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**Magnetic Circuits : **

3. - MAGNETIC CIRCUITS : "3rd PART"

A coil (inductor) must generally have a high inductance (**L**) with respect to its resistive part **R.** For this, a coil is provided with a ferromagnetic core.

Indeed, in the previous theories, it has been noted that the value of the inductance for a coil is also a function of the material located therein. The ferromagnetic core thus makes it possible to significantly increase the inductance while retaining the value of the resistance constituted by the winding of the wire.

To obtain a high inductance, it is necessary that the core is closed on itself so that the set of induction lines is contained in the core.

Examining the coil of Figure 11-a, we see that the induction lines are closed outside the core through the air layers outside the latter.

** **

It is enough to gradually close the nucleus on itself (Figure 11-b) until the two ends are in contact (Figure 11-c) so that all induction lines are **"trapped"** in the nucleus.

Thus, a closed core has been obtained, which is traversed by the entire induction flux produced by the coil. No induction line can close in the air.

**The resulting inductance is equal to the product of the coreless inductor due to the relative permeability of the material constituting the ferromagnetic core.**

For manufacturing reasons, the cores used in practice generally do not have the form shown in Figure 11-c, but the **"column" form** (Figure 12-a), or the **"armored or armored"** form (Figure 12-b), which is the most used.

We can also note that the winding does not perform the complete turn of the core, but is located only in one of its rectilinear parts.

This is related to how the coil is made. At first, only the winding of the wire is performed and in a second step, the metal carcass made of sheets is assembled with the winding of the coil.

As shown in Figure 13, the induction flux closes almost completely in the nucleus. Only a few dotted lines of induction come out of the nucleus.

To explain the fact that the induction lines remain inside the nucleus, we must imagine that the ferromagnetic core consists of very small elementary magnets and put end to end. Thus, the induction lines follow exactly the preferred orientation of these small magnets in the core.

**Since the induction lines preferentially pass through the nucleus, we say that the nucleus is more permeable than the air to the induction lines.**

An image can be given by imagining a water permeable terrain surrounded by impervious terrain. The permeable central ground represents the core, the impermeable outer ground represents the air surrounding the core.

When the rain falls, it is obvious that it is the central, permeable ground that absorbs most of the water. It is the same with the induction lines and the ferromagnetic core. For this reason, the term **"magnetic permeability"** is used for a particular material.
With respect to air, the ferromagnetic core determines an increase in inductance, or induction flux, precisely because it is more easily traversed by induction lines than air.

The second advantage in the use of a kernel lies in the fact that this kernel channels the induction lines, that is to say, it forces them to travel a mandatory "path".

The induction lines leaving the path imposed by the nucleus constitute **the dispersion flux.**

This dispersion flux can generally be neglected in front of the induction flux in the case of a core coil.

As shown in Figure 13 above, the winding and the self-closing core have analogies with an electrical circuit. The set is therefore called **magnetic circuit.**

For each type of magnetic circuit, one can find the corresponding electrical circuit **:** for example, the magnetic circuit of Figure 13-a corresponds to the electric circuit of Figure 13-c constituted by a non-negligible resistance conductor connected to a battery.

Like the **f.e.m.** (**E**) circulates a current (**I**) in the conductor, we can say that the **f.m.m.** (**magnetomotive force**) **"N.I."** crosses the nucleus through the induction flow .

Considering the electrical circuit similar to a given magnetic circuit, the examination of the latter can be facilitated. For example, the magnetic circuit of Figure 13-b corresponds to the electrical circuit of Figure 13-d. The latter consists of two conductors of identical resistance in parallel and connected to the battery. The current (**I**) supplied by the battery is subdivided into two equal parts **I / 2** in each conductor.

The induction flux in a corresponding magnetic circuit exhibits a similar behavior. Indeed, the flow produced by the coil is divided into two equal flows indicated by **/ 2** Figure 13-b, each passing through one of the lateral branches of the core.

We can continue the analogy between magnetic circuits and electrical circuits. For an electric circuit, when we divide the **f.e.m.** by the current, we obtain the resistance (OHM law) of the circuit. For a magnetic circuit, if we divide the **f.m.m.** by the induction flux, a quantity similar to the resistance of the electric circuit is obtained **;** it is the **magnetic reluctance** of the nucleus. The symbol is **R** and this reluctance is expressed in **1 / H.**

The magnetic reluctance indicates the number of ampere-turns required to obtain an induction flux of a Weber (**Wb**).

Since the resistance is a function of length and section for a given conductor, the reluctance is a function of that of the core.

The reluctance is proportional to the length of the core and inversely proportional to its section.

Just as resistivity intervenes in the calculation of the electrical resistance of a given material, absolute permeability intervenes for the calculation of the reluctance of a ferromagnetic core.

The higher the core permeability, the higher the induction flux and the lower the reluctance.

**In conclusion, we can say that the reluctance of a ferromagnetic core is obtained by dividing its length by its section and by its absolute permeability.**

The magnetic circuits considered until now are closed (their core is closed on itself).

Note that there are open circuits. In this case, the kernel has a gap as shown in Figure 14-a. This gap is a small area of space where the nucleus is interrupted.

The direction of the induction lines is practically unaffected by this gap.

If we know the section and the length of the gap and the magnetic permeability of the air, we can calculate the reluctance presented to the induction flux in the gap. This reluctance of the gap is higher than that of a ferromagnetic core of the same dimensions as the gap.

This new magnetic circuit is similar to the circuit of Figure 14-b. The resistor **R** has a much higher resistive value than the conductors that connect to the battery.

This resistance **R** is similar to the air gap of the magnetic circuit while the two electrical conductors are similar to the ferromagnetic core.

**The total reluctance of the magnetic circuit is equal to the sum of the reluctance of the core and that of the gap.**

After having shown these analogies between a magnetic circuit and an electric circuit, it is necessary to present the differences.

As we know, the current flowing through an electric circuit is proportional to the **f.e.m.**, but it is not the same for a magnetic circuit, there is more proportionality between the **f.m.m.** and the induction flow.

Under certain conditions, for an increase in **f.m.m.**, the induction flux does not vary.

This fact is due to the presence of the nucleus and it is necessary to consider the behavior of the constituent material of the nucleus in relation with the variations of the **f.m.m.**.

To examine the behavior of a particular ferromagnetic material, a core is built with it and then a coil is placed around it. A gradually increasing current (**I**) is circulated so as to increase the **f.m.m.** (**N x I**).

For each current **I**, the induction flow is measured using a flowmeter. This makes it possible to draw a curve representing the induction flux as a function of the **f.m.m.** (**N x I**).

The values of the **f.m.m.** are plotted on the horizontal axis of a **Cartesian coordinate system.** The values of the flow are shown on the vertical axis.

Figure 15-a shows this curve for a common ferromagnetic material. At the beginning, at the point **O,** the **f.m.m.** is zero as well as the flow
**; **then the **f.m.m.** increases, we see that the flow
also increases, at first relatively little (at the beginning of the curve), then in a second time, much more **;** in a third step, when approaching point **A,** the variation of the flow
decreases sharply until it vanishes almost beyond point **A.**

At point **A,** there is **magnetic saturation.** It is said that beyond point **A,** **the nucleus is saturated.** Indeed, the higher the **f.m.m. increases**, the more the number of elementary magnets which constitute the nucleus is oriented in the direction of the induction lines.

When we get to point **A,** all the elementary magnets are oriented and therefore, the flow can not grow.

The curve of Figure 15-a is the **first magnetization** curve because it is obtained when a ferromagnetic core is magnetized for the first time.

Now, we have to consider the case of a coil with a core covered by an alternating current. For this, let's start from point **A** at the saturation point previously described.

One would think that when the **f.m.m.** decreases, the flow
resumes the same values as before, but it is not so.

In Figure 15-b, it can be seen that the flow
from point **A** to point **B** takes values greater than those relating to the first magnetization.

In particular, when the current becomes zero, we see that the flow is not (point **B**).

This is the **residual flux or remanent flux.**

Beyond the point **O,** to the left, the values of the **f.m.m.** become negative, that is, current **I** has changed direction. (Figure 15-c).

We realize that when the **f.m.m.** reaches a certain negative value (point **C**), the flow becomes zero.

We see that to cancel the residual flow, it is necessary to circulate in the coil winding a certain current directed in the opposite direction to that used to magnetize the core.

It can be said that the induction flux follows the variations of the alternating current with a certain delay. This phenomenon constitutes **magnetic hysteresis** (hysteresis means delay).

If the **f.m.m.** continues to increase, the current always being in the opposite direction to that of the first magnetization, the flux increases (curve from point **C** to point **D**) but it has changed direction with respect to that of the **first magnetization.**

When we reach the point **D,** the nucleus is saturated, all the elementary magnets are oriented in the opposite direction to that of the **first magnetization.**

When the current decreases again to zero, the flux
decreases from point **D** to point **E.** Therefore, there is still a remanent flux equal in intensity to that seen previously but in opposite directions.

When the **f.m.m.** increases again, we go from the point **E** to the point **F** (**zero flow**) and then we reach the point **A** of saturation. Thus, a complete cycle of hysteresis has been completed.

The arrows in Figure 15-c indicate the direction of travel of the cycle.

For any coil having a ferromagnetic core and traversed by an alternating current, there is such a hysteresis ring in the core. This is the case, in particular, **of power transformers which will be the object of the next theory**

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