NETWORK SYNTHESIS G47 



design constants are the Qk . They are to be small enough so that they 

 do not affect the total phase excursion II (z), when | 2; | = 1. Their 

 specific \'alues are to make | II(z) \ approximately constant, when 

 \ z\ = 1. In general the (required) series in z in the numerator makes 

 it extremely difficult to determine the required values for the Qk . No 

 reasonably simple general procedure has yet been found. 



20. ARBITRARY R.\TIONAL FRACTIONS 



The preceding sections were devoted to the approximation a to a, 

 using n arbitrary natural modes, but no arbitrary frequencies of in- 

 finite loss. Similar techniques may be used when there are n" arbitrary 

 natural modes and 71' arbitrary frequencies of infinite loss. As the appli- 

 cations become more involved, however, routine calculations must be 

 supplemented increasingly with an element of art. 



For simultaneous design of natural modes and frequencies of infinite 

 loss, we must go back from (31) to the a formulation in (21). This we 

 shall now write: 



2^ C2kZ = -log -^ 



The functions N and D are polynomials. The coefficients will be defined 

 as follows: 



,// 2n' 



N = Ko + 7viV ••• K';,, 

 D = 1 -\- K[z^ • • • K'„,0' 



(82) 



By comparison with (21), the zeros of .V, in terms of 2", are the zT". 

 (Note the minus sign in 81.) The zeros of D are then the z'„^. For physical 

 networks, n" ^ n' . 



Equations (50), describing a, may be retained as they stand. Com- 

 bining (50) and (81) gives, in place of (52): 



a - 5 = X (C2. - C.k)T,k 

 2^ (C2fc - C2k)z = -log 



^R{z)R{-z) 



The function R{z)R{ — z) is exactly the same as before. The reciprocal 

 of the function will still be ^ KkZ^'', with the Kk related to C2k by (58), 

 (59), (61). The new rational fraction N/D will appear where previously 

 we had the polynomial Ko -\- • • ■ KnZ ". 



