NETWORK SYNTHESIS 033 



Table I — Z-Plauc Natural i\fodes for Equalization of Two Identical 



Unwanted Modes 



n zMz\ V4N^ zj\zo\ 



1 -.5000 ± i .7071 Non Physical 



2 -.3333 ± i .4714 ±(.3492 ± i .6747) -.3492 ± i .6747 



3 -.6059 ± z .7784 Non Physical 

 -.0720 ± / .6384 ±(.5340 ± i .5977) -.5340 ± i .5977 



4 +.1378 ± i .6782 ±(.6441 ± i .5264) -.6441 ± i .5264 

 -.5378 ± i .3582 ±(.2328 ± / .7695) -.2328 ± i .7695 



5 -.6703 ± / .8187 Non Physical 



+ .2942 ± i .6684 ±(.7157 ± / .4670) -.7167 ± i .4670 



-.3757 ± i .5701 ±(.3918 ± i .7275) -.3918 ± i .7275 



the coefficients may then be chosen to eliminate terms from the series 

 23 {C-ik — C-ikYr-ik , representing a — a, subject to the previous condi- 



ne (n— l)e 



tion that two zeros must be z' = zl . Tliis replaces = by = , in 

 (35), and changes (3G) to: 



E it.^"- = 1 - (n + 1) (l)" + n (^ij 



1 i> In (~1 2\2 



(39) 



If n is odd, all the roots z^ can be physical only if iv„+2 is negative. 

 On the other hand, any finite negative it„+2 leads to a larger error, 

 a — a, than K"„+2 = 0. Reducing /C„+2 to zero is the same as reducing 

 the degree of the polynomial by one, which amounts to reducing n by 1, 

 from an odd to the next smaller even integer. In other words, a physical 

 design with an odd number of natural modes is less effective, for the 

 present application, than a simpler network, with the next smaller even 

 number of modes. 



Note that the z„ in Table I are proportional to Zo . This means that 

 root extraction methods need be used only once for each value of n, 

 after which the roots may be cjuickly adjusted for any value of zq , 

 corresponding to any assigned value of the two identical modes, po . 



13. ACCURACY 



The accuracy of a completed design can be checked by calculating a 

 from the natural modes p„ , and comparing a with a. It is important, 

 however, to have at least some information about accuracy in advance 



