Laws of electromagnetic
induction |
Groupings
of reels |
Parallel
grouping |
|
Return
to the synopsis |
To
contact the author |
Low
of page |
Created it, 05/10/15
Update it, 05/11/22
N° Visitors
ELECTROMAGNETIC INDUCTION “1st PART”
In this lesson, we will examine some very important phenomena generated by an inductance. These phenomena known as of electromagnetic induction were discovered by English Michael FARADAY.
1. - ELECTROMAGNETIC INDUCTION
1. 1. - INDUCED ELECTROMOTIVE FORCE
Figure 1 illustrates the electromagnetic phenomenon of induction. On this figure, we note the presence of a reel traversed by a current of intensity I. This reel thus generates a flow of induction whose lines appear figure 1 ; also let us note the presence of a whorl which can move compared to the reel.
By moving the whorl of the position which it occupies figure 1-a, with its position 1-b, it appears on its terminals a potential difference (or tension) which persists throughout all displacement. The whorl can be compared with a pile and for this reason, it is given the name of electromotive force induced to the tension thus produced.
Since the whorl is an open circuit, it cannot circulate of current in this one, just like a pile does not output any current when it is connected to no apparatus. What we must now determine, it is the cause producing this electromotive force (in summary f.e.m.) induced.
The induced f.e.m. is not directly related to the displacement of the whorl but is the consequence of this displacement.
The direct consequence of the displacement of the whorl, such as it appears figure 1, is that the flow of induction embraced by this one varies. Indeed on the figure 1-a, we realize that the whorl embraced some lines of the flow of induction produced by the reel. When, on the other hand, the whorl is brought in the position which it occupies figure 1-b, it then any more does not embrace any line of induction of the flow produced by the reel. During the displacement of the whorl, the flow embraced by this one thus passed from a certain value to zero.
The illustrated example figure 2 consolidates us on our assumption.
By moving the whorl of the described position figure 2-a, with that represented figure 2-b, we note that, in spite of its displacement, it does not appear on its terminals any f.e.m. induced. Indeed, in this precise case, the variation of embraced flow is null because, some is the position of the whorl during its displacement, it permanently embraces the totality of the flow produced by the reel. We can conclude that :
To induce a f.e.m. in a whorl, it is necessary to vary the flow of induction embraced by this whorl.
In our example of figure 1, the variation of flow consists of a reduction but we obtain the same phenomenon if, on the contrary, flow increases as that occurs if the whorl passes from the position of the figure 1-b to that of the figure 1-a. The f.e.m. induced in the whorl being due to the flux variation, any cause which involves this variation produced an induced f.e.m. On this subject, let us recall that the flow of induction Ø produced by a reel is a function of the intensity (I) current which circulates in its whorls. If we vary this current, the flow of induction varies and if this one is embraced partly or entirely by a fixed whorl, this variation causes the appearance of a f.e.m. induced in the whorl.
This case is illustrated appears 3-a, where the flow produced by a reel traversed by a current (I) is embraced by two whorls A and B.
If we shut off the current (the I) which traverses the reel, flow of induction of this reel disappears as we see it figure 3-b. In this example, there is also variation of the flow embraced by the whorls A and B and appearance of a f.e.m. induced in those.
An identical phenomenon occurs not only when we shut off the current (reduction of the flow of a certain value to that of zero) but also if we turn on the power. In this case, flow passes from a zero value to a given value.
We know now that there are two means of varying the flow embraced by a whorl: either by displacement of the whorl, or by variation of the current which produces flow.
For the continuation of our explanations, we are interested primarily in the flux variations produced by current fluctuations, because this case meets in the majority of the circuits.
1. 2. - LAWS OF ELECTROMAGNETIC INDUCTION
To concretely use the phenomena related to electromagnetic induction, it is necessary to know with which parameters is dependant the induced f.e.m. and in particular how to calculate its value. For this, again let us consider the figure 3-a which is deferred for the circumstance figure 4.
We note on this figure that the flow produced by the reel crosses whorl A entirely, whereas the whorl B embraces that a part. Consequently, at the time when we shut off the current (I) circulating in the reel, the variation of the flow embraced in whorl A is more important than the flux variation embraced in the whorl B. But, as the induced f.e.m. is due to the variation of embraced flow, it is easy to guess that its value is all the more large as the flux variation is important. The f.e.m. induced in whorl A is higher than that induced in the whorl B.
The value of the f.e.m. induced in a whorl depends on the variation of the flow of induction which crosses this whorl and it is all the more high as this variation is important.
We know that the induced f.e.m. is created during all the time that lasts the flux variation embraced. Until now, we supposed only one cancellation of the flow produced by the reel following a cut of the current which crosses it: thus, flow varies very quickly and we obtain the creation of an induced f.e.m. only during one short moment. However, nothing prevents us from varying flow more slowly by putting for example a variable resistor in series between the pile and the reel than it feeds.
We obtain thus the assembly of figure 5.
The variable resistor is provided with a cursor which, while moving, inserts in the circuit a more or less high resistance. When the cursor is in contact with point A, the current (I) does not cross resistance, its intensity is maximum as well as the flow of induction which crosses the whorl placed in front of the reel (figure 5-a). If the cursor is brought between points A and B, as in the case of the figure 5-b, the current (I) crosses part of resistance, its intensity decreases (law of OHM) as well as the flow embraced by the whorl.
When the cursor is brought to the point B, as on the figure 5-c, the current (I) crosses the totality of the variable resistor, its intensity can be regarded as null if resistance is of very strong value and the flow embraced by the whorl is also cancelled.
With such a device, we have the possibility of varying the current, therefore the flow of induction, a maximum to a minimum and this by moving the cursor of point A at the point B of the variable resistor.
Let us suppose initially that the displacement of the cursor of A towards B is carried out in a time of 1 second. The variation of flow will last one second creating during this time an induced f.e.m. of 2 V for example.
So now, after having brought back the cursor of B towards A, we again move it A towards B but in a 10 seconds time. We note that same flux variation that previously is carried out either into 1 second but in 10 seconds, or if we apply the things differently: that in the same time of 1 second, we determine a variation of flow 10 times less than in the first case. Since the variation of flow in 1 second is now 10 times weaker than in the first case, the f.e.m. induced in the whorl has also a value 10 times lower and instead of obtaining 2 V, we do not obtain any more that 0,2 V. the f.e.m. remaining constant lasting all the variation of flow, at the end of 10 seconds it is always of 0,2 V.
From this example, we deduce that :
For a flux variation given, the induced f.e.m. is inversely proportional to the time put by this flow to vary.
After these considerations, it is easy to include/understand the law stated by the German physicist Franz Ernst NEUMANN (1798-1895) according to whom :
The electromotive force (E)
induced in a whorl is obtained by dividing the variation of flow (
f)
by the duration (t)
of this flux variation.
NOTE :
(which
is read “delta”, fourth letter of the
Greek alphabet) is usually employed in the formulas as well physical as
mathematical, to symbolize a variation.
If instead of a whorl, we are in the presence of a rolling up of several whorls crossed by the same flow, to the variation of this flow corresponds an identical induced f.e.m. in each whorl. Since the whorls of the same rolling up are in series between them, the induced f.e.m. in each one of them are added like the f.e.m. several piles connected in series.
At the boundaries of the reel, we have a f.e.m. equal to the product of the induced f.e.m. of a whorl by the number of whorls of the reel.
Until now, we supposed that the f.e.m. is induced in an opened whorl, therefore in whom no current circulates.
Let us consider the same whorl now but connected for example to a resistance, we thus obtain a closed loop. In this circuit, the induced f.e.m. makes circulate a current called running induced.
To determine the direction of the induced current, we must apply the Lenz's law, stated precisely by the Russian physicist Heinrich LENZ (1804-1865) ; according to this law :
The current induces has a direction such as it is opposed to the cause which gave him birth.
Therefore, to determine the direction of flow of the induced current, we must first of all know the cause which generates this current and then consider how this current can be opposed to this cause.
For including/understanding this well, we on figure 6 refer. In this example, we consider the flow produced by a single whorl fed by means of a whorl connected in series with a variable resistor. As we saw with figure 5, this device enables us to vary the current (I1) circulating in the whorl. Since the role of this circuit is to produce the flow of induction, it is called primary circuit.
A second whorl connected to a resistance on the other hand constitutes the induced circuit : it is in this circuit that appears the induced f.e.m. and that circulates the current induces I2.
Let us consider the illustrated case appears 6-a, in which the current (I1) decreases when we move the cursor of the variable resistor of A towards B. the reduction in the current (I1) in the primary circuit causes the reduction in the flow generated by this circuit. As this flow crosses the induced circuit, the variation of this flow produces a f.e.m. induced in this circuit, which makes circulate an induced current (I2) in the whorl.
The cause which gave rise to the current induces I2 is thus the reduction in the flow embraced by the whorl.
In accordance with the Lenz's law to mitigate this cause, the current induces I2 must circulate in the whorl according to the direction such, which it thwarts the reduction in the flow embraced by this whorl.
We know that any whorl traversed by a current generates a flow of induction, in the whorl which constitutes the induced circuit will produce a flow of induction determined by the flow of the current induces I2. This flow of induction compensates for the reduction in the flow produced by the primary circuit and embraced by the whorl. So that that occurs, the lines of induction of flow incipient in the induced circuit, must be directed in the same direction as those of inductive flow so as to reinforce them and thus thwart the reduction of it.
Appear 6-a are indicated two flows in question: that produced by the current (I1) circulating in the primary circuit has its drawn lines of induction milk some continuous while the flow generated by the I2 current circulating in the induced circuit sees its lines of induction drawn in discontinuous features. We note that in accordance with what we have just explained that two flows have their lines of induction directed in the same direction, therefore that produced flows are added. (To facilitate the reading to you, we represent the same circuit like below).
We know now the smell of the lines of induction in the induced circuit, but the direction of those depends on the direction of flow of the current in the whorl. Therefore, like the lines of induction of the primary circuit and the induced circuit are directed in the same direction, that means that the currents I1 and I2 also circulate in the same direction and this in their respective whorl. As we see it figure 6-a, according to the conventional direction, the current (I1) circulates of the positive pole to the negative pole of the pile. The current induces I2 circulates in the same direction as the orientation of the arrows indicates it appearing figure 6-a close to the whorl of the induced circuit.
Let us apply the same explanations to the case of the figure 6-b, where the cause which generates the current induces I2 is not more one reduction of the flow embraced by the induced circuit but its increase.
The direction of the I2 current must such as it be created a flow of induction which is opposed to that produced by the primary circuit.
The lines of induction of two flows as in the preceding case are represented in continuous features for the flow created by the primary circuit and in discontinuous features for the flow created by the induced circuit. As the lines of induction of these two flows are contrary direction, the currents which generate them are also opposite direction.
Also knowing in this case the direction of the current I1 (which did not change besides compared to the preceding case), we materialized figure 6-b the direction of flow of the current induces I2.
From these two examples, we deduce that:
The direction of flow of the induced current depends on the way in which flow embraced by the induced circuit varies, i.e. if it increases or decreases.
In comparison with figure 6, we observe that the lines of induction of the flow produced by the current induces I2 cross not only the whorl of the induced circuit but also the whorl of the primary circuit. We include/understand whereas with any variation of the current I2 induces, therefore flow which it produces, a f.e.m. is induced in the whorl of the primary circuit which embraces this flow.
This phenomenon of interaction of a circuit on the other bears the name of mutual induction.
There is not only when there are two distinct circuits, i.e. a primary circuit and an induced circuit, case of figure 6, but also when there is one circuit.
For controlling this phenomenon well, let us analyze figure 7. In the two cases illustrated by this figure, we vary slowly the current (I1) traversing a reel.
The case of the figure 7-a considers a reduction in the current I1 (displacement of the cursor of the variable resistor of A towards B). The I1 current determines a flow in the reel: if this one decreases, the flow embraced by this reel also decreases. If flow decreases, there is creation of a f.e.m. induced in the reel. It is said that it creates for itself a phenomenon of self-induction.
This f.e.m. induced determines the circulation of a current. This one, named I2 in the figure 7-a is called current of self-induction.
The Lenz's law is valid in this case also ; it enables us to determine the direction of I2 which is opposed to the cause which gave him birth, gold as this cause is the reduction in flow (consequence of the reduction in I1), I2 will create a flow directed in the same direction as that created by I1. In conclusion I2 circulates in the same direction as I1.
These two flows are represented figure 7 in continuous features and discontinuous features, as we did previously.
So on the other hand the I1 current, increases as for the illustrated case appears 7-b (displacement of the cursor of the variable resistor of B towards A), the current of I2 self-induction circulates in the direction opposed to I1. Indeed, in this case for thwarted the increase in the flow created by I1, I2 creates a flow of opposite direction.
We note that in the case of the self-induction, it occurs what we already saw in the case of mutual induction (figure 6) with the difference nevertheless that the two currents, instead of circulating in two distinct circuits (a primary circuit and an induced circuit) circulate in the same one.
In connection with these currents, we can make the two following observations :
When we decrease the intensity of a current
which traverses a reel, a second current occurs which tends to
compensate for the reduction in the first.
When we increase the
intensity of a current which traverses a reel, the second current
created, tends to be opposed to the increase
in the first.
It Appert of these two observations that :
The reel is opposed in all the cases to the variation of the current which crosses it, that this one decreases or increases.
If we send in a reel a current whose intensity sudden of continual variations, this current meets on behalf of the reel a permanent opposition to its variations: in other words, the reel makes obstacle with the passage of this current.
Of this, we understand that the reel can achieve in the circuits a contrary task with that exerted by the condenser. In the preceding lessons, we know that a condenser prevents the passage of a current provided by a pile, i.e. having a constant intensity. On the contrary, the reel offers an obstacle in the passing of a current whose intensity varies constantly (standard of current that we will analyze in the next lesson).
We can make the same observation with resistance, however it should be remembered that this one offers an obstacle in the passing of the current, whether its intensity is constant or that it undergoes continual variations. On the other hand, the reel makes smell its effect only with the fluctuating currents, we can thus be used for to us about it to separate two types of different currents when they are superimposed in the same circuit.
Because of this application, one should not any more regard the reel as an element which produces a flow of induction but like an element able to make obstacle with a fluctuating current. In the same way, we must take account of specific inductance to each reel under a different aspect.
The formula of computation of the induced f.e.m. (E
= Ø
/
t)
resulting from the law of NEUMANN also applies in the case of the f.e.m. of
self-induction. We can say that the f.e.m. self-induction is obtained by
dividing the variation of the flow embraced by the reel, by time that this
variation lasts. In this formula, if we replace the variation of flow embraced Ø
by the product of the inductance and the variation of the current (Ø
= L x I),
we can state the following law :
The f.e.m. of self-induction is obtained by multiplying inductance by the variation of the current and by dividing this product by time that this variation lasts.
We deduce from this formula that the f.e.m. of self-induction is directly related to the inductance of the reel. This f.e.m. is all the more high as inductance is large, and conversely.
As with this f.e.m. is directly dependant, the current of self-induction which is opposed to the variation of the initial current traversing the reel, we can say that:
Inductance indicates the aptitude of a reel to be opposed to the variations of the current which traverses it.
The inductance of a reel playing an important part in the phenomenon of self-induction, for this reason it is also called coefficient of self-induction.
Until now in the description of the phenomena of mutual induction and self-induction, we always considered reels without core, but it is obvious that the same phenomena occur with reels provided with a core. In this case, the phenomena are strongly amplified, because the inductance of a reel equipped with a core is increased considerably compared to the same reel without core.
We will reconsider this precise point in the next lessons and in particular when we analyze the operation of the transformers.
1. 3. - GROUPINGS OF REELS
Like resistances and the condensers, the reels can be associated between them and to form groupings series or parallel. The groupings of reels are used very little in practice, however to be complete on this component we must speak about it.
Figure 8 are given the symbols corresponding to the reels without core and those equipped with cores. The only difference between the two types resides in the presence of a feature above the symbol.
1. 3. 1. - GROUPING SERIES
Two reels without core, connected in series are represented on the figure 9-a.
As in any assembly series, the two L1 reels and L2 are crossed by same current I. the intensity of I can vary according to the displacement of the cursor of the variable resistor. Any variation of intensity of I produces in each reel a f.e.m. of self-induction called E1 for L1 and E2 for L2. The values of E1 and E2 are determined by formulas 1 and 2 :
In the figure 9-b, we replaced the reels L1 and L2 by the equivalent reel Leq. At the boundaries of this reel, any variation of I determines a f.e.m. self-induction (Et) whose value is determined by formula 3 :
The second characteristic of any assembly series and that the total tension on its terminals is equal to the sum of the tensions present at the terminals of each element, but like let us know we it, the f.e.m. self-induction do not escape this characteristic. We can thus write that :
By replacing in this equality the f.e.m. by their value deduced previously from relations 1, 2 and 3, we write :
If we consider in the two exposed cases figure 9
of the current fluctuations I;
identical for one length of time t
equalizes, we can simplify the two members of the equality (4) by I
/ t
and we obtain :
In conclusion, we can affirm that :
The equivalent inductance presented by two or several reels connected in series is obtained by adding inductance with each reel.
1. 3. 2.
- PARALLEL
GROUPING
Two reels without core, connected in parallel are represented on the figure 10-a.
As in any parallel assembly, there is the same terminal voltage of the reels L1 and L2. Therefore, in the event of variation of current I, it will appear at the boundaries of L1 and L2 same the f.e.m. of self-induction E.
In this type of connection, it is necessary for us thus primarily to analyze the behavior of the current. The current (I) is divided into two parts I1 and I2 crossing respectively the reels L1 and L2 :
Any variation
of
I is reflected in the same proportions on I1
and I2.
I
= (I1
+ I2) = I1
+ I2The variations I1
and I2
determine at the boundaries of L1 and L2
f.e.m. of identical self-induction E
(parallel assembly).
We know that I
= I1
+ I2
by replacing I1
and I2
by their value determined previously we obtain the relation (1) :
Figure 10-b, we deduce relation 2 :
Two relations 1 and 2 give same variation I
of current and are thus equal :
By simplifying the two terms of the equality by E
x t,
we obtain :
We thus have just determined the value of inductance (Leq) equivalent to the two L1 reels and L2 connected in parallel.
Extended to the general case, this formula becomes :
When two reels only are connected in parallel, we adopt the following formula which derives from the general formula :
Let us note finally that if the reels connected in parallel have all same inductance L, equivalent inductance Leq is obtained by dividing the value of their inductance L by number (N) of reels, that is to say :
Leq = L / n
It is important to remember that the rules laid down for the assemblies of reels are valid only when the flow of induction of each reel is not embraced by the other reels which are connected to him.
Indeed, in the contrary case, there is also the phenomenon of mutual induction with influence of a reel on the other.
In practice, this phenomenon of mutual induction can be eliminated thanks to the use of reels equipped with an entirely closed core which “channel” flow. One obtains the same result while moving away the two reels sufficiently.
|
Click
here for the following lesson or in the synopsis envisaged to this end. |
High
of page |
 Preceding
page |
Following
page |
