A METHOD OF IMPEDANCE CORRECTION 813 



Once 02^3 are known the individual values of Ui, a2, and as can be found 

 directly from the previous equations. The two radicals on the left side 

 of the equation must be taken as positive in order to secure positive 

 elements, which is the same as saying that the two conic sections 

 must intersect in the first quadrant. The square root on the right 

 hand side may be taken either as positive or negative, the susceptance 

 characteristic obtained with the negative sign being usually preferable. 

 It is also possible to eliminate two of the as directly, obtaining the 

 equation 



[Ai" - iAoAs - iAsW + 8^3 Vl +^1 + ^2 + ^3^1^ 



- [2Ai^ + lA^Az - 4^i^3]a,2 - 8^3 Vl + ^1 + ^2 + ^3 a^ 



+ [(^2 + ^3)2 + 4^3] = 0, 



which can be solved by standard methods. The former method is 

 shorter, however. 



The susceptance is given by 



„ 5o + B,x^ + B^x^ 



B = — X 



1 + Aix"" + A^x^ + Azx* ' 



where 



Bq= ai -\- az — a2, 



Bi = a2 -\- a^a^ — aia^ — laia^az, 



B2 = a-^a'^az — a^az. 



Methods of Choosing Power Series Coefficients 



Having developed the relations between the power series coefficients 

 and the network elements we are now ready to consider methods of 

 choosing the parameters to fit given impedance requirements. Upon 

 rewriting our equation for the real component of the network admit- 

 tance in the form 



1 + AxX^ + A2X^ + • • • AnX"^ = 



vr^ 



X 



2 



Z.G 



we see that the problem reduces to the approximation of the ratio of 



:^Vl — x^ to the desired conductance G, both of which are known, 



by means of the polynomial \ -\- Axx^ -\- . . . + Anx"^". In most 

 practical designs the desired filter impedance will be a constant re- 



