NETWORK SYNTHESIS 



G31 



1)3 



H 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 



_ Fig. 7 — Error when four natural modes equalize two natural modes at 



PO = — 0.75«(;. 



The actual accuracy may be calculated from the final zeros and poles, 

 from non-zero terms in the expansion of a — a, or by an analysis in 

 terms of z which ^dll be described later. 



A sample plot of a — a: is shown in Fig. 7. This corresponds to an 

 equalizer of four natural modes, compensating for an initial loss which 

 rises to about 8 db at the top useful frequency, or a distortion of ±4 db 

 about the median loss. Residual errors are order of ±0.06 db. 



A little later we shall return to the question of accuracy, to take up 

 methods of estimating what can be done with other numbers of arbi- 

 trar}' natural modes, and other values of the assigned modes. First, 

 however, we shovild investigate whether the network singularities z, 

 determined by (36) meet the other necessary conditions. 



12. PROPERTIES OF Z^ 



In the first place, | Zo \ must be >1. Otherwise, the function of z in 

 (31) will have no power series expansion over the useful interval \z\ = 1 ; 

 and (31) will not, in fact, determine the gain a over the useful interval. 

 It turns out, however, that the condition does not give trouble in the 

 synthesis of natural modes, when there are no arbitrary frequencies of 

 infinite loss. This may be demonstrated l)y the argument outlined below. 



The z„ are zeros of the polynomial in (34), which we have given the 

 special form (36), by applying (35). A function theoretic test f or | 2, | < 1 



