802 BELL SYSTEM TECHNICAL JOURNAL 



in which the A's are constants involving the arbitrary quantities ai, a^, 

 etc. which specify the network elements. 



The quantity F{x) is usually either the resistance or conductance 

 component of the load impedance, and in any case is a quantity entirely 

 determined by that impedance. In order to secure the proper resist- 

 ance or conductance from the corrected structure, therefore, it is 

 merely necessary to choose such values of the constants Aq • - • An 

 that the polynomial satisfies the equation 



. , A , A Fix) 



Rior G) 



with sufificient accuracy when R is given the desired value of the cor- 

 rected resistance or conductance. The problem of approximating a 

 given curve by a polynomial of given degree is a well known one in 

 mathematics and such general methods as expansions in power series 

 or Legendrian harmonics exist for its solution. We can, therefore, 

 consider that the choice of these constants presents no particular diffi- 

 culty. Even without the help of these general methods, however, the 

 problem is so simple that suitable approximations can be obtained by 

 cut-and-try methods. 



These polynomial coefficients Aq • • ■ An are merely intermediate 

 parameters which specify the values of the elements in the network 

 implicitly but do not give them directly. In order to determine the 

 relation between these coefficients and the actual element values it is 

 necessary to make a direct computation of the impedance of the net- 

 work in terms of the a's and sort out the various powers of x in the 

 resulting expression. Each of the quantities Ai--- An is thereby 

 expressed as a function of the a's. Our next step must then be to de- 

 termine values of the network elements by solving the set of simul- 

 taneous equations relating them to the numerical values of the poly- 

 nomial coefficients. In accordance with the procedure we have 

 adopted, the design is completed by the computation of the reactance 

 or the susceptance of the network, and its adjustment to the desired 

 value by the addition of a suitable final branch. The discussion of the 

 application of the method to filter impedances given in the next section 

 will illustrate the process in detail. 



Prooj of Properties of Ladder Type Resistance Correctors 



As we observed in a previous paragraph the ratio of the load resist- 

 ance or conductance to the corrected resistance or conductance can be 

 expressed in this simple fashion as a polynomial in x only when the 

 load impedance belongs to a certain mathematical class. Appropriate 



