724 BELL SYSTEM TECHNICAL JOURNAL 



mate a transfer characteristic to a specified degree of precision over the 

 useful band, it is not possible for the transfer function chosen to represent 

 the transfer characteristic to approximate zero outside the useful band in a 

 manner to produce a resistance efficiency of 100 per cent. This limitation is 

 then the prerequisite for modifying the performance which the coupling 

 networks are required to achieve. The usual range of resistance efl6ciencies 

 specified for input and output coupling network applications is approxi- 

 mately 45 to 80 per cent. 



This modification of the final performance of the coupling networks may 

 be examined quantitatively by referring to eqs. (1), (4), and (5). In the first 

 two of these equations the integral may be taken only over the useful fre- 

 quency range, oi to aj2 , provided that the right-hand side of each of these 

 equations is multiplied by the specified resistance efficiency expressed as a 

 fraction. '° In eq. (5) the equal sign holds only in the limiting case when the 

 resistance eflficiency is 100 per cent. If these equations are modified in the 

 manner indicated, the variation of the transfer characteristic outside the 

 useful frequency range may be chosen in any way which satisfies the total 

 area requirements in eqs. (1) and (4) as they stand. 



Following the choice of a satisfactory' transfer characteristic, the next 

 general problem is the realization of a physical network which will approxi- 

 mate this specified characteristic to the required degree of precision over the 

 complete frequency spectrum. The solution of this problem is the main 

 purpose of this paper. 



As is well-known in network theory, the general form of the squared 

 magnitude of the transfer impedance of any physical two-terminal-pair reac- 

 tive network terminated in resistance may be expressed as the quotient of 

 two polynomials in co-. 



Zi2(joo) 



Ro 



Ao -{- AlOf -\- A2O3 + • • • + ^nW " 



(6) 



Before the necessary' and sufficient conditions that the - ^" derived from 



eq. (6) be the transfer impedance of a lossless network terminated in re- 

 sistance are stated, it is appropriate to develop the modifications which must 



be made in eq. (6) if 



Ro 



is to approximate the transfer characteristic, 



e-°', in this problem. This requires that a closer examination be made of the 

 physical limitation that the coupling networks correspond, in part, in struc- 

 ture to the equivalent circuit of the coupling transformer to be used. Figure 8 

 shows the high-side equivalent circuit of either coupling transformer of 

 Figs. 4 and 5. 



'" coi is usually chosen as zero. 



