Transistor configuration

We know that, transistor can be used as an amplifier. For an amplifier, two terminals are required to supply the weak signal and two terminals to collect the amplified signal. Thus four terminals are required but a transistor is said to have only three terminals Therefore, one terminal is used common for both input and output.

This gives rise to three different combinations.

1.      Common base configuration (CB)

2.      Common emitter configuration (CE)

3.      Common collector configuration (CC)

1. CB configuration

A simple circuit arrangement of CB configuration for pnp transistor is shown below.

In this configuration, base is used as common to both input and output. It can be noted that the i/p  section has an a.c. source V_i along with the d.c. source V_EB. The purpose of including V_EB is to keep EB junction always forward biased (because if there is no V_EB then the EB junction is forward biased only during the +ve half-cycle of the i/p and reverse biased during the –ve half cycle). In CB configuration, I_E –i/p current, I_C –o/p current.

TRANSISTORS

A transistor is a sandwich of one type of semiconductor (P-type or n-type) between two layers of other types.

Transistors are classified into two types;

1.      pnp transistor

pnp transistor is obtained when a n-type layer of silicon is sandwiched between two p-type silicon material.

2.      npn transisitor

npn transistor is obtained when a p-type layer of silicon is sandwiched between two n-type silicon materials.

Figure3.1  below shows the schematic representations of a transistor which is equivalent of two diodes connected back to back.

Rhéostat de glissement rotorique

Il n’existe aucun variateur industriel fonctionnant sur ce principe. Le fonctionnement est rappelé ici pour mémoire.

Comme précédemment, le but est de déplacer le point de fonctionnement.

Concrètement, pour obtenir une variation continue de la vitesse, les résistances additionnelles sont remplacées par un gradateur dont l'effet est une modification de la résistance "apparente".

Les bobines rotoriques sont fermées sur des résistances montées en série avec un gradateur de courant. Ce gradateur permet d’ajuster la valeur efficace des courants rotoriques, faisant ainsi varier le glissement g. 

Gradateur

On peut faire varier la vitesse d’un moteur à cage d’écureuil en faisant simplement varier la tension à ses bornes. Pour un couple résistant donné, lorsque la tension d’alimentation diminue, le glissement augmente et, par conséquent, la vitesse est réduite.

Cette méthode de commande de vitesse est intéressante lorsque la charge exerce un couple qui varie sensiblement avec le carré de la vitesse, comme, par exemple, un ventilateur ou une pompe centrifuge.

Une réduction de la tension d'alimentation provoque une réduction du couple moteur. Ceci va provoquer une modification du point de fonctionnement de l'ensemble moteur - machine entraînée.

Cette technique simple présente l'inconvénient de limiter fortement les utilisations. On voit ci-contre que si le ventilateur à couple parabolique trouve une plage assez large de variation, il n'en est pas de même du tapis roulant (à couple constant) qui peut très vite décrocher du fait de la forte diminution du couple moteur.

Le diagramme synoptique d‘un gradateur alimentant un moteur d’induction triphase est représenté ci-dessous :

Pour abaisser la tension, il suffit d’augmenter l’angle d’amorcage. Par exemple, lorsque les impulsions sont retardées de 100°, la tension aux bornes du moteur est environ 50% de la tension nominale.

Onduleur autonome à modulation de largeur d’impulsion (M LI)

a/ Justification :

Les onduleurs à source de tension génèrent des tensions et des courants dont la composante harmonique est relativement élevée. Ces harmoniques produisent des couples pulsatifs.  Quand le moteur tourne à une vitesse relativement élevée, ces pulsations sont amorties par l’inertie mécanique.

Cependant, à basse vitesse, elles peuvent produire une vibration considérable. Dans certaines applications, comme les machines outils, ces vibrations sont inacceptables si la haute précision est recommandée. Dans ce cas, un système d’entraînement utilisant un onduleur à modulation de largeur d’impulsion (MLI) est la solution.

b/ Schéma de principe :

Ce type de variateur est composé d’un pont redresseur qui produit une tension constante, d’un filtre et d‘un onduleur à thyristor ou à transistor. Grâce aux signaux émis par l’unité de commande d’allumage, l’onduleur génère une série d’impulsions de tension positives d’amplitude constante, suivies par une série d’impulsions semblables mais de signe contraire. La largeur de ces impulsions et les intervalles les séparant sont ajustés de sorte que la forme d’onde se rapproche d’une sinusoïde. À basse fréquence, les impulsions sont moins larges ce qui donne une tension efficace moins grande permettant de garder le rapport tension fréquence constant.

Zener voltage regulator

The circuit diagram of Zener voltage regulator is shown below

A zener diode of breakdown voltage V_Z is connected in reverse biased condition across the load R_L such that it operates in breakdown region. Any fluctuations in the current are absorbed by the series resistance R_s. The Zener will maintain a constant voltage V_Z

( equal to V_o) across the load unless the input voltage does not fall below the zener breakdown voltage V_Z.

Case(i) When input voltage V_in varies and R_L is constant

If the input voltage increases, the Zener diode which is in the breakdown region is equivalent to  a battery V_Z as shown in figure. The output voltage remains constant at V_Z (equal to Vo) and the excess voltage is dropped across the series resistance R_S. We know that for a zener diode under breakdown region large change in current produces very small change in voltage, thereby the output voltage remains constant.

Case (ii) When V_in is constant and R_L varies.

If there is a decrease in the load resistance R_L and the input voltage remains constant then there is a increase in load current.

Since V_in is constant the current cannot come from the source. This addition load current is driven from the battery V_Z and we know that even for a large decrease in current the Zener output voltage V_z remains same. Hence the output voltage across the load is also constant..

Zener Diode

The reverse voltage characteristics of a semiconductor diode including the breakdown region is shown below.

Zener diodes are the diodes which are designed to operate in the breakdown region. They are also called as Breakdown diode or Avalanche diodes.

The symbol of Zener diode is shown below

The breakdown in the Zener diode at the voltage V_z may be due to any of the following mechanisms.

1. Avalanche breakdown

• We know that when the diode is reverse biased a small reverse saturation current I₀ flows across the junction because of the minority cariers in the depletion region.

·         The velocity of the minority charge carriers is directly proportional to the applied voltage. Hence when the reverse bias voltage is increased, the velocity of minority charge carriers will also increase and consequently their energy content will also increase.

• When these high energy charge carriers strikes the atom within the depletion region they cause other charge carriers to break away from their atoms and join the flow of current across the junction as shown above. The additional charge carriers generated in this way strikes other atoms and generate  new carriers by making them to break away from their atoms.

• This cumulative process is referred to as avalanche multiplication which results in the flow of large reverse current and this breakdown of the diode is called avalanche breakdown.

2.Zener breakdown

We have electric field strength = Reverse voltage/ Depletion region

From the above relation we see that the reverse voltage is directly proportional to the electric field hence, a small increase in reverse voltage produces a very high intensity electric field with ina narrow Depletion region.

Therefore when the reverse voltage to a diode is increased, under the influence of high intensity electric filed large numbr of electrons within the depletion region break the covalent bonds with their atoms as shown above and thus a large reverse current flows through the diode. This breakdown is referred to as Zener breakdown.

FILTERS

We know that the output of the rectifier is pulsating d.c. ie the output obtained by the rectifier is not pure d.c. but it contains some ac components along with the dc o/p. These ac components are called as Ripples, which are undesirable or unwanted. To minimize the ripples in the rectifier output filter circuits are used. These circuits are normally connected between the rectifier and load as shown below.

Filter is a circuit which converts pulsating dc output from a rectifier to a steady dc output. In otherwords, filters are used to reduce the amplitudes of the unwanted ac components in the rectifier.

Note: A capacitor passes ac signal readily but blocks dc.

2.8.1 Types of Filters

1.      Capacitor Filter (C-Filter)

2.      Inductor Filter

3.      Choke Input Filter (LC-filter)

4.      Capacitor Input Filter (Π-filter)

2.8.2 Capacitor Filter( C-filter)

• When the Input signal rises from o to a the diode is forward biased therefore it starts conducting since the capacitor acts as a short circuit for ac signal it gets charged up to the peak of the input signal and the dc component flows through the load R_L.

• When  the input signal fall from a to b the diode gets reverse biased . This is mainly because of the voltage across the capacitor obtained during the period o to a is more when comapared to V_i. Therefore there is no conduction of current through the diode.

• Now the charged capacitor acts as a battery and it starts discharging through the load R_L. Mean while the input signal passes through b,c,d section. When the signal reaches the point d the diode is still reverse biased since the capacitor voltage is more than the input voltage.

• When the signal reaches point e, the input voltage can be expected to be more than the  capacitor voltage. When the input signal moves from e to f the capacitor gets charged to  its peak value again. The diode gets reverse biased and the capacitor starts discharging. The final output across R_L is shown in Fig. 2.8

The ripple factor for a Half-wave rectifier with C-filer is given by

r= 1/2√3f_CR_L

f-----the line frequency ( Hz)

C-----capacitance ( F)

R_L------- Load resistance (Ω)

Ripple factor for full-wave rectifier with C-filter is given by r = 1/ 4 √3 f _C R_L

2.8.3 Advantages of C-Filter

• low cost, small size and good characteristics.
• It is preferred for small load currents ( upto 50 mA)
• It is commonly used in transistor radio, batteries eliminator etc.

Onduleur autonome à source de tension

a/Principe :

            Ce variateur de vitesse est constitué d’un redresseur à thyristors suivi d’un filtre de tension (inductance et condensateur) et d’un onduleur autonome à thyristors ou à transistors.

L’unité de commande et d’allumage est construite autour d’un module intelligent à microcontrôleur. Cette unité génère les impulsions d’amorçage ou de blocage aux moments appropriés pour ajuster les valeurs de tension ou de fréquence.

b/ Conservation de couple :

Afin de maintenir un couple maximal constant à la charge, le redresseur fournit à l’onduleur une tension qui est  proportionnelle à la fréquence.  Le rapport entre la tension et la fréquence est un paramètre qu’il est possible d’ajuster selon l’application.

VARIATEURS DE VITESSE POUR MACHINE A CA.

INTRODUCTION :

            Avec le développement de l’électronique de puissance, les moteurs à courant alternatif sont de plus en plus utilisés pour les entraînements à vitesse variable. Les moteurs alternatifs présentent certains avantages sur les moteurs à courant continu :

·       construction plus robuste, aucun collecteur, ce qui demande moins d’entretien.

·       coût de construction moindre.

·       plus faible inertie.

·       grande précision de la vitesse de rotation, de l’ordre de 0,5% en boucle ouverte.

·       possibilité de très grande vitesse de rotation pour les petits moteurs.

      Par contre, les variateurs électroniques de vitesse sont plus complexes que ceux des moteurs à courant continu. En effet, la vitesse de rotation des moteurs à courant continu est commandéé par la variation de la tension à leurs bornes, tandis que pour les moteurs à courant alternatif,  la tension et la fréquence doivent-être variables.

Pour faire varier la vitesse d’un moteur à courant alternatif  on utilise:

·       un gradateur ;

·       un onduleur autonome à fréquence variable ;

·       un cycloconvertisseur.

A/ GENERALITES :

Bridge rectifier

·        The circuit diagram of a bridge rectifer is shown above. It uses four diodes and a transformer.

·        During the +ve half-cycle, end A is +ve and end B is –ve thus diodes D₁ and D₃ are forward bias while diodes D₂ and D₄ are reverse biased thus a current flows through diode D_1, load R_L ( C to D) and diode D₃.

·        During the –ve half-cycle, end B is  +ve and end A is –ve thus diodes D₂ and D₄  are forward biased while the diodes D₁ and D₃ are reverse biased. Now the flow of current is through diode D₄ load R_L ( D to C) and diode D₂. Thus, the waveform is same as in the case of center-tapped full wave rectifier.

Centre tapped full –wave rectifier

·         The circuit diagram of a center tapped full wave rectifier is shown in fig. 2.6 above. It employs two diodes and a center tap transformer.  The a.c. signal to be rectified is applied to the primary of the transformer and the d.c. output is taken across the load R_L.

·         During the +ve half-cycle end X is +ve and end Y is –ve this makes diode D₁ forward biased and thus a current i₁ flows through it and load resistor R_L.Diode D₂  is reverse biased and the current i₂ is zero.

• During the –ve half-cycle end Y is +Ve and end X is –Ve. Now diode D₂ is forward biased and thus a current i₂ flows through it and load resistor R_L. Diode D₁ is reversed and the current i₁ = 0.

Disadvantages

• Since, each diode uses only one-half of the transformer secondary voltage the d.c. output is comparatively small.
• It is difficult to locate the center-tap on secondary winding of the transformer.
• The diodes used must have high Peak-inverse voltage.

Half-wave rectifier

The circuit diagram of  a half-wave rectifier is shown in fig. 2.5 below along with the I/P and O/P waveforms.

• The transformer is employed in order to step-down the supply voltage and also to prevent from shocks.
• The diode is used to rectify the a.c. signal while , the pulsating d.c. is taken across the load resistor RL.
• During the +ve half cycle, the end X of the secondary is +ve and end Y is  -ve . Thus , forward biasing the diode. As the diode is forward biased, the current flows through the load RL and a voltage is developed across it.
• During the –ve half-cycle the end Y is +ve and end X is –ve  thus, reverse biasing the diode. As the diode is reverse biased there is no flow of current through RL thereby the output voltage is zero.

Redresseur type parallèle double PD3

Comme pour un redresseur parallèle double à diodes triphasé, la charge voit une tension égale à la différence entre la tension délivrée par le commutateur « plus positif » et celle fournie par le commutateur plus négatif ».

Le thyristor Th1 est susceptible de conduire lorsque la tension V1 est la plus positive des composantes V1, V2 et V3. Il est commandé à l’amorçage après un angle de retard α (retard par rapport à la conduction naturelle des diodes). Le thyristor Th4 st à son tour susceptible de conduire lorsque V2 devient la plus négative. Il est commandé à l’amorçage après un angle de retard à l’amorçage α. Si ces deux thyristors conduisent simultanément, on aura en sortie        Vs= V1-V2=U12.

Redresseurs commandés à thyristors triphasés

1/Considérations générales :

Contrairement à la commande en monophasé, ou l’angle de retard à l’amorçage des thyristors est référencé au zéro de la sinusoïde du secteur, en triphasé le point de référence est l’instant ou deux tensions composant le système triphasé équilibré deviennent égales (instant de conduction des diodes dans un redresseur non commandé).

2/Redresseur type parallèle P3 :

Le thyristor Th1 est susceptible de conduire à l’instant Л/6 (instant de conduction naturelle des diodes) ou la tension V1 devient la plus positive. Contrairement à une diode, le thyristor ne pourra conduire que lorsqu’une impulsion de gâchette lui est délivrée.

Diode equivalent circuit

It is generally profitable to replace a device or system by its equivalent circuit. Once the device is replaced by its equivalent circuit, the resulting network can be solved by traditional circuit analysis technique.

The forward current I_f  flowing through the diode causes a voltage drop in its internal resistance r_f. Therefore the forward voltage V_F applied across the actual diode has to overcome

1. potential barrier Vo
2. internal drop I_f r_f

V_f = Vo + I_f r_f

For silicon diode Vo=0.7V whereas for Germanium diode Vo = 0.3 V.

 For ideal diode r_f =0.

24.1 Basic Definitions

1.Knee voltage or Cut-in Voltage.

It is the forward voltage at which the diode starts conducting.

2. Breakdown voltage

It is the reverse voltage at which the diode (p-n junction) breaks down with sudden rise  in reverse current.

3. Peak-inverse voltage (PIV)

It is the max. reverse voltage that can be applied to a p-n junction without causing damage to the junction.

If the reverse voltage across the junction exceeds its peak-inverse voltage, then the junction exceeds its Peak-inverse voltage, then the junction gets destroyed because of excessive heat. In rectification, one thing to be kept in mind is that care should be taken that reverse voltage across the diode during –ve half cycle of a.c. doesnot exceed the peak-inverse voltage of the diode.

4. Maximum Forward current

It is the Max. instantaneous forward current that a p-n junction can conduct without damaging the junction. If the forward current is more than the specified rating then the junction gets destroyed due to over heating.

5.Maximum Power rating

It is the maximum power that can be dissipated at the junction without damaging it. The power dissipated across the junction is equal to the product of junction current and the voltage across the junction.

SEMICONDUCTOR DIODE

When a p-type semiconductor material is suitably joined to n-type semiconductor the contact surface is called a p-n junction. The p-n junction is also called as semiconductor diode.

• The left side material is a p-type semiconductor having –ve acceptor ions and +vely charged holes. The right side material is n-type semiconductor having +ve donor ions and free electrons.

• Suppose the two pieces are suitably treated to form pn junction, then there is a tendency for the free electrons from n-type to diffuse over to the p-side and holes from p-type to the n-side . This process is called diffusion.

Redresseur double alternance à pont mixte

Il consiste en deux thyristors seulement. Les deux autres sont remplacés par des diodes. Ainsi si le thyristor Th1 est amorcé, la diode D2 se met spontanément en conduction pour fermer le circuit de la charge. La différence par rapport à un pont complet est que la tension Vs ne pourra plus devenir négative (conduction simultanée de D1 et D2).

α =Л/3 :

Redresseur double alternance à pont complet

Nous supposerons que la charge connectée au redresseur est telle que le courant ne s’annule jamais au cours de la période, donc il y a toujours des thyristors en conduction (Hypothèse de conduction continue).

Pendant l’alternance positive, les thyristors Th1 et Th4 sont amorcés à l’instant α ainsi Vs=Ve. Les thyristors Th1 et Th4 continuent à conduire même après l’inversion de la tension du secteur puisque le courant n’est pas interrompu.

A l’instant  Л+α, on envoie une impulsion d’amorçage aux gâchettes de Th2 et Th3. Ceux-ci s’amorcent puisque la tension VAK qui leur est appliquée est positive. Leur amorçage provoque une extraction du courant anodique de Th1 et Th2 et l’inversion de leur tension VAK, ils se bloquent donc. Dans ces conditions, Vs = -Ve.

Redresseur commandé simple alternance

Les impulsions d’amorçage sont envoyées en retard par rapport au zéro du secteur d’un angle α. Ainsi, on amorce le thyristor aux instants  α, 2Л+α, 4Л+α, ….etc.

Lorsque le thyristor s’amorce, on aura  Vs=Ve. Dans le cas d’une charge résistive, le courant Is s’annule lorsque la tension secteur passe par zéro et le thyristor se bloque spontanément.

La valeur moyenne récupérée  est :   Vsmoy =Vmax/(2Л) . (1+cosα), elle est maximale pour α=0°.

HALL EFFECT

HALL EFFECT

If a piece of metal or semiconductor carrying a current I is placed in a transverse magnetic field B then an electric field E is induced in the direction perpendicular to both I and B. This phenomenon is known as Hall effect.

Hall effect is normally used to determine whether a semi-conductor is n-type or p-type.

To find whether the semiconductor is n-type or p-type

i)                    In the figure. above, If I is in the +ve X direction and B is in the +ve  Z direction, then a force will be exerted on the charge carriers (holes and electrons) in the –ve Y direction.

ii)                  This force is independent of whether the charge carriers are electrons or holes. Due to this force the charge carriers ( holes and electrons) will be forced downward towards surface –1 as shown.

iii)                If the semiconductor is N-type, then electrons will be the charge carriers and these electrons will accumulate on surface –1 making that surface –vely charged with respect to surface –2. Hence a potential called Hall voltage appears between the surfaces 1 and 2.

iv)                Similarly when surface –1 is positively charged with respect to surface –2, then the semiconductor is of P-type. In this way, by seeing the polarity of Hall voltage we can determine whether the semiconductor is of P-type or N-type.

Applications of Hall effect

Hall effect is used to determine,

• carrier concentration, conductivity and mobility.

• The sign of  the current carrying charge.

• Charge density.

• It is used as magnetic field meter.

Carrier lifetime (τ)

In a pure semiconductor, we know that number of holes are equal to the number of electrons. Thermal agitation however, continues to produce new hole electron pairs while other hole-electron pair disappear as a result of recombination.

On an average, a hole will exist for τ_p second and an electron will exist for τ_n second before recombination. This time is called the carrier lifetime or Mean lifetime.

The average time an electron or hole can exist in the free state is called carrier lifetime.

Drift and Diffusion current

Drift and Diffusion current

The flow of current through a semiconductor material is normally referred to as one of the two types.

Drift current
• If an electron is subjected to an electric field in free space it will accelerate in a straight line form the –ve terminal to the + ve terminal of the applied voltage.

• However in the case of conductor or semiconductor at room temperature, a free electrons under the influence of electric field will move towards the +ve terminal of the applied voltage but will continuously collide with atoms all the ways as shown in figure 1.9.

 

Each time, when the electron strikes an atom, it rebounds in a random direction but the presence of electric field doesnot stop the collisions and random motion. As a result the electrons drift in a direction of the applied electric field.

•  The current produced in this way is called as Drift current and it is the usual kind of current flow that occurs in a conductor.

Diffusion current

• The directional movement of charge carriers due to their concentration gradient produces a component of current known as Diffusion current.

• The mechanism of transport of charges in a semiconductor when no electric field is applied called diffusion. It is encountered only in semiconductors and is normally absent in conductors.

With no applied voltage if the number of charge carriers (either holes or electrons) in one region of a semiconductor is less compared to the rest of the region then there exist a concentration gradient.

• Since the charge carriers are either all electrons or all holes they sine polarity of charge and thus there is a force of repulsion between them.

• As a result, the carriers tend to move gradually or diffuse from the region of higher concentration to the region of lower concentration. This process is called diffusion and electric current produced due to this process is called diffusion current.

• This process continues until all the carriers are evenly distributed through the material. Hence when there is no applied voltage, the net diffusion current will be zero.

Fermi-level

Fermi level indicates the level of energy in the forbidden gap.

1. Fermi-level for an Intrinsic semiconductor

•
• We know that the Intrinsic semiconductor acts as an insulator at absolute zero temperature because there are free electrons and holes available but as the temperature increases electron hole pairs are generated and hence number of electrons will be equal to number of holes.
• Therefore, the possibility of obtaining an electron in the conduction band will be equal to the probability of obtaining a hole in the valence band.

• If Ec is the lowest energy level of Conduction band and Ev is the highest energy level of the valence band then the fermi level Ef is exactly at the center of these two levels as shown above.

2. Fermi-level in a semiconductors having impurities (Extrinsic)
a) Fermi-level for n-type Semiconductor
• Let a donar impurity be added to an Intrinsic semiconductor then the donar energy level (ED)  shown by the dotted lines is very close to conduction band energy level (Ec).
• Therefore the unbonded valence electrons of the impurity atoms can very easily jump into the conduction band and become free electros thus, at room temperature almost all the extra electrons of pentavalent impurity will jump to the conduction band.

• The donar energy level (ED) is just below conduction band level (Ec) as shown in figure1.10(a). Due to a large number of free electrons, the probability of electrons occupying the energy level towards the conduction band will be more hence, fermi level shifts towards the conduction band.

b) Fermi-level for P-type semiconductor

• Let an acceptor impurity be added to an Intrinsic semiconductor then the acceptor energy level (Ea) shown by dotted lines is very close to the valence band shown by dotted lines is very close to the valence band energy level (Ev).

• Therefore the valence band electrons of the impurity atom can very easily jump into the valence band thereby creating holes in the valence band.

 

• The acceptor energy level (EA) is just above the valence band level as shown in figure 1.11 (b).
•  Due to large number of holes the probability of holes occupying the energy level towards the valence band will be more and hence, the fermi level gets shifted towards the valence band.

 

Redresseurs à diodes triphasés

1/Considérations générales :

Les charges industrielles connectées aux redresseurs sont généralement des récepteurs inductifs (moteurs à courant continu ….). On pourra généralement adopter les considérations suivantes :

       Le courant circule en permanence dans le récepteur.

       Ce courant est presque constant.

Cette situation correspond au cas le plus fréquent des redresseurs industriels qui sont connectés généralement à un réseau triphasé.

2/Redresseur type parallèle P3:

Chaque phase du secondaire du transformateur triphasé est mise en série avec une diode. Les diodes sont montées en cathodes équipotentielles. C’est la diode qui voit la tension la plus positive qui conduit.

Redresseurs à diodes monophasés

1/Redresseur simple alternance :

Le signal d’entrée Ve est sinusoïdal d’amplitude Vmax et fréquence f (généralement f=50Hz) : Ve= Vmax sin(2Лft). Pendant l’alternance positive la diode D conduit et Vs=Ve. Pendant l’alternance négative la diode D se bloque spontanément et Vs=0V (charge résistive). Le signal obtenu possède une valeur moyenne   Vmoy= 1/T ∫ Vs(t) dt = Vmax /Л  et une valeur efficace  Veff= Vmax/2.

Choix de la diode : La diode doit conduire un courant redressé moyen de valeur Vmax/RЛ. Elle doit également supporter une tension inverse de valeur Vmax.