A 50MHz to 100MHz dc-dc power converter using Gallium Arsenide power switches is studied. GaAs Schottky rectifiers with high breakdown voltage and very small Ron x Con switching quality factor have been fabricated. A 10V to 5V (or 8V) prototype with an output power of 2.6 Watts and a power efficiency of 77% has been reported

Ajram, S.

Gaye, M.

Maynadier, P.

Salmer, G.

English

AMS Acta

AN ULTRA-HIGH SWITCHING FREQUENCY STEP-DOWN DC-DC CONVERTERBASED ON GALLIUM ARSENIDE DEVICESM. GAYE, S. AJRAM, P.MAYNADIER* and G. SALMERIEMN, UMR CNRS 8520, Avenue Poincaré, B.P. 69,59652 Villeneuve d'Ascq cedex, France,E-mail : Mamadou.Gaye@iemn.univ-lille1.fr* ALCATEL SPACE INDUSTRIES,  26, Av. Champollion BP 1187  31037Toulouse cedex, France.ABSTRACTA 50MHz to 100MHz dc-dc power converter using Gallium Arsenide power switches is studied.GaAs Schottky rectifiers with high breakdown voltage and very small Ron x Con switching qualityfactor have been fabricated. A 10V to 5V (or 8V) prototype with an output power of 2.6Watts and apower efficiency of 77% has been reported.INTRODUCTIONRegarding the evolution of electronic systems now, toward more and more integrated technologies,it appears a growing need for high switching frequencies and small size DC-DC power converters,especially those used in portable systems and for space applications.In DC-DC converter applications, high efficiency and high switching speed are of criticalimportance. GaAs MESFET devices are similar to Silicon metal-oxide semiconductor field effecttransistor (MOSFETs) in that they are majority carrier devices. They switch quickly without theminority carrier storage seen in bipolar devices. The primary advantages for using GaAs over siliconare low ON resistance, low voltage, fast switching speed, and semi-insulating substrate. The use ofGallium Arsenide power devices like MESFETs and Schottky barrier rectifiers have permitted topush the switching frequencies of DC-DC converters from 1MHz up to 150MHz [1][2]High frequency switching allows first of all to reduce proportionally the size of filtering elementslike inductors and capacitors and thus, to perform a quick control of DC output voltage. Suchcomponents have interesting applications in communication systems using dynamic-bias Class Aamplifiers, where bias voltage level has to vary according to the level of the RF input envelopesignal. This permits to improve power efficiency and to decrease power consumption [3].However, up to now, only step-up converters were used owing to their simplified circuit topology.CIRCUIT IMPLEMENTATIONIn this paper, the study is focused on the implementation of a step-down 10V to 5V (8V) converterusing floating source GaAs power MESFET and appropriate gate driver.The circuit topology is illustrated in figure 1. The main problem encountered was to realise a drivercircuit that permits to turn on/off the power MESFET, for which the source is electrically floatingand switching between 10 and 0VIn addition, taking into account the very short transition times at 100MHz (< 500ps) and thepotentially high di/dt related overshoot, it was necessary to fabricate a specific power GaAsSchottky rectifier that handles 2A-20V with reduced on state junction capacitance and seriesresistance (respectively 32pF and 0.3Ω for a 200x200µm2 area).A low doping level n-GaAs layer has been grown by MBE on an N++ substrate in our laboratory.Back-face ohmic contacts have been realised after the substrate has been thinned to 250µm.Schottky contact, mesa definition, electrolytic metal thickening, and silicon nitride passivationprocess have been done. A cross section is shown and commented in figure 2.An prototype based on hybrid circuit technology was implemented on an alumna substrate. It uses a1A-15V X-band power MESFET, a X-band MESFET based gate driver, a 110nH-HF ferrite basedvery low loss inductor and various DC decoupling chip capacitors (figure 3).RESULTSMeasurements show a high efficiency regime in the whole 40 to 100MHz frequency range (figure4).A 10V to 5V (or 8V) conversion ratio with an output power of 2.56Watts has been performed. Apower efficiency of 77% has been obtained at 40MHz. It decreases linearly when the switchingfrequency increased. This is mainly due to the switching losses in the power MESFET during theON-OFF and OFF-ON transitions where the transistor equals to its ON-state or its OFF-stateresistance. Thus, using MESFET power transistor with an 0.5Ω-ON state resistance (RON ) insteadof a 4Ω one (used in this prototype) will allows us to achieve more than 85% power efficiency at100MHz.In figure 4, we can also observe a large and linear variation of the output voltage Vout when dutyratio D varies from 30% to 80% following the theoretical Vout=D*Vin, where Vin represents theinput voltage. Concerning the power efficiency, it increases to reach 74% for higher values of dutyratio. Losses in lower duty ratio are consequences of the knee voltage of the diode (about 0.8V).On the other hand, time domain simulations using PSPICE permit to emphasise the capability of thecircuit in term of response time of the output voltage by using a Pulse Width Modulation feedbackloop (figure 5). In fact, 300ns transition times are expected, which should permit to modulate theDC output voltage at 3MHz.CONCLUSION AND PERSPECTIVESRegarding the results of this prototype, we can observe a close relationship between theperformances of the circuit and the GaAs devices characteristics. The above prototype has permittedto demonstrate the feasibility of step-down converters and to highlight the pressing need for a smartpower GaAs technology. The PWM feedback loop circuits should in fact be implemented very closeto the gate driver in order to prevent circuit oscillations and voltage overshoots.Currently, power DC-DC converters using high switching frequencies have two principallimitations: the relatively small breakdown voltage of HF switches, and HF losses of passivecomponents.In fact, L and X band power MESFETs, which seem to be the most adapted for these applicationshave a 15V to 20V breakdown voltages which allow only some low voltage DC-DC converters [4]Novel large gap III-V devices technologies such as GaN should also be investigated in order torealise higher voltage conversion systems.AKNOWLEDGEMENTSAuthors are grateful to ALCATEL SPACE INDUTRIES, Toulouse for supporting this study. Wealso are indebted to WimWaanders of Philips Components Dept in Eindhoven, for supplying ferritematerial used in this work.REFERENCES[1] S. Ajram, R. Kozlowski and G. Salmer, “Application of III-V Power Devices for DC/DCconversion”, Electronics Letters, Vol. 32, No. 1, pp. 67-68, Jan. 1996[2] Shenai, C.S. Korman, and B.J. Baliga, “High-Frequency Power Supply Applications”, HFPC,1989, May, pp. 32-61.[3] Gary Haninton, P.F. Chen, V. Radisic, T. Itoh, P.M. Asbeck, “Microwave Power AmplifierEfficiency Improvement with a 10MHz HBT DC-DC Converter”, IEEE MTT Symposium Digest,1998, pp. 589-592.[4] M. Gaye, S. Ajram, J.Y. Lebas, R. Kozlowski and G. Salmer, “A 50-100MHz 5V To –5V, 1WCUK Converter Using Gallium Arsenide Switches ”, IEEE ISCAS’2000 Symposium, May 2000,Vol 1, pp. 264-267.InductorFilteringCapacitorVinSchottkyRectifier(PWM)SwitchingDriverVoutLoadVref VoutMESFETorMOSFETFig. 1 : Square - Wave DC/DC step-down or Buck ConverterElectrolytic gold –plated contactthickeningMesaprofileN+ substrateN−Active layerSchottkycontactFig.2: SEM photography of a Si3N4-passivated Schottky power diode used in the BuckconverterFig.3: Characterisation cell of the Buck converter prototype realised at IEMN. Many RFaccesses are implemented in order to visualize switching signal waveforms5055606570758030 40 50 60 70 80 90 100 110 120Frequency (MHz)Efficiency (%)010203040506070809010020 30 40 50 60 70 80 90Duty Ratio (%)Power Efficiency (%)012345678910Output Voltage (V)Fig 4:Left : Power efficiency variations versus switching frequency (Input Voltage = 10V,Output Voltage = 8V, Load = 25Ω).Right : Power efficiency and output voltage variations versus the duty ratio (Inputvoltage=10V, Load = 25Ω, F=100MHz).Fig 5: Time variations of converter’s output voltage driven by a pulsed reference signal(PSPICE simulation)

An ultra-high switching frequency step-down DC-DC converter based on Gallium Arsenide devices

A 50MHz to 100MHz dc-dc power converter using Gallium Arsenide power switches is studied.GaAs Schottky rectifiers with high breakdown voltage and very small Ron x Con switching qualityfactor have been fabricated. A 10V to 5V (or 8V) prototype with an output power of 2.6Watts and apower efficiency of 77% has been reported

Gaye, Mamadou

Ajram, Sami

Maynadier, Paul

Salmer, Georges

HAL Descartes

An ultra high switching frequency step-down DC-DC converter based on gallium arsenide devices

AN ULTRA-HIGH SWITCHING FREQUENCY STEP-DOWN DC-DC CONVERTER
BASED ON GALLIUM ARSENIDE DEVICES
M. GAYE, S. AJRAM, P.MAYNADIER* and G. SALMER
IEMN, UMR CNRS 8520, Avenue Poincaré, B.P. 69,
59652 Villeneuve d'Ascq cedex, France,
E-mail : Mamadou.Gaye@iemn.univ-lille1.fr
* ALCATEL SPACE INDUSTRIES,  26, Av. Champollion BP 1187  31037
Toulouse cedex, France.
ABSTRACT
A 50MHz to 100MHz dc-dc power converter using Gallium Arsenide power switches is studied.
GaAs Schottky rectifiers with high breakdown voltage and very small Ron x Con switching quality
factor have been fabricated. A 10V to 5V (or 8V) prototype with an output power of 2.6Watts and a
power efficiency of 77% has been reported.
INTRODUCTION
Regarding the evolution of electronic systems now, toward more and more integrated technologies,
it appears a growing need for high switching frequencies and small size DC-DC power converters,
especially those used in portable systems and for space applications.
In DC-DC converter applications, high efficiency and high switching speed are of critical
importance. GaAs MESFET devices are similar to Silicon metal-oxide semiconductor field effect
transistor (MOSFETs) in that they are majority carrier devices. They switch quickly without the
minority carrier storage seen in bipolar devices. The primary advantages for using GaAs over silicon
are low ON resistance, low voltage, fast switching speed, and semi-insulating substrate. The use of
Gallium Arsenide power devices like MESFETs and Schottky barrier rectifiers have permitted to
push the switching frequencies of DC-DC converters from 1MHz up to 150MHz [1][2]
High frequency switching allows first of all to reduce proportionally the size of filtering elements
like inductors and capacitors and thus, to perform a quick control of DC output voltage. Such
components have interesting applications in communication systems using dynamic-bias Class A
amplifiers, where bias voltage level has to vary according to the level of the RF input envelope
signal. This permits to improve power efficiency and to decrease power consumption [3].
However, up to now, only step-up converters were used owing to their simplified circuit topology.
CIRCUIT IMPLEMENTATION
In this paper, the study is focused on the implementation of a step-down 10V to 5V (8V) converter
using floating source GaAs power MESFET and appropriate gate driver.
The circuit topology is illustrated in figure 1. The main problem encountered was to realise a driver
circuit that permits to turn on/off the power MESFET, for which the source is electrically floating
and switching between 10 and 0V
In addition, taking into account the very short transition times at 100MHz (< 500ps) and the
potentially high di/dt related overshoot, it was necessary to fabricate a specific power GaAs
Schottky rectifier that handles 2A-20V with reduced on state junction capacitance and series
resistance (respectively 32pF and 0.3Ω for a 200x200µm2 area).
A low doping level n-GaAs layer has been grown by MBE on an N++ substrate in our laboratory.
Back-face ohmic contacts have been realised after the substrate has been thinned to 250µm.
Schottky contact, mesa definition, electrolytic metal thickening, and silicon nitride passivation
process have been done. A cross section is shown and commented in figure 2.
An prototype based on hybrid circuit technology was implemented on an alumna substrate. It uses a
1A-15V X-band power MESFET, a X-band MESFET based gate driver, a 110nH-HF ferrite based
very low loss inductor and various DC decoupling chip capacitors (figure 3).
RESULTS
Measurements show a high efficiency regime in the whole 40 to 100MHz frequency range (figure
4).
A 10V to 5V (or 8V) conversion ratio with an output power of 2.56Watts has been performed. A
power efficiency of 77% has been obtained at 40MHz. It decreases linearly when the switching
frequency increased. This is mainly due to the switching losses in the power MESFET during the
ON-OFF and OFF-ON transitions where the transistor equals to its ON-state or its OFF-state
resistance. Thus, using MESFET power transistor with an 0.5Ω-ON state resistance (RON ) instead
of a 4Ω one (used in this prototype) will allows us to achieve more than 85% power efficiency at
100MHz.
In figure 4, we can also observe a large and linear variation of the output voltage Vout when duty
ratio D varies from 30% to 80% following the theoretical Vout=D*Vin, where Vin represents the
input voltage. Concerning the power efficiency, it increases to reach 74% for higher values of duty
ratio. Losses in lower duty ratio are consequences of the knee voltage of the diode (about 0.8V).
On the other hand, time domain simulations using PSPICE permit to emphasise the capability of the
circuit in term of response time of the output voltage by using a Pulse Width Modulation feedback
loop (figure 5). In fact, 300ns transition times are expected, which should permit to modulate the
DC output voltage at 3MHz.
CONCLUSION AND PERSPECTIVES
Regarding the results of this prototype, we can observe a close relationship between the
performances of the circuit and the GaAs devices characteristics. The above prototype has permitted
to demonstrate the feasibility of step-down converters and to highlight the pressing need for a smart
power GaAs technology. The PWM feedback loop circuits should in fact be implemented very close
to the gate driver in order to prevent circuit oscillations and voltage overshoots.
Currently, power DC-DC converters using high switching frequencies have two principal
limitations: the relatively small breakdown voltage of HF switches, and HF losses of passive
components.
In fact, L and X band power MESFETs, which seem to be the most adapted for these applications
have a 15V to 20V breakdown voltages which allow only some low voltage DC-DC converters [4]
Novel large gap III-V devices technologies such as GaN should also be investigated in order to
realise higher voltage conversion systems.
AKNOWLEDGEMENTS
Authors are grateful to ALCATEL SPACE INDUTRIES, Toulouse for supporting this study. We
also are indebted to WimWaanders of Philips Components Dept in Eindhoven, for supplying ferrite
material used in this work.
REFERENCES
[1] S. Ajram, R. Kozlowski and G. Salmer, “Application of III-V Power Devices for DC/DC
conversion”, Electronics Letters, Vol. 32, No. 1, pp. 67-68, Jan. 1996
[2] Shenai, C.S. Korman, and B.J. Baliga, “High-Frequency Power Supply Applications”, HFPC,
1989, May, pp. 32-61.
[3] Gary Haninton, P.F. Chen, V. Radisic, T. Itoh, P.M. Asbeck, “Microwave Power Amplifier
Efficiency Improvement with a 10MHz HBT DC-DC Converter”, IEEE MTT Symposium Digest,
1998, pp. 589-592.
[4] M. Gaye, S. Ajram, J.Y. Lebas, R. Kozlowski and G. Salmer, “A 50-100MHz 5V To –5V, 1W
CUK Converter Using Gallium Arsenide Switches ”, IEEE ISCAS’2000 Symposium, May 2000,
Vol 1, pp. 264-267.
Inductor
Filtering
Capacitor
Vin
Schottky
Rectifier
(PWM)
Switching
Driver
Vout
LoadVref Vout
MESFET
or
MOSFET
Fig. 1 : Square - Wave DC/DC step-down or Buck Converter
Electrolytic gold –
plated contact
thickening
Mesa
profile
N+ substrate
N−Active layer
Schottky
contact
Fig.2: SEM photography of a Si3N4-passivated Schottky power diode used in the Buck
converter
Fig.3: Characterisation cell of the Buck converter prototype realised at IEMN. Many RF
accesses are implemented in order to visualize switching signal waveforms
50
55
60
65
70
75
80
30 40 50 60 70 80 90 100 110 120
Frequency (MHz)
E
ff
ic
ie
n
cy
 (
%
)
0
10
20
30
40
50
60
70
80
90
100
20 30 40 50 60 70 80 90
Duty Ratio (%)
P
o
w
er
 E
ff
ic
ie
n
cy
 (
%
)
0
1
2
3
4
5
6
7
8
9
10
O
u
tp
u
t 
V
o
lt
ag
e 
(V
)
Fig 4:Left : Power efficiency variations versus switching frequency (Input Voltage = 10V,
Output Voltage = 8V, Load = 25Ω).
Right : Power efficiency and output voltage variations versus the duty ratio (Input
voltage=10V, Load = 25Ω, F=100MHz).
Fig 5: Time variations of converter’s output voltage driven by a pulsed reference signal
(PSPICE simulation)


AMS Acta - Alm@DL - Università di Bologna

AN ULTRA-HIGH SWITCHING FREQUENCY STEP-DOWN DC-DC CONVERTER
BASED ON GALLIUM ARSENIDE DEVICES
M. GAYE, S. AJRAM, P.MAYNADIER* and G. SALMER
IEMN, UMR CNRS 8520, Avenue Poincaré, B.P. 69,
59652 Villeneuve d'Ascq cedex, France,
E-mail : Mamadou.Gaye@iemn.univ-lille1.fr
* ALCATEL SPACE INDUSTRIES,  26, Av. Champollion BP 1187  31037
Toulouse cedex, France.
ABSTRACT
A 50MHz to 100MHz dc-dc power converter using Gallium Arsenide power switches is studied.
GaAs Schottky rectifiers with high breakdown voltage and very small Ron x Con switching quality
factor have been fabricated. A 10V to 5V (or 8V) prototype with an output power of 2.6Watts and a
power efficiency of 77% has been reported.
INTRODUCTION
Regarding the evolution of electronic systems now, toward more and more integrated technologies,
it appears a growing need for high switching frequencies and small size DC-DC power converters,
especially those used in portable systems and for space applications.
In DC-DC converter applications, high efficiency and high switching speed are of critical
importance. GaAs MESFET devices are similar to Silicon metal-oxide semiconductor field effect
transistor (MOSFETs) in that they are majority carrier devices. They switch quickly without the
minority carrier storage seen in bipolar devices. The primary advantages for using GaAs over silicon
are low ON resistance, low voltage, fast switching speed, and semi-insulating substrate. The use of
Gallium Arsenide power devices like MESFETs and Schottky barrier rectifiers have permitted to
push the switching frequencies of DC-DC converters from 1MHz up to 150MHz [1][2]
High frequency switching allows first of all to reduce proportionally the size of filtering elements
like inductors and capacitors and thus, to perform a quick control of DC output voltage. Such
components have interesting applications in communication systems using dynamic-bias Class A
amplifiers, where bias voltage level has to vary according to the level of the RF input envelope
signal. This permits to improve power efficiency and to decrease power consumption [3].
However, up to now, only step-up converters were used owing to their simplified circuit topology.
CIRCUIT IMPLEMENTATION
In this paper, the study is focused on the implementation of a step-down 10V to 5V (8V) converter
using floating source GaAs power MESFET and appropriate gate driver.
The circuit topology is illustrated in figure 1. The main problem encountered was to realise a driver
circuit that permits to turn on/off the power MESFET, for which the source is electrically floating
and switching between 10 and 0V
In addition, taking into account the very short transition times at 100MHz (< 500ps) and the
potentially high di/dt related overshoot, it was necessary to fabricate a specific power GaAs
Schottky rectifier that handles 2A-20V with reduced on state junction capacitance and series
resistance (respectively 32pF and 0.3Ω for a 200x200µm2 area).
A low doping level n-GaAs layer has been grown by MBE on an N++ substrate in our laboratory.
Back-face ohmic contacts have been realised after the substrate has been thinned to 250µm.
Schottky contact, mesa definition, electrolytic metal thickening, and silicon nitride passivation
process have been done. A cross section is shown and commented in figure 2.
An prototype based on hybrid circuit technology was implemented on an alumna substrate. It uses a
1A-15V X-band power MESFET, a X-band MESFET based gate driver, a 110nH-HF ferrite based
very low loss inductor and various DC decoupling chip capacitors (figure 3).
RESULTS
Measurements show a high efficiency regime in the whole 40 to 100MHz frequency range (figure
4).
A 10V to 5V (or 8V) conversion ratio with an output power of 2.56Watts has been performed. A
power efficiency of 77% has been obtained at 40MHz. It decreases linearly when the switching
frequency increased. This is mainly due to the switching losses in the power MESFET during the
ON-OFF and OFF-ON transitions where the transistor equals to its ON-state or its OFF-state
resistance. Thus, using MESFET power transistor with an 0.5Ω-ON state resistance (RON ) instead
of a 4Ω one (used in this prototype) will allows us to achieve more than 85% power efficiency at
100MHz.
In figure 4, we can also observe a large and linear variation of the output voltage Vout when duty
ratio D varies from 30% to 80% following the theoretical Vout=D*Vin, where Vin represents the
input voltage. Concerning the power efficiency, it increases to reach 74% for higher values of duty
ratio. Losses in lower duty ratio are consequences of the knee voltage of the diode (about 0.8V).
On the other hand, time domain simulations using PSPICE permit to emphasise the capability of the
circuit in term of response time of the output voltage by using a Pulse Width Modulation feedback
loop (figure 5). In fact, 300ns transition times are expected, which should permit to modulate the
DC output voltage at 3MHz.
CONCLUSION AND PERSPECTIVES
Regarding the results of this prototype, we can observe a close relationship between the
performances of the circuit and the GaAs devices characteristics. The above prototype has permitted
to demonstrate the feasibility of step-down converters and to highlight the pressing need for a smart
power GaAs technology. The PWM feedback loop circuits should in fact be implemented very close
to the gate driver in order to prevent circuit oscillations and voltage overshoots.
Currently, power DC-DC converters using high switching frequencies have two principal
limitations: the relatively small breakdown voltage of HF switches, and HF losses of passive
components.
In fact, L and X band power MESFETs, which seem to be the most adapted for these applications
have a 15V to 20V breakdown voltages which allow only some low voltage DC-DC converters [4]
Novel large gap III-V devices technologies such as GaN should also be investigated in order to
realise higher voltage conversion systems.
AKNOWLEDGEMENTS
Authors are grateful to ALCATEL SPACE INDUTRIES, Toulouse for supporting this study. We
also are indebted to WimWaanders of Philips Components Dept in Eindhoven, for supplying ferrite
material used in this work.
REFERENCES
[1] S. Ajram, R. Kozlowski and G. Salmer, “Application of III-V Power Devices for DC/DC
conversion”, Electronics Letters, Vol. 32, No. 1, pp. 67-68, Jan. 1996
[2] Shenai, C.S. Korman, and B.J. Baliga, “High-Frequency Power Supply Applications”, HFPC,
1989, May, pp. 32-61.
[3] Gary Haninton, P.F. Chen, V. Radisic, T. Itoh, P.M. Asbeck, “Microwave Power Amplifier
Efficiency Improvement with a 10MHz HBT DC-DC Converter”, IEEE MTT Symposium Digest,
1998, pp. 589-592.
[4] M. Gaye, S. Ajram, J.Y. Lebas, R. Kozlowski and G. Salmer, “A 50-100MHz 5V To –5V, 1W
CUK Converter Using Gallium Arsenide Switches ”, IEEE ISCAS’2000 Symposium, May 2000,
Vol 1, pp. 264-267.
Inductor
Filtering
Capacitor
Vin
Schottky
Rectifier
(PWM)
Switching
Driver
Vout
LoadVref Vout
MESFET
or
MOSFET
Fig. 1 : Square - Wave DC/DC step-down or Buck Converter
Electrolytic gold –
plated contact
thickening
Mesa
profile
N+ substrate
N−Active layer
Schottky
contact
Fig.2: SEM photography of a Si3N4-passivated Schottky power diode used in the Buck
converter
Fig.3: Characterisation cell of the Buck converter prototype realised at IEMN. Many RF
accesses are implemented in order to visualize switching signal waveforms
50
55
60
65
70
75
80
30 40 50 60 70 80 90 100 110 120
Frequency (MHz)
Ef
fic
ie
nc
y 
(%
)
0
10
20
30
40
50
60
70
80
90
100
20 30 40 50 60 70 80 90
Duty Ratio (%)
Po
w
er
 E
ffi
ci
en
cy
 (%
)
0
1
2
3
4
5
6
7
8
9
10
O
ut
pu
t V
ol
ta
ge
 (V
)
Fig 4:Left : Power efficiency variations versus switching frequency (Input Voltage = 10V,
Output Voltage = 8V, Load = 25Ω).
Right : Power efficiency and output voltage variations versus the duty ratio (Input
voltage=10V, Load = 25Ω, F=100MHz).
Fig 5: Time variations of converter’s output voltage driven by a pulsed reference signal
(PSPICE simulation)


https://hal.archives-ouvertes.fr/hal-00157875

An ultra-high switching frequency step-down DC-DC converter based on Gallium Arsenide devices

Abstract

Similar works

Full text

Available Versions

AMS Acta

HAL Descartes

AMS Acta

AMS Acta - Alm@DL - Università di Bologna