The dielectric one
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Calculation
of the capacitance of a capacitor |
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Created it, 05/10/15
Update it, 05/12/16
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CONDENSERS “1st PART”
After having studied resistances, let us see the operation of the condensers.
Since one needs different quantities of electricity to carry bodies of different size to the same electric potential, we can characterize each body by the quantity of electricity which it must have to reach the potential of a volt ; this quantity of electricity is the electric capacity (symbol C) of the body.
For a body which has a quantity of determined electricity, and which is with a determined potential, one obtains the electric capacity by dividing the quantity of electricity by the potential.
For example, one obtains the capacity of a body which has a quantity of electricity of four coulombs and which is with a potential of eight volts by making division 4 / 8 = 0,5 ; 0,5 are thus needed Coulomb to reach the potential of a volt, i.e. the electric capacity is 0,5 Coulomb per volt.
Thus, the electric capacity is measured in Coulomb per volt, measuring unit to which was given the name of farad (symbol F), in the honor of the scientist Michael FARADAY, already quoted for its research on the electrolytic solutions. The body considered in our example thus has a capacity of 0,5 farad.
It should be noted that a sphere of the dimension of the ground would have a capacity from approximately 1 F ; the farad is thus a measuring unit too much large.
For this reason, in practice, we use mainly the submultiples of the farad is :
The microfarad (symbol
µF) which is equivalent to
one millionth of farad (10-6 F).
The nanofarad
(symbol nF)
which is equivalent to a billionth of farad (10-9
F is 10-3 µF).
The picofarad (symbol
pF) which is
equivalent to one millionth of millionth of farad (10-12 F, is 10-6
µF or 10-3 nF).
The capacity of a body depends initially on the presence in its vicinity of other electrified bodies.
This observation can be made in experiments by considering the two metal plates of figure 1.
These two rigorously identical metal plates are connected each one to a pole of a pile, and take care of positive or negative electricity according to the pole to which they are connected.
With the figure 1-a, we note that the process of load of the two plates is very simple, a certain number of electrons goes from the negative pole of the pile to the plate which is connected to him, charging it negatively, while the positive pole of the pile attracts a number equal of electrons to the plate which is connected to him ; this one takes care positively.
In the drivers which connect each plate to the pile, it creates for itself a movement of electrons whose direction is indicated figure 1-a. The movement of electrons ceases itself when the quantity of electrons present on each plate is such as each plate is with the same potential as the pole of the pile which is connected to him.
Between the two plates exists the same potential difference as at the boundaries of the pile.
As we saw previously, the quantity of electricity which each plate will be able to store depends on its capacity, but, the two plates being identical, they thus have the same capacity they store two equal quantities of electricity but one is positive and the other negative one.
Let us suppose now that we bring closer the two plates by laying out them well opposite one the other, as indicated on the figure 1-b, but by avoiding any contact between them not to put the pile in short-circuit.
This bringing together of the plates, it appears a new circulation of electrons, in the direction indicated figure 1-b and thus an increase in the quantity of electricity contained on each plate.
For the moment, limit we to note this established fact, the explanation will be given by it later.
We note that the quantity of electricity on each plate increased, although the potential of those did not change.
We can thus affirm that by bringing closer two plates, their capacity increases.
Since the capacity changes while varying the distance between the plates, we should not any more take account of only one plate but consider a unit made up of two plates placed vis-a-vis vis-a-vis a determined distance, in the manner indicated figure 2.
This provision represents the simplest type of the condenser, which precisely consists of two plates in glance, called for the circumstance reinforcements, provided with two drivers (called terminals) for their connection with the circuits. On figure 2 the graphic symbol of the condenser is also given to you such as you meet it in the electric diagrams.
Whatever their characteristics or the manufacturer who conceives them, a condenser always consists of two reinforcements separated by an insulator.
To define the capacitance of a capacitor, it is necessary to take account of its two reinforcements and to thus consider the potential difference existing between them.
Although a condenser is composed in all the cases of two reinforcements, only one quantity of electricity takes into consideration. It is consisted of the electrons which, as we saw in figure 1, passed from the reinforcement become positive on the reinforcement become negative and this via the pile.
We must only consider the missing quantity of electrons on a reinforcement or that presents of surplus on the other, since it acts, in any event, of the same quantity of electricity which was transferred from a reinforcement on the other.
The capacitance of a capacitor is obtained by dividing the quantity of electricity present on one of its reinforcements by the potential difference existing between its reinforcements.
The condenser is an element of the electric circuits and is characterized by its capacity, as resistance is characterized by its resistive value.
We know the role of the resistance which is to produce voltage drops, and we will see the role of the condenser later on.
The first condenser was produced by Dutch Pierre MUSSCHENBROCK (1692 - 1761) who discovered the properties almost by chance of them at the same time as German George VON KLEIST (1700 - 1748), during its experiments on electricity.
The experiments of these scientists showed the influence that with on the capacitance of a capacitor insulating matter placed between its reinforcements, and who constitutes his dielectric.
The dielectric one of the condenser of figure 2 is the air and for this reason, this condenser is called condensing with air.
The dielectric one of the condensers can also be another insulating material such as the mica, paraffined paper, polystyrene, certain ceramic substances, etc…
Very quickly, the scientists noted that the capacitance of a capacitor to air increased when they put between his reinforcements a dielectric solid; for example, a mica plate laid out between the plates of a capacitor makes increase its capacity from five to six times, according to the mica employed.
What means that, by having always the same potential difference between the reinforcements of the condenser, the quantity of electricity presents on those becomes five to six times higher if one replaces the air by the mica sheet.
This behavior is due to the fact that the dielectric solid put between the reinforcements of the condenser polarizes, as we will see it.
Let us consider the figure 1-a on which a condenser having is illustrated a dielectric solid. The dielectric one entirely occupies the space ranging between the two reinforcements. In this figure 1-a, also some atoms of the dielectric one appear, which, to simplify our explanation are supposed to consist of four electrons revolving around the core on a single orbit.
As long as no tension is applied at the boundaries of the condenser, the electrons revolve regularly around their respective core (fig. 1-a). So on the other hand we connect the reinforcements of the condenser at the boundaries of a pile as in the figure 1-b, the electrons are attracted by the reinforcement which becomes positive and pushed back by that becoming negative.
As the dielectric one is insulating, the electrons cannot leave their orbit, but on the other hand, they modify it. The electrons pass more close to the positive reinforcement and further from the negative reinforcement during the gravitation around their core (figure 1-b).
If we consider the phenomenon as a whole, we see that it creates for itself a displacement of electrons which, although resident related to their atom, approach nevertheless the left end of the dielectric one. This displacement thus generates a dissymmetry in the distribution of the electric charges inside the dielectric one.
The left end of dielectric towards which the electrons move becomes negative and is called negative pole, while the right end which sees to move away the electrons is called positive pole.
We can say that the dielectric one polarizes because its ends take different electric polarities.
Polarization of dielectric depends on the increase in the loads present on the reinforcements of the condenser and, consequently, this polarization determines an increase in its capacity.
When we analyze energy relating to a condenser, we will give an explanation on this fact. For the moment, it is enough to remember that the capacitance of a capacitor depends on the insulating material of which is made up dielectric sound, and more particularly on its polarization inherent in such or such type of dielectric.
Two Condensers, whose reinforcements are of the same surface and separate consequently distance but from which the dielectric one is different, have different capacities.
The difference between the properties of materials constituting the dielectric ones is characterized by the absolute permittivity of material.
The symbol of the absolute permittivity of a material is e (Greek letter and is read “epsilon”); its unit is the farad per meter (symbol F / m).
Very dielectric has its own permittivity. That of the air, which is regarded besides as identical to that of the vacuum, is called constant dielectric air or vacuum and has as a symbol e0 (epsilon zero). e0 is worth 1 / 36 P x 109 F / m is to facilitate calculations : 8,85 pF / m.
The knowledge of e0 is very important in practice bus, it is not habit to indicate the absolute permittivity (e) of a material and you will rather find the permittivity relative (er) which indicates the report / ratio of the absolute permittivity of material considered and the permittivity of the air or the vacuum.
er = e / e0
The relative permittivity er does not have, as for it, of unit (e0 and e having the same unit that is the F / m).
As an indication, are deferred the values of the relative permittivity er of some materials used for the realization of dielectric of the condensers
|
Material |
Relative permittivity er |
|
Dry air |
1 |
|
Special paper for condenser (KRAFT) |
4,5 |
|
Mica |
5 to 6 |
|
Magnesia titanate |
5,4 to 20 |
|
Glow, Rutile-zirconias, Titanate of calcium |
30 to 220 |
|
Barium titanates and Zirconates |
500 to 15 000 |
|
Polystyrene (Styroflex) |
2,3 |
|
Polytetrafluoretylene (PTFE, Teflon) |
2 |
|
Polymonochlorotrifluoretylene (PCFTE) |
2,3 to 2,8 |
|
Ethylene polyterephthalate (Polyester, Mylar) |
3,1 |
|
Electrolytic with aluminum |
9 |
|
Electrolytic with tantalum |
11 |
NOTE : The relative permittivity is also called relative permittivity, just as the absolute permittivity is also called absolute permittivity.
CALCULATION
OF THE CAPACITANCE OF A CAPACITORWe thus know that the capacitance of a capacitor depends on its dimensions (surface of the reinforcements and distance between them) and on its dielectric: we must thus be able to calculate this capacity according to these elements, just as we could determine the resistance of a driver according to its dimensions and the matter which constitutes it.
We saw that when we increase the surface of the plates of a capacitor, we increase the quantity of electricity present on those and thus also capacity of the condenser: the capacitance of a capacitor is proportional to the surface of its reinforcements thus :
We saw then that the quantity of electricity on the reinforcements increases if we decrease the distance which separates them. We can conclude that the capacitance of a capacitor is inversely proportional to the distance which separates its reinforcements :
Finally the absolute permittivity of material used intervenes. The larger this constant is high, east the capacity of the condenser :
C = f (e)
Combination of the three relations which we have just established, the general formula of computation of a condenser thus becomes :
C = e . (S / d)
C : Capacity out of F
e : Absolute permittivity out of F / m
S : Surface reinforcements in m²
d : Outdistance between the reinforcements (or thickness of dielectric) in m
However, as we generally consider the relative permittivity, the preceding formula becomes :
C = er . e0 . (S / d)
After having examined all the elements of the condenser which influence its capacity, we will analyze the behavior of the condenser when it is inserted in an electric circuit so as to include/understand the reasons for which this component is very much used in practice.
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