42 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



magnet flux supplied by the core to an external path of reluctance (R, 

 using equation (9) and the demagnetization curve for the core material. 

 (R" may be determined from the initial slope (Ri = d\^/d(p at JF = 0. By 

 differentiation of equation (14), this initial slope is given by the equa- 

 tion: 



(R» = i-^ ■) (Si". (15) 



(Sii may be evaluated as the sum of the core reluctance and the external 

 reluctance 01,. In determining the core reluctance, /x should be taken as 

 the incremental permeability, or the slope of the demagnetization curve 

 at B = ipolo,- With (Rj thus evaluated, (Si" is given by equation (15). 



5 MAGNETIC RELUCTANCES AND CONDUCTANCES 



As shown in Section 3, determination of the magnetization or ip — \S 

 relations is equivalent to determining the reluctance (R, or 3^/^, and this 

 in turn reduces to the determination of the component reluctances of 

 the magnetic circuit. Some of the more useful expressions for the evalua- 

 tion of these component reluctances are given in this section. Where 

 parallel paths appear, the computations involve the reciprocals of re- 

 luctances. It is convenient to refer to the reciprocal of a reluctance as a 

 magnetic conductance, or permeance. 



Toroid 



The magnetic field of a toroidal coil is shown in Fig. 8. Provided the 

 medium within the coil is homogeneous, and the section diameter small 

 compared with the toroid's diameter, symmetry requires the field to 

 have the simple character shown. As J? = Ht and B — tiH, 



(R = - = -—=— , (17) 



(p jitLa jia 



which applies whether the path is in iron or air. 



If the toroid is of iron, and the coil is concentrated over part of the 

 length, some of the field appears outside the toroid. This effect is second- 

 ary, and (17) still applies to a close approximation. If a cut is made and 

 an air gap introduced, the total reluctance is greatly increased, but the 

 field within the toroid retains the same character except near the gap. 

 The reluctance of the iron path, now in series with the air gap reluctance, 

 is given by (17). This expression is therefore of general use in determining 

 the reluctance of iron parts of length / and uniform cross-section a. 



