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(16127a) (16127b)
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P is known as the permeance of the ux path, and
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1 / v is its permeability By is the permeance of a ux path of unit length and unit sectional area
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38 Point Relations The mmf (Fig 1633) is entirely consumed in forcing the
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ux against the reluctance of the magnetic path The ux density is uniform, and the magnetic path has a uniform reluctance Since the mmf drop R is uniformly
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distributed throughout the whole length of the lines of force, the mmf drop per unit length may be de ned as
D k1v D A
(16128a) k1vB (16128b)
K is known as the mmf drop per unit length, the eld gradient, or the eld intensity It is a vector whose magnitude is proportional to that of B and whose direction is along B Both B and K are quantities de ned at every point of a magnetic eld Hence Eq (16128b) expresses a point relation and holds true at every point of a eld, irrespective of whether the eld is uniform or not 39 Units The relations developed above are general and hold true independently
of the units used For purposes of numerical computations, however, it is necessary to use a consistent set of units, as listed in Table 165 With this choice of units, the constants k1, , and v in Sec 37a, for air or vacuum, become k1
1 (4 )10 3195
1257 10
henry per cm3
(16129)
henry per in3 0313 103 yrneh per in3
(1 4 )109
103 yrneh per cm3
and the relations of Secs 37 and 38, for an air core, become
M R0
(a) (a )
(b) (b )
l (c) A
A (c ) l
(16130)
The relations in Eqs (16130b) and (16130b ) are point relations and hold true irrespective of whether the eld is uniform or not The remaining equations in Eqs (16130) apply only to a uniform air eld wherein l is the length of the ux path and A is the cross-sectional area perpendicular to the direction of ux
40 Series and Parallel Reluctances and Permeances Two permeances are in parallel if the same mmf acts on them Two reluctances are in series if the same ux path passes through them Expressions for the equivalent permeance of a parallel circuit and the equivalent reluctance of a series circuit are developed as follows
Equivalent Permeance Consider a set of n permeances in parallel Let the ux through each of them be
ELECTRICAL ENGINEERING
TABLE 165 Physical Quantities, Symbols, and Units
Symbol l t or T
Physical quantity Length Time Permeability
Equation Fund unit Fund unit
Dimension [L] [T] [RTL 1] [I] [R] [RI] [L2] [L2] [RT] [R 1T 1L] [RIT] [RITL 2] [I] [IL 1] [RI2] [RI2T] [R 1T 1L 1] [RTL]
Name of unit Centimeter Second Henry per cm cube Ampere Ohm Volt Sq cm Cu cm Henry Yrneh per cm cube Weber Weber per sq cm Ampere-turn Amp-turns per cm Watt Joule Yrneh Henry
I, i R, r E, e A V L
Current Resistance Emf Area Volume Inductance Reluctivity Magnetic ux
Fund unit Fund unit E RI A l1l2 V l1l2l3 di e L dt 1
B M H
d dt A NI
B M H
Flux density Mmf Magnetic intensity Power Energy Reluctance Permeance
l EI PT vl A A l
(16131)
The total ux in the circuit is
1 2 n
M(P1
P2 Pn
(16132a) (16132b)
where
Thus in a parallel circuit the equivalent permeance is the sum of the parallel permeances Equivalent Reluctance Consider next a set of reluctances in series Let the mmf drop across each of them be
M1 R1 M2 R2 Mn Rn
(16133)
CHAPTER SIXTEEN
Then the total mmf of the circuit is
M M1 M2 R R1 Mn R1
R2 Rn
(16134a) (16134b)
where
Thus in a series circuit the equivalent reluctance is the sum of the reluctances
41 Magnetic Properties of Matter According to its magnetic properties, matter falls into either one of the following classes:
1 Diamagnetic matter 2 Paramagnetic matter a Ferromagnetic b Nonferromagnetic Diamagnetic matter includes substances whose magnetic permeability is less than 0 Among these may be mentioned antimony and bismuth The permeability of bismuth, which is one of the most highly diamagnetic substances known, is 1254 10 8 H / cm3, which is not very different from 0 1257 10 8 Paramagnetic matter includes substances whose magnetic permeability is equal to or greater than 0 Air, wood, copper, brass, etc have a permeability which is approximately 0 Iron and iron alloys and nickel and cobalt and their alloys have permeabilities varying from 0 to 105 0, depending on the ux density used and the composition of the alloy This latter class of paramagnetic matter which possesses high permeability is referred to as ferromagnetic in order to distinguish it from nonferromagnetic matter, whose permeability is very close to that of vacuum The absence of materials of very high reluctivity accounts for the fact that there is no known magnetic insulator Such an insulator, if it existed, would have a very low permeability or very high reluctivity The reluctance of a path made of this R would be low In material would be high, and hence in the relation M order to appreciate this fact, compare the ratio of the resistivity of insulators to that of conductors with the ratio of the reluctivity of paramagnetic or diamagnetic materials to that of ferromagnetic materials as follows: Resistivity of silver: 1 163 10 6 Resistivity of gutta-percha: 2 37 1014 Reluctivity of permalloy: v1 2 10 5 Reluctivity of bismuth: v2 8 107
37 163 8 2
1014 10 6 107 10 5 4
227 1012
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