1.2.1.1 Atomic Structure

Metals, like all elements, have an atomic structure. Around the positive-charged nucleus, the negative-charged electrons^* circulate in different orbits. The electron mass equals 9.1095 × 10⁻²⁸ g (Ashcroft [1.1]), that is, 0.000551 of the mass of the smallest of all atoms—the atom of hydrogen. In normal conditions, the number of electrons in the atom equals the number of positive-charged protons in the nucleus. Due to this, the atom as a whole is a neutral particle. According to de Broglie hypothesis (1924) about the wave properties of elementary particles, the electron has the definite wavelength λ depending on its velocity. The particle with the momentum p = mv corresponds to the wavelength of λ = h/p (where h = 6.6262 × 10⁻³⁴ Js is the Planck constant). In the case of the simplest atom—the hydrogen atom, which has a circular electron orbit—the length 2πr of the orbit should be a multiple of the wavelength (2πr = nλ); otherwise, an interferential quenching of electron waves would occur (Figure 1.2c).

Hence, electrons can occupy only strictly defined orbits of discretely changing diameters. The regions between the permitted orbits are the “prohibited” zones for electrons. This phenomenon is called quantization of orbits, where the integer value of n = 1, 2, 3, . . ., ∞ is called the main quantum number.

In reality, in a multielectron atom, the electrons and nucleus are subordinated to complex influences of a Coulomb and centrifugal forces as well as an external (e.g., terrestrial) magnetic field. Due to that, they move on more complex orbits, which have forms of ellipses relocating in the space. The electrons themselves, however, are in a rotating motion with an angular momentum, called spin.

Therefore, the full description of the electron state in the orbit requires a definition of a set of four quantum numbers:

1. The main quantum number, n = 1, 2, 3, . . ., ∞, which defines the large axis of elliptic orbit, that is, the main energy level or the so-called electronic shell (electronic layer). It has been proven by investigations using cathode rays. These shells are typically denoted with the letters K, L, M, N, O, P, Q (Figure 1.3).

2. Azimuthal quantum number, also called the secondary quantum number l = 0, 1, 2, . . ., (n – 1), which separates the electrons of each shell into n sub-shells, each having slightly different energy. This number defines the small axis of the elliptic orbit. These are the energy sublevels, which create sets of trajectories in the frame of every main energy level. They are marked with the letters s, p, d, f (Figure 1.3).

3. Magnetic quantum number, also called the third quantum number, m_l = 0, ±1, ±2, . . ., ±l, which defines the spatial quantization of the plane elliptic orbits. It is connected with the existence of magnetic moment of an electron, which causes directional orientation of the atom in external (e.g., terrestrial) magnetic field.

4. Spin magnetic quantum number, m_s = ±1/2, which defines the orientation of the vector (axis) of the spin (levorotatory or dextrorotatory).

Both the magnetic quantum numbers m_l and m_s define a number of electrons in a subgroup.

According to the principle called Pauli exclusion principle (1925), none of the electrons in the atom can have the same set of values of the quantum numbers mentioned above like any other electron. Since all these quantum numbers are strictly connected to each other, it means that in a given shell and subshell, there can exist only a strictly defined number of electrons “filling” the given energy level (Figure 1.3).

For instance, for n = 1, there can only be the following possible numbers l = 0, m_l = 0, m_s = ±1/2. It means that in the first shell, only up to two electrons can exist. If there is only one electron, it is a chemically active atom of hydrogen. If there are two electrons, it is a chemically inactive helium atom with the complete shell. At the shell n = 2, two subgroups l = 0 and l = 1 are possible, with the numbers m_l = 0, 0 and ±1 and m_s = +1/2 and −1/2; which means together 8 electrons in the shell, and so on. In this way, all sublevels s can hold at the most 2 electrons, subshells p–6 electrons, d–10 electrons, and f–14 electrons. In the normal state of an atom, electrons successively occupy the lowest (nearest to nucleus) energy levels.

After completing a given shell, it starts building a new shell. However, not all sublevels, as shown in Figure 1.3, are fully filled by electrons. Sometimes, in multi-electron atoms, the repulsing forces from other electrons in the atom cause that farther subshells (Figure 1.3) are more advantageous, from the energy point of view, for a new electron, despite the fact that the subgroup nearer the nucleus is not yet completed (see groups M and N in Figures 1.3 and 1.17 [later in the chapter]). The elements with such a structured atom are called transitory. To this class belong all elements with ferromagnetic properties, such as Fe, Co, Ni, and others (Figure 1.4).

The number of electrons in the extreme, outer shell of a given element predominantly decides the chemical properties of a metal. Therefore, these properties show periodicity moving from the atomic number Z = 1–109, corresponding to the number of electrons in the atom. The electrons placed at the outer main orbit (outer shell) are called valence electrons. In the theory of metals, these electrons are called conduction electrons (or charge carriers) and these electrons decide the electric conductivity of a body. According to the simplified P. Drude’s model (1900) of electrical and thermal conductivity ([1.1], p. 23), “metal atoms that assemble into a solid body get rid of (lose) its valence electrons.” These electrons can move freely within the metal creating the so-called electron gas, whereas ions of metal remain unchanged and immovable.

1.2.1.2 Ionization

In order to lift an electron to a higher energy level than what it occupies in the normal state of the atom, it is necessary to deliver to the electron an additional energy, such as a photon, with a certain defined portion of energy (e.g., of light), equal to 1 energy quantum. We then say that the atom has become excited. The easiest to excite are the valence electrons, because the nearest higher energy level is always open for them, and the distances between outer energy levels are smaller than between internal levels.

The excitation of an electron can lead to its full separation from the atom. It causes ionization of the atom. In such a case, a single-positive-charged ion is created. It is possible to create double- and triple-charged ions. In another case, when an atom “catches” an additional electron onto its orbit, a negative ion is created. The ionization potential and the excitation potential (resonant potential) are expressed in electron-volts (eV).

The energy of 1 eV equals the energy that one electron gains while shifted between points of potential different by 1 V. Examples of ionization energy values for various atoms are: 13.5 eV for hydrogen, 24.5 eV for helium, 13.6 eV for oxygen, 21.5 eV for neon, 10.4 eV for mercury, and 14.5 eV for nitrogen. A totally or partly ionized gas is called plasma. The degree of ionization of plasma changes with the change of temperature. Electric conductivity of plasma in the presence of a magnetic field is a tensor quantity (anisotropy).

1.2.1.3 Crystal Structure of Metals

Metals have crystal structure, that is, ions of metals are distributed in space in an ordered mode. Among 14 possible combinations of distribution of ions in the space, the most important in metals are three types of elementary grids presented in Figure 1.5.

The metals sodium, vanadium, chromium, niobium, and wolfram (tungsten) crystallize according to the regular space-centered grid (Figure 1.5a). The metals copper, silver, gold, nickel, and aluminum crystallize according to the regular face-centered crystallographic grid (Figure 1.5b).

Iron, however, can occur in two different crystallographic forms. At normal temperatures, it has the regular space-centered grid (Figure 1.5a), whereas at temperatures higher than 906°C, it adopts the regular face-centered crystallographic grid (Figure 1.5b).

About 30 elements, including α-cobalt, magnesium, neodymium, platinum, titanium, zinc, and zirconium, crystallize according to the hexagonal grid structure (Figure 1.5c) with the densest packing in space [1.1].

Properties of crystalline bodies depend essentially on the spacing of ions in crystal. The spacing of ions along one crystallographic axis can be different than the spacing along another axis. It results in different physical properties of the body in different directions, that is, anisotropy of crystals. Physical properties of metals in solid state depend mainly on their crystallographic structure. For instance, the metal mass density at any temperature can be calculated on the basis of knowing the structure and characteristics of the elementary space grid of the metal.

In reality, metals do not have an ideal crystallographic structure. Their grid is deformed by admixtures, unoccupied nodes, thermal motion of ions around the equilibrium position, and so on. As a result of these deviations, “the electric conductivity of metals is not infinite” ([1.1], p. 167).

1.2.1.4 Electrical Conductivity and Resistivity of Metals

The proximity of particular atoms in a solid body against each other, and especially in crystal, causes mutual penetration of electrons from one atom to another, creating considerable forces between interacting atoms and splitting of stable energy levels into a big number of intermediate energy levels, close to each other, which are permitted for the motion of electrons. This effect obviously appears strongest on the external surface. This mutual interaction of atoms (ions) in the case of metals is so large that for external valence electrons it creates a practically continuous zone of permitted energy levels adherent to each other, called energy band.

In metals, only a part of the allowed energy levels in the valence zone is occupied by electrons. The remaining higher energy levels, which also create a continuous zone (energy band), are free. Therefore, while it is necessary to supply defined discrete (quantum) portions of energy in single atoms for the excitation of a valence electron to a higher energy level, in metal, the continuity of permitted energy levels makes it possible to lift the valence electron to a higher free level by means of any arbitrary small amount of energy. Therefore, if in a sample of metal we produce an electric field, the interaction force of this field causes the excitation of electrons to higher free energy levels of this zone (called conduction band) as well as their motion toward the direction of higher potential of the imposed field. This motion of electrons is the electric current, and the described phenomenon is called electron conductivity.

Hence, any continuous zone of partially filled permitted energy levels is called conduction band. Electrons in fully filled energy levels (as in the valence band) cannot therefore in principle participate in the electron conductivity. This is due to the lack of sufficiently near, free energy level that could be occupied by electron after receiving an additional small portion of energy from the electric field.

Generally, we say that valence electrons in metal are in an “unbounded” state, creating a specific electron gas filling the space created by regularly disposed positive ions. This gas can move in ionic lattice under the influence of imposed electric field. Unbounded electrons, colliding with the ions of the lattice, recoil from them in an elastic way, but cannot leave the metal, because it is prevented by the difference of potentials on metal–vacuum (or metal–dielectric) boundary. This potential difference is related to performing a certain work called electronic work function (L = eV). A number of electrons, which grows with increasing temperature of metal, acquires sufficiently large energy required for an electron to leave the metal. This effect is called electron thermoemission.

Despite the fact that densities of electron gas are a thousand times higher than that of classical gases, in the Drude’s model (1900) of electrical and thermal conductivity ([1.1], p. 23), the electron gas follows the normal kinetic theory of gases, with small modifications. According to this theory, the free electrons, moving with a constant speed under the influence of applied voltage, convey their energy by colliding with the crystal lattice and are subject to scattering. This scattering, appearing as a “phonon” resistivity (or collision resistivity), ρ_i(T), occurs on irregularities of lattice, caused mainly by thermal vibration of ions (Wyatt [1.22], p. 551). By analogy to the quanta of radiation field—photons, the quanta of field of displacement of vibrating ions are called phonons (Ashcroft [1.1], p. 540). Owing to the described collisions, the electron gas brings body to the status of thermal equilibrium with the ambience. It is because the hotter the body region (bigger ion vibrations) where collisions occur, the faster the electron will abandon it. This causes the strong dependence of the resistivity ρ_i(T) on temperature (Figure 1.6).

In the temperature range occurring in electric machines and power equipment, exactly this type of resistivity of electron–phonon interaction dominates. In this range of temperatures, the resistivity is approximated in the known way, with a straight line:

At significant reduction of the temperature T, the dispersion of phonons lessens— at the beginning linearly to about one-third of the Debye temperature (344 K for Cu), and next according to the fifth power of the temperature, up to the steady-state residual value ρ_r. In this way, the next components of resistance come to light, and as a result, in accordance with the Matthiessen’s rule (1864), the total resistance ρ of metal can be expressed (Handbook [1.13], p. 26); ([1.22], p. 555]) as

where ρ_r is the residual resistance in low (helium) temperature, to a large extent dependent on lattice defects and even on trace impurities whose influence at room temperature can be completely neglected.

The ρ_i(T) component of the electron–phonon interaction, as per Bloch and Grüneisen (Smolinski [1.12], p. 27), is expressed as the ideal resistivity:

• At high temperatures

  ρi=AMTD(TTD)5⁢(1.3)

• At very low temperatures

  ρi=AMTD(TTD)5∫0τ/τDz5(ez-1)(1-ez)dz≈{bT5(T⁢<TD10)aT(T>TD)⁢(1.4)

  where T_D is the Debye temperature (426 K for Al, 1160 K for Be) [1.12], M is the atom mass of metals (26.97 for Al, 9.013 for Be), and A is the Bloch’s coefficient.

Since aluminum and beryllium have large T_D and small M and A, only these metals, but not, for example, Cu (344 K; 63.54), should be applied as conductors for operation at low temperatures. Furthermore, they should be very pure metals.

ρ_H(T)—the so-called magnetoresistance, related to the Hall effect described below; it is strong in cadmium, weak in aluminum, and negligible in many others metals [1.12].

ρ_l(T)—resistivity related to the so-called dimensional effect that appears when dimensions of wire, foil, grains, and so on, are comparable with the mean free path of electrons (e.g., about 0.1 mm or more); in such a case, the resistivity increase is inversely proportional to the smallest dimension [1.12].

ρ_eddy(T)—a component resulting from the skin effect and eddy currents at alternating currents (ACs); in engineering, it is taken into account with the help of the coefficient k_ad of additional losses:

ρ_eddy = (k_ad − 1) ρ_i

In the design of a cryoelectric apparatus, where conductors made of high-purity metals and of thickness no more than 50 μm are applied, a compromise should be found between the “dimensional effect” and the “skin effect.”

The flow of electric current of density J under the influence of an external electric field E is described by Ohm’s law:

where the coefficient of proportionality σ = 1/ρ is called conductivity.^*

If N is the number of free electrons of charge e, per unit volume of conductor, and v is the net velocity of electrons as an effect of operation of the electric field E and retarding collisions with atoms oscillating due to heat, the electric current flowing through the surface A will be

After the introduction of mobility of electrons μ_e = v/E, we obtain from Ohm’s law E = J ρ the important formula for the resistivity of metal:

where m = 9.1095 × 10⁻³¹ kg is the rest mass of electron; τ = m/(ρNe²) is the relaxation time, that is, the average time of free motion of electron between the succeeding collisions, or otherwise time constant, of convection of electron; and v_u(t) = (|e|E)/ m (τ(1 − e⁻t/ τ)) (Kozlowski L. [1.7]). The τ values, in the temperature range from 273 to 373 K, vary in the limits: τ_Cu = (2.7 . . . 1.9) × 10⁻¹⁴ s and τ_Fe = (0.24 . . . 0.14) × 10⁻¹⁴ s.

It means that up to the voltage frequency of the order f = 10⁸ MHz, the electron transit time can be neglected, as well as their velocity, which “at the strongest fields appearing in metals is on the order of 0.01 mm/s” ([1.7], p. 187).

1.2.1.5 Influence of Ingredients on Resistivity of Metals

Centers of dissipation of oscillations in the concentration n_osc = rT and the corresponding conductivity σ_osc, plus defects (n_d, σ_d) and impurities (n_imp, σ_imp), reduce the mean free path of electrons, which causes an increase of metal resistivity, according to the formula ([1.7], p. 188)

where m* is the effective mass of the electron, v_F is the Fermi velocity, and r is the coefficient of proportionality to absolute temperature T of lattice oscillation concentration.

The conductivity σ of materials used in electrical engineering is usually expressed in percents of conductivity of the international standard of annealed copper ([1.22], p. 562), IACS (International Annealed Copper Standard), of the value σ = 58.824 × 10⁶ S/m, at 20°C (100% IACS).^* At present, there is available a copper with conductivity σ = 103% IACS. Pure silver has σ = 106% IACS and pure aluminum has σ = 60% IACS. The resistivity of conductors significantly depends on ingredients and impurities. For example, the resistivity of alloy Cu–Ni is ρ ≈ (1.5 + 1.35 × δ_Ni%) × 10⁻⁸ Ωm at the nickel content δ_Ni% from 0 to 3.32% ([1.22], p. 556).

As per J. Linde (Ann. Phys. 1932), at admixture contents 1% dissolved in copper, silver, or gold, the resistivity increase is proportional to (Δz)², where Δz is the difference between the valence of the material dissolved and the solvent.

Pure metals show a regular crystalline structure and have a small resistivity. Plastic deformation and presence of admixtures, even in small amounts, cause the deformation of the crystal lattice and an increase in metal resistivity.

At recrystallization by annealing, the resistivity increased due to plastic processing can be reduced back to the initial value. In Figures 1.7 through 1.9, the influence of different admixtures on the conductivity of copper, aluminum, and iron is shown.

1.2.1.6 Resistivity at Higher Temperatures

At temperatures higher than room temperature, up to about 100°C, the resistivity of bronze, Cu, and Al maintains linear dependence (Figure 1.10), whereas the resistivity of steel increases rapidly.

At transition from the solid to the liquid state in most metals, a jump in resistivity occurs (Figure 1.6). The ratio of this increase is: for mercury, 3.2; for tin and zinc, 2.1; for copper, 2.07; for silver, 1.90; for aluminum, 1.64; and for sodium, 1.45 [1.13].

1.2.1.7 Thermoelectricity

When two different metals come into contact, there appears between them a difference of potentials, which is caused by different values of electron work functions (of exit from metal) and also because the numbers of free electrons in different metals are not equal. Hence, the pressure of electronic “gas” in different metals may be different. This contact difference of potentials for different pairs of metals is in a range from a few tenth (fraction) of volt to several volts. If temperatures of contacts are equal, the sum of potential differences in a closed circuit, consisting of two different conductors, equals zero. If, however, one of the welds in this circuit has a higher temperature T₂ than the second one T₁, in the circuit, a resultant thermoelectric force is created (the Seebeck effect, 1821), equal to

where k is the Boltzmann constant, e is the charge of the electron, and n_A and n_B are the number of electrons per unit volume in metals A and B.

This effect is used for the measurement of temperature with thermoelements (thermocouples). Keeping the temperature of one of the junctions at 0°C, we use the electromotive force E_u of the circuit to read the temperature of the second junction, after corresponding calibration.

For measurements of temperature in different ranges, one uses different pairs of metals [1.22]: from −200°C to 400°C copper–constantan (60%Cu + 40%Ni); from 0°C to 1000°C chromel (90%Ni + 10%Cr)–alumel (90%Ni + 5%Al); up to 1700°C platinium + platinium + 13%rodium; and in very low temperatures, for example, Au + 0.03%Fe (10⁻⁵ V/K) ([1.22], p. 560).

In measurement systems and calibration resistors, one tries to use metals with a thermoelectric force as small as possible with respect to copper, in order to prevent the introduction of an additional error. Such an alloy with very small thermoelectric force in relation to copper (about 1 μV/°C) is manganin, as opposed to constantan (4.05 × 10⁴ μV/°C) used for thermocouples.

Thermoelectric forces E_u of different metals in relation to platinum are given in Table 1.1. In order to calculate the thermoelectric force of a circuit created by two metals, we subtract from each other two corresponding values from Table 1.1. For example, a copper–constantan thermoelement, with the weld temperature of 100°C and its ends at 0°C, produces E_u = 0.75 − (−3.5) = 4.25 mV, and the copper has positive polarity. Thermoelectric forces also appear in uniform conductors if a drop in temperature exists along their length. It means the gradient of the temperature is accompanied by the gradient of the electrical potential (the Thomson effect, 1854).

Table 1.1 Average Thermoelectric Force E_u of Metals with Respect to Platinum in the Temperature Range from 0°C to 100°C

The phenomena described above are reversible, that is, an opposite to the Seebeck effect is the Peltier effect (1834): when a current is made to flow through a junction composed of materials A and B, heat is generated at one junction and absorbed at the other junction. The Peltier effect, due to its small efficiency, is used to cool only small volumes, utilizing semiconductor thermoelements, which demonstrate higher thermoelectric forces and lower thermal conductivity. The thermoelectricity of semiconductors is utilized, among others, in temperature sensors and other devices.

1.2.1.8 Thermal Properties

The high thermal conductivity of metal conductors is connected with their electrical conductivity. It is because heat transfer takes place mainly with the help of free conduction electrons, that is, the electron gas. Between Ohm’s law and Fourier’s equation of thermal conductivity exists a formal analogy.

According to the experimental Wiedemann-Franz law (1853), at a given temperature the thermal conductivity of metal λ is proportional to the electric conductivity σ, that is,

where the constant a, called the Lorentz number, at 373 K equals a_Fe = 2.88 × 10⁻⁸ V²/K², a_Cu = 2.29 × 10⁻⁸ V²/K² [1.1] and is approximately constant for most of the metals, T is the absolute temperature (in K), λ is the thermal conductivity (in W/(K m)), and σ is the electrical conductivity (in S/m). Pure metals have higher thermal conductivity than alloys. Thermal conductivity of dielectrics is much lower because of the absence of free conductive electrons and the thermal energy in dielectrics is transferred only by elastic oscillations of ions. In metals, both types of thermal conductivity exist, but the first of them plays a decisive role.

Table 1.2 Important Electrical, Thermal, and Chemical Properties of Copper

Source: Adapted from Handbook of Electrical Materials. (in Russian) Vol. 2, Moscow: Gosenergoizdat, 1960; Wesolowski K.: Materials Science. Vols. I, II, III. (in Polish) Warsaw: WNT, 1966.

The electrons moving under the influence of a temperature gradient generate at the ends of an open metal conductor the thermoelectromotive force (the Seebeck effect).

In constructional problems, an important role is played by the material’s coefficient of thermal expansion (CTE), especially at cooperation of parts made of different metals, for instance, copper rods in iron slots of electric machines. In Table 1.2, some important electrical, thermal, and chemical properties of copper are collected.

1.2.1.9 Mechanical Properties

The mechanical properties of metals to a large extent depend on their crystallo-graphic structure and temperature (Figure 1.11). Among the metals used in electrical construction, a special attention is paid to copper. Its mechanical properties significantly depend on thermal and plastic processing and on the contents of admixtures.

In the case of copper, the relation of the extension force P versus elongation Δl does not show an evident border of plasticity P_pl, in contrast to the analogical graph for steel (Figure 1.12). Up to the limit of P_s, the elongation is exclusively elastic, and above that limit—it is elastic and plastic. The point P_H (limit of proportionality) is where the curve begins deviation from the straight line. The force P_r represents the limit of strain endurance to lengthening. Modulus of elasticity (Young’s modulus) is the ratio of stress, below the limit of elasticity, to the relative elongation (strain), that is, the slope of the linear portion of the stress–strain curve.

Plastic processing of copper at cold temperatures (squeeze), appearing during draw-out, causes its hardening with a significant increase of endurance to split (up to 400–500 N/mm²), and a small elongation at split (1–2%). Such a wire is strongly elastic at bending and is not suitable for winding.

Annealing, which consists of heating copper to a temperature of several hundred degrees Celsius (above the recrystallization temperature) and then rapid cooling, causes significant softening of copper. Along with the softening, a remarkable reduction of endurance to split takes place (up to about 200 N/mm²) as well as an increase of plasticity. It also lessens the resistivity, by 2–3%.

The recrystallization temperature is the temperature at which the removal of internal stresses of plastic deformation of crystals appears, by increasing the size of some crystals at the cost of others. The recrystallization temperature varies in the range between 280°C and 400°C, for different sorts of copper and for different deformations, and is accompanied by a sudden fall of strain strength and hardness and at the same time an increase of elongation. In the case of pure copper, the initial temperature of recrystallization is about 180°C [1.23]. For most metals, the higher the deformation, the lower the recrystallization temperature. Any impurities and ingredients in copper increase its recrystallization temperature.

In turbine generators of older construction, rotor windings were made of soft electrolytic copper with the modulus of elasticity of 10⁵ N/mm², limit of elasticity of 42 N/mm², and CTE of 17 × 10⁻⁶ K⁻¹. At such small strengths of copper, any faster starting or stopping of large turbogenerators was accompanied by a permanent deformation of conductors in the slots, caused by reciprocal interaction of thermal expansion and the friction of the conductors on slots. It resulted in damages of winding, insulation, or clampings. Application of copper along with the addition of 0.07–0.1% of silver, with cold press treatment, increased its limit of elasticity to 150 N/mm², which reduced the risk of such damages. Also, for windings of large transformers, sometimes a copper with silver additions is used, which, contrary to normal copper, does not lose its increased elasticity obtained by plastic treatment during exploitation. Due to similar reasons, for the rotor windings, sometimes a conducting aluminum alloy is used, for example, Cond-Al (Latek [1.35]) with a elasticity limit of 11 deca-newton (daN)/mm² and a thermal coefficient of expansion of 13.1 × 10⁻⁶ 1/K. In Table 1.3, selected important mechanical properties of copper are collected.

For commutator bars of electric machines, sometimes an abrasion-resistant, sufficiently hard copper is used, with a hardness of at least 75 daN/mm².

Because components of electric circuits and material for construction elements (clamping plates, consoles) of electric machines and transformers are under hazard from strong magnetic leakage fields, various bronzes are used occasionally. These bronzes contain, apart from copper, tin, beryllium, chromium, magnesium, zinc, cadmium, silicon, and other metals. At properly selected composition, these alloys can reach endurance to split in the order of 80–100 daN/mm², and even more. However, they have a higher resistivity than copper. For instance, cadmium bronzes with contents of 0.9% Cd have after broaching the conductivity of 83–90% of the conductivity of copper and tensile strength up to 73 daN/mm². They are used for manufacturing slipper conductors and corresponding contact elements (commutators) due to their high abrasion resistance. Beryllium bronzes (2.25% Be) reach a strength of 110 daN/mm² at a conductivity of 30% [1.23].

Aluminum, often used as a material replacing copper, is about 3.5 times lighter than copper, and its resistivity is 1.65 times higher than the resistivity of copper.

Table 1.3 Basic Mechanical Properties of Copper at Temperature 20°C

Source: After Handbook of Electrical Materials. (in Russian) Vol. 2, Moscow: Gosenergoizdat, 1960.

Aluminum conductors of the same length and resistance as that of copper will be about 2 times lighter than the copper conductor. However, its diameter must be 1.28 times bigger than the diameter of the copper conductor of the same resistance. It is additionally difficult in the case of replacing copper conductors by aluminum conductors, when the space is limited.

In Table 1.4, selected important technical properties of aluminum are presented.

Pure aluminum has a remarkably (2–3 times) lower mechanical strength than copper, but its alloys with magnesium (Mg), silicon (Si), iron (Fe), and so on have much better mechanical properties. For instance, the alloy called aldrey (0.3–0.5% Mg; 0.4–0.7 Si, and 0.2–0.3% Fe) has the mass density and conductivity almost the same as aluminum, and the tensile strength (35 daN/mm²) close to the strength of copper. This material is used for the conductors of overhead power lines.

In the air, aluminum is always covered by a thin layer of oxide (about 0.001 mm) which protects the metal against further corrosion, but also makes it difficult to join aluminum conductors with each other. Welding aluminum elements or aluminum with copper requires a special technology. On a humid contact of Al–Cu appears an electric cell with a current flow from aluminum to copper, which causes a considerable corrosion of the aluminum conductor.

Table 1.4 Selected Important Electric, Mechanical, Thermal, and Chemical Properties of Aluminum

Source: Adapted from Handbook of Electrical Materials. (in Russian) Vol. 2, Moscow: Gosenergoizdat, 1960.

1.2.1.10 Hall Effect and Magnetoresistivity of Metals

In order to calculate the resistivity of metal (1.7), one has to know the density of free electrons of valence and mobility of electrons. In one of the method of determination of these values, the Hall effect (1879) is used. If a current i flows in the x direction through a sample of metal placed in a uniform magnetic field of flux density B directed along the z axis (Figure 1.13), then the Lorentz force that acts on the electrons moving in the metal

directed along the y axis in the negative direction. This force presses the electrons to the lower side of the conductor (Figure 1.13a). This results in a nonuniform distribution of electric charges, which causes the appearance of a transverse electric field E_H (Hall field) directed along the negative direction of the y axis and opposing the gathering of electrons in the lower part of the conductor. This phenomenon is called the Hall effect.

The electromotive force F = eU_H acting on electrons due to the Hall effect should, in equilibrium state, compensate the magnetic force (1.11), which means

Introducing, according to Equation 1.6, the current density J = Nev, we obtain

The value

characterizing the body properties is called the Hall constant. Equation 1.14 allows the experimental determination of this constant, as well as the density of free electrons N in the conductor. Knowing the Hall constant and resistivity of the metal, it is possible then to determine, on the basis of Equation 1.7, the electron mobility μ_e. Considering the dimensions of the plate sample (Figure 1.13b), we can express Equation 1.14 in a more convenient form:

The resistivity of the metal in the direction of the current flow

is called magnetoresistivity. This value, in magnetic fields of flux density B not higher than about 10 T, grows with an increase of the magnetic flux intensity H ([1.22], p. 564) according to the dependence

and at stronger fields, this growth is directly proportional to H.

Since in most metals the density of free electrons N is on the order of 10²⁹ electrons/m³, the Hall constant R_H is not large. At room temperature, R_H equals, for example, 2.5 × 10⁻¹⁰ Vm³/(A ⋅ Wb) for Na and 0.55 × 10⁻¹⁰ Vm³/(A ⋅ Wb) for Cu (Wilkes [1.24], p. 272). In spite of that, the Hall effect in metals has been used for building direct current (DC) generators in the form of a copper disk rotating between magnet poles ([1.22], p. 563). Electrons in it are shifted to the edge of the rotating disk, which causes a creation of voltage between the axis and edge of the disk. This system is reversible and can also operate as a motor. The Hall effect is also utilized for building magnetohydrodynamic generators (Figure 2.6), or pumps of liquid metals, and other equipment. The Hall effect, even more distinctively than in metals, appears in semiconductors (Section 1.2.4).

1.2.2.1 Superconductor Era in Electric Machine Industry

As it was mentioned, Poles K. Olszewski and Z. Wróblewski (1883), by liquefaction of permanent gases (air, oxygen, carbon monoxide, and nitrogen in static state as well as hydrogen in hazy state) initiated cryogenics. 25 years later, Dutchman H. Kammerlingh-Onnes liquefied helium and discovered superconductivity at a temperature of 4.2 K. It was too low and too expensive to be used in industrial technology. A new way was opened with the discovery of the so-called high-temperature superconductors by J. Bednorz and K. A. Müller (Nobel Prize, 1987). It was revolutionary for cryotechnology of liquid nitrogen (N₂) (77.3 K) and hydrogen (H₂). Both gases are very cheap and sufficient to start building modern superconducting machines and transformers.

For example:

According to S.P. Mecht i N. Avers from Waukesha Electric System and M.S. Walker of Intermagnetics General Corp., dimensions of an average distribution transformer of 30 MVA and 138/13.8 kV lessen at the cooling: from (1) conventional (49 ton and 22 800 L of oil), to (2) cryogenic (24 ton), to (3) liquid nitrogen in open cycle LN2 (16 ton).

“The cryo-cooled HTS transformer operates at closed cycle and all refrigeration is self-contained. The open cycle unit has an internal supply of liquid nitrogen refrigerant, which is automatically replenished, periodically, from remotely located liquefiers or storage vessels. Oil-filled transformers in urban location, which superconducting transformers may replace, are often surrounded by sprinkler systems and oil containment structures” (IEEE Spectrum, July 1997, p. 43).

On the other hand, according to R.D. Blaucher from the U.S. National Renewable Energy Laboratory, power losses in two large 300 MVA generators amount to (1) over 5 MW in a conventional construction (η = 98.6%), and (2) about 2 MW in a superconducting construction (η = 99.4%).

“Conventional and low-temperature superconducting (LTS) generators having 100–600 MVA ratings have similar loss profiles but different efficiencies. A 300-MVA LTS generator’s losses total 2 MW or so—almost the same as just the resistive losses with a conventional rotor. The total includes refrigeration power offset by reduced exciter losses. Assume that 50 W of heat is removed by a liquid helium refrigerator, requiring 50 kW for the compressor (room temperature power): refrigerator loss thus represents about 3 percent of the entire LTS machine losses and only 0.02 percent of its efficiency.

In effect, once the rotor is rid of resistive losses, as in this LTS application, the added refrigerating losses barely make a dent in total machine efficiency—at least, for generators with ratings in excess of 100 MW” (IEEE Spectrum, July 1997).

1.2.3.1 Magnetic Polarization and Magnetization

Electrons in atoms, due to the rotation around the nucleus (orbital moment) and around its own axis (spin moment), act as currents flowing in closed circles and therefore produce a magnetic field. Such elementary circulating currents appear in all bodies. They can be substituted by equivalent dipoles with the magnetic moments

where μ_o = 4π × 10⁻⁷ H/m is the magnetic constant, or the permeability of vacuum; i₀ is the elementary current of rotating electrons; s₀ is the vector numerically equal to the surface encircled by this current, directed perpendicularly as per the rule of a right-handed screw; m is the magnetic flux going out from the pole, sometimes called fictitious magnetic mass; and d is the vector of distance between the poles, directed as s₀.

Magnetic properties of bodies are shaped by the nature of these dipoles and their behavior in a magnetic field. Under the influence of a magnetic field, the dipoles existing in any material medium are more or less ordered. In this way, the body becomes magnetically polarized.

In order to describe the level of magnetization of a body, the vector notions of magnetic polarization J_i and magnetization H_i were introduced:

The magnetic polarization is the full magnetic moment of the volumetric unit of a body, which has N accordingly directed dipoles, that is

This is the value expressed in teslas, similar to the flux density B = μH.

Inside a magnetized body, the flux density from an external field (B₀ = μ₀H) adds itself to the vector J_i from the elementary dipoles. Both values add to each other as vectors and yield the effective flux density

1.2.3.2 Ferromagnetics, Paramagnetics, and Diamagnetics

In magnetically isotropic media, the magnetic polarization is proportional to the magnetic field intensity, that is

The coefficient χ = (∂H_i/∂H) is called magnetic susceptibility. It is the measure of changes in body magnetization under the influence of an external field. If we divide both sides of Equation 1.25 by H, and consider the last expression, we shall obtain

The coefficient μ_r = 1 + χ is called relative permeability. Depending on this relationship, all materials can be divided into the following groups:

• Ferromagnetic (χ ≫ 1), like Fe, Co, Ni, Cd, and their alloys

• Paramagnetic (0 < χ < 1), for example, steel over the Curie temperature, and antiferromagnetics (e.g., MnSe, MnTe)

• Diamagnetic (χ < 0) and quasi-diamagnetic, for example, Cu, Al in AC field, or superconductors

From the technical point of view, ferromagnetics may be subdivided, on the basis of their hysteresis loops width (Figure 1.20b), into soft and hard magnetic materials.

Soft ferromagnetics have narrow hysteresis loops, with a small value of the coercive magnetic field intensity H_c (0.6−50 A/m) and rather high remnant magnetic flux density B_r (up to 1.7 T). Soft ferromagnetics are used for the laminated cores of electric machines and transformers with alternating flux, where small iron losses and high flux densities are desired.

Hard magnetic materials, on the contrary, have broad hysteresis loops, with H_c from about 8 to 200 kA/m and, for rare earth magnets, even more than 550 kA/m (Figure 1.22 [later in the chapter]). The latter materials are used for the production of permanent magnets of DC, synchronous, hysteresis, stepping, and other motors. Their energy per unit volume is high.

The excitation in electric machines is produced by electromagnets in large machines and by permanent magnets in small ones. This is due to the fact that the magnetic energy of permanent magnets is proportional to the cubic power (l³) of the linear dimensions l, while the magnetic energy of electromagnets is proportional to the product of flux Φ = Bs and the magnetizing force F_m = IN, that is, to the fourth power of linear dimensions. At the same time, if it is considered that the volume of a permanent magnet in an electric machine is inversely proportional to its maximal energy (BH)_max, and that since the beginning of the twentieth century, the specific energy of permanent magnets has increased more than 30 times (as shown in Figure 1.22 [later in the chapter]), the impact on machine design and weight that has been achieved can be understood. In the same period, the per-unit power loss in laminated cores has been reduced almost 10 times, which again has improved the design parameters. It is expected that amorphous magnetic materials, high-temperature superconductivity, and silicon micromechanics will have further significant impact on modern motors and their performance.

The magnetic permeability μ of a body depends, therefore, on the number N of magnetic dipoles in a unit volume of the body and on the direction of the polarization vector J_i with regard to the vector H of magnetic field intensity.

In a general case of anisotropic medium, the permeability can be a tensor (Equation 2.82). Normally, however, the vectors are parallel or antiparallel to each other (i.e., in opposite directions).

Depending on the values and directions of the vectors, all bodies can be divided into the following groups:

• Ferromagnetic bodies, in which the second component in Equation 1.27 is positive and much bigger than μ₀. This group includes the bodies with clearly evident magnetic properties, that is, iron (Fe), nickel (Ni), cobalt (Co), gadolinium (Gd), and their alloys, with the permeability μ_r higher than 1.1.

• Paramagnetic bodies, in which the second component in Equation 1.27 is positive but much lower than μ₀. Most of these bodies have the relative permeability μ_r = 1.000–1.001, not depending on the external field. They are, for instance, iron at a temperature higher than the Curie point, platinum family, sodium (Na), potassium (K), iron salts, oxygen (O), and others. In some materials, interatomic forces act in a way that magnetic moments of adjusted atoms are antiparallel against each other. This phenomenon is called antiferromagnetism and also shows hysteresis and Curie point. Since the permeability of antiferromagnetics is very small, they are considered as paramagnetic bodies (e.g., MnSe and MnTe).

• Diamagnetic bodies, in which the second component in Equation 1.27 is negative. Their relative permeability μ_r is therefore a little smaller than 1. Placed in a strong external field, they are repelled toward a weaker field. This is due to the fact that in the bodies the external field causes such a change in motion of electrons along orbits, which according to the Lenz rule will produce a field opposite to the excitation field. This effect exists also in ferromagnetics, but in them it is masked by a much stronger opposite action of magnetic moments of spins. This is why a body becomes diamagnetic–when the resulting magnetic moment of particle equals zero, which means that the outer electron states are full.

The differences in the above properties of particular bodies follow from different structures of atoms and particles, as well as from different crystallographic structure.

It should be mentioned that nonferromagnetic conductors placed in an alternating field also behave like diamagnetics and repel external field. It concerns especially the superconducting state, in which the body acts as an ideal diamagnetic. This effect is caused, of course, by the induced eddy currents and not due to the internal atomic structure of a body.

1.2.3.3 Atomic Structure of Ferromagnetics

Magnetization of ferromagnetics follows from the specifics of their atomic and crystallographic structure.

Let us consider an isolated atom of iron. Iron has the atomic number 26. It means that 26 electrons are present in the orbits of the iron atom. The same is the number of protons in its nucleus. Figure 1.17 shows a schematic distribution of these electrons in shells determined by the main quantum number n and in subshells determined by the azimuthal quantum number l. Magnetic properties of iron are determined in principle by the electrons in the subshells (n = 3, l = 2). This subshell has space for 10 electrons (see Figure 1.3), but in iron it contains only 6 electrons; it means that is filled in only partly. In spite of that, in the next shell, there are two electrons filling the subgroup of n = 4, l = 0 (Figure 1.3). In a solid body, these electrons are free and are located in the conductive zone. This is the required condition of participating electrons from a not-filled subgroup (n = 3, l = 2) in ferromagnetic phenomena.

Every electron has a spin moment of quantity of motion with regard to its axis. It is accompanied by the magnetic moment of electron (in Wb ⋅ m), defined by the formula (Kozłowski [1.7], p. 230)

where e is the charge of the electron, μ₀ is the magnetic constant, m is the mass of the electron, and h is the Planck constant.

The value p_s/μ₀ is called Bohr magneton. The total spin magnetic moment of an atom is the sum of spin moments of particular electrons.

These moments, similar to the spins themselves, can be both parallel and anti-parallel. In many bodies, the number of positive spins equals the number of negative spins. In the case of an atom of iron, the spins of particular electrons in all filled shells also compensate each other, but in the unfilled shell n = 3, l = 2, there exist four uncompensated spins, thanks to which the atom as a whole has the resultant spin magnetic moment equal to the four Bohr magnetons (Figure 1.17). Atoms can also have an uncompensated orbital magnetic moment. However, it is much lower than the spin moment and its participation in the total spin magnetic moment of atom in the case of solid bodies is very low.

In a solid ferromagnetic body, adjacent atoms approach each other and as result of their interaction, they change the above-described motion of electrons in particular atoms or in gases of ferromagnetic metals. At crystallization, in the resulting action of interatomic forces of exchange, of electrostatic nature, splitting and mutual overlaps of external energy levels take place. Owing to that, the external electrons are no more tied to a specific atom, but have a tendency to pass from one atom to another, as well as from the last level (n = 4, l = 0) to the one before the last (n = 3, l = 2), and vice versa. The resulting effect of these motions is a partial compensation of nonbalanced spins, which leads to the reduction of the time-averaged magnetic moment of an atom. Consequently, the resultant spin magnetic moment is reduced at crystallization, in the case of iron from 4 to 2.22, and in the case of nickel—from 2 to 0.6 of Bohr magnetons.

According to Heisenberg quantum theory (1928), expanded by Bethe and others, the described interaction between the dipoles is an effect of the so-called energy of exchange W_w, which is a function of the distance (a) between atoms relative to the radius (r) of the unfilled shell 3d (4f for Gd); see Figure 1.18.

The exchange forces put dipoles in order. At big relative distances a/r, the exchange forces are weak and the material is paramagnetic. At decreasing a/r, the exchange forces are growing and cause parallel positioning of adjacent dipoles, which is a characteristic for ferromagnetics. At further reduction of a/r, the interaction becomes negative and causes the creation of antiferromagnetism or ferrimagnetism,^* which also has antiparallel adjacent spins, but not with the same magnetic moments, which causes some unbalanced magnetization. The introduction of alloy ingredients to pure manganese (Mn), which is antiferromagnetic, increases the distance a between atoms and such alloys, for instance, MnBi or MnCuAl [1.22], can become ferromagnetic.

1.2.3.4 Zones of Spontaneous Magnetization

According to Weiss domains theory (1907), ferromagnetic materials consist of a large number of microzones of spontaneous magnetization (the so-called domains) that contain a significant number of atoms. Their magnetic moments (spins) are oriented in parallel. In this way, these zones even in the absence of an external field are always magnetized to the saturation. Magnetic moment of domain is defined by the magnitude and direction of magnetization and the volume of the domain itself. The direction of magnetization depends on the crystallographic structure of the body. In the absence of an external field and external mechanical stresses, the vectors of magnetic polarization J_i of particular domains are directed in one of the six directions of easiest magnetizations. These directions correspond to the edges of a cube of elementary crystal lattice of iron (Figure 1.5a). In an unmagnetized sample of ferromagnetic, the vectors of magnetization of domains are directed in a chaotic, disordered way (Figure 1.19a), so that the resultant magnetic moment of the sample equals zero. Applying an external field causes, in the final phase, a rotation of the magnetic moments of every atom around their axis (Figure 1.19c and d). Inside a domain, the magnetic moments of atoms, remaining parallel to each other, are oriented in a direction closer to the direction of external field.

Dimensions of domains are big enough (from 0.1 to 0.001 mm) so that by using special methods (powder figures), one can observe them with the help of a normal optical microscope of 200× magnification. Domains can be smaller (or rarely bigger) than particular crystals within which they create something like closed magnetic circuits. Bloch (1932) proved that boundaries between particular domains are not sharp, but they occupy a zone of dimensions of the order of 100 radii of an atom, in which a successive rotation of electron spins at passage from atom to atom takes place. These zones are called boundary layers or Bloch walls. The magnetic moment of a domain can also change itself with no change of direction from its magnetization, but only by a shifting of the Bloch wall, causing an increase of volume of the domain (Figure 1.19b).

1.2.3.5 Form of the Magnetization Curve

The initial magnetization curve (1 in Figure 1.20a) represents the dependence of the flux density B on the magnetic field intensity H in a sample, which in the initial state was not magnetized. The magnetization curve can be divided into three main sections (Figure 1.20a).

Section I—initial, where the curve goes out from the origin of coordinates at the angle defined by the initial permeability (dB/dH = μ_in). In this section, the curve is concave toward the bottom and is subject to Rayleigh’s law (per [1.2]):

where υ = dμ/dH is a constant value.

Changes of flux density in this section of the curve are, in principle, reversible. It means that at diminution of the magnetic field intensity, the flux density returns practically to its previous value. In section I, a reversible shifting of borders takes place, causing an increase of these domains whose direction of magnetization is close to the direction of the external field (Figure 1.19b).

Section II of the magnetization curve (Figure 1.20a) proceeds as the steepest. In the scope of this section, irreversible changes of flux density take place, which are accompanied with the irreversible shifting of borders of the spontaneous magnetization (Figure 1.19c) and, next, the irreversible jumping rotation of magnetization vectors toward the direction of easy magnetization, closer to the direction of the applied field. This process of magnetization, progressing in a jump-like manner (4 in Figure 1.20a), is called the Barkhausen effect. The permeability dB/dH in this section is the highest.

Section III, corresponding to saturation, has the smallest inclination and the permeability dB/dH of the sample. At infinite growth of the magnetic field H, the curve tends to the permeability μ₀. In a significant part of this section, the changes of flux density are reversible. It corresponds to reversible rotation of magnetization vector of the sample from the state shown in Figure 1.19c to the state of full saturation (Figure 1.19d). The polarization of the sample, J_i = B − μ₀H, at growth of the external field H is approaching to a certain constant value J_is (saturation). The flux density B instead, according to formula (1.25), increases further to infinity together with an increase of the field intensity H, according to the linear dependence (B = J_is + μ₀H). The slope of this straight line is small after all, in comparison to the course of magnetization curve of iron. Therefore, we can say here also about a constant flux density of saturation B_s ≈ J_is. It equals about 2.16 T for pure and slightly siliconized iron, ca. 1.98 T for hot-rolled and 2.02 T for cold-rolled transformer sheets, 2 T for cast steel, and 1.5 T for cast iron (Tables 1.6 and 1.7).

By diminishing the external field, the vectors of magnetic polarization return to the nearest direction of easy magnetization, which causes the residual flux density (remanence) B_r.

The rotation of a magnetic polarization vector in phase III requires overcoming the energy of magnetic anisotropy in order to rotate magnetic moments from the direction of easiest magnetization toward the direction of more difficult magnetization. This energy is much higher than the energy necessary for irreversible shifting of Bloch walls. In soft magnetic materials (narrow hysteresis loop), the motion of Bloch walls in phase II goes without greater disturbances. Thanks to that, the rotations of magnetic polarization vectors in phase III are so small that they do not pass the “difficult” direction and due to that they are reversible.

In hard magnetic materials, the motions of Bloch walls are more difficult and the magnetic polarization vectors in the course of doing significant rotations are passing the direction of difficult magnetization, and for their return to the previous state an additional energy is needed, which increases the hysteresis loop.

1.2.3.6 Hysteresis

The irreversible processes occurring in section II of the magnetization curve are the cause of the hysteresis loop.

Table 1.6 Conductivity σ, Per-Unit Hysteresis Losses Δp_h1 at 50 Hz and at the Flux Density B_m, Coefficient η in Steinmetz Formula (1.39), Metal Mass Density ρ_m, and Saturation Flux Density B_sat. of Different Types of Steel

Source: Adapted from Bozorth R.M.: Ferromagnetism. New York: Van Nostrand, 1951; Jezierski E.: Transformers. Theory. WNT Warsaw, 1975 (see also [5.2]); Jezierski E. et al., Transformers. Construction and Design. (in Polish) Warsaw: WNT 1963; Neiman L. R. and Zaicev I. A.: Experimental verification of the surface effect in tubular steel bars. (in Russian) Elektrichestvo, 2, 1950, 3–8, and others.

Table 1.7 Magnetic Materials Used in Electronics and Power Electronics

Source: After Handbook of Automation Engineer (in Polish), WNT 1973, and others.

One of the best known methods of a mathematic description of the complicated magnetization processes is the Preisach model of hysteresis from 1935 (published in Zeitschrift für Physik, 1938) in which a ferromagnetic material is represented as a collection of small elementary domains (hysterons) with parallel-connected rectangular hysteresis loops (Figure 1.20a, right). Each hysterons is magnetized to a value of either h or −h. They interact with each other and create a stepped graph with accuracy, depending on the number of elementary loops, N. The Preisach model has been followed and improved by other researchers too (e.g., Atherton [1.28]).

At small values of H_m, the symmetric branches of the hysteresis loops are parabolic. When H_m increases, the loops become longer, reaching the shape of the letter S and their dimensions approach a certain boundary shape called boundary hysteresis loop (Figure 1.21). At yet bigger values of H_m, only the “moustaches” of hysteresis loop lengthen. They run along the normal magnetization curve and tend to the saturation flux density B_sat = J_is + μH ≈ J_is = const (Table 1.6).

Vertexes of hysteresis loop create the so-called commutation (vertex) magnetization curve, usually identified with the initial magnetization curve.

The boundary hysteresis loop in the points of crossing of coordinate axes defines two characteristic parameters: residual (remanence) flux density, ±B_r, and coercive magnetic field intensity, ±H_c (Figure 1.21). The H_c determines the external field necessary for the full demagnetization of a sample. The hysteresis loop shape is the basis for the division of magnetic materials into soft with a narrow hysteresis loop (cores of transformers and electric machines—Figures 1.21 through 1.24) and hard with a broad hysteresis loop (permanent magnets—Figure 1.22). The surface area inside the hysteresis loop, expressing the energy necessary for the remagnetization of the sample, in J/m³

is at the same time equal to the power loss for the hysteresis Δp_h1 during one (1) period. A hysteresis loop measured oscillographically at an alternating current AC (called dynamic) is broader (Figure 1.21) than the loop measured at direct current DC. This is caused by additional power losses from eddy currents induced by changes of magnetization. In soft magnetic materials, one pursues to reach the hysteresis loop as narrow as possible. Such are the so-called silicon steels. Addition of silicon, without changing any other conditions, causes a significant reduction of power losses caused by hysteresis loop phenomena. In addition, the hysteresis losses depend on many other factors, of which some more important ones are ingredients, manufacturing and plastic working, mechanical stresses, and heat treatment (annealing).

There were great improvements to the working process of iron. For instance, transition from what was widely used until the beginning of the 1960s, hot-rolled (4% Si) transformer sheets to anisotropic, cold-rolled (2–3% Si) transformer sheets, caused 4 times reduction of hysteresis losses (at B_m = 1 T) (Table 1.6; Figure 1.23). Next, the discovery and application (around 1980) of the rapidly cooled amorphous strips (Figure 1.24) caused another 4 times reduction of the iron loss in comparison to cold-rolled sheets.

A soft magnetic material of small hysteresis losses should be very pure, with uniform structure, free of internal deformations, and with small anisotropy, so that motions of Bloch walls and the rotation of magnetization vectors can be executed with as little obstacles as possible.

On the other hand, a hard magnetic material (with a high coercion) should have maximally hindered motions of Bloch walls and the rotation of magnetic polarization vectors. This is supported by strong stresses inside crystal lattices, big anisotropy, presence of other phases, and grainy or powdery material structure. Fine crystals of powder may not have Bloch walls at all and therefore magnetization of most of them can be carried out solely by nonreversible rotation of a polarization vector, which requires very strong fields. That is the reason for the large coercive intensity of powdered magnetic materials.

In electronics and power electronics engineering (energoelectronics), the alloys most often used are that of ferromagnetic metals, Fe, Ni, and Co, as well as ferrites.

Ferrites are semiconductors (σ = from 1 to 10⁻⁹ S/m) created from complex compounds of the ferric oxide (Fe₂O₃) with oxides of other bivalent metals with the general formula MeO ⋅ Fe₂O₃, where symbol Me means Ni, Mn, Fe, Co, Li, Mg, Zn, or Cu. They are manufactured by a onefold or twofold burning of mixture of these components in temperatures from 900°C to 1400°C. There are distinguishable soft magnetic ferrites (cores of induction coils and transformers of high frequency) and hard magnetic ferrites (permanent magnets). Ferrites with rectangular hysteresis loops have been used for manufacturing memory elements for computers, magnetic amplifiers, and so on. In magnetic amplifiers (Table 1.7), small power transformers, instrument transformers, and HF coils, apart from silicon sheets, amorphous strips, and ferrites, also so-called permalloys, are used. The permalloys are nickel–iron alloys, with a high relative magnetic permeability and contents of 35–85% of Ni, and the rest is Fe, Mo (molybdenum permalloy), Mn, Cr, or Cu. The great success of the 1980s was the previously mentioned amorphous strips of type METGLASS, so-called “metallic glass” (Table 1.7; Figure 1.24) used for magnetic cores, from low (50 Hz) to high frequency, for example, 100 kHz, and for pulse and signal transformers (alloys 2605 SC, 2605 S-3) for space vehicles of 400 Hz (2605 CO), and others [1.36].

Hard magnetic materials for permanent magnets are characterized by the part of the hysteresis loop situated in the second quadrant, between B_r and H_C (Figures 1.21 and 1.22), called demagnetization curves. The smaller the volume and mass of the permanent magnet, V_m, with other conditions being the same, the bigger is the magnet energy (BH/2) (in J/m³) and is expressed by the dependence (Turowski [1.18]):

where Φ is the magnetic flux of the magnet and ΣU_m is the sum of magnetic voltages in the magnetic circuit. Therefore, dimensions of the circuit are selected so that the operation of the system be executed near the maximum energy (BH)_max, called energy factor of magnets, or its specific energy.

A typical shape of a demagnetization curve is a sloping line from the point (B_r, 0) to (0, H_C)^* (e.g., ALNICO alloys—Figure 1.22) or to a straight line (e.g., ferrites). In the former case, Bozorth ([1.2], p. 277) recommends an analytical approximation of the magnetization curve, after the so-called Frölich–Kenelly law from 1881:

which leads to the formula

Other approximation formulae of magnetization curves and their evaluations are given in Section 7.1.

The shape of a magnetization curve is characterized with the help of the so-called shape factor or convexity of the curve

For the curve approximated with hyperbola (1.32), as per Bozorth [1.2]

Theoretically, the γ range is between 0.25 (rectilinear magnetization curve) and 1.0 (rectangular hysteresis loop). Practically, the γ value lies between 0.25 (barium ferrite FB1) and 0.65 (ALNICO) [1.2].

After many years of exploiting permanent magnets made of alloy or ferrites, in the 1970s–1980s, there appeared revolutionary discoveries and then quick implementations of new magnetic materials with contents of rare earths neodymium (Nd) and samarium (Sm) (Figure 1.22). In 1986, in Poland, the rare earth permanent magnets (REPM) of type Nd–Fe–B were manufactured and put into practice by the Institute of Material Technology (IMT) of the Warsaw University of Technology.

A relatively low cost of manufacturing was achieved, which promised a broad implementation into industrial practice.

The energy factor (BH)_max increased immediately from 5–18 kJ/m³ to 200–225 kJ/m³ (for the Nd−Fe−B magnets from IMT, Warsaw University of Technology), which changed the whole concept of design, especially of small electric machines with permanent magnets.

The largest resources of rare earths (5 times more than the rest of the world) are in China (Zhang [1.47]). Thanks to the large production of neodymium oxide (200–300 tons a year), the price of neodymium fell in 1986 and 1987 by about 20–30% a year. Magnets of (BH)_max up to 45 MGOe^* = 358.2 kJ/m³ have already been reached. Magnets of Nd−Fe−B have been used in China for manufacturing of thin-dimensional loudspeakers of aerophones, for magnetic faces, magneto-mechanical devices, motors, generators, motors for window wipers, wind generators, DC tachometric generators, servomotors, and stepping and synchronous motors. In 1987, Zhang Xi [1.47] stated that the production of neodymium was higher than the demand for it, and there was a need to accelerate the exploitation of the Nd−Fe−B magnets. Some important drawbacks of REPMs are their low Curie points and strong sensitivity to changes of temperature and corrosion. The industrial application of REPM is discussed in the book chapter (J. Turowski, S. Wiak et al. in [1.3], pp. 19–49).

Power losses in core iron, P_Fe, are one of the fundamental characteristics of magnetic materials for laminated cores of AC electromagnetic equipment. Traditionally, we divide them into those from eddy currents (eddy loss) P_eddy and those from hysteresis loops (hysteresis loss) P_h.

The eddy-current losses are further subdivided into so-called classic P_ed.cl—connected with the thickness and resistivity of iron sheets—which are calculated with the methods of electrodynamics (Section 6.2), and the so-called losses from eddy-current anomalies (Turowski [2.34] and Hammond [2.8], p. 224) P_ed.an depending on the crystallographic structure of the sheet, which means

where as per Equation 6.16, the “classic” eddy-current losses are, in W/kg

where ρ_m is the mass density (kg/m³).

According to Nippon Steel Corporation [1.41], the losses (in W/kg) resulting from the eddy-current “anomaly” are^*

(for amorphous magnetics, χ = 10–300), d is the thickness of sheet, ω = 2πf, B_m is the maximum flux density, σ is that electric conductivity (S/m), ρ_m is the density of the metal (kg/m³), 2L is the distance between the domain walls creating parallel strips of thickness d.^† From Pry’s et al. formula (1.38), it follows that the eddy-current losses can be reduced by the refinement of the magnetic domain structure. For this purpose, a laser irradiation of iron sheets crosswise to the direction of rolling has been used [1.41].

For instance, at B_m = 1.7 T, d = 0.30 mm, and f = 50 Hz, before the lasing, the losses were equal to 1.23 W/kg, whereas after the lasing, they were equal to 0.97 W/ kg [1.41]. The hysteresis losses in anisotropic iron sheets typically amount from 18% to 20% of P_Fe.

The hysteresis losses are equal to the area of hysteresis loop (1.30) multiplied by the frequency of remagnetization f. Currently, there is no sufficiently accurate method of theoretical calculation of the hysteresis loop (except the discrete Preisach mathematical model—Figure 1.20). This is why the hysteresis losses are calculated with the help of semiempirical formulae. One of the oldest and most popular formulae for the hysteresis losses is the Steinmetz formula, from 1891, which expresses the hysteresis losses (in W/kg), at the frequency f (in hertz):^*

where η is the semiempirical coefficient depending on the chemical constitution, thermal treatment, and mechanical working of steel (Table 1.6); B_m is the maximum in-time flux density (in T) at sinusoidal remagnetization; x = 1.5–3.0 is the semiempirical exponent depending on the type of steel, which as per E. Jezierski [1.5] is

• For isotropic, hot-rolled electrical steel, at B_m = 0.5–1.0 T, x = 1.6

• For anisotropic, cold-rolled transformer steel, at B_m ≤ 1.45 T, x ≈ 2;

• For anisotropic, cold-rolled transformer steel, at 1.5 ≤ B_m ≤ 1.6 T, x ≈ 2.25

• For anisotropic, cold-rolled transformer steel, at B_m = 1.7 T, x ≈ 2.60

In the range of variation of the maximum flux density and in the case of electrical sheets used in electric machines and transformers, one also uses the simplified, semiempirical Richter formula [1.5]

where P_h is the hysteresis power losses, in W; ε is a constant depending on the type of steel: for sheets without silicon admixture, ε = 4.4–4.8 m⁴/(H kg); for generator sheets with medium siliconizing (about 2% Si), ε = 3.8 m⁴/(H kg), and for transformer sheets with silicon content 4%Si, ε = 1.2–2.0 m⁴/(H kg); f is the frequency (in Hz); B_m is the maximum in-time flux density (in T) at sinusoidal remagnetization; M_Fe is the iron mass (in kg). By adopting the power exponent x = 2, in simplified calculations, one can consider jointly the hysteresis losses and eddy-current losses, which are proportional to Bm2f2 (6.16).^†

In more accurate calculations, however, the total iron losses P_Fe (1.35) should be calculated on the basis of the curves of the power-per-unit loss, in W/kg, p_Fe = f(B_m) of electrical sheets at a given frequency f and a given thickness d of the sheet (Figures 1.23 and 1.24a), where p_Fe = p_eddy + p_h are the per-unit losses, in W/kg, measured and provided by a manufacturer of steel.

The total power losses in iron of laminated core can then be calculated from the dependence

where k_ad (equal from 1.05 to 1.15) is the coefficient of additional losses as a result of constructional and processing factors of building the core, and M_Fe is the mass of iron core, in kg.

1.2.3.7 Superposition of Remagnetization Fields

In present power electronics systems, there often appears an additional overmagnetization of the core by a DC field or by a field of another frequency.

At a superposition of slow-alternating and higher-frequency fields, the total losses for remagnetization are equal to the area of basic hysteresis loop plus a sum of all partial cycles BB₂ (Figure 1.20a) (Bozorth [1.2], p. 441).

Tellinen et al. [1.42] proposed a simple approximate formula for the per-unit hysteresis losses at superposition of AC field B_m and DC field B_a:

where p_h is the per-unit losses at symmetric remagnetization with amplitude of flux density B_m; η = 1–0, for example, for cold-rolled sheet η ≈ 0.9 at B_m > 1.0 T. A similar formula can be found in Ref. [7.2] (p. 23).

The total per-unit losses p_Fe in iron at the superposition of DC and AC fields, as per Ref. [1.42], can be expressed by this simple formula, with accuracy up to ±15%:

for B_1m from 1.0 to 1.6 T.

In a similar way like the power losses in Equation 1.41, one can calculate the apparent power P_app consumed for the excitation of the core. Using corresponding graphs for p_app = f(B_m) (Figures 1.23 and 1.24b), we get

1.2.3.8 Amorphous Strips

In the 1970s–1980s, Allied Corporation [1.36] discovered, and implemented to industrial production, a new magnetic material METGLASS of revolutionary low per-unit losses p_Fe and consumption of reactive power for magnetization (Figure 1.24). One of the early papers on METGLASS application was published in Poland by the author (J. Turowski) jointly with S. Partyga in 1982 [9.6] and very recently by Maria Dems et al. [1.50], [1.51], [1.52].

Allied’s discovery consisted in establishing that a thin layer of fluid metal of type FeBSi, FeBSiC, and others, when cooled at a high rate, about 1,000,000°C/s (Figure 1.25), freezes to a thin (30–50 μm) elastic strip of frozen fluid, with amorphous structure, similar to glass. From it comes its name “METGLASS” (metallic glass). The reason for the rapid solidification of fluid metal is that its atoms cannot follow up their order and create crystals. Amorphous strips demonstrate per-unit loss p_Fe and magnetizability q_Fe about 4 times lower than the best cold-rolled anisotropic iron sheets (M4).

This was the second (after cold-rolled sheets) revolution in the construction of power transformers, reactors, and other electromagnetic apparatus (Table 1.7), especially because the cost of this material dropped down from about 150 $/kg in 1972 to about 3.3 $/kg in 1987. It affected both power supply equipment and power electronics.

1.2.3.9 Rotational Hysteresis

At rotational remagnetization, which appears at rotational fields in electric machines, there can also occur the above-described nonreversible rotation of the vector of magnetic polarization, which for soft magnetic materials is the fourth process, beyond the cycle shown in Figure 1.20.

At the axial remagnetization (Figure 1.20a), the field energy, W_h, dissipated for hysteresis, in J/m³, equals the area of the loop (1.30), but at the rotational remagnetization, the energy W_h0 of rotational hysteresis is determined as the energy necessary for the rotation of the sample by 360°, that is

where M(Θ) is the curve of the torque, in J/m³.

This phenomenon in rotating electric machines causes, depending on the maximal flux density B_m, about 1.8 times the increase of hysteresis losses, in comparison with the axial (AC) remagnetization. When the flux density B_m increases from zero, these losses first grow, reaching a maximum at about 1.5–1.6 T, and then they decrease to almost zero [1.27], [2.3]. The hysteresis angle ∠(B, H) is the biggest in section II of the magnetization curve (Figures 1.19c and 1.20a), and then decreases almost to zero in section III. Recently, a novel test fixture to measure rotational core losses in machine lamination was published [10.35].

1.2.3.10 Types of Magnetization Curves

Magnetization curves B = f(H) of ferromagnetics are determined experimentally. The magnetization curve called initial or primary (curve 1 in Figure 1.20a) describes the magnetization process of a sample after its previous complete demagnetization. Demagnetization can be reached by a periodic remagnetization of the metal at successive reductions of the magnetic field intensity amplitude. The initial curve is situated near the growing (right) branch of the hysteresis loop. Their ordinates, with a pretty good accuracy, can be substituted by an arithmetic mean of the upper and lower ordinates of the maximal loop (curve 2 in Figure 1.20a). The initial magnetic characteristic (curve) practically overlaps with the curve, passing through vertexes of the family of hysteresis loops, but just slightly lower (Figure 1.21). The curve connecting vertexes of symmetric hysteresis loops is called the vertex curve, commutational curve, or basic curve.

Ideal magnetic characteristics (curve 3 in Figure 1.20a) can be obtained by simultaneous magnetization of a sample by a DC field and an AC field with amplitude being reduced progressively to zero. The measurement is carried out in the way that at every value of DC field, the AC field amplitude is being gradually reduced, from the value equivalent to full technical saturation down to zero. The ideal or hysteresisless magnetization curve, obtained in this way, is characterized by lack of inflexion point. The course of an ideal magnetization curve is very close to the line running through the centers of the horizontal chords of the hysteresis loop corresponding to saturation (Figure 1.21).

1.2.3.11 Curie Point

The magnetization process of ferromagnetic bodies described above undergoes changes while increasing to higher temperatures, while the values of μ, J_is ≈ B_sat, B_r and H_C gradually decrease. Initially, these changes are very slow (Figure 1.26), and only after passing a certain temperature T_C, called Curie point, does the body lose its ability to keep the microzones of spontaneous magnetization (domains), and becomes paramagnetic, with the relative permeability μ_r close to one. This phenomenon is reversible. The Curie temperature is 770°C for iron and 690°C for silicon iron (4% Si).

The Curie temperature phenomenon can be utilized for a direct conversion of thermal energy into mechanical energy, by means of the so-called Curie Motor. In the Curie Motor, an iron disk rotating in the field of a permanent magnet is heated from one side of the magnet by a gas burner to the Curie point and, due to that, the magnet attracts the cold part of the disk, situated on the other side of the magnet.

In metrology apparatus, electricity meters, and electric tachogenerators with permanent magnets, the almost linear dependence of the flux density B on temperature [1.33]

can be a reason of significant errors. The so-called thermomagnetics, with the Curie temperature from 65°C to 120°C, made as shunts of external circuit of magnet, enable thermocompensation of the decrease of useful flux density.

Permanent magnets with rare earth have lower Curie temperatures T_C than usual iron (e.g., Japanese Hicorex-Nd from Hitachi Corporation has T_C = 300°C) and higher sensitivity to temperature fluctuations, which requires greater effort for the elimination of this influence. This is an important disadvantage of these materials, and in REPM motors, it is often necessary to apply special magnetic shunt thermocompensators. Thermomagnets used for thermocompensator should have the Curie temperature between 65° and 120°C, corresponding to the maximal rated (nominal) temperature t_N and the nominal flux density B_N of permanent magnet applied in electric machines.^* An increase of Curie temperature can also be achieved, for instance, with the substitution of iron by cobalt.

Iron, depending on the temperature, exists in three allotropic forms. The space lattice of pure iron in normal temperature (α-Fe, ferrite) corresponds to the form of body-centered lattice (Figure 1.5a). This form possesses ferromagnetic characteristics, which it loses at the Curie temperature, transitioning into the paramagnetic form β-Fe without change of the lattice structure. At a temperature of 910°C, β-Fe passes into γ-Fe (austenite) phase-centered lattice (Figure 1.5b) and next, at 1400°C, it passes into the δ-Fe form with the same structure like α-Fe. These transformations of the metal structure are often presented in phase graphs as a function of temperature and contents of admixtures.

1.2.3.12 Nonmagnetic Steel

The structural transformations of iron enable obtaining nonmagnetic steel and cast iron, which have paramagnetic properties, but maintain at the normal (room) temperature the permanent austenite structure γ-Fe. The relative permeability of such materials does not exceed 1.05–1.5, and their electric conductivity does not exceed (0.5–1.4) × 10⁶ S/m.

The nonmagnetic austenite steels [1.13], [1.9] can be divided into

• Manganic, nickelous, nickelous-manganic, and stainless chromic-nickelous steel

• Nonmagnetic cast steel

• Nonmagnetic cast iron: nickelous, manganic, copperized, and nickelous-manganic (nomag) [1.9]

Manganic steels used for bonnets and shrouds of rotors of large electric machines contain, for instance, 0.25–0.35% C, 18–19% Mn, 1% Cr, and 3–5% Ni. The limit of tear resistance is at the same time about 100 daN/mm². Orientational composition of nonmagnetic copperized cast iron used for light section (thin walls) casting is 3.4–3.9% C, 2.6–3% Si, 6.8–7.5% Mn, 1.5–2% Cu, 0.4–0.6% Al, and 0.3–0.7% P [1.13] and the tear resistance limit is 12–16 daN/mm². Some steel types (with small contents of Ni and Mn) can lose the nonmagnetic properties after being heated up to 400–500°C, or at cold working while passing from the form γ to α.

Typical elements made of the nonmagnetic cast iron include [1.13] covers, containers, sleeves, and other parts of oil switch gears, consoles, yoke beams, and other parts of large power transformers, elements in current instrument transformer housings, flanges, tubes and other reinforcement parts of distribution equipment, magnetic core pressing disks of stators and rotors, internal fly wheels, winding consoles and bus-bars in electric machines, covers and parts of welding transformers, parts of crane electromagnets, and so on. The nonmagnetic alloys should preferably be easy to form (or mold), because they are often used for manufacturing of complicated details. From this viewpoint, however, these materials are inferior to the nonferromagnetic metals, like brass, bronze, and aluminum alloys. An advantage of nonmagnetic steels is their big resistivity, which is another factor helping to reduce power losses due to eddy currents, in addition to small permeability.

1.2.3.13 Influence of Various Factors on Properties of Magnetic Materials

Magnetic properties of various materials generally depend on the chemical constitution, way of manufacturing, and thermal working. They can be generally divided into groups of properties structurally sensitive and structurally nonsensitive (Bozorth [1.2]). The properties that are structurally sensitive, that is, sensitive to such factors as general chemical constitution, admixtures, mechanical stresses, temperature, crystallographic structure, and orientation of crystals, include the permeability μ_r, the coercive force H_C, and the hysteresis losses P_h. The properties that are structurally nonsensitive include the polarization of saturation J_is, the Curie point, the magnetostriction at saturation λ_y, and the constant of magnetic anisotropy K.

An advantageous phenomenon from the viewpoint of electromagnetic calculation of constructional elements made of steel is the fact that significant differences in magnetic properties appear only in the region of weak fields (Figure 1.27), and particularly in the growing part of the curve μ = f(H). With the growth of magnetic field intensity, in the dropping part of the curve μ = f(H), differences in magnetic properties of different alloys are gradually decreasing, and at H = 10⁴ A/m all the curves practically join each other into one curve (Figures 1.27 and 1.29 [later]). More information can be found in [10.10].

The purer a metal is, the steeper is its magnetization curve and the bigger its maximum permeability; at higher field intensities, these differences become lesser. This property is important from the viewpoint of generalized results in the calculation of strongly saturated iron parts (e.g., calculation of impulse magnetization current at switching-on of a transformer or a turbine-generator). Adding alloy-forming components to iron causes the reduction of its flux density.

On the basis of Table 1.8, one can approximately evaluate the flux density, which at a given magnetic field intensity corresponds to the sort of constructional steel when the contents of admixtures do not exceed 2%. As it follows from the table, the biggest influence on the worsening of magnetic properties of an alloy is carbon C. This is why cast iron, which always contains more than 1.7% carbon (mainly in the form of graphite) and other admixtures, has the lowest magnetizability (magnetic permeability). For these reasons, among others, the yoke beams of the biggest power transformers are sometimes made of spherical cast iron of high strength, because small permeability is linked with small eddy-current losses, as it follows from the next considerations. The spherical cast iron has a fine-grained structure of graphite (contrary to lamellar microstructure of graphite in ordinary cast iron), which makes its strength properties closer to the cast iron (tear resistance 70 daN/mm² and bending strength 100 daN/mm²). The chemical composition of cast iron used for yoke beams of large power transformers (H.W. Kerr et al. Proc. IEE 4/64) is as follows:

3.5% C, 2% Si, 0.2% Mn, 1.5% Ni, at properties: σ = 1.82 × 10⁶ S/m, μ_r,max = 1400, or an alloy of 3% C, 2.8% Si, 2.4% Mn, 24% Ni, and σ = 0.98 × 10⁶ S/m, μ_r.max = 1.03. The spherical cast iron is used, for example, for the manufacture of automotive parts, such as crankshafts and distribution shafts, cylinders, piston rings, and so on.

Typical changes of iron resistivity as a function of temperature are presented in Figures 1.6 and 1.28. The character of these curves is also approximately the same for most all iron alloys. Knowing the resistivity at a temperature of 20°C, one can approximately evaluate it for any temperature, for example—at hardening, annealing, or induction smelting.

The fact that admixtures reduce at the same time, the magnetic permeability μ and conductivity σ helps reduction of errors in calculating eddy-current power losses and resistivity of steel (Chapters 2 and 3) caused by material dispersion of these values, because in the mentioned formulae, there typically appears the ratio μ/σ. For instance, for the Polish constructional steel (Figure 1.29):

1.2.3.14 Types of Magnetic Permeability

The ratio of flux density to magnetic field intensity, μ = B/H, in a case when the point (B, H) lies on the primary magnetization curve, is called primary permeability or simply, permeability. The limit that approaches the primary permeability (or the amplitude permeability, μ_a = B_m/μ₀H_m [1.37]), at H approaching zero, is called initial permeability, μ_i.

The highest value on the permeability curve (Figure 1.29) is called the maximal permeability, μ_max. It occurs at relatively weak fields.

The permeability at submagnetization, μ_Δ [1.37], also called the permeability of partial cycle characterizes the properties of a body in a direct (submagnetizing) field and in a superimposed field, which can be alternating. If on a sample, apart of the direct field of intensity H_b, there is an additional alternating magnetic field of intensity H_Δ that is causing the alternating flux density B_Δ (Figure 1.20a), then μ_Δ = B_Δ/H_Δ.

Reversible permeability, μ_rev, is the limit to which tends to μ_Δ, when H_Δ tends to zero. In a demagnetized material, at the zero submagnetization field, the reversible permeability μ_rev and initial permeability μ_in are coinciding with each other. The permeability of partial cycle μ_Δ and the reversible permeability μ_rev depend on the value of submagnetization field H_b (at increasing H_b, the permeability μ_Δ gets smaller) as well as they depend on the magnetic history of the sample. The permeability μ_Δ also depends on H_Δ. The permeability of partial cycle μ_Δ serves for the characterization of materials from which one makes cores of output transformers, magnetic amplifiers, and coils for pupinizations of telecommunication lines [1.2].

Return permeability is the notion analogical to the permeability of partial cycle μ_Δ, but with the difference that changes of sample magnetization are not due to the superposition of an alternating field, but by a partial demagnetization and remagnetization of a permanent magnet, at the opening and closing armature of its magnetic circuit. Opening the armature of a fresh magnetized magnet causes reduction of the flux density from value B_r to B₂ (Figure 1.20). All next cycles of opening and closing will run through the recoil loop B₁B₂ (Figure 1.20). The straight line B₁B₂ is called recoil straight line. It can be considered as an approximate parallel to the demagnetization curve in point B_r. The tangent of the angle of inclination of the recoil straight line B₁B₂ is called the recoil permeability. This concept is utilized for the calculation of permanent magnets and synchronous generators with permanent magnets.

Differential (dynamic) permeability (1.48) is utilized sometimes for the characterization of the steepness of the magnetization curve (μ_dyn = dB/dH). It is the coefficient of inclination of a tangent to the magnetization curve. As the flux density increases from zero, the differential permeability quickly increases and then decreases. When the differential permeability drops to a value merely a few times bigger than the permeability of air, then the iron is considered saturated.

Ideal permeability is the ratio B/H found from the ideal magnetization curve. This permeability, as can be seen from curve 3 in Figure 1.20a, has no extreme value (no inflexion point). It has, instead, a very large initial value, at H = 0.

Nonlinear permeability is the value obtained from the differentiation of the right-handed side of the second Maxwell equation 2.2 with consideration of the functional dependence μ = f(H). In a case when investigating the media of varying permeability, the first Maxwell equation 2.1 remains with no change (in metals, at power frequencies, the displacement currents can be ignored), whereas in the second Maxwell equation 2.2, we must differentiate in terms of the time the composite function μ(H) ⋅ H:

Generally, then

where μ_n is called dynamic, nonlinear, or differential magnetic permeability, equal to

The form of Equation 1.48 presents a simple, though cumbersome, way of finding the curve μ_n(H) from the magnetization curve of a given type of steel (Figure 1.29).

1.2.3.15 Permeability at High Frequencies

At the power frequencies, the permeability does not depend on the frequency, but begins to decrease at radio frequencies (RFs). At the frequency of light, it reaches the value of one [1.2].

Another problem is with the dielectric permittivity ε = ε₀ ε_r and the displacement current density (dD/dt) ≈ ε(dE/dt). In electromagnetic literature, it is commonly accepted (with no discussion) that in normal metals, the current density J_metal = σE ≫ ε_metal(dE/dt) ≈ jωε_metalE. It means that it is commonly accepted that the polarization current of particles in metal, represented by ε_metal, is much smaller and masked by “electron gas” of free electrons. This is one of the most interesting questions and still an open problem of solid-state physics, not sufficiently explained. There exist absolutely controversial opinions from “ε_metal → ∞ at f = 0,” as per Ashcroft [1.1, p. 407], to even ε_metal = 0. The authors’ evaluation shows however that ε_metal is rather nearer to ≈1.

Complex permeability is the permeability taking into consideration a phase shift (in time) of the flux density B_m and the magnetic field intensity H_m in case both values are alternating sinusoidally (first harmonics).^* Like any complex number, one can express it by module μ and argument ψ

where μ is the permeability in the top, return point of hysteresis, called amplitude permeability, μ_a = (1/μ₀) (B_m/H_m) = μe^−jψ, at B_m = const [1.37].

Remagnetization of an iron sample needs the delivery of both reactive and active power to the excitation circuit. Results from that apparent power equals the product of the effective (rms) value of flux density and conjugate effective (rms) value of magnetic field intensity, or vice versa. If we assume sinusoidal changes Hm*=Hme−jωt and after (1.49) B_m = μH_m e^−jψ e^jωt, then we obtain the apparent power per volume (VA/m³) of the sample at frequency f hertz (where H_m, B_m–modules)

On the basis of Equation 1.50, knowing the hysteresis losses P_V per volume, or the magnetization power Q_V of the sample, one can calculate the argument ψ of the complex permeability μ¯:

The complex permeability (1.49) can also be represented as

Considering Equation 1.51, we shall obtain the so-called

conservative permeability

consumption permeability

and loss angle (δ)

The above equations do not consider losses from eddy currents, which are also a reason of phase shifts. These phase shifts can be accounted for either by a corresponding increase of power in the above equations or by the introduction of separate arguments derived from the calculation of eddy currents (Section 3.2).

The assumption of sinusoidal alternation in time of H and B is equivalent to the substitution of the real hysteresis loop by a corresponding inclined ellipse with the same area as in the real hysteresis loop.

Equivalent (effective) complex permeability [2.8], [1.37] is an imaginary permeability of a package of sheets at alternating flux density, in which eddy currents, displacing the flux, cause an imaginary reduction of the effective permeability μ¯e and its shift in phase, whereas μ¯e=Bm,average/Hm (see Section 6.2.1). The μ¯e serves, among others, for the evaluation of the quality factor of coils with iron: Q = ωL/R (Bozorth [1.2], pp. 610–612). The μ¯e is also the permeability of a hypothetic uniform core equivalent to a core of the same dimensions, but containing nonmagnetic gaps and parts with various and/or nonuniform materials.

Tensor permeability appears at the analysis of anisotropic media. It will be discussed in Section 2.3 (Equation 2.82).

Imaginary permeability (fictitious, supposed), μ_im. This term is used (Bozorth [1.2], p. 670) for the permeability of a cylindrical iron sample in which, due to finite length, appears as the internal demagnetization field H_demagn (Figure 1.34 [later in the chapter]) taken into account by means of the demagnetization coefficient N (1.58)

The value of the coefficient N depends on the ratio m of the length to the diameter of the sample; for example, at m = 10 and μ > 300, μ_i = 60–70 ≈ const ([1.2], p. 673).

Quasi-permeability (seeming permeability), μ_q. The author (J. Turowski) introduced this notion in 1982 (J. Turowski [1.16], p. 35) to denominate^* an equivalent fictitious permeability of a solid iron, where a conductor with AC is running nearby. This name is useful in the application of the method of mirror images (Section 5.2) for alternating currents and fields, when the “mirror is not fully clean” due to the eddy-current reaction and iron saturation.

1.2.3.16 Magnetic Anisotropy

A ferromagnetic monocrystal, being a body with a uniform, accurate crystal structure, has different properties along various crystallographic axes. The main crystallo-graphic directions in an elementary isometric lattice of crystal of iron are nominated by the axes [100], [110], [111] (Figure 1.30). The monocrystal of iron has the biggest permeability and smallest hysteresis losses at magnetization in the direction parallel to the [100] edge of cube, the smallest permeability and biggest hysteresis losses in the direction of the longest diagonal [111], and average values in the direction of the smaller diagonal [110].

The anisotropic properties of iron are utilized in the manufacturing of low-loss, cold-rolled electrical sheets of high magnetizability and low per-unit losses. The isotropic, hot-rolled transformer sheets (4% Si), formerly used extensively for power transformer cores, but now practically eliminated, as well as the motor and some generator sheets (about 2% Si), have isotropic, polycrystalline structures with the random orientation of crystals. Due to that, their magnetic properties in various directions are practically identical. Higher-quality types include cold-rolled sheets, which are divided into

1. Generator isotropic sheets (in Poland EP—Figure 1.23), with almost the same magnetic properties like hot-rolled sheets or those with minimal anisotropy; the cold-rolling here has mainly the purpose to obtain better smoothness and equalization, which gives better filling factor of iron core.

2. Anisotropic transformer sheets, also called textured or grain-oriented (in Poland ET—Figure 1.23), in which most of the crystals are oriented in the most advantageous direction [100], in accordance with the rolling direction (Bochnia 2006 [1.55]).

Specific features of the textured sheets manufacturing process include the use of iron with the Si content of 2–3%, cold rolling with interoperation annealing for recrystallization, and final thermal processing at a high temperature (1100°C) in hydrogen atmosphere, which causes a reduction of the contents of carbon in the iron to a value of about 0.005% [1.2]. Cold-rolled transformer sheets are typically delivered as double sided, isolated with a thin ceramic layer, called carlit. Generator sheets can also be covered by an electroinsulating organic coating, or not isolated, which helps to punch it.

The industrial manufacturing methods of textured sheets typically do not fully exploit all their capabilities. The texture obtained (crystal layout) and traditionally used gives only one advantageous direction of magnetization, [100] or [001] (Figure 1.31a). The scheme of the most advantageous texture, not yet achieved in industrial practice, is presented in Figure 1.31b. Such texture, achieved already in the laboratory [1.45], [1.46], would allow to totally eliminate the [111] direction with the most difficult magnetic properties, and to achieve the two most advantageous directions along the [100] and crosswise [001] rolling. An industrial production of the latter sheets would open broad possibilities of their application in building of rotating electric machines.

Lately, there has been a trend in building iron cores in a manner to make the flux passing perpendicularly to the rolling direction as small as possible.

As one can see from Figure 1.32, the ratio of the per-unit losses in the most advantageous directions to the per-unit losses in the most difficult direction depends on the flux density, for example, in the range of B_m from 0.5 to 1.5 T the ratio is from 0.20 to 0.25. The relation of the magnetization energy in the direction of the most difficult magnetization to the energy in the direction of the easy magnetization is called anisotropy constant, K [1.22].

1.2.3.17 Magnetostriction

Magnetostriction describes the process of the change of crystals and dimensions of a body under the influence of magnetization (Figure 1.33). It is caused by the change of interatomic distances between the atoms of a lattice. Magnetostriction, which is characterized by the relative elongation, ε = Δl/l, can be positive or negative. It is one of the sources of noise of electric machines and transformers, but it is also exploited in ultrasonic generators.

Ferrites [1.22]. Ferrites belong to the ceramic, magnetic materials obtained from magnetite Fe₂O₄ or from the double oxide FeO ⋅ Fe₂O₃. In contrast to the normal ferromagnetics, ferrites have a very small electric conductivity and can be considered as insulators (Figure 1.35 [later in the chapter]).

Because of that, phenomena appear in ferrites that are hidden in other materials due to the screening effect of eddy currents. These phenomena are related to the so-called ferromagnetic resonance. Spins of electrons, considered as spinning tops with defined magnetic field, gyrating in an external magnetic field H, perform a precession motion. This causes a tensor relation of the B and H vectors (Simonyi [1.11]). Soft ferrites (e.g., silicon ferrites), with a narrow hysteresis loop (Table 1.7), are used for high-frequency transformers, over 100 kHz. Hard ferrites, containing manganese and magnesium, can reach a hysteresis loop form close to square, which allows using them in computer memory storage. Other ferrites, for instance, BaFe₁₂O₁₉ (magnadur) are used as permanent magnets with very large coercive magnetic intensity (Figure 1.22)—for magnets with big air gaps. Ferrites can have an anisotropic structure. They are relatively cheap and are produced on the basis of powder metallurgy [1.22].

1.2.3.18 Demagnetization Coefficient

If an iron rod is placed in an external magnetic field, H_ext, on its ends will be created magnetic poles, which in turn produce in the rod its own field. This field is directed opposite to the external field and is called demagnetization field (Figure 1.34). The magnetic field intensity of the real (internal) magnetic field in any cross section of the sample is the difference between the external field H_ext and the demagnetization field H_dem, that is

In a first-order approximation, the demagnetization field intensity is proportional to the magnetic polarization of the sample J_i

The proportionality factor N is called demagnetization factor. It depends, first of all, on the shape of the sample. The demagnetization factor can be calculated exactly only for ellipsoid of revolution, placed in a uniform field. Inside of such ellipsoids, the field is always uniform and their demagnetization factor is expressed by the formula^*

in which n is the ratio of the ellipsoid revolution axis directed along the lines of the external field, to the axis perpendicular to it.^* The first form of the expression (1.59) for N is more convenient when n > 1, and the second form when n < 1. By changing the ratio of axes, one can obtain approximate values of N for bodies having a shape different than ellipsoid. For instance, in the case of an infinitely extended plate placed across the field (n = 0), we can obtain the biggest possible value of N = 1.

In the case of a cylinder placed across the field, N = 0.5 [1.2], and a sphere (n = 1) we get N = 0.33, and finally, in the case of an infinite rod placed along the field (n → ∞) we get N = 0. For cylindrical iron rods with the ratio of length to diameter n′ = 5, the obtained N was ≈0.5, and at n′ = 100, N = 0.0045 [1.9]. It follows from this, that iron rods placed inside a solenoid, of length some dozens times bigger than its diameter, can be considered as practically infinitely long.

However, at short samples, one has to consider the demagnetization effect. These conclusions are important from the viewpoint of model investigations with long rods and at the induction heating of short pins or steel charges.

The demagnetization effect reveals itself in machines and electric equipment with permanent magnets in the form of the so-called recoil straight line (Turowski [1.18]; [1.2], p. 285). It has a significant effect, among others, on the reduction of the resultant (apparent) permeability μ′ of a sample of finite length (1.56).

1.2.4.1 Hall Effect in Semiconductors

The Hall effect, described earlier for metals (Figure 1.13), occurs even more distinctively in semiconductors, in which electrons and holes gather themselves in opposite ends of a plate, thus increasing the transverse electric field. Formulae (1.14) and (1.15) are also valid. The Hall constant for semiconductors of type p or n (without dopants of the opposite conductivity) equals to

where N′ is the density of electrons or holes, respectively, and e is the elementary charge (of electron or hole).

The Hall constant in the case of germanium plates is on the order of 10⁻³ Vm/(AT).

The Hall effect in semiconductors is utilized for measurements of magnetic field intensity (the so-called hallotrons).

Also, for the measurement of magnetic fields are utilized spirals made of bismuth wire, whose resistance changes under the influence of change of magnetic field intensity (the so-called gaussotrons).

Piezoelectricity, discovered in 1880 by Pierre and Paul Curie, consisting of mechanical deformation of some crystals lacking symmetry center (like quartz SiO₂, Seignette’s salt NaKC₄H₄O₆ ⋅ 4H₂O, or titanate barium BaTiO₃), under the influence of an external electric field, is utilized, among others, for building precise electromechanical elements of automatics and microdrives of power up to 50 W. Various types of “ratchet” mechanisms enable conversion of crystal’s oscillation into unidirectional displacement of a rotor or a linear element. This effect is reversible.

At a mechanical deformation, for instance at strain, distances between ions increase, which causes an increase of dipole moments between pairs of ions. If a crystal does not have a center of symmetry, the dipoles of opposite signs are not self-balancing and some resultant dipole moment appears, which manifests itself by the emergence of opposing electric charges on the opposite surfaces of crystals, and eventually an electric field.