Created it, 05/10/15
Update it, 05/11/21
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3. - MAGNETIC CIRCUITS “3rd PART”
A reel (inductance) must generally have an inductance (L) high compared to its resistive part R. For that, a reel is provided with a ferromagnetic core.
Indeed, in the preceding theories, it was noted that the value of inductance for a reel is also according to material located inside this one. The ferromagnetic core thus makes it possible to increase inductance notably, while preserving the value of the resistance consisted the rolling up of the wire.
To obtain a raised inductance, it is necessary that the core is closed on itself so that the whole of the lines of induction is contained in the core.
By examining the reel of the figure 11-a, one realizes that the lines of induction are closed again outside the core while passing by the layers of surrounding air with this last.
It is enough to gradually close again the core on itself (figure 11-b) until the two ends are in contact (figure 11-c) so that all the lines of induction “are imprisoned” in the core.
Thus, a closed core was obtained, which is crossed by the totality of the flow of induction produced by the reel. No line of induction can be closed again in the air.
Resulting inductance is equal to the product of inductance without core by the relative permeability of material constituting the ferromagnetic core.
For reasons of manufacture, the cores used in practice generally do not have the illustrated form appear 11-c, but the shape “with column” (figure 12-a), or form it “armoured or armoured” (figure 12-b), which is used.
We can note in addition that rolling up does not carry out the full rotation of the core, but is located only in one of its rectilinear parts.
That is related to the manufacturing process of the reel. Initially, only the rolling up of the wire is carried out and in the second time, the metal carcass made up of sheets is assembled with the rolling up of the reel.
As figure 13 indicates it, the flow of induction is closed again almost completely in the core. Only, some lines of induction traced in dotted line leave the core.
To explain the fact that the lines of induction remain inside the core, it should be imagined that the ferromagnetic core consists of very small and end to end put elementary magnets. Thus, the lines of induction follow exactly the privileged orientation of these small magnets in the core.
Since the lines of induction pass preferentially in the core, we say that the core is more permeable than the air with the lines of induction.
An image can be given by imagining a permeable ground to water surrounded by an impermeable ground. The permeable central ground represents the core, the impermeable external ground represents the air surrounding the core.
When the rain falls, it is quite obvious that it is the central ground, permeable which absorbs the essence of water. It is the same with the lines for induction and the ferromagnetic core. For this reason, one employs the term of “magnetic permeability” for a given material.
Compared to the air, the ferromagnetic core determines an increase in inductance, or flow of induction, precisely because it is more easily crossed by the lines of induction than the air.
The second advantage in the use of a core lies in the fact that this core channels the lines of induction, i.e. it them constrained to traverse an obligatory “way”.
The lines of induction leaving the course imposed by the core constitute the flow of dispersion.
This flow of dispersion can generally be neglected in front of the flow of induction in the case of a reel with core.
As shown in the figure 13 above, rolling up and the core closed again on itself present analogies with an electric circuit. The unit is thus called magnetic circuit.
For each type of magnetic circuit, one can find the circuit electric corresponding: for example, the magnetic circuit of the figure 13-a corresponds to the electric circuit of the figure 13-c consisted a considerable driver of resistance connected to a pile.
Like the f.e.m.
(E) made circulate a current (I)
in the driver, one can say that the f.m.m. (magnetomotive
force) “N.I.” fact of crossing the core
by the flow of induction 
.
By considering the electric circuit similar to a given magnetic circuit, the examination of this last can be facilitated. For example, to the magnetic circuit of the figure 13-b the electric circuit of the figure 13-d corresponds. This last consists of two drivers of identical resistance put in parallel and connected to the pile. The current (I) provided by the pile subdivides in two equal parts I / 2 in each driver.
The flow of induction in a magnetic circuit
correspondent presents a similar behavior. Indeed, the flow
produced by the reel is divided in two equal flows indicated by
/ 2 figure 13-b, each one crossing one of the
side branches of the core.
One can continue the analogy between magnetic circuits and electric circuits. For an electric circuit, when one divides the f.e.m. by the current, one obtains the resistance (law of OHM) of the circuit. For a magnetic circuit, if one divides the f.m.m. by the flow of induction, one obtains a size similar to the resistance of the electric circuit; it is about the magnetic reluctance of the core. The symbol is R and this reluctance is expressed into 1 / H.
The magnetic reluctance indicates the number of ampere turns necessary to obtain a flow of induction of a Weber (Wb).
As resistance is a function length and section for a given driver, the reluctance is related to those of the core.
The reluctance is proportional to the length of the core and inversely proportional to its section.
Just as the resistivity intervenes in the calculation of the electric resistance of a given material, the absolute permeability intervenes for the calculation of the reluctance of a ferromagnetic core.
The higher the permeability of the core is, the higher flow of induction is and the weaker the reluctance will be.
In conclusion, one can say that the reluctance of a ferromagnetic core is obtained by dividing its length by its section and its absolute permeability.
The magnetic circuits considered until now are closed (their core is closed on itself).
Let us note that there are open circuits. In this case, the core has an air-gap as that appears figure 14-a. This air-gap is a small area of space where the core is stopped.
The direction of the lines of induction is practically not modified by this air-gap.
If one knows the section and the length of the air-gap as well as the magnetic permeability of the air, one can calculate the reluctance presented at the flow of induction in the air-gap. This reluctance of the air-gap is higher than that of a ferromagnetic core of same dimensions as the air-gap.
This new magnetic circuit is similar to the circuit of the figure 14-b. Resistance R has a resistive value much higher than the drivers which connect to the pile.
This resistance R is similar to the air-gap of the magnetic circuit while the two electric drivers 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 air-gap.
After having shown these analogies between a magnetic circuit and an electric circuit, it is advisable to present the differences of them.
Like let us know we it, the current which traverses an electric circuit is proportional to the f.e.m., but it is not the same for a magnetic circuit, there is no more proportionality between the f.m.m. and flow of induction.
Under certain conditions, for an increase in the f.m.m., the flow of induction does not vary.
This fact is due to the presence of the core and it is thus necessary to consider the behavior of material constitutive of the core in relation to the variations of the f.m.m.
To examine the behavior of a determined ferromagnetic material, one builds a core with him then one lays out a reel around this core. One makes circulate a current (I) gradually growing in order to increase the f.m.m. (N x I).
For each current I, one measures the flow of induction using a fluxmeter. That makes it possible to plot a curve representing the flow of induction according to the f.m.m. (N x I).
The values of the f.m.m.
are deferred on the horizontal axis of a Cartesian reference mark. The values of
flow
are deferred on the vertical axis.
The figure 15-a represents this curve for a
ferromagnetic material running. At the beginning, at the point O,
the f.m.m. is null as well as flow
; then the f.m.m. increases, one
notes that flow
also increases, initially relatively little (at the beginning of the curve),
then in the second time, much more; in the third time, when one
approaches point A, the variation of flow
decreases clearly until being practically cancelled beyond point A.
At point A, there is magnetic saturation. It is said that beyond point A, the core is saturated. Indeed, more the f.m.m. increases, more the number of elementary magnets which constitute the core is directed in the direction of the lines of induction.
When one arrives at point A,
all the elementary magnets are directed and consequently, flow 
cannot increase.
The curve of the figure 15-a is the curve of the first magnetizing because it is obtained when one magnetizes for the first time a ferromagnetic core.
Now, we should consider the case of a reel with core traversed by a AC current. For that, let us leave point A at the point of saturation previously described.
One could think that when the f.m.m.
decreases, flow
takes again the same values that previously, but it of it is nothing.
On the figure 15-b, one sees that the flow
of point A at the point B
takes values higher than those relating to the first magnetizing.
In particular, when the current becomes null, we see that flow is not to it (not B).
It is about residual flow or remanent flow.
Beyond the point O, towards the left, the values of the f.m.m. become negative, i.e. current I changed direction. (Figure 15-c).
One realizes that when the f.m.m.
reached a certain negative value (point C),
flow
becomes null.
We see thus that cancelling residual flow, it is necessary to make circulate in the rolling up of the reel a certain current directed in contrary direction with that having been used to magnetize the core.
One can say that the flow of induction follows the variations of the AC current with a certain delay. This phenomenon constitutes magnetic hysteresis (hysteresis means delay).
If the f.m.m. continues to increase, the current being always in the contrary direction with that of the first magnetizing, flow increases (curve of the point C at the point D) but it changed direction compared to that of the first magnetizing.
When one arrives at the point D, the core is saturated, all the elementary magnets were directed in the contrary direction with that of the first magnetizing.
When the current decreases again until being
cancelled, flow
Thus decrease point D at the point E.
it exists still a remanent flow equal in intensity to that considering
previously but of contrary direction.
When the f.m.m. increases again, one passes from the point E at the point F (null flow) then one joined point A of saturation. Thus, one achieved a complete hysteresis loop.
The arrows on the figure 15-c indicate the direction of course of the cycle.
For any reel having a core ferromagnetic and traversed by a AC current, there is such a hysteresis loop in the core. It is the case, in particular, of the mains transformers which will be the object of the next theory.
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