MAGNETIC DESIGN OF UELAYh- 55 



network shown in Fig. 20(h). The complete magnetic circuit, then, con- 

 sists of an infinite number of these elementary networks < onnectcd 

 end-to-end and terminated in lumped reluctances at the extreme ends. 

 Note that this approximation to the magnetic circuit explicitly recog- 

 nizes the distributed nature of the leakage flux. 



Consider now the several quantities which enter into the approxima- 

 tion. Each of the terminating reluctances ((Ri and (JI2) includes two com- 

 ponent reluctances in parallel. The first component is simply the leakage 

 reluctance across the end of the line. The second component is the sum 

 of the reluctances of the magnetic members and series air gaps which 

 complete the circuit from one side of the line to the other. For any single 

 value of the working gap, (Ri and (R2 rna.y be taken as constants. 



The quantit}^ p, which is the leakage permeance per unit length be- 

 tween the two sides of the line, depends upon the geometry of the struc- 

 ture, and may be calculated by the methods of Section 5. Provided the 

 configuration of the magnetic line is uniform throughout its length, p is 

 taken as a constant. The assumption that p is constant is eciuivalent to 

 assuming that all leakage paths between the two sides of the transmission- 

 line lie in planes that are perpendicular to the core. This condition is 

 not satisfied near the ends of the core, Init correction for the end effects 

 may be made in evaluating the terminal reluctances, (Ri and (R2 ■ 



The quantit}' ^, the series reluctance per unit length of line, involves 

 magnetic material whose permeability varies with flux density. It is a 

 variable whose magnitude depends upon the applied magnetomotive 

 force, upon the terminating reluctances, and upon position along the 

 line, since it is a function of (p. In the low density region, however, its 

 value is substantially independent of <p. Application of the magnetic 

 circuit relations results in the following equations: 



% = h- ^^, (20A) 



dy 



d<p 

 dy 



whence: 



j2 



d (f 

 dy^ 



= -Pf, (20B) 



= -p{h - ^'f). (21) 



Subject to the validity of the original assumptions, the solution of equa- 

 tion (21) describes the way in which the flux ^p ^■uries with y, the distance 

 along the line. 



