OG THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



At low densities, where ^ is constant, (21) may be solved to give a 

 picture of the flux distribution in the magnet. The solution is obtained 

 in terms of hyperbolic functions. The resulting expression is somewhat 

 cumbersome, but reduces to a much simpler form in the special case 

 where ^ is small compared with h. Neglecting ^, equation (20A) reduces 

 to: 



-j- = h or / = hy 

 dy 



In this case then, the magnetomotive force varies linearly along the core, 

 as was assumed in the discussion of Fig. 17 in Section 5. 

 For/ = hy, equation (20B) becomes: 



(I<p 

 dy 



whence : 



-phy, 



^ - fa — —^ , (22; 



where <Po is flux at ^ = (one end), and ph is a constant for a given 

 magnet structure. Thus, for this approximate case, the core flux falls 

 off along its length approximately as the square of the distance from one 

 end. The second term in (22) represents the leakage flux. For the whole 

 core, for which y = f, the leakage flux is given by p hf-/2, or bj^ p iiJ/2. 

 The average flux linked per turn, however, is the integral 



''phy- 



UZ-f'^^- 



corresponding to a leakage reluctance of pf/3. Hence the factor 3 is 

 used to evaluate the average flux linked per turn in Fig. 17. 



From this consideration of the more rigorous transmission line treat- 

 ment, it is apparent that ^ must be small compared with h for the lumped 

 core and leakage reluctance approximation to be valid. If ^^p/h is small, 

 the potential drop in the core, (Rc<p, is small compared to the applied 

 magnetomotive force fF. In most ordinary electromagnets, the ratio 

 (Rc/3^ is small in the low density region. For sensitive relays with long 

 cores of .small cross-section, this is not the case, and the lumped constant 

 treatment is correspondingly limited in accuracy, even at low densities. 



At high densities, however, the core reluctance increases even in 

 ordinary electromagnets. To use the transmission line analogy at high 

 densities, it is necessary to express ^tp in (20A) in terms of the Froehlich- 



