Created it, 05/10/15
Update it, 06/02/19
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ALTERNATING VOLTAGE AND INDUCTIVE CIRCUIT “3rd PART”
On figure 10 is represented the simplest type of inductive circuit, it does not understand that only one reel ; in this case also, if there were a certain number of reels, we could replace them by a single reel of an inductance equal to that presented on the whole by all the reels inserted in the circuit.
We remember moreover than one reel does not have only its characteristic resistance, but which it offers also a resistance due to the driver which constitutes its whorls. For the reels which have few whorls and which are formed by a driver of a rather large section, this resistance is very low, and one can thus neglect it.
We now will precisely see the inductive circuits which include/understand reels of negligible resistance and which thus have only one inductance.
To make circulate a D.C. current in a circuit of this type (figure 10-a), it is enough to apply a very low tension, since the resistance met by the current is almost null.
On the contrary to make circulate in the same circuit a AC current (figure 10-b) of effective value equal to that of the D.C. current, one needs a higher tension, because, like know we it, the reel with the property to be opposed to the variation of the current which crosses it, and consequently, it obstructs the circulation of the AC current which varies precisely continuously.
We point out that when the current increases, the reel produced a f.e.m. of self-induction which tends to make circulate a current in direction opposed to that which is increasing, precisely to fight the increase of it.
When on the contrary, the current decreases, the reel produced a f.e.m. of self-induction which tends to make circulate a current in the same direction as that which is decreasing, to precisely fight the reduction of it.
We also observe that the f.e.m. self-induction produced by the reel must be, at every moment, equal to the power provided by the generator since its poles are connected directly at the ends of the reel.
According to these remarks, we are able to find the form which the power provided by the generator must have to make circulate in the inductive circuit a given current.
Let us suppose, for example, that in the circuit the current represented on the figure 11-c circulates, where the lines of the sinusoid which correspond to the increase in the current are indicated by a strong feature to distinguish them from the finer features which represent the reduction in the current.
One could show that, if this current is sinusoidal, the tension which determines its circulation is also sinusoidal ; in this case, however, the sinusoid which represents the tension is shifted compared to that which indicates the current, but in a way different from that which we saw for the condensers.
Figure 11-c, one deduces that, in the interval of time ranging between the moments t = 0 second and t = 0,05 second, the current is positive and increases, while passing from the zero value to the maximum value from 1,5 A. As it is positive, the current leaves the pole of the generator indicated by A and circulates in the circuit as the arrows of the figure 11-a indicate it. Since this current increases, the f.e.m. self-induction (E) is opposed to its passage, and tends to make circulate a current directed in contrary direction, as the arrow drawn beside the reel indicates it.
All in all, the reel behaves in its turn as a second generator which tends to fight the action of the generator with AC current feeding the circuit ; this reel, as it tends to make circulate a current in the direction of the arrow drawn beside it, offers at its ends the polarities indicated by the figure 11-a. But since the reel is directly connected to the generator, it has the same polarities that him, as one sees it on this same figure.
In the interval of time ranging between the moments t = 0,05 second and t = 0,1 second, the current is still positive but it decreases while passing from the maximum value to the zero value ; as it is still positive, the current continues to leave the pole of the generator indicated by A and to circulate in the circuit as the arrows of the figure 11-b indicate it. Since the current decreases now, the f.e.m. E, to be opposed to this reduction, tends to make circulate a current directed in the same direction, as the arrow drawn beside the reel indicates it.
Since the direction in which the f.e.m. (E) tends to make circulate a current is opposed to that of the figure 11-a, the polarities at the ends of the reel drawn on the figure 11-b are also reversed compared to those of the figure 11-a ; consequently, the polarities of the generator are also reversed since they must always be as those of the reel.
After 0,1 second, the current is again null and inverts its direction of circulation ; therefore, in the interval of time ranging between the moments t = 0,1 second and t = 0,15 second, the current circulates in the direction indicated by the arrows on the figure 11-d, and it is negative because it enters now the generator by the pole indicated by A. Since this current increases, while passing from the zero value to the negative maximum value, the f.e.m. (E) is opposed again to its passage, and it tends to make circulate a current directed in contrary direction, as the arrow drawn close to the reel indicates it.
(We defer the same diagram below for more comprehension).
This arrow is thus directed in contrary direction of that drawn close to the reel on the figure 11-a, owing to the fact that the current changed its direction of circulation; consequently, the polarities indicated on the figure 11-d at the ends of the reel and thus of the generator, are also reversed compared to those of the figure 11-a.
After having reached the negative maximum value, the current starts again to decrease until it cancels, during the interval of time ranging between the moments t = 0,15 second and t = 0,2 second during which it circulates in the circuit with the direction indicated on the figure 11-e.
Since the current decreases again, the f.e.m. E is still opposed to this reduction and tends to make circulate a current directed in the same direction, as the arrow drawn beside the reel indicates it. This arrow is directed in contrary direction of that drawn beside the reel on the figure 11-b, always owing to the fact that the current reversed its direction of circulation; consequently the polarities of the figure 11th, indicated at the ends of the reel and thus of the generator, are also reversed compared to those of the figure 11-b.
Thanks to these remarks, we thus could establish which are the polarities at the ends of the generator; they enable us to know if the power provided by the generator is positive or negative: we remember indeed that, as one established previously, we consider that this tension is positive or negative according to the sign of the pole of the generator indicated by A.
On the figure 11-a, we see that this pole is positive and we can thus deduce from it that between 0 second and 0,05 second the tension is also positive. On the contrary, between 0,05 second and 0,1 second, as between 0,1s and 0,15s, the tension is negative, because pole A is negative, as one sees it on the figure 11-b and the figure 11-d. The tension is again positive between 0,15 second and 0,2 second bus on the figure 11-e, one sees that pole A is again positive.
Now to be able to trace the sinusoid which represents the tension, it is still necessary to know at which moments it is cancelled : to this end, we observe that the tension must be cancelled when the generator reverses its polarities.
On figure 12, one sees that that occurs when the current ceases increasing and that it is ready to decrease, i.e. when it reached its positive with 0,05 second and negative maximum value with 0,15 second.
Consequently, the sinusoid which represents the tension must again cut the horizontal axis at these moments, while, according to what was known as previously, it must be above this axis between 0 second and 0,05 second and below between 0,05 second and 0,15 second, then above between 0,15 second and 0,2 second ; the sinusoid thus has the form of figure 12 where it was supposed that the tension had a maximum value of 20 volts.
It is seen immediately that this sinusoid with the same form as that of figure 7 (above) for the current which circulates in a capacitive circuit, and all that was known as in connection with this current is thus valid maintaining for the tension.
Just as in a capacitive circuit, the current is out of phase in advance of a quarter of period compared to the tension, in the same way we can say now as in an inductive circuit, the tension is out of phase in advance of a quarter of period compared to the current.
We also see that, in a capacitive circuit as in an inductive circuit, one always has a dephasing of a quarter of period between the tension and the current, and these two sizes are one or the other advances some according to the type of circuit.
By adopting the system of vectorial representation, the two vectors which represent the tension and the current are laid out as on figure 13.
The vector representative of the current is laid out horizontally so that the ordinate of its end is null; the vector representative of the tension is on the other hand vertical and the ordinate of its end is thus equal to the maximum value Vmax of the sinusoidal tension which begins at this moment.
The vectorial representation highlights well that in this case, as in that of the condenser, there is a dephasing of 90° between the two sinusoidal sizes. However now, the tension has a lead over the current : indeed, if the vectors of figure 13 are observed and that those turn in anti-clockwise direction, it is noticed that the Vmax vector precedes by 90° the Imax vector.
Now, it does not remain us any more that to see the way in which between them the current and the tension are dependant relating to an inductive circuit; we remember that the reel is opposed to circulation AC current, while reacting to its variations: one thus calls inductive reactance the obstacle erected by inductance to the AC current and one indicates it by the Xl symbol.
Like resistance and the capacitive reactance, the inductive reactance is measured in ohms.
It is thus understood why, in a way similar so that we already saw for the capacitive circuit, it is possible to apply the law of OHM to the inductive circuit, provided that one considers the inductive reactance presented by the circuit, and that one uses the effective values of the tension and the current.
We already saw for the capacitive circuit that its reactance must be calculated according to the elements on which it depends; we thus will seek, for the case of the inductive circuit, on which elements its reactance depends, in order to be able to calculate it.
On this subject, we remember that the reactance presented by a reel is due to the f.e.m. of self-induction produced by the reel even and tending to obstruct the variations of the current: the reactance will thus depend on the elements on which the f.e.m. depends on self-induction.
In one of the preceding lessons, we already saw
that this f.e.m. depends on the product of
the inductance of the reel by the speed with which the current varies which
traverses it ; in addition, as we already saw in the case of the
capacitive circuit, the speed with which varies an alternative size is indicated
by the pulsation 
which is given by the product
= 2 x n x F.
We can thus conclude that the inductive reactance is obtained by multiplying the number 2 x n by the frequency F and inductance L :
Of this relation, one can see that if the frequency F decreases, the Xl reactance decreases and in the case of continuous sizes (null frequency) the reactance is cancelled completely; there is a contrary phenomenon with that which was observed for the capacitive reactance.
The law of OHM can be extended to an inductive circuit by replacing R by Xl, which gives :
where V and I are the effective values of the two sinusoidal sizes.
As example, let us calculate the voltage drop at the boundaries of a reel having an inductance L = 4 H, crossed by an effective current of 0,1 A of frequency F = 100 Hz.
First of all let us calculate the inductive reactance Xl :
from where :
In the next lesson, we will examine the behavior of the inductances equipped with a magnetic core.
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