Magnetism

Magnetism is a fundamental physical phenomenon by which materials and moving electric charges exert attractive or repulsive forces on one another. It is one of the four fundamental forces of nature, alongside gravity, electromagnetism, and the weak and strong nuclear forces. Magnetism is primarily associated with magnetic fields, which arise from the motion of electric charges.

Magnetic materials can be classified on the basis of their magnetic properties and responses to external magnetic fields. The main classes of magnetic materials are:

Ferromagnetic materials are those that can be permanently magnetized. They have a strong attraction to magnetic fields and retain their magnetic properties even after the external field is removed. This strong magnetic behaviour is due to the alignment of magnetic moments in the material. Examples are Iron, Cobalt, and Nickel. They exhibit spontaneous magnetization, have high magnetic permeability and have an important hysteresis.

Paramagnetic materials are weakly attracted to magnetic fields and do not retain magnetic properties when the external field is removed. Their magnetic moments align with the external magnetic field, but the alignment is weak due to thermal agitation. Examples are Aluminum, Platinum, and Manganese. They exhibit a small positive susceptibility to magnetic field, have no hysteresis and magnetic effects disappear when the external magnetic field is removed.

Diamagnetic materials are repelled by magnetic fields. They create an induced magnetic field in a direction opposite to that of the applied magnetic field. This property is present in all materials but is often masked by stronger magnetic effects if present. Examples are Copper, Gold, Bismuth, and Graphite. They don't retain any magnetization in the absence of external fields and exhibit a small negative susceptibility to magnetic fields.

Antiferromagnetic materials have a net magnetic moment of zero because the magnetic moments of atoms or ions align in opposite directions, cancelling each other out. This occurs below a certain temperature called the Néel temperature. Examples are Manganese oxide, Iron oxide, and Chromium. They exhibit no net macroscopic magnetization and above the Néel temperature, they become paramagnetic.

Ferrimagnetic materials have magnetic moments of atoms on different sublattices that are opposed, but unequal, resulting in a net magnetic moment. They are similar to ferromagnetic materials but differ in the arrangement and magnitude of their magnetic moments. Examples are Magnetite, Yttrium iron garnet and ferrites like magnesium ferrite. These materials exhibit spontaneous magnetization like ferromagnetic materials, have high magnetic permeability and show hysteresis behaviour similar to ferromagnetic materials.

$$\mathfrak{F} = \mathfrak{Z} \cdot \phi$$ $$\phi = \mathfrak{Y} \cdot \mathfrak{F} $$

$$\mathfrak{Z} = \frac{l}{A_c} \frac{1}{\mu_0 (\mu_r'-j\mu_r'')}$$ Note that the reluctance depends solely on the physical properties and geometrical shape of the material used as the core. The relationship between reluctance and impedance is the following one $$Z = j \omega L_0 (\mu_r' - j \mu_r'') \ , \ \mathfrak{Z} = \frac{l}{A_c} \frac{1}{\mu_0 (\mu_r'-j\mu_r'')} \ \rightarrow \ Z = \frac{j \omega}{\mathfrak{Z}}$$ where L₀ is the inductance obtained with the same magnetic circuit where the core is not present (mu = mu0) and all flux lines flow as there was the core. $$L_0 = \frac{A_c}{l} \mu_0$$ The two complex and the real parts are the called inductive and loss components respectively and they are $$X = \omega L_0 \mu_r'$$ $$R = \omega L_0 \mu_r''$$ One can define a loss tangent for the magnetic circuit which gives an idea of how much the component is lossy. $$tan \delta = \frac{\mu_r''}{\mu_r'}$$

$$\mathfrak{Z}_{tot} = \mathfrak{Z}_1 + \mathfrak{Z}_2 + \mathfrak{Z}_3 + ... + \mathfrak{Z}_n$$ $$\mathfrak{R}_{tot} = \mathfrak{R}_1 + \mathfrak{R}_2 + \mathfrak{R}_3 + ... + \mathfrak{R}_n$$

$$\frac{1}{\mathfrak{Z}_{tot}} = \frac{1}{\mathfrak{Z}_{1}} + \frac{1}{\mathfrak{Z}_{2}} + \frac{1}{\mathfrak{Z}_{3}} + ... + \frac{1}{\mathfrak{Z}_{n}}$$ $$\frac{1}{\mathfrak{R}_{tot}} = \frac{1}{\mathfrak{R}_{1}} + \frac{1}{\mathfrak{R}_{2}} + \frac{1}{\mathfrak{R}_{3}} + ... + \frac{1}{\mathfrak{R}_{n}}$$

$$\phi_1 + \phi_2 + \phi_3 + ... + \phi_n = 0 \quad \sum_{i=1}^n \phi_i = 0$$

$$\mathfrak{F}_1 + \mathfrak{F}_2 + \mathfrak{F}_3 + ... + \mathfrak{F}_n = 0 \quad \sum_{i=1}^n \mathfrak{F}_i = 0$$

From Faraday's law $$v(t) = \frac{d\phi (t)}{dt}$$ From the magnetic flux definition (n is the number of turns of the coil) $$\phi(t) = n \int_S \bar{B} \cdot d\bar{A}$$ Combining the two previous equations one gets $$v(t) = n A_c \frac{dB(t)}{dt} = n A_c \mu_0 \mu_r \frac{dH(t)}{dt}$$ Remembering that $$n \mathfrak{F}(t) = H(t) l \quad \mathfrak{F}(t)=i(t)$$ one gets $$v(t) = n A_c \mu_0 \mu_r \frac{dH(t)}{dt} = n^2 \frac{A_c}{l} \mu_0 \mu_r \frac{di(t)}{dt} \quad L=n^2 \frac{A_c}{l} \mu_0 \mu_r$$

$$\mathfrak{F}(t) = H(t) \cdot l = \phi(t) \cdot \mathfrak{R} = n i(t)$$