28 BELL SYSTEM TECHNICAL JOURNAL 



Ki, the mid-series iterative impedance of the artificial Hne, to the 

 total impedance on the right of the mid-point of the series impedance 

 Zi. These three relations, which can be written down at once, are: 



Z, + 2' + Z" = Z.-j^^^, = 0, 



Ve-^ Z" 2Z2 - Ko 



V Z' 2Zo + K2 



K, = ^Zi-f Z" = ^Zi -t- ^^'^' 



2 2 2Z2 "i" jtv2 



from which the formulas for T, Ki, and K2, in terms of Zi, Z2, are 

 found to be: 



r = 2sinh-i |-^||i = 2 sinh-i ^7. (1) 



Ki I /TT-^ f -, , Zi \ ±i. , /^ , 1 „\ ±1 . , ( series ,_,. 



K, \ = ^^'^^ V + Izj ^ = H^ + 4 ■'J ' ^' ""^ I shunt, (2) 



and the formulas for Zi and Zo in terms of F and Kx or 7^2 are likewise 

 found to be: 



Zi = 2X1 tanh i r = X2 sinh T, (3) 



Z2 = i^i/sinh r = ^ X2 coth i r. (4) 



Formulas (3) and (4) are in the nature of design formulas in that 

 they determine the impedance Zi and Z2, at assigned frequencies, 

 which will ensure the assigned values of V and K at these frequencies. 

 In general, however, it Avould not be evident how best to secure these 

 required values of Zi and Zo; complicated or even impossible net- 

 works might be called for, even to approximate values of Zi and Z2 

 assigned in an arbitrary manner. Fortunately, practical require- 

 ments are ordinarily satisfied by meeting maximum and minimum 

 values for the attenuation constant throughout assigned frequency 

 bands. Formulas (8) and (9) may be employed for this purpose as 

 explained below. 



It is convenient to have formulas (1) and (2) expressed in a variety 

 of ways, since no one form is well suited for calculation throughout 

 the entire range of the variables. Accordingly, the following analyti- 

 cally equivalent expressions are here collected together for reference: 



