236 BELL SYSTEM TECHNICAL JOURNAL 



The effect of the added network is easily understood from the pre- 

 ceding equations. It will be noticed that although these equations 

 were written under the assumption that i? is a real quantity, they will 

 still be valid if R is complex. We need therefore merely to replace R by 

 the impedance of the auxiliary network terminated by the variable 

 resistance. If we represent this impedance by Zr, the appropriate 

 expression is 



X + ta nh xP 



/Lr — Aoi — j TTT' \'i 



\ -\- X tanh \p 



where i^ is the transfer constant of the added network and x is, as 

 before, the ratio of the variable resistance to i?o. Since reciprocal val- 



Z 



ues of X still correspond to reciprocal values of ^ , all of the preceding 



conditions of symmetry in the resulting family of characteristics are 

 maintained. The simplest formulation for the new (p is secured from 



equation (6). Upon replacing the "a:" of this expression by -5- we 



Rn 

 readily find that the equation becomes 



X — 1 



d = d^i -\ 1 — -e~-'^(p + higher order terms. (8) 



X -\- 1 



The effect of the added network is therefore merely to multiply the 

 original (p by e~'*. 



An example of the use of this device in conjunction with the network 

 of Fig. 3 is given by Fig. 5. The parameter a was chosen equal to 2, 

 which corresponds to a maximum change in attenuation of 12 db. 

 The auxiliary network, as will be seen, is a conventional bridged- 7" 

 equalizer. The characteristics of the structure are shown by Fig. 6. 

 The series of straight lines represents the assumed curves, while the 

 circles show the actual computed points. The scale of the drawing is 

 too small to show the differences between the two very clearly. The 

 actual error, at the worst setting and frequency, amounts, however, to 

 about 0.05 db. Of this total, about half is due to the intrinsic distor- 

 tion of the structure, which in this instance is controlled by the (p^ term, 

 the effect of the imaginary component of <p being negligible. The re- 

 maining half is due to the failure of the bridged-7" network to realize 

 the desired 1/' characteristic with sufficient precision, and could pre- 

 sumably be eliminated by the addition of more elements to the 

 structure. 



Since both ^0 and \p can be controlled by auxiliary networks, the 

 structure of Fig. 3 is by itself theoretically sufficient to meet all require- 

 ments. There exist, however, a number of other circuits which also 



