766 BELL SYSTEM TECHNICAL JOURNAL 



harmonic components, as was done by S. P. Thompson. '' Solutions 

 in this form possess all the advantages and disadvantages of numerical 

 ones in which any change of conditions leads to a new problem; the 

 solution desired is a general one which will describe the phenomena in 

 terms of coefficients characteristic of the magnetic material, and this 

 type of solution is the subject of analysis in the following pages. 



Perhaps the least difficult method of arriving at an analytical 

 solution without making any assumptions as to the form of the loops 

 is the following. A power series for each branch of any loop is 

 formulated, and the two resultant equations are combined in a trigo- 

 nometric series. In this way the solutions, each one valid over but half 

 the cycle, are combined to represent the relation of B to // over the 

 entire cycle. 



The flux density on each branch of the loop may be defined in 

 terms of two values of magnetizing force, one the maximum value of 

 magnetizing force on the particular branch with which we happen to 

 be concerned and the other the instantaneous value of the magnetizing 

 force on that branch. The equation for either branch may then be 

 expressed as a double power series in these two variables, the instan- 

 taneous magnetizing force Qi) which will be expressed in gilberts/cm, 

 and the maximum value of the magnetizing force (//), expressed in 

 the same units; 



BQi, II)-T.T. a„JV"H^\ (1) 



7H = n=0 



where 



._ 1 dBQi, II) 



mini dh'"dH" 



(2) 



The parallelism of this representation with that for the plate current- 

 grid potential curves of a three electrode thermionic tube is evident, — 

 in both cases double power series are involved."* The coefficients fl„,„ 

 are derivatives which are evaluated at the point h = 0, // = 0, and 

 it will be understood in (2) that these particular values are inserted 

 after the derivatives have been taken, in quite the usual manner. 



There is a further interesting parallel here to the vacuum tube case 

 regarding simplification of the general relation. In the case of the 

 vacuum tube a simplification of the double power series due to van 

 der Bijl has been employed which represents the family of tube char- 

 acteristics by an equation in a single variable. This simplification 

 consists in assuming the amplification factor to be constant, and the 

 important point for us here is that it is equivalent to the assumption 



^"On Hysteresis Loops and Lissajous' Figures," Phil. Afiif^., 1910. 

 '' Peterson and Evans, "Modulation in Vacuum Tubes used as Amplifiers," Bell 

 System Technical Journal, July, 1927. 



