40 BELL SYSTEM TECHNICAL JOURNAL 



Simple Analysis of Inductance; Hysteresis Neglected 



The magnetizing force in a thin annular core of mean diameter d 

 (cm.) due to current i (ampere) flowing in a uniform winding of N 

 turns is 



H — -—i — oersted. (1) 



In a core of appreciable radial thickness, the effective magnetic 

 diameter rather than the arithmetical mean diameter must be used 

 in this and following equations, as will be explained in eq. (61). 



When the bridge is balanced with sinusoidal current of peak value 

 im, the peak inductive voltage drop across the standard coil, livfLim, 

 must equal that across the test coil lirfN^m X 10~^, where $„ is the 

 peak magnetic flux in the coil.^ Whence, for an annular coil, 



L = ^^^ X 10-9 henry. (2) 



The flux within an annular coil is composed of that in the core and 

 that in the air space. The expression for inductance can therefore 

 be separated into two terms, giving 



L = -^ (Mm^ + Aa) X 10-9 henry, (3) 



where A and A a are the cross-sectional areas of the core and residual 

 air space, respectively, and Hm is the magnetic permeability of the core, 

 now assumed to be constant throughout the cycle. 

 The inductance due to the core alone is then 



Lm^ L- La' = "^^^^ X 10-9, where L/ = ^^^ X lO'^. (4) 

 a a 



The permeability of the core material can be obtained from this 

 inductance as 



"- = 4^ X 10». (5) 



The peak flux density in the core is derived from eqs. (5) and (1) as 



Bm = iXmHm = j^2 ^ ^^^ gauss, (6) 



where I is the r.m.s. current in the winding. 



' A list of most frequently used symbols will be found in the appendix. 



