778 BELL SYSTEM TECHNICAL JOURNAL 



Part 3. Application of the Analysis 



Air-Gaps and Dilution. Under certain conditions improvement in 

 the operation. of iron core coils toward freedom from harmonic pro- 

 duction may be attained by inserting air-gaps in the magnetic path. 

 The expressions which we have derived up to this point are valid for 

 a material having the constants assigned, and the question now arises 

 as to the parameters which characterize the operation of the iron core 

 including air-gaps, and their relation to the parameters for the original 

 core without air-gaps. With these relations given, our previous work 

 may be applied to cores with air-gaps. 



In establishing the correspondence between the parameters for the 

 two cases, it is instructive to use two methods — one a direct attack,^'* 

 the other resting on an analogy with non-linear vacuum tube circuits. ^^ 

 We may determine the effects sought for by the direct method on 

 consideration of a single branch of a hysteresis loop, which is expressed 

 by equations (4a) or (5a) of Part 1, 



Bill, II) = J^^aJi^H\ (36) 



Now with an air-gap in the magnetic circuit, the magnetomotive force 

 effective is that applied, reduced by the drop acioss the air-gap, or 



m' = m - Pip, , . 



M' = M - p^, ^ ^ 



in which mM, ^$ are instantaneous and maximum values, respec- 

 tively, of the impressed m.m.f. and flux, and m which p represents the 

 air-gap reluctance 



P = \/A, (38) 



X being the length of air-gap and A the core cross-section. 



In order to apply (37) we re-express (36) in terms of magneto- 

 motive force and flux as follows: 



<p{m,AI) = ZZiS^ni-M'\ (39) 



r i' 



/7-+S 



where I is the length of magnetic circuit in the iron. Equations (37) 

 may now be substituted in (39) to yield 



<p(m, M) = i:L4T7(m - p<py{M - pc^)^ (40) 



T S '' 



" Due to Mr. H. P. Evans. 

 " See Appendix 5. 



