advanced technological
and applications issues such as reliable implementation of commutation strategies
(Mahlein et al. 2002b, Wheeler et al. 2002a), overvoltage protection (Nielsen et al.
1997, Mahlein et al. 2002a), packaging (Klumpner et al. 2002b), operation under
abnormal conditions (Blaabjerg et al. 2002, Klumpner and Blaabjerg 2002,
Wiechmann et al. 2002) and filter design (Bernet et al. 2002, Klumpner et al. 2002a).
Several educators in the area of power electronics believe that matrix converters
are a very attractive topic for teaching, from a conceptual point of view, at graduate
level. In addition, MC is becoming a mature technology and, for this reason, should
be included in Electrical Engineering courses. However, excellent and modern books
do not cover this matter with enough depth (Mohan et al. 1995, Trzynadlowski
1998), demanding a higher effort from the student to learn the concepts. There
is therefore a need to develop new tools for teaching the theory of this power
converter. In addition, the operation of the MC entails several practical problems
such as complicated commutation of the switches and possible generation of over-
voltages and overcurrents (Venturini 1980, Mohan et al. 1995, Wheeler et al. 2002b).
These issues make the study of the MC using an experimental approach very
difficult. For this reason, a simulation approach appears to be a very good solution
for the beginner student.
These authors have not found any publication concerning the simulation of MCs.
Nor do popular circuit-oriented simulation software packages such as PSPICE and
PSIM have the simulation of an MC as a standard block in their libraries. It is
 Control of matrix converter 65

possible to simulate the operation of an MC using these circuit analysers, but the
simulation is usually limited to simple control and modulation strategies. On the other
hand, it is possible to use a general equation solver such as MatlabÕ –SimulinkÕ
to study the behaviour of this converter. This approach is more limited in the simula-
tion of the semiconductors, but it permits the consideration of more complicated
control strategies.
The primary objective of this work is to serve as a first approach to matrix
converters, covering both analysis and simulation of them using the direct transfer
function approach.
The paper is organized as follows: Section 2 presents the basic topology and the
working principle of the MC; Section 3 presents the modulation method of the
converter, based on the direct transfer function approach; Section 4 presents some
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application issues such as commutation problems, soft commutation strategy and

overvoltage protection; Section 5 presents a detailed simulation scheme using
MatlabÕ –SimulinkÕ ; and Section 6 discusses some results. Finally, Section 7
provides some concluding remarks.

2. Power circuit and working principle of the converter

This section describes a pedagogical way to explain the MC’s working principle.
In general, the matrix converter is a single-stage converter with m
n bidirec-
tional power switches, designed to connect an m-phase voltage source to an n-phase
load. The MC of 3
3 switches, shown in figure 1, is the most important converter
from a practical point of view, because it connects a three-phase source to a three-
phase load, typically a motor. The high-frequency conversion process can also be
easily used in a AC–DC converter, which is today known as a pulse width modulated
current source rectifier.

v sA Matrix Converter
Lf Rf iA

i sA
Cf S Aa S Ab S Ac
v sB vA
Lf Rf iB
switch array

N i sB
S Ba S Bb S Bc
v sC v
B
Lf Rf iC

i sC
S Ca S Cb S Cc
Power vC
grid vaN vbN vcN
input filter ia ib ic

R
van

n Load

Figure 1. Basic power circuit of the MC with input filter.

 66 J. Rodrı´guez et al.

In the basic topology of the MC shown in figure 1, vsi , i ¼ fA, B, Cg, are the
source voltages, isi , i ¼ fA, B, Cg, are the source currents, vjn , j ¼ fa, b, cg, are the
load voltages with respect to the neutral point of the load n, and ij , j ¼ fa, b, cg,
are the load currents. Additionally, other auxiliary variables have been defined to
be used as a basis of the modulation and control strategies: vi , i ¼ fA, B, Cg, are the
MC input voltages, ii , i ¼ fA, B, Cg, are the MC input currents, and vjN , j ¼ fa, b, cg,
are the load voltages with respect to the neutral point N of the grid.
The filter (Cf , Lf , Rf ) located at the input of the converter has two main
purposes:
(1) It filters the high-frequency components of the matrix converter input
currents (iA , iB , iC ), generating almost sinusoidal source currents (isA , isB , isC ).
(2) It avoids the generation of overvoltage produced by the fast commutation of
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currents iA , iB , iC , due to the presence of the short-circuit reactance of any real

power supply.
It should be noted that Rf is the internal resistance of inductor Lf, not an additional
resistor.
Each switch sij , i ¼ fA, B, Cg, j ¼ fa, b, cg, can connect or disconnect phase i of
the input stage to phase j of the load and, with a proper combination of the
conduction states of these switches, arbitrary output voltages vjN can be synthesized.
Each switch is characterized by a switching function, defined as follows:
(
0 if switch sij is open
Sij ðtÞ ¼ ð1Þ
1 if switch sij is closed

Due to the presence of capacitors at the input of the MC, only one switch of each
column can be closed. Furthermore, the inductive nature of the load makes it
impossible to interrupt the load current suddenly and, therefore, at least one switch
of each column must be closed. Consequently, it is necessary that one and only one
switch per column is closed at each instant. This condition can be stated in a more
compact form as follows:
X
Sij ðtÞ ¼ 1; j ¼ fa, b, cg, 8t ð2Þ
i¼A, B, C

Equation (2) imposes several restrictions on the way in which the switches are turned
on or off, as will be discussed in Section 4.
In order to develop a modulation strategy for the MC, it is necessary to develop
a mathematical model, which can be derived directly from figure 1, as follows:
 Applying Kirchhoff ’s voltage law to the switch array, it can be easily found
that
2 3 2 32 3
vaN ðtÞ SAa ðtÞ SBa ðtÞ SCa ðtÞ vA ðtÞ
6 7 6 76 7
4 vbN ðtÞ 5 ¼ 4 SAb ðtÞ SBb ðtÞ SCb ðtÞ 54 vB ðtÞ 5 ð3Þ
vcN ðtÞ SAc ðtÞ SBc ðtÞ SCc ðtÞ vC ðtÞ

It is worth noting that (3) is only valid if (2) holds. Otherwise, these equations
are inconsistent with the physical element distribution of figure 1.
 Control of matrix converter 67

 Applying Kirchhoff ’s current law to the switch array, it can be found that
2 3 2 32 3
iA ðtÞ SAa ðtÞ SAb ðtÞ SAc ðtÞ ia ðtÞ
6 7 6 76 7
4 iB ðtÞ 5 ¼ 4 SBa ðtÞ SBb ðtÞ SBc ðtÞ 54 ib ðtÞ 5 ð4Þ
iC ðtÞ SCa ðtÞ SCb ðtÞ SCc ðtÞ ic ðtÞ

Equations (3) and (4) are the basis of all modulation methods which consist in
selecting appropriate combinations of open and closed switches to generate the
desired output voltages. It is important to note that the output voltages (vjN , j ¼
fa, b, cg) are synthesized using the three input voltages (vi , i ¼ fA, B, Cg) and that
the input currents (ii , i ¼ fA, B, Cg) are synthesized using the three output currents
(ij , j ¼ fa, b, cg), which are sinusoidal if the load has a low-pass frequency response.
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The filter can be modelled with the aid of the following equations:



d dvi ðtÞ dvi ðtÞ
vsi ðtÞ ¼ vi ðtÞ þ Lf i ðtÞ þ Cf þ Rf ii ðtÞ þ Cf
dt i dt dt
ð5Þ
vsi ðsÞ  ðLf s þ Rf Þii ðsÞ
, vi ðsÞ ¼
Lf Cf s2 þ Rf Cf s þ 1
where x(s) denotes the Laplace transform of x(t).
In addition,
isi ðsÞ ¼ ii ðsÞ þ Cf svi ðsÞ ð6Þ
Substituting (6) in (5) we have
1 sCf
isi ðsÞ ¼ ii ðsÞ þ vsi ð7Þ
Lf Cf s2 þ Rf Cf s þ 1 Lf Cf s2 þ Rf Cf s þ 1
From (5) it can be seen that if the filter parameters are properly selected, the
switch array input voltages will be similar to those at the grid. This is very important,
because the modulation principles work under the assumption that vsi ¼ vi .
Equation (7) states that the input currents isi are simply a filtered version of the
switch array input currents ii, plus a filtered version of the input voltages vsi . The
nature of the former filter is low pass, thus it is easy to eliminate high-frequency
harmonics of ii. The latter filter is a pass-band filter and, because of the low-
frequency nature of vsi , it can be assumed that the effect of this voltage in isi is
negligible.

3. Modulation of the matrix converter

In this section, the basic Venturini modulation strategy for the MC will be presented.
Modulation is the procedure used to generate the appropriate firing pulses to each of
the nine bidirectional switches (sij ) in order to generate the desired output voltage. In
this case, the primary objective of the modulation is to generate variable-frequency
and variable-amplitude sinusoidal output voltages (vjN ) from the fixed-frequency
and fixed-amplitude input voltages (vi). The easiest way of doing this is to consider
time windows in which the instantaneous values of the desired output voltages are
 68 J. Rodrı´guez et al.

sampled and the instantaneous input voltages are used to synthesize a signal whose
low-frequency component is the desired output voltage.
If tij is defined as the time during which switch sij is on and T as the sampling
interval (width of the time window), the synthesis principle described above can be
expressed as

tAj vA ðtÞ þ tBj vB ðtÞ þ tCj vC ðtÞ

vjN ðtÞ ¼ ; j ¼ fa, b, cg ð8Þ
T

where vjN ðtÞ is the low-frequency component (mean value calculated over one
sampling interval) of the jth output phase and changes in each sampling interval.
With this strategy, a high-frequency switched output voltage is generated, but the
fundamental component of the voltage has the desired waveform.
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Obviously, T ¼ tAj þ tBj þ tCj 8j and therefore the following duty cycles can be
defined:
tAj tBj tCj
mAj ðtÞ ¼ , mBj ðtÞ ¼ , mCj ðtÞ ¼ ð9Þ
T T T

Extending (8) to each output phase, and using (9), the following equation can
be derived:
2 3 2 32 3
vaN ðtÞ mAa ðtÞ mBa ðtÞ mCa ðtÞ vA ðtÞ
6 7 6 76 7
6 vbN ðtÞ 7 ¼ 6 mAb ðtÞ mBb ðtÞ mCb ðtÞ 76 vB ðtÞ 7
4 5 4 54 5
vcN ðtÞ mAc ðtÞ mBc ðtÞ mCc ðtÞ vC ðtÞ
, vo ðtÞ ¼ MðtÞ  vi ðtÞ ð10Þ

where vo ðtÞ is the low frequency output voltage vector, vi ðtÞ is the instantaneous input
voltage vector and MðtÞ is the low-frequency transfer matrix of the MC. Using the
fact that the matrix in (4) is the transpose of the matrix in (3), and following an
analogous procedure for the currents, it can be shown that

ii ðtÞ ¼ MT ðtÞ  i o ðtÞ ð11Þ

where ii ðtÞ ¼ ½iA ðtÞ iB ðtÞ iC ðtÞT is the low-frequency component input current
vector, i o ðtÞ ¼ ½ia ðtÞ ib ðtÞ ic ðtÞT is the instantaneous output current vector and
MT ðtÞ is the transpose of MðtÞ.
Equations (10) and (11) are the basis of the Venturini modulation method
(Venturini 1980, Venturini and Alesina 1980) and allow us to conclude that the low-
frequency components of the output voltages are synthesized with the instantaneous
values of the input voltages and that the low-frequency components of the input
currents are synthesized with the instantaneous values of the output currents.
Suppose that the input voltages are given by
2 3 2 3
vA ðtÞ Vi cosð!i tÞ
6 7 6 7
vi ðtÞ ¼ 4 vB ðtÞ 5 ¼ 4 Vi cosð!i t þ 2
=3Þ 5 ð12Þ
vC ðtÞ Vi cosð!i t þ 4
=3Þ
 Control of matrix converter 69

and that due to the low-pass characteristic of the load the output currents are
sinusoidal and can be expressed as
2 3 2 3
ia ðtÞ Io cosð!0 t þ Þ
6 7 6 7
i o ðtÞ ¼ 4 ib ðtÞ 5 ¼ 4 Io cosð!0 t þ 2
=3 þ Þ 5 ð13Þ
ic ðtÞ Io cosð!0 t þ 4
=3 þ Þ

Furthermore, suppose that the desired input current vector is given by

2 3 2 3
iA ðtÞ Ii cosð!i tÞ
6 7 6 7
ii ðtÞ ¼ 4 iB ðtÞ 5 ¼ 4 Ii cosð!i t þ 2
=3Þ 5 ð14Þ
iC ðtÞ Ii cosð!i t þ 4
=3Þ
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that the desired output voltage can be expressed as

2 3 2 3
vaN ðtÞ qVi cosð!0 tÞ
6 7 6 7
vo ðtÞ ¼ 4 vbN ðtÞ 5 ¼ 4 qVi cosð!0 t þ 2
=3Þ 5 ð15Þ
vcN ðtÞ qVi cosð!0 t þ 4
=3Þ

and that the following active power balance equation must be satisfied with
3qVi Io cosðÞ 3Vi Ii
Po ¼ ¼ ¼ Pi ð16Þ
2 2
where Po and Pi are the output and input active power, respectively, and q is the
voltage gain of the MC.
With the previous definitions, the modulation problem is reduced to that of
finding a low-frequency transfer matrix MðtÞ such that (11) and (10) are satisfied,
considering the restrictions in (12)–(16).
The explicit form of matrix MðtÞ can be obtained from Venturini and Alesina
(1980) and Wheeler et al. (2002b) and can be reduced to the following expression:
 
1 vi ðtÞvjN ðtÞ
mij ðtÞ ¼ 1 þ 2 ð17Þ
3 Vi2

where i ¼ fA, B, Cg and j ¼ fa, b, cg.

It is worth noting that the derivation of MðtÞ supposes no sampling of the desired
output voltages (reference), because it is a continuous time solution. In order to
implement a digital simulation of the MC or to develop an experimental setup, it
is necessary to consider a sampled version of (17):
 
1 vi ðkT ÞvjN ðkT Þ
mij ðkT Þ ¼ 1 þ 2 ð18Þ
3 Vi2

where i ¼ fA, B, Cg, j ¼ fa, b, cg, k 2 Z and T is the sampling interval. If T is small
enough, the differences between (17) and (18) will be negligible.
Following the previous discussion, the following MC control procedure can be
proposed:
(1) A sample of the input voltages vi and the desired output voltages vjREF ¼ vjN ,
j ¼ fa, b, cg, must be obtained.
(2) With the aid of (18), matrix MðtÞ must be calculated.
 70 J. Rodrı´guez et al.

1.5
1

Aj
0.5
S
t
Aj
0
−0.5
0 0.5 1 1.5 2
1.5
1
Bj

0.5
S

t
Bj
0
−0.5
0 0.5 1 1.5 2
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1.5
1
Cj

0.5
S

t
0 Cj

−0.5
0 0.5 1 1.5 2
time [ms]
Figure 2. Switching functions of the jth output phase.

(3) Using (9), conducting times tij should be generated.

(4) Nine pulses of duration tij must be generated, according to the pattern shown
in figure 2 for the jth output phase. Note that this corresponds to the genera-
tion of the switching functions Sij ðtÞ (see (1)).
(5) The switching functions Sij ðtÞ must be used to turn on or off the bidirectional
switches of the MC in an appropriate way.
With the previously discussed strategy, the output voltages of figure 3 are
generated. Note that the figure shows only one output voltage with its reference;
it can be appreciated that the output voltage has an important harmonic content due
to the switching synthesis principle. Figure 4 shows a detailed version of figure 3,
which clearly demonstrates that the output voltage is synthesized using a temporal
average of the input voltages: in one switching interval, denoted T in the figure, the
output voltage switches among the three input voltages and the time that each of
these voltages is applied to the output determines its contribution to the output
voltage low-frequency component.
Note that, because of the averaging working principle, the output voltage low-
frequency component cannot exceed the maximum available amplitudes for all
instants. The reference can attain its maximum at an arbitrary time, say, for example,
t ¼ 1:7 ms (see figure 4) and therefore the worst-case maximum available amplitudes
are equal to 0.5Vi and, therefore, the voltage gain of the MC is restricted to be less
than 0.5. It must be clarified, however, that this limit is small since the modulation
under consideration uses the phase-to-neutral voltages to synthesize the output
voltages, i.e. this is a limitation arising from the modulation used, not from the
MC. It is possible to use the line-to-line voltages or space vector modulation to
increase the maximum voltage gain to q ¼ 0.866 (Huber et al. 1989, Huber and
Borojevic 1995, Wheeler et al. 2002b).
 Control of matrix converter 71

vaN ref
1
vA
0.8 vB
0.6 vC
vaN
0.4
amplitude [p.u.]

0.2
0
−0.2
−0.4
−0.6
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−0.8
−1

0 0.005 0.01 0.015 0.02

time [s]

Figure 3. Typical output voltage of an MC (see text for nomenclature and details).

1
v
C v
0.8 aN v
A
0.6
0.4
amplitude [p.u.]

0.2
0
v ref
aN
−0.2
−0.4
−0.6
−0.8 v
B
−1
T
1 1.5 2 2.5 3
time [s] −3
x 10
Figure 4. Working principle of an MC (detail of figure 3).

4. Application issues

Before we describe the simulation strategy it is important to take note of two prob-
lems that arise when implementing an MC in practice. These are the commutation
problem and the overvoltage problem.
 72 J. Rodrı´guez et al.

(a)

TA
D1
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V1 V2
D2
TB
(b)

Figure 5. (a) Diode bridge bidirectional switch and (b) common emitter back-to-back
bidirectional switch.

4.1 The commutation problem

In order to approximate the behaviour of ideal bidirectional switches, many
technically feasible solutions have been proposed, as shown in figure 5. We will
concentrate on the most popular one, and refer the interested reader to Wheeler
et al. (2002a,b) and Mahlein et al. (2002b).
The considered solution uses the topology of figure 5(b) to implement a bidirec-
tional switch. In figure 5(b) TA and TB are simply IGBT transistors and D1 and D2
are diodes. Due to D1 and D2, if both transistors are off there will be no current
circulation; that is, the considered topology can block voltages of any polarity. If, for
example, V1 > V2 , TA is on and TB is off, current will flow in the direction indicated
by figure 5(b). In the contrary case, i.e. if V1 < V2, TB is on and TA is off, current will
flow in the opposite direction. The previous characteristics make the presented
topology a bidirectional switch.
It is worth noting that, although the topology of figure 5(b) is the basis of a
bidirectional switch, the time required to turn on or off an IGBT makes the com-
mutation a nontrivial task. As an example of how the commutation actually takes
place, consider figure 6. It should be noted that in order to obey equation (2), the
following conditions should be avoided:

)
TA1 and TB2 on, with VA > VB
ð19Þ
TA2 and TB1 on, with VA < VB
)
TA1 and TA2 off, with iL > 0
ð20Þ
TB1 and TB2 off, with iL < 0
 Control of matrix converter 73

T A1
VA
s1

T B1 iL

T A2
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s2

VB
T B2

Figure 6. Matrix converter with two-phase grid and single-phase load.

If condition (19) is violated, the sources will be short-circuited and, in the case
of condition (20), an abrupt interruption of the load current will occur and an
overvoltage will appear.
The commutation method that is presented is based on load current measure-
ments and is called soft-switching. It works as follows:
(1) Suppose that initially s1 is on, s2 is off, iL > 0 and that it is necessary to turn
s1 off and s2 on. From (19) and (20), this implies TA1 , TB1 are on and TA2 , TB2
are off. Note that in the previous conditions neither (19) nor (20) is violated.
(2) Turn off TB1 . This brings no overvoltage problems, since there is no current
flow through TB1 and therefore the load current is not interrupted.
(3) Turn on TA2 . Note that there may or may not be current flow through TA2 ,
depending on the magnitudes of V1 and V2.
(4) Turn off TA1 . If TA2 was not actually conducting, there will be an over-
voltage due to the load current flow interruption, and this will turn on
diode D. Since TA2 was on, the load will be connected to source VB, thus
neutralizing the overvoltage. Note that this completes the commutation: s1 is
off, s2 is on.
(5) Turn on TB2 , so that s2 can conduct in either direction.
The previous discussion can be summarised in the diagram shown in figure 7(a).
and the general commutation strategy can be summarised as follows:
(1) Determine the direction of current load iL.
(2) Depending on the direction of iL, turn off the non-conducting transistor in
the active switch (which will be turned off ).
 74 J. Rodrı´guez et al.

on on on off off T A1
on off off off off T B1

off off on on on T A2
off off off off on T B2

end
start transient legend

(a)

on on off
off off off
i L> 0
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off on on
off off off

on off
on off

off on
off on

off off off

on on off
i L< 0

off off off

off on on

start transient end

(b)

Figure 7. State diagram of the soft-switching commutation method: (a) for positive load
current; (b) general case, i.e. load current of any polarity.

(3) Depending on the direction of iL, turn on the transistor that should be
conducting in the switch that will be turned on.
(4) Turn off the transistor that is still on in the active switch.
(5) Turn on the transistor that is still off in the switch that has just been
turned on.
It is important to note that the above procedure does not violate condition
(19) or (20). Figure 7(b) shows the general state diagram of the commutation strategy
under the assumption that initially s1 is on and s2 is off.

4.2 Overvoltage problem

In the previous subsection, a convenient way to commutate the IGBTs was
discussed. It was noted that in some cases there are overvoltages that should be
managed appropriately to avoid destruction of the semiconductor. Another source
of overvoltages is grid perturbations and fault states in the load and therefore it is
important to have a method of dealing with this phenomenon. A standard way of
doing this is to use a clamp circuit as shown in figure 8, where capacitor C is loaded
to the desired clamp level. When overvoltage occurs, the diode conducts and the
RC circuit maintains the voltage level at a safe value. In normal operation, the
 Control of matrix converter 75

Matrix converter
LC- Filter

Grid
(50 Hz)

C R

Diode clamp circuit

IM
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Figure 8. Clamp circuit to manage overvoltages.

1 1/3 1
mij
T 2
1 2
viN Sampling tij
2 interval
vjREF

1/(Vi*Vi)

Figure 9. Generation of the duty cycle mij in the MatlabÕ –SimulinkÕ software package.

diodes are off and the clamp circuit has no influence on the MC operation. It
is important to note that the power level is very low for the clamp circuit
(Klumpner et al. 2002b).

5. Simulation

The converter is simulated using the MatlabÕ –SimulinkÕ package to demonstrate

the basic principle of the MC for the student. Equation (18) is used to obtain the
elements of the low-frequency transfer matrix MðtÞ and times tij . Figure 9 shows the
general structure of the module that generates the components mij of matrix MðtÞ,
taking as inputs the current samples of the MC input voltages (vi ) and of the desired
output voltages (vjN ¼ vjREF ).
The most important part of the simulation is the generation of the switching
functions of the bidirectional switches (Sij ðtÞ). These functions correspond to the
gate drive signals of the power switches in the real converter. Figure 10 presents
the block diagram used to generate these functions in the case of the jth output
phase. This block diagram is more easily understood if we consider the
variables and waveforms shown in figure 11, which corresponds to the following
 76 J. Rodrı´guez et al.

1
tAj

+ A
r out 1
−
SAj
Simulation CompA
clock
2
sampling (T) not(A) and B SBj

− B
out
3
2 +
not(A) and not(B) SCj
tBj CompB
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Figure 10. Pulse generation scheme for one output phase.

3
x 10
1 tAj+tBj
0.5
r

t
0 Aj
0 0.5 1 1.5 2
1.5
1
A=SAj

0.5 t
0 Aj
−0.5
0 0.5 1 1.5 2
1.5
1
0.5
B

0
−0.5
0 0.5 1 1.5 2
1.5
1
SBj

0.5 t
0 Bj
−0.5
0 0.5 1 1.5 2
1.5
1
SCj

0.5 tCj
0
−0.5
0 0.5 1 1.5 2
time [ms]
Figure 11. Variables used for the pulse generation of one output phase.

conditions: switching (sampling) time T ¼ 1 ms and conduction times tAj ¼ 0:4 ms,
tBj ¼ 0:2 ms and tCj ¼ 0:4 ms.y The variable r is a ramp function with slope 1, start-
ing from zero at the beginning of each sampling interval. This variable is compared
with times tAj and tAj þ tBj , using comparators CompA and CompB respectively.
The output of comparator CompA is the required switching function SAj, which

y
These times are given only as an example of what the pulse generation scheme can give as
a particular result.
 Control of matrix converter 77

1
vA
4
SAj

2 1
vB vjN
5
SBj

3
vC
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6
SCj
Figure 12. Generation of the output voltage of the jth phase.

corresponds to a pulse of amplitude 1 with a duration equal to tAj . The following

logic decisions are used to generate the other switching functions:

SBj ¼ notðAÞ and B
ð21Þ
SCj ¼ notðAÞ and notðBÞ

Figure 12 presents the block diagram used to generate the output voltages with
respect to the neutral N of the source. As expressed by (3), these voltages are
obtained from the product of the input voltages and the switching functions.
In general, the neutral of the load n is isolated from the neutral of the source N.
In order to calculate the output currents, it is necessary to obtain previously the
output voltages of the MC with respect to neutral n (vjn ). This is achieved by the
following equation:
vjn ¼ vjN  vnN , with j ¼ fa, b, cg ð22Þ

where the voltage between neutrals is obtained from

vaN þ vbN þ vcN
vnN ¼ ð23Þ
3
The load currents are obtained from voltages vjn using the following equation:
ij ðsÞ 1
¼ , j ¼ fa, b, cg ð24Þ
vjn ðsÞ Ls þ R

where s is the Laplace operator and L , R are the load parameters.

Figure 13 presents in a clear form the block diagram corresponding to
equations (22)–(24), used to generate the different voltages and currents in the
load. The currents at the input side of the MC are generated by the blocks of
figure 14, which is based on equation (4).
In order to make the link between the grid input voltages (vsi ) and currents (isi )
and the MC input voltages (vi ) and currents (ii), the filter model presented in
 78 J. Rodrı´guez et al.

1
vaN VnN
2 1/3
vbN
3
2
vcN
van
1
1
L.s+R
ia

4
vbn
1
3
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L.s+R
ib

6
vcn
1
5
L.s+R
ic
load model

Figure 13. Generation of voltages and currents in the load.

1
ia
4
Sia

2 1
ib ii
5
Sib

3
ic
6
Sic
Figure 14. Generation of the input currents in MatlabÕ .

equations (5) and (7) must be considered. This is easily achieved by considering the
transfer function blocks of the Control System Toolbox of MatlabÕ (The
Mathworks 1999), as shown in figure 15.
Finally, figure A.1 in the appendix presents the general block diagram of the
MC simulation, where block 1 models the input filter, block 2 generates the switch-
ing functions, blocks 3, 4 and 5 generate the output voltages vaN , vbN , vcN , block 6
represents the load, generating its voltages and currents, and blocks 7, 8 and 9
calculate the MC input currents iA, iB and iC.
 Control of matrix converter 79

1
1 1
Lf*Cf.s2+Rf*Cf.s+1
Vsi ViN

Lf.s+Rf
2
Lf*Cf.s2+Rf*Cf.s+1
ii

1
2
Lf*Cf.s2+Rf*Cf.s+1
isj

Cf.s
Lf*Cf.s2+Rf*Cf.s+1

Figure 15. Filter model developed in MatlabÕ .

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6. Results

Some studies have been done using the following parameters: source voltage
amplitude 250 V, 50 Hz, load resistance R ¼ 10
, load inductance L ¼ 30 mH,
voltage gain q ¼ 0.45, output frequency f0 ¼ 50 Hz (this means that the reference
has an amplitude equal to 0.45
250 ¼ 112.5 V and a frequency of 50 Hz), switching
frequency fs ¼ 1/T ¼ 1 kHz and simulation step ¼ 0.01 ms. The parameters of the
input filter are Lf ¼ 30 mH, Rf ¼ 0.1
and Cf ¼ 25 mF.
For the resolution of the equations a five-order fixed-step solver, included in
MatlabÕ –SimulinkÕ (ODE5 (Dormand-Price)), has been used.
Figure 16 shows the output voltage vaN and the load current ia for the above
conditions. The working principle of the MC is clearly demonstrated. The low-pass
characteristic of the load produces an almost sinusoidal current ia. In addition, it can
be observed that the MC can generate output frequencies that are not restricted by
the source frequency, which typically is the case in phase-controlled cycloconverters
(Gyugyi and Pelly 1976).
To validate the simulation results, figure 17 shows the output voltage and output
current for an 18 kW induction motor operating at 30 Hz. The similarity between the
experimental and simulation results is evident and therefore it can be assumed that
the simulation developed is in agreement with experimental results.
Figure 18(a) shows that the input current generated by the MC has the form of
several pulses with a high di/dt, making it necessary to introduce an input filter
to avoid the generation of overvoltages. The frequency spectrum of figure 19(a)
confirms the presence of high-order harmonics in the input current iA. Figure 18(b)
shows that the source current isA is free of high-frequency harmonics, due to the
action of the input filter, which also is confirmed by figure 19(b).
It is important to note that the proposed simulation strategy allows the student
to run simulations in abnormal conditions. As an example, figure 20 shows the
output phase voltage and the output current in the same conditions of figure 16,
but considering a voltage gain of q ¼ 0.9 and an output frequency of f0 ¼ 20 Hz. Note
that the voltage gain is greater than the maximum allowable (q ¼ 0.5) and therefore
the low-frequency component of the generated voltage is heavily distorted, as can
be recognized in the distortion of the load current. There are intervals in which
the input voltage level is not enough to synthesize the desired output voltage (recall
the comments at the end of Section 3 regarding the maximum voltage gain).
 80 J. Rodrı´guez et al.

400 va REF
200

[V]
0

an
v
−200

−400
0 0.01 0.02 0.03 0.04 0.05
time [s]
(a)
10

5
ia [A]
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−5

−10
0 0.01 0.02 0.03 0.04 0.05
time [s]
(b)

Figure 16. (a) Output voltage vaN , its reference (bold line), and (b) output current ia.

400

200
VaN [V]

−200

−400
0 0.01 0.02 0.03 0.04 0.05
time [s]
(a)
40

20
i [A]

0
a

−20

−40
0 0.01 0.02 0.03 0.04 0.05
time [s]
(b)

Figure 17. Waveforms for 18 kW induction motor drive at 415 V, 50 Hz input frequency,
30 Hz output frequency and sampling frequency of 2 kHz: (a) load voltage and (b) output
current. (Experimental result.)

7. Conclusions

The working principle of the MC, controlled with the direct transfer function
approach, has been presented. The modulation strategy and the most important
equations are clearly presented. In addition, an intelligent commutation strategy
is explained, which avoids the generation of overvoltages and overcurrents.
 Control of matrix converter 81

10

iA [A]
0
v
−10 sA

0 0.01 0.02 0.03 0.04 0.05

time [s]
(a)

10
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isA [A]

0
v
sA
−10

0 0.01 0.02 0.03 0.04 0.05

time [s]
(b)

Figure 18. (a) Input current before the filter (THD ¼ 120%) and (b) filtered input current
(source current; THD ¼ 14%).
Relative amplitude

0.5

0
0 1000 2000 3000 4000
frequency [Hz]
(a)
Relative amplitude

0.5

0
0 1000 2000 3000 4000
frequency [Hz]
(b)

Figure 19. (a) Spectrum of input current iA before filtering and (b) after the filter (source
current isA ).
 82 J. Rodrı´guez et al.

400

200

van [V]
0

−200 va REF

−400
0 0.02 0.04 0.06 0.08 0.1
time [s]
(a)
20
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10
ia [A]

−10

−20
0 0.02 0.04 0.06 0.08 0.1
time [s]
(b)

Figure 20. (a) Output voltage vaN , its reference (bold line), and (b) output current ia
with overmodulation.

In addition, the simulation of the MC controlled with the direct trans-

fer function approach has been presented and clearly explained. The model
reproduces in a very good form the behaviour of the converter, including the
effect of the input filter. In addition, other aspects such as operation under
abnormal conditions and overmodulation can be easily and directly simulated. It is
important to note that the simulation results are in good agreement with experi-
mental data.
The authors believe that, after studying this paper, students will clearly under-
stand the most relevant issues of this attractive converter: safe commutation of
power transistors, generation of overvoltages, advanced modulation methods and
generation and filtering of harmonics. In this way, students get prepared to develop
more advanced studies in the area of modern power converters.

Acknowledgment
The authors gratefully acknowledge the financial support provided by the Chilean
Research Fund FONDECYT (grant No. 1030368) and of the Universidad Técnica
Federico Santa Marı́a.

Appendix

Figure A.1 shows the global simulation scheme as described in the text.
 Grid 1 vK 7, 8, 9
SAa
Vsa Va vA vsa
ia
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[iA] ia Isa isa

SBa ib
Filter A
ic
Vsb Vb vB iK
SCa SKa
[iB] ib Isb Input current
SKb
Filter B and source voltage
isj SAb SKc
Vsc Vc vC
[iC] ib Isc ia
SBb
Filter C ib iA iB iC
va ref ic
SCb
iK
SKa
[iC]
SKb
SAc
[iB]
vb ref SKc
[iA]

Control of matrix converter

SBc
ia
ib
vc ref
SCc
ic
iK
vA SKa
q iK
Reference 2 vB SKb
Voltage gain
vC SKc
va_ref vj
SAj
SBj
SCj
va vb vc
Working
principle
vA
ij
vB
t vC
vj vj 6
Clock SAj
SBj va N ia
ib
Help: SCj
ic
vb N
va n ij
q (voltage gain <= 0.5) vb n
vA
frec (output frequency in Hz) vc N vc n
paso (simulation step) vB vj n
Load circuit
h (sampling interval) vC
va_n vb_n vc_n
L y R (load parameters) vj
SAj
Lf, Cf Rf (filter parameters)
SBj
SCj 3, 4, 5

Figure A1. Global simulation scheme for the matrix converter.

83
 84 J. Rodrı´guez et al.

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