Positive Output Luo DC-DC Converters in Power
Management Unit for Small Satellites
(Full text in English)
1
Andrei BĂESCU , Andrei COCOR1, Adriana FLORESCU1, Dan STOICHESCU1
1
University Politehnica of București, Faculty of Electronics, Telecommunications and Information Technology
Abstract
This paper analyses the possibility of using Luo converters in the structure of a power management unit for
small satellites. The efficiency of elementary and self-lift versions of Luo DC-DC converter types is
mathematically and by simulations proved. In the final part, a comparison between the parameters of the two
converters is made. An overview of the battery management system is offered, with the role of each component
being briefly described. A possible design, development phases and the advantages brought by the management
unit are also presented.
Keywords: battery management unit, self-lift Luo converters, small satellites
1. Introduction
Small satellites have become a common
presence in space during the last years [1].
A significant increase in the number of
recent space experiments can also be
observed [2]. As a direct consequence, the
increased demands and technical challenges
for the power system are to be considered
[3]. The miniaturized satellites standards
like the “nanosat”, “picosat” and, more
recently, “femtosat” push the limits even
further [4].
Alternative sources of energy, like solar
or wind energy, also named clean energy,
offer a new perspective for most of the
modern technologies, improving the quality
of life [5]. Being renewable, they are used in
various
domains
ranging
from
telecommunications
[6],
to
hybrid
propulsion systems [7] or to providing energy
for small satellites [8]. An efficient system
must be capable of saving energy, by
managing its own power consumption, just
like managing the energy schedule of a
building [9].
The engineers face the problem of
restricted space, limited amount of the
available energy and limited financial
resources [10]. The challenge for the small
satellite is to maintain cost-effectiveness by
maximizing capabilities and performance
while meeting difficult design and financial
restrictions [11]. Being relatively cheap and
having a simple structure, the positive
output Luo converter fits very well in the
satellite’s battery management unit and
complies all the requirements from financial
and complexity point of view [12].
The next chapter presents a diagram of
the power management system. The two
proposed Luo converter versions are tested
and simulated in chapter three. In the last
chapter, the research and development
phases of the unit are presented along with
the advantages and the comparative results
given by the two possible Luo converters’
implementations.
2. Overview of the Power Management
System
The proposed system incorporates
advanced features that can be usually found
in more complex entities (i.e. MPPT and
battery capacity monitoring), while the
complexity and costs are kept under control
[13]. It will provide as well a reasonable
amount of flexibility for communication with
the other components of the power system
[14].
The simplified block diagram of the
proposed power management system is
presented in Figure 1.
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
43
Figure 1. Block diagram of the power management system (PMS)
The system consists of three functional
units: the Battery Management Unit (BMU),
the Point Of Load (POL) Unit(s) and The
System Interface Unit (SI).
The BMU is responsible with the
management of the incoming energy from
the available sources (photovoltaic panels
PVs, fuel cells FCs, radioactive or thermal
generators, etc.) as well as with the
management of the available storage
devices (batteries, super-capacitors) [15].
This unit implements all the necessary
algorithms for batteries charging/fitness
estimation process, alternative storage
elements management (super-capacitors)
and power-shedding [16]. The Maximum
Power Point Tracking (MPPT) algorithm is
also the responsibility of this unit [17].
The Point Of Load Unit (POL) is the main
access node of the distributed power
architecture. The main role of the POL is to
convert the input DC voltage (from the DC
bus) to another voltage required by the
external load (can be part of the payload).
Each POL unit must be able to communicate
with the BMU and with the SI via the Energy
Management Bus (EMB).
The System Interface Unit (SI) is the main
interface between the energy management
unit and the entity/payload. His main role is
to maintain the communication between the
entity/payload and the other components of
the energy management unit. This unit is
responsible to find alternative energy routes
in case of the failure of one of the POLs.
Each energy-processing unit features a
high performance DC-DC converter and a
digital supervisory system capable to
manage the conversion process and
communicate with the external systems. In
order to maintain the conversion’s high-
efficiency over a wide range of the output
current, the DC-DC converters implement
the
latest
state-of-the-art
control
techniques.
The electronic components of the
converter must be carefully dimensioned in
order to fulfil the volume and the mass
requirements for spacecraft entities [18].
The topology is also of vital importance, as
the system must avoid the catastrophic
failures that can affect the payload [19].
Energy received from the solar cells is
used to power the system during sun-on
periods, and to recharge the battery pack for
sun-off periods. During solar eclipse, the
battery is used as the primary power source.
The main power line (DC bus) feeds a number
of DC-DC power converters (POLs), which
provide the necessary supply voltages for the
satellite’s electronics [20]. A battery
monitor is also integrated into this system to
measure sustainable power consumption and
facilitate the power management.
The BMU provides all the necessary
information regarding the available incoming
energy and the battery pack status in realtime [21]. The main decision regarding the
power-shedding comes from the BMU and is
send to the POL units through EMB.
The battery-charging unit is part of the
BMU and can handle any available chemistry
(Ni-Cd, Ni-MH, Li-ion, etc.). As the
performance of the battery pack decreases
over time, a monitoring system capable to
estimate the fitness/capacity of the
batteries is considered highly valuable [22].
Prognostics
and
health
management
technologies are used to assess the reliability
and performance of an autonomous system
and, under its actual lifecycle conditions, to
determine the advent of failure and mitigate
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
44
system risk. The system provides an
interface with the external components of
the entity.
3. Battery Management Unit block analysis
According to recent technology used by
the European Space Agency (ESA), a typical
value for the voltage that feeds a small
satellite (also called line voltage) is 50 V.
The latest spacecraft battery models are
available in 8 V and 32 V versions as
standard, but also models with custom
voltages in this range are created on
demand [23].
In this case, an essential aspect of the
previous proposed schema is the DC-DC
converter used in the BMU. Choosing the
converter type for such a tricky application
as charging the battery of an isolated
system, often for an undetermined period,
depends a lot on the simplicity, cost and
efficiency of the selected unit.
A. Mathematical Analysis and Simulation of
Positive Output Luo converter
Positive output Luo converter fits this
description and its electronic schema can be
seen in Figure 2.
Figure 2. Electronic diagram of the positive output Luo converter designed in OrCAD Capture
The capacitor C main role is to store and
transfer energy from the power source input
to the output through L1. If the capacity of C
is enough large, the variation of voltage
across C, at steady state, will be ignored.
During the conduction period of the static
contactor, source current is:
i I = i L1 + i L 2
The inductance of coil L1 receives energy
from the source. The currents iL1 and iL2 are
increasing, while the coil inductance L2
absorbs energy from both the source and
capacitor C.
During the lock period of the static
contactor, the source current is equal to
zero. Capacitor C is loaded by current iL1,
which flows through the diode D.
The energy that was stored in L1 is now
transferred to the capacitor C. In this period,
iL2 remains continuous while it crosses the
circuit through R, C0 and D. Both currents iL1
and iL2 are decreasing. Because the ripples of
iL1 and iL2 are small,
i L 1 ≈ I L 1 and i L 2 ≈ I L 2
the static contactor is locked and the
charging of C increases:
Q + = (1 − D )TI L1
(3.1)
This charging decreases during the
conduction period of the static contactor:
Q − = DTI
L2
(3.2)
Because Q+ = Q—:
I L2 =
1− D
I L1
D
(3.3)
Because C0 has the role of a low pass
filter,
I L2 = I 0
(3.4)
i L = iL1 + i L 2
during the start-up, iL is equal to zero during
the lock period.
The average current through the source
is:
I I = Di L = D (i L1 + i L 2 ) = D (1 +
1− D
) I L1 = I L1
D
(3.5)
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
The output current is:
1− D
IO =
II
D
I D = I L1 + I L 2 =
(3.6)
D
V1
1− D
(3.7)
Conversion ratio of the converter in
continuous conduction mode is:
N=
VO
D
=
VI 1 − D
(3.8)
The current IL1 increases and is sustained
by VI during start-up time. In the lock period,
it decreases and is inversely polarized by –
VC.
DTV1 = (1 − D)TVC
(3.9)
D
V I = VO
1− D
(3.10)
VC =
DTVI
L1
(3.11)
Because
I I = Di L = D ( i L 1
1− D
+ i L 2 ) = D (1 +
) I L1 ,
D
the relative ripple of current iL1 is:
ξ1 =
∆i L1 DTVI (1 − D) R
=
=
I L1
L1 I I
NfL1
∆iL2 =
Because I L 2
Q+
1− D
TI
=
C
C
∆vc =
ρ=
(3.12)
Q = COVO , ∆Q = CO∆VO , ∆iL2 =
DTV1
L2
it results:
∆Q =
1 ∆iL2 T T DTV1
=
2 2 2 9 L2
(3.20)
so the voltage ripple of V0 is:
(3.21)
and the relative ripple results:
∆vO
DT 2 VI
D
1
ε=
=
=
2
VO 9CO L2 VO 9 N f CO L2
(3.22)
In discontinuous conduction mode, the
current iD becomes zero during blocking.
Discontinuous conduction mode condition is
ζ ≥ 1, (D2 / N2) and (R / 2R) ≥ 1, so:
R
z
=D N
2 fL
2
VO = D (1 − D )
(3.14)
During the lock state, the next relations
are valid:
i D = i L1 + i L 2
(1 − D)TVO
L
(3.19)
(3.23)
The output voltage in discontinuous mode
is:
1− D
=
I L1 , it results:
D
∆i D = ∆i L1 + ∆i L 2 =
(3.18)
I
As
(3.13)
∆i
DR
ξ 2 = L2 =
I L2
NfL2
(3.17)
∆vC (1 − D)TI I
1
=
=D
vC
CVO
fCR
N≤D
DTVI
L2
(3.16)
∆Q DT 2V1
∆vO =
=
CO 9CO L2
The current’s ripple through L1 is:
∆i L1 =
IO
1− D
∆i D
D2R
ξ=
= 2
ID
N fL
The output voltage is:
VO =
45
(3.15)
while
R
1
≥
2 fL 1 − D
R
VI
2 fL
(3.24)
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
46
Figure 3. Simulation of the positive output Luo converter in PSPICE
For simulating the converter, the
following values were chosen:
VI = 20 V; VO = 30 V; T = 20 us; f = 50 kHz; D
= 0.6; L1 = 10 m; C = 20 u; L2 = 10 m; CO =
20 u; R = 20; the ideal diode with snubber D.
For these values, using the above
formulas of the converter, the following
results were obtained, according to Figure 3:
VO = 30 V; N= 1.5; VC = 30 V; ∆iL1 = 24 mA;
ξ1 = 10.6; ∆iL1 = 24 mA; ξ2 = 16; ∆iD = 48 mA;
ξ = 6.4; ∆vC=0.9 V; ρ= 0,03; ∆vO= 3 mV; ε=
0.1 m.
B. Mathematical Analysis and Simulation of
Self-Lift Positive Output Luo converter
The self-lift positive output Luo converter
(see Figure 4) is an improved version of the
elementary Luo converter, using the voltagelift technique.
Figure 4. Electronic diagram of the positive output self-lift Luo converter designed in OrCAD Capture
The functioning of the circuit is defined
by the two diodes that are conducting
alternatively; the current flows through D1
during the conduction period and diode D is
locked, while during the lock period the
current flows through D and D1 is locked.
From here on, in the same way as for the
previous
converter,
the
functioning
parameters are defined:
N=
VO
1
=
VI 1 − D
(3.27)
II =
1
IO
1− D
(3.28)
Q+ = I C − ON DT = I O DT
(3.29)
Q− = IC−OFF(1− D)T = IL (1− D)T
(3.30)
VC = VCO = VO
(3.25)
During commutation,
1
VI
1− D
(3.26)
Q+ = Q−
VO =
(3.31)
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
D
IO
1− D
IL =
(3.32)
During the lock period,
I D = I L + I LO
D
=
IO
1− D
1 R
ξ1 = 2
2N fL
ξ2 =
ξ=
1 R
2N 2 fLO
1 R
2 N 2 fLeq
(3.33)
(3.34)
(3.35)
Leq = LLO /(L + LO )
The relative ripples of the voltages are:
D 1
ρ=
2 fCR
σ1 =
ε=
N 1
2 fC1R
1
D
2
8 N f C O LO
(3.37)
1
R
=1
2
2 N fLeq
D 2 (1 − D ) R
MDCM = 1 +
2 fLeq
(3.41)
where zN is R/(fLeq)
When N>MB, the circuit functions in DCM
mode. In this case, the current through diode
iD decreases to zero at t = t1 = [D + (1−D) m]T,
where DT < t1 < T and 0 < m < 1 and “m” is
the fill factor, defined as:
M2
D( R / fLeq )
(3.42)
DTV1 = (1 − D)mT (VC − VC1 )
(3.43)
1
ζ
=
VO = [1 +
D
]VI
(1 − D)m
VO = (1 + D 2 (1 − D)
R
)VI
2 fLeq
(3.38)
The transformation report of
converter in discontinuous mode is:
(3.39)
MDCM = 1 +
The variation report of the current iD is 1,
when the circuit functions on the limit:
ξ=
The limit between the continuous mode
(CCM) and discontinuous mode (DCM) is:
m=
(3.36)
47
D 2 (1 − D ) R
2 fLeq
(3.44)
(3.45)
the
(3.46)
The output voltage will linearly increase
as long as the load resistance R is increased.
(3.40)
Figure 5. Simulation of the positive output self-lift Luo converter in Pspice
For simulating the converter, the
following numerical values were chosen:
VI = 20 V; VO = 30 V; T = 20 us; f = 50 kHz; D
= 0.6; L = 10 m; C = 20 u; C1= 20 u; Lo = 1m;
Co = 20 u; R = 40; D and D1 the ideal diodes
with snubber.
48
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
For these values, using the formulas of
the converter, the following results were
obtain, according to fig.5:
VO = 50 V; N = 2.5; VC = 50 V; ξ1 = 6.4; ξ2
= 96; ξ = 64; ρ = 0.0075; σ1 = 0.01875; ε = 1
4. Conclusions and results
For both converters, a 20 V input voltage
was used, for the same period and at the
same frequency. In the case of the
elementary positive output Luo converter,
the output voltage (30 V) is 150 % greater
than the input voltage. When using the selflift version of the converter, the output
voltage is increased up to 250 % of the input
voltage value (Figure 6).
Figure 6. Output voltage comparison for the
proposed Luo converters
The steps of developing an efficient
battery management unit can be resumed at
defining the needed characteristics and
finding the best solution on the market for
the DC-DC converter and for the control and
management unit:
1. For the DC-DC converter:
− Investigate the available topologies for
this system;
− Chose the suitable topology;
− Investigate the large and small signal
response of the proposed topology;
− Propose a control strategy and a
control system for the proposed
topology;
− Investigate the opportunity of
implementing more complex control
strategies in order to improve the
performance of the proposed topology.
2. For the control and management unit:
− Investigate the available MPPT
algorithms
advantages
and
disadvantages.
− Analyse the required computational
power in the context of limited
available energy.
− Investigate the opportunity of
implementing more advanced MPPT
algorithms (i.e. Fuzzy Logic or Neural
Networks).
− Investigate the available batteries
charging algorithms for the specific
chemistry.
− Investigate the available methods for
batteries fitness estimation.
− Following these steps, the main
development phases can be defined:
− Design the proposed DC-DC converter.
− Implement the DC-DC converter using
the available commercial components
− Design the proposed digital control and
management system using commercial
available components.
− Design the battery protection unit.
− Develop the prototype (PCB, etc.).
− Develop the firmware for the digital
unit.
− Validate and verify the prototype.
This paper offers solutions for the first
two main development phases, proposing
two suitable converter types: the Luo
elementary positive output converter and
the modern Luo self-lift positive output
converter. Although both types of converters
are adequate for the battery management
unit, checking all the structure and costs
requirements, the self-lift version of the
converter brings a considerable advantage
by obtaining greater output figures for the
same input parameters.
The design of the proposed digital control
management system and the other block
components will be presented in future
works.
5. Acknowledgment
The work has been funded by the Sectorial
Operational
Program
Human
Resources
Development 2007-2013 of the Ministry of
European Funds through the Financial Agreement
POSDRU/159/1.5/S/134398,
also
through
POSDRU/174/1.3/S/149155, as well as by a grant
of the Romanian National Authority for Scientific
Research, Program for research – Space
Technology and Advanced Research – STAR,
project number 80/29.22.2013, code CDI_ID 230.
6. References
[1] NASA official website, http://www.nasa.gov
/mission_pages/smallsats/ (retrieved Dec.
2014).
[2] Defence Industry Daily, Small Is Beautiful: US
Military Explores Use of Microsatellites,
http://www.defenseindustrydaily.com/Smal
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
l-Is-Beautiful-US-Military-Explores-Use-ofMicrosatellites-06720/ (retrieved June 2011).
[3] LEE J., KIM E., SHIN K., “Design and
Management of Satellite Power Systems”,
The 34th IEEE Real-Time Systems Symposium,
Dept. of EECS, University of Michigan, 2013.
[4] TRISTANCHO
J.,
GUTIERREZ
J.,
“Implementation of a femto-satellite and a
mini-launcher”, Universitat Politecnica de
Catalunya, May 2010.
[5] FLORESCU A., BĂESCU A., POPESCU A.,
STOICHESCU D., “PV charging systems for
remote rural areas with elementary and selflift
SEPIC
converters”,
International
Symposium on Fundamentals of Electrical
Engineering, Nov. 2014, Cat. No.:CFP1493YART ISBN: 978-1-4799-6821-3
[6] COCOR A., FLORESCU A., POPESCU A.,
STOICHESCU D., “Power supply blocks with
Cúk and Self-Lift Cúk converters for
telecommunication sites”, International
Symposium on Fundamentals of Electrical
Engineering, Nov. 2014, Cat. No.:CFP1493YART, ISBN: 978-1-4799-6821-3
[7] STANCIU V., CHEFNEUX M., ANGHELIŢĂ P.,
VĂRĂTICEANU
B.,
“Hybrid
propulsion
conceptual model for watercraft - Eco-Boat”,
EEA, Oct.-Dec. 2012, Vol. 60, nr. 4, Art 11.
[8] TAN B., TSENG K., “Intelligent and reliable
power supply system for small satellites”,
The 25th International Telecommunications
Energy Conference, 2003, Yokohama, Japan,
pp. 249–255, ISBN:4-88552-196-3, INSPEC
Accession No: 8023197
[9] ONOSE B., BOBOC C., “Icpe partner in Energy
IN TIME (EIT) Project”, EEA, April-June 2014,
Vol. 62, nr. 2, Art 11.
[10]
DUNBAR R., HARDMAN G., “A Flexible
Autonomous Power Management System for
Small
Spacecraft”,
Orbital
Sciences
Corporation, 1990.
[11]
http://en.wikipedia.org/wiki/Miniaturiz
ed_satellite, (retrieved Oct. 2014)
[12]
CHILAMBARASAN M., LATHA DEVI M.,
RAMESH BABU M., “Design and Simulation of
Self Lift Positive Output Luo Converter Using
Incremental Conductance Algorithm for
Photovoltaic
Applications”,
Applied
Mechanics and Materials, 2014, Vol. 622.
[13]
PASCA S., TOMESCU N., Electronica
analogică și digitală – Dispozitive și circuite
electronice fundamentale [Analogical and
Digital Electronics – Devices and Fundamental
Circuits], Ed. 2, Editura Albastră, ClujNapoca, 2008, ISBN 978-973-650-227-9
[14]
OANCEA C., GOLOVANOV C., “Concept of
virtual instrument in unconventional energy
monitoring”,
Journal
of
Environment
Protection and Ecology, 2006, vol.7, no.3,
pp.594-599, Sofia, Bulgaria, ISSN 1311-5065
49
[15]
http://en.wikipedia.org/wiki/Battery_m
anagement_system (retrieved Nov. 2014).
[16]
MASSIE L., “Future trends in space power
technology”, Aerospace and Electronic
Systems, 2002, IEEE (Vol.6, Issue: 11), pp. 8–
13, ISSN 0885-8985, INSPEC Accession No:
4079618.
[17]
VARADI Z., GOROCZ V., SZABO J., “Power
subsystem for small satellites with unified
operational
mode
controller”,
2nd
International
Conference
on
Space
Technology (ICST), 2011, pp 1–4, ISBN: 978-14577-1874-8, INSPEC Accession No: 12345484,
Athens.
[18]
VAN DER ZEL V., BLEWETT M., CLARK C.,
HAMILL D., “Three generations of DC power
systems for experimental small satellites”,
Applied Power Electronics Conference and
Exposition, 1996, pp. 664–670, vol.2, Print
ISBN: 0-7803-3044-7, INSPEC Accession No:
5274384, San Jose.
[19]
WILSON
T.,
“A
high-temperature
superconductor energy-momentum control
system for small satellites”, Applied
Superconductivity, 2003, IEEE Transactions
on (Vol. 13, Issue: 2), pp. 2287–2290, ISSN
1051-8223, INSPEC Accession No: 7711408.
[20]
UNO M., KUKITA A., “Multi-port
converter integrating boost and switched
capacitor converters for single-cell battery
power system in small satellite”, ECCE Asia
Downunder (ECCE Asia), 2013, pp. 747–752,
ISBN 978-1-4799-0483-9, INSPEC Accession
No: 13712102, Melbourne.
[21]
VERA E., KINSNER W., “Autonomous
power management system for a small
satellite”,
Conference
Proceedings
Communications, Power and Computing
(WESCANEX 95), 1995, IEEE (Vol.2), pp. 312–
317, Print ISBN 0-7803-2725-X, INSPEC
Accession No: 5317002.
[22]
SYED M., MUHAMMAD S., “Modular &
COTS based power system for small LEO
satellite”, International Conference on
Aerospace Science & Engineering (ICASE),
2013, pp. 1-3, INSPEC Accession No:
14221590, Islamabad
[23]
http://www.clyde-space.com/products
/spacecraft_batteries, (retrieved Nov. 2014).
7. Biography
Andrei BĂESCU was born in
Constanţa (România), on May 18,
1988.
He graduated the Politehnica
University, Faculty of Electronics,
Telecommunications and Information Technology, in București (România), in 2011.
He is a PhD student in applied electronics and
information technology at the Politehnica
50
ELECTROTEHNICĂ, ELECTRONICĂ, AUTOMATICĂ, vol. 63 (2015), nr. 2
University, in București (România), since 2013.
His research interests concern: modern energy
conversion systems, power electronics, clean
energy systems.
Universitatea
Correspondence
address:
Politehnica
din
București,
Splaiul
Independenţei, nr.313, 060042 București,
România, e-mail: andrei.baescu@gmail.com
Andrei COCOR was born in
Constanţa (România), on October
28, 1988.
He graduated the Politehnica
University, Faculty of Electronics,
Telecommunications and Information Technology
in București (România), in 2011.
He is a PhD student in applied electronics and
information technology at the Politehnica
University, in București (România), since 2013.
His research interests concern: modern energy
conversion systems, power electronics, clean
energy systems.
Correspondence
address:
Universitatea
Politehnica
din
București,
Splaiul
Independenţei, nr.313, 060042 București,
România, e-mail: andrei.cocor@gmail.com
Adriana FLORESCU was born in
București (România), on December 3,
1964.
She graduated the Politehnica
University, Faculty of Electronics,
Telecommunications and Information Technology in
București (România), in 1988.
She received the PhD degree in electronic
engineering from the Politehnica University of
București (România), in 2001.
She is Professor at the Politehnica University, in
București (România).
Her research interests concern: power
electronics, artificial intelligence and computer
assisted analysis.
Correspondence
address:
Universitatea
Politehnica
din
București,
Splaiul
Independenţei, nr.313, 060042 București,
România, e-mail: adriana.florescu@yahoo.com
Dan STOICHESCU was born in
București (România), on November 7,
1942.
He
graduated
the
Politehnica
University, Faculty of Electronics,
Telecommunications and Information Technology in
București (România), in 1965.
He received the PhD degree in electronic
engineering from the Politehnica University of
București (România), in 1981.
He is Professor at the Politehnica University, in
București (România).
His research interests concern: automated
systems, power electronics and theory of
information transmission.
Correspondence
address:
Universitatea
Politehnica
din
București,
Splaiul
Independenţei, nr.313, 060042 București,
România, e-mail: stoich@elia.pub.ro