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Electrical and Dielectric Capacitance of a Capacitor :
ELECTRIC CAPACITY :
After having studied the resistances, let us see how the capacitors work.
Since it takes different amounts of electricity to carry bodies of different sizes to the same electrical potential, we can characterize each body by the amount of electricity it must have to reach the potential of a volt ; this quantity of electricity is the electrical capacity (symbol C) of the body.
For a body that has a determined quantity of electricity, and that is at a certain potential, the electrical capacity is obtained by dividing the quantity of electricity by the potential.
C = Q / V
For example, we obtain the capacity of a body which has a quantity of electricity of four coulombs and which is at a potential of eight volts by dividing 4 / 8 = 0.5 ; it is therefore 0.5 coulomb to reach the potential of a volt, that is to say that the electrical capacity is 0.5 coulomb per volt.
Thus, the electrical capacitance is measured in coulomb by volt, unit of measure to which was given the name of farad (symbol F), in honor of the scientist Michaël FARADAY, already cited for his researches on electrolytic solutions. The body considered in our example therefore has a capacity of 0.5 farad.
It should be noted that a sphere of the dimension of the earth would have a capacity of about 1 F ; the farad is therefore a unit of measurement much too big.
For this reason, in practice, we mainly use the sub-multiples of the farad :
The microfarad (symbol µF) which equals one millionth of farad (10^{-}^{6} F).
The nanofarad (symbol nF) which is equivalent to one billionth of farad (10^{-9} F, that to say 10^{-3} µF).
The picofarad (symbol pF) which is equivalent to one millionth of a millionth of farad (10^{-12} F, that to say 10^{-6} µF or 10^{-3} nF).
THE CAPACITOR :
The capacity of a body depends in the first place on the presence in its vicinity of other electrified bodies.
This finding can be made experimentally, considering the two metal plates of Figure 1.
These two rigorously identical metal plates are each connected to a pole of a battery, and charge themselves with positive or negative electricity according to the pole to which they are connected.
In Figure 1-a, we find that the process of charging the two plates is very simple, a number of electrons goes from the negative pole of the battery to the plate that is connected to it, charging it negatively, while the pole positive of the pile attracts an equal number of electrons to the plate connected to it ; it is positively charged.
In the conductors that connect each plate to the battery, a movement of electrons is created, the direction of which is indicated in FIG. 1-a. The movement of electrons ceases by itself when the quantity of electrons present on each plate is such that each plate is at the same potential as the pole of the battery which is connected to it.
Between the two plates exists the same potential difference as the terminals of the battery.
As we have seen previously, the amount of electricity that each plate will be able to store depends on its capacity, and, the two plates being identical, they have the same capacity so they store two equal quantities of electricity but one is positive and the other negative.
Suppose now that we bring the two plates closer together, placing them face to face, as shown in Figure 1-b, but avoiding any contact between them so as not to short the battery.
From this approximation of the plates, it appears a new circulation of electrons, in the direction shown in Figure 1-b and therefore an increase in the amount of electricity contained on each plate.
For the moment, let us limit ourselves to note this state of fact, the explanation will be given later.
We find that the amount of electricity on each plate has increased, although the potential of these has not changed.
We can therefore say that by bringing two plates together, their capacity increases.
Since the capacity changes by varying the distance between the plates, we must no longer consider a single plate but consider a set consisting of two plates placed face to face at a given distance, as shown in Figure 2.
This arrangement represents the simplest type of capacitor, which is precisely made of two facing plates, called for the circumstance armatures, provided with two conductors (called terminals) for their connection to the circuits. In Figure 2 you are also given the graphic symbol of the capacitor as you meet it in the electrical diagrams.
Whatever their characteristics or the manufacturer who designs them, a capacitor always consists of two frames separated by an insulator.
To define the capacity of a capacitor, it is necessary to take into account its two frames and thus consider the difference of potential existing between them.
Although a capacitor is composed in all cases of two frames, a single quantity of electricity is taken into account. It consists of the electrons which, as we have seen in Figure 1, went from the frame became positive on the armature became negative and this through the stack.
We only have to consider the amount of electrons missing on one armature or the extra one on the other, since this is, in any case, the same quantity of electricity that has been transferred from one frame on the other.
The capacity of a capacitor is obtained by dividing the amount of electricity present on one of its armatures by the potential difference existing between its armatures.
The capacitor is an element of the electrical circuits and is characterized by its capacity, as the resistance is characterized by its resistive value.
We know the role of resistance, which is to produce voltage drops, and we will see later the role of the capacitor.
THE DIELECTRIC
The first condenser was made by the Dutch Pierre MUSSCHENBROCK (1692 - 1761) who discovered its properties almost by chance at the same time as the German Georges VON KLEIST (1700 - 1748), during his experiments on electricity.
The experiments of these scientists have shown the influence on the capacitance of a capacitor of the insulating material placed between its armatures, and which constitutes its dielectric.
The dielectric capacitor of Figure 2 is air and for this reason, this capacitor is called air capacitor.
The dielectric capacitors can also be another insulating material such as mica, wax paper, polystyrene, some ceramic substances, etc ...
Very quickly, the scientists found that the capacity of an air capacitor increased when they put between its armatures a solid dielectric; for example, a plate of mica disposed between the armatures of a capacitor increases its capacity by five to six times, depending on the mica employed.
This means that, by always having the same potential difference between the capacitor plates, the amount of electricity present on them becomes five to six times higher if the air is replaced by the mica sheet.
This behavior is due to the fact that the solid dielectric placed between the capacitor plates is polarized, as we will see.
Consider Figure 1-a on which is illustrated a capacitor having a solid dielectric. The dielectric completely occupies the space between the two frames. In this figure 1-a, also appear some atoms of the dielectric, which, to simplify our explanation are supposed to consist of four electrons gravitating around the nucleus in a single orbit.
As long as no voltage is applied across the capacitor, the electrons gravitate regularly around their respective nucleus (Figure 1-a). If, on the other hand, we connect the capacitor plates to the terminals of a cell as in Figure 1-b, the electrons are attracted by the armature, which becomes positive and repulsed by the one becoming negative.
As the insulator, the electrons can not leave their orbit, but they modify it. Electrons move closer to the positive armature and further away from the negative armature during gravitation around their nucleus (Figure 1-b).
If we consider the phenomenon as a whole, we see that a displacement of electrons is created which, although remaining bound to their atom, nevertheless approaches the left end of the dielectric. This displacement thus generates an asymmetry in the distribution of the electrical charges inside the dielectric.
The left end of the dielectric to which the electrons are moving becomes negative and is called the negative pole, while the right end, which sees the electrons moving away, is called the positive pole.
We can say that the dielectric is polarized because its ends take on different electrical polarities.
The polarization of the dielectric depends on the increase of the charges present on the capacitor plates and, consequently, this polarization determines an increase of its capacitance.
When we analyze the energy relative to a capacitor, 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 its dielectric is constituted, and more particularly of its polarization inherent to this or that type of dielectric.
DIELECTRIC CONSTANT
Two capacitors, whose armatures are of the same surface and separated by the same distance but whose dielectric is different, have different capacities.
The difference between the properties of the materials constituting the dielectrics is characterized by the absolute dielectric constant of the material.
The symbol of the absolute dielectric constant of a material is e (Greek letter and reads epsilon) ; his unit is the farad per meter (symbol F / m).
Any dielectric has its own dielectric constant. That of air, which is also considered to be identical to that of vacuum, is called the dielectric constant of air or vacuum and has the symbol e0 (epsilon zero). e0_{ }is 1 / 36 P x 10^{9 }F / m to facilitate the calculations : 8,85 pF / m.
The knowledge of e0 is very important because in practice, it is not customary to indicate the absolute dielectric constant (e) of a material and you will find rather the relative dielectric constant (er) which indicates the ratio of the dielectric constant absolute of the material considered and the dielectric constant of the air or vacuum.
er = e / e0
The relative dielectric constant er ne does not have, for its part, unit (e0 and e having the same unit which is the F / m).
As an indication, the values of the relative dielectric constant and of some materials used for the realization of the capacitor dielectric are reported :
Material |
Relative dielectric constant er |
Dry air |
1 |
Capacitor special paper (KRAFT) |
4,5 |
Mica |
5 à 6 |
Titanate of Magnesia |
5,4 à 20 |
Rutile, Rutile Zirconia, Calcium Titanate |
30 à 220 |
Titanates and Barium Zirconates |
500 à 15 000 |
Polystyrene (Styroflex) |
2,3 |
Polytetrafluoroethylene (PTFE, Teflon) |
2 |
Polymonochlorotrifluorétylène (PCFTE) |
2,3 à 2,8 |
Polytéréphtalate d'éthylène (Polyester, Mylar) |
3,1 |
Aluminum electrolytic |
9 |
Tantalum electrolytic |
11 |
NOTE : The relative dielectric constant is also called relative permittivity, just as the absolute dielectric constant is also called absolute permittivity.
CALCULATION OF THE CAPACITY OF A CAPACITOR
So we know that the capacitance of a capacitor depends on its dimensions (surface of the armatures and distance between them) and its dielectric: we must be able to calculate this capacity according to these elements, just as we have been able to determine the resistance of a driver according to its size and the material that constitutes it.
We have seen that when we increase the surface of the armatures of a capacitor, we increase the quantity of electricity present on them and thus also the capacity of the capacitor : the capacity of a capacitor is proportional to the surface of its reinforcements so :
C = f (S)
We have since seen that the quantity of electricity on the armatures increases if we diminish the distance which separates them. We can conclude that the capacity of a capacitor is inversely proportional to the distance which separates its armatures :
C = f (1 / d)
Finally, the absolute dielectric constant of the material used is involved. The higher this constant, the greater the capacity of the capacitor :
C = f (e)
From the combination of the three relations that we have just established, the general formula for calculating a capacitor thus becomes :
C = e . (S / d)
C : Capacity in F
e : Absolute dielectric constant in F / m
S : Reinforcement surface in m²
d : Distance between reinforcement (or dielectric thickness) in m
However, since we most often consider the relative dielectric constant, the previous formula becomes :
C = er . e0 . (S / d)
After examining all the capacitor elements that affect its capacitance, we will analyze the behavior of the capacitor when it is inserted into an electrical circuit so as to understand the reasons why this component is widely used in practice.
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