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per unit machine volume. In other words, the size of the machine is greatly reduced by the use of magnetic materials. In view of the fact that magnetic materials form a major part in the construction of electric machines, in this chapter properties of magnetic materials are discussed and some methods for analyzing the magnetic cir- cuits are outlined. 1.1 MAGNETIC CIRCUITS In electrical machines, the magnetic circuits may be formed by ferromag- netic materials only (as in transformers) or by ferromagnetic materials in conjunction with an air medium (as in rotating machines). In most electrical machines, except permanent magnet machines, the magnetic field (or flux) is produced by passing an electrical current through coils wound on ferro- magnetic materials. 1.1.1 i-H RELATION We shall first study how the current in a coil is related to the magnetic field intensity (or flux) it produces. When a conductor carries current a magnetic field is produced around it, as shown in Fig. 1.1. The direction of flux lines or magnetic field intensity H can be determined by what is known as the thumb rule, which states that if the conductor is held with the right hand with the thumb indicating the direction of current in the conductor, then 1 2 chapter 1 Magnetic Circuits FIGURE 1.1 Magnetic field around a current-carrying con- ductor. the fingertips will indicate the direction of magnetic field intensity. The relationship between current and field intensity can be obtained by using Ampere's circuit law, which states that the line integral of the magnetic field intensity H around a closed path is equal to the total current linked by the contour. Referring to Fig. 1.2, (1.1) where H is the magnetic field intensity at a point on the contour and dl is the incremental length at that point. If fJ is the angle between vectors H and dl, then p H dl cos {} = L i (1.2) Now, consider a conductor carrying current i as shown in Fig. 1.3. To obtain an expression for the magnetic field intensity H at a distance r from the conductor, draw a circle of radius r. At each point on this circular contour, H and dl are in the same direction, that is, fJ = 0. Because of symmetry, H will be the same at all points on this contour. Therefore, from Eq. 1.2, H PH·dl=i H 21Tr = i H=-z_· 21Tr (1.2a) Closed path FIGURE 1.2 Illustration of Ampere's cir- cuit law. - H Magnetic Circuits 3 FIGURE 1.3 Determirtation of magnetic field intensity H due to a current-carrying conductor. 1.1.2 B-H RELATION The magnetic field intensity H produces a magnetic flux density B everywhere it exists. These quantities are functionally related by B = JJ.H weber 1m2 or tesla B = J.l.rJJ.oH Wb/m2 or T (1.3) (1.4) where fJ. is a characteristic of the medium and is called the permeability of the medium JJ.o is the permeability of free space and is 47rl0-7 henry/meter f.l.r is the relative permeability of the medium For free space or electrical conductors (such as aluminum or copper) or insulators, the value of J.l.r is unity. However, for ferromagnetic materials such as iron, cobalt, and nickel, the value of J.l.r varies from several hundred to several thousand. For materials used in electrical machines, J.l.r varies in the range of 2000 to 6000. A large value of J.l.r implies that a small current can produce a large flux density in the machine. 1.1.3 MAGNETIC EQUIVALENT CIRCUIT Figure 1.4 shows a simple magnetic circuit having a ring-shaped magnetic core, called a toroid, and a coil that extends around the entire circumference. FIGURE 1.4 Toroid magnetic circuit. 4 chapter 1 Magnetic Circuits When current i flows through the coil of N turns, magnetic flux is mostly confined in the core material. The flux outside the toroid, called leakage flux, is so small that for all practical purposes it can be neglected. Consider a path at a radius r. The magnetic intensity on this path isH and, from Ampere's circuit law, ~H · dl = Ni HZ =Ni H21Tr = Ni (1.5) (l.Sa) (1.6) The quantity Ni is called the magnetomotive force (mmf) F, and its unit is ampere-turn. From Eqs. 1.3 and 1.8 HZ= Ni = F H=~ iAt!m (1. 7) (1.8) (1.9) If we assume that all the fluxes are confined in the toroid, that is, there is no magnetic leakage, the flux crossing the cross section of the toroid is cp =I BdA <P =BA Wb ( 1.1 0) (1.11) where B is the average flux density in the core and A is the area of cross section of the toroid. The average flux density may correspond to the path at the mean radius of the toroid. If H is the magnetic intensity for this path, then from Eqs. 1.9 and 1.11 + F (a) cp = pNi A = Ni l l! p,A Ni (jt ~R (b) (1.12) FIGURE 1.5 Analogy between (a) magnetic circuit and (b) electric circuit. Magnetic Circuits 5 TABLE 1.1 Electrical versus Magnetic Circuits Driving force Produces Limited by Electric Circuit Emf (E) Current (i = EIR) Resistance (R = l!aA)a Magnetic Circuit Mmf (F) Flux (<I> = FI!J't) Reluctance (!J't = l! pA.)' • cr, Conductivity; p,, permeability. where F m rzk=-l =_!_ J.LA p (1.13) (1.14) is called the reluctance of the magnetic path and P is called the permeance of the magnetic path. Equations 1.12 and 1.13 suggest that the driving force in the magnetic circuit of Fig. 1.4 is the magnetomotive force F ( = Ni), which produces a flux <I> against a magnetic reluctance rzft. The magnetic circuit of the toroid can therefore be represented by a magnetic equivalent circuit as shown in Fig. l.Sa. Also note that Eq. 1.13 has the form of Ohm's law for an electric circuit (i = E/ R). The analogous electrical circuit is shown in Fig. l.Sb. A magnetic circuit is often looked upon as analogous to an electric circuit. The analogy is illustrated in Table 1.1. 1.1.4 MAGNETIZATION CURVE If the magnetic intensity in the core of Fig. 1.4 is increased by increasing current, the flux density in the core changes in the way shown in Fig. 1.6. The flux density B increases almost linearly in the region of low values of the magnetic intensity H. However, at higher values of H, the change of B is nonlinear. The magnetic material shows the effect of saturation. The B-H curve, shown in Fig. 1.6, is called the magnetization curve. The reluctance of the magnetic path is dependent on the flux density. It is low when B is B High '1fL FIGURE 1.6 B-H characteristic (magne- L£... ___________ ~ H tization curve). 11 6 chapter 1 Magnetic Circuits 1.4 1.3 1.2 1.1 1.0 IV 0.9 Ui ~ 0.8 I:Q :6 0.7 "iii <:: "' 0.6 '0 " "' 0.5 u:: 0.4 Sili~on -f-- ...- I"'"" sheet steel > ,/ 1/ ,..... / Cast steel1 ~ v I v I ./ I I v I v Cast iron .....,. B* ~ -0.3 0.2 0.1 v ..,. v / ./ v ~,/ 0 200 400 500 800 1000 Field intensity, H (At/m) -- FIGURE 1.7 Magnetization curves. low, high when B is high. The magnetic circuit differs from the electric circuit in this respect; resistance is normally independent of current in an electric circuit, whereas reluctance depends on the flux density in the mag- netic circuit. The B-H characteristics of three different types of magnetic cores-cast iron, cast steel, and silicon sheet steel-are shown in Fig. 1. 7. Note that to establish a certain level of flux density B* in the various magnetic materials the values of current required are different. 1.1.5 MAGNETIC CIRCUIT WITH AIR GAP In electric machines, the rotor is physically isolated from the stator by the air gap. A cross-sectional view of a de machine is shown in Fig. 1.8. Practi- cally the same flux is present in the poles (made of magnetic core) and the air gap. To maintain the same flux density, the air gap will require much Stator FIGURE 1.8 Cross section of a rotating ma- chine. - Magnetic Circuits 7 more mmf than the core. If the flux density is high, the core portion of the magnetic circuit may exhibit a saturation effect.