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A magnetic structure excited by a magnetomotive force = Ni
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Figure 1613 Analogy between magnetic and electric circuits
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It is important at this stage to review the assumptions and simpli cations made in analyzing the magnetic structure of Figure 1613:
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1 All of the magnetic ux is linked by all of the turns of the coil 2 The ux is con ned exclusively within the magnetic core 3 The density of the ux is uniform across the cross-sectional area of the core
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You can probably see intuitively that the rst of these assumptions might not hold true near the ends of the coil, but that it might be more reasonable if the coil is tightly wound The second assumption is equivalent to stating that the relative permeability of the core is in nitely higher than that of air (presuming that this is the medium surrounding the core): if this were the case, the ux would indeed be con ned within the core It is worthwhile to note that we make a similar assumption when we treat wires in electric circuits as perfect conductors: the conductivity of copper is substantially greater than that of free space, by a factor of approximately 1015 In the case of magnetic materials, however, even for the best alloys, we have a relative permeability only on the order of 103 to 104 Thus, an approximation that is quite appropriate for electric circuits is not nearly as good in the case of magnetic circuits Some of the ux in a structure such as those of Figures 1612 and 1613 would thus not be con ned within the core (this is usually referred to as leakage ux) Finally, the assumption that the ux is uniform across the core cannot hold for a nite-permeability medium, but it is very helpful in giving an approximate mean behavior of the magnetic circuit The magnetic circuit analogy is therefore far from being exact However, short of employing the tools of electromagnetic eld theory and of vector calculus, or advanced numerical simulation software, it is the most convenient tool at the engineer s disposal for the analysis of magnetic structures In the remainder of this chapter, the approximate analysis based on the electric circuit analogy will be used to obtain approximate solutions to problems involving a variety of useful magnetic circuits, many of which you are already familiar with Among these will be the loudspeaker, solenoids, automotive fuel injectors, sensors for the measurement of linear and angular velocity and position, and other interesting applications
16
Principles of Electromechanics
EXAMPLE 162 Analysis of Magnetic Structure and Equivalent Magnetic Circuit
Problem
i= 01 A N turns h
Calculate the ux, ux density, and eld intensity on the magnetic structure of Figure 1614
Solution
Known Quantities: Relative permeability; number of coil turns; coil current; structure
geometry
Find: ; B; H Schematics, Diagrams, Circuits, and Given Data: r = 1,000; N = 500 turns; i = 01 A The magnetic circuit geometry is de ned in Figures 1614 and 1615 Assumptions: All magnetic ux is linked by the coil; the ux is con ned to the magnetic core; the ux density is uniform Analysis:
l = 01 m, h = 01 m, w = 001 m
Calculation of magnetomotive force From equation 1628, we calculate the magnetomotive force: F = mmf = N i = (500 turns)(01 A) = 50 A t
Mean path
Calculation of mean path Next, we estimate the mean path of the magnetic ux On the basis of the assumptions, we can calculate a mean path that runs through the geometric center of the magnetic structure, as shown in Figure 1615 The path length is: lc = 4 009 m = 036 m
008 m 009 m
The cross sectional area is A = w 2 = (001)2 = 00001 m2 Calculation of reluctance Knowing the magnetic path length and cross sectional area we can calculate the reluctance of the circuit: lc lc 036 R= = = = 2865 106 A t/Wb A r 0 A 1,000 4 10 7 00001 The corresponding equivalent magnetic circuit is shown in Figure 1616 Calculation of magnetic ux, ux density and eld intensity On the basis of the assumptions, we can now calculate the magnetic ux: = 50 A t F = = 175 10 5 Wb R 2865 106 A t/Wb
01 m
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