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**Electromagnetic induction : **

In this lesson, we will examine some very important phenomena generated by inductance. These so-called phenomena of **electromagnetic induction** were discovered by the **English Michael FARADAY.**

1. - ELECTROMAGNETIC INDUCTION **:**

**1. 1. -** **INDUCED ELECTROMOTOR FORCE**

Figure 1 illustrates the phenomenon of electromagnetic induction. In this figure, we note the presence of a coil traversed by a current of intensity **I.** This coil thus generates an induction flux whose lines appear in Figure 1 **;** note also the presence of a coil that can move relative to the coil.

By moving the turn of the position it occupies Figure 1-a, at its position 1-b, there appears at its terminals a potential difference (or voltage) that persists for the duration of the displacement. The coil can be compared to a battery and for this, it is given the name of electromotive force induced to the voltage thus produced.

Since the turn is an open circuit, it can not circulate current in it, just as a battery does not deliver any current when it is not connected to any device. What we must now determine is the cause producing this induced electromotive force (abbreviated **f.e.m.**).

**The f.e.m. induced is not directly related to the movement of the turn but is the consequence of this displacement.**

The direct consequence of the displacement of the turn, as it appears in Figure 1, is that the induction flux embraced by it varies. Indeed in Figure 1-a, we find that the coil embraced a few lines of the induction flux produced by the coil. When, on the other hand, the coil is brought into the position it occupies in Figure 1-b, then it does not embrace any induction line of the flux produced by the coil. During the movement of the turn, the flow embraced by it has thus passed from a certain value to zero.

The example shown in Figure 2 confirms our hypothesis.

By moving the turn of the position described in Figure 2-a, to that shown in Figure 2-b, we find that, despite its movement, it does not appear at its terminals any **f.e.m.** induced. Indeed, in this specific case, the variation of the kissed flux is zero because, whatever the position of the coil during its displacement, it embraces permanently all the flux produced by the coil. We can conclude that **:**

**To induce a f.e.m. in a turn, it is necessary to vary the induction flux embraced by this turn.**

In our example of Figure 1, the flux variation consists of a decrease but we obtain the same phenomenon if, on the contrary, the flux increases as happens if the turn passes from the position of Figure 1-b to that of Figure 1-a. The **f.e.m.** induced in the spiral being due to the variation of flux, any cause which causes this variation produces a **f.e.m.** induced. In this regard, remember that the induction flux (**φ**) produced by a coil is a function of the intensity (**I**) of the current flowing in its turns. If we vary this current, the induction flux varies and if it is partially or totally embraced by a fixed turn, this variation causes the appearance of a **f.e.m.** induced in the spire.

This case is illustrated in Figure 3-a, where the flow produced by a coil traversed by a current (**I**) is embraced by two turns **A** and **B.**

If we cut the current (**I**) running through the coil, the induction flux of this coil disappears as we see in Figure 3-b. In this example, there is also variation of the flux embraced by turns **A** and **B** and appearance of a **f.e.m.** induced in these.

An identical phenomenon occurs not only when we cut the current (decrease the flow of a certain value to that of zero) but also if we restore the current. In this case, the flow changes from a null value to a determined value.

**We now know that there are two ways to vary the flow embraced by a turn : either by displacement of the turn, or by variation of the current that produces the flow.**

For the rest of our explanations, we are mainly interested in the variations of flux produced by current variations, because this case is found in most circuits.

**
1. 2. **- **LAWS
OF ELECTROMAGNETIC INDUCTION**

To concretely use the phenomena related to the electromagnetic induction, it is necessary to know to which parameters the **f.e.m.** induced and in particular how to calculate its value. For this, consider again Figure 3-a which is reported for the occasion figure 4.

We note on this figure that the flux produced by the coil completely crosses the turn **A,** while the turn **B** embraces a part. Consequently, at the moment when we cut the current (**I**) flowing in the coil, the variation of the flux embraced in the turn **A** is greater than the variation of flux embraced in turn **B.** However, like the **f.e.m.** induced is due to the variation of the flow embraced, it is easy to guess that its value is even greater than the variation of flux is important. The **f.e.m.** induced in turn **A** is greater than that induced in turn **B.**

**The value of the f.e.m. induced in a turn depends on the variation of the induction flux which crosses this turn and is all the higher as this variation is important.**

We know that the **f.e.m.** induced is created during the entire duration of the variation of the flow embraced. Until now, we have only supposed a cancellation of the flux produced by the coil following a cut of the current which crosses it **:** thus, the flow varies very quickly and we do not obtain the creation of a **f.e.m.** induced only for a brief moment. However, nothing prevents us from varying the flow more slowly by putting for example a variable resistance in series between the battery and the coil that it feeds.

We thus obtain the assembly of Figure 5.

The variable resistor is provided with a slider which, when moving, inserts into the circuit a more or less high resistance. When the cursor is in contact with the point **A,** the current (**I**) does not cross the resistance, its intensity is maximum as well as the induction flow which passes through the coil placed in front of the coil (Figure 5-a). If the cursor is moved between points **A** and **B,** as in the case of Figure 5-b, the current (**I**) passes through a part of the resistance, its intensity decreases (**OHM LAW**) and the flux embraced by the spire.

When the cursor is moved to point **B,** as shown in Figure 5-c, the current (**I**) crosses the whole variable resistance, its intensity can be regarded as invalid if the resistance is very high value and the flux embraced by the spire also vanishes.

With such a device, we have the possibility of varying the current, and therefore the induction flux, from a maximum to a minimum and this by moving the cursor from point **A** to point **B** of the variable resistor.

Suppose first that the movement of the cursor from **A** to **B** is realized in a time of **1 second.** The variation of the flux will last one second while creating a **f.e.m.** induced by **2 V** for example.

If now, after returning the cursor from **B** to **A,** we move it again from **A** to **B** but in **10 seconds.** We find that the same flux variation as before does not happen in **1 second but in 10 seconds,** or if we apply things differently **:** that at the same time of **1 second,** we determine a flux variation 10 times less only in the first case. Since the change in flux in **1 second** is now 10 times lower than in the first case, the **f.e.m.** induced in the turn also has a value 10 times lower and instead of getting **2 V,** we get more than **0.2 V**. The **f.e.m.** remaining constant throughout the flow variation, after **10 seconds** it is always **0.2 V.**

From this example, we deduce that **:**

**For a given flow variation, the f.e.m. induced is inversely proportional to the time taken by this flow to vary.**

After these considerations, it is easy to understand the law enunciated by the German **physicist Franz Ernst NEUMANN (1798-1895)** according to which **:**

**The electromotive force ( E) induced in a turn is obtained by dividing the flow variation (**

** NOTE : (** which reads "delta", fourth letter of the Greek alphabet) is commonly used in both physical and mathematical formulas, to symbolize a variation.

If instead of a turn, we are in the presence of a winding of several turns crossed by the same flow, the variation of this flow corresponds a **f.e.m.** induced identical in each turn. Since the turns of the same winding are in series with each other, the **f.e.m.** induced in each of them add up as the **f.e.m.** several batteries connected in series.

**At the terminals of the coil, we have a f.e.m. equal to the product of the f.e.m. induced by a turn by the number of turns of the coil.**

So far, we have assumed that the **f.e.m.** is induced in an open turn, therefore in which no current flows.

Consider now the same turn but connected for example to a resistance, we thus obtain a closed circuit. In this circuit, the **f.e.m.** induced circulates a current called induced current.

To determine the direction of the induced current, we must apply the **law of LENZ**, stated precisely by the Russian physicist **Heinrich LENZ (1804-1865) ;** according to this law**:**

**The induced current has a meaning such that it opposes the cause that gave it birth.**

So, to determine the flow direction of the induced current, we must first know the cause that generates this current and then consider how this current can oppose this cause.

To understand this well, refer to Figure 6. In this example, we consider the flow produced by a single turn energized by means of a coil connected in series with a variable resistor. As we have seen with Figure 5, this device allows us to vary the current (**I1**) flowing in the turn. Since the role of this circuit is to produce the induction flux, it is called the **inductive circuit.**

A second turn connected to a resistance is instead the induced circuit **:** it is in this circuit appears the **f.e.m.** induced and circulates the induced current **I2.**

Consider the case illustrated in Figure 6-a, in which the current (**I1**) decreases when we move the slider of the variable resistor from **A** to **B.** The decrease of the current (**I1**) in the inductor circuit causes the decrease of the flux generated by this circuit. As this flux crosses the induced circuit, the variation of this flux produces a **f.e.m.** induced in this circuit, which circulates an induced current (**I2**) in the turn.

The cause that gave rise to the induced current **I2** is therefore the decrease of the flux embraced by the coil.

In accordance with the **law of LENZ** to palliate this cause, the current induces **I2** must circulate in the turn in the sense such that it counteracts the decrease of the flow embraced by this turn.

We know that any turn traversed by a current generates an induction flux, in the turn which constitutes the induced circuit will produce an induction flux determined by the circulation of the induced current **I2.** This induction flux compensates the decrease of the flux produced by the inductor circuit and embraced by the turn. For this to occur, the induction lines of the incipient flux in the induced circuit must be directed in the same direction as those of the inducing flux so as to strengthen them and thus counteract the decrease.

Figure 6-a shows the two flows in question **:** that produced by the current (**I1**) flowing in the induction circuit at its induction lines drawn in continuous lines while the flux generated by the current **I2** flowing in the induced circuit sees his lines of induction drawn in broken lines. We note that according to what we have just explained that the two flows have their induction lines oriented in the same direction, so that the product flows are added. (For your convenience, we represent the same circuit as below)

We now know the direction of the induction lines in the induced circuit, but the meaning of these depends on the direction of flow of the current in the turn. Therefore, since the induction lines of the inductor circuit and the induced circuit are oriented in the same direction, this means that the currents **I1** and **I2** also flow in the same direction and this in their respective turn. As we see in Figure 6-a, in the conventional sense, the current (**I1**) flows from the positive pole to the negative pole of the battery. The induced current **I2** flows in the same direction as indicated by the orientation of the arrows appearing in Figure 6-a near the turn of the induced circuit.

Let us apply the same explanations to the case of Figure 6-b, where the cause that generates the induced current **I2** is no longer a decrease in the flux of the induced circuit but its increase.

The direction of the current **I2** must be such that it creates an induction flux which is opposed to that produced by the inductive circuit.

The induction lines of the two flows are, as in the previous case, represented in continuous lines for the flux created by the inductor circuit and in discontinuous lines for the flux created by the induced circuit. Since the induction lines of these two flows are in opposite directions, the currents which generate them are also of opposite direction.

Also knowing in this case the direction of the current **I1** (which has also not changed compared to the previous case), we have shown Figure 6-b the flow direction of the induced current **I2**.

From these two examples, we deduce that **:**

**The direction of flow of the induced current depends on the manner in which the flux embraced by the induced circuit varies, that is, whether it increases or decreases.**

With regard to Figure 6, we observe that the induction lines of the flux produced by the induced current **I2** pass through not only the turn of the induced circuit but also the turn of the inductive circuit. We then understand that at any variation of the induced current **I2**, therefore of the flux that it produces, a **f.e.m.** is induced in the turn of the inductor circuit which embraces this flow.

This interaction phenomenon from one circuit to the other is called **mutual induction.**

It occurs not only when there are two distinct circuits, ie an inductive circuit and an induced circuit, case of Figure 6, but also when there is only one circuit.

To properly control this phenomenon, let us analyze Figure 7. In both cases illustrated by this figure, we slowly vary the current (**I1**) traveling through a coil.

The case of Figure 7-a considers a decrease in current **I1** (displacement of the slider of the variable resistor from **A** to **B**). The current **I1** determines a flow in the coil **:** if it decreases, the flux embraced by this coil also decreases. If the flow decreases, it creates a **f.e.m. induced in the coil.** It is said that a phenomenon of self-induction is created.

This **f.e.m.** induced determines the flow of a current. This one, named **I2** in Figure 7-a is called self-induction current.

The **law of LENZ** is valid in this case also **;** it allows us to determine the meaning of **I2** which opposes the cause that gave birth to it, but as this cause is the decrease of the flow (consequence of the decrease of **I1**), **I2** will create a flow directed in the same direction than the one created by **I1.** In conclusion **I2** flows in the same direction as **I1.**

These two flows are represented in Figure 7 in continuous lines and in broken lines, as we did previously.

If on the other hand the current **I1,** increases as in the case illustrated in Figure 7-b (displacement of the slider of the variable resistor from **B** to **A**), the self-induction current **I2** flows in the opposite direction to **I1.** Indeed, in this case for thwarted the increase of the flow created by **I1, I2** creates a flow of opposite direction.

We find that in the case of self-induction, what we have already seen is the case of mutual induction (Figure 6) with the difference that the two currents instead of circulating in two circuits distinct (an inductive circuit and an induced circuit) circulate in the same.

About these currents, we can make the following two observations **:**

**When we decrease the intensity of a current that flows through a coil, a second current starts which tends to compensate for the decrease of the first.**

**When we increase the intensity of a current that flows through a coil, the second current created, tends to oppose the increase of the first.**

It appears from these two observations that **:**

**The coil opposes in all cases to the variation of the current passing through it, whether it decreases or increases.**

If we send into a coil a current whose intensity undergoes continual variations, this current meets on the part of the coil a permanent opposition to its variations **;** in other words, the coil obstructs the passage of this current.

From this, we understand that the coil can accomplish in the circuits a task opposite to that exerted by the capacitor. In previous lessons, we know that a capacitor prevents the passage of a current supplied by a battery, that is to say having a constant intensity. On the contrary, the coil offers an obstacle to the passage of a current whose intensity varies constantly (type of current that we will analyze in the next lesson).

We can make the same observation with the resistance, however it must be remembered that this one presents an obstacle to the passage of the current, that its intensity is constant or that it undergoes continual variations. On the other hand, the coil makes its effect only with currents of variable intensity, so we can use it to separate two different types of currents when they are superimposed in the same circuit.

Because of this application, the coil must no longer be considered as an element that produces an induction flux but as an element capable of obstructing a current of variable intensity. Similarly, we must take into account the inductance specific to each coil in a different aspect.

The formula for calculating the **f.e.m.** induced
**(E = Ф / t)
**from the NEUMANN law also applies in the case of the f.e.m. self-induction. We can say that the f.e.m. self-induction is obtained by dividing the variation of the flux embraced by the coil, by the time that this variation lasts. In this formula, if we replace the variation of the flow embraced **Ф**
by the product of the inductance and the variation of the current **(Ф = L x I)**,
we can state the following law **:**

**The f.e.m. self-induction is obtained by multiplying the inductance by the variation of the current and dividing this product by the time that this variation lasts.**

We deduce from this formula that the **f.e.m.** self-induction is directly related to the inductance of the coil. This **f.e.m.** is higher when the inductance is large, and vice versa.

As in this **f.e.m.** is directly related, the current of **self-induction which opposes the variation of the initial current traversing the coil,** we can say that **:**

**Inductance indicates the ability of a coil to oppose variations in the current flowing through it.**

Since the inductance of a coil plays an important role in the phenomenon of self-induction, it is for this reason that it is also called the __coefficient of self-induction__.

So far in the description of the phenomena of mutual induction and self-induction, we have always considered coreless coils, but it is obvious that the same phenomena occur with coils provided with a nucleus. In this case, the phenomena are greatly amplified because the inductance of a coil provided with a core is considerably increased compared to the same coreless coil.

We will come back to this point in the next lessons and especially when we analyze the operation of transformers.

**
1. 3.** -** ****Groupings of
Coils**

Like the resistors and the capacitors, the coils can be associated with each other and form series or parallel groups. The groupings of coils are very little used in practice, however to be complete on this component we must speak about it.

Figure 8 are given the symbols corresponding to coreless coils and those equipped with cores. The only difference between the two types is the presence of a line above the symbol.

**Fig. 8. **-** Symbolic representations of the Coil.**

**1. 3. 1. - GROUPING SERIES**

Two coreless coils connected in series are shown in Figure 9-a.

As in any series assembly, the two coils **L1** and **L2** are traversed by the same current **I.** The intensity of **I** can vary as a function of the displacement of the slider of the variable resistor. Any variation in intensity of **I** produced in each coil a **f.e.m.** auto-induction called **E1** for **L1** and **E2** for **L2.** The values of **E1** and **E2** are determined by **formulas 1 and 2 :**

In Figure 9-b, we replaced the coils **L1** and **L2** by the equivalent coil **Leq.** At the terminals of this coil, any variation of I determines a **f.e.m.** self-induction (**Et**) whose value is determined by formula 3 **:**

The second characteristic of any series assembly and that the total voltage at its terminals is equal to the sum of the voltages present at the terminals of each element, but as we know, the **f.e.m.** self-induction do not escape this characteristic. We can therefore write that **:**

**E1 + E2 = Et**

By replacing in this equality the **f.e.m.** by their value previously deduced from relations 1, 2 and 3, we write **:**

If we consider in both cases shown in Figure 9 current variations
**I**
**;** identical for a duration **t**
equal,we can simplify the two members of the equality (4) by**
I / t**
and we get** :**

**L1 + L2 = Leq**

In conclusion, we can say that **:**

**The equivalent inductance presented by two or more coils connected in series is obtained by adding the inductance of each of the coils.**

**Leq = L1 + L2 + L3 + ...**

**
****1. 3. 2.**
-** GROUPING ****PARALLEL **

Two coreless coils connected in parallel are shown in Figure 10-a.

As in any parallel assembly, there is the same voltage across the coils **L1** and **L2.** Thus, in case of variation of the current **I,** it will appear at the terminals of **L1** and **L2** the same **f.e.m.** self-induction **E.**

In this type of connection, we must therefore essentially analyze the behavior of the current. The current (**I**) is divided into two parts **I1** and **I2** crossing respectively the coils **L1** and **L2 :**

**I = I1 + I2 **

Any variation of ** **de**
I **has the same effect on **I1** and **I2.**

**I = (I1 + I2) = I1 + I2**

The variations **I1**
and **I2**
determine at the terminals of **L1** and **L2** a **f.e.m.** self-induction **E** identical (parallel assembly).

We know that **I = I1 + I2
**en remplaçant **I1
and I2**
by their previously determined value we get the relation (1) **:**

From Figure 10-b, we deduce the relation 2 **:**

87/5000
The two relations 1 and 2 give the same variation **I**
of current and are therefore equal **:**

By simplifying the two terms of equality by
**E x t**,
we obtain: **:**

We have thus determined the value of the inductance (**Leq**) equivalent to the two coils **L1** and **L2** connected in parallel.

Extended to the general case, this formula becomes **:**

When only two coils are connected in parallel, we adopt the following formula which derives from the general formula **:**

**Finally, note that if the coils connected in parallel all have the same inductance L, the equivalent inductance Leq is obtained by dividing the value of their inductance L by the number (n) of coils, is :**

**Leq = L / n**

**It is important to remember that the rules established for coil assemblies are valid only when the induction flux of each coil is not embraced by the other coils connected to it.**

Indeed, in the opposite case, it also occurs the phenomenon of mutual induction with influence of a coil on the other.

In practice, this phenomenon of mutual induction can be eliminated through the use of coils provided with a fully closed core that "channel" the flow. The same result is obtained by moving the two coils sufficiently far apart.

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