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Inductive Reactance of an Electrical Circuit :
3. 2. - INDUCTIVE CIRCUIT
In Figure 10 is shown the simplest type of inductive circuit, it comprises only one coil ; in this case also, if there were a number of coils, we could replace them with a single coil of inductance equal to that presented in total by all the coils inserted into the circuit.
We remember moreover that a coil not only exhibits its characteristic resistance, but that it also offers resistance due to the conductor that constitutes its turns. For coils that have few turns and are formed by a driver of a large enough section, this resistance is very low, and we can neglect it.
We will now see the inductive circuits which include negligible resistor coils and which therefore have only one inductor.
To circulate a DC current in a circuit of this type (Figure 10-a), it is sufficient to apply a very low voltage, since the resistance encountered by the current is almost zero.
On the contrary, in order to circulate in the same circuit an alternating current (Figure 10-b) of Efficient Value equal to that of the direct current, a higher voltage is required because, as we know, the coil has the property of to oppose to the variation of the current which crosses it, and consequently, it hampers the circulation of the alternating current which varies precisely continually.
Remember that when the current increases, the coil produces a f.e.m. self-induction which tends to circulate a current in the opposite direction to that which is increasing, precisely to combat the increase.
When on the contrary, the current decreases, the coil produces a f.e.m. self-induction which tends to circulate a current in the same direction as that which is decreasing, precisely to combat the decrease.
We also observe that the f.e.m. self-induction produced by the coil must be, at each moment, equal to the voltage supplied by the generator since its poles are connected directly to the ends of the coil.
From these remarks, we are able to find the shape that must have the voltage supplied by the generator to circulate in the inductive circuit a specific current.
Suppose, for example, that in the circuit flows the current shown in Figure 11-c, where the lines of the sinusoid which correspond to the increase of the current are indicated by a strong line to distinguish them from the finer lines which represent the decrease of the current.
It could be shown that if this current is sinusoidal, the voltage which determines its circulation is also sinusoidal ; in this case, however, the sinusoid which represents the voltage is shifted from that which indicates the current, but in a different way from that which we saw for the capacitors.
From Figure 11-c, it can be deduced that, in the time interval between the instants t = 0 seconds and t = 0.05 seconds, the current is positive and increases, passing from the zero value to the maximum value of 1.5 A. As it is positive, the current goes out of the pole of the generator designated by A and flows in the circuit as indicated by the arrows of Figure 11-a. As this current increases, the f.e.m. self-induction (E) opposes its passage, and tends to circulate a current directed in opposite direction, as indicated by the arrow drawn next to the coil.
In short, the coil behaves in turn as a second generator that tends to fight the action of the AC generator supplying the circuit ; this coil, as it tends to circulate a current in the direction of the arrow drawn next to it, offers at its ends the polarities indicated by Figure 11-a. But since the coil is directly connected to the generator, it has the same polarities as it, as we see in this same figure.
In the time interval between the instants t = 0.05 seconds and t = 0.1 seconds, the current is still positive but it decreases when passing from the maximum value to the zero value ; as it is still positive, the current continues to exit the pole of the generator designated by A and circulate in the circuit as indicated by the arrows of Figure 11-b. Since the current is now decreasing, the f.e.m. E, to oppose this decrease, tends to circulate a current directed in the same direction, as indicated by the arrow drawn next to the coil.
Since the meaning in which the f.e.m. (E) tends to flow a current is opposite to that of Figure 11-a, the polarities at the ends of the coil drawn in Figure 11-b are also reversed compared to those of Figure 11-a ; therefore, the polarities of the generator are also reversed as long as they must always be like those of the coil.
After 0.1 seconds, the current is again zero and reverses its direction of circulation ; therefore, in the time interval between the instants t = 0.1 seconds and t = 0.15 seconds, the current flows in the direction indicated by the arrows in Figure 11-d, and it is negative because it enters now in the generator by the pole designated by A. Since this current increases, passing from the null value to the maximum negative value, the fem (E) again opposes its passage, and it tends to circulate a current directed in opposite direction, as indicated by the arrow drawn near the coil.
(We report the same diagram below for more understanding).
This arrow is therefore directed in the opposite direction to that drawn near the coil in Figure 11-a, because the current has changed its direction of movement ; therefore, the polarities shown in Figure 11-d at the ends of the coil and therefore the generator, are also reversed compared to those of Figure 11-a.
After reaching the maximum negative value, the current starts to decrease until it is canceled, during the time interval between the instants t = 0.15 seconds and t = 0.2 seconds during which it circulates in the circuit with the direction shown in Figure 11-e.
Since the current decreases again, the f.e.m. E still opposes this decrease and tends to circulate a current directed in the same direction, as indicated by the arrow drawn next to the coil. This arrow is directed in the opposite direction to that drawn next to the coil in Figure 11-b, still because the current has reversed its direction of circulation ; therefore the polarities of Figure 11-e, indicated at the ends of the coil and therefore the generator, are also reversed compared to those of Figure 11-b.
Thanks to these remarks, we were able to establish which are the polarities at the ends of the generator; they allow us to know if the voltage supplied by the generator is positive or negative : we remember that, as we have established above, we consider that this voltage is positive or negative according to the sign of the generator pole designated by A.
In Figure 11-a, we see that this pole is positive and we can deduce that between 0 seconds and 0.05 seconds the voltage is also positive. On the contrary, between 0.05 seconds and 0.1 seconds, as between 0.1s and 0.15s, the voltage is negative because the pole A is negative, as can be seen in Figure 11-b and in Figure 11-d. The voltage is again positive between 0.15 seconds and 0.2 seconds because in Figure 11-e, we see that the pole A is positive again.
Now to be able to draw the sinusoid which represents the tension, it is necessary to know at which moments it is canceled out : for this purpose, we observe that the tension must be canceled when the generator reverses its polarities.
In Figure 12, we see that this occurs when the current stops increasing and it is ready to decrease, that is to say when it reaches its maximum positive value at 0.05 seconds and negative at 0.15 seconds.
As a result, the sinusoid that represents the voltage must cut the horizontal axis at these times, while, according to what has been said above, it must be above this axis between 0 seconds and 0.05 seconds and below between 0.05 seconds and 0.15 seconds, then again above 0.15 seconds to 0.2 seconds ; the sinusoid therefore has the form of Figure 12 where it was assumed that the voltage had a maximum value of 20 volts.
We see immediately that this sinusoid has the same shape as that of Figure 7 (above) for the current flowing in a capacitive circuit, and all that has been said about this current is now valid for the voltage .
As in a capacitive circuit, the current is out of phase by a quarter of a period with respect to the voltage, so we can now say that in an inductive circuit, the voltage is out of phase by a quarter period relative to the current.
We also see that, in a capacitive circuit as in an inductive circuit, there is always a phase shift of a quarter of a period between the voltage and the current, and these two quantities are one or the other in advance according to the type of circuit.
By adopting the vector representation system, the two vectors representing the voltage and the current are arranged as in Figure 13.
The representative vector of the current is arranged horizontally so that the ordinate of its end is zero ; the representative vector of the voltage is against vertical and the ordinate of its end is equal to the maximum value Vmax of the sinusoidal voltage which begins at this time.
The vector representation clearly shows that in this case, as in that of the capacitor, there is a phase shift of 90° between the two sinusoidal quantities. However, now the voltage is ahead of the current : indeed, if we observe the vectors of Figure 13 and they turn counterclockwise, we note that the vector Vmax precedes the 90° vector Imax.
Now, we only have to see how the current and the voltage related to an inductive circuit are connected to each other ; we remember that the coil opposes the flow of the alternating current, reacting to its variations : inductive reactance is thus called the obstacle opposed by the inductor to the alternating current and is indicated by the symbol XL.
Like resistance and capacitive reactance, the inductive reactance is measured in ohms.
We therefore understand why, in a way analogous to what we have already seen for the capacitive circuit, it is possible to apply the OHM law to the inductive circuit, provided that we consider the inductive reactance presented by the circuit, and that the rms values of voltage and current are used.
We have already seen for the capacitive circuit that its reactance must be calculated according to the elements on which it depends ; we will therefore look, for the case of the inductive circuit, which elements depends on its reactance, so as to be able to calculate it.
In this regard, we remember that the reactance presented by a coil is due to the f.e.m. self-induction produced by the coil itself and tending to impede the variations of the current : the reactance will therefore depend on the elements on which depends the f.e.m. self-induction.
In one of the previous lessons, we have already seen that this fem depends on the product of the inductance of the coil by the speed with which the current which traverses it varies ; on the other hand, as we have already seen in the case of the capacitive circuit, the speed with which an alternating quantity varies is indicated by the pulse which is given by the product = 2 x π x F.
We can therefore conclude that the inductive reactance is obtained by multiplying the number 2 x π by the frequency F and by the inductance L :
From this relation, we can see that if the frequency F decreases, the reactance Xl decreases and in the case of continuous quantities (zero frequency) the reactance vanishes completely ; a phenomenon contrary to that observed for capacitive reactance occurs.
The Ohm law can be extended to an inductive circuit by replacing R by Xl, which gives :
V = Xl x I
where V and I are the effective values of the two sinusoidal quantities.
For example, calculate the voltage drop across a coil having an inductance L = 4 H, crossed by an effective current of 0.1 A frequency F = 100 Hz.
First of all, calculate the inductive reactance Xl :
Xl = 2 x π x F x L = 2 x 3,14 x 100 x 4 = 2 512 ohms
from where :
V = Xl x I = 2 512 x 0,1 = 251,2 Veff
In the next lesson, we will examine the behavior of inductors with a magnetic core.
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