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BELL SYSTEM TECHNICAL JOURNAL 



sistance. It is then convenient to rewrite the equation as 





AnX^'') = Vl - X\ 



where i?o denotes the desired constant resistance. The problem thus 

 becomes that of simulating Vl — x^ in the range < x < 1 by means 

 of a polynomial in x~ of degree n, and if we assume that the parameter 

 Zo can be chosen arbitrarily the polynomial is completely unrestricted, 

 since the constant term as well as the coefficients of the various powers 

 of X can be taken at pleasure. 



There are several ways of proceeding from this point. The simplest 

 makes use of the binomial theorem. Upon expanding Vl — x"^ with 

 the help of this theorem we reach the relation 





1 1 2 



1 4 1 ^ 



8 16 



Equating corresponding powers of x gives 



Zq = Rq, 



Ai= - 1/2, 

 A2= - 1/8, 

 ^3= - 1/16, 



Using n branches in the conductance controlling network it is possible 

 to take the first n terms of the binomial expansion into account. The 

 elements corresponding to these values of Ai, Ai, etc. can of course be 

 found by the equations derived previously. The results are summarized 

 in the following table. 



TABLE I 



The conductance characteristics corresponding to these choices of 

 parameters are shown on Fig. 6. The curve w = 0, which corresponds 



