METHOD OF IMPEDANCE CORRECTION 833 



We might, for example, use it to approximate a pure resistance varying 

 in an arbitrary manner with frequency, which would be a valuable 

 impedance element in certain circumstances. 



None of these possibilities has been investigated in detail, and 

 naturally the measure of success which can be achieved with any one 

 of them will depend largely upon the precise conditions of the problem. 

 The mathematical form we have specified for the load impedance of 

 the network is so broad however that if we were to consider only this 

 aspect of the situation we might conclude that the scope of the structure 

 is well nigh universal. For example, the impedance of any finite 

 network of resistances, inductances, and capacities can be written in 

 the appropriate mathematical form. Even when the load impedance 

 is not described in the required manner, either because it is empirically 

 determined or because it has the wrong theoretical formula, the type of 

 algebraic expression we have been considering is so general that it can 

 always be matched approximately. 



Unfortunately, the range of application promised by this rather 

 superficial mathematical discussion may be severely restricted by 

 other considerations. In the general case, for instance, the number of 

 terms in the denominator of the resistance expression will be greater 

 than the number of branches in the correcting network and it will not 

 be possible to choose them all arbitrarily. Moreover, even when the 

 correct number of conditions have been imposed upon the power 

 series coefficients we have no assurance that the resulting system of 

 simultaneous equations between coefficients and element values can 

 be solved, or that the solutions, if obtained, will always correspond to 

 physically realizable elements. Finally, we may observe that although 

 no difficulty was experienced in the reactance or susceptance correction 

 of filters, it seems probable that, in view of the limited range of char- 

 acteristics which can be simulated by physically realizable reactive 

 structures, a straightforward application of the general method of 

 resistance correction will often leave us with a reactive characteristic 

 which cannot be corrected. 



These difficulties may occasionally be overcome by slight modifica- 

 tions in the design process. Among other possibilities for example, we 

 can adjust the lowest powers both in the denominator of the resistance 

 expression and numerator of the reactance expression ^ to desirable 

 values, obtain an approximate value for the effect of higher powers in 

 both expressions by a trial computation and readjust the coefficients 

 of the lower powers to take account of these previously neglected terms. 



8 Since the denominator of the reactance is equal to that of the resistance, whose 

 value is prescribed by the requirements, the reactance expression can be determined 

 completely from its numerator alone. 



