NETWORK SYNTHESIS 



G21 



If we define a new scale factor /v^ by 7v , = Kl , we may write (9) as 

 follows: 



a + ?"/3 = log < 



ame Rational 



n/ i.\f\ Function in — 1/0 



(13) 



Similar expressions for the separate gain and phase functions may be 

 derived from (2) : 



2cL = log< 





n(i-z^ 



JSame Rational 



\ Function in — 1/2 



i2(i = log< 



z z 



1-7 1+7 



n — ?n- ' 



(14) 



1 + 



z 



1 - -, 

 z„ 



Same Rational 

 Function in — 1/z 



Equation (13) holds at all values of p and z, while (14) holds at all 



real frequencies. Simplifications of (14) should be noted, good for the 



useful interval only. When \z\ = 1, l/z = z*. Recalling also that log 



I X f is 2 log \x\, and similar elementary relations, one obtains from (14) : 



When 1 2 I = 1, 



log 



K\ 



n(-l) 



n(i-|- 



z z 



1-7 1 + 77 



/? = Phase n — ; n - 



(15) 



1 + 



z 



1 - -7/ 



Zn 



Comparison with (5) and (7) gives: 



( 1 - - ) ( 1 + — ) = -— (p - p,) 



\ zj \ z,z/ U^Z, 



Thus (9) is equivalent to (1) provided 



K==K 



n - 



n - 





(11) 



(12) 



