DESIGN OF REACTIVE EQUALIZERS 



m 



E_ 



Zn{j(^) 



Rn 



R 8 



Ro' 



(3) 



I''iiuiUy, tlie transmission gain ex (in nepers) is related to the current ratio 



II 

 I 



I he 



, or tlie voltage ratio 



, by e". Hence, the quantitative statement for 

 imitation on the response of these coupling circuits becomes 



f e'" cico = f 

 Jo Jo 



R 



doi = 



TT 



2CnRo 



(4) 



Equation (4) is the general formula which relates the response character- 

 istic over the complete frequency range to the prescribed capacitance C„ 

 and the resistance Ro . This formula is especially helpful in attaching an 

 analytical meaning to the term partial reactive equalization. If a = f(u) 

 is used to describe the attenuation characteristic of a line or cable over a 

 specified finite frequency band, a = kf(u) will be the transmission response, 

 in nepers, which is required to equalize a stated fraction of this loss at every 

 frequency in the specified range, k is then the constant {k < I) which 

 numerically expresses the degree of equalization.^ 



Thus, the a = kf(oi) in eq. (4) is the desired insertion gain characteristic 

 to compensate partially for the line loss characteristic, and is directly related 

 to this loss over a specified frequency range by a constant k. The limitation 

 on the response expressed by eq. (4) will be clear if the transmission a is now 

 defined as « = ao + kf(u}), where aa represents the general response level. 

 Before this expression is substituted in eq. (4), however, it is necessary to 

 change the limits of integration. Thus, the specification of a maximum re- 

 sponse over a finite frequency band requires that the limits become coi and 

 W2 , the extreme frequencies of the useful band. Since R must be positive, 

 this condition requires that e-" be zero everywhere outside the useful range. 

 Carrying out the integration, the result becomes 



m 



< ^In 



2C„Ro / e 



*" oil 



2fc/(w) 



dco 



(5) 



Since ^/(co) is always prescribed, ao is readily computed. 



So far, the equations have considered only the ideal case when the transfer 

 characteristic e'-" is zero outside the useful band. As previously stated, this 

 condition specifies a resistance efficiency of 100 per cent. In practical appli- 

 cations, where a finite number of network elements are employed to approxi- 



* By (1) substituting the equivalent current source for E, (2) applying the principle 

 of reciprocity to the input circuit, and (3) writing the relations for the transfer of power 

 through the circuit, eq. (3) is readily derived. 



' In practice, this constant is called the "slope" of equalization. 



