Download Input Filter Design of a Mains Connected Matrix

1
Input Filter Design of a Mains Connected
Matrix Converter
S. F. Pinto, Member, IEEE, and J. F. Silva, Senior Member, IEEE
Abstract—The aim of this paper is to design a single stage LC
input filter for a three phase matrix converter, guaranteeing low
current harmonic contents as well as low power losses. Based on a
single phase equivalent model, the criteria to design the filter are
established and the best location for the filter damping resistor is
discussed.
To validate the filter design, experimental results are obtained
using a direct controlled matrix converter. These results show
that the input currents present low harmonic content as well as
reduced total harmonic distortion.
Index Terms—Input filter, matrix converters, power quality.
I. INTRODUCTION
ver the last decades, the generalized use of power
electronic converters, specially diode rectifiers and
naturally commutated power converters, in household
appliances, traction systems and in other industry applications
has represented an important source of low order harmonics,
resulting in disturbances in the power quality. Even though
some of these problems may be reduced, usually by means of
large input filters, the use of high frequency switching power
converters such as matrix converters, nearly allows the
elimination of these low order harmonics. As a consequence,
the input filters may be strongly reduced.
The three phase direct (without DC link) AC-AC power
converters, also known as matrix converters, result from the
association of nine bidirectional switches, that allow the
connection of each one of the three input phases to any of the
three output phases (topological constraints must be verified).
Usually, these bidirectional switches are obtained as the
association of two semiconductors with turn-off capability,
(IGBTs), and a pair of diodes (Fig 1). The high frequency
control of matrix converters was introduced in 1980 [1] and,
since then, other control techniques have been presented [2,3].
O
II. INPUT FILTER EQUIVALENT MODEL
Generally, control methods assume that matrix converters
are fed by ideal voltage sources (the mains) and feed ideal
current sources. Assuming that the mains have mainly an
This work was supported by CAUTL – Centro de Automática da
Universidade Técnica de Lisboa.
S. F. Pinto is with the Department of Electrical Engineering, Instituto
Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL (e-mail:
[email protected]).
J. F. Silva is with the Department of Electrical Engineering, Instituto
Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL (e-mail:
[email protected])
inductive behaviour, in order to fulfil the demand of an ideal
voltage source, it will be necessary to connect capacitors as
close as possible to the converter input, in order to allow the
decoupling between the line inductances and the converter
switching circuits. The introduction of these capacitors,
together with the mains inductors, results in a second order
filter [4]. To guarantee that the resultant filter has the adequate
cut-off frequency, in order to filter the matrix converter
impulsive currents, it is introduced an additional inductor
connected in series with the mains inductor. To damp the
oscillations that result from the matrix converter hard
switching, it is necessary to include a resistor in the filter. This
resistor may be connected in series or in parallel with the filter
inductor or in parallel with the capacitor.
The choice of the most adequate location for the input filter
resistance, as well as the determination of all the input filter
components is based on a single phase equivalent model and
should consider the following criteria: a) cut-off frequency; b)
minimization of the input power factor displacement; c)
minimization of the displacement factor between the mains
voltages and the voltages applied to the matrix converter; d)
minimization of the ripple in the line currents and/or the
capacitors voltages; e) minimization of the power losses in the
damping resistor.
Fig. 1. Three-phase matrix converter.
The filter may be represented as a single phase equivalent
model (Fig. 2), in which vi represents the mains voltage and
imatrix is the impulsive current injected by the matrix converter.
Both the input ii current and the capacitor vcf voltage will
depend on them. Assuming that the mains voltage vi may be
considered as a perfect sinusoid with a 50Hz frequency, the
2
filter will have a steady-state response that may be easily
calculated. However, the same line of thought is not valid for
the high frequency commutated imatrix current that depends on
the load currents. For this reason, it is useful to study the filter
response to a step applied at imatrix.
ii
il
r
l
ωn =
1
lC
 r

 r + rr + 1
 rcf

r
l


r
rl
imatrix
icf
rcf
C
vcf
vi
Fig. 2. Equivalent single phase input filter.
According to Fig.2, it is possible to express the input ii(s)
current only as a function of the input voltage vi(s) and the
matrix converter input current imatrix(s).
 1  r
r
1
1  l
 r + 1
+s
+ r C + C +
 lC  rl rcf rcf
rl
lC  rl rcf
rl






1  l
l
1  rr
r
s2 + s
+ + C rr  +
+ r + 1




lC  rcf rl
rl
 lC  rcf

s2
ii ( s ) =
Q=



 v ( s) +
i

1
1  rr
 + 1
+

rl C lC  rl

+
imatrix ( s )
 1  r

1  l
l
2
 r + rr + 1
s +s
+ + C rr  +
 lC  rcf

lC  rcf rl
rl



s
Also, it is possible to express the capacitor vcf(s) voltage
only as a function of the input voltage vi(s) and the matrix
converter input current imatrix(s).

1
1  rr
 + 1
+

rl C lC  rl

vcf ( s ) =
vi ( s ) −


 r

1 l
l
1
r
2
r
r

s +s
+
+ C rr  +
+ + 1
 lC  rcf

lC  rl rcf
rl
(2)



1 rr
s +
C lC
−
imatrix ( s )
 1  r

1  l
l
2
 r + rr + 1
s +s
+
+ C rr  +
 lC  rcf

lC  rl rcf
rl



s
 r

r
2  r + r + 1
 rcf

r
l


(5)
 r

 r + rr + 1
 rcf

rl




1  l
l
+ + C rr 


lC  rcf rl

1
lC
(6)
In order to guarantee: a) the minimization of the resonance
introduced at the cut-off frequency (6); b) the reduction of the
high frequency harmonics, it is desirable that the cut-off
frequency is in a frequency range at least one decade above the
mains frequency ωi and one decade below the commutation
frequency ωs (Fig. 3).
(7)
ωi << ω c << ω s
If the switching frequency is nearly 10kHz, the cut-off
frequency should not be higher than 2kHz and lower than
500Hz.
(8)
2 × 500π < ωc < 2 × 103 π
101
ξ=0,1
ξ=0,3
ξ=0,5
100
ξ=0,707
ξ=1
The denominators of (1) and (2) are in the canonical form
of a second order system (3), where ωn represents the natural
frequency (or cut-off frequency) and ξ represents the damping
factor.
(3)
d ( s ) = s 2 + 2ξω n s + ωn 2
The quality factor Q may be calculated as a function of the
ξ damping factor (4):
-20
1
2ξ
 l

l

+ + C rr 
 rcf

rl


However, in order to reduce the power losses only one of
the three possible locations for the filter resistance will be
chosen: a) series connection with inductor; b) parallel
connection with inductor; c) parallel connection with
capacitor.
Even though most authors consider the resistance connected
in parallel with the inductor, this paper will analyze the three
possibilities [5]. In any of the cases, the cut-off frequency will
be given by:
101
Q=
ξ=
ωc = ωn =
(1)
1
lC
a)
102
0,1 ωc
ωc
10 ωc
0
ξ=0,1
ξ=0,3
-40
ξ=0,707
ξ=0,5
-60
(4)
From (1,2) and (3,4), the ωn natural frequency, the ξ
damping factor and the Q quality factor are given by:
ξ=1
-80
-100
-120
-140
-160
b)
-180
0,1 ωc
ωc
10 ωc
Fig. 3. Bode diagrams for a second order system, with varying damping
factors ξ.: a) Amplitude diagram; b) Phase diagram.
3
III. RESISTANCE VALUE CALCULATION
A. Series connection with inductor
If the resistance is connected in series with the inductor,
according to Fig.2, the following simplifications should be
done:
(9)
rr ≠ 0 1 rl = 0 1 rcf = 0
Substituting (9) in (1) and (2), the frequency response of the
input current and capacitor voltage to the matrix converter
current imatrix(s) is:
1


lC
i
(s)
ii ( s ) =
r
1 matrix
vi ( s ) = 0

s2 + s r +

l lC

1
1

s
+

C rr lC
i
( s)
vcf ( s ) = −rr
r
1 matrix
vi ( s ) = 0

s2 + s r +
l lC

ωn 2
s 2 + 2ξω n s + ω n 2
(12)
l
C
(13)
ξ=0,1
1.6
1.6
1.4
1.4
α=2
α=4
1.2
1.2
1
1
α=1
α=2
0.8
α=100
α=100
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1
2
3
4
5
t [s]
6
7
8
9
10
0
a)
0
1
2
3
4
5
t [s]
6
7
8
9
10
b)
ξ=0,2
1.6
ξ=0,3
1.4
From (18), if the inductor values are in the order of mH,
ξ=0,5
1.2
1
0.8
ξ=0,707
0.6
ξ=1
0.4
0.2
0
0
2
1.8
α=1
Fig. 5. Step response of a second order system, considering ωn=1 and a
variable damping factor ξ.
2
1.8
(18)
α=4
1
1
Q=
=
2ξ rr
C
l
2
0.8
According to (5), the ωn natural frequency, the ξ damping
factor and the Q quality factor are given by:
r
ξ= r
2
2
l
the capacitor voltage transfer function will not produce
significant differences when compared to a second order
system without zeros (Fig. 4, 5).
1.8
ω
s n + ωn 2
αξ
c( s ) =
d ( s)
s 2 + 2ξω n s + ω n 2
⇒ α=
From (18), if the inductor value are in the order of mH, then
Also, the capacitor voltage is in the canonical form:
1
ωn =
lC
ωn
1
=
α ξ C rr
α>100. For that reason, assuming that ξ = 2 2 , the zero in
(11)
b( s )
If α=1, the zero (17) is equal to the real part of poles (14).
As a consequence, for high values of α, it is expected that the
zero does not have a significant influence on the step response
(Fig. 5). However, near α=1, the zero will produce a high
increase in the step response overshoot.
Comparing (1) and (12) and using (13), it is possible to
establish:
(10)
The input current (10) is in the canonical form (11):
a(s ) =
one represented in Fig.4. However, depending on the zero
location, the same line of thought may not be valid for the
capacitor step response. The zero of transfer function (12) will
be located at:
(17)
s z = −αξω n
1
2
3
4
5
6
7
8
9
10
t [s]
Fig. 4. Step response of a second order system, considering ωn=1 and a
variable damping factor ξ.
The poles of transfer functions (11) and (12) are located at:
then α>100. For that reason, assuming that ξ = 2 2 , the zero
of the capacitor voltage transfer function will not produce
significant differences when compared to a second order
system without zeros (Fig. 4, 5).
As far as damping is concerned, the series connected
resistor is a good solution. However, if power losses are
considered (19), it may not be an adequate approach,
especially for high input currents.
(19)
pr r (t ) = rr ii 2 (t )
The typical step response of a second order system, without
zeros, is represented in Fig. 3. In order to guarantee,
simultaneously, fast response times and high damping with low
Also, this solution may not be acceptable because it may
reduce considerably the voltage available at the capacitors
voltages (20), thus reducing the matrix converter maximum
transfer ratio.
(20)
vcf ( s ) = vi ( s ) − (s l + rr )ii ( s )
overshoot, it is usual to consider ξ = 2 2 = 0.707 .
Assuming that C/l is in the range defined by (15), the
resistance value that guarantees ξ=0.707 will be in the range
defined by (16).
(15)
10 −3 < C l < 10 −2
B. Parallel connection with inductor
If the resistance is connected in series with the inductor,
according to Fig.2, the following simplifications should be
done:
(21)
rr = 0 1 rl ≠ 0 1 rcf = 0
(16)
Substituting (21) in (1) and (2), the frequency response of
the input current and capacitor voltage to the matrix converter
s p1, 2 = −ξω n ± ωn ξ 2 − 1
14,1Ω < rr < 44,7Ω
(14)
The input current ii(t) step response will be similar to the
4
current imatrix(s) is:
1
1

s
+

rl C lC
ii ( s ) =
i
( s)
1
1 matrix

2
vi ( s ) = 0
s
s
+
+

C rl lC

1

s

C
i
(s)
vcf ( s ) = −
1
1 matrix
vi ( s ) = 0

s2 + s
+
C rl lC

p r l (t ) = v r2 (t ) rl
For low values of vr l (t ) voltage applied to the inductor, the
(22)
According to (5), the ωn natural frequency, the ξ damping
factor and the Q quality factor are given by:
1
lC
ωn =
1
2rl
ξ=
l
C
Q = rl
C
l
(23)
According to (22), the input current has a zero. Comparing
(22) with (12), the value of α may be determined.
1
ωn
 αξ = r C

l

2ξω = 1
n

C rl
⇒ α=
1
(24)
2ξ 2
According to (14), (17) and Fig. 5, in order to avoid high
overshoot, the value of α should be as high as possible to
guarantee that the ii(s) zero is not coincident or near the real
part of the poles. According to (24), high values of α are
obtained only for low values of ξ. However, in order to
guarantee the adequate damping of the oscillations that result
from the step response, the damping factor should not be two
low. Considering two reference values for the damping factor
and (15), table I is obtained.
TABLE I
CALCULATION OF FILTER RL RESISTANCE, FOR TWO VALUES OF DAMPING
FACTOR ξ.
ξ
α
0.5
0.707
2
1
rl (Ω)
{10 … 31.6}
{7.07 … 22.36}
1
ξ=0,5
0,5
ξ=0,707
0
-0,5
0
1
2
3
4
5
6
7
8
9
(25)
l
10
t [s]
Fig. 6. Step response of a second order system with a zero at the origin,
considering ωn=1 and a damping factor ξ.=0.5 and ξ.=0.707.
The capacitor voltage has a zero at the origin. Its step
response will be similar to the one represented in Fig. 6.
To guarantee that the amplitude of the voltage applied to
the capacitor is nearly equal to the amplitude of the mains
voltage, it is assumed that the voltage drop at the inductance
terminals should not exceed 10% of the input voltage.
According to Fig.2, the power losses depend directly on the
square value of the voltage applied to the filter inductance.
power losses, using this approach, are not expected to be high.
However, result (25) should be compared with the other two
approaches.
C. Parallel connection with capacitor
If the resistance is connected in parallel with the capacitor,
according to Fig.2, the following simplifications should be
done:
(26)
rr = 0 1 rl = 0 1 rcf ≠ 0
Substituting (26) in (1) and (2), the frequency response of
the input current and capacitor voltage to the matrix converter
current imatrix(s) is:
1


lC
i
(s )
ii ( s ) =
1
1 matrix
vi ( s ) = 0

s2 + s
+

C rcf lC

1

s

C
i
( s)
vcf ( s ) = − 2
1
1 matrix
vi ( s ) = 0
s +s
+

C rcf lC

(27)
According to (5), the ωn natural frequency, the ξ damping
factor and the Q quality factor are given by:
ωn =
1
lC
ξ=
1
2 rcf
l
C
Q = rcf
C
l
(28)
The input current has the typical response of a second order
system without zeros. The capacitor voltage transfer function
has one zero at the origin. According to figures 4 and 6, it
should be chosen ξ.=0.707 in order to minimize the
overshoot. As a result (28), (15), the filter resistance should be
in the range defined by:
(29)
7.1Ω < rc f < 22.4Ω
However, even though this solution guarantees an adequate
damping, it does not present good results when power losses
are considered (30), as they depend on the square value of the
input voltage. Besides, to decrease the power losses, the filter
resistance would have to increase. According to (29), this
would reduce the damping of the filter step response.
(30)
p r cf (t ) = vcf2 (t ) rcf
D. Choice of the resistor location
All the three approaches for the input filter resistance
location considered in the previous sections guaranteed a good
damping of the oscillations produced by the matrix converter
commutations. However, the power losses are substantially
different for each one of the three approaches. The best
solution should be chosen comparing the power losses, which
are responsible for the converter efficiency reduction, to the
total matrix converter power.
Assuming that the input current is nearly equal to its
fundamental harmonic, the power delivered by the mains may
be calculated according to (31).
(31)
P = Vi I i cos φi
5
The losses associated to the solutions presented in sections
A and C are compared in (32), and simplified using the Ii
current obtained as a function of the converter power (31), and
considering that the capacitor voltage is nearly equal to the
mains voltage.
Pr cf
=
Pr r
Vcf2 rcf
Pr cf
⇒
rr I i2
Pr r
=
Vi4 cos 2 φi
rr rcf P 2
(32)
From (32) it is possible to conclude that, in general, the
resistance in parallel with the capacitor will result in higher
power losses, especially for high input voltages and low matrix
converter power.
A similar analysis is done in order to compare the losses
associated to the solutions presented in sections B and C
(resistance connected in parallel with the inductance or
capacitor). The inductance voltage is expressed as a function
of the input voltage and, in general, is much lower than the
mains voltage. Assuming that k<0.1 and that the rl and the rcf
vcf ( s )
vi ( s )
sl
ii ( s )
< 0,1 ⇒
vi ( s )
2
Pr c
⇔
2
Pr l
k Vi rl
=
rl
2
(33)
k rcf
(36)
ω l Ii
Vi
< 0,1 ⇔ l <
0,1Vi
ω Ii
(37)
The displacement factor introduced by the inductance in the
capacitor voltage is given by (38).
For a fixed voltage drop at the inductor terminals ( ω l I i ),
expressed as a fraction of the input voltage, it is possible to
calculate the displacement factor between the mains voltage
and the capacitor voltage.
 cos(φi ) ω l I i Vi
 1 + sen (φi ) ω l I i Vi
φc = arctg 
6




(38)
Vl = 0,1 Vi
5
4
φc (degrees)
Pr l
≈
Vi2 rcf
s l rl ii ( s )
i ( s)
≈1− s l i
s l + rl vi ( s )
vi ( s )
Considering that the maximum voltage drop at the
inductance terminals does not exceed 10% of the input
voltage, the filter inductance may be calculated according to
(37). It is assumed that the input voltage is given by
vi ( jω ) = Vi e j 0 and the input current is given by ii ( jω ) = I i e jφi :
resistances have similar values it is possible to conclude that
the power losses associated to the approach in which the
resistance is connected in parallel with the capacitor are still
higher.
Pr cf
=1−
3
Vl = 0,05 Vi
2
The losses associated to the solutions presented in sections
A and B are compared in (34), and simplified using the Ii
current obtained as a function of the converter power (31), and
considering that the capacitor voltage is nearly equal to the
mains voltage.
Pr l
Pr r
=
Vl2 rl
rr I i2
⇔
Pr l
Pr r
=
(k P )2
1
4
rr rl I i cos 2 φi
(34)
Once again, assuming that k<0.1, it is possible to conclude
that only for very low input Ii currents or near zero input
power factors it may be better to choose the resistance in series
with the inductance.
Most researchers [6] choose the resistance connected in
parallel with the inductance because this is the topology that,
in the majority of applications, allows the minimization of the
power losses.
IV. INDUCTION VALUE CALCULATION
Even though the lC product has already been calculated in
(8), it is necessary to consider additional criteria in order to
calculate the inductor value separately. The inductor should be
adequately calculated otherwise it may introduce a high
displacement factor and/or voltage drop between the input
voltages and the capacitor voltages.
Based on Fig. 2 and (2), the capacitor voltage is given by:
vcf ( s ) = vi ( s ) −
sl
ii ( s )
s l rl + 1
(35)
As the resistance rl has high values (table 1), in order to
minimize the inductor voltage drop, at the mains frequency
f=50Hz the inductance impedance should be significantly
lower than the resistance.
1
0
-90
0
90
φi (degrees)
Fig. 7. Displacement factor between the mains voltage and the capacitor
voltage, for a varying lagging/leading input power factor (–90º to 90º),
imposing a fixed voltage drop at the inductor terminals ( ω l I i ), expressed as
a fraction of the input voltage.
Based on Fig. 7, if the maximum value of the inductance
voltage drop represents 10% of the input voltage, the
displacement factor between the mains voltage and the
capacitor voltage, should not exceed 6º.
V. CAPACITOR VALUE CALCULATION
To calculate the capacitor value, it should be taken in
account the fact that, according to the previous chapter, the
inductance was calculated to minimize the voltage drop and
displacement factor at the inductor terminals. For that reason,
it is possible to assume that the capacitor voltage is nearly
equal to the mains voltage. In these conditions, the
displacement factor between the fundamental harmonic of the
matrix converter current i1matrix ( s) and the mains current ii (s )
is the capacitor current (39). This displacement factor can be
minimized choosing an adequate value for the filter capacitor.
(39)
ii ( s ) = s C vc ( s ) + i1matrix ( s )
Based on (39) and assuming that the capacitor voltage is
given by vc ( jω ) = Vc e j 0 and the fundamental harmonic of the
input current is given by i1matrix ( jω ) = I1matrix e jφ1matrix , it is
possible to calculate the displacement factor between the two
6
currents:
φi − φ1matrix
 ω C Vcf

= arctg 
+ sen(φ1matrix ) cos(φ1matrix )  − φ1matrix (40)
I

 1matrix

The displacement factor between the first harmonics of the
input ii(jω) current and the matrix converter i1matrix ( s) current is
represented in Fig.8 for different values of γ = ω C Vcf I1matrix .
80
60
γ=0,5
φi (degrees)
40
20
γ=0,8
γ=0
0
γ=0,1
-20
-40
-60
-80
-90
0
90
be higher than 2kHz.
In the laboratory prototype it was used a filter with values:
l=6mH, C=6µF (delta connected capacitors), r=20Ω.
As expected, the results of Fig. 9 show that the input
current presents a low harmonic content.
VII. CONCLUSIONS
In this work, the design of a single stage LC input filter for
a three phase matrix converter was presented, based on criteria
such as minimization of power losses, minimization of
displacement factors between voltage or currents and
maximization of the voltage transfer ratio. The design was
done using a single phase equivalent model and the best
location for the filter damping resistor was discussed. The
experimental results showed that the input current presented a
low harmonic content.
φ1matrix (degrees)
Fig. 8. Displacement factor between the first harmonics of the input ii(jω)
current and the matrix converter i1matrix ( s ) current, for different values of
VIII. REFERENCES
[1]
γ = ω C Vcf I1matrix .
According to (40) it is necessary to guarantee that when
[2]
φ1matrix = 0 the displacement factor introduced by the capacitor
ϕcond (41) is minimum. The maximum capacitor value is
obtained based on the maximum acceptable displacement
factor between the two currents.
(41)
ϕ cond = arctg (ω C Vcf I1matrix )
Cmax =
I1matrix
ω Vcf
tg (φmax )
[3]
[4]
(42)
It is important to note that the capacitor value (42),
obtained with the equivalent single phase model, is adequate if
the matrix converter filter capacitors are star connected.
However, if the capacitors are delta connected, the result (42)
should be divided by 3: C ∆ max = C max 3 .
Fig. 9. Input current waveform and FFT.
VI. EXPERIMENTAL RESULTS
In order to validate the input filter design, some results
were obtained for a direct controlled three phase matrix
converter. Even though this control approach does not
guarantee fixed switching frequency, it is assumed that it will
[5]
[6]
A. Alesina; M., Venturini; "Solid state power conversion: a Fourier
analysis approach to generalized transformer synthesis"; IEEE
Transactions on Circuits and Systems, vol. CAS-28, no 4, pp. 319-330,
April 1981.
L. Huber; D. Borojevic; N. Burany; "Analysis, design and
implementation of the space-vector modulator for forced-commutated
cycloconverters’; IEE Proceedings-B Electric Power Applications,
vol.139, no 2, pp. 103-113, March 1992.
S. Pinto; F. Silva, "Direct control methods for matrix converters with
input power factor regulation" in Proc. 2004 IEEE Power Electronics
Specialists Conf., pp. 2366-2372.
P. Nielsen, "The matrix converter for an indusction motor drive" Ph.D.
dissertation, Danish Academy of Technical Sciences., Univ. Aalborg,
1996.
S. F. Pinto, "Conversores matriciais trifásicos: generalização do
commando vectorial directo," Ph.D. dissertation, Dept. Electrotchnical.
Eng., Univ. Técnica de Lisboa, IST, 2003.
C. Neft; C. Schauder; "Theory and design of a 30-hp matrix converter";
IEEE Transactions on Industry Applications, vol. 28, no 3, pp.
546-551, May/June 1992.
IX. BIOGRAPHIES
S.F.Pinto (M’2002) was born in Coimbra,
Portugal, in 1969. She graduated from the Instituto
Superior Técnico, Lisboa in 1992, received the Msc
in 1995 and Phd in 2003 from the same University.
Currently she is an Assistant Professor of Power
Electronics of the Energy group at the Department
of EEC, IST, teaching Power Electronics and
Control of Power Converters. She is also a
researcher at Centro de Automática of UTL. Her main research interests are in
the areas of modeling, control and simulation of power converters, with
special emphasis on matrix converters.
J. Fernando Silva (M’90) born in 1956,
Monção Portugal, received the Dipl. Ing. in
Electrical Engineering (1980), the Doctor Degree
in Electrical and Computer Engineering (EEC) in
1990 and the Habil. Degree in Electrical and
Computer Engineering in 2002, from Instituto
Superior Técnico (IST), Universidade Técnica de
Lisboa (UTL), Lisbon, Portugal. Currently he is an
Associate Professor of Power Electronics, of the
Energy group at the Department of EEC, IST,
teaching Power Electronics and Control of Power Converters. As a researcher
at Centro de Automatica of UTL, his main research interests include
modelling, simulation, topologies and advanced control in Power Electronics.
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