INTRODUCTION
Recently, the induction motor fed from three phase matrix converter has established
their importance in industrial drive applications. In reality, the matrix converter
provides important benefits such as bidirectional power flow, sinusoidal input
current with adjustable displacement angle (i.e., controllable input power factor),
and a great potential for size reduction due to the lack of dclink capacitors
for energy storage (Alesina and Venturini, 1989; Casadei
et al., 2002; Wheeler et al., 2002).
Various modulation methods for matrix converters have been investigated. Indirect
ACAC modulation (Hara et al., 2004), PWM patterns
for nine switches of the matrix converter are generated directly from the output
voltage reference and the input current reference. In virtual ACDCAC modulation
(Itoh et al., 2004), the matrix converter is
controlled as a combination of a virtual current source PWM rectifier and a
virtual voltage source PWM inverter, and then switching patterns for the matrix
converter are synthesized. In the latter case, the problem is that when switching
patterns are generated to obtain a zero voltage vector in the virtual inverter,
phase currents do not flow in the virtual rectifier regardless of the switching
patterns for the rectifier, and as a result, the input current waveforms become
distorted. In this problem was solved by changing the carrier wave on the virtual
inverter side (Itoh et al., 2004). Another solution
involves applying space vector modulation to the virtual rectifier inverter
control (Cha and Enjeti, 2003; Helle
et al., 2004; Huber and Borojevic, 1995).
In another method, the maximum input linetoline voltage is discarded in PWM
modulation when the modulation factor is low, thus suppressing harmonics in
the output voltage (Odaka et al., 2009). The
present study deals with Space Vector Modulation (SVM) in virtual rectifier
inverter control within the framework of virtual ACDCAC modulation.
In this study, induction motor fed by direct controlled matrix converter with
a Space Vector Modulated (SVM) was proposed. A complete mathematical analysis
of the power circuit along with the duty cycle calculation (switching algorithm)
is described for both low voltage transfer ratio (0.5) and maximum voltage transfer
ratio (0.866). The whole model is then realized by using Simulink blocks such
as math operators, relational operators, and delay circuits. Finally, the proposed
mathematical model is validated using a passive RL load and active induction
motor load.
SPACE VECTOR MODULATION
Space Vector Modulation refers to a special switching sequence which is based
on the upper switches of a three phase matrix converter. Theoretically, SVM
treats a sinusoidal voltage as a phasor or amplitude vector which rotates at
a constant angular frequency, ω. This amplitude vector is represented in
dq plane where it denotes the real and imaginary axes. As SVM treats all three
modulating signals or voltages as one single unit, the vector summation of three
modulating signals or voltages are known as the reference voltage, V_{oref}
which is related to the magnitude of output voltage of the switching topologies.
The aim of SVM is to approximate the reference voltage vector, V_{oref}
from the switching topologies.
For a balanced three phase sinusoidal system the instantaneous voltages may
be expressed as (1)
This can be analyzed in terms of complex space vector:
where e^{jθ} = cos + j sinθ and represents a phase shift
operator and 2/3 is a scaling factor equal to the ratio between the magnitude
of the output linetoline voltage and that of output voltage vector. The angular
velocity of the vector is ω_{0} and its magnitude V_{0}.
Similarly, the space vector representation of the three phase input voltage
is given by (3)
where V_{i} is the amplitude and ω_{i}, is the constant
input angular velocity.
If a balanced three phase load is connected to the output terminals of the
converter, the space vector forms of the three phase output and input currents
are given by:
Respectively, where
is the lagging phase angle of the output current to the output voltage and
is that of the input current to the input voltage.

Fig. 1: 
Three phase matrix converter 
Switching principle: The three phase matrix converter (MC) topology
is shown in Fig. 1.
Since MC connects load directly to the voltage source by using nine bidirectional
switches, the input phases must never be shorted, and due to the inductive nature
of the load, the output phases must not be left open. If the switching function
of a switch, S_{ij} in Fig. 1, is defined as :
The constraints can be expressed as:
For a three phase MC there are 27 valid switch combinations giving thus 27
voltage vectors as shown in Table 1. The switching combinations
can be classified into three groups which are, synchronously rotating vectors,
stationary vectors and zero vectors.
CONTROL ALGORITHM
The indirect transfer function approach is employed in both voltage source
rectifier (VSR) and voltage source inverter (VSI) parts of the MC. Consider
the VSR part of the circuit in Fig. 2 as a standalone VSR
loaded by a dc current generator, i_{dc}.
Voltage source rectifier space vector modulation: The VSR input current
vector diagram is shown in Fig. 3. The space vector of the
desired input current can be approximated by two adjacent as shown in Figure
4.
Table 1: 
Matrix converter switching vectors 


Fig. 2: 
Indirect matrix conversion 

Fig. 3: 
Input current vector diagram 

Fig. 4: 
Vector diagrams (a) output sextant 2 (b) input sextant 1 
The duty cycles for VSR are calculated as (8)(10).
where m_{i} is the VSR modulation index:
For a switching cycle within the first sector:
Substitute θ_{i} with:
φ_{i} is the arbitrary angle. The transfer matrix of the VSR,
is defined as:
Replacing the modulation index from (11) in (14) resulting the desired input
current phase. The VSR output voltage is determined as:
Voltage space inverter space vector modulation: Consider the VSI part
of the MC in Fig. 2 as a standalone VSI supplied by a dc voltage
source V_{pn}=V_{dc}. The VSI switches can assume only six allowed
combinations which yield nonzero output voltages.

Fig. 5: 
Output voltage vector diagram 
Hence, the resulting output line voltage space vector is defined by Eq.
2 can assume only seven discrete values, V_{0} – V_{6}
in Fig. 5, known as voltage switching state vectors.
The space vector of the desired output line voltages is
can be approximated by two adjacent state vectors V_{d} and V_{q},
and the zero voltage vector, V_{o} using PWM as shown in Fig.
4, where
is the sampled value of
at an instant within the switching cycle T_{g}. The duty cycles of the
switching state vectors are;
where, m_{v} is the VSI modulation index
The sectors of the VSI voltage vector diagram in Figure 5
correspond directly to the six sextants of the three phase output line voltages
shown in Fig. 6.

Fig. 6: 
Six sextants of the output line voltage waveforms 
The averaged output line voltages are:
For the first sextant,
and,
By substitution of (22) in (21):
Substituting the modulation index from (20) in (23), the output line voltages
are obtained:
The VSI averaged input current is determined as:
Output voltage and input current SVM: Direct converter modulation can
be derived from the indirect transfer function. First modulation is carried
out as if the converter is an indirect. The switch control signals for DMC are
then derived based on the relation between the VSR and VSI. The modulation index
of the DMC is given as:
For simplicity, m_{v}=1 and m=m_{i}.The modulation algorithm
is derived similar to VSR and VSI except in the opposite direction. Since, both
the VSR and VSI hexagons contain six sextants; there are 36 combinations or
operating modes. However, only 27 valid switch combinations giving thus 27 voltage
vectors as shown in Table 1. If the first output voltage and
the first input current are active, the transfer matrix become:
The output line voltages are:
which finally yield to:
Where,
As can be seen, the output line voltages are synthesized inside each switching
cycle from samples of two input line voltages, V_{ab} and V_{ac}.
By comparison of (30) and (31), it can be concluded that simultaneous output
voltage and input current SVM can be obtained by employing the standard VSI
SVM sequentially in two VSI sub topologies of the three phase MC.
When the standard VSI SVM is applied in the first VSI sub topology, where V_{pn}=V_{ab},
the duty cycles of the two adjacent voltage switching state vectors are d_{avi}
and d_{βαvi} as defined in (31). The standard VSI SVM in
the second sub topology, with V_{pn}=V_{ac}, results in the
state switching vector duty cycles d_{αβvi} and d_{βvi},
also defined as (31). The remaining part of the switching cycle is given as:
MATHEMATICAL MODELING
The complete model of MC is shown in Fig. 7. It comprises
input modulator, output modulator, MC modulator and MC IGBTs switches.
Figure 8 is the sector identification and reference angle
generation. The angle is generated from the reference output frequency by integrating
it. Based on the angle, the sector can be identified. The result is shown in
Fig. 11. The modulation of input current is shown in Fig.
9; the output voltage modulation is similar to VSI. The switch control signals
for MC are shown in Fig. 10.
RESULT AND DISCUSSION
The simulations of direct matrix converter are carried out using MATLAB/SIMULINK.
The processing took 56 sec for the passive RL load and 45 sec for the induction
machine load. It was loaded by three phase induction motor (3 hp, 200 V, 60
Hz star connected) for 0.5 and 0.866 transfer ratio.
Figure 11 shows the sector identification and reference
angle generation. The angle is generated from the reference output frequency
by integrating it. Based on the angle, the sector can be identified.
The input and output line voltage with loaded passive load is shown in Fig.
13 and 14 for transfer ratio of 0.5 and 0.866. For the
induction machine loaded the simulation result is shown in Fig.
15 and 16. Figure 16 and 17
is the input current for the passive load and induction motor load respectively.
The input currents are mostly sinusoidal for the induction motor load.

Fig. 8: 
Sector identification 

Fig. 10: 
Direct matrix converter modulator 

Fig. 11: 
Result for sector identification 

Fig. 12: 
Input and output voltage with passive load for q=0.5; R=135.95Ω,
L=168.15 mH, V_{im}=100 V, f_{o} = 60 Hz, f_{s}
= 2 kHz 

Fig. 13: 
Input and output voltage with passive load for q=0.866; R=135.95Ω,
L=168.15 mH, V_{im}=100 V, f_{o} = 60 Hz, f_{s}
= 2 kHz 

Fig. 14: 
Input and output voltage with loaded induction motor for q=0.5;
3 hp, R_{s }=0.277Ω, R_{r}=0.183Ω, N_{r}=1766.9
rpm, L_{m}=0.0538H, L_{r}=0.05606H, L_{s}=0.0533H,
f_{o}=60 Hz, f_{s}=2 kHz 

Fig. 15: 
Input and output voltage with loaded induction motor for q=0.866;
3 hp, R_{s }=0.277Ω, R_{r}=0.183Ω, N_{r}=1766.9
rpm, L_{m}=0.0538H, L_{r}=0.05606H, L_{s}=0.0533H,
f_{o}=60 Hz, f_{s}=2 kHz 

Fig. 16: 
Input current with passive load; R=135.95Ω, L=168.15
mH, V_{im}=100 V, f_{o} = 60 Hz, f_{s} = 2 kHz (a)
q=0.5, (b) q = 0.866 

Fig. 17: 
Input current with loaded induction motor for q=0.866; 3
hp, R_{s }=0.277Ω, R_{r}=0.183Ω, N_{r}=1766.9
rpm, L_{m}=0.0538 H, L_{r}=0.05606H, L_{s}=0.0533
H, f_{o}=60 Hz, f_{s}=2 kHz 
CONCLUSION
The main constraint in the theoretical study of matrix converter control is
the computation time it takes for the simulation. This constraint has been overcome
by the mathematical model that resembles the operation of power conversion stage
of matrix converter. This makes the future research on matrix converter easy
and prosperous. The operation of direct control matrix converter was analysed
using mathematical model with induction motor load for 0.866 voltage transfer
ratio.
ACKNOWLEDGMENT
The authors thank the Faculty of Electrical and Electronic Engineering of WHO
that funded the project with resources received for research from both University
of Malaysia Pahang and TATi University College (Short Grant 90019001).