The Building-up of Sinusoidal Currents in Long 

 Periodically Loaded Lines 



By JOHN R. CARSON 



IMPORTAX T iiiforniauon regarding llic oxccUeiiCL' of a signal 

 transmission system is deducible from a knowledge of the mode 

 in which sinusoidal currents "build-up" in response to suddenly 

 applied sinusoidal electromotive forces, since on the character and 

 duration of the "building-up" process depend the speed and iidelit\- 

 with which the circuit transmits rapid signal fluctuations.' The 

 object of this note is to disclose and discuss general formulas and curves 

 which describe the building-up phenomena, as a function of the line 

 characteristics and the frequency of the applied e.m.f., in the ex- 

 tremely important case of long periodically loaded lines. The formulas 

 in question are approximate but give accurate engineering information 

 and are applicable to all types of periodic loading under two restric- 

 tions: (1) the line must be fairly long, that is, comprise at least 100 

 loading sections, and (2) it must be approximately equalized, as 

 regards absolute steady-state values of the recei\ed current, in the 

 neighborhood of the applied frequency. Fortunately these condi- 

 tions are usually satisfied in practice in those cases where the building- 

 up phenomena are of practical engineering importance. Furthermore, 

 the formulas to be discussed suppi\' a means for the accurate and rapid 

 comparison of different types of loading in correctly engineered lines. 

 The building-up process may be precisely defined and formulated 

 as follows: Suppose that an e.m.f., E cos oit, is suddenK- ai)[jlied, al 

 reference time t=o, to a network of transfer impedance 



Z(;co) = I Z(ia)) 1 • exp [;B(a))]. (1) 



The resultant current, /(/), may be written as 



no = 2 |2^ ■' ^^+P^ """^ [<^!-Biu>)] + <r ,sin H-7^(co)] ;■ , (2) 



= 2>/(l+p)= + <T=|2^cosM-/^(a>)+e], (3) 



wiu-re 



1 = tan~'{a p). 



P'videnlly the functions p and a must be — 1 and o respectively for 



negative values of /, and approach the limits +1 and o as / = co . 



' For pulilishcd discussions of the "building-up" of sinusoidal currents in loaded 

 lines, see Clark, Joiirn. A.I.E.E., Jan., 19J3, Kupfniuller, Telegraplien u. Fernsprech- 

 Technik, Nov., 1923; Carson, Trans. .^.I.^:.1•:., 1919. 



558 



