INTRODUCTION
The matrix converter is a direct ACAC converter which uses an array of controlled bidirectional switches to create a controllable output voltage system with unrestricted frequency (Alesina and Venturini, 1981, 1989; Arevalo, 2008; Arevalo et al., 2010; Bradaschia et al., 2009; Cardenas et al., 2009; Huber and Borojevic, 1995; Imayavaramban et al., 2006; Shephered and Zhang, 2004; Vargas et al., 2009, 2010; Wheeler et al., 2002). Recently, matrix converters receive a lot of attention because they have several advantages such as they have simple and compact power circuit, generation of load voltage with arbitrary amplitude and frequency can be obtained, they operate with nearly sinusoidal input and output currents with harmonics only around or above the switching frequency, operation with unity input displacement factor for any load can be obtained and they have bidirectional power flow (Alesina and Venturini, 1981, 1989; Arevalo, 2008; Arevalo et al., 2010; Bradaschia et al., 2009; Cardenas et al., 2009; Huber and Borojevic, 1995; Imayavaramban et al., 2006; Shephered and Zhang, 2004; Vargas et al., 2009, 2010; Wheeler et al., 2002).
Few investigations have used Matlab/Simulink for simulation of the matrix converter to obtain its performance when used in some applications (Altun and Sunter, 2003; Sunter and Altun, 2005; Imayavaramban et al., 2006). However, either the transfer ratios were calculated by mfile in Matlab (Imayavaramban et al., 2006) or a simplified solution algorithm (Altun and Sunter, 2003; Sunter and Altun, 2005) was used. In this study, Matlab/Simulink package is used to simulate the matrix converter in detail when venturini and optimum venturini modulation techniques (Wheeler et al., 2002; Arevalo, 2008) are used. The matrix converter is loaded by an inductive static load.
MATERIALS AND METHODS
Matrix converter model: Figure 1 shows a simplified representation of a matrix converter system. It consists of nine bidirectional power switches which are controlled to connect the input 3phase voltage source to a 3phase load. Controlling of these switches is achieved according to a certain modulation strategy.
The modulation strategy is based on a desired output voltage magnitude and frequency and an input displacement factor (Alesina and Venturini, 1981, 1989; Arevalo, 2008; Huber and Borojevic, 1995; Imayavaramban et al., 2006; Shephered and Zhang, 2004; Wheeler et al., 2002). The matrix converter will be modeled by its output voltages and input currents in terms of switching functions of the switches as follows:
The switching functions in Eq. 1 and 2 are defined as follows (Huber and Borojevic, 1995; Wheeler et al., 2002):

Fig. 1: 
Simplified representation of a matrix converter system 


Fig. 2: 
General form of the switching pattern 

In order to avoid shortcircuited input terminals and opencircuited output
phases, these switching functions should satisfy the following constraint equation
(Huber and Borojevic, 1995; Wheeler et al., 2002):
The shape of the switching functions depends on the switching pattern used; a typical switching pattern is shown in Fig. 2 (Wheeler et al., 2002). By considering that the bidirectional power switches work with high switching frequency, a lowfrequency output voltage of variable amplitude and frequency can be generated by modulating the duty cycle of the switches using their respective switching functions.
The switching frequency is usually >20 times, the output frequency in order to obtain an output voltage with low harmonic content (Imayavaramban et al., 2006). The lowfrequency transfer matrix also known as modulation matrix (Arevalo, 2008; Shephered and Zhang, 2004) is defined by (Wheeler et al., 2002; Imayavaramban et al., 2006):
Where:
The constraint equations for the matrix converter can be written as (Alesina and Venturini, 1981, 1989; Imayavaramban et al., 2006; Wheeler et al., 2002):
There are several modulation techniques used to obtain the modulation matrix, M (t) such as Venturini method, Venturini optimum method, Scalar method and Space vector modulation method (Wheeler et al., 2002). In this study, the Venturini and Venturini optimum methods will be used. When the voltage gain ratio, q is ≤0.5 for unity input displacement factor, the modulation duty cycles can be obtained using the Venturini method from the following compact form (Arevalo, 2008; Wheeler et al., 2002):
For:
where, v_{k} denotes the input voltages which are given by:
and v_{j} denotes the reference output voltages which are given by:
When the voltage gain ratio, q is >0.5 and
for unity displacement factor, the modulation duty cycles can be obtained using
the Venturini optimum method from the following compact form (Arevalo, 2008;
Wheeler et al., 2002):
for K = A, B, C; j = {a, b, c}; β_{K} = 0, 2π/3, 4π/3 for K = A, B, C, respectively. The reference output voltages, v_{j} are obtained from (Alesina and Venturini, 1989; Arevalo, 2008; Imayavaramban et al., 2006; Wheeler et al., 2002):
Inductive load model: The inductive load will be modeled in terms of the load current differential equation given by:
where:
R and L are the load resistance and inductance.
System analysis: The performance of the system under consideration will
be obtained for given voltage gain ratio, q, output frequency, f_{o}
and unity input power factor using the following procedure. When the voltage
gain ratio, q is ≤0.5 for given input voltages (Eq. 7),
the reference output voltages are obtained using Eq. 8. The
input and output voltages are used in Eq. 6 to obtain the
modulation duty cycles of the matrix converter switches. When the voltage gain
ratio q is >0.5 and
, the reference output voltages are obtained from Eq. 10.
The input and output voltages are used in Eq. 9 to obtain
the modulation duty cycles of the switches. The modulation duty cycles are used
to obtain the conduction periods of the switches during a switching interval,
T_{seq} for a given switching frequency, f_{s}. When the switching
pattern shown in Fig. 2 is used the switching functions of
the converter switches, S_{Kj} can be obtained by comparing the modulation
duty cycles with a saw tooth waveform whose frequency equals the switching frequency
and its amplitude is unity.

Fig. 3: 
System Simulink block diagram 


Fig. 4: 
Subsystem matrix converter 


Fig. 6: 
Subsystem 0.5<q≤0.866 

The switching functions and the input voltages are used in Eq.
1 to obtain the 3phase output voltages of the matrix converter. The 3phase
output voltages are used in Eq. 11 to obtain the load currents.
The corresponding input currents can be obtained using Eq. 2.
System simulation: The system under consideration is simulated using
Matlab/Simulink software package. Figure 3 shows the Simulink
block diagram of the system corresponding to the system of Fig.
1. This block diagram consists of three subsystems which are called matrix
converter, RLload and MC input current. The blocks labeled v_{A}, v_{B}
and v_{C} represent the input voltage waveforms which are given in Eq.
7. The details of the subsystem matrix converter are shown in Fig.
4. It consists of two subsystems called q≤ 0.5 and 0.5<q≤ 0.866.
The details of these subsystems are shown in Fig. 5 and 6,
respectively.

Fig. 7: 
Subsystem RLload 


Fig. 8: 
Subsystem MC input current 

In Fig. 5 and 6, the block labeled vst
represents the saw tooth waveform and the block labeled enable is used to enable
the subsystem output to be used when its input equals to unity otherwise the
output will be disabled. Figure 5 represents Eq.
1, 6 and 8. Figure 6
shows Eq. 1, 9 and 10.
Figure 7 shows the details of the subsystem RLload shown
in Fig. 3. It represents Eq. 11. Figure
8 shows the details of the subsystem MC input current of Fig.
3. It represents Eq. 2.
RESULTS AND DISCUSSION
Results are obtained for the system under consideration with a voltage gain
ratio q = 0.4, 100 Hz output frequency, 5 kHz switching frequency, 230 V input
voltage (peak value), 50 Hz input frequency, 10 Ω load resistance and 20
mH load inductance. Figure 9 and 10 show
the output voltage and current waveforms of the matrix converter, respectively.
The obtained output current waveforms are compared with those obtained in reference (Imayavaramban et al., 2006) as shown in Fig. 10 in order to validate the simulation process. It is found that the two sets of results are almost identical.
It can be noted from Fig. 10 that the output current waveforms
are approximately sinusoidal waveforms therefore they have low harmonic content.

Fig. 9: 
Output voltage waveforms at q = 0.4, f_{o} = 100
Hz 


Fig. 10: 
Output current waveforms at q = 0.4, f_{o} = 100
Hz 


Fig. 11: 
Output voltage waveforms at q = 0.85, f_{o} = 30
Hz 

Another set of results is obtained for the system under consideration with
a voltage gain ratio q = 0.85, 30 Hz output frequency, 5 kHz switching frequency,
110 V input voltage (peak value) and with the same values used for previous
case for the input frequency, load resistance and load inductance. Figure
11 and 12 show the output voltage and current waveforms
of the matrix converter, respectively. The obtained output current waveforms
are compared with those obtained in reference (Imayavaramban et al.,
2006) as shown in Fig. 12 in order to validate the simulation
process.

Fig. 12: 
Output current waveforms at q = 0.85, f_{o} = 30
Hz 

It is found that the two sets of results are almost identical. Figure
12 shows that the output current waveforms are more distorted compared to
those obtained for the 1st case (Fig. 10).
CONCLUSION
A Matlab/Simulink block diagram has been constructed in detail for a matrix converter/an inductive load system. The constructed block diagram for the matrix converter can be used to study and analyze the performance of any application in which the matrix converter is used whether the voltage gain ratio q≤0.5 or 0.5<q≤0.866. The obtained results are almost identical with those obtained from the literature which proves the validity of the simulation using Matlab/Simulink package.