8 BELL SYSTEM TECHNICAL JOURNAL 



being studied — or, more generally, large G/uC. If it were not for a 

 need of these exact formulas, the approximate formula (7.4) would 

 have been derived by the mere application of Taylor's theorem to (1). 



Part II 



Impedance of Open-Wire Lines 



It will be recalled that the characteristic impedance of an ordinary 

 open-wire line depends primarily on its inductance and its capacity, 

 only secondarily on its resistance, and far less still on its leakance; 

 and hence that its impedance is at least roughly equal to vL/C. 



Of the quantities defined by equations (2), . . . (7), the four most 

 suitable for describing open-wire lines are F, k, b, and a. F is suitable 

 as the independent variable, approximately proportional to the fre- 

 quency, k is suitable as one parameter. For the other parameter, 

 which evidently must involve the leakance, b or a respectively is the 

 most suitable according as the leakance G is approximately propor- 

 tional to the frequency or is approximately independent of the fre- 

 quency. The corresponding suitable forms of the equation for the 

 characteristic impedance K are then 



K = kJ 1+iL, (8) K = kJl±H. (9) 



\{b+i)F \a+iF 



The quantity k = V L/C which occurs in (8) and (9) as a mere 

 factor is significant as being the value that the impedance approaches 

 when the frequency is indefinitely increased 4 ; it is also the value the 

 impedance would have at all frequencies if, without changing L and C, 

 the line could be rendered non-dissipative. For ordinary open-wire 

 lines at voice frequencies (R/wL small or fairly small compared to 

 unity) it is at least a rough approximation to the value of the im- 

 pedance. This limiting value k = vi/C will be termed the " nominal 

 impedance " or, more fully, the " nominal characteristic impedance." 6 



The amount K — k by which the characteristic impedance K exceeds 

 the nominal characteristic impedance k will be termed the " excess 

 impedance," and hence its two components the " excess resistance " 

 and the " excess reactance "; (or, more fully, for the three: the " excess 

 characteristic impedance," " excess characteristic resistance," and 

 " excess characteristic reactance," respectively). The latter two 



4 Provided that C/f approaches zero. 

 _ * Strictly speaking, k varies slightly with the frequency, because of the varia- 

 tions of L and even C. 



