46 



BELL SYSTEM TECHNICAL JOURNAL 



to odd harmonic voltages which are important from the standpoint of 

 modulation and noise. In a-c. bridge measurements where the balance 

 is made so as to bring the voltages of fundamental frequency to 

 equality, it suffices to carry through the mathematics for this frequency 

 alone. This has been done for sheet and wire cores to an accuracy 

 sufficient for most purposes by W. Cauer.'^ From his results for the 

 inductance and power loss in a laminated core, the apparent per- 

 meability and loss resistance are calculated to be 



Ho sinh d + sin 6 

 d cosh 6 + COS 6 



4^2 



97r 





76^ 26^ 

 60 "^ 457r "^ 



- MO ( 1 - 3Q + 732 



46' 76^ 29' 



— Mo 



1+\B^- 



4\B^d' 



1 + 2 ^^^ 



R/m = 



2irfL(j sinh 6 — sin 6 

 6 cosh 6 + cos 6 



SaHmfLo 



+ 



3mo 



30 "^ 457r 



7r0« 



^4 60 40 



TfUd' 



420 "^ 600 



02 



+ ^X5^Lo/(l +^ - 



60 



40 



(31) 



(32) 

 (33) 



(34) 

 (35) 



The quantity 6 = 2tvH ixof I p, where p is in e.m.u.; and Bm = /jLmHm, 

 where M»n is independent of /. 



The hyperbolic function parts of these equations are valid at any 

 frequency, but they give only those parts of fj. and R which are due 

 to the constant initial permeability mo- The series having a or X as 

 coefficients give the increases due to hysteresis. 



The apparent permeability mm, which is calculated from the meas- 

 ured inductance, decreases as the measuring frequency is increased. 

 Furthermore, at higher frequencies, this permeability rises less rapidly 

 with rise in measuring current than it does at low frequencies, and it 

 will actually decline with increasing H at frequencies higher than 

 that necessary to make 6 > 1.6, approximately. Thus, for the 

 accurate determination of mo, fJ-m, X and a, it is necessary to make 

 measurements at frequencies low enough to suppress these correction 

 terms. 



"\V. Cauer, Arch.f. Rlektrotechnik 15, 308 (1925). 



