Circuit Theory

In electrical circuits, the connection between the Passive elements, is carried out by means of conductive materials that force the electric current to follow certain paths, obeying Kirchhoff's laws. When it comes to studying electrical machines, electromagnets and other electromagnetic devices, a similar problem of channeling and concentrating high densities of magnetic flux arises, in the regions where it is needed, which is achieved by means of ferromagnetic materials. A magnetic circuit is generally formed by an iron structure, on which one or more coils are rolled through which currents circulate, which give rise to the flows that appear in the system. The rigorous calculation of the flows produced is generally very difficult and for a precise determination it would be necessary to correctly use the Maxwell equations and the help of an analog or digital calculator (computer); however, the rules of the magnetic circuits that will be studied in this chapter, allow solving the problem in an approximate way and most of the time sufficient for the applications that occur in Electrotechnics.

As is already known from a basic electromagnetism course, the Macroscopic magnetic properties of a linear, homogeneous and isotropic material are defined according to the value of the magnetic susceptibility Xm› which is a dimensionless coefficient that expresses the proportionality between the magnetization or magnetization M and the intensity of the magnetic field H according to the equation:

M= Xm H

As you also want magnetic induction B is related to the Fields H and M by:

B = Mo (H + M)

B = Ho (H + Xm H) = 4o (1 +Х) H = MOM, H = H

Where u represents the magnetic permeability of the medium (M = MO M.) And M, the relative permeability which in turn is equal to 1+ Xm Mo is the permeability of the vacuum and which in SI units is 4710-7 H/m. According to the value of M, the materials are classified into:

Diamagnetic: if M, ≤ 1 Paramagnetic: if M, ≥ 1 Ferromagnetic: if M, >> 1 (Xm has a high value)

To understand the macroscopic magnetic behavior of A material it is necessary to resort to quantum mechanics. However, a qualitative description of magnetic phenomena can be given based on the classic Bohr-Sommerfeld atomic model. According to this model we can assume that the atom is formed by a fixed central nucleus that contains protons and neutrons around which the electrons rotate describing closed orbits that can be considered as electrical circuits. Each of these "circuits" originates a dipolar magnetic moment m (which is the product of the current through the spiral surface of the circuit), which is associated with an angular momentum L or moment of the amount of motion (L = mr2∞, where m is the mass of the electron, r is the radius of its orbit and ∞ the angular velocity of rotation) It must also be taken into account that the electron rotates on itself: electron spin, which gives rise to a greater angular momentum and an additional dipolar magnetic moment that is incorporated into the atom. The previous effect is called spin-orbital interaction (or L-S bond), thanks to it the orbital moment of the electrons is linked with their magnetic moment of spin, forming the total magnetic moment of the atom.

In a diamagnetic material, the net magnetic moment due to The orbital movements of electrons and their spines in any particular atom is zero in the absence of an external magnetic field. When applying an external field of induction B, a force will appear on the electrons Orbitals according to the Lorentz formula:

Em = q (u x B)

Diagmagnetism

Diamagnetism is mainly due to the orbital movement of Electrons within an atom and is present in all materials. In most of them the effect is very weak and that is why sometimes this phenomenon is masked by stronger ones, as occurs in the paramagnetic and ferromagnetic materials that will be studied later. The diamagnetic materials have no remaining magnetism, which means that the induced magnetic moment disappears when the applied outer field is annulled. The value of Xm in diamagnetic materials is independent of temperature, this fact was discovered experimentally in 1895 by Pierre Curie, and justifies the fact that the Larmor movement of the electrons is established very early and both the thermal movement and the collisions between atoms do not modify the Larmor frequency.

Paramagnetism

In some materials, the magnetic moments due to the Movements of electrons: orbital and spin, are not completely canceled and atoms and molecules have a net magnetic moment. By applying an external magnetic field, in addition to producing a weak diamagnetic effect, the field tends to align the molecular magnetic moments in the direction of it, which causes an increase in induction. The macroscopic effect is then equivalent to a positive magnetization, that is, to a positive magnetic susceptibility. The alignment process is considerably counteracted by the random thermal vibrations of the material. There is little coherent interaction between atoms and therefore the increase in induction is quite small, being Xm of the order of 10^-3.

The materials that present this behavior are called paramagnetic, highlighting among them: aluminum, magnesium, titanium and tungsten.

Paramagnetism is produced mainly by the magnetic dipolar moments of the spines of the electrons. The alignment forces of the field acting on the molecular dipoles are counteracted by the distortion produced by the thermal agitation. Unlike diamagnetism, which is independent of temperature, the paramagnetic effect does depend on it, being stronger at low temperatures, when there is less thermal agitation. Paramagnetic susceptibility follows Curie's law:

Xm = C/T

In which C is a constant and T the absolute temperature. At room temperature the previous value, is as mentioned before the order of 10-3, that is, of the order of one hundred times the diamagnetic susceptibility. This means that in paramagnetic substances the diamagnetic effect can be dispensed with due to its low value.

Citation

Universidad Politecnica de Madrid: Maquinas Electricas