DISTORTION CORRECTION 461 



where mi — cosh- {Ao/2), 



mi = sinh Aq, 

 and mz = 2 coth (^o/2), 



mi being greater than unity, while m^ and ms have the sign of A^). 

 This relation for Zu' stated approximately in words is as follows: To 

 raise the attenuation, magnify the given Zu and add series resistance, then 

 add parallel resistance to the whole; to lower the attenuation, magnify Zu 

 and add such negative resistances. An example is given by Networks 

 \a and 2)a of Appendix IV. 

 An impedance equivalent form of structure for Zi/ is 



zi/ = — ^- T- + m^'R, (28) 



— 1 — 



m/zii mi'R 



where m/ = sech- {A^jl), 

 m^i = 4 cosech A o, 

 and W3' = 2 tanh (^o/2), 



m\ being positive and less than unity, while m^ and mz have the sign 

 of A%. Hence with this form, to raise the attenuation, reduce the given Zw 

 and add parallel resistance, then add series resistance to the whole; to 

 lower the attenuation, reduce Zn and add such negative resistances. An 

 example is given by Networks \b and 36, Appendix IV. 



It will be seen from these relations derived from a physical Zn 

 that when ^0 is positive a physical Zu' always results. When Ao is 

 negative, however, physical impedances would be obtained only under 

 certain conditions, depending upon the given Zu and upon Aq. 



One practical utility of the relations would occur in the following 

 situation. Suppose that a design was being attempted from assumed 

 attenuation values with a network having such a general characteristic 

 and that Zn consists of some structure in series or in parallel with a 

 resistance element. The latter resistance as determined from the 

 linear equations may come out to be negative and give Zn an unphysical 

 structure. In such a case we could apply the above relations and 

 raise all the attenuation values uniformly such an amount Ao that the 

 resulting network Zi/ would be physical. 



Corresponding relations between two networks of the ladder type are 



Zn' = e^'zii + (e-4o _ i)i?; (29) 



and between two of the bridged-T (I) type are 



