496 BELL SYSTEM TECHNICAL JOURNAL 



2. An iterative impedance which equals that of the smooth line at all 

 frequencies. 



Such a network could have a number of uses. For example, it 

 could serve as a substitute for a small length of smooth line where 

 approximately exact simulation is required as in certain laboratory 

 tests, or as part of an artificial line in a balancing network. Leakance 

 changes can be provided for by means of particular adjustable re- 

 sistances. The design can represent the special case of a distortionless 

 line at the lower frequencies and, if non-dissipative, give a phase net- 

 work having a constant time-of-phase-transmission in this frequency 

 range. 



The method of solution differs considerably from those previously 

 used for the other networks and so will be given here. To begin with 

 let 



. 1 impedance of any section of smooth line, or its equivalent, 

 Za — series i ^^ propagation constant 7, iterative impedance k, and 

 ., =shuntj j^^g^j^^_ 



Also let 



X = open-circuit] 



Mmpedance of the smooth Ime section. 

 Y = short-circuit J 



Then 



7/ = Viji^ = tanh-i VF/X, 

 and (76) 



k = V2a26 = VXF. 



{B. S. T. J., October, 1924, p. 617.) 

 From these 



z, = kyl = ^IXY tanh-i ^|Y|X, 



and (77) 



z, = k/yl = VXF/tanh-i VF/X; 



thus Za and Zi> are inverse networks of impedance product k'-. In a 

 physical smooth line z„ is simulated by series resistance and inductance 

 and Zb by parallel resistance and capacity (assuming the line constants 

 to be independent of frequency), both represented by simple physical 

 networks. In other cases they may be realized in desired frequency 

 ranges, more or less approximately, by physical networks. It will 

 be assumed in what follows that Za and Zb are given by the above 

 formulae. 



The structure which is to simulate the smooth line is shown in its 

 general form as Network 18, Appendix IV, wherein s„ and Z), are con- 

 sidered as two types of physical elements whose combinations in 



