14

Matrix Converters
14.1 PRINCIPLE OF THE MATRIX CONVERTER
An arbitrary number of input lines can be connected to an arbitrary number of
output lines directly using bidirectional semiconductor switches, as shown in
Fig. 14.1. The multiple conversion stages and energy storage components of
conventional inverter and cycloconverter circuits can be replaced by one switch-
ing matrix. With ideal switches the matrix is subject to power invariancy so that
the instantaneous input power must always be equal to the instantaneous output
power. The number of input and output phases do not have to be equal so that
rectification, inversion, and frequency conversion are all realizable. The phase
angles between the voltages and currents at the input can be controlled to give
unity displacement factor for any loads.
In the ideal, generalized arrangement of Fig. 14.1 there do require to be
significant constraints on the switching patterns, even with ideal switches. Some
previous discussion of this given in Sec. 9.1. Both sides of the matrix cannot be
voltage sources simultaneously since this would involve the direct connection of
unequal voltages. If the input is a voltage source, then the output must be a current
source, and vice versa. It is a basic requirement that the switching functions must
not short-circuit the voltage sources nor open-circuit the current sources.
When the input lines are connected to an electric power utility then the
source is imperfect and contains both resistance (power loss) and inductance
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 416
FIG. 1 General arrangement of an ideal switching matrix.
(energy storage). Many loads, especially electric motors, are essentially inductive
in nature and may also contain internal emfs or/and currents. The basic premises
of electric circuit theory apply also to matrix convertersit is not possible to
instantaneously change the current in an inductor nor the voltage drop across a
capacitor.
To achieve the operation of an ideal matrix converter, it is necessary to
use ideal bidirectional switches, having controllable bidirectional current flow
and also voltage blocking capability for both polarities of voltage. The detailed
attributes of an ideal switch are listed in Sec. 1.2.
14.2 MATRIX CONVERTER SWITCHES
There is no such thing as an ideal switch in engineering reality. Even the fastest
of semiconductor switches requires finite and different switching times for the
switch-on and switch-off operations. All switching actions involve power dissipa-
tion because the switches contain on-state resistance during continuous conduc-
tion. Various options of single-phase bidirectional switches are given in Fig. 14.2.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 417
FIG. 2 Single-phase bidirectional switches: (a) two SCRs in inverse-parallel, (b) triac
bidirectional switch, (c) two IGBTs in inverse parallel (probably nonviable due to limited
reverse blocking), and (d) IGBT diode switch.
A fast switching pair such as Fig. 14.2c can be employed if the devices have
reverse blocking capability, such as the MCT or the non-punch-through IGBT.
Afast-acting switch that has been reportedly used in matrix converter exper-
iments is given in Fig. 14.3 [34]. Two IGBTs are connected using a common
collector configuration. Since an IGBT does not have reverse blocking capability,
two fast recovery diodes are connected in antiseries, each in inverse parallel
across an IGBT, to sustain a voltage of either polarity when both IGBTs are
switched off. Independent control of the positive and negative currents can be
obtained that permits a safe commutation technique to be implemented.
The common collector configuration has the practical advantage that the
four switching devices, two diodes and two IGBTs, can be mounted, without
isolation, onto the same heat sink. Natural air-cooled heat sinks are used in each
phase to dissipate the estimated losses without exceeding the maximum allowable
junction temperatures. In the reported investigation the devices used included
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 418
FIG. 3 Practical switch for matrix converter operation: (a) heat sink mounting and (b)
equivalent circuit.
JGBT-International Rectifier (IRGBC 30 F, 600 V, 17 A)
Fast recovery diode-SSG. Thomson (STTA3006, 600 V, 18 A)
A separate gate drive circuit transmits the control signal to each IGBT. Electrical
isolation between the control and the power circuits can be achieved using a high-
speed opto coupler to transmit the control signal and a high-frequency transformer
to deliver the power required by a driver integrated circuit.
14.3 MATRIX CONVERTER CIRCUIT
The basic circuit of a three-phase-to-three-phase matrix converter, shown in Fig.
14.4, consists of three-phase groups. Each of the nine switches can either block
or conduct the current in both directions thus allowing any of the output phases
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 419
FIG. 4 Basic circuit of a three-phase matrix converter.
to be connected to any of the input phases. In a practical circuit the nine switches
seen in Fig. 14.4 could each be of the common configuration of Fig. 14.3 [34].
The input side of the converter is a voltage source, and the output is a current
source. Only one of the three switches connected to the same output phase can
be on at any instant of time.
In general, low-pass filters are needed at both the input and output terminals
to filter out the high-frequency ripple due to the PWM carrier. An overall block
diagram of an experimental matrix converter system is given in Fig. 14.5.
Nine PWM signals, generated within a programmable controller, are fed
to switch sequencer circuits via pairs of differential line driver receivers. In the
switch sequencers the PWM signals are logically combined with current direction
signals to produce 18 gating signals. Isolated gate driver circuits then convert
the gating signals to appropriate drive signals capable of turning the power
switches on or off. Each power circuit is protected by a voltage clamp circuit.
Azero crossing (ZC) detector is used to synchronize to the input voltage controller
signals. For three-phase motor loads the output filter may not be necessary.
14.4 SWITCHING CONTROL STRATEGIES FOR
PWM MATRIX CONVERTERS IN THREE-
PHASE MOTOR APPLICATIONS [34,35]
When a PWM matrix converter is used to control the speed of a three-phase ac
motor the control system should possess the following properties:
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 420
FIG. 5 Basic building blocks of the matrix converter.
Provide independent control of the magnitude and frequency of the gener-
ated output voltages (i.e., the motor terminal voltages).
Result in sinusoidal input currents with adjustable phase shift.
Achieve the maximum possible range of output-to-input voltage ratio.
Satisfy the conflicting requirements of minimum low-order output voltage
harmonics and minimum switching losses.
Be computationally efficient.
Many different methods have been considered as the basis of analyzing and
designing a workable matrix converter. Because of the complexity of the neces-
sary switching the associated control logic is also complex and involves large
and complicated algorithms. General requirements for generating PWM control
signals for a matrix converter on-line in real time are
Computation of the switch duty cycles within one switching period
Accurate timing of the control pulses according to some predetermined
pattern
Synchronization of the computational process with the input duty cycle
Versatile hardwave configuration of the PWMcontrol system, which allows
any control algorithm to be implemented by means of the software
Microprocessor-based implementation of a PWM algorithm involves the use of
digital signal processors (DSPs).
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 421
The two principal methods that have been reported for the control of a
matrix converter are discussed separately, in the following subsections.
14.4.1 Venturini Control Method [34]
A generalized high-frequency switching strategy for matrix converters was pro-
posed by Venturini in 1980 [36,37]. The method was further modified to increase
the output-to-input voltage transfer ratio from 0.5 to 0.866. In addition, it can
generate sinusoidal input currents at unity power factor irrespective of the load
power factor.
14.4.1.1 Principle
In the Venturini method a desired set of three-phase output voltages may be
synthesized froma given set of three-phase input sinusoidal voltages by sequential
piecewise sampling. The output voltage waveforms are therefore composed of
segments of the input voltage waves. The lengths of each segments are determined
mathematically to ensure that the average value of the actual output waveform
within each sampling period tracks the required output waveform. The sampling
rate is set much higher than both input and output frequencies, hence the resulting
synthesized waveform displays the same low-frequency spectrum of the desired
waveform.
14.4.1.2 Switching Duty Cycles
The Venturini principle can be explained initially using a single-phase output. A
three-phase output is generated by three independent circuits, and the analytical
expressions for all three waveforms have the same characteristics. Consider a
single output phase using a three-phase input voltage as depicted in Fig. 14.6.
Switching elements S
1
S
3
are bidirectional switches connecting the output phase
to one of the three input phases and are operated according to a switching pattern
shown in Fig. 14.6b. Only one of the three switches is turned on at any given
time, and this ensures that the input of a matrix converter, which is a voltage
source, is not short-circuited while a continuous current is supplied to the load.
As shown in Fig. 14.6a within one sampling period, the output phase is
connected to three input phases in sequence; hence, the output voltage V
o1
is
composed of segments of three input phase voltages and may be mathematically
expressed as
V
t k
T
t k
T
t k
T
V
V
V
o
s s s
i
i
i
1
1 2 3
1
2
3

]
]
]

]
]
]
]
]
( ) ( ) ( )
(14.1)
Symbol k represents the number of sampling intervals, t
n
(k) for n 1, 2, 3 are
the switching on times, t
1
(k) t
2
(k) t
3
(k) equals the sampling period T
s
and
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 422
FIG. 6 One output phase of a matrix converter: (a) equivalent circuit and (b) switching
pattern.
t
n
(k)/T
s
are duty cycles. For a known set of three-phase input voltages, the wave-
shape of V
o1
, within the kth sample, is determined by t
1
(k), t
2
(k), t
3
(k).
It should be noted that the switching control signals m
1
, m
2
, m
3
shown in
Fig. 14.6b can be mathematically represented as functions of time
m t u kT u kT t k
k
s s 1
0
1
( ) ( ) ( ) +
[ ] {

(14.2)
m t u kT t k u kT t k t k
k
s s 2
0
1 1 2
( ) ( ) ( ) ( ) +
[ ]
+ +
[ ] {

(14.3)
m t u kT t k t k u k T
k
s s 3
0
1 2
1 ( ) ( ) ( ) ( ) + +
[ ]
+
[ ] {

(14.4)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 423
where
u(t
0
)
{
1 t t
0
0 t t
0
is a unit step function and the switch is on when m
n
(t) 1 and off when m
n
(t)
0.
For constructing a sinusoidal output wave shape, function m
n
(t) defines a
sequence of rectangular pulses whose widths are sinusoidally modulated. Conse-
quently, its frequency spectrum consists of the required low-frequency compo-
nents and also unwanted harmonics of higher frequencies. Since the required
output voltage may be considered as the product of these functions and the sinusoi-
dal three-phase input voltages, the Fourier spectrum of the synthesized output
voltage contains the desired sinusoidal components plus harmonics of certain
frequencies differing from the required output frequency.
The input current I
i1
equals output current I
o1
when switch S
1
is on and
zero when S
1
is off. Thus the input current I
i1
consists of segments of output
current I
o1
. Likewise, the turn-on and turn-off of switches S
2
and S
3
results in
the input currents I
i2
and I
i3
containing segments of output current I
o1
. The width
of each segment equals the turn-on period of the switch. Corresponding to the
voltage equation (14.1), the three average input currents are given by
I
t k
T
I I
t k
T
I I
t k
T
I
i
s
o i
s
o i
s
o 1
1
1
2
1
3
1
2 3

( ) ( ) ( )
(14.5)
14.4.1.3 Modulation Functions
Applying the above procedure to a three-phase matrix converter, the three output
phase voltages can be expressed in the following matrix form:
V t
V t
V t
m t m t m t
m t m t
o
o
o
1
2
3
11 12 13
21 22
( )
( )
( )
( ) ( ) ( )
( ) (

]
]
]
]
]
)) ( )
( ) ( ) ( )
( )
( )
( )
m t
m t m t m t
V t
V t
V t
i
i
i
23
31 32 33
1
2
3

]
]
]
]
]

]
]
]
]
]
(14.6)
or
[V
o
(t)] [M(t)][V
I
(t)]
and the input currents as
I t
I t
I t
m t m t m t
m t m t
i
i
i
1
2
3
11 12 13
21 22
( )
( )
( )
( ) ( ) ( )
( ) (

]
]
]
]
]
)) ( )
( ) ( ) ( )
( )
( )
( )
m t
m t m t m t
I t
I t
I t
o
o
o
23
31 32 33
1
2
3

]
]
]
]
]

]
]
]
]
]
,
(14.7)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 424
or
[I
I
(t)] [M
T
(t)] [I
0
(t)]
where M(t) is the modulation matrix. Its elements m
ij
(t), i,j 1, 2, 3, represent the
duty cycle t
ij
/T
s
of a switch connecting output phase i to input phase j within one
switching cycle and are called modulation functions. The value of each modulation
function changes from one sample to the next, and their numerical range is
0 m
ij
(t) 1 i,j 1, 2, 3 (14.8)
Bearing in mind the restriction imposed on the control of matrix switches stated
above, functions m
ij
(t) for the same output phase obey the relation
j
ij
m i

1
3
1 1 3
(14.9)
The design aimis to define m
ij
(t) such that the resultant three output phase voltages
expressed in Eq. (14.6) match closely the desired three-phase reference voltages.
14.4.1.4 Three-Phase Reference Voltages
The desired reference-phase voltages should ensure that the maximum output-
to-input voltage transfer ratio is obtained without adding low-order harmonics
into the resultant output voltages. To achieve this, the reference output voltage
waveform to be synthesized must, at any time, remain within an envelope formed
by the three-phase input voltages, as shown in Fig. 14.7a. Thus when the input
frequency
i
is not related to the output frequency
o
, the maximum achievable
output-to-input voltage ratio is restricted to 0.5, as illustrated in Fig. 14.7a.
The area within the input voltage envelope may be enlarged by subtracting
the common-mode, third harmonic of the input frequency from the input phase-
to-neutral voltages. For example, when a voltage of frequency 3
i
and amplitude
equal to V
im
/4 is subtracted from the input phase voltages, the ratio of output to
input voltage becomes 0.75 as shown in Fig. 14.7b. Note that this procedure is
equivalent to adding the third harmonics of the input frequency to the target
output-phase voltage. The introduction of the third-order harmonics of both the
input and output frequencies into the reference output-phase voltages will have
no effect on an isolated-neutral, three-phase load normally used in practice, as
they will be canceled in the line-to-line output voltages.
Further improvement on the output voltage capability can be made by sub-
tracting the third harmonics of the output frequency from the target output phase
voltage. This is to decrease the peak-to-peak value of the output phase voltage,
as illustrated in Fig. 14.7c. With the magnitude of the output-frequency third
harmonics equivalent to V
om
/6, an output-to-input voltage ratio of 0.866 can be
achieved. This figure is the theoretical output voltage limit for this type of con-
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 425
FIG. 7 Third harmonic addition to increase the maximum achievable output-voltage
magnitude of matrix converter: (a) output voltages, V
o
0.5V
in
, (b) output voltages, V
o
0.75V
in
, and (c) output voltages, V
o
0.866V
in
.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 426
verter. The two-step, third-harmonic modulation described above results in the
following output phase voltages expression [38,39]
V t
V t
V t
V
t
t
o
o
o
om
1
2
3
0
0
2
3
( )
( )
( )
cos
cos
cos

]
]
]
]
]

|
.

`
}

00
0
0
0
4
3
6
3
3
3
t
V
t
t
om

|
.

`
}

]
]
]
]
]
]
]
]
]
]

cos
cos
cos tt
V
t
t
t
im
i
i
i

]
]
]
]
]
+

]
]
]
]
]
4
3
3
3
cos
cos
cos

(14.10)
where V
om
and
o
are the magnitude and frequency of the required fundamental
output voltage and V
im
and
i
, are the magnitude and frequency of the input
voltage, respectively. The three terms of Eq. (14.10) can be expressed as
V V V
o
A
o
B
o
C
[ ]
+
[ ]
+
[ ] (14.11)
14.4.1.5 Derivation of the Modulation Matrix
Having defined the three-phase output reference voltage, determination of modu-
lation function M[t] involves solving Eqs. (14.6) and (14.7) simultaneously. The
three-phase input voltages with amplitude V
im
and frequency
i
and the three-
phase output currents with amplitude I
om
and frequency
o
are given by
V t V
t
t
t
i im
i
i
i
( )
cos
cos
cos
[ ]

|
.

`
}

|
.

`
}

2
3
4
3
]
]
]
]
]
]
]
]
]
]
(14.12)
and
I t I
t
t
t
o om
o o
o o
o o
( )
cos( )
cos
cos
[ ]

|
.

`
}

|
.

`





2
3
4
3
}}

]
]
]
]
]
]
]
]
]
]
(14.13)
Assume the desired output voltage of Eq. (14.10) to consist only of the first term,
[V
o
]
A
, then
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 427
V t
V t
V t
V
t
t
o
o
o
om
o
o
1
2
3
2
3
( )
( )
( )
cos
cos
cos

]
]
]
]
]

|
.

`
}

oo
t
|
.

`
}

]
]
]
]
]
]
]
]
]
]
4
3

(14.14)
By eliminating the other two terms of Eq. (14.10), the achievable ratio V
om
/V
im
becomes 0.5. Derivation of the modulation matrix for the output voltages defined
in Eq. (14.10) is given in Ref. 34. Using the three-phase input voltages of Eq.
(14.12) to synthesize the desired output voltage of Eq. (14.14), the modulation
matrix derived must be in the form expressed as either
m t A
t t
t
A
o i o i
o i
( )
cos ( ) cos ( )
cos ( )
[ ]

+ +

]
]
]
+
+
1
2
3
4
3

]
]
]
+

]
]
]
+

]
]
]
+ cos ( ) cos ( ) cos (




o i o i o i
t t
2
3
4
3
))
cos ( ) cos ( ) cos ( )
t
t t t
o i o i o i




+

]
]
]
+ +

]
]
]

4
3
2
3

]
]
]
]
]
]
]
]
]
]
(14.15)
or
m t A
t t t
A
o i o i o i
( )
cos ( ) cos ( ) cos ( )
[ ]

]
]
]

2
4
3
2
3

]
]
]

]
]
]

]
cos ( ) cos ( ) cos ( )



o i o i o i
t t t
2
3
4
3
]]
]

]
]
]

]
]
]

cos ( ) cos ( ) cos ( )

]
]
]
]
]
]
]
]
]
]
(14.16)
where A
1
and A
2
are constants to be determined. Substituting either Eq. (14.15)
or Eq. (14.16) into Eq. (14.6) gives three output voltages which are sinusoidal
and have 120 phase shift between each other.
V m t V t AV
t
t
t
o
A A
i im
o
o
o
[ ]

[ ] [ ]

|
.

`
}

+ +
( ) ( )
cos
cos
cos
3
2
2
3
1

|
.

`
}

]
]
]
]
]
]
]
]
]
]
4
3

(14.17)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 428
or
V m t V t A V
t
t
t
o
A A
i im
o
o
o
[ ]

[ ] [ ]

|
.

`
}

( ) ( )
cos
cos
cos
3
2
2
3
2

|
.

`
}

]
]
]
]
]
]
]
]
]
]
4
3

.
(14.18)
If both [m(t)]
A
and [m(t)]
A
are used to produce the target output voltage, then
[V
o
]
A
[V
o
]
A
[V
o
]
A
, yielding
A A
V
V
om
im
1 2
2
3
+
(14.19)
Applying the same procedure as above to synthesise the input currents, using Eq.
(14.7), yields
I m t I t A I
t
t
i
A A
T
o om
i o
i o
[ ]

[ ] [ ]

+
+
|
.

+ +
( ) ( )
cos ( )
cos
3
2
2
3
1


``
}

+
|
.

`
}

]
]
]
]
]
]
]
]
]
]
cos

i o
t
4
3 (14.20)
or
I m t I t A I
t
t
i
A A
T
o om
i o
i o
[ ]

[ ] [ ]

|
.

( ) ( )
cos ( )
cos
3
2
2
3
2


``
}

|
.

`
}

]
]
]
]
]
]
]
]
]
]
cos

i o
t
4
3 (14.21)
This means that modulation matrix [m(t)]
A
results in an input current having a
leading phase angle
o
, while [m(t)]
A
produces an input current with a lagging
phase angle
o
.
Let the required sinusoidal input currents be defined as
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 429
I t I
t
t
t
i im
i i
i i
i i
( )
cos ( )
cos
cos
[ ]

|
.

`
}

|
.

`





2
3
4
3
}}

]
]
]
]
]
]
]
]
]
]
(14.22)
where
i
is an input displacement angle.
Setting [I
i
(t)] [I
i
]
A
[I
i
]
A
and substituting [I
i
]
A
and [I
i
]
A
defined
by Eqs. (14.20) and (14.21), respectively, the input phase current equation is
given as
3
2 3 3
1 2
I A t A t
I t
om i o i o
im i
cos( ) cos( )
cos

+ +

]
]
]

3
0 2 4
|
.

`
}

i
, ,
As cos(A B) cosA cosB sinA sinB, applying this to both sides of the
equation yields
( ) cos cos cos cos A A t
I
I
t
i o
im
om
i i 1 2
3
2
3 3
+
|
.

`
}

|
.

`
}

(14.23a)
and
( )sin sin sin sin A A t
I
I
t
i o
im
om
i i 2 1
3
2
3 3

|
.

`
}

|
.

`
}

(14.23b)
Neglecting the converter power losses, the input power of the circuit equals the
output power of the circuit; hence,
V
im
I
im
cos
i
V
om
I
om
cos
o
(14.24)
This yields
I
I
V
V
im
om
om
im
o
i

cos
cos

(14.25)
Substituting I
im
/I
om
from Eq. (14.23) into Eq. (14.25) gives
( ) cos cos cos cos A A t
V
V
t
i o
om
im
i o 1 2
3
2
3 3
+
|
.

`
}

|
.

`
}

( )sin sin
tan
tan
sin A A t
V
V
t
i o
om
im
i
o
i 2 1
3
2
3 3

|
.

`
}

..

`
}

sin
o
(14.26)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 430
Solving Eq. (14.26) simultaneously, coefficients A
1
and A
2
are found to be
A Q A Q
i
o
i
o
1 2
1
3
1
1
3
1
|
.

`
}

+
|
.

`
}

tan
tan
tan
tan

(14.27)
where Q V
om
/V
im
is an output-to-input voltage ratio, which also satisfies
Eq. (14.21). Note that the input displacement angle and hence the input power
factor of the matrix converter may be adjusted by varying these coefficients
appropriately.
The overall modulation matrix may be written as the sum of [m(t)]
A
and [m(t)]
A
. As adding three elements on the same row of the modulation
matrix results in zero at any instant, the constant 1/3 must be added to each
element to satisfy the constraint specified in Eq. (14.9) giving
m t m t m t
A A
( ) ( ) ( )
[ ]

]
]
]
]
]
+
[ ]
+
[ ]
+
1
3
1 1 1
1 1 1
1 1 1 (14.28)
Substituting each component in Eq. (14.28) with the results derived above,
the formulas for the overall modulation matrix are expressed by
m t ( )
[ ]

]
]
]
]
]
]
]
]
]
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
1
3
Q
i
tan /tan
3
1
oo
o i o i
o i
t t
t
( )
+
[ ]
+
[ ]
+
[ ]
cos ( ) cos ( ) /
cos ( ) /
cos (

2 3
4 3
oo i o i
o i
o i
t t
t
t
+
[ ]
+
[ ]
+
[ ]
+



) / cos ( ) /
cos ( )
cos ( )
2 3 4 3
4

/ cos ( )
cos ( ) /
3
2 3
[ ]
+
[ ]
+
[ ]

]
]
]
]
]
o i
o i
t
t
+
Q
i
tan /tan +
3
1
oo
o i o i
o i
t t
t
( )

[ ]

[ ]

[ ]
cos ( ) cos ( ) /
cos ( ) /
cos (

4 3
2 3
oo i
o i
o i
o i
t
t
t
t

[ ]
+
[ ]

[ ]
+




) /
cos ( ) /
cos ( )
cos ( )
2 3
4 3 4


/
cos ( )
cos ( ) /
3
4 3
[ ]
+
[ ]
+
[ ]

]
]
]
]
]
o i
o i
t
t
(14.29)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 431
Alternatively, a general and simplified formula for one of the nine elements in
matrix [m(t)] above may be written as [39]
m t Q t i j
Q
ij o i
( ) ( ) cos ( ) ( )
( ) c
+ + +
[ ]

]
]
]

+ +
1
3
1 1 2 4
3
1

oos ( ) ( )

o i
t i j

]
]
]

2
3 (14.30)
where i,j l, 2, 3; Q V
om
/V
im
; and tan
i
/ tan
o
is an input-to-output
phase transfer ratio. As shown in Eq. (14.29), m
ij
(t) is a function of
i
,
o
, Q,
and . Assuming that V
i
and
i
are constants, Q is a variable of V
o
, while
o
,
the phase lag between the load current and voltage, is a nonzero value. Since
unity input power factor operation is always desired,
i
, is zero. This, in turn,
leads to being zero. Subsequently Eq. (14.30) may be simplified as
m t Q t i j
t i
ij o i
o i
( ) cos ( ) ( )
cos ( ) (
+ + +
[ ]

+
1
3
1 2 4
3
2

]
]
]
|
.

`
}

]
]
]
j)

3
(14.31)
For practical implementation, it is important to note that the calculated values of
m
ij
(t) are valid when they satisfy the conditions defined by Eqs. (14.8) and (14.9).
Also Eq. (14.30) gives positive results when V
o
/V
i
3/2
14.4.2 Space Vector Modulation (SVM) Control
Method
The space vector modulation (SMV) technique adopts a different approach to the
Venturini method in that it constructs the desired sinusoidal output three-phase
voltage by selecting the valid switching states of a three-phase matrix converter
and calculating their corresponding on-time durations. The method was initially
presented by Huber [40,41].
14.4.2.1 Space Vector Representation of Three-Phase
Variables
For a balanced three-phase sinusoidal system the instantaneous voltages maybe
expressed as
V t
V t
V t
V
t
t
t
AB
BC
CA
ol
o
o
o
( )
( )
( )
cos
cos( )
cos(

]
]
]
]
]

120

]
]
]
]
]
240 ) (14.32)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 432
This can be analyzed in terms of complex space vector

V V t V t e V t e V e
o AB BC
j
BC
j
ol
j t
o
+ +

]
]

2
3
2 3 4 3
( ) ( ) ( )
/ /
(14.33)
where e
j
cos j sin and represents a phase-shift operator and 2/3 is a
scaling factor equal to the ratio between the magnitude of the output line-to-line
voltage and that of output voltage vector. The angular velocity of the vector is

o
and its magnitude is V
ol
.
Similarly, the space vector representation of the three-phase input voltage
is given by

V Ve
i i
j t
i

( )
(14.34)
where V
i
is the amplitude and
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

I I e
o o
j t
o o

( )
(14.35)

I I e
i i
j t
i i

( )
(14.36)
respectively, where
o
is the lagging phase angle of the output current to the
output voltage and
i
is that of the input current to the input voltage.
In the SVM method, the valid switching states of a matrix converter are
represented as voltage space vectors. Within a sufficiently small time interval a
set of these vectors are chosen to approximate a reference voltage vector with
the desired frequency and amplitude. At the next sample instant, when the refer-
ence voltage vector rotates to a new angular position, a new set of stationary
voltage vectors are selected. Carrying this process onward by sequentially sam-
pling the complete cycle of the desired voltage vector, the average output voltage
emulates closely the reference voltage. Meanwhile, the selected vectors should
also give the desired phase shift between the input voltage and current.
Implementation of the SVM method involves two main procedures: switch-
ing vector selection and vector on-time calculation. These are both discussed in
the following subsections.
14.4.2.2 Definition and Classification of Matrix
Converter Switching Vectors
For a three-phase matrix converter there are 27 valid on-switch combinations
giving thus 27 voltage vectors, as listed in Table 14.1. These can be divided into
three groups.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
M
a
t
r
i
x
C
o
n
v
e
r
t
e
r
s
4
3
3
TABLE 14.1 Valid Switch Combinations of a Matrix Converter and the Stationary Vectors (k 2/3)
Output voltages Input currents Voltage vector Current vector
On switches V
AB
V
BC
V
CA
I
a
I
b
I
c
Magnitude Phase Magnitude Phase
1 S
1
, S
5
, S
9
V
ab
V
bc
V
ca
I
A
I
B
I
C
V
il

i
t I
O

o
t
2 S
1
, S
6
, S
8
V
ca
V
bc
V
ab
I
A
I
C
I
B
V
il

i
t4/3 I
O

o
t
3 S
2
, S
4
, S
9
V
ab
V
ca
V
bc
I
B
I
A
I
C
V
il

i
t I
O

o
t 2/3
4 S
2
, S
6
, S
7
V
bc
V
ca
V
ab
I
C
I
A
I
B
V
il

i
t 4/3 I
O

O
t2/3
5 S
3
, S
4
, S
8
V
ca
V
ab
V
bc
I
B
I
C
I
A
V
il

i
t 2/3 I
O

o
t 4/3
6 S
3
, S
5
, S
7
V
bc
V
ab
V
ca
I
C
I
B
I
A
V
il

i
t 2/3 I
O

o
t 4/3
1P S
1
, S
5
, S
8
V
ab
0 V
ab
I
A
I
A
0 kV
ab
/6 kI
A
/6
1N S
2
, S
4
, S
7
V
ab
0 V
ab
I
A
I
A
0 kV
ab
/6 kI
A
/6
2P S
2
, S
6
, S
9
V
bc
0 V
bc
0 I
A
I
A
kV
bc
/6 kI
A
/2
2N S
3
, S
5
, S
8
V
bc
0 V
bc
0 I
A
I
A
kV
bc
/6 kI
A
/2
3P S
3
, S
4
, S
7
V
ca
0 V
ca
I
A
0 I
A
kV
ca
/6 kI
A
7/6
3N S
1
, S
6
, S
9
V
ca
0 V
ca
I
A
0 I
A
kV
ca
/6 kI
A
7/6
4P S
2
, S
4
, S
8
V
ab
V
ab
0 I
B
I
B
0 kV
ab
5/6 kI
B
/6
4N S
2
, S
5
, S
7
V
ab
V
ab
0 I
B
I
B
0 kV
ab
5/6 kI
B
/6
5P S
3
, S
5
, S
9
V
bc
V
bc
0 0 I
B
I
B
kV
bc
5/6 kI
B
/2
5N S
2
, S
6
, S
8
V
bc
V
bc
0 0 I
B
I
B
kV
bc
5/6 kI
B
/2
6P S
1
, S
6
, S
7
V
ca
V
ca
0 I
B
0 I
B
kV
ca
5/6 kI
B
7/6
6N S
3
, S
4
, S
9
V
ca
V
ca
0 I
B
0 I
B
kV
ca
5/6 kI
B
7/6
7P S
2
, S
5
, S
7
0 V
ab
V
ab
I
C
I
C
0 kV
ab
/2 kI
C
/6
7N S
1
, S
4
, S
8
0 V
ab
V
ab
I
C
I
C
0 kV
ab
/2 kI
C
/6
8P S
3
, S
6
, S
8
0 V
bc
V
bc
0 I
C
I
C
kV
bc
/2 kI
C
/2
8N S
2
, S
5
, S
9
0 V
bc
V
bc
0 I
C
I
C
kV
bc
/2 kI
C
/2
9P S
1
, S
4
, S
9
0 V
ca
V
ca
I
C
0 I
C
kV
ca
/2 kI
C
7/6
9N S
3
, S
6
, S
7
0 V
ca
V
ca
I
C
0 I
C
kV
ca
/2 kI
C
7/6
0A S
1
, S
4
, S
7
0 0 0 0 0 0 0 0 0 0
0B S
2
, S
5
, S
8
0 0 0 0 0 0 0 0 0 0
0C S
3
, S
6
, S
9
0 0 0 0 0 0 0 0 0 0
C
o
p
y
r
i
g
h
t

2
0
0
4
b
y
M
a
r
c
e
l
D
e
k
k
e
r
,
I
n
c
.
A
l
l
R
i
g
h
t
s
R
e
s
e
r
v
e
d
.
Chapter 14 434
1. Group I. Synchronously rotating vectors. This group consists of six
combinations (1 to 6) having each of the three output phases connected
to a different input phase. Each of them generates a three-phase output
voltage having magnitude and frequency equivalent to those of the
input voltages (V
i
, and
i
,) but with a phase sequence altered from that
of the input voltages. As the input frequency is not related to the output
frequency, the SVM method does not use the above listed vectors to
synthesize the reference voltage vector that rotates at a frequency
o
.
2. Group II. Stationary vectors. The second group (1P to 9N) is classified
into three sets. Each of these has six switch combinations and has a
common feature of connecting two output phases to the same input
phase. The corresponding space vectors of these combinations all have
a constant phase angle, thus being named stationary vector. For exam-
ple, in the first set of the group, output phases B and C are switched
simultaneously on to input phases b, a, and c. This results in six switch
combinations all giving a zero output line-to-line voltage V
BC
. For the
second set, the short-circuited output phases are C and A; hence, V
CA
is zero. In the final six, output phases A and B are connected together
and the zero line-to-line voltage is V
AB
. The magnitudes of these vec-
tors, however, vary with changes of the instantaneous input line-to-
line voltages.
3. Group III. Zero vectors. The final three combinations in the table form
the last group. These have three output phases switched simultaneously
on to the same input phase resulting in zero line-to-line voltages and
are called zero voltage vectors. When a three-phase load is connected
to the converter output terminals, a three-phase output current is drawn
fromthe power source. The input currents are equivalent to the instanta-
neous output currents; thus, all input current vectors corresponding to
the 27 output voltage vectors are also listed in Table 14.1.
14.4.2.3 Voltage and Current Hexagons
A complete cycle of a three-phase sinusoidal voltage waveform can be divided
into six sextants as shown in Fig. 14.8. At each transition point from one sextant
to another the magnitude of one phase voltage is zero while the other two have
the same amplitude but opposite polarity. The phase angles of these points are
fixed. Applying this rule to the 18 stationary voltage vectors in Table 14.1, their
phase angles are determined by the converter output line-to-line voltages V
AB
V
BC
,
and V
CA
. The first six, all giving zero V
BC
, may locate either at the transition
point between sextants 1 and 2 (
o
t 30) or that between sextants 4 and 5
(
o
t 210), depending upon the polarity of V
AB
and V
CA
. From the waveform
diagram given in Fig. 14.8 the three having positive V
AB
and negative V
CA
are
at the end of sextant 1; conversely, the other three are at the end of sextant 4.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 435
FIG. 8 Six sextants of the output line-to-line voltage waveforms.
The magnitudes of V
AB
and V
CA
are determined by the switch positions of the
converter and can correspond to any of the input line-to-line voltages, V
ab
V
bc
, or
V
ca
. Similarly, vectors in the second set generating zero V
CA
and nonzero V
AB
and V
BC
are at the end of either sextant 3 or 6. The three having positive V
BC
and negative V
AB
are for the former sextant, the other three are for the latter. The
final set is for sextants 2 and 5.
Projecting the stationary voltage and current vectors onto the plane, the
voltage hexagon obtained is shown in Fig l4.9a. It should be noted that this
voltage vector diagram can also be obtained by considering the magnitudes and
phases of the output voltage vectors associated with the switch combinations
given in Table 14.1. The same principle can be applied to the corresponding 18
input current vectors, leading to the current hexagon depicted in Fig. 14.9b. Both
the output voltage and input current vector diagrams are valid for a certain period
of time since the actual magnitudes of these vectors depend on the instantaneous
values of the input voltages and output currents.
14.4.2.4 Selection of Stationary Vectors
Having arranged the available switch combinations for matrix converter control,
the SVMmethod is designed to choose appropriately four out of 18 switch combi-
nations from the second group at any time instant. The selection process follows
three distinct criteria, namely, that at the instant of sampling, the chosen switch
combinations must simultaneously result in
1. The stationary output voltage vectors being adjacent to the reference
voltage vector in order to enable the adequate output voltage synthesis
2. The input current vectors being adjacent to the reference current vector
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 436
FIG. 9 Output-voltage and input-current vector hexagons: (a) output-voltage vectors and
(b) input-current vectors.
in order that the phase angle between the input line-to-line voltage and
phase current, and hence the input power factor, being the desired value
3. The stationary voltage vectors having the magnitudes corresponding
to the maximum available line-to-line input voltages
To satisfy the first condition above, consider the reference voltage vector that
lies in one of the six sectors at any particular time instant. One of the line-to-
line voltages in this corresponding sector is bound to be either most positive or
most negative, hence being denoted as the peak line. The vectors selected to
synthesis the reference voltage vector should be those that make the voltage of
the peak line nonzeros. For example, when the reference voltage vector
ref
is
in sector 2 as shown in Fig 14.9a, the peak line is V
OCA
, and the stationary vectors
giving nonzero V
CA
are the first and third sets in group II of Table 14.1, thus 12
in total.
Further selection takes both second and third requirements into account.
From Eq. (14.35),
i
is the phase-angle between input line-to-line voltage and
phase current, which, for unity input power factor, must be kept at 30, giving
zero phase shift angle between the input phase voltage and current. Following
the same principle for the reference voltage vector, at a particular time interval
the reference current vector locates in one of six sectors and so does the input
line-to-line voltage vector. Note that the input voltage vector leads the current
vector by 30 and transits from the same sector as that of the current to the
next adjacent one. Consequently, the maximum input voltage value is switched
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 437
between two input line-to-line voltages of these two sectors. Taking the previously
selected 12 vectors and using the maximum available input voltages, there are
four vectors with peak-line voltages equivalent to one of these two line-to-line
voltages, and these are chosen. This can be illustrated using the input line-to-line
voltage waveform in Fig. 14.10b when the reference current vector I

ref
is in sector
1, the input line-to-line voltage vector
i
may lie in either sector 1 or 2. The
maximum input line-to-line voltage in sector 1 is V
ab
, while that in sector 2 is
V
ca
. Among the 12 stationary vectors, those having V
OCA
equivalent to either
V
ab
or V
ca
are the suitable ones and these are vectors 1P, 3N, 7N, and 9P in
Table 14.1. Following the above stated principle, the selected sets of stationary
vectors for reference voltage vector and input current vector in sextants 1 to 6
are listed in Table 14.2
14.4.2.5 Computation of Vector Time Intervals
As described above, the selected voltage vectors are obtained from two subsets
of the stationary vector group. In particular, vectors 1P and 3N are from the zero
V
BC
subset, and 7N and 9P are from the zero V
AB
subset. The sum of these, given
by |
o1P
| |
o3N
| due to the 180 phase angle between them, defines a vector
ou
. Similarly, the solution of |
o7N
| |
o9P
| gives another vector
ov
. As shown
in Fig. 14.10a both
ou
and
ov
are adjacent vectors of the output reference
voltage vector
ref
. Based on the SVM theory, the relation for these voltage
vectors can be written
t
t T
ref ov ov ou ou
s
V dt T V T V
0
0
+

+

(14.37)
where T
ov
and T
ou
represent the time widths for applying vectors
ov
and
ou
,
respectively, t
0
is the initial time, T
s
is a specified sample period.
FIG. 10 Vector diagrams: (a) output sextant 2 and (b) input sextant 1.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
C
h
a
p
t
e
r
1
4
4
3
8
TABLE 14.2 Selected Sets of Switch Combinations
Input sextant 1 Input sextant 2 Input sextant 3 Input sextant 4 Input sextant 5 Input sextant 6
Output sextant 1 1P,4N,6P,3N 5N,2P,3N,6P 2P,5N,4P,1N 6N,3P,1N,4P 3P,6N,5P,2N 4N,1P,2N,5P
Output sextant 2 3N,9P,7N,1P 9P,3N,2P,8N 1N,7P,8N,2P 7P,1N,3P,9N 2N,8P,9N,3P 8P,2N,1P,7N
Output sextant 3 4P,7N,9P,6N 8N,SP,6N,9P 5P,8N,7P,4N 9N,6P,4N,7P 6P,9N,8P,5N 7N,4P,5N,8P
Output sextant 4 6N,3P,1N,4P 3P,6N,5P,2N 4N,1P,2N,5P 1P,4N,6P,3N 5N,2P,3N,6P 2P,5N,4P,1N
Output sextant 5 7P,1N,3P,9N 2N,8P,9N,3P 8P,2N,1P,7N 3N,9P,7N,1P 9P,3N,2P,8N 1N,7P,8N,2P
Output sextant 6 9N,6P,4N,7P 6P,9N,8P,5N 7N,4P,5N,8P 4P,7N,9P,6N 8N,5P,6N,9P 5P,8N,7P,4N
C
o
p
y
r
i
g
h
t

2
0
0
4
b
y
M
a
r
c
e
l
D
e
k
k
e
r
,
I
n
c
.
A
l
l
R
i
g
h
t
s
R
e
s
e
r
v
e
d
.
Matrix Converters 439
For the example given above, the space vector diagrams of output sextant
2 and input sextant 1 are shown in Fig. 14.10. Let the phase angle of the reference
voltage vector,
o
t, be defined by the sextant number (16) and the angle within
a sextant
o
(0
o
60). Similarly, the phase angle of the input current
vector,
i
t
i
30, may be defined by the input sextant number and the
remaining angle
i
. The subsequent derivation is then based on these vector dia-
grams.
For a sufficiently small T
s
, the reference voltage vector can be regarded as
constant, and hence Eq. (14.37) can be expressed in two-dimensional form as
T V t V t V
s ref
o
o
P o P N o P
cos( )
sin( )
cos

+

+

]
]
]

( )
30
30
1 1 3 3
330
30
0
1
7 7 9 9

]
]
]
+
( )

]
]
]
sin
t V t V
N o N P o P


(14.38)
where t
1P
, t
3N
, t
7N
, and t
9P
are the time widths of the associated voltage vectors.
Note that in general these time widths are denoted as t
1
, t
2
, t
3
, and t
4
.
As the magnitude and frequency of the desired output voltage are specified
in advance, the input voltage and phase angle are measurable and relationships
for evaluating the magnitudes of the stationary voltage vectors are given in Table
14.1. Equation (14.38) can then be decomposed into the scalar equations
t
1P
V
il
cos
i
t t
3N
V
il
cos(
i
t 240) V
ol
T
s
sin(60
o
), (14.39)
t
7N
V
il
cos
i
t t
9P
V
il
cos(
i
t 240) V
ol
T
s
sin
o
, (14.40)
where V
il
is the magnitude of the input line-to-line voltage and V
ol
is that of the
output line-to-line voltage.
Applying the SVM principle to control the phase angle of the reference
current vector, the following equation results.
T I
t
t
t I t
s iref
i i
i i
P i P N
cos( )
sin( )

]
]
]

30
30
1 1 7
II
t I t I
i N
N i N P i P
7
3 3 9 9
30
30
30
30
( )

]
]
]
+
( )

cos
sin
cos
sin

]
]
]


(14.41)
Using the magnitudes of the input current vectors given in Table 14.1, the above
equation may be written as
( )
cos( )
sin( )
1
30
30
1 3
+

]
]
]

+

T I
t
t
I
t t
s iref
i i
i i
A
P NN
P N
t t
1
3
1 3
( ) +

]
]
]
]
]

(14.42)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 440

]
]
]

+

T I
t
t
I
t t
s iref
i i
i i
C
N P
cos( )
sin( )
30
30
1
7 9
33
7 9
( ) +

]
]
]
]
]
t t
N P
,

(14.43)
where is an arbitrary variable that enables Eq. (14.41) to be divided into two
parts. Since it is necessary to set only the phase angle of the reference input
current vector, Eq. (14.42) becomes
sin( )
cos( )
( )


i i
i i
N P
N P
t
t
t t
t t






+
30
30
3
3 1
3 1
which can be rearranged to give
t
3N
cos(
i
t
i
30) t
1P
cos(
i
t
i
90) 0 (14.44)
Repeating this procedure with Eq. (14.43) results in
t
9P
cos(
i
t
i
30) t
7N
cos(
i
t
i
90) 0 (14.45)
Solving Eqs. (14.39), (14.40), (14.44), and (14.45) results in expressions for
calculating the four vector time widths
t
QT
P
s
i
o i 1
30
60 60

cos cos
sin( )sin( )

(14.46)
t
QT
N
s
i
o i 3
30
60

cos cos
sin( )sin

(14.47)
t
QT
N
s
i
o i 7
30
60

cos cos
sin sin( )

(14.48)
t
QT
P
s
i
o i 9
30

cos cos
sin sin

(14.49)
where Q V
ol
/ V
il
is the voltage transfer ratio, and
o
and
i
, are the phase
angles of the output voltage and input current vectors, respectively, whose values
are limited within 060 range. The above equations are valid for when the
reference output-voltage vector stays in output sextant 2 while the reference input-
current vector is in input sextant 1. For different sets of vectors the same principle
is applied.
In principle, each of the time widths is restricted by two rules, namely
0 1
t
T
k
s
(14.50)
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 441
and
k
k
s
t
T

1
4
1
(14.51)
In variable speed ac drive applications, unity input power factor is desired. As
a consequence, the sum of the four vector time widths is normally less than a
switching period T
s
when the maximum output-to-input voltage ratio is limited
to 0.866. The residual time within T
s
is then taken by a zero vector; thus,
t T t
s k
k
0
1
4

(14.52)
14.5 SPECIMEN SIMULATED RESULTS [34]
14.5.1 Venturini Method
A specimen simulated result is given in Fig. 14.11 for a balanced, three-phase
supply applied to a symmetrical, three-phase, series R-L load, without the use of
an input filter, assuming ideal switches. With a carrier frequency of 2 kHz the
output current waveform is substantially sinusoidal at 40 Hz but has a 2000/40,
or 50, times ripple. It can be seen that the phase angle between the input phase
voltage and the fundamental input current is zero, resulting in unity displacement
factor.
14.5.2 Space Vector Modulation Method
A specimen simulated result is given in Fig. 14.12 for the case of balanced 240-
V (line), 50-Hz supply with symmetrical, three-phase, series R-L load. The carrier
(switching) frequency is 2 kHz, resulting in a substantially sinusoidal output
current of f
o
40 Hz.
Comparison of Fig. 14.12 with Fig. 14.11 shows that the two methods give
very similar results. The SVM method has the advantage of simpler computation
and lower switching losses. The Venturini method exhibits superior performance
in terms of output voltage level and input current harmonics.
14.6 SUMMARY
The matrix converter holds the promise of being an all-silicon solution for reduc-
ing the use of expensive and bulky passive components, presently used in inverter
and cycloconverter systems. Its essentially single-stage conversion may prove to
be a crucial factor for improving the dynamic performance of system.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 442
FIG. 11 Simulated matrix converter waveforms. V
o
/V
i
0.866, f
o
40 Hz, f
s
2
kHz. (a) Output line-to-line voltage. (b) Output current. (c) Input current (unfiltered). (d)
Input phase voltage and input average current [34].
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Matrix Converters 443
FIG. 12 Simulated matrix converter waveforms using the SVM algorithm. V
o
/V
i

0.866, f
o
40 Hz. (a) Output line-to-line voltage. (b) Output current. (c) Input current
(unfiltered). (d) Input phase voltage and input average current [34].
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.
Chapter 14 444
Matrix converters have not yet (2003) made any impact in the commercial
converter market. The reasons include the difficulties in realizing high-power
bidirectional switches plus the difficulties in controlling these switches to simul-
taneously obtain sinusoidal input currents and output voltages in real time. In
addition, the high device cost and device losses make the matrix converter less
attractive in commercial terms.
Recent advances in power electronic device technology and very large scale
integration (VLSI) electronics have led, however, to renewed interest in direct
ac-ac matrix converters. Ongoing research has resulted in a number of laboratory
prototypes of new bidirectional switches. As device technology continues to im-
prove, it is possible that the matrix converter will become a commercial competi-
tor to the PWM DC link converter.
Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.