A METHOD OF IMPEDANCE CORRECTION 



805 



We wish to show that the impedance after the addition of the w + 2nd 

 branch is 



Zn 



+2 



Fix) 



+ i 



Nn+2{X) 



Dn+2{x) Dn+iix) 



The various N's and D's, of course, represent polynomials. The 

 denominator of the imaginary component of Z„ is accented, to indicate 

 that it is not necessarily equal to the denominator of the real com- 

 ponent. The denominators in the Yn+i expression, however, have been 

 given the same designation, since they are equal in the expression we 

 have set up for the admittance at the terminals of the first network 

 branch. This fact is not essential in the proof which follows, but its 

 use somewhat simplifies the procedure. Direct mesh computation 

 gives 



Fix) 



Zn+2 — 



Dn 



+ 1 



F' , M+i 



+ 



Dn-\-l Dn+1 



+ al^2xWn+i + 2a„+2X 



— I- 



[7V„+1 + an+2xDn+i'] 



D. 



n+1 



F' _^Nl+r 



Dn+1 Dn+i J 



+ al+2X^D„+i + 2an+2X 



Since a„+2 is arbitrary, the resistance component will have the speci- 

 fied form only if 



^"+^1^, 



+ 



N' 



n+l 



D 



n+l - 



is a polynomial in x. If this condition is satisfied the reactance expres- 

 sion can obviously be put in the required form. 



In order to examine the denominator of the resistance expression 

 more closely we state iV„+i, and Dn+i in terms of iV„, Dn, and Z)„. 

 Direct mesh computation, again, gives 



F„+i = 



Fix) 



-[S 



+ ^^ I + a2+i.x-2Z)„ + lan+ixDn 



A'n 



d: 



— I 



an+lX -\- jy Dn 



:.D 



r F'^ N'^ A N ' 



^" D^ + 77^ + «»+i-^''^'' + 2an+,xDn '-^ 



1 = ^n [-, +^J + al+ixW, + 2an+ixD„^ 



